NetDEM - Neural Network Enabled Discret Element Method

[This page is prepared in Chinese, and will be part of a book aiming at beginners of DEM.]

应力 (Stress)

基于连续性介质理论，材料的动量守恒方程可表示为（忽略体力及加速度项）

$\begin{align} \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} = \boldsymbol{0} \end{align}$

采用爱因斯坦求和约定，上式可写作

$\begin{align} \nabla_i \sigma_{ij} = 0_j \end{align}$

利用动量守恒方程可得等式

$\begin{align} \nabla_i (\sigma_{ij} x_k) = \nabla_i \sigma_{ij} x_k + \sigma_{ij} \nabla_i x_k = \nabla_i \sigma_{ij} x_k + \sigma_{ij} \delta_{ik} = \sigma_{jk} \end{align}$

针对某个颗粒 $p$ ，其内部应力的积分可计算为

$\begin{align} \int \sigma_{jk} ~\text{d} \Omega = \int \nabla_i (\sigma_{ij} x_k) ~\text{d} \Omega = \int n_i (\sigma_{ij} x_k) ~\text{d} \Gamma = \int n_i (\sigma_{ij} x_k) ~\text{d} \Gamma = \int f_j x_k ~\text{d} \Gamma = \sum^c f_j x_k \end{align}$

其中， $c$ 表示该颗粒的所有接触、 $f_j$ 表示接触力、 $x_k$ 表示接触位置。进而，利用颗粒在接触力作用下的静力平衡公式

$\begin{align} \sum^c f_j x_k^p = (\sum^c f_j) x_k^p = 0 \end{align}$

可得

$\begin{align} \int \sigma_{jk} ~\text{d} \Omega = \sum^c f_j x_k = \sum^c f_j (x_k - x_k^p) = \sum^c f_j b_k \end{align}$

其中， $x_k^p$ 为颗粒质心、 $b_k$ 为由接触点指向颗粒质心的向量（branch vector）。

针对代表单元体，其域内的平均应力可由其内部所有颗粒内部的应力积分，再除于代表单元体体积得到，表达为

$\begin{align} \bar{\sigma}_{jk} = \frac{1}{V} \int \sigma_{jk} ~\text{d} \Omega = \sum^c (f_j b_k^p - f_j b_k^q) = \sum^c f_j d_k \end{align}$

其中， $c$ 为代表单元体内的所有接触、 $d_k$ 表示颗粒质心相对位置矢量。

应变 (Strain)

假设单元变形梯度（deformation gradient）为 $\boldsymbol{F}$ ，则变形后，颗粒 $p$ （初始位置 $\boldsymbol{X}^p$ ）与颗粒 $q$ （初始位置 $(\boldsymbol{X}^q$ ）质心相对位置矢量 $\boldsymbol{d}$ 可计算为

$\begin{align} d_i = F_{ij} (X^p_j - X^q_j) \end{align}$

本构张量 (Constitutive tangent moduli)

$\begin{align} \text{d} f_i = k_n (\text{d} F_{mn}) d_n n_n n_i + k_t (\text{d} F_{mn}) d_n t_n t_i \end{align}$

$\begin{align} C_{ijkl} = \frac{\partial \sigma_{ij}}{\partial F_{kl}} = \sum^c k_n n_i d_j n_k d_l + k_t t_i d_j t_k d_l \end{align}$

参考文献 ¹ ² ³ ⁴ ⁵

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参考文献

R. I. Borja and J. R. Wren. Micromechanics of granular media Part I: Generation of overall constitutive equation for assemblies of circular disks. Computer Methods in Applied Mechanics and Engineering, 127(1-4):13–36, November 1995. doi:10.1016/0045-7825(95)00846-2. ↩
K. Bagi. Stress and strain in granular assemblies. Mechanics of Materials, 22(3):165–177, March 1996. doi:10.1016/0167-6636(95)00044-5. ↩
J. R. Wren and R. I. Borja. Micromechanics of granular media Part II: Overall tangential moduli and localization model for periodic assemblies of circular disks. Computer Methods in Applied Mechanics and Engineering, 141(3):221–246, February 1997. doi:10.1016/S0045-7825(96)01110-3. ↩
N. P. Kruyt and L. Rothenburg. Statistical theories for the elastic moduli of two-dimensional assemblies of granular materials. International Journal of Engineering Science, 36(10):1127–1142, August 1998. doi:10.1016/S0020-7225(98)00003-2. ↩
S. Luding. Micro–macro transition for anisotropic, frictional granular packings. International Journal of Solids and Structures, 41(21):5821–5836, October 2004. doi:10.1016/j.ijsolstr.2004.05.048. ↩