Simple analytical solutions exist for the deflection by line loads of thin elastic beams that overlie an inviscid fluid substrate. Deflections for more complex rectangular- and triangular-shaped loads can be evaluated by integration. The deflection of a beam that varies in thickness along its length is best solved, however, by numerical rather than analytical methods. Foremost among them are finite difference techniques, and both two- and three-dimensional codes are now widely available. Thin-plate theory and a flat Earth are good approximations to make in most geological modelling applications.

Flexure is an important feature of deep-sea trench–outer rise systems. In particular, the topographic rises seaward of deep-sea trenches can be explained by vertical and horizontal loads that act on the approaching oceanic lithosphere. While the source of these loads is not always clear, they combine to flex the lithosphere downwards by a few kilometres at the trench and upwards by several hundreds of metres in the region of the topographic rise. Many of the differences in the bathymetry of deep-sea trench–outer rise systems can be explained by variations in Te of the approaching plate. Others may be due to variations in the state of stress that exist across the plate boundary. There is evidence that the flexure is large enough at some trenches to locally weaken the lithosphere and cause it to fail through normal faulting. Seamounts and oceanic islands that ‘ride’ the topographic rise will experience uplift and then subsidence as they are carried by plate motions into the trench. This will also be the case for continents that arrive at a trench because of the subduction of oceanic crust. Uplift is in the process of affecting New Caledonia and other neighbouring islands, and flexure seaward of trenches may explain several erosional unconformities that are observed in the ancient rock record.

Irrespective of dynamic topography, flexure influences landscape evolution in many tectonic settings. The shape of a flexural rim flank uplift formed by loading or unloading of the crust, for example, can induce a family of river systems in the landscape including dip, scarp and break-through streams. The strongest evidence of flexure’s influence though is from studies of large river systems in fold and thrust belts which incise sufficiently to unload the crust and form anticlines and in foreland basins where rivers exiting a fold and thrust belt may be redirected by flexure to flow along structure.

Perhaps most importantly, flexure affects structural inheritance and the Wilson Cycle. The Earth’s elastic lithosphere that can retain a memory of a geological event, appear both strong and weak on long timescales, and support stresses of hundreds of MPa ensures that flexure will play a role in explaining structural inheritance. Inheritance plays a major role in evolution of sedimentary basins as they morph from one type to another, in orogenic belts where thin-skin and thick-skin tectonic styles may vary along strike a fold and thrust belt, and in the Wilson Cycle where continents disperse and reassemble and oceans open and close.

The idea that loads on the Earth’s surface may be regionally rather than locally supported can be traced back to Gilbert and Barrell’s work at the turn of the twentieth century. Their work was carried out at a time when the U.S. Coast and Geodetic Survey promoted the use of local, rather than regional, models of isostasy. The geodesists, with the exception of Putnam, showed that regional models of isostasy were not required to explain geodetic data.

While Gilbert perhaps was influenced by the geodesists, Barrell was not. Barrell challenged the geodesists’ conclusions concerning local models of isostasy, invoking instead the idea that topography was supported regionally by the lateral strength of the lithosphere. Following Barrell’s death, there was a vacuum, and the new ideas of regional isostasy gradually gave way again to local models of isostasy, which continued to be championed by the geodesists, notably Hayford, Bowie and Heiskanen

The history thereafter is a punctuated one with “bursts” of activity followed by long periods of quiet. The work centred on the activities of a few individuals: Vening Meinesz, Gunn and Walcott. Vening Meinesz and Gunn were contemporaries but there is little evidence of any contact between them.

The Earth is the yardstick against which the state of isostasy on the terrestrial planets will be assessed in the future. The primary data sets will continue to be gravity anomaly and topography data together with seismic data which have the potential to image the surfaces of flexure directly. We are close to defining the relative contributions of plate flexure and mantle dynamics in contributing to Earth’s topography and gravity fields as well as to its crustal structure and vertical motion history. The acquisition of higher-resolution data will increase the number of estimates of Te of the planets which, in turn, will help us to understand better the complexities of their geodynamical evolution.