Linear response superposition principle

A dynamic system is linear if the Principle of Superposition can be applied. This states that The response y t) of a linear system due to several inputs x t),... 

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. 

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... 

Here p is the set of characteristic values of the parameters i.e. p(x) = p co(jt) where w(x) has values centered on 1. Often we can set p = / p(x) g(x) dx. The proof is really a statement of what linearity means, for if g(x)djt is the input concentration, g(x)dx.A(p(x)) is the output when the parameter values are p(x). Here x serves merely as an identifying mark, being truly an index variable and the integration in equation (14) follows from the superposition principle for linear systems. The same principle allows an obvious extension to multiple input, multiple output linear systems. A becomes a matrix whose elements are the response at one of the out-ports to a unit input at one of the in-ports, the input at all the others being zero. The detail of this case will not be elaborated here, but rather shall we pass to an application. 

The key property in this complex, unpredictable, random-like behavior is nonlinearity. When a system (process, or model, or both) consists only of linear components, the response is proportional to its stimulus and the cumulative effect of two stimuli is equal to the summation of the individual effects of each stimulus. This is the superposition principle, which states that every linear system can be studied by breaking it down into its components (thus reducing complexity). In contrast, for nonlinear systems, the superposition principle does not hold the overall behavior of the system is not at all the same as the summation of the individual behaviors of its components, making complex, unpredictable behavior a possibility. Nevertheless, not every nonlinear system is chaotic, which means that nonlinearity is a necessary but not a sufficient condition for chaos. 

The major features of linear viscoelastic behavior that will be reviewed here are the superposition principle and time-temperature equivalence. Where they are valid, both make it possible to calculate the mechanical response of a material under a wide range of conditions from a limited store of experimental information. 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

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- ... 

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 ... 

In spite of these complications, the viscoelastic response of an amorphous polymer to small stresses turns out to be a relatively simple subject because of two helpful features (1) the behavior is linear in the stress, which permits the application of the powerful superposition principle and (2) the behavior often follows a time-temperature equivalence principle, which permits the rapid viscoelastic response at high temperatures and the slow response at low temperatures to be condensed in a single master curve. 

This idea can be used to formulate an integral representation of linear viscoelasticity. The strategy is to perform a thought experiment in which a step function in strain is applied, e t) = Cq H t), where H t) is the Heaviside step function, and the stress response a t) is measured. Then a stress relaxation modulus can be defined by E t) = <7(t)/ o Note that does not have to be infinitesimal due to the assumed superposition principle. To develop a model capable of predicting the stress response from an arbitrary strain history, start by decomposing the strain history into a sum of infinitesimal strain increments ... 

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... 

We have used the generalized phenomenological Maxwell model or Boltzmann s superposition principle to obtain the basic equation (Eq. (4.22) or (4.23)) for describing linear viscoelastic behavior. For the kind of polymeric liquid studied in this book, this basic equation has been well tested by experimental measurements of viscoelastic responses to different rate-of-strain histories in the linear region. There are several types of rate-of-strain functions A(t) which have often been used to evaluate the viscoelastic properties of the polymer. These different viscoelastic quantities, obtained from different kinds of measurements, are related through the relaxation modulus G t). In the following sections, we shall show how these different viscoelastic quantities are expressed in terms of G(t) by using Eq. (4.22). 

If linear, a sine excitation input results in a sine response. However, the immittance concept can be extended to nonlinear networks, where a sine wave excitation leads to a nonsinusoidal response. Including a separate immittance value for each harmonic component of the response performs the necessary extension. In the linear region, the principle of superposition is valid. This means, for example, that the presence of strong harmonics in the applied current or voltage would not affect immittance determination at the fundamental frequency or a harmonic (Schwan, 1963). Some lock-in amplifiers can measure harmonic components, making it possible to analyze nonlinear phenomena and extend measurement to nonsinusoidal responses. 

Fig. 6.9 Illustration of Boltzmann superposition principle for the strain response linearly integrated with the stepwise increase of stresses...

X serves merely as an identifying mark, being truly an index variable, and the integration in Equation 14 follows from the superposition principle for linear systems. The same principle allows an obvious extension to multiple input, multiple output linear systems. A becomes a matrix whose elements are the response at one of the outports to a unit input at one of the in-ports, the input at all the others being zero. 

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... 

Figure 7b illustrates the way in which the responses add for a two-step history in which each step has the same magnitude. Equation 20a is the general form and is the discrete form of the linear superposition principle cast as a simple shear. It shows the simple linear additivity of the responses. A similar equation... 

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. 

It is now a simple matter to show that the Weyl-form of fractional calculus is an exceedingly powerful mathematical method when treating materials whose internal processes obey algebraic decays. We follow here the description given in [9]. Denoting the response of the system to an external perturbation tf (f) by (f), one can express the relation between these two functions in terms of the response of the system to a step pertinbation 0(t). Namely, because of the superposition principle and of causaUty, in the framework of linear response one obtains ... 

This states in effect that the response of a linear viscoelastic material to stress increments a, applied at different times t is the sum of the responses to the stress increments applied separately and independently. A corollary allows superposition of the stress responses to incrementally applied strain increments. By passing to infinitesimal increments, responses to continually varying stress and strain can be calculated using the Boltzmann superposition principle ... 

Without going into further details the conclusion from the analysis is that one can indeed obtain both 2 and i correlation functions quite directly from the two responses at least for the simple molecular anisotropy and simple field pulses assumed. Further generalization to molecules of arbitrary symmetry presents problems discussed by Rosato and Williams (75) while other pulse shapes require more complicated treatments than for linear effects with simple superposition principles (from unpublished calculations of the writer). There is also the question of local field effects. 

If the material is subjected to a time-dependent strain, the situation becomes more complicated. However, in the case of a linear viscoelastic material (like many food products) the superposition principle can be applied the response of the stress to a strain increment is independent of the already existing strain. The effect of the strain as a function of time can therefore be integrated, and the generalized Hooke s law can be extended to describe the stress-strain behavior of linear viscoelastic materials relatively easily. 

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. 

The linear viscoelastic response of LDPE/LDH nanocomposites has been studied using dynamic oscillatory measurements at constant strain amplitude of 2% and frequency sweep of 0.05-100 rad s . The response of aU nanocomposites is found to be quahtatively similar in the temperature range 160-240 °C. However, the time-temperature superposition principle is not... 

Solution. This problem is solved in Reference 10 (p. 56ff) by direct integration of the differential equation for the Maxwell element. Here, we will apply Boltzmann s superposition principle to obtain the results and, in doing so, again illustrate how information from one type of linear test (stress relaxation) may be used to predict the response in another (dynamic testing). 

MIMO (multiple input, multiple output) process modeling is inherently more complicated than SISO modeling. For linear systems, the Principle of Superposition holds, which allows MIMO models to be developed through a series of single step tests for each input, while holding the other inputs constant. For a process with three inputs (n) and three outputs (y), we can introduce a step change in and record the responses for yi, y2, and 3. The three transfer functions involving u, namely... 

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