Plastic Strain Rate Tensor Research Articles

The effect of strain biaxiality on the fracture of zirconium alloy fuel cladding

• Experimental device for tubular specimen under biaxial load. • Biaxiality effect on the fracture behavior of zirconium alloy cladding. • Fracture mechnisims of zirconium alloy cladding. During a Reactivity Initiated Accident (RIA), nuclear fuel cladding experiences a multiaxial loading state in which the Pellet-Cladding Mechanical Interaction (PCMI) produces a strain biaxiality ratio (ε zz /ε θθ ) of between 0 and 1. This study examines the effect of the strain biaxiality ratio on the fracture of Zircaloy-4 fuel cladding. A new mechanical test has been developed in order to apply several levels of strain biaxiality. Digital Image Correlation (DIC) is used to measure the strain field, to identify the onset of cladding failure, and to obtain the corresponding critical loading state. Results show that the strain biaxiality has a significant effect on the hoop strain at failure, and the smallest fracture strain is obtained for nearly plane strain conditions. A model is proposed for the accumulation of damage due to plastic deformation, based on the Lode parameter of the plastic strain rate tensor. The model is used to predict the failure hoop strain as a function of the strain biaxiality ratio (ε zz /ε θθ ), and good agreement is found between the experimentally measured and predicted failure strains.

Alternative General Disciplines for Plastic Flow, New Phenomenological and Crystal-Plasticity Models Applicable from Quasi-Static to Extreme High Strain Rate Loadings

Traditional plasticity theories, either time-dependent phenomenological plasticity or crystal plasticity, have been discovered to have some inconsistency problems under dynamic loadings. Such discrepancy is essentially arisen from the fundamental postulation that only internal states of the material determine the additive plastic strain rate tensor. However, in logical, except for internal states, the external loading conditions are also close-relevant factors. In this article, we modified the fundamental postulation, and proposed a set of general disciplines for the determination of material plastic flow, which is based on the view that the plastic flow is determined by both internal states and external loading conditions, and is satisfied with the principle of maximal increment of entropy. Then, we proposed a crystal plasticity model and a phenomenological model based on the new disciplines. A few shock experiments on single-crystal zirconium were conducted to validate the new disciplines and models, including two shots of poly-crystal OFHC copper impacting and one shot of high-energy laser ablation. It was demonstrated that our disciplines and models are well-matched and applicable even up to the strain rate at the order of 109s-1.

Numerical analysis of stress-state-dependent damage and failure behavior of ductile steel based on biaxial experiments

The paper deals with a numerical model to investigate the influence of stress state on damage and failure in the ductile steel X5CrNi18-10. The numerical analysis is based on an anisotropic continuum damage model taking into account yield and damage criteria as well as evolution equations for plastic and damage strain rate tensors. Results of numerical simulations of biaxial experiments with the X0- and the H-specimen presented. In the experiments, formation of strain fields are monitored by digital image correlation which can be compared with numerically predicted ones to validate the numerical model. Based on the numerical analysis the strain and stress quantities in selected parts of the specimens are predicted. Analysis of damage strain variables enables prediction of fracture lines observed in the tests. Stress measures are used to explain different stress-state-dependent damage and failure mechanisms on the micro-level visualized on fracture surfaces by scanning electron microscopy.

Nonstationary Axisymmetric Elastic Processes in an Incompressible Medium with Preliminary Finite Irreversible Deformations

The kinematic equations of the multiplicative model of large elastoplastic deformations are generalized for hardening materials with different principal directions of the stress and plastic strain rate tensors. By taking into account the intermediate configuration, an algorithm is proposed for analyzing dynamic elastic processes in a medium with preliminary irreversible deformations. The use of the algorithm is illustrated by the example of cylindrical elastic shock waves in an incompressible elastoplastic medium.

Application of a Non-Coaxial Soil Model with an Anisotropic Yield Criterion in Tunnelling

Soil strength isotropy and coaxiality of the principal axes of stress and plastic strain rate tensors are often assumed in modelling the soils around tunnels, in the conventional plastic theory. However, the complexity of the soil response during the excavation of tunnelling cannot be well exhibited. In this paper, a recently developed two-dimensional (2D) elastoplastic constitutive model incorporating both soil strength anisotropy and non-coaxiality is firstly reviewed and then implemented in the finite element platform ABAQUS through the user-defined material subroutine (UMAT). This model is then numerically applied to analyze tunnel excavation problems. Two case studies, i.e., centrifuge testing and field testing, are carried out and compared with the numerical simulations to investigate the influences of soil anisotropy and non-coaxiality on the stress paths of soils around the tunnel crown and the subsurface settlement induced by tunnel excavations. The results show that the representative soil elements around tunnel crown experience severe principal stress orientations. The predictions of normalized subsurface settlement troughs can be improved by considering initial soil strength anisotropy compared to the conventional Mohr–Coulomb model. A larger value of the non-coaxial coefficient results in a larger magnitude of the maximum vertical displacement. The normalized subsurface settlement troughs are better improved in the case of centrifuge testing than that of the field testing.

Effect of the plasticity model on the yield surface evolution after abrupt strain-path changes

Abrupt strain path changes without elastic unloading have been used in the literature to investigate the yield surface of sheet metals, both experimentally and theoretically. Such pioneering studies emphasized an apparent non-normality of the plastic strain rate tensor with respect to the trace of the yield surface in stress space, following such a strain-path change. They inspired numerous subsequent developments of plasticity models including non-associated flow rules. In this paper, this type of abrupt strain-path changes is investigated using state-of-the-art plasticity models. The aim is to emphasize the respective contributions of elasticity, isotropic-kinematic hardening, and rate sensitivity, to the apparent violation of the normality condition. The results show that these classical ingredients of plasticity models significantly contribute to the apparent vertex and loss of normality. These effects are quantified for typical sheet metals subject to biaxial-to-shear orthogonal strain path change.

How to Realize Volume Conservation During Finite Plastic Deformation

Volume conservation during plastic deformation is the most important feature and should be realized in elastoplastic theories. However, it is found in this paper that an elastoplastic theory is not volume conserved if it improperly sets an arbitrary plastic strain rate tensor to be deviatoric. We discuss how to rigorously realize volume conservation in finite strain regime, especially when the unloading stress free configuration is not adopted in the elastoplastic theories. An accurate condition of volume conservation is first clarified and used in this paper that the density of a volume element after the applied loads are completely removed should be identical to that of the initial stress free states. For the elastoplastic theories that adopt the unloading stress free configuration (i.e., the intermediate configuration), the accurate condition of volume conservation is satisfied only if specific definitions of the plastic strain rate are used among many other different definitions. For the elastoplastic theories that do not adopt the unloading stress free configuration, it is even more difficult to realize volume conservation as the information of the stress free configuration lacks. To find a universal approach of realizing volume conservation for elastoplastic theories whether or not adopt the unloading stress free configuration, we propose a single assumption that the density of material only depends on the trace of the Cauchy stress by using their objectivities. Two strategies are further discussed to satisfy the accurate condition of volume conservation: directly and slightly revising the tangential stiffness tensor or using a properly chosen stress/strain measure and elastic compliance tensor. They are implemented into existing elastoplastic theories, and the volume conservation is demonstrated by both theoretical proof and numerical examples. The potential application of the proposed theories is a better simulation of manufacture process such as metal forming.

Prediction of microscale plastic strain rate fields in two-phase composites subjected to an arbitrary macroscale strain rate using the materials knowledge system framework

In this work, a data-driven reduced-order model is presented to predict the microscale spatial distribution of the plastic strain rate tensor in an isotropic two-phase composite subjected to an arbitrary macroscopically imposed strain rate tensor. This model was built using the framework of localization linkages called Material Knowledge Systems (MKS), which has been demonstrated to exhibit a remarkable combination of accuracy and low computational cost. In prior work, the MKS framework was successfully used to predict the local strain rate fields in multiphase composites subjected to a selected macroscale strain rate tensor. In this work, the MKS framework is extended to include the complete set of all macroscale strain rate tensors that could be applied. This is accomplished by developing novel representations that allow a parametrization of the localization kernel over the complete space of unit symmetric traceless second-rank tensors and implementing them with the required fast computational strategies. The MKS localization linkage produced in this work was calibrated and validated to results from microscale finite element models.

Closed system of coupling effects in generalized thermo-elastoplasticity

In this paper, the field equations of the generalized coupled thermoplasticity theory are derived using the postulates of classical thermodynamics of irreversible processses. Using the Legendre transformations two new thermodynamics potentials <i>P</i> and <i>S</i> depending upon internal thermodynamic forces Π are introduced. The most general form for all the thermodynamics potentials are assumed instead of the usually used additive form. Due to this assumption, it is possible to describe all the effects of thermomechanical couples and also the elastic-plastic coupling effects observed in such materials as rocks, soils, concretes and in some metalic materials. In this paper not only the usual postulate of existence of a dissipation qupotential (the Gyarmati postulate) is used to derive the velocity equation. The plastic flow constitutive equations have the character of non-associated flow laws even when the Gyarmati postulate is assumed. In general formulation, the plastic strain rate tensor is normal to the surface of the generalized function of plastic flow defined in the the space of internal thermodynamic forces Π but is not normal to the yield surface. However, in general formulation and after the use the Gyarmati postulate, the direction of the sum of the plastic strain rate tensor and the coupled elastic strain rate tensor is normal to the yield surface.

A directional modification of the Levkovitch–Svendsen cross-hardening model based on the stress deviator

In the original Levkovitch–Svendsen cross-hardening model parallel and orthogonal projections required for the yield surface evolution with respective dynamic and latent hardening effects are associated with the unit plastic flow direction np=Ėp/|Ėp|. This work gives a detailed investigation regarding the consequences and proposes the use of the so–called radial direction ns=[S-X]/|S-X| instead where S=dev(σ). It is shown that for an initially plastically anisotropic material under load paths with proportional stresses the original model brings a continuous directional change in the plastic strains. Eventually, even if the dynamic hardening component is bypassed, the material model predicts additional strengthening in loading direction due to latent hardening. In this undesired response, the broken coaxiality of the stress deviator and plastic strain rate tensor with initial anisotropy is the cause. This entanglement of isotropic/kinematic hardening and latent hardening creates difficulties – especially in the parameter identification even for the simplest uniaxial loading. The introduced modification to the model remedies this undesired feature and, hence, makes it possible to isolate the hardening sources during parameter identification stage. The discussions are supported by analytically and numerically derived yield loci for various scenarios. Our analytical studies allow definition of critical material parameter limits for the latent hardening parameter sl in terms of the initial anisotropy and the constant stress deviator ratio.

A bipotential-based limit analysis and homogenization of ductile porous materials with non-associated Drucker–Prager matrix

In Gurson's footsteps, different authors have proposed macroscopic plastic models for porous solid with pressure-sensitive dilatant matrix obeying the normality law (associated materials). The main objective of the present paper is to extend this class of models to porous materials in the context of non-associated plasticity. This is the case of Drucker–Prager matrix for which the dilatancy angle is different from the friction one, and classical limit analysis theory cannot be applied. For such materials, the second last author has proposed a relevant modeling approach based on the concept of bipotential, a function of both dual variables, the plastic strain rate and stress tensors. On this ground, after recalling the basic elements of the Drucker–Prager model, we present the corresponding variational principles and the extended limit analysis theorems. Then, we formulate a new variational approach for the homogenization of porous materials with a non-associated matrix. This is implemented by considering the hollow sphere model with a non-associated Drucker–Prager matrix. The proposed procedure delivers a closed-form expression of the macroscopic bifunctional from which the criterion and a non-associated flow rule are readily obtained for the porous material. It is shown that these general results recover several available models as particular cases. Finally, the established results are assessed and validated by comparing their predictions to those obtained from finite element computations carried out on a cell representing the considered class of materials.

Coupled glide-climb diffusion-enhanced crystal plasticity

This paper presents a fully coupled glide-climb crystal plasticity model, whereby climb is controlled by the diffusion of vacancies. An extended strain gradient crystal plasticity model is therefore proposed, which incorporates the climbing of dislocations in the governing transport equations. A global–local approach is adopted to separate the scales and assess the influence of local diffusion on the global plasticity problem. The kinematics of the crystal plasticity model is enriched by incorporating the climb kinematics in the crystallographic split of the plastic strain rate tensor. The potential of the fully coupled theory is illustrated by means of two single slip examples that illustrate the interaction between glide and climb in either bypassing a precipitate or destroying a dislocation pile-up.

Anisotropic ductile damage fully coupled with anisotropic plastic flow: Modeling, experimental validation, and application to metal forming simulation

The main goal of this paper is the modeling, numerical simulation, and experimental validation of the anisotropic ductile damage effects on initially anisotropic plastic flow with mixed (isotropic and kinematic) nonlinear hardening under large plastic strains for metal forming processes simulation. A symmetric second-rank damage tensor together with a symmetrized fourth-rank damage-effect tensor is used to describe the anisotropic ductile damage evolution and its effect on the large plastic flow with hardening. Following the concept of effective state variables in the framework of the total energy equivalence assumption, the “Murakami” fourth-rank damage-effect tensor is chosen to describe the anisotropic damage effect on the elastic-plastic behavior including the mixed hardening. The “Lemaitre” ductile anisotropic damage evolution relationships, where the principal directions of the damage rate tensor are governed by those of the plastic strain rate tensor, are used. As difference with the works cited above, the nonlinear mixed isotropic and kinematic hardening is taken into account considering the full and strong damage effects through the effective state variables deduced from the total energy equivalence assumption initially proposed by Saanouni et al. The non-associative plasticity theory is considered, and the “ Hill 1948 ” quadratic (equivalent) stress norm is used to describe the large plastic anisotropic flow accounting for mixed isotropic and kinematic hardening with anisotropic damage effects. The formulation is performed assuming finite plastic strains and small elastic strains through the so-called rotated frame formulation. The obtained model was implemented into ABAQUS/Explicit® FE software thanks to the user’s developed subroutine VUMAT. The numerical aspects related to the time discretization of the fully coupled anisotropic constitutive equations are carefully described. Finally and for the validation purpose, the model is identified using an appropriate experimental data base concerning the grade 316L stainless steel to simulate numerically some metal forming processes.

Material Model based on Non-associated Flow Rule with Higher-order Yield Function for Anisotropic Metals

A new expression for the plastic constitutive model for materials with initial anisotropy is proposed. A plastic strain rate tensor should be permitted to follow, to a certain extent, the rotation of the stress rate tensor, which rotates instantly from the direction of the current stress tensor as in the case of plastic instability. For this purpose, a non-associated normality model, in which the plastic potential function is defined independently of the yield function, has been adopted in the proposed model. An explicit expression for the equivalent plastic strain rate, which is plastic-work-conjugated with the defined equivalent stress corresponding to the proposed yield function, is also presented. This is important for expressing a generalized work-hardening rule for materials with plastic flow stress anisotropy. The proposed theory is expected to overcome the serious problems in the associated plastic flow theory.

Comment on the use of the associated flow rule for transversely isotropic elasto-viscoplastic materials

The constitutive modeling of rate-dependent plasticity of anisotropic materials is critically examined. Specifically, the paper concentrates on the use of the associated flow rule, also known as the normality rule, which is typically applied to formulate a constitutive rule for the plastic strain rate tensor from a static yield criterion (Bishop and Hill, 1951; Hill, 1948, 1971). For anisotropic yielding, the associated flow rule is based on the yield criterion of Hill (1948). Using the resulting plastic strain-rate tensor in a dynamic model for elasto-viscoplasticity and studying the onset of yielding, one finds that the resulting stresses at yielding are inconsistent with the values of the yield stresses initially contained in the yield criterion of Hill. A procedure is developed to quantify the inconsistency in a compact form and to correct the associated flow rule to achieve consistency. By analyzing experimental yield data for transversely isotropic polypropylene, the magnitude of the required correction term is quantified. In contrast to the associated flow rule, consistency-related corrections are practically obsolete for a recently developed constitutive rule for the plastic strain rate tensor that is of statistical origin and not based on a yield criterion (Hütter and Tervoort, 2008, 2013).

Stress update algorithm for non-associated flow metal plasticity

. In this paper we present the continuum plasticity model based on non-Associated Flow Rule (nonAFR) for Hill’s48 quadratic yield function. In case of non-AFR, Hill’s quadratic function used as plasticpotential function, makes use of plastic strain ratios to determine the direction of effective plastic strain rate.In addition, the yield function uses direction dependent yield stress data. Therefore more accuratepredictions are expected in terms of both yield stress and strain ratios at different orientations. Weimplemented a modified version of the non-associative flow rule originally developed by Stoughton [1] intothe commercial finite element code ABAQUS by means of a user material subroutine UMAT. The mainalgorithm developed includes combined effects of isotropic and kinematic hardening [2]. This paper assumesproportional loading cases and therefore only isotropic hardening effect is considered. In our model theincremental change of plastic strain rate tensor is not equal to the incremental change of the compliancefactor. The validity of the model is demonstrated by comparing stresses and strain ratios obtained from finiteelement simulations with experimentally determined values for deep drawing steel DC06. A criticalcomparison is made between numerical results obtained from AFR and non-AFR based models.

Tangential continuity of elastic/plastic curvature and strain at interfaces

The continuity vs discontinuity of the elastic/plastic curvature & curvature rate, and strain & strain rate tensors is examined at non-moving surfaces of discontinuity, in the context of a field theory of crystal defects (dislocations and disclinations). Tangential continuity of these tensors derives from the conservation of the Burgers and Frank vectors over patches bridging the interface, in the limit where such patches contract onto the interface. However, normal discontinuity of these tensors remains allowed, and Kirchhoff-like compatibility conditions on their normal discontinuities across the concurring interfaces are derived at multiple junctions. In a simple plane case and in the absence of surface-disclinations, the compatibility of the normal discontinuities in the elastic curvatures assumes the form of a Young’s law between the grain-to-grain disorientations and the sines of the dihedral angles. Complete continuity of the plastic strain rate tensor at triple junctions also derives from the compatibility of the normal discontinuities in the plastic strain rates in such conditions.

Evaluation of Associated and Non-Associated Flow Metal Plasticity; Application for DC06 Deep Drawing Steel

In this paper the capabilities of Associated Flow Rule (AFR) and non-AFR based finite element models for sheet metal forming simulations is investigated. In case of non-AFR, Hill’s quadratic function used as plastic potential function, makes use of plastic strain ratios to determine the direction of effective plastic strain rate. In addition, the yield function uses direction dependent yield stress data. Therefore more accurate predictions are expected in terms of both yield stress and strain ratios at different orientations. We implemented a modified version of the non-associative flow rule originally developed by Stoughton [1] into the commercial finite element code ABAQUS by means of a user material subroutine UMAT. The main algorithm developed includes combined effects of isotropic and kinematic hardening [2]. This paper assumes proportional loading cases and therefore only isotropic hardening effect is considered. In our model the incremental change of plastic strain rate tensor is not equal to the incremental change of the compliance factor. The validity of the model is demonstrated by comparing stresses and strain ratios obtained from finite element simulations with experimentally determined values for deep drawing steel DC06. A critical comparison is made between numerical results obtained from AFR and non-AFR based models

J031011 応力増分方向依存性構成式の塑性異方性への展開

It is proposed the expression of a newly extended plastic constitutive model to initial anisotropy materials. Proposed flow rule permits a plastic strain rate tensor to follow up a certain extent a stress rate tensor which rotates instantly from the direction of current stress tensor as is the case of plastic instability phenomenon. It has quite same structure of the non-associated normality model in which the plastic potential function is defined independently from yield function. It is also suggested the expression of the equivalent plastic strain iate which is conjugated with the equivalent stress corresponding to the proposed yield function in plastic work rate. This is important to express generalized work-hardening rule for materials which have plastic flow stress anisotropy. The proposed theory may overcome the serious problems in the associated plastic flow theory.

Comparative study and application of a coupled nonlinear kinematic hardening model for structural metallic material behavior

According to the experimental observation made by Phillips and Lee in 1979, the direction of the movement of the yield surface centre occurs between the stress rate tensor and the plastic strain rate tensor. When the plastic strains dominate the global behavior of the structure in question, the effect of the plastic flow rule can be unwitting overlooked. However, the application of more advanced engineering materials, e.g. high strength steel etc., is increasing and the ambiguity of plastic flow can not be ignored. Take the high strength steel of 420Mpa as an example, a detailed comparative discussion of a coupled nonlinear kinematic hardening model is presented to account for such an experimental observation. A better prediction for the kinematic hardening behavior can be expected by adding a new term originally proposed by Frederick and Armstrong in 1996. In this work, the advantages as well as the accuracy for the coupled nonlinear constitutive model is discussed.