Lithospheric flexure

Lithospheric flexure is bending of a mechanically strong outer shell under loads. A flat elastic plate approximation balances bending, buoyancy and applied loading. In the Fourier domain, bending contributes a term proportional to the flexural rigidity times the fourth power of horizontal wavenumber; it therefore suppresses short-wavelength deflection particularly strongly.

A region of upward deflection beyond the main downward flexure caused by loading an elastic lithosphere. Its position and amplitude depend on flexural rigidity, load geometry and boundary conditions. It is regional bending of the plate, rather than a separate buoyant load.

In the linear flat-elastic plate model, let downward deflection be $w$, applied downward pressure be $q$, flexural rigidity be $D$, and the restoring geophysical density contrast be $\Delta \rho >0$. The static balance isFor a sinusoidal Fourier mode of horizontal wavenumber magnitude $k$, applying the fourth spatial derivative multiplies its amplitude by $k^4$. HenceThe displayed dimensionless ratio compares bending resistance with buoyant restoration. Large $\chi$ means substantial elastic support and suppressed deflection. Small $\chi$ gives nearly local isostasy. The associated length $L_f=(D/\Delta \rho g{)}^{1/4}$ makes the load-width dependence explicit; some conventions define a flexural parameter with an additional factor $4^{1/4}$. At zero wavenumber, bending contributes nothing: total support remains buoyant even when the spatial distribution of subsidence changes.

The effective elastic thickness is the thickness of a homogeneous elastic plate with the same effective bending stiffness as the loaded lithosphere. In an isotropic thin-plate model, $D=E{T}_e^3/[12(1-{\nu }^2)]$, with Young's modulus $E$ and Poisson's ratio $\nu$. It depends on thermal state, material creep, loading history and yielding, and is not generally the crust's thickness or the full thermal thickness.