Elastic constants and microcracks in YBa 2 Cu 3 O 7
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We analyze theoretically the effect of microcracks and voids on the apparent elastic constants of polycrystalline YBa2Cu3O7. Using measurements by Holcomb and Mayo, we calculate crack density and crack aspect ratio. We obtain reasonable intrinsic elastic constants. For the bulk modulus, for example, we predict values close to these obtained by neutron-diffraction studies: 123 GPa for a polycrystal and 122 GPa for a monocrystal.
I. INTRODUCTION We analyze theoretically the effect of microcracks on the effective macroscopic elastic constants of a polycrystalline ceramic material, YBa2Cu3O7, perhaps the most studied high-r c superconductor. Knowing the intrinsic elastic constants is important for many reasons. Elastic constants relate to material characterization, acoustic Debye temperature, loaddeflection behavior, thermoelastic stress, residual stress, plastic deformation, interatomic force constants and potentials, hardness, fracture toughness, and many other properties. This list contains three kinds of relationships. Elastic constants can enter simply as parameters, for example in Hooke's law. From them, we can estimate other difficult-to-measure properties such as surface energy. We can use them to check more fundamental material properties; for example, the second volume derivative of the interatomic potential gives the dilational elastic constant, the bulk modulus. In superconductors, in the BCS model, elastic constants enter both the Debye characteristic temperature dD and the electron-phonon parameter A. Elastic constants represent collective long-wavelength excitations, the acoustic phonons, which couple directly with longwavelength boson excitations. In conventional superconductors, the elastic stiffnesses soften during cooling below Tc} In general, elastic constants provide a sensitive probe of phase transitions.2 Because most oxide superconductors are polycrystals with large thermal-expansion anisotropy, we expect them to contain microcracks. Holcomb and Mayo3 studied YBa2Cu3O7 and found a fivefold increase in the bulk modulus upon applying a 1-GPa isostatic pressure, which presumably closed the cracks, changing both the strain-gage and ultrasonic-velocity response. According to their experiment, we assume that pores
that close under a pressure of 1 GPa are microcracks and the others are voids. Then we use a self-consistent method developed by Budiansky and O'Connell 4 to eliminate the effect of microcracks. We then use a multiple-scattering plane-wave approach developed by Ledbetter and Datta5 to eliminate the effect of voids. A 1-GPa pressure would produce very little effect on the intrinsic elastic stiffnesses. Considering polycrystalline YBaaCusO;,:, Ledbetter6 used ultrasonic methods and found the apparent bulk modulus sometimes as low as fifty percent of that expected either from measurements on monocrystals (presumably crack-free) or expected from ionic-model calculations. Another study, theoretical, by Ledbetter,7 focused on the effects of disk-shaped voids (approximations to cracks) and found
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