Boltzmann superposition principl

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. 

Now consider the situation in which the stress, ai, was applied at time, ti, and an additional stress, Boltzmanns Superposition Principle states that the total strain at time, t, is the algebraic sum of the two independent responses. 

A plastic which behaves like a Kelvin-Voigt model is subjected to the stress history shown in Fig. 2.87. Use the Boltzmanns Superposition Principle to calculate the strain in the material after (a) 90 seconds (b) 150 seconds. The spring constant is 12 GN/m and the dashpot constant is 360 GNs/m. ... 

With crystalline plastics, the main effect of the crystallinity is to broaden the distribution of the relaxation times and extend the relaxation stress too much longer periods. This pattern holds true at both the higher and low extremes of crystallinity (Chapter 6). With some plastics, their degree of crystallinity can change during the course of a stress-relaxation test. This behavior tends to make the Boltzmann superposition principle difficult to apply. 

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. 

The first assumption involved in using the Boltzmann superposition principle is that elongation is proportional to stress, that is, compliance is independent of stress. The second assumption is that the elongation created by a given load is independent of the elongation caused by any previous load. Therefore, deformation resulting from a complex loading history is obtained as the sum of the deformations that can be attributed to each separate load. 

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. 

The Boltzmann superposition principle states that the response of a material to a given load is independent of the response of the material to ary load that is already on the material. Another consequence of this principle is that the deformation of a specimen is directly proportional to the applied stress when all deformations are compared at equivalent, times... 

Figure 7 illustrates the Boltzmann superposition principle for a polymer that obeys a common type of behavior given by the Nutting equation... 

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 ... 

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 ... 

The ideal stress relaxation experiment is one in which the stress is instantaneously applied. We have seen in Section 4.4.2 the exponential relaxation that characterises the response of a Maxwell model. We can consider this experiment in detail as an example of the application of the Boltzmann Superposition Principle. The practical application of an instantaneous strain is very difficult to achieve. In a laboratory experi-... 

The application of a linearly ramped strain can provide information on both the sample elasticity and viscosity. The stress will grow in proportion to the applied strain. The ratio of the strain over the applied time gives the shear rate. Applying the Boltzmann Superposition Principle we obtain the following expression ... 

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... 

Now in order to apply the Boltzmann Superposition Principle (Equation 4.60) we need to express this as a strain rate. Differentiating with respect to time gives us... 

Time Dependence in Flow and the Boltzmann Superposition Principle... 

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) ... 

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) ... 

Stress relaxation tests need not have a second step, although some workers recommend a second step in the opposite direction. The Boltzmann superposition principle for polymers allows for multiple step-change tests of both types (stress or strain) as long as the linear limit of the polymer is not exceeded (Ferry, 1980). 

Polymers are generally assumed to obey the Boltzmann superposition principle in the domain of small strains. When there are changes of loading conditions, the effects of these changes are additive when the corresponding responses are considered at equivalent times. For instance, if different stresses a0, CT, a2, are applied at different times 0, t], t2,, respectively, the... 

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 ... 

The Boltzmann Superposition Principle Apply the Boltzmann superposition principle to obtain the LVE (Eq. 3.3-8) using x t) = y0Ge /x. Consider the applied strain y(t) as being applied discretely in a series of small steps Ay, as shown in the following figure ... 

The Boltzmann Superposition Principle Alternate form of the LVE Equation... 

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. 

This equation, one of many possible forms of expressing the Boltzmann superposition principle, indicates that the effects of mechanical history are linearly additive (12,13). 

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

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... 

As indicated above, cancelation of a given perturbation is interpreted by the material as if a perturbation of opposite sign were applied on it. The Boltzmann superposition principle can be expressed in a generalized way by... 

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. 

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