Rate-independent Plasticity Research Articles

Multiscale modeling of irradiated polycrystalline FCC metals

We propose a set of models for the post-irradiation deformation response of polycrystalline FCC metals. First, a defect- and dislocation-density based evolution model is developed to capture the features of irradiation-induced hardening as well as intra-granular softening. The proposed hardening model is incorporated within a rate-independent single crystal plasticity model. The result is a non-homogeneous deformation model that accounts for defect absorption on the active slip planes during plastic loading. The macroscopic non-linear constitutive response of the polycrystalline aggregate of the single crystal grains is then obtained using a micro–macro transition scheme, which is realized within a Jacobian-free multiscale method (JFMM). The Jacobian-free approach circumvents explicit computation of the tangent matrix at the macroscale by using a Newton–Krylov process. This has a major advantage in terms of storage requirements and computational cost over existing approaches based on homogenized material coefficients in which explicit Jacobian computation is required at every Newton step. The mechanical response of neutron-irradiated single and polycrystalline OFHC copper is studied and it is shown to capture experimentally observed grain-level phenomena.

Failure Mechanisms and Damage Model of Ductile Cast Iron Under Low-Cycle Fatigue Conditions

Strain-controlled low-cycle fatigue (LCF) tests were conducted on ductile cast iron (DCI) at strain rates of 0.02, 0.002, and 0.0002/s in the temperature range from room temperature to 1073 K (800 °C). A constitutive-damage model was developed within the integrated creep-fatigue theory (ICFT) framework on the premise of strain decomposition into rate-independent plasticity and time-dependent creep. Four major damage mechanisms: (i) plasticity-induced fatigue, (ii) intergranular embrittlement (IE), (iii) creep, and (iv) oxidation were considered in a nonlinear creep-fatigue interaction model which represents the overall damage accumulation process consisting of oxidation-assisted fatigue crack nucleation and propagation in coalescence with internally distributed damage (e.g., IE and creep), leading to final fracture. The model was found to agree with the experimental observations of the complex DCI-LCF phenomena, for which the linear damage summation rule would fail.

Analysis of uniaxial ratcheting behavior and cyclic mean stress relaxation of a duplex stainless steel

The growing presence of a non-conventional stainless steel like a super duplex grade S32750 requires clarification of the structural responses to cyclic loading. This study attempted to characterize the uniaxial ratcheting behavior and the cyclic mean stress relaxation of S32750 super duplex stainless steel which had cyclic hardening characteristics under strain-controlled cyclic loading. A constitutive model capable of simulating the uniaxial ratcheting response and the mean stress relaxation was developed in the framework of rate-independent plasticity theory. Verification of the proposed constitutive model was achieved by simulating uniaxial strain- and stress-controlled cyclic loading tests including cyclic mean stress relaxation and comparing the predicted results with the experimental measurements. The influences of mean stress level, stress amplitude and loading history on the uniaxial ratcheting behavior as well as the effects of mean strain and strain amplitude on the mean stress relaxation was further examined through the verified constitutive model.

Particle finite element analysis of the granular column collapse problem

The problem of granular column collapse is investigated by means of an axisymmetric version of the particle finite element method (PFEM). The granular medium is represented by a simple rate-independent plasticity model and the frictional contact between the granular flow and its rigid basal surface is accounted for. In the version of the PFEM developed for this study, the governing equations of the boundary value problem are cast in terms of an optimization problem and solved using mathematical programming tools. The agreement between model and experiment is generally satisfactory, quantitatively as well as qualitatively. However, the friction angle of the granular material, as well as the exact interface conditions between the base and granular material, are shown to have a relatively significant influence on the results.

Plastic Slip Patterns through Rate-Independent and Rate-Dependent Plasticity

Plastic deformation induces various types of dislocation microstructures at different length scales, which eventually results in a heterogeneous deformation field in metallic materials. Development of such structures manifests themselves as macroscopic hardening/softening response and plastic anisotropy during strain path changes, which is often observed during forming processes. In this paper we present two different non-local plasticity models based on non-convex potentials to simulate the intrinsic rate-dependent and rate-independent development of plastic slip patterns, which is the simplified mechanism for the intrinsic microstructure development. For the sake of mechanistic understanding, the formulation and the simulations will be conducted in one-dimension which does not exclude its extension to multi-dimensions resulting in a crystal plasticity framework.

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.

Edelen’s dissipation potentials and the visco-plasticity of particulate media

Following is an elaboration on D. G. B. Edelen’s (1972–1973) nonlinear generalization of the classical Rayleigh-Onsager dissipation potentials and the implications for the models of viscoplasticity. A brief derivation is given via standard vector calculus of Edelen’s potentials and the associated non-dissipative or “gyroscopic” forces and fluxes. It is also shown that certain extensions of Edelen’s formulae can be obtained by means of a recently proposed source-flux relation or “inverse divergence,” a generalization of the classical Gauss-Maxwell construct. The Legendre–Fenchel duality of Edelen’s potentials is explored, with important consequences for rate-independent friction or plasticity. The use of dissipation potentials serves to facilitate the development of viscoplastic constitutive equations, a point illustrated here by the special cases of Stokesian fluid-particle suspensions and granular media. In particular, we consider inhomogeneous systems with particle migration coupled to gradients in particle concentration, strain rate, and fabric. Employing a mixture-theoretic treatment of Stokesian suspensions, one is able to identify particle stress as the work conjugate of the global deformation of the particle phase. However, in contrast to past treatments, this stress is not assumed to be a privileged driving force for particle migration. A comparison is made with models based on extremal dissipation or entropy production. It is shown that such models yield the correct dissipative components of force or flux but generally fail to capture certain non-dissipative, but mechanically relevant components. The significance of Edelen’s gyroscopic forces and their relation to reactive constraints or other reversible couplings is touched upon. When gyroscopic terms are absent, one obtains a class of strongly dissipative or hyperdissipative materials whose quasi-static mechanics are governed by variational principles based on dissipation potential. This provides an interesting analog to elastostatic variational principles based on strain energy for hyperelastic materials and to the associated material instabilities arising from loss of convexity.

Application of fractional continuum mechanics to rate independent plasticity

In the paper the generalisation of classical rate independent plasticity using fractional calculus is presented. This new formulation is non-local due to properties of applied fractional differential operator during definition of kinematics. In the description small fractional strains assumption is hold together with additive decomposition of total fractional strains into elastic and plastic parts. Classical local rate independent plasticity is recovered as a special case.

Discrete micromechanics of elastoplastic crystals in the finite deformation range

We present a rate-independent crystal plasticity theory in the finite deformation range. The formulation revolves around theory of distribution and strong discontinuity concepts applied to the slip systems. Uniform and conforming deformation fields are introduced, from which deformation gradients for the crystal lattice and the crystal itself are derived. For a crystal deforming in single slip, we show that the crystal rotates the active slip system the same way as the lattice does, leading to an elegant and exact stress-point integration algorithm for the overall crystal stresses. For a crystal deforming in multiple slips the crystal no longer rotates the slip systems exactly as the lattice does. For this case, we present a stress-point integration algorithm accounting for the exact push-forward operation induced by the lattice on the active systems. We also consider a simplified stress-point integration algorithm for multislip systems that remains highly accurate for a wide range of stress paths considered. The framework for system activation and the selection of linearly independent slip systems follows a well-established ‘ultimate algorithm’ for rate-independent crystal plasticity developed for infinitesimal deformation.

On the homogenization of metal matrix composites using strain gradient plasticity

The homogenized response of metal matrix composites (MMC) is studied using strain gradient plasticity. The material model employed is a rate independent formulation of energetic strain gradient plasticity at the micro scale and conventional rate independent plasticity at the macro scale. Free energy inside the micro structure is included due to the elastic strains and plastic strain gradients. A unit cell containing a circular elastic fiber is analyzed under macroscopic simple shear in addition to transverse and longitudinal loading. The analyses are carried out under generalized plane strain condition. Micro-macro homogenization is performed observing the Hill-Mandel energy condition, and overall loading is considered such that the homogenized higher order terms vanish. The results highlight the intrinsic size-effects as well as the effect of fiber volume fraction on the overall response curves, plastic strain distributions and homogenized yield surfaces under different loading conditions. It is concluded that composites with smaller reinforcement size have larger initial yield surfaces and furthermore, they exhibit more kinematic hardening.

Gradient single-crystal plasticity within a Mises–Hill framework based on a new formulation of self- and latent-hardening

This paper develops a theory of rate-independent single-crystal plasticity at small length scales. The theory is thermodynamically consistent, and makes provision for power expenditures resulting from vector and scalar microscopic stresses respectively conjugate to slip rates and their tangential gradients on the individual slip systems. Scalar generalized accumulated slips form the basis for a new hardening relation, which takes account of self- and latent-hardening. The resulting initial-boundary value problem is placed in a variational setting in the form of a global variational inequality.

Variational gradient plasticity at finite strains. Part III: Local–global updates and regularization techniques in multiplicative plasticity for single crystals

The third part of our work on variational inelasticity with long-range effects outlines a formulation and finite element implementation of micromechanically-motivated multiplicative gradient plasticity for single crystals. In order to partially overcome the complexity of full multislip scenarios, we suggest a new viscous regularized formulation of rate-independent crystal plasticity, that exploits in a systematic manner the long- and short-range nature of the involved variables. To this end, we outline a multifield scenario, where the macro-deformation and the plastic slips on crystallographic systems are the primary fields. Related to these primary fields, we define as the long-range state the deformation gradient, the plastic slips and their gradients. We then introduce as the short-range plastic state the plastic deformation map, the dislocation density tensor and scalar hardening parameters associated with the slip systems. It is then shown that the evolution of the short range state is fully determined by the evolution of the long-range state. This separation into long- and short-range states is systematically exploited in the algorithmic treatment by a new update structure, where the short-range variables play the role of a local history base. The model problem under consideration accounts in a canonical format for basic effects related to statistically stored and geometrically necessary dislocation flow, yielding micro-force balances including non-convex cross-hardening, kinematic hardening and size effects. Further key ingredients of the proposed algorithmic formulation are geometrically exact updates of the short-range state and a distinct regularization of the rate-independent dissipation function that preserves the range of the elastic domain. The formulation is shown to be fully variational in nature, governed by rate-type continuous and incremental algorithmic variational principles. We demonstrate the modeling capabilities and algorithmic performance by means of representative numerical examples for multislip scenarios in fcc single crystals.

On the Selection of Active Slip Systems in Rate Independent Crystal Plasticity

Non-uniqueness of the set of active slip systems is a crucial issue in crystal plasticity. To avoid this problem one may perform viscoplastic regularization. This introduces a certain rate dependency, while many crystals are known to behave rate independently. One would require very low viscosity parameters in the regularized model to resemble the experimental behavior of rate independent crystals, which in turn entails numerical difficulties. Furthermore, no direct approach is known to model deformation banding using viscoplastically regularized models. Hence, to adequately treat rate independent crystal plasticity an alternative method is needed. The proposed method, Maximum Dissipation Crystal Plasticity (MDCP), achieves uniqueness by selecting the set of active slip systems according to its dissipation. In a finite element calculation, a system of coupled quadratic equations is solved at every integration point to define the material behaviour. This approach is formally equal to the method of incremental energy minimization recently proposed by Petryk et al. It can be shown that a viscoplastically regularized model is a limiting case of MDCP, giving similar results when cross hardening becomes negligible. Nevertheless, recent 3D dislocation dynamics calculations by Devrince et al. show that cross hardening in fcc crystals is far more important than self hardening. In such cases MDCP gives results distinctly different from its rate dependent counterpart. Fewer slip systems are selected by MDCP, resulting in more slip on the individual active systems. The proposed method is numerically implemented as an Abaqus user material subroutine within the large deformation framework, such that the simulation of arbitrary load cases is possible.

Strong tangential discontinuity modeling of shear bands using the extended finite element method

A method is developed for modeling of shear band with strong tangential discontinuity by means of cohesive surfaces within the extended finite element method (XFEM). A rate-independent non-associated plasticity model is incorporated along the strong discontinuity to consider the highly localized regions. Once the localization is occurred, tangential enrichment degrees of freedom are added to the localized elements, and the discontinuity is captured regardless of mesh resolution and alignment. By introducing the tangential enrichment function, the discontinuity is only imposed in the tangential direction, while the continuity across the shear band is automatically fulfilled. Adopting bilinear quadrilateral elements within the context of XFEM allows for the plastic deformation of shear band to be obtained with quadratic distribution within an enriched element. Since the strong discontinuity approach is employed, the singularity of strain field at the position of displacement jump is attained through a Dirac delta distribution. By means of this singularity, the cohesive shear traction is derived for the J2 plasticity model and is applied onto the band interfaces in order to reproduce the dissipative mechanism of the band. Several numerical examples are analyzed to assess the accuracy and robustness of the proposed approach.

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.

Polycrystalline behavior analysis of extruded magnesium alloy AZ31

Uniaxial tensile and compressive tests were performed at room temperature on extruded AZ31 Mg alloy specimens and distinct tensile-compressive anisotropy was detected. Deformed specimens were examined and the results indicate that the generation of {10\(\bar 1\)2}〈10\(\bar 1\)1〉 twin is responsible for the mechanical anisotropy. A rate independent crystal plasticity model, which accounts for both slip and twinning, was developed for polycrystalline hexagonal close packed (HCP) materials. Model predictions for the stress-strain curves and texture evolution were in reasonable agreement with the experimental results. Specifically, the model captured the three stages of strain hardening for uniaxial-compression. By comparing stress-strain curves and texture evolution between model predictions and experimental measures, information about the dominant slip and twinning systems active at room temperature was deduced.

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.

On the macroscopic description of yield surface evolution by means of distortional hardening models: Application to magnesium

Texture evolution in polycrystals due to rotation of the atomic lattice in single grains results in a complex macroscopic mechanical behavior which cannot be reasonably captured only by classical isotropic or kinematic hardening in general. More precisely and focusing on standard rate-independent plasticity theory, the complex interplay at the microscale of a polycrystal leads to an evolving macroscopic anisotropy of the yield surface, also known as distortional or differential hardening. This effect is of utmost importance, if non-radial loading paths such as those associated with forming processes are to be numerically analyzed. In the present paper, different existing distortional hardening models are critically reviewed. For a better comparison, they are reformulated into the modern framework of hyperelastoplasticity, and the same objective time derivative is applied to all evolution equations. Furthermore, since the original models are based on a Hill-type yield function not accounting for the strength differential effect as observed in hcp metals such as magnesium, respective generalizations are also discussed. It is shown that only one of the resulting models can fulfill the second law of thermodynamics. That model predicts a high curvature of the yield function in loading direction, while the opposite region of the yield function is rather flat. Indeed, such a response can be observed for some materials such as aluminum alloys. In the case of magnesium, however, that does not seem to be true. Therefore, a novel constitutive model is presented. Its underlying structure is comparably simple and the model is thermodynamically consistent. Conceptually, distortional hardening is described by an Armstrong–Frederick-type evolution equation. The predictive capabilities of the final model are demonstrated by comparisons to experimentally measured data.

Numerical simulation of finite strain viscoplastic problems

A large strain elastoplastic constitutive model based on hyperelasticity and multiplicative decomposition of deformation gradient tensor is extended to viscous case, in a framework similar to the one that has been proposed by Ponthot in an hypoelastic context. In this way a very useful framework can be obtained, able to deal with both rate dependent and rate independent problems.In this work a review of theoretical details and numerical implementation of the model are discussed. Similarly to what is done in rate independent plasticity, a Newton–Raphson scheme has been used to solve the non linear consistency condition in order to compute the viscoplastic multiplier.A plane strain plate with a central circular hole under tension is simulated in order to test the proposed model. Large deformation effects are considered in all the simulations carried out. Different parameters of the constitutive model are varied in order to study the sensitivity of the proposed algorithm.

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.