Rate-independent Theory Research Articles

Modeling and simulation of chemo-elasto-plastically coupled battery active particles

AbstractAs an anode material for lithium-ion batteries, amorphous silicon offers a significantly higher energy density than the graphite anodes currently used. Alloying reactions of lithium and silicon, however, induce large deformation and lead to volume changes up to 300%. We formulate a thermodynamically consistent continuum model for the chemo-elasto-plastic diffusion-deformation behavior of amorphous silicon and it’s alloy with lithium based on finite deformations. In this paper, two plasticity theories, i.e. a rate-independent theory with linear isotropic hardening and a rate-dependent one, are formulated to allow the evolution of plastic deformations and reduce occurring stresses. Using modern numerical techniques, such as higher order finite element methods as well as efficient space and time adaptive solution algorithms, the diffusion-deformation behavior resulting from both theories is compared. In order to further increase the computational efficiency, an automatic differentiation scheme is used, allowing for a significant speed up in assembling time as compared to an algorithmic linearization for the global finite element Newton scheme. Both plastic approaches lead to a more heterogeneous concentration distribution and to a change to tensile tangential Cauchy stresses at the particle surface at the end of one charging cycle. Different parameter studies show how an amplification of the plastic deformation is affected. Interestingly, an elliptical particle shows only plastic deformation at the smaller half axis. With the demonstrated efficiency of the applied methods, results after five charging cycles are also discussed and can provide indications for the performance of lithium-ion batteries in long term use.

Rate dependent but time independent plasticity formulation with application to sand

A novel rate dependent but time independent plasticity constitutive framework is proposed by rendering the flow rule and the internal variables evolution equations, appropriate functions of the strain rate, while maintaining a rate independent yield surface. This new framework is applied to a full-fledged rate independent and Critical State Theory compatible constitutive model for sands, transforming it into a rate dependent to address the cases where rate dependence is the main interest in comparison with creep and relaxation of secondary importance and manifestation. A qualitative comparison is made with available rate dependent data for sands.

Elastoplastic Analysis of Metallic Parts Employing a Meshless Method.

The use of finite element method-based approaches has been popular in studying the elastoplastic behavior of metal parts. However, there has been a growing demand for meshless methods. In response, researchers have developed a meshless solution for 2D elastoplastic evaluation of metal components. This approach obtains the locally symmetric weak form of the governing elastoplastic integral equations at each node throughout the problem area and boundary. The elastoplastic constitutive relationships consider a small deformation rate independent associative flow theory applicable to isotropic hardening materials. The proposed solution algorithm can handle loading, unloading, and reverse loading. Numerical results were computed using Gaussian and spline weight functions, and the presented meshless solution proved to be robust and accurate for conducting the elastoplastic investigation of metallic parts. Furthermore, the Gaussian weight function was found to be more robust than the spline weight function. In conclusion, this paper presents a reliable meshless solution for elastoplastic analysis and highlights the advantages of using Gaussian weight functions.

A theory of coupled strain-gradient plasticity with species transport at small deformations

This work presents a mechanical isotropic rate-independent theory for plastically deformed materials coupled with species transport. The mass and virtual power balances are natural ingredients used in this work to obtain appropriate local balance laws for species transport, macroscopic, and microscopic forces. The second law of thermodynamics is another key tool used to obtain thermodynamically consistent constitutive relations for the species flux and microscopic stresses. The free energy is approximated as a quadratic form and used to obtain the energetic microscopic stresses as linear combinations of their respective energy conjugates and the species densities. Rate-independent Mises flow rule is deduced in terms of accumulated plastic strain and species density. Furthermore, variational formulation for the coupled theory is obtained as a variational inequality.

On feasibility of rate-independent stress paths under proportional deformations within hypoplastic constitutive model for granular materials

We study stress paths that are obtained under proportional deformations within the rate-independent hypoplasticity theory of Kolymbas type describing granular materials like soil and broken rock. For a particular simplified hypoplastic constitutive model by Bauer, a closed-form solution of the corresponding system of non-linear ordinary differential equations is available. Since only negative principal stresses are relevant for the granular body, the feasibility of the solution consistent with physics is investigated in dependence of the direction of a proportional strain path and constitutive parameters of the model.

Cyclic mean stress relaxation behaviour of P91 steel: Experiments and constitutive modeling

In the present study effect of mean strain on cyclic plastic deformation characteristics of P91 steel is experimentally evaluated and compared with results of finite element simulation considering cyclic plasticity models. P91 steels being used in various components in fossil fired and nuclear power plants are subjected to cyclic loading with mean strains/stresses. Symmetrical and asymmetrical strain controlled tests of P91 steel conducted at room temperature revealed cyclic softening nature during the fatigue cycling process. The influence of mean strain imposed showed strong dependence on stress relaxation behavior and fatigue life of P91 steel. Tensile mean stress was found to relax steeply in initial cycles followed by stabilization during the asymmetric strain cycling. A reduction in fatigue life is observed with increase in mean strain for particular strain amplitude. An attempt has also been made to simulate asymmetric strain controlled behavior of P91 steel through cyclic plastic modeling in the framework of rate independent plasticity theory. Ohno-Wang material model is employed to predict the influence of mean strain on stress relaxation behavior of the investigated steel. The simulated results depicted that Ohno-Wang model captures the cyclic plastic deformation behavior of P91 steel reasonably well.

A Class of Rate-Independent Lower-Order Gradient Plasticity Theories: Implementation and Application to Disc Torsion Problem.

As the characteristic scale of products and production processes decreases, the plasticity phenomena observed start to deviate from those evidenced at the macroscale. The current research aims at investigating this gap using a lower-order gradient enhanced approach both using phenomenological continuum level as well as crystal plasticity models. In the phenomenological approach, a physically based hardening model relates the flow stress to the density of dislocations where it is assumed that the sources of immobile dislocations are both statistically stored (SSDs) as well as geometrically necessary dislocations (GNDs). In the crystal plasticity model, the evolution of the critical resolved shear stress is also defined based on the total number of dislocations. The GNDs are similarly incorporated in the hardening based on projecting the plastic strain gradients through the Burgers tensor on slip systems. A rate-independent formulation is considered that eliminates any artificial inhomogeneous hardening behavior due to numerical stabilization. The behavior of both models is compared in simulations focusing on the effect of structurally imposed gradients versus the inherent gradients arising in crystal plasticity simulations.

Influence of the Non-Schmid Effects on the Ductility Limit of Polycrystalline Sheet Metals.

The yield criterion in rate-independent single crystal plasticity is most often defined by the classical Schmid law. However, various experimental studies have shown that the plastic flow of several single crystals (especially with Body Centered Cubic crystallographic structure) often exhibits some non-Schmid effects. The main objective of the current contribution is to study the impact of these non-Schmid effects on the ductility limit of polycrystalline sheet metals. To this end, the Taylor multiscale scheme is used to determine the mechanical behavior of a volume element that is assumed to be representative of the sheet metal. The mechanical behavior of the single crystals is described by a finite strain rate-independent constitutive theory, where some non-Schmid effects are accounted for in the modeling of the plastic flow. The bifurcation theory is coupled with the Taylor multiscale scheme to predict the onset of localized necking in the polycrystalline aggregate. The impact of the considered non-Schmid effects on both the single crystal behavior and the polycrystal behavior is carefully analyzed. It is shown, in particular, that non-Schmid effects tend to precipitate the occurrence of localized necking in polycrystalline aggregates and they slightly influence the orientation of the localization band.

Rolling and sliding between non-spherical particles

Besides normal and tangential forces, rolling and sliding are also important interactions of particles, and should be considered in the discrete element method. However, there are various definitions of rolling and sliding, some of which are quite different and even contradictory. On the premise of trying not to use definitions, the rolling and sliding velocities between non-spherical particles are derived, and satisfy the objectivity. The sliding and rolling velocities between spherical particles are included in the results as a special case. A rolling resistance model calculated by the rolling velocity is applied to simulate direct shear tests of non-spherical particles. In addition to coinciding with experiments, the shear curves also satisfy the rate independent theory. As a demonstration of the model’s universality, the wave propagation in uniaxial compression tests is also simulated.

Localized necking predictions based on rate-independent self-consistent polycrystal plasticity: Bifurcation analysis versus imperfection approach

The present study focuses on the development of a relevant numerical tool for predicting the onset of localized necking in polycrystalline aggregates. The latter are assumed to be representative of thin metal sheets. In this tool, a micromechanical model, based on the rate-independent self-consistent multi-scale scheme, is developed to accurately describe the mechanical behavior of polycrystalline aggregates from that of their single crystal constituents. In the current paper, the constitutive framework at the single crystal scale follows a finite strain formulation of the rate-independent theory of crystal elasto-plasticity. To predict the occurrence of localized necking in polycrystalline aggregates, this micromechanical modeling is combined with two main strain localization approaches: the bifurcation analysis and the initial imperfection method. The formulation of both strain localization indicators takes into consideration the plane stress conditions to which thin metal sheets are subjected during deformation. From a numerical point of view, strain localization analysis with this crystal plasticity approach can be viewed as a strongly non-linear problem. Hence, several numerical algorithms and techniques are developed and implemented in the aim of efficiently solving this non-linear problem. Various simulation results obtained by the application of the developed numerical tool are presented and extensively discussed. It is demonstrated from these results that the predictions obtained with the Marciniak–Kuczynski procedure tend towards those yielded by the bifurcation theory, when the initial imperfection ratio tends towards zero. Furthermore, the above result is shown to be valid for both scale-transition schemes, namely the full-constraint Taylor model and self-consistent scheme.

Influence of rolling resistance on the shear curve of granular particles

Abstract The rotation of a single particle plays a very important role in the simulation of granular flow through the discrete element method (DEM). Many researchers have noted that the effect of rolling resistance at the contact points should be considered in DEM simulations. The rolling resistance between particles, which is mainly introduced from the force or rolling deformation (velocity) between particles, is indicated in different models. However, various models cause different results. Hence, the most feasible model should be determined. For this purpose, five models are adopted to simulate the simple shear flow by DEM in this work. Simulation indicates that the contraction/dilatancy and force–displacement curves at different volume fractions are in good agreement with the experimental findings. The curves of Models 1 to 3 computed by force and angular velocity separate at different shear speeds, but this result does not satisfy the rate independent theory. The curves of Models 4 and 5, which are calculated by the rolling deformation, are better in keeping with the rate independence than the three previous models. After increasing the particle-size distribution, Model 5 appears to be more reliable. The defects of the four other models are discussed from the physical mechanism perspective. The force–displacement curves of Model 5 at a difference of 20 times in shear speed coincide perfectly. Overall, the findings satisfy both the objectivity and the rate independent theory.

An anisotropic, viscoplastic power law for face-centred cubic crystals: comparative evaluations in the Goss and Brass orientations of channel die compression

An anisotropic, viscoplastic power law is introduced and applied to analysis of the Goss and Brass orientations, and compared with predictions from rate-independent theory and experiment. The structure of the new power law is so chosen that it has the capability of approaching rate-independent results for lattice rotation and crystal shearing, after finite rotation about the load axis, in the range of unstable lattice orientations in (110) channel die compression. (Rate-independent predictions of shear and lattice rotation are in good to very good agreement with experiments on aluminium and copper in that range, whereas classicisotropicpower-law results are not.) It is established that, for sufficiently large power-law exponentn, the new anisotropic, elasto-viscoplastic theory predicts: (i) lattice stability in each of the Goss and Brass orientations, consistent with both experiment and rate-independent theory; (ii) zero crystal shear in the Goss orientation, also consistent with both; (iii) finite shear in the Brass orientation, in very good agreement with experiment and rate-independent theory; and (iv) a lateral-constraint stress that remains essentially elastic in both orientations, as predicted by rate-independent theory and close to experimental measurements for aluminium and copper in the Brass orientation.

Comparative Analysis of Ck- and C0-GFEM Applied to Two-dimensional Problems of Confined Plasticity

For many practical applications in engineering, a complex structure shows linear elastic behavior over almost all its extension, but exhibits confined plasticity contained in some small critical regions, e.g. stress concentrations in fillets and sharp internal corners. The behavior of C0- and Ck-GFEM is investigated in this class of problems. The first goal of this study is to verify the actual formulation of the Ck-GFEM for two-dimensional elastoplasticity, as a modification of the C0-GFEM formulation. The Ck-GFEM is based on a set of basis functions with Ck continuity over the domain. The approximation functions are constructed from a Ck continuous partition of unity, over which polynomial enrichment functions (or any special function) can be applied, in the same fashion as in the usual C0-GFEM. In this way, the finite element approximations show continuous responses for both displacements and stresses across inter-element interfaces. An investigation is performed to assess the behavior of higher-regularity partitions of unity against conventional C0 counterparts. The irreversible response and hardening effects of the material is represented by the rate independent J2 plasticity theory with linear isotropic hardening of material and von Mises yield criteria, being considered only monotonic loading and the kinematics of small displacements and small deformations. The focus herein is to enlighten any possible advantage of smoothness in the presence of plastification phenomena, seeking for improvements in capturing the evolution of the process zone.

On rotation deformation zones for finite-strain Cosserat plasticity

In this article, a numerical solution method for the finite-strain rate-independent Cosserat theory of crystal plasticity is developed. Based on a time-incremental minimization problem of the mechanical energy, a limited-memory Broyden–Fletcher–Goldfarb quasi-Newton method applied to a finite-difference discretization is proposed. First benchmark tests study the convergence to an analytic solution. Further simulations focus on the investigation of rotation localization zones, the bending of a rod, and a torsion experiment.

Rigid Plastic Analysis of Metal Parts using Meshless Approach

Plastic deformation of metal parts has been a matter of concern for investigators in academia, industry and research institutions all over the world. Literature reveals that earlier researchers have applied efforts for predicting plastic deformations using mesh based approach. A truly meshless formulation for rigid plastic analysis of metal parts has been developed in the present study for both plane stress and plane strain cases. In the present formulation, the governing equations are obtained for different set of scattered nodes over the problem domain and the integral equation for rigid plastic behavior is obtained through weak form over a local sub-domain. The meshless solution functions are obtained for different set of scattered nodes through moving least square technique. Essential boundary conditions are enforced through Penalty approach. The rigid plastic constitutive relationships incorporate only small deformation. Material constitutive relationship include Von-Mises yield criterion with rate independent associative flow theory. The solution algorithm for rigid plastic analysis of metal parts using meshless approach is discussed in the present work. Numerical results have been computed through two test functions using both linear and quadratic basis function which shows that presented formulation is accurate and robust for carrying out the rigid plastic analysis of metal parts.

Evaluation of normalisation methods for uniaxial bias extension tests on engineering fabrics

An investigation has been conducted to evaluate the performance of normalisation methods for the uniaxial bias extension test. The predictions of two published theories for rate-independent behaviour are examined and a third heuristic method is proposed. Using hypothetical test data, the predictions of the two rate-independent theories are shown to be equivalent for specimen dimensions of high aspect ratio; in this case the predictions can be well-represented using the simple heuristic formula. The predictions diverge for specimens of low aspect ratio, when specimens are sheared to very high shear angles. In order to examine the significance of this divergence on real data, results from tests on several different engineering fabrics are normalised. Differences in the predictions of the normalisation methods are observed and the question of the significance of these differences is discussed. The paper also examines the applicability of rate-independent theory for normalising rate-dependent materials.

On crystal shear, lattice rotation and constraint stress in (1 1 0) channel die compression: rate-independent and viscoplastic analyses and predictions compared

Rate-independent crystal plasticity theory and a classic viscoplastic power-law are investigated, contrasted and compared for finite deformation analysis of fcc crystals in channel die compression, including full consideration of lattice straining. Both experiment-based anisotropic and isotropic (Taylor) hardenings are evaluated in rate-independent theory; and an unlimited range of power-law exponent n is considered in viscoplasticity. The focus is on predictions of lateral constraint stress, lattice rotation and crystal shear, and their comparison with experiment. General elastic-plastic equations (for both theories) are given for the range of unstable lattice orientations in (1 1 0) compression (‘range I’) and evaluated before and after a finite rotation of the lattice about the load axis. Equations also are given and evaluated for the ‘Brass’ orientation. It is shown that the theories can be in close agreement at the onset of finite deformation in range I, but that viscoplasticity gives results (for any n) after finite rotation that are in sharp contrast to rate-independent theory. The latter’s predictions for crystal shear and lattice rotation are in good to very good agreement with finite deformation experiments on aluminium and copper. The inclusion of lattice elasticity is found to have a negligible effect in range I. In contrast, for finite deformation in the stable Brass orientation, elastic-viscoplastic theory can be made to agree very closely with rate-independent theory and with experiment.

A finite strain model of combined viscoplasticity and rate-independent plasticity without a yield surface

A new internal variable formulation dealing with mechanisms with different characteristic times in solid materials is proposed within a finite deformation framework. The framework relies crucially on the consistent combination of a general viscoplastic theory and a rate-independent theory (generalized plasticity) which does not involve the yield surface concept as a basic ingredient. The formulation is developed initially in a material setting and then is extended to a covariant one by applying some basic elements and results from the tensor analysis on manifolds. The covariant balance of energy is systematically employed for the derivation of the mechanical state equations. It is shown that unlike the case of finite elasticity, for the proposed formulation the covariant balance of energy does not yield the Doyle–Ericksen formula, unless a further assumption is made. As an application, by considering the material (intrinsic) metric as a primary internal variable accounting for both elastic and viscoplastic (dissipative) phenomena within the body, a constitutive model is proposed. The ability of the model in simulating several patterns of the complex response of metals under quasi-static and dynamic loadings is assessed by representative numerical examples.

A new macroscopically anisotropic pressure dependent yield function for metal matrix composite based on strain gradient plasticity for the microstructure

Metal matrix composites with long aligned elastic fibers are studied using an energetic rate independent strain gradient plasticity theory with an isotropic pressure independent yield function at the microscale. The material response is homogenized to obtain a conventional macroscopic model that exhibits anisotropic yield properties with a pressure dependence. At the microscale free energy includes both elastic strains and plastic strain gradients, and the theory demands higher order boundary conditions in terms of plastic strain or work conjugate higher order tractions. The mechanical response is investigated numerically using a unit cell model with periodic boundary conditions containing a single fiber deformed under generalized plane strain conditions. The homogenized response can be modeled by conventional plasticity with an anisotropic yield surface and a free energy depending on plastic strain in addition to the elastic strain. Hill's classical anisotropic yield criterion is extended to cover the composite such that hydrostatic pressure dependency, Bauschinger stress and size-effects are considered. It is found that depending on the fiber volume fraction, the anisotropic yield surface of the composite is inclined compared to a standard pressure independent yield surfaces. The evolution of the macroscopic yield surface is investigated by quantifying both anisotropic hardening (expansion) and kinematic hardening (translation), where the coefficients of anisotropy and the Bauschinger stress are extracted.

Comparative evaluation of a viscoplastic power-law and rate-independent crystal plasticity in channel die compression

General equations in (110) channel die compression are derived for a viscoplastic power-law, and comparative evaluations made with rate-independent theory and experiment. The latter theory has been shown in a series of papers (2007–2012) to predict well the finite-deformation experimental behavior (1966–2007) of fcc crystals in this family of orientations, and to give a rational basis for the elastoplastic transition that precedes the onset of finite multiple-slip. It is established analytically that, during this elastoplastic transition, the power-law equations in the limit of unbounded exponent n are identical with the rate-independent equations for lateral stress-rate and (very small) lattice rotation-rate. Moreover, results for aluminum and copper agree very closely for large n in four initial orientations investigated numerically. At the onset of finite deformation (in general when four or more systems are equally stressed) the respective results for stress-rate differ sharply, with the exception of the experimentally stable Brass orientation. When lattice elasticity is included in the power-law in this orientation (with an n of 100 or greater), it predicts results essentially indistinguishable from rate-independent theory for both aluminum and copper, in good agreement with experiment. In two orientations near the ends of the range, the power-law lattice rotation-rates at the onset of finite deformation for large n also agree closely with the rate-independent results. However, in the specific orientation from which there is large lattice rotation, the power-law significantly under-predicts rate-independent and experimental results after large strains, whatever the value of exponent n.