Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

Quasilinear rate-type constitutive equations and incremental stress waves

The requirement that the leading edge of an incremental loading pulse superimposed on a preloading that has reached a constant state will propagate at the elastic wave speed imposes some restrictions on the material functions describing a quasilinear rate-type material. These restrictions are discussed, and it is shown that the incremental wave behavior implies the existence of at least one relaxation boundary, and that a natural continuous transition from the quasilinear equation to a semilinear equation occurs in the neighborhood of the relaxation boundary.How the knowledge of a relaxation boundary can be used in determining the material functions is also discussed.     

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     

A Property of Equations of Rigid/Plastic Material Obeying a Voce–Type Hardening Law

Using a simple example, the rotation of a rigid cone in rigid/plastic hardening material, the paper shows a qualitative difference between the solutions for two groups of hardening laws. The first group includes hardening laws with no saturation stress. In this case the solution under sticking conditions exists at any rotation angle of the cone up to infinity. The second group includes hardening laws with a saturation stress. For such laws the solution exists up to a finite value of the rotation angle. Once this angle has been reached, the solution breaks down. At the beginning of the process the behavior of the solution for both groups of hardening materials is similar. However, at the final stage the behavior of rigid/plastic hardening materials of the second group is similar to the behavior of rigid perfectly plastic materials. A specific hardening law with a saturation stress is applied to illustrate the general solution and the restrictions imposed by this law, and a priori specified interfacial law (sticking) on existence of the solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

The frictional sliding response of elasto-plastic materials in contact with a conical indenter

Over the past decade, many computational studies have explored the mechanics of normal indentation. Quantitative relationships have been well established between the load–displacement hysteresis response and material properties. By contrast, very few studies have investigated broad quantitative aspects of the effects of material properties, especially plastic deformation characteristics, on the frictional sliding response of metals and alloys. The response to instrumented, depth-sensing frictional sliding, hereafter referred to as a scratch test, could potentially be used for material characterization. In addition, it could reproduce a basic tribological event, such as asperity contact and deformation, at different length scales for the multi-scale modeling of wear processes. For these reasons, a comprehensive study was undertaken to investigate the effect of elasto-plastic properties, such as flow strength and strain hardening, on the response to steady-state frictional sliding. Dimensional analysis was used to define scaling variables and universal functions. The dependence of these functions on material properties was assessed through a detailed parametric study using the finite element method. The strain hardening exponent was found to have a greater influence on the scratch hardness and the pile-up height during frictional sliding than observed in frictionless normal indentation. When normalized by the penetration depth, the pile-up height can be up to three times larger in frictional sliding than in normal indentation. Furthermore, in contrast to normal indentation, sink-in is not observed during frictional sliding over the wide range of material properties examined. Finally, friction between indenter and indented material was introduced in the finite element model, and quantitative relationships were also established for the limited effects of plastic strain hardening and yield strength on the overall friction coefficient. Aspects of the predictions of computational simulations were compared with experiments on carefully selected metallic systems in which the plastic properties were systematically controlled. The level of accuracy of the predicted frictional response is also assessed by recourse to the finite element method and by comparison with experiment.     

Modeling the Bauschinger effect for sheet metals,part I: theory

It is essential to model the Bauschinger effect correctly for sheet metal forming process simulation and subsequent springback prediction when material points are subjected to cyclic loading conditions. The combined nonlinear hardening model for time independent cyclic plasticity, proposed by Chaboche and co-workers, is examined and a simple modification is suggested for the isotropic part of the hardening rule to utilize the conventional tensile test data directly. This modification is useful for the materials whose reverse loading curves saturate to the monotonic loading curve. In addition, an anisotropic nonlinear kinematic hardening model (ANK model) is proposed in an attempt to represent the Bauschinger effect more realistically. Possible offset in flow stress is modeled by treating the back stress evolution during reverse loading differently from the initial loading. This strategy coupled with the modified isotropic hardening rule seems to provide a way to model the Bauschinger effect consistently over multiple cycles. Two types of auto-body alloys are examined in this paper. Associated material parameters are determined by employing available tension-compression test data and multi-cycle bend test data. A developed finite element formulation is applied to analyze simple validation type of problems. The cyclic stress–strain curves generated from the proposed ANK model match remarkably well with measured data.     

Determination of material properties using nanoindentation and multiple indenter tips

Nanoindentation is a useful method to probe the material properties of a solid. Its effective use lies in interpreting the data collected from a nanoindentation experiment with an associated analytical/numerical solution of the corresponding problem configuration. In this paper, a parametric finite element study has been performed to develop a new procedure for extracting elastic–plastic properties of a material through nanoindentation experiments with a substantially improved accuracy for the elastic properties of a elastic–plastic solid. The procedure involves data collected through the use of two, different, nanoindenter tips. Non-dimensional functions were constructed for two different indenter geometries to show that test results from multiple indenters, when appropriately manipulated, deliver superior results, compared to using one indenter. The material was assumed to be an isotropic elastic–plastic solid with power law hardening. Friction between the indenter and the material was included in the cases studied. The ratio of yield strength to elastic modulus was assumed to be in the range 0.0005–0.02 and the hardening coefficient was assumed to be between 0 and 0.4. Poisson’s ratio was fixed at 0.3.     

一般加载规律的弹塑性本构关系

将有关文献给出一般加载规律一维全量理论的简单模型推广到一般加载规律的一维增量理论,进而推广到一般加载规律的多维增量理论,在此基础上,建立了推导一般加载规律的多维增量理论的本构关系的一种途径。应用这种途径,从应力空间的加载函数和应变空间的加载函数出发,推导了等向强化材料和被加热的等向强化材料的一般加载规律的弹塑性本构关系的两种表示形式。理论和实例均表明,这种途径对等向强化材料、随动强化材料和理想弹塑性材料均适用。     

Constitutive analysis of elastic-plastic crystals at arbitrary strain

A time-independent constitutive framework is constructed for crystals capable of crystallographic shearing. No restrictions are placed on their elasticity or on the amounts of slip. In deriving properly-objective relations due account is taken of the separate motions of the material and lattice. A precise specification of the structure of the flow rule leads to conditions for the existence of plastic potentials, as well as to an exact statement of the slip criterion required for normality of the plastic strain-increment in conjugate variables. Objective hardening laws are also examined and sufficiency conditions obtained for uniqueness of the slip magnitudes.     

模型复合材料弹塑性界面应力分析

由纤维增强弹塑性基体所产生的界面具有弹塑性力学行为。考虑到一般材料的塑性变形都遵循幂硬化规律,对模型复合材料的界面进行弹性和应变硬化状态下的变形规律及其应力分析。以纤维拔出试验为研究模型,将界面分成弹性区和塑性区。利用界面应力剪滞理论,分别建立弹性区和塑性区的界面力学基本方程。选择适当的位移函数满足基本方程及埋入纤维的边界条件,再按位移函数求出弹性区和塑性区的界面剪应力。推导出平均界面剪应力与纤维     

A nonlocal strain gradient plasticity theory for finite deformations

Strain gradient plasticity for finite deformations is addressed within the framework of nonlocal continuum thermodynamics, featured by the concepts of (nonlocality) energy residual and globally simple material. The plastic strain gradient is assumed to be physically meaningful in the domain of particle isoclinic configurations (with the director vector triad constant both in space and time), whereas the objective notion of corotational gradient makes it possible to compute the plastic strain gradient in any domain of particle intermediate configurations. A phenomenological elastic–plastic constitutive model is presented, with mixed kinematic/isotropic hardening laws in the form of PDEs and related higher order boundary conditions (including those associated with the moving elastic/plastic boundary). Two fourth-order projection tensor operators, functions of the elastic and plastic strain states, are shown to relate the skew-symmetric parts of the Mandel stress and back stress to the related symmetric parts. Consistent with the thermodynamic restrictions therein derived, the flow laws for rate-independent associative plasticity are formulated in a six-dimensional tensor space in terms of symmetric parts of Mandel stresses and related work-conjugate generalized plastic strain rates. A simple shear problem application is presented for illustrative purposes.     

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。     

On the application of the plastic spin concept for the description of anisotropic hardening in finite deformation plasticity

The aim of the paper is to fill the gap between the general theoretical formulation of the constitutive relations for plastic spin and practical applications for proper prediction of material behavior at finite plastic deformations and anisotropic hardening. An approximation to the representation of the general constitutive equation for plastic spin is considered and the pertinent substructure corotational rate is applied to formulate the relation for rigid-plastic material with kinematic hardening. The simple shear traction problem is analysed and the proposed model is verified with the experimental results of Swift. The merits of the present proposal vis-à-vis the existing theories are discussed.     

A plane stress anisotropic plastic flow theory for orthotropic sheet metals

A phenomenological theory is presented for describing the anisotropic plastic flow of orthotropic polycrystalline aluminum sheet metals under plane stress. The theory uses a stress exponent, a rate-dependent effective flow strength function, and five anisotropic material functions to specify a flow potential, an associated flow rule of plastic strain rates, a flow rule of plastic spin, and an evolution law of isotropic hardening of a sheet metal. Each of the five anisotropic material functions may be represented by a truncated Fourier series based on the orthotropic symmetry of the sheet metal and their Fourier coefficients can be determined using experimental data obtained from uniaxial tension and equal biaxial tension tests. Depending on the number of uniaxial tension tests conducted, three models with various degrees of planar anisotropy are constructed based on the proposed plasticity theory for power-law strain hardening sheet metals. These models are applied successfully to describe the anisotropic plastic flow behavior of 10 commercial aluminum alloy sheet metals reported in the literature.     

Thermodynamic restrictions and wave features of a non-linear Maxwell model

A non-linear rate-type constitutive equation, established by Rajagopal, provides a generalization of the Maxwell fluid. This note embodies such a constitutive equation within the scheme of materials with internal variables thus allowing also for solids with both dissipative and thermoelastic mechanisms. The compatibility with the second law of thermodynamics, expressed by the Clausius–Duhem inequality, is examined and the restrictions on the evolution equations are determined. Next the propagation condition of discontinuity waves is derived, for shock waves and acceleration waves, by regarding the body as a definite conductor. Infinitesimal shock waves and acceleration waves show similar effects. The effective acoustic tensor proves to be the sum of a thermoelastic tensor and a tensor arising from the rate-type equation.     

Pressure dependency,strength-differential effect,and plastic volume expansion in metals

Within the framework of a purely mechanical rate-type theory of finitely deforming elastic-plastic materials, a special set of constitutive equations is introduced. These equations include a dependence both on mean normal stress and on plastic strain, and involve non-normality of plastic strain-rate. They accommodate both the strength-differential effect and plastic volume expansion. Values of the material coefficients in the constitutive equations are approximately determined with the help of published experimental data for AISI 4330 steel. The plastic strain-rate is shown to have a large non-normal component, resulting in a small plastic volume expansion. The theory is in reasonable agreement with experiment for plastic strains up to about 2% in magnitude. Further experiments are suggested so that the constitutive coefficients can be completely and accurately determined.     

Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to sheet metal forming

The paper discusses the derivation and the numerical implementation of a finite strain material model for plastic anisotropy and nonlinear kinematic and isotropic hardening. The model is derived from a thermodynamic framework and is based on the multiplicative split of the deformation gradient in the context of hyperelasticity. The kinematic hardening component represents a continuum extension of the classical rheological model of Armstrong–Frederick kinematic hardening. Introducing the so-called structure tensors as additional tensor-valued arguments, plastic anisotropy can be modelled by representing the yield surface and the plastic flow rule as functions of the structure tensors. The evolution equations are integrated by a new form of the exponential map that preserves plastic incompressibility and uses the spectral decomposition to evaluate the exponential tensor functions in closed form. Finally, the applicability of the model is demonstrated by means of simulations of several deep drawing processes and comparisons with experiments.     

On the Plastic Wave Propagation Along the Specimen Length in SHPB Test

To observe the plastic wave propagation, an experimental setup is designed with a SHPB facility and a high speed digital camera. Two types of OFHC copper were selected as specimen materials: in the cold work condition and after total annealing, which represent non strain hardening and strain hardening material respectively. The rise time of incident impulse in the SHPB test is relevant to bar’s radius. A maximum allowable specimen length and a maximum allowable impact velocity (MAIV) of striker are proposed for the SHPB test. The propagation of plastic waves is observed along specimen length at the beginning of specimen’s plastic deformation in SHPB test. However, for both types of material, no plastic wave motion is caught along specimen length for large plastic strain level. Side confinement effect of friction is found to be significant, even with lubricant in the experiment.