JIAYIN (KAY) LU 卢嘉茵
Applied Mathematician & Computational Artist & Freelance Photographer
Numerical Methods on Simulating Quasi-static Elastoplastic Materials
Spring 2020-Present
Advisor: Prof. Christopher Rycroft
Status: The project led to a paper submitting to Journal of Computational Physics:
In solid mechanics, I used numerical methods to simulate plastic deformation of elastoplastic materials. Many materials of engineering importance have elastoplastic behavior, including metals, granular materials, and amorphous solids such as bulk metallic glasses (BMGs). Therefore, it is important to study their plastic deformation to predict material failure. In many practical scenarios, such as laboratory tests, the material mechanics are in the quasi-static regime. However, few existing numerical methods can simulate this realistic physical regime.
Working with my advisor Prof. Christopher Rycroft, I have improved upon existing numerical methods for quasi-static elastoplastic materials. In the quasi-static limit, the Cauchy stress the Cauchy stress σ of the material
satisfies ∇ · σ = 0. In previous work [1], my advisor discovered an intriguing mathematical correspondence of the system of partial differential equations (PDEs) describing quasi-static elastoplastic materials, and a different type of physical system, the incompressible Navier-Stokes equations for fluid dynamics, as shown in the figure below. Therefore, the quasi-static elastoplastic PDE system can be solved via a projection method analogous to the fluid's projection method [2].
However, the original projection methods for quasi-static elastoplasticty and fluid dynamics are both first order accurate in time. In the last fifty years, improvements on the fluid's projection method with second-order temporal accuracy have been developed [3]. Drawing inspiration from these improvements, I developed a two-stage predictor-corrector projection method with incremental velocity to solve quasi-static elastoplaticity. The method achieves fully second-order temporal accuracy for all solution fields, as shown in the numerical example below.
Figure: Numerical example to test the order of accuracy of the numerical scheme.
(a) The initialization of χ field. χ is a state variable describing disorderliness of the material. Higher χ corresponds to more plastic deformation.
(b) The χ field at the end of time, simulated using the full model; A horizontal shear band developed.
(c) The accuracy plot for the reduced model and the full model for the three fields, velocity v , stress σ and χ. The reduced model neglects advection effects and involves only the plasticity terms in the time evolution of the variables, which helps to separate the plasticity terms out for numerical analysis. oth the reduced and the full models achieve fully second-order accuracy for all field variables.
In addition, I developed an efficient adaptive timestepping scheme for the projection method, by bounding the projection size in each timestep. This numerically forces the system to conform closely to quasi-staticity, and effectively reduces the splitting error of the projection method. The adaptive timestepping scheme allows us to obtain highly accurate solutions with many fewer timesteps, as shown in the plots below.
(a) Comparison of accuracy of solution fields. Both the constant and the adaptive timestepping schemes reached the same level of solution accuracy;
(b) Comparison of the number of timesteps taken for the two schemes to reach the same level of solution accuracy. The adaptive timestepping scheme took many fewer timesteps.
Having higher-order and more accurate numerical simulations can help researchers study the complicated plastic deformation behaviors of the materials, which are critical to a better understanding of the mechanisms that lead to material failures. The project led to a paper currently in submission to Journal of Computational Physics.
Reference
[1] Chris H. Rycroft, Yi Sui, and Eran Bouchbinder. An eulerian projection method for quasi-static elastoplasticity. Journal of Computational Physics, 300:136–166, 2015.
[2] A.J.Chorin. Numerical solution of the navier–stokes equations. Mathematics of Computation, 22:104, 1968.
[3] David L. Brown, Ricardo Cortez, and Michael L. Minion. Accurate projection methods for the
incompressible navier–stokes equations. Journal of Computational Physics, 168(2):464–499, 2001.