## What is/are Isotropic Hardening?

Isotropic Hardening - Moreover, the cyclic loading results demonstrate the isotropic hardening behaviors and the saturation of yielding strength when the maximum strain reaches 10%.^{[1]}These features include kinematic hardening when the proof strain denoting yield is low, isotropic hardening when it is high, formation of a ‘nose’ in the loading direction and flattening of the rear part of the yield surface.

^{[2]}Isotropic hardening plasticity and nonlinear failure criterion were considered for both ferrite and martensite phases.

^{[3]}From plastic work equivalence, an isotropic hardening model can be readily constructed from the tensile and shear test data without inverse finite-element analysis.

^{[4]}Additional simulations were performed on bead on plate model considering JIS-SM400 in order to investigate applicability of isotropic hardening model for residual stress estimation.

^{[5]}Finally, after comparison between four combinations of constitutive behavior, a simple isotropic yield with isotropic hardening model is suggested to predict the tensile residual hoop stress in the rolled joint, conservatively.

^{[6]}— Based on the analysis of the results of experimental studies of 12Х18Н10Т stainless steel specimens under a rigid (controlled deformation) process of deformation, which includes a sequence of monotonic and cyclic loading modes, some features and differences in isotropic and anisotropic hardening processes under monotonic and cyclic loading are revealed.

^{[7]}The proposed yield criterion can accurately predict the anisotropic hardening along 0°, 45°, 90° and equi-biaxial directions under tension and compression, which will make the significant improvements in the stress predictions including 15°, 30°, 60° and 75°.

^{[8]}The theory is based on the concepts of free energy and damage parameter and has the attributes of both rate-dependent and rate-independent plasticity, kinematic and isotropic hardening.

^{[9]}In addition, the material functions for the elastic modulus, the yield function and the isotropic hardening/softening will be modified for the accurate description of the cyclic mobility.

^{[10]}Then the predicted temperature distribution was sequentially coupled to the mechanical analysis considering the isotropic hardening model.

^{[11]}In this work, the description of the anisotropic nature of the hardening of the composite material and the numerical homogenization for the J2 flow with isotropic hardening is proposed.

^{[12]}The relations of the flow theory with kinematic and isotropic hardening are used as equations of state.

^{[13]}In the proposed cyclic hardening law, isotropic hardening was calculated in the first loading cycle.

^{[14]}To treat the bulk, an elastoplastic material model with isotropic hardening as well as different hardening rules for small strains is incorporated into the DG framework.

^{[15]}The isotropic hardening and von-Mises yield criteria are considered to check the plasticity condition.

^{[16]}Furthermore, by assuming that the general formulation obtained for the perfectly plastic matrix remains valid for each loading increment, the BIV model is extended to considering that the solid matrix exhibits an isotropic hardening by using an explicit algorithm.

^{[17]}The paper presents a finite deformation, isotropic hardening, non–associative elastic–plastic constitutive model ( FD _ Milan model) for describing the mechanical behavior of a wide range of bonded natural geomaterials such as stiff overconsonsolidated clays, porous soft rocks or bio–improved soils.

^{[18]}The elastic-plastic constitutive model with isotropic hardening is taken as an example for illustration.

^{[19]}A novel strategy to model the isotropic hardening stagnation is developed within a fully implicit integration scheme in order to speed up the computation and to improve the material description.

^{[20]}Therefore, first, the proportional results are reproduced and then the non-proportionality parameter is incorporated into the isotropic hardening law.

^{[21]}The constitutive relationships are developed for anisotropic materials with an anisotropic hardening assumption.

^{[22]}Exploration is made for quasi-static finite deformations of isotropic hardening elastic–viscoplastic solids.

^{[23]}A commercially available Abaqus/CAE software was used to perform an explicit non-linear analysis with a macroscale modelling approach, assuming the recycled composites as both homogenous and isotropic hardening.

^{[24]}This plastic model bypasses a multistep numerical procedure in calculating the first derivative of homogeneous anisotropic hardening (HAH) [1] function.

^{[25]}Furthermore, we present unified formulations for saturated and unsaturated states in which the isotropic hardening law and the critical state line are described in a bi-logarithmic space defined by the logarithms of the mean effective stress and void ratio.

^{[26]}Results show that QP1180 and QP980 exhibit pronounced strain hardening, strength differential effect and anisotropic hardening rather than DP980.

^{[27]}A modified Drucker-Prager Cap (DPC) model with an elliptical cap surface using the new material characterization method was developed to capture the anisotropic hardening behavior and hydrostatic effect of the powder mixture.

^{[28]}The polymeric matrix is treated as an ideal elastoplastic solid with isotropic hardening behavior, whereas the clay nanoparticles are simplified as stiff, linearly elastic platelets.

^{[29]}Considering thermo-metallurgical-mechanical coupling behaviors, solid phase transition effect, the isotropic hardening model and mobile heat resource were performed on the stress field and distortion in the joints of V-bevel angle 45°, 60°, 75°, and 90°.

^{[30]}Upon these assumptions, along with mathematical and physical laws governing the developments of interfacial damage, kinematic/isotropic hardenings, and frictional sliding, the model is strictly deduced and numerically implemented within the thermodynamic framework.

^{[31]}Further, the elastoplastic behavior of 5083-aluminium alloy is determined through uniaxial tensile tests, so that the anisotropic, the isotropic hardening and the damage parameters are acquired.

^{[32]}Numerical modelling is performed in micro-structures using the concept of representative volume element (RVE) where the matrix is considered as an ideally plastic material governed by the von Mises model with isotropic hardening, while inclusions are adopted as very stiff elastic materials.

^{[33]}The calculations used a plasticity model with isotropic hardening.

^{[34]}At the same time, a plastic behavior law with isotropic hardening for the bolt is proposed.

^{[35]}The enhanced Homogeneous Anisotropic Hardening (e-HAH) is an advanced constitutive model that can take account of the strain path change influence on the material behavior by using a stress-based indicator.

^{[36]}Kinematic hardening is suggested to substitute the isotropic hardening assumption for better prediction of FLCs with strain path changing effect.

^{[37]}Further, the elastic–plastic analysis of FGM is assumed to follow J2-plasticity with isotropic hardening.

^{[38]}Elastic, isotropic hardening and Johnson-Cook material model were selected to account for elasticity, plasticity and stress softening behaviour of the ductile materials as the material model formulations respectively.

^{[39]}On the numerical side, the evolving non-associated Hill48 (enHill48) plasticity model considering anisotropic hardening and plastic strain ratio evolution is employed to describe the anisotropic plastic deformation.

^{[40]}The corresponding macroscopic constitutive equations are based on the second Ohno-Wang model, combined with refined rule of isotropic hardening.

^{[41]}The constitutive models were applied in simulations of single- and multi-stage cold forming processes, revealing the significant effect of anisotropic hardening on the predicted component properties and process forces, originating in the process-intrinsic strain path reversals as well as in strain path reversals between subsequent forming stages.

^{[42]}The relaxation of residual stress under a cyclic load is a transient phenomenon that is described in this paper using classic plasticity theory by employing a combined, non-linear, kinematic and isotropic hardening material model.

^{[43]}New empirical relationships and probabilistic distributions of the optimized model parameters, such as post-yielding hardening ratio, isotropic hardening in compression and tension, plus initial curvature, are presented.

^{[44]}Constitutive relations account for J 2 -flow theory with nonlinear kinematic/isotropic hardening, coupled with isotropic continuum damage mechanics.

^{[45]}The elastoplastic constitutive equations are obtained incrementally using Hoffman’s criterion and isotropic hardening model.

^{[46]}To study the effect of constitutive model on the accuracy of forming simulations, a combined nonlinear isotropic/kinematic hardening model as well as the isotropic hardening part of the same model are calibrated based on five compression-tension-compression uniaxial stress experiments.

^{[47]}Back stress and drag stress are used to describe the kinematic and isotropic hardening/softening behaviour of the material, respectively.

^{[48]}The following models were used in the analysis: linear-elastic, elastic–perfectly plastic (Coulomb–Mohr) and elastic–plastic with isotropic hardening (Modified Cam-Clay).

^{[49]}In this study, the strengths of a steel pipe in different material orientations are predicted using a distortional anisotropic hardening model, namely, the HAH model, implemented in a finite element simulation.

^{[50]}

## associated flow rule

An artificial neural network (ANN) was constructed to replace the nonlinear stress-integration scheme conducted in the conventional theoretical constitutive model under isotropic hardening and associated flow rule.^{[1]}An anisotropic plasticity model is developed based on the non-associated flow rule with anisotropic hardening.

^{[2]}It was found that the associated-flow-rule-based YLD2000-2D and Hu2005 yield criteria under the condition of anisotropic hardening can be used for predictions of yield stress whereas the isotropic hardening model is not as representative.

^{[3]}The associated flow rule and isotropic hardening were assumed.

^{[4]}

## finite element model

53 g/cm 3, close to experimental values of the deformation modulus were obtained using a three-dimensional finite-element model of polylinear isotropic hardening "PLAS (Miso)," in the program "ANSYS.^{[1]}Multilinear isotropic hardening and modified time hardening models are used to create the finite element model.

^{[2]}The finite element model based on the non-linear elastic model and the homogeneous anisotropic hardening model (HAH) is also established for the springback prediction and stress analysis.

^{[3]}

## equivalent plastic strain

The cohesive zone model and equivalent plastic strain criterion with isotropic hardening is respectively employed to describe the interlaminar and metal layer damage.^{[1]}The results show that the proposed yield function maintains great accuracy for anisotropic hardening with increasing equivalent plastic strain (EPS) beyond the initial yield loci.

^{[2]}An arbitrary function of the equivalent plastic strain controls isotropic hardening, and Prager’s law describes kinematic hardening.

^{[3]}

## Bilinear Isotropic Hardening

Three elastoplastic models are evaluated in this study: the bilinear isotropic hardening model with Von Mises yield condition, the bilinear isotropic hardening model with the Hill anisotropic yield condition, and the nonlinear isotropic hardening model with Hill anisotropic yield condition.^{[1]}The heterogeneity of the material was modeled by layer-by-layer elastic-plastic properties with bilinear isotropic hardening.

^{[2]}The finite element analysis (FEA) software ANSYS is used for calculation and analysis, the ANAND constitutive model is adopted to describe the visco-plastic behavior of AU80Sn20 and sintered nano-silver, the bilinear isotropic hardening is adopted to describe the plastic deformation of materials, and the thermal-mechanical coupling is used to realize the multi-scale coupling between the soldering layer and other function layers.

^{[3]}Based on the stress relaxation tests and the equivalent vibration equation of modal analysis, the creep constitutive model and the bilinear isotropic hardening plasticity material model (BISO) are combined to establish the numerical simulation model of TVSR of 7075 aluminum alloy ring part.

^{[4]}For steel reinforcement, a bilinear isotropic hardening model, a linear orthotropic model was used for composite reinforcement.

^{[5]}The Young's modulus, yield strength and tangent modulus (for bilinear isotropic hardening) are varied according to an exponential function.

^{[6]}The analysis was performed for two material models, elastic-perfectly plastic and bilinear isotropic hardening.

^{[7]}

## Multilinear Isotropic Hardening

Multilinear isotropic hardening and modified time hardening models are used to create the finite element model.^{[1]}Numerical calculations were carried out in the Ansys® software and involved the application of bilinear and multilinear isotropic hardening.

^{[2]}In the numerical analysis of this joint type, two different material models including multilinear isotropic hardening (MISO) and cohesive zone model (CZM) were used in the adhesive layer.

^{[3]}For that aim, the model is enriched by estimating a temperature-dependent friction coefficient using theoretical relationships, and by considering a temperature-dependent multilinear isotropic hardening equation as a plasticity model representing the material.

^{[4]}

## Dependent Isotropic Hardening

We outlined a micromechanical model with a lattice-misfit-dependent isotropic hardening that takes a unique description depending on the phase: an Eshelby tensor (sphere or penny) for the precipitates (γ’ phase) and a continuous medium for the matrix (γ phase).^{[1]}A common belief in phenomenological strain gradient plasticity modeling is that including the gradient of scalar variables in the constitutive setting leads to size-dependent isotropic hardening, whereas the gradient of second-order tensors induces size-dependent kinematic hardening.

^{[2]}

## Linear Isotropic Hardening

The von Mises yield criterion of plane-stress with linear isotropic hardening is adopted in constitutive description of elasto-plastic material.^{[1]}The model incorporates damage evolution and healing model, and the von Mises linear isotropic hardening plasticity model and is validated with static tensile and two-cycle tensile tests on pure Surlyn® 8940 specimens.

^{[2]}

## isotropic hardening model

From plastic work equivalence, an isotropic hardening model can be readily constructed from the tensile and shear test data without inverse finite-element analysis.^{[1]}Additional simulations were performed on bead on plate model considering JIS-SM400 in order to investigate applicability of isotropic hardening model for residual stress estimation.

^{[2]}Three elastoplastic models are evaluated in this study: the bilinear isotropic hardening model with Von Mises yield condition, the bilinear isotropic hardening model with the Hill anisotropic yield condition, and the nonlinear isotropic hardening model with Hill anisotropic yield condition.

^{[3]}Finally, after comparison between four combinations of constitutive behavior, a simple isotropic yield with isotropic hardening model is suggested to predict the tensile residual hoop stress in the rolled joint, conservatively.

^{[4]}Then the predicted temperature distribution was sequentially coupled to the mechanical analysis considering the isotropic hardening model.

^{[5]}Considering thermo-metallurgical-mechanical coupling behaviors, solid phase transition effect, the isotropic hardening model and mobile heat resource were performed on the stress field and distortion in the joints of V-bevel angle 45°, 60°, 75°, and 90°.

^{[6]}The elastoplastic constitutive equations are obtained incrementally using Hoffman’s criterion and isotropic hardening model.

^{[7]}In this study, the strengths of a steel pipe in different material orientations are predicted using a distortional anisotropic hardening model, namely, the HAH model, implemented in a finite element simulation.

^{[8]}For steel reinforcement, a bilinear isotropic hardening model, a linear orthotropic model was used for composite reinforcement.

^{[9]}The finite element model based on the non-linear elastic model and the homogeneous anisotropic hardening model (HAH) is also established for the springback prediction and stress analysis.

^{[10]}Presented paper evaluates the time and temperature independent modeling possibilities of Chaboche combined nonlinear kinematic hardening model and isotropic hardening model with von Mises yield criterion.

^{[11]}The self-similar isotropic hardening model developed by Deshpande and Fleck has been widely used.

^{[12]}

## isotropic hardening law

Therefore, first, the proportional results are reproduced and then the non-proportionality parameter is incorporated into the isotropic hardening law.^{[1]}Furthermore, we present unified formulations for saturated and unsaturated states in which the isotropic hardening law and the critical state line are described in a bi-logarithmic space defined by the logarithms of the mean effective stress and void ratio.

^{[2]}A new constitutive model within the framework of Chaboche model was developed by improving the nonlinear isotropic hardening law and kinematic hardening law with cyclic characteristic parameters.

^{[3]}Thus, a multi scale numerical model of the IBF process was established based on the isotropic hardening law at the macro scale and digital material representation concept connected with the crystal plasticity theory at the micro scale.

^{[4]}

## isotropic hardening behavior

Moreover, the cyclic loading results demonstrate the isotropic hardening behaviors and the saturation of yielding strength when the maximum strain reaches 10%.^{[1]}A modified Drucker-Prager Cap (DPC) model with an elliptical cap surface using the new material characterization method was developed to capture the anisotropic hardening behavior and hydrostatic effect of the powder mixture.

^{[2]}The polymeric matrix is treated as an ideal elastoplastic solid with isotropic hardening behavior, whereas the clay nanoparticles are simplified as stiff, linearly elastic platelets.

^{[3]}

## isotropic hardening plasticity

Isotropic hardening plasticity and nonlinear failure criterion were considered for both ferrite and martensite phases.^{[1]}Based on the stress relaxation tests and the equivalent vibration equation of modal analysis, the creep constitutive model and the bilinear isotropic hardening plasticity material model (BISO) are combined to establish the numerical simulation model of TVSR of 7075 aluminum alloy ring part.

^{[2]}The model incorporates damage evolution and healing model, and the von Mises linear isotropic hardening plasticity model and is validated with static tensile and two-cycle tensile tests on pure Surlyn® 8940 specimens.

^{[3]}

## isotropic hardening characteristic

Both kinematic and isotropic hardening characteristics of Fe-SMA are observed, and a combined kinematic/isotropic hardening model with calibrated parameters is shown to adequately capture the hysteretic behavior of the material.^{[1]}[7] proposed a simplified shear behavior model and hysteretic criterion of high strength bolted connection, and reported that the shear behavior presents the isotropic hardening characteristics under cyclic loading.

^{[2]}

## isotropic hardening rule

The traditional isotropic hardening rule is modified as well to capture the accelerated cyclic softening phenomena observed in the prolonged hold time of CFI tests.^{[1]}A visco-plasticity constitutive model based on Ohno-Wang kinematic hardening rule and Marquis isotropic hardening rule is used to characterize non-proportional cyclic behavior.

^{[2]}

## isotropic hardening part

To study the effect of constitutive model on the accuracy of forming simulations, a combined nonlinear isotropic/kinematic hardening model as well as the isotropic hardening part of the same model are calibrated based on five compression-tension-compression uniaxial stress experiments.^{[1]}By introducing an anisotropic distortional hardening function into the isotropic hardening part of the classical Chaboche rate-dependent constitutive model, the plastic-deformation-induced distortional and anisotropic hardening behaviors of subsequent yield surfaces are characterized.

^{[2]}

## isotropic hardening assumption

The constitutive relationships are developed for anisotropic materials with an anisotropic hardening assumption.^{[1]}Kinematic hardening is suggested to substitute the isotropic hardening assumption for better prediction of FLCs with strain path changing effect.

^{[2]}

## isotropic hardening material

In the exampled calculation of the inelastic seismic strain response beyond an elastic regime, precise inelastic seismic analyses with Chaboche’s kinematic and Voce isotropic hardening material models are used.^{[1]}The relaxation of residual stress under a cyclic load is a transient phenomenon that is described in this paper using classic plasticity theory by employing a combined, non-linear, kinematic and isotropic hardening material model.

^{[2]}