On a uniformly-valid asymptotic plate theory

On a consistent rod theory for a linearized anisotropic elastic material: I. Asymptotic reduction method

“…This will be achieved through Taylor series expansions in terms of the rectangular coordinates about the rod axis, while the cylindrical coordinates will be used on the lateral surface. Previously, an asymptotic approach has been introduced by one of the authors (HHD) and co-authors for a dimension reduction from a 3D problem to a 2D problem to derive consistent plate and shell theories, see Dai and Song [18], Song and Dai [19, 20], Wang et al [21] and Yu et al [22]. Here, the dimension reduction is from 3D to 1D, which is actually more difficult.…”

“…Previously, one of the authors (HHD) and co-authors have introduced an asymptotic approach for constructing 2D plate and shell models from the 3D differential formulation, see Dai and Song [18], Song and Dai [19, 20], Wang et al [21] and Yu et al [22]. This approach was used in Chen et al [23] to obtain a plane-stress beam model for a linearized isotropic elastic material with pointwise error estimates and subsequently in Pruchnicki [24] to obtain a beam model in a 3D setting for a beam with rectangular cross-section.…”

A refined dynamic finite-strain shell theory for incompressible hyperelastic materials: equations and two-dimensional shell virtual work principle

“…However, in many practical situations for the traction edge, one only knows four conditions: the cross-thickness force resultant and the bending moment (with direction along the edge tangent), and one does not know how to impose the other two boundary conditions. For a plate theory, these issues were addressed in [14]. Here, with some modifications, those ideas from this previous work will be used for a shell theory.…”

On a consistent rod theory for a linearized anisotropic elastic material II. Verification and parametric study

“…The key idea is the establishment of recursive relations which represent higher-order coefficients in terms of lower-order ones to eliminate most of the unknowns and this can be done just through linear algebraic equations. The basic framework and fundamental idea in [1] were developed by one of the present authors and his co-authors in [2–4] and several subsequent reduction theories have also been established by using this procedure, see [5–8]. Compared with these works [5–8], the feature of [1] is, as a three-dimensional (3D) to one-dimensional (1D) reduction theory, it creatively utilizes some special techniques (such as the Fourier series expansion of the lateral traction) to overcome some non-trivial challenges (such as the inconsistent and unclosed system of equations), which finally leads to a simple and physically meaningful rod theory including four scalar leading rod equations, recursive relations for higher-order coefficients, and six boundary conditions at each end through 3D virtual work principles.…”