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 哈尔滨工程大学学报  2018, Vol. 39 Issue (4): 607-613  DOI: 10.11990/jheu.201612027 0

### 引用本文

XUE Yanzhuo, LU Xikui, WANG Qing, et al. Simulation of three-point bending test of ice based on peridynamic[J]. Journal of Harbin Engineering University, 2018, 39(4), 607-613. DOI: 10.11990/jheu.201612027.

### 文章历史

1. 哈尔滨工程大学 船舶工程学院, 黑龙江 哈尔滨 150001;
2. 大连理工大学 海岸和近海工程国家重点实验室, 辽宁 大连 116024

Simulation of three-point bending test of ice based on peridynamic
XUE Yanzhuo1, LU Xikui1, WANG Qing1, BAI Xiaolong1, LI Zhijun2
1. School of Ship Engineering, Harbin Engineering University, Harbin 150001, China;
2. State key laboratory coastal and offshore engineering, Dalian University of Technology, Dalian 116024, China
Abstract: To study the feasibility of Peridynamic simulating the process from deformation to damage of ice under applied force, the three-point bending test of ice is simulated. A elastic-brittle constitutive model of ice material is established and the Peridynamic numerical algorithms solving dynamic problem and quasi static problem are given by combining with the adaptive dynamic relaxation method. The results show that the Peridynamic method has advantages in the damage simulation, and is very accurate to solve the quasi static problem; the numerical simulation results of three-point bending test were compared with the experimental results, which showed that the relative error of deformation process is smaller than 5%, crack location and specimen fracture results are identical with the experimental results. The result indicates that Peridynamic is suitable for application on deformation and damage process of ice material.
Key words: ice    three-point bending test    deformation    damage    peridynamic    quasi static    adaptive dynamic relaxation    elastic-brittle constitutive model

1 近场动力学理论

 $\begin{array}{l} \rho \mathit{\boldsymbol{\ddot u}}\left( {x, t} \right) = \int_{{H_x}} {\mathit{\boldsymbol{f}}(\mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x}}\prime, t} \right)-} \\ \mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x}}, t} \right), \mathit{\boldsymbol{x}}\prime-\mathit{\boldsymbol{x}}{\rm{)d}}{V_{x\prime }} + \mathit{\boldsymbol{b}}\left( {\mathit{\boldsymbol{x}}, t} \right) \end{array}$ (1)
 Download: 图 1 近场范围内物质点的相互作用 Fig. 1 Interaction between material points in horizon

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

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

 $\mu \left( {t, \mathit{\boldsymbol{\xi }}, \mathit{\boldsymbol{\eta }}} \right) = \left\{ \begin{array}{l} 1, \;\;\;\;\;s < {s_0}\\ 0, \;\;\;\;\;s \ge {s_0} \end{array} \right.$ (4)

 $D\left( {\mathit{\boldsymbol{x}}, t} \right) = 1-\frac{{{\smallint _{{H_x}}}\mu \left( {\mathit{\boldsymbol{x}}, t, \mathit{\boldsymbol{\xi }}, \mathit{\boldsymbol{\eta }}} \right){\rm{d}}{V_\xi }}}{{{\smallint _{{H_x}}}{\rm{d}}{V_\xi }}}$ (5)

2 准静态问题的近场动力学数值计算方法

 $\rho \mathit{\boldsymbol{\ddot u}}_i^n = \sum\limits_p {\mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{u}}_p^n-\mathit{\boldsymbol{u}}_i^n, {\rm{ }}{\mathit{\boldsymbol{x}}_p}-{\mathit{\boldsymbol{x}}_i}} \right){V_p} + \mathit{\boldsymbol{b}}_i^n}$ (6)

 $\mathit{\boldsymbol{L}}_i^n = \sum\limits_p {\mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{u}}_p^n-\mathit{\boldsymbol{u}}_i^n, {\rm{ }}{\mathit{\boldsymbol{x}}_p}-{\mathit{\boldsymbol{x}}_i}} \right){V_p}}$ (7)

L=Ku表示内力的体积力密度，其中K表示等效刚度。方程(6)可简写为

 $\rho \mathit{\boldsymbol{\ddot u}}_i^n = \mathit{\boldsymbol{L}}_i^n + \mathit{\boldsymbol{b}}_i^n$ (8)

 $\mathit{\boldsymbol{ \boldsymbol{\varLambda} \ddot u}} + \mathit{\boldsymbol{C\ddot u}} + \mathit{\boldsymbol{Ku}} = \mathit{\boldsymbol{b}}$ (9)

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot u}}}^{1/2}} = \frac{{h{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}^{-1}}\left( {{\mathit{\boldsymbol{b}}^0}-\mathit{\boldsymbol{K}}{\mathit{\boldsymbol{u}}^0}} \right)}}{2} + \frac{{\left( {2-dh} \right){{\mathit{\boldsymbol{\dot u}}}^0}}}{2}\;\;\;\;\left( {n = 0} \right)\\ {{\mathit{\boldsymbol{\dot u}}}^{n + 1/2}} = \frac{{\left( {2 - dh} \right)}}{{\left( {2 + dh} \right)}}{{\mathit{\boldsymbol{\dot u}}}^{n - 1/2}} + \frac{{2h{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}^{ - 1}}\left( {{\mathit{\boldsymbol{b}}^n} - \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{u}}^n}} \right)}}{{\left( {2 + dh} \right)}}\;\;\;\;\left( {n \ne 0} \right)\\ \;\;\;\;\;\;\;{\mathit{\boldsymbol{u}}^{n + 1}} = {\mathit{\boldsymbol{u}}^n} + h{{\mathit{\boldsymbol{\dot u}}}^{n + 1/2}} \end{array}$ (10)

 $\varepsilon = \frac{{\frac{1}{2}{{\left( {{{\mathit{\boldsymbol{\dot u}}}^{n-1/2}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}{{\mathit{\boldsymbol{\dot u}}}^{n-1/2}}}}{{\frac{1}{2}{{\left( {{\mathit{\boldsymbol{u}}^n}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{L}}^n}}} \le {\varepsilon _{{\rm{tol}}}}$ (11)

 ${\lambda _{ii}} \ge \frac{{{h^2}}}{4}\sum\limits_{j = 1}^n {\left| {{K_{ij}}} \right|}$ (12)

 $\sum\limits_{j = 1}^n {\left| {{K_{ij}}} \right|} = \sum\limits_{j = 1} {\left| {\frac{{\partial \mathit{\boldsymbol{f}}\left( {\xi, \eta } \right){V_j}}}{{\partial \eta }}} \right|}$ (13)

 ${\lambda _{ii}} = \frac{{{h^2}}}{4}\frac{1}{{\left| {{\xi _{{\rm{min}}}}} \right|}}c\alpha {V_\delta }$ (14)

 ${d^n} = 2\sqrt {\left( {{{\left( {{\mathit{\boldsymbol{u}}^n}} \right)}^{{\rm{T1}}}}{\mathit{\boldsymbol{K}}^n}{\mathit{\boldsymbol{u}}^n}} \right)/\left( {{{\left( {{\mathit{\boldsymbol{u}}^n}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{u}}^n}} \right)}$ (15)

3 悬臂梁算例

3.1 计算模型

 Download: 图 2 受集中力的悬臂梁 Fig. 2 A fixed support beam under concentrated force

3.2 计算结果

 ${u_z}\left( L \right) =-F\left( {\frac{{{L^3}}}{{3{\rm{EI}}}} + \frac{L}{{\kappa GA}}} \right)$ (16)

3.3 动态算法模拟对比

3.4 阻尼系数对收敛性的影响

 Download: 图 5 悬臂梁自由端挠度与时间步数的关系 Fig. 5 Relationship between deflection and number of time steps

4 冰三点弯曲数值模拟 4.1 试验

 Download: 图 6 三点弯曲试验原理图 Fig. 6 Schematic diagram of three-point bending test

 $E = \frac{{P{L^3}}}{{4b{h^3}\delta }}$ (17)

4.2 数值模拟 4.2.1 冰的弹脆性近场动力学模型

 Download: 图 8 冰的性质与应变率的关系 Fig. 8 Relationship between properties of ice and strain rate

4.2.2 计算过程

4.2.3 计算结果

5 结论

1) 本文将自适应动态松弛法引入近场动力学算法，提出了一种高效率的准静态算法，并通过算例证明了算法的准确性与高效性。

2) 建立的冰材料的近场动力学模型，对冰的三点弯曲变形和裂纹扩展进行了数值模拟研究，结果与试验结果吻合。

3) 近场动力学方法能很好地模拟冰材料变形过程和破坏过程而不需要为破坏产生引进任何外在的破坏准则，是数值研究冰力学行为的有效手段。

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