Elastoplastic Solids Research Articles

Augmented Incremental Potential Contact for Sticky Interactions.

We introduce a variational formulation for simulating sticky interactions between elastoplastic solids. Our method brings a wider range of material behaviors into the reach of the Incremental Potential Contact (IPC) solver recently developed by (Li et al. 2020). Extending IPC requires several contributions. We first augment IPC with the classical Raous-Cangemi-Cocou (RCC) adhesion model. This allows us to robustly simulate the sticky interactions between arbitrary codimensional-0, 1, and 2 geometries. To enable user-friendly practical adoptions of our method, we further introduce a physically parametrized, easily controllable normal adhesion formulation based on the unsigned distance, which is fully compatible with IPC's barrier formulation. Furthermore, we propose a smoothly clamped tangential adhesion model that naturally models intricate behaviors including debonding. Lastly, we perform benchmark studies comparing our method with the classical models as well as real-world experimental results to demonstrate the efficacy of our method.

Mesoscale smoothed particle hydrodynamics simulation of seizure and flash temperature for dry friction of elastoplastic solids in a newly developed model

This study developed a simulation model using a smoothed particle hydrodynamics (SPH) method targeted to seizure process at the mesoscale. The mechanisms of wear, adhesion, and heat generation leading to seizure at the mesoscale were modelized without assumptions or theories based on empirical rules. In particular, we targeted on flash temperature during seizure process, which is difficult to measure directly in experiment and has not been simulated without using friction theory. Our model consisted of both a macroscopic elastoplastic consideration and a microscopic interfacial interaction consideration, and the heat generation scheme that 90% of the plastic strain energy is converted to heat energy were adopted in the model. The simulation demonstrated the seizure process in which the contact state is maintained by the strong interfacial interaction as the plastic strain progresses and the temperature rapidly rises. The flash temperature by the simulation provided a reasonable quantitative match at order level to a temperature estimated by substituting true contact area and interfacial heat flux obtained by the simulation into a theoretical formula of flash temperature.

Wedge indentation of elastoplastic solids — from single indentation to interaction between indenters

Wedge indentation of elastoplastic solids — from single indentation to interaction between indenters

Recurrent neural network plasticity models: Unveiling their common core through multi-task learning

Recurrent neural network models are known to be particularly suitable for data-driven constitutive modeling due to their built-in memory variables. The main challenge preventing their widespread application to engineering materials lies in their excessive need of data for training. Here, we postulate that RNN models of elasto-plastic solids feature a large common core that is shared by all materials of the same class. The common core is complemented by material-specific layers with parameters that vary from material-to-material. After training RNN models for 25 different von Mises materials, we identify the common core of a multi-task model. Furthermore, we demonstrate through ensemble transfer learning that adding a new material to the multi-task model requires datasets that are two to three orders of magnitude smaller than the datasets needed for training an RNN from scratch. In addition, to multi-task learning, we introduce probabilistic ensembles of RNN plasticity models to quantify the epistemic uncertainty. A deep drawing simulation is performed to demonstrate the superior generalization capabilities of RNNs identified via multi-task learning as compared to those obtained through single task training.

Thermal fatigue behaviors of thin-walled structures with holes: Experiments and phase field fracture modeling

The assessment of thermal fatigue behaviors of thin-walled structures with holes is a challenging task. A novel experimental approach was designed to test the thermal fatigue behaviors of thin-walled structures with holes. The experiments were conducted under three test conditions (300–850/950/1000 ℃) to investigate the behaviors of crack initiation and early propagation. The cyclic temperature loading exhibits high-transient and wide-ranging characteristics, enabling the construction of a non-uniform transient temperature field in the heating step. A thermal fatigue phase field model was developed for elastoplastic solids, and thermal fatigue fracture modeling and life prediction were performed. The phase field effectively reproduces crack morphology, and the experimental results closely match the simulation results when fatigue parameters are properly selected. Raising the upper limit of temperature will significantly diminish thermal fatigue life, accelerate crack growth, and induce a broader range of material damage. The temperature gradient in the heating step enhances the cyclic plastic behavior at the edge of the hole, significantly amplifying the detrimental impact of thermal fatigue.

Computational study on the effect of anisotropy in the transversely isotropic elasticity of elastoplastic solids on depth-sensing indentation with a point-sharp indenter

This study investigated depth-sensing indentation via finite element simulation using a sharp-pointed indenter on elastoplastic solids (EPSs) with transversely isotropic elasticity (TIE). It aims to quantify the effect of elastic anisotropy on the relationship between the indentation load, P, and the penetration depth of the indenter, h, that is, the P-h curve. The anisotropy of both Young’s modulus, E, and shear modulus, G, significantly affected the P-h curve. To quantify this effect, the revised Sneddon’s equation was modified as a function of the elastic anisotropy. The validity of an equation, which was developed elsewhere for isotropic elastoplastic solids, to extract the indentation elastic parameter from the P-h curve for EPSs with TIE was confirmed, where the parameter was given as a function of the relative residual depth and indentation unloading parameter. Using these equations, we obtained both the elastic and plastic deformation resistances in the indentation direction as functions of anisotropy in E and G for the EPSs with TIE. This study provides a method to quantify the influence of elastic anisotropy on the P-h curve, offering valuable insights for designing and predicting the mechanical properties of anisotropic materials in practical engineering applications.

An Asynchronous Variational Integrator for Contact Problems Involving Elastoplastic Solids

An Asynchronous Variational Integrator for Contact Problems Involving Elastoplastic Solids

Phase field fracture in elasto-plastic solids: Considering complex loading history for crushing simulations

Phase field models have gained growing popularity in fracture analysis. However, phase field modelling remains lacking for complex real-life applications such as structural crushing analyses to date. The novelty of this study is to propose an explicit phase field modelling framework considering complex loading history to predict the crushing behaviour of additively manufactured metallic structures. First, the explicit phase field model is formulated by considering stress state dependent fracture initiations with non-proportional loading. Then, five types of material specimens with a wide range of stress states are experimentally tested to calibrate material parameters for additively manufactured 316L steel and Ti-6Al-4V titanium. Last, the square tubes are 3D-printed and tested under three-point bending and axial compression to demonstrate the capacity of the proposed model. The modelling results unveil that the crushing behaviour of both materials could be properly reproduced in terms of force-displacement curves and crack paths. Finally, the fracture mechanism of crushing deformation is analysed. For the three-point bending of Ti-6Al-4V tubes, cracks are mainly induced by the medium stress triaxiality tension and shear loading. For the axial compression of 316L tubes, the stress states of critical elements are relatively diverse yet slightly concentrated at the shear, compression-shear, and high stress triaxiality tension regions. Remarkably, non-proportional loading is significant in crushing deformation as a material point may experience a significant change in stress state with loading. By considering a non-proportional loading dependent threshold, the proposed phase field model can predict the crushing behaviour of structures even when the fracture initiations do not locate on the fracture locus. This study provides an effective explicit phase field framework featured with an insightful fracture mechanism for quasi-static crushing of additively manufactured metallic materials.

A unified phase field solution to fracture analyses on (pseudo) elastoplastic solids and structures

Despite that phase field to brittle fracture has been extensively studied and extension towards solving ductile fracture has been reported, crack evolution in pseudo elastoplastic solids is still an on-going research area. Taking engineered cementitious composites (ECC) as example, it is featured with pseudo ductility, by developing multiple fine cracks to withstand excessive deformation, causing significant numerical difficulties. In this regard, pseudo plasticity is introduced into phase field method to form a novel approach, guided by degrading fracture toughness, to ductile failure analyses. Moreover, a yield surface decay model is proposed to consider the progressive loss of strain-bearing capacity in pseudo elastoplastic solids under damage accumulation by multi-cracking development. Based on the proposed model, a unified solution is achieved for fracture analyses on both conventional and pseudo elastoplastic solids. The model robustness is first verified by a numerical benchmark and an in-situ experiment on metallic materials under quasi-static regime, before applying to study the fracture mechanics of bulk ECCs under both direct tension and flexure. The proposed method is presented with an open-source implementation in COMSOL Multiphysics, and is demonstrated to be capable of assisting future analyses and designs on (pseudo) elastoplastic solids and structures.

Multi-temporal decomposition for elastoplastic ratcheting solids

This paper presents a multi-temporal formulation for simulating elastoplastic solids under cyclic loading. We leverage the proper generalized decomposition (PGD) to decompose the displacements into multiple time scales, separating the spatial and intra-cyclic dependence from the inter-cyclic variation. In contrast with the standard incremental approach, which solves the (non-linear and computationally intensive) mechanical balance equations at every time step, the proposed PGD approach allows the mechanical balance equations to be solved exclusively for the small-time intra-cyclic response, while the large-time inter-cyclic response is described by simple scalar algebraic equations. Numerical simulations exhibiting complex cyclic responses, including a 2D problem and an application to a monopile foundation, demonstrate that PGD solutions with a limited number of space–time degrees of freedom may be obtained numerically, only requiring a few modes to accurately capture the reference response.

Phase field modeling of thermal fatigue crack growth in elastoplastic solids and experimental verification

It is challenging to simulate thermal fatigue crack growth in elastoplastic solids, meanwhile, the phase field model is very attractive for simulating complex cracking problems. A thermal-elastoplastic phase field model for thermal fatigue was developed, and its accuracy and availability were verified by typical examples. Then a verified thermal fatigue experiment of the V-notch specimen was carried out and the crack growth was simulated based on the phase field model. The results indicate that the process of thermal fatigue crack in elastoplastic solids from initiation to propagation is effectively simulated by the proposed phase field model. As the regularization length increases, the simulated crack has a longer length and stronger diffusivity. The appropriate regularization length needs to be determined based on experimental results.

A data-driven reduced-order surrogate model for entire elastoplastic simulations applied to representative volume elements

This contribution discusses surrogate models that emulate the solution field(s) in the entire simulation domain. The surrogate uses the most characteristic modes of the solution field(s), in combination with neural networks to emulate the coefficients of each mode. This type of surrogate is well known to rapidly emulate flow simulations, but rather new for simulations of elastoplastic solids. The surrogate avoids the iterative process of constructing and solving the linearized governing equations of rate-independent elastoplasticity, as necessary for direct numerical simulations or (hyper-)reduced-order-models. Instead, the new plastic variables are computed only once per increment, resulting in substantial time savings. The surrogate uses a recurrent neural network to treat the path dependency of rate-independent elastoplasticity within the neural network itself. Because only a few of these surrogates have been developed for elastoplastic simulations, their potential and limitations are not yet well studied. The aim of this contribution is to shed more light on their numerical capabilities in the context of elastoplasticity. Although more widely applicable, the investigation focuses on a representative volume element, because these surrogates have the ability to both emulate the macroscale stress-deformation relation (which drives the multiscale simulation), as well as to recover all microstructural quantities within each representative volume element.

An elastoplastic phase-field model for dynamic fracture of nickel-based super-alloys

This paper presents a elastoplastic phase-field model for predicting crack nucleation, propagation, and branching, among other things, in elastoplastic solids subjected to dynamic loading. During the entire fracture process of the nickel-based super-alloys, with the accumulation of plastic strain, to some extent the elastic response of the material is reduced and the rate of phase-field evolution is affected, this means that the elastic strength and phase-field evolution of the material will be influenced by the yield surface and the hardening function. Therefore, we construct elastic and plastic energy degradation functions to more accurately reflect the effects of mechanical fields on the damage field, which distinguish between two successive failure mechanisms: (i) elastic strain and (ii) simultaneous elastoplastic localization deformation. In a variational framework, we derive the governing equations and the corresponding weak forms. Then, damage evolution criterion combining elastoplastic and resistance energies and a plastic yielding criterion are derived. From a numerical point of view, we give model parameters’ selection criteria. Finally, the proposed model is applied to some impact scenarios with varying impact velocities or pre-existent cracks. The comparison of numerical and experimental results shows it can be easily used for identifying material gradation profiles that manipulate crack propagation.

A double-phase field method for mixed mode crack modelling in 3D elasto-plastic solids with crack-direction-based strain energy decomposition

Crack-direction-based decomposition of elastic strain energy could effectively control the propagation of tensile and shear cracks in a phase field modelling context. The objective of the proposed double-phase field model is to extend the crack-direction-based decomposition strategy from 2D brittle fracture to complex mixed-mode crack modelling in a 3D setting, with plastic deformation incorporated. Both effective (undamaged) stress and plastic strain are split in the crack-orientation-based coordinate system. The decomposed tensile/shear stress contribute to tensile/shear damage evolution, respectively. The plastic contribution is coupled by relating the decomposed tensile/shear plastic strain to the corresponding tensile/shear crack energy release rates. Crack surface normal direction, represented by two orientation variables in 3D spatial domains, is determined by the F-criterion. The proposed model is implemented via ABAQUS subroutines with a staggered scheme for two phase field variables and crack direction. The simulation of a single-edge notch specimen under shear loading demonstrates that the ratio between shear and tensile crack energy release rates plays a significant role in the crack mode and mechanical response. Numerical results of a group of uniaxial tension, simple shear and tension shear specimens show good agreement with the experimental data in terms of the force–displacement curve and crack path, exhibiting the validity of the proposed model for capturing different crack modes. This model has also been proven effective for complex 3D problems via the third Sandia Challenge example.

A thermal equilibrium approach to modelling multiple solid–fluid interactions with phase transitions, with application to cavitation

This paper presents a new approach to model phase transitions in multi-material problems based on the assumption of thermal equilibrium. The method is used to construct an equation of state for water which captures the liquid–vapour phase transition and is suitable for simulating cavitating flows. A key advantage of this approach is that the physics of the phase transition is entirely encoded within the equation of state. Complex multi-physics models can therefore be augmented to include the physics of phase transitions without further complicating the partial evolution equations. This is substantiated by integrating the model within a diffuse interface scheme for coupled solid–fluid dynamics, and implemented using a structured adaptive mesh refinement (SAMR) strategy. The combined model facilitates the fully-coupled simulation of complex multi-material, multi-physics problems featuring elastoplastic solids, gases, cavitating liquids and reactive materials. Evaluation of the model is shown to add little overhead – for parts of the domain without cavitation the equations reduce to those of the underpinning model with simple stiffened gas equation of state for liquid components; for parts of the domain where cavitation does occur, it requires a non-linear relaxation with only one degree of freedom. The method is applied to several strenuous test cases for cavitating flows and shown to be in good agreement with previously published works.

A thermodynamically consistent finite strain phase field approach to ductile fracture considering multi-axial stress states

Phase field models for ductile fracture have gained significant attention in the last two decades due to their ability in implicitly tracking the nucleation and propagation of cracks. However, most crack phase field formulations for elastoplastic solids focus only on the effects of plastic deformation, and do not consider the different multi-axial stress states that may arise in practical designs. In this work, a thermodynamically consistent phase field approach coupled with finite strain plasticity, considering multi-axial stress states is presented. In order to account for the coupling between plasticity and stress states, the Stress-Weighted Ductile Fracture Model (SWDFM) is utilized. The SWDFM represents a criterion for predicting ductile crack initiation under both monotonic and cyclic loadings based on histories of an internal plastic variable, stress triaxiality, and the Lode angle parameter. The excellent performance of the SWDFM for predicting ductile crack initiation motivates for its incorporation into a phase field approach for predicting both crack initiation and propagation through degradation of the fracture toughness. Moreover, based on the second law of thermodynamics, exact requirements are imposed on the rate at which the fracture toughness can evolve. A novel function for degrading the plastic yield surface during the evolution of damage is introduced. This function, in line with experimental observations, leads to an accumulation of plastic deformation in damaged regions of a solid, and avoids numerical instabilities arising from concentrations of large plastic deformations in severely damaged regions. For validating the proposed model, results of computational simulations are compared to data from selected tests considering different multi-axial stress states. Comparisons of the numerical results with data from laboratory experiments demonstrate the capabilities of the proposed framework.

On Thermodynamically Compatible Finite Volume Schemes for Continuum Mechanics

In this paper we present a new family of semidiscrete and fully discrete finite volume schemes for overdetermined, hyperbolic, and thermodynamically compatible PDE systems. In the following, we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics as well as the more complex Godunov--Peshkov--Romenski (GPR) model of continuum mechanics, which, with the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first-order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the entropy inequality as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper [Dokl. Akad. Nauk SSSR, 139(1961), pp. 521--523] exactly at the discrete level. All other terms in the governing equations of the more general GPR model, including nonconservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing PDEs.

A unified diffuse interface method for the interaction of rigid bodies with elastoplastic solids and multi-phase mixtures

This work outlines a new multi-physics-compatible immersed rigid body method for Eulerian finite-volume simulations. To achieve this, rigid bodies are represented as a diffuse scalar field, and an interface seeding method is employed to mediate the interface boundary conditions. The method is based on an existing multi-material diffuse interface method that is capable of handling an arbitrary mixture of fluids and elastoplastic solids. The underlying method is general and can be extended to a range of different applications including high-strain rate deformation in elastoplastic solids and reactive fluid mixtures. As such, the new method presented here is thoroughly tested through a variety of problems, including fluid–rigid body interaction, elastoplastic–rigid body interaction, and detonation–structure interaction. Comparison is drawn between both experimental work and previous numerical results, with excellent agreement in both cases. The new method is straightforward to implement, highly local, and parallelizable. This allows the method to be employed in three dimensions with multiple levels of adaptive mesh refinement using complex immersed geometries. The rigid body field can be static or dynamic, with the tangent of hyperbola interface capturing reconstruction method being used to keep the interface sharp in the dynamic case.

Numerical Simulation of Conical and Linear-Shaped Charges Using an Eulerian Elasto-Plastic Multi-Material Multi-Phase Flow Model with Detonation.

This study developed a hydrocode to numerically simulate both conical and linear-shaped charges using an Eulerian multi-material and multi-phase flow model. Elasto-plastic solids and the detonation of a high explosive charge were modeled using a Johnson–Cook material model and the programmed burn model, respectively. Further, the plasticity of the solids was calculated using a radial return mapping algorithm. The model was solved using a high-resolution computational fluid dynamics (CFD) technique on Cartesian grids. Material interfaces were tracked using the level-set method, and the boundary conditions were imposed using the ghost fluid method. The developed hydrocode was validated using high-speed impact problems. Consequently, the developed hydrocode was used to successfully simulate the evolution and penetration of metal jets in shaped charges after a detonation.

A generalized finite element method without extra degrees of freedom for large deformation analysis of three-dimensional elastic and elastoplastic solids

The generalized finite element method (GFEM) is versatile and powerful in the numerical analysis of various engineering problems. However, its application to large deformation analysis of elastoplastic solids is rare, especially in three dimensions (3D), since it suffers from singular system matrix and awkward implementation. This issue is caused by the extra degrees of freedom (DOFs) of GFEM. In this work, a nonlinear GFEM for large deformation analysis of elastic and elastoplastic solids in 3D is developed by using the extra-DOF-free enrichments proposed by Tian (2013) Updated Lagrangian (UL) formulation incorporating both geometric and material nonlinearities is employed and hyperelastic and hypoelastic–plastic constitutive models are considered. As a result of the elimination of the extra DOFs, implementation of the developed method is much more convenient than the standard GFEM and singular system matrix does not present. The capability of the proposed nonlinear GFEM in modeling three-dimensional large elastoplastic deformations is investigated by several typical examples. Numerical results demonstrate that, in comparison to the traditional finite element method, the proposed nonlinear GFEM is remarkably more accurate and stable. Particularly, in analysis of extreme deformations, the commercial software ABAQUS fails even when sophisticated elements are used, however the proposed nonlinear extra-DOF-free GFEM is still stable and converged results can be obtained.