Showing papers on "Constitutive equation published in 1989"

An Introduction To Rheology

Abstract: 1) What is rheology? historical perspective the importance of non-linearity solids and liquids rheology is a difficult subject components of rheological research. 2) Viscosity practical ranges of variables which affect viscosity the shear-dependent viscosity of non-Newtonian liquids viscometers for measuring shear viscosity. 3) Linear viscoelasticity the meaning and consequences of linearity the Kelvin and Maxwell models the relaxation spectrum oscillatory shear relationships between functions of linear viscoelasticity methods of measurement. 4) Normal stresses the nature and origin of normal stresses typical behaviour of N 1 and N 2 observable consequences of N 1 and N 2 methods of measuring N 1 and N 2 relationships between viscometric functions and linear viscoelastic functions. 5) extensional viscosity importance of extensional flow theoretical considerations experimental methods experimental results some demonstrations of high extensional viscosity behaviour. 6) Rheology of polymeric liquids general behaviour effect of temperature on polymer rheology effect of molecular weight on polymer rheology effect of concentration on the rheology of polymer solutions polymer gels liquid crystal polymers. molecular theories the method of reduced variables empirical relations between rheological functions practical applications. 7) Rheology of suspensions the viscosity of suspensions of solid particles in Newtonian liquids the colloidal contribution to viscosity viscoelastic properties of suspensions suspensions of deformable particles the interaction of suspended particles with polymer molecules also present in the continuous phase computer simulation studies of suspension rheology. 8. Theoretical rheology basic principles of continuum mechanics successful applications of the formulation principles some general constitutive equations constitutive equations for restricted classes of flows simple constitutive equations of the Oldroyd/Maxwell type solution of flow problems.

Solidification theory for concrete creep. I: formulation

Abstract: A new general constitutive law for creep is presented, in which the aging due to continuing hydration of cement is taken into account in a simple manner, and better justified than in existing theories. Microchemical analysis of the solidification process is used to show that the aging may be modeled as a growth of the volume fraction of load-bearing solidified matter (hydrated cement), which may be described as a nonaging viscoelastic material. It was found that a history integral should be used to express the rate, rather than the total value of the viscoelastic strain component. Other study findings are also discussed.

Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface

Abstract: Paper 1 [1988 g][1] of this series began an investigation whose goal is a thermomechanics of two-phase continua based on Gibbs’s notion of a sharp phase-interface endowed with thermomechanical structure. In that paper a balance law, balance of capillary forces, was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics; the chief results are thermodynamic restrictions on constitutive equations, exact and approximate free-boundary conditions at the interface, and a hierarchy of free-boundary problems. The simplest versions of these problems (the Mullins-Sekerka problems) are essentially the classical Stefan problem with the free-boundary condition u = 0 for the temperature replaced by the condition u = h K, where K is the curvature of the free-boundary and h > 0 is a material constant. This dependence on curvature renders the problem difficult, and apart from numerical studies involving linearization stability, there are almost no supporting theoretical results.

Numerical experiments on localization in frictional materials

Abstract: Two types of numerical experiment are performed in order to elucidate the nature of localization in a frictional material. In the first type, a continuum calculation is done with a strain-hardening constitutive model. Localization is shown to occur when the value of the strength parameter has a random distribution in space. In the second type of numerical experiment, the distinct element method is used to conduct a shear test on a simulated sample of 1000 disks. Localization is seen to occur: measurements are made of shear band thickness, distribution of particle spins, contact forces and stress components.

Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials

Abstract: The first part of the paper is devoted to the motivation of the constitutive equation of the masonry-like material. It is proved that this equation is the result of three fundamental constitutive assumptions: infinitesimal elasticity, no tensile strength, and a postulate of normality. A necessary and sufficient condition for the existence of a strain energy function is also supplied.

Solidification Theory for Concrete Creep. II: Verification and Application

Abstract: The theory that was formulated in the preceding paper is verified and calibrated by comparison with important test data from the literature pertaining to constant as well as variable stress at no (or negligible) simultaneous drying. Excellent agreement is achieved. The formulation describing both elastic and creep deformations contains only four free material parameters, which can be identified from test data by linear regression, thus simplifying the task of data fitting. For numerical structural analysis, the creep law is approximated in a rate-type form, which corresponds to describing the solidified matter by a Kelvin chain with nonaging elastic moduli and viscosities. This age-independence on the microlevel makes it possible to develop for the present model a simple version of the exponential algorithm.

Constitutive Modeling of Ratchetting Effects—Part I: Experimental Facts and Properties of the Classical Models

Abstract: Etude des effets de rochetage, soit l'accumulation progressive de deformation, cycle apres cycle, induite par une charge cyclique secondaire superposee a une charge primaire constante. Discussion de faits experimentaux et de l'exactitude des equations constitutives classiques

Elasticity of Particulate Materials

Abstract: Problems associated with isolation, measurement, and modeling the elastic behavior of particulate materials are discussed. Because of pervasive yielding, exclusively elastic behavior of particulate materials is restricted to infinitesimal increments of unloading. Three‐dimensional elastic constitutive equations are formulated in terms of two scalar functions representing effects of void ratio and stress history, a reference fabric matrix, a stress‐compliance matrix, and a Poisson's ratio matrix. The stress‐compliance matrix accounts for the discovery in 1979 that elastic shear stiffness is independent of stress normal to the plane of shear. Transformation of coordinates and the definition of reference fabric are carefully considered in the developed formulation.

Force budget: I. Theory and numerical methods

Abstract: A practical method is developed for calculating stresses and velocities at depth using field measurements of the geometry and surface velocity of glaciers. To do this, it is convenient to partition full stresses into lithostatic and resistive components. The horizontal gradient in vertically integrated lithostatic stress is the driving stress and it describes the horizontal action of gravity. The horizontal resistive stress gradients describe the reactions. Resistive stresses are simply related to deviatoric stresses and hence to strain-rates through a constitutive relation. A numerical scheme can be used to calculate stresses and velocities from surface velocities and slope, and from ice thickness. There is no mathematical requirement that the variations in these quantities be small.

A cohesive zone model for dynamic shear faulting based on experimentally inferred constitutive relation and strong motion source parameters

Abstract: To understand constitutive behavior near the rupture front during an earthquake source shear failure along a preexisting fault in terms of physics, local breakdown processes near the propagating tip of the slipping zone under mode II crack growth condition have been investigated experimentally and theoretically. A physically reasonable constitutive relation between cohesive stress τ and slip displacement D, τ = (τi − τd)[1 + α log (1 + βD)] exp (−ηD) + τd, is put forward to describe dynamic breakdown processes during earthquake source failure in quantitative terms. In the above equation, τi is the initial shear stress on the verge of slip, τd is the dynamic friction stress, and α, β, and η are constants. This relation is based on the constitutive features during slip failure instabilities revealed in the careful laboratory experiments. These experiments show that the shear stress first increases with ongoing slip during the dynamic breakdown process, the peak stress is attained at a very (usually negligibly) small but nonzero value of the slip displacement, and then the slip-weakening instability proceeds. The model leads to bounded slip acceleration and stresses at and near the dynamically propagating tip of the slipping zone along the fault in an elastic continuum. The dynamic behavior near the propagating tip of the slipping zone calculated from the theoretical model agrees with those observed during slip failure along the preexisting fault much larger than the cohesive zone. The model predicts that the maximum slip acceleration be related to the maximum slip velocity and the critical displacement Dc by , where k is a numerical parameter, taking a value ranging from 4.9 to 7.2 according to a value of τi/τp (τp being the peak shear stress) in the present model. The model further predicts that be expressed in terms of and the cut off frequency fmaxs of the power spectral density of the slip acceleration on the fault plane as and that in terms of Dc and fmaxs as . These theoretical relations agree well with the experimental observations and can explain interrelations between strong motion source parameters for earthquakes. The pulse width of slip acceleration on the fault plane is directly proportional to the time Tc required for the crack tip to break down, and fmaxs is inversely proportional to Tc.

A Gradient-Dependent Flow Theory of Plasticity: Application to Metal and Soil Instabilities

Abstract: We propose a gradient-dependent flow theory of plasticity for metals and granular soils and apply it to the problems of shear banding and liquefaction. We incorporate higher order strain gradients either into the constitutive equation for the flow stress or into the dilantancy condition. We examine the effect of these gradients on the onset of instabilities in the form of shear banding in metals or shear banding and liquefaction in soils under both quasi-static and dynamic conditions. It is shown that the higher order gradients affect the critical conditions and allow for a wavelength selection analysis leading to estimates for the width or spacing of shear bands and liquefying strips. Finally, a nonlinear analysis is given for the evolution of shear bands in soils deformed in the post-localization regime.

Fracture Energy‐Based Plasticity Formulation of Plain Concrete

Abstract: A comprehensive constitutive model is developed for the triaxial behavior of plain concrete. The formulation covers the full load-response spectrum in tension as well as in compression. The concrete model is based on the nonassociated flow theory of plasticity with hardening in the prepeak regime and fracture energy-based softening in the post-peak regime. The constitutive development is guided by recent experimental observations on laboratory specimens which are loaded under “mixed” control in tension, and in triaxial compression at different levels of lateral confinements. The constitutive parameters are calibrated from a series of stroke-controlled laboratory experiments which include direct tension and triaxial compression tests at different confinements. The predictive quality of the fracture energy-based plasticity model is ascertained with a number of separate load-history experiments for which triaxial test data are available in the concrete literature. Pertinent remarks on the numerical implementation of the elastic-plastic constitutive relations with comments on relevant issues related to the uniqueness and stability of nonassociated strain-softening computations, are also presented.

Fluid pressures in deforming porous rocks

Abstract: The constitutive relations describing the fluid pressure response of a porous medium to changes in stress and temperature must reflect the microscopic processes that are operative over the time scale allowed for the deformation. Short-duration deformations are readily described by undrained moduli, and intermediate duration deformations by drained moduli, both of which are formulated through linear elastic theory. Long-term deformations that operate over geologic time are normally dominated by irreversible processes and result in considerably larger deformations, for the same applied stress conditions, than would be expected from their elastic counterparts. Model constitutive equations are developed for both elastic and irreversible processes and the magnitude and interpretation of the relevant material properties examined. Although the theory is presented in general terms, a sample calculation shows that for sandstone the inelastic deformation is one and one half orders of magnitude greater than the elastic deformation at the same applied stress. This difference in magnitude has a significant effect on the effective hydraulic diffusivity, various pore pressure coefficients, and the prospective fluid pressure development of the sediment.

On the application of dual variables in continuum mechanics

Abstract: Stress and strain tensors that arise in the expression of the stress power are called “conjugate variables”. More special is the term “dual variables” which has been introduced in connection with incremental constitutive relations of hypoelasticity and plasticity, where the rates of both tensors arise. We propose a rational rule from which the most natural form of tensor-valued kinematic and dynamic variables (strain and stress tensors) including their corresponding time rates can be deduced. Dual variables and their associated dual derivatives are characterized by the property that apart from the stress power also the incremental stress power is invariant under a group of transformations that corresponds to a set of physically reasonable intermediate configurations. We outline the precursory history of these concepts and then discuss in detail how the invariance properties can be realized in the various stress and strain measures. We finally demonstrate the concept in three different applications: The rate form of the principle of virtual work, the formulation of constitutive relations in viscoelasticity and the formulation of incremental constitutive assumptions of rate-independent plasticity.

An exact derivation of the thin plate equation

Abstract: It is shown that, when the traditional assumptions of thin plate theory are taken as exact methematical hypotheses, the desired field and boundary equations can be obtained by mere integration over the thickness of the corresponding equations for a three-dimensional cylindrical body made of a homogeneous, linearly elastictransversely isotropic, constrained material, yet avoiding some inconsistencies usually to be found in textbooks of structural mechanics.

Shear-banding and liquefaction in granular materials on the basis of a Cosserat continuum theory

Abstract: The constitutive equations of a two-dimensional flow theory of plasticity for granular materials with Cosserat structure are outlined. The post-failure behavior is modeled by means of softening rule that is based on a two fractions mixtures theory for sand. The paper is focusing on the two major bifurcation phenomena observed in element tests, i.e. shear-banding and liquefaction.

Plasticity: Theory and Engineering Applications

Abstract: 1. Basic Concepts and Notations. The mechanical properties of materials. Material models. Historical remarks. Notations. 2. Fundamental Relations and Governing Equations. Stress state. Strain state. Stress-strain relations. Constitutive equations of elastic-perfectly plastic bodies. The governing equations for linearly elastic-perfectly plastic bodies. 3. Variational Principles. Principle of virtual work and complementary virtual work. Variational principles of elasticity. Variational principles of plasticity. Discontinuity of stress and velocity fields. 4. Elasto-Plastic Analysis of Some Simple Problems. Analysis of a simple bar structure. Torsion of prismatic bars. Thick-walled spheres and cylinders under internal pressure. 5. Plastic Analysis of Plane Strain and Plane Stress Problems. Basic relationships. Plane plastic flow. Plane stress problems. Application of the finite element method. 6. Plastic Analysis and Design of Bar Structures. Yield condition for bar members. Elasto-plastic analysis. Limit analysis. Deflections. Plastic design. Application of mathematical programming. 7. Plastic Analysis and Design of Plates and Shells. Assumptions. Limit analysis of plates. Limit analysis of shells. Optimal design. 8. Various Inelastic Material Properties. Plastic hardening materials. Granular materials. Viscous materials. 9. Dynamic Plastic Response of Structures. Fundamental relationships. Simple rigid-plastic theory. Effect of strain rate sensitivity. Effect of large displacements. Effect of elastic deformations. Applications. Concluding remarks. References. Subject Index.

Analysis of large-strain shear in rate dependent face-centred cubic polycrystals: correlation of micro- and macromechanics

Abstract: Micro- and macroscopic aspects of large-strain deformation are examined through analyses of shear by using physical and phenomenological models. Past experiments and analyses are first reviewed to reveal current issues and put the present work in perspective. These issues are addressed by a complete set of simulations of large-strain shear with a finite-strain, rate-dependent polycrystal model. The model is based on a rigorous constitutive theory for crystallographic slip that accounts for the development of crystallographic texture and the effects of texture on constitutive response. The influences of strain hardening, latent hardening, strain-rate sensitivity, boundary constraints, and initial textures on texture evolution and constitutive response are studied. Coupled stress and strain effects such as axial elongation during unconstrained shear and the development of normal stresses during constrained shear are related to material properties, boundary constraint and texture. The formation of ideal textures and their role in determining polycrystalline behaviour is discussed in quantitative terms. Large-strain shear is also studied by using several phenomenological constitutive theories including J 2 -flow theory, J 2 -corner theory, and two versions of finite-strain kinematic hardening theory. The behaviours predicted by these phenomenological theories and the physically based polycrystal model are directly compared. A noteworthy outcome is the close correspondence found between the predictions of J 2 -corner theory and those of the micromechanically based physical model.

Introduction to Continuum Mechanics for Engineers

Abstract: 1. One-Dimensional Continuum Mechanics.- 1.1. Kinematics of Motion and Strain.- 1.2. Balance of Mass.- 1.3. Balance of Linear Momentum.- 1.4. Balance of Energy.- 1.5. General Balance.- 1.6. The Entropy Inequality.- 1.7. Example Constitutive Equations.- 1.8. Thermodynamic Restrictions.- 1.9. Small Departures from Thermodynamic Equilibrium.- 1.10. Small Departures from Static Equilibrium.- 1.11. Some Features of the Linear Model.- 2. Kinematics of Motion.- 2.1. Bodies and Deformations.- 2.2. Velocity, Acceleration, and Deformation Gradients.- 2.3. Transformation of Linear, Surface, and Volume Elements.- 2.4. Strain Kinematics.- 2.5. Infinitesimal Strain Kinematics.- References.- 3. Equations of Balance.- 3.1. Balance of Mass.- 3.2. Balance of Linear Momentum.- 3.3. Balance of Angular Momentum.- 3.4. Balance of Energy.- 3.5. The Entropy Inequality.- 3.6. Jump Equations of Balance-Material Versions.- References.- 4. Models of Material Behavior.- 4.1. Examples.- 4.2. Isothermal Elasticity-Thermodynamic Restrictions.- 4.3. Isothermal Elasticity-Material Frame Indifference.- 4.4. Isothermal Elasticity-Material Symmetry.- 4.5. Incompressible Isothermal Elasticity.- 4.6. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Constitutive Assumptions.- 4.7. Thermoelastic Material with Heat Conduction and Viscous Dissipation-General Thermodynamic Restrictions.- 4.8. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Equilibrium Thermodynamic Restrictions.- 4.9. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Material Frame Indifference.- 4.10. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Material Symmetry.- 4.11. Constitutive Equations for a Compressible, Conducting, Viscous Fluid.- 4.12. Constitutive Equations for an Isotropic Linear Thermoelastic Solid with Heat Conduction.- References.- 5. Materials with Internal State Variables.- 5.1. Constitutive Assumptions and Thermodynamic Results.- 5.2. Maxwell-Cattaneo Heat Conductor.- 5.3. Maxwellian Materials.- 5.4. Closing Remarks-Alternate Forms of the Entropy Inequality.- References.- Appendix A. Mathematical Preliminaries.- A.1. Vector Spaces.- A.2. Linear Transformations.- A.3. Inner Product Spaces.- A.4. Components of Vectors and Linear Transformations.- A.5. Cross Products, Determinants, and the Polar Decomposition Theorem.- A.6. Multilinear Functionals and Tensor Algebra.- A.7. Euclidean Point Spaces, Coordinate Systems.- A.8. Vector Analysis.- Appendix B. Representation Theorems.