Fading memory

The stress depends on the extent of reaction, p(tf), which progresses with time. However, it is not enough to enter the instantaneous value of p(t ). Needed is some integral over the crosslinking history. The solution of the mutation problem would require a constitutive model for the fading memory functional Gf Zflt, t p(t") which is not yet available. This restricts the applicability of dynamic mechanical experiments to slowly crosslinking systems. 

Coleman,B D., Noll,W. Simple fluids with fading memory, pp. 530-552. In Proc. of the International Symposium on Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, Haifa. New York Pergamon Press 1962. 

Day W.A., The Thermodynamics of Simple Materials with Fading Memory, Springer Tracks on Natural Philosophy, 22, Berlin (1972). 

One can notice that the dissipative terms in the dynamic equation (3.11) (taken for the case of zero velocity gradients, z/jj = 0) have the form of the resistance force (D.3) for a particle moving in a viscoelastic liquid, while the memory functions are (with approximation to the numerical factor) fading memory functions of the viscoelastic liquid. The macromolecule can be considered as moving in a viscoelastic continuum. In the case of choice of memory functions (3.15), the medium has a single relaxation time and is characterised by the dynamic modulus... 

A fading memory function rj(s) can be represented as a sum of exponential functions... 

This differential form can be integrated to give the integral form of the model which can also be derived from the Boltzman superposition principle using the concept of fading memory of viscoelastic liquids ... 

J.U. Kim, Global smooth solutions for the equations of motion of a nonlinear fluid with fading memory,, rch. Rat. Mech. Anal., 79 (1982) 97-130. 

The time-dependent growth of Nx after start-up of steady shearing for a polyethylene melt is shown in Fig. 1-10. Note that at steady state the first normal stress difference is larger than the shear stress at this particular shear rate. The normal stress differences usually are more shear-rate-dependent than the shear stress. In fact, if the isotropic liquid belongs to a fairly general class known as viscoelastic simple fluids with fading memory (Coleman and Noll 1961), then at low shear rates the normal stress differences depend quadratically... 

These coefficients, along with the shear viscosity rj = cyii/y, often approach constant values at low shear rates these are called the zero-shear values, rjo, 4>i,o, and 4 2,o- Figure 1-9 shows for a polyethylene melt that the zero-shear constant values of rj r]o and 4 1 — 4/1,0 are approached at low shear rates. For a viscoelastic simple liquid with fading memory, the zero-shear values of the viscosity and first normal stress coefficient are related to the zero-frequency values of the dynamic moduli by... 

At low shear rates, a viscoelastic simple liquid with fading memory satisfies the constitutive equation of a Newtonian liquid, that is,... 

Figure 5.15 Form of the fading memory of Eq. (5.40) in a shear creep experiment.

Figure 5.16 Schematic sketch showing fading memory for an arbitrary shear stress history.

Though the stress at time t in memory fluids is expected to depend on the history of the deformation, the dependence is stronger for recent deformations than for ancient ones. In other words, these fluids exhibit fading memory (23). The slow flow and small deformation approximations have been used to establish constitutive equations for memory fluids. In the slow flow approximation (23), a sequence of deformation histories is assumed in which each history differs from a reference history in that the time scale is slowed by a... 

By the close of the twentieth century the golden age of drug discovery research had become a fading memory, to be passed on by research old-timers to fresh new faces in the... 

Constitutive principles of memory give more detailed description of the influence of the past on the response. Section 2.3 describes the role of time scales (cf. Sect. 1.1) for this description and the expression of the influence of the past (or history) through internal variables. The influence of the past frequently expected from the experience is contained in the constitutive principle of fading memory the more the past is remote, the less influence it has on the response. Mathematical formulation of this principle need not be unique [24-29] (for introduction see [2, 4, 6, 30]). In limit of equilibrium there is some time interval (natural time scale or relaxation time, cf. Rem. 11 in Chap. 1 Sects. 2.4, 3.8, 4.7) after which the (practical) influence of the past is negligible. 

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]. ... 

Let the response/ be given by the function g(.) through the following example of functional with fading memory tF... 

For the general procedure of such an approximation of fading memory to a differential one (eliminating exceptional fading exponential exp(-3.t)) see [31]. 

The problem is rather in the physical identification of these hidden fSj but if they are known the use of internal variables with evolution Eq. (2.72) is another way to describe materials with (long range) memory. Solutions of (2.72) is facilitated in the case of fading memory (where always // < 0 [61]), but also more intricate cases may be described as phase changes [65], plasticity and generally any hysteresis [66], cf. also Rem. 31 in Chap. 3 important are applications in chemically reacting mixtures (where [3j may be, e.g., extent of reaction ) which will be discussed here in Sect. 2.4 and in Chap. 4. 

Lubliner, J. On fading memory in materials of evolutionary type. Acta Mech. 8,75-81 (1969) BataiUe, J., Kestin, J. Irreversible processes and physical interpretation of rational thermodynamics. J. Non-Equilib. Thermodyn. 4,229-258 (1979)... 

In summary, polymeric materials exhibit rubber elasticity if they satisfy three requirements (a) the polymer must be composed of long-chain molecules, (b) the secondary bond forces between molecules must be weak, and (c) there must be some occasional interlocking of the molecules along the chain lengths to form three-dimensional networks. Should the interlocking arrangements be absent, then elastomers lack memory, or have a fading memory and are not capable of completely reversible elastic deformations. 

In the last two centuries, a lot of attempts and discussion have been made on the elucidation and development of the various constitutive models of liquids. Some of the theoretical models that can be mentioned here are Boltzmann, Maxwell (UCM, LCM, COM, 1PM), Voight or Kelvin, Jeffrey, Reiner-Rivelin, Newton, Oldroyd, Giesekus, graded fluids, composite fluids, retarded fluids with a strong backbone and fading memory, and so on. Further and deeper knowledge related to the physical and mathematical consequences of the structural models of liquids and of the elasticity of liquids can be found in Ref. [6]. 

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