Macroscopic theory and nonlinear finite element analysis of micromechanics of single crystals at finite strains

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the macroscopic rate-independent and rate-dependent analysis of micromechanics of metal single crystals undergoing finite elastic-plastic deformations which is based on the assumption that inelastic deformation is solely due to crystallographic slip. The formulation relies on a multiplicative decomposition of the material deformation gradient into incompressible elastic and plastic as well as a scalar valued volumetric part. Furthermore, the crystal deformation is described as arising from two distinct physical mechanisms, elastic deformation due to distortion of the lattice and crystallographic slip due to shearing along certain preferred lattice planes in certain preferred lattice directions. Macro- and microscopic stress measures are related to Green’s macroscopic strains via a hyperelastic constitutive law based on a free energy potential function, whereas plastic potentials expressed in terms of the generalized Schmid stress lead to a normality rule for the macroscopic plastic strain rate. Estimates of the microscopic stress and strain histories are obtained via a highly stable and very accurate semi-implicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step, and, furthermore, the development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical simulation of finite strain elastic-plastic tension tests is presented to demonstrate the efficiency of the algorithm.     

Constitutive model and finite element formulation for large strain elasto-plastic analysis of shells

This contribution presents a refined constitutive and finite element formulation for arbitrary shell structures undergoing large elasto-plastic deformations. An elasto-plastic material model is developed by using the multiplicative decomposition of the deformation gradient and by considering isotropic as well as kinematic hardening phenomena in general form. A plastic anisotropy induced by kinematic hardening is taken into account by modifying the flow direction. The elastic part of deformations is considered by the neo-Hookean type of a material model able to deal with large strains. For an accurate prediction of complex through-thickness stress distributions a multi-layer shell kinematics is used built on the basis of a six-parametric shell theory capable to deal with large strains as well as finite rotations. To avoid membrane locking in bending dominated cases as well as volume locking caused by material incompressibility in the full plastic range the displacement based finite element formulation is improved by means of the enhanced assumed strain concept. The capability of the algorithms proposed is demonstrated by various numerical examples involving large elasto-plastic strains, finite rotations and complex through-thickness stress distributions.     

Incremental constitutive equation for large strain elasto plasticity

Starting from the standard theory with intermediate configuration for finite deformations of an isotropic elasto-plastic material with isotropic hardening, rate type constitutive equations are obtained. The small elastic strain approximation is then discussed and it is shown that, in this approximation, these equations reduce to Hill's formalism of large strain elasto-plasticity obtained from the classical Prandtl-Reuss relations of infinitesimal plasticity by substituting for the infinitesimal strain rate, stress and stress rate respectively the rate of deformation tensor, the Cauchy stress tensor and the Jaumann stress rate tensor. The limiting case of perfect plasticity is also investigated.     

Nonlinear static/dynamic analysis for elasto-plastic laminated plates with interfacial damage evolution

A new analysis model, which includes the effects of interfacial damage, geometrical nonlinearity and material nonlinearity, is presented for elasto-plastic laminated plates. Based on the model, the nonlinear equilibrium differential equations for elasto-plastic laminated plates with interfacial damage are established. The finite difference method and iteration method are adopted to solve these equations. The nonlinear static and dynamic behaviors for the elasto-plastic laminated plates under the action of transverse loads are analyzed. Effects of interfacial damage on the stress and displacement distribution and nonlinear dynamic response are discussed in the numerical examples together with the comparison of nonlinear mechanical behaviors between the elastic and elasto-plastic laminated plates. Numerical results show that both the interfacial damage and plastic deformation put obvious influence on the mechanical properties of structures.     

Design sensitivity analysis and optimization of non‐linear transient dynamics. Part I—sizing design

A continuum‐based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non‐linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non‐linear dynamic problems, such as crashworthiness design. The first‐order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path‐dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate‐based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.     

Large strain elastic-plastic theory and nonlinear finite element analysis based on metric transformation tensors

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-plastic deformations. The formulation relies on the introduction of a mixed-variant metric transformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure whose rate is shown to be additively decomposed into elastic and plastic strain rate tensors. The mixed-variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response in the elastic-plastic solid. Additionally, the plastic material behavior is assumed to be governed by a generalized J ₂ 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 1st and higher order Padé approximations. Estimates of the stress and strain histories are obtained via a highly stable and accurate explicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step. The development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical solution of finite strain elastic-plastic problems is presented to demonstrate the efficiency of the algorithm. Received: 17 May 1998     

Micromechanical modeling of local field distribution for a planar composite under plastic deformation

Summary Due to statistical distribution of local material property, local stress and strain fields in a composite are random in nature. Classical micromechanical methods can only predict the average value of these local fields in different phases. An analytical method, which combines the maximum entropy theory and secant moduli method, is proposed in this paper. The distribution of the local field for a planar composite with plastic deformation is examined by the proposed method. The results show that with increase of plastic deformation the stress field in the matrix becomes more and more inhomogeneous. The predicted results on the stress distribution are in reasonable agreement with finite element simulation. Some salient features near the elastic and plastic deformation transition revealed by finite element simulation are also discussed.     

Strain Measures for Rigid Crushable Foam in Uniaxial Compression

Abstract:  Progressive collapse deformation mechanisms in Rohacell-51WF foam during uniaxial compression has been studied. Measures of a macroscopic engineering strain are identified. The elastic and plastic parts of a macroscopic engineering strain can be predicted by using the compression failure strain, lock-up strain, and time dependent elastic and plastic parts of lock-up strain, which are material parameters. Identification of strain measures in a uniaxial compression test is essential to get material parameters for an elastoplastic model. The viscoelastic recovery property of Rohacell-51WF foam is also described.     

An additive theory for finite elastic–plastic deformations of the micropolar continuous media

In this paper, the method of additive plasticity at finite deformations is generalized to the micropolar continuous media. It is shown that the non-symmetric rate of deformation tensor and gradient of gyration vector could be decomposed into elastic and plastic parts. For the finite elastic deformation, the micropolar hypo-elastic constitutive equations for isotropic micropolar materials are considered. Concerning the additive decomposition and the micropolar hypo-elasticity as the basic tools, an elastic–plastic formulation consisting of an arbitrary number of internal variables and arbitrary form of plastic flow rule is derived. The localization conditions for the micropolar material obeying the developed elastic–plastic constitutive equations are investigated. It is shown that in the proposed formulation, the rate of skew-symmetric part of the stress tensor does not exhibit any jump across the singular surface. As an example, a generalization of the Drucker–Prager yield criterion to the micropolar continuum through a generalized form of the J ₂-flow theory incorporating isotropic and kinematic hardenings is introduced.     

Plastic deformation in multilayered thin films during indentation unloading: a modeling analysis incorporating viscoplastic response

The indentation behavior of metal/ceramic multilayered thin films is studied numerically using the finite element method. The axisymmetric model consists of alternating aluminum (Al) and silicon carbide (SiC) layers above a silicon (Si) substrate, with the rate-dependent viscoplastic response of Al accounted for. Different unloading rates, with and without a hold period at the peak indentation load, are considered. Attention is devoted to plastic deformation in the Al layers during the unloading phase of indentation. It is found that the hold period stabilizes the deformation so the unloading response becomes insensitive to the unloading rate. However, significant parts of the Al layers under indentation still experience plastic deformation during unloading, due to the mechanical constraint imposed by the hard SiC layers. Consequently unloading is no longer an elastic event in this heterogeneous material system. The unloading induced plasticity is further analyzed by tracking the stress and strain histories inside the material throughout the course of indentation.     

Numerical simulation of the plastic flow properties of iron single crystals

Summary The present paper deals with the numerical simulation of the plastic flow properties of iron single crystals as well as their influence on the macroscopic elastic-plastic deformation and localization behavior affected by superimposed hydrostatic pressure. Based on experimental observations the onset of plastic yielding on the microscale is described by an extended microscopic yield condition taking into account various microscopic stress components acting on the respective slip systems. In addition, to be able to compute inelastic deformations from a plastic potential, the latter is expressed in terms of workconjugate microscopic stress and strain measures which leads to a non-associated flow rule for the macroscopic plastic strain rate. On the numerical side, generalized functions for constitutive parameters will be used to be able to simulate the single crystal's microscopic deformation behavior observed in experiments. Estimates of the current microscopic stresses and strains are obtained via an efficient and remarkably stable plastic predictor-elastic corrector technique which is incorporated into a nonlinear finite element program. Numerical simulations of uniaxial tests show quantitatively the influence of hydrostatic pressure on current material data. Further numerical studies on the additional constitutive non-Schmid terms elucidate their effect on iron single crystal's macroscopic deformation and localization behavior.     

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.     

Plastic behavior of composites and porous media under isotropic stress

The macroscopic relation between isotropic stress and dilatation is analyzed for a composite consisting of elasto-plastic matrix and elastic particles or empty cavities (porous material), on the basis of the composite spheres assemblage model. Results show that plastic macro-dilatation is not significant for elastic particles but is very significant for porous materials.     

Formulation of implicit finite element methods for multiplicative finite deformation plasticity

Some constitutive and computational aspects of finite deformation plasticity are discussed. Attention is restricted to multiplicative theories of plasticity, in which the deformation gradients are assumed to be decomposable into elastic and plastic terms. It is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non-symmetric even for associated plastic flow. Both explicit (i.e. no equilibrium iteration) and implicit finite element formulations are considered. An assumed strain formulation is used to accommodate the near-incompressibility associated with fully developed isochoric plastic flow. As an example of explicit integration, the rate tangent modulus method is reviewed in some detail. An implicit scheme is derived for which the consistent tangents, resulting in quadratic convergence of the equilibrium iterations, can be written out in closed form for arbitrary material models. All the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly. Examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method.     

Simulation of cyclic stress/strain evolutions for multiaxial fatigue life prediction

This study deals with simulation for cyclic stress/strain evolutions and redistributions, and evaluation of fatigue parameters suitable for estimating fatigue lives under multiaxial loadings. The local cyclic elastic–plastic stress–strain responses were analyzed using the incremental plasticity procedures of ABAQUS finite element code for both smooth and notched specimens made of three materials: a medium carbon steel in the normalized condition, an alloy steel quenched and tempered and a stainless steel, respectively. Emphasis is on the studying of ‘intelligent’ material behaviors to resist fracture, such as stress redistribution and relaxation through plastic deformations, etc. For experimental verifications, a series of tests of biaxial low cycle fatigue composed of tension/compression with static and cyclic torsion were carried out on a biaxial servo-hydraulic testing machine (Instron 8800). Different multiaxial loading paths were used to verify their effects on the additional cyclic hardening. The comparisons between numerical simulations and experimental observations show that the FEM simulations allow better understanding on the evolutions of the local cyclic stress–strain and it is shown that strong interactions exist between the most stressed material element and its neighboring material elements in the plastic deformations and stress redistributions. Based on the local cyclic elastic–plastic stress–strain responses, the energy-based multiaxial fatigue damage parameters are applied to correlating the experimentally obtained lives. Improved correlations between the predicted and the experimental results are shown. It is concluded that the improvement of fatigue life prediction depends not only on the fatigue damage models, but also on the accurate evaluations of the cyclic elasto-plastic stress/strain responses.     

On the nature of plastic deformation generated by hydrostatic pressure in silicon single crystals

Macroscopic plastic deformation of silicon single crystals, caused by annealing at hydrostatic pressure and high temperature, was studied by X-ray topography and transmission electron microscopy. The analysis is given of elastic and thermal properties of material around surface cracks and scratches from which deformation process is propagated. The idea of elastic misfit between damaged self-strained material at cracks and scratches and defect-free silicon matrix, is introduced. On the basis of theoretical and experimental data it is concluded that the plastic deformation of silicon at high pressure consists of two processes. The first is a loss of coherency of cracks and scratches by the emission of dislocations at misfitting second phase precipitates present in silicon. The second is the macroscopic yielding from incoherent cracks and scratches at lower elastic strain energies.The presented mechanism explains also the deformation behaviour of silicon crystals subjected to tensile stress at high temperatures; the generation and propagation of dislocations at oxide precipitates before the macroscopic yielding [3].     

Linear and elastic–plastic fracture mechanics revisited by use of Fourier transforms – theory and application

This paper investigates whether and how discrete Fourier transforms (DFT) can be used to compute the local stress/strain distribution around holes in externally loaded two-dimensional representative volume elements (RVEs). To this end, the properties of DFT are first reviewed and then applied to the solution of linear elastic and time-dependent elastic plastic material response. The equivalent inclusion method is used to derive a functional equation which allows for the numerical computation of stresses and strains within an RVE with heterogeneities of arbitrary shape and stiffness. This functional equation is then specialized to the case of circular and elliptical holes of different minor axes which eventually degenerate into Griffith cracks. The numerically predicted stresses and strains are compared to the corresponding analytical solutions for a single circular as well as an elliptical hole in an infinitely large plate under tension as well as to finite element calculations (for time-independent elastic/plastic material response).     

Analysis of forming rate sensitive materials

Summary To study how the rate of deformation effects forming rate sensitive materials, the Bodner-Partom elastic-viscoplastic constitutive law is incorporated into a finite element program. This law postulates both elastic and plastic components of deformation at any stress level. The stress is a Hookean function of the elastic strain, while the plastic deformation rate is a function of the deviatoric stress and an internal state variable defining the load history. The finite element derivation adopts the small strain assumption with updated coordinates. The equilibrium rate equation is formulated using total velocities with the nonlinearities incorporated into an equivalent plastic load vector depending upon the current stress. The resulting equation explicitly includes time and is a true rate equation. The program calculates the current stress field using the incremental equilibrium equation. An iterative technique is used to ensure that the assumed current load rate used to calculate the current stress field is correct. Convergence of the iteration procedure needs only be monitored at the velocity specified nodes. To demonstrate the applicability of this method, two plane strain problems, a tensile bar and strip rolling, with rate sensitive materials are investigated.With 12 Figures     

钢管混凝土结构材料非线性的一种有限元分析方法

为了更简单地考虑梁单元的材料非线性受力性能,把断面广义力和广义应变的概念运用于单元分析中,将单元的弹塑性刚度矩阵分离为弹性刚度矩阵和塑性刚度矩阵。这样,梁单元的变形可以由弹性变形和塑性变形简单地迭加,结构内力可通过弹性应变能的斜率(弹性刚度矩阵)与位移的乘积求得,从而在增量-迭代计算时可较准确且较快地计算出结构变形后的不平衡力。应用这一计算方法,推导了基于纤维模型的三维梁单元的钢管混凝土结构的有限元基本公式,并将其植入能考虑几何非线性的三维梁单元非线性计算程序NL_Beam3D中以计算结构的双重非线性问题。算例分析表明该方法和程序能较准确地反映钢管混凝土结构的双重非线性特性。