X-FEM simulation for two-unequal-collinear cracks in 2-D finite piezoelectric specimen

Abstract

A study for two-unequal-collinear cracks in a 2-D finite piezoelectric specimen is carried out using a new set of six crack-tip enrichment functions proposed here for piezoelectric media in the X-FEM framework. The intensity factors and energy release rate are calculated using interaction integral in conjugation with the near tip behavior given by the Stroh formalism. Effect of finite size of the specimen is analyzed with respect to offset distances of the cracks from the specimen boundaries. ERR variations are investigated with respect to inter- crack space, crack lengths and electrical/mechanical loadings. Hence, two-unequal-collinear cracks in an infinite domain problem is simulated, analyzed and validated using X-FEM. Further, ERR at the crack tips for the asymmetric case of two collinear equal and unequal cracks, is also computed. It is concluded through this investigation that the proposed enrichment functions could be used to handle the problems of fracture mechanics in 2-D piezoelectric media within a good accuracy.

Appendix 1

Assuming the plain strain conditions, the constitutive equations can be written as

$$ \left\{\begin{array}{l} \varepsilon_{xx} \\ \varepsilon_{yy}\\ 2 \varepsilon_{xy} \end{array} \right\} = \left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]\left\{\begin{array}{l} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy}\\ \end{array} \right\} + \left[\begin{array}{ll} b_{11} & b_{21}\\ b_{12} & b_{22} \\ b_{13} & b_{23} \end{array}\right] \left\{ \begin{array}{l} D_{x} \\ D_{y} \\ \end{array} \right\}$$

(45)

$$ \left\{\begin{array}{l} E_{x} \\ E_{y} \end{array} \right\} = - \left[\begin{array}{lll} b_{11} &b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{array} \right]\left\{\begin{array}{ll} \sigma_{xx}\\ \sigma_{yy} \\ \sigma_{xy} \end{array} \right\} + \left[\begin{array}{ll} d_{11} &d_{12} \\ d_{12} & d_{22} \end{array} \right]\left\{\begin{array}{l} D_{x} \\ D_{y} \end{array}\right\} $$

(46)

where coefficients $a_{ij},\,b_{ij} \,{\text{and}}\,d_{ij}$ are reduced elastic, piezoelectric and dielectric constants, respectively defined by Sosa (1992).

Now using extended Lekhnitskii’s formalism to piezoelectric solids, the following potential function representation is introduced

$$ \begin{aligned} \sigma_{xx} & = \frac{{\partial^{2}F}}{{\partial y^{2} }};\;\sigma_{yy} = \frac{{\partial^{2}F}}{{\partial x^{2} }};\sigma_{xy} = - \frac{{\partial^{2}F}}{\partial x\partial y};\\ D_{x} & = \frac{\partial \psi}{\partial y};\;D_{y} = - \frac{\partial \psi }{\partial x}\\ \end{aligned} $$

(47)

where $F\left( {x,y} \right)\,{\text{and}}\,\,\psi \left( {\text{x,y}} \right)$ are complex potential functions.

It can be shown that the equilibrium equations are automatically satisfied by Eq. 47. Using the strain and electric field compatibility equations, the following sixth order differential equation can be derived for $F\left( {x,y} \right)$

$$ D_{1} D_{2} D_{3} D_{4} D_{5} D_{6} F = 0 $$

(48)

where $D_{n} = \left( {\frac{\partial }{\partial y} - a_{n} \frac{\partial }{\partial x}} \right),\,\,{\text{and}}\;a_{n} \left( {n = 1, \ldots,6} \right)$ are the roots of the characteristic equation

$$ P\left( {a_{t} } \right) = l_{1} \left( {a_{t} } \right)l_{3} \left( {a_{t} } \right) + l_{2}^{2} \left( {a_{t} } \right) = 0 $$

(49)

and

$$ \begin{aligned} l_{1} \left( {a_{t} } \right) & = d_{11} a_{t}^{2} - 2d_{12} a_{t} + d_{22} ; \hfill \\ l_{2} \left( {a_{t} } \right) & = b_{11} a_{t}^{3} - (b_{21} + b_{13} )a_{t}^{2} + \left( {b_{12} + b_{23} } \right)a_{t} - b_{22} ; \hfill \\ l_{3} \left( {a_{t} } \right) & = a_{11} a_{t}^{4} - 2a_{13} a_{t}^{3} + \left( {2a_{12} + a_{33} } \right)a_{t}^{2} - 2a_{23} a_{t} + a_{22} . \end{aligned} $$

(50)

The other complex variables defined in Sect. 4.3 are given below

$$ \begin{aligned} \delta_{n} & = \frac{{l_{2} \left( {a_{n} } \right)}}{{l_{1} \left( {a_{n} } \right)}}; \hfill \\ p_{n} & = a_{11} a_{n}^{2} + a_{12} - a_{13} a_{n} + \delta_{n} \left( {b_{11} a_{n} - b_{21} } \right); \hfill \\ q_{n} & = \left( {a_{12} a_{n}^{2} + a_{22} - a_{23} a_{n} + \delta_{n} b_{12} a_{n} - \delta_{n} b_{22} } \right)/a_{n}; \hfill \\ s_{n} & = b_{11} a_{n}^{2} + b_{12} - b_{13} a_{n} - \delta_{n} \left( {d_{11} a_{n} - d_{12} } \right); \hfill \\ t_{n} & = b_{21} a_{n}^{2} + b_{22} - b_{23} a_{n} - \delta_{n} \left( {d_{12} a_{n} - d_{22} } \right); \hfill \\ \end{aligned} $$

(51)

$$ \left[ {\Uplambda_{ij} } \right] = \frac{1}{\Updelta }\left[\begin{array} {lll}a_{2} \delta_{3} - a_{3} \delta_{2}& \delta_{2} - \delta_{3} & a_{3} - a_{2} \hfill \\ a_{3} \delta_{1} - a_{1} \delta_{3} & \delta_{3} - \delta_{1} & a_{1} -a_{3} \hfill \\ a_{1} \delta_{2} - a_{2} \delta_{1} & \delta_{1} - \delta_{2} & a_{2} - a_{1} \hfill \\ \end{array} \right], $$

(52)

where $\Updelta = a_{1} \left( {\delta_{2} - \delta_{3} } \right) + a_{2} \left( {\delta_{3} - \delta_{1} } \right) + a_{3} \left( {\delta_{1} - \delta_{2} } \right)$ and a ₁, a ₂ and a ₃ are the roots of the Eq. 49 with positive imaginary parts.