Boltzmann superposition integral

Let us suppose the strain applied at time t0 increases over a time v to a maximum value y. At times less than to — v no strain is applied and at times greater than t0 the strain is constant. This gives the limits to the Boltzmann superposition integral ... 

Now we can apply these conditions to the Boltzmann superposition integral (Equation 4.60) ... 

So far we have seen that if we begin with the Boltzmann superposition integral and include in that expression a mathematical representation for the stress or strain we apply, it is possible to derive a relationship between the instrumental response and the properties of the material. For an oscillating strain the problem can be solved either using complex number theory or simple trigonometric functions for the deformation applied. Suppose we apply a strain described by a sine wave ... 

Now the Boltzmann superposition integral is given by Equation (4.60). Substituting for the strain and replacing t by t in Equation (4.88) gives... 

Whilst the flow curves of materials have received widespread consideration, with the development of many models, the same cannot be said of the temporal changes seen with constant shear rate or stress. Moreover we could argue that after the apparent complexity of linear viscoeleastic systems the non-linear models developed above are very poor cousins. However, it is possible to introduce a little more phenomenological rigour by starting with the Boltzmann superposition integral given in Chapter 4, Equation (4.60). This represents the stress at time t for an applied strain history ... 

In steady simple shear, the top plate in Fig. 7.23 is moved at a constant velocity v. The shear rate 7 = v]//t is a time-independent constant that can be pulled out of the Boltzmann superposition integral ... 

Use the Boltzmann superposition integral to derive the storage modulus of a viscoelastic liquid as a sine transform of the stress relaxation modulus G(t) [Eq. (7.149) with 6 eq = 0)]. Also derive the loss modulus as a cosine transform of G(t) [Eq. (7.150) with Ggq = 0] for a viscoelastic liquid. 

The first expression gives the Boltzmann superposition principle for the special case that the rate of shear q is a constant3. For the derivation of eq. (2.2), the convolutional integral is used (48) ... 

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. 

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... 

Fig. 6.9 Illustration of Boltzmann superposition principle for the strain response linearly integrated with the stepwise increase of stresses...

The general approach to discussing linear viscoelasticity comes from the Boltzmann superposition principle represented as a hereditary integral. For the shear stress as a function of shear strain, one obtains... 

The integral in Equation (4.3) is called a Duhamel integral, and it is a useful illustration of the consequences of the Boltzmann superposition principle to evaluate the response for a number of simple loading programmes. Recalling the development that leads to Equation (5.2) it can be seen that the Duhamel integral is most simply evaluated by treating it as the summation of a number of response terms. Consider two specific cases ... 

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. 

Since the Zener model is a linear viscoelastic model, it obeys the Boltzmann superposition principle. In this problem we are concerned with a strain history which is a smoothly varying function of time, with y undergoing sinusoidal oscillations. Therefore the integral form of the BSP is the most straightforward one to apply... 

Equation (26) is what is known as a rate-type constitutive equation, and it gives the extra stress in implicit form. Integral equations, on the other hand, are explicit in the stress, and the simplest of these can be derived based on the Boltzmann superposition principle. According to this principle, the stress in a material at any time can be obtained by adding stress tbatjndivjduallv result from earh Vio... 

A general single integral constitutive equation results if the Boltzmann superposition principle is applied to a nonrspecified tensor functional, of the macroscopic strain, represented by the Finger... 

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. 

Here the strain y f) is taken relative to the current state, so y(t) = 0. This memory integral formulation is a special case of Boltzmann s superposition integral,... 

The Boltzmann superposition principle is one starting point for a theory of linear viscoelastic behaviour, and is sometimes called the integral representation of linear viscoelasticity , because it defines an integral equation. An equally valid starting point is to relate the stress to the strain by a linear differential equation, which leads to a differential representation of linear viscoelasticity. In its most general form, the equation is expressed as... 

Non-linear viseoelastie theories can also be created by generalising the Boltzmann superposition prineiple (see Chapter 5). Leaderman [18], working on polymer fibres, was the first to do this and Findley and Lai [19] have adopted a similar approach. Non-linearity is introdueed into the Boltzmann integral by ineluding strain or stress dependence into the integrand. Leaderman s integral takes the form... 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

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). 

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