Equilibrium stress-relaxation force

The method (27) can best be explained with reference to Figure 2. After stretching to 10, the force f is measured as a function of time. The strain is kept constant throughout the entire experiment. At a certain time, the sample is quenched to a temperature well below the glass-transition temperature, Tg, and cross-linked. Then the temperature is raised to the relaxation temperature, and the equilibrium force is determined. A direct comparison of the equilibrium force to the non-equilibrium stress-relaxation force can then be made. The experimental set-up is shown in Figure 4. 

At equilibrium and at a particular RVP, the amount of adsorbed water held by a cellulose generally will be greater if it has been obtained following desorption from a higher RVP and not by adsorption from a lower RVP. The cause of this hysteresis is not fully established [303]. One explanation is based on the internal forces generated when dry cellulose swells, limiting the amount of moisture adsorbed whereas when swollen cellulose shrinks, stress relaxation occurs since the cellulose is plastic and permits a higher uptake of moisture. 

On the other hand the nature of the retractive forces in the yield and post-yield regions has been the subject of much controversy. Bull (1945), Woods (1946a,b), Astbury (1947), Elod and Zahn (1949a), and Breuer (1962) have concluded from the effects of temperature on the retractive forces that entropy contributes very little to retractive forces at strains up to 30 %. Meyer and Haselbach (1949) and Meyer et al. (1952), however, consider that the fibers must reach an equilibrium condition before measurements are made and conclude that the forces are entirely entropic. There can be no doubt that after stress relaxation at high temperatures the residual force is largely entropic (Feughelman and Mitchell, 1959), but this force is only a fraction of the initial force. 

In the experiments, two main types of time-dependent flows have been studied start-up flows and stress relaxation. In the start-up flow experiments, shear flows with constant shear rates and elongational flows with constant elongational rates are started in the system in equilibrium under no external force, and the time-dependent stress build-up in the system is measured. In the stress relaxation experiments, constant deformations are applied to or removed from the system, and the time-dependent relaxation of the stress is measured. In this section, we study these two types within the framework of transient network theory. 

Suppose that the equilibrium stress field resulting from the relaxation step shown in part (c) of Figure 6.23 is denoted by alj(x,y). An expression for this stress field can be written immediately by appeal to the representation theorem for stress in terms of the elastic Green s function for a concentrated force under plane strain conditions. Suppose a stress field ijk(x, y) can be found which, for fixed k, is the stress field in the plane due to a concentrated force of unit magnitude applied at the origin x = Q, y = Q and acting in the A —th direction. This singular solution is known, so the solution for any distribution of concentrated forces can be constructed by... 

To initiate a chemical relaxation it is necessary to perturb the system from its initial equilibrium position. This is done by applying a forcing function, which is an appropriate experimental stress to which the system responds with a shift in equilibrium configuration. Forcing functions can be transient (a sudden, essentially discontinuous Jolt ) or periodic (a cyclic stress of constant frequency). 

In summary, one may stress that the two-time-scale description on which the Kramers approach is based (see previously) clearly appears here in the time and spatial domains. During the first stage, the system relaxes rapidly and nonexponentially on a time scale rqs t and behaves as if there is no external force. On the longer time scale t", the system is characterized by the well-defined spatial equilibrium distribution, developed equilibrium values for the dynamical variables, and relaxes exponentially. 

Figure 8.2. Force relaxation of tendon under an applied load when a connective tissue such as tendon is kept at a fixed length (strain), the force (stress) is observed to decrease with time. The elastic component of the force is the force that remains at equilibrium Fn and is time independent. The elastic fraction is the force stored that is recovered when the load is removed.

The adsorbed surfactant film is assumed to control the mechanical-dynamical properties of the surface layers by virtue of its surface viscosity and elasticity. This concept may be true for thick films (>100 run) whereby intermolecular forces are less dominant (i.e., foam stability under dynamic conditions). Surface viscosity reflects the speed of the relaxation process which restores the equilibrium in the system after imposing a stress on it. Surface elasticity is a measure of the energy stored in the surface layer as a result of an external stress. 

In many cases a sloping plateau appears, possibly due to different equilibrium pressure lattice expansions, relaxation of residual forces to relieve the stress in the metal matrix are attributed to this distinctive feature of metal hydride systems as shown in Fig. 12.1(a), the slope of the plateau has been defined as dlnpd/dhi(H/M). 

The fundamental difference between mechanical stresses and tliermal stresses lies in the nature of the loading. Thermal stresses as previously stated are a result of restraint or temperature distribution. The fibers at high temperature are compressed and those at lower temperatures are stretched. The stress pattern must only satisfy the requirements for equilibrium of the internal forces. The result being that yielding will relax the thermal stress. If a part is loaded mechanically beyond its yield strength, the part will continue to yield until it breaks, unless the deflection is limited by strain hardening or stress redistribution. The external load remains constant, thus the internal stresses cannot relax. 

The correspondence principle states that for problems of a statically determinate nature involving bodies of viscoelastic materials subjected to boundary forces and moments, which are applied initially and then held constant, the distribution of stresses in the body can be obtained from corresponding linear elastic solutions for the same body subjected to the same sets of boundary forces and moments. This is because the equations of equilibrium and compatibility that are satisfied by the linear elastic solution subject to the same force and moment boundary conditions of the viscoelastic body will also be satisfied by the linear viscoelastic body. Then the displacement field and the strains derivable from the stresses in the linear elastic body would correspond to the velocity field and strain rates in the linear viscoelastic body derivable from the same stresses. The actual displacements and strains in the linear viscoelastic body at any given time after the application of the forces and moments can then be obtained through the use of the shift properties of the relaxation moduli of the viscoelastic body. Below we furnish a simple example. 

When a polymer chain stretches, entropy tends to return the coil to its equilibrium configuration, leading to an elastic restoring force. Thus, elastic stresses are generated as the polymer chain stretches and relaxes in response to flow. A liquid exhibiting both elastic and viscous stresses is viscoelastic. We note that the ratio of a viscosity r/ to an elastic modulus G yields a characteristic relaxation time X rj/G characterizing the memory of a fluid of its past deformation history or the timescale for a stretched polymer chain to relax toward equilibrium. 

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