Showing papers on "Constitutive equation published in 1979"

Modeling of rock friction: 1. Experimental results and constitutive equations

Abstract: Direct shear experiments on ground surfaces of a granodiorite from Raymond, California, at normal stresses of ∼6 MPa demonstrate that competing time, displacement, and velocity effects control rock friction. It is proposed that the strength of the population of points of contacts between sliding surfaces determines frictional strength and that the population of contacts changes continuously with displacements. Previous experiments demonstrate that the strength of the contacts increases with the age of the contacts. The present experiments establish that a characteristic displacement, proportional to surface roughness, is required to change the population of contacts. Hence during slip the average age of the points of contact and therefore frictional strength decrease as slip velocity increases. Displacement weakening and consequently the potential for unstable slip occur whenever displacement reduces the average age of the contacts. In addition to this velocity dependency, which arises from displacement dependency and time dependency, the experiments also show a competing but transient increase in friction whenever slip velocity increases. Creep of the sliding surface at stresses below that for steady state slip is also observed. Constitutive relationships are developed that permit quantitative simulation of the friction versus displacement data as a function of surface roughness and for different time and velocity histories. Unstable slip in experiments is controlled by these constitutive effects and by the stiffness of the experimental system. It is argued that analogous properties control earthquake instability.

Stress and temperature in the bending lithosphere as constrained by experimental rock mechanics

Abstract: Summary. Previous attempts to deduce the stress distribution in the bending lithosphere near a consuming plate margin have relied on the observed bathymetry and an assumed constitutive relation for lithospheric behaviour, eg. perfectly elastic, viscous/perfectly plastic, or elastic perfectly plastic. From the point of view of rock mechanics, each of these approximations fails to describe one or more of several basic phenomena, including brittle failure of rock, temperature dependence of elasticity, and temperature and/or strain rate dependence of ductile behaviour. In order to formulate a more realistic constitutive relation, a limiting yield strength curve, which is primarily a function of temperature, is constructed from data from brittle failure and ductile flow experiments. The moments which can be supported by plates with this constitutive behaviour are compared to the moments calculated from bathymetric profiles. The comparison indicates that moments required by the bathymetric data are consistent with moments supported by plates with experimentally determined constitutive laws as extrapolated to geo- logically reasonable temperatures and strain rates. The stresses developed in such models are required to reach values greater than 100 MPat in the depth range 25-45 km. Geotherms necessary for strength curves consistent with moments calculated from the bathymetric data match those derived from heat flow data for the Aleutian, Bonin, Mariana and Tonga trenches. Of the trenches studied, only the geotherm inferred from the Kuril trench data is significantly different, perhaps implying that the Kuril plate is weaker than the others. The strength curves show that as a first approximation it is better to assume that bending moment is independent of curvature of the plate than to assume that bending moment and curvature are linearly related.

Gravity flow of cohesionless granular materials in chutes and channels

Abstract: A constitutive equation appropriate for flow of cohesionless granular materials at high deformation rates and low stress levels is proposed. It consists of an extension and a reinterpretation of the theory of Goodman & Cowin (1972), and accounts for the non-Newtonian nature of the flow as evidenced by Bagnold's (1954) experiments. The theory is applied to analyses of gravity flows in inclined chutes and vertical channels. Experiments were set up in an attempt to generate two-dimensional shear flows corresponding to these analyses. Velocity profiles measured by a technique which makes use of fibre optic probes agree qualitatively with the theoretical predictions, but direct comparison is inappropriate because of unavoidable side-wall friction effects in the experiments. The existing measure of agreement suggests that the most prominent effects have been included in the proposed constitutive relations. Tests in the inclined chute revealed the possible existence of surge waves and granular jumps analogous to hydraulic jumps.

Dynamics of concentrated polymer systems. Part 4.—Rheological properties

Abstract: The constitutive equation for polymer melts and concentrated solutions derived in the previous papers is applied to two typical rheometrical flows: steady and transient shear flow, oscillatory shear flow superposed on steady flow, steady and transient uniaxial elongational flow. The stress responses predicted are qualitatively in good agreement with experiments except for one case (the first normal stress in the transient shear flow). A particularly interesting result is that the constitutive equation suggests instability in steady shear flow and uniaxial elongational flow.

A constitutive model for sand in triaxial compression

Abstract: Although there is a large number of constitutive models for sand available in the literature it is believed that a fresh approach, striking a balance between complexity and theoretical rigour, is desirable. The approach here has certain conceptual links with the Cam Clay series of elastic–plastic models, but includes the more general starting assumption that the yield function, plastic potential and failure locus should be given quite distinct mathematical expressions. Possible physical bases for the proposed forms are discussed. Ways in which the parameters required to define the model may be determined are suggested and the use of the model is then demonstrated. Firstly, it is shown that, where a limited set of experimental data is available, the model is flexible enough to be able to match the test results. Secondly, it is shown that, where a wide range of test results has been produced, it is possible to determine the model constitutive parameters from a small number of tests and proceed to make satisfactory predictionsfor other, quite different, types of test. The model is developed for sand at a single initial density, but the way in which the constitutive parameters might be expected to vary with density is discussed. The model is described for conditions of triaxial compression, and extension to more general stress states will be needed before it can be put to the test of incorporation in, for example, a finite element program.

Determination of flow stress: Part 1 constitutive equation for aluminium alloys at elevated temperatures

Abstract: An alternative form of the hyperbolic sine relationship is developed for use as a constitutive equation at elevated temperatures. It is shown how an optimization technique may be applied to results from torsion tests to determine the constants in this equation and that this method must be more accurate than those presented previously. Experimental results for two aluminium alloys indicate that the activation energy during hot torsion testing is the same as the activation energy for bulk self-diffusion. It is shown that thermal changes occurring during testing affect results significantly.

Plastic-Fracturing Theory for Concrete

Abstract: Incremental plasticity and fracturing (microcracking) material theory are combined to obtain a nonlinear triaxial constitutive relation that is incrementally linear. The theory combines the plastic stress decrements with the fracturing stress decrements, which reflect microcracking, and accounts for internal friction, pressure sensitivity, inelastic dilatancy due to microcracking, strain softening, degradation of elastic moduli due to microcracking, and the hydrostatic nonlinearity due to pore collapse. Failure envelopes are obtained from the constitutive law as a collection of the peak points of the stress strain response curves. Six scalar material functions are needed to fully define the monotonic response. One function, the dilatancy due to microcracking, is determined theoretically based on Budianski-O’Connell’s calculation of the effective elastic contents of a randomly microcracked elastic material by the self consistent method for composites.

A 3D Hypoelastic Concrete Constitutive Relationship

Abstract: A three-dimensional constitutive relationship for concrete is developed and results are compared to experimental data available in the literature. The model uses the concept of equivalent uniaxial strain, the nonlinear stress-strain equation of Saenz, and the Argyris failure surface for concrete strength. The technique of evaluating the parameters to adjust the constitutive relationship for various concretes is examined. Comparison is made with the biaxial experimental data of Kupfer, Hilsdorf, and Rusch, and with the three-dimensional experimental data of Schickert and Winkler. The model appears capable of simulating the stress-strain response of concrete under a large range of conditions and is easily incorporated into finite element programs. Flow charts for implementation of the constitutive relationship are presented.

Constitutive Model for Short-Time Loading of Concrete

Abstract: A constitutive model based on nonlinear elasticity is proposed, where the secant values of Young’s modulus and Poisson’s ratio are changed appropriately. This alteration is obtained through the use of a nonlinearity index that relates the actual stress state to the failure surface. The model simulates the strain hardening before failure, the failure itself and the strain softening in the post-failure region. The dilation of the concrete and the influence of all three stress invariants are considered. All stress states including those where there are tensile stresses can be dealt with; however, the model is calibrated using experimental data obtained by a uniaxial compressive and tensile test only. The model predictions are demonstrated to be in good agreement with experimental results involving a wide range of stress states and different types of concrete.

The motion of small particles in non-Newtonian fluids

Abstract: There are many problems in which the motion of a small particle, bubble or drop in a non-Newtonian fluid differs in an important qualitative way from its corresponding motion in a Newtonian fluid. From a theoretical point of view such problems are conveniently separated into two groups. In the first, some aspect of the particle's motion only exists, for small Reynolds number, because the suspending fluid is non-Newtonian. Examples of this class include the cross-stream (or lateral) motion of spherical particles in a unidirectional shear flow, rotational motion of an orthotropic particle in sedimentation (leading to a deterministic equilibrium orientation), and cross-orbital drift in the rotation of an axisymmetric particle in shear flow. In these cases, a major change in the orientation or position of the particle can result from small instantaneous contributions of non-Newtonian rheology to the particle's motion, provided that these act “cumulatively” over a sufficiently long period of time. An analytical description of the fluid mechanics relevant to this process may thus be based on the asymptotic limit of a nearly-Newtonian fluid using the so-called “retarded-motion” expansion, and a relevant constitutive model for viscoelastic materials is the Rivlin—Ericksen nth-order fluid. Comparison between theory and experiment shows excellent qualitative (and frequently quantitative) agreement for such problems even when the flow is too rapid, in a rheological sense, for strict adherence to the requirements of a retarded-motion expansion. The second major class of problems is that in which the observed difference between Newtonian and non-Newtonian behavior is due to an important, O(1) change in the fluid motion at all times. In this case, the only possible theoretical description which is valid in more than an asymptotic sense is one based on a full non-linear constitutive model, including “memory”, and thus a solution of the equations of motion is generally possible only via numerical methods. Unlike the first class of problems, an important determining factor in successful match between experiment and theory is therefore a judicious (fortunate?) choice of the constitutive model. In the second part of this paper, I shall discuss some examples of numerical and experimental studies which pertain to particle motions in the regime of strongly viscoelastic flows.

A uniaxial viscoplastic model based on total strain and overstress

Abstract: A previously proposed, uniaxial differential constitutive equation (E.R. C ernocky and E. K rempl , 1979a, b), nonlinear in the Cauchy stress and the engineering strain but linear in the stress and strain-rates, is specialized to an overstress model. It is shown by qualitative arguments that the solutions correspond to typical room-temperature viscoplastic behavior of AISI type 304 stainless steel. Two unknown coefficient functions are determined by extrapolation of room temperature relaxation data for this steel. The stiff first-order nonlinear differential equations are then numerically integrated for a variety of test histories. These include strain control with strain-rates from 10 −6 to 800s −1 , stress control with stressrates from 1.95 kPa s −1 to 19.5 MPa s −1 , instantaneous large changes in strain-rate and stress-rate, and partial unloading and reloading in strain and stress control and tension-tension cyclic creep. The computed results show good qualitative agreement with tests. Based on these results we consider that the model is a good representation of metal deformation behavior as long as the overstress does not change sign.

The Irreversibility Assumption of Network Disentanglement in Flowing Polymer Melts and its Effects on Elastic Recoil Predictions

Abstract: Tanner has suggested a constitutive equation for polymer melts and solutions based on a network rupture hypothesis: entanglements are lost irreversibly in the process of deformation as soon as a limiting strain magnitude is reached Although this model shows serious defects in describing time‐dependent material properties it has the appealing feature of irreversibility On the other hand the BKZ model (considered here in the simplified version of a separable memory function), being more useful in describing startup experiments, does not posses this feature of irreversibility, leading to largely erroneous results in recovery calculations An improved version of a single integral constitutive equation is presented, where the irreversibility of the process of network disentanglement is incorporated Predictions of elastic recoil behavior in elongation and shear are compared with experimental results for a well‐characterized LDPE melt

A non-linear uniaxial integral constitutive equation incorporating rate effects, creep and relaxation

Abstract: A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.

Uniaxial Cyclic Loading of Elastic-Viscoplastic Materials

Abstract: : Elastic-viscoplastic constitutive equations based on two internal state variables are utilized to determine material response for uniaxial cyclic loading conditions. These equations are capable of representing the principal features of cyclic loading behavior including softening upon stress reversal, cyclic hardening or softening, tendency towards a stable limit cycle, cyclic relaxation, and cyclic creep. Calculations were performed for various stress and strain controlled conditions using material constants intended to represent commercially pure titanium and aluminum and OFHC copper. Capabilities and limitations of the analytical formulations are discussed in relation to computed results and corresponding test data. (Author)

Applications of Generalized Derivatives to Viscoelasticity.

Abstract: : Generalized derivatives of fractional order are used to construct stress-strain constitutive relations for viscoelastic materials, based on the observed sinusoidal behavior of the materials The non-periodic behavior of one material is observed in the laboratory and compares favorably with the non-periodic behavior of the material predicted by its generalized derivative constitutive relation Having established that the generalized derivative constitutive relation is an appropriate mathematical model for the general motion of at least one viscoelastic material, the tools for the analysis of structures of engineering interest are put forward In particular, attention is focused on a finite element formulation of and solutions to the equations of motion for structures containing elastic and viscoelastic components (Author)

One dimensional dynamic electromechanical constitutive relations of ferroelectric materials

Abstract: We present a macroscopic theory for the dynamic response of a poled polycrystalline ferroelectric material describing its coupled electromechanical interactions. The treatment is restricted to an idealized material representing the simplest system capable of displaying ferroelectricity; it includes changes in both the magnitude of the electric dipoles and the orientation of the domains. Coupling between the electrical and mechanical properties of the material is considered and the constitutive equations are linearized to illustrate the resulting dynamic response.

Elongational behaviour of polymer melts in constant elongation-rate, constant tensile stress, and constant tensile force experiments

Abstract: The extensibility of polymer melts is of great practical importance for polymer processing. On the basis of a single-integral constitutive equation with a straindepent memory functional verified for two similar well-characterized LDPE melts, predictions are made on material behaviour in uniaxial extension under constant strain-rate, tensile stress, and tensile force conditions. It is found that experiments at constant tensile force are more adequately described by assuming purely viscous response of the polymer melts than by assuming Maxwell-model type of behaviour.

Combining Phenomenology and Physics in Describing the High Temperature Mechanical Behavior of Crystalline Solids

Abstract: It is now possible to predict quantitatively the high temperature mechanical behavior of pure metals, solid solution alloys and dispersion hardened alloys, based on an understanding of a number of physical factors influencing power law creep, including: (a) atom mobility by lattice diffusion and by dislocation pipe diffusion, (b) elastic constants of the matrix material, (c) subgrain size, (d) stacking fault energy, and (e) crystallographic texture. This quantitative picture can be extended and generalized to transient situations using the work hardening-recovery approach, and strengthening due to back stresses, solutes, and irradiation can be incorporated within the same framework. The resulting set of constitutive equations for creep rests on a firm physical foundation and yet can predict the high-temperature behavior of materials under the complex histories typical of technological applications. 72 references.

Nonlinear creep of concrete adaptation and flow

Abstract: A nonlinear integral-type creep law is developed by generalizing the linear superposition integral for the creep rate rather than the total strain. At low service stress level there is a significant (though previously overlooked) nonlinearity that consists in gradual stiffening or adaptation to a sustained compressive stress. It is modeled by a stress-dependent acceleration of the age-dependence of stiffness, and by an adaptation parameter whose rate is a function of the stress and age. The high-stress nonlinearity that consists of a weakening of the stiffness, is essentially without memory and is described by an additive rate-type flow term. Its stress dependence and the flow rate decay is modeled by kinematic hardening. An extension to elevated temperatures, which agrees with recovery data, is indicated. Although uniaxial creep is of primary interest, a rational triaxial generalization involving proper stress invariants is derived.

Thermodynamics of Solidifying or Melting Viscoelastic Material

Abstract: Investigated are thermodynamic restrictions on rate-type creep laws for porous materials which slowly solidify while carrying load (aging of concrete, i.e., gradual filling of pores in concrete by hydration products) or slowly melt (gradual dehydration of concrete at high temperature). Thermodynamic potentials (Helmholtz free energy and complementary or Gibbs free energy) are determined. The chemical dissipation of elastic energy is calculated and the condition of its positiveness is proposed; this requires that elastic relations be introduced in terms of stress and strain rates for solidifying materials and in terms of stresses and strains for melting materials. Some creep laws used in practice are found to have thermodynamically inadmissible form. Creep laws of thermodynamically correct form are shown. The known forms of such laws often cannot, however, fit available long-time creep test data for concrete very well, unless some material parameters or rates are allowed to have thermodynamically inadmissible negative values for short periods of time.