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The surface energy of a liquid may be measured by stretching a liquid membrane (which increases the surface area and hence the surface energy). In that case, in order to increase the surface area of a mass of liquid by an amount, δA, a quantity of work, γ δA, is needed (where γ is the surface energy density of the liquid).
Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue) The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments .
Different deformation modes may occur under different conditions, as can be depicted using a deformation mechanism map. Permanent deformation is irreversible; the deformation stays even after removal of the applied forces, while the temporary deformation is recoverable as it disappears after the removal of applied forces.
Comparison of surface energy, creating new surface on the left, and surface stress due to elastic deformation. Surface stress was first defined by Josiah Willard Gibbs [1] (1839–1903) as the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface. Depending upon the convention used, the area is either ...
The isochoric deformation gradient is defined as ¯:= /, resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free. Using this one can subsequently define the isochoric left Cauchy–Green deformation tensor B ¯ := F ¯ ⋅ F ¯ T = J − 2 / 3 B {\displaystyle {\bar {\boldsymbol {B ...
Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue) The Kirchhoff–Love theory is an extension of Euler–Bernoulli beam theory to thin plates. The theory was developed in 1888 by Love [2] using assumptions proposed by Kirchhoff. It is assumed that a mid-surface plane can be ...
This provides a comparison between bending stiffness (elasticity) and surface tension (capillarity). An elastic structure will be significantly deformed once its length is larger than the elasto-capillary length, which can be explained when gain of surface energy of a material is larger than stored elastic energy while bending.
For thick plates, we have to consider the effect of through-the-thickness shears on the orientation of the normal to the mid-surface after deformation. Raymond D. Mindlin's theory provides one approach for find the deformation and stresses in such plates. Solutions to Mindlin's theory can be derived from the equivalent Kirchhoff-Love solutions ...