Maxwell solid

Calculate the longitudinal strain of a viscoelastic rod of a material that behaves like (a) a Maxwell model, (b) a Maxwell solid in shear but an elastic solid in bulk and (c) a viscoelastic solid standard in shear but an elastic solid in bulk. The material is constrained in such a way that the lateral dimensions cannot vary when it is under uniform forces of compression at both ends of the rod. 

The observation of viscous flow in SiOz films was first reported by EerNisse(23,24). Essentially, a compressive intrinsic stress was found to exist in Si02 films grown below 1000°C and this stress was relieved at higher temperatures. The densification of SiO films was reported(20,21) and a unified model that explains both the occurrence of stress in Si0 and the higher density was published(25). This model utilized the concept of viscous relaxation in a Maxwell solid. The main idea is depicted in Fig.7 where the molar volume change is seen to cause the stress and density increase which are both relieved via viscous flow at sufficiently high temperatures. These ideas were recently incorporated into a revised oxidation mode1(26). Part of this revision modifies the oxidation expression to include the stress driven viscous relaxation. From a consideration of SiC as a simple Maxwell solid the expression for F becomes ... 

Figure 6.17 Kelvin and Maxwell solids and related creep and stress relaxation.

Consider for instance a Maxwell solid submitted to a constant deformation y(.t) = yo so that y = 0. Shear stress relaxation with time is foimd solving... 

Hence, it is found that a Kelvin solid flows under constant stress once time is of the order of Tr (Figure 6.17a). Some authors refer to Kelvin Tr as retardation time. As discussed below, glass delayed elasticity and flow can be captured with a Burger solid that combines in series a Kelvin and a Maxwell solid. A Kelvin solid yields retardation while a Maxwell one yields relaxation. Relaxation time informs on the time scales at which a viscoelastic solid will behave elastically or relax. Let us consider glass transition the viscosity is 10 -Pas while shear modulus of most glasses scales with Pa so that relaxation time is of the... 

Let us illustrate this behaviour for the case of a Maxwell solid. The expression for G can be found taking again Eq. (6.42) with the complex notation (with f = iw T ) ... 

The basis for the familiar non-slip boundary condition is a kinetic theory argument originally presented by Maxwell [23]. For a pure gas Maxwell showed that the tangential velocity v and its derivative nornial to a plane solid surface should be related by... 

Maxwell considered the motion of a gas in the neighborhood of a plane solid wall, in che presence of a temperature gradient. In particular, when Che velocity field is one dimensional and everywhere parallel to the wall, and the temperature gradient is parallel to the velocity field, he found that... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

It is one of the wonders of the history of physics that a rigorous theory of the behaviour of a chaotic assembly of molecules - a gas - preceded by several decades the experimental uncovering of the structure of regular, crystalline solids. Attempts to create a kinetic theory of gases go all the way back to the Swiss mathematician, Daniel Bernouilli, in 1738, followed by John Herapath in 1820 and John James Waterston in 1845. But it fell to the great James Clerk Maxwell in the 1860s to take... 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

Worse was to come. Boltzmann in 1872 made the same weird statistical equality hold for every mode in a dynamical system. It must, for example, apply to any internal motions that molecules might have. Assuming, as most physicists did by then, that the sharp lines seen in the spectra of chemical elements originate in just such internal motions, any calculation now of Cp/C would yield a figure even lower than 1.333. Worse yet, as Maxwell shatteringly remarked to one student, equipartition must apply to solids and liquids as well as gases Boltzmann has proved too much. ... 

Now let the silver be supposed to be surrounded by a monatomic gas, such as helium. The atoms of the gas, when they collide with those of the solid, will impart kinetic energy to them. According to Maxwell s Law of Equipartition of Energy, two bodies are in temperature equilibrium when the mean kinetic energy for each degree of freedom of motion of each atom is the same in both. The atoms of the solid and those of the gas have three degrees of freedom each to each degree corresponds the kinetic... 

FIGURE 9.8 Typical stress-strain plots for a strip of recombinant resilin tested in phosphate-buffered saline (PBS). Sample cycled to 225%, showing resilience of 97% (solid curve) and later tested to failure showing extension at break of 313% (dotted curve). (FromElvin, C.M., Carr, A.G., Huson, M.G., Maxwell, J.M., Pearson, R.D., Vuocolo 1, T., Liyon, N.E., Wong, D.C.C., Merritt, D.J., and Dixon, N.E., Nature, 437, 999, 2005.)... 

FIGURE 9.14 Typical approach force curve (solid line) for a sample which is penetrated by the scanning probe microscope (SPM) tip. Also shown is the force curve (dashed line) when the tip encounters a hard surface (glass) and schematic drawings of the relative positions of the SPM tip and the sample surface as related to the force curves. (From Huson, M.G. and Maxwell, J.M., Polym. Test., 25, 2, 2006.)... 

It is likely that most biomaterials possess non-linear elastic properties. However, in the absence of detailed measurements of the relevant properties it is not necessary to resort to complicated non-linear theories of viscoelasticity. A simple dashpot-and-spring Maxwell model of viscoelasticity will provide a good basis to consider the main features of the behaviour of the soft-solid walls of most biomaterials in the flow field of a typical bioprocess equipment. 

FIG. 21 Effective diffusion coefficients from Refs. 337 and 193 showing comparison of volume average results (Ryan) with models of Maxwell, Weisberg, Wakao, and Smith for isotropic systems (a), and volume averaging calculations (solid lines) and comparison with data for anisotropic systems (b). (Reproduced with kind permission of Kluwer Academic Publishers from Ref. 193, Fig. 3 and 12, Copyright Kluwer Academic Publishers.)... 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

The Maxwell theory of X-ray scattering by stable systems, both solids and liquids, is described in many textbooks. A simple and compact presentation is given in Chapter 15 of Electrodynamics of Continuous Media [20]. The incident electric and magnetic X-ray helds are plane waves Ex(r, f) = Exo exp[i(q r — fixO] H(r, t) = H o exp[/(q r — fixO] with a spatially and temporally constant amplitude. The electric field Ex(r, t) induces a forced oscillation of the electrons in the body. They then act as elementary antennas emitting the scattered X-ray radiation. For many purposes, the electrons may be considered to be free. One then finds that the intensity /x(q) of the X-ray radiation scattered along the wavevector q is... 

Figure 2. Comparison of the simulated velocity distribution (histogram) with the Maxwell— Boltzmann distribution function (solid line) for kgT —. The system had volume V — 1003 cells of unit length and N = 107 particles with mass m = 1. Rotations (b were selected from the set Q — tt/2, — ti/2 about axes whose directions were chosen uniformly on the surface of a sphere.

The theory on the level of the electrode and on the electrochemical cell is sufficiently advanced [4-7]. In this connection, it is necessary to mention the works of J.Newman and R.White s group [8-12], In the majority of publications, the macroscopical approach is used. The authors take into account the transport process and material balance within the system in a proper way. The analysis of the flows in the porous matrix or in the cell takes generally into consideration the diffusion, migration and convection processes. While computing transport processes in the concentrated electrolytes the Stefan-Maxwell equations are used. To calculate electron transfer in a solid phase the Ohm s law in its differential form is used. The electrochemical transformations within the electrodes are described by the Batler-Volmer equation. The internal surface of the electrode, where electrochemical process runs, is frequently presented as a certain function of the porosity or as a certain state of the reagents transformation. To describe this function, various modeling or empirical equations are offered, and they... 

Water absorption can also cause significant changes in the permittivity and must be considered when describing dielectric behavior. Water, with a dielectric constant of 78 at 25°C, can easily impact the dielectric properties at relatively low absorptions owing to the dipolar polarizability contribution. However, the electronic polarizability is actually lower than solid state polymers. The index of refraction at 25°C for pure water is 1.33, which, applying Maxwell s relationship, yields a dielectric constant of 1.76. Therefore, water absorption may actually act to decrease the dielectric constant at optical frequencies. This is an area that will be explored with future experiments involving water absorption and index measurements. 

Figure 14.11 Elastic (<30 and loss (G 0 components of the dynamic modulus of the first seven generations of bulk PAMAM dendrimers as a function of shifted shear frequency [aT co] at 40°C. Solid lines indicate a fit for each curve to the generalized Maxwell model...

This material is a linear viscoelastic solid and is described by the multiple Maxwell model with an additional term, the spring elasticity... 

For a viscoelastic solid the situation is more complex because the solid component will never flow. As the strain is applied with time the stress will increase continually with time. The sample will show no plateau viscosity, although there may be a low shear viscous contribution. This applies to both a single Maxwell model and one with a spectrum of processes ... 

Finally, in the quantum approximation the radiation is no longer treated classically (i.e., using Maxwell s equation), and so both radiation and matter are described by quantum methods. For most of the features in the spectra of solids, this approach is not necessary and it will not be invoked. However, this approximation also leads to important aspects, such as zero-point fluctuations, which are relevant in the theory of lasers and Optical Parametic Oscillators (Chapter 3). 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

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