王晓军. 陶瓷和混凝土冲击问题的物质点法研究. 清华大学硕士学位论文，2012.5

冲击问题会引起结构的大变形甚至破坏,涉及到几何非线性、材料非线 性和边界条件非线性。数值模拟手段为解决这类问题提供了有力工具。物质 点法采用质点对变形体进行离散,质点携带位置、速度、应力、应变等信息, 可以描述物体的受力和变形,适合处理与变形历史相关的本构关系。MPM 在规则的背景网格上求解动量方程和空间导数,避免了拉格朗日法的网格畸 变以及欧拉法的物质界面追踪和对流项处理,可以有效模拟冲击问题。陶瓷 和混凝土属于脆性材料,在受压状态下强度很高,广泛应用于装甲和防护结 构。本文将陶瓷和混凝土的本构方程引入物质点法中,对冲击问题进行数值 模拟,分析结构的防护性能。
本文对三类本构关系做了介绍,推导了超弹性材料的应力应变关系,得 出了材料参数的数量关系。由本构关系的客观性得出了次弹性本构方程中的 应力率需为客观张量,并以 Drucker-Prager 模型为例介绍了应力更新步骤和 径向返回法。
陶瓷的本构方程为 JH-2 模型,本文先将该模型在有限元程序中实现, 求解一个单元模型在不同工况下的应力、应变值,与商用软件 LS-DYNA 的 结果非常吻合,验证了模型实现方式的正确性。然后基于物质点法模拟了陶 瓷材料的碰撞和侵彻问题,分析了材料的力学性能。
本文介绍了两种常用的混凝土模型 HJC 和 RHT,针对 HJC 模型做了与 JH-2 模型类似的验证,发现 LS-DYNA 在受拉段对静水拉应力进行了截断, 此时对损伤量不做处理,指出了 LS-DYNA 在求解屈服应力时出现的错误。 对于 RHT 模型,本文采用 P-α状态方程求解压力,在本课题组开发的物质 点法程序 MPM3D 中实现该模型,模拟了弹体对混凝土靶板的侵彻问题,结 果与实验值较吻合。

Impact problems involve large deformation and damage of structures, along with geometric nonlinearity, material nonlinearity and boundary condition nonlinearity. Nu- merical simulation is a powerful tool for solving these problems. Material point method (MPM) discretizes a material domain by using a collection of particles, which carry all state variables such as position, velocity, stress, strain, etc and can describe load carry- ing and deformation of the material domain, so this method is suited for his- tory-dependent material models. MPM, which can effectively simulate impact problems, solves momentum equations and computes spatial derivative using regular background grid, so it eliminates the drawbacks of mesh distortion in Lagrangian description and difficulties of tracking material interface and dealing with convection term in Eulerian description. The brittle materials, ceramics and concrete, whose strength increases as the pressure increases, are widely used for armor and protective structures. In this thesis, ceramics and concrete material models are implemented in MPM and then numerical simulation is made for impact problems, so as to analyze the protective property of structures.
Three types of constitutive models, including hyper-elasticity and hypo-elasticity, are introduced. For hyper-elasticity stress-strain equation is derived and quantitative re- lations of material parameters are shown. When it comes to hypo-elasticity, stress rate must be an objective tensor to meet the principle of objectivity of the constitutive equa- tions. The stress update procedure and radial return of Drucker-Prager material model, for example, are also introduced.
In order to validate the stress update procedure, JH-2 model, for ceramics material, is implemented in our explicit FE code EFEM, and the comparison of results, such as stress, strain, damage and so on, between EFEM and commercial finite element soft- ware LS-DYNA, is made by using single element model under different loading cases. Then crash and penetration problems are simulated to analyze material property of ce- ramics based on MPM.
HJC and RHT material models are adopted to describe the behavior of concrete. The results show that in LS-DYNA pressure is cut off without dealing with damage and there is an obvious error of calculating the yield stress when validation is done for HJC model in the similar way for JH-2. The RHT material model, together with P-α EOS (Equation of State), is implemented in our three-dimensional explicit material point method code MPM3D, and the results from simulating projectile pene- trating targets are in good agreement with those from experiments.