Modeling multiaxial stress states in forming simulation of woven fabrics

. During the forming of woven fabrics, different multiaxial stress states can occur, depending on the given process conditions and the complex deformation mechanisms of the interwoven structure of the textile. Particularly under constrained forming conditions, induced e.g. by blank holders or by adjacent metal layers in fiber-metal-laminate (FML) forming, the multiaxial stress states may include in-plane compression. A hyperelastic, invariant-based constitutive model has been proposed in previous work to consider such biaxial and normal-shear coupling for both positive and also negative strains. In the present work, this constitutive model is applied to forming simulation at component scale to investigate the significance of individual coupling aspects for the prediction of the forming behavior under different multiaxial stress states. For that purpose, FMLs and pure fabric laminates are formed to a tetrahedron geometry. In a comparative simulation study, the individual strain couplings of the invariant-based material model are differently activated or suppressed. The simulation results reveal that biaxial coupling has a significant effect on the draping behavior, if the draping is partially


Introduction
In the past two decades, a large number of material models have been developed to describe the forming behavior of textile materials at meso and macro scale [1,2]. However, only macroscale models are suitable for forming simulation at component level [3] and enable virtual validation and optimization of forming processes [4,5]. Since textiles are prone to large shear, non-orthogonal material models are required [6][7][8]. This can be realized by hypoelastic or hyperelastic material modeling. Hypoelastic approaches [9][10][11] describe stress rates as direct functions of strain rates, depending on the stiffness. At each timestep, stress increments can be calculated directly from strain increments, which makes hypoelastic approaches convenient for implementation in nonlinear finite element approaches. For sufficient accuracy, however, they require sufficiently small timesteps. The advantage of hyperelastic approaches [12][13][14] is the integral representation of the stress-strain relationship, which enables timestep-independent solutions. The stress-strain relation is based on the strain energy density and their differentiation with respect to strain measures.
For woven fabrics that exhibit pronounced orthotropy with two preferred directions, the strain energy potential can be easily decomposed into distinct strain energy components, depending on different invariants of the deformation state [15][16][17]. The invariant-based energy formulation has the additional advantage of being independent of the choice of the basis reference system. Since experimental studies have shown interdependencies between distinct strain components, hyperelastic models with invariant-coupling, e.g., biaxial-tension [18] and tension-shear coupling [19], have been proposed. A recent review on modelling coupling mechanisms in woven fabrics, with particular focus on invariant-based constitutive modeling is given in [20].
In the context of fiber-metal-laminate (FML) forming, Schäfer et al. [21] have introduced an invariant-based model, which considers both biaxial and normal-shear coupling, but not just for positive strains (i.e. tension), but also for negative strains. The additional consideration of compressive strains in biaxial and normal-shear coupling becomes necessary, if wrinkling is suppressed during forming, e.g. due to adjacent metal layers as in FML forming, but also due to blank holders or grippers [21]. Under such constrained condition in thickness direction, the bending and wrinkling behavior becomes insignificant compared to the complexly interacting components of the membrane behavior, namely shear, tension and compression [22,23].
In the present work, the invariant-based constitutive model presented by Schäfer et al. [21] is used to evaluate the significance of different strain couplings in forming simulation at component scale. For that purpose, the invariants that control the couplings in the different components of the strain energy densities are activated or suppressed in different combinations. Additionally, the effects are compared for strongly constrained forming of FML and less constrained forming of a pure fabric laminate. First, the material [21] is summarized with the most essential equations to indicate the interactions that are considered in the strain energy density components. Subsequently, the numerical test case, a tetrahedron geometry, and the applied materials are introduced. Finally, the forming simulation results are presented and discussed in terms of the coupling aspects and their significance on the forming results for varying forming conditions.

Hyperelastic Multiaxially Coupled Invariant-Based Constitutive Model
The hyperelastic invariant-based model, examined in this work for multiaxial coupling effects, was originally proposed and verified in our previous work [21]. For the sake of brevity, only the most important equations of the model are presented in the following. For more details, the reader is kindly referred to [21]. The invariant-based model was implemented in a user subroutine VUMAT for forming simulation in ABAQUS/EXPLICIT. Membrane and bending behavior are modeled in a decoupled way by superimposed finite membrane elements M3D3 and finite shell elements S3R. The membrane elements use the new VUMAT, while the shell elements use a bending idealization with section integration and deactivated membrane properties.
The basic assumption of the membrane model is the additive decomposition of the total strain energy density = ( 4 , 8 ) + , ( 10 , 4 , 8 ) (1) into a tension-compression component ( 4 , 8 ) and a tension-compression-dependent shear component , ( 10 , 4 , 8 ). The components are based on invariants of the right Cauchy-Green (RCG) tensor to be independent of the choice of the coordinate system.
Tension-compression coupling. The tension-compression-dependent strain energy density component is composed of two components , and , according to the two fiber directions and , where the indices and represent tension ( ) or compression ( ). The composition depends on the current strain state: = for 4 ≤ 1; = for 8 > 1; and = for 8 ≤ 1 [21]. The invariants 4 and 8 are the quadratic stretches in fiber directions and , respectively.
For tension, the strain energy densities are with the uniaxial strain energy densities where and are material parameters to be determined by unidirectional tensile tests. In Eq. (3), the influence of transverse strains is considered by the fiber-fiber-interaction factors taking values greater than 1 for transverse tension and values between 0 and 1 for transverse compression. The tension-compression parameter , is introduced to limit the influence of the transverse strains on the tensile strain energy density, which is particular important in the case of negative transverse strains [21].
, can take values between 0 (complete strain energy reduction due to compression is possible) and 1 (no influence). In this work, , is set to 0.75 for negative and positive transverse strains. Primarily, the biaxial coupling coefficient provides a strictly monotonous increase or decrease of the strain energy density for positive or negative transverse strains, respectively. It can be determined by biaxial tension tests.
Compression is typically not considered in composite forming simulation, since the textile material can easily buckle and wrinkling can occur. However, under constrained conditions, e.g. by blank holders or adjacent metal layers, wrinkling is suppressed and the resulting multiaxial stress states may include in-plane compression. In this case, a linear stress-strain relation and no strain components coupling is assumed, leading to the compressive strain energy densities where are material parameters that are proportional to the compressive [21]. To date, there is no test setup available for a reliable characterization of the compressive stiffness, since the fabrics are prone to buckling. Therefore, the compression stiffness is assumed according to numerical parameter studies performed by Werner et al. [23].
Normal-shear coupling. Shear is a prevalent deformation mechanism in woven fabrics, and thus, crucial for material modeling. According to the mesoscopic nature of the interwoven fiber bundles, pure shear can be divided in three zones with very low, medium and high shear stiffness. To capture this steadily increasing shear stiffness, a polynomial of third order is assumed for the stress-strain relation, which leads to the shear strain energy density where the shear invariant 10 represents the shear angle, and material parameters can be determined by pure shear tests [21]. The shear strain energy density is increased for superimposed fiber tension and decreased for superimposed fiber compression. In the case of positive fiber strains, additional strain energy densities , ( 10 , 4 ) = are added, where , are material parameters that can be determined by combined tension-shear tests [21]. In the case of negative fiber strains, the shear strain energy densities (8) and possibly (9) are multiplied by exponential reduction factors with the compression-shear parameter , to achieve a smooth asymptotic reduction of the shear strain energy density [21]. Thus, , can take values less or equal to 1, but always greater than zero.
Total strain energy density.
Combining the tension strain energy density (3) and the compression strain energy (7) with the shear strain energy density (8), influenced by fiber tension (9) or fiber compression (10), yields: Model verification. The prediction capability of the new multiaxially-coupled invariant-based constitutive model was verified by means of tensile and bias-extension tests [21], partly with comparison to experimental results. The simulation results of these verification tests at coupon level revealed that the new model is capable to capture transverse coupling effects, i.e. transverse stresses in the case of constrained loading and, vice versa, transverse strains in the case of free loading [21]. This is in contrast to the ABAQUS built-in FABRIC model, which is not able to predict transverse effects. In Fig. 2, the effect of differently activated coupling mechanisms is illustrated at the example of a simple uniaxial tensile test, where the resulting transverse strain component 22 is compared to experimental results from Cherouat and Bourouchaki [24]. The results in Fig. 2 and particularly in Fig. 2 (b) confirm that the biaxial normal strain coupling is responsible for correctly predicting the deformed fabric shape [21], which is clearly concave in transverse direction, as observed in experiments [24].  [21].

Parametric Study on Individual Multiaxial Couplings and their Effects on Forming Results
To investigate the significance of the different tension-compression and normal-shear interactions, the proposed constitutive model is applied to forming simulation at component scale. Furthermore, forming simulation of a fiber-metal laminate (FML) is compared to forming simulation of a pure fabric to analyze the effect of different forming constraints on the multiaxial interactions. Numerical test case and materials.
A 280 x 280 mm FML with three woven fabric layers and two metallic cover layers is formed into a tetrahedron shape, following the work of Yao et al. [19]. For comparison, three fabric layers without metal blanks are formed into the same geometry. In both cases, a blank holder pressure of 2 MPa is applied to prevent wrinkling. The inter-ply behavior between the rigid-body tools, the metal sheets and the fabric plies is modelled as COULOMB friction with the same friction parameters as used in [21]. The model setup including dimensions and boundary conditions is given in Fig. 3.
The metal material is steel DC04 and is assumed to behave elastic-plastic, as characterized experimentally and modelled by Werner et al [22,23].The fabric material is a plain weave fabric according to the industrial TWINTEX® TPP60N22P-060 with comingled glass/PP fibers, used and characterized by Komeili et al. [25]. The material behavior of the woven fabric is assumed nonlinear elastic according to the equations given in Section 2. The material parameters of the nonlinear models for steel and woven fabric are all the same as in our previous work [21], partly taken from the mentioned literature [22,23,25] and partly assumed. Failure and damage are not considered, neither in the metal sheets nor in the woven fabric. It needs to be noted that the forming simulation example is a purely numerical test case to expose the effect of different interactions. Some of the material parameters are assumed, the blank holder pressure is comparatively high to avoid wrinkling and the forming simulation is conducted up to a rather large tool stroke with strains reaching values beyond failure or yielding. Nevertheless, the test case is chosen and suited to illustrate the importance of the distinct multiaxial coupling aspects for stronger and weaker forming constraints.
Results for FML forming. Fig. 4 shows the distribution of the resulting shear angle 12 for the deformed FML, where the Fabric material model available in Abaqus (Fig. 4a) is compared to four different configurations of the new hyperelastic material model with differently activated strain couplings. The results reveal that the normal-shear coupling (Fig. 4e) has only marginal effect on the forming results, since the shear angle distribution is comparable to the FABRIC model (Fig. 4a) and the hyperelastic model without coupling (Fig. 4b). In contrast, the fully coupled hyperelastic model (Fig. 4c) and the model with biaxial normal coupling (Fig. 4d) lead to considerably higher shear angles. The same conclusion can be drawn from Fig. 5, where the shear angle is plotted along the dotted line in Fig. 4a and is compared for all five configurations. An explanation for the strong influence of biaxial normal coupling is the presence of high normal stresses due to the encasing metal layers that hinder formation of wrinkles and promote deformation in shear. Interestingly, just a minor effect of normal-shear coupling can be observed by the green only normal-shear coupling curve in Fig. 5. It is located just slightly below the gray no-coupling curve, because the higher tensile strains partly impede shear deformation. The same is observed for the fully coupled VUMAT model, which is slightly below the only biaxial coupling curve that neglects normal-shear coupling. Results for pure fabric forming.
To investigate the relevance of biaxial and normal-shear coupling for weakened forming constraints, forming simulations are performed for three fabric layers without metal blanks. Furthermore, the friction coefficient between fabric and tool is smaller than the friction coefficient between fabric and metal sheet, which further reduces the resistance against deformation in the fabric. These modifications are implemented to facilitate shear deformation and material draw-in, and thus, to reduce normal stresses in fiber direction.
The resulting shear angles 12 of the deformed three fabric layers are depicted in Fig. 6. Again, the ABAQUS/EXPLICIT built-in FABRIC model (Fig. 6a) is compared to four configurations of the new hyperelastic material model with differently activated strain couplings. As Fig. 6 reveals, the influence of interacting strain components on the resulting shear deformation is much smaller than in the more strongly constrained FML forming case. This conclusion is again confirmed by Fig. 7, where the shear angle is plotted along the dotted line in Fig. 6a. Again, the results of the fully coupled hyperelastic model (Fig. 6c) and the biaxial normal coupling (Fig. 6d) are close together and are above the results of the other configurations, but the difference between the curves is much smaller than for the constrained forming results shown in Fig. 5. Overall, the shear deformation is, as expected, larger than in the case of FML forming with metal blanks, which results from reduced friction, smaller normal strains and less coupling effects.

Summary
Forming simulation at component scale was performed to evaluate the significance of individual coupling effects in constitutive modeling of the fabric plies. For this purpose, a previously proposed invariant-based constitutive model, which includes biaxial and normal-shear coupling for positive and negative strains, was examined at the example of forming simulation to a tetrahedron. To compare the influence of strong and weak forming constraints on the multiaxial coupling effects, a fiber-metal laminate (FML) and a pure fabric laminate were formed to the tetrahedron geometry. Based on the simulation results, it could be shown that biaxial coupling has a significant effect on the forming results, if forming is constrained by adjacent metal layers. In the case of free or almost free draping conditions, this coupling effect is much smaller. Nevertheless, blank holder forces also cause forming constraints that can partially prevent wrinkling and promote the importance of biaxial strain coupling. The results also revealed that biaxial coupling has a larger influence on the forming results than normal-shear coupling.