Topology optimization considering fracture mechanics behaviors at specified locations

Abstract

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications.

Appendix A

Details of derivation of discrete form of the J integral expression are as follows.

In a discrete form, the equilibrium equation reads

$$ \mathbf{K}\mathbf{d}=\mathbf{p}, $$

(A.1)

where d is the nodal displacement vector, p is the external force vector and K is the global stiffness matrix, which is expressed by

$$ \mathbf{K}={\displaystyle \sum_{i=1}^N{{\mathbf{G}}_i}^{\mathrm{T}}\left({\displaystyle {\int}_{\Omega_i}{{\mathbf{B}}_i}^{\mathrm{T}}{\mathbf{D}}_i{\mathbf{B}}_i\mathrm{d}\Omega}\right){\mathbf{G}}_i}. $$

(A.2)

After solving Equation A.1, the J integral can be calculated along a path composed entirely of element edges, as expressed in the matrix form

$$ J={\displaystyle \sum_{p=1}^M{\displaystyle {\int}_{\varGamma_p}\left(\frac{1}{2}{\boldsymbol{\upvarepsilon}}^{\mathrm{T}}\mathbf{D}\boldsymbol{\upvarepsilon } \mathrm{d}y-{\mathbf{T}}^{\mathrm{T}}\frac{\partial \mathbf{u}}{\partial x}\mathrm{d}s\right)}}, $$

(A.3)

where M is the total number of element edges which constitute the path of integral and Γ _p is the p th element edge, T is the vector of traction on the contour and u is the displacement vector. For the element edges at which the strain and stress are discontinuous, the strain and traction are taken as their average values of two neighboring elements (for the pth element edge shown in Fig. 32, the two neighboring elements are EL  ⁱ _p (inside the contour) and EL  ^o _p (outside the contour)).

Fig. 32

A schematic illustration of a finite element model of a crack

The first term of the J integral in Equation A.3 can be further written as

$$ \begin{array}{c}\hfill {J}_{{\mathrm{T}}_{\mathrm{I}}}=\frac{1}{8}{\displaystyle \sum_{p=1}^M{\displaystyle {\int}_{\varGamma_p}{\left({\mathbf{B}}_p^{\mathrm{i}}{\mathbf{d}}_p^{\mathrm{i}}+{\mathbf{B}}_p^{\mathrm{o}}{\mathbf{d}}_p^{\mathrm{o}}\right)}^{\mathrm{T}}{\mathbf{D}}_p\left({\mathbf{B}}_p^{\mathrm{i}}{\mathbf{d}}_p^{\mathrm{i}}+{\mathbf{B}}_p^{\mathrm{o}}{\mathbf{d}}_p^{\mathrm{o}}\right)\mathrm{d}y}}\hfill \\ {}\hfill ={\mathbf{d}}^{\mathrm{T}}\left({\mathbf{K}}_{\mathrm{I}}^{\mathrm{i}\mathrm{i}}+{\mathbf{K}}_{\mathrm{I}}^{\mathrm{o}\mathrm{o}}+{\mathbf{K}}_{\mathrm{I}}^{\mathrm{i}\mathrm{o}}\right)\mathbf{d}\hfill \\ {}\hfill ={\mathbf{d}}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{I}}\mathbf{d}\hfill \end{array}, $$

(A.4)

where the superscripts “i” and “o” indicate the inner and outer side of the contour, respectively as shown in Fig. 32; B ⁱ _p and B ^o _p are the displacement–strain matrix of the two neighboring elements on the inner and outer side of the p th element edge, respectively. The expressions of K ⁱⁱ_I , K ^oo_I , K ^io_I are given as

$$ \left\{\begin{array}{l}{\mathbf{K}}_{\mathrm{I}}^{ii}=\frac{1}{8}{\displaystyle \sum_{p=1}^M{{\mathbf{G}}_p^{\mathrm{i}}}^{\mathrm{T}}\left({\displaystyle {\int}_{\varGamma_p}{{\mathbf{B}}_p^{\mathrm{i}}}^{\mathrm{T}}\mathbf{D}{\mathbf{B}}_p^{\mathrm{i}}\mathrm{d}y}\right){\mathbf{G}}_p^{\mathrm{i}}}\\ {}{\mathbf{K}}_{\mathrm{I}}^{\mathrm{o}\mathrm{o}}=\frac{1}{8}{\displaystyle \sum_{p=1}^M{{\mathbf{G}}_p^{\mathrm{o}}}^{\mathrm{T}}\left({\displaystyle {\int}_{\varGamma_p}{{\mathbf{B}}_p^{\mathrm{o}}}^{\mathrm{T}}\mathbf{D}{\mathbf{B}}_p^{\mathrm{o}}\mathrm{d}y}\right){\mathbf{G}}_p^{\mathrm{o}}}\\ {}{\mathbf{K}}_{\mathrm{I}}^{\mathrm{i}\mathrm{o}}=\frac{1}{4}{\displaystyle \sum_{p=1}^M{{\mathbf{G}}_p^{\mathrm{i}}}^{\mathrm{T}}\left({\displaystyle {\int}_{\varGamma_p}{{\mathbf{B}}_p^{\mathrm{i}}}^{\mathrm{T}}\mathbf{D}{\mathbf{B}}_p^{\mathrm{o}}\mathrm{d}y}\right){\mathbf{G}}_p^{\mathrm{o}}}\end{array}\right. $$

(A.5)

Similarly, the second term of the J integral is expressed as

$$ \begin{array}{c}\hfill {J}_{{\operatorname{T}}_{II}}=-\frac{1}{4}{\displaystyle \sum_{p=1}^M{\displaystyle {\int}_{\varGamma_p}{\left({\mathbf{B}}_p^{\mathrm{i}}{\mathbf{d}}_p^{\mathrm{i}}+{\mathbf{B}}_p^{\mathrm{o}}{\mathbf{d}}_p^{\mathrm{o}}\right)}^{\mathrm{T}}{\mathbf{D}}_p\mathbf{n}\left(\frac{\partial {\mathbf{N}}_p^{\mathrm{i}}}{\partial x}{\mathbf{d}}_p^{\mathrm{i}}+\frac{\partial {\mathbf{N}}_p^{\mathrm{o}}}{\partial x}{\mathbf{d}}_p^{\mathrm{o}}\right)\mathrm{d}s}}\hfill \\ {}\hfill =-{\mathbf{d}}^{\mathrm{T}}\left({\mathbf{K}}_{II}^{ii}+{\mathbf{K}}_{II}^{oo}+{\mathbf{K}}_{II}^{io}+{\mathbf{K}}_{II}^{oi}\right)\mathbf{d}\hfill \\ {}\hfill =-{\mathbf{d}}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{II}}\mathbf{d}\hfill \end{array}, $$

(A.6)

where N is the shape function matrix; n is a matrix that consists of components of the direction vector of the integration contour; the expressions of n, K ⁱⁱ _II , K ^oo _II , K ^io _II and K ^oi _II are given as

$$ \mathbf{n}=\left[\begin{array}{l}{n}_1\kern0.5em 0\\ {}0\kern0.75em {n}_2\\ {}{n}_2\kern0.5em {n}_1\end{array}\right], $$

(A.7)

$$ \left\{\begin{array}{l}{\mathbf{K}}_{II}^{ii}=\frac{1}{4}{\displaystyle \sum_{p=1}^M{{\mathbf{G}}_p^{\mathrm{i}}}^{\mathrm{T}}\left({\displaystyle {\int}_{\varGamma_p}{{\mathbf{B}}_p^{\mathrm{i}}}^{\mathrm{T}}\mathbf{D}\mathbf{n}\frac{\partial {\mathbf{N}}_p^{\mathrm{i}}}{\partial x}\mathrm{d}s}\right){\mathbf{G}}_p^{\mathrm{i}}}\\ {}{\mathbf{K}}_{II}^{oo}=\frac{1}{4}{\displaystyle \sum_{p=1}^M{{\mathbf{G}}_p^{\mathrm{o}}}^{\mathrm{T}}\left({\displaystyle {\int}_{\varGamma_p}{{\mathbf{B}}_p^{\mathrm{o}}}^{\mathrm{T}}\mathbf{D}\mathbf{n}\frac{\partial {\mathbf{N}}_p^{\mathrm{o}}}{\partial x}\mathrm{d}s}\right){\mathbf{G}}_p^{\mathrm{o}}}\\ {}{\mathbf{K}}_{II}^{io}=\frac{1}{4}{\displaystyle \sum_{p=1}^M{{\mathbf{G}}_p^{\mathrm{i}}}^{\mathrm{T}}\left({\displaystyle {\int}_{\varGamma_p}{{\mathbf{B}}_p^{\mathrm{i}}}^{\mathrm{T}}\mathbf{D}\mathbf{n}\frac{\partial {\mathbf{N}}_p^{\mathrm{o}}}{\partial x}\mathrm{d}s}\right){\mathbf{G}}_p^{\mathrm{o}}}\\ {}{\mathbf{K}}_{II}^{oi}=\frac{1}{4}{\displaystyle \sum_{p=1}^M{{\mathbf{G}}_p^{\mathrm{o}}}^{\mathrm{T}}\left({\displaystyle {\int}_{\varGamma_p}{{\mathbf{B}}_p^{\mathrm{o}}}^{\mathrm{T}}\mathbf{D}\mathbf{n}\frac{\partial {\mathbf{N}}_p^{\mathrm{i}}}{\partial x}\mathrm{d}s}\right){\mathbf{G}}_p^{\mathrm{i}}}\end{array}\right. $$

(A.8)

Finally, the J integral is expressed in matrix form as

$$ J={J}_{{\operatorname{T}}_I}+{J}_{{\operatorname{T}}_{II}}={\mathbf{d}}^{\mathrm{T}}\left({\mathbf{K}}_{\mathrm{I}}-{\mathbf{K}}_{\mathrm{I}\mathrm{I}}\right)\mathbf{d}. $$

(A.9)

It is easy to prove that K _I is symmetric and K _II is asymmetric.