Some theoretical results about second-order work,uniqueness, existence and controllability independent of the constitutive equation

When they are studied as continuum media, granular materials and other soils and rocks exhibit a complex behavior. Contrary to metals, their isotropic and deviatoric behavior are coupled. This implies some mathematical difficulties concerning boundary-value problems solved with constitutive equations modelling the salient features of such geomaterials. One of the well-known consequences is that the so-called second-order work can be negative long before theoretical failure occurs. Keeping this in mind, the starting point of this work is the pioneering and illuminating work of Nova (1994), who proved that using an isotropic hardening elasto-plastic model not obeying the normality rule, it is possible to exhibit either loss of uniqueness or loss of existence of the solution of a boundary-value problem as soon as the second-order work is negative. Because the geomaterial behavior is quite difficult to model, in practice many different constitutive equations are used. It is then important to study the point raised by Nova for other constitutive equations. In this paper, his result is generalized for any inelastic rate-independent constitutive equation. Similarly the link between localization and controllability proved by Nova (1989) is extended to some extent to a general inelastic model.     

An elastic-viscoplastic constitutive model for the hot-forming of aluminum alloys

Under hot-forming conditions characterized by high homologous temperatures and strain-rates, metals usually exhibit rate-dependent inelastic behavior. An elastic-viscoplastic constitutive model is presented here to describe metal behavior during hot-forming. The model uses an isotropic internal variable to represent the resistance offered to plastic deformation by the microstructure. Evolution equations are developed for the inelastic strain and the deformation resistance based on experimental results. A methodology is presented for extracting model parameters from constant true strain-rate compression tests performed at different temperatures. Model parameters are determined for an Al-1Mn alloy and an Al-Mg-Si alloy, and the predictions of the model are shown to be in good agreement with the experimental data.     

Uniqueness in the boundary-value problems for the static equilibrium equations of a mixture of two elastic solids occupying an unbounded domain

In this paper the Authors study the uniqueness of the solutions to the most important boundary-value problems for the static equilibrium equations of a mixture of two linear elastic solids. Some uniqueness theorems concerning the mixed boundary-value problem and the displacement problem are proved for unbounded domains. If the mixture is anisotropic, mild assumptions are imposed on the displacement fields at infinity. If the mixture is isotropic, uniqueness is proved for exterior domains without artificial restrictions upon the behavior of the unknown fields at infinity.     

Wave speeds, shear bands and the second-order work for incrementally nonlinear constitutive models

The paper discusses the consequences of the incremental nonlinearity of a constitutive model of a solid for the analyses of characteristic wave speeds, acceleration waves, the second-order work criterion and shear band formation. Incremental nonlinearity may entail qualitative changes in the results as compared to incrementally linear models. Certain well-known correlations cannot be established if the constitutive equation is assumed to be incrementally nonlinear. In particular, the spectra of the characteristic wave speeds and the acceleration wave speeds become continuous and are described by different equations. The second-order work criterion as a sufficient condition of uniqueness of the incremental boundary value problem loses its applicability in bifurcation analyses, unless the applicability can be proved for a particular type of nonlinearity. The singularity of the acoustic tensor in the general nonlinear case correlates neither with the vanishing of the second-order work nor with the shear band formation.     

Three-dimensional, finite deformation, viscoplastic constitutive models for polymeric materials

A general methodology for developing three-dimensional. finite deformation, viscoplastic constitutive models for polymeric materials is presented. The development begins with the presentation of a one-dimensional spring and dashpot construction which exhibits behavior typical of polymeric materials, namely strain-rate dependence, stress relaxation, and creep. The one-dimensional construction serves as a starting point for the development of a three-dimensional, finite deformation, viscoplastic constitutive model which also exhibits typical polymeric behavior. Furthermore, the three-dimensional constitutive model may be easily generalized to incorporate an arbitrary number of inelastic processes, representing (inelastic) microstructural deformation mechanisms operating on different time scales. Strain-rate dependence, stress relaxation, and creep phenomena are discussed in detail for a simple version of the constitutive model. Test data for a particular polymer is used to validate the simple model. It is concluded that the methodology provides a flexible approach to modeling polymeric materials over a wide range of loading conditions.     

Entropic thermoelasticity of thin polymeric films

Summary An asymptotic membrane theory for thin, heat-conducting, finitely-deforming elastic films is presented. Constitutive equations for the stress and heat flux are obtained from their three-dimensional antecedents and applied to the study of isotropic films that exhibit entropic elasticity. The model is illustrated by solving coupled axisymmetric boundary-value problems for the deformation and temperature fields.     

Modeling of continuum damage for application in elastic-viscoplastic constitutive equations

A procedure is described for including isotropic and directional damage as load-history dependent softening variables in a set of elastic-viscoplastic constitutive equations. The evolution equation proposed for isotropic damage integrates to an exponential form for the case of constant stress. Directional damage is represented as a second-order symmetric tensor with a scalar effective value used in the constitutive equations. A method is proposed for treating directional damage in the case of non-proportional loading histories. Comparisons are given of uniaxial creep test results for an alloy at high temperatures with calculations based on the constitutive equations with the inclusion of isotropic damage.     

Numerical aspects of non-local modeling of the damage evolution in elastic–plastic materials

The design of mechanical systems in modern industrial plants requires reliable and efficient methods to predict the behavior of structural materials. For complex loading conditions, the behavior of the structural materials is determined by damage evolution, strain rate and temperature. The subject of the article is the modeling of the damage evolution in elastic–plastic materials of structural components, which are utilized at various temperatures. To achieve this goal, a hybrid model of steel cracking is applied. The hybrid model uses a finite element simulation combined with an experimental test realized in the macroscale. By using the hybrid model, the modeling of the damage evolution affords possibilities of determining macroscopic effects of the steel micro-defects. An essence of solving the predicting behavior of structural materials with micro-defects consists in time integration procedures for constitutive equations. In the article a semi-implicit time integration procedure is presented. The semi-implicit time integration procedure is suitable for the inelastic materials (compressible or incompressible) with the combined kinematic–isotropic hardening behavior. Its numerical solutions are stable, namely without the oscillatory behavior. By spatial averaging over a representative volume (RV), the homogenization technique (HT) is used for the defining of non-local variables in the constitutive equations. Evolutionary algorithms (EAs) based on local selections are applied to perform the homogenization technique. Within the framework of the large strain theory, the non-local continuum satisfies the objectivity requirements. Limitations on applicability of the -integral approach to construct crack growth resistance curves are also presented.     

Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation

A constitutive model for orthotropic elastoplasticity at finite plastic strains is discussed and basic concepts of its numerical implementation are presented. The essential features are the multiplicative decomposition of the deformation gradient in elastic and inelastic parts, the definition of a convex elastic domain in stress space and a representation of the constitutive equations related to the intermediate configuration. The elastic free energy function and the yield function are formulated in an invariant setting by means of the introduction of structural tensors reflecting the privileged directions of the material. The model accounts for kinematic and isotropic hardening. The associated flow rule is integrated using the so-called exponential map which preserves exactly the plastic incompressibility condition. The constitutive equations are implemented in a brick-type shell element. Representative numerical simulations demonstrate the suitability of the proposed formulations.     

Large strain constitutive modelling and computation for isotropic,creep elastoplastic damage solids

A three-dimensional fully coupled creep elastoplastic damage model at finite strain for isotropic non-linear material is developed. The model is based on the thermodynamics of an irreversible process and the internal state variable theory. A hyperelastic form of stress–strain constitutive relation in conjunction with the multiplicative decomposition of the deformation gradient into elastic and inelastic parts is employed. The pressure-dependent plasticity with strain hardening and the damage model with two damage internal variables are particularly considered. The rounding of stress–strain curves appearing in cycling loading is reproduced by introduction of the creep mechanism into the model. A numerical integration procedure for the coupled constitutive equations with three hierarchical phases is proposed. A consistent tangent matrix with consideration of the fully coupled effects at finite strain is derived. Numerical examples are tested to demonstrate the capability and performance of the present model at large strain.     

Multi-axial thermo-mechanical analysis of power plant components from 9–12% Cr steels at high temperature

The aim of this paper is to analyze local changes of stress and strain states in a power plant component under a transient thermal environment. A robust constitutive model is developed to describe inelastic behavior of advanced 9–12% Cr heat-resistant steels at high temperature and in a multi-axial stress state. The model includes the constitutive equation for the inelastic strain rate tensor, the evolution equation for a tensor-valued state variable to reflect hardening/recovery processes and two evolution equations for two scalar-valued variables that characterize softening and damage states. The model is calibrated against experimental creep curves and verified for inelastic responses under different isothermal and non-isothermal loading paths. Steam temperature and loading profiles that correspond to an idealized start-up, holding and shut-down sequence of a power plant component are assumed. To estimate the thermal fields, transient heat transfer analysis is performed. The results are applied in the subsequent structural analysis using the developed inelastic constitutive model. The outcome is a multi-axial thermo-mechanical fatigue loop which can be used for damage assessment.     

A nonlinear CDM model for ductile failure analysis of steel bridge columns under cyclic loading

A nonlinear cyclic plasticity damage model for ductile metals, which is able to take large deformation effects into consideration, has been developed using a new damage dissipation potential formulation in order to predict the cyclic inelastic behavior of steel bridge piers. The cyclic constitutive equations that employ the combined isotropic–kinematic hardening rule for plastic deformation is incorporated into the damage mechanics in conjunction with the large strain formulation. The damage growth law is based on the experimental observations that the evolution of microvoids results in nonlinear damage accumulation with plastic deformation. The damage model parameters and the procedure for their identification are presented. The proposed model has been validated and successfully applied to thin-walled steel bridge tubular columns subjected to alternating lateral displacements to evaluate the seismic performance.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Modeling a smooth elastic–inelastic transition with a strongly objective numerical integrator needing no iteration

Large deformation evolution equations for elastic distortional deformation and isotropic hardening/softening have been developed that model a smooth elastic–inelastic transition for both rate-independent and rate-dependent response with no need for loading–unloading conditions. A novel special case is a rate-independent overstress model. Specific simplified constitutive equations are proposed that capture the main effects of elastic-plastic and elastic-viscoplastic materials with only a few material parameters. Moreover, a robust and strongly objective numerical integrator for these simplified evolution equations has been developed which needs no iteration. Examples show the influence of the various parameters on the predicted material response. The smoothness of the elastic–inelastic transition in the proposed model, with the associated overstress, tends to spread the inelastic region. This side effect prevents severe deformation from being localized in an element region that continues to reduce in size with mesh refinement. However, preliminary calculations indicate the need for additional modeling of a material characteristic length that independently controls the size of a localized severely deformed region.     

Creep and recovery of nonlinear viscoelastic materials of the differential type

In this work, the creep and recovery properties of rubberlike viscoelastic materials in simple shear are studied by two special constitutive equations for isotropic, nonlinear incompressible viscoelastic material of the differential type. The creep and recovery processes are of significant importance to both the mechanics analysis and engineering applications. The constitutive equations introduced in this work generalize the Voigt-Kelvin solid and the 3-parameter model of classical linear viscoelasticity. They describe the uncoupled non-Newtonian viscous and nonlinear elastic response of an isotropic, incompressible material. The creep and recovery processes are treated for simple shear deformation superimposed on a longitudinal static stretch. Closed form solutions are provided and both processes are described effectively by the exponential function.     

Numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage

This paper deals with a class of rate-independent metal plasticity models which exhibit non-linear isotropic hardening, non-linear kinematic hardening (Chaboche-Marquis model) and ductile damage (Lemaitre-Chaboche model). The backward Euler scheme is used to integrate the rate constitutive relations. The non-linear equations obtained are solved by the Newton method. The consistent tangent operator is obtained by exact linearization of the algorithm. Despite the complexity of the constitutive equations, closed-form expressions are derived, without any approximations. Analytical, numerical and experimental results are presented and discussed.     

低屈服点钢材与Q345B和Q460D钢材本构关系对比研究

为进一步探讨材料本构行为对构件及结构受力性能的影响,首先,进行了LYP100低屈服点钢材的本构关系试验研究,分析此材料的单调性能、滞回性能、耗能能力及循环本构模型等。在此基础上,全面对比LYP100和LYP160低屈服点钢材、普通钢材（Q345B）及高强度钢材（Q460D）的本构关系。最后,通过对比不同钢材的循环本构模型以及理想弹塑性模型对结构构件滞回行为的预测结果,深入研究材料本构关系对构件及结构的重要影响。结果表明:低屈服点钢材单调以及循环强屈比均在2.0~3.0以上,是普通钢材以及高强度钢材的2.0倍~3.0倍。同时,低屈服点钢材具有更好的延性和耗能能力。由于低屈服点钢材具有显著的各向同性强化行为,其采用循环本构模型和理想弹塑性模型的计算结果差异更大。因此,在结构计算分析中,需要根据所采用的钢材选取适当的本构关系模型。     

Constitutive relations for cracked metal matrix composites

A method is given for the prediction of the overall stress-strain response of elastoplastic solids containing a doubly periodic rectangular array of cracks. The behavior of the undamaged inelastic material is represented by constitutive laws for metal matrix composites based on a micromechanics analysis. The effect of cracking is incorporated by adopting a second order expansion of the displacement vector, in conjunction with the equations of equilibrium and displacement and traction continuity conditions. Due to the nonlinearity and path dependence of deformation, the stresses need to be represented by a higher order expansion than the second. The method is implemented to predict the overall stress-strain response of a cracked isotropic inelastic material, as well as cracked unidirectional and laminated metal matrix composites.     

各向同性弹性介质非线性本构方程

从张量函数出发,围绕共轭应力、应变变量,研究了各向同性非线性弹性介质各种形式的本构方程以及各种形式方程之间的关系。推导出用张量不变量,标量不变量表示的两种形式非线性Green弹性介质本构方程。证明了方程是完备的,不可约的。作为应用举例,研究了橡胶材料的工程应用问题。     

Thermo-mechanical fatigue modeling of advanced metal matrix composites in the presence of microstructural details

An analytical model developed for predicting the inelastic response of metal matrix composites subjected to axisymmetric loading is employed to investigate the behavior of SiC---Ti composites under thermo-mechanical fatigue loading. The model is based on the concentric cylinder assemblage consisting of arbitrary numbers of elastic or inelastic sublayers with isotropic, transversely isotropic, or orthotropic, temperature-dependent properties. In the present work, the inelastic response of the titanium matrix is modeled by the Bodner-Partom unified viscoplastic theory. These features of the model allow the investigation of microstructural effects (such as the layered morphology of the SCS-6 fiber, including the weak carbon coating, and matrix microstructure) and rate-dependent response of the matrix on the fatigue behavior.

In this paper, we employ the predictions of the multiple concentric cylinder model to study the effects of the layered morphology of the SCS-6 SiC fiber and two-phase microstructure of the Ti-15-3 matrix on the response of a SiC---Ti composite under thermo-mechanical fatigue loading. It is shown that ignoring the microstructure can lead to significant errors in the predictions of the internal stress and inelastic strain distributions.