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 浙江大学学报(工学版)  2017, Vol. 51 Issue (7): 1324-1330  DOI:10.3785/j.issn.1008-973X.2017.07.008 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2017.07.008
[复制中文]
YU Yang-tian, ZHANG Qing, GU Xin. Hybrid model of peridynamics and finite element method under implicit schemes[J]. Journal of Zhejiang University(Engineering Science), 2017, 51(7): 1324-1330.
dx.doi.org/10.3785/j.issn.1008-973X.2017.07.008
[复制英文]

### 作者简介

orcid/org/0000-0002-6397-4158.
Email: yuyangtian@hhu.edu.cn

### 通信联系人

orcid/org/0000-0002-2819-5976.
Email: lxzhangqing@hhu.edu.cn

### 文章历史

Hybrid model of peridynamics and finite element method under implicit schemes
YU Yang-tian , ZHANG Qing , GU Xin
Department of Engineering Mechanics, Hohai University, Nanjing 210098, China
Abstract: A hybrid model of peridynamics (PD) and finite element method (FEM) was proposed and applied to solve problems of fracture mechanics in order to combine the unique advantage of PD in solving discontinuities and the computational efficiency of FEM. The improved prototype microelastic brittle (PMB) model of peridynamics was utilized for the regions where material failure was expected. The region without failure was discretized by FEM. The truss element was introduced to bridge peridynamic subregions and finite element subregions. The hybrid model is based on the implicit schemes, and it need not consider a fictitious damping term in solving static problems. The computational efficiency and accuracy of the model were improved. The static elastic deformation of a simply supported beam and the propagation process of mode Ⅰ fracture in a three points bend beam were simulated to verify the accuracy and utility of the presented model. Results obtained by the model agreed well with the theoretical solutions.
Key words: peridynamics    finite element method (FEM)    hybrid model    improved prototype microelastic brittle model    implicit scheme

1 理论简介 1.1 近场动力学方法

 图 1 物质点间的相互作用 Fig. 1 Interaction between material points
 $\begin{array}{l} \rho \mathit{\boldsymbol{\ddot u}}\left( {\mathit{\boldsymbol{x}},t} \right) = \int_{{{\rm{H}}_x}} {f\left( {\mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x'}},\mathit{\boldsymbol{t}}} \right) - \mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{t}}} \right),\mathit{\boldsymbol{x'}} - \mathit{\boldsymbol{x}}} \right){\rm{d}}{V_{x'}}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;b\left( {\mathit{\boldsymbol{x}},t} \right). \end{array}$ (1)

 $f\left( {\mathit{\boldsymbol{\eta }},\mathit{\boldsymbol{\xi }}} \right) = \frac{{\mathit{\boldsymbol{\xi + \eta }}}}{{\left\| {\mathit{\boldsymbol{\xi + \eta }}} \right\|}}\mathit{\boldsymbol{c}}\left( {\mathit{\boldsymbol{\xi }},\delta } \right)s\mu \left( {t,\mathit{\boldsymbol{\xi }}} \right).$ (2)

 $s = \frac{{\left\| {\mathit{\boldsymbol{\xi + \eta }}} \right\| - \left\| \mathit{\boldsymbol{\xi }} \right\|}}{{\left\| \mathit{\boldsymbol{\xi }} \right\|}};$ (3)

μ为一标量函数,

 $\mu \left( {t,\mathit{\boldsymbol{\xi }}} \right) = \left\{ \begin{array}{l} 1,\;\;\;\;\;s\left( {t',\mathit{\boldsymbol{\xi }}} \right) < {s_0},0 \le t' \le t;\\ 0,\;\;\;\;其他. \end{array} \right.$ (4)

 ${s_0} = \frac{{{f_{\rm{t}}}}}{E}.$ (5)

s超过s0时, 点对间不再发生作用.

 $\varphi \left( {\mathit{\boldsymbol{x}},t} \right) = 1 - \frac{{\int_H {\mu \left( {\mathit{\boldsymbol{x}},t,\mathit{\boldsymbol{\xi }}} \right){\rm{d}}{V_\xi }} }}{{\int_H {{\rm{d}}{V_\xi }} }}.$ (6)

1.2 改进的PMB模型

 $c\left( {\mathit{\boldsymbol{\xi }},\delta } \right) = c\left( {0,\delta } \right)g\left( {\mathit{\boldsymbol{\xi }},\delta } \right).$ (7)

 $g\left( {\mathit{\boldsymbol{\xi }},\delta } \right) = \left\{ \begin{array}{l} 1,\;\;\;\;\left\| \mathit{\boldsymbol{\xi }} \right\| \le \delta ;\\ 0,\;\;\;\;\left\| \mathit{\boldsymbol{\xi }} \right\| > \delta . \end{array} \right.$ (8)

 $g\left( {\mathit{\boldsymbol{\xi }},\delta } \right) = \left\{ \begin{array}{l} {\left( {1 - {{\left( {\frac{{\left\| \mathit{\boldsymbol{\xi }} \right\|}}{\delta }} \right)}^2}} \right)^2},\;\;\;\;\;\;\left\| \mathit{\boldsymbol{\xi }} \right\| \le \delta ;\\ \;\;\;\;\;\;\;\;\;0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left\| \mathit{\boldsymbol{\xi }} \right\| > \delta . \end{array} \right.$ (9)

 $c\left( {0,\delta } \right) = \left\{ \begin{array}{l} \frac{{72E}}{{{\rm{ \mathsf{ π} }}{\delta ^4}}},\;\;\;\;\;\;\;三维;\\ \frac{{315E}}{{{\rm{8 \mathsf{ π} }}{\delta ^3}}},\;\;\;\;\;\;平面应力;\\ \frac{{210E}}{{{\rm{5 \mathsf{ π} }}{\delta ^3}}},\;\;\;\;\;\;平面应变. \end{array} \right.$ (10)
1.3 有限元方程

 $\mathit{\boldsymbol{Ku = F}}.$ (11)

 $\left. \begin{array}{l} \mathit{\boldsymbol{K = }}\sum\limits_e {\mathit{\boldsymbol{C}}_e^{\rm{T}}\mathit{\boldsymbol{k}}{\mathit{\boldsymbol{C}}_e}} ,\\ \mathit{\boldsymbol{F = }}\sum\limits_e {\mathit{\boldsymbol{C}}_e^{\rm{T}}{\mathit{\boldsymbol{F}}^e}} . \end{array} \right\}$ (12)

2 数值计算格式 2.1 近场动力学隐式求解格式

 $\rho \mathit{\boldsymbol{\ddot u}}_i^n = \sum\limits_j {\mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{u}}_j^n - \mathit{\boldsymbol{u}}_i^n,{\mathit{\boldsymbol{x}}_j} - {\mathit{\boldsymbol{x}}_i}} \right){V_j} + \mathit{\boldsymbol{b}}_i^n} .$ (13)

 $\sum\limits_j {\mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{u}}_j^n - \mathit{\boldsymbol{u}}_i^n,{\mathit{\boldsymbol{x}}_j} - {\mathit{\boldsymbol{x}}_i}} \right){V_j} + \mathit{\boldsymbol{b}}_i^n} = 0.$ (14)

 $\mathit{\boldsymbol{f}} = {\mathit{\boldsymbol{K}}_{\rm{P}}}\mathit{\boldsymbol{u}} \cdot \mu \left( s \right).$ (15)

 图 2 局部坐标系下物质点对间的相互作用 Fig. 2 Interactions between material points in local coordinates

 $\begin{array}{l} \mathit{\boldsymbol{f'}} = {\left[ {{f_{x'i}},\;\;\;{f_{y'i}},\;\;{f_{z'i}},\;\;{f_{x'j}},\;\;\;{f_{y'j}},\;\;{f_{z'j}}} \right]^{\rm{T}}},\\ \mathit{\boldsymbol{u' = }}{\left[ {{{u'}_i},\;\;{{v'}_i},\;\;{{w'}_i},\;\;{{u'}_j},\;\;{{v'}_j},\;\;{{w'}_j}} \right]^{\rm{T}}}. \end{array}$ (16)

 $\mathit{\boldsymbol{f'}} = {{\mathit{\boldsymbol{K'}}}_{\rm{P}}}\mathit{\boldsymbol{u'}} \cdot \mu \left( s \right).$ (17)

 ${{\mathit{\boldsymbol{K'}}}_{\rm{P}}} = \frac{c}{{\left\| \mathit{\boldsymbol{\xi }} \right\|}}\left[ {\begin{array}{*{20}{c}} 1&0&0&{ - 1}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ { - 1}&0&0&1&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0 \end{array}} \right],$ (18)

 图 3 整体坐标系下的物质点对 Fig. 3 Material points in global coordinates

 $\left. \begin{array}{l} l = \cos \alpha = \frac{{{x_j} - {x_i}}}{{\left\| \mathit{\boldsymbol{\xi }} \right\|}},\\ m = \cos \beta = \frac{{{y_j} - {y_i}}}{{\left\| \mathit{\boldsymbol{\xi }} \right\|}},\\ n = \cos \gamma = \frac{{{z_j} - {z_i}}}{{\left\| \mathit{\boldsymbol{\xi }} \right\|}}. \end{array} \right\}$ (19)

 ${\mathit{\boldsymbol{K}}_{\rm{P}}} = \frac{c}{{\left\| \mathit{\boldsymbol{\xi }} \right\|}}\left[ {\begin{array}{*{20}{c}} {{l^2}}&{}&{}&{}&{}&{}\\ {lm}&{{m^2}}&{}&{}&{{\rm{sym}}}&{}\\ {\mathit{ln}}&{mn}&{{n^2}}&{}&{}&{}\\ { - {l^2}}&{ - lm}&{ - \mathit{ln}}&{{l^2}}&{}&{}\\ { - lm}&{ - {m^2}}&{ - mn}&{lm}&{{m^2}}&{}\\ { - \mathit{ln}}&{ - mn}&{ - {n^2}}&{\mathit{ln}}&{mn}&{{n^2}} \end{array}} \right].$ (20)

 $\sum\limits_j {{\mathit{\boldsymbol{K}}_{\rm{P}}}\left( {\left| {{\mathit{\boldsymbol{x}}_j} - {\mathit{\boldsymbol{x}}_i}} \right|} \right)\left( {\mathit{\boldsymbol{u}}_j^n - \mathit{\boldsymbol{u}}_i^n} \right){V_j} + \mathit{\boldsymbol{b}}_i^n = {\bf{0}}.}$ (21)

2.2 近场动力学与有限元的混合模型

 图 4 有限元(FE)和近场动力学(PD)子区域 Fig. 4 Finite element (FE) and peridynamic (PD) regions

 图 5 结点与物质点连接示意图 Fig. 5 Interactions between FE nodes and PD points through trusses

 $k\left( {\xi ',\delta } \right) = \left\{ \begin{array}{l} \frac{{72E}}{{{\rm{ \mathsf{ π} }}{\delta ^4}}}{\left( {1 - {{\left( {\frac{{\xi '}}{\delta }} \right)}^2}} \right)^2},\;\;\;\;\;\;\;三维;\\ \frac{{315E}}{{{\rm{8 \mathsf{ π} }}{\delta ^3}}}{\left( {1 - {{\left( {\frac{{\xi '}}{\delta }} \right)}^2}} \right)^2},\;\;\;\;\;\;平面应力;\\ \frac{{210E}}{{{\rm{5 \mathsf{ π} }}{\delta ^3}}}{\left( {1 - {{\left( {\frac{{\xi '}}{\delta }} \right)}^2}} \right)^2},\;\;\;\;\;\;平面应变. \end{array} \right.$ (22)

 图 6 PD与FEM混合建模计算流程图 Fig. 6 Flowchart for hybrid model of PD and FEM
3 算例分析 3.1 简支梁的弹性变形

 图 7 简支梁模型示意图 Fig. 7 Geometry of specimen

 图 8 中性轴竖向位移曲线 Fig. 8 Vertical displacement curves of neutral axis

3.2 含初始裂缝三点弯曲梁的破坏分析

 图 9 三点弯曲梁几何尺寸 Fig. 9 Geometry of specimen

 ${\rm{CMOD = }}\frac{{4\sigma a}}{E} \cdot {V_1}\left( \alpha \right).$ (23)

 $\begin{array}{*{20}{c}} {{V_1}\left( \alpha \right) = 0.76 - 2.28\alpha + 3.87{\alpha ^2} - 2.04{\alpha ^3} + \frac{{0.66}}{{{{\left( {1 - \alpha } \right)}^2}}},}\\ {\alpha = a/D,\sigma = \frac{{3PL}}{{2{D^2}}}.} \end{array}$

 图 10 荷载-CMOD曲线 Fig. 10 Curves of load-CMOD
 图 11 梁破坏过程的模拟结果 Fig. 11 Failure process simulation of beam
4 结语