﻿ 功能梯度材料动态断裂力学的径向积分边界元法
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 力学学报  2015, Vol. 47 Issue (5): 868-873  DOI: 10.6052/0459-1879-15-150 0

### 引用本文 [复制中英文]

[复制中文]
Gao Xiaowei, Zheng Baojing, Liu Jian. DYNAMIC FRACTURE ANALYSIS OF FUNCTIONALLY GRADED MATERIALS BY RADIAL INTEGRATION BEM[J]. Chinese Journal of Ship Research, 2015, 47(5): 868-873. DOI: 10.6052/0459-1879-15-150.
[复制英文]

### 文章历史

2015-04-29收稿
2015-06-24录用
2015–07–02网络版发表

1 功能梯度材料弹性动力学问题的边界-域积分方程

 ${\sigma _{ij,j}} + {f_i} = \rho {\ddot u_i}$ (1)
 ${\sigma _{ij}} = {c_{ijkl}}{u_{k,l}}$ (2)
 图 1 含有裂纹的非均质弹性体 Fig.1 A non-homogeneous solid with a crack

 ${c_{ijkl}} = \mu (x)c_{ijkl}^0$ (3)

 ${u_i} = {\bar u_i}{\mkern 1mu} ,\;在边界\Gamma {}_u上$ (4a)
 ${t_i} = {\sigma _{ij}}{n_j} = {\bar t_i}{\mkern 1mu} ,在边界{\Gamma _t}上$ (4b)

 $u(x,{t_0}) = {u_0}(x){\mkern 1mu} ,\;\;x \in \Omega$ (4c)
 $\mathop u\limits^. (x,{t_0}) = {v_0}(x){\mkern 1mu} ,\;\;x \in \Omega$ (4d)

 $\int_\Omega {{U_{ij}}({\sigma _{jk,k}} - \rho {{\ddot u}_j})} d\Omega = 0$ (5)

 $\begin{array}{l} \int_\gamma {{U_{ij}}{\sigma _{jk}}{n_k}d\Gamma } - \int_\gamma {{U_{ij,k}}c_{jkrs}^0{n_s}\mu {u_r}d\Omega } + \\ \int_\Omega {{U_{ij,ks}}c_{jkrs}^0\mu {u_r}d\Omega } + \int_\Omega {{U_{ij,k}}c_{jkrs}^0{\mu _{,s}}{u_r}d\Omega } - \\ \int_\Omega {\rho {U_{ij}}{{\ddot u}_j}d\Omega } = 0 \end{array}$ (6)

 ${U_{ij,ks}}c_{jkrs}^0 + {\delta _{ir}}\delta (x,{x^p}) = 0$ (7)

 $\begin{array}{*{20}{l}} {{{\tilde u}_i}({x^p}) = \int_\gamma {{U_{ij}}(x,{x^p}){t_j}(x)d\Gamma } - }\\ {\int_\gamma {{T_{ij}}(x,{x^p}){{\tilde u}_j}(x)d\Gamma } + }\\ {\int_\Omega {{V_{ij}}(x,{x^p}){{\tilde u}_j}(x)d\Omega } - }\\ {\int_\Omega {\frac{\rho }{{\mu (x)}}{U_{ij}}(x,{x^p}){{\tilde u}_j}(x)d\Omega } } \end{array}$ (8)

 $\begin{array}{l} {V_{ij}} = {U_{il,k}}c_{lkjs}^0{{\tilde \mu }_{,s}} = \\ \frac{{ - 1}}{{4\pi \alpha (1 - v){r^\alpha }}}\{ {{\tilde \mu }_{,k}}{r_{,k}}\left[{(1 - 2v){\delta _{ij}} + \beta {r_{,i}}{r_{,j}}} \right] + \\ (1 - 2v)({{\tilde \mu }_{,i}}{r_{,j}} - {{\tilde \mu }_{,j}}{r_{,i}})\} \end{array}$ (9)

2 域积分转化为边界积分

 $u(x) = \sum\limits_A {\alpha _i^A{\phi ^A}(R)} + a_i^k{x_k} + a_i^0$ (10)

 $u(x) = \Phi (x)\left\{ u \right\} = \sum\limits_{i = 1}^n {{\phi _i}{u_i}}$ (11)

 $\begin{array}{l} c{{\tilde u}_i}({x^p}) = \int_\gamma {{U_{ij}}(x,{x^p}){t_j}(x)d\Gamma } - \\ \int_\gamma {{T_{ij}}(x,{x^p}){{\tilde u}_j}(x)d\Gamma } + \\ \{ {{\tilde u}_j}\} \int_\gamma {\frac{1}{{{r^\alpha }}}\frac{{\partial r}}{{\partial n}}} F_{ij}^1d\Gamma - \\ \{ {{\tilde u}_j}\} \int_\gamma {\frac{1}{{{r^\alpha }}}\frac{{\partial r}}{{\partial n}}F_{ij}^2d\Gamma } \end{array}$ (12)

 $F_{ij}^1 = \int_0^{r({x^p},{x^q})} {{V_{ij}}(x,{x^p})\Phi {r^\alpha }dr}$ (13a)
 $F_{ij}^2 = \int_0^{r({x^p},{x^q})} {\frac{\rho }{{\mu (x)}}{U_{ij}}(x,{x^p})\Phi {r^\alpha }dr}$ (13b)

3 数值实施

 ${H_b} \cdot {\widetilde u_b} = {G_b} \cdot {t_b} + {V_b} \cdot {\widetilde u_t} - {M_b} \cdot {\mathop {\widetilde u}\limits^{..} _t}$ (14)
 $I \cdot {\widetilde u_i} + {H_i} \cdot {\widetilde u_i} = {G_i} \cdot {t_b} + {V_i} \cdot {\widetilde u_t} - {M_i} \cdot {\mathop {\widetilde u}\limits^{..} _t}$ (15)

 $\begin{array}{l} \left( {\left[{\begin{array}{*{20}{c}} {{H_b}}&{\bf{0}}\\ {{H_i}}&I \end{array}} \right] - \left[{\begin{array}{*{20}{c}} \begin{array}{l} {V_b}\\ {V_i} \end{array} \end{array}} \right]} \right)\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\widetilde u_b}\\ {\widetilde u_i} \end{array} \end{array}} \right\} = \\ \left[{\begin{array}{*{20}{c}} \begin{array}{l} {G_b}\\ {G_i} \end{array} \end{array}} \right]\left\{ {{t_b}} \right\} - \left[{\begin{array}{*{20}{c}} \begin{array}{l} {M_b}\\ {M_i} \end{array} \end{array}} \right]\left\{ {{{\mathop {\widetilde u}\limits^{..} }_t}} \right\} \end{array}$ (16)

 $M\mathop x\limits^{..} + Kx = F$ (17)

 ${\mathop x\limits^{..} _{t + \Delta t}} = \frac{{2{x_{t + \Delta t}} - 5{x_t} + 4{x_{t - \Delta t}} - {x_{t - 2\Delta t}}}}{{\Delta {t^2}}}$ (18)

 ${\mathop x\limits^. _{t + \Delta t}} = \frac{{{x_{t + \Delta t}} - {x_t}}}{{\Delta t}}$ (19)

 $\begin{array}{l} \left( {\frac{2}{{\Delta {t^2}}}M + K} \right){x_{t + \Delta t}} = \\ \frac{5}{{\Delta {t^2}}}{x_t} + \frac{1}{{\Delta {t^2}}}M\left( { - 4{x_{t - \Delta t}} + {x_{t - 2\Delta t}}} \right) + {F_{t + \Delta t}} \end{array}$ (20)
4 应力强度因子计算

 $\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {K_{\rm{I}}}\\ {K_{{\rm{II}}}}\\ {K_{{\rm{III}}}} \end{array} \end{array}} \right\} = \mathop {\lim }\limits_{r \to 0} \sqrt {\frac{{2\pi }}{r}} \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {H^{({\rm{I}})}}\Delta {u_2}(r)\\ {H^{({\rm{II}})}}\Delta {u_1}(r)\\ {H^{({\rm{III}})}}\Delta {u_3}(r) \end{array} \end{array}} \right\}$ (21)

 $\kappa = \left\{ \begin{array}{l} 3 - 4v{\mkern 1mu} ,平面应变\\ \frac{{3 - v}}{{1 + v}},{\mkern 1mu} 平面应力 \end{array} \right.$ (22)

 $\Delta {u_i}(r) = {u_i}(x \in \Gamma _{\rm{c}}^ + ) - {u_i}(x \in \Gamma _{\rm{c}}^ - )$ (23)
5 算例与分析

 图 2 含中心直裂纹的功能梯度材料平板 Fig.2 FGM plate with a vertical central crack

 图 3 规则动态应力强度因子 Fig.3 Regular dynamic stress intensity factor

 图 4 含中心斜裂纹的功能梯度材料平板 Fig.4 FGM plate with an inclined central crack
 图 5 规则化动态应力强度因子 Fig.5 Regular dynamic stress intensity factor
6 结 论