Boltzmann Superposition Principle relaxation

With crystalline plastics, the main effect of the crystallinity is to broaden the distribution of the relaxation times and extend the relaxation stress too much longer periods. This pattern holds true at both the higher and low extremes of crystallinity (Chapter 6). With some plastics, their degree of crystallinity can change during the course of a stress-relaxation test. This behavior tends to make the Boltzmann superposition principle difficult to apply. 

If the Boltzmann superposition principle holds, the creep strain is directly proportional to the stress at any given time, f Similarly, the stress at any given lime is directly proportional to the strain in stress relaxation. That is. the creep compliance and the stress relaxation modulus arc independent of the stress and slrai . respectively. This is generally true for small stresses or strains, but the principle is not exact. If large loads are applied in creep experiments or large strains in stress relaxation, as can occur in practical structural applications, nonlinear effects come into play. One result is that the response (0 l,r relaxation times can also change, and so can ar... 

The ideal stress relaxation experiment is one in which the stress is instantaneously applied. We have seen in Section 4.4.2 the exponential relaxation that characterises the response of a Maxwell model. We can consider this experiment in detail as an example of the application of the Boltzmann Superposition Principle. The practical application of an instantaneous strain is very difficult to achieve. In a laboratory experi-... 

Stress relaxation tests need not have a second step, although some workers recommend a second step in the opposite direction. The Boltzmann superposition principle for polymers allows for multiple step-change tests of both types (stress or strain) as long as the linear limit of the polymer is not exceeded (Ferry, 1980). 

Chapters 5 and 6 discuss how the mechanical characteristics of a material (solid, liquid, or viscoelastic) can be defined by comparing the mean relaxation time and the time scale of both creep and relaxation experiments, in which the transient creep compliance function and the transient relaxation modulus for viscoelastic materials can be determined. These chapters explain how the Boltzmann superposition principle can be applied to predict the evolution of either the deformation or the stress for continuous and discontinuous mechanical histories in linear viscoelasticity. Mathematical relationships between transient compliance functions and transient relaxation moduli are obtained, and interrelations between viscoelastic functions in the time and frequency domains are given. 

The Boltzmann superposition principle can be used to relate the steady state compliance to the stress relaxation modulus (see Problem 7.44) ... 

One of the direct consequences of the Boltzmann superposition principle is that there is a relationship between the stress relaxation modulus and the creep compliance. We have already seen that when dealing with time-independent... 

Still another relationship between experimental parameters is a direct consequence of the Boltzmann superposition principle. We will derive the equations relating the shear stress relaxation modulus G(t) to the in-phase and out-of-phase dynamic shear moduli G oi) and G"(co) starting from equation (2-46)... 

This convolution integral expresses the relationship between the creep compliance and the stress relaxation modulus. It is exact and depends only on the applicability of the Boltzmann superposition principle. 

Linear viscoelasticity is the simplest viscoelastic behavior in which the ratio of stress to strain is a function of time alone and not of the strain or stress magnitude. Under a sufficiently small strain, the molecular structure will be practically unaffected, and linear viscoelastic behavior will be observed. At this sufficiently small strain (within the linear range), a general equation that describes all types of linear viscoelastic behavior can be developed by using the Boltzmann superposition principle (Dealy and Wiss-brun, 1990). For a sufficiently small strain (yo) in the experiment, the relaxation modulus is given by... 

Thus viscoelasticity is characterized by dependencies on temperature and time, the complexities of which may be considerably simplified by the time-temperature superposition principle. Similarly the response to successively loadings can be simply represented using the applied Boltzmann superposition principle. Experimentally viscoelasticity is characterized by creep compliance quantified by creep compliance (for example), stress relaxation (quantified by stress relaxation modulus), and by dynamic mechanical response. 

GPa, respectively, with relaxation time r 5 s. The pofymer is subjected to a constant rate of tensOe strain e = 10" s". Derive the stress-strain relation Boltzmann superposition principle. 

Use the integral form of the Boltzmann superposition principle to show that the creep compliance and stress relaxation modulus of any linear viscoelastic material are related through... 

In linear viscoelasticity the stress relaxation test is often used, along with the time-temperature superposition principle and the Boltzmann superposition principle,... 

Upon a large shear rate, the polymer flow exhibits nonlinear viscoelasticity. In this case, the Boltzmann superposition principle becomes invalid, and the fluid appears as a non-Newtonian fluid. A typical treatment is to consider the nonlinear resptmse as separate processes at two different time scales the first one is the rapid elastic recovery in association with the shear rate, which can relax part of the stress instantaneously the second one is the slow relaxation of the rest stress in associa-ti(Mi with time. Thus, the nonlinear relaxation modulus can be expressed as... 

In this chapter we describe the common forms of viscoelastic behaviour and discuss the phenomena in terms of the deformation characteristics of elastic solids and viscous fluids. The discussion is confined to linear viscoelasticity, for which the Boltzmann superposition principle enables the response to multistep loading processes to be determined from simpler creep and relaxation experiments. Phenomenological mechanical models are considered and used to derive retardation and relaxation spectra, which describe the time-scale of the response to an applied deformation. Finally we show that in alternating strain experiments the presence of the viscous component leads to a phase difference between stress and strain. 

Stress relaxation behaviour can be represented in an exactly complementary fashion using the Boltzmann superposition principle. Consider a stress relaxation programme in which incremental strains Aei, Ae2, Ae3, etc. are added at times T, T2, Tj, etc., respectively. The total stress at time t is then given by... 

Once modifications to functions of this kind have been made, the Boltzmann superposition principle can no longer be assumed to apply, and there is no simple replacement for it. This marks a significant change in the level of difficulty when moving from linear to non-linear theory. In the linear case, the material behaviour is defined fully by single-step creep and stress relaxation, and the result of any other stress or strain history then can be calculated using the Boltzmann integral. In the non-linear case we have lost the Boltzmann equation, and it is not even clear what measurements are needed for a full definition of the material. 

Another simple adaptation of the Boltzmann superposition principle is that of Findlay and Lai [14], who worked with step stress histories applied to specimens of poly(vinylchloride). Their theory was reformulated by Pipkin and Rogers [15] for general stress and strain histories. Pipkin and Rogers took a non-linear stress relaxation modulus R t, e), defined somewhat differently from G in Equation (10.4) ... 

The linear viscoelastic properties G(t)md J t) are closely related. Both the stress-relaxation modulus and the creep compliance are manifestations of the same dynamic processes at the molecular level in the liquid at equilibrium, and they are closely related. It is not the simple reciprocal relationship G t) = 1/J t) that applies to Newtonian liquids and Hookean solids. They are related through an integral equation obtained by means of the Boltzmann superposition principle [1], a link between such linear response functions. An example of such a relationship is given below. 

Nonlinearity in creep is associated with severe deviations from the Boltzmann superposition principle in creep recovery. An example of extreme effects in a crystalline polymer is shown in Fig. 16-18 for recovery of polyethylene following partial stress relaxation at constant strain for various times and strain magnitudes. It is clear that recovery is much slower at large strains but is somewhat faster for shorter durations of the initial straining. In general, strains less than 0.01% appear to be required for conformity to the Boltzmann superposition principle in this system. ... 

In Chapter 5, we introduced linear viscoelasticity. In this scheme, the observed creep or stress relaxation behaviour can be viewed as the defining characteristic of the material. The creep compliance function - the ratio of creep strain e t) to the constant stress a - is a function of time only and is denoted as J t). Similarly and necessarily, the stress relaxation modulus, the ratio of stress to the constant strain, is the function G(r). Any system in which these two conditions do not apply is non-linear. Then, the many useful and elegant properties associated with the linear theory, notably the Boltzmann superposition principle, no longer apply and theories to predict stress or strain are approximations that must be supported by experiment. 

Example 15.6 Solve Example 15.1 by applying the Boltzmann superposition principle, thereby demonstrating how the stress-time response in an engineering stress-strain test may be predicted from stress-relaxation data. 

It is possible to analyse stress relaxation in a similar way using the Boltzmann superposition principle. In this case it is necessary to define a stress relaxation modulus G(t) which relates the time-dependent stress a(t) to the strain e through the relationship... 

The correspondence principle following the Boltzmann superposition principle allows the conversion of the common mechanical relationships of linear elasticity theory into linear viscoelasticity simply by replacing cr by time-dependent a t) and e by time-dependent e(t). Young s modulus E or the relaxation modulus Ej (f)= cr(f)/e is accordingly transformed to the creep modulus c(f) = cile t) orthe creep compliance/(f) = s(f)/(7,respectively. These time-dependent parameters can be determined from tensile creep and relaxation experiments. In compression or shear mode, the corresponding parameters of moduli are calculated in a similar manner. 

It is not convenient to represent the result of a creep experiment in terms of the relaxation modulus, since it is the stress that is controlled rather than the strain. For this purpose we can use an alternative expression of the Boltzmann superposition principle in which stress is the independent variable. For the case of simple shear, this is shown by Eq. 4.20. 

Extensional flows yield information about rheological behavior that cannot be inferred from shear flow data. The test most widely used is start-up of steady, uniaxial extension. It is common practice to compare the transient tensile stress with the response predicted by the Boltzmann superposition principle using the linear relaxation spectrum a nonlinear response should approach this curve at short times or low strain rates. A transient response that rises significantly above this curve is said to reflect strain-hardening behavior, while a material whose stress falls... 

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