scispace - formally typeset
Search or ask a question

Showing papers on "Constitutive equation published in 1970"


Journal ArticleDOI
TL;DR: In this article, it was shown that the Onsager reciprocal relations lead to the relation α2 + α 3 = α6 - α5, and hence there are only five independent viscosity coefficients for a nematic liquid crystal.
Abstract: Constitutive equations for nematic liquid crystals were first established by Ericksen and Leslie. They used a stress tensor with six viscosity coefficients αi (i = 1...6). It is shown in this paper that the Onsager reciprocal relations lead to the relation α2 + α 3 = α6 - α5 . Hence there are only five independent viscosity coefficients for a nematic liquid crystal.

404 citations


Journal ArticleDOI
TL;DR: In this article, a constitutive equation for dilute emulsions is developed by considering the deformations, assumed infinitesimal, of a small droplet freely suspended in a time-dependent shearing flow.
Abstract: A constitutive equation for dilute emulsions is developed by considering the deformations, assumed infinitesimal, of a small droplet freely suspended in a time-dependent shearing flow. This equation is non-linear in the kinematic variables and gives rise to ‘fluid memory’ effects attributable to the droplet surface dynamics. Furthermore, it has the same form as the corresponding expression for a dilute suspension of Hookean elastic spheres (Goddard & Miller 1967), and reduces to a relation previously proposed by Schowalter, Chaffey & Brenner (1968) when time-dependent effects become small.Numerical solutions are also presented for the case of a small bubble in a steady extensional flow for the purpose of estimating the range of validity of the small deformation analysis. It is shown that, unlike the drag of a bubble which, in creeping motion, is known to be relatively insensitive to its exact shape, the macroscopic stress field in an emulsion is not well described by the present analysis unless the shapes of the deformed bubbles agree closely with those given by the first-order theory. Thus, the present rheological equation should prove of value in a qualitative rather than a quantitative sense.

312 citations


Journal ArticleDOI
01 Jan 1970
TL;DR: In this article, the constitutive equation for creep, shrinkage and thermal expansion of concrete and concrete is derived, which reflects correctly the effect of variable humidity and temperature, including the effects of size, shape and stress distribution.
Abstract: The constitutive equation for creep, shrinkage and thermal expansion, which reflects correctly the effect of variable humidity and temperature, including the effect of size, shape and stress distribution, is derived. Cement paste and concrete are treated as a multi-phase composite material, in which both the static and thermodynamic conditions of equilibrium must be considered.

90 citations


Journal ArticleDOI
TL;DR: In this article, complete constitutive equations are given for the slow flow of a linear viscous fluid through an anisotropic linear elastic solid, and a uniqueness theorem is stated for this model.
Abstract: The discussions of constitutive equations and their use in related studies by a number of writers for interacting continua, which is based on the form of theory given byGreen andNaghdi [1], is shown to be satisfactory for many purposes even though the expressions given for the partial stresses are incomplete. Complete constitutive equations are given in this paper, and a uniqueness theorem is stated, for the slow flow of a linear viscous fluid through an anisotropic linear elastic solid.

51 citations


Journal ArticleDOI
TL;DR: In this paper, a constitutive equation for incompressible materials under isothermal conditions is proposed, which gives very satisfactory results for the styrene-butadiene rubber that was investigated.
Abstract: A constitutive equation is proposed for incompressible materials under isothermal conditions which gives very satisfactory results for the styrene‐butadiene rubber that was investigated. The material functions in this equation are evaluated from the results of uniaxial and equal biaxial stress relaxation tests. Theoretical expressions are developed for a number of deformation histories and compared to experimental results. The uniaxial histories which are investigated are the constant stretch‐rate loading, the exponential loading, and the constant stretch‐rate unloading.

47 citations


Journal ArticleDOI
L. B. Freund1
TL;DR: In this article, a special kinematical viewpoint is taken, so that the elastic and plastic deformation processes can be considered separately, and this separation is also accommodated by a simplified thermodynamic theory of the deformation process.

40 citations


Journal ArticleDOI
TL;DR: In this article, an experimental method for measuring stresses in the extensional flow of polymer solutions is presented, and results are discussed, and the data obtained are well correlated by a simple integral equation and the need for more complicated "straingth" theories is not supported.

37 citations


Journal ArticleDOI
TL;DR: In this paper, a theory of thermomechanical materials with memory is developed, where response functionals are assumed to depend on the histories of the deformation gradient temperature and the integrated history of the temperature gradient.

36 citations


Journal ArticleDOI
TL;DR: In this article, a variational principle is formulated which yields the balance laws and constitutive equations of a nonconducting, charge-free elastic solid interacting with electromagnetic fields, and it is found that the form of the total energymomentum tensor and the constitutive equation that follow from a Lagrangian action which depends arbitrarily on the inverse deformation gradients and the electromagnetic field tensor are identical to those obtained by formulating a constitutive theory of a nondissipative material based on the basic mechanical, thermodynamical, and electromagnetic balance laws of a continuum
Abstract: A variational principle is formulated which yields the balance laws and constitutive equations of a nonconducting, charge‐free elastic solid interacting with electromagnetic fields. It is found that the form of the total energy‐momentum tensor and the constitutive equations that follow from a Lagrangian action which depends arbitrarily on the inverse deformation gradients and the electromagnetic field tensor are identical to those obtained by formulating a constitutive theory of a nondissipative material based on the basic mechanical, thermodynamical, and electromagnetic balance laws of a continuum.

22 citations


Journal ArticleDOI
TL;DR: In this article, a homogeneous continuum model is presented to describe the dynamic behavior of a laminated medium, including the effects of temperature variations, on the basis of assumed two-term expansions of the displacements and the temperature increments across the thicknesses of the layers, the state of deformation and temperature distribution in the composite are described by six fields, i.e., gross displacements, local deformations, gross temperatures and local temperature variations.
Abstract: A homogeneous continuum model is presented to describe the dynamic behavior of a laminated medium, including the effects of temperature variations. On the basis of assumed two-term expansions of the displacements and the temperature increments across the thicknesses of the layers, the state of deformation and the temperature distribution in the composite are described by six fields, i. e., gross displacements, local deformations, gross temperatures and local temperature variations. Balance equations are derived for the stress resultants and the first moments of the stresses across the thicknesses of the layers, as well as for resultant heat fluxes and their moments. A set of constitutive equations is presented for a laminated medium composed of layers of two anisotropic thermoviscoelastic solids. The special cases of isotropic thermoviscoelastic layers, anisotropic thermoelastic layers, and isotropic thermoelastic layers are discussed briefly.

18 citations


Journal ArticleDOI
TL;DR: In this article, an approximate nonlinear theory is derived to describe the mechanical behavior of a laminated composite consisting of alternating layers of two homogeneous materials subjected to large deformations, based on two-term expansions of the motion across the thicknesses of the undeformed layers.

Journal ArticleDOI
TL;DR: In this paper, conservation laws appropriate for a class of oriented fluids, at each point of which are associated three deformable directors, are derived and discussed, and constitutive equations relevant to this class of fluids are given.
Abstract: The development of a theory for the mechanical behavior of a fluid with deforming substructure is considered. Conservation laws appropriate for a class of oriented fluids, at each point of which are associated three deformable directors, are derived and discussed, and constitutive equations relevant to this class of fluids are given. These equations are then applied to study the behavior of dilute suspensions of both rigid and deformable substructure particles. It is shown that some of the results obtained by previous authors can be derived in a simple manner.

01 Jan 1970
TL;DR: In this article, the effect of very small nonaxisymmetric initial deviations from the exact shape upon the deformations and the critical time of a thin-walled circular cylindrical shell was investigated.
Abstract: : The purpose of the investigation described in this paper is the study of the effect of very small nonaxisymmetric initial deviations from the exact shape upon the deformations and the critical time of a thin-walled circular cylindrical shell which was manufactured with larger initial deviations of an axisymmetric type. The calculations are carried out in a manner similar to that of a recent paper by the senior author. It is assumed that all the deformations are due to nonlinear steady creep governed by Odqvist's law. In consequence of the nonlinearity of the constitutive equation and the use of three terms in the expressions for the deformations and the stresses the trigonometric calculations become so complicated that they must be carried out by means of the high-speed digital computer. For this purpose use is made of the 'REDUCE' program. It is found that in a particular case the critical time of the shell is reduced to about one-half the original value when one adds to the small axially symmetric component of the initial deviations a nonaxisymmetric component which is ten orders of magnitude smaller. The reduction in critical time is represented by a factor of about 1/15 when the amplitudes of the axisymmetric and nonaxisymmetric initial deviations are equal. (Author)

DOI
01 Jan 1970
TL;DR: In this paper, a generalised triaxial, strain rate sensitive, history and temperature dependent constitutive model has been implemented in the finite element code CAPA-3D. Explicit procedures have been formulated for the experimental determination of the model parameters.
Abstract: An extensive experimental and analytical investigation is currently being carried out on the mechanisms leading to the initiation and propagation of damage in viscoplastic materials. One of the major goals of the investigation is the development and the finite elements implementation of a generalised triaxial, strain rate sensitive, history and temperature dependent constitutive model. Explicit procedures have been formulated for the experimental determination of the model parameters. As a minimum, only uniaxial test results are needed for determination of the basic parameters. The model has been implemented in the finite element code CAPA-3D. Results of the utilization of CAPA-3D for the investigation of the dynamic non-linear response of a road pavement are reviewed in the last part of this contribution.

Book ChapterDOI
01 Jan 1970
TL;DR: In this article, a non-isothermal finite linear theory of viscoelasticity with infinitesimal deformation is presented for "thermo-rheologically simple" solids.
Abstract: Starting with the results for a non-isothermal finite linear theory of viscoelasticity, a systematic derivation of a linearized theory with infinitesimal deformation is presented for ‘thermo-rheologically simple’ solids. A comparison of the resulting constitutive equations and the internal dissipation with those given previously is included.

Journal ArticleDOI
TL;DR: In this article, constitutive equations for non-linear viscoelastic materials under variable stress loading are presented in terms of nonlinear models and analytical solutions are obtained for stress, velocity and strain at the wave front in an impulsively loaded semi-infinite rod of material described by these nonlinear constitutive equation within the assumptions of small deformation theory.
Abstract: Constitutive equations are presented for non-linear viscoelastic materials under variable stress loading nd are interpreted in terms of non-linear viscoelastic models Analytical solutions are obtained for stress, velocity and strain at the wave front in an impulsively loaded semi-infinite rod of material described by these non-linear constitutive equations within the assumptions of small deformation theory

Journal ArticleDOI
TL;DR: In this article, the results of uniaxial, equal biaxonial, and unequal homogeneous biaoxial tension of viscoelastic materials under isothermal conditions were compared with predictions based on the approximate constitutive equations of Refs. 1 and 2.
Abstract: In this paper we present the results of uniaxial, equal biaxial, and unequal homogeneous biaxial tension of viscoelastic materials under isothermal conditions and compare them with predictions based on the approximate constitutive equations of Refs. 1 and 2. Theoretical expressions for uniaxial and equal biaxial constant and logarithmic stretch‐rates and for unequal homogeneous biaxial single‐step and double‐step relaxation and constant stretch‐rate are developed and compared with experimental results for a styrene‐butadiene rubber (SBR); agreement is found to be satisfactory. The uniaxial and equal biaxial constant stretch results for very short ramp times are used to predict the behavior of actual single‐step relaxation tests and to study the effects of fast motion on the constitutive equations of Refs. 1 and 2.

01 Jun 1970
TL;DR: In this paper, a nonlinear homogeneous constitutive equation for highly filled polymeric materials such as solid propellants is developed for stress analysis, and a series of correspondence principles are derived wherein half of the solution can be obtained by solving an equivalent linear elastic problem.
Abstract: : Nonlinear homogeneous constitutive equations are developed in this thesis for highly filled polymeric materials such as solid propellants. In the range of strains below vacuole dilatation these materials obey the homogeneity rule of linearity but do not obey the superposition rule. Such materials typically exhibit an irreversible 'stress softening' called the 'Mullins' Effect.' The development in this dissertation stems from attempting to mathematically describe the failing microstructure of these composite materials in terms of a linear cumulative damage model. It is demonstrated that pth order Lebesgue norms of the strain history can be used to describe the state of damage in these materials and can also be used in the constitutive equation to characterize their time dependent mechanical response to strain disturbances. Stress analysis procedures for materials having nonlinear homogeneous constitutive equations are developed for two and three dimensional proportional boundary value problems. A series of correspondence principles are derived wherein half of the solution, either the stresses or the strains, can be obtained by solving an equivalent linear elastic problem. The remaining half of the solution can be obtained by substituting the linear elastic solution into the nonlinear homogeneous constitutive equation.

Journal ArticleDOI
01 Dec 1970
TL;DR: In this paper, a linearized stability analysis of a fluid flowing in a gravity field between horizontal planes in Couette flow under conditions such that the temperature of the bottom plane exceeds that of the top is analyzed.
Abstract: A linearized stability analysis has been applied to a fluid flowing in a gravity field between horizontal planes in Couette flow under conditions such that the temperature of the bottom plane exceeds that of the top. The analysis was carried out for two different forms of constitutive equations: (1) a “generalized second‐order equation,” which is a differential model usually associated with continuum theories, and (2) an integral equation which has its basis in a molecular model and accounts for network junctions which rupture at a certain critical strain. Both models lead, after application of a number of simplifying assumptions, to the same set of differential equations which must be satisfied at a condition of nonoscillatory marginal stability. In contrast to the case for Newtonian fluids, there is coupling between the equations describing momentum and energy disturbances. This fact leads, for viscoelastic fluids, to a critical Rayleigh number which is dependent upon flow properties. In particular, the critical Rayleigh number is highly sensitive to the sign and magnitude of the second normal stress difference, a result shared with earlier studies of some other stability problems. Significance of the results is discussed.

Journal Article
TL;DR: A review of LABORATORY test data as mentioned in this paper reveals that most of the most common highway mediums, under conditions REPRESENTATIVE of moving traffic on an INSERVICE PAVEMENT, exhibit a non-linear response to stress.
Abstract: A REVIEW OF LABORATORY TEST DATA REVEALS THAT MOST COMMON HIGHWAY MATERIALS, UNDER CONDITIONS REPRESENTATIVE OF MOVING TRAFFIC ON AN INSERVICE PAVEMENT, EXHIBIT A NONLINEAR RESPONSE TO STRESS. THE REPORTED STRESS-STRAIN RESPONSE OF PAVEMENTS CONSTRUCTED WITH SUCH MATERIALS VARIES FROM THE STRESS-SOFTENING TO THE STRESS-STIFFENING TYPE, IN ACCORDANCE WITH THE RESPONSE OF THE CONSTITUTENT MATERIALS. A NONLINEAR ELASTIC INCREMENTAL FINITE ELEMENT ANALYSIS OF A UNIFORM SAND MASS SUBJECTED TO A UNIFORM CIRCULAR SURFACE LOAD , USING A CONSTITUTIVE EQUATION BASED ON PUBLISHED LABORATORY DATA, REVEALED A PRONOUNCED STIFFENING RELATIONSHIP BETWEEN THE APPLIED PRESSURE AND SURFACE DEFLECTION AND SLIGHTLY NONLINEAR RELATIONSHIPS BETWEEN THE APPLIED PRESSURE AND THE VERTICAL STRESSES INDUCED IN THE MASS. AN APPROXIMATE NONLINEAR ELASTIC ANALYSIS OF A FULL-DEPTH ASLPHAT CONCRETE PAVEMENT OVER A SANDY CLAY SUBGRADE, USING STRESS-STRAIN COEFFICIENT MATRICES MEASURED IN LABORATORY TRIAXIAL TESTS ON THE MATERIALS, GAVE ALMOST LINEAR RELATIONSHIPS BETWEEN THE APPLIED PRESSURE AND THE RESULTING DEFLECTION, AND DISTRIBUTIONS OF STRESSES AND STRAINS WITH THE STRUCTURE VERY SIMILAR TO THOSE YIELDED BY A LINEAR ELASTIC ANALYSIS USING STRESS-STRAIN COEFFICIENTS AT REALISTIC STRESS LEVELS. TO AN ENGINEERING APPROXIMATION, A LINEAR ANALYSIS WAS SUFFICIENTLY ACCURATE IN THE CASE OF THIS PARTICULAR FULL-DEPTH ASPHALT CONCRETE PAVEMENT BUT APPEARED UNACCEPTABLE IN THE CASE OF A PAVEMENT WITH UNBOUND GRANULAR MATERIALS CLOSE TO THE SURFACE. /AUTHOR/

Journal ArticleDOI
TL;DR: In this article, the authors review the rational foundations of the generalized Newtonian fluid (GNF) and discuss the problems involved in its practical application to real materials, including the types of material constants which must appear and the dimensionless groups which govern the solutions of boundary value problems.
Abstract: The concept of the generalized Newtonian fluid (GNF) provides a useful basis for the formulation of constitutive equations for real materials. The purpose of this paper is to review the rational foundations of the generalized Newtonian fluid, and to discuss the problems involved in its practical application to real materials. Of special interest are the types of material constants which must appear and the dimensionless groups which govern the solutions of boundary value problems. It is demonstrated that, if the material function of a GNF contains one material constant with units of viscosity, it must also contain at least one material constant with units of time (or reciprocal time). The role of this characteristic time for both purely‐viscous and elastic fluids is discussed.

Journal ArticleDOI
TL;DR: In this article, a mathematical analysis of the shock wave structure incorporating Maxwell moment methods and Boltzmann equation for stress and heat flux constitutive relations has been presented for the first time.
Abstract: Mathematical analysis of shock wave structure incorporating Maxwell moment methods and Boltzmann equation for stress and heat flux constitutive relations

Journal ArticleDOI
TL;DR: A thermodynamic foundation for isothermal plasticity is laid on the basis of the hypotheses that during plastic deformation the infinitesimal increment Δγ of irreversible entropy depends on the current state of stress and the stress increment.
Abstract: A thermodynamic foundation for isothermal plasticity is laid on the basis of the hypotheses that (a) during plastic deformation the infinitesimal increment Δγ of irreversible entropy depends on the current state of stress and the stress increment and (b) that Δγ>0 for plastic deformation, whereas Δγ=0 for elastic deformation. The existence of a yield locus is established as a consequence. Constitutive eaqutions for small and large deformations are then derived. The small deformation constitutive equations coincide with the classical forms that rest on the grounds of mechanical stability.

Journal ArticleDOI
TL;DR: In this paper, an outline is given of the phenomenological theory of fading memory recently explored by V. J. Mizel and the author, which provides a general framework in which one can derive the restrictions which the second law of thermodynamics places on the constitutive equations of materials with memory.
Abstract: An outline is given of the phenomenological theory of fading memory recently explored by V. J. Mizel and the author. The theory provides a general framework in which one can derive the restrictions which the second law of thermodynamics places on the constitutive equations of materials with memory.

Dissertation
01 Jan 1970
TL;DR: In this article, the authors present a statistical theory for the characterization of time-dependent properties of nonlinear viscoelastic materials, including the concept of the duration of the memory, a method for its determination, and its usefulness.
Abstract: CLASSICAL AND STATISTICAL THEORIES FOR THE DETERMINATION OF CONSTITUTIVE EQUATIONS by JOSEPH ELIAS SOUSSOU Submitted to the Department of Civil Engineering on May 15, 1970 in partial fulfillment of the requirements for the degree of Doctor of Phil".osophy. Various aspects of the determination of the Constitutive Equation of materials fulfilling the Fading Memory principle are studied. These materials are considered in isothermal conditions and many of the derivations are limited to the one-dimensional case. A short review discusses the different mathematical representations which are used to describe the Constitutive Equation of this class of materials. Section 2 discusses the special case of linear viscoelastic materials. The discussion concentrates on the treatment and analysis of data obtained for such materials. More specifically the time-temperature superposition principle is discussed as well as the methods of curve-fitting which are useful in representing the measured viscoelastic functions in algebraic forms. Finally a method is presented for the comparison and evaluation of the consistency of creep and relaxation data obtained by a set of independent experiments. Section 3 deals with the problems associated with the determination of the Constitutive Equation of nonlinear viscoelastic materials. The concept of the "duration of the memory", a method for its determination, and its usefulness are presented. Section 4 presents a statistical theory for the characterization of time-dependent properties. This theory was used previously for nonlinear electrical systems and is applied to the determination of nonlinear Constitutive Equations. Thesis Supervisor: Fred Moavenzadeh Title: Associate Professor of Civil Engineering =NOR

Journal ArticleDOI
TL;DR: In this paper, a new theory for the constitutive equations in Cosserat elasticity is proposed based on the assumption that the rotation vector depending on the displacement vector should be coupled with a rotation vector independent of the displacement vectors.
Abstract: A new theory for the constitutive equations in Cosserat elasticity is proposed. It is based on the assumption that the rotation vector depending on the displacement vector should be coupled with a rotation vector independent of the displacement vector. This eliminates the indeterminancies in stress and couple-stress encountered earlier.


01 Dec 1970
TL;DR: In this article, a rate-dependent constitutive model for isotropic metal deformation was developed, which describes a broad spectrum of elastic-plastic response in isotropics.
Abstract: : A rate-dependent constitutive model is developed which describes a broad spectrum of elastic-plastic response in isotropic metals, ranging from quasi-static behavior through the thermally activated intermediate strain rate regime, up to the high strain rate region where phonon viscosity and relativistic effects appear to control the flow process. Upon reverse straining from a plastically prestrained state, the constitutive model exhibits a rate- dependent Bauschinger effect. An attempt has been made to utilize, wherever possible, current knowledge in the theory of dislocation dynamics in formulating the constitutive model. In most cases, only simple models of governing deformation mechanisms can be constructed and, even to accomplish this, considerable speculation sis required. Where dislocation theory is unable to provide guidance in defining and characterizing a particular mechanism, a phenomenological approach has been followed. The advanced constitutive model developed here has been incorporated into the one-dimensional, finite-difference RIP code. The application of this model to 6061-T6 aluminum is described.

ReportDOI
01 Mar 1970
TL;DR: In this paper, a thin sheet of natural rubber is assumed to be incompressible, isotropic, and perfectly elastic and the strain energy function and constitutive equations have been determined, and the material is classified as a Generalized Rivlin-Mooney type.
Abstract: : Biaxial and uniaxial experiments have been conducted on a thin sheet of natural rubber, which can be assumed to be incompressible, isotropic, and perfectly elastic. The strain energy function and constitutive equations have been determined, and the material is classified as a Generalized Rivlin-Mooney type. Biaxial experiments were then conducted on the same sheet with a circular cutout and stress concentration factors were obtained. Results indicate a significant increase in the factor with increased displacements. A modified Particle-In-Cell (P.I.C.) method has been developed and analytical results were obtained for a sheet with a rigid circular inclusion. It is shown that the stress concentration factor for a Rivlin-Mooney material increases with increasing deformations, a result which is in qualitative agreement with solutions obtained by other methods. The use of the Generalized Rivlin-Mooney material, however, leads to a decrease in stress concentration with increasing deformations.

Journal ArticleDOI
TL;DR: In this paper, short cylindrical specimens of Armco iron were tested at room temperature under compressive axial loads at compressive, axial load at strains up to 0·6% and strain rates up to 103 in/in sec during loading, the axial stress was measured with a thin piezoelectric disk inserted between the specimen and the loading bar.