Optimal Design of Brittle Composite Materials: a Nonsmooth Approach

Abstract

Our goal is to design brittle composite materials yielding maximal energy dissipation for a given static load case. We focus on the effect of variation of fiber shapes on resulting crack paths and thus on the fracture energy. To this end, we formulate a shape optimization problem, in which the cost function is the fracture energy and the state problem consists in the determination of the potentially discontinuous displacement field in the two-dimensional domain. Thereby, the behavior at the crack surfaces is modeled by cohesive laws. We impose a nonpenetration condition to avoid interpenetration of opposite crack sides. Accordingly, the state problem is formulated as variational inequality. This leads to potential nondifferentiability of the shape-state mapping. For the numerical solution, we derive first-order information in the form of subgradients. We conclude the article by numerical results.

Appendix: Shape Derivatives of Different Quantities

In the following, we denote by c the number of continuous elements in the finite element mesh. Furthermore, we denote the four nodes of one rectangular (possibly cohesive) element by X ₁,X ₂,X ₃,X ₄ with X _i =(X _i1,X _i2)^T, i=1,…,4. Because of the chosen discretization of the design space, the values of $X^{k}_{ij}$, i=1,…,4, j=1,2, k=1,…,c+m depend linearly, and thus continuously differentiable on the shape variable γ. Therefore, the constant shape derivatives $X^{\prime k}_{ij}$, i=1,…,4, j=1,2, k=1,…,c+m can be easily determined at the beginning of the shape optimization process. We follow [25] for the derivation of the shape sensitivities for τ′, ν′ as well as for W′, $\boldsymbol{A}'_{\mathrm{stif}}$, $\boldsymbol{f}'_{\mathrm{coh}}$ and $\boldsymbol{f}'_{\lambda}$. We notice that our problem structure is more involved than pure elasticity problems discussed in [25].

The unit vector tangential to a cohesive element is determined by its four nodes as

(52)

Thus, the unit normal vector ν=(−τ ₂,τ ₁)^T is dependent on the four nodes by

(53)

with

(54)

The shape derivative of the normal vector ν of a cohesive element with the nodes (X ₁,X ₂,X ₃,X ₄) can be calculated as

(55)

with

(56)

and $\frac{d\nu_{i}}{dX_{3j}}=\frac{d\nu_{i}}{dX_{1j}}$, $\frac{d\nu_{i}}{dX_{4j}}=\frac{d\nu_{i}}{dX_{2j}}$, i=1,2, j=1,2. The shape derivative of the tangential vector τ′ can be calculated in a similar fashion.

The determinant of the Jacobian of the transformation from a cohesive element to the reference element is given by

(57)

and its shape derivative is

(58)

The shape derivative of the stiffness matrix $\boldsymbol{A}'_{\mathrm{stif}}$ is assembled elementwise from the local element matrices $\boldsymbol{A}^{\prime e}_{\mathrm{stif}}$. Writing Hooke’s law σ _ij (u)=c _ijkl e _ij (u) in a compact form

(59)

with

(60)

the local stiffness matrices $\boldsymbol{A}^{e}_{\mathrm{stif}}$ are calculated as

(61)

Here, the matrices L contain the derivatives of the basis functions (compare [34]). These matrices L are evaluated at four integration points χ _ei , i=1,…,4, respectively, and |I _e | is the determinant of the Jacobian matrix for the transformation to the continuous reference element (compare [25]). The shape derivatives $\boldsymbol{A}^{\prime e}_{\mathrm{stif}}$ are then given by

(62)

with

(63)

and

(64)

(65)

Moreover, the shape derivative of the determinant of the Jacobian of the transformation map from a continuous element to the reference element is given by

(66)

The shape derivative of the fracture energy W for given solution $(\bar{\boldsymbol{u}}_{d},\bar{\boldsymbol{\lambda}}_{d})$ of the state problem can be calculated as

(67)

The shape derivative of f _coh can be assembled elementwise from the local force terms $\boldsymbol{f}^{\prime e}_{\mathrm{coh}}$, with

(68)

Finally, the shape derivative of f _λ is given by

(69)

We notice that the matrices B do not depend on the shape variable because of the use of isoparametric elements (see [25] for further details).