Maximum friction law in plasticity

The maximum friction law postulates that the friction stress is equal to the maximum possible shear stress admissible by the constitutive equations. The boundary value problems including the maximum friction stress as a boundary condition reveal special mathematical features which are of interest in the development of theories and models, numerical simulation and engineering applications. The present paper shortly reviews the singularity in velocity fields that can occur in the vicinity of maximum friction surfaces. A large class of rigid plastic (in a broad sense that the elastic portion of the strain rate tensor is neglected) is considered. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A time integration procedure for plastic deformation in elastic-viscoplastic metals

A simple unconditionally stable numerical procedure for time integration of the flow rule for large plastic deformation of an elastic-viscoplastic metal is developed. Specific attention is focused on a unified set of constitutive equations which represents a generalization (for large deformation and thermomechanical response) of the Bodner-Partom model [6, 7]. An analytical solution is obtained for large deformation simple shear at constant shear rate. Numerical examples of simple shear, a corner test exhibiting the transition from uniaxial compression to shear, and simple tension are considered which demonstrate the stability and accuracy of the procedure. It is shown that the same procedure can be used for a rate insensitive metal characterized by a yield function as well as for a rate sensitive metal characterized by an overstress model. Finally, an appendix is provided which records the basic equations associated with the small deformation theory.     

简单剪切振荡现象及弹塑性本构的限制条件

基于在真应力空间刻划弹塑性物质的强化、软化和理想塑性特性,本文以刚塑性随动强化模型为例说明产生简单剪切振荡现象,是与模型在简单剪切变形下,其强化和软化特性发生交替变化的现象有关。为使弹塑性本构模型更符合实际,要求它们必须满足如下条件:即对任意弹塑性加载变形过程,本构模型所给出的应力应该是非振荡的,所描述的强化和软化特性,不存在从软化阶段至强化阶段的过渡。     

Kinematics of finite elastoplastic deformations

In this paper we review various approaches to the decomposition of total strains into elastic and nonelastic (plastic) components in the multiplicative representation of the deformation gradient tensor. We briefly describe the kinematics of finite deformations and arbitrary plastic flows. We show that products of principal values of distortion tensors for elastic and plastic deformations define principal values of the distortion tensor for total deformations. We describe two groups of methods for decomposing deformations and their rates into elastic and nonelastic components. The methods of the first group additively decompose specially built tensors defined in a common basis (initial, current, or “intermediate”). The second group implies a certain relation connecting tensors that describe elastic and plastic deformations. We adduce an example of constructing constitutive relations for elastoplastic continuums at large deformations from thermodynamic equations.     

Granular materials: Parameter identification and modelling of localisation problems

The prediction of landsliding requires an exact knowledge of the mechanical behaviour of granular materials. This kind of materials, e. g., sand, have a very complex deformation behaviour, which depend on the stress state and on the loading history. In this work, the deformation behaviour of the solid skeleton is characterised via homogeneous triaxial tests on dry sand specimens. Additionally, an appropriate elasto-plastic material law to describe the solid skeleton in the frame of Theory of Porous Media (TPM) is used, which is implemented in the FE tool PANDAS. Furthermore, a single-surface yield criterion with isotropic hardening, which limits the elastic domain, and a non-associated plastic flow are employed. The determination of the material parameters of the linear elasticity law as well as the single-surface yield criterion are based on test data of triaxial experiments. The material parameters are identified using a derivative-based optimisation method (donlp2), which is coupled with PANDAS. Finally, a simulation of a benchmark test is presented to show shear band localisation effects, where the material behaviour is described by a triphasic porous media model based on the TPM, where the constituents are a deformable solid skeleton and two pore fluids, water and air. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental Investigations on Polymeric Foams

The knowledge of the material behaviour of polymeric foams and their experimental investigation is the starting point for the structure of the chosen constitutive equations and for the following identification of the material constants therein. Especially for the parameter identification, it is necessary to make an adequate set of experimental data available. In this regard, it is important that the experiments make the different kinds of material behaviour visible like elastic, plastic or viscous material properties. For this reason, the foam is observed under uniaxial tension and compression and under simple shear tests combined with different deformation states in axial direction. Unfortunately, due to different reasons, e.g., the foam must be sticked on the fastener to realize the tests mentioned above, it is very difficult to initialize a homogenous deformation state in the specimen. Therefore, the experiments are recorded with a standard digital camcorder to get local information of the deformation state by tracking single points with algorithms of the digital image processing. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

正交各向异性材料的混合硬化弹塑性本构方程

建立了混合硬化正交各向异性材料的屈服准则,进而推导了与之相关的塑性流动法则.根据简单应力状态的实验曲线,可得到广义等效应力-应变关系.初始屈服曲面与材料的弹性常数有关,材料退化为各向同性且只考虑各向同性硬化时,屈服函数退化为Huber-Mises屈服函数,相关的本构方程退化为Prandtl-Reuss方程.     

Analysis of adiabatic shear bands in thermo-elasto-viscoplastic materials by using piece-wise discontinuous basis functions

An adiabatic shear band (ASB) is a narrow region of intense plastic deformation that forms when some metallic alloys and some polymers are deformed at high strain rates and there is not enough time for the heat generated by plastic deformations to diffuse away. The study of ASBs is important because an ASB is a precursor to shear/ductile fractures. Initial-boundary-value problems simulating the initiation and propagation of an ASB have been analyzed usually using the finite element method (FEM). Because of the large plastic strains involved, the FE mesh needs to be refined several times to delineate the ASB width. Each refinement requires, in turn, interpolation of data from the previous mesh to the new one which causes a smoothening of the sharp gradients of the deformation fields, and affects characteristics of the ASB. In this paper, we propose the application of the finite element method with piecewise discontinuous basis functions for studying the occurrence of ASBs in simple shearing deformations of a body composed of an isotropic and homogeneous thermo-elastoviscoplastic material. The mathematical model of the problem is defined by a system of coupled nonlinear partial differential equations and an inequality constraint associated with the plastic strain rates admissibility.     

Geometrical super-elements for elasto-plastic shells with large deformation

This paper presents the application of the so-called Geometrical Elements Method (Lukasiewicz and Szyszkowski, 1974; Pogorelov, 1967) to the solution of elasto-plastic problems of shells. The approach is based on the observation that, during large deformations, the shell structure deforms in a nearly isometrical manner. Therefore, its deformed shape can be determined and analysed making use of the Gauss theorem according to which the Gaussian curvature of the isometrically deformed surface remains unchanged. The shell structure is subdivided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. The equilibrium of the whole structure is defined by the stationary value of the Hamiltonian function which requires the calculation of the strain energy in the elements. This can easily be obtained if we recognize that the isometrically deformed elements contain only bending energy. Using the method described, we are able to significantly the number of unknown values defining the shape of the deformed structure. The problem is reduced to the numerical evaluation of the minimum of a function of many variables. The elasto-plastic state of stress of the plastic material in the structure canbe determined by using the deformation theory of plasticity or the theory of plastic flow. Also, the strains and stresses in the plastic regions are the only functions of the assumed displacements field. The corresponding energy of the plastic deformation can easily be evaluated and added to the minimized functionals. For example, the elasto-plastic behaviour of a spherical shell under a concentrated load is studied. The solution obtained defines the large deformation behaviour and the motion of the plastic zones on the surface of the shell.     

The choice of constitutive relations for an ice cover

The constitutive relations between the internal stresses and the deformation parameters of a sea ice cover, which are used in the AIDJEX elastoplastic model and Hibler's non-linearly viscous model, are investigated. It is shown that the structural instability of the ice cover with respect to plastic shear deformations is a consequence of the associated flow rule used in these models. The use of constitutive relations which violate the associated flow rule, but which are in good agreement with the physical properties of granular media, is suggested. An ice cover damage parameter and an empirical equation which describes the change in this parameter are introduced into the treatment. Energy relations are investigated.     

Using distributed contacts in DEM

Granular flows in shear cells have been extensively studied using the Discrete Element Method (DEM) over the last two decades. These studies have typically been performed using the soft-sphere approach where deformation is assumed elastic and small relative to the characteristic grain scale. Consequently internal stresses and strains are not able to be modelled. As a first step towards addressing these limitations, we introduce a variant of DEM, the Distributed Contact DEM (DCDEM). This method models distributed normal and frictional contacts. In this initial implementation plastic deformation is not simulated and elastic deformation is simulated by permitting overlap as in traditional DEM. The method is compared against traditional DEM for a normal and oblique impact and a granular shear cell in the small deformation limit.     

Modelling Metals At Finite Deformation

During metal forming processes, substantial microstructural changes occur in the material due to large plastic deformations leading to different mechanical properties. It is of great interest to predict the behaviour of these materials at different fabriction stages and of the final product. At first glance, the behaviour of metals can be approached by an elastoplastic isotropic material model with a volumetric-deviatoric split and isotropic hardening. In order to perform the calculations, a logarithmic strain is considered in the principal directions of stress and strain space, allowing to make predictions even at finite deformations. Because of the actual nature of metals, the crystalline structure, the deformation at the microstructural level is much more complex. Due to the mathematically algorithmic form of an elastic predictor and a plastic corrector, the elastoplastic model can be extended to crystal plasticity which is similarly handled in terms of a critical resolved shear stress on defined slip planes in the crystal. Hardening can be modelled through a viscoplastic power law. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

弹性动脉血管中血液流动特性的模拟和分析

研究了两个不同的非牛顿血液流动模型:低粘性剪切简单幂律模型和低粘性剪切及粘弹性振荡流的广义Maxwell模型．同时利用这两个非牛顿模型和牛顿模型,研究了磁场中刚性和弹性直血管中血液的正弦型脉动．在生理学条件下,大动脉中血液的弹性对其流动性态似乎并不产生影响,单纯低粘性剪切模型可以逼真地模拟这种血液流动．利用高剪切幂律模型模拟弹性血管中的正弦型脉动流,发现在同一压力梯度下,与牛顿流体相比较,幂律流体的平均流率和流率变化幅度都更小．控制方程用Crank-Niclson方法求解．弹性动脉中血液受磁场作用是产生此结果的直观原因．在主动脉生物流的模拟中,与牛顿流体模型比较,发现在匹配流率曲线上,幂律模型的平均壁面剪切应力增大,峰值壁面剪切应力减小．讨论了弹性血管横切磁场时的血液流动,评估了血管形状和表面不规则等因素的影响．     

Finite strain inelasticity for isotropy,a simple and efficient finite element formulation

We consider a formulation of associative isotropic J₂‐elastoplasticity at finite inelastic strains and aspects of its numerical implementation. The essential ingredients include the multiplicative decomposition of the deformation gradient in elastic and inelastic parts, the definition of a convex elastic domain in stress space and a material representation of the constitutive equations for general non‐Cartesian coordinate charts. On the numerical side we propose a stress update algorithm for elasto‐plastic response, including isotropic hardening. The finite element formulation is based on assumed strain and enhanced strain variational principles, for a complete outline see [3]. Remarkably the formulation is very similar to the case of infinitesimal plasticity: (i) The scheme of linear return mapping algorithm takes the form of standard return mapping of the infinitesimal theory for the case of isotropic elastic response. (ii) The algorithmic elastoplastic moduli have a similar structure as in the linear case. Together with an exact fulfillment of plastic incompressibility by means of a simple correction one achieves an advantageously efficient finite element formulation. Its performance is documented by a numerical example.     

Elastoplastic stability of glass-reinforced plastic plates with a central metal layer

The problem of the stability of a three-layer plate with a central plastic layer of metal sandwiched between elastic glass-reinforced plastic outer layers is considered. The presence of a metal layer restrains the development of creep strains in the glass-reinforced plastic and makes it possible to neglect the viscous strain components. The general equations of the problem are obtained, and the approximate Il'yushin formulation [1] is considered. An example is presented for a rectangular plate in pure shear. It is shown that the elastic anisotropic layers play the part of a load-relieving system for the central plastic layer [3], which results in an increase in the over-all critical load for the layered plate.Kalinin Polytechnic Institute. Translated from Mekhanika Polimerov, No. 5, pp. 909–915, September–October, 1969.     

Elastoplastic deformation during projectile–wall collision

The Smoothed Particle Hydrodynamics method for elastic solid deformation is modified to include von Mises plasticity with linear isotropic hardening and is then used to investigate high speed collisions of elastic and elastoplastic bodies. The Lagrangian mesh-free nature of SPH makes is very well suited to these extreme deformation problems eliminating issues relating to poor element quality at high strains that limits finite element usage for these types of problems. It demonstrates excellent numerical stability at very high strains (of more than 200%). SPH can naturally track history dependent material properties such as the cumulative plastic strain and the degree of work hardening produced by its strain history. The high speed collisions modelled here demonstrate that the method can cope easily with collisions of multiple bodies and can also naturally resolve self-collisions of bodies undergoing high levels of plastic strain. The nature and the extent of the elastic and plastic deformation of a rectangular body impacting on an elastic wall and of an elastic projectile impacting on a thin elastic wall are investigated. The final plastically deformed shapes of the projectile and wall are compared for a range of material properties and the evolution of the maximum plastic strain throughout each collision and the coefficient of restitution are used to make quantitative comparisons. Both the elastoplastic projectile–elastic wall and the elastic projectile–elastoplastic wall type collisions have two distinct plastic flow regimes that create complex relationships between the yield stress and the responses of the solid bodies.     

A numerical framework for rheological models based on the decomposition of the deformation rate tensor

In this contribution, the rheological models based on the decomposition of the deformation rate tensor, as proposed by Palmow (1984), is combined with the numerical solution of objective tensorial ODEs as published by Rashid (1993). The resulting framework is suitable to model the complex inelastic properties of many materials at large strains. As an example, the Schwedoff model, which is appropriate for the simulation of metal forming processes, is analyzed within a cyclic simple shear test. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)