The Young's modulus E and the internal friction of polycrystalline ceramic La2-xSrxCuO4 were measured over the composition range 0.05 x 0.30. A striking elastic constant anomaly, reported previously, was observed, in which E falls by 40% or more and the internal friction increases with decreasing temperature. The temperature region of the elastic softening falls rapidly with increasing Sr concentration x and occurs near the tetragonal-orthorhombic (T-O) transition temperature. The soft modulus does not recover at lower temperatures but persists to at least 100 K. This elastic softening and the accompanying internal friction are attributed to stress-induced migration of martensitic domain boundaries produced by the T-O transition. It is also found that there is no correlation between the elastic softening and the occurrence of superconductivity. © 1990 The American Physical Society.