Boltzmann superposition theory

The Phenomenology of the Linear Theory of Viscoelasticity. One of the powers of the linear viscoelasticity theory is that it is predictive. The constitutive law that comes from Boltzmann superposition theory requires simply that the material functions discussed above be known for a given material. Then, for an arbitrary stress or deformation history, the material response can be obtained. In addition, the elastic-viscoelastic correspondence principle can be used so that boundary value problems such as beam bending, for which an elastic solution exists, can be solved for linear viscoelastic materials as well. Both of these subjects are treated in this section. 

There are two superposition principles that are important in the theory of Viscoelasticity. The first of these is the Boltzmann superposition principle, which describes the response of a material to different loading histories (22). The second is the time-temperature superposition principle or WLF (Williams, Landel, and Ferry) equation, which describes the effect of temperature on the time scale of the response. 

There are many types of deformation and forces that can be applied to material. One of the foundations of viscoelastic theory is the Boltzmann Superposition Principle. This principle is based on the assumption that the effects of a series of applied stresses acting on a sample results in a strain which is related to the sum of the stresses. The same argument applies to the application of a strain. For example we could apply an instantaneous stress to a body and maintain that stress constant. For a viscoelastic material the strain will increase with time. The ratio of the strain to the stress defines the compliance of the body ... 

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 ... 

Boltzmann superposition principle A basis for the description of all linear viscoelastic phenomena. No such theor) is available to serve as a basis for the interpretation of nonlinear phenomena—to describe flows in which neither the strain nor the strain rate is small. As a result, no general valid formula exists for calculating values for one material function on the basis of experimental data from another. However, limited theories have been developed. See kinetic theory viscoelasticity, nonlinear, bomb See plasticator safety. 

Mechanical property characterisation of artificial polymers (fibrous and non-fibrous) is often preceded by a mechanical conditioning treatment (Ward and Hadley, 1993) if the material is vi.scoelastic. This treatment is designed to provide a standard, reproducible microstructural state, so that results from different experiments, materials and laboratories can be compared easily. The conditioning treatment is deemed necessary because the mechanical properties of viscoelastic materials are affected by their entire previous mechanical history, as articulated in the Boltzmann superposition principle (Ward and Hadley, 1993). To predict mechanical behaviour accurately, one ought in theory to know the entire loading history of specimens since their manufacture Under practical conditions, only comparatively recent history is relevant, so specimens can be... 

Boltzmann Superposition and the Constitutive Law for Linear Viscoelasticity. The underlying assumption of the Boltzmann superposition principle is that responses to loads or deformations applied to a material at different times are linearly additive. This set of assumptions leads to the constitutive laws of linear viscoelasticity theory which can be considered as a linear response theory. For discussion purposes, consider a Maxwell material that is subjected to a two-step deformation history. The history is such that a deformation yi = Ayi... 

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) ... 

All the relations in Sections B to F above may be regarded as originating in the Boltzmann superposition principle or the constitutive equation of Chapter 1, equation 7. The foundation of the theory has also been related to the principles of linear irreversible thermodynamics. It has been pointed out by Meixner that certain other postulates are taken for granted. Many of the specific predictions... 

In the course of tensile creep, the form of the time dependence of strain (as expressed by the stretch ratio X, for example) depends on the magnitude of tensile stress at high stresses." " Recovery is considerably more rapid than would be predicted from the Boltzmann superposition principle, as illustrated in Fig. 13-23 for polyisobutylene of high molecular weight. " The course of recovery is predicted successfully by the theory of Bernstein, Kearsley, and Zapas. 2 - 22 -pije stress-dependent recoverable steady-state compliance D = which is equal to Z) at low stresses, decreases with increasing Ot- This effect, moderate when the tensile strain e is defined as X — 1, is more pronounced when it is replaced by the Hencky strain, defined as In X. The stress dependence of steady-state compliance in shear will be discussed in Chapter 17. The reader is referred to the review by Petrie" for more details. 

In defining the constitutive relations for an elastic solid, we have assumed that the strains are small and that there are linear relationships between stress and strain. We now ask how the principle of linearity can be extended to materials where the deformations are time dependent. The basis of the discussion is the Boltzmann superposition principle. This states that in linear viscoelasticity effects are simply additive, as in classical elasticity, the difference being that in linear viscoelasticity it matters at which instant an effect is created. Although the application of stress may now cause a time-dependent deformation, it can still be assumed that each increment of stress makes an independent contribution. From the present discussion, it can be seen that the linear viscoelastic theory must also contain the additional assumption that the strains are small. In Chapter 11, we will deal with attempts to extend linear viscoelastic theory either to take into account non-linear effects at small strains or to deal with the situation at large strains. 

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... 

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. 

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 Boltzmann superposition principle represents the stress as a result of changes in the state of strain at previous times. In the linear theory valid for small strains, these can be represented by the linear strain tensor. In Lodge s equation the changes in the latter are substituted by changes in the time dependent Finger tensor... 

A corner-stone of the theory of linear viscoelasticity is the Boltzmann superposition principle. It allows the state of stress or strain in a viscoelastic body to be determined from knowledge of its entire deformation history. The basic assumption is that during viscoelastic deformation in which the applied stress is varied, the overall deformation can be determined from the algebraic sum of strains due to each loading step. Before the use of the principle can be demonstrated it is necessary, first of all, to define a parameter known as the creep compliance J(t) which is a function only of time. It allows the strain after a given time e(t) to be related to the applied stress or for a linear viscoelastic material since... 

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. 

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11. 

The Boltzmann superposition principle is the embodiment of the theory of linear viscoelasticity, and it is valid for both steady and transient deformations, provided that the extent of deformation is low. A specific form of the stress-relaxation modulus may be obtained by permitting the stress response in a polymer to be made up of an elastic contribution and a viscous contribution. Thus, if we again use the Maxwell element encountered previously in Example 12.2 and Figure 12.4 of Chapter 12, the total strain y in the spring and dashpot combination is (at any time) a sum of the individual strains that is,... 

This may contradict one of the basic laws of electrostatics, that is the linear superposition of fields. This is a fundamental problem for the theory and to any other theory or development, such as that due to Guggenheim below (Section 10.13.1), which makes use of any such combination. In fact this is one of the big problems in the theory. The other big problem is that the x/tjS in the Poisson equation and in the Maxwell-Boltzmann distribution are different and have a different physical basis (see Section 10.6.5). This is believed by many to be yet another fundamental problem for the theory. 

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

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."... 

According to the second postulate of the Boltzmann adopted in his theory of the elastic aftereffect, and the underlying Boltzmann-Volterra model that describes the relaxation phenomena, using a function of heredity [13] action occurred in the past few strains on the stresses caused by deformation of the body at any given time, do not depend on each other and therefore algebraically added. This position has received also the name of a principle of the Boltzmann s superposition. It should be noted that the polymer body superposition principle holds in the upper-bounded the range of deformation, stress and rate of change. 

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