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以颗粒二体接触力模型为基础和出发点的软球离散元模拟是当前颗粒物理和力学领域广泛应用的研究手段.但文献上经常使用的、包括著名的Hertz-Mindlin和Luding在内的力模型并没有完全明确弹性势能或耗散热的计算方法，故从热力学层面看它们还需要完善.考虑到机械能的耗散行为是这类材料的重要物理内容，本文借鉴近年来提出的颗粒固体流体动力学（GSH）思路，提出一种具有明确势能和热功率的接触力建模方法.该理论除明确给出了机械能和热能的计算公式外，还能具体描述能量守恒、热力学平衡态和熵增加等基本原理，解决了传统接触力模型在这些方面的欠缺问题.初步计算显示本文模型的恢复系数可以随碰撞速率的增加而减弱，这比现有的其他模型更符合实验观测.虽然为简单起见这些公式仅局限于二维和忽略颗粒转动运动情况，文中讨论了如何推广到三维含转动情形，以及所涉及的滚动和扭转接触力的热力学处理问题.鉴于是否在Onsager非平衡热力学基础上建模是本文给出的接触力公式有别于当前其他模型的关键所在，文中强调了这里的主要建模对象应该是热力学特征函数和Onsager迁移系数，而接触力是它们的推导结果.这是一个与目前直接针对接触力进行建模的不同思路.文中对颗粒物质特有的、反映样品几何变形与弹应变之间联络的一个非对角迁移系数做了详细介绍，并且认为它与打滑等复杂力学现象关系密切，无论宏观GSH尺度上，还是细观接触力尺度上都不可忽略.The starting premise of any soft discrete element method simulation, widely used in granular physics and granular mechanics, is the modelling of grain-grain contact force. Most of models often used in the literature including the famous ones by Hertz-Mindlin and Luding, do not present the algorigthy of total elastic potential, or the rate of dissipation which is mainly due to the partially frictional character of the forces. This renders the question of thermodynamic consistency unsettled. A model that possesses explicit expressions for both is proposed here. It is conceptually closely related to the continuum-mechanical theory of granular solid hydrodynarmics (GSH). This theory contains expressions for the total elastic potential and the thermal energy, it accounts for energy conservation and the positivity of entropy production, and it clarifies the equilibrium properties of granular media. All these are lacking (or hidden) in the contact models widely used in the literature. A preliminary calculation shows that the restitution coefficient varies with the impact velocity, which is an added bonus, and demonstrates the model's increased realism. For simplicity, the equations presented in this work are limited to the 2D-case and neglect granular rotations. Nevertheless, the generalization to the 3D-case and the inclusion of granular rotations are carefully discussed, clarifying how to treat rolling and the torsional forces in a thermodynamically consistent fashion. A key point of the present approach, and the major difference to other force models, is the fact that, starting from the characteristic thermodynamic potential, we employ the Onsager reciprocity relation to set up the transport coefficients. The contact forces (usually postulated) are then derived from them. This difference is both conceptually and methodologically relevant. We discussed in detail off-diagonal transport coefficients, especially the so called gear ratio that is particular to granular matter. It reflects the difference between the elastic and the total strain, and is closely related to the slip movement of contact surface, which occur during shear, rolling and torsional deformations. It is relevant to both the macroscopic GSH scales, and the mesoscopic granular scale.
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