Plastic spin associated with a non-normality theory of plasticity

A plasticity model using a vertex-type plastic flow rule on a smooth yield surface for an anisotropic solid has been proposed recently. This model is here completed by incorporating the effect of plastic spin. Simple shear with a large shear strain is one of the hardest tests on finite strain anisotropic plasticity models, and it is here shown which plastic spin expression is needed to avoid unrealistic oscillatory behavior of the shear stress under large shear strains. The idea of using non-normality with a smooth yield surface originates from a recent proposal of using an abrupt strain path change to determine the subsequent yield surface shape. For this method both polycrystal plasticity calculations and experiments have shown a vertex-type response on the apparently smooth yield surface.     

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 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 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.     

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...     

A cyclic plasticity model for single crystals

The Armstrong–Frederick type kinematic hardening rule was invoked to capture the Bauschinger effect of the cyclic plastic deformation of a single crystal. The yield criterion and flow rule were built on individual slip systems. Material memory was introduced to describe strain range dependent cyclic hardening. The experimental results of copper single crystals were used to evaluate the cyclic plasticity model. It was found that the model was able to accurately describe the cyclic plastic deformation and properly reflect the dislocation substructure evolution. The well-known three distinctive regimes in the cyclic stress–strain curve of the copper single crystals oriented for single slip can be reproduced by using the model. The model can predict the enhanced hardening for crystals oriented for multislip, showing the model's ability to describe anisotropic cyclic plasticity. For a given loading history, the model was able to capture not only the saturated stress–strain response but also the detailed transient stress–strain evolution. The model was used to predict the cyclic plasticity under a high–low loading sequence. Both the stress–strain responses and the microstructural evolution can be appropriately described through the slip system activation.     

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。     

Mechanism-based multi-surface plasticity model for ideal truss lattice materials

The mechanical behavior of ideal truss lattice materials is controlled by the so-called direct action mechanism at the microscale which involves the uniform stretching and compressing of individual truss members. Standard homogenization techniques have been employed to develop a general micromechanics-based finite-strain constitutive model for truss lattice materials. Furthermore, a specialized small-strain plasticity model has been derived. Both models have been implemented in a finite-element program and used to simulate the anisotropic plastic behavior of the octet-truss lattice material in various applications including cyclic uniaxial loading, pure shear, and three-point bending. The constitutive model predictions agree well with the results obtained from discrete finite element models. Regarding the plasticity of the octet-truss lattice material, it has been found that the elastic domain is constrained by twelve pairwise parallel hyperplanes in the six-dimensional stress space. Moreover, the mechanism-based small-strain formulation reveals that the direction of plastic flow is normal to the pressure-dependent macroscopic yield surfaces.     

考虑路径相关性的非比例循环塑性本构模型

根据非比例加载下金属材料响应的延迟特性及加载路径相关性,选取沿应力迹法向的塑性应变的累积量作为非比例加载影响的度量,相应给出反映非比例附加强化的变量,并假设其模量和强化率与加载路径的几何参数相关．为反映由于非比例加载而引起的材料强化的异向效应,在Valanis的塑性内时响应方程中引入与加载路径几何性质有关的应力项,构成非比例循环塑性本构关系．对316和304不锈钢材料在一些典型非比例循环加载路径下的应力响应进行了理论预测,与Benallal等及McDowell的实验结果取得了良好的一致．     

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.     

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.     

Reconcile the perfectly elastoplastic model to simulate the cyclic behavior and ratcheting

Perfectly elastoplastic constitutive model is modified through a smoothing factor introduced by Liu [Liu, C.-S., 2003. Smoothing elastoplastic stress–strain curves obtained by a critical modification of conventional models. Int. J. Solids Struct. 40, 2121–2145]. The new model allows plasticity to happen in a non-zero-measure yield volume in stress space, rather than that of conventional zero-measure yield surface, and within the yield volume the plastic modulus is varying continuously. It endows a specific strain-hardening rule of flow stress and is able to describe the phenomena of strain hardening, cyclic hardening, the Bauschinger effect, mean-stress relaxation, strain ratcheting, out-of-phase hardening, as well as erasure-of-memory. In order to suppress the over prediction of ratcheting we consider a scalar function of smoothing factor, which can simulate the saturation behavior of uniaxial/multiaxial strain ratcheting. These effects are demonstrated through numerical examples. The existence of stress equilibrium point and limiting surface is a natural result without requiring an extra design. Moreover, the non-linear constitutive equations can be converted into a linear system for augmented stress in the Minkowski space, of which the symmetry group is a proper orthochronous Lorentz group SO_o(5, 1). The augmented stress is a time-like vector, moving on hyperboloids inside the cone. When taking the Prager kinematic hardening rule into account we can simulate some cyclic behaviors of SAE 4340 and grade 60 steels within a certain accuracy through the use of only three material constants and a fixed smoothing factor. To simulate the ratcheting behaviors of SS304 stainless steel we allow the smoothing factor to be an exponential decaying function of λ.     

A simple plasticity model for prediction of non-coaxial flow of sand

A bounding surface plasticity model for non-coaxiality, another aspect of anisotropic behavior of sands under rotation of principal stress axes; is developed in the critical state framework. Numerous experimental evidences exist that corroborate dependence of plastic shear strain rate direction on inherent fabric anisotropy. At first, general expressions for plastic strain rate with respect to possible emerge of non-coaxial flow are obtained. Consequently, using an anisotropy state parameter that is specially developed for this model and accounts for the interaction between imposed loading and soil fabric; effect of anisotropy on plastic flow direction is taken into account. Besides, novel circumstances are proposed for plastic modulus and dilatancy under rotation of principal stress axes. Finally, it is shown that the model is able to simulate successfully the non-coaxial behavior of sands subjected to principal stress axes rotation.     

A critical state sand model with elastic–plastic coupling

Soil elastic moduli are highly pressure-dependent. Experimental findings have indicated that the elastic shear modulus of sands depends on p^χ, where p is mean principal effective stress and χ is a non-dimensional parameter. χ practically remains unchanged for shear strains less than 10⁻⁵ where the mechanical behavior is purely elastic. However, experiments have revealed that the emergence of plasticity for shear strains larger than 10⁻⁵ provokes a gradual increase in χ. Technically, this observation is an elastic–plastic coupling effect in which plasticity causes to change the elastic characteristics. Here, this issue is considered in hyper-elasticity framework in conjunction with a critical state compatible bounding surface plasticity platform for granular soils. To this aim, constitutive equations linking χ to a proper kinematic hardening parameter are presented. Then, using the proposed approach, a hyper-elastic theory is modified to consider the mentioned elastic–plastic coupling effect in the whole domain of the elastoplastic behavior. Adopting the improved hyper-elasticity necessitates the modification of a number of basic plasticity platform elements. In this regard, dilatancy and plastic hardening modulus of the bounding surface platform are modified. Successful performance of the modified constitutive model is presented against experimental data of loading/unloading triaxial tests.     

Cyclic ratchetting of 1070 steel under multiaxial stress states

Cyclic ratchetting behavior of 1070 steel is studied under proportional and nonproportional loading with specific emphasis on the ratchetting rate decay mechanisms for large numbers of loading cycles. Under proportional loading, where the principal stress directions are unchanged, the ratchetting evolves in the mean stress direction. Under nonproportional loading, however, the ratchetting direction is determined by the loading path and can be different from the mean stress direction. The ratchetting rate decreases with increasing loading cycles, displaying a power law relationship with the number of loading cycles. The experimental ratchetting results indicate that under cyclic loading the material exhibits a tendency toward complying with a linear hardening rule with concomitant hysteresis loop closure. Based on the fundamental framework of plasticity theory and detailed evaluation of the stress-strain behaviors, the ratchetting can be classified into two basic types; Type I, which is identifiable with proportional loading where the ratchetting is due to the different values of the plastic modulus function at the symmetric loading points with respect to the mean stress state, and Type II, which represents nonproportional loading where the ratchetting is driven by the noncoincidence of the plastic strain rate vector and the translation direction of the yield surface (backstress rate vector). The Armstrong-Frederick-based plasticity models modified by Chaboche et al. and Bower are ill-suited for describing the experimental results of both types of ratchetting. The Ohno-Wang model, which introduces a threshold concept, can account for the ratchetting rate decay of Type II ratchetting, providing results that agree with experimental observations. Modification may be needed for the Ohno-Wang model so that the model can better describe Type I ratchetting.     

饱和砂土的循环边界面本构模

饱和砂土在循环加卸载过程中,土体发生显著的组构变化和塑性变形的累积。试验现象分析表明,循环过程中,砂土的首次与再循环塑性模量既有区别又有联系。因此,论文引入考虑组构变化的剪胀内变量,并提出循环塑性模量的关系式,真实描述上述两种现象;进而基于已有本构模型的基础上,建立饱和砂土的循环边界面本构模型。最后,将饱和砂土的模型预测结果与三轴试验结果进行验证对比,得到较好地吻合,这表明该模型能够合理反映饱和砂土循环加卸载的变形行为。     

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.     

On the implementation of a multi-surface kinematic hardening plasticity and its applications

This paper is concerned with an application of the multi-surface plasticity in solid mechanics and geotechnical problems. The model is of a von-Mises type with associated flow rule, originally proposed by Montans. The Mroz translation rule is implemented to the movements of the yield surfaces and the fully implicit scheme with radial mapping method is applied in numerical computations. Algorithmic consistent tangent modulus with numerical integration algorithm of constitutive equations is extracted. The model is developed in the class of kinematic hardening models, so the ‘Masing’ rule is preserved. The model is able to consider the plastic strain accumulation in constant axial stress state, such as ratcheting. The implementation is validated by means of a simple deformation path of combined extension and compression test, a pure shear test with pseudo-random loading, a test which demonstrates the capabilities of the model in simulation of cyclic loading and ratcheting, a cyclic shear test in saturated undrained sand and finally, the analysis of a plate with holes, which presents the shear band using the multi-surface plasticity model.     

Multiaxial ratcheting with advanced kinematic and directional distortional hardening rules

Ratcheting is defined as the accumulation of plastic strains during cyclic plastic loading. Modeling this behavior is extremely difficult because any small error in plastic strain during a single cycle will add to become a large error after many cycles. As is typical with metals, most constitutive models use the associative flow rule which states that the plastic strain increment is in the direction normal to the yield surface. When the associative flow rule is used, it is important to have the shape of the yield surface modeled accurately because small deviations in shape may result in large deviations in the normal to the yield surface and thus the plastic strain increment in multi-axial loading. During cyclic plastic loading these deviations will accumulate and may result in large errors to predicted strains.This paper compares the bi-axial ratcheting simulations of two classes of plasticity models. The first class of models consists of the classical von Mises model with various kinematic hardening (KH) rules. The second class of models introduce directional distortional hardening (DDH) in addition to these various kinematic hardening rules. Directional distortion describes the formation of a region of high curvature on the yield surface approximately in the direction of loading and a region of flattened curvature approximately in the opposite direction. Results indicate that the addition of directional distortional hardening improves ratcheting predictions, particularly under biaxial stress controlled loading, over kinematic hardening alone.     

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.