Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Constitutive modeling of temperature and strain rate dependent elastoplastic hardening materials using a corotational rate associated with the plastic deformation

In this paper, a constitutive model with a temperature and strain rate dependent flow stress (Bergstrom hardening rule) and modified Armstrong-Frederick kinematic evolution equation for elastoplastic hardening materials is introduced. Based on the multiplicative decomposition of the deformation gradient,new kinematic relations for the elastic and plastic left stretch tensors as well as the plastic deformation-dependent spin tensor are proposed. Also, a closed-form solution has been obtained for the elastic and plastic left stretch tensors for the simple shear problem.To evaluate model validity, results are compared with known experimental data for SUS 304 stainless steel, which shows a good agreement with the results of the proposed theoretical model.Finally, the stress-deformation curve, as predicted by the model, is plotted for the simple shear problem at room and elevated temperatures using the same material properties for AA5754-O aluminium alloy.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C_e. The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.     

Analysis of anisotropy and plastic spin effects on localization phenomena

Summary The main objective of the paper is the investigation of the influence of the anisotrophy and plastic spin effects on criteria for adiabatic shear band localization of plastic deformation. A theory of thermoplasticity is formulated within a framework of the rate-type covariance material structure with a finite set of internal state variables. The theory takes into consideration such effects as plastic non-normality, plastic-induced anisotropy (kinematic hardening), micro-damage mechanism, thermomechanical coupling and plastic spin. The next objective of the paper is to focus attention on cooperative phenomena in presence of the plastic spin, and the discussion on the influence of synergetic effects on localization criteria. A particular constitutive law for the plastic spin is assumed. The necessary condition for a localized plastic deformation region to be formed is obtained. This condition is accomplished by the assumption that some eigenvalues of the instantaneous adiabatic acoustic tensor vanish. A procedure has been developed which allows us to discuss two separate groups of effects on the localization phenomenon along a shear band. Plastic spin, spatial covariance and kinematic hardening effects are investigated at an isothermal process in an undamaged solid. In the second case, an adiabatic process in a damaged solid is discussed when the spatial covariance terms and the plastic spin are neglected. Here the thermomechanical coupling, micro-damage mechanism and kinematic hardening effects are examined. For both cases, the criteria for adiabatic shear band localization are obtained in an exact analytical form. Particular attention is focused on the analysis of the following effects: (i) plastic non-normality; (ii) plastic spin; (iii) covariant terms; (iv) plastic strain-induced anisotropy; (v) micro-damage mechanism; (vi) thermomechanical couplings. Cooperative phenomena are considered, and synergetic effects are investigated. A discussion of the influence of the plastic spin, kinematic hardening and covariant terms on the shear band localization conditions is presented. A numerical estimation of the effects discussed is given. Received 24 April 1997; accepted for publication 23 December 1997     

Modeling and numerical issues in hyperelasto-plasticity with anisotropy

In the paper we discuss modeling and numerical issues that arise in conjunction with anisotropic hyperelastic–plastic response. Both elastic and plastic anisotropy are included. A particular kinematic hardening rule is proposed and its predictive capability is investigated. From the numerical viewpoint, we are concerned with the algorithmic consequences of the loss of coaxiality that arises from anisotropy. The numerical investigation shows that significant truncation errors are introduced if commonly used linearizations of the (classical) exponential Backward Euler rule are utilized in the presence of non-coaxiality.     

Evaluation of advanced anisotropic models with mixed hardening for general associated and non-associated flow metal plasticity

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948). The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003). The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989). Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

Modelling inelastic behaviour of orthotropic metals in a unique alignment of deviatoric plane within the stress space

A finite strain constitutive model to predict the deformation behaviour of orthotropic metals is developed in this paper. The important features of this constitutive model are the multiplicative decomposition of the deformation gradient and a new Mandel stress tensor combined with the new stress tensor decomposition generalized into deviatoric and spherical parts. The elastic free energy function and the yield function are defined within an invariant theory by means of the structural tensors. The Hill’s yield criterion is adopted to characterize plastic orthotropy, and the thermally micromechanical-based model, Mechanical Threshold Model (MTS) is used as a referential curve to control the yield surface expansion using an isotropic plastic hardening assumption. The model complexity is further extended by coupling the formulation with the shock equation of state (EOS). The proposed formulation is integrated in the isoclinic configuration and allows for a unique treatment for elastic and plastic anisotropy. The effects of elastic anisotropy are taken into account through the stress tensor decomposition and plastic anisotropy through yield surface defined in the generalized deviatoric plane perpendicular to the generalized pressure. The proposed formulation of this work is implemented into the Lawrence Livermore National Laboratory-DYNA3D code by the modification of several subroutines in the code. The capability of the new constitutive model to capture strain rate and temperature sensitivity is then validated. The final part of this process is a comparison of the results generated by the proposed constitutive model against the available experimental data from both the Plate Impact test and Taylor Cylinder Impact test. A good agreement between experimental and simulation is obtained in each test.     

Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening

This work addresses the formulation of the thermodynamics of nonlocal plasticity using the gradient theory. The formulation is based on the nonlocality energy residual introduced by Eringen and Edelen (1972). Gradients are introduced for those variables associated with isotropic and kinematic hardening. The formulation applies to small strain gradient plasticity and makes use of the evanescent memory model for kinematic hardening. This is accomplished using the kinematic flux evolution as developed by Zbib and Aifantis (1988). Therefore, the present theory is a four nonlocal parameter-based theory that accounts for the influence of large variations in the plastic strain, accumulated plastic strain, accumulated plastic strain gradients, and the micromechanical evolution of the kinematic flux. Using the principle of virtual power and the laws of thermodynamics, thermodynamically-consistent equations are derived for the nonlocal plasticity yield criterion and associated flow rule. The presence of higher-order gradients in the plastic strain is shown to enhance a corresponding history variable which arises from the accumulation of the plastic strain gradients. Furthermore, anisotropy is introduced by plastic strain gradients in the form of kinematic hardening. Plastic strain gradients can be attributed to the net Burgers vector, while gradients in the accumulation of plastic strain are responsible for the introduction of isotropic hardening. The equilibrium between internal Cauchy stress and the microstresses conjugate to the higher-order gradients frames the yield criterion, which is obtained from the principle of virtual power. Microscopic boundary conditions, associated with plastic flow, are introduced to supplement the macroscopic boundary conditions of classical plasticity. The nonlocal formulation developed here preserves the classical assumption of local plasticity, wherein plastic flow direction is governed by the deviatoric Cauchy stress. The theory is applied to the problems of thin films on both soft and hard substrates. Numerical solutions are presented for bi-axial tension and simple shear loading of thin films on substrates.     

A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

Description of plastic anisotropy effects at large deformations. Part II: the case of transverse isotropy

In a previous paper (see Tsakmakis, 1999) a general thermodynamically consistent (visco-) plasticity theory has been developed, which accounts for anisotropy effects. For simplicity, isotropic hardening has not be regarded, while anisotropy arises from kinematic hardening and orientational evolution of the underlying substructure. In the present paper the capabilities of this theory are discussed for the study case of transverse isotropy. Anisotropy effects are elaborated in the free energy and the yield function by means of structural tensors. Characteristic features of the transversely isotropic model are illustrated for the case of homogeneous simple shear.     

Anisotropic plasticity for sheet metals using the concept of combined isotropic-kinematic hardening

Hill's 1948 anisotropic theory of plasticity (Hill, R., 1948. A theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. London A193, 281–297) is extended to include the concept of combined isotropic-kinematic hardening, and the objective of this paper is to validate the model so that it may be useful for analyses of sheet metal forming. Isotropic hardening and kinematic hardening may be experimentally observed in sheet metals, if yielding is defined by the proportional limit or by a small proof strain. In this paper, a single exponential term is used to describe isotropic hardening and Prager's linear kinematic hardening rule is applied for simplicity. It is shown that this model can satisfactorily describe both the yield stress and the plastic strain ratio, the R-ratio, observed in tension test of specimens cut at various angles measured from the rolling direction of the sheet. Kinematic hardening leads to a gradual change in the direction of the plastic strain increment, as the axial strain increases in the tension test; while in the traditional approach for sheet metal, this direction does not change due to the use of isotropic hardening.     

Kinematic hardening in large strain plasticity

A finite strain hyper elasto-plastic constitutive model capable to describe non-linear kinematic hardening as well as non-linear isotropic hardening is presented. In addition to the intermediate configuration and in order to model kinematic hardening, an additional configuration is introduced – the center configuration; both configurations are chosen to be isoclinic. The yield condition is formulated in terms of the Mandel stress and a back-stress with a structure similar to the Mandel stress.It is shown that the non-dissipative part of the plastic velocity gradient not governed by the thermodynamical framework and the corresponding quantity associated with the kinematic hardening influence the material behaviour to a large extent when kinematic hardening is present. However, for isotropic elasticity and isotropic hardening plasticity it is shown that the non-dissipative quantities have no influence upon the stress–strain relation.As an example, kinematic hardening von Mises plasticity is considered, which fulfils the plastic incompressibility condition and is independent of the hydrostatic pressure. To evaluate the response and to examine the influence of the non-dissipative quantities, simple shear is considered; no stress oscillations occur.     

Necking of anisotropic micro-films with strain-gradient effects

Necking of stubby micro-films of aluminum is investigated numerically by considering tension of a specimen with an initial imperfection used to onset localisation. Plastic anisotropy is represented by two different yield criteria and strain-gradient effects are accounted for using the visco-plastic finite strain model. Furthermore, the model is extended to isotropic anisotropic hardening （evolving anisotropy）. For isotropic hardening plastic anisotropy affects the predicted overall nominal stress level, while the peak stress remains at an overall logarithmic strain corresponding to the hardening exponent. This holds true for both local and nonlocal materials. Anisotropic hardening delays the point of maximum overall nominal stress.     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

Micromorphic continuum. Part II: Finite deformation plasticity coupled with damage

It is demonstrated how a micromorphic plasticity theory may be formulated on the basis of multiplicative decompositions of the macro- and microdeformation gradient tensor, respectively. The theory exhibits non-linear isotropic and non-linear kinematic hardening. The yield function is expressed in terms of Mandel stress and double stress tensors, appropriately defined for micromorphic continua. Flow rules are derived from the postulate of Il’iushin and represent generalized normality conditions. Evolution equations for isotropic and kinematic hardening are introduced as sufficient conditions for the validity of the second law of thermodynamics in every admissible process. Finally, it is sketched how isotropic damage effects may be incorporated in the theory. This is done for the concept of effective stress combined with the hypothesis of strain equivalence.     

The constitutive equations for mixed hardening orthotropic material

IntroductionAyieldingcriterionandtheassociatedflowtheorywereproposedbyHill[1]forinitiallyorthotropicmetalsin 1 948,whichareusedbroadly .AquadraticformofstressesisusedastheplasticpotentialthatisindependentofhydrostaticstressinHillplasticitytheory .Butcompressedbyhydrostaticstress,considerabledeformationwillbeproducedinorthotropicmaterials.Inthecaseofcyclicloading ,duetoBauschingereffect,thekinematichardeningcannotbeneglected .Inthispaper,kinematichardeningandproportionalhardeningareconsidered…     

Deformation gradient based kinematic hardening model

A kinematic hardening model applicable to finite strains is presented. The kinematic hardening concept is based on the residual stresses that evolve due to different obstacles that are present in a polycrystalline material, such as grain boundaries, cross slips, etc. Since these residual stresses are a manifestation of the distortion of the crystal lattice a corresponding deformation gradient is introduced to represent this distortion. The residual stresses are interpreted in terms of the form of a back-stress tensor, i.e. the kinematic hardening model is based on a deformation gradient which determines the back-stress tensor. A set of evolution equations is used to describe the evolution of the deformation gradient. Non-dissipative quantities are allowed in the model and the implications of these are discussed. Von Mises plasticity for which the uniaxial stress–strain relation can be obtained in closed form serves as a model problem. For uniaxial loading, this model yields a kinematic hardening identical to the hardening produced by isotropic exponential hardening. The numerical implementation of the model is discussed. Finite element simulations showing the capabilities of the model are presented.