A constitutive model in finite viscoelasticity

A new constitutive model is suggested for the viscoelastic behavior of rubber-like materials at finite strains. The model treats a viscoelastic medium as a system with a variable number of purely elastic links, which can arise and collapse due to micro-Brownian motion of molecules.Assuming that the processes of birth and death for elastic links are independent of stresses, we obtain operator linear constitutive equations in finite viscoelasticity. According to this model, elastic and viscous effects may be distinguished and described independently of each other by a relaxation measure and a strain energy density.The potential energy of deformations is assumed to depend on the principal invariants of the relative Finger tensor of strains. Unlike the standard approach, we do not suggest any expression for the strain energy densitya priori, but suppose that this function is presented as a sum of two functions of one variable which are found by fitting experimental data.The proposed approach allows results of several experiments (uniaxial tension, biaxial tension, and torsion) for styrene butadiene rubber and butyl rubber to be predicted correctly.     

Finite element simulations of martensitic phase transitions and microstructures based on a strain softening model

A new approach for modeling multivariant martensitic phase transitions (PT) and martensitic microstructure (MM) in elastic materials is proposed. It is based on a thermomechanical model for PT that includes strain softening and the corresponding strain localization during PT. Mesh sensitivity in numerical simulations is avoided by using rate-dependent constitutive equations in the model. Due to strain softening, a microstructure comprised of pure martensitic and austenitic domains separated by narrow transition zones is obtained as the solution of the corresponding boundary value problem. In contrast to Landau-Ginzburg models, which are limited in practice to nanoscale specimens, this new phase field model is valid for scales greater than 100 nm and without upper bound. A finite element algorithm for the solution of elastic problems with multivariant martensitic PT is developed and implemented into the software ABAQUS. Simulated microstructures in elastic single crystals and polycrystals under uniaxial loading are in qualitative agreement with those observed experimentally.     

Thermodynamic nature and control of the elastic limit in solids

The elastic limit of a solid is implicit in its thermo-elastic properties and can be determined from the constitutive equations of internal energy and entropy in the elastic range. The second law of thermodynamics is responsible for this, as it sets an upper bound to the internal energy that a material can store during isothermal elastic deformation processes. A link between irreversibility and elasticity can thus be established, which allows for a better control of the properties of strength, ductility and elastic limit of the material. For elastic-plastic materials of practical interest it implies that the yield limit cannot be assigned independently of the elastic constitutive equations, although the current approaches do so. An application to elastic-plastic materials with linear thermo-elastic properties reveals that, in the one-dimensional case, all information on the entropy of the material can be drawn from standard uniaxial tests. An easy procedure can then be devised to design the preparation process of the material so that the desired combination of strength, ductility and elastic limit can be achieved within the admissible values for these quantities.     

半结晶聚合物损伤演化的实验表征与数值模拟

损伤本构模型对研究材料的断裂失效行为有重要意义, 但聚合物材料损伤演化的定量表征实验研究相对匮乏. 通过4种高密度聚乙烯(high density polythylene, HDPE)缺口圆棒试样的单轴拉伸实验获得了各类试样的载荷-位移曲线和真应力-应变曲线, 采用实验和有限元模拟相结合的方法确定了HDPE材料不同应力状态下的本构关系, 并建立了缺口半径与应力三轴度之间的关系;采用两阶段实验法定量描述了4种HDPE试样单轴拉伸过程中的弹性模量变化, 并建立了基于弹性模量衰减的损伤演化方程, 结合中断实验和扫描电子显微镜分析了应力状态对HDPE材料微观结构演化的影响. 结果表明缺口半径越小, 应力三轴度越大, 损伤起始越早、演化越快; 微观表现为: 高应力三轴度促进孔洞的萌生和发展, 但抑制纤维状结构的产生;基于实验和有限元模拟获得的断裂应变、应力三轴度、损伤演化方程等信息提出了一种适用于聚合物的损伤模型参数确定方法, 最后将本文获得的本构关系和损伤模型用于HDPE平板的冲压成形模拟, 模拟结果与实验结果吻合良好.      

New integral formulation and self-consistent modeling of elastic–viscoplastic heterogeneous materials

Predicting the overall behavior of heterogeneous materials, from their local properties at the scale of heterogeneities, represents a critical step in the design and modeling of new materials. Within this framework, an internal variables approach for scale transition problem in elastic–viscoplastic case is introduced. The proposed micromechanical model is based on establishing a new system of field equations from which two Navier’s equations are obtained. Combining these equations leads to a single integral equation which contains, on the one hand, modified Green operators associated with elastic and viscoplastic reference homogeneous media, and secondly, elastic and viscoplastic fluctuations. This new integral equation is thus adapted to self-consistent scale transition methods. By using the self-consistent approximation we obtain the concentration law and the overall elastic–viscoplastic behavior of the material. The model is first applied to the case of two-phase materials with isotropic, linear and compressible viscoelastic properties. Results for elastic–viscoplastic two-phase materials are also presented and compared with exact results and variational methods.     

大变形下初始斜交异性本构方程

采用材料主轴法,建立了初始斜交异性材料在变形构形（Euler描述）下的斜交异性本构方程,以及在初始构形（Lagrange描述）下的形式。具体给出了斜交异性线弹性材料方程的显式,它在Lagrange描述下形式简洁,可方便地用于有限元计算。文中指出,在变形构形下是线弹性的材料,在Lagrange描述下其本构方程一般已成为非线性,我们称之为本构转换非线性。这种非线性在实际的有限元计算中还未引起重视。为理论简明,本构方程是对二维给出的。     

Direct determination of multi-axial elastic potentials for incompressible elastomeric solids: an accurate,explicit approach based on rational interpolation

With a novel approach based on certain logarithmic invariants, we demonstrate that a multi-axial elastic potential for incompressible, isotropic rubber-like materials may be obtained directly from two one-dimensional elastic potentials for uniaxial case and simple shear case, in a sense of exactly matching finite strain data for four benchmark tests, including uniaxial extension, simple shear, bi-axial extension, and plane-strain extension. As such, determination of multi-axial elastic potentials may be reduced to that of two one-dimensional elastic potentials. We further demonstrate that the latter two may be obtained by means of rational interpolating procedures for uniaxial data and shear data displaying strain-stiffening effects. Numerical examples are presented in fitting Treloar’s data and other data.     

Generalized variational principles for linear elastic materials with voids

In this paper, the idea of variational principles of linear elastic theory is used to establish generalized variational principles for linear elastic materials with voids. The fundamental equations of linear elastic materials with voids used have already been established in Ref. [5].     

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

Deformations of an elastic,internally constrained material. Part 1: Homogeneous deformations

The nonlinear elastic response of a class of materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite deformation of a variety of metals is investigated. The purely kinematical consequences of the Bell constraint are discussed, and restrictions on the full range of compatible deformations are presented in geometrical terms. Then various forms of the constitutive equation relating the stress and stretch tensors for an isotropic elastic Bell material are presented. Inequalities on the mechanical response functions are introduced. The importance of these in applications is demonstrated in several examples throughout the paper.This paper focuses on homogeneous deformations. In a simple illustration of the theory, a generalized form of Bell's empirical rule for uniaxial loading is derived, and some peculiarities in the response under all-around compressive loading are discussed. General formulae for universal relations possible in an isotropic elastic, Bell constrained material are presented. A simple method for the determination of the left stretch tensor for essentially plane problems is illustrated in the solution of the problem of pure shear of a materially uniform rectangular block. A general formula which includes the empirical rule found in pure shear experiments by Bell is derived as a special case. The whole apparatus is then applied in the solution of the general problem of a homogeneous simple shear superimposed on a uniform triaxial stretch; and the great variety of results possible in an isotropic, elastic Bell material is illustrated. The problem of the finite torsion and extension of a thin-walled cylindrical tube is investigated. The results are shown to be consistent with Bell's data for which the rigid body rotation is found to be quite small compared with the gross deformation of the tube. Several universal formulas relating various kinds of stress components to the deformation independently of the material response functions are derived, including a universal rule relating the axial force to the torque.Constitutive equations for hyperelastic Bell materials are derived. The empirical work function studied by Bell is introduced; and a new constitutive equation is derived, which we name Bell's law. On the basis of this law, we then derive exactly Bell's parabolic laws for uniaxial loading and for pure shear. Also, form Bell's law, a simple constitutive equation relating Bell's deviatoric stress tensor to his finite deviatoric strain tensor is obtained. We thereby derive Bell's invariant parabolic law relating the deviatoric stress intensity to the corresponding strain intensity; and, finally, Bell's fundamental law for the work function expressed in these terms is recovered. This rule is the foundation for all of Bell's own theoretical study of the isotropic materials cataloged in his finite strain experiments on metals, all consistent with the internal material constraint studied here.     

Optimization of the strain energy density in linear anisotropic elasticity

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.     

Waves in Constrained Linear Elastic Materials

This paper deals with the propagation of acceleration waves in constrained linear elastic materials, within the framework of the so-called linearized finite theory of elasticity, as defined by Hoger and Johnson in [12, 13]. In this theory, the constitutive equations are obtained by linearization of the corresponding finite constitutive equations with respect to the displacement gradient and significantly differ from those of the classical linear theory of elasticity. First, following the same procedure used for the constitutive equations, the amplitude condition for a general constraint is obtained. Explicit results for the amplitude condition for incompressible and inextensible materials are also given and compared with those of the classical linear theory of elasticity. In particular, it is shown that for the constraint of incompressibility the classical linear elasticity provides an amplitude condition that, coincidently, is correct, while for the constraint of inextensibility the disagreement is first order in the displacement gradient. Then, the propagation condition for the constraints of incompressibility and inextensibility is studied. For incompressible materials the propagation condition is solved and explicit values for the squares of the speeds of propagation are obtained. For inextensible materials the propagation condition is solved for plane acceleration waves propagating into a homogeneously strained material. For both constraints, it is shown that the squares of the speeds of propagation depend by terms that are first order in the displacement gradient, while in classical linear elasticity they are constant. This revised version was published online in July 2006 with corrections to the Cover Date.     

Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

An approximate solution to the problem of cone or wedge indentation of elastoplastic solids

The model developed in this Note makes it possible to determine the value of the mean indentation pressure usually named hardness from the elastoplastic properties of materials and also the shape of the cone or that of the wedge. The approximation rests upon the definition of a linear elastic solid which has the same indentation pressure as the material actually indented. Cases of cone and wedge indentation are studied. A method to determine the uniaxial stress–strain curve of materials from indentation tests is given. The results are validated using finite element simulations. To cite this article: G. Kermouche et al., C. R. Mecanique 333 (2005).     

Spherical indentation of incompressible rubber-like materials

In recent years, indentation tests have been proven very useful in probing mechanical properties of small volumes of materials. However, a class of materials that very little has been done in this direction is rubber-like materials (elastomers). The present work investigates the spherical indentation of incompressible rubber-like materials. The analysis is performed in the context of second-order hyperelasticity and is accompanied by finite element computations and an extensive experimental program with spherical indentors of different radii. Uniaxial tensile tests were also performed and it was found that the initial elastic modulus correlates well with the indentation response. The experiments suggest stiffer indentation response than that predicted by linear elasticity, which is somehow counter-intuitive, if the uniaxial material response is to be considered. Regarding the uniqueness of the inverse problem, that is to establish material properties from spherical indentation tests, the answer is disappointing. We prove that the inverse problem does not give unique answer regarding the constitutive relation, except for the initial stiffness.     

Micropolar hypo-elasticity

In this paper, the concept of hypo-elasticity is generalized to the micropolar continuum theory, and the general forms of the constitutive equations of the micropolar hypo-elastic materials are presented. A new co-rotational objective rate whose spin is the micropolar gyration tensor is introduced which describes the deformation of the material in view of an observer attached to the micro-structure. As special case, simplified versions of the proposed constitutive equations are given in which the same fourth-order elasticity tensors are used as in the micropolar linear elasticity. A 2-D finite element formulation for large elastic deformation of micropolar hypo-elastic media based on the simplified constitutive equations in conjunction with Jaumann and gyration rates is presented. As an example, buckling of a shallow arc is examined, and it is shown that an increase in the micropolar material parameters results in an increase in the buckling load of the arc. Also, it is shown that micropolar effects become important for deformations taking place at small scales.     

Constitutive equations in finite viscoplasticity of semicrystalline polymers

Three series of uniaxial tensile tests with constant strain rates are performed at room temperature on isotactic polypropylene and two commercial grades of low-density polyethylene with different molecular weights. Constitutive equations are derived for the viscoplastic behavior of semicrystalline polymers at finite strains. A polymer is treated as an equivalent network of strands bridged by permanent junctions. Two types of junctions are introduced: affine whose micro-deformation coincides with the macro-deformation of a polymer, and non-affine that slide with respect to their reference positions. The elastic response of the network is attributed to elongation of strands, whereas its viscoplastic behavior is associated with sliding of junctions. The rate of sliding is proportional to the average stress in strands linked to non-affine junctions. Stress–strain relations in finite viscoplasticity of semicrystalline polymers are developed by using the laws of thermodynamics. The constitutive equations are applied to the analysis of uniaxial tension, uniaxial compression and simple shear of an incompressible medium. These relations involve three adjustable parameters that are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is revealed that the viscoplastic response of low-density polyethylene in simple shear is strongly affected by its molecular weight.     

An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations

The initial yield surface of 2D lattice materials is investigated under biaxial loading using finite element analyses as well as by analytical means. The sensitivity of initial yield surface to the dominant deformation mode is explored by using both low- and high-connectivity topologies whose dominant deformation mode is either local bending or strut stretching, respectively. The effect of microstructural irregularity on the initial yield surface is also examined for both topologies. A pressure-dependent anisotropic yield criterion, which is based on total elastic strain energy density, is proposed for 2D lattice structures, which can be easily extended for application to 3D cellular solids. Proposed criterion uses elastic constants and yield strengths under uniaxial loading, and does not rely on any arbitrary parameter. The analytical framework developed allows the introduction of new scalar measures of characteristic stresses and strains that are capable of representing the elastic response of anisotropic materials with a single elastic master line under multiaxial loading.     

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.     

OBTAINING MULTI-AXIAL ELASTIC POTENTIALS FOR RUBBER-LIKE MATERIALS VIA AN EXPLICIT,EXACT APPROACH BASED ON SPLINE INTERPOLATION

An explicit, exact approach is proposed to obtain multi-axial elastic potentials for isotropic rubber-like materials undergoing large incompressible deformations. By means of two direct, explicit procedures, this approach reduces the problem of determining multi-axial poten- tials to that of determining one-dimensional elastic potentials. To this end, two one-dimensional potentials for uniaxial case and simple shear case are respectively determined via spline inter- polation and, then, the two potentials are extended to generate a multi-axial elastic potential using a novel method based on certain logarithmic invariants. Eventually, each of the multi-axial potentials will exactly match the finite strain data from four benchmark tests.