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应力 (Stress)

[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）

∇⋅σ=0\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

∇iσij=0j\begin{align} \nabla_i \sigma_{ij} = 0_j \end{align}

利用动量守恒方程可得等式

∇i(σijxk)=∇iσijxk+σij∇ixk=∇iσijxk+σijδik=σjk\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 pp，其内部应力的积分可计算为

∫σjk dΩ=∫∇i(σijxk) dΩ=∫ni(σijxk) dΓ=∫ni(σijxk) dΓ=∫fjxk dΓ=∑cfjxk\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，cc 表示该颗粒的所有接触、fjf_j 表示接触力、 xkx_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

∑cfjxkp=(∑cfj)xkp=0\begin{align} \sum^c f_j x_k^p = (\sum^c f_j) x_k^p = 0 \end{align}

可得

∫σjk dΩ=∑cfjxk=∑cfj(xk−xkp)=∑cfjbk\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，xkpx_k^p 为颗粒质心、bkb_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

σˉjk=1V∫σjk dΩ=∑c(fjbkp−fjbkq)=∑cfjdk\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，cc 为代表单元体内的所有接触、dkd_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 F\boldsymbol{F}，则变形后，颗粒 pp（初始位置 Xp\boldsymbol{X}^p）与 颗粒 qq（初始位置 (Xq(\boldsymbol{X}^q）质心相对位置矢量 d\boldsymbol{d} 可计算为

di=Fij(Xjp−Xjq)\begin{align} d_i = F_{ij} (X^p_j - X^q_j) \end{align}

本构张量 (Constitutive tangent moduli)​

dfi=kn(dFmn)dnnnni+kt(dFmn)dntnti\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} Cijkl=∂σij∂Fkl=∑cknnidjnkdl+kttidjtkdl\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 (Borja & Wren, 1995) (Bagi, 1996) (Wren & Borja, 1997) (Kruyt & Rothenburg, 1998) (Luding, 2004)

Bagi, K. (1996). Stress and strain in granular assemblies. Mechanics of Materials, 22(3), 165–177. https://doi.org/10.1016/0167-6636(95)00044-5
Borja, R. I., & Wren, J. R. (1995). 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. https://doi.org/10.1016/0045-7825(95)00846-2
Kruyt, N. P., & Rothenburg, L. (1998). Statistical theories for the elastic moduli of two-dimensional assemblies of granular materials. International Journal of Engineering Science, 36(10), 1127–1142. https://doi.org/10.1016/S0020-7225(98)00003-2
Luding, S. (2004). Micro–macro transition for anisotropic, frictional granular packings. International Journal of Solids and Structures, 41(21), 5821–5836. https://doi.org/10.1016/j.ijsolstr.2004.05.048
Wren, J. R., & Borja, R. I. (1997). 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. https://doi.org/10.1016/S0045-7825(96)01110-3