An endochronic theory accounting for deformation induced anisotropy of metals under biaxial load

Using the experimental results of yield surfaces obtained by Wu and Yeh [1991] (Int. J. Plasticity, 7, 803) for 304 stainless steel, this work provides a verification of the endochronic theory of plasticity accounting for deformation induced anisotropy. The experiments were performed under proportional loading conditions. The main difference between this paper and other papers that attempt to describe the distortion of a yield surface is that, in addition to distortion, motion of yield surface (kinematic hardening) has also been addressed by this paper. The result has shown that the theory predicts the experimental data with substantial accuracy. However, since in this theory the plastic strain increment, although normal to the initial yield surface, is in the radial direction emanating from the center of the subsequent yield surface, validity of the present model must be further studied for the case involving nonproportional loading conditions.     

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.     

An endochronic model of yield surface accounting for deformation induced anisotropy

In this article, an endochronic model of yield surface is proposed. Based on this model, the yield surface is simulated such that the forward and rear parts of the yield surface are described by different ellipses which are characterized by corresponding aspect ratio functions, respectively. Verification of the endochronic theory used the experimental results of yield surfaces obtained by Wu and Yeh for 304 stainless steel (Wu, H.C., Yeh, W.C., 1991. On the experimental determination of yield surfaces and some results of annealed 304 stainless steel. Int. J. Plasticity 7, 803–826). The experiments were performed cyclically under uniaxial, torsional, and combined axial–torsional loading conditions. The result has shown that the agreement between the prediction and experiments is quite satisfactory. In addition to the distortion of the yield surface plastically behaving a sharp front accompanied by a blunt rear, the anisotropic kinematic hardening effect has been addressed in this investigation. Although the experimental results of yield surfaces subjected to non-proportional loading conditions can be found in the literature, lack of information about the plastic strain history makes it impossible to verify the theory under such complicated loading conditions. The domain of applicability and validity of the theory, which is defined in terms of plastic strain increments, need be further investigated with the aim to set up related experiments.     

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.     

岩石非弹性特性的内时理论模型

针对岩石的非弹性特性,将内时理论引入到岩石特性的研究中来。首先对内时理论的方程作了简要推导和说明。将核函数作四阶的Prony展开,将积分方程用一组微分方程来代替,最后得出可以进行数值运算的内时方程。在MTS压机上进行了砂岩、花岗岩等不同岩石样品的多种循环加卸载实验,得到一系列应力—应变曲线。根据滞回曲线的拐点,用最小二乘拟合法来具体确定内时方程中的有关参数。将理论产生的各种滞回曲线与实验滞回曲线作比较,可以看出在滞回曲线和残余应变这二个岩石非弹性特性最明显的标志上,非常一致。说明在中等应变强度时,利用内时理论可以对岩石非弹性特性作很好的研究。与塑性理论比较,可以免去确定屈服面的困难。最后对岩石滞回特性的微观机制用P-M模型作了简要讨论和分析。     

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.     

Localization in elasto-plastic materials: Influence of the plasticity yield surface in biaxial loading conditions

The influence of the plasticity yield surface on the development of instabilities in plane plates in biaxial loading is analyzed in order to understand and simulate the localization pattern observed in an expanding hemisphere experiment. First, a criterion for the activation of slip bands is formulated in the form of a critical hardening coefficient: it is particularized to the Von Mises and Tresca surfaces. In the Von Mises case, the criterion gives a strongly negative hardening coefficient in biaxial loading conditions different from the ones of plane strain. In the Tresca case, the criterion is fulfilled for a perfectly plastic material in uniaxial and biaxial loading; besides, in equi-biaxial loading, two possible orientations for slip bands are exhibited; this can be understood, with a few approximations, by the existence of a vertex point on the Tresca yield surface which give additive degrees of freedom for the direction of the plastic strain rate. Second, the development of localization in the loading conditions met in an expanding hemisphere experiment is simulated using both plasticity yield surfaces; whereas the Von Mises simulation does not localize, the Tresca simulation exhibits a pattern composed of a network of shear bands of different orientations; this pattern is not far from the pattern observed experimentally.     

Strain path theory of plastic constitutive relation

Based on Illiushin's two hypotheses, a new plastic constitutive equation of integral type is proposed. The deviatoric stress induced by a strain path consists of two parts which vary according to different laws. Comparison of theoretical calculation with recent experiments is satisfactory. The present theory is not an endochronic theory.     

各向同性率无关材料本构关系的不变性表示

在内变量理论的框架下,针对各向同性率无关材料,使用张量函数表示理论建立了塑性应变全量及增量本构关系的最一般的张量不变性表示. 它们均由3个完备不可约的基张量组合构成,这3个基张量分别是应力的零次幂、一次幂和二次幂. 因此得出,塑性应变、塑性应变增量与应力三者共主轴. 通过对基张量的正交化,给出了本构关系式在主应力空间中的几何解释. 进一步,全量(或增量)本构关系中3个组合因子被表达为应力、塑性应变(或塑性应变增量)的不变量的函数. 当塑性应变(或塑性应变增量)的3个不变量之间满足一定关系时,所给出的本构关系将退化为经典的形变理论(或塑性势理论).最后,还讨论它与奇异屈服面理论的关系,当满足一定条件时,两者是一致的.      

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.     

YIELD SURFACES AND PLASTIC FLOW OF 45 STEEL UNDER TENSION-TORSION LOADING PATHS

An experimental analysis on the subsequent yield-surfaces evolution using multiple specimens is presented for a 45 steel after a prescribed pre-strain loading in three different directions respectively, and the yielding is defined by a designated offsetting strain. The size of the subsequent yield surface is found smaller than the initial yield surface; the negative cross effects are observed in the normal loading direction, its shape is not a Mises circle but has a rather blunt nose in loading direction and flat in the opposite. These results strongly depend on the loading path and the prescribed offset plastic strain. The plastic flow direction to the subsequent yield surface is investigated, and it is found that the plastic flow direction deviates from the normal flow rule. The deviation differs from preloading case to preloading case. And the plastic flow direction would have a larger deviation from the normal of the yield surface, if the subsequent yield was defined by a smaller offset strain. Furthermore, the experiments are simulated using the Chaboche model, and the results show that it can rationally predict yield-surface only when yield is defined by a fairly large offset strain.     

Effect of yield surface evolution on bending induced cross sectional deformation of thin-walled sections

Bend–stretch forming is commonly used to shape extruded tubular aluminum parts for automotive and other applications. The tubes are pre-stretched, pressurized and bent over rigid curved dies. Tension prevents buckling of the compressed side and helps reduce springback. An unwanted byproduct of the process is distortion of the cross section. Small amounts of pressure applied during forming can reduce this distortion. The problem was studied through a combination of experiment and analysis. In numerical models of the process the inelastic behavior of the aluminum alloy was modeled through isotropically hardening plasticity. With the limitations imposed by this model, use of a J₂-type yield surface resulted in uniform underprediction of cross sectional deformation. The predictions matched the measurements when a non-quadratic yield function appropriately “calibrated,” was used instead. The change in yield function shape alters the instantaneous normals to the yield surface which, in turn, affect the calculated strain increments. This paper demonstrates how suitably calibrated nonlinear kinematic hardening models can have the same corrective effect. The calibration involves selection of a suitable kinematic hardening rule. Changes in the hardening direction alter the instantaneous normals and therefore alter the plastic strain increments resulting in approximately the same net effect as the switch from the von Mises to the non-quadratic yield function.     

Characterization studies of a potential photoelastoplastic material

A controlled-strain apparatus capable of performing the tests essential to establishing the yield locus of polycarbonate was designed and built. Material characterization tests were performed in the form of uniaxial strain rate, creep, isotropy, isoclinic and reloading tests. The yield locus for polycarbonate was determined and an analytical expression in the form of a hypotrochoid was developed. Yield locus results indicate that, although polycarbonate is not a Mises material, the yield locus may be approximated by a circle over a substantial range of Lode’s variable. Tests were run on large plates with holes. Good correlation with two-dimensional theory was obtained in the elastic zone and in the plastic zone where plastic strains were small.     

An empirical methodlogy to estimate a local yield stress in work-hardened surface layers

A methodology is proposed for estimating the local yield stress in work-hardened surface layers. It is based on the concept of in-depth normalized variation of hardness and x-ray diffraction peak width, both of which measure the strain-hardening attained by the materials' surface-treated layers due to, for example, shot-peening. Its principle is directly founded on the classical hardness theory. To study the evolution of those values with plastic deformation, specimens of five steels with different mechanical properties were subjected to interrupted tensile tests. The tests were performed at successive increments of plastic strain, until fracture occurred. The specimens were loaded and unloaded in increments of about 2% true strain. After each plastic strain increment, hardness and diffraction peak width were measured. It was observed that the variations of diffraction peak width and hardness are related to the material's strain-hardening, and their normalized variations can be considered proportional to the normalized variation of the material's yield stress. Thus, where the yield stress of the bulk material, its hardness or a characteristic diffraction peak width value, and their relative variations along the hardened layers, are known, an empirical expression could be used to estimate the local yield stress as a function of the treated depth.     

On the Basic Relationships of the Flow Theory for Strain-Hardening Plastic Solids

The paper presents a new form of the basic relationships for the theory of plastic bodies with arbitrary anisotropic hardening in the neighborhood of a regular point of an arbitrary loading surface. In these relationships, the increment of the strain components is expressed in terms of the increment of the stress components, plastic-strain components, and stress components     

Effect of deviatoric nonassociativity on the failure prediction for elastic–plastic materials

This paper presents a theoretical study of the effect of nonassociativity of the plastic flow rule on the critical plastic modulus for discontinuous bifurcation in an elastic–plastic material. Nonassociativity in both the spherical and the deviatoric spaces are considered, with an emphasis on the effect of nonassociativity in the deviatoric space. A particular form of nonassociativity in the deviatoric space is introduced, where the projections of the plastic flow direction and the normal to the yield surface are assumed to have the same length but the projection of plastic flow direction is allowed to lag that of the normal by an angle. It is shown that even for the simple yield surface of von Mises, nonassociativity in the deviatoric space can lead to a bifurcation for a load parameter significantly lower than the value predicted with an associated flow rule.     

Integral constitutive equations of elastic-plastic materials

In this paper the integral constitutive equations of elastic-plastic materials are studied. The endochronic theory can be deduced from this theory. It is shown that the endochronic should be selected compatible with the yield function of the calssical plasticity and this can be considered as a principle of selecting endochronic. Applying this principle the appropriate endochronics of the plastically compressible materials and the orthotropic materials are derived. The second approximate theory of the integral constitutive equation is also discussed in this paper.This paper was reported by the National Natural Science Foundation of China.     

混凝土率型内时损伤本构模型

混凝土是一种典型的率敏感材料,为了更好地描述混凝土结构在动力、冲击荷载作用下的强度和变形特征,本文结合内时理论和损伤理论建立了一种考虑混凝土率效应的内时损伤本构模型。该模型的特点:将混凝土材料的受力软化效应分解为密实状态的塑性效应和由微裂缝扩展引起的刚度退化效应。前者由内时理论来描述,这使该模型摆脱了一般弹塑性模型中屈服面的概念,从而更符合混凝土的变形特性,并且简化了非线性计算过程;后者由损伤理论来描述,根据混凝土的动力试验结果建立了增量型的损伤演变方程,从而使该模型能够较好地反映混凝土的动力特性。最后,应用本文建议的模型对一钢筋混凝土简支梁进行了非线性分析,结果表明:当结构承受快速荷载作用时,应变率对结构的受力性能影响较大,在进行结构分析时必须予以考虑。     

含主应力轴旋转的广义塑性位势理论

大量岩土实验与工程实践表明传统塑性位势理论无法合理反映岩土材料的基本变形机制。从塑性位势理论角度来看,当存在主应力轴旋转时,塑性应变增量与应力不共主轴,此时最一般情况下的塑性应变增量须用六个线性无关的塑性势函数来表述,从而提出含主应力轴旋转的广义塑性位势理论一般表达式。通过矩阵分析,将一般应力增量分解成共轴分量与旋转分量之和。在应力增量分解的基础上,提出含主应力轴旋转的广义塑性位势理论的分解表达式     

AN INVESTIGATION ON ASYMPTOTIC NEAR-TIP FIELDS OF DYNAMIC CRACK IN AN ELASTIC-PERFECTLY PLASTIC COMPRESSIBLE MATERIAL

The stress and deformation fields near the tip of a mode-I dynamic crack steadilypropagating in an elastic-perfectly plastic compressible material are considered under plane strain condi-tions. Within the framework of infinitesimal displacement gradient theory, the material is character-ized by the Von Mises yield criterion and the associated J_2 flow theory of plasticity. Through rigorousmathematical analysis, this paper eliminates the possibilities of elastic unloading and continuousasymptotic fields with singular deformation, and then constructs a fully continuous and boundedasymptotic stress and strain field. It is found that in this solution there exists a parameter (?)_0 whichcannot be determined by asymptotic analysis but may characterize the effect of the far field. Lastly thevariations of continuous stresses, velocities and strains around the crack tip are given numerically fordifferent values of (?)_0.