Nonlinear elastic constitutive relations of residually stressed composites with stiff curved fibres

Applied Mathematics and Mechanics - Mechanical models of residually stressed fibre-reinforced solids, which do not resist bending, have been developed in the literature. However, in some residually...     

Solutions of some simple boundary value problems within the context of a new class of elastic materials

Some simple boundary value problems are studied, for a new class of elastic materials, wherein deformations are expressed as non-linear functions of the stresses. Problems involving ‘homogeneous’ stress distributions and one-dimensional stress distributions are considered. For such problems, deformations are calculated corresponding to the assumed stress distributions. In some of the situations, it is found that non-unique solutions are possible. Interestingly, non-monotonic response of the deformation is possible corresponding to monotonic increase in loading. For a subclass of models, the strain-stress relationship leads to a pronounced strain-gradient concentration domain in the body in that the strains increase tremendously with the stress for small range of the stress (or put differently, the gradient of the strain with respect to the stress is very large in a narrow domain), and they remain practically constant as the stress increases further. Most importantly, we find that for a large subclass of the models considered, the strain remains bounded as the stresses become arbitrarily large, an impossibility in the case of the classical linearized elastic model. This last result has relevance to important problems in which singularities in stresses develop, such as fracture mechanics and other problems involving the application of concentrated loads.     

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.     

Acoustomechanical constitutive theory for soft materials

Acoustic wave propagation from surrounding medium into a soft material can generate acoustic radiation stress due to acoustic momentum transfer inside the medium and material, as well as at the interface between the two. To analyze acoustic-induced deformation of soft materials, we establish an acoustomechanical constitutive theory by com-bining the acoustic radiation stress theory and the nonlinear elasticity theory for soft materials. The acoustic radiation stress tensor is formulated by time averaging the momen-tum equation of particle motion, which is then introduced into the nonlinear elasticity constitutive relation to construct the acoustomechanical constitutive theory for soft materials. Considering a specified case of soft material sheet subjected to two counter-propagating acoustic waves, we demonstrate the nonlinear large deformation of the soft material and ana-lyze the interaction between acoustic waves and material deformation under the conditions of total reflection, acoustic transparency, and acoustic mismatch.     

A relation between shear flow material functions for a single-integral constitutive equation

This contribution suggests some relations between normal-stress differences and shear stress, both in the steady state and in the transient regimes. It shows that the general form of the BKZ-type equation can be used for the discussion of the relations between several material functions. In addition, a geometric interpretation is offered and some examples are discussed.     

Evaluation of a simple nonlinear elastic constitutive form

A nonlinear, two constant stress-deformation form is deduced for elastic materials. At very large stretch ratios of greater than about 3 or 4, the model exhibits the strain stiffening behavior common to many elastomers. The constitutive form is very simple since the two material constants enter it as multiplying constants times certain nonlinear deformation terms. The model is evaluated with respect to data upon natural rubber under both uniaxial and bi-axial stress conditions. The model is also used to evaluate data obtained from a nonlinear membrane inflation experiment. The latter experimental capability and corresponding data are new.Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.     

Exact integration of constitutive equations of kinematic hardening materials and its extended applications

In this paper, an exact formula for the integration of the constitutive equations of kinematic hardening material is presented. Its algorithms are simple and clear. For isotropic hardening or mixed hardening material, the formula is still an exact solution for the case of radial loading, and it is an approximate solution with reasonable accuracy for the case of non-radial loading. The computation results show that the procedure proposed in this paper improves both accuracy and efficiency of numerical integration schemes adopted widely in elastic-plastic finite element analysis.     

The constitutive equation resemblance of elastic — Plastic multiphase solid

Based on composite materials, the equivalent elastic-plastic constitutive equations of multiphase solid are researched. According to the suggested definition of constitutive equivalence it is demonstrated that the multiphase solid, composed of several kinds of homogeneous elastic-plastic media that conform to the generalized normality rule, has the same type of constitutive equations as its constituents have that also conform to the generalized normality rule.     

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.     

The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach

After recalling the constitutive equations of finite strain poroelasticity formulated at the macroscopic level, we adopt a microscopic point of view which consists of describing the fluid-saturated porous medium at a space scale on which the fluid and solid phases are geometrically distinct. The constitutive equations of poroelasticity are recovered from the analysis conducted on a representative elementary volume of porous material open to fluid mass exchange. The procedure relies upon the solution of a boundary value problem defined on the solid domain of the representative volume undergoing large elastic strains. The macroscopic potential, computed as the integral of the free energy density over the solid domain, is shown to depend on the macroscopic deformation gradient and the porous space volume as relevant variables. The corresponding stress-type variables obtained through the differentiation of this potential turn out to be the macroscopic Boussinesq stress tensor and the pore pressure. Furthermore, such a procedure makes it possible to establish the necessary and sufficient conditions to ensure the validity of an ‘effective stress’ formulation of the constitutive equations of finite strain poroelasticity. Such conditions are notably satisfied in the important case of an incompressible solid matrix.     

A mathematical theory of materials with elastic range and the definition of back stress tensor

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CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY (Ⅱ)-LINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

The linear constitutive equations and field equations of unsaturated soils were obtained through linearizing the nonlinear equations given in the first part of this work. The linear equations were expressed in the forms similar to Biot’s equations for saturated porous media. The Darcy’s laws of unsaturated soil were proved. It is shown that Biot’s equations of saturated porous media are the simplification of the theory. All these illustrate that constructing constitutive relation of unsaturated soil on the base of mixture theory is rational.     

Some tests of constitutive equations for entangled polymers based on planar-extension flow histories in a periodically constricted channel

Stress and velocity were determined locally by birefringence measurements and laser Doppler velocimetry for a mildly entangled polystyrene solution flowing at steady state in a rectangular channel with sinusoidally varying wall spacing. Having measured both the velocity and stress fields, we were able to test constitutive equations locally, i.e., without solving the equations of motion for the entire flow. Four were examined for the periodic planar extensions on the channel centerplane: the Newtonian model, the Lodge network model, the Doi-Edwards tube model, and the Wagner-Schaeffer modification of Doi-Edwards. High enough Weissenberg and Deborah numbers were reached to produce sizable departures from the Newtonian predictions. The Doi-Edwards model underpredicted the stress, as did Wagner-Schaeffer, although to a lesser extent. Predictions of the Lodge model were best of all, a surprising result in view of its inadequacy for simple shear deformations. The predictions of the Lodge model, without parameter adjustment, agreed remarkably well with the planar extension data over the accessible range for our apparatus: Deborah numbers up to 2.0, extensional Weissenberg numbers up to 6.5, and a maximum extension ratio of about 2.3.     

动脉壁静态非线性力学性质的实验和理论研究

在动脉血管壁力学实验及已有的拟弹性理论研究基础上,提出了一个理论模型来分析具有残余应力动脉壁的非线性力学性质． 在动脉壁被模拟为均质、正交各向异性、不可压缩和具有初应力材料的前提下,建立了一个表达有残余应力动脉壁静态三维非线性拟弹性性质的ｅ指数型本构方程． 动脉壁本构方程中的十个拟弹性参数是用我们的动脉实验数据及所发展的多曲线联合逼近算法数据拟合来确定．     

Creep damage constitutive equations for multi-axial states of stress for 0.5Cr0.5Mo0.25V ferritic steel at 590°C

Multi-axial creep damage constitutive equations are developed for 0.5Cr0.5Mo0.25V ferritic steel at 590°C. It is aimed to overcome the deficiency of inconsistency in predicted and observed creep strains existed in previous formulations [Q. Xu, in: European Conference on Advances in Mechanical Behaviour, Plasticity and Damage, 7–9 November 2000, Tours, France; Creep Damage Mechanics Research Report, The School of Engineering, The University of Huddersfield, 2000; I.J. Perrin, D.R. Hayhurst, J. Strain Anal. 31 (4) (1996) 299]. In this paper, the lifetime and strain at failure are qualitatively validated in a wider range of states of stress by which the capability of producing consistent results is demonstrated.     

Dynamic stress intensity factors of two 3D rectangular cracks in a transversely isotropic elastic material under a time-harmonic elastic P-wave

The dynamic stress intensity factors (DSIFs) of two 3D rectangular cracks in a transversely isotropic elastic material under an incident harmonic stress wave are investigated by generalized Almansi’s theorem and the Schmidt method in the present paper. Using 2D Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack, the characteristics of the harmonic wave and the distance between two rectangular cracks on the DSIFs of the transversely isotropic elastic material.     

Dilatational plastic constitutive equations and their application to ductile fracture analysis of axisymmetric specimens

In many ductile materials voids nucleate and grow under large strain and triaxial stress, which yield volumetric plastic expansion. A constitutive equation is presented, which accounts for this plastic dilatancy. The plastic moduli involved in this equation can be calibrated by using necking tests of axisymmetric bars, void model analysis and computer simulation. To verify the rationality of such a constitutive equation and adjust the values of plastic moduli, the constitutive equation with its moduli to be determined is applied to analyse the ductile fracture behaviour of axisymmetric bars.     

Limiting velocity of crack propagation in elastic materials

A crack is represented as a continuous set of linear dislocations. Simple analytical expressions are obtained for the potential and kinetic energies of the environment of moving cracks and the attached mass of cracks for an arbitrary form of the stress applied to the crack P(x). It is shown that the indicated analytical expressions are bilinear integrals of the functions P(x) and ∂P(x)/∂x. These integrals are calculated in explicit form for a constant stress over the entire crack length and the stress due to the action of molecular adhesion forces in a narrow region near the crack openings. It is shown that the calculation results does not depend on the form of molecular adhesion forces. The proposed approach to describing cracks and calculations based on it has made it possible for the first time to obtain a complete analytical expression for the limiting crack propagation velocity in elastic materials as a function of the main mechanical characteristics of such materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 158–166, July–August, 2009.     

多胞材料的动态应力应变状态及其一致近似关系

多胞材料在高速冲击下呈现出逐层压溃的变形模式,塑性冲击波模型可以用来表征这种集中变形带的传播行为。本文中采用截面应力计算方法得到了随机蜂窝在恒速冲击下的一维应力分布,进而对冲击波的传播规律进行了分析。比较了高速冲击下由不同方法得到的冲击波速度与冲击速度的关系,结果表明R-PP-L（率无关,刚性-理想塑性-锁定）模型高估了冲击波速度,但R-PH（率无关,刚性-塑性硬化）模型以及一维冲击波理论得到的冲击波速度与有限元结果比较接近。冲击波速度与冲击速度在高速情形下趋于线性关系,但随着冲击速度的减小,冲击波速度不断减少并趋于常数。根据这一特征和塑性冲击波模型,发展了可以表征冲击波速度与冲击速度的关系、动态应力应变关系的一致近似模型。

     

Microdamage of a nonlinear elastic material in combined stress state

The structural theory of short-term damage is used to study the coupled processes of deformation and microdamage of a physically nonlinear material in a combined stress state. The basis for the analysis is the stochastic elasticity equations for a physically nonlinear porous medium. Damage in a microvolume of the material is assumed to occur in accordance with the Huber-Mises failure criterion. The balance equation for damaged microvolumes is derived and added to the macrostress-macrostrain relations to produce a closed-form system of equations. It describes the coupled processes of nonlinear deformation and microdamage of the porous material. Algorithms are developed for calculating the dependence of microdamage on macrostresses and macrostrains and plotting stress-strain curves for a homogeneous material under either biaxial normal loading or combined normal and tangential loading. The plots are analyzed depending on the type of stress state __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 30–39, November 2006.