Continuity violation trajectories in perfectly plastic bodies

The consistency equations for the increments of small strains in a triorthogonal isostatic coordinate system supplemented with additional relations between the physical components of the inconsistency tensor are considered. There are six significant consistency equations. It is proved that, for the stress states corresponding to the edge of the Coulomb-Tresca prism, there are only three independent consistency equations. Systems of independent consistency equations written in the isostatic coordinate mesh are explicitly indicated and studied. Sufficient conditions for the remaining three consistency equations to be satisfied if the three independent consistency equations are satisfied are obtained. It is shown that the continuity violations on the surface of a perfectly plastic body propagate into the depth of the body along asymptotic lines on the layers of a vector field indicating the directions of the maximum principal normal stress. Since the asymptotic lines are less curved than any other lines on the surface (in the sense that the normal curvature of the asymptotic lines is zero), the continuity violations propagate into a perfectly plastic body along the least curved trajectories, which permits one to speak of the minimum curving of crack propagation trajectories in perfectly plastic rigid bodies.     

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.      

Plane stress state problems for notched bodies whose plastic properties depend on the form of the stress state

We consider one possible approach to the problem of describing the dependence of material plastic strain characteristics on the stress hydrostatic component arising in many porous, fractured, and other inhomogeneous materials. The plastic strain of the media under study is investigated under the plasticity assumption in the corresponding generalized form with the use of the form parameter of the stress state. The plasticity constitutive relations are stated on the basis of the plastic flow law associated with the accepted plasticity condition. For the conditions of plane stress state in the framework of the material rigid-plastic model, a system of partial differential equations is obtained and conditions for its hyperbolicity are determined. The relations for determining the stress fields and velocity fields in plastic domains are obtained, and their properties are investigated. The problem of tension of a strip with symmetric angular notches is solved, where the stress fields are determined and the continuous displacement rate field is constructed. The problem of uniform symmetric tension of a plane with a circular hole is considered. The stress fields in a strip with symmetric circular notches are examined. A comparison with solutions for plastically incompressible media whose properties are invariant with respect to the form of the stress state is performed.     

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I

A Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM) model is developed, extending continuum plasticity theory for studying initial-boundary value problems of small-scale plasticity. PMFDM results from an elementary space-time averaging of the equations of Field Dislocation Mechanics (FDM), followed by a closure assumption from any strain-gradient plasticity model that attempts to account for effects of geometrically necessary dislocations (GNDs) only in work hardening. The specific lower-order gradient plasticity model chosen to substantiate this work requires one additional material parameter compared to its conventional continuum plasticity counterpart. The further addition of dislocation mechanics requires no additional material parameters. The model (a) retains the constitutive dependence of the free-energy only on elastic strain as in conventional continuum plasticity with no explicit dependence on dislocation density, (b) does not require higher-order stresses, and (c) does not require a constitutive specification of a ‘back-stress’ in the expression for average dislocation velocity/plastic strain rate. However, long-range stress effects of average dislocation distributions are predicted by the model in a mechanistically rigorous sense. Plausible boundary conditions (with obvious implication for corresponding interface conditions) are discussed in some detail from a physical point of view. Energetic and dissipative aspects of the model are also discussed. The developed framework is a continuous-time model of averaged dislocation plasticity, without having to rely on the notion of incremental work functions, their convexity properties, or their minimization. The tangent modulus relating stress rate and total strain rate in the model is the positive-definite tensor of linear elasticity, and this is not an impediment to the development of idealized microstructure in the theory and computations, even when such a convexity property is preserved in a computational scheme. A model of finite deformation, mesoscopic single crystal plasticity is also presented, motivated by the above considerations.Lower-order gradient plasticity appears as a constitutive limit of PMFDM, and the development suggests a plausible boundary condition on the plastic strain rate for this limit that is appropriate for the modeling of constrained plastic flow in three-dimensional situations.     

软化Mohr-Coulomb材料强度参数的测定方法

为了得到试件的粘聚力和内摩擦角随轴向塑性压应变变化的曲线提出本方法。试件的弹塑性本构关系遵循相关联的Mohr-Coulomb强度准则;对常规三轴试验,试件受力进入塑性状态后,处在棱椎状屈服面的棱上,加载过程遵循Koiter流动法则。按经典塑性力学理论,推导得到轴向塑性压应变与轴向应力与轴向应变的关系;在常规三轴试验机上获得不同围压下试件的全程应力-应变曲线,进而可得到各自围压下轴向塑性压应变随加载过程的变化曲线;把来自不同围压下对应同一轴向塑性压应变的应力分别代入屈服面方程,即可求得对应的粘聚力和内摩擦角。结果表明,Mohr-Coulomb材料的两个强度参数的变化由轴向塑性压应变确定。轴向塑性压应变可以作为塑性变形的状态参数,它和试件的受力过程可以唯一确定试件的变形过程。     

Couple stresses in crystalline solids: origins from plastic slip gradients, dislocation core distortions, and three-body interatomic potentials

In the presence of plastic slip gradients, compatibility requires gradients in elastic rotation and stretch tensors. In a crystal lattice the gradient in elastic rotation can be related to bond angle changes at cores of so-called geometrically necessary dislocations. The corresponding continuum strain energy density can be obtained from an interatomic potential that includes two- and three-body terms. The three-body terms induce restoring moments that lead to a couple stress tensor in the continuum limit. The resulting stress and couple stress jointly satisfy a balance law. Boundary conditions are obtained upon stress, couple stress, strain and strain gradient tensors. This higher-order continuum theory was formulated by Toupin (Arch. Ration. Mech. Anal. 11 (1962) 385). Toupin's theory has been extended in this work to incorporate constitutive relations for the stress and couple stress under multiplicative elastoplasticity. The higher-order continuum theory is exploited to solve a boundary value problem of relevance to single crystal and polycrystalline nano-devices. It is demonstrated that certain slip-dominated deformation mechanisms increase the compliance of nanostructures in bending-dominated situations. The significance of these ideas in the context of continuum plasticity models is also dwelt upon.     

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.     

On attainable lower boundary of the three-dimensional Coulomb-Tresca invariant

An attainable lower boundary of the three-dimensional Coulomb-Tresca invariant is constructed to facilitate the search of plasticity conditions for isotropic bodies which, just as the Tresca-Saint Venant plasticity condition, ensure the hyperbolic analytic type of three-dimensional equations of the mathematical theories of plasticity based on the generalized associate flow law. The construction is performed by using the system of three “two-dimensional” tangential stresses related to the given three-dimensional stress state. It is proved that the Coulomb-Tresca invariant attains its lower bound in any plane strain state where the out-of-plane principal normal stress is intermediate (or median).     

黏弹-Perzyna黏塑性有限元法应力更新隐式算法

黏弹-黏塑性耦合模型的黏弹性部分由弹簧、黏壶和Kelvin链串联而成,黏塑性部分为双曲线型DruckerPrager屈服函数、各向同性硬化和Perzyna黏塑性流动模型。基于黏弹性蠕变柔度,通过定义与弹性问题相对应的与时间增量相关的黏弹性剪切模量和体积模量,导出增量递推形式的本构方程。为保证算法的收敛和稳定性,把Perzyna黏塑性流动方程转化为与弹塑性相似的一致性条件,建立黏塑性增量因子单侧逼近其收敛值的N-R迭代算法。最后,给出应力更新完全隐式算法和最终计算公式。分别采用黏弹性、黏弹-塑性和黏弹-黏塑性本构关系对一地基蠕变模型进行三维有限元分析和比较,结果表明,本文算法具有较高的计算效率和稳定性。     

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.     

An exact semi-analytic solution for residual stresses and strains within a thin hollow disc of pressure-sensitive material subject to thermal loading

There is considerable current interest in the development of constitutive equations for pressure-dependent plastic materials. In particular, in contrast to classical plasticity there is no commonly accepted relation to connect stress and strain or strain rate for such materials. Analytic and semi-analytic solutions are convenient to compare qualitative features of boundary value problems solved for different models. Such comparative studies can be useful to choose this or that model for specific applications. Analytic and semi-analytic solutions are also necessary to verify numerical codes. In the present paper, a new semi-analytic solution for a thin hollow disc subject to thermal loading is developed. A numerical method is only necessary to solve transcendental equations. The constitutive equations for connecting the plastic portion of the strain rate tensor and the stress tensor consist of the Drucker-Prager yield criterion and its associated flow rule. Therefore, the main distinguished feature of the solution is that the material is plastically compressible.     

Elastoplastic model and theory of slipping for the three-dimensional stress state

On the basis of concepts of the Batdorf-Budyanskii theory of slipping, we construct a model of elastoplastic medium for the case of three-dimensional stress state. The slipping conditions on the unit site take into account the local yield criterion and the local loading criterion. Under certain assumptions, one can integrate the increments of plastic shears over all possible sites of slipping in the case of an arbitrary three-dimensional stress state and obtain the constitutive relations for the elastoplastic model, which is a version of the theory of plastic flow.     

A note on incremental equations for a new class of constitutive relations for elastic bodies

Some new classes of constitutive relations for elastic bodies have been proposed in the literature, wherein the stresses and strains are obtained from implicit constitutive relations. A special case of the above relations corresponds to a class of constitutive equations where the linearized strain tensor is given as a nonlinear function of the stresses. For such constitutive equations we consider the problem of decomposing the stresses into two parts: one corresponds to a time-independent solution of the boundary value problem, plus a small (in comparison with the above) time-dependent stress tensor. The effect of this initial time-independent stress in the propagation of a small wave motion is studied for an infinite medium.     

Generalizing J 2 flow theory: Fundamental issues in strain gradient plasticity

It has not been a simple matter to obtain a sound extension of the classical J₂ flow theory of plasticity that in- corporates a dependence on plastic strain gradients and that is capable of capturing size-dependent behaviour of metals at the micron scale. Two classes of basic extensions of clas- sical J₂ theory have been proposed: one with increments in higher order stresses related to increments of strain gradi- ents and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gra- dients. The theories proposed by Muhlhaus and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy ther- modynamic requirements on plastic dissipation. On the other hand, theories of the second class proposed by Gudmundson in 2004 and Gurtin and Anand in 2009 have the physical deficiency that the higher order stress quantities can change discontinuously for bodies subject to arbitrarily small load changes. The present paper lays out this background to the quest for a sound phenomenological extension of the rate- independent J₂ flow theory of plasticity to include a de- pendence on gradients of plastic strain. A modification of the Fleck-Hutchinson formulation that ensures its thermo- dynamic integrity is presented and contrasted with a compa- rable formulation of the second class where in the higher or- der stresses are expressed in terms of the plastic strain rate. Both versions are constructed to reduce to the classical J₂ flow theory of plasticity when the gradients can be neglected and to coincide with the simpler and more readily formulated J₂ deformation theory of gradient plasticity for deformation histories characterized by proportional straining.     

Analysis for polycrystal nonproportional cyclic plasticity with a dissipated energy based rule for cross-hardening

Based on a nonclassical hardening law and the Hill’s self-consistent scheme, a new approach is proposed for the analysis of polycrystal nonproportional cyclic plasticity. A novel parameter related to the plastic dissipation on each slip system is proposed and embedded in the Bassani’s definition of cross-hardening. The tangential elastoplastic tensor relating the increments of stress and strain in a single crystal is derived and the corresponding numerical algorithm for polycrystal plasticity is developed. The elastoplastic response of 316 stainless steel subjected to typical biaxial nonproportional strain cycling is analyzed, and the main features are well replicated. The validity of the proposed approach is demonstrated by the satisfactory agreement between the computed results and experimental observation.     

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。     

Variational methods for the steady state response of elastic–plastic solids subjected to cyclic loads

Solids (or structures) of elastic–plastic internal variable material models and subjected to cyclic loads are considered. A minimum net resistant power theorem, direct consequence of the classical maximum intrinsic dissipation theorem of plasticity theory, is envisioned which describes the material behavior by determining the plastic flow mechanism (if any) corresponding to a given stress/hardening state. A maximum principle is provided which characterizes the optimal initial stress/hardening state of a cyclically loaded structure as the one such that the plastic strain and kinematic internal variable increments produced over a cycle are kinematically admissible. A steady cycle minimum principle, integrated form of the aforementioned minimum net resistant power theorem, is provided, which characterizes the structure’s steady state response (steady cycle) and proves to be an extension to the present context of known principles of perfect plasticity. The optimality equations of this minimum principle are studied and two particular cases are considered: (i) loads not exceeding the shakedown limit (so recovering known results of shakedown theory) and (ii) specimen under uniform cyclic stress (or strain). Criteria to assess the structure’s ratchet limit loads are given. These, together with some insensitivity features of the structure’s alternating plasticity state, provide the basis to the ratchet limit load analysis problem, for which solution procedures are discussed.     

Orthogonality hypothesis and gradientality principle in plasticity theory

We present the orthogonality hypothesis and the generalized Il’yushin gradientality principle in the plasticity theory. We use the orthogonality hypothesis to obtain local three-dimensional constitutive relations of the theory of plastic strain processes. We also discuss a model of the limit complex loading surface for a plastically deformable medium.