Multiscale modeling of composite materials and Structures
with DIGIMAT & ANSYS

Executive summary
In this paper, we briefly introduce two multi-scale modeling approaches, namely the Mean-Field Homogenization (MFH) and Finite Element Homogenization (FEH) methods. These powerful techniques relate the microscopic and macroscopic stress and strain fields when modeling material behaviors and hence can capture the influence of the material microstructure (i.e. fiber orientation, fiber content, fiber length, etc.) on its macroscopic response. To illustrate these techniques, we also present (i) an application of finite element homogenization to a nanostructure and (ii) the study of an injected glass fiber reinforced plastic neon light clasp using finite element computations at the macro scale coupled with MF homogenization at the micro scale.

Material multi-scale modeling
Consider a plastic part made up of a thermoplastic polymer reinforced with short glass fibers. As typical of the injection molding manufacturing process, the fiber distribution inside the final product will vary widely in terms of orientation and length (Fig. 1). The composite material will be both anisotropic and heterogeneous, which makes it extremely difficult to perform a reliable simulation of the product using a classical FE approach based on macroscopic constitutive models. However, a predictive simulation is possible via a multiscale approach, which can be described in a rather general setting as follows.

Figure 1: (left) Fiber orientation distribution (from injection molding); (right) Nonlinear structural ANSYS FEA of the pedal where the nonlinear and anisotropic behavior of the material is described by DIGIMAT, taking into account the local fiber orientation.. Models courtesy of Rhodia & Trelleborg.

Let us study a heterogeneous solid body whose microstructure consists of a matrix material and multiple phases of so-called “inclusions”, which can be short fibers, platelets, particles, micro-cavities or micro-cracks. Our objective is to predict the response of the body under given loads and boundary conditions (BCs), based on its microstructure. We can distinguish two scales, the microscopic and macroscopic levels, respectively. The former corresponds to the scale of the heterogeneities, while at the macro scale the solid can be seen as locally homogeneous. In practice, it would be computationally impossible to solve the mechanical problem at the fine micro scale. Therefore, we consider the macro scale and assume that each material point is the center of a representative volume element (RVE), which contains the underlying heterogeneous microstructure.

Figure 2: Comparison between the classical FE and the coupled FE/MFH approaches.

Classical solid mechanics analysis is carried out at the macro scale, except that at each computation point, strain or stress values are transmitted as BCs to the underlying RVE. In other words, a numerical zoom is realized at each macro point. The RVE problems are solved and each returns stress and stiffness values, which are used at the macro scale. Now the only difficulty in this two-scale (and more generally multiscale) approach is to solve the RVE problems. It can be shown that for a RVE under classical BCs, the macro strains and stresses are equal to the volume averages over the RVE of the unknown micro strain and stress fields inside the RVE. In linear elasticity, relating those two mean values gives the effective or overall stiffness of the composite at the macro scale.

Figure 3: Microstructure with uniformly distributed inclusions (center). S11 stress distribution in the inclusions (right) and in the matrix (left) for randomly placed inclusions.

In order to solve the RVE problem, one can use the well-known finite element (FE) method. It offers the advantages of being very general and extremely accurate, when proper care is taken. However, it has two major drawbacks, which are: serious meshing difficulties for realistic microstructures and a large CPU time for nonlinear problems, such as for inelastic material behaviour. Another completely different method is mean-field homogenization (MFH), which is based on assumed relations between volume averages of stress or strain fields in each phase of a RVE. Compared to the direct FE method, and actually to all other existing scale transition methods, MFH is both the easiest to use and the fastest in terms of CPU time. However, two shortcomings of MFH are that it is unable to give detailed strain and stress fields in each phase and it is restricted to ellipsoidal inclusion shapes. A typical example of MFH is the Mori-Tanaka model (1973) which is applicable to and highly successful for two-phase composites with identical and aligned ellipsoidal inclusions. The model assumes that each inclusion of the RVE behaves as if it were alone in an infinite body made of the real matrix material. The BCs in the single inclusion problem correspond to the volume average of the strain field in the matrix phase of the real RVE. The single inclusion problem was solved analytically by J.D. Eshelby (1957) in a landmark paper, which is the cornerstone of MFH models. Mori-Tanaka and other MFH models were generalized to other cases, such as thermoelastic coupling, two-phase composites with misaligned fibers (using a multi-step approach) or multi-phase composites (using a multi-level method).

Figure 4: S33 stress (left) distribution in the nano phases and in the RVE for a z-direction uniaxial loading and Young's moduli comparison (right) for clustered and non clustered geometries (Ematrix = 2195 MPa, Efiller = 7000 MPa).

The predictions have been extensively verified against direct FE simulation of RVEs or validated against experimental results. As a general conclusion, it was found that in linear (thermo)elasticity, MFH can give extremely accurate predictions of effective properties, although for distributed orientations, progress in closure approximation will be welcome. Note also that MFH can be used for UD, and for each yarn in woven composites. An important and still ongoing effort both in theoretical modeling and in computational methods is the generalization of MFH to the material or geometric nonlinear realms. Such extension involves some major difficulties. Within a coupled multi-scale analysis, FE method is used at macro scale, while at each Gauss integration point, MFH computation is carried out, either in the linear or nonlinear regime. This is the most feasible approach in practice (Fig. 2).

Extensive verification and validation results show that MFH can be used in practice for nonlinear problems and leads to good predictions in general, while work continues on improving accuracy in some situations (and reducing CPU time for coupled multi-scale analysis).

FE homogenization: an application to nanocomposites
Most likely will nanomaterials be the materials of tomorrow, as they offer new horizons of applications in a wide variety of fields, e.g. nanoelectronics, bio-nanotechnology and nanomedicine. As such, more and more effort is put in understanding and modeling their behavior as well as acquiring know-how about nanoeffects. While new tools are being developed to tackle this engineering challenge, some are already available to the engineer of today. Among them: Finite Element Homogenization (FEH).

Modeling the effect of nano-inclusions
Material scientists face several challenges related to the design and the processing of nanocomposites as, at the nano scale, new physics and phenomena that are negligible at the macro scale enter the picture. For instance, uniform dispersion of the nanofiller inside the composite matrix is sought to improve the material mechanical properties, while clustering and percolation are desired when the conductivity of a base material, thermal or electrical, needs to be increased. Achieving one or the other nowadays constitutes a challenge in terms of both material processing and study. FEH, as it requires the studied geometry to be explicitly generated and meshed, allows an accurate modeling of nanofiller distribution inside a matrix material. As an illustration, we present the effect of nano-inclusions on the elastic mechanical properties of a macroscopic material point.
Figure 3 presents a periodic nanostructure, also referred to as Representative Volume Element (RVE), that has been generated using DIGIMAT-FE. The volume fraction of the inclusion phase is 5% and the inclusions are spherical. Once meshed, this geometry will be subjected to uniaxial tensile conditions in the RVE x-, y- and z-directions and the finite element problem will be solved using the ANSYS finite element solver.

Figure 3 also illustrates the stress distribution in the matrix and inclusion phases, in the case of the x-axis uniaxial tensile test. Due to the proximity of the inclusions, stress concentrations appear. As such, tensile stresses follow a certain distribution which, depending on the presence of clusters or not, can be quite widespread around the average stress. Figure 4 compares the stress distribution in the phases and at the composite level for both a clustered and non clustered geometry. Higher stress levels are observed for the clustered case. These stress levels could prematurely lead to debonding. At low volume fraction of inclusions, it is thus preferable to avoid clustering, as it does not significantly affect the mechanical response of the material.

FE/MFH coupled computation: an application to an industrial part
For many reasons (manufacturing costs and flexibility, processing methods, high strength vs. lightness ratio, etc.), injected parts made up of short glass fiber reinforced plastics have become omnipresent in our daily life. But when it gets to model such materials, can macroscopic constitutive material models capture effects such as the injection process? The answer is no, as they do not capture the influence of the fiber orientation on the material mechanical behavior, which depends on the injection process.

Figure 5: Coupled analysis process. DIGIMAT takes the fiber orientation tensor obtained from the injection molding as input, in addition to the material properties and serves as material modeler for the ANSYS finite element simulation.

Figure 6: Representation of the neon light clasp and of the contacts between the four independent parts. Courtesy of Trilux and CADFEM GmbH.

Figure 7: Modeling of the Bergamid material. Tensile response for the isotropic case, fixed fiber orientation (1D), random 2d orientation (2D) and random 3d orientation (3D). Courtesy of Trilux and CADFEM GmbH.

The following example, which consists of a neon light clasp subjected to loading, introduces the process of a coupled analysis between Moldex3D, DIGIMAT-MF and ANSYS. This process (Fig. 5) consists of the following steps:

1. The injection molding process is simulated using Moldex3D. Among the available results are the fiber orientation tensors that will serve as input to DIGIMAT in the structural simulation.
   
2. The orientation tensors computed in 1. are mapped from the injection mesh onto the coarser structural one using MAP.
   
3. The structural simulation is run using the ANSYS finite element solver coupled with DIGIMAT-MF, the multi-scale material modeler that performs MFH at each integration point of the structural mesh.
   

Problem description
The light clasp consists of several parts made up of steel and glass fiber reinforced polyamide (Fig. 6). Closure of the clasp is simulated by imposing a displacement to the slide while blocking the support and part of the inner part.

Figure 8: S11 stress [MPa] distribution in the clasp for the isotropic linear elastic (left) and nonlinear anisotropic models (right). Courtesy of Trilux and CADFEM GmbH.

The goal of the simulation is to compare the response obtained using a linear elastic model of the material and using DIGIMAT-MF to perform MFH with elastic glass fibers and an elasto-plastic model for the PA (Fig. 7).

Simulation results
While the FEH approach offers the advantage of yielding an accurate description of the strain/stress fields in the RVE, MFH only yields the average stresses and strains at the micro level. It also yields the plastic deformation in the elasto-plastic matrix phase. Figure 8 compares the S11 stress level computed using the conventional and multi-scale FE approaches. Up to 21% difference is observed in the stress magnitude, with the stiffer linear elastic model yielding the higher stresses.
This case study illustrates the superiority of the multi-scale nonlinear approach on the linear elastic homogeneous one to model the material, as both accounting for the fiber orientation and the material nonlinearity help predict more accurately the mechanical response of the clasp under loading.