Energy-conjugate

The inverse of H determines the geometric compliance matrix (Nalewajski, 1993, 1995, 1997, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008) describing the open system in the Qi,F)-representation. The relevant thermodynamic potential is defined by the total Legendre transform of the system BO potential, which replaces the state-parameters (N, Q) with their energy conjugates (/a, F), respectively ... 

Eliminating A/r from the first Equation 30.14 and inserting it into the second Equation 30.14 then gives the following transformation of the representation independent displacements (AN, AF) into the linear responses of their energy conjugates... 

We summarize in Table 2.1 the physical dimensions of various energy forms. We emphasize that the list is not complete. In square brackets the physical dimension of the extensive variable has been given. The intensive variable has the energy-conjugated physical dimension. In the following sections, we discuss briefly some issues of the enagy forms detailed in Table 2.1. 

Obviously, the general procedure to get a capacity is to formulate a thermodynamic function initially in its natural variables, say (7(5, V, n). Then a variable is replaced by its energy-conjugated variable, such as U(T, V,n), and the partial differential with respect to this variable is formed to get the capacity, namely dU T, V, n)/dT. 

From the foregoing it is appealing to relate generalized susceptibilities xj as the differential of an arbitrary extensive variable X with respect to its energy-conjugated intensive variable... 

The entropy and the temperature are energy-conjugated variables, in the same way as the volume and the pressure. Thus, Eq. (6.67) resembles somehow the osmotic... 

We made here again use of the law of Schwarz. We can represent the process now in refined representation either in F—h diagram or in tx-n diagram, cf. Fig. 9.2. As the process of the paddle wheel can be represented with two different energy conjugated variables, also, for example, the Carnot process can be represented in... 

It is understood that the density function of the second Piola-Kirchhoff stress = TjjEI 0 is the energy-conjugate or dual stress to E under the definition of the increment of internal energy dua = Lagrangian description (cf. Sect. 2.7). 

As mentioned in Sect. 3.4.3, only the isotropic material body can form the rate of internal energy d Ua t, s) together with the rate e of the generalized Eulerian strain measure and its energy-conjugate corotational Euler stress =... 

There is a second characteristic property which all cases have in common One always deals with a pair of energy conjugated variables, that is to say, the displacement caused by the field results in work. More specifically, if a field gives rise to a displacement dx, then the work per unit volume is... 

The mean-squared fluctuations in thermal equilibrium can be calculated as for any pair of energy conjugated variables, by applying Boltzmann statistics. The probability distribution for (j)k is... 

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