Crystal Elasticity

9Thin Plate Bending

Beam bending tests are inappropriate for the experimental study of materials in the thin form, such as metal thin sheets. Indeed, if samples of thickness ranging from 0.1 to 0.3 mm, instead of ranging from 1.5 to 2 mm, are considered, a free bending test generates a transverse bending in addition to the longitudinal bending. The vibration mode then becomes a mixed plate-beam state. An experimental study of such geometries requires thin plate bending tests, which will be considered square in order to simplify the process. This chapter focuses on this new configuration.

9.1. Bending vibrations of a homogeneous thin plate

Consider a homogeneous plate of length L along the x and y directions, of thickness B along the z direction, and Young’s modulus E, Poisson’s ratio ν, density ρ and having a bending resonance frequency F_FP. According to the Euler–Bernoulli kinematic hypothesis (shears γ_xz and γ_yz are zero), three deformations generated can be written as follows:

[9.1]

[9.2]

[9.3]

It can be seen that the double deflection in the x and y directions generates a shear. The deformation energy can be written as:

[9.4]

According to Hooke’s law, the relations between ...