﻿ 静态渐近解在动态断裂问题中的适用性分析
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 哈尔滨工程大学学报  2020, Vol. 41 Issue (6): 824-831  DOI: 10.11990/jheu.201903081 0

### 引用本文

CHEN He, ZOU Guangping. Discussion on the applicability of static asymptotic solutions in dynamic fracture[J]. Journal of Harbin Engineering University, 2020, 41(6): 824-831. DOI: 10.11990/jheu.201903081.

### 文章历史

Discussion on the applicability of static asymptotic solutions in dynamic fracture
CHEN He , ZOU Guangping
College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: Dynamic stress intensity factor-test methods, such as the strain gauge method, are mostly based on the static asymptotic solution of a strain field. However, the application of an asymptotic solution in the dynamic fracture is limited. In this work, the simulation of a modified compact tension shear specimen loaded by the split-Hopkinson tension bar is carried out. The difference between the numerical solution and asymptotic solution of a strain field near the crack tip is investigated to identify the proper location of strain gauges in dynamic fracture testing. The numerical solution of the strain agrees well with the asymptotic solution in the limited region and largely varies in other regions. In these regions, the errors caused by the solution of the dynamic stress intensity factor and the strain components are of the same order. In summary, the static asymptotic solution cannot be directly applied in the dynamic fracture without a detailed analysis of certain problems.
Keywords: dynamic fracture    static asymptotic solution    dynamic stress intensity factor    strain gauge method    numerical simulation    asymptatic solution    compact tension shear specimen loaded

1 裂尖应力应变场渐近解

 $\left\{ \begin{array}{l} {\sigma _{ij}} = \frac{1}{{\sqrt {2{\rm \pi} r} }}\begin{array}{*{20}{l}} {[{K_{\rm I}}(t)f_{ij}^{\rm I}(\theta ) + {K_{{\rm I}{\rm I}}}(t)f_{ij}^{{\rm I}{\rm I}}(\theta )]} \end{array}\\ {\varepsilon _{ij}} = \frac{1}{{\sqrt {2{\rm \pi} r} }}\begin{array}{*{20}{l}} {[{K_{\rm I}}(t)g_{ij}^{\rm I}(\theta ) + {K_{{\rm I}{\rm I}}}(t)g_{ij}^{{\rm I}{\rm I}}(\theta )]} \end{array} \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} f_{11}^{\rm I} = \cos \frac{\theta }{2}\left( {1 - \sin \frac{\theta }{2}\sin \frac{{3\theta }}{2}} \right)\\ f_{11}^{{\rm I}{\rm I}} = - \sin \frac{\theta }{2}\left( {2 + \cos \frac{\theta }{2}\cos \frac{{3\theta }}{2}} \right)\\ f_{22}^{\rm I} = \cos \frac{\theta }{2}\left( {1 + \sin \frac{\theta }{2}\sin \frac{{3\theta }}{2}} \right)\\ f_{22}^{{\rm I}{\rm I}} = \sin \frac{\theta }{2}\cos \frac{\theta }{2}\cos \frac{{3\theta }}{2} \end{array} \right.$ (2)
 $\left\{ \begin{array}{l} g_{11}^{\rm I} = \frac{1}{E}\cos \frac{\theta }{2}\left( {1 - \nu - (1 + \nu )\sin \frac{\theta }{2}\sin \frac{{3\theta }}{2}} \right)\\ g_{11}^{{\rm I}{\rm I}} = - \frac{1}{E}\sin \frac{\theta }{2}\left( {2 + (1 + \nu )\cos \frac{\theta }{2}\cos \frac{{3\theta }}{2}} \right)\\ g_{22}^{\rm I} = \frac{1}{E}\cos \frac{\theta }{2}\left( {1 - \nu + (1 + \nu )\sin \frac{\theta }{2}\sin \frac{{3\theta }}{2}} \right)\\ g_{22}^{{\rm I}{\rm I}} = \frac{1}{E}\sin \frac{\theta }{2}\left[ {2\nu + (1 + \nu )\cos \frac{\theta }{2}\cos \frac{{3\theta }}{2}} \right] \end{array} \right.$ (3)

Sun[18]指出，裂纹尖端区域应力具有奇异性，应力对坐标的导数也是奇异的，而位移是有限的，因此平衡方程左边远远大于右边，基于上述假设，在裂尖附近将静态问题的渐近解应用于动态问题是合理的；而在距离裂尖较远处渐近解不适用。文献[18]虽然给出定性的解释，但并未提及裂尖“附近”的具体范围。

2 改进的紧凑拉伸剪切试样数值模拟

 Download: 图 1 改进的紧凑拉伸剪切试样几何尺寸 Fig. 1 Geometry of MCTS specimen
 Download: 图 2 改进的紧凑拉伸剪切试样装配图 Fig. 2 Assembly of MCTS specimen

 Download: 图 3 SHTB入射波 Fig. 3 Incident wave of SHTB
 Download: 图 4 改进的紧凑拉伸剪切试样裂纹尖端网格划分 Fig. 4 Mesh at crack tip of MCTS specimen
3 动态应力强度因子计算

 $\left\{ \begin{array}{l} {K_{\rm I}} = \mathop {\lim }\limits_{r \to 0} \sqrt {2{\rm \pi} r} {\left. {{\sigma _{22}}} \right|_{\theta = 0}}\\ {K_{{\rm I}{\rm I}}} = \mathop {\lim }\limits_{r \to 0} \sqrt {2{\rm \pi} r} {\left. {{\sigma _{12}}} \right|_{\theta = 0}} \end{array} \right.$ (4)

 Download: 图 5 裂纹尖端塑性区 Fig. 5 Plastic zone at crack tip

 Download: 图 6 动态应力强度因子数值解 Fig. 6 Numerical solution of DSIF
4 裂尖应变场分析

 $\left[ \begin{array}{l} \varepsilon _{11}^{(1)}\\ \varepsilon _{22}^{(2)} \end{array} \right] = \left[ \begin{array}{l} \frac{{g_{11}^{\rm I}({\theta _1})}}{{\sqrt {2{\rm \pi} {r_1}} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{g_{11}^{{\rm I}{\rm I}}({\theta _1})}}{{\sqrt {2{\rm \pi} {r_1}} }}\\ \frac{{g_{22}^{\rm I}({\theta _2})}}{{\sqrt {2{\rm \pi} {r_2}} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{g_{22}^{{\rm I}{\rm I}}({\theta _2})}}{{\sqrt {2{\rm \pi} {r_2}} }} \end{array} \right]\left[ \begin{array}{l} {K_{\rm I}}\\ {K_{{\rm I}{\rm I}}} \end{array} \right]$ (5)

Rittel[21]提出一种双应变片求解动态应力强度因子的实验方法，其中r1θ1r2θ2分别为应变片1、2在裂尖局部坐标系内的坐标，反解式(5)即可求出动态应力强度因子。本文将通过式(5)计算的值称为渐近解，通过有限元分析得到的解称为数值解，二者之间的关系如图 7所示。可以看出，若节点应变渐近解与数值解不相同，则动态应力强度因子渐近解与数值解也会存在误差。本节在不同方向上选取节点，分析其应变渐近解与数值解之间的差异。

 Download: 图 7 应变分析流程 Fig. 7 Flow graph of strain analysis

ε11ε22θ=0°及θ=±22.5°方向上相对误差平均值小于20%。ε11误差最小的节点位于θ=0°，r=2.52 mm处，编号为1 023；除该节点外，ε22误差最小的节点位于θ=-22.5°，r=3.02 mm处，编号为1 051。其应变曲线分别如图 8(a)(b)所示。将节点1 023的应变分量ε11，节点1 051的应变分量ε22代入式(5)反解出动态应力强度因子，并与数值解对比。如图 8(c)所示。可见K渐近解与数值解误差很小，而K误差很大。由于K数值解约等于零，且系数矩阵存在零元素，故K渐近解明显偏离数值解。

 Download: 图 8 Ⅰ型载荷工况节点应变及动态应力强度因子渐近解与数值解对比 Fig. 8 Nodal strain and comparison of asymptotic solution and numerical solution of DSIF for mode Ⅰ loading

 Download: 图 9 Ⅱ型载荷工况节点应变及DSIF渐近解与数值解对比 Fig. 9 Nodal strain and comparison of asymptotic solution and numerical solution of DSIF for mode Ⅱ loading

 Download: 图 10 应变理想区 Fig. 10 Suitable zone of strain solution

5 结论

1) 裂尖应变场的解析解在特定方向上与数值解较为接近(Δε≤10%)，而在其他方向上有较大差异。对于I型载荷工况，误差较小的方向是0°附近；对于II型载荷工况则是±45°附近。

2) DSIF解析解与数值解的相对误差取决于节点应变的相对误差，2者在相同数量级。

3) 静态渐近解在动态断裂问题中的适用范围非常有限，且与试样载荷工况有关。应针对具体问题分别讨论。

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