The Uniaxial Compression Test

transcript

Report 7-01-117-7 THE UNIAXIAL TENSION TEST

ISSN 0169-9288 Asphalt Concrete Response (ACRe)

May 2001 ir. S.M.J.G. Erkens and ing. M.R. Poot

Report 7-00-117-7 THE UNIAXIAL TENSION TEST

ISSN 0169-9288 Asphalt Concrete Response (ACRe)

May 2001 ir. S.M.J.G. Erkens and ing. M.R. Poot

This project was carried out as an assignment for the Technology Foundation STW, Applied Science Division of NWO and the Technology Programme of the Ministry of Economic Affairs as part of the NWO Priority Programme for Materials Research, co-financed by the Shell Research Laboratory in Rouen. Assignment number: DCT 4090-4

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ACKNOWLEDGEMENTS

I want to thank all my colleagues from the Road and Railroad Research Laboratory for their advice and support. Special thanks are due to G. Galjé, who took care of the complete specimen production process. He organised this so well that even when he became ill, we did not have to stop testing because of a lack of specimens. The contribution of J.A.M. Kalf, who glued all the strain gauges onto the specimens and was ready to help us throughout this part of the project, and the assistance of J.W. Bientjes with respect to the hydraulic system are also very much appreciated. The financial support from the Technology Foundation STW, Applied Science Division of NWO via the Technology Programme of the Ministry of Economic Affairs and the Shell Research and Technology in Rouen, France, is also gratefully acknowledged. Sandra Erkens

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SUMMARY

In the Asphalt Concrete Response (ACRe) project, a three dimensional, non-linear material model for Asphalt Concrete is being developed. This model, which is based on the flow surface proposed by Desai, describes three types of material response, namely: elasticity, visco-plasticity and cracking. In the ACRe project also the test set-ups and procedures necessary for the model parameter determination are being developed. This requires an extensive test program, in which firstly the uniaxial compression test was developed. Secondly, the tension set-up described in this report was developed. To differentiate between opening of the crack and unloading of the undamaged parts of the specimen after the peak load, a special specimen shape is used. The diameter at the top and bottom of the specimen is 80 mm and this gradually reduces until 50 mm at the centre. This shape ensures that the crack will occur near the centre, without the stress concentrations that would be caused by notches. The specimens are compacted in this shape with a gyratory compactor, using a split mould inside the standard gyratory mould. The mixture used in this project is a down-scaled Dense Asphalt Concrete (DAC) with a maximum aggregate size of 5 mm and a 45/60 bitumen. In developing the set-up a lot of attention was paid to ensuring uniaxial loading. This led to a set-up fitted with three hinges for proper alignment. During the test a continuously increasing deformation is applied. This is controlled via a set of three displacement transducers (LVDT’s) connected to the specimen at 120o intervals. Because of the variation in material response, two sets of LVDT’s are used. One set with a range of ±1 mm is as used for the brittle conditions while a set of ± 5 mm was used for those conditions at which the response is more plastic. The unloading is registered with cross-shaped strain gauges. These provide information on the unloading in both the vertical and the horizontal directions. Because of the specimen shape, the vertical gauge registers a combination of radial and axial strain. This total, tangential strain is decomposed in to its components to obtain the axial strain values. The strength values found are used to develop a general relation that expresses the tensile strength as a function of strain rate and temperature. Using this relation in combination with the one found for the compressive strength, four of the five model parameters are determined. The procedures used for this as well as the general expression developed for the model parameters are discussed in detail.

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SAMENVATTING

Het Asphalt Concrete Response (ACRe) project is erop gericht om een driedimensionaal, niet-lineair materiaalmodel voor asfalt te ontwikkelen. Dit model, dat is gebaseerd op de vloeifunctie die is ontwikkeld door Desai, beschrijft drie typen materiaalgedrag, namelijk elasticiteit, visco-plasticiteit en scheurvorming. Binnen het project worden ook de opstellingen en testprocedures ontwikkeld die nodig zijn om de modelparameters te bepalen. Dit vereist een uitgebreid experimenteel programma, waarbinnen in eerste instantie de uniaxiale drukproef is ontwikkeld. In tweede instantie is de uniaxiale trekproef die in dit rapport wordt beschreven ontwikkeld. Om na de piek het openen van de scheur te kunnen scheiden van het ontlasten van de onbeschadigde proefstukdelen, wordt een speciale proefstukvorm gebruikt. Aan de boven- en onderkant heeft het proefstuk een diameter van 80 mm en dat verloopt geleidelijk naar 50 mm in het midden. Deze vorm zorgt ervoor dat de scheur in het midden ontstaat, zonder de spanningsconcentraties die het gevolg zijn van zaagsneden. De proefstukken worden in deze vorm verdicht in een gyrator, door een splitmal te gebruiken in de standaard gyratormal. Het asfaltmengsel dat wordt gebruikt is een verschaald Dicht Asfalt Beton (DAB) met een maximale korrelgrootte van 5 mm en een 45/60 bitumen. Bij het ontwikkelen van de opstelling is veel aandacht besteed aan het realiseren van een uniaxiale belasting. Om dit te bereiken is de opstelling voorzien van een drietal scharnieren. De proeven worden verplaatsingsgestuurd gedaan, waarbij een continue toenemende vervorming wordt opgelegd. Deze vervorming wordt aangestuurd door middel van drie op 120o afstand op het proefstuk bevestigde verplaastsingsopnemers (LVDT’s). Vanwege de variatie in materiaalgedrag, zijn er twee sets van LVDT’s gebruikt. Een set had een bereik van ±1 mm en werd gebruikt voor het brossere gedrag terwijl een set met een bereik van ±5 mm is gebruikt voor test condities die aanleiding gaven tot een meer plastisch gedrag. Het ontlasten wordt gemeten met kruisvormige rekstroken. Deze meten in zowel vertikale als horizontale richting. Vanwege de proefstukvorm registreert de verticale rekstrook een soort tangentiële rek, die bestaat uit een radiale en een axiale rekcomponent. Deze tangentiële rek wordt ontbonden om de axiale rek informatie te verkrijgen. De treksterktes die uit de proeven komen zijn gebruikt om een algemene uitdrukking voor de treksterkte als functie van temperatuur en reksnelheid te verkrijgen. Op basis van die relatie en de eerder gevonden soortgelijke uitdrukking voor de druksterkte zijn vervolgens vier van de vijf modelparameters bepaald. De hiervoor gebruikte methoden en de algemene relaties die zijn opgesteld voor de modelparameters worden uitgebreid beschreven.

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CONTENTS

1. INTRODUCTION .................................................................................................... 9

2. SPECIMEN COMPOSITION, PRODUCTION AND PREPARATION ......... 10

2.1 MIX COMPOSITION IN MASS PERCENTAGES AND MIX DENSITY (MIX) ......................... 10 2.2 SPECIMEN PRODUCTION PROCEDURE ......................................................................... 11

2.2.1 Principle of gyratory compaction ........................................................................ 11 2.2.2 The specimen shape ............................................................................................. 12 2.2.3 Specimen production ............................................................................................ 15

2.2.3.1 Compaction .................................................................................................. 15 2.2.3.2 Detaching the specimen from the mould ..................................................... 15 2.2.3.3 Specimen composition ................................................................................. 16 2.2.3.4 Specimen preparation ................................................................................... 17

3. TEST SET-UP ....................................................................................................... 19

3.1.1 Uniaxial tension ................................................................................................... 19 3.2 TENSION SET-UP IN DETAIL ....................................................................................... 22 3.3 STRAIN GAUGES ........................................................................................................ 23

3.3.1 Introduction.......................................................................................................... 23 3.3.2 Type and position ................................................................................................. 24 3.3.3 Checking the strain gauges values ....................................................................... 25

4. TEST PROCEDURES, CONDITIONS AND RESULTS .................................. 29

4.1 TEST PROCEDURES .................................................................................................... 29 4.2 TEST CONDITIONS ..................................................................................................... 29

4.2.1 Overview of the test conditions ............................................................................ 29 4.2.2 Relation between strain rate and deformation rate ............................................. 30

4.3 TEST RESULTS ........................................................................................................... 32 4.4 DATA-ANALYSIS ................................................. ERROR! BOOKMARK NOT DEFINED.

4.4.1 General expression for the tensile strength ......................................................... 42

5. MODEL PARAMETER DETERMINATION .................................................... 45

5.1 THE MATERIAL MODEL .............................................................................................. 45 5.2 INFLUENCE OF THE MODEL PARAMETERS ON THE FLOW SURFACE ............................. 46

5.2.1 Influence of ....................................................................................................... 47 5.2.2 Influence of ........................................................................................................ 48 5.2.3 Influence of n........................................................................................................ 49 5.2.4 Influence of ....................................................................................................... 49 5.2.5 Influence of R ....................................................................................................... 50

5.3 DETERMINE MODEL PARAMETERS FROM UNIAXIAL DATA .......................................... 51 5.3.1 Determine R and ................................................................................................ 51 5.3.2 Determine n .......................................................................................................... 54 5.3.3 Determine ......................................................................................................... 60

5.4 FINAL REMARKS ........................................................................................................ 65

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6. CONCLUSIONS AND RECOMMENDATIONS ............................................... 66

6.1 CONCLUSIONS ........................................................................................................... 66 6.2 RECOMMENDATIONS ................................................................................................. 67

REFERENCES……………….……………………………………………………………..64

APPENDIX 1: FILLER COMPOSITION……...………………………………………....65

APPENDIX 2: SIEVE CURVE CRUSHED ROCK 0/5 AND FILLER…………………66

APPENDIX 3: BITUMEN CHARACTERISTICS…...…………………………………..67

APPENDIX 4: STRAINGAUGES, TYPE, EQUIPMENT & CALCULATIONS……...68

APPENDIX 5: DISPLACEMENT TRANSDUCERS…………………………………….75

APPENDIX 6: CALIBRATION DATA LOADCELL …………………………………...82

APPENDIX 7: TEST DATA & DETERMINATION AVERAGE RESPONSE………..83

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1. INTRODUCTION

In the Asphalt Concrete Response (ACRe) project, a three dimensional, non-linear material model for asphalt concrete (A.C.) is being developed. Such a model describes the material response, from virgin till failed, to any three dimensional state of stress. The ACRe model is based on Desai’s flow surface, combined with relations that govern the different types of response (elasticity, cracking, plastic deformation) exhibited by asphalt concrete. These types of response and the flow surface are expressed in mathematical expressions. These are generally applicable, since they characterise a type of behaviour, the exact characteristics (strength, stiffness, rate of degradation) are controlled by parameters in these expressions. These parameters are material-specific and have to be determined through a series of tests. The parameters are determined from the response of the material to a given state of stress, it does not really matter what state of stress this is, as long as it is the same state of stress throughout the specimen. Tests that result in such a uniform state of stress usually require sophisticated set-ups and test procedures. In the ACRe projects a series of such test are being developed to facilitate the parameter determination for the model development. This requires an extensive test programme, in which firstly the uniaxial monotonic compression test was developed and used. Secondly, the uniaxial tension set-up described in this report was developed. In Chapter 2 the material used in the ACRe project, a down-scaled Dense Asphalt Concret, is discussed. Also, the specimen preparation is presented. Since it was considered important to know the approximate location of the crack beforehand, without the stress concentrations caused by notches, a special specimen shape was used. As shown in this chapter, this shape influenced the specimen production considerably. The tension set-up itself is described in Chapter 3. In this chapter also the specimen instrumentation, which is also distinctly influenced by the specimen shape, is discussed. The test conditions, data-analysis and test results are presented in Chapter 4. The test conditions are compared to those used for the compression test and the differences, which result from the difference in tension and compression response, are explained. On the basis of the results, a general expression for the tensile strength as a function of temperature and strain rate is developed. Four out five of the model parameters can be determined on the basis of tension and compression results. Now that these are available, the determination of these model parameters is carried out. This is described in Chapter 5, along with the development of general relations for the model parameters. Finally, the conclusions and recommendations that resulted from this part of the project are presented in Chapter 6.

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2. SPECIMEN COMPOSITION, PRODUCTION AND PREPARATION

2.1 MIX COMPOSITION IN MASS PERCENTAGES AND MIX DENSITY (MIX) The asphalt mixture used in the experiments is the standard ACRe-mixture, which was previously discussed in Erkens et al. 1998 and 2000. This mixture is a down-scaled Dense Asphalt Concrete (DAC) with a maximum aggregate size of 5 mm. The mixture consists of 3 components: 1. Filler: The filler was a weak filler of 100 % calcium powder with a density of 2770 kg/m3 for

the first batch and 2760 kg/m3 for the second batch (Appendix 1). 2. Crushed rock: used in the mix was a crushed rock with a maximum grain size of 5 mm and a

density of 2675 kg/m3. The sieve curve of this crushed rock and that of the aggregate (rock+filler) is shown in Appendix 2.

3. Bitumen: The bitumen was 45/60 bitumen with a density of 1020 kg/m3. From the binder two batches were used, the first had a penetration of 47 x 0.1 mm and TRK= 51 C ( PI = -1.10) and the second had a penetration of 47x0.1 mm and TR&K = 52 C ( PI = -0.83; see Appendix 3). The samples were obtained directly from stock.

The specific densities of the components are summarised in table 2.1. More information about the uncertainties in table 2.1 is presented in Erkens et al. 1998 and 2000. Component Density [kg/m3] Sand 2675 ± 15 Filler 2770 resp. 2760 ± 10 Bitumen 1025 ± 5

Table 2.1: Density of the mix components For the production of the gyratory parabolic specimens a 1.267 kg mix was used consisting of: 0.985 kg crushed sand 0/5 0.167 kg Wigro filler (weak) 0.115 kg bitumen 45/60 Then the mass percentages are: 77.7 % crushed sand (compression: ms = 77.1 %) 13.2 % filler (compression: mf = 14.3 %) 9.1 % bitumen (compression: mb = 8.6 %) These mass percentages differ somewhat from those in the specimens used in the compression test. This is the result from the different quantities of material that were needed for both specimens: 1.267 kg for the tension test and 3.5 kg for the gyratory cores from which the compression specimens were drilled. Because of the larger quantities needed for the latter, another scale and procedure were used, which led to the observed difference. The mass percentages found for the compression specimens are shown between brackets behind those for the tension specimens. It can be seen that the compression specimens contained 0.5% less bitumen and 1.1% more filler than the

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tension specimens. The density of the mix can be calculated using the above mass percentages (mj) and the specific density of each component, according to Test 67: Specimen Density (C.R.O.W. 1990):

Where: mj = mass percentage of component j [%] j = density of component j [kg/m3] index s = crushed sand index f = filler index b = bitumen This yields a mix density of mix = 2345 10 kg/m3. The uncertainty in this density is calculated from the uncertainties of its components (Erkens and Poot 1998).

2.2 SPECIMEN PRODUCTION PROCEDURE

2.2.1 PRINCIPLE OF GYRATORY COMPACTION

The specimens have been compacted by means of the “GYROPAC” gyratory compactor (Figure 2.1). In the gyratory compactor, the asphalt mix is subjected to compressive and shearing forces similar in nature to those encountered under a roller during the road construction process. To achieve this effect, the asphalt mix is put in a cylindrical steel mould with a diameter of 150 mm and a height of 190 mm. During compaction a constant vertical compressive stress of 240 kPa is applied to the asphalt mix, while the mould is rotated about its longitudinal axis with a speed of 60 rounds per minute. The angle of tilt is specified and the applied compression is maintained constant throughout the compaction process.

Figure 2.1: The gyratory compactor

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The gyratory shear on the specimen is caused by the combination of the movement of the mould and that of the top and bottom plates. According to the manual of the gyrator (IPC 1993) the geometry is such that the mould applies 70 % of the shear whilst the top and bottom plates contribute the remaining 30 %. The vertical position of the bottom plate is fixed in relation to the mould, whilst that of the top plate varies as the specimen is compacted. Hence the top plate contributes a greater amount of kneading at the commencement of compaction than at the end of compaction. Figure 2.2 shows the principle of gyratory compaction.

P - Pressure

- Gyratory angle

Figure 2.2: Principle of gyratory compaction The gyratory compactor is capable of compacting specimens for a pre-set number of cycles. At each cycle the height of the specimen is measured and displayed. For the production of the specimen the settings of the gyrator were as follows: An air pressure of 240 kPa A tilt angle of 2

2.2.2 THE SPECIMEN SHAPE

In a tension test failure is a local phenomenon, for specimen instrumentation it is therefore important to know where failure will occur, to prevent losing data due to a crack which disables the instruments. The asphalt mixture that is used in the ACRe project is very homogeneous and as a result, fracture in cylindrical specimens can occur anywhere, but most likely near the caps where the load is transferred onto the specimen and peak stresses occur. In any case, there is no guarantee that fracture will occur under controlled circumstances, in case of failure in the specimen the position is unknown and in case of failure at the caps the stress distribution is unknown. Both these problems can be solved by creating a geometry with a preference for failure at a certain position. This can be achieved by a specimen diameter that is maximal at the caps and reduces towards the centre of the specimen. This type of geometry had been used before in asphalt concrete research, but in those cases the shape was achieved by cutting originally cylindrical specimens. That approach could not be used in this case, since cutting might induce weak spots in the area of decreased diameter, causing an underestimation of the tensile strength. To prevent this, a completely new method was developed, based on the gyratory compactor. Using an additional splitting mould inside the standard gyratory mould with 100 mm diameter, the specimens were compacted in their parabolic, axi-symmetric shape (Figure 2.5). Based on the results from the compression tests (Erkens and Poot 1998 and 2000a), it was decided to limit the minimum specimen diameter to 50 mm. The smallest gyratory mould has a diameter of 100 mm, this meant that the diameter had to double over half the height of the mould.

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It soon became obvious that a parabolic transition could not increase the diameter so much in the limited height that was available. This was solved by using a parabolic transition to go from D=50 mm to D=80 mm and a linear transition to get from D=80 mm to D=100 mm. Initially, several shapes were considered for this transition. These shapes are shown in Figure 2.2 , where the height of the gyratory mould is plotted on the horizontal axis and the diameter on the vertical axis. The centre of the mould is shown also, as a continuous line. The arrows and the dotted black lines indicate the position to which the bottom plate is moved when the mould is placed in the gyrator. The material also moves upwards due to this displacement of the bottom plate and to prevent spilling the material from the top, initially the material must remain as much beneath the top of the mould as the bottom plate moves upwards. The dotted grey line, just to the left of the legend indicates this position. The eventual position of the bottom plate and the maximum level to which the mould can be filled define the area over which the diameter of the specimen can change. The several options presented in this graph are named for the remaining thickness (“a”) at that position.

0 25 50 75 100 125 150h [mm]

D [mm]

a=10mm

a=15mm

a=20mm

shift of the bottom plate

Figure 2.2: Different split mould geometries were considered

Naturally, there can be no “remaining thickness” of the split mould. This would provide a platform on which the asphalt mixture would be compacted, rather than sliding into the mould itself. For the same reason, the slope of the parabolic part of the mould mustnot be to large in order to allow the mixture to be pushed toward the centre of the split mould during compaction. If this doesnot happen, the specimen centre will not be sufficiently compacted. Furthermore, the mixture will initially occupy more space than after compaction. For that reason, the volume in the mould must be larger than the eventual specimen in order to store all the material prior to compacting it. Finally, on both sides of the split mould there must be some extra height to allow proper compaction. This is necessary because the compaction plates are 100 mm in diameter and at the end of the compaction there must be a layer of asphalt concrete between each compaction plate and the split mould. If this is not the case, the compaction plate will be pushed against the split mould, which will carry the entire load, thus preventing the mixture from being compacted further. This means that, from the gyratory mould height of 154 mm, about 20 mm cannot be used for the split mould because of the displacement of the bottom plate. Another 20 mm cannot be used to ensure enough “compaction space” on both sides of the mould. As a result, the split mould can take up only 114 mm, for practical purposes this was rounded down to 110 mm.

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a=0: d= -0.0083x 2 + 0.9091x - 4E-14

a=10: d= -0.0074x2 + 0.8148x + 2.5926

a=15: d= -0.0062x2 + 0.6875x + 6.0938

a=20:d = -0.0041x2 + 0.449x + 12.653

0 10 20 30 40 50 60 70 80 90 100 110 120

h [mm]

d [mm] a=0 mm

a=10 mm

a=15 mm

a=20 mm

Figure 2.3: Different solutions with using a mould height of only 110 mm

Using the same a-values, but now where necessary with a linear cut-off (under 45o) to ensure that the split mould length was limited to 110 mm, the shapes shown in Figure 2.3 were obtained. To emphasize the shape of the split mould, in this graph only the cross section of the split mould wall is shown. Because any sudden change in geometry will cause stress concentrations, the linear transitions from the a-values to zero split mould thickness would have to be cut off. As a result, the actual specimen length for each alternative shape would be limited to the distance between the markers in Figure 2.3. The shapes obtained with a=15 and a=20 mm would have resulted in lengths of 80 and 70 mm respectively, which was considered too small. The remaining shapes are rather similar, a=0 results in a longer specimen (l=110 mm for a=0 mm, versus l=90 mm for a=10 mm), but for a=10 mm the slope of the split mould is less steep. This might present less problems with compaction. Since a specimen length of 90 mm was deemed acceptable while there was no indication about what slope would lead to compaction problems, it was decided not to take unnecessary risks with respect to the slope. As a result the solution with a=10 mm was selected.

l =l/3

(a) (b)

Figure 2.4: Shape used in the ACRe project (a) and that with a cylindrical part at the centre (b)

A side effect of the height limitations is that there was no room to use a cylindrical area at the centre of the specimen. Only at the centre the diameter is the minimum value of 50 mm, to simplify instrumentation it would have been practical to have some height at the centre of the specimen over which the diameter would have been 50 mm. On the other hand, creating a smooth transition between a curved and a flat surface is notoriously difficult and a non-smooth transition leads to stress concentrations. For these reasons, it was decided to settle for specimens without a cylindrical area and accept the difficulties involved in instrumenting it. Later on, work published by van Mier (1997) showed that even in a smooth transition (numerical modelling) stress concentrations will occur in the rounded corners, initiating cracking at those locations.

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Figure 2.5: Parabolic split mould (right) which was used inside the gyratory mould (left)

2.2.3 SPECIMEN PRODUCTION

2.2.3.1 Compaction Originally, the split mould was rigidly connected to the gyratory mould by means of screws (Figure 2.5, left), but this caused a problem at the beginning of the compaction process. When the mould is placed in the gyrator, the bottom plate moves upward (20-25mm), usually this results in the mixture moving up as well. But due to the shape of the split mould, in this case it requires quite some pre-compaction of the asphalt in the bottom part of the mould. As a result it takes a few minutes to get the mould in place. Until the mix is poured in the mould, it is continuously kept at the right temperature for compaction, but once it is inside the mould, the temperature is no longer maintained. During the attempts to place the mould, the mix temperature drops, which causes a less effective compaction. To solve this problem, the inner mould is left loose inside the gyratory mould. That means that the mould can be placed easily in the gyrator, preventing unnecessary loss of heat. Despite the energy that is wasted in moving the inner mould inside the gyratory mould, the compaction achieved in the centre of the specimens using this approach is considerably better than that found with the fixed inner mould (Figure 2.6).

Figure 2.6: Difference in achieved compaction (expressed as the percentage of air voids, va) for standard gyratory cores and parabolic ones with the mould fixed and loose

2.2.3.2 Detaching the specimen from the mould After the specimen is cooled down, the split mould is pressed out of the gyratory mould by means of a press. The inner mould remains in place. After this, the split mould must be detached from the

Rigid connection between split and gyratory moulds

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specimen, which is difficult due to the sticky characteristics of bitumen. Many coatings were used to try to solve this problem: teflon spray, wax, temperature resistant tape, a mixture of glycerine and talc but it still required excessive force to loosen the mould. Eventually impreglon, a factory-applied surface treatment, seemed to work. However, this got damaged quickly and was eventually rubbed off by the sand particles in the mix during compaction. The problem was finally solved by the development of a device that allowed the two halves of the mould to be loosened slowly by using the excess material of the specimen to deliver the necessary resistance (Figure 2.7) in combination with the use of a mixture of glycerine and talc on the split mould surface. The specimen and the mould are placed inside a frame that is connected to the specimen by two screws at the top and bottom. Since there is a layer of 20 to 30 mm of excess material on both sides, this does not damage the eventual specimen. Two smaller frames are connected to the two halves of the mould. By means of especially long screws, these frames are slowly pushed back from the central frame, thus loosening the mould. In Figure 2.7 an already cut specimen (without excess material) is used to show the frame and mould more clearly.

Figure 2.7: A device was developed which allowed loosening the inner mould by using the specimen itself to provide the necessary resistance

Later on, an additional split mould that splits into four instead of two parts was developed. For this mould a ring was made that provided the necessary resistance to take of the parts of the split mould. Even when three parts are loose, these can be put back in place and when the fourth part is pulled off, the specimen is pushed into the three other split parts, thus protecting it and maintaining its shape.

2.2.3.3 Specimen composition Once the specimen is removed from the mould, the excess material (all the material outside the parabolic shape) is cut-off, which results in a specimen with 80 mm diameter at top and bottom, 50 mm at the centre and a parabolic shape in between. The degree of compaction and void percentage varies over the height of this specimen (Figure 2.6). Since failure occurs locally, it is not correct to use the average mix composition to indicate the composition of the failure zone. Therefore, a slice of material (approximately 20 mm thick) at each side of the failure zone is be cut-off and analysed. This provides information about the mix composition in the failure zone, which will eventually be used to relate the mix composition to the material characteristics. In case a specific mix composition is required, for example to get the volumetric composition specified by a design manual, the void percentage must be rigidly controlled. To achieve this, the

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mix temperature must remain high during compaction. That the mix temperature is the main parameter in the achieved compaction can be concluded from the results from compacting with the parabolic mould fixed and with the mould loose, as was presented above. Ideally, the mould should be heated throughout the compaction process, but this might prove difficult, since it has to be able to move in the gyrator. A plausible alternative is the use of an insulating cover for the mould. This will effectively reduce the rate of cooling, thus providing a longer period to achieve the desired compaction. At the time these specimens were produced, neither of these options was available and as a result the void percentage in the centre of the specimens differs from that realised in the specimens used in the compression tests. Fortunately, the compression results proved to be quite independent of the void percentage (Erkens and Poot 1998 and 2000a), so this will not cause problems in establishing a relation between tension and compression characteristics for the model parameter determination. Because the specimen composition was not determined prior to testing and the geometry (shape) was the same for all specimens due to the use of the mould, there is no overview of specimen composition and geometry presented here.

2.2.3.4 Specimen preparation Once the specimen is removed from the mould, the excess material has to be cut off. To prevent that the specimen (and especially the vulnerable mid section) would be damaged in the process, the specimen was placed in a cutting mould. The cutting mould is a Perspex mould with the same shape as the split mould (Figure 2.8). The cut mould is bolted to the sawing table during sawing and provides continuous support to the specimen. The non-parabolic parts of the specimen are supported at the bottom, but not covered by the top of the mould. These parts are cut off, leaving a specimen of 90 mm high.

Figure 2.8: Two views of the Perspex cutting mould used to protect the specimen during cutting

After cutting, the specimen is glued to the end caps that are used to transfer the load to the specimen in the tension set-up. Despite the use of the cutting mould, the cutting surfaces can still be somewhat non-parallel due to movements of the cutting blade and some freedom in positioning it. To prevent gluing the caps non-parallel, since that would lead to bending stresses in the specimen, a gluing mould is used. In this mould the centre of the specimen, rather than one of the cutting surfaces is used as a reference. The specimen is clamped at its centre, using a PVC ring of 40 mm

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high. The ring has the same shape as the split mould and the cutting mould and the caps are placed parallel to the centre of the specimen. The glue that is used is Araldite SW404 (an epoxy resin) with HY2404 hardener. This is a two-component filling glue and it is smeared on both the specimen surface and the cap to ensure proper contact with both. To ensure that the glue can correct any deviation of the cutting surface, more glue than necessary is used. This allows the glue to fill any gap there might exist between cutting surface and cap (which is held parallel to the centre of the specimen by the mould). Excess glue spills out between cap and specimen and to prevent a layer of glue on the top side of the specimen and the bottom cap, both are covered with tape. The glue spills onto the tape and is later on removed, together with the tape (Figure 2.9).

Figure 2.9: The gluing mould ensures that the caps are parallel to the centre of the specimen

When the central clamping ring of the gluing mould is attached to the specimen, its top is used to mark the specimen with horizontal lines at 120o intervals. These markers are later on used to position the strain gauges (Section 3.3). This way, all strain gauges are placed at the same height with respect to the centre of the specimen.

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3. TEST SET-UP

3.1.1 UNIAXIAL TENSION

To perform a true uniaxial tension test the alignment of set-up and specimen is of great importance. Two hinges, at either side of the specimen, are needed and the force has to be applied along the specimen axis. Furthermore, the system must be flexible enough to accommodate imperfect specimens and that means that two hinges are not enough. Usually, the specimens are connected to the hinges via caps, which are glued onto the specimen ends screw fastened to the hinges. The screw thread ensures that the caps are placed perpendicular to the working line, making it impossible to accommodate deviations in the specimens prior to fracture. Once a crack appears, the separating parts of the specimen can rotate with respect to each other, but before that both caps have to be placed exactly in line. If a cap is slightly misaligned (or the specimen skewed), the specimen will not fit in the set-up. This is due to the fact that at both hinges the screw thread requires the cap to be placed perpendicular to the working line of the set-up. If the caps are not parallel, this is not possible.

loadcell

traverse

specimen

actuator

Figure 3.1: Two hinges give rotational freedom once a crack occurs, but they do not compensate slight misalignments

To compensate for misalignment one of the hinges must be able to move horizontally. In the set-up developed for the ACRe project this is realised by means of a third hinge (Figure 3.2). Naturally, the specimen preparation procedure aims at preventing misalignment in the specimens (Chapter 2), but it is impossible to achieve perfect alignment. The three-hinge construction is used to accommodate any remaining (small) deviations. If the deviations would be large, the specimen would be loaded eccentrically (the work line of the force is the line between the hinges). In that case the specimen would not be loaded in uniaxial tension, but in bending.

Lab W&S ACRe 20

2 hinges and horizontal displacement

( A ) ( B ) ( C )

Figure 3.2: Alignment is important in an uniaxial tension test

The tension set-up that was developed consists of a rigid frame which is placed in a temperature controlled cabinet. The upper hinge is placed directly on the traverse of this frame. The loadcell is positioned underneath this hinge and immediately below the loadcell is the second hinge. The actuator is connected to the bottom part of the frame, outside the temperature controlled cabinet. The third hinge is placed on the actuator. The specimens are connected between the two lower hinges. The fact that the loadcell is also placed between two hinges ensures that the registered force is the actual force applied on the specimen (Figure 3.3).

Figure 3.3: The TU Delft tension set-up uses 3 hinges to provide the required degrees of freedom

HINGES

LOAD CELL

TU Delft ACRe 21

Unlike in a compression test, failure in a tension test is a local phenomenon. As a result, measuring the overall response gives a combination of responses. Since the objective in this project is to register the response due to a known state of stress, it is important to separate the different stress (and thus response) situations. In a tension test, the following responses occur from the peak onwards: in the failure zone the material is cracking at both sides of the failure zone the material is unloading By registering both the overall response and that of the unloading, not-cracked parts it is possible to reconstruct the individual responses (Figure 3.4). However, in order to do that it must be known beforehand where the crack will occur (actually: where it will not occur). This is be achieved by the special specimen shape, which was discussed in the previous chapter.

la=l/3

lb=l/3

lc=l/3

la=lc =l/3

lb=l/3 + wla, lc

F [kN]

u [mm]

Figure 3.4: When the failure location is known, the different types of response in a tension test can be separated.

Such a peculiar specimen shape also has a disadvantage: it is rather difficult to instrument. For the overall measurements, this can be solved by connecting displacement transducers between the caps instead of on the specimen. The influence of the glue and caps on the measurements is considered negligible due to the thin layer and high stiffness, respectively. The connection between the caps is realised by a set of rings. In the upper ring there are three holes at 120o angles, in which the LVDT’s are clamped. The bottom plate is a massive ring on which the LVDT’s are placed. These LVDT’s are not only used to register the deformation, but also to apply it (on-specimen control). The local instrumentation, to register the unloading, is even more difficult. Several methods were considered for registering this behaviour (Section 3.3 ) and eventually strain gauges turned out to give the best results. Independent of the measurement system that is used, the (approximate) location of the crack must be known because the region in which unloading is registered must be outside the cracking region. If this is not the case, the crack will invalidate the unloading measurement. Hence the shape of the specimen, it ensures that the crack will occur near the centre of the specimen and the areas with larger diameters can be used to register the unloading. Another aspect of the localised failure under tension is that the deformations are rather small compared to those found in a compression test. Also, in tension the deformations up to failure vary with the test conditions (temperature and strain rate). The overall deformations vary between 0.03

Lab W&S ACRe 22

mm and 3 mm. For stable experiments in which the softening behaviour can be registered, the energy required to increase the specimen deformation (crack width) must remain larger than the energy released by the unloading of both the uncracked parts of the specimen and the loading frame. If this is not the case, the crack will grow explosively and no information on the softening behaviour is obtained. To minimise the chance that this happens, the loading frame should be stiff compared to the specimen (to limit the energy stored in the frame). This demand resulted in a large and stiff loading frame (Figure 3.3). Another option is to use only a part of the specimen deformation as the test control parameter, for example only the middle half of the specimen. This was not used, for two reasons, first of all it was not known beforehand to which part of the specimen the cracks would be confined. The shape ensures fracture near the centre, but it does not guarantee that it will be exactly at the centre. If a crack would occur at such a position that the connection points of the control transducers would fail, the test would get out of control and all information would be lost. Second of all, the shape of the specimen makes it very difficult to attach anything to it. This will be discussed in more detail in the section about the unloading measurements (Section 3.3). Furthermore, it was thought that the limited specimen height (90 mm) and the increasing diameter towards the caps would sufficiently limit the energy stored in the specimen to ensure a stable test.

3.2 TENSION SET-UP IN DETAIL The tension test set-up consists of a rigid loading frame in which an MTS 50 kN hydraulic actuator is mounted. The actuator is connected rigidly to the bottom plate. The frame itself is placed on a frame that places the bottom plate at 1 m above the floor. The bottom plate is a sandwich construction of a 600x330x100 mm steel plate bolted onto a 600x600x50 mm steel plate the top plate of this sandwich construction provides the connection to the vertical bars and stiffens the bottom plate to the extend necessary to support a 50 kN actuator. The vertical bars are 100 mm in diameter, made from stainless steel. They are connected to the bottom plate via 8 M10 bolts each. The steel traverse has dimensions 530x150x200 mm. The actuator piston passes through the bottom plate. Directly on this piston the first of three hinges is screwed. The other side of this hinge is screwed into the thread of one of the specimen end caps. Between the other cap and the load cell, placed above the specimen, is a second hinge and the third hinge is positioned between the load cell and the traverse. During the test the load, the total specimen deformation and the axial and radial strain are registered. The measurement systems are connected to a PC-based data acquisition system, which produces a single ASCII output file for each test. These files are the basis for the data-analysis procedures discussed in Chapter 4. The force applied with the 50 kN MTS actuator is measured with a Lebow 50 kN loadcell (model 3157). The calibration data from the loadcell is shown in Appendix 6 The actuator is controlled by an MTS testcontroller (model 407) which allows the selection of the appropriate force range for the test conditions. The test is displacement-controlled and a programmable function generator (MTS microprofiler model 418.91) generates its rate. Since some of the tests were performed at rather high deformation rates and low temperatures, which resulted in high stiffnesses, the system was fitted with a 3.8 l/min high response valve (MTS, model 252.41), which allowed the actuator to apply the required forces at a high speed. The specimen is put in a temperature-controlled Weis-Enet cabinet with inside dimensions 1.0x1.0x1.0 m. The unit can control the temperature between –40 and 100 oC with an accuracy of 0.1 C.

TU Delft ACRe 23

The axial deformation of the specimen was measured by three LVDT’s. Depending on the test conditions ±5 mm or ±1 mm LVDT’s were used. The specimen deformation was also controlled via these LVDT’s. This on-specimen control system ensured that the applied deformation was the actual deformation of the specimen. The calibration data of both sets of Solartron LVDT’s are given in Appendix 5. The displacement transducers that are used to control the test and provide an overall deformation rate are connected to a set of rings placed on the caps directly at the end of the specimen. The influence of these caps and the glue that connects the specimen to them, is considered to be negligible since both components are considerably stiffer than the specimen, even under the extreme conditions: Ecap=210000 MPa (for steel) or 71000 MPa (for aluminum), Eglue ≈ 9000 MPa and Especimen ≈ 10000 max. The axial and radial strains are registered at a fixed height (Section 3.3) using cross shaped strain gauges. These gauges are connected to a series of Peekel amplifiers (Appendix 4). Because of the specimen shape, the axial strains register a combination of vertical and radial strains, in Section 3.3 and Appendix 4 the way in which this signal is decomposed to obtain the actual vertical strain is shown. All measuring devices are connected to an input/output board, which consists of a BNC connection panel with 16 single ended or 8 differential connections. These channels are connected to the data acquisition board, a PCMCIA-12AI card from Keithley metrabyte, which is placed in a Pentium II portable PC. For the tension test 10 single ended connections are used (force, 3 LVDT’s, 3 axial and 3 radial strain gauges). The data acquisition board was controlled through the Windows based data acquisition program TestPoint. Through this program the number of scans per channel and the scan rate are defined. The latter was varied with the test conditions, while the number of samples was kept constant at 10.000. For the more brittle conditions, the maximum scan rate of the measurement card. The data acquisition was triggered by the microprofiler shortly before the test started. During a test, the output of the measurement systems was registered in voltages and saved as columns in an ASCII file. These files were then further analysed by means of Microsoft Excel.

3.3 STRAIN GAUGES

3.3.1 INTRODUCTION

The parabolic shape of the specimen is necessary to fix the location of fracture without introducing stress concentrations, but leaves no straight parts that can easily be instrumented. Nevertheless information on the unloading of the non-damaged parts of the specimen had to be obtained. For this strain gauges were used. Unlike displacement transducers, strain gauges do not have to be calibrated since the parameters that determine their output are provided directly by the manufacturer. Calibration of strain gauges is therefore a matter of checking whether you register what you intended to. A problem that can occur is that the strain gauge is glued on the outside of the specimen, a place where the characteristics and thus the response might differ from those at the core. This is due to the method of compacting, the specimens are gyratory compacted in their parabolic shape. As a result the outer surface is covered by a thin layer of bitumen that doesn’t represent the core of the specimen. This bitumen skin was scratched off at the position where the strain gauges were glued, to limit the influence. Whether this was sufficient, had to be verified. Other problems that can occur are incorrect positioning of the strain gauges, so the gauge is registering under an angle with the intended direction, or is placed at a different height and, thus, at a different diameter than assumed. Also, the type of gauge and glue has to be correct for the material on which the gauge is placed. These aspects of strain measurement verification are discussed in Section 3.3.3.

Lab W&S ACRe 24

3.3.2 TYPE AND POSITION

The strain gauges used here are 90˚ 2-element cross gauges (type PFC-10-11) from Tokyo Sokki Kenkyujo. These are foil-etched gauges with a polyester backing. The gauge length is 10 mm and k=2.13. The gauges are glued on the specimen with a cyanoacrylate single-component adhesive and coated with PU120 from Hottinger Baldwin Messtechnik. The gauge signal is amplified by a Peekel CA 110 universal precision amplifier. The CA-110 offers a total balancing range of ± 5000 μV/V. Three 2-element cross strain gauges are glued directly onto the specimen surface at the same height at 120˚ intervals. The signal of the radial gauges could effectively be considered to register at a given height, registering directly the radial strains at those positions. The tangential gauges, however, necessarily cover a range of heights in which the change in diameter was not negligible. As a result, these gauges registered a combination of radial and axial deformations (Figure 3.5). To obtain the “true” axial strains, these had to be extracted from the tangential gauge signal.

Figure 3.5: The tangential strain gauge registers a combination of radial and axial strain

Before determining the way to calculate the axial strains from the tangential signal, the shape of the signal over the gauge length had to be known. Since a straight line can be used to approximate the shape over a limited height, it was assumed that this would also be the case for the strains. If the specimen is considered to behave linear elastically, the same Young’s modulus and Poisson’s ratio can be used overall and in that case there is a direct relation between the specimen cross section, stress and strain, see Figure 4.1. Using the radial strain, the axial strain can be determined from the tangential strain gauge signal. The relation between the axial and the radial component in the tangential signal depends on the angle with the vertical axis under which the gauge is glued. Since this angle varies with the height, the gauges should be glued at a fixed height on the specimen. In determining the optimal place for the gauges, two things had to be considered:

TU Delft ACRe 25

1 The gauges should be glued some distance from the caps, since the deformation at the caps is restraint and there occurs a change in stiffness, both leading to stress concentrations/disturbances in the stress distribution

2 The gauges should be placed at some distance from the specimen centre, since that is the area at which cracking will occur and the specimen will fail. Naturally, this position is not completely fixed; the crack is not always straight so the gauges should be far enough from the centre to prevent cracking through them.

The gauges themselves had a length of 10 mm. The length was chosen such that they had a length larger than the maximum aggregate size while it still allowed the shape of the glued tangential gauge to be approximated by a straight line over that length. Since the height of the specimens was approximately 90 mm and the length of the gauges was 10 mm, this left a margin of 35 mm to be divided in distance to the cap and the centre, respectively. It was decided to glue the gauges at 15 mm from the centre, which left 20 mm between the gauge and the cap. To ensure that the three 2-element cross strain gauges are always glued on the same distance from the centre for all specimens, the clamping ring (Section 2.2.3.4) is used to mark these positions. This mark is 20 mm from the centre of the specimen. The radial gauges are aligned along this mark, so that the tangential strain gauge is at 15-25 mm from the centre. Since the geometry at this position is known, the axial deformation can now be determined from the tangential gauge signal. The method used is presented in Appendix 4. This calculation is based on some assumptions and simplifications: 1 For linear elastic behaviour (low stress levels as well as unloading) the assumption is true,

but during loading at any cross section (over the height of the gauge) non-linearity is induced the assumption becomes invalid. Since it appears that the gauges record remaining deformations, the assumption does not hold true throughout a test. It is therefore necessary to establish whether the error induced by this assumption is acceptably small.

2 Due to the variable cross section, the strains and thus the strain rates vary over the height. Because of the strain rate sensitivity of the material, this will induce variations in stiffness, which trigger different stress distributions, which in turn will have an effect on the strains…. Since it is not possible to take such a diffuse effect into account, it is once again a matter of establishing whether the error is acceptable. Fortunately, however, this strain rate effect may be due to the viscous characteristics, in which case the elastic parameters that control the unloading are independent of this phenomenon.

This yields the following expression for the axial strain:

22 22 2 2 2 * 2

1 2 ,1 2

1 12 2 4 ( )

gauge R C C

h h h h L R R RR R

Using this relation, the axial strain was calculated from the tangential strain gauge data.

3.3.3 CHECKING THE STRAIN GAUGES VALUES

As stated in the introduction, the strain gauge data had to be verified. This was menat to ensure that various possible disturbances did not significantly influece the measurements. These disturbances were: The glue and the carrier of the strain gauge are rather stiff, which may have an effect on what

the gauge registers

Lab W&S ACRe 26

The temperature and strain rate sensitivity of the material itself will lead to a wide variety of stiffness’ throughout the range of test conditions and it had to be checked if all these conditions could be covered by (one type/combination of) glue and strain gauges

The bitumen skin on the outside of the specimen might cause the strain gauges to register a response that differed from that of the main body of the specimen

The first two points were addressed using a compression specimen (cylinder without bitumen skin) in the compression set-up. The specimen was instrumented with strain gauges in the radial direction, the signal of these gauges, as well as that of a bridge of Wheatstone, were compared to that of the circumferential kitt. This is a very accurate system of a chain with an extensometer (a reliable calibrated transducer which consists of a full bridge of straingauges) that was used in the compression tests (Erkens and Poot 1998). From these tests it became clear that strain gauges were more accurate that the bridges of Wheatstone and that the gauges with the glue could be used at all the intended test temperatures. In Figure 3.6 the measured radial strains of a straingauge and an extensometer are compared under an uniaxial compressive stress of ca. 1.5 MPa. .

0 0.5 1 1.5 2

time[s]

rad []Extensometer

straingauge

Figure 3.6: Comparison of straingauge value with extensometer signal under compression

To address the third point, the influence of the bitumen skin, a tensile specimen was instrumented with a clip on gauge over its diameter at the height of the radial strain gauges. Due to its shape, it was not possible to attach a clip-on-gauge in the vertical direction. The radial strains found from the strain gauges and the clip-on-gauge matched very well, which led to the conclusion that the bitumen skin did not significantly influence the measurements. The two signals are shown in Figure 3.7, the variation in the clip-on-gauge signal is due to its larger range, the registered strains are very small and the noise in the signal is very pronounced. More information about this part of the strain gauge verification is shown in Appendix 4.

TU Delft ACRe 27

time [ms]

Figure 3.7: Radial strains determined using a strain gauge and a clip-on-gauge respectively

During these tests it was also seen that there was no difference between the overall signal obtained from the displacement transducers and that computed on the basis of the strain gauges at 0oC, while there was an increasing mis-match at higher temperatures. On the basis of additional tests (Appendix 4) it could be concluded that this was due to the increased plastic nature of the material at higher temperatures. This leads to more plastic deformations, which concentrate at the centre of the specimen (where the diameter is smaller and, consequently, the stresses are higher). As a result a relatively large part of the overall deformations occur at or near the centre while the deformations at the positions of the strain gauges remains smaller. The relation between the overall deformations and the strain at the strain gauges is discussed in Section 4.2.2. The additional tests use to ensure that the larger plastic deformations at the centre were the cause of the increasing difference between overall and strain gauge signal involved tests with strain gauges at different heights. Some results are shown in Figure 3.8. The test was performed in force control with an amplitude of 1.5 kN and at T=15 °C. The second strain gauge was placed at the height used for the gauges in the actual experiments, the first one was place 10 mm closer to the centre of the specimen while the third one was placed 10 mm closer to the cap. The steeper slope of the average signal for the gauges closer to the centre corroborates the impression that the difference between gauges and LVDT’s in figure 4 at T=30 °C is due to increasing plastic deformations near the centre. The dynamic response of the gauges at the original position (2) however hardly differs from the ones closer to the centre (1) which means that the position of the strain gauge is far enough from the cap to obtain a reliable elastic response. The amplitude of the dynamic signal for the thrid gauge, placed closes to the cap is smaller, so it appears that that gauge is indeed influenced by the proximity of the caps.

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0 1 2 3 4 5 6 7 8 9 10 11 12

time [s]

Figure 3.8: Axial and radial strain signals for gauges at the original position (2), closer to the centre (1) and closer to the cap (3).

TU Delft ACRe 29

4. TEST PROCEDURES, CONDITIONS AND RESULTS

4.1 TEST PROCEDURES The specimens were stored in a storeroom at the test temperature for at least 24 hours prior to testing. This storeroom was placed next to the set-up and to ensure that the temperature is the storeroom and the set-up was the same, an electronic reference thermometer was used to check the temperature in both. Before testing the specimen was put in the temperature controlled set-up in a box of sand, the sand provided continuous support which prevented damage to the specimen (due to its shape the specimen is extremely sensitive to bending under its own weight). In this box the strain gauges were connected to the precision amplifiers by soldering wires to the soldering terminals on the specimen. After this the specimen was placed between the hinges and instrumented with 3 axial LVDT's for on-specimen control. A plastic curtain at the front of the temperature cabinet ensured that during specimen instrumentation the temperature was kept more or less constant. Despite this, the specimen was left in the set-up for about half an hour to account for slight temperature variations before starting the test. After the instrumentation the LVDT's, the loadcell and the straingauges could be set in range and the appropriate settings of the programmable function-generator and data-acquisition system could be set (i.e. strain rate and scan rate). To be able to measure the unloading of the specimen after cracking the actuator must not move back upwards after specimen failure. This was accomplished by setting the 407 MTS servo controller to valve clamping at +50 % of its full stroke. This setting was activated when a pre-set error-level was detected. It was attempted to apply a small pre-load just before the test to ensure that there would be no play in the set-up components (particularly the hinges). For 0 and 15oC this could readily be achieved, but for 30oC the pre-load relaxed immediately. This causes a disturbance at the beginning of the response curves at this temperature.

4.2 TEST CONDITIONS

4.2.1 OVERVIEW OF THE TEST CONDITIONS

The tests were performed in the displacement-control mode, applying a continuous axial deformation rate. Since asphalt concrete response is temperature and strain rate sensitive, these parameters are varied within the test programme. The test conditions can therefore be specified by a combination of temperature and strain rate, or temperature and deformation rate. The latter parameter, the applied deformation rate of the actuator, is used to control the test but it is the strain rate in the material that determines the response. For the specimen shape used in these tests the relation between the two is not straightforward, therefore it will be discussed in more detail in Section 4.2.2. The results from the tension tests will be combined with those from the compression (Erkens and Poot 1998 and 2000a), to determine model parameters. Therefore, the results from both tests under similar conditions (temperature and strain rate) must be known. This can be achieved by performing

Lab W&S ACRe 30

the tests at the same conditions or by generalising the results by expressing them as a function of strain rate and temperature. The latter step (generalisation of the results) is done anyway, to allow implementation of the test results in the model. Initially the same test conditions were chosen for the tension tests as for the compression tests. During the preliminary tests, however, it became apparent that the response in tension is so different from that in compression that simply maintaining the same conditions would not result in the best characterisation of the material response under tension. This is particularly the case for higher strain rates, in the compression tests the plateau value of the strength was not reached even with the highest strain rates. The existence of a plateau could only be deduced, on the basis of theoretical as well as experimental observations. In tension it turned out that, especially at lower temperatures, most of the rates used in compression sufficed to actually reach the plateau strength and, thus, they did not provide any information on the changes in the strength as a function of (temperature and) strain rate. Because of this and because the strain rates for which the plateau was reached varied with temperature, the test conditions were adapted and the strain rates selected for each temperature separately. As a result, the temperatures are the same as for the compression tests, but the strain rates are different. These rates were selected for each temperature in such a way as to ensure for what strain rate the plateau was reached and to get as much information as possible on the transition to that plateau. The resulting condition are shown in Table 4.1, together with the originally intended (compression) conditions.

Tension Compression v [mm/s]

[%/s] T [oC] # samples v [mm/s]

[%/s] T [oC] # samples

0.000664 0.001 0 4 0.1 0.1 0 5 0.00664 0.01 0 5 1 1 0 5 0.0664 0.1 0 5 5 5 0 5 0.664 1 0 4 10 10 0 5 0.0664 0.1 15 5 0.1 0.1 15 5 0.33 0.5 15 4 1 1 15 5 0.664 1 15 4 5 5 15 5 0.0664 0.1 30 4 10 10 15 5 0.664 1 30 4 0.1 0.1 30 5 1.99 3 30 4 1 1 30 5 3.32 5 30 5 5 5 30 5 6.64 10 30 3 10 10 30 5

Table 4.1: Overview of the test conditions

4.2.2 RELATION BETWEEN STRAIN RATE AND DEFORMATION RATE

As stated above, the relation between the overall deformation rate ant the strain rate at the position where cracking occurs is not straightforward, due to the specimen shape. This shape causes the stresses to vary over the specimen height. If a single stiffness is assumed for the whole specimen, the strains can be found on the basis of the stress distribution (geometry). This yields a relation between the overall deformation (rate) and the strain (rate) at the centre of the specimen.

1F Fu dh dh dh

EA E A

Where: u= deformation F= applied force E= Youngs’ modulus

TU Delft ACRe 31

A= specimen cross section (at height h)

0.0005

0.0015

0.0025

0.0035

0 10 20 30 40 50 60 70 80 90

h [mm]

[mm/mm]

0.25u [mm]

Figure 4.1: Strain and overall deformation over the specimen

The overall deformation is found by integrating over the specimen height and the deformations over the strain gauges by integrating over the strain gauge height. The ratio between the two for any combination of F and E is:

hgauge

htotal

Where: h0=height of the bottom of the strain gauge (with respect to the specimen) h1=height at the top of the strain gauge (with respect to the specimen) hspec=total specimen height The strain registered by the strain gauges is the deformation that occurs over the gauges divided by the gauge length. So if the gauge signal is multiplied by the gauge length and divided by the above-mentioned ratio, it yields the overall specimen deformation. The gauge length is 10 mm which is placed at an angle with the cylinder axis, so the “effective height (along that axis) is 9.59 mm. As a result, the relation between the total specimen deformation and the strain gauge data is:

9.5983.22

0.11521

gauge gauge gaugetotal gaugeh

Using the same approach, a relation between the strain at the centre of the specimen (where the crack will occur) and the overall deformation can be established:

0.01511 1spec spec

centre centre centrecentre total totalh h

A Au u

udh dh

Lab W&S ACRe 32

66.4111

total centre centre

centre

These same relations hold between the strain rate at the centre and the overall deformation rate. Since the crack will occur at the centre, the strain rate at that position is the one to which the strength will be related. The overall deformation rate should therefore be determined on the basis of the intended strain rate at the centre. For the strain rates mentioned in Table 4.1 this means that they must be multiplied by 0.66411 to obtain the deformation rate (strain rates were given in %/s). This yields the deformation rates shown in the table.

It must, however, be noted that this is an approximation of the real situation. In reality, the relation between strain at the centre and overall deformation is complicated because the response does not remain linear elastic. The regions with higher stresses will first start to exhibit inelastic response. The corresponding inelastic deformations will change the relation between overall and centre strain, since now the strains at the centre will be larger. Another aspect that complicates the relation between overall strain and strain at the centre is the fact that the stiffness of asphalt concrete is a strain rate dependent quantity. As such, the stiffness will vary over the specimen height, just like the strains. The higher strains at the centre lead to a higher stiffness, which in turn leads to smaller strains. This is a continuous process that is to complicated to account for. For that reason, despite the complications mentioned here, it was decided to use the relation between strains and overall deformation that can be determined on the basis of linear elasticity.

4.3 TEST RESULTS

4.3.1 INTRODUCTION

During each test the axial and radial deformations and the forces were recorded. The accuracy of these signals was known from the calibration data of the displacement transducers and loadcell, respectively. Using the specimen geometry, these signals were transformed into strains and stresses, from which the apparent tensile strength (ft), the axial strain at peak stress (ax,max) and the radial strain at peak stress (rad,max) could be determined. The variability in these computed quantities was obtained through a standard error analysis procedure (Applied physics 1986 and Anderson 1986). The variability of any computed quantity can be obtained by partial differentiation of its expression with respect to the input parameters and using the variability of each of these parameters, to find the composite variability. The variability obtained via this procedure is related solely to the measured quantities, it does not express the effect of assumptions used (such as the use of linear elasticity to relate the strain gauge data to the overall deformations) in the analysis or the effect of “weaknesses” in a specimen. The additional variation, above that computed using the error analysis procedure, can be contributed to these effects. For a more elaborate description of the error analysis procedure the reader is referred to Erkens et al. 2000. Where possible, the elasticity parameters, Young’s modulus (E) and Poisson’s ratio () were also determined for each individual test. In some cases, where the test itself was slow but failure occurred brittle, the scan rate was slow to capture the total response (necessary because of a limited total number of scans) and this resulted in a lack of information about the unloading behaviour. Since unloading, by definition, occurs elastically, the elasticity parameters were determined on the

TU Delft ACRe 33

basis of of the unloading branches. In case of the combination of large overall deformations and brittle failure the information about unloading was insufficient to determine these parameters. If the unloading information was captured, linear regression analysis was used to determine the steepest part of the stress-strain and radial-axial strain curves, respectively. The standard errors from these regression analyses turned out to be larger than the variability based on the above mentioned error analysis procedure. For that reason, the standard errors from the regression analyses are used to express the variability in Poisson’s ratio and Young’s modulus.

4.3.2 NUMBER OF REPITITIONS

On the basis of the number of tests that were performed for each test condition, average values were determined for all the test conditions. The variability in these averages was also determined. For this the variation in the tests at each condition was used. On the basis of the standard deviation in these values a 95% reliability interval was constructed for the predicted value. For a standard normal distribution, such an interval is obtained by multiplying the standard deviation by 1.96. However, this requires an infinite number of samples. In case of a limited number of samples, the 95% reliability is found from:

0.025 1x t nn

with: x = half the length of the 95% reliability interval for x t0.025(n-1) = the right hand critical value of the Student t-distribution with (n-1) degrees

of freedom and a 2.5% chance of over-estimation standard deviation in the sample n = number of observations in the sample The value of t0.025(n-1) for a number of n’s is shown in Table 4.2.

N t0.025(n-1) 2 12.7 3 4.30 4 3.18 5 2.78 6 2.57 7 2.45 8 2.37 9 2.31 10 2.26 15 2.15 20 2.09 ∞ 1.96

Table 4.2: Right hand critical values of the Student t-distribution with a 2.5% chance of over-estimation as a function of the number of degrees of freedom

In this case, n is varies from three to five. Obviously, for three values the 95% reliability interval will be wider than for five samples. The variation in the number of tests per condition resulted from two things, the first was a limited number of specimens (because of the split mould, only two specimens per day could be produced) and the second was the large variation in response for the various conditions. The objective was to find, for each temperature, one strain rate at the plateau and two rates that corresponded to tensile strengths below the plateau to capture the transition. For this reason, initially two strain rates were used at each temperature. At those conditions three repetitions were performed. On the basis of these results a third strain rate was selected and, if

Lab W&S ACRe 34

necessary a fourth. Once the trend was captured for each temperature, the remaining specimens could be used for additional repetitions. Because there were not enough specimens to upgrade all conditions to five repetitions, it was decided to go to four repetitions per condition and five repetitions for those conditions that exhibited the most variation. At two conditions, however, a problem occurred during the fourth test. As a result, at those conditions only three repetitions are available.

4.3.3 DATA ANALYSIS AND INTERPRETATION

4.3.3.1 Control parameter and measuring length For stable experiments in which the post-peak response can be registered, the energy required for crack growth must remain larger than the energy released by the unloading of both the uncracked parts of the specimen and the loading frame. If this is not the case, the crack will grow explosively, resulting in a vertical unloading branch or even a snap-back (Figure 4.2). The latter is a phenomenon known from rock and concrete testing. To minimise the chance that this happens, the loading frame should be stiff compared to the specimen, to limit the energy stored in the frame. Another way to minimise the change of unstable crack growth is the selection of a proper control variable. If the applied force is used to control the test, which is the classical way, the test can not be continued after the peak. The reason is that the control variable should be a continuously increasing quantity. After the peak, the specimen may still have remaining strength but it is less than the peak value. Trying to increase the load past the peak value leads to sudden failure. Using the axial deformation allows the registration of the post-peak response and provides information on the remaining strength in this region, which is an important factor in the design of structures since it decreases the necessary safety margins. If however the experiment becomes unstable, the relation between force and deformation is no longer unique (one deformation level corresponds to several force levels, depending on the stage of the test). In that case, in a displacement controlled test, always a vertical drop of the force signal is registered.

stress

deformation

Snap back

Registered in displacement control

Registered in force control

Figure 4.2: If too much elastic energy is released a snap back can occur

In the case of a uniaxial tension test, also the length over which the specimen deformation is registered is important. By using on-specimen control, the effect of the unloading of the set-up on the control signal is precluded. Because a crack is very local and all the material on either side is unloading, the released energy that effects the control variable increases with increasing registration length (Figure 4.3). If the location of the crack is known, it is possible to use only a small part on either side of that location to control the test. Another option is the use of many LVDT’s and a

TU Delft ACRe 35

sophisticated system to ensure that the one that registers the largest deformation is used as a control signal (Mier, van 1997). This ensures a continuously increasing control signal (Vliet, van 2000). In case of the highly strain rate sensitive nature of asphalt concrete the latter solution might cause problems, during the post-peak response local failure will disturb the strain rate that is imposed. Switching between LVDT’s at various positions where the strain rates may vary, will complicate the situation. Using the overall deformation at least ensures a consistent definition of the overall strain rate.

Figure 4.3: Influence measuring length (Hordijk, 1991)

With respect to the measuring length, for concrete values of 35 mm (Mier, van 1997) and 70 mm (Vliet, van 2000) have been reported. In the latter case, it also involved direct tension specimens with a gradually decreasing cross section. The specimens used in this project are 90 mm in height and because, unlike the specimens used in concrete research, they have a circular cross section, it is difficult to attach instrumentation to it (there are no straight parts). Furthermore, asphalt concrete exhibits not just elastic deformation and fracture but also delayed elastic and permanent deformations in ratios that depend on the temperature and strain rate, which makes it difficult to determine the amount of energy dissipation beforehand. For these reasons it was decided to use the full specimen length as measuring length for the overall response.

4.3.3.2 The different response components Beside the overall response, in the post-peak regime also the unloading of the uncracked parts was registered using the strain gauges. The total deformation that is registered during the tests consists of several components, namely instantaneous elastic, delayed elastic (viscous) and permanent (plastic) deformation. These different components of resonse are in reality not as easy to separate as in theory. This can be seen from Figure 4.4 where the force and axial strain data are shown as function of the number of scans (which is equivalent to a time base) for two different test conditions. The graph at the top shown an example of sudden failure and the one at the bottom illustrates ductile failure. From these graphs it can be seen that even in the case of sudden failure the load diminishes gradually at first. In case of ductile failure this gradual unloading continuous to the end of the test. The fact that unloading, at least initially, occurs gradual rather than instantaneous makes the distinction between instantaneous and delayed elastic response more difficult. Similarly, the recovery of delayed elastic deformation continuous well after the force has returned to zero, reducing the remaining deformation.

Lab W&S ACRe 36

sudden failure

1 501 1001 1501 2001 2501 3001 3501 4001 4501

scan #

micro strain

10force [kN]

strain

ductile failure

1 501 1001 1501 2001 2501 3001 3501 4001

scan #

micro strain

5force [kN]

strain

Figure 4.4: Signals for sudden (above) and ductile (underneath) failure

Because of these complications it was necessary to establish applicable definitions for the different components of response, to ensure that these were determined in the same way for all test conditions, despite the different strain and scan rates that were used. For this reason, the steepest part of the unloading branch is defined as the one corresponding to “instantaneous” unloading. Similarly, the point at which the force is zero is used to define the end of the test and the remaining deformation at this point is considered the permanent deformation. These definitions are illustrated in Figure 4.5, where a force-deformation diagram for ductile failure is shown. If sudden failure occurred, the gradual descend of the load-displacement diagram is suddenly disturbed, the deformation increases and the load decreases rapidly (Figure 4.6).

TU Delft ACRe 37

15_0.1_T200

0 0.5 1 1.5 2 2.5 3

utot [mm]

F [kN]

permanent delayed instantaneous

Figure 4.5: Different components of deformation in a test with ductile failure

0 0.5 1 1.5 2 2.5

utot [mm]

F [kN]

permanent instantaneous

crack width (w)

Figure 4.6: Different components of deformation in a test with sudden failure

4.3.3.3 Elasticity parameters and fracture energy Since unloading, by definition, occurs elastically, the elasticity parameters, Young’s modulus (E) and Poisson’s ratio (), were determined on the basis of the unloading branches whenever they were registered. Linear regression analysis was used to determine the slope of the steepest part of the unloading path of the stress-strain and the corresponding part of the radial-axial strain curve, respectively. For the tests were sudden unloading occurred this procedure could not be used, but from several conditions for which both the unloading and the ascending branches of the load-strain (and load-deformation) diagram were captured (at 30oC it was not possible to apply a pre-load, because it relaxed immediately, which lead to a disturbance in the ascending branch ). From these results it became apparent that the slope of the ascending branch was quite similar to that of the steepest part of the unloading branch (Table 4.3), which is in good agreement with theory. The agreement for the Poisson’s ratio was not as good as for the elasticity modulus, but it was still reasonable. For this reason, if the unloading branch was not captured accurately, the ascending branch was used to determine the elasticity parameters.

crack width (w)

Lab W&S ACRe 38

unloading branch ascending branch code code T ['C] e' [%/s] E [MPa] poisson E [MPa] poisson

0_0.001_T75 T75 0 0.0010 2913 0.20 3262 0.24 0_0.001_T179 T179 0 0.0006 4389 0.22 4629 0.31 0_0.001_T102 T102 0 0.0010 2192 0.18 2413 0.39

average 0.0009 3165 0.20 3435 0.31 stdev 1120 0.02 1118 0.08

15_0.1_T101 T101 15 0.10 1902 0.32 2569 0.39 15_0.1_T200 T200 15 0.10 1557 0.27 1902 0.29 15_0.1_T205 T205 15 0.10 2210 0.21 2167 0.22 15_0.1_T166 T166 15 0.10 1685 0.28 1495 0.37 15_0.1_T105 T105 15 0.10 1991 0.25 2759 0.31

average 0.10 1869 0.27 2178 0.32 stdev 257 0.04 508 0.07

15_0.5_T201 T201 15 0.50 4633 0.30 4506 0.35 15_0.5_T208 T208 15 0.51 5673 0.18 5289 0.28 15_0.5_T203 T203 15 0.50 3826 0.24 4068 0.27 15_0.5_T122 T122 15 0.50 *** *** 5053 0.31

average 0.50 4711 0.24 4729 0.30 stdev 926 0.06 549 0.04

Table 4.3: Elasticity parameters obtained from the unloading and ascending branches

The area under the load-crack width diagram is usually considered as the fracture energy. This is a parameter used in models for tensile fracture and the registration of the unloading response was done to determine this parameter. If the deformation of the unloading specimen halves is subtracted from the overall deformation, what remains is the crack width. In Figure 4.5 this is the distance between the overall and the unloading branches. The area between these branches is the fracture energy (Figure 4.7). The crack width at which the stress is again zero is defined as the maximum crack width (wmax).

15_0.1_T200

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

w [mm]

[N/mm2]

Figure 4.7: The fracture energy is the area under the stress-crack width diagram

4.3.4 TEST RESULTS

In Table 4.4 the individual test results are shown numerically. As stated above, the unloading branch could not be captured for some test conditions. For those conditions the stiffness modulus (E) and Poisson’s ratio () were determined from the ascending branch of the stress-strain diagram, these values are printed in italic. Table 4.4 also contains information about the composition of the

TU Delft ACRe 39

specimens. For these specimens the degree of compaction and void percentage varies over the height of this specimen. Since failure occurs locally, it is not correct to use the average mix composition to indicate the composition of the failure zone. For this reason, slices of material (approximately 20 mm thick) at each side of the failure zone were cut-off and analysed after the test (Figure 4.8). The strength values (ft) shown are determined using the diameter at the height of the crack. This information could be obtained from these slices, by using the differences in weight to determine the distance of the crack to the actual centre of the specimen. In most cases, the difference in diameter was negligible and it was never large, since the cracks all occurred very close to the actual specimen centre. On basis of these slices also the void percentage in the failure area (V%) was determined. The specimen density shown was determined in the usual way by weighting the complete specimens dry, wet and under water . And finally the fracture energy (Gf) and maximum crack width (wmax) that were determined from the unloading branches and overall response in the way that was discussed earlier (Sections 4.3.3.2 and 4.3.3.3) are shown for each test.

Figure 4.8: Slices from both sides of the crack give the composition of the failure zone

Lab W&S ACRe 40

filename code T ddt ddt prfstk prfstk V% V% ft fc SE E SE(E) Gf Gf wmax wmax

[oC] [%/s] [kg/m3] [N/mm2] [N/mm2] [N/mm] [mm]0_0.001_T75 T75 0 0.0010 2126 ± 2.34 11.8 ± 0.4 2.7 ± 0.1 0.20 ± 0.66 2913 ± 27 2.2E-01 ± 0.87 8.7 ± 4.150_0.001_T179T179 0 0.0006 2240 ± 1.17 7.4 ± 0.4 2.5 ± 0.1 0.22 ± 0.47 4389 ± 33 2.2E-01 ± 0.87 0.4 ± 4.150_0.001_T102T102 0 0.0010 2163 ± 1.17 8.2 ± 0.4 2.8 ± 0.1 0.18 ± 0.72 2192 ± 0 1.7E+00 ± 0.87 4.0 ± 4.15average 0 0.0009 ± 5.E-04 2177 ± 145 9.1 ± 5.8 2.7 ± 0.3 0.20 ± 0.05 3165 ± 2780 7.2E-01 ± 2.16 4.4 ± 10.290_0.01_T119 T119 0 0.010 2258 ± 1.19 6.6 ± 0.4 6.3 ± 0.2 0.35 ± 0.00 5843 ± 2 3.5E-01 ± 3.72 0.4 ± 0.770_0.01_T123 T123 0 0.010 2222 ± 1.18 8.4 ± 0.4 5.2 ± 0.1 0.24 ± 0.00 5872 ± 12 4.5E+00 ± 3.72 0.9 ± 0.770_0.01_T211 T211 0 0.010 2271 ± 2.31 4.3 ± 0.4 5.3 ± 0.1 0.31 ± 0.00 5755 ± 23 8.1E+00 ± 3.72 2.0 ± 0.770_0.01_T177 T177 0 0.010 2202 ± 1.16 9.6 ± 0.4 4.2 ± 0.1 0.25 ± 0.01 7107 ± 166 *** ± 3.72 *** ± 0.770_0.01_T167 T167 0 0.010 2209 ± 2.29 8.8 ± 0.4 4.3 ± 0.1 0.31 ± 0.00 6722 ± 7 3.4E-01 ± 3.72 0.2 ± 0.77average 0 0.010 ± 4.E-05 2232 ± 40 7.5 ± 2.6 5.0 ± 1.1 0.29 ± 0.06 6260 ± 765 3.3E+00 ± 4.63 0.9 ± 0.9630_0.1_T162 T162 30 0.10 2237 ± 1.17 9.2 ± 0.4 0.3 ± 0.1 0.33 ± 0.03 182 ± 1 2.1E-01 ± 0.15 2.2 ± 0.9330_0.1_T163 T163 30 0.10 2187 ± 1.16 11.1 ± 0.4 0.3 ± 0.1 0.35 ± 0.03 210 ± 1 1.3E-01 ± 0.15 3.0 ± 0.9330_0.1_T171 T171 30 0.10 2195 ± 1.16 12.6 ± 0.4 0.3 ± 0.1 0.37 ± 0.02 187 ± 1 2.4E-01 ± 0.15 2.4 ± 0.9330_0.1_T62 T62 30 0.10 2246 ± 1.18 9.3 ± 0.4 0.5 ± 0.1 0.38 ± 0.00 259 ± 4 4.8E-01 ± 0.15 4.2 ± 0.93average 30 0.10 ± 1.E-04 2216 ± 50 10.6 ± 2.6 0.3 ± 0.2 0.36 ± 0.04 210 ± 60 2.6E-01 ± 0.24 2.9 ± 1.4815_0.1_T101 T101 15 0.10 2243 ± 1.18 8.2 ± 0.4 2.3 ± 0.1 0.32 ± 0.63 1902 ± 45 7.1E-01 ± 0.19 0.7 ± 0.3715_0.1_T200 T200 15 0.10 2239 ± 1.18 6.9 ± 0.4 2.4 ± 0.1 0.27 ± 0.40 1557 ± 71 1.2E+00 ± 0.19 1.5 ± 0.3715_0.1_T205 T205 15 0.10 2216 ± 1.17 8.1 ± 0.4 2.0 ± 0.1 0.21 ± 0.55 2210 ± 45 9.0E-01 ± 0.19 0.8 ± 0.3715_0.1_T166 T166 15 0.10 2245 ± 1.94 9.0 ± 0.4 1.7 ± 0.1 0.28 ± 0.69 1685 ± 34 8.9E-01 ± 0.19 1.3 ± 0.3715_0.1_T105 T105 15 0.10 2255 ± 1.19 7.6 ± 0.4 2.6 ± 0.1 0.25 ± 0.53 1991 ± 74 1.1E+00 ± 0.19 1.4 ± 0.37average 15 0.10 ± 1.E-04 2240 ± 15 8.0 ± 1.0 2.2 ± 0.5 0.27 ± 0.05 1869 ± 320 9.6E-01 ± 0.23 1.1 ± 0.460_0.1_T108 T108 0 0.10 2213 ± 2.32 8.1 ± 0.4 4.9 ± 0.1 0.00 ± 0.00 *** ± 0 *** ± 0.02 *** ± 0.010_0.1_T121 T121 0 0.10 2214 ± 1.18 8.5 ± 0.4 4.2 ± 0.1 0.20 ± 0.00 12878 ± 3 2.0E-02 ± 0.02 0.01 ± 0.010_0.1_T125 T125 0 0.10 2190 ± 1.18 8.5 ± 0.4 4.5 ± 0.1 0.19 ± 0.00 10537 ± 3 1.9E-02 ± 0.02 0.02 ± 0.010_0.1_T148 T148 0 0.10 2281 ± 1.18 1.5 ± 0.4 7.1 ± 0.1 0.23 ± 0.00 14851 ± 5 6.2E-02 ± 0.02 0.03 ± 0.010_0.1_T115 T115 0 0.10 2248 ± 1.19 7.1 ± 0.4 5.1 ± 1.6 0.20 ± 0.00 14157 ± 305 4.5E-02 ± 0.02 0.01 ± 0.01average 0 0.10 ± 4.E-04 2229 ± 45 6.7 ± 3.7 5.2 ± 1.4 0.20 ± 0.03 13106 ± 3017 3.7E-02 ± 0.03 0.0 ± 0.0115_0.5_T201 T201 15 0.50 2276 ± 1.18 4.6 ± 0.4 4.6 ± 0.1 0.30 ± 1.09 4633 ± 27 8.1E-01 ± 0.52 0.3 ± 1.1915_0.5_T208 T208 15 0.51 2195 ± 1.17 10.3 ± 0.4 3.7 ± 0.1 0.18 ± 0.59 5673 ± 12 8.1E-01 ± 0.52 1.4 ± 1.1915_0.5_T203 T203 15 0.50 2258 ± 1.18 6.4 ± 0.4 4.5 ± 0.1 0.24 ± 0.65 3826 ± 103 1.7E+00 ± 0.52 2.7 ± 1.1915_0.5_T122 T122 15 0.50 2238 ± 1.19 7.8 ± 0.4 4.5 ± 1.3 0.31 ± 0.00 5053 ± 0 4.6E-01 ± 0.52 0.1 ± 1.19average 15 0.50 ± 5.E-03 2242 ± 60 7.3 ± 3.8 4.3 ± 0.6 0.26 ± 0.09 4796 ± 1235 9.4E-01 ± 0.83 1.1 ± 1.9030_1_T70 T70 30 1 2230 ± 2.32 7.0 ± 0.4 1.1 ± 0.0 0.25 ± 0.00 551 ± 0 1.1E+00 ± 0.16 2.6 ± 0.4730_1_T128 T128 30 1 2247 ± 1.19 7.8 ± 0.4 1.2 ± 0.1 0.46 ± 0.00 622 ± 14 9.7E-01 ± 0.16 3.6 ± 0.4730_1_T173 T173 30 1 2219 ± 1.16 8.7 ± 0.4 0.7 ± 0.1 0.26 ± 0.00 209 ± 9 8.2E-01 ± 0.16 2.7 ± 0.4730_1_T69 T69 30 1 2233 ± 2.37 7.7 ± 0.4 0.7 ± 0.1 0.37 ± 0.00 863 ± 20 1.2E+00 ± 0.16 2.7 ± 0.47average 30 1 ± 3.E-03 2232 ± 20 7.8 ± 1.1 0.9 ± 0.4 0.33 ± 0.16 561 ± 430 1.0E+00 ± 0.26 2.9 ± 0.7415_1_T207 T207 15 1 2209 ± 1.15 6.1 ± 0.4 5.0 ± 0.0 0.31 ± 0.08 14912 ± 12 3.1E-01 ± 0.17 0.07 ± 0.0315_1_T209 T209 15 1 2245 ± 1.18 5.2 ± 0.4 5.2 ± 0.1 0.29 ± 0.00 17353 ± 9 5.2E-02 ± 0.17 0.02 ± 0.0315_1_T206 T206 15 1 2217 ± 1.18 8.6 ± 0.4 4.9 ± 0.1 0.25 ± 0.00 15624 ± 81 2.6E-01 ± 0.17 0.05 ± 0.0315_1_T155 T155 15 1 2253 ± 2.00 5.7 ± 0.4 5.6 ± 0.1 0.28 ± 0.00 15075 ± 15 4.7E-01 ± 0.17 0.10 ± 0.03average 15 1 ± 2.E-01 2231 ± 35 6.4 ± 2.4 5.2 ± 0.4 0.28 ± 0.04 15741 ± 1780 2.7E-01 ± 0.27 0.06 ± 0.050_1_T64 T64 0 1 2229 ± 1.18 7.4 ± 0.4 4.5 ± 0.0 0.21 ± 0.00 16986 ± 5 1.7E-02 ± 0.01 0.01 ± 0.000_1_T124 T124 0 1 2272 ± 2.38 3.7 ± 0.4 6.5 ± 0.0 0.27 ± 0.01 13540 ± 0 3.1E-03 ± 0.01 0.00 ± 0.000_1_T126 T126 0 1 2243 ± 2.35 7.2 ± 0.4 5.4 ± 0.1 0.20 ± 0.00 13888 ± 5 2.5E-02 ± 0.01 0.01 ± 0.000_1_T80 T80 0 1 2251 ± 1.19 6.1 ± 0.4 5.8 ± 0.1 0.20 ± 0.00 17283 ± 4 2.8E-02 ± 0.01 0.01 ± 0.00average 0 1 ± 2.E-01 2249 ± 30 6.1 ± 2.7 5.5 ± 1.4 0.22 ± 0.05 15424 ± 3155 1.8E-02 ± 0.02 0.01 ± 0.0130_3_T67 T67 30 3 2225 ± 1.74 7.8 ± 0.4 1.9 ± 0.0 0.25 ± 0.00 1696 ± 17 8.7E-01 ± 0.16 2.5 ± 0.1830_3_T158 T158 30 3 2262 ± 1.17 6.3 ± 0.4 1.8 ± 0.1 0.27 ± 0.00 1710 ± 33 *** ± 0.16 *** ± 0.1830_3_T113 T113 30 3 2219 ± 1.19 8.2 ± 0.4 2.2 ± 0.1 0.26 ± 0.00 1746 ± 20 1.1E+00 ± 0.16 2.2 ± 0.1830_3_T160 T160 30 3 2228 ± 1.95 9.3 ± 0.4 1.3 ± 0.1 0.19 ± 0.00 879 ± 20 8.6E-01 ± 0.16 2.3 ± 0.18average 30 3 ± 3.E-02 2234 ± 35 7.9 ± 2.0 1.8 ± 0.6 0.24 ± 0.06 1508 ± 670 9.6E-01 ± 0.25 2.3 ± 0.2830_5_T127 T127 30 5 2248 ± 1.18 6.1 ± 0.4 2.2 ± 0.0 0.30 ± 0.00 1719 ± 6 1.6E+00 ± 0.69 3.1 ± 0.8530_5_T175 T175 30 5 2191 ± 1.15 10.9 ± 0.4 1.5 ± 0.1 0.24 ± 0.00 1000 ± 0 1.0E+00 ± 0.69 2.6 ± 0.8530_5_T210 T210 30 5 2245 ± 1.18 6.4 ± 0.4 2.1 ± 0.1 0.28 ± 0.00 1193 ± 6 2.3E+00 ± 0.69 2.4 ± 0.8530_5_T149 T149 30 5 2246 ± 1.18 5.8 ± 0.4 1.8 ± 0.1 0.30 ± 0.00 644 ± 13 2.8E+00 ± 0.69 4.4 ± 0.8530_5_T74 T74 30 5 2252 ± 1.19 7.4 ± 0.4 2.6 ± 0.5 0.34 ± 0.00 1770 ± 13 1.5E+00 ± 0.69 2.6 ± 0.85average 30 5 ± 3.E-02 2236 ± 35 7.3 ± 2.6 2.0 ± 0.5 0.29 ± 0.05 1265 ± 600 1.8E+00 ± 0.86 3.0 ± 1.0530_10_T169 T169 30 52 2223 ± 1.93 0.0 ± 0.4 4.6 ± 0.1 0.33 ± 0.00 4077 ± 9 9.6E-02 ± 0.03 0.04 ± 0.0130_10_T170 T170 30 41 2178 ± 1.16 11.2 ± 0.4 4.1 ± 0.1 0.34 ± 0.00 4701 ± 22 4.4E-02 ± 0.03 0.02 ± 0.0130_10_T76 T76 30 46 2245 ± 2.35 7.0 ± 0.4 5.5 ± 0.1 0.36 ± 0.00 5792 ± 0 1.0E-01 ± 0.03 0.04 ± 0.01average 30 46 ± 13 2215 ± 85 6.1 ± 14.0 4.7 ± 1.8 0.34 ± 0.04 4857 ± 2160 8.1E-02 ± 0.08 0.04 ± 0.03

Table 4.4: Overview test conditions and results for the individual specimens

As expected, the results from the tension tests show more variation than those from the compression tests. This can easily be understood by the kind of failure that occurs in both tests, in compression failure is due to overall material degradation, which results in a kind of average response from the specimen. In tension failure is localised and thus more sensitive to local weaknesses in the specimen. Combined with the fact that the variation in specimen composition was larger in the tension than in the compression specimens (due to the shape and production procedure, Chapter 2), this explains the larger variation. Fortunately, this did not prohibit the construction of average response curves for each test condition once differences in the scan rates were taken into account. For the more plastic conditions the results could be averaged directly, if brittle failure occurred the

TU Delft ACRe 41

procedure had to be adapted somewhat. This was necessary to prevent a sudden change in the average values due to the disappearance of one of the signals after sudden failure (in that case the stress level of that signal drops to zero). The procedure used to arrive at the average signals is discussed in detail in Appendix 7. In this appendix also the individual curves and the average curve that was determined are shown for each condition. Graphic representations of the average results are shown, grouped per temperature, in Figure 4.9, Figure 4.10 and Figure 4.11. In the graphs, on the right side the stress (at the specimen centre) is plotted against the axial deformation. From the peak, there are two responses plotted, the overall deformation and the unloading path. This unloading of the specimen is calculated from the strain gauge measurements on the basis of the elastic strain distribution in the specimen, as described in Section 4.2.2. On the left side the stress is plotted versus the radial deformations at the centre, also computed from the strain gauge registration using the elastic strain distribution. The kinks that can be seen in some of the curves are the effect of averaging, they are not present in the individual curves.

-0.1 0.1 0.3 0.5 0.7 0.9

uax [mm]

[N/mm2]

0.001%/s

0.01%/s

0.1%/s1%/s

urad [mm]

Figure 4.9: Average response curves at 0oC and several strain rates

-0.15 0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 1.2 1.35 1.5

uax [mm]

[N/mm2]

0.1%/s1%/s0.5%/s

urad [mm]

Figure 4.10: Average response curves at 15oC and several strain rates (the changes in slope are due to averaging, at those points one specimen exhibited brittle failure)

Lab W&S ACRe 42

-0.1 0.4 0.9 1.4 1.9uax [mm]

[N/mm2]10%/s

5%/s3%/s1%/s

0.1%/s

urad [mm]

Figure 4.11: Average response curves at 30oC and several strain rates

A numerical summary of the average results for each test condition is provided in Table 4.5, where the reliability intervals have been rounded off. This rounding, which in some cases also leads to a rounding off of the average values, creates differences between the values in Table 4.5 and the original values given Table 4.4.

Table 4.5: Average results per test condition, with 95% reliability intervals

4.4 GENERAL EXPRESSION FOR THE TENSILE STRENGTH Eventually, the model will be used to analyse laboratory tests and road constructions. For these applications, it is necessary to have general expressions for the strength and the model parameters. In this way, the required input is limited to the type of mix, or maybe the mix composition, the temperature profile and the loading rate. From this input the finite element program CAPA-3D will determine the temperature and loading rate at each integration point in the structure under analysis and with that information the model parameters can be determined. For this reason a general expression for the tensile strength as a function of temperature and strain rate will be developed on the basis of the test results. Basically, the relation must be able to describe the physical trends, in this case the authors assume these trends to be:

=0: this corresponds to not loading the specimen, which means that the specimen is not loaded, ft=0

T [oC] [%/s] prfstk [kg/m

3] V% ft [N/mm

2] E [N/mm

0 0.001 ± 5.E-04 2175 ± 145 7.1 ± 6.5 2.7 ± 0.5 0.05 ± 0.56 2616 ± 9400 0.01 ± 4.E-05 2230 ± 40 4.7 ± 2.0 5.1 ± 1.5 *** ***

30 0.1 ± 1.E-04 2215 ± 50 5.4 ± 2.0 0.3 ± 0.5 0.36 ± 0.05 210 ± 6015 0.1 ± 1.E-04 2240 ± 15 4.4 ± 1.0 2.4 ± 0.5 0.25 ± 0.10 2070 ± 4450 0.1 ± 4.E-04 2230 ± 45 4.8 ± 2.0 5.2 ± 1.5 *** ***

15 0.5 ± 5.E-03 2240 ± 60 4.3 ± 2.5 4.3 ± 1.0 0.24 ± 0.10 5955 ± 479530 1 ± 3.E-03 2230 ± 20 4.7 ± 1.0 0.9 ± 0.5 0.33 ± 0.20 560 ± 43015 1 ± 2.E-01 2230 ± 35 4.8 ± 1.5 5.2 ± 0.5 *** ***0 1 ± 2.E-01 2250 ± 30 4.0 ± 1.5 5.5 ± 1.5 *** ***

30 3 ± 3.E-02 2235 ± 35 4.6 ± 1.5 1.8 ± 1.0 0.24 ± 0.10 1510 ± 67030 5 ± 3.E-02 2235 ± 35 4.5 ± 1.5 2.0 ± 0.5 0.29 ± 0.05 1265 ± 60030 46 ± 13 2215 ± 85 5.4 ± 4.0 4.7 ± 2.0 *** ***

TU Delft ACRe 43

: the strength will not increase indefinitely, but reach a limit: ft=C. Based on the “glass modulus” of bitumens, this limit is expected to be temperature independent. Yet it will be reached sooner (at lower strain rates)for lower temperatures. T-: for extremely low temperatures, asphalt will exhibit glass-like, linear-elastic behaviour until sudden, brittle fracture occurs. This ultimate tensile strength will be

independent of the loading rate: ft=C, in contrast with the apparent compressive strength which is the part of the ultimate strength that can be realised for a specific combination of temperature and loading rate.

T: for extremely high temperatures (approximately 160oC) the bitumen will become a fluid, which has no uniaxial strength: ft=0

These considerations were also used to develop an expression for the uniaxial compressive strength (Erkens and Poot 1998 and 2000a), where they resulted in an non-linear relation (Equation 4.7).

Where is the strain rate in s-1, T the temperature in Kelvin en a,b,c,d are regression constants. The same general relation was fitted to the tension data, shown in Figure 4.12. The non-linear regression analysis was carried out using the statistical package SPSS 9.0. In these analyses iterative estimation algorithms are used to find the most appropriate set of regression constants. Due to the non-linear nature of the relations, there are local minima and maxima in the solution, as a result the solution found is not necessarily the best one. The quality of the results can depend on the analysis method selected and on the choice of the starting values for the regression constants. To asses if the solution was really the optimal one, several sets of starting values were used. These starting values as well as the corresponding solutions are shown in Table 4.6. Initial values Result

a b c d a b c d Set 1 1 0 1 1 Error about size of exponent Set 2 5 -80 25400 0.9 5.555 -80.2096 25047 0.864 Set 3 5 0 1 1 Error about size of exponent Set 4 1 -80 1 1 Error about size of exponent Set 5 1 0 25400 1 5.555 -80.2096 25047 0.864 Set 6 1 0 1 0.9 Error about size of exponent Set 7 1 0 1000 1 No convergence Set 8 1 0 1200 1 5.555 -80.2096 25047 0.864 Set 9 50 0 1200 1 5.555 -80.2096 25047 0.864

Table 4.6: The results of non-linear regression for Equation 4.7 using different starting values

The algorithm used in the analyses was the Levenberg-Marquardt algorithm, the Sequential Quadratic Programming algorithm did not work for this data set. The starting values were based on “hand-fitting” the relation to the tension data in Excel. From the overview it can be seen that if a solution is found, it is always the same one. As a result, the authors feel confident that this is the most suitable description for the tension data as a function of temperature and strain rate. That the

Lab W&S ACRe 44

relation described the test results rather well can be seen from Figure 4.11, where the lines are the predicted strength values and the markers the individual test results.

0.0 1.0 2.0 3.0 4.0 5.0 6.0

strain rate centre [% / s]

ft [N/mm2] Test data 0'C Eq. 2.3 for T=273K (0'C)

Test data 15'C Eq. 2.3 for T=288K (15'C)

Test data 30'C Eq. 2.3 for T=303K (30'C)

Figure 4.12: Tension test results (markers) and Equation 4.7

TU Delft ACRe 45

5. MODEL PARAMETER DETERMINATION

5.1 THE MATERIAL MODEL A material model consists of a yield surface and constitutive relations. The yield surface separates elastic from inelastic response and the constitutive relations describe the strains as a function of the stresses for both these types of response. The most well known example is that of uniaxial ideal plasticity (Figure 5.1). For stresses smaller than the yield stress the material behaves elastically, according to Hooke’s law, and for a stress equal to the yield stress the material starts to yield. In other words, it exhibits increasing deformations at he same stress level, which means that the relation between stresses and strains has changed. The state of stress at which the transition between the two types of response occurs is determined by the flow surface. This surface is the collection of all the stress combinations that will cause this transition from one type of response to the other. In the uniaxial case, this transition is a point, in a two-dimensional (plane stress) situation it is a line and in the three-dimensional case it is a surface. The flow surface is expressed as a function of the state of stress (f()), if this function is negative (f()<0) the state of stress is within the yield surface and the response is elastic. If the function for the yield surface for the applied stress is zero ((f()=0), that state of stress is positioned on the yield surface and the response is inelastic. States of stress outside the yield surface cannot exist.

Constitutive relations:1) =E

2) =E( ) = f/

Inside the ellipse: constitutive relation 1)On the ellipse:constitutive relation 2)

Inside the cylinder: constitutive relation 1)On the :constitutive relation 2)

cylinder

Figure 5.1: Example of ideal plasticity, showing both constitutive relations. Also showing the yield surface in 2D and 3D.

The material model used in the ACRe project is based on the flow surface that was proposed by Desai et al. (1986). The model formulation is given in Equation 5.1.

Lab W&S ACRe 46

)3cos(1

Jf (5.1)

Where: invariant stressfirst the1 zzyyxxI

2 2 22 2 22 2 3 1 2 1 3 2 3

2 2 2 2 2 2

6the second deviatoric stress invariant

2 ( ) ( ) ( )

xx yy yy zz zz xx xy yz xz

3 xx yy zz xy yz xz xx yz yy xz zz xy

J s s s

J p p p p p p

he third deviatoric stress invariant

the isotropic stress3

3 3cos(3 )

i - th principle deviator stress

xx yy zz

51 10 [Pa]=0.1 [MPa]=atmospheric pressure

, , ,n,R= model parameters, depending on material characteristicsap x

The flow surface is a so-called single surface flow model, which means that it covers all stress conditions. Furthermore, the surface can be used as a limiting surface for soils, granular materials and concrete as well as asphalt. The exact characteristics (size, shape and position in the three-dimensional stress space) depend on the material parameters, which will be different for different materials, but the mathematical expression for the surface is the same. Within the ACRe project this flow surface is combined with constitutive relations that describe elastic, visco-plastic and fracturing response. Thus, the whole range of response observed in pavement materials can be described. From hereon the term ACRe-model will be used to refer to this combination of flow surface and constitutive relations. For more information on the implementation of this model and the precise constitutive relations that were used, the reader is referred to Al-Khoury (1993) van Breda (1994), Scarpas et al (1997b and 1998) and Erkens et al. (2000b). In these publications the model is described in various states of development. During this period the model underwent continuous changes based on the input from both the tests and the numerical developments, with the objective of obtaining a refined material model especially suited for asphalt-type materials. Undoubtedly, future developments in testing and material modeling will continue to initiate changes in the model, always enhancing its capabilities.

5.2 INFLUENCE OF THE MODEL PARAMETERS ON THE FLOW SURFACE The model parameters , n and R each govern a specific aspect of the model. In this section the role of each parameter will be discussed briefly. This description is largely the same as the one presented in the report about the compressions tests (Erkens and Poot 1998) but on details it is different because of the continuing development of the model. Throughout this section the model is plotted in the I1-J2 space, which allows the 3D model to be represented in 2D.

TU Delft ACRe 47

5.2.1 INFLUENCE OF

The model parameter determines the size of the flow surface, the size increases with decreasing so this parameter controls the hardening of the material. For elastic states of stress it retains its original value, but as soon as non-linear states of stress occur decreases until=0 at peak stress, when the hardening stops (Figure 5.2).

0 -0.2-0.15-0.1-0.050m/m]

[N/mm2]

C 0 C =0

Figure 5.2: The model parameter controls the hardening response

The influence of on the size of the flow surface is shown in Figure 5.3. At peak stress, for , the surface reduces to a straight line at the I1-J2 surface. At this point it can be compared to the well-known Mohr-Coulomb flow surface, although the cross-section of the Mohr-Coulomb flow surface is six-facetted rather than smooth, or the Drucker-Prager surface (Figure 5.4).

-8 -6 -4 -2 0 2I1 [MPa]

J2 [MPa]

Figure 5.3: Influence of on the shape of the flow surface, plotted in I1-J2 space

Lab W&S ACRe 48

1) Mohr-Coulomb2) Drucker-Prager3) von Mises

Figure 5.4: For =0 zero the Desai flow surface reduces to the Drucker-Prager surface, which is similar to the well-known Mohr-Coulomb flow surface

5.2.2 INFLUENCE OF The model parameter determines the slope of the (ultimate) surface, the slope increases with increasing It is stress-state independent, but it can vary as a function of, for example, temperature and loading rate (Figure 5.5).

-8 -6 -4 -2 0 2I1 [MPa]

nR=MPa

J2 [MPa]

Figure 5.5: Influence of on the shape of the flow surface, plotted in I1-J2 space

After the peak load, once =0, degradation of can be used to obtain an overall reduction in strength via isotropic softening. In this case will be expressed as some decreasing function of the non-linear strains.

TU Delft ACRe 49

5.2.3 INFLUENCE OF n

The model parameter n determines the apex of the surface, it expresses the state of stress after which the material starts to dilate. The apex is defined as that point of the flow surface where the tangent is a horizontal line (f/I1=0, see Figure 5.6), indicating a fully deviatoric state of stress. How this condition can be used to determine n, will be discussed later on. Changes in the value of n do not only influence the shape, but also the size of the surface. This is shown in Figure 5.6.

-7 -6 -5 -4 -3 -2 -1 0 1 2

I1 [MPa]

n=3n=2.5

n=3.25n=3.5

J2 [MPa]

Figure 5.6 Influence of n on the shape of the flow surface, plotted in I1-J2 space

5.2.4 INFLUENCE OF The model parameter beta determines the shape of the model on the -plane1. For it’s circular and with increasing it becomes triangular. Since cos(3) is 1 for uniaxial states of stress, the square root term in Equation 5.1 reduces to: (1-). The effect of uniaxiality on is:

2 2 4cos(3 ) 1 0 0, or +2k , with k=1,2,3..

3 3 3k

These values of correspond to the position of the , and axes on the -plane, indicating that uniaxial test results are related to a state of stress where two principal stresses are zero and the third is not. Whether or not the shape of the cross section is triangular is determined by the multiaxial states of stress, between the axes (Figure 5.7). Therefore, uniaxial test results can not be used to determine this aspect of material behaviour and, until multiaxial tests confirm or deny it, the behaviour is assumed to be independent of . This assumption means that is set to zero until it can be determined from the results of multiaxial tests. Basically, this means that the cross section on the -lane becomes a circle, like it is for the Drucker-Prager flow surface (Figure 5.4).

1 The -plane is the plane perpendicular to the isotropic stress axis (1=2=3) which passes through the origin in the principal stress space. Any state of stress can be described by cylinder co-ordinates using its projection on the -plane. These co-ordinates are the distance from the original point to its projection along the isotropic stress axis (=I1/3), the distance from the origin to the projection (r=(2J2)) and the angle between r and the 1 axis (). The advantage of this representation of the state of stress is that it immediately separates the influence of the isotropic stresses, which cause volume changes, and the deviatoric stresses, which cause distortion.

tangent

Lab W&S ACRe 50

Sqrt(J2) [MPa]

Figure 5.7: Influence of and on the flow surface plotted in the I1-J2 and -plane

5.2.5 INFLUENCE OF R

R is the three dimensional tensile strength, which is an indication of cohesion. For cohesionless materials R=0, which means that only states of stress for which I10 are possible within that material. For increasing R values, the flow surface moves in the direction of the positive I1 axis (Figure 5.8).

Figure 5.8: Influence of R on the shape of the flow surface plotted in the I1-J2 space

Strain rate and temperature sensitivity can be incorporated into the classical Desai surface by specifying the model parameters as functions of strain rate and temperature. Since the parameters are determined from tests at different temperatures and strain rates (see next section), the model parameters are automatically obtained for various conditions.

TU Delft ACRe 51

5.3 DETERMINE MODEL PARAMETERS FROM UNIAXIAL DATA For uniaxial states of stress, the expression for the flow surface simplifies considerably. In that case:

1 2 3, 0

1 1 2 3I (5.3) 2 2 2 2 2 2

21 1 12 1 2 3

1 1 2 1

2 3 3 3 2 3 3 3 3

I I IJ

31 1 13 1 2 3

3 3 3 27

32 222

2 23 3 3 3 2727 27cos(3 ) 1

2 2 2 11273

Substitution in Equation 5.1 yields:

As was stated in the previous section, is assumed to be zero until the results from multiaxial tests are available, so Equation 5.3 reduces to:

p (5.8)

5.3.1 DETERMINE R AND It was already stated that the hardening parameter is zero at peak stress. As a result, the first term within brackets in Equation 5.8 becomes zero and the expression becomes:

1Rff cc (5.9)

The only model parameters left in this expression are R, which is the three dimensional tensile strength, and . R can be found from uniaxial tension and compression data as the intercept with the I1 axis of a line through the tensile and compressive strength, plotted in I1-J2 space (Figure 5.9).

Lab W&S ACRe 52

-11 -9 -7 -5 -3 -1 1 3 5

x=|ft-fc|

J2 [MPa]

y=|sqrt(1/3*fc2)-sqrt(1/3*ft

Figure 5.9: R can be determined from uniaxial tension and compression test results

In previous publications, no or only preliminary tension data were available and as a result only approximate values for the model parameters could be obtained. Now that the results from both the compression and the tension test are available, four out of five model parameters can accurately be determined. Using the results from the tension and compression tests, R can be found as:

(5.10)

And is:

1 13 3

f R ff f

(5.11)

Since general expressions for ft and fc as functions of strain rate and temperature are available, both R and can now be expressed in a similar way. Basically, R and are the intercept and slope of the ultimate surface, the line through fc and ft on the I1-√J2 plane. For the conditions tested, some of these lines cross (Figure 5.10) and as a result the relations for R and as a function of temperature and strain rate are not straightforward. These relations, found by substituting the general expressions for fc and ft in Equations 5.10 and 5.11, are plotted in and . Both the data points and the relations shown in these graphs are based on the analytical relations presented above (Equations 5.10 and 5.11), for the data points the average tensile and compressive strength values from the experiments were substituted, for the general relations (Equation 4.7) for the strength values were used.

3/1Uniaxial compressive strength (fc, (1/3fc

Uniaxial tensile strength (ft, (1/3ft

TU Delft ACRe 53

-60.0 -50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0

SQRT(J2) [N/mm2]

I1 [N/mm2]

0 & 10 0 & 5 0 & 1 0 & 0.10 & 0.01 0 & 0.001 15 & 10 15 & 515 & 1 15 & 0.5 15 & 0.1 30 & 1030 & 5 30 & 3 30 & 1 30 & 0.1

Figure 5.10: Ultimate slopes (lines through fc and ft ) for the test conditions

0 1 2 3 4 5 6 7 8 9 10 11 12

strain rate [%/s]

R [N/mm2] 30'C15'C0'CEq. 5.10

Figure 5.11: Data points and general expressions for R

Lab W&S ACRe 54

0 1 2 3 4 5 6 7 8 9 10 11 12

strain rate [%/s]

30'C15'C0'CEq. 5.11

Figure 5.12: Data points and general expressions for

5.3.2 DETERMINE n

The model parameter n is related to the onset of dilation in the specimen (Section 5.2.3). Dilation is the increase in volume that results from the opening of internal cracks. At the beginning of a compression test, the axial strain is larger than the radial strain, which leads to a decrease in volume. Once dilation starts, the volume increases. Therefore, the onset of dilation is related to the point at which the volumetric strain changes from decreasing to increasing. Like stresses, strains can be decomposed in a volumetric and a deviatoric part. The volumetric strains are related to changes in volume and the deviatoric strains are related to changes in shape. In this case, only the volumetric strains are of interest and in particular the plastic volumetric strains, since dilation does not occur in the elastic region. The beginning of dilation can therefore be defined as the minimum of the plastic volumetric strain curve, or mathematically:

0pvol (5.12)

(5.13)

where: plastic multiplier (has to be determined experimentally) plastic flow function, the McCauley brackets indicate that plastic flow only occurs for

positive values of the flow function (<for and<for

:an indication of the direction of plastic straining

ij :Kronecker delta, 0 for ij and 1 for i=j Since the flow surface f is defined as a function of the stress invariants rather than the stress vector, Equation 5.9 becomes:

TU Delft ACRe 55

pvol ij

ij ij ij

Jf I f J f

(5.14)

The derivatives of J2 and J3 to the stress vector reduce to zero after multiplication with ij. This leads to:

3 0pvol

(5.15)

If (I1-R)0, the derivative of the flow function to the first stress invariant is zero for:

211 (5.16)

The left-hand solution is a trivial one, so the right-hand one will be used in the remainder of this derivation. Rewriting the expression for the flow surface results in:

)3cos(1

Jf (5.17)

Substitute Equation 5.12 in 5.13 to get:

2 )3cos(11

2 (5.18)

Evaluating Equation 5.15 for uniaxial states of stress yields:

*( )133

f fR f

(5.19)

The state of stress in this expression must be evaluated at the beginning of dilation (dil). This point can be determined by means of the axial versus volumetric strain plot and the axial strain versus stress plot. The first plot can be used to determine the axial strain at minimum volumetric strain and

Lab W&S ACRe 56

with this strain the corresponding state of stress can be determined using the second plot (Figure 5.13).

-0.015-0.01-0.0050

vol[m/

axial m/m

-0.003

-0.002

-0.001

-1.0mm

Figure 5.13: Determine the state of stress at the onset of dilation

This information is obtained from the compression tests (Erkens and Poot 1998 and 2000a) but it could not be used to evaluate Equation 5.19 until the results from the tension tests were available. From Equation 5.19 it is clear that the only term in the expression for n that is not yet known as a function of the temperature and strain rate is the stress at the onset of dilation, dil. For this reason, it was attempted to express dil as a function of strain rate and temperature, using the same general relation as for the tension and compression strength. The result is shown in Figure 5.14, where Equation 5.20 is plotted through the data points.

0.432781396

1 11 56 1

dil dcb

(5.20)

Where:

=strain rate [m/m]

= stress at the onset of dilation [N/mm ]

T = temperature [K]

a,b,c,d= regression constants

TU Delft ACRe 57

0 1 2 3 4 5 6 7 8 9 10 11 12 [%/s]

dil [N/mm2]

30'C&0.1%/s 15'C&0.1%/s 0'C&0.1%/s30'C&1%/s 15'C&1%/s 0'C&1%/s30'C&5%/s 15'C&5%/s 0'C&5%/s30'C&10%/s 15'C&10%/s 0'C&10%/s303K 288K 273K

Figure 5.14: Individual values of the stress at the onset of dilation, combined with the general expression (Equation5.20 )

The expression for n (Equation 5.19), which exhibits a vertical asymptote for ≈fc after a long horizontal branch, leads to very high n-values if dilatation starts at a stress close to the uniaxial compressive strength. A plot of n as a function of the stress is shown in Figure 5.15. This plot shows that for states of stress close to the compressive strength a small variation in the stress at the beginning of dilation results in a large variation in n. On the other hand, if the state of stress at the onset of dilation differs considerably from the compressive strength, a variation in the state of stress causes hardly any variation in n.

Figure 5.15: Plot of n as a function of the stress at the onset of dilation (Eq. 5.16), the asymptote near fc explains the larger variation in n-values for states of stress near fc

An overview of the state of stress at the onset of dilation for he various test conditions is shown in Table 5.1 (a) and (b). To illustrate the effect of small differences between dil and fc also the n-values are shown. Because R and are needed to determine n, these parameters are also incorporated in this table. To get a good indication of the variation, the values are determined for the individual compression tests, using the average tensile strength at each condition for the required tensile input. The average tensile strength has to be used because it is not possible to relate individual compression results to individual tension results since the tests were not performed on the same specimen.

Lab W&S ACRe 58

file code T [oC] ' [%/s] fc [N/mm2] dil [N/mm2] dil-ft R n*

com01_1 D66 30 0.10 -2.0 -0.2 88.2 0.82 0.16 2.2

com01_2 D209 30 0.10 -1.9 -0.5 72.6 0.83 0.16 2.9

com01_3 D344 30 0.10 -1.8 -0.8 52.8 0.84 0.15 4.4

com01_4 D211 30 0.10 -1.8 -0.2 91.3 0.84 0.16 2.1

com01_5 D345 30 0.10 -1.9 -1.1 43.6 0.82 0.16 5.9

average -1.9 -0.6

stdev 0.1 0.3

com01_6 D206 15 0.1 -6.1 -4.1 33.0 7.8 0.06 5.2com01_7 D210 15 0.1 -5.8 -3.9 33.7 8.0 0.06 5.0com01_8 D221 15 0.1 -5.4 -3.5 34.6 8.5 0.05 4.7com01_9 D207 15 0.1 -5.7 -3.5 37.7 8.2 0.06 4.4com01_10 D215 15 0.1 -5.3 -3.7 29.8 8.6 0.05 5.4average -5.7 -3.8stdev 0.3 0.2com01_11 D63 0 0.1 -21.5 -19.5 9.2 13.6 0.13 27.0com01_12 D111 0 0.1 -22.2 -20.7 6.9 13.4 0.13 37.4com01_13 D120 0 0.1 -21.1 -19.8 6.4 13.6 0.12 38.7

com01_14 D117 0 0.1 -21.9 -20.7 5.3 13.5 0.13 48.6

com01_15 D122 0 0.1 -21.5 -19.7 8.3 13.6 0.13 30.2

average -21.6 -20.1

stdev 0.4 0.6

com1_1 D202 30 1 -3.5 -0.7 81.5 2.5 0.12 2.3

com1_2 D362 30 1 -3.2 -0.7 79.0 2.5 0.10 2.3

com1_3 D67 30 1 -3.4 -0.8 76.0 2.5 0.11 2.4

com1_4 D363 30 1 -3.0 -0.9 71.4 2.6 0.10 2.5

com1_5 D62 30 1 -3.8 -2.5 32.7 2.4 0.12 6.8

average -3.4 -1.1

stdev 0.3 0.8

com1_6 D219 15 1 -11.2 -8.0 27.9 19.2 0.04 5.6

com1_7 D113 15 1 -11.3 -9.8 13.5 19.0 0.05 11.7

com1_8 D203 15 1 -10.9 -9.0 17.4 19.7 0.04 8.9

com1_9 D48 15 1 -12.7 -10.6 17.1 17.4 0.06 9.9

com1_10 D204 15 1 -11.6 -10.1 13.2 18.6 0.05 12.2

average -11.5 -9.5

stdev 0.7 1.0

com1_11 D133 0 1 -37.6 -35.6 5.4 13.0 0.18 69.1

com1_12 D118 0 1 -37.6 -36.2 3.6 13.0 0.18 105.5

com1_13 D131 0 1 -38.4 -36.6 4.7 13.0 0.19 82.5

com1_14 D127 0 1 -37.9 -35.8 5.5 13.0 0.18 68.3

com1_15 D201 0 1 -37.7 -35.9 5.0 13.0 0.18 76.0

average -37.8 -36.0

stdev 0.3 0.4

Table 5.1a: n-related stress values for the individual tests

TU Delft ACRe 59

file code T [oC] ' [%/s] fc [N/mm

2] dil [N/mm

2] dil-ft R n

com5_1 D364 30 5 -5.1 -1.1 77.6 6.8 0.06 2.3

com5_2 D353 30 5 -5.3 -1.4 74.6 6.5 0.07 2.3

com5_3 D346 30 5 -5.8 -1.5 75.0 6.2 0.08 2.4

com5_4 D365 30 5 -5.2 -2.0 61.9 6.6 0.06 2.8

com5_5 D342 30 5 -4.5 -0.9 79.0 7.4 0.05 2.2

average -5.2 -1.4

stdev 0.5 0.4

com5_6 D368 15 5 -17.1 -13.1 23.5 15.4 0.09 8.4

com5_7 D360 15 5 -16.3 -13.4 17.9 15.7 0.09 10.9

com5_8 D359 15 5 -16.9 -14.9 11.8 15.4 0.09 17.2

com5_9 D356 15 5 -16.7 -13.9 16.6 15.5 0.09 11.9

com5_10 D354 15 5 -17.3 -13.7 20.4 15.3 0.09 9.8

average -16.8 -13.8

stdev 0.4 0.7

com5_11 D366 0 5 -52.0 -44.0 15.4 12.3 0.22 30.2

com5_12 D352 0 5 -50.8 -46.5 8.5 12.3 0.22 56.9

com5_13 D349 0 5 -50.9 -48.6 4.5 12.3 0.22 109.2

com5_14 D351 0 5 -44.0 -40.0 9.1 12.6 0.20 46.3

com5_15 D369 0 5 -49.4 -43.1 12.7 12.4 0.21 35.8

average -49.4 -44.4

stdev 3.2 3.3

com10_1 D50 30 10 -8.6 -4.7 45.5 9.0 0.08 3.9

com10_2 D125 30 10 -6.8 -2.7 61.1 10.4 0.05 2.7

com10_3 D140 30 10 -8.0 -5.3 33.9 9.4 0.07 5.2

com10_4 D212 30 10 -6.5 -3.1 52.8 10.8 0.05 3.1

com10_5 D138 30 10 -7.5 -3.1 59.4 9.7 0.06 2.9

average -7.5 -3.7

stdev 0.9 1.2

com10_6 D137 15 10 -23.0 -17.0 26.1 14.1 0.13 9.0

com10_7 D130 15 10 -21.6 -17.2 20.0 14.4 0.12 11.5

com10_8 D65 15 10 -23.0 -15.6 32.0 14.1 0.13 7.1

com10_9 D132 15 10 -22.0 -15.8 28.2 14.3 0.12 8.0

com10_10 D135 15 10 -21.9 -17.5 20.2 14.3 0.12 11.5

average -22.3 -16.6

stdev 0.7 0.9

com10_11 D116 0 10 -54.6 -43.1 20.9 12.2 0.22 22.1

com10_12 D123 0 10 -57.0 -42.5 25.5 12.2 0.23 18.1

com10_13 D61 0 10 -58.4 -47.4 18.7 12.1 0.23 26.7

com10_14 D114 0 10 -57.1 -44.9 21.4 12.2 0.23 22.4

com10_15 D218 0 10 -56.6 -43.1 23.9 12.2 0.23 19.5

average -56.7 -44.2

stdev 1.4 2.0

Table 5.2b: n-related stress values for the individual tests (continued)

From these tables it can be seen that for small differences between dil and fc the n-values increase. Also, for these conditions small variations in dil result in large differences in n-values. In some cases, the n-values become extremely large. Since this is due to the asymptote, it seems reasonable

Lab W&S ACRe 60

to use a cut-off value to prevent “infinite” n-values. In the next section about a the relation between n and a is used to find a reasonable cut-of value.

5.3.3 DETERMINE

The last parameter to be determined, , controls the hardening. Since all other parameters are known ( is assumed to be zero, as stated before) can be computed for each stress level in a compression or tension test using the expression for the flow surface (Equation 5.21).

)3cos(1

0)3cos(1

(5.21)

Based on Equation 5.21 values for for all states of stress throughout the stress strain curves can be found. However, based on the definition for , only the values between the onset of non-linearity and peak strength have to be determined. The value found for the state of stress at the onset of non-linearity gives 0 and from the peak stress until complete annihilation of strength, =0. To find 0,

Equation 5.21 must be evaluated at the state of stress at which non-linearity is initiated (plas). All the parameters in the equation are known for any combination of temperature and strain rate, with the exception of the state of stress. If an expression for plas can be found, Equation 5.21 can be used to determine the 0-values for every combination of temperature and strain rate. Similarly as for dil in the previous section, the same general relation as for the tension and compression strengths was used to describe plas. The result is shown in , where the individual data points are shown along with the relation that was found (Equation 5.22).

0 1 2 3 4 5 6 7 8 9 10 11 12 [%/s]

plas [N/mm2]

30'C&0.1%/s 15'C&0.1%/s 0'C&0.1%/s30'C&1%/s 15'C&1%/s 0'C&1%/s30'C&5%/s 15'C&5%/s 0'C&5%/s30'C&10%/s 15'C&10%/s 0'C&10%/s303K 288K 273K

Figure 5.16: Individual values of the stress at the onset of dilation, combined with the general expression (Equation 5.22)

TU Delft ACRe 61

0.482654192

1 11 42 1

plas dcb

(5.22)

Where:

=strain rate [m/m]

= stress at the onset of dilation [N/mm ]

T = temperature [K]

a,b,c,d= regression constants

The degradation of from 0 at the onset of plasticity to zero at the peak can be expressed as either a function of the equivalent plastic strain

Tp,ij p, ij ; i, j x, y, z x e e (5.23)

or the plastic work Wp:

p pW d s e (5.24)

In earlier publications, based on the preliminary tests, Wp was used since this parameter does not require information on the radial deformations. Currently, is controlled via The relation used to describe the degradation is:

0 e kxa a (5.25)

Where:

the equivalent plastic strain

the initial value (material parameter)

controls the rate of degradation (material parameter)

Both the parameters in this expression are heavily influenced by the n-values. Especially 0 becomes very small for large n-values. These small 0 are equal to zero for all practical purposes. Especially in any numerical application, values smaller than 10-16 are equivalent to zero. In the original values found for n and 0 on the basis of the test results and the analytical relations for the model parameters are shown (printed in italics) as well as the values found when a cut-off value of 10 is used for the n-values. This cut-off consisgts of two requirements, if n is found to be larger than 10 or is the difference between fc and dil is smaller than 20%, n is set to 10. The latter requirement is used because the expressions for fc and dil as functions of the temperature and strain rate can, for some conditions lead to dil > fc. This is physically equivalent to a situation where no hardening occurs. In the expression for n, this yields negative n-values, which is now prevented by setting n to 10. When n is set to 10, 0 is set to zero. This means that for those conditions there is no hardening response.

Lab W&S ACRe 62

file code T [oC] ' [%/s] fc [N/mm2] R n*

com01_1 D66 30 0.10 -2.0 0.82 0.16 2.2 2.2 9.71E-02 9.71E-02

com01_2 D209 30 0.10 -1.9 0.83 0.16 2.9 2.9 2.14E-02 2.14E-02

com01_3 D344 30 0.10 -1.8 0.84 0.15 4.4 4.4 5.04E-04 5.04E-04

com01_4 D211 30 0.10 -1.8 0.84 0.16 2.1 2.1 1.16E-01 1.16E-01

com01_5 D345 30 0.10 -1.9 0.82 0.16 5.9 5.9 1.77E-05 1.77E-05

average -1.9

stdev 0.1

com01_6 D206 15 0.1 -6.1 7.8 0.06 5.2 5.2 1.66E-08 1.66E-08com01_7 D210 15 0.1 -5.8 8.0 0.06 5.0 5.0 5.56E-08 5.56E-08com01_8 D221 15 0.1 -5.4 8.5 0.05 4.7 4.7 1.24E-07 1.24E-07com01_9 D207 15 0.1 -5.7 8.2 0.06 4.4 4.4 6.33E-07 6.33E-07com01_10 D215 15 0.1 -5.3 8.6 0.05 5.4 5.4 6.09E-09 6.09E-09average -5.7stdev 0.3com01_11 D63 0 0.1 -21.5 13.6 0.13 27.0 10.0 7.15E-62 0.00E+00com01_12 D111 0 0.1 -22.2 13.4 0.13 37.4 10.0 5.77E-88 0.00E+00com01_13 D120 0 0.1 -21.1 13.6 0.12 38.7 10.0 3.02E-90 0.00E+00

com01_14 D117 0 0.1 -21.9 13.5 0.13 48.6 10.0 3.92E-114 0.00E+00

com01_15 D122 0 0.1 -21.5 13.6 0.13 30.2 10.0 5.99E-70 0.00E+00

average -21.6

stdev 0.4

com1_1 D202 30 1 -3.5 2.5 0.12 2.3 2.3 4.06E-02 4.06E-02

com1_2 D362 30 1 -3.2 2.5 0.10 2.3 2.3 3.20E-02 3.20E-02

com1_3 D67 30 1 -3.4 2.5 0.11 2.4 2.4 2.06E-02 2.06E-02

com1_4 D363 30 1 -3.0 2.6 0.10 2.5 2.5 1.40E-02 1.40E-02

com1_5 D62 30 1 -3.8 2.4 0.12 6.8 6.8 8.13E-09 8.13E-09

average -3.4

stdev 0.3

com1_6 D219 15 1 -11.2 19.2 0.04 5.6 5.6 1.17E-10 1.17E-10

com1_7 D113 15 1 -11.3 19.0 0.05 11.7 10.0 2.58E-25 0.00E+00

com1_8 D203 15 1 -10.9 19.7 0.04 8.9 10.0 1.82E-18 0.00E+00

com1_9 D48 15 1 -12.7 17.4 0.06 9.9 10.0 7.80E-21 0.00E+00

com1_10 D204 15 1 -11.6 18.6 0.05 12.2 10.0 5.45E-26 0.00E+00

average -11.5

stdev 0.7

com1_11 D133 0 1 -37.6 13.0 0.18 69.1 10.0 5.39E-176 0.00E+00

com1_12 D118 0 1 -37.6 13.0 0.18 105.5 10.0 3.79E-267 0.00E+00

com1_13 D131 0 1 -38.4 13.0 0.19 82.5 10.0 6.61E-209 0.00E+00

com1_14 D127 0 1 -37.9 13.0 0.18 68.3 10.0 6.51E-171 0.00E+00

com1_15 D201 0 1 -37.7 13.0 0.18 76.0 10.0 7.73E-191 0.00E+00

average -37.8

stdev 0.3

Table 5.3a: Model parameters for individual tests, showing n and a0 before and after imposing the cut-off limit

TU Delft ACRe 63

file code T [oC] ' [%/s] fc [N/mm

2] R n

com5_1 D364 30 5 -5.1 6.8 0.06 2.3 2.3 1.77E-02 1.77E-02

com5_2 D353 30 5 -5.3 6.5 0.07 2.3 2.3 1.26E-02 1.26E-02

com5_3 D346 30 5 -5.8 6.2 0.08 2.4 2.4 1.42E-02 1.42E-02

com5_4 D365 30 5 -5.2 6.6 0.06 2.8 2.8 2.29E-03 2.29E-03

com5_5 D342 30 5 -4.5 7.4 0.05 2.2 2.2 1.89E-02 1.89E-02

average -5.2

stdev 0.5

com5_6 D368 15 5 -17.1 15.4 0.09 8.4 8.4 5.87E-17 5.87E-17

com5_7 D360 15 5 -16.3 15.7 0.09 10.9 10.0 2.58E-23 0.00E+00

com5_8 D359 15 5 -16.9 15.4 0.09 17.2 10.0 2.56E-38 0.00E+00

com5_9 D356 15 5 -16.7 15.5 0.09 11.9 10.0 6.07E-25 0.00E+00

com5_10 D354 15 5 -17.3 15.3 0.09 9.8 9.8 5.67E-20 5.67E-20

average -16.8

stdev 0.4

com5_11 D366 0 5 -52.0 12.3 0.22 30.2 10.0 3.52E-78 0.00E+00

com5_12 D352 0 5 -50.8 12.3 0.22 56.9 10.0 1.05E-144 0.00E+00

com5_13 D349 0 5 -50.9 12.3 0.22 109.2 10.0 6.27E-283 0.00E+00

com5_14 D351 0 5 -44.0 12.6 0.20 46.3 10.0 1.35E-115 0.00E+00

com5_15 D369 0 5 -49.4 12.4 0.21 35.8 10.0 3.87E-92 0.00E+00

average -49.4

stdev 3.2

com10_1 D50 30 10 -8.6 9.0 0.08 3.9 3.9 7.72E-06 7.72E-06

com10_2 D125 30 10 -6.8 10.4 0.05 2.7 2.7 1.15E-03 1.15E-03

com10_3 D140 30 10 -8.0 9.4 0.07 5.2 5.2 1.27E-08 1.27E-08

com10_4 D212 30 10 -6.5 10.8 0.05 3.1 3.1 2.19E-04 2.19E-04

com10_5 D138 30 10 -7.5 9.7 0.06 2.9 2.9 8.39E-04 8.39E-04

average -7.5

stdev 0.9

com10_6 D137 15 10 -23.0 14.1 0.13 9.0 9.0 5.72E-19 5.72E-19

com10_7 D130 15 10 -21.6 14.4 0.12 11.5 10.0 1.19E-24 0.00E+00

com10_8 D65 15 10 -23.0 14.1 0.13 7.1 7.1 2.20E-14 2.20E-14

com10_9 D132 15 10 -22.0 14.3 0.12 8.0 8.0 2.16E-16 2.16E-16

com10_10 D135 15 10 -21.9 14.3 0.12 11.5 10.0 1.83E-24 0.00E+00

average -22.3

stdev 0.7

com10_11 D116 0 10 -54.6 12.2 0.22 22.1 10.0 3.63E-56 0.00E+00

com10_12 D123 0 10 -57.0 12.2 0.23 18.1 10.0 5.42E-45 0.00E+00

com10_13 D61 0 10 -58.4 12.1 0.23 26.7 10.0 6.61E-68 0.00E+00

com10_14 D114 0 10 -57.1 12.2 0.23 22.4 10.0 2.12E-55 0.00E+00

com10_15 D218 0 10 -56.6 12.2 0.23 19.5 10.0 2.71E-48 0.00E+00

average -56.7

stdev 1.4

Table 5.4b: Model parameters for individual tests, showing n and a0 before and after imposing the cut-off limit (continued)

The second parameter in the expression for the hardening parameter is important only when o is unequal to zero. As a result, only the conditions for which this is the case were used to find an

Lab W&S ACRe 64

expression for . The data points used to establish this relations are shown in Figure 5.17, from this picture it can be seen that only -values for 30 and 15 oC are used. A numerical overview of the -values for all conditions is presented in Table 5.5.

-100 0 1 2 3 4 5 6 7 8 9 10 11

strain rate [%/s]

Figure 5.17: -values as a function of the strain rate and temperature (Eq. 5.26)

1 1 0.097

(5.26)

With: n: the model parameter related to dilation (see previous section) a,b: regression coefficients

T [oC] [%/s] fc [N/mm

2] ft [N/mm

2] R dil n

0 0.001 -5.8 2.7 ± 0.5 9.7 0.050 0.01 -11.5 5.1 ± 1.5 18.0 0.05

30 0.1 -1.9 ± 0.1 0.3 ± 0.5 0.8 0.16 -0.35 2.4 5.62E-02 -111 -0.1315 0.1 -5.7 ± 0.5 2.4 ± 0.5 8.2 0.06 -3.83 5.0 5.18E-08 -168 -1.110 0.1 -21.5 ± 1 5.2 ± 1.5 13.6 0.13 -19.68 10.0 0.00E+00 -3000 -10.90

15 0.5 -9.4 4.3 ± 1.0 16.0 0.05 2.030 1 -3.5 ± 0.5 0.9 ± 0.5 2.5 0.12 -0.77 2.4 2.69E-02 -102 -0.5915 1 -11.5 ± 1 5.2 ± 0.5 18.8 0.05 -9.24 8.1 2.83E-16 -413 -2.100 1 -38.0 ± 1 5.5 ± 1.5 13.0 0.19 -36.45 10.0 0.00E+00 -1900 -23.88

30 3 -4.6 1.8 ± 1.0 5.9 0.06 2.030 5 -5.0 ± 1 2.0 ± 0.5 6.8 0.06 -1.48 2.4 8.10E-03 -119 -1.1215 5 -17.0 ± 1 5.3 ± 15.4 0.09 -14.22 10.0 0.00E+00 -949 -5.550 5 -49.5 ± 4.5 5.5 ± 12.4 0.21 -44.21 10.0 0.00E+00 -8290 -30.59

30 10 -7.5 ± 1.5 3.0 ± 2.0 9.7 0.06 -3.66 3.3 8.72E-05 -156 -2.7015 10 -22.3 ± 1 5.4 ± 14.2 0.12 -17.37 10.0 0.00E+00 -1259 -12.060 10 -56.5 ± 2 5.5 ± 12.2 0.23 -44.24 10.0 0.00E+00 -3200 -18.56

Table 5.5: Overview of the model parameters for the various test conditions

The bold lines refer to conditions that were tested both in compression and tension, the other conditions were tested solely in tension or compression. In those cases, the other strength at that condition is determined using the general expressions for ft and fc. Besides the model parameters, also the stress at the onset of dilation and that at the onset of plasticity are shown.

TU Delft ACRe 65

To illustrate the effect of Equation 5.26, the average normalized degradation curves for the test conditions shown in Table 5.5 are compared to the curves found using the -values from the equation. This is shown in Figure 5.18, the n-values that correspond to the test conditions are shown in Table 5.5.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

15'C&1%/s 15'C&0.1%/s30'C&10%/s 30'C&5%/s30'C&1%/s 30'C&0.1%/sn=8 n=5n=3.3 n=2.4n=2

Figure 5.18: Normalised degradation curves for the hardening parameter (), test results and Equation 5.26

5.4 FINAL REMARKS The relations shown for the various model parameters are based on the test results. Whether these relations, such as the incorporation of general expressions for dil and plas and the cut-off introduced for n will prove applicable in numerical simulations will have to be determined on the basis of actual simulations. The experience gained from those activities may very well lead to the development of other, numerically more appropriate, relations. That does however, not invalidate the applicability of the relations presented here, it only involves practical issues.

Lab W&S ACRe 66

6. CONCLUSIONS AND RECOMMENDATIONS

6.1 CONCLUSIONS The test program described in this report leads to a series of conclusions concerning uniaxial tension testing and the response of asphalt concrete under tension loading. Because of the localised nature of tensile failure, after the peak the specimen exhibits a combination of responses. The crack is opening, while the undamaged specimen halves are unloading. The specimen shape that was developed to allow separation of these different kinds of response works very well. The specimens all failed in the middle part of the specimen, close to the actual centre. Since all the fracture surfaces were rather flat, it can also be concluded that the set-up itself, with its three hinges, works well. A disadvantage of the specimen shape is that there no longer exists a direct relation between the overall deformation and the strain at the height where fracture occurs when non-linear effect play a role. In this report a relation between overall deformations and strain at the centre plane is derived on the basis of the elastic stress and strain distribution. When inelasticity occurs, and from the test results it is plain that this usually does occur prior to fracture, this relation is not valid. As a result the strain rate at the location of fracture is known only by approximation. On the basis of the comparison between the strain gauge data and the overall relations it is assumed that this approximation is reasonable. The variation in the test results is larger than for the compression tests, but this was expected because the localised failure in tension makes the test much more sensitive local weaknesses in the specimen. This was further emphasized by the larger variation in specimen composition that resulted from the specimen shape. This shape makes compaction of the centre part of the specimen, where the crack will occur, difficult. Because it was not possible to maintain the mixture temperature during compaction, this complication led to an increased variation in specimen composition compared to cylindrical specimens. Despite the variations in response, if the response curves are normalised with respect to the stress and deformation values at the peak, the individual curves are very similar for the different conditions. Strain gauges turned out to be the most appropriate tool to register the unloading of the undamaged specimen parts. Despite this, the unloading behaviour could not be captured accurately for all conditions, due to limitations in the data-acquisition equipment. For very brittle conditions the maximum scan rate (100kHz) was insufficient to capture the unloading branch and for test conditions were brittle failure occurred in a relatively slow test (where a slow scan rate had to be used), the unloading could also not be captured. Fortunately, for most test conditions the unloading information was registered. In the unloading branch the elastic (instantaneous) and viscous (delayed) response of the material can be distinguished. The general trend in tensile strength values is similar to that observed for the compressive strength. Only for tension, the plateau strength is actually reached in the tests. Because this happens for slower strain rates at low temperatures, the strain rates were selected separately for each

TU Delft ACRe 67

temperature. The resulting test program provided enough information to develop a relation for the tensile strength as a function of temperature and strain rate. Using this relation for the tensile strength in combination with the one found for the compression tests allowed the model parameters to be determined. These parameters are also expressed as functions of temperature and strain rate to facilitate implementation in the finite elements package CAPA-3D.

6.2 RECOMMENDATIONS In order to capture the brittle unloading behaviour in relatively slow tests, the registration (or at least the scan rates) of the overall data acquisition and the strain gauge registration should some how be separated. This would facilitate the registration of relatively slow overall signal as well as the sudden, fast unloading at fracture. It is, however, known that it is extremely difficult to develop a fast triggering system that facilitates the registration of such brittle fracture response. Strain gauge registration proved very successful, but it is also labour intensive and, thus, time consuming. For this reason, the developments in laser equipment should be kept in mind. At the moment, those systems can not provide the same accuracy as the strain gauges, but in the future this is bound to change. Especially if open graded or coarse mixtures are to be used, a non-contacting system offers direct advantages. For coarser mixtures, if a reliable unloading registration system is available, cylindrical specimens might be used instead of the parabolic shape discussed in this report. For some mixtures tests on cylinders, using PVC rings around the specimen at the gluing plane at the caps (to ensure a larger gluing surface) have already proven to be effective. For relatively homogeneous mixtures as the ACRe mixture, however, it does not seem to work. In a series of three tests, performed to see if the cylinders would yield the same strength and stiffness values, all specimens failed at the caps. If the parabolic specimen shape is to be used in combination with prescribed degrees of compaction in the failure zone, it is highly recommended to develop a means of maintaining a high mixture temperature during the compaction process. That would help to overcome the friction along the inner mould and achieve a sufficient level of compaction in the centre region.

Lab W&S ACRe 68

REFERENCES

Al-khoury, R.I.N., (1993), 3-Dimensional Non-Linear Soil Modelling, International Institute for

Infrastructural, Hydraulical and Environmental Engineering in Delft, Report IP029 Anderson, H., (1986), Statistische Technieken en hun Toepassingen, Nijgh & van Ditmar educatief,

s Gravenhage Breda, M.v., (1994), Finite Elements Simulation of Interfaces in Pavements, C.R.O.W., (1990), RAW Standard Conditions of Contract for Works of Civil Engineering

Construction, VanderLoeff/Drukker B.V., Enschede Desai, C.S., Somasundaram, S. and Frantziskonis, G., (1986), a Hierarchical approach for

Constitutive Modelling of Geologic Materials, International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 10, No. 3, Pg. 225-257

Erkens, S.M.J.G. and Poot, M.R., (1998), The Uniaxial Compression Test - Asphalt Concrete Response (ACRe),

Erkens, S.M.J.G. and Poot, M.R., (2000a), Additional Compression Tests - Asphalt Concrete Response (ACRe),

Erkens, S.M.J.G., Scarpas, A. and Liu, X., (2000b), 3D Finite Element Model for Asphalt Concrete Response Simulation, Paper presented at the 2nd International Symposium on 3D Finite Elements in Pavement Engineering, Charleston,West-Virginia, USA

IPC, (1993), Gyropac Materials Compaction Machine, Operating and maintenance manual, IPC, Australia

Mier, J.G.M.van., (1997a), Fracture Processes of Concrete, CRC Press, Boca Raton New York London Tokyo

Physics, (1986), Handleiding Natuurkundig Practicum, deel 4 foutenanalyse (in Dutch) Scarpas, A., Blaauwendraad, J., Al-Khoury, R. and Gurp, C.v., (1997b), Experimental calibration

of a viscoplastic-fracturing computational model, Paper presented at the Computational Methods and Experimental Measurements (CMEM), Rhodos, Greece, May 1997b

Scarpas, A. and Blaauwendraad, J., (1998), Experimental Calibration of a Constitutive Model for Asphaltic Concrete, Paper presented at the Euro-C conference on the Computational Modelling of Concrete Structures, Badgastein, Oostenrijk, 31 march- 3 april 1998

TU Delft ACRe 69

APPENDIX 1: FILLER COMPOSITION

RL210-01

Steengroeve Laboratory Analysis-report Sample-name Wigro Sample no. 95-134(I);96-

134(II);97153(III) Sample-type Filler Date of arrival 1/09/95(I);22/01/97(II);

11/12/97(III) Sample-code 73300 Document no. Source WSK Principal GD Destination TU-Delft Contact person Verhoeven Transport by G&L Installation Relation Analyse date 01/09/95(I);22/01/97(II);

11/12/97(III) Copy rapport RBL281/TU-Delft Remarks Laboratory stock for external laboratories WI

ANALYSES SI Value of batch no. Warranty

FILLERANALYSES I II III Min. Max. 231 GRAIN SIZE DISTRIBUTION

ALPINE

+ 63 m %(m/m) 19 15 14 5 25 + 90 m %(m/m) 11 9 9 0 15 + 2 mm %(m/m) 0 0 0 0 232 BITUMEN NUMBER ml/100g 47 44 46 42 48 235 MASS LOSS 150 C %(m/m) 0.34 0.6 0.1 0.0 1.5 ANALYSES AFTER DRYING AT 110 C

222 DRYING AT 110 C %(m/m) X X X 236 DENSITY kg/m3 2780 2770 2770 2675 2875 237 VOIDS %(v/v) 38 38 37 36 42 238 SOLVABILITY H2O %(m/m) 2.0 2.2 0.8 0 10 239 SOLVABILITY HCL %(m/m) 63.3 65.5 67.7 55 75

Lab W&S ACRe 70

APPENDIX 2: SIEVE CURVE CRUSHED ROCK 0/5 AND FILLER 00

0.01 0.1 1 10

Sieve size [mm]per

crushed rock

aggregate (crushedrock + filler)

Dry sieving Crushed rock 0/5 Aggregate (sand + filler) Sieve [mm] Percentage passing

[% m/m] Percentage passing

[% m/m] 4 97.5 96.8

2.8 94.9 94.4 2 90.6 90.8 1 60.4 65.3

0.5 34.5 43.4 0.355 26.4 36.6 0.25 18.1 29.6 0.18 11.3 25.2

0.125 5.9 20.6 0.063 1.8 14.6

From the previous Appendix it is known that the filler was sieved on the 63 m, 90 m and 2 mm sieves. It is assumed that the material that remained on the 90 m sieve and passed through the 2 mm sieve would have passed the 125 m sieve used to analyse the sand. The combined sieve data is found by computing the mass percentage on each sieve (difference between two adjacent values) and expressing that as a percentage of the combined mass (m%*=m% * Ms/(Ms+Mf). Where necessary, the percentages for the filler and crushed rock are combined. For example: Sieve 63 m: 1.8% x 2.7/3.2+ 84% x0.5/3.2=14.6% (2.7 kg rock+ 0.5 kg filler=3.2 kg aggregate) Sieve 125 m: (5.9%-1.8%)x2.7/3.2+16%x0.5/3.2+14.6%=20.6% Sieve 180 m: (11.3%-5.9%)x2.7/3.2+20.6%=25.2%, etc.

TU Delft ACRe 71

APPENDIX 3: BITUMEN CHARACTERISTICS 00 1. Penetration test

Pen.45/60 (batch I) Pen.45/60 (batch II)

1e 2e 3e average 1e 2e 3e average

47 46 48 47 47 48 47 47.3

2. Ring & Ball

R&B (batch I) R&B (batch II)

Level thermostat

Level boiling ring

Starting temperature

Result Level thermostat

Level boiling ring

Starting temperature

Result

45 9 8.6 51/51 45 9 7.2 52/52.1

120)log(50)&(

1952)log(500)&(20

penBRT

penBRTPI (A3.1)

Result batch I: PI = -1.10 Result batch II: PI = -0.83 Density bitumen: b= (1020 5) kg/m3

Lab W&S ACRe 72

APPENDIX 4: STRAINGAUGES, TYPE, EQUIPMENT & CALCULATIONS A4.1 MEASURING PRINCIPLE The strain generated in the specimen causes a variation in resistance in the strain gauge, which is proportionally to this strain as follows:

ε = strain in specimen ΔR= resistance change due to strain [Ω] R = gauge resistance [Ω] K = gauge factor This resistance change is very small and requires a Wheatstone bridge to convert it to voltage output:

Figure A4.1: Straingauge on specimen (R1) and dummy strain gauge (R4) in a Wheatstone bridge (half bridge configuration)

In this bridge a dummy straingauge (R4) is used to compensate for thermal expansion of both the straingauge and the specimen, together with the thermal coefficient of resistance of the gauge material. This dummy straingauge is glued on another parabolic specimen in the same way as the specimen under test and put in the temperature cabinet of the test set-up. For this compensation a half-bridge configuration is needed. Now the change in resistance of R1 is only caused by the stress due to loading and can be measured:

And using equation (1) the strain follows from:

TU Delft ACRe 73

Where: K= gauge factor of strain gauge U= bridge voltage [V] Um= measured voltage[V] Um is amplified by a Peekel CA 110 universal precision amplifier. The CA-110 offers a total balancing range of ± 5000 μV/V. The strain gauges used here are 90˚ 2-element cross gauges (type PFC-10-11) from Tokyo Sokki Kenkyujo. These are foil-etched gauges with a polyester backing. The gauge length is 10 mm and k=2.13. The gauges are glued on the specimen with a cyanoacrylate single-component adhesive and coated with PU120 from Hottinger Baldwin Messtechnik. A4.2 CALCULATION OF THE AXIAL STRAIN FROM THE RADIAL SIGNAL To ensure that the three 2-element cross strain gauges are always glued on the same distance from the centre for all specimens, the clamping ring (section ????) is used to mark these positions. This mark is 20 mm from the centre of the specimen. The radial gauges are aligned along this mark (figure 3), so that the tangential strain gauge is at 15-25 mm from the centre. Using this position, the axial deformation was determined from the tangential gauge signal in the following way: h Lgauge=10 mm R R Approximating the shape of the specimen by a straight line over the height of the gauge gives a

slope of 2.96 mm on 10 mm of height. This gives:

22 2 2 2

109.59 , 2.84

gaugeL R h h h

h h mm R mm

In which Lgauge is the original length of the gauge. At any time during a test the gauge length is:

2 2* * *(1 )gauge gauge gaugeL L R h ,

with R* and h* the vertical and horizontal components of the strain gauge at that point. Both these values are unknown.

* * *1 2

; with 1,2 indicators of the top and bottom of the vertical strain gauge

Lab W&S ACRe 74

In order to determine h*, it is necessary to know R*. This value is, however, not known. What is known is the radial strain at half the length of the gauge:

Unfortunately, this does not provide information on the strain values at top and bottom. To find those values, an additional assumption/equation is needed. Therefore, it is assumed that the stiffness as well as the relation between the radial and axial strains is constant over the height. These assumptions mean that the stresses and strains vary proportionally with the cross section, which gives the necessary information on the slope over the height of the gauge. It must, however, be noted that this assumption implies some simplifications of the real situation; 3 For linear elastic behaviour (low stress levels as well as unloading) the assumption is true,

but during loading at any cross section (over the height of the gauge) non-linearity is induced the assumption becomes invalid. Since it appears that the gauges record remaining deformations, the assumption does not hold true throughout a test. It is therefore necessary to establish whether the error induced by this assumption is acceptably small.

4 Due to the variable cross section, the strains and thus the strain rates vary over the height. Because of the strain rate sensitivity of the material, this will induce variations in stiffness, which trigger different stress distributions, which in turn will have an effect on the strains…. Since it is not possible to take such a diffuse effect into account, it is once again a matter of establishing whether the error is acceptable. Fortunately, however, this strain rate effect may be due to the viscous characteristics, in which case the elastic parameters that control the unloading are independent of this phenomenon.

Using the assumption yields:

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( )( )

( ) ( )

F FR h h R h h R h R h h

EA h E R h

FR h h R h

Fh R h

This should be valid for any h, including the one at the centre of the gauge. Substituting this gives: 2

If the term F

in the expression for ΔR(h) is replaced by 2C CR it gives:

( )R C CR h RR h

Since: * * *

1 2 1 1 2 2

2 2, , 2

1 2 1 2 ,1 2 1 2

1 1( ) ( )R C C R C C

R R R R R R R

R RR R R R R

R R R R

TU Delft ACRe 75

Remembering that the total signal from the gauges could be split according to:

2 2 2 2* * * *(1 ) (1 )gauge gauge gauge axialL L R h R h

gives, in combination with the above mentioned expression for R*:

2* 21 2 ,

1 1( ) (1 )gauge R C C axialL R R R hR R

Rewriting yields:

22 2* 2

1 2 ,1 2

222 2 2 2 * 2

1 2 ,1 2

2 22 2 2 2 *1

1 1( ) (1 )

1 12 ( ) 0

gauge R C C axial

axial gauge R C Ca b

L R R R hR R

h h h L R R RR R

b b ac

h h h h L R

1 1( )

R C CR RR R

Since the axial strain should be zero if the gauge signal is, indicating that there are no deformations, and since in that case: R,C = 0 as well, R1-R2 = R and (L*

gauge)2 = R2 + h2 , the term within accolades becomes zero.

As a result, the only relevant solution of the above expression is the plus variant. A4.3 VERIFICATION OF THE STRAIN GAUGE SIGNALS As stated in Chapter 3, the strain gauge signal had to be verified with respect to some potential errors: The glue and the carrier of the strain gauge are rather stiff, which may have an effect on what

the gauge registers The temperature and strain rate sensitivity of the material itself will lead to a wide variety of

stiffness’ throughout the range of test conditions and it had to be checked if all these conditions could be covered by (one type/combination of) glue and strain gauges

The bitumen skin on the outside of the specimen might cause the strain gauges to register a response that differed from that of the main body of the specimen

The first two points were addressed using a compression specimen (cylinder without bitumen skin) in the compression set-up. This is discussed in Chapter 3, the third point is addressed here. In the tension set-up the average strain of the axial straingauges was compared with the LVDT overall values and the average radial strain was compared with a calibrated clip-on gauge. The overall LVDT-signal was transformed to an “equivalent strain” at the position of the gauges by multiplying it with the factor between the deformations at gauge level versus the total in case of elastic behaviour.

Lab W&S ACRe 76

Some results are shown in the pictures underneath. They show the response to a force controlled sine load at 0, 15 and 30˚ C. All tests were performed using a frequency of 1 Hz. In the first graph, the individual strain gauge signals are shown as well. The strain values are given on the left-hand axis and the force values are shown on the right hand axis. The different signals are labelled in the graph, underneath a description of the various signals is given, as well as the label used for each signal (between brackets): applied force (Force) the axial strain at gauge level, computed on the basis of the average transducer signal (=overall

deformation) (εLVDT(ax)) the three individual tangential strain gauge signals, (εtan) (only in the first graph) the average tangential strain gauge signal, (εtan, av) (only in the first graph) average axial strain gauge signal that is corrected for the radial influences (calculated axial

strain) (εgauge(ax)) the three individual radial strain gauge signals, (εrad) (only in the first graph) the average radial strain gauge signal (εgauge(rad)) radial signal of a clip-on gauge (εclip(rad))

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

time [ms]stra

0 100 200 300 400 500 600 700 800 900

time [ms]

εLVDT(ax)

εgauge(ax)Force

εgauge(rad)εclip(rad)

εLVDT(ax)

εclip(rad)

εgauge(rad)

εgauge(ax)

TU Delft ACRe 77

0 500 1000

time [ms]

Figure 5: Strain signals for different temperatures in force control As expected, the phase lag between the force and strain signals increases with increasing temperature. Also, the plastic deformations (slope of the line around which the strain signals vary) increases with the temperature. Another effect that can be observed is that there exists a difference between the measured LVDT axial strain and the axial strain of the strain gauges. This is especially the case for T= 30 ˚C and mainly is a plastic strain. This difference can be due to an error in the gauge registration (e.g. the “stiffening” effect), which leads to an underestimation of the (plastic) strains. But it can also be due to the fact that the “back-calculated” strains from the transducers are based on the relation between the elastic deformations over the specimen. In reality, plastic deformations will occur, especially at the centre of the specimen, as a result deformations will be larger near the centre. If this is neglected the back-calculated plastic strains will be larger than those actually occurring at the position of the gauges. To establish which possible explanation was the correct one, a clip-on gauge was used. The red lines in the graph give the results of a clip-on gauge clipped at the same height of the radial gauges to determine the changes in specimen diameter. This would also give an indication whether the strains at the surface differed from the specimen response. As can be seen from Figure , the correspondence between the radial strains from the gauges and that from the clip-on gauge is fairly good. It was tried to also check the axial signals with those clips, but the specimen shape made it impossible to connect the clips satisfactorily in this direction. From these results, it appeared that the difference between the strains obtained from the LVDT signal and those from the strain gauges was mainly due to the non-linear elastic deformation that occur at and near the centre. As an additional test, a specimen was instrumented with strain gauges at different heights. Usually, three gauges are glued at the same height at 120o intervals, but in this case they were shifted one gauge length, so one gauge was glued 10 mm higher and one 10 mm lower than usual. In this way, the top of one gauge was at the same height as the bottom of another. An example of the results is shown in Figure .

εLVDT(ax)

εclip(rad)

εgauge(rad)

εgauge(ax)

Lab W&S ACRe 78

0 1 2 3 4 5 6 7 8 9 10 11 12

time [s]

Figure A4.3: Axial and radial strain signals for gauges at the original position (2), closer to the centre (1) and closer to the cap (3).

The test was done in force control with an amplitude of 1.5 kN and at T=15 °C. The steeper slope of the average signal for the gauges closer to the centre also corroborates the impression that the difference between gauges and LVDT’s in figure 4 at T=30 °C may be due to increasing plastic deformations near the centre. The dynamic response of the gauges at the original position (2) however hardly differs from the ones closer to the centre (1) which means that the position of the strain gauge is far enough from the cap to obtain a reliable elastic response.

TU Delft ACRe 79

APPENDIX 5: DISPLACEMENT TRANSDUCERS A set of three Solartron displacement transducers is used to control the applied deformation signal and register the overall deformation of the specimen. The magnitude of the overall deformations varies with the test conditions (temperature and strain rate), which requires the use of different sets of displacement transducers. For the more brittle response transducers with a range of ± 1 mm are used and when the deformations were larger a set of LVDT’s with a range of ± 5 mm was used. The accuracy of the transducers was determined by calibrating them in the set-up. Using a calibration unit, the transducers were set to a given deformation and the corresponding voltage was registered. This was repeated three times and on the basis of the deformation-voltage data a linear relation between the two was determined for each transducer. The maximum deviation between the registered data and this calibration line, expressed as a percentage of the full range of the transducer is the non-linearity. The calibration sheets of all the transducers are shown in the rest of this appendix.

Lab W&S ACRe 80

Calibratie solartron transducer range: +/- 1Transducer 1input [mm]signal[V] cal.line error

a1 -0.100379 -0.004756 a0-1 9.99 -1.007539272 -0.007539272 sa(a1) 0.000115 0.000691 sa(a0)

-0.9 8.95 -0.903145422 -0.003145422 r^2 0.999948 0.00448 sa(y)-0.8 7.93 -0.800759146 -0.000759146 F 767146.1 40 vg-0.7 6.92 -0.699376657 0.000623343 ks(reg) 15.3992 0.000803 ks(resid)-0.6 5.93 -0.600001742 -1.74187E-06-0.5 4.95 -0.501630614 -0.001630614-0.4 3.92 -0.398240551 0.001759449-0.3 2.92 -0.297861849 0.002138151-0.2 1.94 -0.199490721 0.000509279-0.1 0.95 -0.100115805 -0.000115805

0 0 -0.004756039 -0.0047560390.1 -0.99 0.094618877 -0.0053811230.2 -2.01 0.197005153 -0.0029948470.3 -2.99 0.295376281 -0.0046237190.4 -3.97 0.393747409 -0.0062525910.5 -4.95 0.492118537 -0.0078814630.6 -5.96 0.593501026 -0.0064989740.7 -7.01 0.698898663 -0.0011013370.8 -8.01 0.799277365 -0.0007226350.9 -9.02 0.900659854 0.000659854

1 -10.03 1.002042343 0.002042343-1 9.99 -1.007539272 -0.007539272

-0.9 8.91 -0.899130274 0.000869726-0.8 7.88 -0.795740211 0.004259789-0.7 6.87 -0.694357722 0.005642278-0.6 5.86 -0.592975233 0.007024767-0.5 4.93 -0.49962304 0.00037696-0.4 3.9 -0.396232977 0.003767023-0.3 2.86 -0.291839126 0.008160874-0.2 1.86 -0.191460424 0.008539576-0.1 0.91 -0.096100657 0.003899343

0 -0.01 -0.003752251 -0.0037522510.1 -1.05 0.100641599 0.0006415990.2 -2.07 0.203027875 0.0030278750.3 -3.05 0.301399003 0.0013990030.4 -4.02 0.398766344 -0.0012336560.5 -4.96 0.493122324 -0.0068776760.6 -6.02 0.599523748 -0.0004762520.7 -7.06 0.703917598 0.0039175980.8 -8.07 0.805300087 0.0053000870.9 -9.08 0.906682576 0.006682576

1 -10.03 1.002042343 0.002042343max: 0.008539576non linearity 0.43 % of full scale

LVDT 1

-10 -5 0 5 10

Voltage [V]d

signal[V]

cal.line

LVDT 1

-0.008

-0.006

-0.004

-0.002

-1 -0.5 0 0.5 1

distance[mm]

TU Delft ACRe 81

Calibratie solartron transducer range: +/- 1Transducer 3input[mm] signal[V] cal.line error

a1 -0.100645 -0.008531 a0-1 9.92 -1.006926326 -0.006926326 sa(a1) 0.000138 0.000831 sa(a0)

-0.9 8.87 -0.901249384 -0.001249384 r^2 0.999925 0.005382 sa(y)-0.8 7.85 -0.798591784 0.001408216 F 531635.1 40 vg-0.7 6.84 -0.69694063 0.00305937 ks(reg) 15.39884 0.001159 ks(resid)-0.6 5.88 -0.600321712 -0.000321712-0.5 4.9 -0.501689899 -0.001689899-0.4 3.87 -0.398025852 0.001974148-0.3 2.9 -0.300400486 -0.000400486-0.2 1.92 -0.201768674 -0.001768674-0.1 0.96 -0.105149755 -0.005149755

0 0 -0.008530837 -0.0085308370.1 -1.02 0.094126764 -0.0058732360.2 -2 0.192758576 -0.0072414240.3 -3 0.293403283 -0.0065967170.4 -3.97 0.391028648 -0.0089713520.5 -4.95 0.489660461 -0.0103395390.6 -6 0.595337403 -0.0046625970.7 -7.02 0.697995003 -0.0020049970.8 -8.05 0.801659051 0.0016590510.9 -9.06 0.903310205 0.003310205

1 -10.09 1.006974252 0.006974252-1 9.88 -1.002900538 -0.002900538

-0.9 8.82 -0.896217149 0.003782851-0.8 7.8 -0.793559548 0.006440452-0.7 6.79 -0.691908395 0.008091605-0.6 5.84 -0.596295924 0.003704076-0.5 4.84 -0.495651217 0.004348783-0.4 3.82 -0.392993616 0.007006384-0.3 2.84 -0.294361804 0.005638196-0.2 1.84 -0.193717097 0.006282903-0.1 0.94 -0.103136861 -0.003136861

0 -0.06 -0.002492155 -0.0024921550.1 -1.09 0.101171893 0.0011718930.2 -2.06 0.198797259 -0.0012027410.3 -3.05 0.298435518 -0.0015644820.4 -4 0.394047989 -0.0059520110.5 -5 0.494692696 -0.0053073040.6 -6.06 0.601376085 0.0013760850.7 -7.08 0.704033686 0.0040336860.8 -8.12 0.80870418 0.008704180.9 -9.09 0.906329546 0.006329546

LVDT 3

-0.015

-0.005

-1 -0.5 0 0.5 1

distance[mm]

LVDT 3

-10 -5 0 5 10

Voltage [V]d

signal[V]

cal.line

Lab W&S ACRe 82

a1 -0.10054 9.58E-05 a0-1 9.96 -1.001280144 -0.001280144 sa(a1) 9.31E-05 0.000561 sa(a0)

-0.9 8.97 -0.901745792 -0.001745792 r^2 0.999966 0.003632 sa(y)-0.8 7.99 -0.803216839 -0.003216839 F 1167085 40 vg-0.7 6.95 -0.6986555 0.0013445 ks(reg) 15.39947 0.000528 ks(resid)-0.6 5.94 -0.597110354 0.002889646-0.5 4.97 -0.499586798 0.000413202-0.4 4.01 -0.40306864 -0.00306864-0.3 3.03 -0.304539686 -0.004539686-0.2 1.97 -0.197967553 0.002032447-0.1 1 -0.100443996 -0.000443996

0 0 9.57521E-05 9.57521E-050.1 -0.95 0.095608513 -0.0043914870.2 -1.93 0.194137467 -0.0058625330.3 -2.96 0.297693408 -0.0023065920.4 -3.95 0.397227759 -0.0027722410.5 -4.93 0.495756713 -0.0042432870.6 -5.91 0.594285666 -0.0057143340.7 -6.91 0.694825415 -0.0051745850.8 -7.95 0.799386753 -0.0006132470.9 -8.98 0.902942694 0.002942694

1 -10 1.005493238 0.005493238-1 9.89 -0.994242361 0.005757639

-0.9 8.94 -0.8987296 0.0012704-0.8 7.98 -0.802211441 -0.002211441-0.7 6.92 -0.695639308 0.004360692-0.6 5.93 -0.596104957 0.003895043-0.5 4.94 -0.496570606 0.003429394-0.4 3.97 -0.39904705 0.00095295-0.3 3.01 -0.302528891 -0.002528891-0.2 1.95 -0.195956758 0.004043242-0.1 0.99 -0.099438599 0.000561401

0 0 9.57521E-05 9.57521E-050.1 -1 0.100635501 0.0006355010.2 -1.94 0.195142864 -0.0048571360.3 -3 0.301714998 0.0017149980.4 -3.99 0.401249349 0.0012493490.5 -4.98 0.5007837 0.00078370.6 -5.95 0.598307256 -0.0016927440.7 -6.91 0.694825415 -0.0051745850.8 -8 0.804413741 0.0044137410.9 -9 0.904953489 0.004953489

LVDT 110

-0.008

-0.006

-0.004

-0.002

-1 -0.5 0 0.5 1

distance[mm]

LVDT 110

-10 -5 0 5 10

Voltage [V]d

input[mm]

TU Delft ACRe 83

a1 -0.505478 -0.005777 a0-5 9.7 -4.9089103 0.09108965 sa(a1) 0.000585 0.003503 sa(a0)

-4.5 8.93 -4.5196925 -0.0196925 r^2 0.999946 0.022704 sa(y)-4 7.92 -4.0091601 -0.0091601 F 746828.4 40 vg

-3.5 6.94 -3.513792 -0.013792 ks(reg) 384.9794 0.020619 ks(resid)-3 5.94 -3.0083143 -0.0083143

-2.5 4.96 -2.5129462 -0.0129462-2 3.96 -2.0074685 -0.0074685

-1.5 2.96 -1.5019908 -0.0019908-1 1.97 -1.0015679 -0.0015679

-0.5 0.98 -0.501145 -0.0011450 0 -0.0057769 -0.0057769

0.5 -0.99 0.49464601 -0.0053541 -1.98 0.99506891 -0.0049311

1.5 -2.97 1.49549181 -0.00450822 -3.96 1.99591471 -0.0040853

2.5 -4.96 2.50139239 0.001392393 -5.95 3.00181529 0.00181529

3.5 -6.94 3.50223819 0.002238194 -7.93 4.00266109 0.00266109

4.5 -8.92 4.50308399 0.003083995 -9.92 5.00856166 0.00856166

-5 9.7 -4.9089103 0.09108965-4.5 8.94 -4.5247473 -0.0247473

-4 7.95 -4.0243244 -0.0243244-3.5 6.95 -3.5188467 -0.0188467

-3 5.95 -3.0133691 -0.0133691-2.5 4.96 -2.5129462 -0.0129462

-2 3.97 -2.0125233 -0.0125233-1.5 2.98 -1.5121004 -0.0121004

-1 1.98 -1.0066227 -0.0066227-0.5 0.99 -0.5061998 -0.0061998

0 -0.01 -0.0007221 -0.00072210.5 -0.98 0.48959124 -0.0104088

1 -1.98 0.99506891 -0.00493111.5 -2.98 1.50054659 0.00054659

2 -3.97 2.00096949 0.000969492.5 -4.96 2.50139239 0.00139239

3 -5.96 3.00687006 0.006870063.5 -6.95 3.50729296 0.00729296

4 -7.94 4.00771586 0.007715864.5 -8.93 4.50813876 0.00813876

LVDT 157

-5 -3 -1 1 3 5

distance[mm]

LVDT 157

-10 -5 0 5 10

Voltage [V]d

input[mm]

Lab W&S ACRe 84

a1 -0.498348 -0.009611 a0-5 10.01 -4.998079167 0.001920833 sa(a1) 0.00012 0.000728 sa(a0)

-4.5 9.02 -4.504714184 -0.004714184 r^2 0.999998 0.004721 sa(y)-4 8.01 -4.001382232 -0.001382232 F 17277612 40 vg

-3.5 7.01 -3.503033764 -0.003033764 ks(reg) 384.9991 0.000891 ks(resid)-3 6 -2.999701812 0.000298188

-2.5 5 -2.501353344 -0.001353344-2 3.99 -1.998021392 0.001978608

-1.5 2.99 -1.499672924 0.000327076-1 1.99 -1.001324457 -0.001324457

-0.5 0.99 -0.502975989 -0.0029759890 0 -0.009611006 -0.009611006

0.5 -1.01 0.493720946 -0.0062790541 -2.01 0.992069414 -0.007930586

1.5 -3.01 1.490417881 -0.0095821192 -4.02 1.993749834 -0.006250166

2.5 -5.03 2.497081786 -0.0029182143 -6.05 3.005397223 0.005397223

3.5 -7.05 3.503745691 0.0037456914 -8.05 4.002094158 0.002094158

4.5 -9.06 4.505426111 0.0054261115 -10.05 4.998791094 -0.001208906

-5 10 -4.993095683 0.006904317-4.5 9 -4.494747215 0.005252785

-4 8 -3.996398747 0.003601253-3.5 6.99 -3.493066795 0.006933205

-3 6 -2.999701812 0.000298188-2.5 5 -2.501353344 -0.001353344

-2 3.99 -1.998021392 0.001978608-1.5 2.98 -1.49468944 0.00531056

-1 1.98 -0.996340972 0.003659028-0.5 0.99 -0.502975989 -0.002975989

0 -0.02 0.000355963 0.0003559630.5 -1.01 0.493720946 -0.006279054

1 -2.02 0.997052898 -0.0029471021.5 -3.02 1.495401366 -0.004598634

2 -4.03 1.998733318 -0.0012666822.5 -5.04 2.502065271 0.002065271

3 -6.04 3.000413738 0.0004137383.5 -7.06 3.508729175 0.008729175

4 -8.06 4.007077643 0.0070776434.5 -9.06 4.505426111 0.005426111

5 -10.05 4.998791094 -0.001208906max: 0.009611006non linearity 0.10 % of full scale

LVDT 60

-0.015

-0.005

-5 -3 -1 1 3 5

distance[mm]

LVDT 60

-10 -5 0 5 10

Voltage [V]d

input[mm]

TU Delft ACRe 85

a1 -0.493001 -0.001475 a0-5 10.17 -5.015292599 -0.015292599 sa(a1) 0.0002 0.001223 sa(a0)

-4.5 9.15 -4.51243186 -0.01243186 r^2 0.999993 0.007925 sa(y)-4 8.12 -4.004641114 -0.004641114 F 6098324 40 vg

-3.5 7.09 -3.496850368 0.003149632 ks(reg) 383.0165 0.002512 ks(resid)-3 6.07 -2.993989629 0.006010371

-2.5 5.07 -2.500988905 -0.000988905-2 4.05 -1.998128166 0.001871834

-1.5 3.04 -1.500197435 -0.000197435-1 2.02 -0.997336696 0.002663304

-0.5 1.02 -0.504335972 -0.0043359720 0 -0.001475233 -0.001475233

0.5 -1.01 0.496455498 -0.0035445021 -2.02 0.99438623 -0.00561377

1.5 -3.03 1.492316961 -0.0076830392 -4.05 1.9951777 -0.0048223

2.5 -5.07 2.498038439 -0.0019615613 -6.1 3.005829185 0.005829185

3.5 -7.12 3.508689923 0.0086899234 -8.13 4.006620655 0.006620655

4.5 -9.13 4.499621379 -0.0003786214.9 -9.9 4.879231937 -0.020768063-5 10.16 -5.010362592 -0.010362592

-4.5 9.13 -4.502571846 -0.002571846-4 8.1 -3.9947811 0.0052189

-3.5 7.08 -3.491920361 0.008079639-3 6.06 -2.989059622 0.010940378

-2.5 5.05 -2.491128891 0.008871109-2 4.04 -1.993198159 0.006801841

-1.5 3.03 -1.495267428 0.004732572-1 2.01 -0.992406689 0.007593311

-0.5 1 -0.494475958 0.0055240420 0 -0.001475233 -0.001475233

0.5 -1.01 0.496455498 -0.0035445021 -2.03 0.999316237 -0.000683763

1.5 -3.04 1.497246968 -0.0027530322 -4.06 2.000107707 0.000107707

2.5 -5.08 2.502968446 0.0029684463 -6.1 3.005829185 0.005829185

3.5 -7.13 3.513619931 0.0136199314 -8.13 4.006620655 0.006620655

4.5 -9.14 4.504551386 0.0045513864.9 -9.9 4.879231937 -0.020768063

max: 0.020768063non linearity 0.21 % of full scale

LVDT 61

-0.025

-0.015

-0.005

-5 -3 -1 1 3 5

distance[mm]

LVDT 61

-10 -5 0 5 10

Voltage [V]d

input[mm]

Lab W&S ACRe 86

APPENDIX 6: CALIBRATION DATA LOADCELL 00 CALIBRATION FORM FOR LOADCELLS

Customer’s name: ing. M.R. Poot/ Verkeersbouwkunde filename D050-3157809-Loadcell no. Lebow 3157/809 Date: 15-10-1999 By: M. Van der MeererSection: DA / tel. 015-2785523 / fax 015-2782324Testmachine: Amsler Machine range: 100 kN Force: 50 kN Mode: Tension Strain amp: Kompensator Mk ser.no. 5169 (Hottinger)Connections: Chan. 2 1=(E)green, 2=(G) Yellow, 3=(D) brown, 4=(F) White Remarks: Only tested in Tension

CALIBRATION FACTORS Calibrationfactor @ 50 kN Connection A = 0 Str. (Compression)Calibrationfactor A = 0 V/V (Compression)Connection A = 6437.875 Str. (Tension)Calibrationfactor A = 3218.938 V/V (Tension) Connection B = 0 Str. (Compression)Calibrationfactor B = 0 V/V (Compression)Connection B = 6558.846 Str. (Tension)Calibrationfactor B = 3279.423 V/V (Tension)

Connection B -Tension Measured [kN] Measured Str Non. Lin. % Series 1 0 26020 10 27320 -0.378

Non. lin.% St. V. Connection B

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

0 10 20 30 40 50 60

20 28636 -0.027 30 29949 0.013 40 31262 0.033 50 32575 0.045 40 31259 -0.024 30 29947 -0.038 20 28635 -0.065 10 27320 -0.378Series 2 0 26020 10 27321 -0.301 20 28637 0.011 30 29948 -0.012 40 31262 0.033 50 32575 0.045 40 31258 -0.043 30 29946 -0.063 20 28635 -0.065 10 27320 -0.378 Calibration @ 50 kNSeries 3 0 26020 10 27320 -0.378 Connection B = 6558.8465 Str (Tension) 20 28636 -0.027 Calibrationfactor B = 3279.4232 V/V 30 29950 0.039 40 31262 0.033 50 32576 0.061 Max. non.-linearity: -0.4% 40 31259 -0.024 30 29947 -0.038 20 28635 -0.065 10 27321 -0.0301 0 26020 MStr./kN Calc. Zero 131.18 26013.17

TU Delft ACRe 87

APPENDIX 7: TEST DATA & DETERMINATION AVERAGE RESPONSE In this appendix the response curves of the individual tests are shown, as well as the average curve for each test condition. For this reason, also the approach used to obtain these average response curves is discussed in detail. A7.1 DETERMINING THE AVERAGE RESPONSE CURVES The data-acquisition procedures for the tension test required the determination of the optimal scan rate. For every condition, the total deformation and the type of failure (plastic or brittle) was different. Between the hardware limits (maximum scan rate supported by the data acquisition board on the one and maximum number of scans that could be stored without accessing the disk on the other hand) the best scan rate for each condition had to be determined. In some cases, the optimal became apparent only after a number of tests. During these tests the response was captured, but the program was still registering after the test had ended or the unloading was not captured accurately. In both cases the scan rate was adapted to get a more optimal acquisition in the other tests. Since determining the average in excel requires averaging columns of data points, differences in scan rate had to be taken into account. If, for example, one test was registered with a scan rate twice that of another test only one in every two data points of the latter was taken into account. For most conditions, this sufficed and the signals could be averaged directly after differences in scan rate were taken into account (Figure A7.1).

Unequal scan rates, use every other data point of example 2 toget the average curve (bold numbers, skip the italic ones).

example 1 example 20 0 0 01 1.8 0.5 12 2.7 1 23 3.3 1.55 2.64 3.2 2.1 2.95 2.4 2.7 3.16 1.8 3.3 3.17 1.2 3.6 2.88 0.9 3.9 2.6

4.35 2.14.8 1.75.4 1.5

6 1.26.55 1.17.1 17.5 0.97.9 0.8

0 2 4 6 8 10

u [mm]

[N/mm2] example 1 example 2

Figure A7.1: In determining the average response differences in scan rate were accomodated In some cases, however, additional work was needed. For these conditions the individual curves either had their peak values at different deformation levels or they exhibited brittle failure. In the first case, the average value of the strength will be lower than the average of the peak values () and in the latter case, a sudden change in the average value will occur anytime an individual signal “drops out”. In the first case, the shape of the average curve is allright, but the peak strength is underestimated. Multiplying the stress values of the average curve by the quotient of the average of the individual peaks stresses and the peak of the average signal solves the first problem. In case of brittle failure, the signal is left out of the average curve after failure and the average signal obtained after leaving out this signal is offset in such a way that it starts at the level where the average was at just before brittle failure in one of the signals occurred.

Lab W&S ACRe 88

Different peak locations lead to an underestimation of the peak strength. Average of the peak valuesis, in this case, 3.45. The peak in the average curve is 3.25. Multiplying with 3.45/3.25 yields the corrected average.

example 1 example 20 0 0 0

1.5 1.6 2.1 1.62.9 2.6 3.9 2.54.7 3.4 6.3 36.4 3.2 8.1 3.37.4 2.4 11.6 3.58.4 1.6 13.4 39.4 0.9 15.2 2.5

10.4 0.5 17 0.9

0 5 10 15 20

u [mm]

[N/mm2]example 1

example 2

average

"corrected average"

Figuur A7.2: If the maxima are far apart, the average signal is corrected

example 1 example 20 0 0 0

1.5 2.1 1.2 1.22.9 4.06 2.4 2.44.7 6.58 3.6 3.64.7 6.58 4.8 4.85.7 7.98 4.8 06.7 9.38 6 07.7 10.78 7 07.7 10.78 8 0

0 2 4 6 8 10

u [mm]

[N/mm2]example 1 example 2

average "corrected average"

The effect of brittle failure of an individual signal can be corrected by leaving out that signal after its drop and multiplying the remaining signal by the average value of the peaks (peak stress as well as deformationsat the peak) divided by the value of the average signal just before the brittle failure.

Figure A7.3: In case of brittle failure, the effect on the average signal has to be corrected Using these techniques, average signals for test conditions were obtained. In Table A7.1 the corrections used for all the conditions are listed.

TU Delft ACRe 89

ε [%/s] T [oC] Scan rate Peak value brittleness

0.001 0 YES YES NO 0.01 0 YES NO YES 0.1 0 NO YES YES 1 0 NO YES NO

0.1 15 NO NO NO 0.5 15 YES NO NO 1 15 YES NO NO

0.1 30 YES NO NO 1 30 NO NO NO 3 30 NO YES NO 5 30 YES YES NO 10 30 NO YES YES

Table A7.1: Overview of the techniques used to obtain an average response curve for various conditions A7.2 THE INDIVIDUAL TEST RESULTS AND THEIR AVERAGES

0oC & 0.001%/s

-0.2 0.3 0.8 1.3 1.8 2.3

uax [mm]

[N/mm2]

T75T102T179average

urad [mm]

Lab W&S ACRe 90

0oC & 0.01%/s

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3

uax [mm]

[N/mm2]

average

urad [mm]

0oC & 0.1%/s

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

uax [mm]

[N/mm2]

Reeks2

urad [mm]

TU Delft ACRe 91

0oC & 1%/s

-0.005 0.005 0.015 0.025 0.035 0.045

uax [mm]

[N/mm2]T64T80T124T126average

urad [mm]

15oC & 0.1%/s

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

uax [mm]

[N/mm2]T101

average

Lab W&S ACRe 92

15oC & 0.5%/s

-0.15 0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 1.2 1.35 1.5

uax [mm]

[N/mm2]

T122T201T203T208average

urad [mm]

15oC & 1%/s

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

uax [mm]

[N/mm2] T206T207T209average

TU Delft ACRe 93

30oC & 0.1%/s

-0.05 0.45 0.95 1.45 1.95 2.45 2.95

uax [mm]

[N/mm2]T163T171T62T162average

30oC & 1%/s

-0.2 0.3 0.8 1.3 1.8 2.3 2.8 3.3

uax [mm]

[N/mm2]

T128T173T69T70average

Lab W&S ACRe 94

30oC & 3%/s

-0.125 0.375 0.875 1.375 1.875

uax [mm]

[N/mm2] T113

average

30oC & 5%/s

-0.2 0.3 0.8 1.3 1.8 2.3

uax [mm]

[N/mm2]

T127T149T175T210T74average

TU Delft ACRe 95

30oC & 10%/s

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

uax [mm]

[N/mm2] T169T170T76average

Lab W&S ACRe 96

A7.3 NORMALISED RESPONSE CURVES

0oC & 0.001%/s NORM

-2 -1 0 1 2 3 4

uax,norm

T75 T179 T102

0oC & 0.01%/s

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

TU Delft ACRe 97

0oC & 0.1%/s

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0oC & 1%/s

-1 -0.5 0 0.5 1

Lab W&S ACRe 98

15oC & 0.1%/s

-1.5 -1.3 -1 -0.8 -0.5 -0.3 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3

15oC & 0.5%/s

00.10.20.30.40.50.60.70.80.9

-1.5 -0.5 0.5 1.5 2.5 3.5

TU Delft ACRe 99

15oC & 1%/s

-1.5 -1 -0.5 0 0.5 1 1.5 2

uax [mm]

30oC & 0.1%/s

-1 -0.5 0 0.5 1 1.5 2 2.5 3

Lab W&S ACRe 100

30oC & 1%/s

-1.25 -0.25 0.75 1.75 2.75 3.75

30oC & 3%/s

-1.5 -0.5 0.5 1.5 2.5 3.5 4.5

F [kN]T113

TU Delft ACRe 101

30oC & 5%/s

-2 -1 0 1 2 3 4 5

30oC & 10%/s

-1.5 -1 -0.5 0 0.5 1 1.5

The Uniaxial Compression Test

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