Elastic-plastic Solids Research Articles

Temperature Rise in Frictional Sliding Contact of Elastic–Plastic Solids with Fractal Surface

Temperature Rise in Frictional Sliding Contact of Elastic–Plastic Solids with Fractal Surface

Elastic inclusion effects on deformation behavior of indented elastic–plastic solids

Elastic inclusion effects on deformation behavior of indented elastic–plastic solids

A high-order, localized-artificial-diffusivity method for Eulerian simulation of multi-material elastic-plastic deformation with strain hardening

A high-order method for Eulerian simulation of material undergoing large elastic–plastic deformation is developed. Thermodynamically consistent hyperelastic constitutive relations are assumed, facilitating the treatment of solids, liquids, and gases in a unified manner. The method enables the simulation of multi-material interactions using a diffuse interface approach. Numerical capturing of material interfaces, shock waves, contact surfaces, and elastic-plastic strain discontinuities using high-order compact-difference schemes is assisted by Localized Artificial Diffusivity (LAD). In the new setting involving elastic–plastic deformation, the previously established terms for the artificial properties are verified to effectively regularize normal shocks. Additional LAD terms are introduced to the elastic and plastic kinematic equations to regularize shear shocks and other strain discontinuities, improving solution stability. Other important features of the method that improve robustness include the numerical treatment of compatibility terms in the kinematic equations, and the treatment of rotation. Particular emphasis is focused toward new advancements of the methods for plastic-deformation integration and the associated strain hardening of the material, including rate-dependent plasticity. The method is demonstrated on a variety of test problems, including 1-D impacts, a variant of the Shu-Osher problem, a Taylor impact, and a Richtmyer-Meshkov instability between two elastic–plastic solids with strain hardening.

Experimental studies on the interfacial separation and stiffness of rough elastic-plastic solids

The direct measurement of interfacial separation between rough solids is hitherto one of the most challenging tasks in contact mechanics. This paper develops an optical interferometric technique for in-situ measuring interfacial separation with nanoscale precision. For the contact of metallic samples, the evolutions of interfacial separation and stiffness with load are measured. In the loading process, the dependence of interfacial stiffness on load can be described by a power law with the exponent smaller than unity at low loads and larger than unity at large loads. Similar relations have been found for unloading but interrupted with a linear relation at intermediate loads. These characteristics are significantly different from those known for rough elastic solids, and provide new insights in tribology.

From crack line field analysis method to symmetric line analysis method: Elastic-plastic analysis of symmetry problems

The mathematical difficulty encountered in the elastic–plastic analysis of solids lies in solving the partial differential equations analytically in the theory of plasticity. Such a mathematical dilemma has existed for decades and analytical solutions are still waiting to be obtained. This paper extends the crack line field analysis method and puts forward a symmetric line analysis method, which uses the parity of the Taylor series expansion of the plastic stress field near the symmetric line to transform the problem of solving partial differential equations into ordinary ones, and so obtain the general Taylor series solution of the plastic stress field near the symmetric line. The symmetric line analysis method can give elastic–plastic solutions to plane and anti-plane symmetry problems that have not been analytically solved before, which achieves an essential leap forward in the analytical analysis of elastic–plastic solids in the theory of plasticity. In this paper, three types of symmetry problems are defined, and examples are given for the elastic–plastic analysis and solution of each type of symmetry problems of elastic-perfectly plastic solids with the symmetric line analysis method.

A Unified MPM Framework Supporting Phase-field Models and Elastic-viscoplastic Phase Transition

Recent years have witnessed the rapid deployment of numerous physics-based modeling and simulation algorithms and techniques for fluids, solids, and their delicate coupling in computer animation. However, it still remains a challenging problem to model the complex elastic-viscoplastic behaviors during fluid–solid phase transitions and facilitate their seamless interactions inside the same framework. In this article, we propose a practical method capable of simulating granular flows, viscoplastic liquids, elastic-plastic solids, rigid bodies, and interacting with each other, to support novel phenomena all heavily involving realistic phase transitions, including dissolution, melting, cooling, expansion, shrinking, and so on. At the physics level, we propose to combine and morph von Mises with Drucker–Prager and Cam–Clay yield models to establish a unified phase-field-driven EVP model, capable of describing the behaviors of granular, elastic, plastic, viscous materials, liquid, non-Newtonian fluids, and their smooth evolution. At the numerical level, we derive the discretization form of Cahn–Hilliard and Allen–Cahn equations with the material point method to effectively track the phase-field evolution, so as to avoid explicit handling of the boundary conditions at the interface. At the application level, we design a novel heuristic strategy to control specialized behaviors via user-defined schemes, including chemical potential, density curve, and so on. We exhibit a set of numerous experimental results consisting of challenging scenarios to validate the effectiveness and versatility of the new unified approach. This flexible and highly stable framework, founded upon the unified treatment and seamless coupling among various phases, and effective numerical discretization, has its unique advantage in animation creation toward novel phenomena heavily involving phase transitions with artistic creativity and guidance.

Dynamic mode decomposition of deformation fields in elastic and elastic–plastic solids

Without recourse to constitutive assumptions, without knowledge of material properties, and without solving any of the conservation equations, we construct deformation fields for linear solids using Dynamic Mode Decomposition (DMD). Originally developed for analyzing experimental or computational field variables in fluid mechanics, it takes as input vectors of flow variables assembled as columns of a data matrix, and requires a remarkably few lines of code. It is a natural fit for solid mechanics wherein surface displacements can be used for the analysis. Vectors of surface displacements are arrayed in columns to created a data matrix. Singular Value Decomposition of the time-shifted data matrix affords selection of dominant modes (rank) in the deformation field. The DMD algorithm operates on time-shifted data matrices, obtains a reduced-order model that can reconstruct the deformation field, and also provides the dominant temporal and spatial modes of the deformation. In the case of linear elastic solids, DMD can be used to: reconstruct or predict the displacement states. In elastic–plastic solids, the transition from elastic to plastic results in the eigenvalues of the low-rank data matrix going out of the unit circle (implying an unstable growth mode). Using a combination of finite element analyses and displacement measurements obtained using Digital Image Correlation (DIC), we make the case for using DMD for state-estimation and state prediction in elastic solids and identifying onset of plasticity in elastic–plastic solids.

Proposal of new crack-tip-opening-displacement as a mechanical driving force of elastic-plastic fracture mechanics

In this study, the current status of linear and nonlinear fracture mechanics was surveyed, and the conditions required for elastic–plastic fracture mechanics (EPFM) as the next step of fracture mechanics were listed. Then, the imaginary crack tip opening displacement (I-CTOD), expressed by the integral of the plastic strain in a two-dimensional plastic zone, was proposed as a driving force of EPFM. Elastic-plastic FEMs of cracked solids in various work-hardened materials were performed under both the plane stress and plane strain states to understand the singular field characteristics of elastic and plastic strains near a crack tip. The I-CTOD was found to represent the intensity of the plastic strain singular field, the elastic strain singular field, and the imaginary plastic deformation field near the crack tip in elastic–plastic solids under both the plane stress and plane strain states. This I-CTOD is promising as a mechanical driving force parameter of EPFM because the analyzable and visualizable characteristics of the behavior and its clear relationship to actual material behavior make it easy to develop prediction methods for the actual crack behavior.

An Incremental Contact Model for Rough Surfaces of Strain-Hardening Solids

The load–area relation of rough surfaces is of great interest in tribology. For elastic–plastic solids with strain hardening, an incremental model is adopted to analyze the contact of rough surfaces, in which the contact is modeled by accumulation of equivalent circular contacts with varying radius. For three typical rough surfaces with various material properties, comparisons with direct finite element calculations demonstrate the efficiency of this incremental contact model. An approximate linear relation between load and contact area is predicted by both methods up to a contact fraction of 15%. The influence of yield stress and strain-hardening exponent on the load-area proportionality is presented. This work gives a simple while effective method to calculate the load-area relation for rough contact of strain-hardening materials.

A generalised, multi-phase-field theory for dissolution-driven stress corrosion cracking and hydrogen embrittlement

We present a phase field-based electro-chemo-mechanical formulation for modelling mechanics-enhanced corrosion and hydrogen-assisted cracking in elastic–plastic solids. A multi-phase-field approach is used to present, for the first time, a general framework for stress corrosion cracking, incorporating both anodic dissolution and hydrogen embrittlement mechanisms. We numerically implement our theory using the finite element method and defining as primary fields the displacement components, the phase field corrosion order parameter, the metal ion concentration, the phase field fracture order parameter and the hydrogen concentration. Representative case studies are addressed to showcase the predictive capabilities of the model in various materials and environments, attaining a promising agreement with benchmark tests and experimental observations. We show that the generalised formulation presented can capture, as a function of the environment, the interplay between anodic dissolution- and hydrogen-driven failure mechanisms; including the transition from one to the other, their synergistic action and their individual occurrence. Such a generalised framework can bring new insight into environment–material interactions and the understanding of stress corrosion cracking, as demonstrated here by providing the first simulation results for Gruhl’s seminal experiments.

A novel analytical procedure for estimating the solids’ energy dissipation capacity from Depth-Sensing indentation tests

In emergent nanotechnology areas like Micro Electro-Mechanical Systems (MEMS) the components capacity to dissipate strain energy is a key factor to evaluate their potential lifespan. In this context, we present the direct derivation of an analytical expression to estimate the elastic–plastic solids energy dissipation capacity from Depth Sensing Indentation (DSI) data. An expression to the fast calculation of the elastic (Ue)-to-total (Ut) strain energy ratios (Ue/Ut) from loading and unloading curves was particularly proposed. Assuming reasonable power law behaviors for these curves this expression allows calculating the strain energy ratios only considering the maximum displacement (hm) and plastic hysteresis (hp), which are experimentally easy to estimate. The simplifications and assumptions taken in this work ensure, in the worst of the cases, Ue/Ut ratios with uncertainties about 8 %. Despite this, the results could represent an interesting improvement in the time spending-to-accuracy relation for estimating these ratios.

A diffuse-interface compact difference method for compressible multimaterial elastic–plastic flows

In this paper, a novel high-order numerical method is proposed for the simulation of compressible multimaterial solid–fluid problems involving elastic–plastic solid behaviors. The solid, liquid, and gas behaviors are described by an Eulerian interface-capturing model derived from hyperelasticity theory. The deformations in each component of the solids are tracked using a single deviatoric strain tensor. The numerical algorithm employs a tenth-order compact finite difference scheme and a fourth-order Runge–Kutta time-stepping scheme. To prevent numerical oscillations near the material interface, a mechanical equilibrium assumption is introduced for the elastic–plastic problems, and a localized artificial diffusivity method satisfying this assumption is designed and applied in the numerical algorithm. Numerical tests in one and two dimensions are shown to demonstrate the accuracy and feasibility of the proposed approach. In particular, the accuracy and numerical resolution of the method are verified using problems with analytical solutions. Numerical tests of interface advection problems demonstrate that our method preserves the velocity, pressure, and elastic stress equilibria at the material interface and prevents overshoots in the rest of the domain. The two-dimensional Richtmyer–Meshkov instabilities between two elastic or two elastic–plastic solids are simulated to verify the suitability of the numerical method for solving problems involving large deformations. Impact of a 2D Taylor bar to a rigid wall is also simulated to test the robustness of the numerical method.

A generalised phase field model for fatigue crack growth in elastic–plastic solids with an efficient monolithic solver

We present a generalised phase field-based formulation for predicting fatigue crack growth in metals. The theoretical framework aims at covering a wide range of material behaviour. Different fatigue degradation functions are considered and their influence is benchmarked against experiments. The phase field constitutive theory accommodates the so-called AT1, AT2 and phase field-cohesive zone (PF-CZM) models. In regards to material deformation, both non-linear kinematic and isotropic hardening are considered, as well as the combination of the two. Moreover, a monolithic solution scheme based on quasi-Newton algorithms is presented and shown to significantly outperform staggered approaches. The potential of the computational framework is demonstrated by investigating several 2D and 3D boundary value problems of particular interest. Constitutive and numerical choices are compared and insight is gained into their differences and similarities. The framework enables predicting fatigue crack growth in arbitrary geometries and for materials exhibiting complex (cyclic) deformation and damage responses. The finite element code developed is made freely available at www.empaneda.com/codes.

A mechanism-based gradient damage model for metallic fracture

A new gradient-based formulation for predicting fracture in elastic–plastic solids is presented. Damage is captured by means of a phase field model that considers both the elastic and plastic works as driving forces for fracture. Material deformation is characterised by a mechanism-based strain gradient constitutive model. This non-local plastic-damage formulation is numerically implemented and used to simulate fracture in several paradigmatic boundary value problems. The case studies aim at shedding light into the role of the plastic and fracture length scales. It is found that the role of plastic strain gradients is twofold. When dealing with sharp defects like cracks, plastic strain gradients elevate local stresses and facilitate fracture. However, in the presence of non-sharp defects failure is driven by the localisation of plastic flow, which is delayed due to the additional work hardening introduced by plastic strain gradients.

A variational minimization formulation for hydraulically induced fracturing in elastic-plastic solids

A variational modeling framework for hydraulically induced fracturing of elastic-plastic solids is developed in the present work. The developed variational structure provides a global minimization problem. While fracture propagation is modeled by means of a phase-field approach to fracture, plastic effects are taken into account by using a Drucker–Prager-type yield-criterion function. This yield-criterion function governs the plastic evolution of the fluid-solid mixture. Fluid storage and transport are described by a Darcy–Biot-type formulation. Thereby the fluid storage is decomposed into a contribution due to the elastic deformations and one due to the plastic deformations. A local return mapping scheme is used for the update of the plastic quantities. The global minimization structure demands a $$H(\mathrm{div})$$ -conforming finite-element formulation. Furthermore this is combined with an enhanced-assumed-strain formulation in order to overcome locking phenomena arising from the plastic deformations. The robustness and capabilities of the presented framework will be shown in a sequence of numerical examples.

Investigation on cross-scale indentation scaling relationships of elastic–plastic solids

Indentation scaling relationships provide normalized guidance for measuring and predicting mechanical properties in indentation experiments. At the nano-scale, the material size-effect is significant, while conventional scaling relationships fail to depict this phenomenon in nanoindentation precisely. In the present research, cross-scale indentation scaling relationships are investigated using a strain gradient theory. The nanoindentation response is found to be sensitive to different material parameters, including the material intrinsic length, yield stress, and work-hardening exponent across size-scales. If the strain gradient effect is ignored, the nanoindentation scaling relationships approach the macroscopic conventional ones. The cross-scale indentation scaling relationships obtained in the form of dimensionless functions in this work provide quantitative references to instrumented indentation tests on multiple size-scales, coinciding well with experimental results. The understanding of nanoindentation hardness is enhanced by the present work.

Finite Element Analysis of Conical Indentation: Proposing a Contact Area Independent Procedure Based on Strain Energy Analysis to Estimate Materials Mechanical Properties

A systematic finite element analysis (FEA) of conical indentation in different isotropic elastic-plastic bulk solids was realized. In particular, the relation between the indentation Hardness (H)-to-Reduced Elastic Modulus (Er) ratio and the Elastic (Ue)-to-Total (Ut) strain energy ratio was studied. Results showed a linear dependency between both ratios as predicted by analytical studies, being the proportionality constant mainly dependent on cone semi-angle(𝜃) and material Poisson´s ratio (𝑣). A well-agreement with experimental data reported in the literature was observed for a wide variety of materials. By a simple analysis of the Oliver & Pharr method and our results, we proposed two analytical equations to calculate the H and E parameters overcoming the indenter contact area (AC) dependence in indentation data analysis. AC independent analytical procedures can become interesting to simplify the data analysis and overcome considerable uncertainties due to possible pile-up effects associated to different plastic deformation phenomena.

Multiphysics modeling of the initiating capability of detonators. I. The underwater test

Detonators are explosive devices used for the initiation of secondary explosives in commercial and military applications. They are characterized by their initiating capability, which is a critical factor for their safe and effective use but challenging to assess accurately. In this two-part study, we employ numerical simulations to investigate the blast wave generated by detonators and examine their initiating capability. The first part, presented here, follows the European underwater test of initiating capability, which evaluates detonators in isolation (direct method) and the second part considers detonators placed within a receiving explosive charge (indirect method). In the underwater test, the detonator is ignited inside a water tank and the initiating capability is assessed through pressure measurements in the far field. We employ a multiphysics methodology that allows the use of distinct mathematical models for each component such as two-phase reactive materials, elastic–plastic solids, and inert fluids. The computational implementation is validated against underwater experiments and is employed for the simulation of the blast wave generated by different types of detonators. The initial focus is on the general characteristics of the blast wave and subsequently on the differences between detonators of different shell material and thickness. Results show that the blast wave in the near field is asymmetric and varies significantly between detonators, but these features do not persist in the far field. The underwater test considers only the far field and is thus unable to capture the near field differences, which have a significant impact on the initiation of secondary explosives.

A phase field formulation for dissolution-driven stress corrosion cracking

We present a new theoretical and numerical framework for modelling mechanically-assisted corrosion in elastic–plastic solids. Both pitting and stress corrosion cracking (SCC) can be captured, as well as the pit-to-crack transition. Localised corrosion is assumed to be dissolution-driven and a formulation grounded upon the film rupture–dissolution–repassivation mechanism is presented to incorporate the influence of film passivation. The model incorporates, for the first time, the role of mechanical straining as the electrochemical driving force, accelerating corrosion kinetics. The computational complexities associated with tracking the evolving metal–electrolyte interface are resolved by making use of a phase field paradigm, enabling an accurate approximation of complex SCC morphologies. The coupled electro-chemo-mechanical formulation is numerically implemented using the finite element method and an implicit time integration scheme; displacements, phase field order parameter and concentration are the primary variables. Five case studies of particular interest are addressed to showcase the predictive capabilities of the model, revealing an excellent agreement with analytical solutions and experimental measurements. By modelling these paradigmatic 2D and 3D boundary value problems we show that our formulation can capture: (i) the transition from activation-controlled corrosion to diffusion-controlled corrosion, (ii) the sensitivity of interface kinetics to mechanical stresses and strains, (iii) the role of film passivation in reducing corrosion rates, and (iv) the dependence of the stability of the passive film to local strain rates. The influence of these factors in driving the shape change of SCC defects, including the pit-to-crack transition, is a natural outcome of the model, laying the foundations for a mechanistic assessment of engineering materials and structures.

An assessment of shallow neural networks for stress updates in computational solid mechanics

How do computational costs of shallow Artificial Neural Networks (ANNs) compare with conventional schemes for three-dimensional stress updates in elastic, nonlinear elastic, and elastic-plastic solids? We implement ANNs in an explicit finite element solver using strain increments and total strains as inputs for the first two materials, respectively. For elastic-plastic solids, ANN inputs are derivable from three-dimensional displacement fields. Though we minimized size of network architectures and input data sets, we conclude that obtaining reliable multi-axial training data remains a substantial challenge; and for all three materials, computational costs for ANNs exceed costs of conventional methods.