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Generalizations of the Young–Laplace equation for the pressure of a mechanically stable gas bubble in a soft elastic material
1.E. A. Guggenheim, Thermodynamics. An Advanced Treatment for Chemists and Physicists (North-Holland, Amsterdam, 1967), Chap. 1.
7.R. Ball, J. Himm, L. D. Homer, and E. D. Thalmann, Undersea Hyperbaric Med. 22, 263 (1995).
8.P. K. Weathersby, L. D. Homer, and E. T. Flynn, J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 53, 940 (1982).
9.R. D. Vann, in Bove and Davis’ Diving Medicine, 4th ed., edited by A. A. Bove (Saunders, Philadelphia, 2004), Chaps. 4 and 7 (and references therein).
11.M. L. Gernhardt, “Development and evaluation of a decompression stress index based on tissue bubble dynamics,” Ph.D. thesis, University of Pennsylvania Press, 1991.
12.W. A. Gerth and R. D. Vann, Undersea Hyperbaric Med. 24, 275 (1997).
13.R. S. Srinivasan, W. A. Gerth, and M. R. Powell, J. Appl. Physiol. 86, 732 (1999).
15.L. D. Landua and E. M. Lifshits, Theory of Elasticity (Pergamon, New York, 1959), Chap. I.
16.I. S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd ed. (Krieger, Malabar, FL, 1983).
17.The modulus of hydrostatic compression (or the modulus of compression) , the modulus of rigidity (or the shear modulus) , Young’s modulus , and Poisson’s ratio are related as follows (Refs. 15 and 16): , where is a measure of stiffness of an isotropic elastic material, and , , and are related by , which can be taken as the definition of . Also, and are called Lamé’s first and second parameters, respectively. Physically, and , respectively, provide the pure compression and the pure shear contributions to , and is simply the reciprocal of the familiar coefficient of isothermal compressibility , where . For materials with , the elastic characteristics of ordinary fluids (that resist compression but not shear) are recovered. This limit, in which (exactly), is also known as the “isotropic upper limit of ” (Ref. 16). Our interest here is in materials for which , or equivalently, . These are also materials for which is very slightly less than 0.5. Soft rubber and gelatin (e.g., “jello”) are examples.
18.D. A. McQuarrie, Statistical Mechanics (Harper & Row, New York, 1976), Chap. 12.
19.Most determinations of the relative volumetric strain are for highly incompressible condensed phase materials for which the full equation of state is not known. Here the value of , the (extrapolated) zero-pressure volume, is usually unknown, so that cannot be determined directly. For these cases, which are the norm, is estimated from a truncated first-order expansion of the volume in terms of the applied pressure under the assumption that both the applied pressure and the resultant volume change are small (Ref. 15). Thus, . The compressibility functions are presumed known at the required pressure and temperature . It turns out that the effect of the two approximations made above—first-order truncation (since is small) and subsequently replacing by (since is small) exactly cancel one another for an ideal gas. Thus, if is evaluated for an ideal gas, it gives exactly the same value (−1) as is obtained directly for . This is a consequence of the functional form of the ideal gas equation of state .
20.I. I. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955), p. 208.
23.In Refs. 11–14, the symbol is used for what is there (and elsewhere) referred to as the “bulk modulus.” This function is here (and in Ref. 15) referred to either as the “modulus of compression” or as the “modulus of hydrostatic compression.” The symbol used here for this function is (Ref. 17).
24.M. B. Strauss, S. S. Miller, A. J. Lewis, J. E. Bozanic, and I. V. Aksenov, Undersea Hyperbaric Med. 35, 241 (2008).
25.E. D. Thalmann, E. C. Parker, S. S. Survanshi, and P. K. Weathersby, Undersea Hyperbaric Med. 24, 255 (1997) (and references therein).
26.P. Tikuisis, R. Y. Nishi, and P. K. Weathersby, Undersea Biomed. Res. 15, 301 (1988) (and references therein).
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