Modelling of Electronic Textiles
M.T.J. Fonteyn
MT10.07 Supervisors: Dr. P.C.P. Bouten Dr. Ir. R.H.J. Peerlings Prof. Dr. Ir. M.G.D. Geers Philips Research, Eindhoven Photonic Materials and Devices University of Technology, Eindhoven Department of Mechanical Engineering Mechanics of Materials March, 2010
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Abstract
A model is presented for prediction of stresses and strains in electronic textile. Characteristic structural behaviour of textile is implemented in the model by using a combination of truss elements and a continuum material. The rotation of the truss elements results in a non-linearity and evolving anisotropy in the response, which is closely related to that of the true material.
Tensile tests are performed in three orientations to determine material parameters used in the model. These tests are performed in combination with image processing to acquire global and local deformation information of the textile. Tests are performed in these and other orientations and compared and with numerical tensile tests as validation.
A study on the effect of integration of stiff two-dimensional components in fabric is performed. The influence of component stiffness, size, shape and the distance between the components on issues as stress concentrations and substrate response are studied making use of analytical and numerical calculations. Influence of component size with respect to these issues is derived analytically.
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Contents 1 Introduction
1.1 Problem description…………………………………………………… 1 1.2 Research Goal…………………………………………………………. 2 1.3 Current state of art…………………………………………………….. 2 1.4 Strategy followed……………………………………………………… 3 1.5 Outline………………………………………………………………… 4
2 Characteristics of the used textile
2.1 Material……………………………………………………………….. 5 2.2 Mechanical response …………………………………………………. 6
3 Modelling of textile 3.1 Structure of the model……………………………………………….... 9
3.2 Response in diagonal direction………………………………………... 11 3.3 Finite element modelling……………………………………………… 14 3.4 Validation……………………………………………………………… 15
4 Parameter identification
4.1 Test setup and equipment……………………………………………… 17 4.2 Optical method………………………………………………………… 19 4.3 Fitting experimental results……………………………………………. 20
5 Numerical tensile tests 5.1 Marc-Mentat model……………………………………………………. 23 5.2 0 and 90 degrees (Warp and weft)…………………………………….. 25 5.3 45 Degrees (Diagonal) 5.3 .1 Overall comparison………………………………………..… 25 5.3 .2 Contraction of the sample and orientation of the yarns…..….. 27 5.3 .3 Out of plane deformation……………………..……………… 28 5.3 .4 Predicting strain of yarns……………………………..……… 29 5.3 .5 Influence of truss stiffness……………………….………….. 31 5.4 15 and 75 degrees……………………………………………………….. 32 5.5 30 and 60 degrees...…………………………………………………….. 32
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6 Modelling of textile with Components 6.1 Unit cell model………………………………………………………… 35 6.2 Results reference model…………………………………………….… 37 6.3 Results of parameter variations 6.3.1 Variation of component stiffness…………………………….. 39 6.3.2 Variation of component size……………………………….... 40 6.4 Analytical model, deformation in diagonal direction ………………... 41
6.5 General……………………………………………………………….. 43 6.5.1 Influence truss properties.………………. ………………..… 43 6.5.2 Influence truss spacing.………………. …………..………… 45
6.6 Comparison of analytical and numerical model………………………. 46 6.7 Influence of component pattern……………….………………………. 47 6.8 Discussion…………………….……………….………………………. 47
7 Conclusion 50 References 53 Appendices A ………………………………………………………………………….. 55 B ………………………………………………………………………….. 57 C …………………………………………………………………………. 61 D …………………………………………………………………………. 64 E …………………………………………………………………………. 65 E …………………………………………………………………………. 73
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Stresses and strains e Engineering strain
ε True strain
S Engineering stress σ True stress Γ Shear strain Material parameters E Young’s modulus v Poisson’s ratio G Shear modulus Dimensions l Length in unit cell/element h Thickness of sample and model A Cross section area L Size of unit cell with component D Component size
d Spacing between trusses = 2/l Force F Force in truss Vectors
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Tensors E Strain tensor S Stress tensor α Half of the angle between warp and weft yarns. Warp 0 Degrees Weft 90 Degrees Diagobal 45 degrees
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1 Introduction Since the invention of incandescent light bulb light sources and other electronics have undergone tremendous developments which have made electronics applicable at extremely small scales and for many purposes. Over the last decade there is a growing trend towards flexible electronics and electronic components attached on or integrated in flexible substrates like textiles. Many applications are imaginable with flexible electronics, as they can be integrated in for example clothes or furniture. However, integration of electronics in flexible substrates induces new challenges concerning reliability and mechanical failure which have to be overcome. As a first step towards addressing these challenges the mechanical behaviour of a textile substrate with and without components is investigated. 1.1 Problem description Textiles are complex materials which consist of one or more layers of yarns that are woven in a more or less periodic structure. Textiles on which electronic components like LEDs are attached and in which the conductive wires are an integral part of the (woven) substrate are so called electronic textiles. Textile is used as a compliant substrate on which relatively stiff components such as e.g. LEDs and ICs are fixated. An example of electronic textile with LEDs is shown in figure 1.1. The conductive wires usually have material properties that are different from those of the non-conductive yarns in the textile. Particularly the stiffness mismatch between components and substrate results in stress concentrations and may compromise the integrity of the substrate and the connections of the components. This may cause problems concerning reliability and manufacturing. The material used in this study is an orthogonally woven multilayered textile with a matrix of conductive wires interwoven with the warp and weft yarns. The conductive Elitex™ wires are polyester wires with a metal coating. A matrix structure is created in which the conductive warp yarns are woven in the layer at the opposite side of the layer with the conductive weft yarns and these layers are separated by a layer of non-conductive yarns. Via connections are made by interweaving conductive warp and weft yarns at specific points within the matrix.
Fig. 1.1 LEDs attached to textile with conductive yarns. Source: Philips Electronics.
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In order to be able to predict deformations and stress concentrations near components qualitatively as well as quantitatively, and to predict the influence of material and geometric properties of the components, it is necessary to have a model of the textile substrate. This model would allow one to optimize component properties such as geometry the used material, and the component location with respect to other components and conductive wires, without producing and testing all possible variations. 1.2 Research goal The long term research goal towards which the present study contributes is the development and optimization of textile substrates and the fixation of components on them in order to develop robust and durable applications with electronic textiles. The goal in this study is the development of a numerical material model which describes the material behaviour of the textile properly in a qualitative and quantitative way. The model is to be used to predict stresses and strains around components attached to a textile. Essential parameters in the description of the textile’s behaviour will be obtained from experiments. 1.3 Current state of the art Textiles are quite an old field of research. However, during the last decade numerical modelling of textile has become a possibility. Initially these materials were modelled as a continuum, but developments in computer science made multi scale modelling and modelling of geometrical details possible. The smaller the scale, the more detailed the model can be. Modelling at the micro level concerns the internal structure of the yarns in terms of fibres and the interaction between the fibres and perhaps a few yarns in a relatively small part of the textile. An example of micro scale modelling is the multi chain element modelling as done by Zhou (2004). This model describes the deformation of yarns by modelling the mutual contact between fibres in a yarn and the contact between these fibers and those in the orthogonal underlying yarns. Mesoscopic models are mostly FEM models in which the yarns are modelled without geometrical details such as fibres. An example of a mesoscopic program is Wisetex, which models a 3-dimensional unit cell of the textile and computes the shape using the principle of minimum bending energy (Verpoest & Lomov, 2005), or FEM models, which model the compression of solid yarns under shear loading or frictional contact models (Lin et al., 2008, Cherouat et al., 2008). For special applications such as impacts, e.g. for armours, also visco-elastic models are available. (Liu et al., 2006) At the macroscopic level textile is modelled as a continuum in which no yarns and fibres are distinguished, but the textile is modelled as a uniform material. Non-orthogonal continuum models with a covariant coordinate system have been developed by Xue et al. (2003) and Peng et al. (2004). These orthotropic material models are mainly linear elastic. Also isotropic hyper-elastic models are available (Ruíz & González, 2006). A macroscopic model has been developed earlier at Philips Research (Feron, 2008b). This model is an orthotropic elastoplastic model. It uses a Hill48 yield criterion combined with a Nadai-Ludwik hardening law. Unloading behaviour is linear elastic as no plastic deformation occurs during unloading. The constitutive model has its origin in sheet metal forming. An advantage of the model is that it is a continuum model, thus computationally efficient. A disadvantage of the model is the large number of parameters, nine in total, which have to be determined
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experimentally and do not all have a clear physical meaning for textile. A major limitation of the model is that the non-linearity of the loading curves can not be fitted direction independent as the power of the hardening law determines the (isotropic) hardening behaviour in all directions. Once the power of the hardening law is fitted the overall stiffness can be fitted by changing the yield stress but the linearity of the curve cannot be tuned. As result of this the loading behaviour cannot be described properly in all directions, which is illustrated in figures 1.2 and 1.3. In textile the non-linearity is strongly dependent on the orientation.
Fig. 1.2 Fit with Nadai-Ludwik hardening in warp direction. Fig. 1.3 Insufficient description of non-linear loading behaviour.
1.4 Strategy followed For a better description of the direction dependency of the non-linearity a new material model is developed in which the structural behaviour of the textile is incorporated. As a model at macro- or meso-level is the most appropriate for this study the model must not contain much geometrical detail. In contrary to the continuum model developed by Feron the new model consists of a combination of truss elements which models the interaction of yarns, and a compliant continuum material resulting in a model which is a mix of a mesoscopic and a macroscopic description. Combinations of truss and continuum elements are used before by Cherouat (2008), but without the link to physical or measurable properties of textile. For the new model no more than 4 parameters are necessary as the anisotropy and non-linearity parameters are captured naturally by the truss elements. Measurements on textile are performed in three directions, i.e. 0, 90 and 45 degrees with respect to the warp yarns in order to acquire input parameters for the material model. Local strain data from these experiments is obtained with optical analysis and input parameters for the model are identified making use of least square fits and an analytical derivation of the model response in 45 degrees direction. Tensile tests at other orientations are performed and also these tests are compared with numerical tensile tests to validate the model. Finally numerical calculations are performed to model textile with stiff components attached to it and study the effect of component properties and component patterns on the resulting stresses and strains in the substrate.
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1.5 Outline In chapter 2 the properties of the textile used in this study and its mechanical response are described. The model is presented in chapter 3 and the response of it is derived analytically. The experimental methods and parameter fitting method are described in chapter 4 and the model is validated on experiments in chapter 5. In chapter 6 a numerical study to the response of textile with components is presented. The last chapter contains conclusions, discussion and recommendations.
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2 Characteristics of the used textile 2.1 Material There are many types of textile, among them knitted, woven and non-woven textiles can be distinguished. Only woven textiles are within the scope of this study, so knitted and non-woven textiles will not be discussed. Within the category of woven textiles orthogonal and non-orthogonal woven textiles can be distinguished. Both are quite common, but the orthogonal woven textiles are more natural choice for the conductive matrix structures desired in electronic textiles. Woven textiles can either be mono-layered or multi-layered, but here too multi-layered structures are needed for electronic textiles. During production of the textile, the warp yarns are stretched in a loom and repeatedly lifted in a certain combination and order while the weft yarns pass through. The order of passing and lifting determines the structure and number of layers of the textile. Processing conditions like yarn tension and yarn temperature have a major influence on the properties of the textile. However, the production of textile is not within the scope of this study.
Fig. 2.1 A top view of a backlit multilayered orthogonal weave. Fig. 2.2 Cross sections of impregnated textile.
The textile used in this study is an orthogonally woven three layered textile. This textile is shown in figure 2.1. In this image the orthogonal weft and warp directions are clearly visible. Figure 2.2 shows cross sections of the textile at two orientations. To make the cross section images the material was first impregnated with a silicon rubber to fix the yarn and fibre positions during cutting. In these images the three different layers can be clearly distinguished and it is clearly visible that the warp yarns are dense and straight bundles of fibres and the weft yarns are loose bundles curved around the warp yarns.
Fig. 2.3 Connection between the different layers. Fig. 2.4 Cross section of on conductive Elitex wire.
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The connection between the different layers is made by weft yarns wrapped around the warp yarns of the opposing layer at several places. In figure 2.3 such connection has been visualized by indicating the warp yarns with circles and a weft yarn with a dashed curve. These connection points are also visible as the light spots in figure 2.1. In the used textiles also conductive yarns are integrated. Figure 2.4 shows a cross section of an impregnated textile with such a conductive Elitex™ wire. The conductive yarns are clearly thicker than the other yarns. Some properties of the textile are listed in table 2.1. The number of warp yarns is 1.9 per millimetre and the number of weft yarns is 3.2 per millimetre in each layer, which equals 5.7 warp yarns and 9.6 weft yarns per millimetre in the textile. The measured thickness is about 0.6 mm, however, it cannot be measured accurately as the material consists of loose bundles of fibres and yarns as visualized in figure 2.2. The thickness of the samples has been measured with a Heidenhain style with a flat tip at several places and the found values are averaged.
Propteries textile
Orthogonal plain weave, Three layers
Material Polyester
Areal Density 212 g/m2 Thickness 0.6 mm Warp Yarns /mm /layer 1.9 Weft Yarns /mm /layer 3.2
Table 2.1 Properties of the textile used in this study. 2.2 Mechanical response To illustrate the mechanical behaviour of the textile tensile tests have been performed in warp, weft and diagonal direction. For a fair comparison it is preferable to test samples with a length which is much larger than the width so that the effect of inhomogeneous deformations near the clamps on the total measured deformation is negligible. At the same time the sample width must be large enough so that boundary effects at the cut edges have limited influence. Samples used in this study are limited to a length of 100 mm due to the dimensions of the available material. This is smaller than desired, however it is not unreasonably small as the textile has a density of 5.7 and 9.7 yarns per millimeter in respectively warp and weft direction. The width of samples is 26 mm and the tested length is 60 mm. The samples were loaded in 5 cycles with maximum forces of 5N, 10N, and 15N (3x). The unloading response gives an impression of the inelastic behaviour of the material.
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Fig. 2.5 Stress-strain response in warp and weft direction. Fig. 2.6 Stress-strain response in diagonal direction.
To reduce viscoelastic effects the strain rates were kept low by using extension rates of 0.01 mm/s in warp direction, 0.02 mm/s in weft direction and 0.20 mm/s in diagonal direction. The duration of the tests was approximately 400 to 600 seconds in all directions. The responses obtained in the tensile tests, expressed in engineering stresses and strains, are shown in figures 2.5 and 2.6. For the calculation of the engineering stress in the textile the initial cross section area is used. The initial width times the thickness is used as reference area for the engineering stress. It has to be mentioned that the engineering stress is not a measure for the real stress in the fibres and yarns. The curves shown in figures 2.5 and 2.6 clearly demonstrate the anisotropy of the material. The overall strain in weft direction is about three times as high as in warp direction at an equal engineering stress of nearly 1 MPa. The stiffer response in warp direction is due to the fact that the warp yarns are initially virtually straight and have a higher fibre density compared to the weft yarns. The overall strain at maximum load in diagonal direction is about thirty times higher than in warp direction. Also notable are the high non-linearity of the loading behaviour in diagonal direction and the larger residual deformation compared to the warp and weft directions.
Fig. 2.7 Three different deformation modes can be distinguished in a tensile test in diagonal direction: tension in region A, shear in both warp and weft direction in area C, and shear in either warp or weft direction in areas B and B’. Deformations as result of tension in regions A are relatively small.
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The stress-strain curves of the tensile tests are based on the measured clamp displacement and the initial sample dimensions. However, the engineering strain is an average value as the deformation between the clamps is non-uniform, particularly in diagonal direction. In figure 2.8 an image of a sample tested in diagonal direction is shown. Structural effects can be observed in this image. The triangular areas marked “A” behave as relatively stiff areas due to the constrained introduced by the clamps. The loading in these areas is mainly tension. In the middle of the sample, which is marked “C”, the deformations are large. In figure 2.8 a close-up of area C is shown in the undeformed state. The deformed state of this area is shown in figure 2.9. As result of the prescribed elongation, the yarns in this area rotate towards the tensile direction. The dominating deformation mode is therefore shear. The strongly non-linear loading behaviour observed in figure 2.6 is caused by this deformation mode. The transitional areas B and B’ are loaded in both tension and shear.
Fig. 2.8 Undeformed textile. Yarns are orthogonally oriented. Fig. 2.9 Deformed sample, yarns rotated towards loading direction
The difference between the responses in warp, weft and diagonal directions is large. This is not surprising as textile is in fact a structure. The large deformations and the non-linear response in diagonal direction are caused by the large rotation of the yarns. Incorporating this structural behaviour in a material model could result in a better description of the mechanical behaviour without using many fitting parameters.
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3 Modelling of the textile In this chapter the material model used in our numerical study is described in detail. Analytical derivations of the stress-strain behaviour are presented, which are useful for the parameter identification and to get an impression of the response of the model. This analytically derived response is also compared with numerical calculations to validate some assumptions made in the derivation. 3.1 Structure of the model The developed model is a combination of elastic truss elements, which represent the tensile response of the yarns in the textile, and elastic continuum elements to model other effects such as friction between the yarns and compression of the yarns. Figure 3.1 illustrates how the model is constructed. The truss elements are connected at the corners of the continuum elements and are aligned with the edges of the continuum elements. The connections at the corners, which can be compared with pin joints, link the displacements of the continuum elements and the truss elements without constraining the rotations. The combination of a continuum material and truss elements implies that the model is partly a constitutive model and partly a structure. Merely introduce the truss elements increased stiffness in warp and weft direction. They do not necessarily represent individual yarns so the model is an intermediate solution between a mesoscopic and a continuum model.
Fig. 3.1 The material model is a combination of continuum elements and truss elements which are connected at the corner nodes. Stretching in diagonal direction results in rotation of the trusses with respect to each other, mimicking the relative rotation of yarns.
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The model is kept relatively simple by choosing linear elastic material properties for the trusses as well as for the continuum, but if it is desired, non-linear (visco-) elastic or plastic properties can be implemented. The stiffness of the trusses is significantly higher than that of the continuum. As a result, the trusses act as a mechanism which restricts the deformation of the continuum. Deformation of the continuum becomes large compared to that in the trusses in loading directions different from the warp and weft direction. Non-linear response is obtained by using large deformation theory with logarithmic stresses and strains in the continuum material. Contrary to yarns in textile the trusses can withstand compression loading if no additional criterion is implemented. Only numerical calculations of textile with components will be performed with additional criterion. Figure 3.1 shows the model’s undeformed and deformed state in diagonal direction, and should be compared with the undeformed and deformed material as shown in figures 2.8 and 2.9. In this diagonal deformation mode, the stress-strain response is governed by the compliant continuum material as the relatively stiff trusses hardly deform. Note that in this mode the (lateral) strains in the continuum are determined by the orientation of the truss elements. In other deformation directions compression or elongation of the stiffer truss elements will be necessary and thus a relatively stiff response is observed. The rotation of the truss elements results in a nonlinearity and evolving anisotropy in the response, which is closely related to that of the true material. Three specific deformation modes can be distinguished, in each of which the response is dominated by one of the three sets of elements, so their properties can be fitted independently. In figure 3.2 these three modes are illustrated, with the red color indicating the elements which dominate the stress-strain response.
Warp Weft Diagonal
Fig 3.2 Different elements determine the stress-strain response in these three specific deformation modes.
In warp and weft direction the trusses in respectively warp and weft direction govern the stress-strain response, since the continuum elements are compliant compared to them and the lateral trusses are loaded only indirectly via contraction of the continuum. On the other hand, in the diagonal directions the truss properties have little influence on the response as they can be seen as rigid bars in this mode. As a result the stress response is governed by the continuum elements’ response to the deformation imposed on them by the truss network. This expectation will be verified by analytical derivations and numerical calculations in the next sections. To determine the material properties of the truss elements and the continuum material, the cross section area of the truss elements and the thickness and modulus of the continuum have to be defined. The stiffness of the truss elements, which is a product of the Young’s-modulus and the cross section area, determines the response. This means in principle that the area or modulus can be chosen arbitrarily. This is also the case for the thickness of the continuum material. As long as
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only the in plane stiffness is important and out of plane deformations (bending) are not considered. In figure 3.3 it is illustrated how the cross section areas are defined. The dimensions of the continuum material are chosen equal to the dimensions of the sample S, which means that the thickness of the continuum is equal to the measured sample thickness h. The sum of the cross section areas of the individual truss elements is set equal to the cross section area of the sample S, and thus equal to the cross section area of the continuum material. The cross section area of an individual truss is therefore defined to be equal to the spacing between the trusses, d, multiplied by the sample thickness h. Note that this means that the Young’s modulus of the trusses is different from the Young’s modulus of the yarns, even though the stiffness of the yarns is accounted for correctly .
Fig 3.3 The sum of the areas of the elements Ti equals the area of the sample AS.
3.2 Response in diagonal direction In this section the response of the model is derived analytically in diagonal direction. The derivation of the response in diagonal direction is necessary to obtain the properties of the continuum material in the model. The derivation is based on the assumption that the truss elements are stiff compared to the continuum material in order that the trusses can be assumed to be rigid. This assumption is numerically verified in paragraph 3.4.
Fig 3.4 The unit cell with truss elements can be simplified as it is symmetric in 2 directions.
In figure 3.4 a quarter unit cell, is shown in undeformed and deformed state. As result of symmetry only a quarter of this element has to be described. This quarter cell consists of a continuum material with a Young’s modulus Ec and Poisson’s ratio ν and one truss. The truss is rigid, thus the length of the truss remains constant. This enables calculations of stresses and strains directly as function of the strain in tensile direction.
1l and 2l are the initial dimensions of the quarter cell. These dimensions are directly related to
the truss orientation. In undeformed state, as warp and weft yarns are orthogonally oriented, are l1 and l2 equal. Initial deformation can be described with values for l1 and l2 which are not equal.
SAdhAn
iTT ii
=⇒= ∑=1
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The length of the truss is assumed to be constant as the truss is assumed to be rigid:
( )[ ] ( )[ ] 22
21
222
211 11 llelel +=+++ [3.1]
Definition logarithmic strains in the continuum:
( )11 1ln e+=ε [3.2]
( )22 1ln e+=ε [3.3] In initial undeformed situation holds:
lll == 21 [3.4]
(Note that this assumption determines a truss spacing and cross section of 2l ) Making use of relations [4], [5] and [6] equation [3] can be written as function of logarithmic strains:
2)2exp()2exp( 21 =+ εε [3.5]
The strain in lateral direction 2ε can be expressed as function of 1ε . [3.6]
( ))2exp(2ln21
12 εε −= [3.7]
The in-plane strains are known, so the stresses 1σ and 2σ and the strain 3ε in the continuum
material can be calculated with the plane stress equations for large strains. As result of symmetry the 1 and 2 orientations or the continuum do not rotate.
)(1 2121 εεσ v
v
Ec +−
= [3.8]
)(1 2122 εεσ +
−= v
v
Ec [3.9]
)(1 213 εεε +
−−=
v
v [3.10]
The stresses and strains in the continuum are expressed with the formulas above. However, desired is a stress-strain relation in which the contribution of the truss elements on the resulting stress is included. The engineering stresses of the combination of the continuum material and the truss can be expressed as function of the true stress in the continuum and the still unknown forces in the truss.
2
)exp()exp( 1
3211h
FS
εεεσ ++= [3.11]
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2
)exp()exp( 2
3122h
FS
εεεσ ++= [3.12]
The first terms in these equations are the true stresses in the continuum multiplied by the actual strains, to obtain engineering stress. The second part of these expressions are the contributions of the still unknown force F in the truss acting 1 and 2 direction, devided by the cross section area of
the continuum 1 x h. The ratio 2/)exp(ε is the ratio between the force in the truss and its component in 1 or 2 direction. As in uniaxial tension no external forces act on the model in lateral direction S2 has to be zero. Making use of this equilibrium the force F in the truss in lateral direction can be expressed as function of the known stresses and strains in the continuum:
)exp(2 2312 εεεσ −+−= hF [3.13]
Substitution of this force in equation [12] results in the following expression for the engineering stress in tensile direction:
[ ]2211321 )22exp()exp( σεεσεε −−+=S [3.14] The first term between the square brackets is the direct contribution of the continuum to the overall stress in tensile direction, the second part is the contribution of the trusses and is strongly non-linear. By using equations [3.2], [3.7], [3.8], [3.9], [3.10] and [3.14], indicated by blue numbers the overall engineering stress strain response of the model in diagonal direction can be calculated. The response is linearly dependent on the Young’s modulus of the continuum material and non linearly dependent on Poisson’s ratio. Note also that the stress S1 approaches infinity when
1e
approaches 12 − , i.e., when the truss mechanism is fully stretched. The lateral strain response, described by equation [3.7] and is shown in figure 3.5. The resulting stress in tensile direction is shown in figure 3.6 for a Poisson’s ratio of 0.5, normalised by the Young’s modulus Ec of the
continuum. The characteristic behaviour when approaching the strain limit of 41.012 ≈− is clearly visible in both figures. It is also instructive to derive the linearized stress strain response, i.e. that for small strains e1 for which 11 e≈ε and the rotation of the trusses can be neglected. This derivation is given in appendix C and results in:
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2e
v
ES c
lin += = 14Ge [3.15]
where G is the shear modulus of the continuum. This expression shows that for small strains the diagonal response of the model is governed by the shear deformation of the continuum. The linear derivation of this formula is written in appendix C. This linear derivation has also been plotted in figure 3.6. It clearly captures the early response of the non-linear model, but fails to capture the locking effect due to rotation of the trusses.
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Fig. 3.5 Lateral strain as function of tensile strain. Fig. 3.6. Stress-strain response in diagonal direction
The influence of Poisson’s ratio on the response, at a constant shear modulus of one is visualized in figure 3.7. The increading non-linearity with increasing Poisson’s ratio is related to the strain in out of plane direction of the continuum material, in which the material can expand without any restriction.
Fig. 3.7 Response in diagonal direction for different values of ν. Fig. 3.8 Unit cell as implemented in Marc-Mentat.
3.3 Finite element modelling A unit cell of the full model is constructed in Marc Mentat 2008r1 by using one four-node quadrilateral shell element of type 139 and two truss elements type 9 as illustrated in figure 3.8. Displacements of the truss elements are coupled to the displacements of the continuum material as they share the same nodes. The bottom left node is fixed in all directions. Displacement of the top left node is fixed in horizontal direction and the displacement of the bottom right node is fixed in vertical direction. Horizontal displacement of the bottom right and top right node and vertical displacement of the top left and top right node are coupled. The shape of the unit cell remains rectangular and can elongate and contract without restriction, thus a uniaxial stress state is created. Both the truss elements and the continuum material have linear elastic material properties so the model is computationally efficient (Cherouat 2008). Not that the behaviour is non-linear as large deformation theory is used.
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The cross section areas of the trusses are equal to cross section area of the continuum element, as explained in 3.1. Numerical Modelling in Marc Mentat limits the Poisson’s ratio between 0 and nearly 0.5, however this limitation does not exist in the analytical derivations presented in the previous section. Solution control is set to non-positive definite as the combination of truss and continuum elements could result in non-positive definite matrices. Large strain analysis and updated Lagrange method are used. The default setting for the convergence tolerance of 0.1 is changed to 0.01. The load increments step size is set to automatic, with a maximum relative step size of 0.03. 3.4 Validation First the analytical derived response in diagonal direction is compared with the response of the numerical unit cell implemented in Marc-Mentat. To check if the Marc Mentat calculation matches with the analytical calculation the trusses in the marc Mentat model are first made extremely stiff compared to the continuum. For the properties of the continuum a Young’s modulus of 1 MPa and a Poisson’s ratio of 0.4 are chosen. The Young’s modulus of the trusses is set to 1 x 106 MPa and tolerance of the solution control in Marc-Mentat is set to 1%. In figure 3.9 the stress strain response of the Marc-Mentat model is compared with the analytical calculation for rigid trusses. The results of the Marc Mentat calculation and the analytical derivation are nearly identical, so the Marc-Mentat model has been implemented in Marc Mentat correctly.
Fig. 3.9 Marc-Mentat calculation with stiff trusses compared with analytical calculation.
To check for which stiffness values the assumptions of rigid trusses in diagonal loading, and the assumption of negligible influence of the continuum in warp and weft direction are valid, two series of calculations with different values for the Young’s modulus are performed in Marc Mentat. An engineering strain of 1% is prescribed in warp/weft direction and an engineering strain of 10% in diagonal direction. In warp/weft direction the stiffness of the trusses is varied from 0.01 to 40 times the Young’s modulus of the continuum material. The engineering stress for the prescribed strain is calculated with Marc-Mentat. In figure 3.10 the relative contribution of the continuum as function of the relative truss stiffness is plotted, normalized on a truss stiffness of zero. At a truss stiffness of 30 times the stiffness of the continuum or more, the influence of the continuum on the total response is lower than 5%.
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Fig. 3.10 Marc-Mentat calculations warp/weft direction. Fig. 3.11 Marc-Mentat calculations diagonal direction. Relative contribution of the continuum. At engineering stain of 10%, Poisson’s ratio of 0.49.
The results of he calculations in diagonal direction are plotted in figure 3.11. Here the stress is normalized by dividing the values by the result of the calculation with very stiff trusses. The curve reaches a value just above 0.5 for compliant trusses. In case of small strains and a truss stiffness of zero this value would be exactly 0.5 as the (shear) strain in the continuum is doubled as result of the trusses. The difference is only 5% between the response with rigid trusses and trusses with a modulus of truss elements of 30 times of that the continuum. In chapter 4 it will turn out that the difference in stiffness is significantly higher than a factor 30, so the assumptions in warp and weft and diagonal direction are correct and no additional corrections are necessary with fitting the material parameters.
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4 Parameter identification The textile is tested at orientations of 0, 45 and 90 degrees with respect to the warp yarns to obtain the input parameters for the material model. Locally measured strain data is required to determine the desired parameters. Therefore a camera setup and image analysis software is used to measure strains in uniformly deforming parts of the samples. The advantage of optical methods is that there is no physical contact with the sample. From the loading data of the tensile tester in combination with strain data, obtained by image analysis, the model parameters are derived based on tests at 0, 45 and 90 degrees. 4.1 Test setup and equipment Figure 4.1 shows the test setup. The used testing equipment is an Instron 5566 tensile tester and a pc with Instron software. Local strains are obtained with an AVT stingray black and white camera in combination with Labview image analysis software. During testing images are stored on a second PC together with the actual time, clamp position and the load measured by the tensile tester. The sample is lit from behind through a narrow window to reduce glare and disturbing reflections. To keep the camera focussed on the centre of the sample the camera is attached to a linear guide which moves with half of the displacement of the moving clamp. The advantage of the moving focus point is that a larger part of the recorded area is useful for optical analysis.
Fig. 4.1 Test setup, showing tensile tester, sample and the camera mounted on a linear guide.
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Samples are cut from the textile at different orientations. The tested size is equal for all samples, i.e. 60 x 26 x 0.60 mm. The tested sample orientations are 0, 45, 90 15, 30, 60 and 75 degrees. Tests at the first three orientations are used for parameter identification. Only strains measured at the centre of the sample are important for acquiring the parameters of the material model. Therefore the camera captures only the area of interest on the sample to obtain images with maximum resolution of this area. As the structural behaviour of the samples is interesting as well (although not for parameter identification) tests are also performed where the entire sample is recorded, so that effects such as slip at the clamps and failure of yarns are captured. Properties of the test equipment and samples are listed in the table 4.1.
Test equipment Instron 5566 tensile tester 100 N load cell; Compliance 1151 N/mm; acc. 10^-5 Clamps with steel jaws, serrated faces Loading cycles 5,10,15,15,15 N Camera: Stingray AVT F201B. 1624 x 1234 pixels Pulley for camera fixation, ½ x clamp displacement
Table. 4.1 Test equipment. All samples are loaded by five loading-unloading cycles with maximum loads of respectively 5, 10, 15, 15 and 15 N. The unloading behaviour gives insight in the degree to which inelastic behaviour occurs, but is not used for parameter identification. At each orientation at least three samples are tested. The first sample provides information used to estimate an appropriate strain rate, the interval of the captured images and the data output settings on the tensile tester. The other samples provide the data used for the parameter identification. The extension rate of tensile tester is set to a value so that the total test duration is about 500 ± 100 seconds, as shown in table 4.2
Table. 4.2 Average extension rates and test durations for warp, weft and diagonal orientation.
Test conditions, displacement controlled Extension rate Average test duration Warp 0.01 mm/s 400 s Weft 0.02 mm/s 550 s Diagonal 0.20 mm/s 570 s
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4.2 Optical method The area of the sample which is recorded is indicated by a red rectangle in figure 4.2. The method which is used to obtain deformation information of this area is a pattern recognition method developed at Philips Research and programmed in Labview. This method changes the recorded images into binary images and tracks the “centres of gravity” of manually selected bright spots in the textile. More about this method can be found the report of Feron (2008b). In figure 4.3 it is illustrated how the spots are selected for strain calculation. These spots are selected in the first images of the series and are tracked by the software in the following images. Their positions are written to a data file. For a proper tracking of these spots it is important that the contrast between the dots and the surrounding textile is as high as possible and the time interval between subsequent images is not too large. Average light intensity has to be as constant as possible for the individual images, but also for all images in the series.
Fig. 4.2 Red box is indicating the area from which optical data is recorded during the tensile tests.
Fig. 4.3. Tracking bright spots with pattern recognition method.
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For a high contrast the sample is lit from the back by a bright light shining trough a narrow window in order that no light can enter the camera objective directly from the light source and to reduce glare and disturbing reflections. The more spots are tracked the better inaccuracies and inhomogeneities are averaged out, but nine sports seems to be sufficient. The data files with the dot positions are imported in Matlab together with the files containing corresponding load and clamp positions of the tensile tester. Strains are calculated by calculating the average relative change in distance between the selected dots. It has to be mentioned that the results can be influenced by out of plane deformation of the samples. Therefore some care has to be taken in the interpretation of the results. 4.3 Fitting experimental results After analyzing the data acquired with the optical method, local strain data from the uniformly deformed parts of the samples is available in tensile direction and in lateral direction. This local strain data is used to obtain the parameters for the model and is compared with the global strains to detect effects such as failure or slip of yarns from the clamps. In figure 4.4 engineering stress is plotted as function of the global engineering strain derived from the clamp displacement of the tensile tester for a sample orientation of 45 degrees. Figure 4.5 shows the engineering stress as function of the locally measured engineering strain. The shape of the curves is similar but the locally measured strain in the centre of the sample is larger than the engineering strain measured on the clamps. This is due to that fact that deformation of the sample is non-uniform and relatively high at the centre of the sample as the rotation of the yarns, thus elongation of the textile, is constrained near the clamps. This is visible in figure 4.2.
Fig. 4.4 Diagonal orientation, global engineering strain. Fig. 4.5 Diagonal orientation, local engineering strain.
In the figures 4.6 and 4.7 the local strains and the global strains in weft and diagonal direction are plotted as function of time. The global strains measured by the tensile tester are corrected for the compliance of the load cell. In weft direction the curves coincide perfectly, which means that the local and global strains in tensile direction are identical at this orientation. The lateral contraction is less than 18% of the strain in tensile direction, so the deformation of the sample is quite uniform. The match of the curves also means that the optical method is accurate, although some noise is visible in lateral direction as this deformation is small compared to the spatial resolution
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of the camera images. Figure 4.7 visualizes the difference between local strain and global strain in diagonal direction.
Fig. 4.6 Weft orientation, local strain versus global strain. Fig. 4.7 Diagonal orientation, local strain versus global strain.
Figure 4.8 shows local loading-unloading data in weft direction. In the fitting procedures only the loading data is used so the other data is removed by a simple script which removes data from a load which is lower than the highest previous load. What remains is shown in figure 4.9.
Fig. 4.8 Sequence of five loading-unloading cycles. Fig. 4.9. Loading part of the cycles is used for data fitting.
For the parameter identification in warp and weft direction least square fits of a linear curve are applied on the locally measured data. The curves are fitted on two tests for both the warp and weft direction. The fit does not have to pass the origin as only the slope of the curve is a measure for the modulus. The found values are shown in table 4.3.
Material parameters model Warp elements E1 = 88.7 MPa Weft elements E2 = 30.3 MPa Continuum material Ec = 0.20 MPa, ν = 0.49
Table 4.3 Properties for warp, weft and continuum material in model.
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Fig. 4.10 Least square fit in weft direction. Fig 4.11. Least square fit in warp direction. Contrary to the warp and weft directions the analytical model is used for fitting the experiments in diagonal direction. In figure 4.12 local stress strains curves of two tests are shown. The curves do not coincide as a result of different initial states of the textile. Analysis on the images showed that the angle between the warp and weft yarns in test 1 was initially 90.5 degrees. In test 2 the initial angle was 89.5 degrees, resulting in a stiffer behaviour. These differences in initial angle correspond to initial stains of -0.008 and +0.008 respectively for test 1 and 2. When a correction for these initial strains is made by shifting the curves horizontally they coincide perfectly as is visible in figure 4.13. A fitting script has been written which converges to the Young’s modulus for which the least square sum of the local fitting errors is minimized. The Poisson’s ratio determines the linearity of the curve and is set manually. The best fit for this material is obtained with a Poisson’s ratio of approximately 0.51, but it is fitted with a ratio of 0.49 as this simplifies the implementation in Marc-Mentat. The value found for the Young’s modulus is 0.20 MPa.
Fig. 4.12 Tensile tests without correction. Fig. 4.13 Least square fit on both tests after correction.
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5 Numerical tensile tests The input parameters for the material model have been obtained from locally measured strains in the previous chapter and have been implemented in a numerical Marc Mentat model. The Marc-Mentat model is described in detail below. The purpose of the model is not only modelling of uniform stress conditions. Also the non-uniform structural behaviour of the material must be captured properly as the model is to be used to predict stresses around components. Properties of the model such as the pin joints between the weft- and warp trusses, which fix all translations between intersecting trusses, may result in a response of the model which deviates from the response of the textile. To validate the response of the model, the tensile tests in warp, weft and diagonal direction, as well as tensile tests at orientations of 15, 30, 60 and 75 degrees, are compared in this chapter with numerical tensile experiments in the corresponding directions. The comparison is made by confronting both the load-displacement behaviour and the shapes of the deformed samples with those of the numerical model. Subsequently, where differences are observed, the response of the model and tested samples is studied in more detail. 5.1 Marc Mentat model To build a Marc-Mentat model the element size must be defined. Where detailed modelling is required modelling accuracy and calculation effort are often conflicting. For modelling local effects the mesh must be relatively dense, but as only the global response is important for the numerical tensile tests presented in this chapter, a relatively coarse mesh suffices. In appendix D it is shown that the effect of element size on the global behaviour is small. For the tensile test models an element size of 0.5 mm x 0.5 mm and a thickness of 0.6 mm is chosen for the shell elements of type 139. The truss elements, of type 9, have a length of 0.5 mm and a cross section area of 0.3 mm2. This element size implies 2 trusses per millimetre, which is less than the number of yarns in the textile, but accurate enough to model the 26 mm wide sample. The continuum elements are square, which means that the density of trusses is equal in warp and weft direction. Note that these values in principle can be different in the model. The thickness of the continuum material equals the sample thickness of 0.6 mm so the corresponding truss cross section area is 0.5 x 0.6 = 0.3 mm2. For the orientations of 0, 15, 30, 45, 60, 75 and 90 degrees, different Marc-Mentat models are made. The model dimensions are 60 x 26 mm for all orientations and are the same as the tested dimensions. A convenient way to create the models with different orientations in Marc-Mentat is starting with one large sheet of e.g. 200 mm x 200 elements, rotating it to the right orientation and subsequently removing all elements outside the 60 x 26 mm rectangle. The boundary conditions are applied as follows. The displacements of the nodes on the top and bottom edge are prescribed in y-direction and fixed in x- and z-direction, representing the influence of the edges of the clamps. Out of plane deformations in the model are suppressed as these deformations make the simulations unnecessarily complicated. Only in 45 degrees a calculation with allowed out of plane deformation are performed to study the influence of this effect. An example of a model, in 45 degrees is shown in figure 5.1 As deformations are large an updated Lagrange procedure is used in Marc Mentat. The solution control allows non-positive definite matrices as they may indeed become non-positive definite due to large deformations. Furthermore an automatic time step procedure is selected with a maximum relative increment size of 0.02, and a relative force tolerance of 0.5%. This tolerance is
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small but proved to be necessary when the load instead of displacement is prescribed in combination with a very non linear response. Properties of the models are shown in tables 5.1 and 5.2.
Table 5.1 Properties of the Marc-Mentat model
Fig. 5.1 Marc-Mentat tensile test model
Table 5.2 Element properties of the Marc-Mentat model
Dimensions 60 x 26 mm
Number of elements ± 19000
Number of increments ± 40
Tolerance solution control 0.5 %
Calculation time ± 170 s
Warp
elements Weft
elements Continuum elements
Material properties
88.7 MPa 30.3 MPa 0.20 MPa nu = 0.49
Element length
0.5 mm 0.5 mm 0.5 x 0.5 mm
Cross section/ thickness
0.3 mm^2
0.3 mm^2
0.6 mm
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The results of the Marc Mentat calculations are compared below with the experiments in corresponding pairs of orientations of 0 and 90, 15 and 75, and 30 and 60 degrees. The 45 degree orientation is discussed separately. Images of the deformed sample and the model are shown only for the directions were they provide useful information. The results in 0, 90 and 45 degrees are particularly interesting as these directions have been used to fit the model parameters. 5.2 0 and 90 degrees (Warp and weft) In figures 5.2 and 5.3 the numerically and experimentally determined loading curves in warp and weft direction (0 and 90 degrees) are plotted. The load-displacement data obtained from the tensile tester is plotted in blue and the calculated load-displacement behaviour is plotted in red. The response of the model is virtually linear since the truss elements which represent the yarns are linear elastic and the strains are small. The experimental response of the samples is non-linear but the match is reasonable. Deformed samples and models are not shown as the deformations in these orientations are so small that the deformed sample and model cannot be distinguished from the undeformed state.
Fig. 5.2 Tensile test in warp versus numerical tensile test. Fig. 5.3 Tensile test in weft versus numerical tensile test. 5.3 45 Degrees (Diagonal) 5.3.1 Overall comparison The loading curves in the 45 degrees direction are shown in figure 5.4. The Marc-Mentat calculation predicts a 5 % smaller ultimate displacement than measured in the experiment but the overall match of the loading curves is good. When the deformations of the samples are studied in detail some interesting effects can be observed. In figure 5.5 the global strain measured at the clamps is plotted together with the local strain measured in the centre of the sample. The local strain in tensile direction is plotted in blue and the local strain in lateral direction in green. Figure 5.6 shows an image of the deformed sample, in which the area in which local strains are measured is indicated by the letter ‘C’. Strains in this area are higher than the overall strain in the sample as the areas indicated by ‘A’ behave relatively stiff. The global strain is directly coupled to the prescribed clamp displacements, thus the global strain response, plotted with the red curve in figure 5.5, is perfectly (piecewise) linear in time. However, the local strain in tensile direction increases in time in a non-linear fashion. The difference between local and global strain decreases
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in the last three loading cycles, which have the same maximum load. This is the result of slip of the sample in the clamps of the tensile tester.
Fig. 5.4 Tensile test compared with numerical tensile test. Fig. 5.5 Local strain versus global strain.
To visualize this effect other samples are tested with a camera coverage of the entire sample to record the deformation of the sample near the clamps. A careful study of the series of images showed that the yarns slip in the clamps of the tensile tester during loading. The slip occurs at the corners of the sample. Figure 5.6 shows a deformed sample at a maximum load of 15N in the third loading cycle, i.e. the first loading cycle at 15N. The locations where slip of the yarns occurred are indicated by green circles. The areas A, B and C, with different deformation modes, can be clearly distinguished at higher strains by the difference in intensity of light transmitted through the sample. Figure 5.7 shows the deformed Marc Mental model at the same elongation. Comparing the model with the tested sample it can be immediately seen that the size of the compliant area is larger in the tested sample than in the Marc Mentat model. As result of slip the size of areas A decreases and consequently the size compliant area C increases. To indicate the effect of slip the actual size of these stiff areas is indicated by red triangles and the initial size by blue triangles. The ratio between the initial size and actual size of these areas will be used to estimate the effect of slip on the total elongation of the sample.
Fig. 5.6 Zones in which yarns slip out of the yaws are indicated by green circles. This slip results in a decrease of the size of the stiff areas A and an increase of the compliant area C. Near the boundaries of the sample, particularly in area C, the lateral strain is lower than average as a result of slip of the yarns.
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Fig. 5.7 Local strains in the model are larger in area C in the numerical model than in the tested sample at the same global strain. It is assumed that the strain in areas A can be neglected, so that the additional elongation of the sample as a result of slip can be estimated by calculating the change in ratio of the lengths of areas A and C. This is an upper bound estimation as yarns which slip may still carry load. The initial length of area C is 60-26 = 34 mm. As result of slip 26-21.3 = 4.7 mm (undeformed) length is added to the compliant area. Without slip of yarns the displacement is expected to be 1-(60-26)/(60-21.3) = 12% smaller than the measured elongation. Possible effects of the change of the with and length of areas A on the deformation of area C are unknown and are not taken into account. The Marc-Mentat calculation predicts a behaviour that is only 5% stiffer and thus well below the upper bound of 12%. It has to be mentioned that some care has to be taken with interpreting the magnitude of this effect, as the magnitude of slip in this sample might differ from the in the samples from which local strains or model parameters are obtained. 5.3.2 Contraction of the sample and orientation of the yarns Engineering stress is used for model fitting, so the initial cross section area is used for stress calculation in both the model and the samples. The advantage is that the lateral contraction of the sample does not have to be measured. This does not influence the quality of fit when the model and the sample have the same response in contraction. To check if this is true the locally measured lateral contraction and the angle between the yarns of tested samples are compared with analytical calculations and the Marc-Mentat model. In figure 5.8 the lateral strain as a function of the strain in tensile direction is plotted in red for the tested sample and in blue an analytical calculation for rigid trusses is shown. A comparison between the measured and calculated angle results in an identical plot at a different scale so this plot is not shown. At higher strain levels some deviation between the model and sample was expected as result of hindered rotation and elongation of the yarns (Hamila et al. 2009). However the curves match perfectly, which indicates that yarns can still rotate freely. The lateral contraction of the Marc-Mentat model shown in figure 5.7 seems 5 á 10 % higher than the contraction of the experimental sample in figure 5.6. This is the result of boundary effects in the tested sample. The lateral strain measured in the centre of area C (in the sample used for the strain measurements) is equal to the lateral strain in the Marc-Mentat model. Figure 5.9 shows close ups of the boundaries of the model and the tested samples. The angle of the yarns and the trusses is the same at the interior of the sample but at the free boundaries of the sample it deviates. Warp and weft yarns can detach and slip at the boundaries in the experiment, but not in the model.
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Fig. 5.8 Lateral contraction as function of elongation. Fig. 5.9 Close up of model and boundary effects in sample. 5.3.3 Out of plane deformation During testing a small amount of out of plane deformation is observed, but not studied in detail. Images from tests and of the model with out of plane deformation are shown in figures 5.10, 5.11 and 5.12. The maximum observed out of plane deformation is about 0.5 mm for this sample geometry. The strain at which out of plane deformation starts is not exactly the same in each test and is. Slip of yarns reduces the out of plane deformation as the stiff areas near the clamps become smaller. The effect of the out of plane deformation on lateral contraction is visible in figure 5.13. This deformation introduces some noise on the measurements but its effect on the measured lateral strain is small as the magnitude of deformation in out of plane direction is also small.
Fig. 5.10 OOP deformation, sidelit. Fig. 5.11 OOP deformation, backlit. Fig. 5.12 OOP def. model at larger strain.
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In Marc-Mentat out of plane deformation can be initiated by positioning randomly chosen nodes slightly, e.g. 0.001 mm, out of the mid-plane of the sample. This sometimes results in out of plane deformation with two folds, as shown in figure 5.12. This looks similar to the out of plane deformation which occurred in some tensile tests as shown in figure 5.10. The out of plane deformation pattern is dependent on the initial position of the nodes in the Marc-Mentat model. Deformations with one fold and three folds in out of plane direction are also found, which is shown in figure 5.14. No significant influence of out of plane deformation on the load-displacement behaviour was found with Marc-Mentat. The total elongation of the model with out of plane deformation is 16.59 mm and 16.32 mm with suppressed out of plane deformation. The difference in lateral strain between simulations with and without out of plane deformation is about 3%. The influence of out of plane deformation on the global properties is random and small in both the model and the tested samples. The deformation in the model might be tunable by changing the thickness and the modulus of the continuum material as the bending stiffness has influence on wrinkling behaviour (Hamila et al., 2009).
Fig 5.13 Lateral strain is influenced by OOP deformation. Fig 5.14 OOP deformation depends on random initial deformation 5.3.4 Predicting strain of yarns The results shown in figures 5.8 and 5.13, where it is shown that the lateral contraction of the sample and the angle between the yarns match the calculated values, imply that the influence of strains in the yarns can be neglected. It is checked if the assumption of rigid yarn behaviour is in this compliant area right and the influence of strains in the yarns is checked and an attempt is made to measure the stains in the yarns The analytical derivation presented in chapter 3 can be used for an estimation of the maximum strain in the trusses with the parameters obtained from the experiments. The force transmitted by the rigid truss can be calculated by equation [3.13]. When the truss is deformable but the elongation of the truss is small compared to the deformation of the continuum, the strain in the truss can be approximated with linear elastic theory.
The truss cross section area is defined as: 2hlAtruss = [5.1]
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Using small deformation theory we have for the truss strain:
)exp(1
3212 εεεσ +−−=truss
truss Ee [5.2]
This strain can be written as function of the global strain by substituting formulas [3.2], [3.7], [3.9] and [3.10]. The predicted strain in the warp and weft yarns is shown in figure 5.15 and the optically measured strains are shown in figure 5.16. This strain is measured by analysing the change in distance between the bright spots on lines oriented in direction of the warp and weft yarns.
Fig. 5.15 Estimation of average strain in truss elements. Fig. 5.16 Measured strain in warp and weft yarns. Same order of
magnitude as prediction. Measurement is inaccurate.
The estimated strain in the trusses is 0.5% in warp direction and about 2% in weft direction at a maximum overall strain of 32%, which is in the same order of magnitude as the measured strain shown in figure 5.16. The strain in the truss elements is also small at large deformations. However, due to large deformations of the textile in diagonal direction, it is difficult to determine the relatively small yarn strains accurately. Therefore the effect of deformation of the trusses on the total strain is also estimated with Marc Mentat. The uniform deformation of a model with rigid trusses is compared with a deformed model with trusses with properties as derived from the experiments. The comparison is made at a stress equal to the maximum stress in the experiments and is shown in table 5.3. The relative difference in total strain is small, about 3 %, so the assumption that the deformation of the yarns can be neglected seems to be justified.
Stress Continuum Stiffness truss Total strain
0.96 MPa 0.2 MPa, ν=0.49 30 + 89 MPa 0.348 [-]
0.96 MPa 0.2 MPa, ν=0.49 1e6 MPa 0.335 [-] Table 5.3 Comparison in total strain between model with extremely stiff trusses and deformable trusses.
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5.3.5 Influence of truss stiffness Even if the yarns can be regarded as rigid in the uniformly deforming area this does not mean that the deformation of yarns can be completely neglected in the stiff areas near the clamps of the tensile tester. The Marc Mentat model allows to check the influence of truss stiffness on the deformation of these stiff areas. Two Marc-Mentat models at an engineering stress of approximately 1 MPa or a force of 15 N are shown in figure 5.17. The model on the left hand side has truss properties as derived from the experiments. Truss stiffness in the model at the right hand side is extremely high with as stiffness of 106 MPa. The continuum properties are the same in both models. The elongations of the models are 16.36 mm and 12.24 mm, which implies a difference of 4.1 mm. The black dots in the model on the left indicate the intersection of the trusses representing the yarns passing through the corners of the model. The distance between these dots, and thus the length of the compliant area C, is only 0.8 mm larger than the length of the compliant area in the model with the stiff trusses.
Fig. 5.17 Two Marc Mentat models at a load of 15N with elongations of 16.36 mm and 12.24 mm. The model at the left has truss properties as derived from the experiments. The truss stiffness in the model at the right is very high (1 x 106 MPa). Continuum properties are the same in both models. The black dots in the model at the left indicate the intersection of the trusses, representing the yarns passing the corners of the model. The distance between these dots is nearly equal to length of the compliant area in the model on the right hand side.
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The influence of truss properties on deformation of this compliant area is relatively small. However, the displacement of the black dots relative to the edges of the model is 1.7 mm. Thus the strain of the areas A at the centre of the model is 13% in tensile direction and contributes to the total elongation of the model by 3.4 mm. This is 20% of the total elongation of the model. This shows that the strains in trusses do have a significant influence on the total elongation of the model and this is mainly caused by deformation in the stiff areas A. Deformations near the boundaries of areas A are strongly influenced by the truss properties, resulting in significant effect on the total elongation of the model. The trusses in the model stretch up to 9% in area A near the boundaries of areas B. Strains of 25% are observed in the few elements at the corners of the model. In the textile these yarns would slip, or break. It can be concluded that in compliant area C the yarns behave very stiff with strains of maximum 2 %. The assumption that trusses can be assumed rigid for the analytical derivation in chapter 3 is correct as the influence of these strains on the behaviour of the model is small. However, elongation of the trusses has a significant influence on the global deformation when trusses are locally loaded as result of non uniform deformation. Proper fitting in of the properties weft and warp direction is therefore important. 5.4 15 and 75 degrees In figure 5.18 and 5.19 the results in the 15 and 75 degree directions are shown. The numerical loading curves are very linear, but in the experiments the non-linearity becomes significant. The strains are 2 or 3 times higher than the strains in warp and weft direction. This is the result of the fact that only a few yarns transmit the major part of the force from clamp to clamp, so the deformation in these yarns is significantly larger than in warp and weft direction.
Fig. 5.18 15 degrees orientation, numerical versus experimental. Fig. 5.19 75 degrees orientation, numerical versus
experimental.
5.5 30 and 60 degrees Figure 5.20 depicts the results for 30 degrees. The Marc Mentat calculation predicts a maximum displacement which is about 19% lower than in the experiment. The experimental curve is not very smooth and seems to show a kink at a load of 7.5 N. In 60 degrees direction, shown in figure 5.21, the match is better.
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Fig. 5.20 30 degrees, numerical versus experimental. Fig. 5.21 60 degrees, numerical versus experimental.
Similar to the test in 45 degrees direction, yarn slip has an influence on the tensile tests in 30 and 60 degrees, but the effect on the global displacement is even larger in these directions. Slip in 30 degrees direction is larger than in 60 degrees direction and this results in a relatively large increase of the size of the compliant shear band, which is illustrated in figure 5.22. Slip is dominant at the corners which are opposite in the stiff warp direction. The weft yarns are much more compliant and seem to slip less in the clamps. As result slip affects the test in 30 degrees direction more than that in 60 degrees direction. The shear band visible in the model in figure 5.23 has an equal size as the blue band shown in figure 5.22 and is 40% smaller than the shear band in the experimental sample at maximum load, indicated by red lines. A 40% stiffer behaviour is expected, but the difference between the test and calculation is smaller than that. Thus may be due to the fact that the edges of the larger shear band are not very well defined and not very clear and the slipped yarns still carry load.
Fig. 5.22 30 degrees. Slip of yarns results in increase of the size of the shear band.
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Fig. 5.23 30 degrees, Marc-Mentat model at a load of 15N.
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6 Modelling of textile with components
Our model is used to investigate the effect of components on the textile. Issues such as component shape, size and their distribution on the textile substrate have influence on the overall stiffness of the electronic textile and the maximum stresses in the yarns and components. These properties can be related to comfort and reliability of the electronic textile. However, reliability is a complex property which is not straightforward defined and not easy to measure and to model. As measure for reliability the maximum stresses and strains in the trusses, the equivalent stress in the component are compared. The influence of the component stiffness, component size and component distance on these values is studied making use of Marc-Mentat and the trends observed in these simulations are explained using a simplified analytical model. Also the overall stiffness of the electronic textile is of our interest as this important for the comfort of the product. 6.1 Unit cell Model The considered electronic textile can be seen as a textile substrate with a regular matrix of electronic components as shown in figure 1.1. Such regular structures can be subdivided into small identical unit cells from which the whole structure can be built up. See for example the structure shown in figure 6.3, which consists of cells as illustrated in figure 6.1. This cell is a unit cell and is an example of a cell used in this chapter. This cell contains three length scales. The first is the size of the cell, L, the second the component diameter, D, and the third is the truss spacing d. As before, the cross section area of the individual trusses is set equal to the truss spacing d times the thickness of the model h. The cell size L is equal to the centre to centre component distance. Two of these lengths, d and D are varied with respect to L. Similar holds for the orientation of the component, trusses and unit cell. The two independent variables at a given truss orientation are the orientation of the component and of the unit cell with respect to the yarns. The first variable, the orientation of the component is not of interest when round components are used. The second variable, the orientation of the unit cell determines the orientation of the component pattern on the textile and is indicated by ψ in figure 6.2 and 6.3.
Fig. 6.1 Three different length scales. Fig. 6.2 Diagonally oriented pattern. Fig. 6.3 Component pattern aligned with
yarns.
The stiffness of the trusses in warp and weft direction is equal to 80 MPa to achieve symmetric deformation of the unit cell with respect to the diagonal direction. This stiffness is approximately the stiffness of the textile in warp direction as obtained in chapter 4. Also the properties of the
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continuum material are equal to those derived in chapter 4. Components are implemented in the model, at the centre of the unit cell, by locally increasing the stiffness of the continuum. As result the component thickness is equal to the model thickness of 0.6 mm. Yarns in textile have virtually no stiffness in compression as result of buckling. In the model this behaviour is mimicked by using the linear Mohr-Coulomb yield option in Marc-Mentat, with a value of
3/2−=α and a yield stress of 1 x 10-9 MPa. (Marc-Mentat manual, 2008 r1). One model is defined as reference model and only one parameter is varied at once with respect to this reference model. The properties of this reference model are listed in table 6.1. Reference model Material properties/ value Dimensions
Trusses weft 80 Mpa in tension 0 Mpa in compression d/L = 0.01
Trusses warp 80 Mpa in tension 0 Mpa in compression d/L = 0.01
Continuum material 0.20 Mpa, ν =0.49
Component 800 Mpa D/L = 0.1
Component pattern orientation - ψ = 0 degrees
Model thickness 0.6 mm
Table 6.1 Properties of the reference model.
With periodic boundary conditions the mechanical behaviour of an infinitely large repetitive structure can be modelled with only one cell. This periodicity is achieved by coupling the displacement of the opposite boundaries as shown below the unit cell in figure 6.4. These conditions enforce that the distance between opposite points on the boundary is equal to the distance of the corner nodes in both horizontal and vertical direction. As result, the opposite boundaries always have the same shape and length, so the cell remains repetitive. Illustrations of an undeformed and a deformed structure of nine unit cells are shown in figures 6.5 and 6.6.
Fig. 6.4 Unit cell with coupled Fig. 6.5 Undeformed structure of unit cells. Fig. 6.6 Deformed structure of unit cells. boundary displacements.
The average strain or stress are prescribed on the cell with periodic boundary conditions via the displacements and forces of the corners C1, C2 and C4. Here we consider only uniaxial stress states in 0, 45 and 90 degrees with respect to the cell. For the 0 degree case the stress is applied via a horizontal force on node C2. Rigid body motion is suppressed by imposing a zero displacement on node C1 and a zero horizontal displacement on C4. The vertical displacement of C2 and C4 is free. Similar conditions are used for the 90º loading. At 45 degrees a force at 45 º is applied to C2 and C4 and rigid rotation is suppressed by coupling the horizontal displacement of
37
corner C4 to the vertical displacement of corner C2. Note that this is only possible with symmetric unit cells, with the same stiffness in warp and weft direction. In other orientations than 0, 90 or 45 degrees a uniaxial stress state cannot be achieved in a straightforward fashion. 6.2 Results reference model For a comparison of the numerical calculations three different values are studied as a function of the varied properties. The fist is the overall stress strain behaviour of the unit cell, i.e. the overall stiffness of the cell. The second is the maximum stress in the truss elements, which can be related quantitatively to maximum stress in the yarns. The third observed value is the equivalent stress at the centre of the component. The reference model loaded in 0 or 90 degrees is shown in figure 6.8. In these orientations the model is loaded in truss direction and the contribution of the continuum is small, resulting in the linear stress strain response as shown in figure 6.7. The stresses in the yarns and component are low, up to a maximum of approximately two times the applied stress in case of round components with size D/L =0.5. Therefore this loading orientation is not studied further.
Fig. 6.7 Linear stress strain response in 0/90 degrees. Fig. 6.8 Stress in trusses in 0 degrees loading. The stress-strain response of the reference model loaded at 45º is shown in figure 6.9. The maximum strain is approximately 18.37 % at a stress of 0.11 MPa. This is slightly less than the strain of 19.57%, in textile without component at an equal stress of 0.11 MPa. The equivalent stress distribution in the continuum material of the model is shown in figure 6.8. Stresses in the compliant continuum around the component are low. The stresses in the continuum in the component are significantly higher than the applied stress. In case of square components stresses are relatively high at the corners on the diagonal in tension direction. For practical reasons the values at the centre of the component instead of the average stress in the components are compared in the series of calculations with varied parameters. When the deformed model is observed more in detail it can be seen that the unit cell contains four quadrangles, between the corners of the component and the corners of the cell, that are deformed in diagonal direction as uniform textile. These four quadrangles are separated by bands between the sides of the component and the centres of the unit cell boundaries. This deformation pattern is discussed more in detail in paragraph 6.4.
38
Fig. 6.9 Overall stress-strain response reference model. Fig. 6.10 Equivalent stress in continuum, square component. Deformed unit cells of the reference model are shown in figure 6.11 and 6.12, indicating the stresses in the trusses. Figure 6.11 shows the model with a round component and figure 6.12 for a square component. The maximum stresses in the trusses occur in the trusses which are directly connected to the components and are coloured yellow in the figures. Both images use the same colour scale. The maximum stress in the trusses is about 30% higher in the model with round component and is in the same order of magnitude as the stresses at the corner of the components with values of 10 to 20 times the applied stress.
Fig. 6.11 Stress in yarns, round component of D/L = 0.2. Fig. 6.12 Stress in yarns, square component of D/L = 0.2. 6.3 Results of parameter variations Several parameters are varied with respect to the reference model. First the component stiffness is varied. Subsequently the component distance or relative component size D/L is varied. The variations are listed in table 6.2
39
Variations
Component stiffness 0.2 – 3200 MPa Component size D/L = 0.05 - 0.94
Load 0.11 MPa; φ = 45 degrees
Table 6.2 Variations on reference model.
6.3.1 Variation of component stiffness The component stiffness is varied from 0.2 MPa, i.e. the stiffness of the continuum in the uniform textile, to 3200 MPa to study the effect on the global response of the cell, the maximum stress in the trusses and the strain in the component. In figure 6.13 the overall stress strain curves for two different values of the component stiffness are shown, together with the stress-strain curve of the textile without component. This plot shows that a component of size D/L = 0.1 has only little influence on the global response, even for a relatively stiff component. Figure 6.14 shows the relation between the overall cell deformation and component stiffness for a round and a square component at a stress of 0.11 MPa. It becomes clear that a round component results in a slightly more compliant response than a square component. Global strain reaches the asymptotic value at a component stiffness of about 1 x 104 times the stiffness of the continuum material. The reference model is close to this asymptotic value with a stiffness ratio of 4000, indicated by “Reference” in this figure.
Fig. 6.13 Stress-strains response for 2 different values of Fig 6.14 Overall strain divided by strain uniform textile. component stiffness and for the model without component. The maximum stress in the trusses as function of the component stiffness is plotted in figure 6.15. It is remarkable that the curves for round and square components intersect at a component stiffness which is close to the truss stiffness. Figure 6.16 shows the equivalent stress at the centre of the square component as function of the component stiffness. Equivalent stress in the component increases with component stiffness but seems to converge to a maximum of approximately 5 times the applied stress.
40
Fig. 6.15 Maximum stress in truss elements. Fig. 6.16 Equivalent strain at centre component.
6.3.2 Variation of component size The same values are compared as function of the relative component size D/L. Figure 6.17 shows the global stress strain responses of models with different component size. The stresses and strains are normalized by the maximum strain of the uniform model. Overall stiffness increases clearly with increasing component size. In figure 6.18 the maximum overall strains for different component are plotted as function of the relative component size, also normalized on strain of the model without component. A nearly linear dependence between component size and maximum strain is shown for both round and square components. The global response is nearly the same for square and round components.
Fig. 6.17 Overall response with round component, normalized Fig. 6.18 Compliance as function of relative component size on uniform model without component (component size 0). for both round and square components.
Figure 6.19 shows the maximum stress in truss as function of the relative component size for square and round components. It is clear that the strain in the trusses decreases with increasing component size for both round and square components, but the stresses are higher for the model with round component. The equivalent stress in the centre of the square components is shown in figure 6.19. This stress decreases non-linear with increasing components size, and is relatively high for components smaller than D/L < 0.2.
41
Fig. 6.19 Maximum stress in truss elements. Fig. 6.20 Equivalent strain in centre component. 6.4 Analytical model, deformation in diagonal direction For both round and square components the same typical deformation of the diagonally loaded unit cell is observed as illustrated in figure 6.21. In this figure a deformed model with a round component of size D/L = 0.2 is shown. Clearly four different areas with sharp boundaries can be distinguished. In area 4 the deformation is restricted by the relatively stiff component and this region is assumed to be rigid. Note that in area 4 the trusses in the top left and bottom right corners are slightly compressed. In area 2 the shear deformation occurs in vertical direction and in area 3 in horizontal direction. Area 1 deforms identical to the uniform model in diagonal direction. The shear strains in area 2 and 3 have half the magnitude of the strains in area 1, so all (shear) strains can be expressed as a function of the average or overall strain of the cell. Note that these areas can be compared to the areas observed in the tensile tests in diagonal direction in chapter 4 and 5. Area 4 can be related to area A, area 1 to area C a and areas 2 and 3 to areas B and B’ as indicated in figure 5.6
Fig. 6.21 Cell wit round component. This typical deformation pattern is the same for round and square components.
42
For a simplified model a try is given to derive the overall stress strain response of the unit cell analytically for small deformations. A different unit cell is considered to illustrate the stresses and strains that occur in the cell. This cell is indicated by a black rectangle in figure 6.22 and shown in detail in figure 6.23. The trusses at the interfaces are illustrated by the black lines in figure 6.23 To justify the derivations some assumptions have to be made. It is assumed that stresses and strains within the different areas are uniform, and that the shear strains in area 2 and 3 have exactly half the magnitude of the shear strain in area 1 and can be expressed as function of the overall strain of the unit cell One of the assumptions of the simplified model is that the trusses are extremely stiff. Stiff trusses result in stress concentrations in these trusses at the boundaries between the different areas, as observed in chapter 5. These stress concentrations are visible as red and blue coloured trusses in figure 6.22. Contrary to the calculations with varied properties, for this simplified model it is assumed that trusses have stiffness in compression. It is assumed that the differences in shear stresses between areas 1, 2 and 3 are compensated by forces in these trusses, as illustrated in figure 6.23. Thus the resulting (maximum) forces or stresses in the trusses can be derived as function of the overall deformation of the unit cell. The shear stress in the component can be related to the shear stress in areas 2 and 3 and the forces in the trusses, thus also to the overall deformation of the cell. For the stresses and strains in horizontal and vertical direction it is assumed that the contribution of the continuum can be neglected and that stresses in the trusses and component are continuous.
Fig. 6.22 Considered area. Numerical calculation with stiff trusses. Fig. 6.23 Definition of areas with the trusses at interfaces.
The detailed derivation can be found in appendix E. The average stress and strain tensors of the unit cell can be expressed in the local contributions of the 4 areas as function of the cell size L and component size D:
[ ])4(2)3()2()1(22
)()()(1
EEEEE DDLDDLDDLL
+−+−+−= [6.1]
[ ])4(2)3()2()1(22
)()()(1
SSSSS DDLDDLDDLL
+−+−+−= [6.2]
43
As the deformations of the individual areas can be expressed as function of the overall deformation of the cell and their individual stiffnesses are known the overall stiffness of the cell can be derived. The overall compliance of the cell is expressed as:
−−+
−+−
=DL
DL
GDL
DL
DL
L
ES truss 5.04
1
2
1ε [6.3]
Strains in trusses are relatively small compared to the shear strain in the compliant continuum so the overall stress strain relation can be simplified to equation: Compliance
−−≈
DL
DL
GS 5.04
1ε [6.4]
Maximum stress in trusses expressed as function of the applied stress:
−−+
+−=
DL
DL
d
L
DDLL
L
S 5.04)(2
12
2maxσ
[6.5]
Equivalent stress in component can be calculated with equations divided by the applied stress:
2222
2)4(
)5.0(
3
])([
4
4 DLDDDLL
L
S
Seq
−+
+−= [6.6]
6.5 General 6.5.1 Influence truss properties Before the analytical derivations are compared with the series of calculations as discussed in section 6.3 the derivations are compared with some Marc-Mentat calculations. First the derivations are compared for the numerical model with the assumptions as described in paragraph 6.4. Subsequently the influence of compression of trusses and truss stiffness are checked. The Marc-Mentat calculation with the assumptions made in paragraph 6.4, i.e. is shown in figure 6.27. The large strain option is used in Marc-Mentat and the applied stress is 1.10 x 10-3 MPa, 1% of the stress applied in the large strain calculations. The overall strain of the marc mental model is 3.89 x 10-3 and is nearly equal to the analytically calculated strain of 3.88 x 10-3, calculated with equation [6.4]. The stresses in the trusses are relatively high compared to the applied stress in both in compression and tension. The maximum stress is the strusses is 1.356 x 10-2 MPa and the minimum stress is - 1.235 x 10-2 MPa. Also the analytically calculated maximum stress, using equation [6.5], is nearly the same as the stress in the numerical model with a value of 1.363 x 10-2 MPa.
44
In the series of large strain calculations with components compression stresses in the trusses are not allowed. To check the influence of this criterion a calculation is performed in which the trusses have no stiffness in compression and is illustrated in figure 6.28 The effect of this criterion on overall response is negligible. The minimum stress is the trusses is nearly zero and the maximum stress is raised so that the difference between maximum and minimum stress at the interface remains the same. A better approximation for the maximum stress is:
−−=
DL
DL
d
L
S 5.02maxσ
[6.7]
The maximum stress of 2.580 x 10-2 MPa matches with the analytically derived value 2.6053 x 10-2 MPa, using equation above. As the overall response of the cell in which no compression forces in the trusses are allowed is the same as for the simplified model, and also the maximum stress in the trusses can be calculated, the analytical predictions can be used for the series of calculations.
Fig. 6.27 Very stiff trusses. Compression force in trusses allowed. Fig. 6.28 Very stiff trusses. Compression force in trusses not allowed. In the simplified model the trusses are extremely stiff so the stresses are concentrated in single chains of trusses. However in the series of calculations with components the truss stiffness is much lower, so deformation in trusses becomes important. As result of elongation of the trusses the stresses are not concentrated in a single chain of trusses but the stresses are “smeared out” over neighbouring trusses, which is visible in figures 6.29 and 6.30. As result the maximum stress is significantly lower. Maximum stress is less concentrated at the corners of the components. As result lower shear stresses are expected in the component.
45
Fig. 6.29 Relative truss spacing of d/L =0.01. Fig. 6.30 Relative truss spacing of d/L =0.005. 6.5.2 Influence of truss spacing The get more insight in the model, and to estimate the value of the Marc-Mentat prediction of maximum stresses in the trusses, the truss spacing is varied with respect to the reference model. The truss density of the model is varied from L/d =10 to L/d = 240. The sum of cross section areas of all trusses remains constant. Figure 6.29 shows the reference model with a truss spacing of d/L = 0.01. Figure 6.30 shows the model with a truss spacing of 0.005. In this situation the maximum stress in the trusses near the component is higher as the cross section of the yarns is smaller, but increases less than with factor 2 as relatively more stress is carried by neighbouring yarns. The stress in the trusses at a distance from the component remains the same.
Fig. 6.31 1st component of stress in truss elements, S = 0.11 MPa. Fig. 6.32 Maximum stress scales linear with power of 0.43.
Figure 6.31 shows a plot of the maximum stress in the yarns as function of the truss spacing L/d When trusses are very stiff and stress would be concentrated at a single chain of trusses and the maximum stress would scale linear with 1 over the cross section area of the trusses, thus also L/d as the force in the trusses remains the same. The numerically calculated stress scales almost linear with 1 over d to the power of 0.43 as shown in figure 6.32. The value of 1 - 0.43 = 0.57 could be seen as a measure for the stress intensity for a truss stiffness of 80 MPa. It can be concluded that stresses predicted with the numerical model
46
cannot be translated straightforwardly to stresses in yarns in textile when the yarn density is different from the truss density. 6.6 Comparison of analytical and numerical model The analytical expressions are compared with numerical calculations for round and square components as a function of component size. The analytical derivations are based on small deformation theory, so deviations are expected at larger strains. But qualitative comparisons will give insight in the trends observed in the calculations. Figure 6.33 shows the predicted global strain divided by the strain of the uniform model as function the component size. The black and red marks are the results of Marc Mentat calculations for round and square components at 0.11 MPa. The stiffer response for these large strain calculations with components is due to the non-linear behaviour of the model, according to which stiffness increases with strain. However, the trend is predicted very well. The prediction of the maximum stress is plotted in figure 6.34. The blue line is the analytical prediction (Equation [6.7]) for rigid trusses that have no stiffness in compression. This prediction does not match with the numerical values quantitatively as trusses are not rigid in the model and stressed are smeared out over neighbouring trusses. The numerically calculated stresses scale by the power of 1 over 0.43 with the analytical prediction.
Fig. 6.33 Compliance of unit cell as a function of component size. Fig. 6.34 Maximum stress in truss as a function of component
size.
Figure 6.35 shows the equivalent stress at the centres of the square components, divided by the applied stress, as function of the components size together with the analytically calculated average equivalent stress in the component. The values match well except for the largest and smallest components. The trend is catched well.
47
Fig. 6.35 Equivalent stress at centre of square component.
6.7 Influence of component pattern To study the influence of the component pattern some calculations with different patterns are performed. Three situations are illustrated in figure 6.36, with the maximum stresses in the trusses indicated above the illustration. The component fraction is 4 % in the first two situations, and 2 % in the third situation. These three configurations are expected to give an equal overall stress strain response when trusses are undeformable. The ratio of areas 1, 2, 3 and 4 is the same for the first two configurations. The intersections between components in the third configuration behave as stiff areas. The second configuration can in fact be seen as 16 smaller unit cells, with relatively large yarns. Maximum stresses in the trusses are lower in this configuration, so placement of more smaller components on textile instead of fewer large components would be beneficent for the maximum stresses in the yarns Max stress in truss 1.36453 Max stress in truss 0.5964 Max stress in truss 1.2123
Fig. 6.36 Three configurations which would have an equal global stiffness with linearized deformation theory and very stiff trusses.
Figure 6.37 shows the load displacement curves of the three configurations illustrated above, plotted as solid lines. The thirst situation is in fact a model with a component pattern which is rotated 45 degrees with respect to the yarn, but with a component fraction of 2% instead of 4%. The dashed line represents a calculation with a component fraction of 2% with an orientation of 0 degrees. This gives a more compliant response as configuration 3, so a 0 degrees pattern orientation is preferable for the overall response.
48
Fig. 6.37 The solid lines are the overall stress-strain curves of the three cells illustrated in figure 3.36.
At lower strains the load displacement curve for the third case matches the load displacement curves of the first two cases. At higher strains, when strain in the yarns becomes important, the curve tends towards the dashed curve, which can be seen as an lower bound calculation.
6.8 Discussion
- The calculations of the variation of component stiffness show that a component stiffness of about 0.25 times the fabric stiffness in yarn direction, i.e. 20 MPa (with a component thickness that is equal to the fabric thickness of 0.6mm), behave as rigid when the textile is deformed in diagonal direction. This means that components of virtually all solid materials behave as rigid in this loading direction.
- Stiffer components result in higher stresses in the yarns and in the components.
- A component pattern aligned with the yarns is the most advantageous configuration to
achieve a compliant overall stress-strain behaviour. - It can be concluded stresses in yarns en components concentrate in a few yarns and at the
corners of square components. For reliability it is important that stress concentrations to not occur in the conductive yarns or at electronic contacts of the components. Placement and shape of components with respect to the conductive yarns and electronic connections have to be well considered. Round component corners can reduce stress concentrations in the component and at the interface between textile and component at stresses concentrate at four instead of two points.
- Equivalent stress in components and overall stiffness of the electronic textile are
conflicting issues. A smaller component fraction is beneficent for the overall response but unfavourable for the stresses in the components. At the same component fraction more smaller components are beneficial for the maximum stresses in the yarns. The optimum depends on the desired stiffness or comfort, and the limitations of the component material, yarn strength or the connection or the interface between the textile and the component.
49
- Sillicon rubber, e.g. PDMS, with a Young’s Modulus in the range of 0.5 to 10 MPa, is an
example a suitable non-rigid material and could be useful to reduce stresses in yarns at edges of components or electronic contacts or via connections. For example by locally impregnating the fabric or surrounding a small component with low modulus silicon rubber, or by creating a rubber like interface between the solid components and the textile.
50
7 Conclusion For the description of the mechanical behaviour of textile an anisotropic mesoscopic material model is presented which consists of truss elements and a continuum material. The truss elements represent/model the structural response of the material, but do not represent individual yarns, and the continuum material models the resistance against relative rotation of the yarns. The material properties are linear elastic and large deformation theory is used. As a result of the close relation between the structural behaviour of the model and textile only four parameters are needed to model the response. Despite the low numbers of parameters, the model describes the material response better than the continuum model which was used for this purpose previously (Feron, 2008a). Another advantage is that the model is easy to understand and the parameters have an intuitive meaning. Conductive or stiffer yarns can be modelled easily by increasing the stiffness of trusses in the model. This model is implemented in Marc Mentat with the experimentally measured and locally obtained properties and proved to model both the local as global response well. Also the structural behaviour of the textile is captured by the model. Structural deformations of the model are the same as the structural deformation in the textile. Qualitative prediction of stresses in yarns is realistic and failure of yarns can be predicted qualitatively. In textile in tensile tests is failure of yarns is observed at locations where the model predicts maximum stresses. Out of plane deformations are observed in this model but the shape and the number of folds are dependent on the initiation. Modelling of stiff components can be easily implemented by increasing stiffness of the continuum material in the model. The results in structural behaviour, can be compared with structural effects observed near clamps in the tensile tests. Trends in effects of component size on the global behaviour are described analytically for the simplified case and translated to effects observed in the Marc Mentat calculations. The model also still has some limitations:
- Locations of maximum stress in yarns can be predicted with the model, quantitative stress prediction and stress intensity is not studied in detail. For quantitative stress prediction each yarn has to represented by trusses, or the effect of yarn/truss spacing on the stress has to be studied more in detail.
- Truss elements are fixed at the nodes but in real textile sliding between warp and weft
yarns is possible. This could result in overestimation of the stiffness near boundaries.
- The truss elements are linear elastic. The properties of trusses in warp and weft direction have significant influence on the diagonal direction and probably also other directions in cases that the stress is localised in a few yarns. In fact this behaviour is dominated by a few yarns that are subject the large strains. Fitting the warp and weft direction at higher strains could result in better predictions in orientations close to warp and weft direction.
- Fibers in textile can buckle easily. Truss elements in the model, which cannot transmit
compression loading deform plastic in compression. This will be a problem in case of cyclic loading.
51
In this study components are modelled two-dimensionally The model can also be used for modelling of 3 dimensional components, for example silicon rubber components, as shown in figure 7.1.
Fig. 7.1 Tensile model with three- dimensional compliant component.
Plasticity, modelling unloading The purpose of the continuum material is in fact modelling of the resistance against rotations of yarns. This resistance is partially elastic compression of the yarns but an important part is friction. However, in the model as presented, the continuum material is linear elastic. This results in fully reversible loading-unloading behaviour, where in practice the internal friction in the textile determines the loading behaviour. To implement the friction in the unloading behaviour of the model the continuum material can be made plastic with isotropic hardening behaviour. Therefore two new parameters have to be implemented, the yield stress, and a Young’s modulus different from that used in the elastic model. The hardening parameter H replaces the modulus as derived for the elastic model. Derivation of the right plasticity parameters is not studied in detail, but this work is just to indicate the possibilities. The E modulus is important when the stresses are below yield stress, which is a very small part in the loading – unloading cycle.
52
Figure 7.2 shows a close up of the loading curve of a tensile test in diagonal direction. A kind of yield stress is observed and this yield stress and the Young’s modulus E can be determined as illustrated in the figure. H is defined as:
εσ y
cEH += 1
Where Ec is the elastic Young’s modulus as derived in chapter 4. The plastic stress strain response with these parameters is shown in figure 7.3. The loading behaviour is slightly stiffer than expected.
Fig. 7.2 Estimation of Young’s modulus and yield stress. Fig. 7.3 Plastic loading and unloading behaviour.
53
Literature Cherouat, A., Radi, B., El Hami, A. (2008). The frictional contact of the composite fabric’s shaping, 2008 Chow, C.L., Yang, X.J. (2004). A generalised Mixed Isotropic-kinematic Hardening Plastic Model Coupled with Anisotropic Damage for sheet metal Forming. International Journal of damage mechanics. 13;81
Eidel, B. & Gruttmann, F. (2003). Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation. Computational Materials Science 28 (2003) 732–742. Institut für Werkstoffe und Mechanik im Bauwesen (IWMB), TU Darmstadt, Alexanderstr. 7, 64283 Darmstadt, Germany
Feron, M.J.M. (2008a). Mechanics of electronic textiles. Experimental analysis and numerical modelling. Technical report Philips Research Eindhoven. Feron, M.J.M. (2008b). Mechanics of electronic textiles. Deformation analysis using image processing. Technical report Philips Research Eindhoven. Hahm, J., Kim, K., Yin, J., (2000). Hardening of steel sheets with orthotropy axes rotations and kinematic hardening. International journal of Korean society of precision engineering vol 1 no, 1. June 2000. Hamila, N., Boisse, P., Sabourin, F., Brunet, M. (2009). A semi-discrete shell finite element for textile composite reinforcement forming simulation. International Journal for numerical methods in engineering 2009, 79:1443:1466. Lin, H., Long, A.C., Sherburn, M., Clifford, M. J. (2008). Modelling of mechanical behaviour for woven fabrics under combined loading. School of Mechanical and Manufacturing Engineering, University of Nottingham Liu, G.R., Ching, T.W., Tan, V.B.C. (2006) Modelling ballistic impact on woven fabric with LS-DYNA. Computational methods 978-1-4020-3952-2 Page 1879-1884 Department of Mechanical Engineering, National University of Singapore Peng, X.Q. & Cao, J. (2004). A continuum-based non-orthogonal constitutive model for woven composite fabrics. Composites part A: Applied Science and manufacturing, volume 36, issue 6, June 2005, 859-874 Ruíz, M.J.G. & González, L.Y.S. (2006). Comparison of hyperelastic material models in the analysis of fabrics. CAD/CAM/CAE Laboratory, EAFIT University, Medellín, Colombia. Verpoest, I. & Lomov, S.V. (2005). Virtual texttile composites software Wisetex: Integration with mico-mechanical permeability and structural analysis. Katholieke Universiteit Leuven. Xue, P., Peng, X.Q., Cao, J., (2003). A Non-orthogonal constitutive model for characterizing woven composite. Composites part A: Applied Science and manufacturing, Volume 32 issue 2. February 2003. 183-193(11)
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Zhou, G., Sun, X., Wang, Y. (2004). Multi-chain digital analysis in textile mechanics. Composites science and technology. 64:239-244
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Appendix A Non linear analytical derivation in diagonal direction for lll == 21
The length of the truss is assumed to be constant as the truss is assumed to be rigid:
( )[ ] ( )[ ] 22
21
222
211 11 llelel +=+++ [A 1]
Definition logarithmic strains in the continuum:
( )11 1ln e+=ε [A 2]
( )22 1ln e+=ε [A 3] In initial undeformed situation holds:
222222
21
2
2
1
1llelel +=+ εε [A 4]
lll == 21 [A 5]
(Note that this assumption determines a truss spacing and cross section of 2l Making use of relations [4], [5] and [6] equation [3] can be written as function of logarithmic strains:
22222 221 lelel =+ εε [A 6]
2)2exp()2exp( 21 =+ εε [A 7]
The strain in lateral direction 2ε can be expressed as function of 1ε . [A 8]
( ))2exp(2ln2
112 εε −= [A 9]
The in-plane strains are known, so the stresses 1σ and 2σ and the strain 3ε in the continuum
material can be calculated with the plane stress equations for large strains. As result of symmetry the 1 and 2 orientations or the continuum do not rotate.
)(1 2121 εεσ v
v
Ec +−
= [A 10]
)(1 2122 εεσ +
−= v
v
Ec [A 11]
56
)(1 213 εεε +
−−=
v
v [A 12]
The stresses and strains in the continuum are expressed with the formulas above. However, desired is a stress-strain relation in which the contribution of the truss elements on the resulting stress is included. The engineering stresses of the combination of the continuum material and the truss can be expressed as function of the true stress in the continuum and the still unknown forces in the truss.
2
)exp()exp( 1
3211h
FS
εεεσ ++= [A 13]
2
)exp()exp( 2
3122h
FS
εεεσ ++= [A 14]
The first terms in these equations are the true stresses in the continuum multiplied by the actual strains, to obtain engineering stress. The second part of these expressions are the contributions of the still unknown force F in the truss acting 1 and 2 direction, divided by the cross section area of
the continuum 1 x h. The ratio 2/)exp(ε is the ratio between the force in the truss and its component in 1 or 2 direction. As in uniaxial tension no external forces act on the model in lateral direction S2 has to be zero. Making use of this equilibrium the force F in the truss in lateral direction can be expressed as function of the known stresses and strains in the continuum: S2 = 0 [A 15]
)exp(2 2312 εεεσ −+−= hF [A 16]
Substitution of this force in equation [12] results in the following expression for the engineering stress in tensile direction:
[ ]2211321 )22exp()exp( σεεσεε −−+=S [A 17]
The engineering stress expressed as function of the strains iε is:
[ ][ ]2121212321 )22exp(1
)exp( εεεεεεεε +−++−
+= vvv
ES [A 18]
57
Appendix B Non linear analytical derivation in diagonal direction.
ie Engineering strain
iε True strain
iS Engineer stresses in continuum + trusses
iσ True stress in continuum
iF Reaction forces of truss
The length of the truss is assumed to be constant ⇒
( ) ( ) 22
21
222
211 llulul +=+++ [B 1]
Definition logarithmic strains:
( )11
1
1
111 1ln1lnln e
l
u
l
ul+=
+=
+=ε [B 2]
( )22
2
2
222 1ln1lnln e
l
u
l
ul+=
+=
+=ε [B 3]
( ) ( )1111
111 lnlnln lul
l
ul −+=
+=ε ( ) ( ) 1111 lnln ε+=+ lul [B 4]
( ) ( )2222
222 lnlnln lul
l
ul −+=
+=ε ( ) ( ) 2222 lnln ε+=+ lul [B 5]
Actual length and height of the element:
( )( ) 1111
ln11
εε eleul l ==+ + [B 6]
( )( ) 2222
ln22
εε eleul l ==+ + [B 7] Substitute [6] and [7] in [1]
58
222222
21
2
2
1
1llelel +=+ εε [B 8]
1
121
2
2
222222 εε elllel −+= [B 9]
( )1
2
11
2
1
2
212 22
22
2
2
2
222 11 εεε e
l
le
l
l
l
lle −+=−
+= [B 10]
2ε can be written as function of 1ε
( )
−+= 1
2
1 22
2
2 11ln2
1 εε el
l [B 11]
1ε and 2ε are known, so the plane stresses in the continuum can be derived as function of 1ε .
)(1 2121 εεσ v
v
E +−
= [B 12]
)(1 2122 εεσ +
−= v
v
E [B 13]
The strain in out of plane direction can be calculated with the following formula:
Ev 21
3
σσε +−= [B 14]
(or as function of the strain)
Ev
v 213 1
εεε +−
−= [B 15]
The engineering stresses of the combination of the continuum material and the truss can be expressed as function of the true stress in the continuum and the still unknown reaction forces of the truss.
tl
FeeS
2
111
32 += εεσ [B 16]
tl
FeeS
1
222
31 += εεσ [B 17]
59
The ratio between the reaction forces in 1- and 2-direction is determined by the position of the truss, so by the ratio of the strains in 1- and 2-direction.
22
11 2
1
Fel
elF ε
ε
= [B 18]
In uniaxial tensile no external forces act in lateral direction, so there is equilibrium between the lateral stress in the continuum and the lateral force applied by the truss. So equation [17] is equal to 0.
02 =S [B 19] Making use of this equilibrium the reaction force of the truss in lateral direction can be expressed as:
2
1
31
2
122 ε
εεεσ
el
eleeF = [B 20]
The reaction force of the truss in tensile direction is
2
1
31
2
21
21 ε
εεεσ
el
eleetF = [B 21]
The resulting engineering stress in tensile direction is
2
1
332
22
221
211 ε
εεεε σσ
el
eleeeS += [B 22]
The first part of the expression is the contribution of the continuum in tensile direction, the second part is the contribution of the truss. The formula can be rewritten as:
+= 222
2
221
11 2
1
32 σσ ε
εεε
el
eleeS [B 23]
The axial force in the truss can be expressed as a function of 1σ or 2σ and the actual angle f the truss.
The angle of the truss is
−1
2
1
21tan ε
ε
el
el [B 24]
60
2
1
21
1
1
21
2
1
2
31
1
2
tansintansin
σ
ε
ε
εε
ε
ε
−=
=
−−
el
el
teel
el
el
FF [B 25]
The engineering stress expressed as function of the strain is:
[ ]
+−++
−= 21212
2
21
2121 )22exp(1
32 εεεεεεεε vl
lv
v
EeeS [B 26]
S1 expressed as function of 1ε , with )exp(x necessarily written as xe to save space.
( ) ( )( )
( )( )
−++−
−++
−=
−+
−++
−
−+
1
2
1
1222
21
2
1
11
2
1
1222
21
11222
21
2
2
2
111ln
2
222
2
2
1
11ln2
1
111ln
2
1
21 11ln2
111ln
2
1
1ε
εε
ε
εεε
εε
el
lv
el
ele
l
lvee
v
ES
el
l
el
l
v
ve
l
l
[B 27]
61
Appendix C Linear derivation in diagonal direction L is assumed to be constant ⇒
( ) ( )222
211
22
21 ululll +++=+ [C 1]
Engineering strains are defined as:
1
11 l
ue = [C 2]
2
22 l
ue = [C 3]
22
21
2222
22
2111
21 22 lluulluull +=+++++ [C 4]
02211 =+ ulul [C 5]
1122 ulul −= [C 6]
1
122
21
2
2
l
u
l
l
l
u−= [C 7]
Strain 2e as function of 1e
122
21
2 el
le −= [C 8]
Plane stress equations:
)(1 2121 vee
v
ESc +
−= [C 9]
)(1 2122 eve
v
ESc +
−= [C 10]
Stresses in the continuum material in 1- and 2-direction:
62
)(1 12
2
21
121 el
lve
v
ES −
−= +
2
1
l
F [C 11]
)(1 12
2
21
122 el
lve
v
ES −
−= +
1
2
l
F [C 12]
2u and 2ε are directly a result of displacement u1 transmitted by truss L
The reaction force in truss L is a result of2S Deformations are small, so the next assumptions are made:
111 lul ≈+ [C 13]
222 lul ≈+ [C 14] Y-reaction force transmitted by truss to x direction.
F=)sin()sin(
122
ααlSF
−=− [C 15]
F1= )cos(α F [C 16]
Contribution of truss element to the stress in 1-direction:
22
11 )tan(
1F
l
lF
α−= [C 17]
⇒
= −
1
21tanl
lα [C 18]
22
21
21 l
lFF −= [C 19]
The total stress in 1-direction sum of stress in continuum and truss.
22
21
211 l
lSSS cc −= [C 20]
1S =
−−−
−)()(
1 122
21
122
21
122
21
12 el
lve
l
le
l
lve
v
E [C 21]
63
This derivation can be simplified further when 21 ll =
121 )1(1
evv
ESc −
−= = 11
ev
E
+ [C 22]
122 )1(1
evv
ESc −
−= = 11
ev
E
+− [C 23]
111 412
Geev
ES =
+= [C 24]
64
Appendix D Mesh size as only little influence on global response.
With relatively course meshes the global response can be modeled properly, as long as the dimensions are correct
65
Appendix E
Fig 6.20 The typical deformation patterns. Above for round component, the same for square components
Strains in trusses when these compression forces are allowed.
Fig 6.21 Considered area. Fig 6.22 Definition of areas with the trusses at interfaces.
66
Average stress and strain
The dimensions of the different areas )(iA are as follows:
2)1( )( DLA −= , )()2( DLDA −= , )()3( DLDA −= , 2)4( DA = [E.1]
2)4()3()2()1( LAAAA =+++ [E.2] The sum of these areas is by definition equal to the cell size L2. The global or average stress and strain of the unit cell can be expressed in the local contributions of the 4 areas:
)4()3()2()1(
)4()4()3()3()2()2()1()1(
AAAA
AAAAE
++++++= εεεε
[E.3]
[ ])4(2)3()2()1(22
)()()(1
EDEDLDEDLDEDLL
E +−+−+−= [E.4]
Similar for the stresses
[ ])4(2)3()2()1(22
)()()(1
SDSDLDSDLDSDLL
S +−+−+−= [E.5]
The three stress and strain components in region 1,2 and 3 can can be expressed as:
)( 1221)(
1222)(
2211)(
11)( eeeeeeeeE iiii vvvvvvvv +Γ++= εε [E.6]
)( 1221)(
1222)(
2211)(
11)( eeeeGeeEeeES iitrussitrussi vvvvvvvv +Γ++= εε [E.7]
In area 4, the rigid component, the strains are zero and the stresses not yet determined.
0)4( =E 11eevv
+ 0 22eevv
+ 0 )( 1221 eeeevvvv + [E.8]
To solve the equations the average stress and strain the local stresses and strains have to be known. 11 direction Stress continuous at interfaces:
)2(11
)4(11 SS = , )1(
11)3(
11 SS = [E.9]
67
)1(
11S = )3(11S = )1(
11εtrussE [E.10] )2(
11S = )4(11S = )2(
11εtrussE [E.11]
)1(11
)2(11 S
DL
LS
−= [E.12]
Strain constraints 11 direction As result of periodic boundary conditions the strains must be equal
)2(11
)4(11
)3(11
)1(11 )()( εεεε DLDDDL −+=+− [E.13]
0)4(11 =ε => )1(
11)2(
11 εεDL
L
−= [E.14]
)3(11
)1(11 εε = [E.15]
)1(11
)3(11
)1(11
11
)( εεεε =+−
=L
DDL [E.16]
Substitution of stress and strain constraints.
[ ]trusstrusstrusstruss EDEDLDEDLDEDLL
S )2(11
211
)2(1111
2211 )()()(
1 εεεε +−+−+−= [E.17]
The global stress strain relation in 11 direction can be written as:
1111 ε
−+−=
DL
D
L
DLES truss [E.18]
The 11 component of stess in area 2 and 4 can be expressed as in the equation below and will be used for calculation of the eqiavalent stress in the component and maximum stress in the yarns
[ ])2(11
2)1(11
)2(11
)1(11
2211 )()()(
1SDSDLDSDLDSDL
LS +−+−+−= [E.19]
The equation for the 11-component of stress in the component is:
112)2(
11)4(
11 )(
)(S
DDLL
DLLSS
+−−== [E.20]
22 direction, similar to 11-direction
2222 ε
−+−=
DL
D
L
DLES truss [E.21]
68
The 22 component of stress in area 3 and 4 can be expressed as
112
2)3(
22)4(
22 )(S
DDLL
LSS
+−= [E.22]
12 direction Kinematical constraints for the shear deformation are:
)1()2()2(
2
1 Γ=Γ=Γ and 04 =Γ [E.23]
All shear strains are known so global shear deformation 12Γ can be expressed as function of local
shear strains e.g. )1(Γ
[ ])4(12
2)3(12
)2(12
)1(12
2212 )()()(
1 Γ+Γ−+Γ−+Γ−=Γ DDLDDLDDLL
[E.24]
Substitution of the contraints [E.23] in [E.24] results in the expression for the overall strain as function of the local strain in area 1.
)1(1212 Γ−=Γ
L
DL [E.25]
At a similar way the global stress can be expresses ad function of the local stresses. The shear stresses in the continuüm in areas 1, 2 and 3 are directly related to the strains in these areas.
)(12
iS = )(12
iGΓ [E.26]
To determine the stress in the component, )4(12S some effort has to be done.
Figure 6.22 shows the unit cell with the 4 different areas and the trusses at the interfaces at which stresses are concentrated. Shear stress in the component, area 4, is the result of the difference in shearstress between areas 1 and 2 and 1 and 3 The difference in shear stresses between area 1 and 2 and area 1 and 3 is compensated by forces in the trusses between these boundaries. These forces are equal to the length of the boundary times the height, h, times the difference in stress at the interface.
12)1(
12 2)(
2
12 Γ=Γ−= LG
hGDLhF [E.27]
These forces transmitted trough the trusses result in a shear stress in the component, area four additional to the shear stresses directly transmitted between area 2 and 4 and 3 and 4. This results in the expression for the shear stress in area 4:
69
The shear stress in the component can be expresses as function of shear strain:
=+=hD
FSS
2)3(12
)4(12 =Γ−+Γ )1(
12)1(
12 2
)(
2
1
D
DLGG 12
2)1(
12 )(2
1
2
1 Γ−
=ΓDLD
LGG
D
L [E.28]
(Or as function of the applied stress making use of equations [E.30} and [E.42]]:
SDLD
LS
DLD
L
)2
1(4
1
)2
1(2
1 2
12
2
−=
− NB!
Local strains are expressed as function of global strain and local stresses as function of the local strains. Substitution of the expressions for local stress [14] and [16] in the relation between global local stress [6] results in the relation between global stress and global strain in shear.
[ ])4(12
2)3(12
)2(12
)1(12
2212 )()()(
1SDSDLDSDLDSDL
LS +−+−+−= [E.29]
The global stress strain relation in shear is:
12)1(
12122
1
2
1
Γ−
−=Γ
−=
DL
DLGG
L
DLS [E.30]
Global stress strain relations in three directions Three stress strain relations
12
1
1212 2
1112
1
Γ
−+=Γ−
−=
−
L
DG
DL
DLGS [E.31]
[50]
1111 ε
−+−=
DL
D
L
DLES truss [E.32]
2222 ε
−+−=
DL
D
L
DLES truss [E.33]
A uniaxial stress S is applied in diagonal direction, n
v, on a unit cell with trusses in 11 and 22
direction. The tress tensor S can be written as function of 3 components:
70
S = )( 12211222221111 eeeeSeeSeeSvvvvvvvv +++ [E.34]
S = nnSvv
with )(2
121 eenvvv += [E.35]
The three stress components can be expressed as function of the applied stress S:
S = )(2
112212211 eeeeeeeeSvvvvvvvv +++ [E.36]
� SS2
111 = , SS
2
122 = , SS
2
112 = [E.37]
The strain tensor is defined as:
E = )(2
112211222221111 eeeeeeeevvvvvvvv +Γ++ εε [E.38]
The strain in diagonal direction: ε = ⋅n
vE n
v⋅ [E.39]
)(2
1)](
2
1)[(
2
12112212222111121 eeeeeeeeeeeevvvvvvvvvvvv ++Γ+++= εεε [E.40]
)(2
1122211 Γ++= εεε [E.41]
SStiffnessStiffnessStiffness
)111
(4
1
122211
++=ε [E.42]
As result of symmetry 2211 εε = The global stress strain relation is:
SDL
DL
GDL
DL
DL
L
Etruss
−−+
−+−
=5.04
1
2
1ε [E.43]
Strains in trusses are relatively small compared to the shear strain in the compliant continuum. The equation can be simplified to:
SDL
DL
G
−−≈
5.04
1ε [E.44]
71
Maximum stress in yarn versus global strain and global stress
The force difference at the interface equals. 122
12 Γ= LGF [E.45]
The stress difference in the yarn at the interface is directly related to the yarn cross section area d
SDL
DL
d
LG
d
LG
d
L
hd
F
−−==Γ==∆
5.022
1212 εσ [E.46]
When compression stress in the yarns is allowed the maximum stress is equal to
)4(11max 2
S+∆= σσ and )4(11min 2
S+∆−= σσ [E.47]
SDL
DL
d
LS
DDLL
L
−−+
+−=
5.04)(2
12
2
maxσ [E.48]
SDL
DL
d
LS
DDLL
L
−−+
+−−=
5.04)(2
12
2
minσ [E.49]
When compression stress in the yarns is NOT allowed the maximum stress is equal to
hd
F2max =σ = ⇒∆σ S
DL
DL
d
L
−−=
5.02maxσ [E.50]
0min =σ [E.51]
Equivalent stress in component
[ ] [ ] [ ]( )2)4(12
2)4(22
2)4(11
)4( 62
1SSSSeq ++= [E.52]
Substitution of the stress components results in:
2222
2)4(
)5.0(
3
])([
4
4 DLDDDLL
L
S
Seq
−+
+−= [E.53]
72
Fig E1 Global stiffness as function of relative component size Fig E2 Global stiffness as function of relative component size Solid line for trusses with stiffness in compression, dashed line for trusses without stiffness in compression.
Fig E3 11 and 22 component of stress in component Fig E2 Equivalent stress in components
73
Appendix F Global strain and maximum stresses are plotted as function of component size for stiff trusses with stiffness in compression. The match with the analytical derivations is very good with a maximum deviation of 4%
Fig. F1 Global strain prediction rigid (very stiff) trusses. Fig F2 Maximum stress prediction rigid (very stiff) trusses.
Fig. F3 Global strain prediction Fig F2 Maximum stress prediction
