We suggest a modelreentrant elastic softeningthat achieves three useful results. First, and principally, it reconciles existing sound-velocity-elastic-constant measurements with thermodynamics.Second, it leads to Debye characteristic temperatures that agree with those from specific-heat and phonon density-of-states determinations.Third, it links elastic-constanttemperature behavior in Y-Ba-Cu-0 and La-Sr-Cu-O. The model predicts a superconductingstate elastic stiiTness lower than the normal state.Recently, several authors' reported either soundvelocity or elastic-constant changes during the normalsuperconducting phase transition in Y-Ba-Cu-O. These results appear to differ from those reported for La-Sr-Cu-0 and La-Ba-Cu-0. 9 '3 Moreover, because these results appear to show an elastic stiffening below the critical temperature T"they appear to violate thermodynamic requirements.(Notably, Mathias et al. 5 found a large elastic-stiffness softening. ) Thermodynamics of superconductors requires that cooling through the normal-superconducting transition causes an increase in specific heat 4tr ' dT2 dT Here V denotes volume, T temperature, and H, critical magnetic-field intensity.For Y-Ba-Cu-O, several authors'6 ' reported a positive hC. This means that d 2H, /dT2 is not large and negative.We can extend the thermodynamics to the bulkmodulus change M by invoking the Ehrenfest relationship " m -a, a"-TV(-P)'a'-/~C .Here P denotes volume thermal expansivity (1/V)x(8V/8T)p. Thus, SB differs in sign from hC; with the above proviso about d2H /dT, thermodynamics requires that the bulk modulus decreases upon cooling through T,. Pippard extended the thermodynamics to the shearmodulus change: G Ho d Hii T4 G-G, -G. --1 -. (3) T, Here Ho denotes critical magnetic-field intensity at zero temperature and r shear stress. Equation (3) follows from two assumptions: H, varies parabolically with T; shearing produces no change in electromc specific heat. Pippard emphasized that the shear modulus "cannot increase when a metal becomes su perconducting. " Pippard's relationship provided a theoretical thermodynamic basis for~reexisting experimental results of Lan-dauer2' and Olsen 2 for tin: AG/G -3. 5[1 -(T/T, )41 X10 Thus, from both observation and theory, we expect AG to have the same (negative) sign as M and to be small.[We note that the Landau-Lifshitz~3 theory of secondorder-phase-transition thermodynamics predicts the same results as in Eqs.(1)-(3). Their theory contains an order parameter and a transition (on cooling) from a symmetrical to a nonsymmetrical state. Superconductivity represents a special type of second-order phase transition, a condensation without symmetry change, where the order parameter is the energy gap h. ]As an example of our model, consider previously report-ed4 shear-modulus measurements shown in Fig.