Nonlinear stress relaxation

In defining the material functions that describe responses to simple-shear deformations, a standard frame of reference has been adopted. This is shown in Fig. 10.4. The shear stress 7is the component i (equal to 7i2 because of the symmetry of the stress tensor), and the three normal stresses are 7u, in the direction of flow (xj), Gjj in the direction of the gradient and 733, in the neutral (x ) direction. As this is by definition a two-dimensional flow, there is no velocity and no velocity gradient in the Xj direction. However, in describing shear flow behavior, we will follow the conventional practice of referring to the shear stress as 7, and the shear strain as y, where neither symbol is in bold or has subscripts. 

This section treats the time evolution of the stress after a sudden deformation is given to an equilibrium state of the network. The deformation, being followed by a constant strain, creates a stress which gradually relaxes with time. The long-term behavior of stress relaxation following a large stepwise deformation is frequently measured in rheological experiments. It is known as the nonlinear stress relaxation. 

The integral equation (9.41), together with the specific forms of and v°(t), 

Due to the affineness assumption, the stress propagator reduces to an isotropic form 

The stress supported by the chains that are initially active decays according to the law 

We focus on Gaussian chains with a constant recombination rate a in this subsection. Consider a shear deformation 

---

As we have seen the theory has predicted many aspects of the nonlinear stress relaxation. However, there are some experimental results which are not in accordance with the theory and need some discussion. 

Fig. 9.17 Nonlinear stress relaxation of the transient network model with a quadratic chain dissociation rate under a constant shear deformation for y = 0.5. The decay rate is fixed as (a) /3q = 0 and (b) /3q = 1. The total number Ve of active chains and the number Vg of chains that remain active from the initial state are shown on a logarithmic scale. These are normalized by the stationary value of Ve. The shear stress hxy, the first normal stress difference N, and the second normal stress difference N2 are shown in the unit of Ve B T. (Reprinted with permission from Ref. [19].)...

In order to study how the nonlinear stress relaxation depends on the type of deformation, we next consider an elongational strain for which the strain tensor is given by... 

The relative magnitudes of C- and Cpj, are expressed by the ratio = Cp / C- + Cp ) which is aetermined experimentally from nonlinear stress relaxation experiments on the uncross-linked polymer. The contributions to the true stress a of the dual network at any stretch ratio X from the two individual networks are then... 

Fig. 25. Reduced nonlinear stress relaxation modulus as a function of strain for a polymer solution. This plot illustrates the strongly nonlinear or strain-dependent behavior of entangled polymers. After Osaki et al. (80), with permission.

The Knauss-Emri model captures some of the nonlinear stress relaxation response of materials and looks like linear viscoelasticity in the reduced time variables, and hence is relatively straightforward to implement. However, the observation that material nonlinearities occur in shearing deformations as well as in compression, where the free-volume mechanisms predict decreasing mobility suggest that the model is limited in its usefulness (164,165). 

H. G. Merriman and J. M. Caruthers, Nonlinear Stress Relaxation of a Styrene-Butadiene Random Copolymers/ Polym. ScL, Polym. Phys. Ed. 19,1055-1071 (1981). 

An example of nonlinear stress relaxation is shown in Fig. 16-17, where the ratio of time-dependent tensile stress to tensile strain is plotted logarithmically against time for different strains for cellulose monofilaments. (In this case the structure is no doubt preoriented.) The differences can be interpreted as due to a decrease in relaxation times with increasing stress, and the curves can be combined approximately into a composite curve by plotting with reduced variables, with a shift factor Os which decreases very rapidly with increasing strain. It is doubtful, how-ever, 2 whether the latter can be entirely related to fractional free volume in crystalline polymers as it is for amorphous polymers (Section Cl of Chapter 15). 

Bair, S. S. Bearder, H. E. Kern, J. T. Ryan Analysis of Nonlinear Stress Relaxation in Polymeric Glasses, Bell Laboratories, Murray Hill, New Jersey 07974. 

---