Showing papers on "Constitutive equation published in 1968"
1,429 citations
TL;DR: In this article, the authors developed a general theory of heat conduction for nonlinear materials with memory, a theory which has associated with it finite propagation speeds, i.e., a thermal disturbance at any point in the body is felt instantly at every other point; or in terms more suggestive than precise, the speed of propagation of disturbances is infinite.
Abstract: > 0 is a constant. This equation, which is parabolic, has a very unpleasant feature: a thermal disturbance at any point in the body is felt instantly at every other point; or in terms more suggestive than precise, the speed of propagation of disturbances is infinite. In this paper we develop a general theory of heat conduction for nonlinear materials with memory, a theory which has associated with it finite propagation speeds. In Section 3 we determine the restrictions that thermodynamics places on our constitutive relations. We show that our theory differs f rom other theories of heat conduction in that the heat-flux, like the entropy, is determined by the functional for the free-energy. In Section 6 we study the propagation of certain types of weak discontinuities. We show that in certain circumstances waves travelling in the direction of the heat-flux vector propagate faster than waves travelling in the opposite direction. In Section 7 we deduce the linearized theory appropriate to infinitesimal temperature gradients. We show that the linearized constitutive equation for the heat-flux q has the form: 1
1,064 citations
TL;DR: In this article, constitutive equations based on the network models of Yamamoto, Lodge, and Kaye are re-derived in a common notation involving the use of base vectors embedded in the deforming macroscopic continuum.
Abstract: In this mainly expository paper, constitutive equations based on the network models ofYamamoto,Lodge, andKaye are re-derived in a common notation involving the use of base vectors embedded in the deforming macroscopic continuum. The derivations are thereby simplified in some respects and the differences of detail between the models are clarified. InLodges theory, the sub-network superposition assumption is replaced by alternative assumptions concerning the creation and loss of network segments, and the theory is extended to non-Gaussian networks.Kayes theory is extended to allow for the presence of entanglement junctions of different complexities.
211 citations
TL;DR: In this article, it is argued that Reynolds stress evolves in a continuously changing mean field according to a viscoelastic constitutive law, relating stress to deformation history by means of a scalar memory function.
Abstract: A number of shear-flow phenomena can be explained qualitatively if turbulence is regarded as a continuous viscoelastic medium with respect to its action on a mean field. Conditions are sought under which the analogy is quantitative, and it is found that the turbulence must be fine-grained and the mean field weak. For geometrical convenience the turbulence is assumed to be nearly homogeneous and isotropic so that body forces are required to maintain it. The turbulence is found to respond initially to an arbitrary deformation as an elastic medium, in which Reynolds stress is linearly proportional to strain. Three processes that cause the resulting Reynolds stress to relax are distinguished: viscous diffusion, body-force agitation and non-linear scrambling. It is argued that, regardless of which process dominates, Reynolds stress evolves in a continuously changing mean field according to a viscoelastic constitutive law, relating stress to deformation history by means of a scalar memory function. The argument is carried through analytically for weak turbulence, in which non-linear scrambling is negligible, and the memory function is computed in terms of the wave-number-frequency spectrum of the background turbulence. In the course of the analysis, a new type of Reynolds stress arises related to the passage of the turbulence through its sustaining environment of body forces. It is found that the mean field must be surprisingly weak for this ‘translation stress’ to be negligible. Applications of the viscoelasticity theory of turbulent shear flow are discussed in which body forces and therefore translation stress are absent.
183 citations
TL;DR: In this paper, a new constitutive relation for rubber-like materials is developed, based on the work of previous investigators and experiments performed by the author on neoprene film, which accurately represents the response throughout the entire range of deformations and is a generalization of most earlier theories.
151 citations
TL;DR: In this paper, the authors developed the linear theory of micropolar thermoelasticity, and the constitutive equations were constructed and the displacement, micro-rotation and the stress, couple stress formulations were given.
113 citations
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TL;DR: In this article, a clarification concerning constitutive equations for mixtures is provided, which is supplementary to an earlier paper of the authors on interacting continua, and is also related to this paper.
109 citations
TL;DR: The influences of certain classes of mechanical loads on the configuration of a growing body are studied for a special class of linear material and the resulting effects of stress on growth are discussed.
82 citations
77 citations
TL;DR: In this article, a new constitutive equation is proposed for viscoelastic fluids based on the time derivatives of the left Cauchy-Green strain tensor and is thus believed to characterize a new material.
Abstract: In this paper a new constitutive equation is proposed for viscoelastic fluids. It is based on the time derivatives of the left Cauchy-Green strain tensor and is thus believed to characterize a new material. From this constitutive equation, a second-order approximation is derived and it is shown that this second order fluid has a bounded and unique solution to the initial value problem of cessation of steady shear flow. This stability is in direct contrast with the presently well known second order models. Further, the natural time of the fluid is positive, making the theory consistent with the intuitive notion that phenomena in dissipative materials should depend on the past rather than the future. Finally, a test is proposed to distinguish the present second-order model from the earlier version.
50 citations
TL;DR: In this paper, a constitutive equation using a multiple integral functional relationship for stress relaxation of nonlinear viscoelastic material has been investigated for uniaxial stress, and the relaxational functions for constant strain were determined from experiments on polyurethane at constant strain.
Abstract: A constitutive equation using a multiple integral functional relationship for stress relaxation of nonlinear viscoelastic material has been investigated for uniaxial stress. The first three integrals were retained and the relaxational functions for constant strain were determined from experiments on polyurethane at constant strain. Using this information the behavior during a multistep strain history was computed using two approximate procedures (a) assuming a product form for the kernel function and (b) employing a modified superposition principle. The latter yielded the more accurate description, although both gave good results.
01 Jan 1968
TL;DR: In this article, an elastic-viscoplastic theory based on dislocation dynamics is presented for an internally pressurized thick walled sphere, where the total deformation consists of elastic and plastic components throughout the deformation history.
Abstract: The constitutive equations that have been developed for the dynamic behavior of materials presuppose the existence of a reference “static” yield criterion. An alternative formulation motivated by the work on dislocation dynamics considers the total deformation to consist of elastic and plastic components throughout the deformation history. This procedure permits the consideration of large deformations (finite strains) in a direct manner. The present paper outlines an elastic-viscoplastic theory based on this approach and includes numerical results for an internally pressurized thick walled sphere.
TL;DR: In this paper, the effect of viscoelasticity on the hydrodynamic stability of a plane Poiseuille flow is investigated by the linearized method of small two-dimensional disturbances.
Abstract: The effect of slight viscoelasticity on the hydrodynamic stability of a plane Poiseuille flow is investigated by the linearized method of small two‐dimensional disturbances. The resulting equation is that of Orr‐Sommerfeld, modified however by a non‐Newtonian term. The constitutive equation used to represent the response of the fluid is that of a second‐order fluid. The equations are solved numerically, and compared with the results of Thomas. The critical Reynolds number is found to be lowered as the magnitude of a non‐Newtonian parameter increased.
01 Jan 1968
TL;DR: In this article, the deformation of aluminum at strain rates from 10−3 sec−1 to 103sec−1 and temperatures from 300° K to 700° K was studied experimentally under a range of stress states including tension, compression, torsion, and combined tension and torsions.
Abstract: The deformation of aluminum at strain rates from 10−3 sec−1 to 103 sec−1 and temperatures from 300° K to 700° K is studied experimentally under a range of stress states including tension, compression, torsion, and combined tension and torsion. The results from these tests are compared with a generalized constitutive equation developed from the thermally-activated dislocation model of deformation. Predicted functional relationships between the stress, strain and strain-rate invariants and temperature are supported by the experimental data.
TL;DR: In this paper, an alternative constitutive relation, which takes account of the entire strain-history of the motion, leads to the more reasonable result that the equilibrium is stable whenever the fluid has a "fading memory".
Abstract: A recent analysis by Gupta (1967) suggests that a layer of elastico-viscous fluid at rest between parallel plane boundaries may be in unstable equilibrium. This surprising result is attributable to the inadequacy of the constitutive equation adopted by Gupta as the basis for his analysis. An alternative constitutive relation, which takes account of the entire strain-history of the motion, leads to the more reasonable result that the equilibrium is stable whenever the fluid has a ‘fading memory’.
TL;DR: In this article, a nonlinear theory of elastic-plastic continua in the presence of simple force and stress dipoles is proposed and general constitutive equations for such continua are developed and explicit restrictions are derived from thermodynamics.
TL;DR: In this article, nonlinear constitutive equations for a mixture of two elastic solids are discussed and relations between certain of the material constants related to each constituent and the corresponding constants associated with the mixture are determined.
TL;DR: In this paper, it is shown that for certain broad classes of motions the requirements of material symmetry and frame-indifference greatly simplify the form of constitutive equations and these simplifications are derived without invoking integral expansions or other special hypotheses of smoothness for material response.
Abstract: This paper is concerned with general, compressible, isotropic materials, solid or fluid, characterized by functionals which give the stress when the history of the strain is specified. It is shown that for certain broad classes of motions the requirements of material symmetry and frame-indifference greatly simplify the form of constitutive equations. These simplifications are derived without invoking integral expansions or other special hypotheses of smoothness for material response. Among the motions considered in detail are those which are locally equivalent to pure extensions and sheared extensions.
TL;DR: In this paper, an analysis of instability in tension tests is presented, where the flow stress is a unique function of strain, strain-rate and temperature, and the critical impact velocity is discussed and some theoretical predictions are compared with experimental results.
TL;DR: In this article, the steady state, non-isothermal behavior of rigid polyvinyl chloride melt, flowing in capillaries of circular cross-section, was investigated by solving, with the aid of a digital computer, the momentum and energy balance equations.
Abstract: The steady state, non-isothermal behavior of rigid polyvinyl chloride melt, flowing in capillaries of circular cross-section, was investigated by solving, with the aid of a digital computer, the momentum and energy balance equations. It was assumed that the polymer melt can be described by the “Power Law” constitutive equation. The shear rate, temperature and pressure dependent properties of the fluid were obtained experimentally. The effects of the thermal degradation of PVC on its viscosity, were also introduced in the equations of momentum and energy.
The velocity, temperature and pressure profiles, obtained for both adiabatic flow and flow through a tube of constant wall temperature, indicate that considerable heating of the melt, due to viscous dissipation, can be achieved at moderate flow rates. Thermal degradation occurs in the capillary under certain conditions of temperature history and residence time of the fluid. The results of this work are in fair agreement with experimental results in this area.
Journal Article•
TL;DR: The Boltzmann superposition principle as discussed by the authors states that the total stress at any time can be computed by simple summation of the stresses (strains) due to all of the individual strain (stress) increments that have been applied to the body in its past history.
Abstract: •ANALYSIS of stresses and displacements within a layered system subjected to a variety of surface loading is an essential step in the development of a rational method of design for flexible pavements. Such an analysis is generally achieved through the solution of partial differential equations of equilibrium, compatibility, and constitutive equations. The equations of equilibrium are obtained from the conservation of momentum, and the continuity assumption of the body leads through some geometric logic and definition of the strain tensor, to the linear strain-displacement equations. The constitutive equations, which characterize the materials, are generally assumed to be in linear form, thus avoiding the mathematical complications that may otherwise arise. The Boltzmann superposition principle is a consequence of this assumption. This principle states that the total stress (strain) at any time can be computed by simple summation of the stresses (strains) due to all of the individual strain (stress) increments that have been applied to the body in its past history. In the limit of a continuous strain (stress) history, the summation process becomes integration. This principle is often used as an alternate definition of linearity, and constitutive equations can accordingly be written as
TL;DR: The dispersion attenuation of waves in viscoelastic tubes filled with a viscous incompressible fluid are analyzed, with special attention to two cases: where the constitutive relation of the wall is that of a Voigt material, and of a standard linear solid.
Abstract: The dispersion attenuation of waves in viscoelastic tubes filled with a viscous incompressible fluid are analyzed, with special attention to two cases: where the constitutive relation of the wall is that (1) of a Voigt material, and (2) of a standard linear solid. Explicit expressions show the dependence of the phase‐velocity and damping factor on the various parameters characterizing the fluid and the viscoelastic tube. In comparison with elastic tubes, viscoelastic tubes have a higher phase velocity and attenuation coefficient. Also, as the ratio of the relaxation time of the viscoelastic material to the period of the oscillation increases, there is a marked difference between the behavior of the waves for the two types of materials considered. The frequency equation for a stretched elastic tube is also derived. The analysis is restricted to waves whose amplitude is infinitesimal with the wavelength large and the wall thickness small as compared with the radius of the tube.
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TL;DR: In this paper, it was shown that a circular jet in the atmosphere can move with uniform speed in a straight line and twist in either sense about its axis, and the special kinematics of a jet in motion along a fixed curve.
TL;DR: In this article, the general theory of structured fluids is discussed from a physical standpoint, and constraints on the phenomenological coefficients imposed by the second law of thermodynamics are obtained in the context of constitutive equations.
Abstract: The general theory of structured fluids is discussed from a physical standpoint. Two new physical principles are utilized in developing constitutive equations. Restrictions on the phenomenological coefficients imposed by the second law of thermodynamics are obtained.
TL;DR: In this article, a nonlinear theory of an isotropic, elastic-perfectly plastic Cosserat surface and a linear theory of a plate is presented, and the main developments are carried out in the context of nonlinear theories.
01 Jan 1968
TL;DR: In this article, wave front profiles, in terms of the axial strain, are computed for each constitutive relationship, and for different values of the constants that appear in the constitutive equations.
Abstract: A uniform normal stress is suddenly applied at the end of a semi-infinite rod. The stress then remains constant. The bar is made of a hypothetical material with a yield stress significantly lower than the applied stress. Various constitutive relationships are assigned to the bar material. Wave front profiles, in terms of the axial strain, are computed for each constitutive relationship, and for different values of the constants that appear in the constitutive equations. Profiles obtained in this way are compared to determine the sensitivity of the characteristic features of the profiles to the form of the constitutive equation. Special computation techniques for minimizing error are described. Results of the computations are discussed in terms of their relevance to the interpretation of experimental measurements.
01 Jan 1968
TL;DR: In this paper, the authors discuss the thermodynamic approach to combined treatment of rheologic and plastic phenomena and construct a thermodynamic theory of non-linear viscoplastic materials which may be used to describe the behavior of metals under dynamic loads.
Abstract: The aim of the present paper is to discuss the thermodynamic approach to combined treatment of rheologic and plastic phenomena and to construct a thermodynamic theory of non-linear viscoplastic materials which may be used to describe the behavior of metals under dynamic loads.
TL;DR: In this paper, die nichtlinearen Konstitutionsgleichungenfur ein richtungsorientiertes viskoelastisches material with Gedachtnis entwickelt.
Abstract: Es werden die nichtlinearen Konstitutionsgleichungenfur ein richtungsorientiertes viskoelastisches Material mit Gedachtnis entwickelt. Die Beziehungen zwischen dieser Theorie und derjenigen der klassischen Elastizitat und Viskoelastizitat und einige andere Arbeiten auf dem Gebiete der Theorie der richtungsorientierten Medien werden besprochen.
TL;DR: In this article, a nonlinear stress-strain relation for an isotropic, homogeneous material is presented, which provides a realistic nonlinear uniaxial stressstrain curve and accounts for the apparent variation of Poisson's ratio found experimentally when classical linear theory is assumed.