Conservation laws for arbitrary objectives with application to fracture resistant design

Abstract

In this paper, we pose a configurational optimization problem to derive the sensitivity of an arbitrary objective to arbitrary motions of one or more finite-sized heterogeneities inserted into a homogeneous domain. In the derivation, we pose an adjoint boundary value problem and utilize the adjoint fields as well as the definition of a generalized Eshelby energy-momentum tensor for arbitrary objectives to express the final result. The resulting sensitivity may be expressed as surface integrals with jump terms across the heterogeneity boundaries that vanish on homogeneous domains yielding generalized conservation laws for arbitrary objectives. We then derive the specific path-independent forms of the sensitivity of the objective to arbitrary translation, rotation or scaling of the inserted heterogeneities. We next illustrate the application of the derived sensitivities to specific objectives common to fracture mechanics as well as to structural optimization. The chosen objectives include strain energy, trade-off between structural compliance and mass, and an arbitrary objective defined entirely on the boundary of the domain. We show that for the strain energy objective, the derived sensitivities naturally yield the classical J-, L- and M-integrals of fracture mechanics. The theory is implemented within an Isogeometric computational framework for fracture modeling termed Enriched Isogeometric analysis (EIGA). The EIGA computational technique is used to optimally identify worst-case locations for line cracks that are inserted into the domain as well as to optimally mitigate the risk of fracture due to a crack at its worst-case location by sequentially inserting and optimizing the configurations of circular/elliptical stiff/soft inclusions.

Appendix A

1.1 A.1 The Divergence of Generalized Eshelby Energy-Momentum Tensor

We derive below the general form of the divergence $\nabla \cdot \varvec{\Sigma }$ and show that the result reduces to $\nabla \cdot \varvec{\Sigma }=0$ if $\mathbf {C}$ and $\mathbf {b}$ are homogeneous in their domains.

$$\begin{aligned} \begin{aligned} \nabla \cdot \varvec{\Sigma }&= \nabla \cdot \left[ \left( \psi - \varvec{\sigma }:\varvec{\varepsilon }^{a} + \mathbf {b}\cdot \mathbf {u}^{a} \right) \mathbf {I} + \varvec{\sigma }\cdot {\nabla \mathbf {u}^{a}}^{T} + \varvec{\sigma }^{a} \cdot \nabla \mathbf {u}^{T} \right] \\&= \nabla \psi - \nabla (\varvec{\sigma }:\varvec{\varepsilon }^{a}) + \nabla (\mathbf {b}\cdot \mathbf {u}^{a}) + \nabla \cdot (\varvec{\sigma }\cdot {\nabla \mathbf {u}^{a}}^{T}) + \nabla \cdot (\varvec{\sigma }^{a} \cdot \nabla \mathbf {u}^{T}) \\&= \nabla \psi - \nabla (\varvec{\sigma }:\varvec{\varepsilon }^{a}) + \nabla (\mathbf {b}\cdot \mathbf {u}^{a}) + (\nabla \cdot \varvec{\sigma }) \cdot ({\nabla \mathbf {u}^{a}}^{T}) + \varvec{\sigma }:\nabla ({\nabla \mathbf {u}^{a}}^{T}) + (\nabla \cdot \varvec{\sigma }^{a}) \cdot (\nabla \mathbf {u}^{T}) + \varvec{\sigma }^{a}:\nabla (\nabla \mathbf {u}^{T}) \\&= \nabla \psi - \nabla \varvec{\sigma }: \varvec{\varepsilon }^{a} - \varvec{\sigma }:\nabla \varvec{\varepsilon }^{a} + \nabla \mathbf {b}\cdot \mathbf {u}^{a} + \mathbf {b}\cdot ({\nabla \mathbf {u}^{a}}^{T}) - \mathbf {b}\cdot ({\nabla \mathbf {u}^{a}}^{T}) + \varvec{\sigma }:\nabla \varvec{\varepsilon }^{a} - \mathbf {b}^{a} \cdot (\nabla \mathbf {u}^{T}) + \varvec{\sigma }^{a}:\nabla \varvec{\varepsilon }\\&= \nabla \psi - \nabla \varvec{\sigma }: \varvec{\varepsilon }^{a} + \nabla \mathbf {b}\cdot \mathbf {u}^{a} - \mathbf {b}^{a} \cdot (\nabla \mathbf {u}^{T}) + \varvec{\sigma }^{a}:\nabla \varvec{\varepsilon }\\&= \frac{\partial {\psi }}{\partial {\mathbf {u}}} \cdot \nabla \mathbf {u}^{T} + \frac{\partial {\psi }}{\partial {\varvec{\varepsilon }}} : \nabla \varvec{\varepsilon }- \nabla \varvec{\sigma }:\varvec{\varepsilon }^{a} + \nabla \mathbf {b}\cdot \mathbf {u}^{a} - \frac{\partial {\psi }}{\partial {\mathbf {u}}} \cdot (\nabla \mathbf {u}^{T}) + (\mathbf {C}:\varvec{\varepsilon }^{a} - \frac{\partial {\psi }}{\partial {\varvec{\varepsilon }}}):\nabla \varvec{\varepsilon }\\&= - \varvec{\varepsilon }: \nabla \mathbf {C}: \varvec{\varepsilon }^{a} + \nabla \mathbf {b}\cdot \mathbf {u}^{a} \end{aligned} \end{aligned}$$

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1.2 A.2 Simplification for Rotational Transformation

Rotation is described by $\mathbf {v}=\mathbf {W}\mathbf {x}= \mathbf {w}\times \mathbf {x}$ and $\nabla \mathbf {v}= \mathbf {W}^{T}$ in $\Omega _s$.

$$\begin{aligned} \begin{aligned}&\int _{\Omega _{s}} \varvec{\Sigma }: \nabla \mathbf {v} \; d\Omega = \int _{\Omega _{s}} \left( \psi - \varvec{\sigma }:\varvec{\varepsilon }^{a} + \mathbf {b}\cdot \mathbf {u}^{a} \right) \mathbf {I}:\mathbf {W}^{T} \; d\Omega \\&+ \int _{\Omega _{s}} \left( \varvec{\sigma }\cdot {\nabla \mathbf {u}^{a}}^{T} \right) : \mathbf {W}^{T} \; d\Omega + \int _{\Omega _{s}} \left( \varvec{\sigma }^{a} \cdot \nabla \mathbf {u}^{T} \right) : \mathbf {W}^{T} \; d\Omega \\&= \int _{\Omega _{s}} \left( \varvec{\sigma }\cdot {\nabla \mathbf {u}^{a}}^{T} + \varvec{\sigma }^{T} \cdot \nabla \mathbf {u}^{a} \right) : \mathbf {W}^{T} \; d\Omega \\&+ \int _{\Omega _{s}} \left( \varvec{\sigma }^{a} \cdot \nabla \mathbf {u}^{T} + {\varvec{\sigma }^{a}}^{T} \cdot \nabla \mathbf {u}\right) : \mathbf {W}^{T} \; d\Omega \\&+ \int _{\Omega _{s}} \left[ \left( \nabla \cdot \varvec{\sigma }\right) \mathbf {u}^{a} - \nabla \cdot \left( \varvec{\sigma }\mathbf {u}^{a} \right) \right] : \mathbf {W}^{T} \; d\Omega \\&+ \int _{\Omega _{s}} \left[ \left( \nabla \cdot \varvec{\sigma }^{a} \right) \mathbf {u}- \nabla \cdot \left( \varvec{\sigma }^{a} \mathbf {u}\right) \right] : \mathbf {W}^{T} \; d\Omega \\&= \int _{\Omega _{s}} \left( \varvec{\sigma }\cdot \varvec{\varepsilon }^{a} + \varvec{\sigma }^{a} \cdot \varvec{\varepsilon }\right) : \mathbf {W}^{T} \; d\Omega \\ {}&- \int _{\Gamma _{s}} \left( \mathbf {t}\mathbf {u}^{a} + \mathbf {t}^{a} \mathbf {u}\right) : \mathbf {W}^{T} \; d\Gamma - \int _{\Omega _{s}} \left( \mathbf {b}\mathbf {u}^{a} + \mathbf {b}^{a} \mathbf {u}\right) : \mathbf {W}^{T} \; d\Omega \end{aligned} \end{aligned}$$

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1.3 A.3 Simplification for Scaling Transformation

Scaling results when $\mathbf {v}= \alpha \mathbf {x}$ and $\nabla \mathbf {v}= \alpha \mathbf {I}$ in $\Omega _s$, where $\alpha$ is an expansion parameter.

$$\begin{aligned} \begin{aligned}&\int _{\Omega _{s}} \varvec{\Sigma }: \nabla \mathbf {v} \; d\Omega = \int _{\Omega _{s}} \left[ \left( \psi - \varvec{\sigma }:\varvec{\varepsilon }^{a}\right. \right. \\&\left. \left. + \mathbf {b}\cdot \mathbf {u}^{a} \right) \mathbf {I} + \varvec{\sigma }\cdot {\nabla \mathbf {u}^{a}}^{T} + \varvec{\sigma }^{a} \cdot \nabla \mathbf {u}^{T} \right] :\alpha \mathbf {I} \; d\Omega \\&= \alpha \left[ d_m\int _{\Omega _{s}} \psi \; d\Omega - d_m\int _{\Omega _{s}} \left( \varvec{\sigma }:\varvec{\varepsilon }^{a} - \mathbf {b}\cdot \mathbf {u}^{a} \right) \; d\Omega \right. \\&\left. + \int _{\Omega _{s}} \varvec{\sigma }:\varvec{\varepsilon }^{a} \; d\Omega + \int _{\Omega _{s}} \varvec{\sigma }^{a}:\varvec{\varepsilon } \; d\Omega \right] \\&= \alpha \left[ d_m\int _{\Omega _{s}} \psi \; d\Omega - d_m\int _{\Gamma _{s}} \mathbf {t}\cdot \mathbf {u}^{a} \; d\Gamma \right. \\&\left. + \int _{\Gamma _{s}} \left( \mathbf {t}\cdot \mathbf {u}^{a} + \mathbf {t}^{a} \cdot \mathbf {u}\right) \; d\Gamma + \int _{\Omega _{s}} \left( \mathbf {b}\cdot \mathbf {u}^{a} + \mathbf {b}^{a} \cdot \mathbf {u}\right) \; d\Omega \right] \end{aligned} \end{aligned}$$

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where, $d_m$ is the problem dimension (2 or 3).