Set Of Constitutive Equations Research Articles

Continuum rheology of muscle contraction and its application to cardiac contractility

A set of constitutive equations is proposed to describe the mechanics of contraction of skeletal and heart muscle. Fiber tension is assumed to depend on the degree of chemical activation, the stretch ratio, and the rate of stretching of the fibers. The time rate of change of activation is governed by a differential equation. The proposed constitutive equations are used to model the time courses of isotonic and isometric twitches during contraction and relaxation phases of the muscle response to stimulation. Various contractility indices of the left ventricle are considered next by using the proposed constitutive equations. The present analysis introduces a new interpretation of the index of contractility (dP/dt)/P used in cardiac literature. It is shown that this index may not be related at all to the maximum speed of shortening and that it may be dependent on both preload and afterload. The development of pressure during isovolumetric contraction of the left ventricle is shown to be governed by a differential equation describing the time rate of change of tension during isometric contraction of myocardium fibers.

Pressure dependency, strength-differential effect, and plastic volume expansion in metals

Within the framework of a purely mechanical rate-type theory of finitely deforming elastic-plastic materials, a special set of constitutive equations is introduced. These equations include a dependence both on mean normal stress and on plastic strain, and involve non-normality of plastic strain-rate. They accommodate both the strength-differential effect and plastic volume expansion. Values of the material coefficients in the constitutive equations are approximately determined with the help of published experimental data for AISI 4330 steel. The plastic strain-rate is shown to have a large non-normal component, resulting in a small plastic volume expansion. The theory is in reasonable agreement with experiment for plastic strains up to about 2% in magnitude. Further experiments are suggested so that the constitutive coefficients can be completely and accurately determined.

Calculated hardening, softening and perfectly plastic responses of a special class of materials

A special set of constitutive equations is used to calculate possible hardening, softening and perfectly plastic responses under a combination of shearing stress and pressure.

The strength-differential effect in plasticity

In the context of a purely mechanical rate-type theory of elastic-plastic materials, a special set of constitutive equations is discussed. In particular, a yield function is chosen which includes a dependence on mean normal stress. The constitutive equations are capable of describing the strength-differential effect and also predict a plastic volume change. The values of the material coefficients in the constitutive equations are determined from published experimental data for AISI 4330 steel. The predicted results are in good agreement with experiments, except for the magnitude of the predicted plastic volume change, which is too large.

Strain-Hardening Topography of Elastic-Plastic Materials

In the context of a purely mechanical rate-type theory of plasticity, a special set of constitutive equations is discussed. A method [1,2] of characterizing strain-hardening behavior is utilized to examine the different types of response that may be exhibited. Loci of constant strain-hardening behavior in stress space and regions of hardening, softening, and perfectly plastic behavior are determined.

A continuum theory for fully saturated porous elastic materials

A continuum theory for a fully saturated poroelastic material is presented. This continuum description first introduces distributed fluid and solid bodies in an average sense and defines a set of appropriate kinematical and dynamical variables. The conservation laws are then considered for each phase along with appropriate entropy inequalities. These inequalities are then exploited along with the conservation laws and a method of Langrangian multipliers to derive a set of constitutive equations for the medium under consideration. The derived governing equations of motions of solid and fluid phases then indicate that the present formulation contains Biot's theory of poroelastic media as well us Brinkman's theory for flow through porous media as special cases. Finally, a problem on wave propagation in such media is also investigated.

A continuum theory for two phase media

A continuum theory of a two phase solid-fluid media is formulated. The basic balance laws for the solid phase as well as for the fluid phase are presented. Based on thermodynamical consideration a set of constitutive equations are derived and the basic equations of motions of the distributed solid and fluid continua are obtained and discussed. It is shown that the theory contains as its special cases, Mohr-Coulomb criterion of limiting equilibrium of granular materials, Saffman theory of dusty gas, as well as Darcy's law of flow through porous media. It is then concluded that the present theory covers the full spectrum of two phase solid-fluid media from low porosity granular media with Darcy's law of fluid motion to low and high concentration two phase flows such as dusty gas and blood flow.

A generalized continuum theory for granular materials

A generalized continuum theory for granular media is formulated by allowing for the possibility of rotation of granules. The basic balance laws are presented and based on thermodynamical consideration a set of constitutive equations are derived. The theory naturally gives rise to the generation of antisymmetric stress tensor and existence of couple stresses. The basic equations of motion are derived and it is shown that the theory contains Mohr-Coulomb criterion of limiting equilibrium as a special case. The problem of coupled porosity and microrotational wave propagation is investigated and the rectilinear shear flow of granular materials is discussed.

Steady and Transient Creep Behavior Based on Unified Constitutive Equations

A set of constitutive equations, which has been used to describe a variety of quasi-static and dynamic viscoplastic phenomena, are applied to creep problems. Simulations of steady and transient creep at constant stress are obtained which compare well with experimental results both qualitatively and quantitatively. Simulations of creep resulting from rapid changes in the applied stress also compare well with observations. The constitutive equations can be used to describe logarithmic creep. These latter predictions are compatible with experimental results in considerable detail.

Further development of generalized constitutive relations for metal deformation

A set of constitutive equations is presented to describe the history dependent plastic deformation behavior of anisotropic metals under multiaxial loading conditions. The primary variables that characterize the material behavior with increasing deformation are the effective flow strength,k; the residual or back stress vectorαi; and the anisotropy matrix,Mtj. The strain rate is given in terms of an equivalent plastic strain rate, the gradient of a plastic potential, which incorporates key material variables and the stress state. All the material parameters have been determined for a recrystallized Zircaloy-2 from 25 to 450 °C and a solution treated 304 stainless steel from 25 to 650 °C based on experiments that included monotonic, load reversal, and load relaxation tests. The analytical model has been used to simulate the deformation behavior of both metals under a variety of testing conditions, including cyclic and plane strain loading conditions. The general form of the constitutive relation is shown to be consistent with experimental data obtained for various material orientations under different loading conditions.

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

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.

A unified approach to predicting interactions among creep, cyclic plasticity, and recovery

A phenomenological set of constitutive equations has been developed in recent years for high temperature structural analysis. The equations provide a unified description of deformation in both the so-called “short-time plastic” and “long-time creep” regimes for monotonic and cyclic behavior. In particular, a single strain-rate equation is utilized in both regimes, and a single set of internal variables (which reflect such metallurgical features as dislocation pileups, subgrains, and Cottrell atmospheres) governs the non-elastic strain rate under all conditions. These internal variables thus provide the means for calculating interactions such as the effect of creep on subsequent cyclic response or the effect of cold work, recovery, and/or cyclic straining on subsequent creep response. Specific examples of such simulations are given.

Nonlinear Anelastic Behavior of a Synthetic Rubber at Finite Strains

A set of constitutive equations has been developed to represent nonlinear anelastic behavior, i.e., energy losses and rate‐dependent moduli for geometrically large, reversible deformations. These equations are in incremental form and can be used to solve boundary value problems by numerical methods. Two problems were considered: thick‐walled spheres subjected to cycles of internal pressure, and a long rod subjected to imposed axial velocity (the simple tension test). Experiments were carried out for both conditions on specimens made of a synthetic rubber at three different temperatures (each under isothermal conditions) and over two decades of loading rates. Response curves were calculated based on the constitutive equations and chosen values of the material constants. Generally good agreement was obtained between the predicted and experimental results.

A simplified phenomenological model for non-elastic deformation: Predictions of pure aluminum behavior and incorporation of solute strengthening effects

From a previously-presented phenomenological model for monotonic and cyclic deformation, a simplified set of constitutive equations is derived. Quantitative predictions for pure aluminum are compared with the corresponding experimental data for a variety of test situations, including (1) the effect of temperature and strain level on the strain-rate sensitivity, (2) the effect of applied stress on the amount of primary creep, and (3) response to complex loadings such as abrupt changes in stress. A major addition is made to the equations to incorporate solute strengthening effects; the resulting model can generate fairly realistic predictions of (1) the effect of temperature on the yield strength, (2) the effect of solute additions on strain-rate sensitivity, (3) the effect of work hardening on flow stress and strain-rate sensitivity, (4) the effect of solute additions on steady-state creep behavior and (5) peaks in apparent activation energy.

The effects of Brownian rotations in a dilute suspension of rigid particles of arbitrary shape

A set of constitutive equations is derived to describe the time-dependent flow of a dilute suspension of identical rigid particles of arbitrary shape which are influenced by Brownian couples. The form of the orientation probability distribution for small departures from isotropy is found. The case of weak flows with strong Brownian effects is studied in detail, and the viscoelastic approximation and second-order-fluid limit of the constitutive equation are derived for a general particle shape. A full numerical solution is given for ellipsoids. The general nearly spherical particle is also considered and constitutive equations for general flow strengths are obtained for this case.

On the Theory of Extensible Nematic Liquid Crystals

Abstract The theory of heat conducting nematic liquid crystals with stretchable director is considered. The theory of micropolar media with stretch is reviewed and the appropriate field equations are presented. The thermodynamics of the nematic liquid crystals is studied and a set of constitutive equations are derived. The explicit equations of nematodynamics are obtained and some of their features are discussed. The equilibrium state and the rectilinear plane flows are also considered and some simple solutions are presented.

A probabilistic approach to polycrystalline plasticity part I: theory

A set of constitutive equations for polycrystalline plasticity is derived using arguments based directly on the dislocation processes involved. Distributed glide-plane orientations and Burgers-vector directions facilitate handling of the polycrystalline structure, and they yield equations involving probability distributions for variables that are directly related to measurable dislocation quantities. When the motion of the dislocations is isochoric, the tensorial character of the plastic strain rate is shown to be entirely determined by a second-rank symmetric tensor directly related to ordinary elements of crystallographic glide. This same tensor is also shown to determine the resolved shear stress acting on a dislocation in the direction of its Burgers vector, a quantity critical to the determination of the dislocation speed. Evolutionary equations for the dislocation density and the mobile fraction of dislocations are developed to complete the material description. The resulting theory, which allows for the production and interaction of non-uniform dislocation distributions, can model such phenomena as the development of anisotropy with plastic deformation, and material hardening or softening.

Correlation and Molecular Interpretation of Data Obtained in Elongational Flow

Elongational and shear flow experiments on solutions of polybutadiene in dekalin and polyacrylonitrile in dimethyl formamide have been examined. It has been found that by using the total strain in elongation rather than strain rate the data obtained at different rates of extrusion and take-up rate in the elongational flow experiments can be correlated. The use of the concept of total strain can be shown to allow the data to be considered in terms of constant macromolecular entanglement. Moreover, by plotting the variation of the normalized elastic modulus in elongation, Ḡn=G/Ḡ0, against the normalized elongational viscosity, η̃n=η̃/3η0, it becomes possible to obtain the total strain at which the elongational response of a polymer solution changes from mainly viscous to mainly elastic. The results have been considered in terms of a set of constitutive equations developed by Marrucci et al. Remarkable similarity exists between their theoretical predictions and the experimental results, in elongational flow, on the above solutions. These are discussed on the molecular level and it is proposed that the anomalous situation that exists between the proposal that chain entanglement decreases and the change in response of the fluid from viscous to elastic as elongation progresses, can be resolved. This is achieved by postulating the existence of a network of very-long-lifetime entanglements between the macromolecules.

Mechanics of a second-order micro-elastic solid

The field equations of a second-order microcontinuum are discussed and the equation for the local entropy production of a second-order micro-elastic solid is obtained. A set of constitutive equations are also presented and discussed.

Constitutive Equations for Elastic-Viscoplastic Strain-Hardening Materials

A set of constitutive equations has been formulated to represent elastic-viscoplastic strain-hardening material behavior for large deformations and arbitrary loading histories. An essential feature of the formulation is that the total deformation rate is considered to be separable into elastic and inelastic components which are functions of state variables at all stages of loading and unloading. The theory, therefore, is independent of a yield criterion or loading and unloading conditions. The deformation rate components are determinable from the current state which permits an incremental formulation of problems. Strain hardening is considered in the equations by introducing plastic work as the representative state variable. The problem of tensile straining has been examined for a number of histories that included straining at various rates, rapid changes of strain rate, unloading and reloading, and stress relaxation. The calculations were based on material constants chosen to represent commercially pure titanium. The results are in good agreement with corresponding experiments on titanium specimens.