Analytical solutions for 3-D flexural deformation of semi-infinite elastic plates

Abstract

The theory of elastic-plate flexure has played a prominent role in isostatic studies and investigations of lithospheric deformation caused by vertical tectonism. Simple analytical, axisymmetrical solutions exist when the boundaries of the plate are far removed from the area where the load is applied. When deformation is driven by end loads (i.e. bending moments and shear forces) the problem is usually simplified to yield a 2-D solution that approximates a cross-section of the 3-D solution. We present analytical Green's functions for the point-force response of semi-infinite elastic plates resting on an inviscid substratum and being acted upon by constant in-plane forces and arbitrary end loads. This solution may be used to study deformation and stresses caused by loads close to a free geological boundary (e.g. a trench or weak fracture zone). We also present Green's functions for the case when two semi-infinite plates of different rigidities are mechanically coupled and subject to a point load. This solution may be useful when studying deformation near mechanically strong fracture zones or at ocean-continent transitions. The analytical solutions provide simple alternatives to (and calibration of) complex numerical finite-element or finite-difference solutions, and furthermore give additional physical insight into the geological process to be examined. The analytical methods are used to demonstrate the different stress patterns that may arise near weak and strong geological boundaries. We also illustrate how the observed scatter in 2-D determinations of elastic thickness at subduction zones may partly have its origin in 3-D effects.