Formulation and implementation of stress‐driven and/or strain‐driven computational homogenization for finite strain

In this paper, we present a homogenization approach that can be used in the geometrically nonlinear regime for stress‐driven and strain‐driven homogenization and even a combination of both. Special attention is paid to the straightforward implementation in combination with the finite‐element method. The formulation follows directly from the principle of virtual work, the periodic boundary conditions, and the Hill–Mandel principle of macro‐homogeneity. The periodic boundary conditions are implemented using the Lagrange multiplier method to link macroscopic strain to the boundary displacements of the computational model of a representative volume element. We include the macroscopic strain as a set of additional degrees of freedom in the formulation. Via the Lagrange multipliers, the macroscopic stress naturally arises as the associated ‘forces’ that are conjugate to the macroscopic strain ‘displacements’. In contrast to most homogenization schemes, the second Piola–Kirchhoff stress and Green–Lagrange strain have been chosen for the macroscopic stress and strain measures in this formulation. The usage of other stress and strain measures such as the first Piola–Kirchhoff stress and the deformation gradient is discussed in the Appendix. Copyright © 2015 John Wiley & Sons, Ltd.     

Adaptive reproducing kernel particle method using gradient indicator for elasto-plastic deformation

An adaptive meshless method based on the multi-scale Reproducing Kernel Particle Method (RKPM) for analysis of nonlinear elasto-plastic deformation is proposed in this research. In the proposed method, the equivalent strain, stress, and the second invariant of the Cauchy–Green deformation tensor are decomposed into two scale components, viz., high- and low-scale components by deriving them from the multi-scale decomposed displacement. Through combining the high-scale components of strain and the stress update algorithm, the equivalent stress is decomposed into two scale components. An adaptive algorithm is proposed to locate the high gradient region and enrich the nodes in the region to improve the computational accuracy of RKPM. Using the algorithm, the high-scale components of strain and stress and the second invariant of the Cauchy–Green deformation tensor are normalized and used as criteria to implement the adaptive analysis. To verify the validity of the proposed adaptive meshless method in nonlinear elasto-plastic deformation, four case studies are calculated by the multi-scale RKPM. The patch test results show that the used multi-scale RKPM is reliable in analysis of the regular and irregular nodal distribution. The results of other three cases show that the proposed adaptive algorithm can not only locate the high gradient region well, but also improve the computational accuracy in analysis of the nonlinear elasto-plastic deformation.     

Multiple decomposition in finite deformation theory

Summary A strain decomposition method is proposed in finite strain deformation theory. The method is based on the multiplicative decomposition of the deformation gradient with the assumption of intermediate configurations. Kinematically correct additive decomposition of the strain is developed. The strain and stress measures are calculated by way of the dual variables. Geometric linear consitutive models are generalized to finite strain theory. An application and an example are also included for thermoelastoplastic analysis.departed August 4, 2000     

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.     

Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme

A gradient‐enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non‐linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non‐uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher‐order stress tensor are derived based on an extension of the Hill–Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher‐order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects. Copyright © 2002 John Wiley & Sons, Ltd.     

Analysis of laminated crack defect in the upsetting process of heavy disk-shaped forgings

Using nonlinear finite element method, a thermo-mechanical coupled simulation model for the formation mechanism of the laminated crack defect has been established in the upsetting of heavy disk-shaped forgings. Through numerical simulation, the distributions of stress, equivalent strain and strain rate were analysed. Meanwhile the distribution diagram of stress state evolution was obtained, and the uncoordinated deformation, under tri-lateral compression, is determined as the main reason leading to laminated crack defect. To reveal the characteristics of the uncoordinated deformation, the variations of each variable and its gradient in numerical simulation were presented, and a combined prediction model of laminated crack defect were proposed based on degree of deformation and gradient of deformation speed. Subsequently, the morphology and distribution of laminated crack were obtained in the centre of forging using the prediction model. Comparison of calculation results and experimental data indicates that both of them match well. In addition, the effect of friction coefficient on the deformation is also presented. The results show that the decreasing of friction coefficient is an effective measure to restrain the laminated crack defect.     

Triaxial compression deformation for artificial frozen clay with thermal gradient

In order to study the deformation characteristics of artificial frozen soil with thermal gradient, such as the stress-strain relationship, a series of triaxial compression tests for frozen clay had been conducted by K₀DCGF (K₀ consolidation, freezing with non-uniform temperature under loading) method and GFC (freezing with non-uniform temperature, isotropic consolidation) method at various consolidation pressures and thermal gradients. Stress-strain curves in K₀DCGF test present strain softening during shearing process and the elastic strain is approximately 0.001;but which present the strain hardening characteristics in GFC tests and the elastic strain is approximately 0.01. The elastic modulus and peak stress for frozen clay decrease as the thermal gradient increased at different consolidation pressure both in K₀DCGF test and GFC test. The peak stress and elastic modulus in K₀DCGF test are significant independent on the pressure melting and crushing phenomena occurring in GFC test. To describe the shear deformation characteristics for frozen clay with thermal gradient, the exponent and power equations considering the correction equation on thermal gradient and model parameters from frozen clay with uniform temperature are developed .The results indicated that the proposed equations can reproduce the shear deformation well both in K₀DCGF test and GFC test.     

Stress and strain partition in elastic and plastic deformation of two phase alloys

A stress and strain partition theory for two phase alloys was developed on the basis of the modified rules of mixtures. The extreme value condition of macroscopic strain energy density was found through Lagrangian multiplier method. Expressions for macroscopic elastic constants of two phase alloys were derived from the extreme value condition by assuming the strain linearity between constituent phases. Governing equation for stress and strain partition in plastic deformation was also obtained from the extreme value condition. The calculated elastic constants of WC-Co alloys fell invariably within the Hashin and Shtrikman's bounds. According to the governing equation the stress ratio between constituent phases was plotted as a function of strain increment ratio. By applying the governing equation to spheroidized carbon steel and duplex stainless steel, it was shown that the stress ratios, strain ratios, macroscopic stress-strain curves, and internal stresses could be evaluated from thein situ stress-strain curves of constituent phases.     

Effective behavior of porous elastomers containing aligned spheroidal voids

The theoretical need to recognize the link between the basic microstructure of nonlinear porous materials and their macroscopic mechanical behavior is continuously rising owing to the existing engineering applications. In this regard, a semi-analytical homogenization model is proposed to establish an overall, continuum-level constitutive law for nonlinear elastic materials containing prolate/oblate spheroidal voids undergoing finite axisymmetric deformations. The microgeometry of the porous materials is taken to be voided spheroid assemblage consisting of confocally voided spheroids of all sizes having the same orientation. Following a kinematically admissible deformation field for a confocally voided spheroid, which is the basic constituent of the microstructure, we make use of an energy-averaging procedure to obtain a constitutive relation between the macroscopic nominal stress and deformation gradient. In this work, both prolate and oblate voids are considered. As a numerical example, we study macroscopic nominal stress components for a hyperelastic porous material consisting of a neo-Hookean matrix and prolate/oblate voids subjected to 3-D and plane strain dilatational loadings. In this numerical study, the relation between the relevant microstructural variables (i.e., initial porosity and void aspect ratio) for a rather large range of applied stretch is put into evidence for two types of loading. Finally, a finite element (FE) simulation is presented, and the homogenization model is assessed through comparison of its predictions with the corresponding FE results. The illustrated agreement between the results demonstrates a good accuracy of the model up to rather large deformations.     

Geometrically nonlinear stress–strain behavior of hyperelastic solid foams

The present study is concerned with an effective stress analysis of cellular solids in the finite strain regime. The homogenization of the microstructure is performed by means of a strain energy based RVE-procedure which assumes macroscopic equivalence of a representative volume element for the given microstructure and a similar volume element consisting of the effective medium, if the average strain energy density in both volume elements is equal provided that the deformation gradient with respect to both elements is equal in a volume average sense. Disordered microstructures are considered by means of a randomized periodic model in conjunction with a stochastic approach. The model is applied to an analysis of the effective stress–strain behavior of two-dimensional model foams with periodic and disordered microstructure. Special interest is directed to effects of the geometric nonlinearity.     

残余应力对复合材料弹2塑性变形的影响

从细观力学的角度给出了分析残余应力对一般复合材料塑性性能影响的一种解析方法, 该方法基于应力二阶矩的割线模量法及Ponte Castaneda 和W illis 给出的弹性细观模型。有残余应力时, 所提的细观解析模型能够同时考虑纤维形状, 体积百分比, 纤维取向及纤维的分布对复合材料变形的影响。计算结果表明, 残余应力的存在会引起复合材料拉压变形的不对称, 材料宏观的拉压硬化曲线又与复合材料的细观结构参数密切相关。对单向复合材料, 本文作者对其等效割线热膨胀系数, 拉压应力-应变曲线的有限元分析结果与给出的细观解析模型定量吻合。      

Stress gradient due to incremental forming of bonded metallic laminates

The residual stress and associated gradient can affect the performance of a material/component during service. Sheet-metal in incremental sheet forming (ISF) due to missing back support may experience high stress gradient across the thickness. The current work is aimed at experimentally analyzing the through-thickness stress gradient in the Cu/steel bonded laminates after ISF deformation. It is found that ISF induces compressive stress gradient, which can be a way greater (about 18 times) than that the rolling process induces in the parent laminates while bonding, specifically when the deformation angle is high. Further, the tool imposes more stress gradient (1–50% depending on the forming conditions) in its motion direction than that in the transverse (or stretching) direction. Moreover, un-strained Cu/steel laminated sheet experiences higher (25%–68%) gradient than that the pre-strained/rolled sheet endures. Regarding the role of technological parameters, high angle, small tool, average step-size and spindle rotation, and low flow-stress induce high stress gradient. The tension tests of the ISFed samples reveal that the post-ISF tensile strength of laminated sheet increases as the stress gradient increases, thus showing a direct relationship between stress gradient and strain hardening in ISF. Finally, models are proposed to predict the stress gradient in the ISFed Cu/steel components.     

Modeling of Nonlinear Mechanical Behavior for 3D Needled C/C-SiC Composites Under Tensile Load

This paper established a macroscopic constitutive model to describe the nonlinear stress–strain behavior of 3D needled C/C-SiC composites under tensile load. Extensive on- and off-axis tensile tests were performed to investigate the macroscopic mechanical behavior and damage characteristics of the composites. The nonlinear mechanical behavior of the material was mainly induced by matrix tensile cracking and fiber/matrix debonding. Permanent deformations and secant modulus degradation were observed in cyclic loading-unloading tests. The nonlinear stress–strain relationship of the material could be described macroscopically by plasticity deformation and stiffness degradation. In the proposed model, we employed a plasticity theory with associated plastic flow rule to describe the evolution of plastic strains. A novel damage variable was also introduced to characterize the stiffness degradation of the material. The damage evolution law was derived from the statistical distribution of material strength. Parameters of the proposed model can be determined from off-axis tensile tests. Stress–strain curves predicted by this model showed reasonable agreement with experimental results.     

On the Analysis of an EFG Method Under Large Deformations and Volumetric Locking

This work investigates a modified element-free Galerkin (MEFG) method when applied to large deformation processes. The proposed EFG method enables the direct imposition of the essential boundary conditions, as a result of the kronecker delta property of the special shape functions, constructed in the neighborhood of the essential boundary. The plasticity model assumes a multiplicative decomposition of the deformation gradient into an elastic and a plastic part and considers a J ₂ elasto-plastic constitutive relation that accounts for a nonlinear isotropic hardening. The constitutive model is written in terms of the rotated Kirchhoff stress and of the conjugate logarithmic strain measure. A total Lagrangian formulation is considered in order to improve the computational performance of the proposed algorithm. Here, aspects related to the volumetric locking are numerically investigated and an F-bar approach is considered. Some numerical results are presented, under axisymmetric and plane strain assumption, in order to attest the performance of the proposed method.     

Non‐linear version of stabilized conforming nodal integration for Galerkin mesh‐free methods

A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh‐free approximation, is generalized for non‐linear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed configuration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed configuration. The proposed method is independent to the path dependency of the materials. An assumed strain method is employed to formulate the discrete equilibrium equations, and the smoothed deformation gradient serves as the stabilization mechanism in the nodally integrated variational equation. Eigenvalue analysis demonstrated that the proposed strain smoothing provides a stabilization to the nodally integrated discrete equations. By employing Lagrangian shape functions, the computation of smoothed gradient matrix for deformation gradient is only necessary in the initial stage, and it can be stored and reused in the subsequent load steps. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison with the mesh‐free solution using Gauss integration. The performance of the proposed method is shown to be quite robust in dealing with non‐uniform discretization. Copyright © 2002 John Wiley & Sons, Ltd.     

A localizing gradient damage enhancement with micromorphic stress-based anisotropic nonlocal interactions

This article presents a localizing gradient damage model with evolving micromorphic stress-based anisotropic nonlocal interactions. The objective is to model mesh independent fracture behavior of quasi-brittle materials, and to avoid the issues associated with the existing gradient-enhanced damage models. In the proposed model, an evolving anisotropic nonlocal interaction domain governs the spatial diffusive behavior, which helps to maintain a localized damage bandwidth during the final stages of loading. The anisotropy in nonlocal interactions is captured through an anisotropic gradient tensor, which defines the orientation of the diffusive interaction domain based on the principal stresses at a given material point. In this article, a smooth micromorphic stress tensor is utilized for the determination of principal stress states, to enforce a properly oriented interaction across the bandwidth of the damage process zone throughout the loading process. The proposed approach also enables the usage of low order finite elements without any oscillatory micromorphic or nonlocal equivalent strain response in the later stages of deformation. The accuracy and performance of the proposed model are demonstrated numerically in plane strain/stress for mode-I, mode-II, and mixed-mode loading conditions.     

Cyclic inelasticity and high-cycle fatigue of metals with consideration of the stress gradient effect*

Cyclic deformation behavior of metals and alloys under high-cycle loading in the nonuniform stress state is considered. A significant effect of the stress gradient on the cyclic inelastic deformation of the metal surface layers is shown. A model explaining the difference between the fatigue limits in the uniform and nonuniform stress states is proposed, which is based on accounting for the distinctions between cyclic stress–strain diagrams in the uniform and nonuniform stress states and for the fact that the fatigue limit is equal to the cyclic elasticity limit found for inelastic deformation typical of this class of materials.