# Fast Fourier transform solver for damage modeling of composite materials

## Abstract

In the case of highly heterogeneous microstructures, such as textile composites herein, conformal FE meshes are difficult to generate for image-based modeling. As an alternative way regular meshing based on the initial image discretization can be used. However, it requires a large number of elements to reduce undesirable effects due to the voxelized discretization of the phase, such as so-called checkerboard pattern. In this work FAST Fourier transform (FFT) based method has been employed by virtue of its simplicity of meshing and efficiency of parallel computation. One of the major contributions is the extension of the FFT method to nonlinear material modeling based on continuous damage mechanics (CDM). Simple test cases are provided to validate the model. In the last part, the FFT with CDM modeling is applied to a real mesostructure of 3D interlock composite from X-ray computed tomography image.

## Keywords

Continuous damage mechanics Fast Fourier transform Interlock composites## 1 Introduction

By virtue of the development of advanced imaging techniques such as 3D Electron Back-Scattered Diffraction (EBSD), X-ray diffraction and X-ray/neutron tomography, image-based modeling is increasingly used to predict the properties of composite materials. Such a modeling strategy has a great advantage of providing the real description of complex 3D microstructure, including irregular geometrical fluctuations and random manufacturing defects, hence making the prediction of material properties more reliable compared with that from an idealized/virtual microstructure. The image-based modeling consists of four main steps: (i) imaging of the considered material; (ii) meshing, i.e., generating a discretize mesh/grid of the microstructure from the digital image; (iii) application of prediction models (elastic, damage, etc.) to the local constitutive materials; (iv) computation using an appropriate solver. Generally, the imaging step involves multidisciplinary technologies and is conducted by specialists, while the material scientists take over the acquired images to complete the subsequent three steps. The meshing step is closely linked to the numerical method to be used, and the most common one is finite element method (FEM) by dint of its versatility. However, meshing for FEM is generally a tedious step especially for the complex microstructure of composites, such as textile composites. In this context, conformal FE meshing with a good quality is a difficult task and has already been the subject of many researches [1, 2, 3]. Regular meshing based on the initial image discretization is an alternative way to simplify this processing. However, it requires a large number of elements to reduce undesirable effects due to the structured discretization of the phase interfaces (e.g., see [4, 5] for an illustration).

As an alternative to FEM, fast Fourier transform (FFT) based methods [6, 7, 8, 9, 10] become more and more attractive for image-driven modeling of material properties, because they provide great advantages in meshing procedure and parallel computation. The FFT-based methods intrinsically use regular grids to describe the microstructural geometry. Hence, the initial image voxels are directly used as discretized elements without any additional meshing treatment.

Even though the FFT-based methods have been extended to the modeling of nonlinear material behaviors such as viscoelasticity, plasticity, etc., their application with damage models is not yet very popular. It has been applied to non-local fracture model using a simple criterion on maximum stress [11]. Recently, the FFT-based method was used within a specific algorithm to take into account continuum damage model (CDM) [12]: for each loading step, a linear elastic problem was solved with an FFT-based solver and the resultant stresses were used to evaluate the damage field and update the elastic stiffness fields for the next loading step.

We incorporate a CDM into a generic FFT method to predict the mechanical properties of 3D textile composite material from its micro-architecture, the properties of its constituents and the manufacturing defects such as voids. A massive parallel FFT-based solver (AMITEX) interfaces with CDM through a generic UMAT subroutine (common to ABAQUS and CAST3 M). After a brief description of the proposed model, several numerical test cases are presented to discuss the efficiency of the proposed model. The effect of boundary conditions is investigated using a real 3D interlock composite image by X-ray computed tomography.

## 2 FFT-based methods

### 2.1 Numerical algorithm

*Γ*

_{0}is the Green operator and

*E*is the prescribed macroscopic strain. In the basic scheme proposed in [6] (Table 1), Eq. 2 is solved with fixed-point algorithm. The variables to be solved in FFT-based methods are strains or stresses, instead of displacements for FEM. With periodic boundary conditions, the stress/strain fields over a regular grid can be transformed into discretized Fourier space. The convolution product with the Green operator (Eq. 2) is converted to a simple multiplication in Fourier space. Such an algorithm (Table 1) is very suitable for parallel computation, because:

The basic scheme proposed by [6]

| \( \varepsilon^{0} \left( x \right) = E \) | (T1) |

Loop | ||

| \( \sigma^{k} = \sigma \left( {\varepsilon^{k} \left( x \right),d_{\text{i}} } \right) \) | (T2) |

Convergence test | ||

\( \tau^{k} \left( x \right) = \sigma^{k} \left( x \right) - {\mathbb{C}}_{0} :\varepsilon^{k} \left( x \right) \) | (T3) | |

FFT: \( \tau^{k} \left( x \right) \to \hat{\tau }^{k} \left( \xi \right) \) | (T4) | |

| \( \hat{\varepsilon }^{k + 1} \left( \xi \right) = - \left( {\hat{\varGamma }_{0} \cdot \hat{\tau }^{k} } \right)\left( \xi \right) \) | (T5) |

Inverse FFT: \( \varepsilon^{(k + 1)} (\xi )\varepsilon^{(k + 1)} (x) \) | (T6) |

i. (Eqs. T2, T3, T5) are local and can be solved separately for each material point;

ii. The FFT and inverse FFT are not local, but various packages (e.g., parallel FFTW) are available to efficiently complete these two steps in a parallel way.

A code based on this basic scheme (AMITEX [13]) was employed in the present work. This code allows massive parallel simulations on a large number of processors. In addition, it incorporates a procedure accelerating the convergence of the fixed-point algorithm, inspired from the FEM code CAST3M [14], as well as a modified Green operator, as proposed by [15]. As a result, the convergence in AMITEX is faster than the original basic scheme and much less sensitive to the choice of reference material. This FFT algorithm converges efficiently when a progressive damage develops. However, a convergence issue is observed as soon as instable propagation occurs. To address this issue, the FFT algorithm was embedded in an additional loop. If the number of iterations of the FFT algorithm reached a predefined maximum number, the damage variables evaluated during the last iteration were updated and the FFT algorithm (Table 1) was restarted for the same time step. This procedure was also inspired from the FEM code CAST3M.

### 2.2 Material behavior model

Material properties

Yarn | Young’s modulus | 44.4 |

Young’s modulus | 11.64 | |

In-plane shear modulus | 4.48 | |

Poisson ratio | 0.26 | |

Poisson ratio | 0.33 | |

Longitudinal strength in tension | 1500 | |

Longitudinal strength in compression | 1000 | |

Transverse strength in tension | 60 | |

Transverse strength in compression | 150 | |

Shear strength | 50 | |

Fracture energy of yarns in longitudinal direction (kJ) | 11 | |

Fracture energy of yarns in transverse direction (kJ) | 0.2 | |

Matrix | Modulus of the matrix | 3 |

Tension/compression strength of matrix | 80 | |

Shear strength | 40 | |

Fracture energy of matrix (kJ) | 2 |

This model was implemented in a user subroutine UMAT supported by AMITEX. It should be noted that the damage model incorporates a regularization scheme based on Bazant’s crack band model [20], so that the mesh dependency on dissipated fracture energy can be minimized.

## 3 Results and discussion

### 3.1 Test cases

*y*- and

*z*-axes) led to a quasi-brittle behavior, which was consistent with the fact that once the transverse yarn started to be damaged, the matrix could not provide any more resistance, and the damage would propagate rapidly in the matrix phase. The damage field of the unit cell under tension in

*Z*direction (\( \overline{{\varepsilon_{\text{zz}} }} = 2.4\% \)) is shown in Fig. 2b. The damage paths in matrix were inclined to the loading direction, suggesting that the matrix was easier to be fractured under shear mode than under tension mode. It should be noted that this property obtained by simulations is related to the damage functions used in the damage model and the chosen material properties (Table 2).

*d*

_{1}) in the longitudinal yarn (green) and transverse damage (

*d*

_{2}) in the transverse yarn (yellow).

## 4 Application to 3D interlock composites

### 4.1 Imaging and unit cell generation

A 3D interlock composite plate was scanned under the X-ray computed tomography of the ISIS4D platform (LML/LaMcube, France). The 3D images were reconstructed and provided a field of view (FOV) of 13.71 × 3.14 × 9.99 mm^{3} with a voxel size of 8.8 µm. This FOV contained one pseudo-periodic representative volume element (RVE) of the interlock architecture. For unit cell generation, we used the structure-tensor based method proposed by [21]. Warps, wefts matrix and macrovoids were identified using supervised clustering with the gray-level and local orientation information.

#### 4.1.1 Edge effect

Periodic boundary conditions (PBC) are intrinsically prescribed by the FFT-based methods. However, a real microstructure is never perfectly periodic. If the local response of the unit cell is of interest, which is more and more demanded for image-based modeling, the edge effect due to the PBC on a non-periodic real microstructure should be considered. To reduce the edge effect, we added marginal layers around the unit cell (see Fig. 4). These marginal layers were assigned with different elastic (non-damageable) properties according to the loading direction: for the loading in *x*-axis (weft direction), the elastic modulus of margin *Y* would be set to zero, while that of margin *X* was greater than zero; vice versa for loading in *y*-axis (warp direction). The stiffness of the marginal layers (called marginal stiffness hereafter) should be as high as possible because they could act as the role of uniformly transmitting the load between the two corresponding borders.

Simulation of such a unit cell (50,715 elements) was parallelized over 12 processors and each iteration took about 0.88 s.

The simulation using no marginal layer led to the prediction of the damage strongly localized at the unit cell borders due to the geometrical discontinuity, whereas those with high marginal stiffness resulted in the prediction of the damage inside the composite. The damage fields obtained from C0x20 and C0x100 were very similar with each other, which again confirmed that the damage prediction became stable and independent on the marginal stiffness if the marginal stiffness became greater than 60 GPa (C0x20). On the other hand, a higher marginal stiffness (> 150 GPa, C0x50) led to a slow convergence rate of the computation once the damage was initiated. Therefore, as a compromise between the prediction accuracy and computation speed, we chose 60 GPa (C0x20) as the elastic modulus of the marginal layers for further simulations.

## 5 Conclusion

The CDM model has been successfully introduced into the generic FFT-based code. Several test cases showed the damage anisotropy was well considered in the model. The model was then applied to a real mesostructure of a 3D interlock composite to evaluate the effective properties of composites considering the void defects. To reduce the edge effect, elastic marginal layers were added around the unit cell. The effect of the marginal stiffness was investigated, and we proved that the marginal layers were particularly necessary for damage modeling in a real microstructure which was not perfectly periodic. The efficiency of the FFT-based methods for the damage modeling was demonstrated with respect to parallel computation, which cannot be easily achieved by FEM.

## Notes

### Acknowledgements

The Nord-Pas-de-Calais Region and the European Community (FEDER funds) partly funds the X-ray tomography equipment ISIS4D platform (LML/LaMcube, France).

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