﻿ Peridynamic含初始缺陷中心平行双裂纹扩展分析
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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)
 Download: 图 1 近场动力学质点的运动 Fig. 1 The movement of peridynamic particles

 ${w_{\left( k \right)\left( j \right)}} = {w_{\left( k \right)\left( j \right)}}\left( {{y_{\left( {{1^k}} \right)}} - {y_{\left( k \right)}},{y_{\left( {{2^k}} \right)}} - {y_{\left( k \right)}}, \cdots } \right)$ (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 3 无缺陷的双裂纹 Fig. 3 Double cracks without defects
 Download: 图 4 无缺陷双裂纹扩展路径 Fig. 4 Double crack propagation path without defects

 Download: 图 5 随机初始缺陷分布 Fig. 5 Random initial defect distribution
 Download: 图 6 含随机缺陷双裂纹扩展路径 Fig. 6 Double crack propagation path with random defects
4 结论

1) 无缺陷的中心双平行裂纹在外荷载作用下，随着双裂纹纵向间距的变化，裂纹扩展路径也随之改变。

2) 裂纹间距小于3 mm时，双裂纹融合为一条水平裂纹；当平行裂纹间距增大后，裂纹扩展为两条间距不断变大的裂纹，且初始裂纹间距越大，裂纹尖端就越早改变扩展路径的方向。

3) 含随机缺陷(气孔、夹杂、未熔合)的不同纵向间距的平行双裂纹扩展路径与无缺陷裂纹扩展路径对比可知，初始缺陷不仅改变了中心双裂纹扩展路径，同时对中心双裂纹扩展也有一定的抑制作用。

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