The extended finite element method with novel crack-tip enrichment functions for dynamic fracture analysis of interfacial cracks in piezoelectric–piezomagnetic bi-layered structures

Abstract

This paper investigates the dynamic fracture problems of interfacial cracks in piezoelectric–piezomagnetic (PE–PM) bi-layered composite structures under in-plane coupled electro-magneto-mechanical impact loadings by means of the extended finite element method (X-FEM). Considering the magnetoelectrically impermeable crack-face conditions and multi-filed coupled properties in the PE–PM composites, novel and more suitable crack-tip enrichment functions for interfacial cracks in PE–PM bi-layered composite structures are newly derived and implemented in the X-FEM, where the Newmark method is applied and proved to be effective. As the fracture parameter, the J-integral is evaluated using the domain-form of the path-independent contour integral. For dynamic analysis of interfacial cracks in infinite PE–PM bi-layered composite structures, absorbing layers based on the Sarma absorbing boundary conditions are adopted and applied to avoid the unphysical wave reflections at the artificially introduced boundaries in the X-FEM meshes. In the numerical examples, the validity of the proposed scheme is verified by comparing the numerical solutions provided by the X-FEM with either analytical results obtained by solving the corresponding singular integral equations or possible stationary values obtained by introducing the corresponding absorbing layers. Finally, by the numerical examples, the effects of the applied dynamic loadings, time variable and structural geometries on the dynamic J-integral are analyzed and discussed in detail. Some important conclusions are drawn, which should be helpful for the design and applications of the PE–PM layered composite structures.

Appendix: Crack-tip fields in PE–PM bimaterials

According to the Stroh’s formalism, the general solutions for the 2D problem of the linear magnetoelectroelasticity can be expressed as [20, 27]

$$ {\mathbf{u}}^{\left( m \right)} = \left[ {\begin{array}{*{20}c} {u_{1}^{\left( m \right)} } & {u_{3}^{\left( m \right)} } & {\varphi^{\left( m \right)} } & {\phi^{\left( m \right)} } \\ \end{array} } \right]^{\text{T}} = {\mathbf{A}}^{\left( m \right)} {\mathbf{g}}^{\left( m \right)} \left( {z_{s} } \right) + \overline{{{\mathbf{A}}^{\left( m \right)} {\mathbf{g}}^{\left( m \right)} \left( {z_{s} } \right)}} , $$

(A.1a)

$$ {\mathbf{t}}^{\left( m \right)} = \left[ {\begin{array}{*{20}c} {\sigma_{31}^{\left( m \right)} } & {\sigma_{33}^{\left( m \right)} } & {D_{3}^{\left( m \right)} } & {B_{3}^{\left( m \right)} } \\ \end{array} } \right]^{\text{T}} = {\mathbf{B}}^{\left( m \right)} {\mathbf{g}}^{\prime \left( m \right)} \left( {z_{s} } \right) + \overline{{{\mathbf{A}}^{\left( m \right)} {\mathbf{g}}^{\prime \left( m \right)} \left( {z_{s} } \right)}} , $$

(A.1b)

where ${\mathbf{g}}\left( {z_{s} } \right)$ are the unknown complex functions, ${\mathbf{g^{\prime}}}\left( {z_{s} } \right)$ are the derivative of ${\mathbf{g}}\left( {z_{s} } \right)$ with respect to $z_{s} = x_{1} + p_{s} x_{3} \, \left( {s = 1,\;2,\;3,\;4} \right)$ with $p_{s}$ being the Stroh’s eigenvalues with positive imaginary parts, an over bar denotes the complex conjugate, and ${\mathbf{A}} = \left[ {\begin{array}{*{20}c} {{\mathbf{a}}_{1} } & {{\mathbf{a}}_{2} } & {{\mathbf{a}}_{3} } & {{\mathbf{a}}_{4} } \\ \end{array} } \right]$ is a complex matrix with ${\mathbf{a}}_{s}$ being the Stroh’s eigenvectors. The Stroh’s eigenvalues and eigenvectors $p_{s}$ and ${\mathbf{a}}_{s}$ are determined by the eigen-equations $\left[ {{\mathbf{Q}} + p\left( {{\mathbf{R}} + {\mathbf{R}}^{\text{T}} } \right) + p^{2} {\mathbf{T}}} \right]{\mathbf{a}} = 0$,where

$$ {\mathbf{Q}}^{\left( m \right)} = \left[ {\begin{array}{*{20}c} {c_{11}^{\left( m \right)} } & 0 & 0 & 0 \\ 0 & {c_{44}^{\left( m \right)} } & {\delta_{1m} e_{15} } & {\delta_{2m} h_{15} } \\ 0 & {\delta_{1m} e_{15} } & { - \,\alpha_{11}^{\left( m \right)} } & 0 \\ 0 & {\delta_{2m} h_{15} } & 0 & { - \,\mu_{11}^{\left( m \right)} } \\ \end{array} } \right], $$

(A.2a)

$$ \, {\mathbf{R}}^{\left( m \right)} = \left[ {\begin{array}{*{20}c} 0 & {c_{13}^{\left( m \right)} } & {\delta_{1m} e_{31} } & {\delta_{2m} h_{31} } \\ {c_{44}^{\left( m \right)} } & 0 & 0 & 0 \\ {\delta_{1m} e_{15} } & 0 & 0 & 0 \\ {\delta_{2m} h_{15} } & 0 & 0 & 0 \\ \end{array} } \right], $$

(A.2b)

$$ {\mathbf{T}}^{\left( m \right)} = \left[ {\begin{array}{*{20}c} {c_{44}^{\left( m \right)} } & 0 & 0 & 0 \\ 0 & {c_{33}^{\left( m \right)} } & {\delta_{1m} e_{33} } & {\delta_{2m} h_{33} } \\ 0 & {\delta_{1m} e_{33} } & { - \,\alpha_{33}^{\left( m \right)} } & 0 \\ 0 & {\delta_{2m} h_{33} } & 0 & { - \,\mu_{33}^{\left( m \right)} } \\ \end{array} } \right], $$

(A.2c)

and ${\mathbf{B}} = {\mathbf{R}}^{\text{T}} {\mathbf{A}} + {\mathbf{TAP}}, \, {\mathbf{P}} = {\text{diag}}\left[ {p_{1} , \, p_{2} , \, p_{3} , \, p_{4} } \right]$.

With the assumption of the traction-free and magnetoelectrically impermeable crack-face boundary conditions, the continuity and boundary conditions at the interface are given by [20]

$$ {\mathbf{u}}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = {\mathbf{u}}^{\left( 2 \right)} \left( {x_{1} ,0} \right), \, {\mathbf{t}}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = {\mathbf{t}}^{\left( 2 \right)} \left( {x_{1} ,0} \right), \, x_{1} \notin \Gamma_{c} , $$

(A.3a)

$$ {\mathbf{t}}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = {\mathbf{t}}^{\left( 2 \right)} \left( {x_{1} ,0} \right) = 0, \, x_{1} \in \Gamma_{c} . $$

(A.3b)

According to Ref. [28], the unknown complex functions ${\mathbf{g}}^{\left( m \right)}$ can be taken as ${\mathbf{g}}^{\left( m \right)} \left( {z_{s}^{\left( m \right)} } \right) = {{\left\langle {\left( {z_{s}^{\left( m \right)} } \right)^{1 + \delta } } \right\rangle {\mathbf{q}}^{\left( m \right)} } \mathord{\left/ {\vphantom {{\left\langle {\left( {z_{s}^{\left( m \right)} } \right)^{1 + \delta } } \right\rangle {\mathbf{q}}^{\left( m \right)} } {\left( {1 + \delta } \right)}}} \right. \kern-0pt} {\left( {1 + \delta } \right)}}$, where $\langle \rangle$ indicates the diagonal matrix with each component varying with $s$. Using the expression for ${\mathbf{g}}^{\left( m \right)}$ and $z_{s}^{\left( m \right)} = r\left( {\cos \theta + p_{s}^{\left( m \right)} \sin \theta } \right)$, Eq. (A.1) can be rewritten as

$$ \begin{aligned} {\mathbf{u}}^{\left( m \right)} &= r^{{\left( {1 + \delta }\right)}} \left[ {\mathbf{A}}^{\left( m \right)} \left\langle{\left( {\cos \theta + p_{s}^{\left( m \right)} \sin \theta }\right)^{1 + \delta } } \right\rangle {\mathbf{q}}^{\left( m\right)} \right.\\ &\quad\left.+\, {\bar{\mathbf{A}}}^{\left( m \right)} \left\langle{\left( {\cos \theta + \bar{p}_{s}^{\left( m \right)} \sin \theta }\right)^{1 + \delta } } \right\rangle {\bar{\mathbf{q}}}^{\left( m\right)} \right] \mathord{\left/ {\vphantom {{r^{{\left( {1 +\delta } \right)}} \left[ {{\mathbf{A}}^{\left( m \right)}\left\langle {\left( {\cos \theta + p_{s}^{\left( m \right)} \sin\theta } \right)^{1 + \delta } } \right\rangle {\mathbf{q}}^{\left(m \right)} + {\bar{\mathbf{A}}}^{\left( m \right)} \left\langle{\left( {\cos \theta + \bar{p}_{s}^{\left( m \right)} \sin \theta }\right)^{1 + \delta } } \right\rangle {\bar{\mathbf{q}}}^{\left( m\right)} } \right]} {\left( {1 + \delta } \right)}}} \right.\kern-0pt} {\left( {1 + \delta } \right)},\end{aligned} $$

(A.4a)

$$\begin{aligned} {\mathbf{t}}^{\left( m \right)} &= r^{\delta } \left[ {\mathbf{B}}^{\left( m \right)} \left\langle {\left( {\cos \theta + p_{s}^{\left( m \right)} \sin \theta } \right)^{\delta } } \right\rangle {\mathbf{q}}^{\left( m \right)} \right. \\ &\left.\quad+ \,{\bar{\mathbf{B}}}^{\left( m \right)} \left\langle {\left( {\cos \theta + \bar{p}_{s}^{\left( m \right)} \sin \theta } \right)^{\delta } } \right\rangle {\bar{\mathbf{q}}}^{\left( m \right)} \right],\end{aligned} $$

(A.4b)

where $\delta$ is the singularity parameter to represent the singularity of the generalized stresses at the crack-tip. Substituting Eq. (A.4) into Eq. (A.3) and considering the following relation

$$ z_{s} = \left\{ {\begin{array}{*{20}l} {r{\text{e}}^{{ \pm {\text{i}}\pi }} ,} \hfill & {\theta = \pm \pi ,} \hfill \\ {r,} \hfill & {\theta = 0,} \hfill \\ \end{array} } \right. $$

(A.5)

we obtain [28]

$$ \left[ {{\mathbf{H}} + {\text{e}}^{{2\pi \delta {\text{i}}}} {\bar{\mathbf{H}}}} \right]{\mathbf{B}}^{\left( 1 \right)} {\mathbf{q}}^{\left( 1 \right)} = 0, $$

(A.6)

in which ${\mathbf{H}} = {\mathbf{Y}}^{\left( 1 \right)} + {\bar{\mathbf{Y}}}^{\left( 2 \right)} = {\mathbf{D}} + {\text{i}}{\mathbf{W}}$, and ${\mathbf{Y}}^{\left( m \right)} = {\text{i}}{\mathbf{A}}^{\left( m \right)} \left( {{\mathbf{B}}^{\left( m \right)} } \right)^{ - 1}$.

In order to ensure a nontrivial solution for Eq. (A.6), $\left| {{\mathbf{H}} + {\text{e}}^{{2\pi \delta {\text{i}}}} {\bar{\mathbf{H}}}} \right| = 0$ should be satisfied. By assuming $\delta = - \,1/2 + {\text{i}}\varepsilon$, the following equation can be obtained [27,28,29]

$$ \left| {{\bar{\mathbf{H}}} - e^{2\pi \varepsilon } {\mathbf{H}}} \right| = \left| {{\mathbf{W}} + {\text{i}}\beta {\mathbf{D}}} \right| = 0, $$

(A.7)

where $\left| {\varvec{\Delta}} \right|$ is the determinant of matrix ${\varvec{\Delta}}$, $\varepsilon$ is the oscillating index and $\beta = {{\left( {1 - {\text{e}}^{2\pi \varepsilon } } \right)} \mathord{\left/ {\vphantom {{\left( {1 - {\text{e}}^{2\pi \varepsilon } } \right)} {\left( {1 + {\text{e}}^{2\pi \varepsilon } } \right)}}} \right. \kern-0pt} {\left( {1 + {\text{e}}^{2\pi \varepsilon } } \right)}}$ or $\varepsilon = {{ - \left( {\tanh^{ - 1} \beta } \right)} \mathord{\left/ {\vphantom {{ - \left( {\tanh^{ - 1} \beta } \right)} \pi }} \right. \kern-0pt} \pi }$.

Since ${\mathbf{H}}$ is a Hermitian matrix, if $\beta$ and $\bar{\beta }$ are the roots of Eq. (A.7), then $- \beta$ and $- \bar{\beta }$ are also the roots of Eq. (A.7). Therefore, Eq. (A.7) should have the form

$$ \left( {{\text{i}}\beta } \right)^{4} + 2c_{2} \left( {{\text{i}}\beta } \right)^{2} + c_{4} = 0, $$

(A.8)

where $c_{2} = - \,\frac{1}{4}{\text{tr}}\left( {{\mathbf{D}}^{ - 1} {\mathbf{W}}} \right)^{2} ,\;c_{4} = \left| {{\mathbf{D}}^{ - 1} {\mathbf{W}}} \right|$. Finally, the roots of Eq. (A.8) can be expressed as follows

$$ \beta_{1,2} = \pm \sqrt {c_{2} + \sqrt {\left( {c_{2} } \right)^{2} - c_{4} } } , \, \beta_{3,4} = \pm \sqrt {c_{2} - \sqrt {\left( {c_{2} } \right)^{2} - c_{4} } } . $$

(A.9)

In order to analyze the crack-tip singularity described by $\delta$, it is required to investigate the values of $\beta_{s}$. With a similar analysis as in the previous studies [27,28,29], we obtain for transversely isotropic PE–PM bimaterials that $\beta_{1} = - \,\beta_{2}$ are two real numbers and $\beta_{3} = \beta_{4} = 0$. Thus, the crack-tip singularity in this paper can be expressed as $\delta_{1,2} = - 1/2 \pm {\text{i}}\varepsilon_{1}$ and $\delta_{3,4} = - \,1/2$, where $\varepsilon_{1} = {{ - \,\left( {\tanh^{ - 1} \beta_{1} } \right)} \mathord{\left/ {\vphantom {{ - \,\left( {\tanh^{ - 1} \beta_{1} } \right)} \pi }} \right. \kern-0pt} \pi }$.