General Constitutive Relations Research Articles

The gauge theory of dislocations: Static solutions of screw and edge dislocations

We investigate the T(3)-gauge theory of static dislocations in continuous solids. We use the most general linear constitutive relations in terms of the elastic distortion tensor and dislocation density tensor for the force and pseudomoment stresses of an isotropic solid. The constitutive relations contain six material parameters. In this theory, both the force and pseudomoment stresses are asymmetric. The theory possesses four characteristic lengths ℓ1, ℓ2, ℓ3 and ℓ4, which are given explicitly. We first derive the three-dimensional Green tensor of the master equation for the force stresses in the translational gauge theory of dislocations. We then investigate the situation of generalized plane strain (anti-plane strain and plane strain). Using the stress function method, we find modified stress functions for screw and edge dislocations. The solution of the screw dislocation is given in terms of one independent length ℓ1 = ℓ4. For the problem of an edge dislocation, only two characteristic lengths ℓ2 and ℓ3 arise with one of them being the same ℓ2 = ℓ1 as for the screw dislocation. Thus, this theory possesses only two independent lengths for generalized plane strain. If the two lengths ℓ2 and ℓ3 of an edge dislocation are equal, we obtain an edge dislocation, which is the gauge theoretical version of a modified Volterra edge dislocation. In the case of symmetric stresses, we recover well-known results obtained earlier.

Isotropic linear constitutive relations for nonsimple fluids

We investigate the general constitutive relation of an isotropic linear fluid when the stress tensor can depend on higher-order spatial gradients of the velocity. We apply the results to the case of second-grade and third-grade fluids, be they compressible or not. However, the expression of the general isotropic tensor can be a matter of interest also for other classes of nonsimple material.

A general constitutive relation for linear elastic foams

A three-dimensional linear elastic constitutive relation is formulated based on a representative unit cell of foam using elasticity theory and micromechanics homogenization scheme. The displacement and strain fields of the unit cell are obtained from elasticity theory and used to derive the macroscopic strain field defined on the outer surface of unit cell through homogenization scheme. By assuming a uniform macroscopic stress on the unit cell surface and the existence of strain energy potential, the constitutive relation of linear elastic foams is obtained. The newly derived constitutive relation is a function of mechanical property of solid constituent, the geometry of cell struts, and the porosity of foams and is able to characterize the anisotropic behavior of foams due to non-uniform strut geometry. The linear elastic response of open-celled foams with both low- and medium-relative densities can be studied using the derived constitutive relation. The effective elastic modulus for uniform strut geometry is reduced from the constitutive relation and agrees well with Gibson and Ashby's semi-empirical equation, Warren and Kraynik's, and Zhu's analytical models within relative density ranging from 0 to 0.35. For non-uniform strut geometry, the calculated effective elastic moduli in three axial directions are different and the material displays anisotropic behavior. The bulk modulus shows less dependence on the variation of the strut geometry. Poisson's ratios are also reduced from the compliance matrix.

Constitutive relation of weakly anisotropic polycrystal with microstructure and initial stress

Man (Nondestr Test Eval 15:191–214, 1999) derived the constitutive relation of a weakly-textured orthorhombic aggregate of cubic crystallites with effects of microstructure and initial stress. In this paper, a computational expression on the integration \({{\int_{\rm SO(3)} Q^{\otimes h}D^{l}_{m0}{\rm dg}}}\) is given. Then, by means of the computational expression, the general constitutive relation of a weakly-textured anisotropic polycrystal with the consideration of microstructure and initial stress is derived. As special cases of our general constitutive relation, two constitutive relations are given for an isotropic polycrystal and a weakly-textured anisotropic aggregate of cubic crystallites. The acoustoelastic tensor of the reference cubic crystal is derived to determine the material constants of the polycrystal. Two examples are given for understanding the physical meaning of the texture coefficients and the constitutive relations.

Flutter of viscoelastic strips

Using linear model, the parametric sensitivity analysis of viscoelastic panel flutter with an arbitrary function of relaxation, is examined by the Laplace integral transform method. The critical values of free stream velocities and frequencies of vibrations are determined from the condition that the real parts of the poles of integrand must be zero, which correspond to harmonic motion. Approximate and exact values of critical speed and corresponding frequencies for a general isotropic viscoelastic constitutive relations are obtained. The solutions are analyzed for critical, subcritical and supercritical cases. It is shown that the viscoelastic flutter speed is smaller than the corresponding elastic one if elastic moduli of material is equal to the initial value of relaxation function. Influence of aerodynamical damper is studied assuming that the parameter of viscous property of material is small enough in comparison with the parameter of aerodynamical damper and vice versa.

Challenges in developing a general constitutive relation for cyclic loading*

Abstract Deformation of most metals and alloys is time-dependent and this dependency becomes more pronounced at temperatures exceeding a third of the material's melting point. Furthermore, the deformation response of a material to cyclic loading involves transients prior to stabilization. Some manifestations of the transient responses are: strain hardening or softening, loading sequence effect, strain-rate and strain-path history dependency, creep-plasticity interaction, strain ratcheting, among others. In addition, a constitutive model formulated in terms of macroscopic parameters must have a certain underlying microscopic rationalization. In this paper rate-dependent constitutive relations for an inelastic material are presented. This constitutive model is of a coupled nature, in the sense that the effect of prior creep on the subsequent plastic deformation and vice versa, are taken into account. Thus, the framework for these constitutive relations is based on the concept that any loading sequence can be predicted by two separate (elastoplastic and creep) but coupled models. A number of examples covering a wide range of cyclic loading types are presented.

Towards a general state-variable constitutive relation to describe granular deformation

Triaxial-compression experiments were performed on cylindrical quartz sand-packs to explore the universality of state-variable constitutive relations used to describe granular deformation. Samples were subjected to relaxation-creep experiments similar to slide–hold–slide tests from friction studies. For relaxation-creep tests, a constant rate of axial shortening was interrupted by time intervals for which the applied shortening rate was set to zero. During holds, loadforce decayed exponentially with time and the total stress-relaxation scales with the logarithm of hold time, consistent with friction tests. On reloading after holds, stress monotonically increases to the pre-hold steady-state levels without displaying the restrengthening of static friction often observed from shear tests on granular layers. We find that the existing rate- and state-variable friction laws can model the transient strength behavior during relaxation creep and subsequent reloading by using simple assumptions about the deformation of cylindrical sand-packs in triaxial compression. By analogy with the rate- and state-friction constitutive relations, we propose a generalized state-variable expression for granular deformation in simple and pure shear. Although additional experiments under a variety of load boundary geometries and load paths will be needed, such a general constitutive relation may ultimately describe granular deformation under a broad range of conditions (e.g., for large variations in loading rate, shear stress, normal stress, mean stress) and may be applicable to a variety of research problems that involve deformation within granular media.

The decomposition F = FeFp, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous

This study develops a general framework for discussing both isotropic-viscoplastic materials and amorphous materials. The framework, which allows for large deformations, is based on the Kröner–Lee decomposition of the deformation gradient into elastic and inelastic parts, a system of microforces consistent with its own balance, and a mechanical version of the second law that includes, via the microforces, work performed during inelastic flow. The constitutive theory allows for dependences on the elastic and inelastic parts of the deformation gradient and on the inelastic stretch-rate, but dependences on the inelastic spin are not included. The constitutive equation for the microstress T p conjugate to inelastic flow – suitably restricted by the second law – and the microforce balance are shown to be together equivalent to a flow rule that includes a back stress due to the variation in the free energy with inelastic deformation. The introduction of a concept of material microstability reduces this flow rule to one of classical Mises-type. In a theory based on the Kröner–Lee decomposition, there are two classes of symmetry transformations available: transformations of the reference configuration and transformations of the relaxed spaces. We discuss the notion of material symmetry for a general class of materials that includes, as special cases, isotropic-viscoplastic solids, and amorphous solids. Essential to this discussion of symmetry is a general constitutive relation for the microstress T p. The symmetry-based framework allows us to show that for typical boundary-value problems involving isotropic, viscoplastic solids or amorphous solids, if a problem has a solution, then every time- and space-dependent rotation of the relaxed spaces also yields a solution, and it is possible to choose this rotation such that the transformed solution is inelastically spin-free: W p ≡ 0. Thus, when discussing such materials, we may, without loss in generality, restrict attention to flow rules that are inelastically irrotational.

압전 벤더의 효과적인 모델링 기법

In this article, the constitutive relations of three types of piezoelectric benders, which are a unimorph bender, a bimorph bender and a triple-layer bender, are derived based on the beam theory under the quasi-static equilibrium condition. The relation coefficients are described as the geometry and material properties of the benders. More general constitutive relations involving fixed-free, fixed-roll, and fixed-simply supported boundary conditions under the inconsistent length condition between the piezoelectric layer and the nonpiezoelectric one are discussed. The complicated constitutive relations can be easily calculated and checked by using the symbolic function in ‘Mathematica’. The relation coefficients for the benders are plotted in three dimensional graph using the developed program.

Problems on electrorheological fluid flows

In this paper, we develop and analyze a new model describing electrorheological fluid flow. In contrast to existing models, which assume the electric field to be perpendicular to the velocity field and are thus restricted to simple shear flow and flows close to it, we consider the fluid as anisotropic and introduce a general constitutive relation based on a viscosity function that depends on the shear rate, the electric field strength, and on the angle between the electric and the velocity vectors. We study general flow problems under nonhomogeneous mixed boundary conditions with given values of velocities and surface forces on different parts of the boundary. We investigate both the case where the viscosity function is continuous and the case where it is singular for vanishing shear rate. In the latter case, the problem reduces to a variational inequality. Using methods of nonlinear analysis such as fixed point theory, monotonicity, and compactness, we establish existence results for the problems under consideration. Some efficient methods for the numerical solution of the problems are presented, and numerical results for the simulation of the fluid flow in electrorheological shock absorbers are given.

Sol – Gel Modelling Associated with the Rheology of Polymeric Precursors of Ceramic Materials

Abstract A general constitutive relation describing the change of viscoelastic behavior during the liquid - solid (sol - gel) transition which takes place in preceramic polymers is derived on the basis of Jeffrey’s 3-constants model with time dependent viscosities and elasticity. It is postulated that the sol - gel - transition can be analyzed analogous to the solutions of the Avrami equation used for modeling crystallization processes. Two different polymer systems used as precursor for the production of ceramic materials are investigated here: i) a mixture based on polysiloxane, alumatrane and isopropanol; ii) a non-oxidic carbodiimide gel based on the reaction of chlorosi-lanes with bis(trimethylsilyl)carbodiimide. Continuous measurements of the dynamic moduli versus reaction time, as well as creep tests at constant shear stress, evidenced both qualitative similarities and quantitative differences associated with the sol - gel transition of the two polymer systems. The shear rate and viscosity dependence of reaction time in creep tests, respectively the evolution of Lissajous figures associated with oscillatory experiments, are found to be consistent with the numerical simulations of the proposed constitutive relation.

The effective stress principle

The behaviour of soil at the macroscopic scale depends on the behaviour of its components and on the interactions between them at the microscopic scale. Here, assumptions are made only at the microscopic scale and, by use of a suitable averaging procedure, consequences at the macroscopic scale are derived. At the microscopic scale, the soil is assumed to be made of bulk phases and interfaces, both having very general constitutive relations. At the microscopic scale, the incompressibility of all bulk phases, incorporated by use of a thermodynamic theory of constrained materials, implies that a given thermokinetic process experienced by the soil does not depend on an arbitrary and uniform variation of the spherical part of the stress tensor field at all points of the soil. At the macroscopic scale, that thermokinetic process appears not to be influenced by certain variations of the macroscopic stresses and exchanges of momentum of the bulk phases. An extension to the case that one bulk phase is compressible is also presented. Finally, versions of the effective stress principle commonly used in soil mechanics for saturated and unsaturated soils are derived as particular cases.

On the importance of material frame-indifference and lift forces in multiphase flows

In this paper, we discuss the importance of the Principle of Material Frame-Indifference, sometimes referred to as Objectivity, in multiphase flows. We first give a brief account of the formulation of a two-phase mixture and then indicate where and how this principle should be utilized. We will then give a brief review of the meaning of this principle in continuum mechanics and how it has been used in various fields of physics. We also give examples of frame-indifferent and frame-dependent constitutive relations which have been used in multiphase flow studies. The importance of lift forces is also discussed. Finally, we derive a general constitutive relation which is frame-invariant and general enough to be used in many applications.

Hopf bifurcation in thermal convection of viscoelastic fluids within finite domains

Hopf bifurcation in thermal convection is an interesting phenomenon peculiar to the viscoelastic fluids. Employing a general constitutive relation which encompasses Maxwell model, Jeffreys model, Oldroyd-B model and Phan-Thien-Tanner model, the range of viscoelastic parameters where the Hopf bifurcation arises is determined when the viscoelastic fluid is confined in finite domains, which are compatible with the experimental situations. Employing the power series method, Landau equation is derived which determines the temporal evolution of the convection intensity in finite domains with nonslip sidewalls when the Hopf bifurcation occurs. The comparison of these results with the experimental data may be used to guide the selection of constitutive equations and to estimate viscoelastic parameter values.

Simple flows of fluids with pressure–dependent viscosities

In his seminal paper on fluid motion, Stokes developed a general constitutive relation which admitted the possibility that the viscosity could depend on the pressure. Such an assumption is particul...

Inversion of Residual Stress

ABSTRACTA technique for inverting residual stress based on a theory of acoustoelasticity is presented. A general incremental constitutive relation is first derived for a pre-stressed material subjected to an additional infinitesimal elastic deformation. The theory is then employed on using ultrasonic means to evaluate residual stresses of residually stressed materials. The residual stresses are assumed to be homogeneous in materials as usual. The only major assumption in this formulation is that the additional deformations caused by ultrasonic evaluating process are infinitesimal and elastic. No assumption on the origin of residual stresses is needed, nor the assumption on the possible existence of “natural state” of the materials. Successful inversion of residual stresses are demonstrated through a preliminary numerical experiment.

Constitutive modelling of ferroelectric composites with a PSZ matrix

Constitutive relations of ferroelectric composites with a transformable partially stabilized–zirconia (PSZ) matrix are developed based on the mean field theory. The effect of phase transformation of the matrix on the electromechanical coupling behaviour is highlighted. The general constitutive relations are specialized to a model ferroelectric composite in which lead–zirconate–titanate (PZT) inclusions are oriented in three mutually orthogonal directions, in order to illustrate the interaction. The analysis shows that if the composite is partly or completely polarized, t → m phase transformation can be induced by an applied electric field due to electromechanical coupling. t → m phase transformation can increase the toughness of the ferroelectric composite if the absolute values of the residual stress/strain and the residual electric displacement in the composite after the t m phase transformation are larger than those before it.

The vibration and dynamic stability of viscoelastic plates

In this research, the static and dynamic behavior of plates, with a general isotropic viscoelastic constitutive relation, is examined The viscoelastic constitutive equations are written in the Boltzmann–Volterra form. It is well-known that in this form the operator has two different kernels, which make the solution somewhat difficult. In this research, constitutive equations existing in literature are expressed in operator form .The solution technique systematizes the solution of the viscoelastic problem and gives better results in comparison with existing methods in literature. The developed solution technique is applied to triangular plates under static and dynamic loads.

Spherically Symmetric Static Solutions of the Einstein Equations with Elastic Matter Source

The paper is concerned with the Einstein equations for a spherically symmetric static distribution of anisotropic matter. The equations are cast into a system of Fuchsian type ODE for certain scalar invariants of the strain. And then the existence and regularity of this ODE is studied under general constitutive relation. In the case the constitutive relation is given by a quadratic form of strain, it is also shown that the solutions stay regular up to the boundary of the material ball.

General Constitutive Relations of Anorthotropic 6061-T6 Aluminum and Its Fatigue Life under Proportional and Non-Proportional Loadings

Abstract Anisotropic constitutive relations with respect to the principal axes of an orthotropic solid are formulated using the macroscopic continuum plasticity theory of Shih and Lee. The general anisotropic constitutive relations are derived for predicting the plastic deformation behavior of a new material orientation. Straincontrolled cyclic tests were conducted along the 45° direction in the plane of the plate to verify the general anisotropic constitutive relations. The predictions agree very well with the experimental yield and plastic flow behavior. The axial-torsional coupling terms in the general constitutive relations are also determined and explained. The constitutive equations were used to predict the low-cycle fatigue life of this material under both proportional and non-proportional loadings. An anisotropic shear cracking model, developed earlier, provided a good prediction of the fatigue life of this material under both proportional and non-proportional loadings.