Anisotropic parameter identification using inhomogeneous tensile test

In this contribution, an inverse identification strategy of constitutive laws for elastoplastic behaviour is presented. The proposed inverse algorithm is composed on an appropriate finite element calculation combined with an optimisation procedure. It is applied to identify material anisotropic coefficients using a set up of easy performed laboratory tests. The used experimental data are the plane tensile test and the off axes tensile tests. The identified behaviour models are mainly based on Hill's quadratic yield criterion. Two cases of this yield criterion have been considered: the transverse isotropic and the orthotropic one under an associated and non-associated flow rule assumptions for each case. The yield surface has been assumed to expand isotropically (isotropic strain hardening law) as a function of the plastic work.In order to better describe anisotropic plastic properties of the studied materials, a recently planar anisotropic yield function is used. It is a non-quadratic yield criterion which takes account of anisotropic yield stresses as well as anisotropic strain ratios. It is subsequently shown that the agreement between inverse identification results and experimental measurements were improved.We prove also that the presented strategy is a good alternative to the simplified homogeneous tests assumption, especially for the plane tensile test.     

A non-associated flow rule for sheet metal forming

An improved model of material behavior is proposed that shows good agreement with experimental data for both yield and plastic strain ratios in uniaxial, equi-biaxial, and plane-strain tension under proportional loading for steel, aluminum and possibly other alloys. This model is based on a non-associated flow rule in which the plastic potential and yield surface functions are defined by quadratic functions of the stress tensor. The plastic potential aspect of the model is identical to that proposed by Hill for a quadratic anisotropic plastic potential defined in terms of measured r values. The new model differs in that the yield surface, although also defined by a quadratic function of the stress tensor, is defined independently of the plastic potential in terms of measured yield stresses. The model is developed and implemented in an FEM code that is based on a convected coordinate system. Since the associated flow rule, which assumes equivalency between the plastic potential and yield functions, is commonly accepted as a valid law in the theory of plastic deformation of most metals, the arguments for the associated flow rule are also discussed.     

Spring-back evaluation of automotive sheets based on isotropic-kinematic hardening laws and non-quadratic anisotropic yield functions: Part I: theory and formulation

In order to improve the prediction capability of spring-back in automotive sheet forming processes, the modified Chaboche type combined isotropic-kinematic hardening law was formulated based on the modified equivalent plastic work principle to account for the Bauschinger effect and transient behavior. As for the yield stress function, the non-quadratic anisotropic yield potential, Yld2000-2d, was utilized under the plane stress condition. Besides the theoretical aspect of the constitutive law including the general plastic work principle for monotonously proportional loading, the method to determine hardening parameters as well as numerical formulations to update stresses were developed based on the incremental deformation theory and the consistency requirement as summarized in Part I, while the characterization of material properties and verifications with experiments are discussed in Part II and III, respectively.     

Modified Burzynski criterion with non-associated flow rule for anisotropic asymmetric metals in plane stress problems

The Burzynski criterion is developed for anisotropic asymmetric metals with the non-associated flow rule (NAFR) for plane stress problems. The presented pressure depending on the yield criterion can be calibrated with ten experimental data, i.e., the tensile yield stresses at 0°, 45°, and 90°, the compressive yield stresses at 0°, 15°, 30°, 45°, 75°, and 90° from the rolling direction, and the biaxial tensile yield stress. The corresponding pressure independent plastic potential function can be calibrated with six experimental data, i.e., the tensile R-values at 0°, 15°, 45°, 75°, and 90° from the rolling direction and the tensile biaxial R-value. The downhill simplex method is used to solve these ten and six high nonlinear equations for the yield and plastic potential functions, re- spectively. The results show that the presented new criterion is appropriate for anisotropic asymmetric metals.     

Endochronic description of plastic anisotropy in sheet metal

It is shown in this paper that an extended form of Hills quadratic yield criterion for anisotropic sheet metal can be derived from an endochronic theory of plasticity. The extended form considers the combined isotropic–kinematic hardening and the anomalous behavior observed in the anisotropic plastic behavior of sheet metals can be accounted for by the concept of kinematic hardening.This form of anisotropic endochronic theory can accommodate the usual requirement of normality between the plastic strain rate and the yield function. In addition, the theory leads naturally to the expressions for back stresses. This work provides an additional example to show that the form of the intrinsic time is directly related to the form of the yield function.It is suggested that the coefficients of the quadratic yield function be determined from the yield stresses obtained from a set of tension tests.     

Anisotropic hardening and non-associated flow in proportional loading of sheet metals

Conventional isotropic hardening models constrain the shape of the yield function to remain fixed throughout plastic deformation. However, experiments show that hardening is only approximately isotropic under conditions of proportional loading, giving rise to systematic errors in calculation of stresses based on models that impose the constraint. Five different material data for aluminum and stainless steel alloys are used to calibrate and evaluate five material models, ranging in complexity from a von Mises’ model based on isotropic hardening to a non- associated flow rule (AFR) model based on anisotropic hardening. A new model is described in which four stress–strain functions are explicitly integrated into the yield criterion in closed form definition of the yield condition. The model is based on a non-AFR so that this integration does not affect the accuracy of the plastic strain components defined by the gradient of a separate plastic potential function. The model not only enables the elimination of systematic errors for loading along the four loading conditions, but also leads to a significant reduction of systematic errors in other loading conditions to no higher than 1.5% of the magnitude of the predicted stresses, far less that errors obtained under isotropic hardening, and at a level comparable to experimental uncertainty in the stress measurement. The model is expected to lead to a significant improvement in stress prediction under conditions dominated by proportional loading, and this is expected to directly improve the accuracy of springback, tearing, and earing predictions for these processes. In addition, it is shown that there is no consequence on MK necking localization due to the saturation of the yield surface in pure shear that occurs with the aluminum alloys using the present model.     

A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials

In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials is derived. The evolution equation for the active yield surface with reference to the memory yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function for pressure insensitive and sensitive materials. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress–plastic strain curves are also derived and plotted for materials under cyclic loading conditions based on the three yield functions. In addition, the closed-form solutions for one-dimensional stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function under cyclic loading conditions are summarized and presented. For materials based on the Mises and the Hill anisotropic yield functions, the stress–plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker–Prager and Cazacu–Barlat yield functions, the stress–plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. With these closed-form solutions, the important effects of the yield surface geometry on the cyclic plastic behavior due to the pressure-sensitive yielding or the unsymmetric behavior in tension and compression can be shown unambiguously. The closed form solutions for the Drucker–Prager and Cazacu–Barlat yield functions with the associated flow rule also suggest that a more general anisotropic hardening theory needs to be developed to address the ratcheting effects for a given stress range.     

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.     

Analysis of stress-strain relations by use of an anisotropic hardening plastic potential

A method of analyzing plastic behavior by use of an anisotropic hardening plastic potential is proposed. The plastic potential surface in deviatoric stress space is assumed to be the same as the equi-plastic-strain surface. Stress-strain relations in combined loading and in multi-axial cyclic loading are calculated by use of the anisotropic hardening plastic potential and the normality rule of the plastic strain increment vector to the plastic potential surface, which are experimentally determined or confirmed by subjecting thinwalled tubular test specimens of $\text{60}\text{40}$ brass to combined axial load, internal pressure and torsion. The calculated results agree fairly well with the experimental observations.     

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.     

An alternative approach to integrating plasticity relations

A new plasticity integration algorithm is proposed based upon observations from the closed form integration of a generalized quadratic yield function over a single time step. The key to the approach is specification of the normal to the plastic flow potential as a function of the current state and strain increment. This uniquely defines the direction of the stress tensor for a convex, non-faceted flow potential. The stress magnitude and plastic strain increment are computed to satisfy the yield function. A non-quadratic, isotropic, associative flow model is coded to demonstrate accuracy and time step convergence following a step change in loading path. The model is used in additional simulations of strain localization in an expanding ring and a perforated plate.     

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。     

Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals

A Taylor-like polycrystal model is adopted here to investigate the plastic behavior of body centered cubic (b.c.c.) sheet metals under plane-strain compression and the subsequent in-plane biaxial stretching conditions. The <111> pencil glide system is chosen for the slip mechanism for b.c.c. sheet metals. The {110} <111> and {112} <111> slip systems are also considered. Plane-strain compression is used to simulate the cold rolling processes of a low-carbon steel sheet. Based on the polycrystal model, pole figures for the sheet metal after plane-strain compression are obtained and compared with the corresponding experimental results. Also, the simulated plane-strain stress—strain relations are compared with the corresponding experimental results. For the sheet metal subjected to the subsequent in-plane biaxial stretching and shear, plastic potential surfaces are determined at a given small amount of plastic work. With the assumption of the equivalence of the plastic potential and the yield function with normality flow, the yield surfaces based on the simulations for the sheet metal are compared with those based on several phenomenological planar anisotropic yield criteria. The effects of the slip system and the magnitude of plastic work on the shape and size of the yield surfaces are shown. The plastic anisotropy of the sheet metal is investigated in terms of the uniaxial yield stresses in different planar orientations and the corresponding values of the anisotropy parameter R, defined as the ratio of the width plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial tensile loading. The uniaxial yield stresses and the values of R at different planar orientations from the polycrystal model can be fitted well by a yield function recently proposed by Barlat et al. (1997b).     

QUASI－FLOW CORNER THEORY ON LARGE PLASTIC DEFORMATION OF DUCTILE METALS AND ITS APPLICATIONS

ＱＵＡＳＩ－ＦＬＯＷＣＯＲＮＥＲＴＨＥＯＲＹＯＮＬＡＲＧＥＰＬＡＳＴＩＣＤＥＦＯＲＭＡＴＩＯＮＯＦＤＵＣＴＩＬＥＭＥＴＡＬＳＡＮＤＩＴＳＡＰＰＬＩＣＡＴＩＯＮＳＨｕＰｉｎｇ（胡平）ＬｉｕＹｕｑｉ（柳玉启）ＧｕｏＷｅｉ（郭威）ＴａｉＦｅｎｇ（台风）（Ｒ...     

Crack growth resistance for anisotropic plasticity with non-normality effects

For a plastically anisotropic solid a plasticity model using a plastic flow rule with non-normality is applied to predict crack growth. The fracture process is modelled in terms of a traction–separation law specified on the crack plane. A phenomenological elastic–viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy, and in each case the effect of the normality flow rule is compared with the effect of non-normality. Conditions of small scale yielding are assumed, with mode I loading conditions far from the crack-tip, and various directions of the crack plane relative to the principal axes of the anisotropy are considered. It is found that the steady-state fracture toughness is significantly reduced when the non-normality flow rule is used. Furthermore, it is shown that the predictions are quite sensitive to the value of the maximum angle of deviation from normality in the non-normality flow rule.     

Variational Principles for EIastoplastic Continua∗

ABSTRACT

A mixed variational principle is constrained by a homogeneous yield function using a Lagrange multiplier. The Lagrange factor corresponds to the scalar factor in Prager's normality rule for the plastic strain increments. Several reduced functionals and their associated constitutive equations are derived by eliminating some variables.     

Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity

The paper is devoted to the dissipation postulate in anisotropic finite elastoplasticity, properly formulated in terms of the total strain histories, on small cycles only. The equivalence between the dissipation postulate and the existence of the stress potential together with the dissipation inequality is proved. The modified flow rules compatible with the dissipation postulate follow as a necessary condition. The convexity and normality properties can be treated as an equivalent issue of the dissipation postulate only within the framework of Σ models. We identify such classes of Σ-models based on the pre-image theorem. The difficulties arise from the non-injectivity of the Mandel's stress measure, as dependent on the elastic strain. We define the yield stress function and the admissible elastic stress range in Σ-space. The equivalence is achieved only if it possible to construct the elastic range in strain space, having just the topological properties originally assumed as a basis of the dissipation postulate. The normality to the admissible elastic stress range does not mean an associative flow rule. The results are exemplified for transversely isotropic elasto–plastic materials as well as for models with small elastic strains.     

A novel quadratic yield model to describe the feature of multi-yield-surface of rolled sheet metals

Since Hill’s quadratic yield model [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. Lond. A193, 281–297] cannot address enough experimental results for fairly describing the “anomalous” yield behavior as observed in some of rolled sheet metals, a new quadratic yield model is proposed. As the concept of multiple yielding systems is introduced into the new quadratic yield model, seven commonly used experimental results, three uniaxial tension stresses, one equibiaxial tension stress and three strain ratios, can all be taken into account for characterizing the anisotropy of rolled sheet metals. If more experimental results are extra needed for further improving the prediction, this yield model is still workable. As the experimental parameters are defined as functions of loading direction of corresponding test separately from the major part of yield model, the increase of experimental results regarding the same test does not vary the quadratic form of yield model. The representation of this yield model with axes of principal stresses demonstrates the similar form to Hill’s quadratic model. Therefore, many previous studies developed from Hill’s quadratic yield model can be directly upgraded by the new model to reach a higher accurate level.     

Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials

Novel (non-quadratic) plasticity criteria based on Kelvin modes are formulated here for anisotropic materials. As an example, such a macroscopic criterion is applied with success to the case of FCC nickel-base single crystals. Indeed, relying on the cubic symmetry of the material, the Kelvin decomposition of elasticity tensor easily allows for the definition of an objective and loading independent criterion. The criterion identification is performed from different loading cases for CMSX2 single crystal superalloy. Tension-torsion yield surfaces at room temperature and yield stress dependence on crystal orientation are modeled. The Kelvin modes based criterion is compared to experimental data, to Hill and Barlat and coworkers macroscopic criteria and to Schmid law predictions. The results show that a simple three-parameter yield function built thanks to von Mises equivalent Kelvin stresses accounts for a satisfying plasticity criterion for such alloys.Non-quadratic norm ∥·∥_a plasticity framework is addressed. Intrinsic generalizations of Hershey-Hosford criterion are proposed for cubic material symmetry.