Truesdell function

The functions (x a) with a = 312,512 appear in the study of Bose-Einstein condensation of ideal quantum particles [21] that obey Bose-Einstein statistics. Their mathematical properties were studied by Truesdell [22] in detail, so that it is called the Truesdell functions. Their radius of convergence is given by x = 1. Both the functions O (x 3/2) and d> (x 5/2) remain at a finite value at x = 1, but diverge as soon as x exceeds unity. 

While c in (5.112) is a linear function of d, it may be an arbitrary function of s. Truesdell considered cases where c is a polynomial in s, terming (5.112) a hypoelastic equation of grade n, where n is the power of the highest-order term in the polynomial. For a hypoelastic equation of grade zero, the elastic modulus c is independent of s and linear in dand therefore has the representation (A.89). It is convenient to nondimensionalize the stress by defining s = sjljx. Since the stress rate must vanish when d is zero, Cq = 0 and the result is... 

The notion of state was derived from the theory of mechanics and of thermodynamics and generalised by mathematical system theory. Following the tradition of the field theory of continua (Truesdell Noll, 1965) quantities of a theory can be classified into two categories state variables and constitutive quantities. State variables are functions such that their values... 

Thus, we can see that the singularity at x=x is identical to those in Truesdell s functions... 

Note 3.16 (On the Fundamental Identity (Truesdell and Toupin 1960, pp. 469- 74)). For a function q>a we have the following material time-differential with respect to the velocity v ... 

Truesdell, C.(1945) On a Function Which Occurs in the Theory of the Structure of Polymers, Annals of Mathematics, Vol.46,144-157. 

Note that since m(s) and a( i, 2) are functions only of time y, then t]q, y3, and v are constants. A material that can be represented by the constitutive equation given in Eq. (3.76) is called a Coleman-Noll second-order fluid (Coleman and Markovitz 1964 Truesdell and Noll 1965). For steady-state simple shear flow, Eq. (3.76) yields... 

Here, in the commonly accepted notation, we have set the scalar function Lagrange multiplier to be A = — A2. The stress vector t and generalised stress vector s take the forms (cf. Truesdell and Noll [270, p.40], Goodbody [115, p.207])... 

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