The Boltzmann superposition principle

The creep is a function of the entire past loading history of the specimen. 

Each loading step makes an independent contribution to the final deformation, so that the total deformation can be obtained by the addition of all the contributions. 

It is usual to separate out the instantaneous elastic response in terms of the unrelaxed modulus G, giving 

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  

The additional creep e jj — ti) produced by the second loading step is given by 

The raison d itre of this book is that rheological properties of the melt are very sensitive to the molecular structure of a polymer. Rheological properties describe how stress develops in a sample undergoing a prescribed deformation. They also describe the deformation that is caused by a prescribed stress. The most fundamental rheological experiment for a viscoelastic material is a step-strain test, and for melts this nearly always means a step shear strain. In a step shear-strain test, a sample is subjected to a sudden shear strain of magnitude, % at time t=0. The shear stress is measured as a fimction of time, and the ratio of the stress to the applied strain defines the relaxation modulus, G t). 

If the experiment is repeated, with the amount of strain doubled to 2 %, another result will be obtained. If the resulting stress at any given value of t is exactly twice that measured in the first test at the same value of t, the relaxation modulus determined in the two experiments will be identical to each other. From an experimental point of view this is a key feature of linear viscoelastic behavior. The implication is that in both experiments the strain is sufficiently small that the departure of the molecules from their equilibrium state is negligible. Thus, both experiments reveal the behavior of the polymer in its unstrained state. This, in turn, implies that the response to a series of small, step strains will be simply the sum of the responses to each step, where the same relaxation modulus governs each response. 

The additivity of responses can be expressed quantitatively by Eq. 4.2, which gives the stress as a function of time that results from a sequence of small shearing deformations, yff ), occurring at times, q. 

Ludwig Boltzmann generalized this to give the response to a continuously varying shear deformation, rather than a series of step strains, by letting 5/approach zero and writing 

dy(r ) is the shear strain that occurs between t and, df, and / is the shear rate during this period. Equation 4.3 is the special form of the Boltzmann superposition principle for simple shearing deformations. 

Consider two strain increments Asx Aei applied to an initially undeformed body, at times /j, ti. From (1.2.29) the resulting stress at a subsequent time t is given by 

We finally observe that delayed response phenomena akin to creep and relaxation occur in other areas of Mechanics and Physics, and are attributable to the same fundamental cause, namely (usually internal) frictional losses. The mathematical techniques used for analyzing such phenomena are similar to those used in analyzing the properties of the viscoelastic functions. Such a close analogy exists between certain phenomena in the theory of Dielectrics and Linear Viscoelasticity, as emphasized by Gross (1953). 

FIGURE 15.12 Boltzmann superposition principle (a) applied strain history (b) resulting stress history. The experiment can also be reversed, with an applied stress history causing a strain history. The shear stresses and strains shown can also be replaced with tensile stresses and strains. 

Here At, is the stress increment that results from the strain increment Ay,-. The argument t — ti is the time after the application of a particular strain increment Ay,. This behavior is 

According to Boltzmann, the stress in the material at any time t depends on its entire past strain history, although since Gr(t) is a decreasing function of time, the further back a Ay(t,) has occurred, the smaller will be its influence in the present. This leads to the anthropomorphic concept of viscoelastic materials having a fading memory (like an aging professor), with Grit) sometimes known as the memory function. (The concept, of course, is valid even in the absence of linear additivity—it is just much more difficult to quantify.)  

Solution. The relaxation modulus (memory function) Grit) for a Maxwell element is given by Equation 15.7b. For this particular strain history, Ay(/o) = +yo and Ay(/i) = —2y (remember, we need the increment, not the absolute value). Plugging these values into Equation 15.31 gives 

Of course, not aU strain histories consist of a nice series of finite step changes. No matter how an applied strain varies with time, however, it can always be approximated by a series of differential step changes, for which Equation 15.31 becomes 

Another manifestation of linear response is the Boltzmann superposition principle. The stress from any combination of small step strains is simply the linear combination of the stresses resulting from each individual step 7i applied at time q-  

This equation simply states that, for linear response, the stress resulting from each step is independent of all the other steps. The system remembers the deformations that were imposed on it earlier, and continues to relax from each earlier deformation as new ones are applied. The stress relaxation modulus tells how much stress remains at time t from each past deformation 7, through the elapsed time t — ti that has passed since that deformation was applied at time t,-. 

Using the definition of the shear rate [Eq. (7,99)] the summation increment can be transformed into time, since 57,- = 7,5t,-. 

The stress from any smooth strain history can be written as an integral over the strain history, by replacing the above summation with an integration  

The lower integration limit is t — oo because we must integrate over all past times (not just those starting at some arbitrarily defined zero point) to ensure that all past deformations are accounted for. Equation (7.115) can be used to relate many different linear response experiments. 

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]. 

When a mechanical part is made from a polymer, and when it is to be used as a loadcarrying component, obviously it is not necessarily always going to be subject to a constant stress as in the creep test. It generally has to be designed to withstand some history of stress variation. How will the polymer respond to the stress history Can its response be predicted Fortunately, for hnear viscoelastic behavior, predicting the response is possible, because of the principle of superposition of solutions to linear differential equations. The student, of course, remembers that if y,(Ji ) and y2(x) are both solutions of an ordinary differential equation for y x), then the sum y (x) + yj (x) is also a solution. This is the basis of the Boltzmann Superposition Principle for linear polymer behavior. 

FIGURE 3.8 Creep test with step increase in stress. 

Often a conservative approximation to the maximum deflection can be obtained by approximation of a general stress history by a series of step and linear functions. This estimation has the advantage that the displacement response to step and linear functions can be determined analytically. 

FIGURE 3.10 Example applications of the Boltzman superposition principle. Example 3.4 

A creep strain i from the original stress is calculated for f = 3 s  

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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. 

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. 

The Boltzmann superposition principle can be used to relate the steady state compliance to the stress relaxation modulus (see Problem 7.44) ... 

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. 

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... 

Consider now the application of two stress increments Boltzmann superposition principle asserts that the two stresses act independently and the resultant strains add linearly. This situation is illustrated in Figure 2-14. Thus... 

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