A note on lithospheric bending calculations

Summary. This paper derives exact solutions to the equations of static plane strain by means of propagator matrices for homogeneous, gravitating and non- gravitating elastic media. These solutions are immediately verifiable and are flexible under a variety of boundary conditions. Propagator matrices are eminently suitable for computer encoding and, through their multiplication, are applicable to depth-dependent structures. Attention is focused upon the bending of floating plates which are loaded to simulate the deflection of oceanic lithosphere in the vicinity of trenches. By comparing responses computed with and without body forces, I find that gravity does not meaningfully change deflection profiles; however, it can influence important aspects of the internal stress state. Gravitational stresses are proportional to the gradient of the vertical deformation and amount to about 10 per cent of the bending stresses in these models. Propagators which include gravity are used to investigate the effect of regional horizontal stresses upon bending plates. I conclude that applied compressive forces can transport the neutral surface through 10–20 km of depth without significantly deforming the plate profile or increasing the maximum internal stress by more than 30 percent. These calculations support the contention that variable compressive stresses resulting from interplate coupling could account for observed regional differences in neutral surface height. For elastic-plastic material, the fundamental equations of motion become non-linear; however, there appears to be no a priori objection to their linearization. I speculate that the propagator formalism, when applied in an iterative approach, could be a powerful method for computing the deformation of such media.