Integrating Peridynamics with Material Point Method for Elastoplastic Material Modeling

© Springer Nature Switzerland AG 2019. We present a novel integral-based Material Point Method (MPM) using state based peridynamics structure for modeling elastoplastic material and fracture animation. Previous partial derivative based MPM studies face challenges of underlying instability issues of particle distribution and the complexity of modeling discontinuities. To alleviate these problems, we integrate the strain metric in the basic elastic constitutive model by using material point truss structure, which outweighs differential-based methods in both accuracy and stability. To model plasticity, we incorporate our constitutive model with deviatoric flow theory and a simple yield function. It is straightforward to handle the problem of cracking in our hybrid framework. Our method adopts two time integration ways to update crack interface and fracture inner parts, which overcome the unnecessary grid duplication. Our work can create a wide range of material phenomenon including elasticity, plasticity, and fracture. Our framework provides an attractive method for producing elastoplastic materials and fracture with visual realism and high stability

A nonlocal physics-informed deep learning framework using the peridynamic differential operator

The Physics-Informed Neural Network (PINN) framework introduced recently incorporates physics into deep learning, and offers a promising avenue for the solution of partial differential equations (PDEs) as well as identification of the equation parameters. The performance of existing PINN approaches, however, may degrade in the presence of sharp gradients, as a result of the inability of the network to capture the solution behavior globally. We posit that this shortcoming may be remedied by introducing long-range (nonlocal) interactions into the network's input, in addition to the short-range (local) space and time variables. Following this ansatz, here we develop a nonlocal PINN approach using the Peridynamic Differential Operator (PDDO)---a numerical method which incorporates long-range interactions and removes spatial derivatives in the governing equations. Because the PDDO functions can be readily incorporated in the neural network architecture, the nonlocality does not degrade the performance of modern deep-learning algorithms. We apply nonlocal PDDO-PINN to the solution and identification of material parameters in solid mechanics and, specifically, to elastoplastic deformation in a domain subjected to indentation by a rigid punch, for which the mixed displacement--traction boundary condition leads to localized deformation and sharp gradients in the solution. We document the superior behavior of nonlocal PINN with respect to local PINN in both solution accuracy and parameter inference, illustrating its potential for simulation and discovery of partial differential equations whose solution develops sharp gradients

A numerical study on ice failure process and ice-ship interactions by Smoothed Particle Hydrodynamics

In this paper, a Smoothed Particle Hydrodynamics (SPH) method is extended to simulate the ice failure process and ice-ship interactions. The softening elastoplastic model integrating Drucker-Prager yield criterion is embedded into the SPH method to simulate the failure progress of ice. To verify the accuracy of the proposed SPH method, two benchmarks are presented, which include the elastic vibration of a cantilever beam and three-point bending failure of the ice beam. The good agreement between the obtained numerical results and experimental data indicates that the presented SPH method can give the reliable and accurate results for simulating the ice failure progress. On this basis, the extended SPH method is employed to simulate level ice interacting with sloping structure and three-dimensional ice- ship interaction in level ice, and the numerical data is validated through comparing with experimental results of a 1:20 scaled Araon icebreaker model. It is shown the proposed SPH model can satisfactorily predict the ice breaking process and ice breaking resistance on ships in ice-ship interaction

Multiscale Modeling with Meshfree Methods

Multiscale modeling has become an important tool in material mechanics because material behavior can exhibit varied properties across different length scales. The use of multiscale modeling is essential for accurately capturing these characteristics and predicting material behavior. Mesh-free methods have also been gaining attention in recent years due to their innate ability to handle complex geometries and large deformations. These methods provide greater flexibility and efficiency in modeling complex material behavior, especially for problems involving discontinuities, such as fractures and cracks. Moreover, mesh-free methods can be easily extended to multiple lengths and time scales, making them particularly suitable for multiscale modeling. The thesis focuses on two specific problems of multiscale modeling with mesh-free methods. The first problem is the atomistically informed constitutive model for the study of high-pressure induced densification of silica glass. Molecular Dynamics (MD) simulations are carried out to study the atomistic level responses of fused silica under different pressure and strain-rate levels, Based on the data obtained from the MD simulations, a novel continuum-based multiplicative hyper-elasto-plasticity model that accounts for the anomalous densification behavior is developed and then parameterized using polynomial regression and deep learning techniques. To incorporate dynamic damage evolution, a plasticity-damage variable that controls the shrinkage of the yield surface is introduced and integrated into the elasto-plasticity model. The resulting coupled elasto-plasticity-damage model is reformulated to a non-ordinary state-based peridynamics (NOSB-PD) model for the computational efficiency of impact simulations. The developed peridynamics (PD) model reproduces coarse-scale quantities of interest found in MD simulations and can simulate at a component level. Finally, the proposed atomistically-informed multiplicative hyper-elasto-plasticity-damage model has been validated against limited available experimental results for the simulation of hyper-velocity impact simulation of projectiles on silica glass targets. The second problem addressed in the thesis involves the upscaling approach for multi-porosity media, analyzed using the so-called MultiSPH method, which is a sequential SPH (Smoothed Particle Hydrodynamics) solver across multiple scales. Multi-porosity media is commonly found in natural and industrial materials, and their behavior is not easily captured with traditional numerical methods. The upscaling approach presented in the thesis is demonstrated on a porous medium consisting of three scales, it involves using SPH methods to characterize the behavior of individual pores at the microscopic scale and then using a homogenization technique to upscale to the meso and macroscopic level. The accuracy of the MultiSPH approach is confirmed by comparing the results with analytical solutions for simple microstructures, as well as detailed single-scale SPH simulations and experimental data for more complex microstructures

A Material Point Method for Elastoplasticity with Ductile Fracture and Frictional Contact

Simulating physical materials with dynamic movements to photo-realistic resolution has always been one of the most crucial and challenging topics in Computer Graphics. This dissertation considers large-strain elastoplasticity theory applied to the low-to-medium stiffness regime, with topological changes and codimensional objects incorporated. We introduce improvements to the Material Point Method (MPM) for two particular objectives, simulating fracturing ductile materials and incorporation of MPM and Lagrangian Finite Element Method (FEM).Our first contribution, simulating ductile fracture, utilizes traditional particle-based MPM [SSC13, SCS94] as well as the Lagrangian energy formulation of [JSS15] which uses a tetrahedron mesh, rather than particle-based estimation of the deformation gradient and potential energy. We model failure and fracture via elastoplasticity with damage. The material is elastic until its deformation exceeds a Rankine or von Mises yield condition. At that point, we use a softening model that shrinks the yield surface until it reaches the damage thresh- old. Once damaged, the material Lam ́e coefficients are modified to represent failed material. This approach to simulating ductile fracture with MPM is successful, as MPM naturally captures the topological changes coming from the fracture. However, rendering the crack surfaces can be challenging. We design a novel visualization technique dedicated to rendering the material’s boundary and its intersection with the evolving crack surfaces. Our approach uses a simple and efficient element splitting strategy for tetrahedron meshes to create crack surfaces. It employs an extrapolation technique based on the MPM simulation. For traditional particle-based MPM, we use an initial Delaunay tetrahedralization to connect randomly sampled MPM particles. Our visualization technique is a post-process and can run after the MPM simulation for efficiency. We demonstrate our method with several challenging simulations of ductile failure with considerable and persistent self-contact and applications with thermomechanical models for baking and cooking.Our second contribution, hybrid MPM–Lagrangian-FEM, aims to simulate elastic objects like hair, rubber, and soft tissues. It utilizes a Lagrangian mesh for internal force computation and a Eulerian grid for self-collision, as well as coupling with external materials. While recent MPM techniques allow for natural simulation of hyperelastic materials represented with Lagrangian meshes, they utilize an updated Lagrangian discretization and use the Eulerian grid degrees of freedom to take variations of the potential energy. It often coarsens the degrees of freedom of the Lagrangian mesh and can lead to artifacts. We develop a hybrid approach that retains Lagrangian degrees of freedom while still allowing for natural coupling with other materials simulated with traditional MPM, e.g., sand, snow, etc. Furthermore, while recent MPM advances allow for resolution of frictional contact with codimensional simulation of hyperelasticity, they do not generalize to the case of volumetric materials. We show that our hybrid approach resolves these issues. We demonstrate the efficacy of our technique with examples that involve elastic soft tissues coupled with kinematic skeletons, extreme deformation, and coupling with various elastoplastic materials. Our approach also naturally allows for two-way rigid body coupling

An energy-based peridynamic model for fatigue cracking

Fatigue design assessment is a crucial step in the design process of ships and offshore structures. To date, the stochastic approach is commonly used to calculate the total lifetime accumulated fatigue damage and the probability of fatigue failure for the structures. Meanwhile, the details of damage initiation and propagation are infrequently investigated. In terms of predicting crack growth, the traditional approaches face conceptual and mathematical difficulties in terms of predicting crack nucleation and growth, especially for multiple crack paths because the equations in classical continuum mechanics are derived by using spatial derivatives. Peridynamics is a non-local theory using the integral equations rather than differential equations which makes it suitable for damage prediction. In this study, a novel energy-based peridynamic model for fatigue cracking is proposed. The definition of cyclic bond energy release rate range and the energy-based peridynamic fatigue equations for both phases crack initiation and crack growth phases are introduced. For validation, first, a problem of mode-I fatigue crack growth is investigated. Next, different mixed-mode fatigue damages are also investigated and the peridynamic results are compared with the experimental results

Simulations for the explosion and granular impact problems using the SPH method

Simulations of explosions and granular impacts are challenging tasks to tackle using conventional mesh-based methods. In this thesis, a mesh-free technique called smoothed particle hydrodynamics (SPH) in conjunction with the Open-MP and CUDA parallel pro- gramming interfaces is introduced to tackle three-dimensional (3D) problems with large deformations. Chapter 1 gives an introduction of the SPH method and a literature review of the the- oretical improvement of SPH, landmine detonations, underwater explosions, and granular impacts. A research outline of the thesis is also presented at the end of this chapter. The basic ideas of the SPH method and some techniques which are relevant to improve the accuracy and stability of SPH, including the artificial viscosity, artificial stress, boundary implementation, neighboring particles search, and kernel gradient correction, are described in Chapter 2. In order to solve the governing equations, an elaboration of the constitutive models to update the stress tensor of soil and solid and the equation of states (EOSs) is given in Chapter 3. The simulations of the detonation and granular impact problems using the SPH method are thoroughly presented in chapters 4-7. In Chapter 4, in order to tackle 3D problems with large number of particles, the in-house SPH code is parallelized by the Open-MP programming interface. The parallel efficiency is tested by the 3D shaped charge detonation. The simulations of the 2D soil explosion and its effects on structures are investigated in Chapter 5. Based on the parallelization of the SPH code and the simulation of 2D soil explosion, the physical process of the 3D landmine detonation is studied further. The simulations of the 3D underwater explosion within cylindrical rigid and aluminium (Al) tubes including cavitation phenomenon are presented in Chapter 6. The simulations of the 3D granular impacts using GPU acceleration are presented in Chapter 7. The numerical results of SPH are compared against the experimental and other available numerical data, and it is shown that the SPH method is capable of predicting landmine detonations, underwater explosions, and granular impacts. The conclusions, novelties, and future plan of SPH research are summarized in Chapter 8