﻿ 冲击载荷作用下颗粒材料动态力学响应的近场动力学模拟
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 力学学报  2016, Vol. 48 Issue (1): 56-63  DOI: 10.6052/0459-1879-15-291 0

### 引用本文 [复制中英文]

[复制中文]
Zhang Qing, Gu Xin, Yu Yangtian. PERIDYNAMICS SIMULATION FOR DYNAMIC RESPONSE OF GRANULAR MATERIALS UNDER IMPACT LOADING[J]. Chinese Journal of Ship Research, 2016, 48(1): 56-63. DOI: 10.6052/0459-1879-15-291.
[复制英文]

### 文章历史

2015-07-30 收稿
2015-08-07 录用
2015-08-08 网络版发表

1 基于键作用的近场动力学方法 1.1 基本理论与PMB模型

 图 1 近场范围内物质点间相互作用 Fig.1 Schematic of interaction between material points
 $\rho \ddot u\left( {x,t} \right) = \int_{{H_x}} {f\left( {u\left( {x',t} \right),u\left( {x,t} \right),x',x} \right)d{V_{x'}} + b\left( {x,t} \right)}$ (1)

 ${\pmb f}({\pmb \xi ,\pmb\eta }) = cs\mu \dfrac{{\pmb \xi} + {\pmb\eta} }{\left| { {\pmb \xi }+{\pmb \eta } } \right|}$ (2)

 $s_0 = \dfrac{f_{\rm t} }{E}$ (3)

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

 $\varphi (x,t) = 1 - \frac{{\int_H \mu (x,t,\xi )d{V_{x'}}}}{{\int_H d {V_{x'}}}}$ (5)

1.2 改进的PMB模型的本构力函数

Huang等[21, 22]和顾鑫等[13]指出PMB模型中微观模量$c$为常数，无法反映长程力随距离增大而递减的空间分布规律，影响计算精度，提出如下改进的本构力函数

 ${\pmb f}({\pmb \xi},{\pmb\eta }) = c(0,\delta )g({\pmb \xi },\delta )s\mu \dfrac{{\pmb \xi }+{\pmb \eta }}{\left| {{\pmb \xi} +{\pmb \eta }} \right|}$ (6)

 $V_{\rm el} = \dfrac{1}{2}\left[{\lambda \left( {\varepsilon _1 \mbox{ + }\varepsilon _2 + \varepsilon _3 } \right)^2 + 2G(\varepsilon _1^2 + \varepsilon _2^2 + \varepsilon _3^2 )} \right]$ (7)

 ${V_{{\rm{el}}}} = \left\{ \begin{array}{l} \frac{{3E}}{{2\left( {1 - 2\nu } \right)}}\varepsilon _0^2,\;\;\;\;\;\;三维\\ \frac{E}{{1 - \nu }}\varepsilon _0^2,\;\;\;\;\;\;\;\;\;\;\;\;\;\;平面应力\\ \frac{E}{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}}\varepsilon _0^2,\;\;平面应变 \end{array} \right.$ (8)

 $\begin{array}{l} {V_{{\rm{pd}}}} = \frac{1}{2}\int_H {wd{V_\xi }} = \\ \;\;\;\;\;\;\;\frac{1}{2}\int_0^\delta {\int_0^{2\pi } {\int_0^\pi {\left( {\frac{{c\left( {\xi ,\delta } \right)\varepsilon _0^2\xi }}{2}} \right)} } } {\xi ^2}\sin \varphi d\xi d\theta d\varphi \end{array}$ (9)

 ${V_{{\rm{pd}}}} = \frac{1}{2}\int_H {wd{V_\xi }} = \;\frac{1}{2}\int_0^\delta {\int_0^{2\pi } {\left( {\frac{{c\left( {\xi ,\delta } \right)\varepsilon _0^2\xi }}{2}} \right)\xi d\theta d\xi } } t$ (10)

 $c(0,\delta ) = \left\{ \begin{array}{l} \frac{{36E}}{{\pi {\delta ^4}(1 - 2\nu )}},\;\;\;\;\;三维\\ \frac{{105E}}{{4\pi {\delta ^3}\left( {1 - \nu } \right)}},\;\;\;\;\;平面应力\\ \frac{{105E}}{{4\pi {\delta ^3}\left( {1 + \nu } \right)(1 - 2\nu )}},\;\;\;平面应变 \end{array} \right.$ (11)
1.3 边界效应''的修正

 $\begin{array}{l} {V_{{\rm{pd}}}}(x) = \frac{1}{2}\int_{{H_\delta }} {w\left( {\eta ,\xi } \right)} = \frac{1}{2}\sum\limits_{j \in {H_i}} {\omega \left( {\eta ,\xi } \right)} {V_j} = \\ \;\;\;\;\frac{1}{4}\sum\limits_{j \in {H_i}} {c\left( {\eta ,\xi } \right)} ){\left[ {1 - {{(\frac{{\left| {{\xi _{ij}}} \right|}}{\delta })}^2}} \right]^2}\varepsilon _0^2\left| {{\xi _{ij}}} \right|{V_j} \end{array}$ (12)

 ${c_i} = \left\{ \begin{array}{l} \frac{{6E}}{{(1 - 2\upsilon )\sum\limits_{_{j \in {H_i}}} {{{\left[ {1 - {{(\frac{{\left| {{\xi _{ij}}} \right|}}{\delta })}^2}} \right]}^2}} \left| {{\xi _{ij}}} \right|{V_j}}},\;\;\;\;\;\;三维\\ ]\frac{{4E}}{{(1 - \upsilon )\sum\limits_{j \in {H_i}} {{{\left[ {1 - {{(\frac{{\left| {{\xi _{ij}}} \right|}}{\delta })}^2}} \right]}^2}\left| {{\xi _{ij}}} \right|} {V_j}}},\;\;\;\;\;\;平面应力\\ \frac{{4E}}{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)\sum\limits_{j \in {H_i}} {{{\left[ {1 - {{(\frac{{\left| {{\xi _{ij}}} \right|}}{\delta })}^2}} \right]}^2}\left| {{\xi _{ij}}} \right|{V_j}} }},\;\;\;\;\;平面应变 \end{array} \right.$ (13)

 $c_{ij} = \dfrac{c_i + c_j }{2}$ (14)

 图 2 微观模量$c(0,\delta )$值的分布 Fig.2 Distribution of the $c(0,\delta )$ value
2 颗粒间接触作用的短程排斥力

2.1 黏性颗粒材料

2.2 无黏性颗粒材料

 ${\pmb F}_r ({\pmb y}',{\pmb y}) = \min \left\{ {0,c_{ r} \dfrac{\left( {\left\| {{\pmb y}'- {\pmb y}} \right\|} \right)- d}{d}} \right\} \dfrac{{\pmb y }'-{\pmb y }} {\left\| {{\pmb y}'- {\pmb y} } \right\|}$ (15)

 $\left| {{\pmb F}_r ({\pmb y}',{\pmb y})} \right| > 0 \ \ {\rm only if } \ \ \left| {{\pmb y}' - {\pmb y}} \right| < d = \left| {\Delta {\pmb x} } \right|$ (16)

2.3 排斥力常数$c_r$的率定方法

(1)对具有某特征尺度的给定颗粒材料，在选定适合PD计算的物质点尺寸和稳定时间步长后，通过下式计算$c_r$的可能量级范围

 ${\pmb F}_r ({\pmb y}',{\pmb y}) < 2\rho \dfrac{\left| {\Delta {\pmb x}} \right|}{\Delta t^2}$ (17)

(2)分析模拟中颗粒的运动与破碎形态，缩小$c_r$范围. 合适的排斥力常数范围，应使得颗粒体系存在合理破碎，即使得颗粒体系不发生快速溃散''现象，不发生破碎后的单一物质点与颗粒的嵌入或渗透，此时排斥力过大或过小.

(3)比较数值计算的冲击波速与物理试验测定的冲击波速数据，准确率定排斥力常数$c_r$.

3 数值计算方法 3.1 离散格式

 $\rho \ddot u_i^n = \sum\limits_p f (u_p^n - u_i^n,x_p^n - x_i^n){V_p} + b(x_i^n)$ (18)

 $\dot{\pmb u}_i^{n + 1} = \dot{\pmb u}_i^n + \dfrac{\Delta t}{2\rho }({\pmb L}_u + {\pmb b})^n + \dfrac{\Delta t}{2\rho }({\pmb L}_u + {\pmb b})^{n + 1}$ (19)
 ${\pmb u}_i^{n + 1} = {\pmb u}_i^n + \dot{\pmb u}_i^n \Delta t + \dfrac{(\Delta t)^2}{2\rho }({\pmb L}_u + {\pmb b})^n$ (20)

4 算例与分析

Vogler等[26]和Borg等[27, 28]试验测定了刚性板冲击封闭容器中碳化钨陶瓷颗粒的冲击波速响应，碳化钨颗粒具有脆性大、易开裂破碎的特点，为说明PD方法在颗粒材料领域的适用性与可行性，对该试验进行了数值模拟.

4.1 模型建立

 图 3 刚性板冲击陶瓷颗粒试验的二维模型 Fig.3 Two-dimensional simulation model for the impact experiment

PD方法将单颗粒视为大量物质点组成，方便对任意形状颗粒进行数值分析. 采用数值样本生成技术，可建立具有任意颗粒形状、粒径和空间分布规律的数值样本. 为节省篇幅，本文仅采用单一样本进行多组数值试验. 该模拟区域内共含有1 368个颗粒，采用单一粒径的圆形颗粒，并将其离散为37个方形物质点(选取离散尺寸时，考虑计算量与颗粒特征尺寸两方面的平衡)，物质点尺寸为32 / 7 μm，共离散为50 616个物质点. 采用固定2层4 384个物质点的方法模拟固壁边界条件，容器与颗粒间的接触作用通过物质点间的排斥力实施，固壁边界条件会对颗粒材料波速产生轻微影响[26, 27]. 刚性驱动板采用给定速度边界条件实现. 计算时间步长取为$\Delta t = 4e - 10s$，共计算4 μs.

4.2 排斥力常数$c_r$的率定

 图 4 刚性板以245 m/s速度冲击颗粒体系时，选取不同排斥力试常数计算得到的波速 Fig.4 Wave speed of granular system calculated with distinct repulsive force coefficients under velocity 245 m/s
4.3 结果分析

 $\sigma _{\rm H} = \rho _0 U_{\rm S} u_{\rm p }$ (21)

 图 5 不同时刻颗粒体系纵向与横向速度分布(m/s) Fig.5 Longitudinal and transverse velocity fields of granular system at distinct time
 图 6 颗粒旋转示意图(以$A, B, C$三个颗粒为例) Fig.6 Schematic of particle rotation (take $A, B, C$ for example)

 图 7 不同时刻颗粒体系的损伤破坏情况 Fig.7 Damage distribution of granular system at distinct time
5 结 语

(1)在基于键作用的近场动力学理论框架下，本文提出了反映非局部长程力特征和消除边界效应''的改进PMB模型，发展了适用于颗粒材料动力破坏分析的近场动力学方法，给出了描述颗粒接触作用的物质点层次的排斥力模型，提出了排斥力模型中排斥力常数的确定方法. 并应用该方法计算了冲击载荷作用下碳化钨陶瓷颗粒体系的动态力学行为. 研究结果表明，本文提出的改进PMB模型与发展的颗粒材料PD分析方法可以有效求解和分析颗粒材料的动态力学行为.

(2)本文提出的描述颗粒接触作用的排斥力模型只有一个排斥力常数，并可通过数值试验与已有试验数据的对比确定，降低了人为因素对计算结果的影响. 刚性板冲击碳化钨陶瓷颗粒的计算结果表明，不同冲击速度下，PD计算得到的颗粒体系的波速响应与试验结果高度一致；通过颗粒物质点尺度的相互作用描述单颗粒尺度的接触作用，较好地反映颗粒的摩擦、挤压作用效应；自然展现颗粒发生平移与旋转复杂组合的运动、颗粒变形与破碎现象.

(3)本文基于PD理论和方法初步分析了冲击载荷作用下颗粒材料的冲击波速、颗粒体系的运动、变形与破碎规律. 今后将进一步完善适用颗粒材料的近场动力学模型和计算方法，如完善黏性颗粒材料的分析方法，并考虑颗粒间的黏滞特性建模；可以预见，PD方法在颗粒材料计算力学领域中的研究范围将延伸至颗粒自然堆积过程、剪切带的形成与发展，快速流以及颗粒体系的爆炸冲击等问题.

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