Multiscale crack band model for eigenstrain based reduced order homogenization

Abstract

A multiscale crack band model is proposed to alleviate spurious mesh size dependence in the solution for a strain softening constitutive model. A continuum damage mechanics-based material model for the constituent phases of a carbon fiber reinforced polymer (CFRP) based composite material is applied to the Eigenstrain based Reduced-order Homogenization framework for multiscale modeling. This paper presents the formulation and demonstrates dissipated energy regularization within the multiscale modeling framework. A key contribution is computationally efficient implementation of multiscale crack band model as user defined subroutine for the commercial software Abaqus. The accuracy of the multiscale crack band model is demonstrated by critical evaluation of the numerical results for CFRP laminates.

Appendix: Derivation of multiscale crack band model

In order to simplify the computation of the fracture strain, the part of the fracture energy within the softening regime is approximated in a linear fashion:

$$\begin{aligned} \widetilde{G}_\text {f}^{\text {N}(2)}(\epsilon ^\text {N}_\text {f})=\frac{1}{2}\big (\epsilon ^\text {N}_\text {f}-\epsilon ^\text {N}_\text {tr}\big )\bigg [1-\Phi (\upsilon ^\text {N}_\text {tr})\bigg ]E\epsilon ^\text {N}_\text {tr} \end{aligned}$$

(A.1)

The fracture energy for a reference mesh size, $h_{0}$, for uniaxial loading case is expressed as:

$$\begin{aligned} G^\text {N}_\text {f}\Big |_{h_0}=\frac{wh_{0}}{l}\bigg [\widetilde{G}^{\text {N}(1)}_\text {f} +\widetilde{G}^{\text {N}(2)}_\text {f} \Big (\epsilon ^\text {N}_\text {f}\Big |_{h_0}\Big )\bigg ] \end{aligned}$$

(A.2)

Similarly, the fracture energy for any arbitrary mesh size, h, can be given by:

$$\begin{aligned} G^\text {N}_\text {f}=\frac{wh}{l}\bigg [\widetilde{G}^{\text {N}(1)}_\text {f} +\widetilde{G}^{\text {N}(2)}_\text {f} \bigg ] \end{aligned}$$

(A.3)

The area in softening regime for any arbitrary mesh size, h, is scaled with a parameter $\widetilde{c}$, as the fracture energy remains conserved. Hence, for any mesh size, h, the fracture energy due to softening can be expressed by Eq. A.4.

$$\begin{aligned} \widetilde{G}^{\text {N}(2)}_\text {f}=\frac{1}{\widetilde{c}}\bigg [ \frac{l}{wh}{G}^\text {N}_\text {f}-\widetilde{G}_\text {f}^\text {N(1)} \bigg ] \end{aligned}$$

(A.4)

Following the fact that, energy in the hardening regime remains same, substitution of $\widetilde{G}^\text {N(1)}_\text {f}$ from Eqs. A.2 to A.4 leads to:

$$\begin{aligned} \widetilde{G}^{\text {N}(2)}_\text {f}=\frac{1}{\widetilde{c}}\bigg [ \frac{l}{wh}{G}^\text {N}_\text {f}- \frac{l}{wh_0}{G}^\text {N}_\text {f} + \widetilde{c}~\tilde{G}^{\text {N}(2)}_\text {f} \bigg ] \end{aligned}$$

(A.5)

Now, substituting ${G}^\text {N}_\text {f}$ from Eq. A.2 into Eq. A.5, the fracture energy under softening for any mesh size h will lead to Eq. A.6

$$\begin{aligned} \widetilde{G}^{\text {N}(2)}_\text {f}= & {} \dfrac{1}{\widetilde{c}}\frac{h_0}{h}~\bigg (\widetilde{G}^{\text {N}(1)}_\text {f} +\widetilde{G}^{\text {N}(2)}_\text {f}\Big (\epsilon ^\text {N}_\text {f}\Big |_{h_0}\Big )\bigg )\nonumber \\&\quad - \dfrac{1}{\widetilde{c}}~\bigg (\tilde{G}^{\text {N}(1)}_\text {f} +\widetilde{G}^{\text {N}(2)}_\text {f}\Big (\epsilon ^\text {N}_\text {f}\Big |_{h_0}\Big )\bigg ) + \widetilde{G}^{\text {N}(2)}_\text {f}\Big (\epsilon ^\text {N}_\text {f}\Big |_{h_0}\Big )\nonumber \\ \end{aligned}$$

(A.6)

Rearranging the fracture energy terms in Eq. A.6 under conditions of hardening and softening, yields Eq. A.7.

$$\begin{aligned} \widetilde{G}^{\text {N}(2)}_\text {f}= & {} \dfrac{1}{\widetilde{c}}\bigg (\frac{h_{0}}{h}-1\bigg )\widetilde{G}^{\text {N}(1)}_\text {f}\nonumber \\&\quad +\Bigg [1+\dfrac{1}{\tilde{c}}\bigg (\frac{h_{0}}{h}-1\bigg )\Bigg ]\widetilde{G}_\text {f}^{\text {N}(2)}\Big (\epsilon ^\text {N}_\text {f}\Big |_{h_0}\Big ) \end{aligned}$$

(A.7)

Finally, the dissipated energy regularized failure strains can be expressed in terms of $\widetilde{G}^\text {N(2)}_\text {f}$ as:

$$\begin{aligned} \epsilon ^\text {N}_\text {f}=\epsilon ^\text {N}_\text {tr} + 2\dfrac{ \widetilde{G}_\text {f}^{\text {N}(2)}}{\Big [1-\Phi (\upsilon ^\text {N}_\text {tr})\Big ]E\epsilon ^\text {N}_\text {tr}} \end{aligned}$$

(A.8)