Active macro‐zone approach for incremental elastoplastic‐contact analysis

The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact‐detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem. A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem‐elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic‐contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro‐zones, and uses the well‐known concept of self‐equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self‐stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment. Copyright © 2013 John Wiley & Sons, Ltd.     

A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis

A mathematical programming formulation of strain‐driven path‐following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in an FEM context is presented. From the optimization point of view, standard arc‐length strain‐driven elastoplastic analyses, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other non‐linear programming methods, simpler convergence proofs and duality to be exploited. Owing to the unified approach in terms of total stresses, the strain‐driven algorithms become more effective and less non‐linear with respect to a self‐equilibrated stress formulation and easier to implement in the existing codes performing elastoplastic analysis. The elastic domain is represented avoiding any linearization of the yield function so improving both the accuracy and the performance. Better results are obtained using two different finite elements, one with a good behavior in the elastic range and the other suitable for performing elastoplastic analysis. The proposed formulation is compared with a specialized implementation of the primal–dual interior point method suitable to solve the problems at hand. Copyright © 2011 John Wiley & Sons, Ltd.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part II: 3‐D non‐linear behaviour

In a companion paper, the effects of approximations in the flexural‐torsional stability analysis of beams was studied, and it was shown that a second‐order rotation matrix was sufficiently accurate for a flexural‐torsional stability analysis. However, the second‐order rotation matrix is not necessarily accurate in formulating finite element model for a 3‐D non‐linear analysis of thin‐walled beams of open cross‐section. The approximations in the second‐order rotation matrix may introduce ‘self‐straining’ due to superimposed rigid‐body motions, which may lead to physically incorrect predictions of the 3‐D non‐linear behaviour of beams. In a 3‐D non‐linear elastic–plastic analysis, numerical integration over the cross‐section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross‐section that is not consistent with the strain distributions may lead to incorrect predictions of the 3‐D non‐linear elastic–plastic behaviour of beams. This paper investigates the effects of approximations on the 3‐D non‐linear analysis of beams. It is found that a finite element model for 3‐D non‐linear analysis based on the second‐order rotation matrix leads to over‐stiff predictions of the flexural‐torsional buckling and postbuckling response and to an overestimate of the maximum load‐carrying capacities of beams in some cases. To perform a correct 3‐D non‐linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross‐section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3‐D non‐linear elastic–plastic behaviour of beams. Copyright © 2001 John Wiley & Sons, Ltd.     

剪力墙非线性宏观单元的研究与单元开发

罕遇地震作用下剪力墙构件易进入塑性阶段而发生弹塑性损伤,准确地模拟剪力墙构件进入非线性状态后的力学行为对评价剪力墙结构的抗震安全性及基于性能的抗震设计具有重要的意义.通过面向对象语言编制了基于宏观单元的结构弹塑性分析软件平台MESAP及多竖向弹簧剪力墙单元(MVLEM),采用材料本构及Fischinger学者提出的轴向...     

三种非线性梁柱单元的研究及单元开发

罕遇地震作用下混凝土梁柱构件易进入塑性阶段而发生弹塑性损伤,正确地模拟结构进入非线性状态后的力学行为对评价结构的抗震安全性具有重要的意义.通过面向对象语言编制了基于宏观单元的结构弹塑性分析软件平台MESAP,增加了三种非线性梁柱单元:基于刚度法纤维单元、基于柔度法纤维单元及基于柔度法的塑性铰单元.通过算例分析三种非线性...     

Experimental and numerical rolling contact fatigue study on the 32CrMoV13 steel

The aim of this work is to study pure rolling contact fatigue in 32CrMoV13 quenching and tempering steel. The study involves both experimental and numerical work. The influence of the roughness and the residual stresses on the mechanisms and zones of cracking were studied. The results show a rapid reduction in roughness during the first minute of test but even so there will be specimen deterioration. The residual stress profile after rolling contact tests have high compression values in the surface and at a depth of approximately 240 μm, which is related with the Hertzian maximal shear stress. The numerical simulation of the Hertzian loading was used both to determine the elastic shakedown of the material and to apply a high‐cycle multiaxial fatigue criterion. The three‐dimensional finite element analysis used in the numerical calculation includes elastic‐linear kinematic hardening plastic material and allows the introduction of an initial residual stress state. Taking into account the elastoplastic load induced by the Hertz pressure, low‐cycle fatigue tests were used to characterize the mechanical properties of the material. In order to validate the numerical simulation, the results of the calculation after elastic shakedown were compared with the values measured by X‐ray diffraction after rolling contact tests. The results showed a reasonable agreement between calculated and measured stresses. The Dang Van high‐cycle multiaxial fatigue criterion showed a good relationship with the experimental findings.     

Elasto‐plastic deformation of upsetting by boundary element method

Abstract

This paper presents the utilization of boundary element method (BEM) to analyze the elasto‐plastic deformation of upsetting problems. Method of successive elastic solutions is used in the nonlinear analysis; both the linear strain hardening and the power law relation are used as constitutive equations of the material. For the later model the slope of strain hardening at each step is modified to a more correct prediction to make the deformation step larger and to obtain better convergence. The result may verify the stress‐strain curve as it does, and verify the similar pattern of the plastic zone propagation as Roll's result by finite element method. It is shown that various frictional conditions and width‐height ratios of the workpiece also influence the propagation behavior of plastic zones.     

Three‐dimensional mixed‐mode (I and II) crack‐front fields in ductile thin plates — effects of T‐stress

It has been well‐established that the non‐singular T‐stress provides a first‐order estimate of geometry and loading mode (e.g. tension versus bending) effects on elastic–plastic crack‐front field under mode I loading conditions. The objective of this paper is to exam the T‐stress effect on three‐dimensional (3D) crack‐front fields under mixed‐mode (modes I and II) loading. To this end, detailed 3D small strain, elastic–plastic simulations are carried out using a 3D boundary layer (small‐scale yielding) formulation. Characteristics of near crack‐front fields are investigated for a wide range of T‐stresses (T/σ₀ = ?0.8, ?0.4, 0.0, 0.4, 0.8). The plastic zones and thickness and angular and radial variations of the stresses are studied, corresponding to two values of the remote elastic mixity parameters M_e = 0.3 and 0.7, under both low and high levels of applied loads. It is found that different T‐stresses have a significant effect on the plastic zones size and shapes, regardless of the mode mixity and load level. The thickness, angular and radial distributions of stresses are also affected markedly by T‐stress. It is important to include these effects when investigating the mixed‐mode ductile fracture failure process in thin‐walled structural components.     

Topology optimization of material‐nonlinear continuum structures by the element connectivity parameterization

The application of the element density‐based topology optimization method to nonlinear continuum structures is limited to relatively simple problems such as bilinear elastoplastic material problems. Furthermore, it is very difficult to use analytic sensitivity when a commercial nonlinear finite element code is used. As an alternative to the element density formulation, the element connectivity parameterization (ECP) formulation is developed for the topology optimization of isotropic‐hardening elastoplastic or hyperelastic continua by using commercial software. ECP varies the stiffness of zero‐length linear elastic links that connect design domain‐discretizing finite elements. Unloading was not considered. But the advantages of ECP in material‐nonlinear problems were demonstrated: considerably simple analytic sensitivity calculation using a commercial code and simple link stiffness penalization regardless of nonlinear material behaviour. Copyright © 2006 John Wiley & Sons, Ltd.     

Modeling of Plastic Zones in the Process of Shear Deformation of a Half Layer Containing a Crack

We study the evolution of plastic zones in a perfectly elastoplastic half layer containing a crack in its median plane under the conditions of antiplane deformation caused by displacements of the rigid faces of the half layer, deduce equations for the boundary of the continual plastic zone and its length in the plane of the crack as functions of the load, and perform the approximate analysis of the stress intensity factors of the elastic solution for the zones (the so-called linear model of the plastic zone).     

Formulation and numerical treatment of incompressibility constraints in large strain elastic–plastic analysis

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate‐independent finite strain analysis of solids undergoing large elastic‐isochoric plastic deformations. The formulation relies on the introduction of a mixed‐variant metric deformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure which can be additively decomposed into the exact isochoric (deviatoric) and volumetric (spheric) strain measures. This fact may be seen as the basic idea in the formulation of appropriate mixed finite elements which guarantee the accurate computation of isochoric strains. The mixed‐variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response whereas the plastic material behavior is assumed to be governed by a generalized J₂ yield criterion and rate‐independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on higher‐order Padé approximations. To be able to take into account the plastic incompressibility constraint a modified mixed variational principle is considered which leads to a quasi‐displacement finite element procedure. Finally, the numerical solution of finite strain elastic‐plastic problems is presented to demonstrate the efficiency and the accuracy of the algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

The effect of T‐stress on crack‐tip plastic zones under mixed‐mode loading conditions

In this paper, the influence of T‐stress on crack‐tip plastic zones under mixed‐mode I and II loading conditions is examined. The crack‐tip stress field is defined in terms of the mixed‐mode stress intensity factors and the T‐stress using William's series expansion. The crack‐tip stress field is incorporated into the Von Mises yield criteria to develop an expression that determines the crack‐tip plastic zone. Using the resultant expression, the plastic zone is plotted for various combinations of mode II to mode I stress intensity factor ratios and levels of T‐stress. The properties of the plastic zone affected by T‐stress and mixed‐mode phase angle are discussed. The observations obtained on plastic zones variations are important for further fatigue and fracture analyses for defects in engineering structures under mixed‐mode loading conditions.     

Boundary element analysis for an interface crack between dissimilar elastoplastic materials

The boundary element method (BEM) is presented for elastoplastic analysis of cracks between two dissimilar materials. The boundary integral equations and integral representation of stress rates are written in such a form that all integrals can be evaluated by the regular Gaussian quadrature rule. An advanced multidomain BEM formulation is suggested for the solution of analysed problems where the substantial reduction of stiffness matrix is observed. The elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the stresses. The boundary and the yielding zone are discretized by elements with quadratic approximations. In numerical examples the path independence of the J- and L-integrals for a straight interface crack and a circular arc-shaped interface crack are investigated, respectively. The influence of the different values of Young's modulus on the J-integral, shape and size of plastic zones is treated too.     

An incremental procedure for deformation analysis of elastic-plastic frames

In the present paper an incremental procedure is formulated for first-order elastoplastic analysis of plane frame structures discretized into a finite number of beam elements and described by piecewise linear elastic-perfectly plastic constitutive laws, under the assumption of both reversible and irreversible behaviour of material and using piecewise linear yield conditions at any desired degree of discretization in the space of the active stress resultants (axial force, shear force and bending moment). The proposed method, by using the independent elastic-plastic kinematical compatibility equations, restrains the problem sizes within not more than twice the number of the redundant unknowns in the complete elastic frame, regardless of the degree of discretization of the piecewise linear yield conditions, still maintaining the advantage exhibited by the Mathematical Programming methods of requiring only one factorization of the matrix governing the problem when no local unloading occurs. Furthermore, the outlined algorithm allows the additional computational effort to be restrained in the case of local unloading, inasmuch as it requires a new factorization to be performed of a part of the matrix governing the problem, whose size is small with respect to the total size of the matrix.     

A study of crack-tip plastic zone by elastoplastic finite element analysis

An elastoplastic finite element analysis has been carried out on a thin centre cracked panel using plane eight noded quadratic quadrilateral isoparametric elements with nonsingular displacement formulation. The linearised total strain method has been employed for the solution and the material data including those pertaining to postyield behaviour were used for the evaluation of the crack-tip stress field. The shape and size of the plastic zones at the crack tip corresponding to four stress levels have been obtained, which compare favourably with those obtained by other researchers.     

A new boundary element technique without domain integrals for elastoplastic solids

A simple idea is proposed to solve boundary value problems for elastoplastic solids via boundary elements, namely, to use the Green's functions corresponding to both the loading and unloading branches of the tangent constitutive operator to solve for plastic and elastic regions, respectively. In this way, domain integrals are completely avoided in the boundary integral equations. Though a discretization of the region where plastic flow occurs still remains necessary to account for the inhomogeneity of plastic deformation, the elastoplastic analysis reduces, in essence, to a straightforward adaptation of techniques valid for anisotropic linear elastic constitutive equations (the loading branch of the elastoplastic constitutive operator may be viewed formally as a type of anisotropic elastic law). Numerical examples, using J₂‐flow theory with linear hardening, demonstrate that the proposed method retains all the advantages related to boundary element formulations, is stable and performs well. The method presented is for simplicity developed for the associative flow rule; however, a full derivation of Green's function and boundary integral equations is also given for the general case of non‐associative flow rule. It is shown that in the non‐associative case, a domain integral unavoidably arises in the formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

高温载荷下金属蜂窝夹芯结构的相变和塑性分析

为了分析高温下的相变和塑性变形对金属蜂窝夹芯结构固有性能的影响,以含相变的多层板瞬态热传递和热弹塑性理论为依据,采用有限元数值求解方法,对安装有发热元件的金属蜂窝夹芯结构进行了热和结构的非线性耦合分析。通过温度场分析,可以快速确定外胶层发生的固液和气液相变,及其开始出现的时间、位置、严重程度、发展趋势等。结合温度场和应力场的计算结果,可以方便地判定外蒙皮是否产生了塑性变形,并能预测其随加载时间的动态演化规律。