Strain memory function

These functions are independent of the strain-memory function, and only kfc and Gfc can be determined from dynamic data of the viscoelastic moduli. As an example. 

The strain-memory function is derived from the first and second invariants of the Finger strain tensor. For simple shear flow, the strain-memory function is given as... 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

Finally it is worth noting an alternate form for the stress dependence of a series of strains. Some microstructural models utilise the memory function m t). This is the rate of change of the stress relaxation function ... 

The term y(t,t ) is the shear strain at time t relative to the strain at time t. The use of a memory function has been adopted in polymer modelling. For example this approach is used by Doi and Edwards11 to describe linear responses of solution polymers which they extended to non-linear viscoelastic responses in both shear and extension. 

The pressure P0 represents the arbitrary additive contribution to the normal components of stress in an incompressible system, 8i is the Kronecker delta, C[ j 1(t t) is the inverse of the Cauchy-Green strain tensor for the configuration of material at t with respect to the configuration at the current time t [a description of the motion (221)], and M(t) is the junction age distribution or memory function of the fluid. 

Lodge and Meissner have recently examined in detail the stresses during start-up of steady elongational and shear flows (374). The Lodge equation [Eq.(6.15)], with its flow-independent memory function, described the build-up of stress rather well (even for rapid deformations) until a critical strain was reached. Beyond the critical strain (which differed somewhat in shear and... 

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)... 

H. M. Laun, Description of the Non-linear Shear Behaviour of a Low Density Polyethylene Melt by Means of an Experimentally Determined Strain Dependent Memory Function, Rheol. Acta, 17, 1-15 (1978). 

It is worth mentioning that the strain function is not temperature dependent and that the influence of temperature is only applied on the memory function or relaxation modulus through the shortening of the relaxation times with increasing temperatures. 

The equation leads to the definition of a time and strain-dependent memory fimction which can be further factorized into a time-dependent part (the linear memory function) and a strain-dependent damping function. Though on one hand, there is some experimental evidence for this in limited time ranges, on the other hand, a few experiments might question this strong hypothesis since, for example, the damping function obtained fi um step shear rate data is found to be different from that in step shear strain. 

H.M.Laun, Description of the non-linear shear behaviour of a low density polyethylene melt by means of an experimentally determined strain dependent memory function, Rheol. Acta 12 (1978), 1-15. 

Derived firom e Lodge s rubbeilike Hquid theory, the Wagner model is based on a concept of separabiHty, since it is assumed that the memory function is the product of a time-dependent hnear function by a strain-dependent nonlinear fimction. 

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. 

Figure 6.20 shows the evolution of recovery strain with time for the composite with CP contents of 3%, 6%, and 9%, respectively. The shape recovery (ability for the composite to recover the strain at point G) is larger than 98% for all three groups, indicating that the CP-PSMP composite retained its shape recovery ability and shape memory functionality with various CP contents. However, it can be seen that after the hft-off point L, 3% CP composite shows a steeper and sharper slope of recovery (higher recovery rate) as compared to the other two specimens. This indicates that the speed of recovery or shape recovery rate reduces with increasing CP content. For instance, when the time is equal to 3.5 hours, the 3% CP composite has already recovered 86% of the strain while the 6% and 9% specimens have only recovered... 

It is interesting to note that, while the CP content has a significant effect on the stress recovery, its effect on shape recovery is comparatively small. The reason is that shape recovery is basically a global or macroscopic behavior, while stress recovery is dependent more on the microstmeture and internal parameters. For instance, the recovery stress depends on the stiffness at the recovery temperature, while the recovery strain does not. Therefore, the shape recovery ratio and the stress recovery ratio usually do not have the same value and the stress recovery ratio is usually lower than the strain recovery ratio. In this sense, the stress recovery ratio is a more rigorous indicator of shape memory functionality. 

The light-induced shape-memory functionality of the polymers is quantified by cyclic, photomechanical experiments luider stress-controlled and/or strain-controlled conditions. These cyclic experiments have been designed in analogy to those characterizing a thermally-induced SME. 

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