Low-temperature elastic moduli and dilational and shear internal frictions of superconducting ceramic ${\mathrm{GdBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$

Longitudinal and transverse wave velocities, six kinds of elastic parameters, and dilational and shear internal frictions have been simultaneously measured as functions of temperature between room temperature and 77 K in the high-${\mathit{T}}_{\mathit{c}}$ superconductor ${\mathrm{GdBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$. An abrupt increase from 117 K in bulk modulus and Lamè parameter suggests that the order parameter has a much stronger coupling with volume-nonpreserving distortions than with volume-preserving distortions. The 227 and 120 K peaks in dilational friction might be related to structural change and second-order phase transition due to variation of potential energy, respectively. For shear friction, the 214 and 106 K peaks could be attributed to barium atom relaxation with an apparent activation energy of 0.62 eV, and lattice instability, respectively.