Classical plate buckling theory as the small‐thickness limit of three‐dimensional incremental elasticity

Abstract

Classical plate buckling theory is obtained systematically as the small-thickness limit of the three-dimensional linear theory of incremental elasticity with null incremental data. Various a priori assumptions associated with classical treatments of plate buckling, including the Kirchhoff-Love hypothesis, are here derived rather than imposed, and the conditions under which they emerge are stated precisely.

Size and curvature regulate pattern selection in the mammalian brain

“…In light of the previous remarks, we expand the model problem of the curved beam to a fully three-dimensional, nonlinear continuum model [24]. We characterize local kinematic changes through the deformation gradient F = ∂x/∂X, the partial derivative of points in the grown configuration x with respect to their initial position in the ungrown reference configuration X.…”

A reduced model for electrodes-coated dielectric plates

“…Our current research is motivated by the observation that there exist very few reduced models for electroelastic plates and what has been achieved in the purely mechanical case has not yet been fully extended to the electroelastic case. Our current study provides a first such extension of the methodology employed by [40,41,42,43], although in contrast with the purely mechanical case we have chosen not to derive the associated edge conditions; this is because in practice electrodes are rarely extended to the plate edge. The power expansion approach has also been used in the derivation of dynamic plate theories; see [25,33] for instance.…”

“…the coefficient of e 3 in (40)) should be an odd function of x 3 . We observe that similar expansions were used in [42,43,5,46], but in [11,55] the authors only included the terms v + x 3 a 3 e 3 in their expression for u. One of the aims of the present study is to assess the stability implications of this additional approximation.…”

“…The central idea is to assume a power expansion for u,φ andṗ in terms of x 3 , and then to expand the above energy functional further, in a consistent manner, up to and including terms cubic in h. The associated Euler-Lagrange equations then yield an approximate theory for thin plates. Such a reduction has been carried out in [43] for a generally anisotropic, compressible and unstressed plate, and in [7] for the flexural/bending deformation of an incompressible pre-stressed plate.…”

A well-posed finite-strain model for thin elastic sheets with bending stiffness

“…In this way well-posedness is restored in a model based entirely on the three-dimensional theory. To support this position, we note that in classical plate-buckling theory [10,14] the stress scales as  2 whenever the deformation is such that the order - 3 term in the strain energy is non-trivial; i.e., whenever bending occurs in the absence of transverse forces. Further, when transverse forces scale as  3 , the exact eqn.…”