Determination of the ultimate states of elastoplastic bodies

The equations of quasistatic deformation of elastoplastic bodies are considered in a geometrical linear formulation. After discretization of the equations with respect to spatial variables by the finite-element method, the problem of determining equilibrium onfigurations reduces to integration of a system of nonlinear ordinary differential equations. In the ultimate state of a body of an ideal elastoplastic material, the matrix of the system degenerates and the problem becomes singular. A regularization algorithm for determining solutions of the problems for the ultimate states of bodies is proposed. Numerical solutions of test problems of determining the ultimate loads and equilibrium configurations for ideal elastoplastic bodies confirm the reliability of the regularization algorithm proposed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 196–204, September–October, 2000.     

Applying the Explicit Time Central Difference Method for Numerical Simulation of the Dynamic Behavior of Elastoplastic Flexible Reinforced Plates

Based on a stepwise algorithm involving central finite differences for the approximation in time, a mathematical model is developed for elastoplastic deformation of cross-reinforced plates with isotropically hardening materials of components of the composition. The model allows obtaining the solution of elastoplastic problems at discrete points in time by an explicit scheme. The initial boundary value problem of the dynamic behavior of flexible plates reinforced in their own plane is formulated in the von Kármán approximation with allowance for their weakened resistance to the transverse shear. With a common approach, the resolving equations corresponding to two variants of the Timoshenko theory are obtained. An explicit “cross” scheme for numerical integration of the posed initial boundary value problem has been constructed. The scheme is consistent with the incremental algorithm used for simulating the elastoplastic behavior of a reinforced medium. Calculations of the dynamic behavior have been performed for elastoplastic cylindrical bending of differently reinforced fiberglass rectangular elongated plates. It is shown that the reinforcement structure significantly affects their elastoplastic dynamic behavior. It has been found that the classical theory of plates is as a rule unacceptable for carrying out the required calculations (except for very thin plates), and the first version of the Timoshenko theory yields reasonable results only in cases of relatively thin constructions reinforced by lowmodulus fibers. Proceeding from the results of the work, it is recommended to use the second variant of the Timoshenko theory (as a more accurate one) for calculations of the elastoplastic behavior of reinforced plates.     

基于数字散斑相关实验测量的材料构型力的计算方法

材料构型力学主要研究材料中的缺陷(夹杂、空穴、位错、裂纹、塑性区等)的构型(形状、尺寸和位置)改变时,所引起的系统自由能的变化。本研究将基于数字散斑相关技术,实验测量材料试件的位移场分布,随后通过材料构型力的定义式,计算求得弹塑性材料中缺陷构型力的分布。其方法概括如下:位移场通过数字图像相关技术测得;应变及位移梯度场利用三次样条拟合获得;线弹性材料应力通过简单线弹性本构方程获取,而塑性材料的表面应力场通过Ramberg-Osgood本构方程计算求得;弹塑性应变能密度分布则由应力-应变曲线数值积分获得。该方法对普通弹性材料或者弹塑性材料均适用,可以用于各种不同的缺陷及缺陷群的材料构型力测量。     

任意光滑梯度变化的功能梯度材料纯弯曲梁的弹塑性分析

假设功能梯度材料为理想弹塑性材料,其屈服强度和弹性模量均沿梁的高度方向按任意光滑函数连续变化,在小变形及平截面假定下,导出了功能梯度材料纯弯曲梁弹性极限弯矩及塑性极限弯矩的解析表达式,建立了弹塑性应力状态下截面弯矩和截面的弹、塑性应力分布之间的解析关系．研究表明,功能梯度材料梁存在多种可能的屈服模式,其最先屈服的点不一定位于截面应力最大处,而可能位于截面的其他任意位置;屈服强度及弹性模量的梯度变化对梁的弹塑性力学性能有很大影响．研究结果可为功能梯度材料纯弯曲梁的弹塑性问题研究提供一定的参考．     

Stress-strain state and durability of mechanically inhomogeneous welds under low-cycle loading

Relations are proposed for the determination of the stress-strain state, strength, and life of butt welds with mild and hard interlayers under cyclic elastoplastic tension-compression. The accumulation of cyclic and quasistatic damages is determined with allowance for the redistribution of the cyclic elastoplastic strains and hardness of the stress state due to changes in the cyclic properties of separate regions of welds. The theoretical distribution of cyclic strains and the durability of welds under cyclic elastoplastic loading are supported by experimental data __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 29–38, February 2008.     

Constitutive equations for describing the elastoplastic deformation of elements of a body along small-curvature paths in view of the stress mode

Equations relating the components of the stress and strain tensors (constitutive equations) are formulated in terms of Euler coordinates. The equations describe the finite elastoplastic deformation of an isotropic body along paths of small curvature. It is assumed that the stress deviator is coaxial with the plastic-strain differential deviator. The relationships between the first and second invariants of the stress and strain tensors in the case of complex elastoplastic deformation of the body’s elements are determined from base tests on tubular specimens loaded along rectilinear paths for several values of the stress mode angle. Methods for specification of these relationships are proposed. The assumptions adopted to derive the constitutive equations are validated experimentally __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 62–72, April 2006.     

On Uniqueness in the Elastoplastic Analysis of Frames

ABSTRACT

A very simple example of elastoplastic analysis is adduced to illustrate how more than one deformed configuration can be associated with a single prescribed load state. The encoding of holonomic elastoplastic analysis in quadratic programming form leads to an uniqueness theorem for the bending moments (static solution), and yet, even for holonomic elastoplasticity, the uniqueness of the deformed configuration (kinematic solution) cannot be guaranteed. Different kinematic solutions, when they exist, are seen from the simple example to be physically induced by different load paths terminating at the same prescribed load state, and thus it is proved that holonomic elastoplasticity is not strictly path independent. Any two valid kinematic solutions can only differ by the motion of a special mechanism, called a pseudo-mechanism. This device is used to establish a uniqueness theorem for the kinematic solution obtained by quadratic programming or any other form of holonomic elastoplastic analysis. Finally, it is shown that load states exhibiting non-unique kinematic solutions always He on pseudomechanism lines in load space, and that a load path which lies along some part of such a line produces a completely indeterminate deformed configuration.     

On the uniqueness of measuring elastoplastic properties from indentation: The indistinguishable mystical materials

Indentation is widely used to extract material elastoplastic properties from the measured force-displacement curves. One of the most well-established indentation techniques utilizes dual (or plural) sharp indenters (which have different apex angles) to deduce key parameters such as the elastic modulus, yield stress, and work-hardening exponent for materials that obey the power-law constitutive relationship. However, the uniqueness of such analysis is not yet systematically studied or challenged. Here we show the existence of “mystical materials”, which have distinct elastoplastic properties yet they yield almost identical indentation behaviors, even when the indenter angle is varied in a large range. These mystical materials are, therefore, indistinguishable by many existing indentation analyses unless extreme (and often impractical) indenter angles are used. Explicit procedures of deriving these mystical materials are established, and the general characteristics of the mystical materials are discussed. In many cases, for a given indenter angle range, a material would have infinite numbers of mystical siblings, and the existence maps of the mystical materials are also obtained. Furthermore, we propose two alternative techniques to effectively distinguish these mystical materials. The study in this paper addresses the important question of the uniqueness of indentation test, as well as providing useful guidelines to properly use the indentation technique to measure material elastoplastic properties.     

Nonaxisymmetric Thermoelastoplastic Analysis of Branched Shells of Revolution by the Semianalytic Finite-Element Method

A technique is proposed to solve elastoplastic deformation problems for branched shells of revolution under the action of asymmetric forces and a temperature field. The kinematic equations are derived within the framework of the linear Kirchhoff–Love theory of shells and the thermoelastic relations within the framework of the theory of small elastoplastic strains. The problem is given a variational formulation based on the virtual-displacement principle and the Fourier-series expansion of the unknown functions and loads with respect to the circumferential coordinate. The additional-load method is used to solve a nonlinear problem and the finite-elements method is used to carry out a numerical analysis. As an example, an asymmetric stress–strain analysis is performed for a cylindrical shell reinforced by a ring plate.     

Bending of an elastoplastic Hencky bar-chain: from discrete to nonlocal continuous beam models

The static behavior of an elastoplastic one-dimensional lattice system in bending, also called a microstructured elastoplastic beam or elastoplastic Hencky bar-chain (HBC) system, is investigated. The lattice beam is loaded by concentrated or distributed transverse monotonic forces up to the complete collapse. The phenomenon of softening localization is also included. The lattice system is composed of piecewise linear hardening–softening elastoplastic hinges connected via rigid elements. This physical system can be viewed as the generalization of the elastic HBC model to the nonlinear elastoplasticity range. This lattice problem is demonstrated to be equivalent to the finite difference formulation of a continuous elastoplastic beam in bending. Solutions to the lattice problem may be obtained from the resolution of piecewise linear difference equations. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process. The new nonlocal elastoplastic theory associated with both a distributed nonlocal elastoplastic law coupled to a cohesive elastoplastic model depends on length scales calibrated from the spacing of the lattice model. Differential equations of the nonlocal engineering model are solved for the structural configurations investigated in the lattice problem. It is shown that the new micromechanics-based nonlocal elastoplastic beam model efficiently captures the scale effects of the elastoplastic lattice model, used as the reference. The hardening–softening localization process of the nonlocal continuous model strongly depends on the lattice spacing which controls the size of the nonlocal length scales.     

Effective pressure-sensitive elastoplastic behavior of particle-reinforced composites and porous media under isotropic loading

The composite under investigation consists of an elastoplastic matrix reinforced by elastic particles or weakened by pores. The material forming the matrix is pressure-sensitive. The Drucker–Prager yield criterion and a one-parameter non-associated flow rule are employed to formulate the yield behavior of the matrix. The objective of this work is to estimate the effective elastoplastic behavior of the composite under isotropic tensile and compressive loadings. To achieve this objective, the composite sphere assemblage model of Hashin [Z. Hashin, The elastic moduli of heterogeneous materials, ASME J. Appl. Mech. 29 (1962) 143–150] is used. Exact solutions are thus derived as estimations for the effective secant and tangent bulk moduli of the composite. The effects of the loading modes and phase properties on the effective elastoplastic behavior of the composite are analytically and numerically evaluated.     

Thermoelastoplastic Deformation of a Multilayer Ball

The problem of centrally symmetric deformation of a multilayer elastoplastic ball in the process of successive accretion of preheated layers to its outer surface is considered in the framework of small elastoplastic deformations. The problems of residual stress formation in the elastoplastic ball with an inclusion and a cavity are solved under various mechanical boundary conditions on the inner surface and for prescribed thermal compression distributions. The graphs of residual stress and displacement fields are constructed.     

A novel semi-inverse solution method for elastoplastic torsion of heat treated rods

Torsion rods are a primary component of many power transmission and other mechanical systems. The behavior of these rods under elastoplastic torsion is of major concern for designers. Different methods have so far been proposed which deal with the elastoplastic torsion of rods, most of which assume constant yield stress. This assumption produces rough and inaccurate results when the rods are heat treated, since in the process of heat treatment the form of yield stress distribution across the rod cross section changes. We propose a new method for calculating elastoplastic torsion of rods of simply connected cross section which is based on heat treatment observations. In our method the full plastic stress function is obtained by using the semi-inverse method. Elastoplastic stress function is obtained by generalizing the idea of the membrane analogy and using a piecewise continuous stress function. Since the proposed form of yield stress distribution can not be handled by the current Finite Element packages, we produce a computer package with a 3D graphical interface capable of calculating and displaying the 3D elastoplastic stress function, shear stress contours, and torque-angle of rotation per unit length. We show that our method produces excellent agreement for several known cross sections in comparison to methods which assume constant yield stress.     

Analytical Solutions of Nonlinear Static Problems for Threads of Finite Stiffness Under Active Loading

The behavior of elastoplastic threads of finite stiffness under lateral bending is analyzed. Geometrical and physical nonlinearities are taken into account. The material is assumed to be elastoplastic. The nonlinear equations describing the stress—strain state of threads are derived using the virtual-displacement principle. Numerical results are discussed __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 121–129, June 2005.     

Micro–macro approach for the rock salt behaviour

Rock salt is considered as a pure aggregate of halite (mineral NaCl) crystals and its behaviour is investigated by a micro–macro approach. The behaviour of the polycrystalline aggregate is deduced from the properties of the constituent halite crystals. A model for the elastoplastic behaviour of halite crystal has been deduced from experimental data available in the literature. The basic equations of the micro–macro model for the polycrystalline medium and the calculation method are then presented and the elastoplastic behaviour of rock salt is investigated by this method. The hardening effects obtained for the polycrystal are found to be very different from those obtained for FCC metal polycrystals. The differences are explained as a consequence of differences of families of glide systems in these crystals. Finally, the internal stresses in the polycrystal are studied in order to elucidate the origin of cracking and damage of the rock salt.     

Deformation and fracture resistance of butt-welded joints with mild interlayer under elastoplastic tension

Relations are proposed for evaluating the stress-strain state and strength of a butt weld with a mild interlayer under elastoplastic tension in plane-and axisymmetric-strain conditions. The relations allow for the distribution of transverse normal stresses across the interlayer thickness and the interaction and hardening of the metals in the elastoplastic domain. The predicted strain distribution in certain areas of the weld and its strength are verified experimentally __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 119–127, April 2006.     

Elastoplastic modeling of circular fiber-reinforced ductile matrix composites considering a finite RVE

A micromechanical elastoplastic damage model considering a finite RVE is proposed to predict the overall elastoplastic damage behavior of circular fiber-reinforced ductile (matrix) composites. The constitutive damage model proposed in our preceding work (Kim and Lee, 2009) considering a finite Eshelby’s tensor (Li et al., 2005, Wang et al., 2005) is extended to accommodate the elastoplastic behavior of the composites. On the basis of the exterior-point Eshelby’s tensor for circular inclusions and the ensemble-averaged effective yield criterion, a micromechanical framework for predicting the effective elastoplastic damage behavior of ductile composites is derived. A series of numerical simulations are carried out to illustrate stress–strain response of the proposed micromechanical framework and to examine the influence of a Weibull parameter on the elastoplastic behavior of the composites. Furthermore, comparisons between the present predictions and experimental data available in the literature are made to further assess the predictive capability of the proposed model.     

On the measurement of elastoplastic stresses

In principle, one should be able to measure elastoplastic stresses in the same manner as one does elastic stresses; i.e., measure the strains and compute the stresses from the constitutive law. In practice, this is rarely done because of the more complicated material response and the anisotropy of the plastic behavior. Further, elastoplastic stresses should be computed incrementally in the general case. This paper presents procedures for computing stresses from elastoplastic strains measured incrementally in a test under microcomputer control. The approach is evaluated for four different materials—two obeying the assumptions of classical plasticity and two showing anisotropic behavior—by computing the stresses in a smooth specimen from measured principal strains. A useful application is presented by computing the stresses at a notch root from biaxial strains measured with laser-based interferometry. The general conclusion is that even in situations where the material is clearly anisotropic, this approach can give a reasonableestimate of the largest local principal stress. Paper was presented at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WI on June 9–13.     

弹塑性力学问题的虚单元法

具有有限差分法特征的虚单元法,可视为是有限元法向任意多边形单元的扩展。在材料细观力学性能表征、非均质材料力学分析等非线性问题方面,传统的弹塑性有限元法具有网格数目多、效率低下等不足之处,而虚单元法使网格划分更加灵活,为材料的弹塑性力学分析等非线性问题提供了新的思路。基于增量法弹塑性力学原理和双线性投影算子,建立了弹塑性力学问题的虚单元法求解技术,提出了弹塑性力学问题虚单元法的应力更新方案,研究了弹性力学问题虚单元法的精度和收敛性,讨论了虚单元法求解弹塑性力学问题的网格依赖性。同时,开展了任意多边形和凹多边形单元的数值试验研究,结果表明,虚单元法无须分割多边形,仅需节点自由度便可求得单元刚度矩阵和应力等效荷载,程序实现简单,计算精度高,改善了传统有限元的网格依赖性和塑性区的网格奇异性。     

Hoek–Brown criterion applied to circular tunnel using elastoplasticity and in situ axial stress

Based on the nonlinear Hoek–Brown failure criterion, elastoplastic analytical solutions are developed for the elastoplastic stresses, strains and plastic zones around a circular tunnel subjected to different value of the axial in situ stress. Effects of the transverse in situ stress, the axial in situ stress and the strength parameters of rock masses on the elastoplastic stresses, strains and plastic zones in the surrounding rock masses are investigated. It is found from the numerical results that the stresses, strains, and plastic zones in the surrounding rock depend not only on the transverse in situ stress but also on the axial in situ stress as well as the mechanical parameters of rock masses.