Maxwell equations response

The spectra s (v) will be described here in terms of a linear-response theory. We shall employ the specific form [GT, VIG] of this theory, called the ACF method, which previously was termed the dynamic method. The latter is based on the Maxwell equations and classical dynamics. A more detailed description of this method is given in Section II. Taking into attention the central role of the model suggested here, we, for the sake of completeness, give below a brief list of the main assumptions employed in our variant of the ACF method. 

In order to determine fluxes and current density i, it is necessary to know Vp or E For their deflnition it is necessary to use the Maxwell equations. In general case the external electric held induces secondary electric and magnetic fields (the medium s response), which in turn influence the external field. However, if the external magnetic field is absent, and the external electric field is quasi-stationary, then the electrodynamical problem reduces to electrostatic one, namely, to determining of the electric potential distribution in liquid, described by Poisson equation... 

When the solid feature dominates the mechanical response of a shear deformation, the shear stress cr is proportional to the shear strain y, and the proportionality coefficient is the shear modulus E. On the other hand, when the liquid feature dominates the response, the shear stress cr is proportional to the shear rate y, the proportionality coefficient is the shear viscosity 77. Maxwell equation of linear viscoelasticity can be applied to describe the continuous switching between the solid and the liquid (Maxwell 1867),... 

It can be seen that the first 2 terms in the first square brackets correspond to the Maxwell element response and the other terms correspond to the Kelvin element response. The multiparameter model describes the response observed for rPET polymer concrete under a constant load. To obtain the creep strain using Equation 4.33 for a given stress, it is necessary to compute multiparameters, which are the 3 elastic constants ( i, 2 and 3) for the springs and the 3 viscous constants rp, % and 7/3) for the dashpots. Eigure 4.14 shows graphically how to obtain these values from the experimental results. 

This is the governing equation of the Maxwell Model. It is interesting to consider the response that this model predicts under three common time-dependent modes of deformation. 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

It is interesting to consider the response of a Maxwell fluid to an arbitrary shear rate history. Denoting the shear rate as y(t), an arbitrary function of time, the equivalent of equation 3.83 is... 

The spring is elastically storing energy. With time this energy is dissipated by flow within the dashpot. An experiment performed using the application of rapid stress in which the stress is monitored with time is called a stress relaxation experiment. For a single Maxwell model we require only two of the three model parameters to describe the decay of stress with time. These three parameters are the elastic modulus G, the viscosity r and the relaxation time rm. The exponential decay described in Equation (4.16) represents a linear response. As the strain is increased past a critical value this simple decay is lost. 

This result, that the low frequency limit of the in phase component of the viscosity equates to the viscosity of the dashpot, means that for a single Maxwell model it is possible to replace rj by rj(0). Thus far we have concentrated on the description of experimental responses to the application of a strain. Similar constructions can be developed for the application of a stress. For example the application of an oscillating stress to a sample gives rise to an oscillating strain. We can define a complex compliance J which is the ratio of the strain to the stress. We will explore the relationship between different experiments and the resulting models in Section 4.6. 

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

This book has been written for the practitioner, as well as researchers seeking to either predict the optical response of complex liquids or to interpret optical data in terms of microstructural attributes. For these purposes, the book is meant to be self contained, beginning with sections on the fundamental Maxwell field equations describing the interaction of electromagnetic waves with anisotropic media. These interactions include... 

If we now perform a creep experiment, applying a constant stress, a0 at time t = 0 and removing it after a time f, then the strain/ time plot shown at the top of Figure 13-89 is obtained. First, the elastic component of the model (spring) deforms instantaneously a certain amount, then the viscous component (dashpot) deforms linearly with time. When the stress is removed only the elastic part of the deformation is regained. Mathematically, we can take Maxwell s equation (Equation 13-85) and impose the creep experiment condition of constant stress da/dt = 0, which gives us Equation 13-84. In other words, the Maxwell model predicts that creep should be constant with time, which it isn t Creep is characterized by a retarded elastic response. 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

Wilding and Ward showed that the creep and recovery behaviour of the low molecular weight samples could be represented to a good approximation by the model representation shown in Fig. 35(b), which consists of a Maxwell and Voigt element in series, on the basis that the parameters E, E, r and r), are dependent on the stress level. Data for the creep response of the samples under discussion at a constant applied stress Op were therefore fitted to the equation... 

This equation is derived by integrating Eq.( 11-29) with boundary condition)/ = 0, T = To at r = 0. Although the model has some elastic character the viscous response dominates at all but short times. For this reason, the element is known as a Maxwell fluid. 

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