Constitutive relation

The set of constitutive equations for the dilute polymer solution consists of the definition of the stress tensor (6.16), which is expressed in terms of the second-order moments of co-ordinates, and the set of relaxation equations (2.39) for the moments. The usage of a special notation for the ratio, namely 

In some cases, if we consider, for example, the slow motion of a solution of very long macromolecules, the effect of internal viscosity is negligible, so that the set of constitutive equations can be simplified and written as 

For the steady-state case, both equations (9.1)-(9.2) and (9.3)-(9.4) are followed by the steady-state form of the stress tensor 

This equation makes it possible to calculate stresses for low velocity gradients to within third-order terms in the velocity gradient if one knows the moments to within second-order terms in the velocity gradients. Due to the approximations, used earlier in Chapter 2, the results are applicable for small extensions of the macromolecular coil and hence for low velocity gradients the results for the moments are valid to within second-order terms in the velocity gradients. 

We can see that a set of constitutive equations for dilute polymer solutions contains a large number of relaxation equations. It is clear that the relaxation processes with the largest relaxation times are essential to describe the slowly changing motion of solutions. In the simplest approximation, we can use the only relaxation variable, which can be the gyration tensor (S i5J), defined by (4.48), or we can assume the macromolecule to be schematised by a subchain model with two particles. The last case, which is considered in Appendix F in more detail, is a particular case of equations (9.3) and (9.4), which is followed at N = 1, Ai = 2, 

we proceed to the evaluation of the collisional integrals Pc, qc, and y by using Eq. (5.293) as the form of a collisional pair distribution function. Use the coordinates in Fig. 5.12, in which ez is chosen to be parallel to the relative velocity vn. 9 and 0 are the polar angles of k with respect to ez and the plane of ez and ex, respectively. ex, ey, and ez are the three mutually perpendicular unit vectors corresponding to each coordinate in Fig. 5.12. k, as mentioned in the previous section, is the unit normal on the collision point directed from the center of particle 1 to the center of particle 2. Thus, we have 

It is noted that the bouncing velocities of the colliding particles may be expressed by 

The first-order approximation of the collisional stress tensor Pc may be simplified from Eq. (5.277) and Eq. (5.297) as 

after some manipulation, the constitutive relations can be obtained [Jenkins and Savage, 1983 Lun etal., 1984] as follows  

So far, we are able to construct the constitutive equations for qc, Pc, and y. For moderate solids concentrations, we can neglect the kinetic contributions in comparison to the collisional ones. Thus, we can assume P Pc and qk qc. Substituting the constitutive relations into Eqs. (5.274), (5.275), and (5.281), after neglecting the kinetic contributions, yields five equations for the five unknowns ap, Up, and Tc (or ( 2)). Hence, the closure problem is resolved. 

K set of referential internal state variables I velocity gradient tensor 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Specific applications of the theory are not considered in this chapter. Only one example, that of small deformation classical plasticity, is worked out in Section 5.3. The set of internal state variables k is taken to be comprised of 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

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The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

Substituting here the corresponding geometrical and constitutive relations of Sections 1.1.3 and 1.1.4, we obtain H = H(17, w). The set of admissible displacements K is defined by the boundary conditions at F and nonpenetration conditions at the crack F, stated in Section 1.1.7. The variational form of the equilibrium problem is the following ... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

Pseudoplastic fluids are the most commonly encountered non-Newtonian fluids. Examples are polymeric solutions, some polymer melts, and suspensions of paper pulps. In simple shear flow, the constitutive relation for such fluids is... 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

Inelastic Constitutive Relations 169 spatial terms by using (A.26i), (A.46), and a similar relation for k... 

Atluri, S.N., On Constitutive Relations at Finite Strain Hypo-Elasticity and Elasto-Plasticity with Isotropic or Kinematic Hardening, Comput. Methods Appl. Mech. Engrg. 43, 137-171 (1984). 

J.N. Johnson, O.E. Jones, and T.E. Michaels, Dislocation Dynamics and Single-Crystal Constitutive Relations Shock-Wave Propagation and Precursor Decay, J. Appl. Phys. 41, 2330-2339 (1970). 

Fowles, G.R., Determination of Constitutive Relations from Plane Wave Experiments, Shock Dynamics Laboratory, Washington State University Report No. WSU SDL 70-01, Pullman, WA, 72 pp., April 1970. 

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