Showing papers on "Constitutive equation published in 1969"

Introduction to the mechanics of a continuous medium

Abstract: 1. Vectors and Tensors. 2. Strain and Deformation. 3. General Principles. 4. Constitutive Equations. 5. Fluid Mechanics. 6. Linearized Theory of Elasticity. Appendix I: Tensors. Appendix II: Orthogonal Curvilinear.

On the characterization of nonlinear viscoelastic materials

Abstract: Starting with specific constitutive equations, methods of evaluating material properties from experimental data are outlined and then illustrated for some polymeric materials; these equations have been derived from thermodynamic principles, and are very similar to the Boltzmann superposition integral form of linear theory. The experimental basis for two equations under uniaxial loading and the influence of environmental factors on the properties are first examined. It is then shown that creep and recovery data can be conveiently used to evaluate properties in one equation, while two-step relaxation data serve the same purpose for the second equation. Methods of reducing data to accomplish this characterization and to determine the accuracy of the theory are illustrated using existing data on nitrocellulose film, fiber-reinforced phenolic resin, and polyisobutylene. Finally, a set of three-dimensional constitutive equations is proposed which is consistent with nonlinear behavior of some metals and plastics, and which enables all properties to be evaluated from uniaxial creep and recovery data.

Constitutive Equation for the Dynamic Compaction of Ductile Porous Materials

Abstract: A simple phenomenological constitutive equation for porous materials is proposed, which allows a detailed description of irreversible compaction behavior at low pressures and reduces to the correct Hugoniot description at high pressures. The theory is compared to some existing data on Hugoniots of porous aluminum and iron, and fairly simple functional forms of the compaction relation are found to be adequate to fit the data. The constitutive relation is suitable for use with finite difference methods of solution of the one‐ and two‐dimensional equations of motion governing stress wave propagation. Examples of such solutions in one dimension are given to illustrate some of the features of the theory.

Single-phase flow through porous media

Abstract: The local volume average of the equation of motion is taken for an incompressible fluid flowing through a porous structure under conditions such that inertial effects may be neglected. The result has two terms beyond a pressure gradient: g, the force per unit volume which a flowing fluid exerts on a porous structure, and the divergence of the local volume-averaged extra stress tensor (viscous portion of the stress tensor). Constitutive equations for g are examined with the aid of the principle of material indifference. When g is assumed to be a function of the velocity of the fluid relative to the solid as well as various scalars, the usual results for a nonoriented (isotropic) porous structure are obtained. When g is assumed to be a function of the local porosity gradient as well, we derive a new expression for g applicable to oriented (anisotropic) porous structures. For a Newtonian fluid with a constant viscosity, the divergence of the local volume-averaged extra stress tensor is proportional to the Laplacian of the averaged velocity vector. Boundary conditions for the averaged velocity vector are discussed. Three problems are solved for the flow of an incompressible Newtonian fluid in a nonoriented permeable medium. These solutions, as well as an order-of-magnitude analysis, suggest that we may often neglect both the Laplacian of average velocity and the boundary conditions for the tangential components of averaged velocity at an impermeable wall. Two specific constitutive equations for g are proposed for the flow of incompressible Noll simple fluids in nonoriented porous structures. Flow through a porous medium bounded by an impermeable cylindrical surface is solved for these two constitutive equations, and the results are compared with previously available experimental data.

Thermodynamics of Elastic‐Plastic Materials as a Theory with Internal State Variables

Abstract: We develop an analytical framework for the theory of plasticity in which the constitutive equations are single valued and which at the same time represents the proper physical concepts. A Coleman‐Gurtintype thermodynamics which utilizes internal state variables is used for the study of an elastic‐plastic substance. Quantities which are related to the dislocation motion and the dislocation arrangement in the material, play the role of the internal state variables in this thermodynamics. Rate‐independent plasticity is studied as a limiting case of the present theory. We illustrate the theory with a special case: a one‐dimensional, homogeneous, cyclic deformation of a rate‐independent elastic‐plastic body which exhibits a Bauschinger effect.

A mixture of viscous elastic materials with different constituent temperatures

Abstract: : Constitutive equations are discussed for a mixture of any number of materials with elastic and viscous properties in which the constituents may have different temperatures.

A micromorphic approach to dislocation theory and its relation to several existing theories.

Abstract: : Two separate continuum dislocation theories are presented; one dealing with static incompatible, micropolar dislocations and disclinations, as encountered in initial stress problems, and the other with a dynamical theory of micromorphic solids containing continuous distributions of dislocations Relationships between several continuum dislocation theories and micromorphic mechanics are established by providing extensions and new interpretations of the micromorphic theory First both micromorphic and micropolar theories of elastic solids are summarized, and then the theories of Kroner, Fox, and Berdichevskii and Sedov are discussed in some detail within this framework In the last section, by use of micromorphic kinematics, dislocation density, strain, and microstrain tensors are introduced and constitutive equations are constructed Together with the balance laws this constitutes a complete dynamical theory The theory is intended for predictions of motions and micromotions of a solid containing dislocations undergoing elastic deformations From the micromotion, the dislocation density and first stress moments can be calculated (Author)

Some constitutive equations applicable to problems of large dynamic plastic deformation

Abstract: Some fairly general thermodynamic constitutive equations are derived for an elastic-plastic body, which are then reduced to fairly simple forms by making plausible physical assumptions. The reduced equations retain sufficient flexibility to facilitate fitting to experimental results but are also sufficiently simple to offer the prospect of solving problems. An illustrative example is given to support this latter statement.

Convective Instability by Active Stress

Abstract: Composition-dependent stress fields in continuous and mechanically isolated material can, it is shown, initiate and maintain conversion of chemical to kinetic energy. Though the process is analogous to natural convection, neither the body force of the well-known buoyancy mechanism nor the singular inhomogeneity and anisotropy of the interfacial tension mechanism is required. In the cases examined, the material is represented by the constitutive relation for incompressible Newtonian fluid augmented by an active stress which must be anisotropic or nonlinear in concentration gradients (or other, equivalent gradients) in accordance with the oft-misquoted Curie principle. The concentration gradients are supposed to be sustained by steady chemical reaction (or an equivalent transformation process) throughout the material and by exchange with surroundings. Conventional linear analysis of asymptotic stability is used to identify types of stress/concentration-gradient coupling that can render a quiescent state of reaction and diffusion unstable when concentration gradients exceed critical values. It is found that both deviatoric (pure shear) and antisymmetric active stress can support two-dimensional convective instability in a cylinder of material in which the quiescent state is circularly symmetric. Certain cases of stationary instability are solved exactly. Others involving both stationary and oscillatory instability are treated by a new version of the Galerkin method. The results establish the possibility of generating fluid motion by mechanochemical means in continuous material having appropriate subcontinuum structure. Whatever their relevance to protoplasmic movement in biological systems they do contain challenges for experimental and theoretical fluid mechanics and related areas of rheology and chemistry.

On mathematical forms for the material functions in nonlinear viscoelasticity

Abstract: Three special forms of material functions are introduced into the Green-Rivlin constitutive equations of nonlinear viscoelasticity. Two of these reduce the third-order equations to single integral forms, rationally and experimentally derived by others. Published creep and stressrelaxation data are analysed using both the proposed and previously published constitutive equations, particular attention being given to multi-step data. Specific results show the limits of applicability of the third-order constitutive relations for the materials considered.

Transient Response of an Elastomer to Large Shearing and Stretching Deformations

Abstract: Data are presented, using undiluted polyisobutylene, for the transient response to the sudden imposition of simple shearing and constant speed elongation. A constitutive equation due to Bogue describes the basic features of the observed responses.

Network rupture and the flow of concentrated polymer solutions

Abstract: Network rupture hypothesis and molecular structure considerations used in obtaining constitutive equations for flow of polymer melts and solutions

On thermal effects in viscoplasticity

Abstract: The basic object of the present paper is the description of the behaviour of an elastic/viscoplastic material during thermodynamic process within the framework of thermodynamics with internal state variables1.

Constitutive Relation for Rate-Dependent Plastic Flow in Polycrystalline Metals

Abstract: The rate‐dependent constitutive relation developed by Taylor, which considers dislocation motion only in glide directions and on glide planes for which the shear stress is maximum, is extended to include dislocation motion in all glide directions on all glide planes. The theory requires that grain orientation be random, that expanding dislocation loops be rectangular, and that the velocity of edge dislocations be much greater than the velocity of screw dislocations. The velocity of screw dislocations is assumed to be given by vs=vm exp (‐B/τ), where τ is the applied shear stress, B is a constant, and vm is the elastic shear wave velocity. On the basis of this theory, elastic wave attenuation in Armco iron is calculated and compared with the experimental data of Taylor and Rice. It is found that the mobile dislocation density necessary for the theoretical calculation to agree with experimental data is five times greater than that obtained by Taylor on the basis of the simpler theory. Likewise, for a given ...

On the formulation of constitutive equations for living soft tissues

Abstract: : Soft living tissues deform freely under negligible stresses until a certain strain level is reached at which their stiffness increases sharply. Constitutive equations are developed that describe this kind of mechanical behavior and include Hooke's law as a limiting case. It is shown that, similar to Hooke's law, these constitutive equations assure uniqueness of solution for a broad class of boundary value problems. Possible extensions of the theory are briefly indicated. (Author)

Eine Feldtheorie der Versetzungen imCosserat-Kontinuum

Abstract: There is a mathematical analogy between the continuum theory of moving dislocations and electrodynamics. This remarkable analogy points out that the theory of dislocations is incomplete with regard to missing constitutive equations. The generalized electrodynamic theory byMie shows a way to complete the theory of moving dislocations, leading finally to the formulation of a Lagrangian density.

Viscoelastic Behavior of a Nonlinear Fiber-Reinforced Plastic.

Abstract: : The nonlinear mechanical properties of a unidirectional, glass fiber-epoxy composite at 164F are characterized by using creep and recovery tests together with a constitutive equation based on a thermodynamic theory. The results show that the composite is linearly elastic for the load range studied, but nonlinear viscoelastic behavior is observed for the other orientations. Guided by the nonlinear constitutive equation, the creep and recovery data are plotted on double-logarithmic paper, and the material properties are found by shifting the data to form a 'master curve' for each fiber orientation. Four principal creep compliances are estimated by using the master curves and tensor transformation relations. Finally, we use the nonlinear equation to predict strain response due to multiple-step loading and unloading. The Appendix contains two abstracts of papers which were completed and published during the period covered by this report. (Author)

Non-linear solid mechanics, past, present and future

Abstract: First of all, the subject of “Applied Mechanics”, or more particularly that of “Applied Solid Mechanics” must be somehow defined and its borders outlined. In what follows, when speaking about Solid Mechanics, I shall mean Solid Applied Mechanics. Such definition proves to be far from easy and has certainly resulted in different border lines from time to time.