Showing papers on "Constitutive equation published in 1986"
TL;DR: In this article, an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations was carried out, where attention was confined to infinitesimal deformations.
Abstract: An accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors, is carried out in this paper. For simplicity, attention is confined to infinitesimal deformations. The integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations. General rate-independent and rate-dependent behaviour, with plastic hardening or softening, associated or non-associated flow rules and nonlinear elastic response can be efficiently treated within the present framework. Isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding.
800 citations
TL;DR: An elastic constitutive relation for cancellous bone tissue is developed and it is shown that the principal axes of the stress, strain and fabric tensors all coincide at remodeling equilibrium.
Abstract: An elastic constitutive relation for cancellous bone tissue is developed. This relationship involves the stress tensor T, the strain tensor E and the fabric tensor H for cancellous bone. The fabric tensor is a symmetric second rank tensor that is a quantitative stereological measure of the microstructural arrangement of trabeculae and pores in the cancellous bone tissue. The constitutive relation obtained is part of an algebraic formulation of Wolff's law of trabecular architecture in remodeling equilibrium. In particular, with the general constitutive relationship between T, H and E, the statement of Wolff's law at remodeling equilibrium is simply the requirement of the commutativity of the matrix multiplication of the stress tensor and the fabric tensor at remodeling equilibrium, T*H* = H*T*. The asterisk on the stress and fabric tensor indicates their values in remodeling equilibrium. It is shown that the constitutive relation also requires that E*H* = H*E*. Thus, the principal axes of the stress, strain and fabric tensors all coincide at remodeling equilibrium.
399 citations
TL;DR: In this article, a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials is proposed, where strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density, leading to equilibrium equations which remain always elliptic.
Abstract: By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of pre- and post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear.
361 citations
TL;DR: In this article, the relativistic thermodynamics of degenerate gases is presented as a field theory of the 14 fields of particle density and stress, and the field equations are based on the conservation laws of particle numbers and energy-momentum and on a balance of fluxes.
262 citations
TL;DR: In this paper, the constitutive parameters for frictional sliding of initially bare surfaces of Westerly granite, using a recently developed high pressure rotary shear apparatus that allows long distances of sliding and therefore a greater assurance of attaining steady state behavior, were determined.
Abstract: An understanding of the frictional sliding on faults that can lead to earthquakes requires a knowledge of both constitutive behavior of the sliding surfaces and its mechanical interaction with the loading system. We have determined the constitutive parameters for frictional sliding of initially bare surfaces of Westerly granite, using a recently developed high pressure rotary shear apparatus that allows long distances of sliding and therefore a greater assurance of attaining steady state behavior. From experiments conducted at room temperature and normal stresses of 27–84 MPa several important results have been found. (1) A gouge layer 100 to 200 μm thick was developed from the initially bare rock surfaces after 18 to 70 mm of sliding. (2) The steady state frictional resistance, attained after about 10 mm of sliding, is proportional to the negative of the logarithm of the sliding velocity. (3) Abrupt changes in the velocity of sliding result in initial changes in the frictional resistance, which have the same sign as the velocity change, and are followed by a gradual decay to a new steady state value over a characteristic distance of sliding. This velocity weakening behavior is essentially identical with that found by several previous workers on the same material at lower normal stress. (4) Our results are well described by a two state variable constitutive law. The values of the constitutive parameters are quite similar to those found previously at low normal stress, but the characteristic distance is about an order of magnitude smaller than that found at 10 MPa normal stress with thicker layers of coarser gouge. (5) We have approximated our results with a one state variable constitutive law and compared the results with the predictions of existing nonlinear stability analysis; in addition, we have extended the stability analysis to systems possessing two state variables. With such formulations good agreement is found between the experimentally observed and theoretically predicted transitions between stable and unstable sliding. These results allow a better understanding of the instabilities that lead to earthquakes.
255 citations
TL;DR: In this article, the authors measured transient and steady-state recoverable strain in oscillatory shear and uniaxial elongation with rheometers of different type, using a single integral constitutive equation with strain-dependent memory function.
Abstract: Transient and steady‐state recoverable strains in shear and uniaxial elongation were measured with rheometers of different type. Results on melts of linear and branched polyethylenes, polypropylene, polystyrene, polyamide 6, and a PIB solution are reported. To represent the linear viscoelastic behavior, the essential basis of the theoretical treatment, discrete relaxation time spectra have been determined from the dynamic moduli in oscillatory shear. Calculations of the primary normal stress and of the shear and elongational viscosities in the non‐Newtonian range are based on a single integral constitutive equation with strain‐dependent memory function. Recoverable strains observed after unloading the samples from steady‐state flow are calculated by means of a quasilinear treatment of the materials, making use of deformation rate dependent effective relaxation strengths. The agreement of predicted and measured recoverable strains over a very wide range of shear and elongation rates strongly supports this new concept, which does not involve an inversion of the integral constitutive equation. In addition, the applicability of two empirical equations that relate the primary normal stress coefficient and the recoverable strain in steady shear flow to the dynamic moduli are checked on the same materials.
232 citations
TL;DR: In this paper, a moment method is used to obtain the approximate form for the single particle velocity distribution function for the case of smooth, slightly inelastic, uniform spherical particles in which the coefficient of restitutione depends upon the particle impact velocity.
Abstract: Following the granular flow kinetic theory of Lun, Savage, Jeffrey and Chepurniy, a moment method is used to obtain the approximate form for the single particle velocity distribution function for the case of smooth, slightly inelastic, uniform spherical particles in which the coefficient of restitutione depends upon the particle impact velocity. Constitutive equations for stress are derived and the theory is applied to the case of a simple shear flow. Theoretical predictions of stresses are compared with experimental results. The effect of the impact velocity dependente is to cause the stresses to vary with the shear rate raised to a power less than two; this is consistent with the experimental observations. On the basis of the present theory and comparisons with experimental data it is concluded that theoretical models which include both surface friction and an impact velocity dependente will lead to improved agreement between the theoretical predictions and the measurements.
212 citations
TL;DR: In this article, the conservation laws, the constitutive equations, and the equation of state for path-dependent materials are formulated for an arbitrary Lagrangian-Eulerian finite element method.
Abstract: The conservation laws, the constitutive equations, and the equation of state for path-dependent materials are formulated for an arbitrary Lagrangian-Eulerian finite element method. Both the geometrical and material nonlinearities are included in this setting. Computer implementations are presented and an elastic-plastic wave propagation problem is used to examine some features of the proposed method.
182 citations
TL;DR: In this paper, the second part of a series of two papers on the development of a general thermodynamic basis for the study of transport phenomena in porous media is presented, where the porous medium is modelled as a superposition of one solid continuum coexiting and interacting with an N-component fluid-phase continuum.
175 citations
TL;DR: It is shown how to use the results of this analysis to obtain the constitutive relations, as well as the associated material parameters, from the corresponding experiments, to give a detailed account of the deformation and interaction of the fluid and solid phases in the tissue.
Abstract: The dynamic finite deformational behavior of a biphasic model for soft hydrated tissue is examined. In the case of uni-axial confined compression the displacement and stress fields are derived for steady-state permeation, creep, and stress-relaxation. It is shown how to use the results of this analysis to obtain the constitutive relations, as well as the associated material parameters, from the corresponding experiments. It is also shown that the solutions from the theory go much farther, giving a detailed account of the deformation and interaction of the fluid and solid phases in the tissue.
165 citations
TL;DR: In this article, a self-consistent averaging scheme is proposed for estimating the overall, finite deformation response of polycrystalline aggregates consisting of single crystals which undergo plastic flow by rate-dependent crystallographic slip, accompanied by elastic lattice distortion.
Abstract: Based on Hill’s method, a self-consistent averaging scheme is proposed for estimating the overall, finite deformation response of polycrystalline aggregates consisting of single crystals which undergo plastic flow by rate-dependent crystallographic slip, accompanied by elastic lattice distortion. First, constitutive relations for such single crystals are developed assuming that the slip-rate and the associated resolved shear stress are governed by: (1) a power-law relation, and (2) a viscoplastic relation. Then, Hill’s idea that the constraint imposed on a single crystal by the remaining aggregates may be represented by embedding the single crystal in a homogeneous, infinitely extended matrix having the instantaneous overall moduli, is used to formulate a completely self-consistent averaging procedure, valid for rate-dependent materials at finite strains and rotations. This method includes both the Hill and the Krӧner‒Budiansky‒Wu (K. B. W.) methods as limiting cases; when rate-effects are negligible, it reduces to Hill’s self-consistent method as formulated by Iwakuma and Nemat-Nasser for finite deformations, while it reduces to a generalized finite deformation version of the K. B. W. method for strongly rate-dependent materials. Illustrative numerical examples are presented for a plane uniaxial deformation, using a two-dimensional poly crystalline model. These examples clearly show that the rate-dependent crystallographic slip on the level of single crystals produces a more stable overall behaviour of poly crystals. This supports similar results arrived at by other investigators for single crystals and for polycrystals, by using the Taylor averaging scheme. It is shown that, while Taylor’s averaging scheme gives accurate estimates of the incremental quantities at large strains, the total overall quantities differ considerably from the ones obtained by the self-consistent method.
TL;DR: In this paper, the authors derived a constitutive law for the inelastic behavior of a second phase ceramic which undergoes a martensitic phase transformation when a function of the macroscopic stress state attains a critical value.
TL;DR: In this paper, a finite element method for using integral constitutive models in viscoelastic flow simulation is presented, which is based on a streamline element scheme (S.E.K.Z).
Abstract: A finite element method for using integral constitutive models in viscoelastic flow simulation is presented. This method is based on a streamline element scheme (S.E.S) which was reported in Part I of this paper. The technique of particle tracking and strain history calculation is discussed in detail. In calculating the infinite memory integral, either Gaussian or Laguerre numerical quadrature formulae are used in our scheme. Some simple and complex flow problems involving the upper-convected Maxwell integral model are solved as test problems and afterwards the effort is concentrated on the K.B.K.Z. model. Much numerical work is devoted to simulating the axial extrusion swell experiments with LDPE sample A of the IUPAC Working Party on Structure and Properties of Commercial Polymers, using a specific version of the K.B.K.Z. model with multiple relaxation times in the memory function designed by Papanastasiou, Scriven and Macosko. It is shown by the numerical results that if the shear viscosity function of the model is kept unchanged, the calculated swelling ratio is very sensitive to the elongational behaviour of the model; it increases (or decreases) monotonically with the increase (or decrease) of the elongational viscosity in the corresponding stretching rate region. When both the shear and elongational response of this model agree well with experiments, the numerical predictions of the swelling ratio also agree well with experimental data at low and high apparent shear rates, while in the medium region the numerical calculation underestimates the swelling ratio. It is also seen that, using this model in our method, the extrusion calculation is surprisingly stable, even at very high Weissenberg numbers or very high extrusion swelling ratios, thus showing the very promising potential of integral models in the field of viscoelastic flow computation.
TL;DR: In this paper, the onset of convection in a layer of viscoelastic liquid heated from below is investigated, and it is shown that the nature of the convective solution depends strongly on the particular constitutive relation used to characterize the viscousity.
Abstract: The onset of convection in a layer of viscoelastic liquid heated from below is investigated. It is shown that the nature of the convective solution depends strongly on the particular constitutive relation used to characterize the viscoelasticity. For certain models and certain parameter ranges the convection is supercritical and stable, while for other models and parameter ranges it can be subcritical and unstable. It is suggested that observations of convective behavior can provide a test for constitutive relations proposed for a particular liquid. A Fourier representation of the solution to the nonlinear problem is developed which is shown to admit aperiodic, or chaotic, solutions in a specific truncation that generalizes the classical Lorenz system for the Newtonian Benard problem.
TL;DR: In this paper, the effect of material path dependent hardening on neck development and the onset of ductile failure is analyzed numerically using an elastic-viscoplastic constitutive relation that accounts for the weakening due to the growth of micro-voids.
Abstract: The effect of material path dependent hardening on neck development and the onset of ductile failure is analyzed numerically. The calculations are carried out using an elastic-viscoplastic constitutive relation that has isotropic hardening and kinematic hardening behaviors as limiting cases and that accounts for the weakening due to the growth of micro-voids. Final material failure is incorporated into the constitutive model by the dependence of the plastic potential on void volume fraction. Results are obtained for both axisymmetric and plane strain tension. Failure is found to initiate by void coalescence at the neck center in axisymmetric tension and within a shear band in plane strain tension. The increased curvature of flow potential surfaces associated with the kinematic hardening solid leads to somewhat more rapid diffuse neck development than occurs for the isotropic hardening solid. However, a much greater difference between the predictions of the two constitutive models is found for the onset of ductile failure.
TL;DR: In this paper, the authors discuss concepts associated with viscosity, elasticity, hyperbolicity, Hadamard instability and ill posedness of Cauchy problems in the flow of viscoelastic fluids.
Abstract: In this paper we discuss concepts associated with viscosity, elasticity, hyperbolicity, Hadamard instability and ill posedness of Cauchy problems in the flow of viscoelastic fluids. We frame the analysis in terms of vorticity and develop relations between change of type in steady flow and the ill posedness of the unsteady problem. We also consider the problem of regularizing Hadamard instabilities by the addition of Newtonian contributions to the constitutive equations.
TL;DR: In this paper, a simple and general elastoplastic constitutive model for clay is proposed that describes the stress-strain behavior of clay under various stress paths in three-dimensional stresses.
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18 Sep 1986
TL;DR: In this article, a finite element formulation of transient heat conduction in axisymmetric solids is presented, where the authors use the concept of discretization of the finite element method.
Abstract: 1 Fundamentals of the Finite Element Method.- 1.1 Introduction.- 1.2 The concept of discretization.- 1.3 Steps in the finite element method.- References.- 2 Finite Element Analysis in Heat Conduction.- 2.1 Introduction.- 2.2 Review of basic formulations.- 2.3 Finite element formulation of transient heat conduction in solids.- 2.4 Transient heat conduction in axisymmetric solids.- 2.5 Computation of the thermal conductivity matrix.- 2.6 Computation of the heat capacitance matrix.- 2.7 Computation of thermal force matrix.- 2.8 Transient heat conduction in the time domain.- 2.9 Boundary conditions 45 2.10 Solution procedures for axisymmetric structures.- References.- 3 Thermoelastic-Plastic Stress Analysis.- 3.1 Introduction.- 3.2 Mechanical behavior of materials.- 3.3 Review of basic formulations in linear elasticity theory.- 3.4 Basic formulations in nonlinear elasticity.- 3.5 Elements of plasticity theory.- 3.6 Strain hardening.- 3.7 Plastic potential (yield) function.- 3.8 Prandtl-Reuss relation.- 3.9 Derivation of plastic stress-strain relations.- 3.10 Constitutive equations for thermoelastic-plastic stress analysis.- 3.11 Derivation of the [Cep] matrix.- 3.12 Determination of material stiffness (H').- 3.13 Thermoelastic-plastic stress analysis with kinematic hardening rule.- 3.14 Finite element formulation of thermoelastic-plastic stress analysis.- 3.15 Finite element formulation for the base TEPSAC code.- 3.16 Solution procedure for the base TEPSA code.- References.- 4 Creep Deformation of Solids by Finite Element Analysis.- 4.1 Introduction.- 4.2 Theoretical background.- 4.3 Constitutive equations for thermoelastic-plastic creep stress analysis.- 4.4 Finite element formulation of thermoelastic-plastic creep stress analysis.- 4.5 Integration schemes.- 4.6 Solution algorithm.- 4.7 Code verification.- 4.8 Closing remarks.- References.- 5 Elastic-Plastic stress analysis with Fourier Series.- 5.1 Introduction.- 5.2 Element equation for elastic axisymmetric solids subject to nonaxisymmetric loadings.- 5.3 Stiffness matrix for elastic solids subject to nonaxisymmetric loadings.- 5.4 Elastic-plastic stress analysis of axisymmetric solids subject to nonaxisymmetric loadings.- 5.5 Derivation of element equation.- 5.6 Mode mixing stiffness equations.- 5.7 Circumferential integration scheme.- 5.8 Numerical example.- 5.9 Discussion of the numerical example.- 5.10 Summary.- References.- 6 Elastodynamic stress analysis with Thermal Effects.- 6.1 Introduction.- 6.2 Theoretical background.- 6.3 Hamilton's variational principle.- 6.4 Finite element formulation.- 6.5 Direct time integration scheme.- 6.6 Solution algorithm.- 6.7 Numerical illustration.- References.- 7 Thermofracture Mechanics.- 1: Review of fracture mechanics concept.- 7.1 Introduction.- 7.2 Linear elastic fracture mechanics.- 7.3 Elastic-plastic fracture mechanics.- 7.4 Application of the finite element method to fracture mechanics.- 2: Thermoelastic-plastic fracture analysis page.- 7.5 Introduction.- 7.6 Fracture criteria.- 7.7 J integral with thermal effect.- 7.8 Numerical illustrations of J integrals with thermal effect.- 7.9 The "breakable element".- 7.10 Numerical illustrations of stable crack growth.- 3: Thermoelastic-plastic creep fracture analysis.- 7.11 Literature review.- 7.12 Generalized creep fracture model.- 7.13 Path dependence of the Cg* integral.- 7.14 Creep crack growth simulated by "breakable element"1 Fundamentals of the Finite Element Method.- 1.1 Introduction.- 1.2 The concept of discretization.- 1.3 Steps in the finite element method.- References.- 2 Finite Element Analysis in Heat Conduction.- 2.1 Introduction.- 2.2 Review of basic formulations.- 2.3 Finite element formulation of transient heat conduction in solids.- 2.4 Transient heat conduction in axisymmetric solids.- 2.5 Computation of the thermal conductivity matrix.- 2.6 Computation of the heat capacitance matrix.- 2.7 Computation of thermal force matrix.- 2.8 Transient heat conduction in the time domain.- 2.9 Boundary conditions 45 2.10 Solution procedures for axisymmetric structures.- References.- 3 Thermoelastic-Plastic Stress Analysis.- 3.1 Introduction.- 3.2 Mechanical behavior of materials.- 3.3 Review of basic formulations in linear elasticity theory.- 3.4 Basic formulations in nonlinear elasticity.- 3.5 Elements of plasticity theory.- 3.6 Strain hardening.- 3.7 Plastic potential (yield) function.- 3.8 Prandtl-Reuss relation.- 3.9 Derivation of plastic stress-strain relations.- 3.10 Constitutive equations for thermoelastic-plastic stress analysis.- 3.11 Derivation of the [Cep] matrix.- 3.12 Determination of material stiffness (H').- 3.13 Thermoelastic-plastic stress analysis with kinematic hardening rule.- 3.14 Finite element formulation of thermoelastic-plastic stress analysis.- 3.15 Finite element formulation for the base TEPSAC code.- 3.16 Solution procedure for the base TEPSA code.- References.- 4 Creep Deformation of Solids by Finite Element Analysis.- 4.1 Introduction.- 4.2 Theoretical background.- 4.3 Constitutive equations for thermoelastic-plastic creep stress analysis.- 4.4 Finite element formulation of thermoelastic-plastic creep stress analysis.- 4.5 Integration schemes.- 4.6 Solution algorithm.- 4.7 Code verification.- 4.8 Closing remarks.- References.- 5 Elastic-Plastic stress analysis with Fourier Series.- 5.1 Introduction.- 5.2 Element equation for elastic axisymmetric solids subject to nonaxisymmetric loadings.- 5.3 Stiffness matrix for elastic solids subject to nonaxisymmetric loadings.- 5.4 Elastic-plastic stress analysis of axisymmetric solids subject to nonaxisymmetric loadings.- 5.5 Derivation of element equation.- 5.6 Mode mixing stiffness equations.- 5.7 Circumferential integration scheme.- 5.8 Numerical example.- 5.9 Discussion of the numerical example.- 5.10 Summary.- References.- 6 Elastodynamic stress analysis with Thermal Effects.- 6.1 Introduction.- 6.2 Theoretical background.- 6.3 Hamilton's variational principle.- 6.4 Finite element formulation.- 6.5 Direct time integration scheme.- 6.6 Solution algorithm.- 6.7 Numerical illustration.- References.- 7 Thermofracture Mechanics.- 1: Review of fracture mechanics concept.- 7.1 Introduction.- 7.2 Linear elastic fracture mechanics.- 7.3 Elastic-plastic fracture mechanics.- 7.4 Application of the finite element method to fracture mechanics.- 2: Thermoelastic-plastic fracture analysis page.- 7.5 Introduction.- 7.6 Fracture criteria.- 7.7 J integral with thermal effect.- 7.8 Numerical illustrations of J integrals with thermal effect.- 7.9 The "breakable element".- 7.10 Numerical illustrations of stable crack growth.- 3: Thermoelastic-plastic creep fracture analysis.- 7.11 Literature review.- 7.12 Generalized creep fracture model.- 7.13 Path dependence of the Cg* integral.- 7.14 Creep crack growth simulated by "breakable element" algorithm.- References.- 8 Thermoelastic-Plastic Stress Analysis By Finite Strain Theory.- 8.1 Introduction.- 8.2 Lagrangian and Eulerian coordinate systems.- 8.3 Green and Almansi strain tensors.- 8.4 Lagrangian and Kirchhoff stress tensors.- 8.5 Equilibrium in the large.- 8.6 Equilibrium in the small.- 8.7 The boundary conditions.- 8.8 The constitutive equation.- 8.9 Equations of equilibrium by the principle of virtual work.- 8.10 Finite element formulation.- 8.11 Stiffness matrix [K2].- 8.12 Stiffness matrix [K3].- 8.13 Constitutive equations for thermoelastic-plastic stress analysis.- 8.14 The finite element formulation.- 8.15 The computer program.- 8.16 Numerical examples.- References.- 9 Coupled Thermoelastic-Plastic Stress Analysis.- 9.1 Introduction.- 9.2 The energy balance concept.- 9.3 Derivation of the coupled heat conduction equation.- 9.4 Coupled thermoelastic-plastic stress analysis.- 9.5 Finite element formulation.- 9.6 The y matrix.- 9.7 The thermal moduli matrix ?.- 9.8 The internal dissipation factor.- 9.9 Computation algorithm.- 9.10 Numerical illustration.- 9.11 Concluding remarks.- References.- 10 Application of Thermomechanical Analyses in Industry.- 10.1 Introduction.- 10.2 Thermal analysis involving phase change.- 10.3 Thermoelastic-plastic stress analysis.- 10.4 Thermoelastic-plastic stress analysis by TEPSAC code.- 10.5 Simulation of thermomechanical behavior of nuclear reactor fuel elements.- References.- Appendix 1 Area coordinate system for triangular simplex elements.- Appendix 2 Numerical illustration on the implementation of thermal boundary conditions.- Appendix 3 Integrands of the mode-mixing stiffness matrix.- Appendix 4 User's guide for TEPSAC.- Appendix 5 Listing of TEPSAC code.- Author Index.
TL;DR: A finite element program for solving viscoelastic flow problems in plane and axisymmetric geometries based on a streamline element scheme that employs an iterative process to treat elastic stresses as pseudo-body forces and integrates Maxwell-type constitutive equations along the streamlines forming the element boundaries.
Abstract: A finite element program for solving viscoelastic flow problems in plane and axisymmetric geometries is described. The program is based on a streamline element scheme (S.E.S.). One of the principal features of the program is that it employs an iterative process to treat elastic stresses as pseudo-body forces; during each iteration it generates updated streamline-based elements and integrates Maxwell-type constitutive equations along the streamlines forming the element boundaries. The performance of this method was satisfactory, considering both accuracy and efficiency. Results of two simple test problems and two complex flow problems are shown. These problems are simple shearing flow, uniaxial and biaxial elongation, extrudate swelling and flow around a sphere in a long cylinder. In the simple test problems the results were very accurate and were compared with exact solutions and inthe more complex flows the results were in close agreement with data from other programs based on different methods. Some limitations and the current state of development of the program are also discussed.
TL;DR: In this article, the authors developed constitutive relations for liquid and/or gas saturated elastic porous media and formulated intrinsic stress tensors and densities in terms of the partial stress tensor, partial densities, and actual volume fractions occupied by each component.
Abstract: Concepts from the theory of interacting continua are employed to develop constitutive relations for liquid and/or gas saturated elastic porous media. The model is formulated by defining intrinsic stress tensors and densities in terms of the partial stress tensors, partial densities, and actual volume fractions occupied by each component. It is assumed that the constitutive law for each component as a single continuum relates intrinsic pressure to intrinsic deformation. Relative motion between the constituents is allowed through simple Darcy‐type expressions. The governing equations together with the constitutive relations are used to investigate the propagation of both harmonic and transient pulses. In general three modes of wave propagation exist. In the case of a transient pulse, these modes lead to a three‐wave structure. Laplace transform techniques are used to derive closed‐form solutions for transient loading for two limiting values of viscous coupling (i.e., weak viscous coupling, strong viscous co...
01 Mar 1986
TL;DR: In this article, a constitutive model for confined and unconfined concrete subjected to dynamic compression is proposed, based on numer- ous measurements of the maximum stress and the strain at maximum stress in tests performed at different strain rates.
Abstract: A constitutive model for confined and unconfined concrete subjected ro dynamic compression is proposed. This model is based on numer ous measurements of the maximum stress and the strain at maximum stress in tests performed at different strain rates. The proposed model distinguishes between the strain rate effects on dry and wet concrete, and compares well with the available test results.
TL;DR: In this article, a differential stress-strain relation for elastoplasticity is derived, such that the concept of a yield-surface is retained; the definitions of elastic and plastic processes are analogous to those in classical plasticity theory; and its computational implementation, via a 'tangent-stiffness' finite element method and a generalized-midpoint-radial-return' stressintegration algorithm, is simple and efficient.
TL;DR: In this paper, the deformation and strength characteristics of clay in three-dimensional stresses are experimentally discussed, and the test results are compared with the analytical results by the well-known Cam-clay model and those by the tij-clays model which has recently been developed with reference to the extended concept of the Spatially Mobilized Plane.
TL;DR: In this paper, the authors derived governing equations for the location of the neutral axis of a beam under bending which does not in general pass through the centroid of the cross-section, and for the creep response in terms of both curvature rate and load-point displacement rate as functions of the applied moment and power-law creep parameters.
Abstract: Power-law creep parameters of brittle ceramic materials are commonly deduced from load-point displacement data generated by four-point bend experiments, under the assumption that tensile and compressive behaviours obey the same constitutive law. However, because of microcracking and cavitation, it is now well recognized that this premise may not always be valid. The present paper presents an analysis which takes the differences into account. Governing equations are first derived for the location of the neutral axis of a beam under bending which does not in general pass through the centroid of the cross-section, and for the creep response in terms of both curvature rate and load-point displacement rate as functions of the applied moment and power-law creep parameters. Numerical solutions are obtained for any given set of material constants over a wide range of applied moments. It is shown from the plots of creep response against applied moment on a logarithmic scale that even linear curves over a narrow range of applied moment do not necessarily imply identical stress exponents, and that non-linear curves concave upward signify a profound difference in stress exponent between tension and compression. An example is given of applying the present analysis to a set of load-point displacement data on glass-alumina beam specimens crept at 1100° C. The results show that the conventional method over/underestimates the creep rates in compression/tension by two orders of magnitude, indicating a need for using the more accurate analysis presented here. Several recommendations are offered to improve the estimation of power-law creep parameters from bend test data.
TL;DR: In this paper, a new constitutive equation was developed for describing the nonlinear viscoelastic properties, including yield, of polymer solids, which is the logical extension of the standard three-dimensional linear VCE to include the effects of deformationinduced changes in the thermodynamic state of the polymer on the rate of visco-elastic relaxation.
Abstract: A new constitutive equation has been developed for describing the nonlinear viscoelastic properties, including yield, of polymer solids. The nonlinear viscoelastic constitutive equation is the logical extension of the standard three‐dimensional linear viscoelastic constitutive equation to include the effects of deformation‐induced changes in the thermodynamic state of the polymer on the rate of viscoelastic relaxation. The rate of viscoelastic relaxation depends upon the fraction of unoccupied lattice sites (i.e., holes) in the Simha‐Somcynsky statistical thermodynamic equation of state, and the hole fraction is a known function of the temperature, pressure, and specific volume. The nonlinear viscoelastic yield and postyield behavior are a direct consequence of the deformation‐induced dilation. All material constants contained in the constitutive equation can be determined from independent elastic, linear viscoelastic, and pressure‐volume‐temperature measurements—there are no adjustable constants. The con...
TL;DR: In this paper, the general conservation equations for the rapid flow of a binary mixture of smooth, inelastic, spherical granular particles are derived by making simple approximations for the particle velocity distribution functions.
Abstract: Following the approach of the kinetic theory for mixtures of dense gases, the general conservation equations for the rapid flow of a binary mixture of smooth, inelastic, spherical granular particles are derived. Explicit constitutive relations for stress and rate of energy dissipation are obtained by making simple approximations for the particle velocity distribution functions. These approximations are appropriate for cases where collisional interactions are the dominant mechanism for momentum and energy exchange in the system. The theory is applied to the case of simple shear flow. In general, the theory predicts that stresses decrease with increasing concentration of the small particles and decreasing diameter ratio of small to large particles. Theoretical predictions of stresses are compared with experimental results and reasonable agreement is found.
TL;DR: In this article, a two-dimensional consolidation analysis of clay deposits was made with an elasto-viscoplastic constitutive model and Biot's consolidation theory, which can describe the effect of sample thickness and aging on consolidation phenomena.
Abstract: A two-dimensional consolidation analysis of clay deposits was made with an elasto-viscoplastic constitutive model and Biot's consolidation theory. One- and two-dimensional consolidation problems were analysed numerically by the finite element and finite difference methods. Results show that the proposed method can describe the effect of sample thickness and aging on consolidation phenomena. The two-dimensional behaviour of a clay foundation during the construction of embankments also was simulated.
TL;DR: In this article, the tensor function theory is applied to the formulation of constitutive equations of isotropic and anisotropic materials in the secondary and tertiary creep stage.
TL;DR: A continuum mixture theory for the slow flow of a dilute suspension of solid particles in a viscous fluid is outlined in this article, where explicit constitutive equations are posed that specialize the drag to include Stokes and Faxen forces, and the lift to accommodate the "slip-shear" force identified by Saffman and the "disturbance-curvature" forces treated by Ho and Leal.
Abstract: A continuum mixture theory for the slow flow of a dilute suspension of solid particles in a viscous fluid is outlined. The momentum exchange between the fluid and disperse particulate phases accounts for buoyancy, drag and lift forces, and the additional viscous transport associated with the presence of the particles in the fluid. Explicit constitutive equations are posed that specialize the drag to include Stokes and Faxen forces, and the lift to accommodate the ‘‘slip–shear’’ force identified by Saffman and the ‘‘disturbance–shear’’ and ‘‘disturbance–curvature’’ forces treated by Ho and Leal. The additional viscous transport is fixed by comparison with the well‐known Einstein ‘‘effective viscosity’’ correction. Finally, the pressure difference between the phases is specified by a constitutive equation that accounts for Brownian diffusion, local inertia, and bulk viscous effects. Plane Poiseuille flow is examined to illustrate the features of the model. Both approximate analytical solutions and numerical solutions for the full, coupled, nonlinear governing equations show nonuniform distributions for the particles across the channel. This affects the apparent viscosity of the mixture according to whether the particles concentrate in regions of lesser or greater mean shear rate. Finally, the inhomogeneous particle distributions cause the mixture to appear non‐Newtonian, because the form of the distribution (and, consequently, the apparent viscosity) is rate‐dependent.