Time-dependent stress relaxation

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Substituting the relation between relaxation time and mode wavelength [Eq. (8.109)] into the expression for modulus [Eq. (8.116)] leads to the time-dependent stress relaxation modulus that decays as the - 3/4 power of time ... 

These effects have the potential to influence the important time dependent stress relaxation characteristics of foamed components, possibly accelerating compression set. Clearly evident from this work is the need to minimise residual catalyst and cure associated acid levels within the rubber so as to maximise material reliability over extended timescales. 

For experiments performed in shear, there is a rather complicated relation between the time-dependent stress relaxation shear modulus G(t) defined by Equation 3.19 and the time-dependent creep compliance J t) defined by Equation 3.21. But if the slope of log G(r) versus log r is — m, then, to a good approximation. 

In a stress relaxation test a sample is quickly placed under a strain that is then held constant, and the resulting stress is recorded as a function of time. The response of an ideal elastic solid in stress relaxation is a stress that remains constant with time, while an ideal liquid responds with an immediate return to zero stress as soon as the test strain is imposed. Viscoelastic materials respond with a stress that decays with time. Stress relaxation data are commonly reported as a time-dependent stress relaxation modulus ... 

Keywords— Human Spine, Time Dependent, Stress-Relaxation, Spinal Column, Loading. 

One of the simplest ways of demonstrating a time dependent Mullins Effect is through the strain endurance test [1,12], In this test, a sample is strained to some level and held there for several days or longer. The only measurement taken is the time to failure, if the sample fails within the test period. The point of interest here is that samples fail while held at conditions of constant strain when the stress is slowly relaxing or at most constant. This type of failure is clear evidence of a time dependent Mullins Effect and also demonstrates that some portion of the time dependent stress relaxation must be due to chain failure. 

Equations analogous to those for s(t) above, can be written for the time-dependent stress relaxation modulus, Gy(t), of a linearly viscoelastic body subjected to successive shear strains,... 

The most characteristic features of viscoelastic materials are that they exhibit a time dependent strain response to a constant stress (creep) and a time dependent stress response to a constant strain (relaxation). In addition when the... 

Relaxation is the time-dependent stress resulting from a constant strain. The stress is a function of the strain level, the application time and the temperature. The results of tests at a defined temperature can be presented as a load versus time curve or a stress retention versus time curve. 

The data are not usually reported as a stress/time plot, but as a modulus/time plot. This time-dependent modulus, called the relaxation modulus, is simply the time-dependent stress divided by the (constant) strain (Equation 13-71) ... 

Dynamic mechanical and dynamic stress-optical measurements have been performed on side-chain LCEs with different crosslinking densities [8]. It was found that the relaxation strengths depend strongly on the crosslinking density demonstrating that the vicinity of the crosslinking points perturbs the liquid crystalline order. In this experiment the samples were exposed to a time-dependent stress <7(t), and the time-dependent responses, that is the time-dependent strain e(t) and the time-dependent birefringence An (t), were measured. The real part... 

Opposite to the regulations of the standard ISO 899-1 the compression-creep modulus is named with the symbol Fee- The frequently used values in the following Table are the modulus at 1 hour Ecd, 100 hours Fccioo and at 1000 hours Fcciooo- In the case of stress relaxation experiments the compression-relaxation modulus Ere can be determined from the time dependent stress constant strain level Sco-... 

Physically, it can be seen from Figure 4.14(b) that G yg is the time-independent stress in the element G. The time-dependent stress in the other arm starts off at r = 0 with value G yg, because the total deflection is instantaneously in the spring G. As r increases, the dashpot relaxes this stress to zero at r = . Using eqn 4.11 in eqn 4.52,... 

Stress-relaxation curve Graphical representation of the time-dependent stress of solid materials caused by constant strain. 

Where /o is the imposed fixed strain, (p(t) is the relaxation function decreasing from 0) = 1 at f = 0 to (p(°°) = 0att = °°, and is the equilibrium modulus, which is finite for a viscoelastic solid and zero for viscoelastic liquid. The time-dependent stresses arising from different molecular mechanisms are not additive, and hence it is difficult if not impossible to isolate and characterize each one of them individually. Nevertheless, G(t) is connected to J(t) by the convolution integral equation, SoG s)J(t - s)ds = 1, from which one function can be calculated from the other by a numerical procedure [22]. 

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. 

In stress relaxation measurements an instantaneous strain is imposed on the sample and maintained constant, and the time-dependent stress is measured at constant temperature. Stress relaxation can be measured by TMA, provided the instrument is capable of maintaining a constant strain and continuously monitoring the stress. 

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