Mixed Hardening Research Articles

Time Integration Algorithm for a Cyclic Damage Coupled Thermo-Viscoplasticity Model for 63Sn-37Pb Solder Applications

In this paper, a semi-implicit time integration scheme has been developed for a damage-coupled constitutive model to characterize the mechanical behavior of 63Sn-37Pb solder material under thermo-mechanical fatigue (TMF) loading. The scheme is developed to provide an efficient numerical procedure of integration and iteration for calculating stress and other associated state variables within a strain-driven format. In particular, a novel Newton-Raphson iteration algorithm for the damage coupled constitutive material model involving von Mises viscoplastic potential function with nonlinear mixed hardening is formulated. An algorithmic tangent stiffness tensor is derived and the model is implemented numerically into a commercial finite element (FE) code ABAQUS through its user-defined material subroutine. Several numerical simulations are conducted for validation of the proposed algorithm.

Prediction of ductile fracture using isotropic and kinematic hardening rules including void nucleation

ABSTRACTA formulation combining kinematic translation and isotropic expansion for a yield surface based on Gurson–Tvergaard function is used to describe void growth. By adopting a criterion of internal necking of the ligaments between voids, fracture strains for tensile bars made of conventional alloys and powder metal compacts—with micromechanical parameters mostly identified from experiments—are predicted according to kinematic, isotropic and mixed hardening models. Fracture strains predicted by the kinematic‐hardening model are in closer agreement with experiments whereas those estimated according to isotropic‐hardening model are overestimated. The consideration of either step‐like or continuous void nucleation models indicates its great influence on fracture strains and emphasizes a further need to quantify the statistical parameters involved in these models.

A Generalized Mixed Isotropic-Kinematic Hardening Plastic Model Coupled with Anisotropic Damage for Sheet Metal Forming

The paper presents a generalized mixed isotropic-kinematic hardening plastic model coupled with anisotropic damage for sheet metal forming. A nonlinear anisotropic kinematic hardening is developed. For the predication of limit strains at localized necking in stamping under complex strain history, the model and its associated damage criterion for localized necking are established and implemented into LS-DYNA3D by compiling it as a user subroutine. The finite element simulation of LS-DYNA3D based on the present model is carried out. The location of localized necking for sheet metal forming has been successfully identified.

On modeling mixed hardening in computational plasticity

We address herein the calculation of Prager’s hardening parameter in computational plasticity when mixed hardening is considered. We consider two approaches to evaluate the mixed hardening response; namely, based on splitting the plastic strains and based on splitting the plastic modulus. For a one-dimensional stress–strain curve with nonlinear hardening, the proper calculation of Prager’s hardening parameter is demonstrated and some comparisons and insight are provided.

Orthotropic damage in high-strength steel sheets. An elasto-viscoplastic material model with mixed hardening

Within the scope of thermodynamics with internal variables, constitutive and evolution equations (representing ductile deformation of sheets made of high strength steel alloys) with mixed hardening and damage have been derived. As a result of the derivation, the rate-dependent elasto-plastic constitutive model is identified. The material is assumed to be oriented in the principal damage directions, indicating orthotropic damage. Owing to postulates within continuum damage mechanics, a general expression for degradation of elastic properties in materials has been obtained. A numerical algorithm for the integration of the constitutive equations has been developed as well, based on an elastic predictor - plastic/damage corrector procedure. The plastic/damage corrector is based on a fully implicit backward Euler scheme. In order to consider viscoplastic material properties, the overstress (in the definition of the plastic multiplier) is a function of the plastic yield function.

Orthotropic damage in high-strength steel sheets. An elasto-viscoplastic material model with mixed hardening

Orthotropic damage in high-strength steel sheets. An elasto-viscoplastic material model with mixed hardening

The constitutive equations for mixed hardening orthotropic material

A dimensionless stress yield criterion is proposed to describe the mixed hardening of orthotropic material, including kinematic hardening and proportional hardening, and the associated plastic flow law is derived. The generalized effective stress-strain formulae can be obtained correspondingly based on the experimental stress-strain curves in various simple stress states. The initial plastic anisotropy is influenced by the elastic anisotropy. The yield criterion can be reduced to Huber-Mises Criterion for isotropic materials and associated constitutive equations can be degenerated into Prandtl-Reuss equations.

Simulation of Evolution of the Residual Stress in the Surface Layer under Tension and Torsion

It is attempted to combine the method of approaching the problem of surface layer from the point of view of mechanics with that of production technology. The real layer formed by production engineers is described by considering the continuous elastic-plastic model with mixed type strain hardening. The theoretical results thus obtained make it possible to construct a computation program for the state of stress in an axially symmetric element with a surface layer under tension and torsion loading. An application of the model for the computation of the residual stress state produced in a real layer as a result of induction hardening and burnishing, for a stabilized loop of cyclic strain, has been shown.

Identification of a microstructural model for steels subjected to large tensile and/or simple shear deformations

The aim of this paper is to study and to model the anisotropic work-hardening behaviour of a low-carbon interstitial free steel under strain-path changes. The model used is a mixed hardening model in which the texture anisotropy is represented by Hill criterion, whereas both the isotropic and the kinematic hardening are associated with the dislocation structures. In the present investigation, the texture evolution is neglected. Specifically, for an initially orthotropic material, it is assumed that the texture preserves its orthotropic symmetry with respect to a frame that rotates during the deformation. An experimental study, numerical implementation, identification of this microstructural model is proposed. The computation shows that the model provides a satisfactory quantitative description of the influence of a predeformation on the subsequent work-hardening and this for different mechanical tests.

Erratum to “Modeling mixed hardening of alkali halides with a modified version of an internal state variables model” [International Journal of Plasticity 15 (1999) 1067–1088

Erratum to “Modeling mixed hardening of alkali halides with a modified version of an internal state variables model” [International Journal of Plasticity 15 (1999) 1067–1088

935 有限要素法のための繰返し塑性モデルの陰的積分法

This paper describes the stress integration and the consistent tangent modulus for the elasto-plastic finite element method. The return mapping algorithm is applied to the constitutive equations discretized by the backward Euler method for the stress integration, and which are linearized in order to derive the consistent tangent modulus. The plasticity model employed which is based on the assumptions of small deformation, isotropic elasticity, flow rule associated with the von Mises yield surface, and mixed hardening is effective especially for analysis of ratcheting and cyclic stress relaxation due to the kinematic hardening rule suggested recently by Ohno and Karim.

A new cyclic anisotropic model for plane strain sheet metal forming

A finite difference model was developed for sheet metal subjected to plane strain cyclic bending under tension. The model was used to describe the effects on the mechanics of the deformation of the sheet due to mixed isotropic-kinematic cyclic hardening curves. The analytical results were compared with experimental results obtained under testing conditions closely representative of the analytical model. Two tests were used: a pure bending moment device and a bending/unbending under tension device consisting of three cylindrical pins. The model was used to determine a constitutive curve that best characterize the cyclic behavior of the material tested, as compared with the experimental results. The significance of these results were discussed in relation to the prediction of the restraining forces in the sheet as it is drawn through the blank holder drawbeads.

A rate-dependent model for the ductile behavior of salt rocks

Salt rocks, such as potash and rock salt, show a complex mechanical behavior that is rate and history dependent. Various approaches exist to represent the main features of the response of these rocks through constitutive models. The approach proposed in this paper is to formulate the constitutive equations using internal state variables (ISV) in the context of the unified theory of inelastic flow. To illustrate the application of this approach, the authors use an updated version of a viscoplastic model with reduced expressions for laboratory triaxial tests at relatively high confining pressure. It is shown that, under these conditions, mixed hardening of salt rocks can be described through a combination of scalar and directional ISV. The authors also describe in some detail a procedure to obtain the material parameters for the model equations. The paper shows some characteristics of the model response for various loading paths, illustrating the good agreement with experimental observations and highlighting some of the differences with a classical isotropic hardening model.

Numerical determination of yield surfaces of Al/SiC laminated composite plates

In this study the elastic-plastic properties of an aluminum alloy reinforced by unidirectional SiC-fibers are computed by using nonlinear finite element simulations. In order to carry out these computations, a representative volume element assuming a regular quadratic arrangement of the fibers is used. The plastic properties of the composite are described by a quadratic flow criterion which is able to consider both direction and tension-compression anisotropy. Additionally, the flow rule takes into account plastic volume changes. The hardening of the material is defined by a mixed isotropic-kinematic hardening model. By using this modeling yield surfaces of multi-layer laminated composite plates under an in-plane loading are determined.

The B-bar method and the limitation principles

The generalized elastic material provides a reference model to cast in a unitary framework many structural models which are based on nonlinear monotone multivalued relations such as viscoelasticity, plasticity and unilateral models. The modified forms of the Hu-Washizu and Hellinger-Reissner principles and the displacement-based variational formulation are recovered for the generalized elastic material starting from a functional in the complete set of state variables. The related limitation principles are derived and their specialization to elasticity and elastoplasticity with mixed hardening are provided. It is shown that the interpolating fields for the pressure and the volumetric strain usually adopted in the B-bar method lead to a limitation principle. Accordingly the same elastic and elastoplastic solutions can be obtained by means of an approximate mixed displacement⧸pressure variational principle. A second application is concerned with the conditions ensuring the coincidence of the solutions between an approximate two-field mixed formulation and the displacement-based method. Numerical examples are provided to show the coincidence of the solutions obtained from different mixed finite element formulations, in elasticity or elastoplasticity, under the validity of the limitation principles.

Constitutive equations of creep under changing multiaxial stresses

Creep constitutive equations are derived here to describe creep behaviour of metals at non-proportional loading by generalisation of non-linear viscoelaticity equations equipped with temporal analogy of time-stress type. To obtain the new equations, integral dependences for the case of degenerative integral operators are first transformed into differential ones. It is shown that the latters imply the kinematic hardening law. In the proposed constitutive equations, mixed hardening is adopted as a composition of three hardening mechanisms connected with translation, size and shape of the potential surfaces respectively. Weight factors of these mechanisms are defined by two material constants which are obtained from the creep data of non-proportional loading. Two measures of isotropic hardening are introduced: the first is the equivalent strain and the second is a non-decreasing parameter. The constitutive equations introduced are evaluated on the base of experimental data on creep behaviour of D16T aluminium alloy and VT3-1 titanium alloy under non-proportional step loading. Comparison of the theoretical results with the experimental data indicates that the former give good predictions of the material responses. Noticeable differences between the two isotropic measures on predicting the isotropic hardening effect appear when the preceding stress vector rotation is not less than 90°. The non-decreasing parameters are better in rotations of the kind.

Modeling mixed hardening of alkali halides with a modified version of an internal state variables model

Abstract In the ductile regime of inelastic flow, alkali halides behave similarly to various metals. Their behavior is fully plastic, shows strong rate and history dependencies, and exhibits anisotropic hardening. In this paper, the authors present the most recent version of SUVIC, a viscoplastic model with internal state variables (ISV). After recalling the main features of alkali halides inelastic response, the modified equations of the model are presented. They are then used to represent the behavior of polycrystalline sodium chloride submitted to conventional triaxial compression (CTC) and reduced triaxial extension (RTE) tests, using results that highlight mixed (kinematic and isotropic) hardening of the material, hence showing a type of Bauschinger effect. A discussion follows.

A general approach to the evaluation of consistent tangent operators for rate-independent elastoplasticity

We present a general approach to the derivation of the explicit expression of tangent operators, consistent with the finite-step integration schemes of the flow rule, for rate-independent elastoplastic models with mixed hardening. By exploiting the model of generalized standard material, we show that the algorithmic tangent operator can be obtained by inverting a suitable positive definite matrix. Use of the Sherman-Morrison-Woodbury formula allows us to reduce to a half the burden associated with such inversion and to derive an expression of the consistent tangent operator prone to computer implementation. The application of the proposed approach to the von Mises yield criterion with linear kinematic and isotropic hardening shows that the closed-form expression of the operator can be obtained by a straightforward procedure.

Modelling of plastic anisotropy based on texture and dislocation structure

Anisotropy and work hardening behaviour are the most significant material properties in sheet metal forming. Beside the initial anisotropy due to crystallographic texture, anisotropic hardening due to the evolution of the dislocation structure is observed at large prestrains. The developed model is a mixed hardening model in which the shape of the yield locus is generated from the texture and the dislocation structures are responsible for isotropic hardening and kinematic hardening. This microstructural evolution is modelled with three internal state variables and their evolution equations. Furthermore an elastoplastic tangent modulus is derived from the plasticity model. The model is constructed in the framework of dual potentials both in deviatoric stress space and plastic strain rate space. Finally, the flow stress evolution is computed for several strain-path changes.

The Governing Parameter Method for implicit integration of viscoplastic constitutive relations for isotropic and orthotropic metals

A general algorithm of implicit stress integration in viscoplasticity, based on the governing parameter method (GPM) is briefly presented. It is assumed that the associative viscoplastic constitutive relations are governed by the Perzyna formulation with a generalization suggested by Simo and Hughes. The algorithm is first applied to isotropic metals obeying the von Mises yield condition with mixed hardening and then, to orthotropic metals with a generalized Hill's yield condition including a mixed hardening assumption. Derivation of consistent tangent moduli is presented for both viscoplastic material models. The proposed computational procedures are efficient, since they reduce the problem of stress integration to the solution of one nonlinear equation, can use large time steps and are applicable to 2-D, 3-D, shell and beam structures. The tangent elastic viscoplastic matrix provides high convergence rate in the overall equilibrium iterations. Numerical examples illustrate the main characteristics of the developed computational procedures.