Thin-sheet approximations are widely used in geodynamics because of
their potential for fast computation of 3-D lithospheric deformations
using simple numerical techniques. However, this simplicity imposes
limits to boundary conditions, rheological settings and accuracy of
results. This paper presents a new approach to reduce these
restrictions. The mathematical formulation of the model involves the
construction of the depth distributions of stress and velocity fields
using asymptotic approximations of 3-D force balance and rheological
relations. The asymptotic treatment is performed on the basis of a small
geometry parameter epsilon (thickness to width ratio of the thin sheet)
with a high accuracy while keeping terms which are capable of generating
strong singularities due to possible large variations in material
properties in layered systems. The depth profiles are verified by a
condition of exact equilibrium in the depth-integrated force balance and
by an asymptotic approach to the boundary conditions. The set of
analytical depth profiles of velocities and stresses, together with the
2-D equations representing the integrated force balance, result in an
extended thin-sheet approximation (ETSA). The potential of the ETSA is
demonstrated by applications to problems with different types of
boundary conditions and consideration of the types of systems of
equations governing each case. These studies have not found any strong
limitations to the boundary conditions considered and demonstrate the
greater generality and higher accuracy of ETSA in comparison with the
previous generation of thin-sheet approximations. The accompanying paper
demonstrates the results of 2-D experiments based on ETSA.