Showing papers on "Constitutive equation published in 1975"

Rheological properties of nondilute suspensions of deformable particles

Abstract: The creeping flow equations have been solved to account for effects of shear on drop shape when the influence of neighboring drops is considered. A cell model is used to account for finite concentrations of the dispersed phase. From these results a constitutive equation is derived and implications for several flows are discussed. The shape and orientation of drops are computed and compared with existing data. Finally, dynamic viscosity and rigidity are obtained for nondilute suspensions of viscoelastic particles in an oscillatory shear flow.

Curing Stresses in Composite Laminates

Abstract: Analysis of curing stresses in resin matrix composite laminates involves a temperature range over which variation of elastic moduli is appreciable. A method based on total stress-strain-temperature relations is formulated and applied to determine the curing stresses in boron/epoxy composite laminates. This method is shown to be preferable to the incremental method because the former requires the thermal strains and the stress-strain relations only at the final temperature of interest. The use of incremental constitutive equations is also discussed, and it is shown that proper care must be exercised to insure inclusion of interaction terms that have no counterparts in linear theories.

Constitutive relations for concrete

Abstract: A constitutive relation and failure criterion for concrete material under general three-dimensional stress states have been formulated using the work-hardening theory of plasticity. The formulation considers the weak tensile strength, strain-hardening, and fracture behavior of concrete. The resulting stress-strain relationships are compared with existing biaxial loading experimental results and good agreement is generally observed. The relation is suitable for use in the finite element analysis of reinforced or prestressed concrete.

Constitutive equations in suspension mechanics. Part 1. General formulation

Abstract: Neither the phenomenological nor the structural approach to the determination of constitutive equations has yet shown itself to be capable of producing useful and predictive descriptions of the majority of technologically important complex fluids. In the present paper we explore the suggestion that significant progress can be made when these two complementary approaches to rheology are combined. For this initial study we restrict our attention to materials which can be modelled as a suspension of particles in a Newtonian fluid, thereby including most polymer solutions while excluding polymer melts. By applying phenomenological techniques to the basic formulation of suspension mechanics we are able to deduce a common simplified constitutive model for all suspension-like materials and to reveal its physical origin. The present analysis demonstrates that the constitutive model of Hand (1962), involving a single second-order tensor, is not sufficiently general for a rigorous description of the majority of suspension-like materials. Consideration of the constitutive forms for the limiting cases of near-equilibrium and strongly non-equilibrium microstructure suggests, however, that Hand's model may provide a reasonable approximation to the exact constitutive behaviour which is useful over the whole range of flow strengths.

Restrictions on constitutive equations of finitely deformed elastic-plastic materials

Abstract: : The paper is concerned with restrictions on the constitutive equations of elastic-plastic materials in the presence of large deformation. The development is confined to the purely mechanical theory in which the response of the medium is characterized by nonlinear constitutive equations of rate-type. Starting with a physically plausible assumption concerning non-negative work in a closed cycle of deformation, tha authors derive two local inequalities which hold in all motions and which place restrictions on the general constitutive equations of elastic-plastic materials. An implication of one of the inequalities is then used to express the constitutive equation for the plastic strain rate in terms of the yield (or loading) function, as well as the stress response and a measure of work hardening. (Author)

Electrodynamics for moving elastic solids and viscous fluids

Abstract: A general theory of electrodynamics of moving and deformable media is developed on the basis of the Chu formulation of the Maxwell equations and the general principles of continuum mechanics. From the two-dipole model for polarization and magnetization and the concept of the Lorentz force on electric and magnetic dipoles, the body force, the body couple, and the energy supply of electromagnetic origin are determined explicitly in terms of the dectromagnetic variables. These kinetic quantities appear in the balance equations for the mechanics of deformable bodies, which are coupled to the Maxwell equations. Boundary conditions and constitutive equations which satisfy the principle of objectivity are obtained for polarizable and magnetizable elastic solids and viscous fluids.

Nonlinear electroelastic equations cubic in the small field variables

Abstract: The nonlinear differential equations and boundary conditions containing terms up to cubic in the small field variables are obtained from general rotationally invariant nonlinear electroelastic equations derived previously. The electroelastic equations cubic in the small field variables are considerably more tractable than the general electroelastic equations and are applicable in the description of such phenomena as the dependence of wave velocities on wave amplitudes and resonant frequencies on vibration amplitudes in addition to a host of other nonlinear phenomena. The nonlinear constitutive equations for an isotropic purely elastic solid containing terms up to cubic in the small mechanical displacement gradients are presented. The nonlinear equations for the extensional motion of thin isotropic plates containing terms up to cubic in the small mechanical displacement gradients in the plane of the plate are obtained and the influence of the vertical inertia is included, in addition to the extensional sti...

On the theory of heat-conducting micropolar fluids with microtemperatures

Abstract: Microthermopolar fluids are introduced as a subclass of microthermofluids. In this subclass micromotion consists of rigid rotation only. The local balance laws and the linear constitutive equations are extracted from the theory of microthermofluids. Restriction regarding the material coefficients appearing in the constitutive equations are deduced by thermodynamical considerations. The problem of Poiseuille flow between two parallel plates is solved.

Stresses in Dilute Solutions of Bead‐Nonlinear‐Spring Macromolecules. II. Unsteady Flows and Approximate Constitutive Relations

Abstract: Some numerical calculations of the distribution function for bead positions in a linear‐locked‐spring dumbbell are presented together with stresses for transient (start‐of‐shearing) flows. Based upon the idea that this distribution function is sharply peaked in these and other results obtained previously, an approximate constitutive equation of novel form is deduced. The results of stress calculations from the approximate constitutive equations are then compared to those from the numerical work. Some further results for another spring law similar in form to the Langevin spring are also given for comparison. It is concluded that constitutive equations of the type presented are useful for certain classes of flows.

5.—Qualitative Aspects of the Spatial Deformation of Non-linearly Elastic Rods.§

Abstract: In this article we examine the qualitative behaviour of non-planar equilibrium states ofnon-linearly elastic rods subject to terminal loads. In our geometrically exact theory, a rod is endowed with enough geometric structure for it to undergo flexure, torsion, axial extension, and shear. The constitutive equations give appropriate stress resultants and couples as non-linear functions of appropriate strains. These constitutive relations must meet minimal conditions ensuring that they be physically reasonable. It turns out that the equilibrium states of such a rod are governed by a boundary value problem for a quasilinear fifteenth-order system of ordinary differential equations.

Shock−wave propagation in fluid−saturated porous media

Abstract: Mechanical constitutive relations for fluid−saturated porous rocks as previously formulated by Morland and Garg within the theory of interacting continua (TINC) have been generalized to include thermodynamic effects. The model is developed by defining effective stress tensors and effective densities in terms of the partial stress tensors, partial densities, and actual volume fractions occupied by each component. It is postulated that the constitutive law for each component as a single continuum relates effective pressure to effective deformation. Relative motion between the constituents is allowed through a simple Darcy−type law. The governing conservation equations together with the constitutive relations for a binary mixture are solved numerically using a so−called ’’leap−frog’’ finite−difference technique. The model is applied to study shock−wave propagation in a mixture of tuff and water.

On the interaction of rate and history dependence in structural metals

Abstract: In slow motions, such as occur in repeated creep, relaxation and low-cycle fatigue loadings, rate and history dependence interact and the conventional metallic material idealizations do not appear to be appropriate. For later use an operational definition of aging, rate and history dependence is given. Constitutive equations capable of reproducing history dependence must not only depend on the forcing function in the present time interval but must also have a not completely fading memory of prior loadings. It is demonstrated that certain forms of integral constitutive equations and defined history dependence. The history dependence of structural metals is examined and it is shown that prior deformation can change the response of structural metals to the same forcing function permanently but only in degree and not in kind; the variable hereditary property of structural metals is limited and is caused by deformation-induced changes of the microstructure. Variant and invariant response properties under prior loading are stated.

Wave propagation in a condensed medium with N transforming phases: Application to solid‐I–solid‐II–liquid bismuth

Abstract: Constitutive assumptions of local thermal and pressure equilibrium in a mixture of N transforming phases allow construction of a constitutive equation of a generalized Maxwellian form suitable for studying the kinetics of phase transformations induced by shock‐wave loading This equation relates the pressure rate to the strain rate and the phase change rate, the latter being expressed as the time derivative of a vector which has components equal to the mass fractions of the constituent phases A separate kinetic equation is required for evolution of this composition vector Coefficients appearing in the constitutive equation depend only on properties that the mixture displays with a frozen composition which in turn depend directly on properties of the pure‐phase components of the mixture The constitutive equation is applied to solid‐I–solid‐II–liquid bismuth (N=3) When wave propagation calculations on bismuth are compared to previous theory and experiments, a lower bound on the melting rate of 4 μsec−1

On the Nature of Normality of Plastic Strain Rate and Convexity of Yield Surfaces in Plasticity

Abstract: Within the scope of the purely mechanical theory of plasticity, in a previous paper we have derived two inequalities which place restrictions on the constitutive equation for the rate of plastic strain in a finitely deformed elastic-plastic material. Here we take up the matter further, elaborate on the nature of the previously derived restrictions and obtain some additional results pertaining to the normality of the plastic strain rate and convexity of yeild surfaces. Although in the main our discussions are carried out in the context of finite deformation, the nature of the restrictions for infinitesimal deformation is also examined. A special case of an elastic-plastic material in which the stress response is characterized by the stress rate and that of a rigid-plastic material are given detailed attention.

Stress dependence of elastic‐wave attenuation in LiF

Abstract: Shock‐wave propagation studies along the 〈100〉 direction in LiF single crystals are presented for three elastic impact stresses, namely, 39, 29, and 15 kbar. Present results demonstrate a stress threshold for a large increase in plastic strain rate and establish stress relaxation as a bulk property. Analysis of the data reveals a one‐to‐one correspondence between plastic strain rate and stress at the elastic‐shock front. This is in agreement with the constitutive relation generally used for stress‐relaxing solids under plate‐impact loading.

On the theory of boundary conditions

Abstract: The question which boundary conditions are appropriate for the given differential equations of a continuum theory, is treated by the method of nonequilibrium thermodynamics. In the first part the previous investigation1) which had been confined to ordinary hydro(aero-)dynamics is generalized by taking into account higher derivatives in the continuum constitutive laws. This kind of generalization becomes important when a rarefied gas with boundary is treated phenomenologically (wall influence, slip-flow regime). In the second part an even more general scheme is discussed. The underlying applications now are rarefied polyatomic gases within walls, perhaps in an external field. So, the starting point is a general set of transport-relaxation equations with more variables than the hydrodynamical ones. By considering the corresponding entropy production, especially its part due to the boundary, it is again possible to set up constitutive laws, i.e., matching or boundary conditions, at an interface or a surface.

A Spherical Model Calculation for Volumetric Response of Porous Rocks

Abstract: The problem of external pressurization and release is solved for a hollow sphere of incompressible material obeying Coulomb's law of failure. The resulting relations between the applied pressure and the porosity of the sphere are used in formulating constitutive relations for the volumetric response of porous rocks. The theoretical predictions are compared with experimental data for a sandstone and a tuff. (auth)

Linear theory of non-local viscoelasticity

Abstract: The theory of non-local viscoelastic media is studied. The constitutive equation of strain and strain rate dependent, stress and stress rate dependent as well as continuous memory dependent are derived and discussed. The thermodynamical restrictions are also studied.

The strain-rate behavior of ductile polymers

Abstract: The stress-strain-strain-rate behavior of polycarbonate is presented. It is demonstrated that the material does not exhibit a double-yield-point phenomenon, as others have reported, if true stress is plotted against actual strain. Also, the behavior of polycarbonate is presented under constant strain rate and relaxation conditions. The observed behavior of the material is discussed in relation to elements of recent theories of viscoelasto-plasticity due to Nagdi and Murch and Crochet. Simple mechanical models of the Bingham type are presented and are discussed with respect to the constitutive equation characterization of polycarbonte. The advantages and disadvantages of more general models are mentioned. Finally, the strain-rate behavior of PMMA (Polymethylmethacrylate) and a poylester given by others is presented and discussed, relative to polycarbonate and the characterization procedures used.

Interaction of electromagnetic and elastic fields in solids

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