A new computational approach for three-dimensional singular stress analysis of interface voids

Abstract

Defects in terms of three-dimensional voids are commonly encountered at bi-material interfaces. In the current study, the singular stress field near the circumferential corner line of a three-dimensional axisymmetric interfacial void is analyzed using our newly established singular interface edge elements. Under the premise that $\rho {\ll }R$, the proposed singular element method does not depend on the size of the element; thereby, it is not necessary to use refined elements at the interface corner line. The numerical results reveal the intensity of the stress singularity at the interface line of the three-dimensional axisymmetric voids. The obtained stress intensity parameters can be used to judge the local fatigue crack initiation. The geometry effect of the void on the singular stress field at the circumferential interface corner line is studied and discussed in detail.

Appendix

Similar to the interface crack problem, the orders of stress singularities for the singular stress terms in the interface corner domain may also be complex numbers, and the angular variations of singular stress fields change with the material mismatch. If one assumes $\rho {\ll }R$, the displacement field $u_{i} (\rho ,\phi ,\theta )$ including the components $u_{\rho }$, $u_{\phi }$ and $u_{\theta }$ in the vicinity of the interface corner in the local coordinate system can be expressed as [30, 31]

$$\begin{aligned} u_{i} (\rho ,\phi ,\theta )=\hbox {Re}\left\{ {\sum \limits _{n=1}^{N+M} {\rho ^{\lambda _{n} +1}u_{i}^{(n)} (\phi ,\theta )\beta ^{(n)}\;} } \right\} =\hbox {Re}\left\{ {\sum \limits _{n=1}^{N+M} {u_{i}^{(n)} (\rho ,\phi ,\theta )\beta ^{(n)}} } \right\} , \end{aligned}$$

(A1)

in which $\hbox {Re}\{\}$ represents the real part of an expression. $\beta ^{(n)}$ is a stress intensity coefficient related to the external boundary conditions and loads, and is a function of $\theta$, while N and M denotes the number of complex numbers and real numbers intercepted, respectively, from the asymptotic series expressions. $\lambda _{n}$ is an order of the stress singularity, and the tensor $u_{i}^{(n)} (\phi ,\theta )$ contains the angular variation of the displacement components corresponding to the nth terms. The third-order tensor $u_{i}^{(n)} (\rho ,\phi ,\theta )$ is defined by $\rho ^{\lambda _{n} +1}u_{i}^{(n)} (\phi ,\theta )$. Only the terms corresponding to $\hbox {Re}(\lambda _{n} )>-1$ are included in the present asymptotic expressions.

The strain tensor $\varepsilon _{ij} (\rho ,\phi ,\theta )$ including $\varepsilon _{\rho }$, $\varepsilon _{\phi }$, $\varepsilon _{\theta }$, $\varepsilon _{\rho \phi }$, $\varepsilon _{\rho \theta }$ and $\varepsilon _{\phi \theta }$ components with respect to the displacement components in Eq. (A1) can then be obtained from the strain–displacement relations in local coordinates $\rho$, $\phi$ and $\theta$, and is expressed as follows:

$$\begin{aligned} \varepsilon _{ij} (\rho ,\phi ,\theta )=\hbox {Re}\left\{ {\sum \limits _{n=1}^{N+M} {\rho ^{\lambda _{n} }\varepsilon _{ij}^{(n)} (\phi ,\theta )\beta ^{(n)}} } \right\} =\hbox {Re}\left\{ {\sum \limits _{n=1}^{N+M} {\varepsilon _{ij}^{(n)} (\rho ,\phi ,\theta )\beta ^{(n)}} } \right\} , \end{aligned}$$

(A2)

where $\varepsilon _{ij}^{(n)} (\phi ,\theta )$ is a tensor containing the angular variations of the strain components. The stress tensor $\sigma _{ij} (\rho ,\phi ,\theta )$ with components $\sigma _{\rho }$, $\sigma _{\phi }$, $\sigma _{\theta }$, $\tau _{\rho \phi }$, $\tau _{\rho \theta }$ and $\tau _{\phi \theta }$ can be expressed by the strain tensor $\varepsilon _{ij} (\rho ,\phi ,\theta )$ according to the constitutive relation, i.e.,

$$\begin{aligned} \sigma _{ij} (\rho ,\phi ,\theta )=D_{ijkl} \hbox {Re}\left\{ {\sum \limits _{n=1}^{N+M} {\rho ^{\lambda _{n} }\varepsilon _{kl}^{(n)} (\phi ,\theta )\beta ^{(n)}} } \right\} =D_{ijkl} \hbox {Re}\left\{ {\sum \limits _{n=1}^{N+M} {\varepsilon _{kl}^{(n)} (\rho ,\phi ,\theta )\beta ^{(n)}} } \right\} , \end{aligned}$$

(A3)

in which $D_{ijkl}$ is the stiffness tensor of a linear elastic material.

A simple FEM for a 3D sectorial domain (Sze and Wang [29]) can be used to solve the 3D asymptotic singular stress fields in the vicinity of an interface corner herein. The eigenpair (i.e., the order of stress singularity $\lambda _{n}$ and the corresponding angular variation functions) in Eqs. (A1)–(A3) can be solved numerically using this FEM.

When $n\le N$, $\lambda _{n}$ and $\beta _{n}$ are both complex numbers, and $u_{i}^{(n)} (\rho ,\phi ,\theta )$ and $\varepsilon _{ij}^{(n)} (\rho ,\phi ,\theta )$ are both the tensors of displacement and strain including complex numbers. In order to obtain the real variable expressions of the displacement, strain and stress, we first define a matrix containing intermediate variables as follows:

$$\begin{aligned}&{{\hat{{\varvec{u}}}}}_{i}^{(n)} (\rho ,\phi ,\theta )=\left[ {{\begin{array}{cc} {\hbox {Re}\left\{ {u_{i}^{(n)} (\rho ,\phi ,\theta )} \right\} }&{}\quad {\hbox {Im}\left\{ {u_{i}^{(n)} (\rho ,\phi ,\theta )} \right\} }\\ \end{array} }} \right] , (n\le N), \end{aligned}$$

(A4)

$$\begin{aligned}&{{\hat{{{\varvec{\varepsilon }} }}}}_{i}^{(n)} (\rho ,\phi ,\theta )=\left[ {{\begin{array}{cc} {\hbox {Re}\left\{ {\varepsilon _{i}^{(n)} (\rho ,\phi ,\theta )} \right\} } &{}\quad {\hbox {Im}\left\{ {\varepsilon _{i}^{(n)} (\rho ,\phi ,\theta )} \right\} } \\ \end{array} }} \right] , (n\le N), \end{aligned}$$

(A5)

$$\begin{aligned}&{{\hat{{{\varvec{\beta }} }}}}^{(n)}=\left[ {{\begin{array}{cc} {\hbox {Re}\{\beta ^{(n)}\}} &{} \quad {\hbox {Im}\{\beta ^{(n)}\}} \\ \end{array} }} \right] ^{\mathrm{T}}, (n\le N), \end{aligned}$$

(A6)

where Im$\{\}$ denotes the imaginary part of a complex number.

Next, we further establish two matrixes and a vector in terms of ${{\hat{{{\varvec{u}}}}}}_{i}^{(n)},$ ${{\hat{{{\varvec{\varepsilon }} }}}}_{ij}^{(n)}$ and ${{\hat{{{\varvec{\beta }}}}}}^{(n)},$ respectively, as follows:

$$\begin{aligned} {{{\varvec{U}}}}_{i}= & {} \left[ {{{\hat{{{\varvec{u}}}}}}_{i}^{(1)} (\rho ,\phi ,\theta )\;\cdots {{ \hat{{{\varvec{u}}}}}}_{i}^{(N)} (\rho ,\phi ,\theta )\;u_{i}^{(N+1)} (\rho ,\phi ,\theta )\cdots u_{i}^{(N+M)} (\rho ,\phi ,\theta )} \right] , \end{aligned}$$

(A7)

$$\begin{aligned} {{{{\varvec{E}}}}}_{ij}= & {} \left[ {{{\hat{{{\varvec{\varepsilon }} }}}}_{ij}^{(1)} (\rho ,\phi ,\theta )\;\cdots \;{{\hat{{{\varvec{\varepsilon }}}}}} _{ij}^{(N)} (\rho ,\phi ,\theta )\;\varepsilon _{ij}^{(N+1)} (\rho ,\phi ,\theta )\cdots \varepsilon _{ij}^{(N+M)} (\rho ,\phi ,\theta )} \right] ,\end{aligned}$$

(A8)

$$\begin{aligned} {{{\varvec{\beta }}}}= & {} \left[ {{{\hat{{{\varvec{\beta }} }}}}^{(1)\mathrm{T}}\;\cdot \cdot \cdot {{\hat{{{\varvec{\beta }} }}}}^{(N)\text{ T }}\;\beta ^{(N+1)\text{ T }}\;\cdots \beta ^{(N+M)\text{ T }}} \right] ^{T}. \end{aligned}$$

(A9)

Thus, the real variable expression of displacement and stress in Eqs. (A1) and (A3) can be rewritten as:

$$\begin{aligned} {{{\varvec{u}}}}(\rho ,\phi ,\theta )= & {} {{{{\varvec{U}}}}{\varvec{\beta }}}, \end{aligned}$$

(A10)

$$\begin{aligned} {{{\varvec{\sigma }} }}(\rho ,\phi ,\theta )= & {} {{{\varvec{DE}}}{\varvec{\beta }}}, \end{aligned}$$

(A11)

in which the vectors ${{{\varvec{u}}}}(\rho ,\phi ,\theta )$ and ${{ {\varvec{\sigma }} }}(\rho ,\phi ,\theta )$, and the matrixes ${{{\varvec{U}}}}$ and ${{{\varvec{E}}}}$ are defined as

$$\begin{aligned} {{{\varvec{u}}}}(\rho ,\phi ,\theta )= & {} \left\{ {u_{\rho } ,u_{\phi } ,u_{\theta } } \right\} ^{\mathrm{T}}, \end{aligned}$$

(A12)

$$\begin{aligned} {{{\varvec{\sigma }} }}(\rho ,\phi ,\theta )= & {} \left\{ {\sigma _{\rho } ,\sigma _{\phi } ,\sigma _{\theta } ,\sigma _{\rho \phi } ,\sigma _{\rho \theta } ,\sigma _{\phi \theta } } \right\} ^{\mathrm{T}}, \end{aligned}$$

(A13)

$$\begin{aligned} {{{\varvec{U}}}}= & {} \left\{ {{{{\varvec{U}}}}_{\rho } ,\;{{{\varvec{U}}}}_{\phi } ,\;{{{\varvec{U}}}}_{\theta } } \right\} ^{\mathrm{T}}, \end{aligned}$$

(A14)

$$\begin{aligned} {{{\varvec{E}}}}= & {} \left\{ {{{{\varvec{E}}}}_{\rho } ,\;{{{\varvec{E}}}}_{\phi } ,\;{{{\varvec{E}}}}_{\theta } ,\;{{{\varvec{E}}}}_{\rho \phi } ,\;{{{\varvec{E}}}}_{\rho \theta } ,\;{{{\varvec{E}}}}_{\phi \theta } } \right\} ^{\mathrm{T}}, \end{aligned}$$

(A15)

and ${{{\varvec{D}}}}$ is the stiffness matrix of a linear elastic material.

The displacement ${{{\varvec{u}}}}(x,y,z)$ and stress ${{{\varvec{\sigma }} }}(x,y,z)$ in the Cartesian coordinate system (x, y, z) can be directly represented by those in the local coordinate system ($\rho , \phi , \theta )$ through the coordinate transformation (refer to Fig. 2a)

$$\begin{aligned} {{{\varvec{u}}}}(x,y,z)= & {} {{{\varvec{T}}}}_{\mathrm{u}} {{{\varvec{u}}}}(\rho ,\phi ,\theta )={{{\varvec{T}}}}_{\mathrm{u}} {{{\varvec{U}}{\varvec{\beta }}}}, \end{aligned}$$

(A16)

$$\begin{aligned} {{{\varvec{\sigma }}}}(x,y,z)= & {} {{{\varvec{DT}}}}_{\mathrm{s}} {{{\varvec{\varepsilon }} }}(\rho ,\phi ,\theta )={{{\varvec{DT}}}}_{\mathrm{s}} \mathrm{{{\varvec{E}}{\varvec{\beta }} }}, \end{aligned}$$

(A17)

in which ${{{\varvec{u}}}}(x,y,z)$ contains the components $u_{x}, u_{y}$ and $u_{\mathrm {z}}$, and ${{{\varvec{\sigma }}}}(x,y,z)$ includes the components $\sigma _{x}$, $\sigma _{y}$, $\sigma _{z}$, $\tau _{xy}$, $\sigma _{xz}$ and $\sigma _{yz}$. ${{{\varvec{T}}}}_{\mathrm{u}}$ and ${{{\varvec{T}}}}_{\mathrm{s}}$ are the coordinate transformation matrixes for the displacement and strain vectors from the local coordinate system ($\rho , \phi , \theta$) to the Cartesian coordinate system (x, y, z). Numerical solutions of U and E in Eqs. (A16) and (A17) can be obtained by using the FEM of Sze and Wang [29], i.e., they can be discretized using a natural coordinate $\xi$ with respect to the angular coordinate $\phi$. The unknown stress intensity parameters, ${{{\varvec{\beta }}}}$, need to be solved according to the external loads and boundary conditions.