Analysis of axial strain in one-dimensional loading by different models

Different phenomenological equations based on plasticity, primary creep （as a viscoplastic mechanism）, secondary creep （as another viscoplastic mechanism） and different combinations of these equations are presented and used to describe the material inelastic deformation in uniaxial test. Agreement of the models with experimental results and with the theoretical concepts and physical realities is the criterion of choosing the most appropriate formulation for uniaxial test. A model is thus proposed in which plastic deformation, primary creep and secondary creep contribute to the inelastic deformation. However, it is believed that the hardening parameter is composed of plastic and primary creep parts. Accordingly, the axial plastic strain in a uniaxial test may no longer be considered as the hardening parameter. Therefore, a proportionality concept is proposed to calculate the plastic contribution of deformation.     

A consistent eulerian formulation of large deformation problems in statics and dynamics

A concise survey of formulation methods of geometric and material non-linearity problems is given. The survey is concerned mainly with the differences between updated Lagrangian and Eulerian formulations, and with the specific nature and basic characteristics of each. The underlying mechanics and the spatial discretisation for an Eulerian formulation are discussed. An Eulerian formulation with the final equilibrium equations suitable for static and/or dynamic structural analysis is presented. Explicit forms for stiffness matrices and load vectors are given. Differences between the present formulation, the existing Lagrangian formulation, the updated Langrangian formulation and other attempted Eulerian formulations are discussed within the framework of a consistent classification of formulation methods.     

Nonlinear micromechanical formulation of the high fidelity generalized method of cells

The recent High Fidelity Generalized Method of Cells (HFGMC) micromechnical modeling framework of multiphase composites is formulated in a new form which facilitates its computational efficiency that allows an effective multiscale material–structural analysis. Towards this goal, incremental and total formulations of the governing equations are derived. A new stress update computational method is established to solve for the nonlinear material constituents along with the micromechanical equations. The method is well-suited for multiaxial finite increments of applied average stress or strain fields. Explicit matrix form of the HFGMC model is presented which allows an immediate and convenient computer implementation of the offered method. In particular, the offered derivations provide for the residual field vector (error) in its incremental and total forms along with an explicit expression for the Jacobian matrix. This enables the efficient iterative computational implementation of the HFGMC as a stand alone. Furthermore, the new formulation of the HFGMC is used to generate a nested local-global nonlinear finite element analysis of composite materials and structures. Applications are presented to demonstrate the efficiency of the proposed approach. These include the behavior of multiphase composites with nonlinearly elastic, elastoplastic and viscoplastic constituents.     

Solution of strongly nonlinear problems in the mechanics of a solid deformable body

A new algorithm for constructing a Lagrangian formulation of the incremental theory on the basis of finite-element digitization of the equilibrium equations of strongly nonlinear mechanical systems subject to static loads is proposed. Example solutions of problems of nonlinear deformation of a solid that confirm the validity of the applied relations between increments in stresses and increments in deformations is presented. Technical University of Construction and Architecture, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 22–26, March, 1999.     

Design Sensitivity Analysis of Truss Structures with Elastoplastic Material*

ABSTRACT

A continuum-based design sensitivity analysis (DSA) method is presented for configuration (or layout) design of nonlinear structural systems with rate-independent elastoplastic material. Configuration design variables are characterized by shape and orientation changes of the structural component. A continuum-based shape DSA method that utilizes the material derivative of continuum mechanics is extended to account for effects of shape and orientation variations. The incremental analysis method, with updated Lagrangian formulation, is used to derive the design sensitivity for the nonlinear structural system.

To derive the design sensitivity, incremental energy and load forms are utilized. The first variations of energy and load forms and the static response with respect to configuration design variables are described using the material derivative. Direct differentiation is utilized to obtain the first variation of the performance measure explicitly in terms of variations of configuration design variables. With the consistent tangent stiffness matrix employed at the end of each load step to compute the design sensitivity, it is found that no iterations are necessary to compute design sensitivity. In addition, the linear design velocity is used to account for configuration design changes, with the velocity field being updated at each load step of the incremental analysis.     

Impact of a finite elastic-viscoplastic bar

A method for predicting the response of strain-rate sensitive structures under dynamic loading is developed. It is based on a finite difference method, the incremental theory of plasticity, and an elastic work-hardening viscoplastic material idealization. The strain-rate effect, loading and unloading conditions, and wave interactions are automatically accounted for, and adjusted if necessary, as the deformation proceeds. No iteration is required even if the field equations are nonlinear (e.g. non-linear constitutive equations, large deformation, or complicated geometry). We solve as an example the small deflection of a finite bar with a concentrated tip mass. The accuracy is comparable to that obtained by the well-known method of characteristics, a powerful tool for solving elastic-viscoplastic wave problems but which is restricted to small deflections and simple geometry. Because of the form of the constitutive relation selected (elastic work-hardening visco-plastic), several important new features of the dynamics response are brought out. These features are not revealed when simpler, computationally-convenient constitutive relations, such as rigid ideal-viscoplastic, rigid work-hardening viscoplastic and elastic ideal-viscoplastic are used.     

Integral formulation and self-consistent modelling of elastoviscoplastic behavior of heterogeneous materials

Summary Based on projection operators, an integral formulation is proposed for elastoviscoplastic heterogeneous materials. The problem requires a complete mechanical formulation, including the static equilibrium property concerning the known field σ, in addition to the classical field equations concerning the unknown fields ɛ˙ and σ˙. The formulation leads to an integral equation, in which elasticity and viscoplasticity effects interact through an homogeneous elastoviscoplastic medium with elastic moduli C and viscoplastic moduli B. To approximate the integral equation, the self-consistent scheme is followed. In order to obtain consistent approximation conditions, we introduce fluctuations of elastic and viscoplastic strain rate fields with respect to known kinematically compatible fields. It results in a strain rate concentration relation connecting the strain rate at each point to the macroscopic loading conditions and the local stress field. The results are presented and compared with other models and with experimental data in the case of a two-phase material. Received 26 August 1997; accepted for publication 2 July 1998     

饱和多孔介质大变形分析的一种有限元-有限体积混合方法

建立了饱和多孔介质大变形分析的一种有限元-有限体积混合计算方法.将饱和多孔介质视为由固体骨架和孔隙水组成的两相体,其基本方程包括动力平衡方程和渗流连续方程.基于u-p假定和更新的Lagrange方法,饱和多孔介质的动力平衡方程在空间域内采用有限元方法进行离散,而渗流连续方程在空阃域内则采用有限体积法进行离散.通过两个数值算例,一维有限弹性固结和动力荷载作用下堤坝动力响应的计算,验证了该方法的有效性.     

Modeling the thermoelastic–viscoplastic response of polycrystals using a continuum representation over the orientation space

Conventional methodologies towards polycrystal plasticity use an aggregate of single crystals and this choice of the aggregate affects the response of the polycrystal. In order to address this issue, a continuum approach is presented for the representation of polycrystals through an orientation distribution function over the orientation space. Additionally, a constitutive framework for thermoelastic–viscoplastic response of metals based on polycrystal plasticity is presented along with a coupled macro–micro, fully implicit Lagrangian finite element algorithm. Numerical examples that highlight the accuracy, performance and benefits of the proposed approach are presented.     

Finite-element formulations for problems of large elastic-plastic deformation

An Eulerian finite element formulation is presented for problems of large elastic-plastic flow. The method is based on Hill's variational principle for incremental deformations, and is ideally suited to isotropically hardening Prandtl-Reuss materials. Further, the formulation is given in a manner which allows any conventional finite element program, for “small strain” elastic-plastic analysis, to be simply and rigorously adapted to problems involving arbitrary amounts of deformation and arbitrary levels of stress in comparison to plastic deformation moduli. The method is applied to a necking bifurcation analysis of a bar in plane-strain tension.The paper closes with a unified general formulation of finite element equations, both Lagrangian and Eulerian, for large deformations, with arbitrary choice of the conjugate stress and strain measures. Further, a discussion is given of other proposed formulations for elastic-plastic finite element analysis at large strain, and the inadequacies of some of these are commented upon.     

Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization

Evolution of properties during processing of materials depends on the underlying material microstructure. A finite element homogenization approach is presented for calculating the evolution of macro-scale properties during processing of microstructures. A mathematically rigorous sensitivity analysis of homogenization is presented that is used to identify optimal forging rates in processes that would lead to a desired microstructure response. Macro-scale parameters such as forging rates are linked with microstructure deformation using boundary conditions drawn from the theory of multi-scale homogenization. Homogenized stresses at the macro-scale are obtained through volume-averaging laws. A constitutive framework for thermo-elastic–viscoplastic response of single crystals is utilized along with a fully-implicit Lagrangian finite element algorithm for modelling microstructure evolution. The continuum sensitivity method (CSM) used for designing processes involves differentiation of the governing field equations of homogenization with respect to the processing parameters and development of the weak forms for the corresponding sensitivity equations that are solved using finite element analysis. The sensitivity of the deformation field within the microstructure is exactly defined and an averaging principle is developed to compute the sensitivity of homogenized stresses at the macro-scale due to perturbations in the process parameters. Computed sensitivities are used within a gradient-based optimization framework for controlling the response of the microstructure. Development of texture and stress–strain response in 2D and 3D FCC aluminum polycrystalline aggregates using the homogenization algorithm is compared with both Taylor-based simulations and published experimental results. Processing parameters that would lead to a desired equivalent stress–strain curve in a sample poly-crystalline microstructure are identified for single and two-stage loading using the design algorithm.     

A viscoplastic strain gradient analysis of materials with voids or inclusions

A finite strain viscoplastic nonlocal plasticity model is formulated and implemented numerically within a finite element framework. The model is a viscoplastic generalisation of the finite strain generalisation by Niordson and Redanz (2004) [Journal of the Mechanics and Physics of Solids 52, 2431–2454] of the strain gradient plasticity theory proposed by Fleck and Hutchinson (2001) [Journal of the Mechanics and Physics of Solids 49, 2245–2271]. The formulation is based on a viscoplastic potential that enables the formulation of the model so that it reduces to the strain gradient plasticity theory in the absence of viscous effects. The numerical implementation uses increments of the effective plastic strain rate as degrees of freedom in addition to increments of displacement. To illustrate predictions of the model, results are presented for materials containing either voids or rigid inclusions. It is shown how the model predicts increased overall yield strength, as compared to conventional predictions, when voids or inclusions are in the micron range. Furthermore, it is illustrated how the higher order boundary conditions at the interface between inclusions and matrix material are important to the overall yield strength as well as the material hardening.     

A variational formulation for the incremental homogenization of elasto-plastic composites

This work addresses the micro–macro modeling of composites having elasto-plastic constituents. A new model is proposed to compute the effective stress–strain relation along arbitrary loading paths. The proposed model is based on an incremental variational principle (Ortiz, M., Stainier, L., 1999. The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171, 419–444) according to which the local stress–strain relation derives from a single incremental potential at each time step. The effective incremental potential of the composite is then estimated based on a linear comparison composite (LCC) with an effective behavior computed using available schemes in linear elasticity. Algorithmic elegance of the time-integration of J₂ elasto-plasticity is exploited in order to define the LCC. In particular, the elastic predictor strain is used explicitly. The method yields a homogenized yield criterion and radial return equation for each phase, as well as a homogenized plastic flow rule. The predictive capabilities of the proposed method are assessed against reference full-field finite element results for several particle-reinforced composites.     

Design sensitivity analysis for parameters affecting geometry,elastic–viscoplastic material constant and boundary condition by consistent tangent operator-based boundary element method

This paper presents a design sensitivity analysis method by the consistent tangent operator concept-based boundary element implicit algorithm. The design variables for sensitivity analysis include geometry parameters, elastic–viscoplastic material parameters and boundary condition parameters. Based on small strain theory, Perzyna’s elastic–viscoplastic material constitutive relation with a mixed hardening model and two flow functions is considered in the sensitivity analysis. The related elastic–viscoplastic radial return algorithm and the formula of elastic–viscoplastic consistent tangent operator are derived and discussed. Based on the direct differentiation approach, the incremental boundary integral equations and related algorithms for both geometric and elastic–viscoplastic sensitivity analysis are developed. A 2D boundary element program for geometry sensitivity, elastic–viscoplastic material constant sensitivity and boundary condition sensitivity has been developed. Comparison and discussion with the results of this paper, analytical solution and finite element code ANSYS for four plane strain numerical examples are presented finally.     

Finite element analysis of the piezocone test in cohesive soils using an elastoplastic–viscoplastic model and updated Lagrangian formulation

In this research, the finite element analysis of piezocone penetration has been conducted using the elastoplastic–viscoplastic bounding surface model in the updated Lagrangian reference frame. A finite element formulation has been performed considering the viscoplastic contribution of the model and the theory of mixtures has been incorporated to explain the behavior of the soil. The formulated model has been implemented into a finite element program, EPVPCS-S (elastoplastic–viscoplastic coupled system-soil), to analyze the mechanism of piezocone penetration. The results of the finite element analysis have been compared and investigated with the experimental results from the piezocone penetration and dissipation tests conducted using LSU/CALCHAS (Louisiana State University Calibration Chamber System).     

Incremental Magnetoelastic Deformations,with Application to Surface Instability

In this paper the equations governing the deformations of infinitesimal (incremental) disturbances superimposed on finite static deformation fields involving magnetic and elastic interactions are presented. The coupling between the equations of mechanical equilibrium and Maxwell’s equations complicates the incremental formulation and particular attention is therefore paid to the derivation of the incremental equations, of the tensors of magnetoelastic moduli and of the incremental boundary conditions at a magnetoelastic/vacuum interface. The problem of surface stability for a solid half-space under plane strain with a magnetic field normal to its surface is used to illustrate the general results. The analysis involved leads to the simultaneous resolution of a bicubic and vanishing of a 7×7 determinant. In order to provide specific demonstration of the effect of the magnetic field, the material model is specialized to that of a “magnetoelastic Mooney–Rivlin solid”. Depending on the magnitudes of the magnetic field and the magnetoelastic coupling parameters, this shows that the half-space may become either more stable or less stable than in the absence of a magnetic field.      

An efficient Galerkin meshfree formulation for shear deformable beam under finite deformation

This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam. The present nonlinear analysis encompasses the fully geometric nonlinearities due to large deflection, large deformation as well as finite rotation. The incremental equilibrium equation is obtained by the consistent linearization of the nonlinear variational equation. The Lagrangian meshfree shape function is utilized to discretize the variational equation. Subsequently to resolve the shear and membrane locking issues and accelerate the computation, the method of stabilized conforming nodal integration is systematically implemented through the Lagrangian gradient smoothing operation. Numerical results reveal that the present formulation is very effective.     

Analysis of shear deformable plates with combined geometric and material nonlinearities by boundary element method

In this paper a boundary element formulation for analysis of shear deformable plates with combined geometric and material nonlinearities by boundary element method is presented. The dual reciprocity method is used in dealing with the geometric nonlinearity and domain discretization is implemented in dealing with material nonlinearity. The material is assumed to undergo large deflection with small strains. The von Mises criteria is used to evaluate the plastic zone and an elastic perfectly plastic material behaviour is assumed. An initial stress formulation is used to formulate the boundary integral equations. A total incremental method is applied to solve the nonlinear boundary integral equations. Numerical examples are presented to demonstrate the validity and the accuracy of the proposed method.