Showing papers on "Constitutive equation published in 1972"

Rheological Equations from Molecular Network Theories

Abstract: Lodge's molecular network theories are quite successful in describing the linear viscoelastic behavior of polymer solutions and melts, but cannot account for the rate‐of‐strain dependence of various material functions By allowing the junction‐creation rate and the probability of loss of junctions to depend on the second invariant of the rate‐of‐strain tensor, more realistic constitutive equations were obtained Two rheological models are proposed by assuming two different mechanisms for the effect of the rate of strain on the kinetics of the network The experimental data on three fluids (representative of eight viscoelastic fluids) are used to test the models in various flow situations For steady simple shearing and small‐amplitude, sinusoidal simple shearing, both model A and model B are capable of fitting the four functions η, −(τ11−τ22), η′, and G′ rather well over many decades of shear rate or frequency For suddenly changing flow experiments model A is inadequate Model B however appears to be the

On one-dimensional finite-strain beam theory: the plane problem

Abstract: The paper formulates a one-dimensional large-strain beam theory for plane deformations of plane beams, with rigorous consistency of dynamics and kinematics via application of the principle of virtual work. This formulation is complemented by considerations on how to obtain constitutive equations, and applied to the problem of buckling of circular rings, including the effects of axial normal strain and transverse shearing strain.

A Large Deformation Elastic-Viscoplastic Analysis of a Thick-Walled Spherical Shell

Abstract: : A large deformation elastic-viscoplastic theory is formulated which considers both elastic and inelastic deformations to be present at all stages of loading and unloading The theory does not require the assumption of a yield criterion or the prior determination of whether the material is loading or unloading The theory is based on relating the essential parameters to state variables; the particular constitutive relations are motivated by the equations of dislocation dynamics A numerical scheme for calculating deformations is developed and applied to a thick walled spherical shell under internal pressure Various numerical examples are presented (Author)

The rheology of a suspension of nearly spherical particles subject to Brownian rotations

Abstract: A set of constitutive equations, valid for arbitrary linear bulk flows, is derived for a dilute suspension of nearly spherical, rigid particles which are subject to rotary Brownian couples. These constitutive equations are subsequently applied to find the resulting stress patterns for a variety of time-dependent bulk flow fields. The rheological responses are found to exhibit many of the same qualitative features as have been observed in recent experimental investigations of polymeric solutions and other complex materials.

The Free Energy Constitutive Equation for Polymer Solutions from the Dumbbell Model

Abstract: The simplest model of flexible macromolecules in a dilute solution is the elastic dumbbell (or bead‐spring) model. This has been widely used for purely mechanical theories of the stress. In this work the model is used to determine the constitutive equation for the free energy and for the stress under nonisothermal conditions. The results are shown to be consistent with the general thermodynamic theory of simple fluids with fading memory.

Biaxial Stress-Strain Relations for Concrete

Abstract: On the basis of an extensive experimental investigation reported more fully elsewhere, an analytical form of stress-strain relationship for plain concrete in biaxial compression is proposed. This plane-stress orthotropic constitutive relationship accounts for the influence of microcrack confinement as well as Poison's ratio. Other equations found in the literature are shown to be special cases of the one proposed. Also based on experimental results, a simple equation is offered which accounts for the increase in compressive strength of plain concrete when biaxial compressive stress is introduced. Both the biaxial stress-strain equation and the strength equation are shown to give good agreement with the experimental results obtained by the writers as well as by others. Finally, a constitutive law, stated in matrix terms, is given for concrete in biaxial plane stress. This relation is suitable for use in the finite element analysis of reinforced or presstressed concrete.

Deformable Magnetically Saturated Media. I. Field Equations

Abstract: In this article we propose a variational approach to the study of nonlinear elastic solids in which magnetization is constant in magnitude The emphasis is placed upon the application of the different invariances used in modern continuum mechanics: Euclidean invariance, objectivity, and material symmetry In Part I, a variational treatment is given in the spirit of ``oriented media theory'' A comparison is made with the results of a direct treatment starting with the postulation of balance laws Part II is devoted to the development of constitutive equations for a variety of material classes

A Bubble Inflation Technique for the Measurement of Viscoelastic Properties in Equal Biaxial Extensional Flow

Abstract: A bubble inflation technique for establishing equal biaxial extensional flow in viscoelastic materials is presented. This technique was used to measure biaxial extensional viscosity and elastic properties of a polyisobutylene at room temperature (23°C). A theoretical experimental procedure, based on certain idealizations, was developed for establishing bubble growth under constant stress. Modifications of the experimental design were introduced to correct for the nonidealities encountered in practice. The accuracy and reliability of the measurements are tested, and sources of error, and possibilities for future work are discussed. Several constitutive equations are examined with respect to biaxial extensional flow. Predicted behavior patterns are compared with experimental results.

The line spring model for surface flaws.

Abstract: A model is discussed for the analysis of long part-through surface cracks in the walls of plate or shell structures. Such problems are formulated within the context of two dimensional plate and shell theory with the part-cracked section represented as a line-spring in the middle surface. The spring allows relative separations and rotations of the middle surface, and constitutive laws relating these discontinuities to the prevailing force and moment per unit length at any point are taken from the plane strain solution for a strip in combined tension and bending, which contains an edge crack of a corresponding depth. Prior work is reviewed and further line spring constitutive laws are discussed as appropriate to elastic analysis with thermal or residual stresses and to elastic-plastic analysis, with yielding in the ligament between the crack front and far wall in the latter case.

Comparison of rubberlike-liquid theory with stress-growth data for elongation of a low-density branched polyethylene melt

Abstract: The non-linear behavior observed byMeissner can be qualitatively described by the rubberlike-liquid constitutive equations when the constants in the memory function are chosen to fit the data in the linear region at low elongation rates.

Nonlinear Stability of Plane Poiseuille Flow of Viscoelastic Liquids

Abstract: The stability of plane Poiseuille flow to infinitesimal perturbations was studied for the second order and Maxwell fluid rheological models. When the Deborah number based disturbance propagation is small the second order fluid is a consistent constitutive equation and the two models give identical results. This occurs for elasticity number (E) less than 5×10−4. At higher values of E the second order fluid cannot be used. The critical Reynolds number is a slowly decreasing function of E up to E≈10−4 and a rapidly decreasing function subsequently in the region where fluid relaxation effects become important. For a sufficiently elastic fluid the flow transition is governed by a new mode of the Orr‐Sommerfeld equation and differs qualitatively from that for a Newtonian liquid. The results suggest the likelihood of low Reynolds number instability in highly elastic liquids.

Pure Bending, Stretching, and Twisting of Anisotropic Cylindrical Shells

Abstract: The paper considers the problem of determining stresses and deformations in elastic thin-walled, prismatical beams, subject to axial end forces and end bending and twisting moments, within the range of applicability of linear theory. The technically most significant aspect of the work has to do with the analysis of the effect of anisotropy of the material, which is associated with previously not determined modes of coupling between stretching, bending, and twisting. Use of the general formulas of the theory is illustrated for a class of shells consisting of an 'ordinary' material (unable to support stress moments with axes normal to the middle surface of the shell, and unable to undergo transverse shear deformation). Here explicit formulas are obtained for certain types of open as well as of closed-cross-section beams.

On the deformation of solid and tubular cylinders of incompressible isotropic elastic material

Abstract: A particular class of strain-energy functions is discussed in relation to the finite deformation of solid and tubular cylinders of incompressible isotropic elastic material. It is shown that the predictions of the theory correspond closely with data from experiments on the combined torsion and extension of a solid cylinder of natural rubber. This deformation is universal, i.e. it can be maintained in any isotropic elastic solid by the application of suitable surface tractions. The axial and torsional shear deformations of a circular cylindrical tube, however, cannot be so maintained unless the constitutive law conforms with certain conditions. The material constants occurring in the strain-energy function considered here are shown to satisfy inequalities ensuring the existence of these deformations.

Deformable Magnetically Saturated Media. II. Constitutive Theory

Abstract: This article is devoted to the development of constitutive equations of deformable magnetically saturated media in three dimensions. In Sec. 1 we recapitulate the local balance laws and jump conditions derived previously. A thorough study of the consequences of the objectivity requirement is given in Sec. 2. In the following sections, the material symmetry restrictions are examined and exact and approximate constitutive equations are obtained for a variety of material classes.

Materials with Memory

Abstract: The characteristic property of viscoelastic solids which distinguishes them from perfectly elastic solids is the fact that, if they are subjected to a deformation which varies with time, the stress measured at time t, say, depends not only on the instantaneous value of the deformation gradients, but also on the whole previous history of the deformation gradients. In a series of papers, Green and Rivlin1,2 and Green, Rivlin, and Spencer3 have developed constitutive equations for such materials, in which the stress is expressed in terms of the deformation in the form of series of multiple integrals. It is the object of this paper to recapitulate this development, with particular emphasis on the physical assumptions regarding the material which are implied by the mathematical assumptions made in the theory. Such a development is perhaps timely in view of the extensive attempts in recent years to represent the behavior of actual materials in the form given by the theory.

Stability of rotational couette flow of polymer solutions

Abstract: The onset of secondary flow between rotating cylinders (Taylor vortices) was observed for a dilute polymer solution whose viscometric flow properties were characterized rheogoniometrically. The critical Taylor number (flow onset) was predicted accurately by linear stability theory with a stress constitutive equation describing viscometric behavior. The cell spacing differed significantly from that predicted by linear theory. A nonlinear analysis shows that Linear theory will predict the ultimate cell size only for an inelastic liquid. For an elastic liquid a larger wave number (closer spacing) is a lower energy configuration than the linear theory spacing. This is consistent with experiment.

Finite Amplitude Dynamic Motion of Viscoelastic Materials

Abstract: It is shown that an integral constitutive relation containing a memory function depending on strain tensor invariants can describe the rheological behavior of finite amplitude oscillatory motion of polymer solutions both qualitatively and quantitatively Values of the material constants are obtained by a numerical technique of simultaneously curve fitting simple shearing viscosity, first normal stress difference, and small amplitude oscillatory motion data