Discussion: “Closed Form Expression for the Vibration Problem of a Transversely Isotropic Magneto-Electro-Elastic Plate” (Liu, M. F., and Chang, T. P., 2010, ASME J. Appl. Mech., 77, 024502)

Abstract

In Ref. [1] the authors developed a plate theory for bending and vibration of magnetoelectroelastic (MEE) thin plates by employing the Kirchhoff hypothesis, and numerical results are presented and compared with certain existing solutions. The work should be interesting and valuable since MEE plate structures have a wide application prospect in various fields. However, in the derivation of the constitutive relations for thin plates, i.e., Eqs. (15)–(17) in Ref. [1], the authors improperly substituted the zero normal strain (ɛz=0) in the three-dimensional constitutive equations. It can be easily shown that, the presented moment-deflection relations, i.e., Eqs. (33)–(35) in Ref. [1], cannot be reduced to the classical results for the degenerated case of an isotropic elastic material.The correct way to derive the proper constitutive relations for thin plate actually starts from the relation σz=0, since the plate is very thin and a plane-stress state dominates [2–4]. From this relation, one can solve for ɛz in terms of other strain components. Substituting ɛz back into the 3D constitutive equations of MEE materials, the ones for MEE thin plates can be given as follows(1)σx=c¯11(-z∂2w∂x2)+c¯12(-z∂2w∂y2)+e¯31∂ϕ∂z+q¯31∂ψ∂zσy=c¯12(-z∂2w∂x2)+c¯11(-z∂2w∂y2)+e¯31∂ϕ∂z+q¯31∂ψ∂zτxy=-2c66z∂2w∂x∂yDz=e¯31∂u∂x+e¯31∂v∂y-ɛ¯33∂ϕ∂z-d¯33∂ψ∂zBz=q¯31∂u∂x+q¯31∂v∂y-d¯33∂ϕ∂z-μ¯33∂ψ∂zwhere(2)c¯11=c11-c13c13/c33, c¯12=c12-c13c13/c33e¯31=e31-c13e33/c33, q¯31=q31-c13q33/c33ɛ¯33=ɛ33+e33e33/c33, μ¯33=μ33+q33q33/c33d¯33=d33+e33q33/c33Obviously, Eq. (1) is different from Eqs. (15)–(17) presented in Ref. [1], and can be reduced to the classical results for an isotropic elastic plate. A systematic deduction of the two-dimensional theories for MEE beams, plates and shells as well as their applications may be found in Refs. [5–], to name a few.Furthermore, to the writer's knowledge, it is impossible to write the natural frequency of a rectangular plate with an arbitrary combination of boundary conditions in the simple form of Eq. (42) in Ref. [1]. On the other hand, comprehensive discussions on vibration of elastic rectangular plates, as documented in Ref. [9], can be similarly applied to MEE thin plates.