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 哈尔滨工程大学学报  2018, Vol. 39 Issue (10): 1612-1616  DOI: 10.11990/jheu.201704077 0

### 引用本文

ZHAO Jinhai, TANG Hesheng, XUE Songtao. Analysis of the center parallel double-crack propagation with initial casting defects by peridynamics[J]. Journal of Harbin Engineering University, 2018, 39(10), 1612-1616. DOI: 10.11990/jheu.201704077.

### 文章历史

Peridynamic含初始缺陷中心平行双裂纹扩展分析

1. 同济大学 土木工程学院, 上海 200092;
2. 同济大学 土木工程防灾国家重点实验室, 上海 200092

Analysis of the center parallel double-crack propagation with initial casting defects by peridynamics
ZHAO Jinhai1, TANG Hesheng1,2, XUE Songtao1
1. College of Civil Engineering, Tongji University, Shanghai 200092, China;
2. State Key Laboratory of Disaster Prevention in Civil Engineering, Tongji University, Shanghai 200092, China
Abstract: To better simulate the crack propagation, a non-local peridynamics (PD) theory was proposed. The PD method uses the integral method to avoid the problem of traditional methods and uses the distance between adjacent particles and the critical elongation change to describe the force variation law. It simulates the behavior of crack propagation and multi-crack fusion better than the traditional method. In this paper, we first simulate defect-free centrally parallel double-crack propagation paths with different initial distances by PD and then reduce the strength of the bonds between the particles to simulate the centrally parallel double-crack propagation path with random distribution initial defects (porosity, inclusion, and incomplete fusion). Finally, we obtain the influence of initial defects on the fusion and deviation path of the center parallel double crack propagation with different longitudinal spacings.
Keywords: non-local theory    peridynamic    initial defect    crack propagation    center parallel double crack

1 近场动力学理论

 ${\mathit{\boldsymbol{\xi }}_{\left( k \right)\left( j \right)}} = {x_{\left( j \right)}} - {x_{\left( k \right)}}$ (1)

 $\left( {{y_{\left( j \right)}} - {y_{\left( k \right)}}} \right) = \mathit{\boldsymbol{\underline Y}} \left( {{x_{\left( k \right)}},t} \right) = \left[ {\begin{array}{*{20}{c}} {{y_{\left( 1 \right)}} - {y_{\left( k \right)}}}\\ {{y_{\left( 2 \right)}} - {y_{\left( k \right)}}}\\ \vdots \\ {{y_{\left( \infty \right)}} - {y_{\left( k \right)}}} \end{array}} \right]$ (2)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{t}}_{\left( k \right)\left( j \right)}}\left( {{\mathit{\boldsymbol{u}}_{\left( j \right)}} - {\mathit{\boldsymbol{u}}_{\left( k \right)}},{x_{\left( j \right)}} - {x_{\left( k \right)}},t} \right) = }\\ {\mathit{\boldsymbol{\underline T}} \left( {{x_{\left( k \right)}},t} \right)\left\langle {{x_{\left( j \right)}} - {x_{\left( k \right)}}} \right\rangle } \end{array}$ (3)

 $\begin{array}{l} {s_{\left( k \right)\left( j \right)}} = s\left( {{\mathit{\boldsymbol{u}}_{\left( j \right)}} - {\mathit{\boldsymbol{u}}_{\left( k \right)}},{x_{\left( j \right)}} - {x_{\left( k \right)}}} \right) = \\ \frac{{\left( {\left| {{y_{\left( j \right)}} - {y_{\left( k \right)}}} \right| - \left| {{x_{\left( j \right)}} - {x_{\left( k \right)}}} \right|} \right)}}{{\left| {{x_{\left( j \right)}} - {x_{\left( k \right)}}} \right|}} \end{array}$ (4)

 ${w_{\left( k \right)\left( j \right)}} = {w_{\left( k \right)\left( j \right)}}\left( {{y_{\left( {{1^k}} \right)}} - {y_{\left( k \right)}},{y_{\left( {{2^k}} \right)}} - {y_{\left( k \right)}}, \cdots } \right)$ (5)

 $\begin{array}{l} {W_{\left( k \right)}} = \frac{1}{2}\sum\limits_{j = 1}^\infty {\left( {\frac{1}{2}{w_{\left( k \right)\left( j \right)}}\left( {{y_{\left( {{1^k}} \right)}} - {y_{\left( k \right)}},{y_{\left( {{2^k}} \right)}} - {y_{\left( k \right)}}, \cdots } \right) + } \right.} \\ \left. {\frac{1}{2}{w_{\left( j \right)\left( k \right)}}\left( {{y_{\left( {{1^j}} \right)}} - {y_{\left( j \right)}},{y_{\left( {{2^j}} \right)}} - {y_{\left( j \right)}}, \cdots } \right)} \right) \cdot {V_{\left( j \right)}} \end{array}$ (6)

 $\begin{array}{*{20}{c}} {{\rho _{\left( k \right)}}{{\mathit{\boldsymbol{\ddot u}}}_{\left( k \right)}} = \sum\limits_{j = 1}^\infty {\left[ {{\mathit{\boldsymbol{t}}_{\left( k \right)\left( j \right)}}\left( {{\mathit{\boldsymbol{u}}_{\left( j \right)}} - {\mathit{\boldsymbol{u}}_{\left( k \right)}},{x_{\left( j \right)}} - {x_{\left( k \right)}},t} \right) - } \right.} }\\ {\left. {{\mathit{\boldsymbol{t}}_{\left( j \right)\left( k \right)}}\left( {{\mathit{\boldsymbol{u}}_{\left( k \right)}} - {\mathit{\boldsymbol{u}}_{\left( j \right)}},{x_{\left( k \right)}} - {x_{\left( j \right)}},t} \right)} \right]{V_{\left( j \right)}} + {\mathit{\boldsymbol{b}}_{\left( k \right)}}} \end{array}$ (7)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{t}}_{\left( k \right)\left( j \right)}}\left( {{\mathit{\boldsymbol{u}}_{\left( j \right)}} - {\mathit{\boldsymbol{u}}_{\left( k \right)}},{x_{\left( j \right)}} - {x_{\left( k \right)}},t} \right) = }\\ {\frac{1}{2} \cdot \frac{1}{{{V_{\left( k \right)}}}}\left( {\sum\limits_{i = 1}^\infty {\frac{{\partial {w_{\left( k \right)\left( i \right)}}}}{{\partial \left( {{y_{\left( j \right)}} - {y_{\left( k \right)}}} \right)}}{V_{\left( i \right)}}} } \right)} \end{array}$ (8)
 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{t}}_{\left( j \right)\left( k \right)}}\left( {{\mathit{\boldsymbol{u}}_{\left( k \right)}} - {\mathit{\boldsymbol{u}}_{\left( j \right)}},{x_{\left( k \right)}} - {x_{\left( j \right)}},t} \right) = }\\ {\frac{1}{2} \cdot \frac{1}{{{V_{\left( j \right)}}}}\left( {\sum\limits_{i = 1}^\infty {\frac{{\partial {w_{\left( i \right)\left( k \right)}}}}{{\partial \left( {{y_{\left( k \right)}} - {y_{\left( j \right)}}} \right)}}{V_{\left( i \right)}}} } \right)} \end{array}$ (9)

 $\begin{array}{*{20}{c}} {{\rho _{\left( k \right)}}{{\mathit{\boldsymbol{\ddot u}}}_{\left( k \right)}} = \int_H {\mathit{\boldsymbol{t}}\left( {\mathit{\boldsymbol{u'}} - \mathit{\boldsymbol{u}},x' - x,t} \right)} - }\\ {\mathit{\boldsymbol{t'}}\left( {\mathit{\boldsymbol{u}} - \mathit{\boldsymbol{u'}},x - x',t} \right){\rm{d}}H + \mathit{\boldsymbol{b}}\left( {x,t} \right)} \end{array}$ (10)
2 近场动力学物体损伤理论

 $\mu \left( {{x_{\left( j \right)}} - {x_{\left( k \right)}},t} \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 1,\\ 0, \end{array}&\begin{array}{l} {s_{\left( k \right)\left( j \right)}} < {s_c}\\ 其他 \end{array} \end{array}} \right.$ (11)

 $\varphi \left( {x,t} \right) = 1 - \frac{{\int\limits_H {\mu \left( {{x_{\left( j \right)}} - {x_{\left( k \right)}},t} \right){\rm{d}}V'} }}{{\int\limits_H {{\rm{d}}V'} }}$ (12)

3 含缺陷中心平行双裂纹扩展分析

 Download: 图 2 二维板质点位置分布 Fig. 2 The particles position distribution in 2D plate

 Download: 图 4 无缺陷双裂纹扩展路径 Fig. 4 Double crack propagation path without defects