Incompressible body

The form of S(U, V, n) corresponds to the isodynamic expansion of a gas as done in the Gay-Lussac process. It is typical for a gas that it expands deliberately. Therefore, we demand /c 0 for any gas. 

the isentropic temperature change of a van der Waals gas on the volume V is dependent solely on the van der Waals parameter b. For example, this relation is relevant for diesel engines or for the impact sensitivity of explosives. 

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Considering an example of isothermal, incompressible body in elastic contact with the presumption that there are adequate molecular layers on the minimum film thickness spot, we will get the governing equations as follows... 

The foregoing models considered incompressible bodies however, this is never the case in practice. The following section discusses models that specifically consider contact between deformable solids. 

As the heat conduction is being studied in a solid body, the small change in the density as a result of the temperature and pressure variations can be neglected. The model of an incompressible body g = const is therefore used. Under this assumption... 

In the derivation of the heat conduction equation in (2.8) we presumed an incompressible body, g = const. The temperature dependence of both the thermal conductivity A and the specific heat capacity c was also neglected. These assumptions have to be made if a mathematical solution to the heat conduction equation is to be obtained. This type of closed solution is commonly known as the exact solution. The solution possibilities for a material which has temperature dependent properties will be discussed in section 2.1.4. 

The model considered here is that of an incompressible body. This is defined by the fact that the density of a volume element in the material does not change during its movement, i.e. g = g(x,t) = const and therefore dg/dt = 0, which is fulfilled in our case, because due to w = 0 and dg/dt = 0... 

For the case of a homogeneous incompressible body, Love (1927) showed... 

However, Love (1911) had earlier derived a more general result, valid for arbitrary compressibility. Compressibility is characterized by the Lame constant A A increases as a material gets more incompressible to a limiting case of A —> oo for an incompressible body. The equation governing the deformation is re-derived in the appendix, derivation of the Governing Equation. 

Here we have rearranged the constants into a more compact notation. The application of the method becomes even more simple, when the Taylor expansion ends after some terms, as is the case for the incompressible body, where V = % +a T—To)) and Vp = 0. 

We have to distinguish between an isothermal incompressible body and an isentropic incompressible body. We characterize... 

The fundamental equations for an incompressible body have been derived [1], In the following, we suppress the mol number as a variable. 

Since for the isentropic incompressible body we have the independent variables S and p as starting point, the enthalpy is a convenient thermodynamic function to handle this problem. We put... 

Therefore, the free energy is a function of the temperature alone. We emphasize that for the isothermal incompressible body the heat capacity at constant volume is not accessible. When we change the temperature, which is needed to measure the C then we would also change the volume, due to Eq. (4.46). An exception is the special case dy(r)/dr = 0 that occurs in liquid water at the density maximum. Therefore, C is not defined in general however, Cp is readily accessible. 

We discuss finally the case of a thermal expansion coefficient of zero. In this case, a pressure change does not effect a temperature change of the material and also no change in volume. So no input of compression energy is achieved. In contrary to the adiabatic application of pressure in the isothermal case in fact no change in volume will occur. Therefore, the incompressible body should be more accurately addressed as the isothermal incompressible body. 

An important prerequisite is that the machine can exchange the energy forms under consideration. Otherwise, such a process is not possible. To illustrate the above statement we want to perform the Carnot process with an incompressible body. The incompressible body has the function of state... 

If we apply pressure to an incompressible body in the course of an adiabatic process, the body may unexpectedly expand. The increase of pressure leaves even an incompressible body literally not cool. The body can accept energy and can warm up when thermally isolated. Since the pressure has no effect on the dimensions, but the temperature has effect, the body may expand on heating. 

More precisely we must differentiate between an adiabatically incompressible boy and an isothermally incompressible body. Since the generalized susceptibilities must be positive for the sake of stability of matter, we have... 

Here v represents the Poisson ratio, defined as the ratio between the linear contraction and the elongation in the axis of stretching. In the case of constant volume (an incompressible body like rubber), i> = 0.5 and therefore E = 3G for rigid materials p < 0.3. There is also a direct test for tear strength, mainly in the case of thin films, similar to those used in the paper industry. 

As we can see, the second and third invariants, IIb and IIIb are not included. The third invariant relates generally to the relative volume change. For incompressible bodies, this is equal to unity... 

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