Viscoelasticity Boltzmann superposition

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle. Basically this principle proposes that for a linear viscoelastic material, the strain response to a complex loading history is simply the algebraic sum of the strains due to each step in load. Implied in this principle is the idea that the behaviour of a plastic is a function of its entire loading history. There are two situations to consider. 

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 ... 

An important and sometimes overlooked feature of all linear viscoelastic liquids that follow a Maxwell response is that they exhibit anti-thixo-tropic behaviour. That is if a constant shear rate is applied to a material that behaves as a Maxwell model the viscosity increases with time up to a constant value. We have seen in the previous examples that as the shear rate is applied the stress progressively increases to a maximum value. The approach we should adopt is to use the Boltzmann Superposition Principle. Initially we apply a continuous shear rate until a steady state... 

Viscoelastic behavior is classified as linear or non-linear according to the manner by which the stress depends upon the imposed deformation history (SO). Insteady shear flows, for example, the shear rate dependence of viscosity and the normal stress functions are non-linear properties. Linear viscoelastic behavior is obtained for simple fluids if the deformation is sufficiently small for all past times (infinitesimal deformations) or if it is imposed sufficiently slowly (infinitesimal rate of deformation) (80,83). In shear flow under these circumstances, the normal stress differences are small compared to the shear stress, and the expression for the shear stress reduces to a statement of the Boltzmann superposition principle (15,81) ... 

A viscoelastic solid is characterized by the fact that its modulus E is a function of time. Thus, the response of the material to a loading program, s(t) or d(t) needs the application of the Boltzmann superposition principle (Sec. 11.1). In the case of programmed strain ... 

Apply the Boltzmann superposition principle for the case of a continuous stress application on a linear viscoelastic material to obtain the resulting strain y(t) in terms of J(t — t ) and ih/dt, the stress history. Consider the applied stress in terms of small applied At,-, as shown on the accompanying figure. 

The Boltzmann superposition principle applied to a viscoelastic material that has undergone a history of pressures or tensile stresses can be written as... 

According to the Boltzmann superposition principle, the shear strain of a solid viscoelastic material under the action of a harmonic shear stress can be written as (2)... 

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. 

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. 

Find the relation between creep compliance J(t) and recoverable compliance /R(f) using the Boltzmann superposition principle. Dielectric spectroscopy indicates that water molecules respond to an oscillating electric field at a frequency of 17 GHz at room temperature. Is water still a Newtonian liquid at this high a frequency or is it viscoelastic If... 

Inherent in the mathematical treatment of linear viscoelasticity is the Boltzmann superposition principle (15), which, in simple terms, states that the deformation resulting at any time is directly proportional to the applied stress. This is illustrated in Figure 10.5. 

Any of equations (2-45), (2-46), (2-49), or (2-50) is sufficient as a statement of the Boltzmann superposition principle for linear viscoelastic response of a material. Often in particular applications, however, it is more convenient to use one form than another. All can be extended to three dimensions by using the same forms with the strains given by equation (2-18). Thus, for example, equation (2-46) becomes ... 

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... 

The most commonly used model is the Boltzmann superposition principle, which proposes that for a linear viscoelastic material the entire loading history contributes to the strain response, and the latter is simply given by the algebraic sum of the strains due to each step in the load. The principle may be expressed as follows. If an equation for the strain is obtained as a function of time under a constant stress, then the modulus as a function of time may be expressed as... 

First, we need a rule to predict the effect of time-varying loads on a viscoelastic model. When a combination of loads is applied to an elastic material, the stress (and strain) components caused by each load in turn can be added. This addition concept is extended to linear viscoelastic materials. The Boltzmann superposition principle states that if a creep stress ai is... 

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. 

Nonlinear viscoelasticity is the behavior in which the relationship of stress, strain, and time are not linear so that the ratios of stress to strain are dependent on the value of stress. (The Boltzmann superposition principle does not hold). Such behavior is very common in plastic systems, non-linearity being found especially at high strains or in crystalline plastics. 

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. 

Find the tensOe strain at the following times t (a) 1500 s (b) 2500 s. Assume that under these conditions polyprpylene is linear viscoelastic and therefore obeys the Boltzmann superposition prindple. 

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... 

Viscoelasticity D W AUBREY Transient and dynamic viscoelastic functions Boltzmann superposition principle... 

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... 

This superposition principle states that the response of a viscoelastic plastic to a load is independent of any other load already apphed to the plastic. Further, strain is directly proportional to apphed stress when the strains are observed at equal time intervals. The Boltzmann superposition principle quantifies creep strain as a function of stress and time at a given temperature. Constitutive equations express the relationships among stress, strain, and time [12]. 

The linear viscoelastic materials obey the so-called Boltzmann Superposition Principle. As noted by Tschoegl (13), this was the only foray of the Viennese statistical physicist Ludwig Boltzmann into mechanics. The principle states that in linear viscoelasticity effects are simply additive it matters at which instant an effect is created and it is assumed that each increment of stress makes an independent contribution. 

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 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. 

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... 

The Schapery Model. One of the earliest models of the nonlinear viscoelastic response of pol5nners to use the concept of a reduced time is due to Schapery (147-149). The model is based on thermodynamic considerations and has a form similar to the Boltzmann superposition principal described previously. The model time dependences, except for the shift factors, are the same as those obtained in the linear response regime. Hence, the model is relatively easy to implement and to determine the relevant material parameters. It results in a generalization of the generalized superposition principal developed by Leaderman (150). 

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. 

For elastic solids Hooke s law is valid only at small strains, and Newton s law of viscosity is restricted to relatively low flow rates, as only when the stress is proportional either to the strain or the strain rate is analysis of the deformation feasible in simple form. A comparable limitation holds for viscoelastic materials general quantitative predictions are possible only in the case of linear viscoelasticity, for which the results of changing stresses or strains are simply additive, but the time at which the change is made must be taken into account. For a single loading process there will be a linear relation between stress and strain at a given time. Multistep loading can be analysed in terms of the Boltzmann superposition principle (Section 4.2.1) because each increment of stress can be assumed to make an independent contribution to the overall strain. 

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