Truesdell

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

Since elastic behavior is important for inelastic constitutive equations, it is instructive to examine the behavior of the hypoelastic constitutive equation (5.112) in some detail. This has been addressed variously by Truesdell and Noll [20], Eringen [16], Atluri [17], and others. 

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... 

Jaumann s stress rate is not the only indifferent rate which could be used to render (5.117) objective. Truesdell s rate defined by (A.42) is indifferent, as shown in (A.70), and can serve just as well. Inserting Truesdell s rate instead of Jaumann s rate, (5.117) reduces to the three ordinary differential equations... 

Jaumann s stress rate, Truesdell s stress rate could have been used. Substi-... 

Since simple shear is a constant volume deformation, the solution does not depend on coefficients of terms involving tr(various values of a are shown in Fig. 5.9. The solution for a grade zero material using Jaumann s stress rate (5.120) corresponds to = Ug = Ug = 0 so that a = -1, while the solution using Truesdell s stress rate (5.122) corresponds to = 0 and Ug = 1 so that a = 0. The shear... 

The objectivity of the spatial stress rate relation (5.154) may be investigated by applying the coordinate transformation (A.50) representing a rotation and translation of the coordinate frame. The spatial strain and its convected rate are indifferent by (A.58) and (A.62). So are the stress and its Truesdell rate. It is readily verified from (5.151), (5.152), and the fact that K has been assumed to be invariant, that k and its Truesdell rate are also indifferent. Using these facts together with (A.53) in (5.154) with c and b given by (5.155)... 

Truesdell, C. and Toupin, R.A., The Classical Field Theories, in Encyclopedia of Physics, Vol. III/l (edited by Fliigge, S.), Springer-Verlag, Berlin, 1960. 

Kinematical relations in large deformations are given here for reference. Most of the material is well known, and may be extracted or deduced from the comprehensive expositions of Truesdell and Toupin [19], Truesdell and Noll [20], or other texts in continuum mechanics, where further details may be found. 

A. 12) having been used in the last equation. The stress rate s was introduced by Truesdell. Differentiating (A.40i) and rearranging by similar steps... 

C. Truesdell and R.A. Toupin, The Classical Field Theories, Encyclopedia of Physics, Volume III/l, Springer-Verlag, New York, 1960. 

From this kind of continuum mechanics one can move further towards the domain of almost pure mathematics until one reaches the field of rational mechanics, which harks back to Joseph Lagrange s (1736-1813) mechanics of rigid bodies and to earlier mathematicians such as Leonhard Euler (1707-1783) and later ones such as Augustin Cauchy (1789-1857), who developed the mechanics of deformable bodies. The preeminent exponent of this kind of continuum mechanics was probably Clifford Truesdell in Baltimore. An example of his extensive writings is A First Course in... 

I cannot judge whether Truesdell s kind of continuum mechanics is of use to mechanical engineers who have to design structures to withstand specific demands, but the total absence of diagrams causes me to wonder. In any case, I understand (Walters 1998, Tanner and Walters 1998) that rational mechanics was effectively Truesdell s invention and is likely to end with him. The birth and death of would-be disciplines go on all the time. 

Truesdell, C.A. (1977, 1991) A First Course in Rational Continuum Mechanics (Academic Press, Boston). 

Truesdell, C. (1980). The Tragicomical History of Thermodynamics, 1822—1854. New York Spriuger-Verlag. 

Gangue minerals and salinity give constraints on the pH range. The thermochemical stability field of adularia, sericite and kaolinite depends on temperature, ionic strength, pH and potassium ion concentration of the aqueous phase. The potassium ion concentration is estimated from the empirical relation of Na+/K+ obtained from analyses of geothermal waters (White, 1965 Ellis, 1969 Fournier and Truesdell, 1973), experimental data on rock-water interactions (e.g., Mottl and Holland, 1978) and assuming that salinity of inclusion fluids is equal to ffZNa+ -h m + in which m is molal concentration. From these data potassium ion concentration was assumed to be 0.1 and 0.2 mol/kg H2O for 200°C and 250°C. 

Fournier, R.O. and Truesdell, A.H. (1973) An empirical Na-K-Ca geothermometer for natural waters. Geochim. Cosmochim. Acta, 37, 1255-1275. 

Fournier and Truesdell (1973) showed that logmNa+/wK+ +4/31ogm(-22+/mNa+ is constant at constant temperature over 100°C. This value is about 0.8 at 250°C. 

Fig. 2.14. The variation of concentration of with concentration of CP in aqueous solution in equilibrium with a given mineral assemblage at 250°C. I Equilibrium curve based on albite-sericite-Na-montmorillonite-quartz-aqueous solution equilibrium and Na-K-Ca relationship obtained by Fournier and Truesdell (1973). 2 Equilibrium curve based on albite-K-feldspar-aqueous solution equilibrium and Na-K-Ca relationship obtained by Fournier and Truesdell (1973). 3 Wairakite-albite-sericite-K-feldspar-quartz. 4 Calcite-albite-sericite-K-feldspar-quartz (/jjhjCO, = 10 ). 5 Calcite-albite-sericite-Na-montmorillonite-quartz (mH2C03 = 10 ). 6 Ca-montmorillonite-albite-sericite-Na-montmorillonite-quartz. 7 Calcite-albite-sericite-K-feld-spar-quartz (mnjCOj = 10 ). 8 Calcite-albite-sericite-Na-montmorillonite-quartz (mHjCOj = 10 ). 9 Ca-montmorillonite-albite-sericite-K-feldspar-quartz. 10 Anhydrite = 10 ). (Shikazono, 1976)...

Therefore, the empirical relationship obtained by Fournier and Truesdell (1973) is changed into,... 

Fig. 2.39. Na /K+ atomic ratios of well discharges plotted at measured downhole temperatures. Curve A is the least squares fit of the data points above 80°C. Curve B is another emperical curve (from Truesdell, 1976). Curves C and D show the approximate locations of the low albite-microcline and high albite-sanidine lines derived from thermodynamic data (from Fournier, 1981). Small solid subaerial geothermal water Solid square Okinawa Jade Open square South Mariana Through Solid circle East Pacific Rise 11°N Open circle Mid Atlantic Ridge, TAG.

Garrels and Thompson s calculation, computed by hand, is the basis for a class of geochemical models that predict species distributions, mineral saturation states, and gas fugacities from chemical analyses. This class of models stems from the distinction between a chemical analysis, which reflects a solution s bulk composition, and the actual distribution of species in a solution. Such equilibrium models have become widely applied, thanks in part to the dissemination of reliable computer programs such as SOLMNEQ (Kharaka and Barnes, 1973) and WATEQ (Truesdell and Jones, 1974). 

Several chemical geothermometers are in widespread use. The silica geothermometer (Fournier and Rowe, 1966) works because the solubilities of the various silica minerals (e.g., quartz and chalcedony, Si02) increase monotonically with temperature. The concentration of dissolved silica, therefore, defines a unique equilibrium temperature for each silica mineral. The Na-K (White, 1970) and Na-K-Ca (Fournier and Truesdell, 1973) geothermometers take advantage of the fact that the equilibrium points of cation exchange reactions among various minerals (principally, the feldspars) vary with temperature. 

Truesdell, A. H. and B.F Jones, 1974, WATEQ, a computer program for calculating chemical equilibria of natural waters. US Geological Survey Journal of Research 2, 233-248. 

Several computer-based techniques have been developed for more specific applications. Truesdell (45) describes a computer program for calculating equilibrium distributions in natural water systems, given concentrations and pH. Edwards, et al. (31, Z2) have developed computer programs for treating volatile weak electrolytes such as ammonia, carbon dioxide and hydrogen sulfide systems however, in their present state these programs (presumably) do not accommodate metallic species in solutions. 

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