Abstract. 1 Introduction

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1 A multi-scale approach for modeling mechanical behavior of2d and 3D textile-reinforced composites D. Bigaud & P. Hamelin Laboratoire Mecanique Materiaux - IUT A Genie Civil - Universite Claude Bernard, Lyon 1-43 Boulevard du 11 Novembre 1918, Villeurbanne Cedex, France hamelin@iutal2m.univ-lyonl.fr Abstract A procedure for the prediction of the elastic properties and failure behavior of 2D and 3D textile-reinforced composites is presented in this paper. Homogenization techniques based on energetical approaches are performed on multi-scale sub-elements in order to distinguish elastic from fracture behaviors. Scale details range from the filament to the yarn and the unit-cell. The reliability of our procedure is proved in the case of woven and SD-braided composites and the sensitivity to geometrical parameters are simulated. 1 Introduction The potential of 2D and 3D textile-reinforced composites is high but the challenges involved in predicting their mechanical characteristics do not facilitate their full exploitation. The main difficulty is to deal with their important material heterogeneity that implies anisotropy of the stress and strainfields.a geometrical description of the textile reinforcement must constitute the basis of a mechanical modeling which aims at dealing with this heterogeneity. The graphical techniques implicated in computeraided design tools find an appropriate way in the description of textile reinforcements geometry.

2 128 Computer Methods in Composite Materials The objective of this paper is to present a computer program dedicated to both the geometric and mechanical modeling of textile-reinforced composites, In section 2, we briefly present the software main features before giving the theoretical aspects of the model. Experimental data are compared to elastic and failure properties simulations of woven composites in the section 3. The case of a 3D-braided composite is studied in the last section. The effect of geometrical parameters on mechanical characteristics is investigated for both these textilecomposites. 2 Textile-reinforced composites modeling 2.1 General presentation of TIS3D TIS3D is divided in two main parts; a graphical and a mechanical one. The objective of the graphical section is to rebuild the textilereinforcement geometry through basic volumes. Reduced number of parameters allows the idealized geometry description of 2D and 3D reinforcement unit-cells (local representation of the reinforcement). We can talk about "geometrical homogenization" (bigaud*). The mechanical section aims at predicting the effective stiffness of the textile-composite and simulating the internal strains and stresses. So as, the unit-cell is considered as an aggregate of multi-scale elements. At the first scale, the mixture fiber + matrix is represented by a bi-dimensional network of micro-cells. The tows present the properties of this assembly (a meso-cell). At the upper scale, we consider the volume assembly of eight meso-cells. Such an intermediate-cell is created in order to improve the calculation capacity and rapidity. Finally, at the last scale, the unitcell is regarded as an aggregate of NxNyNz intermediate-cells. 2.2 Theoretical aspects For each scale, we have to define the mechanical characteristics of 2D or 3D elements as a function of sub-elements' individual properties. A numerical procedure based on stationary functionals derived from complementary and strain energy is developed. Its general aspects are presented below General aspects The main idea of the method consists in making stationary both the complementary U* and the strain Ug energy. This is done in order to define the lower and upper bounds of the effective stiffness and the

3 Computer Methods in Composite Materials 129 internal strain and stress values of the N%NyNz-cells assembly. In the case of the meso-cells, N%=Ny=2 and Nz=l. In the case of the intermediatecells, N%=Ny=Nz=2. fiber+matrix meso-cell meso-cell 2D assembly of 4 micro-cells and Ug are expressed as: intermediate-cell 3D assembly of 8 meso-cells The choice of meso-cell type depends on whether the intermediate-cell nodes belong to the yam volume or not Figure 1. Unit-cell multi-scale unit-cell 3D assembly of NxNyNz intermediate-cells description. L p=l L (1) Vp is the volume ratio of the p-cell. [s]«and [c]«represent the compliance matrix of the a entity. By means of Lagrangian multipliers, one can take into account the conditions of continuity and averaging of stress (U<j) and strain (Ug) and write the new functional LL* and UP*: *=Uo+X,f(^;(Tj HI,-.,6) *=Ug +ajg(ej ;S p) A.J,a,: Lagrangian multipliers (2)

4 130 Computer Methods in Composite Materials One can then write the conditions 9U_*=0 and 9U,*=0 as: Id! Solving eq.(3) for the sub-elements stress and strain allows to relate the sub and the total average vectors (Chen^): Then the upper and lower bounds of the N%NyNz-cells assembly effective stiffness can be written as: By writing eq(4) as: and by using eq(5), one can estimate the bounds of the internal strain and stress according to imposed external strain and stress up to the microcells scale. The knowledge of yarns' internal stresses and strains is directly exploitable to anticipate the composite failure behavior that is considered as a sequence of yarns local fractures Details on numerical procedure for progressive failure simulation The procedure for failure simulation of the NJSfyNz-cells assembly according to an external mono-axial stress is detailed in figure 2. The bounds of the undamaged composite stiffness matrix are calculated in the first step. Next, by using eq(6), the local stresses within the subelements are determined according to an arbitrary external stress. They are compared to the critical stresses (statistically assigned to each mesocell) in order to investigate the macroscopic stress value that would yield to the failure of thefirstmeso-cell (yarn segment). The stiffness matrix of the meso-cells is reduced by the selective RC method (rows and columns

5 Computer Methods in Composite Materials 131 of the matrix [c]p set to zero according to the critical stress which was exceeded). The elastic analysis is carried on taking into account of the drop of stress (.-».,) and the change of [C]. I Meso-cells' strength assignation ; Solving of the first elastic problem on the undamaged unit ceil Arbitrary macroscopic stress Determination of the local mesoscopic stresses comparisons to the critical stress values i Determination of the macroscopic stress Z* which leads to the failure of the i*h meso-cell Induced macroscopic strain E* Matrix reduction of damaged meso-cell the 1 Drop of to z I from i* Failure behavior modeling is considered as a sequence of elastic studies Figure 2. Illustration of the progressive failure simulation procedure. 3 Model predictions in the case of woven composites 3.1 Geometrical and mechanical characteristics The cases of two plain-woven (plain 1, plain2) and one satin-woven reinforcements are studied. Both these textile composites are glass/epoxy materials. The constituents properties and the main geometrical characteristics of the unit-cells are given in table 1. The main geometrical characteristics are shown in figure Elasticity The reliability of the elastic model in the case of woven composites has been already discussed in a previous paper (bigaud*). The table 2 gives an example of comparisons between the experimental and simulated results and a finite element method (Lene*, Chouchaoui*). The model rapidity and its sensitivity to geometrical parameters allow the study of unit-cell dimensions influence on compliance terms. In figure 4, we present the effect of the weft yarn waviness, assimilated to the unitcell height H, in the case of a plain-woven composite presenting the initial parameters of plain2. The variations of the compliance terms qy

6 132 Computer Methods in Composite Materials (average value of U* and Ug approaches) are described in the case of H varying from 160 to 480 iim. These geometrical bounds corresponds to the minimum and maximum weft yarn waviness. The variations of the total volume of fiber Vfg are also shown. For H=320 im, the warp and weft yarns present the same waviness (H=4b); the value of Vfg is then maximum. For H ranging from 160 to 320 jam, a decrease of qyy is observed (Aqyy=37%). This seems to be inconsistent with the waviness increase along the Y-direction (weft direction). Nevertheless, this can be explained by the increase of Vfg. When Vfg decreases again, qyy increases more rapidly than he decreased (increase of 86 % between H=320 and 480 jam). The variations of q%% can be deduced from those of qyy symmetrically around H=320 jam. We add that q%y et qss respectively decrease of 54 and 66% between H=160 and 320 jam. Glass Epoxy Carbon PA 12 E,, E22-& Gi2=Gn G Table 1. Constituents properties. Vi2=Vi3=V r volum vplurnc e -tm H a b wp H L Plain Plain >wf ^^r Figure 3. Plain-woven fabric unit-cell geometrical parameters. Satin Plain 1 Satin Exact Tis3D Exp. Exact Tis3D Table 2. Comparisons between experimental and numerical results Height H [^m] Figure 4. Influence of H on the compliance terms qy.

7 3.3 Failure Computer Methods in Composite Materials 133 Bi-axial tensile tests have been carried out on composite plates with plain2 reinforcement. In this case, we assume that the composite failure originates from local failures of the tows ((%*=! 240 MPa, (?s*=80 MPa) and the matrix (at*=60 MPa, <Js*=10 MPa) due to tensile and shear stresses. The experimental results and the tensile-tensile strength envelopes simulations (U<, and UE) are shown in figure 5a. First, we must notice that the two simulations give very different values. We also remark that the strength is greater along the X-direction that presents a less waviness then along the Y-direction. Finally, we emphasize that the experimental values are included between the two simulated bounds. This confirms the interest to carry out both the U^ and Ug approaches. In the figure 5b, we show a parametrical study done in order to assess the effect of H on the strength envelopes (average ofu* and UE). For HN320 jam, the strength envelope is symmetric respect to the line Oxx-Cyy- We also observe that the strength <jyy decreases when H (or the waviness) increases. As an example, variation of H from 320 to 480 (am results in a 35%-decrease of the strength in the case of a mono-axial load along the Y-direction. i?nft » H=320 pm " H=360 pm * * 1000 A H=400 nm * x H=440 urn ^800-". 1? 800 «. *H=480pm o " 600; : : : j «*.. x k $ 400- i * * u? : I! * *.. A X. comp. energy AX * strain energy * ^^ AX * o experiments ^ ^ ^^ MAX * 0 1 _.-.,- n ^ yi r r- i»» * i " i ' /. Q (\\\ Figures 5. (a) Comparison between simulated and experimental tensiletensile strength envelopes, (b) Influence of H on the strength envelopes. 4 Model predictions in the case of a 3D-braided composites We propose to test the validity of our numerical procedure for the failure behavior from experiments carried out on Carbon/PA12 3D rectangular Cartesian braided fabrics. The mechanical properties of the constituents are summed up in table 1.

8 134 Computer Methods in Composite Materials 4.1 Geometrical and material characteristics In order to determine the unit-cell geometry of the 3D rectangular Cartesian braided composites, the sectioning of specimens is carried out (figure 6). It allows the path reconstitution of each braiding yarn along the repetition unit. The orientations of each of yarn elements within the subcells and the volume of fiber in the composite can be determined from the process parameters. 4.2 Elasticity Model predictions and experimental values of compliance terms deduced from tensile and shear tests are compared in the table 3. First, we observe important differences between the two energetical approaches. They are respectively of 25, 14 and 25% for S%%, Sss and S%y. We also notice that the experimental value of 8%% is included between the two bounds. The term Sss is underestimated (ASss=15.4%) and the term S%y is overestimated (AS%y=7.3%). These low differences between experimental and simulated values reveal the correct reliability of our model in the case of the elastic behavior of 3D-braided composites. In figure 7, we complete the elastic study by showing the influence of the braiding angle 6 on the compliance terms. We remark that the term S%% increases with the braiding angle that is rather logical since the yarn move away from the X-direction. The term Sss decreases with 0 down to a value included between 40 and 50. Digitization sectioning, \ idealized geometry Figure 6. 3D braided unit-cell reconstitution by specimens sectioning. 4.3 Progressive failure Tensile tests are carried out on forty-three-yarn-braided-composites. The figure 8a shows a comparison between three experimental and two simulated progressive damage functions. In this case, data used to study the braided composite failure behavior are restricted to axial tension and shear strengths of the braiding yarns.

9 Computer Methods in Composite Materials 135 :600r &SS s*y U U(J Exp (a) Figure 7. (a) simulated and experimental compliance terms in the case of a 3D-braided composite, (b) Influence of the braiding angle. The meso-cells strength values are individually assigned according to Weibull laws which parameters are obtained from experiments carried out on yarn elements (bigaud^). The cumulative distribution for tensile and shear strength are: F^=l-exp F,=l-exp L: yarn element length (3 5=reference length) a: axial stress i: shear stress In the case of the U<? approach, the first simulated meso-element failure occurs before the one given by the Ug approach (130 MPa according to Uc and 180 MPa according to Us). The failure of the whole braided composite is then predicted before that determined from the Ug approach. We conclude the study of 3D-braided composites failure by the influence of the braiding angle on the tensile strength. We notice the significant effect of this parameter. The tensile strength value decreases from 805 to 87 MPa when 6 varies from 10 to ^ " (a) e [pni/m] Figures 8. (a) Comparisons between experimental and simulated damage functions in the case of a 3D-braided composite, (b) Influence of the braiding angle.

10 136 Computer Methods in Composite Materials 5 Conclusions A numerical procedure based on the knowledge of textile geometry is proposed in order to simulate the elastic and the failure behavior of woven and braided composites. The elastic predictions are shown to be quite close to usual F.E.M. and experimental results. The reliability of the model can be used to carry out parametrical studies. As far as the damage simulations are concerned, we have observed that experimental strength values are mostly included between the bounds predicted by our two energetical approaches. All these observations are rather encouraging since TIS3D can deal with more or less complex textile composite. In the future, this numerical procedure will be used in order to design the properties of tubular interlock braided composites. References [1] Bigaud, D. & Hamelin, P. TIS3D Software: From geometrical description to mechanical prediction - Application to woven fabric reinforced composites, Proc. ofcadcomp96, pp , [2] Bigaud, D. & Hamelin, P. Mechanical properties prediction of textile reinforced composite materials using a multi-scale energetical approach, Composite Structures, Vol. 38, N 4, pp , [3] Chen, D. and Cheng, S. Analysis of composite materials: A micromechanical approach, Journal of Reinforced Plastics and Composites, 12, pp , [4] Chouchaoui, C.S. Modelisation du comportement des materiaux composites a renforts tisses et a matrice organique, Ph.D. thesis, Universite de technologic de Compiegne, France, [5] Lene, F., Hassim, A. & Paumelle, P. Homogenized behaviour of woven fabric composites, Proc. of "Comportement des composites a renforts tissus - Comportement dynamique des composites", Ed. Pluralis, pp , 1991.