The lithosphere acts as a rigid plate that floats on a ductile mantle (asthenosphere), and we have seen how (Airy) isostasy can be used to predict how deep the lithosphere floats under the weight of a load. Isostasy is a simplification that ignores the possible bending of the lithospheric plate due to lateral variations in the surface load. Isostasy applies for loads of large lateral extent, like for instance a mountain range or a continent. On the other hand, “small” scale features like a valley or a mountain peak do not lead to any isostatic uplift or subsidence because they are completely supported by the elastic strength of the lithosphere. There is a length scale in between where surface loads are partly supported by the lithosphere, and partly supported by the buoyancy of the displaced mantle. This length scale can be estimated assuming that the lithosphere acts like a thin linear elastic plate floating on the mantle. The Hawaiian islands, which are piles of volcanic rock on the oceanic plate, are good examples of loads that bend the lithosphere. The flexure is in this case reproduced by simple solutions of the equation for the deflection of a thin elastic plate. The solutions also predict uplift in the form of a flexural bulge, a feature that isostasy alone cannot predict. Furthermore, observation of flexure allows for estimations of the thickness of the elastic part of the lithosphere.

In this chapter we will first derive the equation for the deflection of a thin elastic plate under loads that are only x-dependent. Some simple solutions for the deflection of a plate are presented and discussed.