Buckling instability in type-II superconductors with strong pinning

In high magnetic fields a noticeable deformation of superconductors occurs in the critical state because of the magnetic force density f5j3B, where j is the current density and B is the magnetic field. This results in an anomalous irreversible magnetostriction ~‘‘suprastriction’’! and shape distortion of type-II superconductors with strong pinning. Similar as in magnetic fluid dynamics the stress tensor of a superconductor in a magnetic field includes an additional term, the Maxwell stress tensor of the magnetic field with components of order B/m0. Since this is quadratic in B, the Maxwell stress tensor in the critical state can be important for elasticity in strong magnetic field. However, even in a field of 10 T the value of B/m0 is small compared to the Young modulus E of the material. We estimate the ratio B/m0E'10 23 for B'10 T and E'100 GPa which is a typical value for YBa2Cu3O72x high-temperature superconductors. The effect of the magnetic field on the elastic behavior may be much higher if one considers bending of thin samples since the effective elastic modulus for bending Ẽ is much less than the Young modulus. In particular, for a long rectangular strip of extension l3w3d (l@w@d) one has Ẽ'E(d/l)!E . If for instance d/l'10 and E'100 GPa, then B/m0 is of the order of the effective bending modulus Ẽ at B'3.5 T. An important consequence of a small value of the effective elastic modulus for bending Ẽ is the classical Euler buckling instability. This elastic instability occurs for rods and thin strips when the longitudinal compression force F at the edges of the sample exceeds a critical value Fb}Ẽ . In particular, one has Fb5p Ewd/48l for a long rectangular strip with one edge clamped and the other edge free as shown in Fig. 1. The buckling instability manifests itself at F>Fb by a sudden bending with amplitude s}AF2Fb. The magnetization of type-II superconductors with strong pinning and the associated magnetic forces are successfully described by the Bean critical state model using a critical current density j c which decreases with increasing temperature and magnetic field. In the transverse geometry of a thin strip in a perpendicular field one has j5 j c in the region where the magnetic flux has penetrated and screening sheet currents J with 0,J, j cd in the flux-free region. 9–11 This