Deformation of an elastic shell with variable thickness: a comparison of different methods

Abstract

SUMMARY Deformation of the outermost parts of single-plate planetary bodies is often modelled in terms of the response of a spherical elastic shell to surface or basal loading. As the thickness of such elastic lithosphere is usually much smaller than the radius of the body, the deformation is commonly approximated by that obtained for a thin elastic shell of uniform thickness. The main advantage of the thin shell approximation is its simplicity—the solution can be expressed analytically if the thickness of the shell is uniform, but even in the case of a thin shell of variable thickness, when the problem must be solved numerically, the computational costs are much lower than in a fully 3-D case. Here we analyse the error of the thin shell approximation by comparing it with the solution obtained for a shell of finite thickness using finite element methods. Special attention is paid to a shell of variable thickness and, in general, to the effect of elastic thickness variations on local deformation. For a shell of uniform thickness with the outer radius corresponding to Mars, we find that the error in radial displacement at low harmonic degrees (l≤ 20) does not exceed 5 per cent for small shell thicknesses (d≤ 50 km) and 10 per cent for thick shells (d∼ 250 km). Similar accuracy is also found for a shell of variable thickness if the thin shell approximation is used. Our numerical tests indicate that local deformation of a shell is mostly sensitive to the average thickness of the shell in the near zone while the effect of thickness variations in the far zone can be neglected in the first approximation. Consequently, the extremely simple thin shell method, designed for shells of uniform thickness, can be effectively used to obtain a reasonably accurate estimate of deflection even in the case of a shell with varying thickness. Finally, we investigate the deformation of an elastic lithosphere due to viscous flow beneath the shell, and we propose an extension of the concept, recently developed to correct dynamic topography for the effect of an elastic lithosphere, to the case of a shell of variable thickness.