Constitutive equation Memory function

Here we confine ourselves to the short range memory expressed by the constitutive principle of differential memory constitutive equations are functions of values and time derivatives of the thermokinetic process (2.5) taken in the present instant. Such constitutive equations may be expected in the material with fading memory where the history (2.5) is developed in the Taylor series about the present instant with restriction to the slow processes (2.5) when higher time derivatives are negligible, cf. [31]. ... 

It has been shown that the thermodynamic foundations of plasticity may be considered within the framework of the continuum mechanics of materials with memory. A nonlinear material with memory is defined by a system of constitutive equations in which some state functions such as the stress tension or the internal energy, the heat flux, etc., are determined as functionals of a function which represents the time history of the local configuration of a material particle. 

Constitutive equations for the Rouse and Zimm models have been derived, and are found to be expressible in the form of Lodge s elastic liquid equation [Eq.(6.15)], with memory function given by (101) ... 

A memory function M t —t ), which is often applied and which leads to commonly used constitutive equations, is written as... 

Equations 3.4-3 and 3.4-4 form the molecular theory origins of the Lodge rubberlike liquid constitutive Eq. 3.3-15 (23). For large strains, characteristic of processing flows, the nonlinear relaxation spectrum is used in the memory function, which is the product of the linear spectrum and the damping function h(y), obtained from the stress relaxation melt behavior after a series of strains applied in stepwise fashion (53)... 

However, the very first attempt to justify DET starting from the general multiparticle approach to the problem led to a surprising result it revealed the integral kinetic equation rather than differential one [33], This equation constitutes the basis of the so-called integral encounter theory (IET), which is a kind of memory function formalism often applied to transport phenomena [34] or spectroscopy [35], but never before to chemical kinetics. The memory... 

One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the material inder study. Furthermore, linear viscoelasticity and nonlinear viscoelasticity are not different fields that would be disconnected in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation that would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

The development of molecular constitutive equations for commercial melts is still a challenging unsolved problem in polymer rheology. Nevertheless, it has been found that for many melts, especially those without long-chain branching, the rheological behavior can be described by empirical or semiempirical constitutive equations, such as the separable K-BKZ equation, Eq. (3-72), discussed in Section 3.7.4.4 (Larson 1988). To use the separable K-BKZ equation, the memory function m(t) and the strain-energy function U, or its strain derivatives dU/dli and W jdh, must be obtained empirically from rheological data. 

An often used method describing the influence of the (long) past and expressing the natural scales more explicitly is the method of internal variables [17, 56-64]. Constitutive equations (for simplicity we discuss the uniform fluid body) even with long range memory are functions of external (i.e., from outside controlled) variables like V, T and of (even several) internal variables (3i. Each / , is controlled by evolution equation for its rate / ,... 

Summary. A procedure really specific for the rational thermodynamics is introduced in this section in the form of several principles put forward to derive the thermodynamically consistent constitutive equations. In their most general form, the constitutive equations were proposed as functions (3.118) on the basis of the principles of determinism, local action, differential memory, and equipresence. They were further reduced to the form (3.121) considering the same material throughout the body and applying the principle of objectivity. Because of our interest in fluids only, the constitutive equations were further modified to this material type by means of the principle of material symmetry giving the final form (3.127). Two special types of fluid were defined by (3.129) and (3.130). 

Because fields (4.121) are controlled from the outside (of the mixture), constitutive equations are relations between (4.119), (4.120) according to the constitutive principle of determinism their independent variables form the thermokinetic process (4.119) giving their values as responses (4.120). For simplicity, we restrict to recent past and nearest surroundings of the considered response by constitutive principles of differential memory and local action. Constitutive equations for responses (4.120) are then functions of the following values of thermokinetic process (4.119) and their (time and space) derivatives taken in a considered instant and place of response (in referential description introduced in Sect.4.1 similarly as in Sect. 3.1, i.e. as (4.1), Py = PyOi-y, t), T = T(Ky, t)), namely... 

So the first-order equation of the dynamics of a macromolecule in very concentrated solutions and melts of polymers has the form (41) where the memory functions defined by relations (42) and (45). This linear equation does not include the reptation dynamics of a macromolecule introduced by de Gennes [8] as a special type of anisotropic motion a macromolecule moves along its contour like a snake. Unbounded lateral motion is assumed to be completely suppressed due to the entanglement of the tagged macromolecule with its many neighbouring coils which, it is assumed, effectively constitutes a tube of radius The reptation of a macromolecule is considered to be important to describe the dynamics of solutions and melts of polymer [9]. 

It is evident in the foregoing examples that deviations from linear viscoelastic behavior are evoked by both large strains and large strain rates. Phenomenological constitutive equations have been developed in which one or the other has a dominant role, as described for example by memory functions which depend either on strain invariants or on strain rate invariants. - " In critical comparisons of pre-... 

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. 

Let us now look at steady-state shear flow properties that the constitutive equation, Eq. (4.158), predicts. Figure 4.15 gives log j//jo versus logr y plots, which were obtained by numerical integration of Eq. (4.167) with the aid of (4.159) for the memory function m(t) and Eq. (4.165) for Fiiyt). Also plotted in Figure 4.15 are. 

The first constitutive equation to be derived from a tube model was that of Doi and Edwards [1]. It can be obtained from Eq. 10.5 by first using Eq. 10.13 for the memory function ... 

Equation (4.4) represents a viscoelastic material where all the time effects come from the history of the scalar invariants of strain and also from aging effects, which can be eliminated by removing the variable t. Depending on the form of the functionals 4 3, this particular constitutive equation can describe both permanent and fading memory viscoelasticity with strain coupling. When the history dependence is eliminated from the functionals then... 

This Lodge network model result is a special case of the Lodge elastic liquid, in that the memory function is a sum of exponentials it is also of the same form as the constitutive equation for the Rouse and Zimm models, except that here the constants Aj and f]j are free parameters to be determined from the experimental data. If these quantities are both taken to be proportional to then the zero-shear-rate viscosity is proportional to and the first normal-stress... 

Either of these Vineyard-like approximations, along with an additional closure relation, will allow the exact results for A (t), F(k, t), and F %k, t) to constitute a closed set of equations. The closure relation consists of an independent approximate determination of the self irreducible memory function O Kk, t). One inmitive notion behind the proposed closure relation is the expectation that the -dependent self-diffusion properties, such as F k, t) itself or its memory function O k, t), should... 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

In equation 16, t is the time when the change in temperature occurs, is the exponent, and r is the characteristic relaxation time. Nonexponentiality memory effect) is reflected in the value of < 1. When treating nonisothermal situations arbitrary thermal history) the relaxation function can be represented by the superposition of responses to a series of temperature jumps constituting the actual thermal history. The Active temperature is defined as the actual... 

---