Formulation Of Constitutive Relations Research Articles

Finite-element analysis of flows of non-Newtonian fluids in three-dimensional enclosures

A finite-element model based on the penalty function formulation of the Navier-Stokes equations governing flows of unsteady, incompressible, non-isothermal, non-Newtonian fluids in three-dimensional enclosures is presented. Power-law and Carreau constitutive relations are used, and the viscosity is assumed to be temperature dependent. The resulting non-linear equations are solved by Picard's method (i.e. direct iteration). The finite-element model is used to analyze several problems, such as flows through circular pipe, cubical cavity, and tubular and square contractions. Wherever possible, comparisons are made between present numerical solutions and available analytical or numerical solutions. The present results are in good agreement with other solutions. The finite-element model developed here can be modified to accommodate other forms of constitutive relations.

Progress in the bond graph representations of economics and population dynamics

The dynamics of the marketplace are amenable to a bond graph description. Franksen pointed out that the economic laws of Walras are the equivalents of the Kirchhoff laws and that this implied a network or structural aspect of the marketplace. We begin with this observation and fully develop the sign and signal-flow conventions. More importantly, we conduct a survey of the economics and related literatures in order to recast the conventional theory into the form of constitutive relations. That work is summarized and extended in this paper. Other new contributions described are associated with economic populations. (An instance of such a population is a group of machines, each of which undergoes a life cycle and then is replaced, refueled or maintained.) The discussion is illustrated with examples. One example is a simple market whose dynamics are characterized by a chaotic attractor, another example is a bond graph of the recycling of a commodity.

Load transfer in a composite containing a broken fiber with imperfect bonding

The effect of interface behavior on the local deformation field is studied by analyzing a composite consisting of a broken fiber imperfectly bonded to its surronding matrix. The fiber-matrix interface is modelled by constitutive relations that specify the value of the shear stress in terms of the relative displacements of the fiber and the matrix along the interface. Two different forms of constitutive relations are considered, one corresponding to an elastic interface model, another to a plastic interface model with possible debonding at sufficiently large interface displacement. The latter model is used to study the process of interface crack growth and fiber pull out. Analytic and numerical solutions are presented for the both models.

The strain energy function in axial plant growth

A model for axial plant growth is formulated based on conservation of energy. The model derivation assumes that a strain energy function exists to describe the dissipation of potential energy associated with water uptake, mechanical deformation, and biosynthesis during growth. The derivation does not, however, make any further assumption on the mathematical form of this constitutive relation. The model is employed to investigate possible forms of the strain energy function as applied to steady root growth. Solutions of the nonlinear partial differential equations governing growth are given for cases when the third derivative of the strain energy function is >, <, or =0. These three cases encompass a multitude of mathematical forms of the strain energy function. The resulting solutions are compared with the realization of steady axial root growth. The results of this analysis indicate that a quadratic form of the strain energy function best described steady growth. This conclusion is consistent with previous assumptions on the form of constitutive relations for growth, and allows further interpretation on the water relations, mechanical, and biosynthetic energies associated with plant growth.

On the application of dual variables in continuum mechanics

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.

Finite element procedures for large strain elastic-plastic theories

In a unified approach, the virtual work equations in rate form and in incremental forms are derived rigorously for elastic-plastic continuum subjected to large strains. The finite element procedures for the analyses of elastic-plastic solid based on Lee's theory and the Green-Naghdi theory are presented. Also, it is shown that transformations can be performed among the Eulerian, the Total-Lagrangian, and the Updated-Lagrangian formulations, and among different forms of constitutive relations, without any approximation.

Misconception which led to the "material frame-indifference" controversy.

The confusion over the nature of the useful restriction on the allowable forms of constitutive relations (``material frame-indifference'', or ``objectivity'') is due to the vague language of its formulation. In the final analysis, the confusion arises because the concept of general covariance of physical laws is applied in the inappropriate setting of the three-dimensional space instead of the four-dimensional space-time.

Specular light scattering from a chiral medium: Unambiguous test of gyrotropic constitutive relations

Amplitudes for light reflection from an isotropic chiral medium are derived for both the Condon and Born forms of constitutive relations. Contrary to general belief, the two forms lead to very different physical predictions. The measurement of differential reflection of left and right circularly polarized light provides a decisive experimental test.

On hardening anisotropy of K0‐consolidated clays

AbstractThe combined isotropic‐kinematic hardening rule for clays is presented using the concept of a multisurface model discussed previously in Reference 1. The mechanical behaviour of clays after one‐dimensional consolidation is next simulated for the axisymmetric stress stress state. A general three‐dimensional formulation of constitutive relations is also presented.

On generalization of uniaxial stress-strain relation

Different forms of constitutive relations have been advanced for elastic, plastic and elastic-plastic behaviour of materials. It is shown that the various forms of the stress-strain relationship are specialized forms of generalization of a single stress-strain relation. For example, it is shown how the laws of elastic deformation, and the incremental and total deformation relationship for plastic behaviour are derivable from the Ramberg-Osgood relation.

Determination of Third- and Fourth-Order Longitudinal Elastic Constants by Shock Compression Techniques — Application to Sapphire and Fused Quartz

A number of solids sustain large elastic compressions under shock-wave loading. In these solids, measurements of the stress and compression in the direction of shock propagation can be used to calculate both third- and fourth-order longitudinal elastic constants if measurements are carried out over a wide range of compressions. Only limited measurements of fourth-order constants have been previously determined by other techniques. Determinations of third-order constants under these large elastic compressions afford the opportunity to test the applicability of the finite-strain formulation of constitutive relations. A general method for calculating these third- and fourth-order constants is presented and applied to shock compression data for sapphire and fused quartz. For sapphire, it is found that C111 ≈ C333 = − (3.3±0.3) × 104 kbar and C1111 ≈ C3333 = + (5.0±1.5) × 105 kbar. For fused quartz, it is found that C111 = + (5.5±0.1) × 103 kbar and C1111 = + (110±10) × 103 kbar. The technique and method of analysis seem generally applicable to solids that exhibit elastic limits of a few percent of their longitudinal elastic constants.