Showing papers on "Constitutive equation published in 1974"

The Scalar Equations of Infinitesimal Elastic-Gravitational Motion for a Rotating, Slightly Elliptical Earth

Abstract: Summary We derive the infinite set of coupled ordinary differential equations over radius that govern the infinitesimal free elastic-gravitational oscillations of a rotating, slightly elliptical Earth with an isotropic perfectly elastic constitutive relation and a hydrostatic prestress field. We show how the symmetries of such a body restrict the most general form of the displacement eigenfunctions. We discuss situations in which finite sets of coupled equations may yield good approximate eigenfunctions and describe the sense in which the solution to such a set approximates the solution of the infinite set. Finally we describe in some detail the equations governing a particular class of finite expansions and show how these may be put in a form convenient for numerical solution. The method used to convert tensor equations to scalar equations is an extension of the generalized surface spherical harmonic expansion of Phinney & Burridge and should be useful in other non spherically-symmetric applications.

Thermo-mechanics of rubberlike materials

Abstract: A theoretical study is made in this paper of the formulation of constitutive equations describing the thermo-mechanical response of solid polymers in the temperature range in which they exhibit rubberlike behaviour. An expression for the Helmholtz free energy of such a material is first constructed on the basis of two assumptions which are motivated by physical arguments concerning the relation of the molecular structure of a cross-linked polymer to its bulk response. The constitutive equations for the stress and the entropy generated by the proposed form of the free energy function are then employed m a detailed investigation of the extension of a cylindrical specimen under prescribed conditions of temperature and pressure, a situation which serves as a model of the experimental arrangement most frequently used in laboratory studies of the mechanical and thermo-mechanical properties of rubberlike solids. Qualitative consistency of the theoretical predictions with observed behaviour is shown to be assured, over the full range of circumstances for which measurements have been reported, by two simple inequalities affecting one of the three response functions appearing in the stress-deformation-temperature relations. The function concerned is closely related to the strain energy governing isothermal deformations of the material at a selected reference temperature and it is associated, by the physical considerations referred to above, with the contribution to the stress of the polymer network. This conclusion shows that a rational macroscopic theory of rubberlike thermoelasticity can be developed in rather general terms. In particular, the requirement that the constitutive equations shall reproduce the anomalous thermo-mechanical effects which are characteristic of solid elastomers imposes restrictions on the response functions no more severe than those which ensure that the purely mechanical behaviour of the material is physically realistic. In the remainder of the paper the capability of the basic theory for furnishing results quantitatively agreeing with experiment is examined. Empirical forms of the three response functions are presented which accurately represent measurements made in tests involving compression at different fixed temperatures and stretching at the reference temperature. Numerical calculations relating to the analysis of the extension of a cylinder, given earlier, are then described and compared with the results of experiments in which thermoelastic inversion phenomena occur. Satisfactory agreement is secured, but it is noted that insufficiency of material data for the rubbers used in the tests precludes an exact correlation of theory and experiment. The final section of the paper is concerned with isothermal deformations of rubberlike materials which are mechanically incompressible (in the sense that volume changes can be brought about by thermal expansion but not by loading at fixed temperature). This property closely approximates the typical behaviour of natural and synthetic rubbers, but its incorporation into a general treatment of rubberlike thermoelasticity presents difficulties and places an undesirable limitation on the scope of the theory. An analogue is shown to exist between the constitutive equations for deformations at the reference temperature and their counterparts in respect of isothermal deformations at other temperatures, and with its aid the problem of the combined extension, torsion and uniform heating of a circular cylinder is solved. Again, a numerical evaluation of the solution is compared with available experimental data.

On the theory of rods I. Derivations from the three-dimensional equations

Abstract: Starting with the three-dimensional theory of classical continuum mechanics, some aspects of both the nonlinear and the linear theories of elastic rods are discussed. Detailed attention is given to the derivation of constitutive equations for the linear isothermal theory of elastic rods of an isotropic material and of variable cross-section, deduced by an approximation procedure from the three-dimensional equations. Explicit linear constitutive relations are obtained for straight isotropic circular rods of non-uniform cross-section; the explicit calculation is carried out (in terms of an approximate specific Gibbs free energy function) in four distinct parts, since the complete system of equations involved separate into those appropriate for extensional, torsional and two flexural modes of deformation. A system of displacement differential equations is derived for flexure of a beam of variable circular cross-section; they reduce to those of the Timoshenko beam theory when the radius of the cross-section is constant.

On the overall moduli of non-linear elastic composite materials

Abstract: F rom the work of R. Hill on constitutive macro-variables it is known that for an inhomogeneous elastic solid under finite strain an overall elastic constitutive law may be defined. In particular, the volume average of the strain energy of the solid is a function only of the volume-averaged deformation gradient. In view of the importance of this result it is re-derived in this paper as a prelude to a discussion of composite materials. A composite material consisting of a dilute suspension of initially spherical inclusions embedded in a matrix of different material is considered. For second-order isotropic elasticity theory an expression for the overall bulk modulus of the composite material is obtained in terms of the moduli of the constituents. When the inclusions are vacuous a known result for the bulk modulus of porous materials is recovered. In certain situations the strengthening/ weakening effects of the inclusions are less pronounced in the second-order theory than in the linear theory.

Lagrangian analysis for multiple stress or velocity gages in attenuating waves

Abstract: Numerical techniques are presented for extending and implementing the Lagrangian analysis originated by Fowles, Williams, and Cowperthwaite. The analysis derives internal energy, specific volume, and stress and particle velocity histories from either stress or particle velocity records from a series of gages embedded in material undergoing uniaxial strain flow. The material may have an arbitrarily complex constitutive relation. The accuracy of the analysis for attenuating waves is studied by handling analytically derived stress and velocity histories. Waves with attenuation of 40–75% can be treated, depending upon the accuracy desired. The analysis is applied to waves measured in graphite, sandstone, asbestos phenolic composite, and porous alumina.

Constitutive relations that imply a generalized Mohr-Coulomb criterion

Abstract: A class of constitutive equations are shown to imply a generalized Mohr-Coulomb criterion for impending flow. The stress is assumed to have an isotropic dependence upon a vector argument in equilibrium and this assumption leads to a constrained stress state which is of the type characterized by the Mohr-Coulomb criterion.

Nonlocal Elasticity and Waves

Abstract: The theory of nonlocal elasticity is developed. Balance laws, jump conditions and the second law of thermodynamics are given. By means of axioms of nonlocality, objectivity and the entropy inequality, the constitutive equations are derived for nonlocal elastic solids. The field equations are obtained and applied to study the propagation of body and surface waves.

Hydrodynamic response of inhomogeneous metallic systems

Abstract: The response of an inhomogeneous metallic system is treated within the hydrodynamical theory in which the constitutive equation is obtained via a quasi-static extension of the theory of inhomogeneous electron gas Formal expression for the frequency-dependent dielectric function of a metallic surface is derived and applications to other systems such as electron-hole drops in semiconductors are discussed

The First Pressure Derivative of the Shear Modulus of Porous Materials

Abstract: Summary A general theory for the calculation of the second order effective elastic moduli of porous materials in which the porosity is in the form of isolated cavities is presented. The particular case of spherical cavities distributed randomly within an isotropic matrix in such a manner that the material is macroscopically isotropic is then considered in detail and an expression for the first pressure derivative of the effective shear modulus of such a material is obtained correct to first order in the porosity. 1. Introduction In a paper by Walton (1973), hereafter referred to as Paper I, the first pressure derivative of the effective bulk modulus of a porous material was calculated. The particular porous medium considered was that of a homogeneous isotropic matrix containing a dilute distribution of spherical cavities, not necessarily of the same size but such that the total porosity c (that is, the ratio of cavity volume to total volume) is so small that terms of order c2 may be neglected in comparison with unity. Furthermore, the distribution was assumed to be random and such that the material is macroscopically homogeneous and isotropic. The aim of the present paper is to extend the method used in Paper I to the calculation of the first pressure derivative of the effective shear modulus of such a material. 2. Second order effective moduli The method is based on considerations of the overall constitutive law and, in the spirit of Hill (1963), the problem of the calculation of the effective elastic moduli of porous materials may be formulated as follows. The model to be considered is that of a large volume V of some porous material subjected to a uniform strain in its outer boundary. The matrix material is assumed both perfectly elastic and homogeneous, although not necessarily isotropic. The porosity, on the other hand, is assumed to be in the form of isolated cavities distributed throughout the matrix in such a manner that the material is macroscopically homogeneous, although not necessarily isotropic. Finally, there is no restriction at this stage on the size of the porosity c. With the 9-vectors S and D denoting the nominal stress and displacement gradient respectively and with superscripts (m) and (c) referring respectively to the solid matrix and the cavities, the constitutive law for the matrix material may be written, correct to second order in D(m),

A dynamic theory for magnetizable elastic solids with thermal and electrical conduction

Abstract: A general dynamical theory of magnetizable, electrically and thermally conducting media is developed for soft ferromagnetic or paramagnetic materials in external electromagnetic fields. The general equations are linearized by assuming infinitesimal strains, linear constitutive equations and that all field variables may be divided into two parts: a "rigid body state" and a "perturbation state". The former is the same as the one in rigid body electrodynamics, and the latter which accounts for electromagnetic interaction with the deformable continuum is coupled with stress and strain through linearized field equations. The theory is developed for general anisotropy but specialized for materials with uniaxial, or higher, symmetry.

Polymeric Equations of State

Abstract: I. INTRODUCTION Equations of state or constitutive equations are a necessary addition to the science of continuum physics. Certain physical laws such as conservation of mass, momentum, and energy apply to all continuous bodies, irrespective of their constitution. In general, these laws are insufficient for the proper specification of the material variables. Hence, constitutive assumptions or equations of state are necessary to make the problem determinate.

On physical measures of rheological responses of some materials in wide ranges of temperature and spectral frequency

Abstract: The major problem in applied sciences, with regard to the response of materials to loads, is the degree of correlation between actual material response and the predictions of corresponding mathematical model or models presented in a form of constitutive equations.

Cyclic creep: an interpretive literature survey

Abstract: An analysis is made of the inelastic deformation of structural materials under cyclic creep conditions such as loading, unloading, reloading, cyclic creep, cyclic relaxation, and low-cycle fatigue with and without hold times under uniaxial and multiaxial states of stress. Constitutive equations such as linear and nonlinear viscoelasticity, plasticity, creep theory and equation of state are examined. It is concluded that an accurate cycle-by-cycle analysis for ratchetting and creep-fatigue interaction is not now possible. For cyclic strain hardening materials, such as stainless steel but not Cr--Mo--V steels, a conservative prediction can be made. 26 figures, 132 references. (DLC)

Finite-element analysis of slow incompressible viscous fluid motion

Abstract: The governing field equations and the constitutive relation are specialized to the boundary value formulation of incompressible viscous fluid motion excluding thermal effects. When choosing the velocities and the hydrostatic pressures as variables, the established non-linear matrix equation in terms of finite-element properties becomes valid for both two- and three-dimensional application. It is shown that a restricted class of problems readily fits within the scope of existing finite-element software designed for conventional structural mechanics analysis provided an effective Lagrange multiplier technique can be incorporated.

Thermodynamics of dissipative materials with entropic elasticity

Abstract: Polymer solutions and melts can both dissipate mechanical energy in flow, as well as accumulate elastic energy. If the assumption is made that elastic energy can be accumulated only through a decrease of conformational entropy, the general thermodynamic theory for non-linear viscoelastic materials simplifies considerably. In particular, though no generality is lost as far as the constitutive equation for stress is concerned, the energy equation, which allows in principle a calculation of temperature distributions arising from frictional heating and heat removal, reduces to the usual form which is valid for viscous materials.

Nonlinear Thermo-Elastic-Plastic and Creep Analysis by the Finite-Element Method

Abstract: Consistent with a Lagrangian displacement formulation, an incremental constitutive relation for uncoupled thermoelastic-plastic and creep deformations is presented. The nonisothermal von Mises yield function and its associated flow rule are utilized, together with both isotropic and kinematic hardening rules. Steady-state creep deformations are considered using Norton-Odqvist's power law. This development is particularly applicable to the nonlinear finite element analysis of three-dimensional structures with timeand temperature-dependent material properties. Using a nonlinear general-purpose computer program which has been developed on the basis of this formulation, a number of numerical examples are solved and the results compared with the closed-form solutions.

On the higher order theories of piezoelectric crystal surfaces

Abstract: This paper presents a higher order theory of crystal finite surfaces within the frame of the three‐dimensional theory of linear piezoelectricity. First, by modifying Hamilton's principle, a variational theorem is deduced. Then, this theorem together with a method of series expansion is employed to establish the theory in a systematic and consistent manner. The theory consists of a hierarchy of two‐dimensional equations of motion, charge equations of electrostatics, initial and boundary conditions, strain‐displacement and electric field‐electric potential relations, and macroscopic constitutive equations. It governs the extensional and flexural as well as torsional motions of piezoelectric crystal shells and plates of uniform thickness. Further, theorems of uniqueness in this theory are presented.

Constitutive equations for heat conduction in general relativity

Abstract: A heat flux constitutive equation is derived in three approximations from a general functional constitutive equation which describes heat conduction in so-called 'simple' thermodeformable media in general relativity. The three approximations correspond to materials having a so-called 'fading memory', and infinitely short memory, and materials of the 'rate-type', respectively. The third approximation may contain the other two as particular cases. Within the frame of the approximations made for isotropic materials, it is shown that interactions between the different transport phenomena, eg, heat flow and viscosity, cannot be accounted for.

On Finite Elastic-Plastic Deformation of Metals

Abstract: : The paper is concerned with a special class of response functions for some of the constitutive equations in the nonlinear isothermal theory of elastic-plastic materials. Detailed attention is given to the development of special forms for the free energy and the stress response, motivated mainly by the mechanical behavior of ductile metals in the plastic range and in the presence of finite strains. After obtaining a properly invariant representation for the free energy response (and hence also for the stress) as a function of certain (easily interpretable) measures of deformation, the terms of the invariants of kinematic measures. Some special cases are elaborated upon and, by way of illustration, the influence of plastic deformation on the material properties of the stress response in a simple tension test is discussed. (Author)

Non-linear viscoelastic behaviour of polymers: An implicit equation approach

Abstract: I t is proposed that a comprehensive description of the non-linear viscoelastic behaviour of polymers can be most readily obtained using an implicit form of the constitutive relation. Extensive creep, constant strain-rate and stress relaxation measurements on oriented polyethylene terephthalate and creep measurements on isotropic polypropylene and polymethylmethacrylate have tested the usefulness of the proposed implicit constitutive equation. First, this equation has been shown to describe accurately the stress dependence of the relaxed and unrelaxed creep response. Secondly, it enables the strain dependence of the stress relaxation response to be accurately predicted from the observed creep behaviour. Finally, making simplifying assumptions regarding the form of the two response functions involved, good first-order predictions were obtained for the creep curves at any stress, the stress relaxation behaviour at any strain and the constant strain-rate behaviour at any strain-rate using only six parameters.

Computational Representation of Constitutive Relations for Porous Material

Abstract: : Porous materials are used as a protection against x-radiation because of their ability to minimize the stress generated by the radiation and to attenuate that stress as it propagates. For accurate design of this protection, wave propagation calculations are made to simulate the radiation deposition, stress generation, propagation, and spallation caused by stress waves. For such a calculation it is necessary to have a constitutive relation (stress-strain- energy relation, or equation of state) that describes the material's response to heating and to compressive and tensile loading. The objective of this report is to document a set of constitutive relations that provide for: Elastic and plastic compaction loading with rate dependence; Heating or cooling that can occur simultaneously with loading; Unloading and rate-dependent fracture; and, Melting and vaporization, with explicit treatment of solid, liquid, vapor, and mixed phases. Accompanying these relations is a user's manual that includes a derivation of the equations for the model and procedures for using it in Lagrangian wave propagation computer programs.