Creep Boltzmann superposition principle

A creep test can be carried out with an imposed stress, then after a time have its stress suddenly changed to a new value and have the test continued. This type of change in loading allows the creep curve to be predicted. The simple law referred to earlier as the Boltzmann superposition principle, hold for most materials, so that their creep curves can thus be predicted. 

Figure 7 Creep of a material that obeys the Boltzmann superposition principle. The load is doubled after 400 s.

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... 

Assuming that the Boltzmann superposition principle holds for the polymer in Problem I, what would the creep elongation be from 100 to 10,000 min if the load were doubled after 100 min ... 

Assuming thai the Boltzmann superposition principle holds and that all of the creep is recoverable, what would the creep recovery curve be for I he polymer in Problem 1 if the load were removed after lO.(KM) min ... 

This is the Boltzmann superposition principle for creep experiments expressed in continuous form. If the stress is a continuous function of time in the interval —oo < < 8i, constant in the interval 0i < / < 02, and again a continuous function for t > 02 (see Fig. 5.14), then Eq. (5.35) cannot be used to obtain e because the contribution of the stress to the strain in the interval 0i < t < 02 would be zero. The response for this stress history is given by... 

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. 

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... 

FIGURE 10.5 Application of the Boltzmann superposition principle to a creep experiment. (Modified fromVasquez-Torres, H. and Cruz-Ramos, C.A., J. Appl. Polym. Sci. 54,1141,1994.)... 

The Boltzmann superposition principle is one of the simplest but most powerful principles of polymer physics.2 We have previously defined the shear creep compliance as relating the stress and strain in a creep experiment. Solving equation (2-6) for strain gives... 

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... 

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. 

The initial stress on a polymer in a creep experiment is 10 N/m. This load is increased by 10 N/m and 10 N/m after 10 and 10 s, respectively. Assuming that the Boltzmann superposition principle holds for this material, find the strain after 10 s. The creep comphance for the material is given by lO" (1... 

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... 

FIGURE 13.17 Application of the Boltzmann superposition principle to a creep experiment. 

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. 

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... 

Provided that the strains are sufficiently small, the Boltzmann superposition principle can be applied to this problem. We can treat the creep as the resultant of two responses one due to a stress a applied over the year (3.15 X 10 s) and the other due to a stress - cr applied over the final 8 months (2.10 x 10 s). Noting the restriction to small stresses and strains, we shall neglect the term in exp(0.9o ) in the creep equation, and use the simplified equation... 

According to the Boltzmann superposition principle, the final creep deformation caused by a series of step loading and unloading increments such as those of Fig. 2 is predictable by the summation of the individual creep responses from each increment. 

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]. 

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. 

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. 

Figure 10.4 Comparison of creep compliance (a) and recovery compliance (b) at three load levels cTi, <72 and <73 for a non-linear viscoelastic material obeying Leaderman s modified Boltzmann superposition principle. Note that the creep and recovery ciuves for a given load level are identical...

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. 

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. 

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. 

The creep compliance /(f) = e(f)/time-dependent strain and a is the constant stress is, at very small strains, less than 1%, approximately independent of stress. The material is said to be linear anelastic or linear visco-elastic. Materials of this category follow the so-called Boltzmann superposition principle which can be expressed as follows ... 

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