Effective models of composite periodic plates—I. Asymptotic solution

Justification of the Bending-Gradient Theory Through Asymptotic Expansions

“…Actually, when looking at the definition of strains in Equation (1.69), only the first derivatives of U 3 and Φ are involved. Having low-order interpolation is a serious advantage compared to "strain-gradient-like" approaches given in [7,21]. Now, let us recall that the derivation of the Bending-Gradient theory through asymptotic expansions was purely formal.…”

“…Now that the asymptotic expansion framework is set, we detail the explicit resolution which is classically performed (see [5,7] for instance). Basically it starts with the derivation of low order displacements which do not gerenate local strain but are related to purely macroscopic displacement fields.…”

“…Hence one needs to seek higher orders for bringing out the effect of shear forces. However, in the cases of laminated plates, this procedure does not lead to Reissner-Mindlin plate theory [7,8].…”

On the Modelling of Prestressed Medium Thickness Uniperiodic Plates

“…That is why in the recent literature these plates are described, in an approximate manner by certain averaged equations which have constant coefficients. To this end we can use a homogenization method [3]. In this contribution the tolerance averaging technique [4] of 2D -plate equations is applied.…”

Asymptotics of the solution of the problem of deformation of an arbitrary locally periodic thin plate

“…Theorems on convergence of normalized elastic fields were proved in these papers under certain restrictions (we note at once that in subsequent publications ( [13], [21]- [26], and others) the restrictions on the geometric level were removed only in part). It should be underscored that, first, passing to the limit leads to a loss of information about the correct asymptotic behavior of the stresses and strains in a three-dimensional plate (see the comments in [15], [18], and also [13, § 6.2]) and, second, the method developed nevertheless does not enable us to derive a qualified error estimate for the two-dimensional model constructed.…”