Constitutive equations higher order

Both E, in ideal solids, and rj, in ideal liquids, are material functions independent of the size and shape of the material they describe. This holds for isotropic and homogeneous materials, that is, materials for which a property is the same at all directions at any point. Isotropic materials are so characterized because their degree of symmetry is infinite. In contrast, anisotropic materials present a limited number of elements of symmetry, and the lower the number of these elements, the higher the number of material functions necessary to describe the response of the material to a given perturbation. Even isotropic materials need two material functions to describe in a generalized way the relationship between the perturbation and the response. In order to formulate the mechanical behavior of ideal solids and ideal liquids in terms of constitutive equations, it is necessary to establish the concepts of strain and stress. 

The Rivlin-Ericksen constitutive equation gives a good account of some characteristics of both the time dependence of the viscoelastic behavior and the normal stress effects. This relationship is based on the assumption that the stress depends not only on the velocity (x ) and the shear rate gradient (dxi/dx ) but also on derivatives of higher order (%, dXp/dXq. .. 8xf /8xi). As a consequence of the principle of material... 

C. Fluid with a higher order memory in volume [4] has constitutive equations (2.8) and therefore by (2.12)... 

Most of the previous developments have proceeded from Maxwell s boundary conditions which are known now to be in error. WALDMANN and co-workers have derived thermodynamically consistent boundary conditions for higher-order constitutive equations. For monatomic gases and Maxwell molecules, these constitutive equations reduce to Grad s 13 moment equations. However, WALDMANN has pointed out that Grad s boundary conditions are thermodynamically inconsistent. For the drag force problem, VESTNER and WALDMANN derived... 

From Waldmann s derivation of thermodynamically consistent boundary conditions for higher-order constitutive equations (valid for small Kn ) [2.103], it appears that the theoretical developments just cited, including those of DERJAGUIN and co-workers, have proceeded from either thermodynamically inconsistent boundary conditions or inaccurate (e.g.. Maxwell s) boundary conditions. Therefore, the apparent agreement suggested, for example by SPRINGER [2.134], PHILLIPS [2.128], or ANNIS and MASON [2.135] may be fortuitous, particularly for the slip regime. 

As an example of this approach let us consider the constitutive equation arrived at (a) by adopting unchanged the field equations and boundary conditions of the linear theory, and (b) introducing cubic and higher order terms in the polynomial representation... ... 

The substantive grade 1 is generally used referring to such models for remarking the dependence of stress on the strain rate only [108]. Constitutive equations with higher order time derivatives are also used [165]. 

To summarize, the QCISD noodel constitutes a size-extensive revision of the CISD model closely related to the CCSD model, from which it may be obtained by omitting some of the higher-order terms in the linked equations. The terms omitt in the (JCISD equations are not computationally demanding and the cost of the QCISD model is the same as for CCSD. For the calculation of total... 

So far the boundary value problem has been considered in terms of a second order differential equation. It is know however from the previous chapter that such a second order equation can also be written as two first order differential equations in two dependent variables by introducing the first derivative as a second variable. In addition one can have a single differential equation of higher order than two with a set of conditions for the variable and derivatives specified at two points and this again constitutes a boundary value problem. Thus the general boundary value problem can be specified as a set of first order differential equations ... 

---