Z angew Math Phys 43 (1992) 0044-2275/92/060943-31 $ 1.50 + 0.20 (ZAMP) 1992 Birkh/iuser Verlag, Basel Elastostatics in the presence of a temperature distribution or inhomogeneity By J. M. Ball, P. K. Jimack* and Tang Qi~', Dept of Mathematics, Heriot-Watt University, Edinburgh, EHI4 4AS, U.K. Dedicated to Klaus Kirchgdssner on the occasion of his 60th birthday 1. Introduction We consider the problem of the equilibrium of an elastic body subjected to a temperature gradient. The boundary of the body is free to move, and there are no applied body or surface forces. Suppose the body occupies in a stress-free reference configuration the slab ~ = co x (0, 6), where co c R 2 is bounded and c~ > 0, and that the temperature gradient is then imposed in the vertical x3-direction, so that the temperature 0 is a given function 0 = 0(x3) with 0"(x3) < 0. Due to thermal expansion the layers co x {x3} will tend to expand with respect to those layers co x {x;} with x; > x3, produc- ing in the case of a thin slab of isotropic material with a small imposed temperature gradient an approximately spherical deformed shape. If we', consider the corresponding two-dimensional problem of the deformation of an elastic strip f~' = (0, l) x (0, 6) under an imposed temper- ature gradient in the vertical x2-direction, an interesting difference emerges. In this two-dimensional problem (which is similar to that of the deforma- tion of a bimetallic strip) an equilibrium solution in which each line (0, l)x {x2} is uniformly stretched to form a circular arc is possible. However, as observed by Davies [16], no surface x3 =constant can be deformed so that its principal stretches are equal and independent of (xl, x2) e co, while at the same time its principal curvatures kl, k2 are equal, positive, and independent of (Xl, x2) e co. Up to a dilation, such a deforma- tion would be an isometry, and thus preserve the Gaussian curvature; but the Gaussian curvature is zero in the reference configuration and equals klk2 > 0 in the deformed configuration. (This is the familiar difficulty encountered when trying to wrap a sphere with paper without forming * Present address: School of Computer Studies, University of Leeds, UK. ~ Present address: Mathematics Division, University of Sussex, UK.