Constitutive equations media

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

Having obtained the elastic equations in terms of shifted entities, and reverting to total entities, the constitutive equations express the total stress cr, the chemical potentials of the extrafibrillar water p,wE and of the salt psE, and the hydration potential of the intrafibrillar water //hydl . in terms of the generalized strains, namely the strain of the porous medium e, the mass-contents of the extrafibrillar water mWE and of the cations sodium mNae, and the mass-content of intrafibrillar water mwi. The interested reader is directed to [3]. 

Let us now obtain a functional, which represents the free energy density of a linear, homogeneous, and isotropic medium, that satisfies the constitutive equation P = e0(e — 1 )E. To obtain the classical result for the free energy density, (l/2)E(eoE -+- P), the function /(P) must acquire the form... 

The stress tensor a in the perfectly elastic and isotropic solid phase of the porous medium is described by the constitutive equation... 

The constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions,... 

Note that under the conditions fixed at the beginning of this sub-section, the vectors E, D, and P are all parallel and proportional. Later we will see that this is not the case when the material is no longer isotropic. With Maxwell s and the constitutive equations, one can derive for an isotropic medium the following wave equation ... 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by... 

The stress-strain relationships of elastomeric (rubbery) networks at low extension (or draw) ratios (k, which is the length of the deformed specimen divided by the length of the initial undeformed specimen) can be described in terms of Equation 11.37 [29], which is the simplest possible constitutive equation for the deformation of an isotropic incompressible medium. 

This reduces from four to three the number of independent phenomenological coefficients required to describe the intrinsic resistance of the porous medium. Inasmuch as c Vq = — V x q, the constitutive equations thus become... 

It is shown that the development of the equations governing THM processes in elastic media with double porosity can be approached in a systematic manner where all the constitutive equations governing deformability, fluid flow and heat transfer are combined with the relevant conservation laws. The double porosity nature of the medium requires the introduction of dependent variables applicable to the deformable solid, and the fluid phases in the two void spaces. The governing partial differential equations are linear in view of the linearized forms of the constitutive assumptions invoked in the formulations. The linearity of these governing equations makes them amenable to solution through conventional mathematical techniques applicable to the study of initial boundary value problems in mathematical physics (Selvadurai, 2000). Such solutions should serve as benchmarks for appropriate computational developments. 

The governing equations for the THM problem are balance equations and constitutive equations. Mass conservation equations apply to water and oil. When the porous medium is deformable, the momentum balance equation (mechanical equilibrium) is also taken into account. In non-isothermal problems, the internal energy balance for the total porous medium must be considered. 

Conservation equations together with constitutive equations describe a phenomenological model of the continuous medium (continuum). 

These laws provide the basis for the continuum model and must be coupled with the appropriate constitutive equations and the equations of state to provide all the equations necessary for solving a continuum problem. The continuum method relates the deformation of a continuous medium to... 

Example 3 illustrates the use of thermodynamics principles in formulating constitutive equations for a poro-viscoelastic medium. The ultimate purpose here is also to develop solutions for a long horizontally aligned tunnel with a circular cross-section embedded in a poro-viscoelastic massif. The setting of the problem is similar to Example 2 discussed above except that the spherical cavity is replaced by a long lined tunnel (Dufour et al. 2009). We start by restricting to small strain problems where the strain tensor of a viscoelastic material can be decomposed into an elastic part (denoted by superscript e ) and a viscoelastic part (superscript V ) ... 

ABSTRACT In the present paper a multiphase model including a hypoplastic formulation of the solid phase is presented and its application to earthquake engineering problems discussed. The macroscopic soil model, which is based on the theory of porous media, comprises three distinct phases namely, solid, fluid and gas phase. For each of these the compressibility of the respective medium is taken into account in the mathematical formulation of the model. The solid phase is modelled using the hypoplastic constitutive equation including intergranular strain to allow for a realistic description of material behaviour of cohesionless soils even under cyclic loading. The model was implemented into the finite element package ANSYS via the user interface and also allows the simulation of soil-structure interaction problems. 

There will be one integration with respect to x and two with respect to y, so we will need to provide one piece of boundary information in the x direction and two in the y direction. The x condition appears to be straightforward We assume that at X = 0 the melt is uniformly at the reservoir temperature, which we denote T (for initial). The thermal boundary condition at a wall is typically written as an equality between the heat flux into the wall from conduction in the fluid and the heat flux from the wall to the surrounding heat transfer medium. It is an equality because the wall is assumed to have no thermal capacitance, so the flux into the wall must equal the flux out. The heat flux in the fluid is equal to -KdT/dy. (This is known as Fourier s law, but it is an empirical constitutive equation, not a law of nature.) The flux to the surroundings is usually written as U T - To), where Ta is the temperature of the ambient environment, which might be air or a heat exchange fluid. U is an overall heat transfer coefficient, which is characteristic of the particular geometry, materials, and flow. The appropriate boundary conditions are then... 

They bring in the stress tensor [S], which is a symmetrical 3x3 matrix. The mechanical behavior of a material is determined by the mathematical expression of the six terms in the stress tensor. The purpose of rheology is to estabhsh these relations (called constitutive equations) between the stresses apphed inside the material and the strains they cause. As such, rheology appears as a disciphne situated upstream of mechanics. The equations for the fundamental law of dynamics can only be solved after determining the rheological behavior of the medium. 

When only elastic behavior is considered, the constitutive equations for the equivalent homogeneous medium may be written in terms of ... 

Additional information about the homogeneous balances with electromagnetic fields is presented in the Appendix. The primary objective is to establish the constitutive relations of these conductive homogeneous media, which we can only do if we specify the type of medium in which we are interested metal, and then homogeneous plasma. We will need these constitutive equations in Chapter 4, when we look at interfaces, because on both sides of these interfaces, we have homogeneous media. 

This chapter deals with fundamental definitions, constitutive equations of a viscoelastic medium subject to infinitesimal strain, and the nature and properties of the associated viscoelastic functions. General dynamical equations are written down. Also, the boundary value problems that will be discussed in later chapters are stated in general terms. Familiar concepts from the Theory of Linear Elasticity are introduced in a summary manner. For a fuller discussion of these, we refer to standard references (Love (1934), Sokolnikoff (1956), Green and Zerna (1968), Gurtin (1972)). Coleman and Noll (1961) have shown that the theory described here may be considered to be a limit, for infinitesimal deformations, of the general (non-linear) theory of materials with memory. 

The calculation of the scattering from the nonatic mesophase requires the use of an appropriate constitutive equation for the stress tensor. The static depolarized light scattering of a defect-free nematic mesof ase reflects fluctuations in the curvature elasticity of the medium. The contribution to the stress from the curvature elasticity involves a curvature free eiMrgy density W. For small distortions in the nematic fleld, W may be expressed in the form [134,135,139, 140]... 

---