Solution existence conditions for elastoplastic constitutive models of granular materials

In this paper, the conditions of solution existence for stress rates under given strain rates are investigated. The focus of the solution existence investigation is on the non-associated flow rule and elastic stress–strain relationship. Granular materials characterized with strong non-associated plastic flows are used as a particular example for analysis. Various flow rules for granular materials are analyzed, including Rowe’s, Roscoe’s flow rules and their modified versions. In the elastic stress–strain relationships of materials, the effects of Poisson’s ratio on solution existence are investigated. Both isotropic and anisotropic elasticity are considered. Given a granular material and its states, it is found that there exists a critical Poisson’s ratio for a particular non-associated flow rule. When the Poisson’s ratio of a material is above this critical Poisson’s ratio, its constitutive model is susceptible to solution non-existence. It is suggested that special attentions should be paid to the selection of material Poisson’s ratio and non-associated flow rule to ensure the existence of elastoplastic solutions.     

数据驱动梯度结构材料弹塑性本构

梯度结构材料因其优异的力学性能被广泛应用于工程结构中.论文整合塑性理论和人工神经网络技术,发展了一种构建梯度结构材料弹塑性本构模型的新方法.该方法基于梯度结构材料不同位置的微结构,构建不同代表性体积单元,进而生成应力应变数据,应用生成的数据训练人工神经网络,建立基于神经网络的材料本构模型.应用该方法,论文开展了针对实际...     

An elastoplastic constitutive relation of whisker-reinforced composite for meso damage: Part I – formulation

An elastoplastic constitutive relation is developed for meso damage of whisker-reinforced composites. A model is constructed that includes orientation distribution of whiskers and slip systems as well as interface and crystal sliding. Evolution of damage will be addressed. Given in Part I is the formulation while examples will be illustrated in Part II.     

Modelling shock waves in composite materials using generalised orthotropic pressure

Continuum Mechanics and Thermodynamics - Excellent mechanical properties of composite materials have numerous engineering applications, especially in aerospace structures. The main characteristics...     

A strong contrast homogenization formulation for multi-phase anisotropic materials

Homogenization relations, linking a material's properties at the mesoscale to those at the macroscale, are fundamental tools for design and analysis of microstructure. Recent advances in this field have successfully applied spectral techniques to Kroner-type perturbation expansions for polycrystalline and composite materials to provide efficient inverse relations for materials design. These expansions have been termed ‘weak-contrast’ expansions due to the conditionally convergent integrals, and the reliance upon only small perturbations from the reference property. In 1955, Brown suggested a different expansion for electrical conductivity that resulted in absolutely convergent integrals. Torquato subsequently applied the method to elasticity, with good results even for high-contrast materials; thus it is commonly referred to as a ‘strong contrast’ expansion. The methodology has been applied to elasticity for two phases of isotropic material, generally assuming macroscopic isotropy (with noted exceptions), thus resulting in a rather elegant form of the solution.

More recently, a multi-phase form of the solution was developed for conductivity. This paper builds upon this result to apply the method to elasticity of polycrystalline materials with both local and global anisotropy. New spectral formulations are subsequently developed for both the weak and strong contrast solutions. These form the basis for efficient microstructure analysis using these frameworks, and subsequently for inverse design applications. The process is taken through to demonstration of a property closure, which acts as the basis for materials design; the closure delineates the envelope of all physically realizable property combinations for the chosen properties, based on the particular homogenization relation being used.     

On minimal representations for constitutive equations of anisotropic elastic materials

The problem of determining minimal representations for anisotropic elastic constitutive equations is proposed and investigated. For elastic constitutive equations in any given case of anisotropy, it is shown that there exist generating sets consisting of six generators and such generating sets are minimal in all possible generating sets. This fact implies that most of the established results for representations of elastic constitutive equations are not minimal and remain to be sharpened. For elastic constitutive equations in some cases of anisotropy, including orthotropy, transverse isotropy, the trigonal crystal class S ₆, and the classes C _2mh , m=1, 2, 3,..., etc., representations in terms of minimal generating sets are presented for the first time.     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state.The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.     

An elastoplastic model for granular materials taking into account grain breakage

Granular materials are constituted of an assembly of particles. In spite of the simplicity of this assembly, its mechanical behaviour is complex. In the first stage we propose a framework to establish correlations between parameters of the supposedly continuous medium and grain properties which are assumed to be constant. However, this hypothesis is no longer valid in the case where physical (shape, size…) or mechanical properties (Young modulus E_g, Poisson's ratio ν_g…) of grains evolve during loading, causing the behaviour of the assembly to modify. We study the influence of the physical and mechanical parameters on grain breakage. We subsequently propose a way to model the influence of the grain breakage on granular materials and we introduce this influence in an elastoplastic constitutive model. Validations are made on two kinds of sands under isotropic and triaxial loading. Since the results of numerical computations corresponded well with the experimental data, we believe that the new model is capable of accurately simulating the behaviour of granular materials under a wide range of stresses and of taking into account, through new parameters, the individual strength of grains.     

Applications of tensor functions to the formulation of yield criteria for anisotropic materials

Yielding of anisotropic materials can be characterized by yield criteria which are scalar-valued functions of the stress tensor and of material tensors, for instance, of rank two or four, characterizing the anisotropic properties of the material. Because of the requirement of invariance, a yield criterion can be expressed as a single-valued function of the integrity basis. In finding an integrity basis involving the stress tensor and material tensors, the constitutive equations are first formulated based on the tensor function theory. Since the plastic work characterizes the yield process, we read from this scalar expression the essential invariants to formulate a yield criterion. Some examples for practical use are discussed in detail.     

An admissible form of an inelastic constitutive equation—II. Examples of elastoplastic constitutive equation

Some examples of elastoplastic constitutive equation are presented using the general theory reported in the preceding paper (Part I). Some examinations of them are given to show that the theory is self-consistent and useful especially for anisotropic materials or materials with anisotropy resulting from plastic deformation. Mises' and Yoshimura's yield functions and a kind of quadratic function are adopted as the yield function. Formulae of r-value after arbitrary pre-straining are given which are of paramount importance in the field of press-forming of sheet metals. Several examples of stress-strain curves for various loading paths are also given.     

The constitutive equations for mixed hardening orthotropic material

IntroductionAyieldingcriterionandtheassociatedflowtheorywereproposedbyHill[1]forinitiallyorthotropicmetalsin 1 948,whichareusedbroadly .AquadraticformofstressesisusedastheplasticpotentialthatisindependentofhydrostaticstressinHillplasticitytheory .Butcompressedbyhydrostaticstress,considerabledeformationwillbeproducedinorthotropicmaterials.Inthecaseofcyclicloading ,duetoBauschingereffect,thekinematichardeningcannotbeneglected .Inthispaper,kinematichardeningandproportionalhardeningareconsidered…     

An approximate explicit constitutive relation for non-flowing granular materials

An approximate stress-strain relation is derived for a granular material composed of spherical elastic granules in contact. The material is assumed to be statistically homogeneous so that the effective stress tensor can be obtained by a volume average. The resulting stress-strain relation is markedly non-linear and begins with the term $\text{∥ε∥}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, where ε is the classical infinitesimal strain tensor. Some simple deformation fields are worked out.     

An improved formulation for domain stress evaluation by boundary element method in elastoplastic problems

An integral expression for domain stresses in the elastoplastic boundary element method is presented. Thus it is not necessary to evaluate the Cauchy. principal value in domain integral. This method shows some improvement in the accuracy and efficiency for the numerical process. A numerical result of axisymmetric elastoplastic problem has been worked out and a comparison with analytical result is given.This work had been supported by National Science Foundation of China.