Numerical and Experimental Analysis of the Directional Stability on Crack Propagation Under Biaxial...

8/9/2019 Numerical and Experimental Analysis of the Directional Stability on Crack Propagation Under Biaxial Stresses

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Numerical and experimental analysis of the directionalstability on crack propagation under biaxial stresses

Rodrguez-Martnez, R.1, Urriolagoitia-Caldern, G. 1, Urriolagoitia-Sosa, G.1,Hernndez-Gmez, L. H.1, Merchn-Cruz, E. A.2, Rodrguez-Caizo, R. G.2,Sandoval-Pineda, J. M.2

1INSTITUTO POLITCNICO NACIONAL Seccin de Estudios de Posgrado eInvestigacin (SEPI), Escuela Superior de Ingeniera Mecnica y Elctrica (ESIME),

Edificio 5. 2do Piso, Unidad Profesional Adolfo Lpez Mateos Zacatenco Col.

Lindavista, C.P. 07738, Mxico, D.F. Mxico.

2INSTITUTO POLITCNICO NACIONAL Seccin de Estudios de Posgrado eInvestigacin (SEPI), Escuela Superior de Ingeniera Mecnica y Elctrica (ESIME).

Unidad profesional, AZCAPOTZALCO, Av. de las Granjas No. 682, Col. Sta.

Catarina Azcapotzalco, C.P. 02550, Mxico D.F. Mxico.

[email protected], [email protected], [email protected].,

[email protected] , [email protected], [email protected],

[email protected]

Abstract. In this paper, the case of Single Edge Notch (SEN) specimens subject toopening/compressive loading was analyzed; The loads are applied in several ratios to evaluatethe influence of the specimen geometry, and the Stress Intensity Factor (SIF)K1values on the

directional stability of crack propagation. The main purpose of this work is to evaluate thebehaviour of the fracture propagation, when modifying the geometry of the SEN specimen and

different relationships of load tensin/compressin are applied.Additionally, the precision ofthe numerical and experimental analysis is evaluated to determine its reliability when solving

this type of problems. The specimens are subjected to biaxial opening/compression loading;both results (numerical and experimental) are compared in order to evaluate the condition of

directional stability on crack propagation. Finally, an apparent transition point related to the

length of specimens was identified, in which the behaviour of values of SIF changes for

different loading ratios.

1. IntroductionMany studies of fracture mechanics [1-4] have shown how the form of the stress field at the crack tip

in Single Edge Notch Specimens (SEN), obey global conditions such as the specimen and applied

loads. Initially this statement was based on single loading, however the problem is more complicated

when mixed loading is involved.

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When considering mixed loading, the crack propagation stability is an engineering problem that has

attracted great attention. Since the pioneer work of Erdogan and Sih [5], where was considered the

crack initiation direction for an angled crack which predicted to be driven under a maximum

circumferential stress criterion, since then several additional factors that are involved in the fracture

process have been found.

Several methodologies have been developed for the understanding of crack propagation under

mixed loading. For example crack extension stability was studied by Cotterell [6-7] applying

eigenfunction series expansion. Regarding crack path extension, Kipp and Sih [8] investigated the

effects caused by the applied load direction, the curvature crack radius and the complete stress (energy

field). Additionally, applications of numerical and experimental techniques have been proposed to

evaluate crack growth instability [1-3], all these work is based on SEN specimens under biaxialloading. An approach using an energetic criterion based on S-Theory trying to understand crack

initiation and propagation in beams with edge crack under mixed loading was analyzed by Nobile [9]

and Sih [10].

From the investigations presented above, it has been proven that there are many factors involved.

To be able to evaluate these factors, it is very important then to be separated in order to establish their

individual contribution to the fracture process. There are two sets of factors that influence the fracture

process (global and local). The global conditions involve geometry, loading conditions, environment,

etc. while the local conditions involve flaws and material imperfections that developed in the

neighborhood of the crack path.

Directional stability of crack extension is highly dependent on global conditions, making the

fracture initiation to be highly stable, stable or unstable. As the loading rises, the local conditions mayshow its influence [3]. So, if the fracture process is dominated by the global conditions, the path crack

growth could be predicted. However, if the local conditions dominate, the crack path could become

indeterminable. For the SENspecimens, on square geometry show a better directional stability than a

narrow one. Nevertheless, compressive and transversely applied stresses could serve as stabilizing

factors.

This work is focused on a SEN specimen subject to a mixed opening and compressive loading.

Where the loads are applied at several different ratios, from which it could be possible to evaluate the

influence of the specimen geometry on the propagation of the crack. Additionally, the precision of the

numerical and experimental analysis is evaluated and compared to determine its reliability when

solving this type of problems. Finally, an apparent transition length of specimens was also identified,

where the behaviour of SIFvalues changes for different loading ratios. Cases presented in this workhave practical application, like in the analysis of cracked structural components, which are subjected

to combined loads. Such are the cases presented in biomechanical analysis, for bones, arteries and

tissues that become affected by cracks.

2. Study casesTo carry out numerical-experimental analysis, seventeen photoelastic models were built with the

dimensions and the applied loads shown in table 1. The parameters L, w, a, Popen y Pcomp are

defined in figures 1 and 2.

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TABLE 1.

CASES OF STUDYLoading condition (N)Geometry of plates

Case 1a Case 1b Case 1c Case 1dL (m) w (m) a (m) Popen Pcomp Popen Pcomp Popen Pcomp Popen Pcomp0.1016

0.1143

0.1270

0.1397

0.1524

0.1651

0.1778

0.1905

0.2032

0.2159

0.2286

0.2413

0.2540

0.2667

0.2794

0.29210.3048

0.1016 0.0635 446 0 446 223 446 2230 446 4460

3. Numerical simulation and resultsNumerical simulations of sixty eight cases (see table 1) were carried out. Different geometric

conditions with different loading ratios were considered. The compressive and opening load values

and loading ratios are presented in table 2. In table 3 are shown the mechanical properties for

policarbonate PSM-1 used in the numerical evaluation, the analysis was performed in a 2D manner

and under elastic conditions.

TABLE 2.

LOADING RELATION

CompressiveLoad (N)

OpeningLoad (N)

RatioPcomp/Popen

0 446 0

223 446 0.5

2230 446 5

4460 446 10

TABLE 3.

MECHANICAL PROPERTIES OF SPECIMENS

Material Propertie Value

Elastic modulus E 2.50 GPaPolicarbonate

PSM-1Poissons ratio 0.38

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3.1 Finite element modelFor this work, FEMwas used to determine the SIF in the components under loading conditions that

are stated as follows. Each specimen was simulated considering conditions as follows; constant width

w = 101.6 mm and a variable length L from 101.6 up to 304.8 mm (figure 1). The crack size a is

considered with a constant value of 63.5 mm for all cases. The loads were applied with a constant

opening value of 446 N and four different transverse compressive load valuated at 0, 223, 2230 and

4460 N respectively for the different study cases (see table 1). It is important to highlight that although

a biaxial load mode was applied, shear load doesn't take place at the crack tip due to the symmetry of

the specimen in regard to compressive load [11]. Finite element analysis was carried out using the

computational programANSYS V 7.0described in refs. [11-12] and the finite element model is shown

in the figure 2, where the boundary and loading conditions can be observed.

Figure 1Model of cracked plate (SENspecimen). Figure 2Finite element model.

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

0 .0 00 0. 10 0 0 .20 0 0 .3 00 0 .4 00 0 .5 00 0 .6 00 0 .7 00

StressIntensityFactorK1

(MPam)

a/L

Stability

of

K1fordifferentratiosof

Pcomp/Popen.Numericalanalysis

Pcomp/Popen=0

Pcomp/Popen=0.5

Pcomp/Popen=5

Pcomp/Popen=10

Figure 3Numerical solution ofK1for different ratios of compressive/opening loads.

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Figure 3 shows the SIF values evaluated for a loading conditions given in table 2, against the

relation of crack length/specimen length a/Lof the specimen. Figure 4 shows the results obtained at

the zone in which the behaviour of K1 is unstable and it is possible to observe how fast the values

change. It seems to be a transition length Lof the specimen that coincides for all study cases, which

has a value of 0.18161m and a geometric ratio of a/L=0.3497. It can be observed as well in figure 4,

that the transition zone seems to be common for all loading condition cases. Results shown in figure 4

seem to be are significant only in a qualitative way, since it is in this zone where seemingly occurs a

change in the tendency of the values ofK1.

0.9380

0.9385

0.9390

0.9395

0.9400

0.9405

0.9410

0.3460 0.3470 0.3480 0.3490 0.3500 0.3510 0.3520 0.3530

StressIntensityFactorK1

(MPa

m)

a/L

BehaviourofK1insidetherange0.1800.183m

Numericalanalysis

Pcomp/Popen=0

Pcomp/Popen=0.5

Pcomp/Popen=5

Pcomp/Popen=10

Figure 4Numerical solution of K1in the transition zone. Length range from 0.180 to 0.183 m.

4. Experimental procedure and resultsTo carry out this work an experimental loading rig was used, it was designed and built by Urriolagoitia

et. al. [2] (figure 5). The rig is a metallic structure designed to apply uniform loads on a rectangular

SEN specimen made with a photo-elastic material. The specimen was prepared with a crack along the

longitudinal axis. The applied loads are of two kinds; an opening load and a longitudinal compressive

pressure. A Schematic drawing of the rig is shown in figure 6. The opening load it is applied to the

specimen through a group of eigth fastener blocks. These fastener blocks have were instrumented with

strain gauges. These blocks are forced to separate by means of a screw, which produces the opening

load in the specimen. This opening load can be measured using strain gauges that have been placed on

the blocks (figure 6). The compressive force required for this experimental work is applied by means

of eight individual fingers placed at the right end of the plate (figure 6). These fingers are operated by

horizontal screws. Each finger is provided with a couple of strain gauges placed in such a way that anykind of flexural stress is suppressed. This characteristic allows a good control of the complete stress

field. It is very important to state, that this study was carried out based on the results reported in other

similar works and on the numerical analysis shown previously [1-4, 13-14]. Actually, other methodshave been used for determining fracture parameters, which are very interesting and provide important

information on the stress field at the crack tip [15].

4.1Experimental tests and computing ofKIProcedure for experimental tests was to mount the plate in the loading rig, which was held by means

of screws. The next steps was placing the loading rig in the polariscope and proceed to connect strain

gages to strain indicator, to be able to measure the loads applied to the specimen through the

deformation of the load cells. The study cases were the same ones that were analyzed by means of

numerical simulation (see tables 1 and 2) where dimensions and loads applied to the specimens are

shown. The crack was introduced making a cut by means of a very thin hacksaw blade, and the crack

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tip was made by means of a sharp blade. For experimental determination of KI the following model

was used, which is described in ref.[16].

( ) ( )

( )

=

2/1

maxmax

1

222

ji

ji

iIrr

rK

(1)

To apply this equation it is necessary to measure two radios that correspond to two consecutive

isochromatic loops riand rj, (fig. 7) as well as their respective maximum shear stresses (mx)iy (mx)j,

which are determined by means of the optical stress law [16] given as

tNf

2max

= (2)

Now, different values of stress intensity factors are calculated establishing all possible

permutations for any couple of isochromatic fringes. From these results it is obtained the stocking IK

and the standard deviation KS . Finally, values that are outside of KI SK are eliminated and another

value IK is computed. This procedure is carried out for all study cases.

Figure 5Experimental loading rig. Figure 6Schematic representation of the rig.

Experimental results are shown in figure 8. It is important to point out that it was not possible to

carried out experimental tests in the transition zone of L, due to the small variation in the magnitude of

this parameter (smaller than 1mm), which didn't allow to build these specimens. At the end of eachexperiment, the opening load was increased to the point of producing fracture so as to observe the

direction of propagation which coincides with the stability approach presented for KI values. The

values of the failure loads for some specimens are shown in figures 9 and 10.

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Figure 7Isochromatic fringe pattern in a photoelastic specimen that defines shear stress state.

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.7 00

StressIntensityFactorK1(MPa

m)

a/L

StabilityofK1fordifferentratiosof

Pcomp/Popen.ExperimentalAnalysis

Pcomp/Popen=0

Pcomp/Popen=0.5

Pcomp/Popen=5

Pcomp/Popen=10

Figure 8Experimental solution ofK1for different ratios of compressive/opening loads.

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Figure 9Stable fracture propagation in specimens with relationship a/L< 0.3497. Figure (a) shows the stressstate at the crack tip when failure occurs, that is 1670 N opening load and 4460 N compressive load. Figure (b)shows the stress state at the crack tip when failure occurs, that is 1480 N opening load and 4460 N compressive

load.

Figure 10Unstable fracture propagation in specimens with relationships a/L> 0.3497. Figure (a) shows thestress state at the crack tip when failure occurs, that is 1290 N opening load and 4460 N compressive load.

Figure (b) shows the stress state at the crack tip when failure occurs, that is 1115 N opening load and 4460 Ncompressive load.

5. DiscussionThe results that were obtained for the values of the SIF have shown clearly the influence of

fundamental parameters such as the specimen geometry and the relation between the opening and

transverse loads. Zanganeh et. al. [17] mentions that this effect is due to the stress term known as T-

stress which is defined as the second non-singular term in Williamss crack tip stress field solution.

Also refers to the Cotterell studies [18] where he says that the T-stress controls the stability of the

crack direction as well as assumed that the ideal direction for crack growth is along the line of

symmetry of the stress distribution ahead of the crack tip and concluded that if the sign of the T-stress

is negative then the crack path has a tendency to return to its original ideal path (referred to as

directionally stable). However, if the T-stress is positive the crack path does not return to this

original ideal path. Additionally, Pacey [19] says that a fringe pattern in the crack tip may be producedin specimens by subjected them to compressive loads in line with de crack growth direction and can

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also be attributed to the specimen geometry and the mode of loading. He also affirms that must be

some point in specimen where a transition occurs from tensile to compressive loading and where the

fringe order is zero.

In figures 3, 8 and 9 it can be observed, as if the length of the specimen is increased, more stable is

the determined values ofK1, and the directional stability of crack propagation is also stable. Numerical

and experimental results were compared and it can be observed like experimental results are very

similar to numerical results, although some values differ due to unavoidable errors in the experimental

process.The findings obtained were consistent in a general manner for all rates of loading conditions

Pcomp/Popen, which also show that the conditions of fracture propagation are stable. On the other hand,

the shorter specimens are more sensitive to changes in the compressive loading. Characteristic can be

observed clearly for the relationship of loadsPcomp/Popen= 10. Seems to exist a transition length of thespecimen that coincides for all cases of study, which has a value of 0.18161 m and a geometric ratio of

a/L = 0.3497. For the geometric relationships of a/Lgreater than this value, and loading conditions of

Pcomp/Popen= 5 and Pcomp/Popen= 10, the values of K1 increase drastically, as well as the direction of

fracture propagation which seem to become more unstable. Analyses using the finite element method

was carried out on transition length, for this studies the results are shown in figure 4 and it can be

corroborated the influence that of the load and geometry ratios in each case of study. By FEM the

exact size of transition length is determined as well which is of 0.18161m and a geometric relationship

of a/L=0.3497.

ConclusionsWhen concluding the numeric and experimental work, as well as when analyzing the results, quite

interesting conclusions were reached regarding the tendency of theK1values for specimens subjected

to biaxial tensile-compressive stresses as well as their influence on the fracture propagation stability.The conclusions of the work are:

1. The SEN specimens with length L > 0.18161 m and with geometric relationships a / L 0.3497are

unstable in the direction of fracture propagation.

3. The Yfactor in SENspecimens subjected to compressive and opening load, is an a / Lfunction

as well.

4. The geometry of the rectangular specimens influences directly theK1values and the directional

stability of fracture propagation.

5. The compressive load increases the stress intensity factor K1considerably in short specimens

up to 45%forPcomp/Popen = 10, while in longer specimensK1is increased by 1.6%for the sameratio of loads.

6. The uncertainty in theK1values for short specimens and big rates of load Pcomp/Popen, produce

uncertainty in the direction of fracture propagation, this seems to indicate that both parameters

are related in a direct way.

7. Without compressive load, a long specimen is more stable for the direction of fracture

propagation.

8.- AcknowledgementThe authors gratefully acknowledge the financial support from the Mexican government by de

Consejo Nacional de Ciencia y Tecnologa and the Instituto Politcnico Nacional, Escuela Superior de

Ingeniera Mecnica y Elctrica.

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9.- References[1] Urriolagoitia-Caldern G and Hernndez-Gmez L H, 1996, Evaluation of crack propagation

stability with the Williams stress function-I. Stress field analysis, International Journal

Computers and Structures,Vol. 61, No. 4, pp 775-780.

[2] Urriolagoitia-Caldern G and Hernndez-Gmez L H, 1997, Evaluation of crack propagation

stability-II. Numerical Analysis, International Journal Computers and Structures, Vol. 63,

No. 5, pp 1007-1014.

[3] Urriolagoitia-Caldern G and Hernndez-Gmez L H, 1997, Experimental analysis of crack

propagation stability in single edge notch specimens, Theoretical and Applied Fracture

Mechanics, Vol. 28, pp 57-68.

[4] Sauceda-Meza I, 2000, Anlisis Numrico-Experimental de la Estabilidad Direccional de

Propagacin de Fractura bajo Campos Biaxiales de Esfuerzos, Ph D Thesis, InstitutoPolitcnico Nacional SEPI-ESIME.

[5] F. Erdogan, G.C. Sih, 1963, On crack extension in plates under plane loading and tranasverse

shear,J. Basic Eng. Trans. ASME85(D), pp. 519-527

[6] B. Cotterell, 1965, On brittle fracture paths,Int. J. Fract. Mech. I, pp. 96-103

[7] B. Cotterell. 1966, Notes on the paths and stability of cracks, Int. J. Fract. Technol. 2, pp. 526-

533

[8] M. E. Kipp. G. C. Sih, 1975, The strain energy density failure criterion applied to notched

elastic solids,Int. J. Solids Struct., 11, pp. 153-173

[9] L. Nobile, 2000, Mixed mode crack initiation and direction in beams with edge crack, Theor.

Appl. Fract. Mech., 33, pp. 107-116

[10] G. C. Sih, 1974, Strain density factor applied to mixed mode crack problems, Int. J. Fract., 10,

pp. 305-321

[11]

ANSYS Structural Analysis Guide, 2004, Chapter 10, Fracture Mechanics, SASP IP Inc, Ed.[12]

ANSYS Element Reference, 2004, Chapter 4, Element Library, SAS IP Inc.

[13] Rodrguez-Martnez R, 2004, Anlisis Numrico-Experimental de Estabilidad Direccional de

Fractura bajo Esfuerzos Biaxiales, MSc Thesis, Instituto Tecnolgico de Tlalnepantla,

Mxico.

[14] Urriolagoitia-Caldern G, 1976, Directional Stability of Cracks Under Biaxial Stresses, Ph D

Thesis, Department of Mechanical Engineering, Imperial College of Science and

Technology, University of London.

[15] Wang C H and Luo P F, 2008, An Experimental Study on the Elastic-Plastic Fracture in a

Ductile Material Under Mixed-Mode loading, Strain, Vol. 44, No. 3, pp 223-230, June.

[16] Dally J. W. and Riley W. F., 1991, Experimental Stress Analysis, McGraw-Hill, Inc., p.428 and

p.511

[17]

Zanganeh, M, Tomlinson, R A, and Yates J. R., 2008, T-stress determination usingthermoelastic stress analysis,Journal of Strain Analysis for Egineering Design, 43, pp. 529-

537

[18] Cotterell, B, 1966, Notes on the paths and stability of cracks, International Journal Fracture,

2(3), pp. 526-533

[19] Pacey, M N, James, M N, and Patterson, E A, 2005, On the use of photoelasticity to study crack

growth mechanisms,Experimental Mechanics, 45

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