A dynamic coupling model of peridynamics and finite elements for progressive damage analysis

Abstract

This study presents a new coupling model (CM) of finite element (FE) and peridynamics (PD), in which only very few PD nodes exist. In the coupling model, PD subregion is directly coupled with FE subregion without an overlapped zone, and the force is transferred between PD nodes and finite elements by a connection stiffness matrix. Since dynamic transformation technique is implemented, PD subregion is adaptively generated and evolved, and ensure that the whole damage process is completed. In addition, as an optimization of the coupling model, a densified-material-point model (DMPM) is achieved, which can remove the limitation of element type and enhance the flexibility of the coupling model. As a result, the computational efficiency of coupling algorithms will be greatly improved, and numerical error can be overcome in inferring the damage region. The capability of the developed coupling model was demonstrated by the stretch examples of plates with different discrete cases, and damage analysis was further conducted to demonstrate the strong capability of the DMPM in capturing failure mode.

Appendix

The element stiffness matrix for quadrilateral elements can be expressed as

$$ \left[ {\mathbf{K}} \right] = \int_{ - 1}^{1} {\int_{ - 1}^{1} {\left[ {\mathbf{B}} \right]^{T} \left[ {\mathbf{D}} \right]\left[ {\mathbf{B}} \right]t\left| {\mathbf{J}} \right|d\xi d\eta } } $$

(66)

where $\left[ {\mathbf{B}} \right]$ is the strain matrix, $\left[ {\mathbf{D}} \right]$ is the elastic matrix, $\left[ {\mathbf{J}} \right]$ is the Jacobian matrix, and t is the thickness of the plate. $(\xi ,\eta )$ is the local coordinate, as shown in Fig. 

Fig. 36

Quadrilateral elements in different coordinate systems. a Global coordinate system; b local coordinate system

36.

$$ \left[ {\mathbf{B}} \right] = \left[ {\begin{array}{*{20}c} {B_{1} } & {B_{2} } & {B_{3} } & {B_{4} } \\ \end{array} } \right] $$

(67)

Each sub-block of the strain matrix and the elastic matrix can be written as

$$ \left[ {B_{i} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial x}} & 0 \\ 0 & {\frac{{\partial N_{i} }}{\partial y}} \\ {\frac{{\partial N_{i} }}{\partial y}} & {\frac{{\partial N_{i} }}{\partial x}} \\ \end{array} } \right];\,\left[ {\text{D}} \right] = \left[ {\begin{array}{*{20}c} {\frac{E}{{1 - v^{2} }}} & {\frac{vE}{{1 - v^{2} }}} & 0 \\ {\frac{vE}{{1 - v^{2} }}} & {\frac{E}{{1 - v^{2} }}} & 0 \\ 0 & 0 & {\frac{E}{2(1 + v)}} \\ \end{array} } \right] $$

(68)

where the determinant of $\left[ {\mathbf{J}} \right]$ can be expressed as

$$ \left| J \right| = \left| {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{4} {\frac{{\partial N_{i} }}{\partial \xi }x_{i} } } & {\sum\limits_{i = 1}^{4} {\frac{{\partial N_{i} }}{\partial \xi }y_{i} } } \\ {\sum\limits_{i = 1}^{4} {\frac{{\partial N_{i} }}{\partial \eta }x_{i} } } & {\sum\limits_{i = 1}^{4} {\frac{{\partial N_{i} }}{\partial \eta }y_{i} } } \\ \end{array} } \right| $$

(69)

where $N_{i}$ represents the shape function and its expression can be obtained as

$$ N_{i} \left( {\xi ,\eta } \right) = \frac{{\left( {1 + \xi_{i} \xi } \right)\left( {1 + \eta_{i} \eta } \right)}}{4}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {i = 1,2,3,4} \right) $$

(70)

where $(x_{i} ,y_{i} )$ and $(\xi_{i} ,\eta_{i} )$ represent the coordinate of element node i in reference to the global coordinate system and the local coordinate system, respectively.

$$ \begin{array}{*{20}l} {\left( {\xi_{1} ,\,\eta_{1} } \right) = \left( { - 1,\,1} \right)} \hfill \\ {\left( {\xi_{3} ,\,\eta_{3} } \right) = \left( {1,\, - 1} \right)} \hfill \\ \end{array} ,\,\begin{array}{*{20}l} {\left( {\xi_{2} ,\,\eta_{2} } \right) = \left( { - 1,\, - 1} \right)} \hfill \\ {\left( {\xi_{4} ,\,\eta_{4} } \right) = \left( {1,\,1} \right)} \hfill \\ \end{array} $$

(71)