A generalized finite element method for three-dimensional fractures in fiber-reinforced composites

Abstract

This paper presents a methodology for the analysis of three-dimensional static fractures in fiber-reinforced materials. Fibers are discretely modeled using a modification of the embedded reinforcement method with bond Slip (mERS) that allows its combination with a generalized finite element method (GFEM) for three-dimensional fractures. Since the GFEM mesh does not need to fit fracture surfaces or fibers, the GFEM–mERS can handle fibers bridging across crack faces at arbitrary angles. The method is verified against three-dimensional FEM solutions using conformal discretizations for crack surfaces and fiber boundaries. The comparison of the method against experimental data and convergence studies of the h- and p-version of the method is also presented.

Appendix 1

1.1 Appendix 1.1: 3DFEM Reference Solution Computed with Abaqus

This appendix presents the main steps adopted in the definition of 3DFEM models that provide reference solutions for the proposed GFEM–mERS. The models are defined and solved using Abaqus  [1]. The Compact Tension specimen with five inclined fibers shown in Fig. 35 is used as an example. All other 3DFEM models are defined using the same procedure and element types.

The fibers and the matrix are independent Abaqus Parts of the problem. Each Part of the geometry is assembled into the Model as a second step in Abaqus. Therefore, the final model is composed by the two components. As an example, Fig. 35 depicts these two components (fibers and matrix) for CT specimen described in Sect. 5.5.

Fig. 35

Abaqus model of CT specimen with inclined fibers

After creating the geometry and setting up the material properties, boundary conditions are applied as shown in Fig. 23. Nodal Dirichlet boundary conditions are prescribed at the specimen openings. Additional point constraints are used to prevent rigid body motions. Other approaches to model the boundary conditions at the specimen openings are available in the literature  [15, 30]. Numerical experiments not reported here show that they lead to nearly identical results as those from the adopted approach.

After prescribing the boundary conditions, the tetrahedron mesh of the matrix domain is refined. Fiber and matrix meshes are conforming along their interface. Abaqus cohesive elements COH3D4 are used at the interface between 3-D matrix and fiber elements. The crack is modeled using double nodes in order to represent the displacement discontinuity across the crack surface. The singular behavior at the crack front is approximated using a highly refined mesh around the crack front as shown in Fig. 23. A very refined mesh is also used in the fiber domain. Figure 27 shows the details of the tetrahedron meshes used for the fibers.

1.2 Appendix 1.2: Derivation of axial strain transformation

The strain tensor of the matrix can be transformed to the fiber coordinate system using

$$\begin{aligned} \hat{{\varvec{\epsilon }}}' = {{\varvec{Q}}} \hat{{\varvec{\epsilon }}} {{\varvec{Q}}}^T \end{aligned}$$

(A.1)

where $\hat{{\varvec{\epsilon }}}'$ is the strain tensor in the fiber coordinate system, $\hat{{\varvec{\epsilon }}}$ is the strain tensor in the matrix coordinate system and ${{\varvec{Q}}}$ is the rotation matrix between these systems. The components of these tensors are given by

$$\begin{aligned} \hat{{\varvec{\epsilon }}}'= & {} \left[ \begin{array}{ccc} \hat{\epsilon }_{11}' &{}\quad \hat{\epsilon }_{12}' &{}\quad \hat{\epsilon }_{13}' \\ \hat{\epsilon }_{12}' &{}\quad \hat{\epsilon }_{22}' &{}\quad \hat{\epsilon }_{32}' \\ \hat{\epsilon }_{13}' &{}\quad \hat{\epsilon }_{23}' &{}\quad \hat{\epsilon }_{33}' \\ \end{array} \right] \end{aligned}$$

(A.2)

$$\begin{aligned} \hat{{\varvec{\epsilon }}}= & {} \left[ \begin{array}{ccc} \hat{\epsilon }_{11} &{}\quad \hat{\epsilon }_{12} &{}\quad \hat{\epsilon }_{13} \\ \hat{\epsilon }_{12} &{}\quad \hat{\epsilon }_{22} &{}\quad \hat{\epsilon }_{32} \\ \hat{\epsilon }_{13} &{}\quad \hat{\epsilon }_{23} &{}\quad \hat{\epsilon }_{33} \\ \end{array} \right] \end{aligned}$$

(A.3)

$$\begin{aligned} {{\varvec{Q}}}= & {} \left[ \begin{array}{ccc} Q_{11} &{}\quad Q_{12} &{}\quad Q_{13} \\ Q_{21} &{}\quad Q_{22} &{}\quad Q_{23} \\ Q_{31} &{}\quad Q_{32} &{}\quad Q_{33} \\ \end{array} \right] = \left[ \begin{array}{ccc} \cos (\theta _x^{x'}) &{}\quad \cos (\theta _y^{x'}) &{}\quad \cos (\theta _z^{x'}) \\ \cos (\theta _x^{y'}) &{}\quad \cos (\theta _y^{y'}) &{}\quad \cos (\theta _z^{y'}) \\ \cos (\theta _x^{z'}) &{}\quad \cos (\theta _y^{z'}) &{}\quad \cos (\theta _z^{z'}) \\ \end{array} \right] \end{aligned}$$

(A.4)

In (A.4), $\theta$ is the angle between a fiber coordinate system direction and a global coordinate system direction. For example, $\theta ^{x'}_x$, is the angle between coordinate axes $x'$ and x. Equation (A.1) in index notation is given by

$$\begin{aligned} \hat{\epsilon }_{ij}' = Q_{ik} \hat{\epsilon }_{kl} Q_{lk}^T \end{aligned}$$

(A.5)

Since the only component of interest is $\hat{\epsilon }_{11}'$ (strain in the fiber direction), from (A.5) we have

$$\begin{aligned} \hat{\epsilon }_{11}' = Q_{1k} \hat{\epsilon }_{kl} Q_{l1}^T = Q_{1k} \hat{\epsilon }_{kl} Q_{1l} \end{aligned}$$

(A.6)

Expanding the previous expression

$$\begin{aligned} \begin{array}{cccccc} \hat{\epsilon }_{11}' = &{} Q_{11} \hat{\epsilon }_{11} Q_{11} &{} + &{} Q_{12} \hat{\epsilon }_{21} Q_{11} &{} + &{} Q_{13} \hat{\epsilon }_{31} Q_{11} \\ &{} +Q_{11} \hat{\epsilon }_{11} Q_{11} &{} + &{} Q_{12} \hat{\epsilon }_{21} Q_{11} &{} + &{} Q_{13} \hat{\epsilon }_{31} Q_{11} \\ &{} +Q_{11} \hat{\epsilon }_{11} Q_{11} &{} + &{} Q_{12} \hat{\epsilon }_{21} Q_{11} &{} + &{} Q_{13} \hat{\epsilon }_{31} Q_{11} \\ \end{array} \end{aligned}$$

(A.7)

Writing it in matrix notation leads to

$$\begin{aligned} \hat{\epsilon }_{11}' = {\varvec{R \hat{{\varvec{\epsilon }}}^{*}}} \end{aligned}$$

(A.8)

where

$$\begin{aligned} {{\varvec{R}}}=\left[ \begin{array}{c} \cos ^2(\theta ^{x'}_x) \\ \cos ^2(\theta ^{x'}_y) \\ \cos ^2(\theta ^{x'}_z) \\ 2\cos (\theta ^{x'}_x)\cos (\theta ^{x'}_y) \\ 2\cos (\theta ^{x'}_y)\cos (\theta ^{x'}_z) \\ 2\cos (\theta ^{x'}_x)\cos (\theta ^{x'}_z) \end{array} \right] ^T \end{aligned}$$

(A.9)

and

$$\begin{aligned} {\varvec{\epsilon }}^{*}=\left[ \begin{array}{c} \epsilon _{11} \\ \epsilon _{22} \\ \epsilon _{33} \\ \epsilon _{12} \\ \epsilon _{13} \\ \epsilon _{23} \end{array} \right] \end{aligned}$$

(A.10)