Boltzmann’s Superposition Principle

Fig. 2.44(b) Predicted strain response using Boltzmann s superposition principle... 

A plastic with a time dependent creep modulus as in the previous example is stressed at a linear rate to 40 MN/m in 100 seconds. At this time the stress in reduced to 30 MN/m and kept constant at this level. If the elastic and viscous components of the modulus are 3.5 GN/m and 50 x 10 Ns/m, use Boltzmann s Superposition Principle to calculate the strain after (a) 60 seconds and (b) 130 seconds. 

Object in this section is to review how rheological knowledge combined with laboratory data can be used to predict stresses developed in plastics undergoing strains at different rates and at different temperatures. The procedure of using laboratory experimental data for the prediction of mechanical behavior under a prescribed use condition involves two principles that are familiar to rheologists one is Boltzmann s superposition principle which enables one to utilize basic experimental data such as a stress relaxation modulus in predicting stresses under any strain history the other is the principle of reduced variables which by a temperature-log time shift allows the time scale of such a prediction to be extended substantially beyond the limits of the time scale of the original experiment. 

The reduced creep curves in Figure 7.6 (e/o or D(t)) should coincide if Boltzmann s superposition principle would hold for each level of stress, D(t) should be the same. However, creep behaviour is non-linear, linearity only occurs at very low values of <7 and , as a limiting case. Therefore, D(t) is D(t, a). 

According to Boltzmann s superposition principle for such linear systems, if the extra field is added at time f1 the total displacement at times t > fj will be... 

The basic foundation of linear viscoelasticity theory is the Boltzmann s superposition principle which states ... 

Instead of using the generalized Maxwell picture, Eq. (4.22) can be obtained from a more formal argument, namely Boltzmann s superposition principle. It assumes that if the stress at the present time t is caused by a step strain at an earlier time t, the stress is linearly proportional to the strain, and the proportionality (the modulus) decreases with the separation of the time, t - t. The modulus, a decaying function of t - t, is denoted by G t — t ). Consider a system which has been inflicted by small step strains at different times, <1, 2, before the present time t. According to... 

Boltzmann s superposition principle, all the stresses as individually caused by these small step strains are independent of each other. As a result, the total stress at t is simply the sum of these stresses. 

We have used the generalized phenomenological Maxwell model or Boltzmann s superposition principle to obtain the basic equation (Eq. (4.22) or (4.23)) for describing linear viscoelastic behavior. For the kind of polymeric liquid studied in this book, this basic equation has been well tested by experimental measurements of viscoelastic responses to different rate-of-strain histories in the linear region. There are several types of rate-of-strain functions A(t) which have often been used to evaluate the viscoelastic properties of the polymer. These different viscoelastic quantities, obtained from different kinds of measurements, are related through the relaxation modulus G t). In the following sections, we shall show how these different viscoelastic quantities are expressed in terms of G(t) by using Eq. (4.22). 

Thus, G t) = —a t)/Xo. This result is expected from the definition of G t) as used in Boltzmann s superposition principle. The way in which G t) is obtained from Eq. (4.27) also illustrates an experimental problem encountered in the measurement of G t). Experimentally the application of a strain involves the movement of a mechanical device, often a motor, which has a rate limit. Thus, e cannot be infinitely small experimentally. At Ao 0.1, the order of 0.05 s for e is basically the state of the art. How an experiment is affected by a finite e is a relative matter. If the relaxation times of G t), which are the interest of study, are sufficiently larger than e, errors caused by the finite e are negligible. 

In summary, if G t), which is contained in Eqs. (4.30), (4.34)-(4.37), (4.49)-(4.51), (4.63) and (4.73), is known, all the linear viscoelastic quantities can be calculated. In other words, all the various viscoelastic properties of the polymer are related to each other through the relaxation modulus G t). This result is of course the consequence of the generalized Maxwell equation or equivalently Boltzmann s superposition principle. The experimental results of linear viscoelastic properties of various polymers support the phenomenological principle. Some viscoelastic properties play more important roles than the others in certain rheological processes related to... 

Linear viscoelasticity is an extension of linear elasticity and hyperelasticity that enables predictions of time dependence and viscoelastic flow. Linear viscoelasticity has been extensively studied both mathematically (Christensen 2003) and experimentally (Ward and Hadley 1993), and can be very useful when applied under the appropriate conditions. Linear viscoelasticity models are available in all major commercial FE packages and are relatively easy to use. The basic foundation of linear viscoelasticity theory is the Boltzmann s superposition principle, which states, "Every loading step makes an independent contribution to the final state."... 

The first relation follows immediately from Boltzmann s superposition principle in the form of Eq. (5.38) when applied to the case of a deformation with constant shear rate Czx- We have... 

To derive the second equation, we consider a dynamic-mechanical experiment and treat it again on the basis of Boltzmann s superposition principle, writing... 

Representing a combination of the equation of state of ideal rubbers and Boltzmann s superposition principle, Lodge s equation provides an interpolation between the properties of rubbers and viscous liquids. The limiting cases of an elastic rubber and the Newtonian liquid are represented by... 

We can interrelate the relaxation and creep functions and the dynamic moduli and compliances via Boltzmann s superposition principle which states that all effects of past history can be considered independently in their contributions to the present state of the (linear) viscoelastic material. Thus, if one subjects the material to, say, incremental strains yo — 0), (y — y ),.. . , (y y -1) at times , ... 

Solution. This problem is solved in Reference 10 (p. 56ff) by direct integration of the differential equation for the Maxwell element. Here, we will apply Boltzmann s superposition principle to obtain the results and, in doing so, again illustrate how information from one type of linear test (stress relaxation) may be used to predict the response in another (dynamic testing). 

The other importance is the additivity of the memory. That is, using shear stress relaxation as an example, the cumulative memory of stresses imposed at different times may be calculated with a theory of Boltzmann s superposition principle. 

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