Showing papers on "Constitutive equation published in 1971"

Viscoelastic Characterization of a Nonlinear Fiber-Reinforced Plastic

Abstract: The nonlinear, viscoelastic behavior of a unidirectional, glass fiber-epoxy composite material is characterized by using isothermal, uniaxial creep and recovery tests together with a constitutive equation based on thermodynamic theory. The nonlinear constitutive equation for uniaxial loading is described first, and then fourth-order tensor transformations relating principal linear viscoelastic creep compli ances, uniaxial creep compliance, and fiber angle are summarized. Following a discussion of experimental aspects, creep and recovery data obtained from several different specimens (each having a differ ent fiber orientation relative to the loading axis) are reduced using a graphical shifting procedure and tensor transformations to evaluate all material properties, including the principal creep compliances. As a check on the constitutive theory, the data are shown to be in ternally consistent. Some simplicity in the analytical representation of the data is found; viz. the nonlinear, uniaxial creep compli...

Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm

Abstract: The theory of dilute polymer solutions based on the bead/spring model in the form given by Zimm is reformulated for arbitrary homogeneous flow histories. It is shown that the center of resistance of a polymer molecule moves with the solvent. By a preliminary transformation [3.4], the equations for the center-of- resistance motion are separated from those for the motion of the N spring vectors. The spring-vector equations involve a symmetric non-singular matrix B [3.14] whose characteristic values equal the non-zero characteristic values of Zimms singular matrix HA. A further transformation [4.2] which diagonalizes B yields separate differential eq. [5.4] for pq*, the polymer contribution to the stress tensor associated with the q normal mode. Transformation to an embedded basis enables one to integrate these equations so as to obtain pq* in terms of the flow history ([5.6], [5.9]), and summation over q then gives the required constitutive eq. [5.17] for the polymer solution. These are of the same form as the “rubber-like liquid” constitutive equations (with addition of a solvent-contribution term) derived from the network theory of Lodge, but the memory function is determined to within three constants (e. g. N, h*, τ1). Peterlina solution for the normal-coordinate distribution function in steady shear flow is generalized for an arbitrary homogeneous (time-dependent or steady) flow and expressed in terms of pq* which can be evaluated when the flow history is given.

A theory of plasticity with non-coincident yield and loading surfaces

Abstract: A generalization of the classical plasticity theories which accounts for the observed phenomenon of noncoincident yield and loading surfaces is proposed. Constitutive relations are developed for arbitrary hardening laws and detailed results are presented and applied for isotropically hardening materials.

Constitutive equations for elastomers

Abstract: General relations were derived by expanding the strain-energy density function in terms of the invariants of the deformation tensor. Some constitutive equations obtained by keeping a third term in the expanson in addition to the two terms retained in the Mooney-Rivlin equation were tested in the light of currently available experimental data. It is shown that by the retention of the third term the upswing in the Mooney stress at low values of γ−1 is successfully predicted, and the stress—strain behavior can be described with excellent accuracy up to break, even in carbon black-filled rubber which is difficult to describe by the Mooney-Rivlin equation.

Rate‐Dependent Relaxation Spectra and Their Determination

Abstract: Stress overshoot at the start of a flow, stress relaxation after the stoppage of a flow, and superposition of a small oscillation upon a steady shear flow are treated by the use of the concept of a deformation rate‐dependent relaxation spectra These analyses are used as the basis of methods to determine the spectrum of relaxation times Although the treatment is restricted to specific types of constitutive equations in which the response function depends on the rate of deformation invariants, it is not necessary to specify any functional form of the response function Experimental determination of the rate‐dependent spectrum may play an important role in the discussion of the relative applicability of different forms of constitutive equations recently proposed in the literature

Nonlinear Continuum Mechanics of Viscoelastic Fluids

Abstract: In the last twenty-five years there has been great progress in the development of nonlinear continuum theories in mechanics and other branches of physics, and in the use of such theories to explain and predict phenomena that cannot be explained rationally by the classical linear theories.

Wave propagation effects in a fluid‐saturated porous solid

Abstract: Constitutive relations for a fluid-saturated elastic-plastic porous solid have been developed within the framework of the theory of interacting continua (TINC) by defining effective stress tensors and effective densities in the composite in terms of the actual volume fractions occupied by each component, partial stress tensors, and partial densities. The model is formulated by postulating that the constitutive law for each component as a single continuum relates effective stress tensor and effective deformation. In contradistinction to the classical homogenized models, TINC allows relative motion between the constituents. The governing conservation equations, together with the constitutive relations for a binary mixture, are solved by the Lax-Wendroff finite difference procedure. The model is applied to study finite-amplitude wave propagation in a porous tuff completely saturated with water. The results are compared with those obtained from a homogenized model. The present model predicts lower pressures and shock velocities than those given by the homogenized model.

On the orthogonal superposition of simple shearing and small-strain oscillatory motions

Abstract: Certain limiting high-and low-frequency relations peculiar to the general Kaye-Bernstein-Kearsley-Zapas single integral constitutive equation are developed for orthogonally superposed simple shearing and small-strain time-sinusoidal shearing. The relations do not hold for a general simple fluid nor for single-integral constitutive equations in which arbitrary dependence of the kernel function on the rates of deformation is allowed. It is shown that solutions of polyisobutylene in cetane (4.4%, 6.8%) and solutions of polyethylene-oxide (0.7%) and J-100 (0.1%) in water obey the relations quite closely.

Dynamic response of rapidly heated plate elements

Abstract: through the constitutive law. Indirect application of the variational principle yields the classical equations of motion, the force-displacement boundary conditions, and the constitutive relationships between the distortions and bending (and twisting) stresses. A semidirect application of the variational principle that eliminates spatial-coordinate dependence yields generalized time-dependent ordinary differential equations of dynamic equilibrium and constitutive relations between the generalized force and displacement parameters of assumed spatial distributions. Results obtained using the dynamic thermoelastic variational principle demonstrate displacement and bending-moment-convergence characteristics far superior to conventional solutions. A study of the influence of temperature-dependent material properties shows that neglect of temperature dependence is an unconservative assumption; further it is demonstrated that incomplete consideration of temperature dependence can lead to dangerously unconservative results.

An Integral Constitutive Equation Based on Molecular Network Theory

Abstract: A new constitutive equation for concentrated polymer solutions and melts is presented that is based on the entanglement theory of Lodge. The strain rate dependence of the memory function is determined using a physical hypothesis of interacting spheres where the spheres represent spheres of influence of the network junctions. The resulting equation has one constant that can be estimated theoretically in addition to the natural relaxation spectrum. At high strain rates, a second empirical constant is introduced to account for the orientation of the spheres of influence. Predictions of the equation and the equations of Bogue, Bird‐Carreau, and Tanner were compared to steady and transient shear stress and normal stress data obtained on a Weissenberg rheogoniometer. The new equation fits nonlinear transient data more satisfactorily than other equations of similar complexity.

Shock‐Wave Structure in Porous Solids

Abstract: In this paper several variations of a simple theory of dynamic compaction of porous solids are presented and discussed. This theory elaborates the conventional theory of shock propagation in such a way that the shock structures observed to propagate in these materials can be described. Steady‐wave profiles are calculated for several compaction models, and the inference of constitutive equations from experimental data is discussed. It is shown that the theory can be made to reproduce steady‐wave profiles observed in the usual plate‐impact experiments exactly.

Applicability of the Oldroyd Constitutive Equation to Flow of Dilute Polymer Solutions

Abstract: It is shown that the equation relating stress and deformation in a dilute solution of polymer molecules (according to the dumbell model) may be placed in the form of the Oldroyd constitutive equation with an appropriate interpretation of constants.

Stagnation point flow of a second-order viscoelastic fluid

Abstract: The boundary-layer equations are solved for the case of two-dimensional flow of a second-order viscoelastic fluid near a stagnation point. It is shown that the effect of viscoelasticity is not only to increase the wall-shear stress but also to cause oscillations in the velocity profile. It is further shown that the constitutive equation for the second-order viscoelastic fluid is not applicable to the analysis of stagnation point flow for Weissenberg numbers greater than approximately 0.32.

A Theory of Viscoplasticity with a Yield Surface. Part 2. Application to Mechanical Behavior of Metals

Abstract: : The endochronic theory of viscoplasticity developed previously by the author is used to give quantitative analytical predictions on the mechanical response of aluminum and copper under conditions of complex strain histories. One single constitutive equation describes with remarkable accuracy and ease of calculation diverse phenomena, such as cross-hardening, loading and unloading loops, cyclic hardening as well as behavior in tension in the presence of a shearing stress, which have been observed experimentally by four different authors.

Thermodynamical Theory of Directed Curves

Abstract: A nonlinear thermodynamical theory of directed curves is obtained by postulating an energy balance, an entropy production inequality, and invariance requirements under superposed rigid body motions. Constitutive equations for a class of simple materials without memory are presented.

Linear theory of cosserat surface and elastic plates of variable thickness

Abstract: : Within the scope of the linear isothermal theory of an elastic Cosserat surface, constitutive equations are derived for an initially flat Cosserat surface in which the initial director (along the normal to the initial surface) is allowed to depend on the surface coordinates. These constitutive equations correspond to those for bending and stretching of a transversely isotropic three dimensional plate. Special attention is given to the relevance and applicability of the results to bending of (three dimensional) plates of variable thickness and comparison is made with a set of equations for elastic plates of variable thickness obtained, by an approximation procedure, from the three dimensional equations. (Author)

Recoverable Deformation of Cohesionless Soils

Abstract: Herein is shown that available experimental data on the recoverable behavior of cohesionless soils may be fitted closely using constitutive relations corresponding to a special case of the first order theory of hypoelasticity. Four forms of first order isotropic constitutive law are first presented and conditions are determined under which each law is integrable to an elastic law. Constants for the isotropic hypoelastic law are then determined which fit available data for triaxial states of stress and it is shown that the resulting law is inelastic. Consideration is then given to fitting the data by means of an anisotropic first order incremental law. Integrability conditions for triaxial loading states are obtained and are shown to be satisfied by both isotropic and anisotropic laws. The integrated law for triaxial loading for the isotropic case is obtained and demonstrated by application to the anisotropic consolidation test. Tests are described to investigate the elastic character of the recoverable behavior.

HELP, A Multi-Material Eulerian Program for Compressible Fluid and Elastic-Plastic Flows in Two Space Dimensions and Time. Volume 1

Abstract: : Volume I details a new numerical model for solving multi-material flows which are functions of two space dimensions and time. The program, called HELP, is basically Eulerian but also contains Lagrangian features for the explicit definition of interface positions. The program is general in the sense that any number of materials can be present in a given Eulerian cell and no special difficulties arise for flows involving extreme material distortions. The program is suitable for compressible media with strength dependence. Compressibility is included by an equation of state giving pressure as a function of density and specific internal energy. Strength is included by means of a constitutive equation giving the deviatoric stresses as functions of the elastic and plastic strains. In this Volume I report, the basic equations are developed, the computer program is documented and sample results are given from six applications.

Theory of induced birefringence in materials with memory

Abstract: Isotropic materials with memory, discussing induced birefringence theory to account for memory and nonlinearity effects in dielectric properties dependence on deformation history

Principal- and slip-line methods of numerical analysis in plane and axially-symmetric deformations of rigid/plastic media

Abstract: Some new principal- and slip-line methods of numerical analysis are presented for plane and axiallysymmetric deformations of rigid/plastic media. These methods are applicable to the solution of problems of incompressible isotropic, or incompressible anisotropic, or (special) compressible isotropic media obeying generalized Tresca/von Mises constitutive equations. The essential idea in the numerical methods considered is an artifice of the reduction of elliptic (and similar) problems of systems of partial differential equations to hyperbolic ones by means of a procedure based upon the initial estimation of one of the field quantities followed by iterations to determine all field quantities. Solutions for the particular problem of combined extension and expansion of a thick-walled circular cylinder under axial force and internal pressure are discussed.

The Stress-Strain Behavior of Mechanically Degradable Polymers

Abstract: Non-linear constitutive equations are developed for highly filled polymeric materials. These materials typically exhibit an irreversible stress softening called the “Mullins’ Effect.” The development stems from attempting to mathematically model 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 deformation history can be used to describe the state of damage in these materials and can also be used in the constitutive equations to characterize their time dependent response to strain distrubances. This method of analysis produces time dependent constitutive equations, yet they need not contain any internal viscosity contributions. This theory is applied to experimental data and shown to yield accurate stress predictions for a variety of strain inputs. Included in the development are analysis methods for proportional stress boundary valued problems for special cases of the non-linear constitutive equation.

Similarity solutions for the converging or diverging steady flow of non-newtonian elastic power law fluids with wall suction or injection. Part I: Two-dimensional channel flow

Abstract: Similarity solutions are determined for the steady flow of an incompressible elastic power law fluid in a two-dimensional channel with nonparallel walls. The possibility of wall suction or injection is considered. Solutions are found to exist only for power law indices of unity. A method is developed for estimating the pressure drop in the naturally converging flow field before the entrance to a capillary. In diverging flows a singularity is found to arise due to the elasticity of the fluid. The singularity corresponds to a Deborah number of unity. It is postulated that the singularity is, for the constitutive equation used here, a possible source of the flow instability commonly referred to as melt fracture.

A Thermodynamical Theory of Fluid Suspensions

Abstract: A development is given of a thermodynamical theory of fluid suspensions of deformable particles. Assuming particles are spherical when undeformed and become ellipsoidal when sheared, linear constitutive equations are extracted from the general theory and restrictions on the material coefficients which appear in these linear constitutive equations are deduced by thermodynamical considerations. Applying this linear theory to steady shear flow of solutions for which particle interactions are negligible, it is found, solely by use of the thermodynamical restrictions on material coefficients, that the theory qualitatively predicts observed behavior in steady shear flow of dilute solutions of random coiling macromolecules, such as solutions of polyisobutylene. In particular, the resulting expressions for apparent viscosity t21/K and normal stress difference t11 − t22 are exactly those known to characterize experimental results for K≤103sec−1. For low shear rates K, the predicted value for the second normal stre...