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

Abstract

<jats:p>Based on previous work for the static problem, in this paper, we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.</jats:p>

Asymptotic derivation of refined dynamic equations for a thin elastic annulus

“…It is also worth noting that the well-expected assumption on leading-order uniform variation of the normal stress across the thickness for which L (0) 2 ≡ 0 in equation (58) would lead to a contradiction in equation (78). In fact, in the latter case, we would have (1 + 2ν)ζ 2 1 − ν P…”

“…A fresh interest in thin elastic shells is inspired by modern advanced applications, including the modelling of soft structures [1,2] and carbon nanotubes [3][4][5][6][7]. In particular, for an elongated nanotube in the form of a circular cylindrical shell, a specialised low-frequency formulation incorporating the peculiarities of near cut-off behaviour has been developed using both ad-hoc energy arguments [8] and asymptotic considerations within the classical Kirchhoff-Love theory [9]; see also more recent publications taking into account non-linearity and anisotropy [10][11][12][13].…”

“…(100) where = η −1 ωR/c 2 . Then, inserting the two-term expansion2 = 2 0 + η 2 2 1 + • • • into this relation, we obtain 2 0 = 2 3(1 − ν) n 2 (1 − n 2 ) 2 1 + n 2(101)…”

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.…”