On the justification of Koiter's model for generalised membrane shells of the "first kind'' confined in a half-space

TL;DR

This work provides a rigorous justification of Koiter's generalised membrane shell theory for shells of the first kind confined in a half-space. By formulating a three-dimensional obstacle problem under a confinement constraint and performing a careful thickness-to-zero asymptotic analysis, the authors derive a two-dimensional variational inequality on an abstract completed space, and show it matches the limit obtained from Koiter's approach. They establish strong convergence of the full 3D solution to a 2D limit and prove that, under suitable geometric conditions, the 2D limits from the 3D obstacle problem and from Koiter's model coincide, thereby validating Koiter's reduction in this constrained setting. A key technical contribution is a density result that identifies the two admissible limit sets, enabling the two pathways to the same reduced problem. The results have implications for the mathematical justification of dimensionally reduced shell models in contact/obstacle contexts, including under Kirchhoff-Love assumptions.

Abstract

In this paper we justify Koiter's model for linearly elastic generalised membrane shells of the first kind subjected to remaining confined in a prescribed half-space. After showing that the confinement condition considered in this paper is in general stronger, under the validity of the Kirchhoff-Love assumptions, than the classical Signorini condition, we formulate the corresponding obstacle problem for a three-dimensional linearly elastic generalised membrane shell of the first kind, and we conduct a rigorous asymptotic analysis as the thickness approaches to zero on the unique solution for one such model. We show that the solution to the three-dimensional obstacle problem converges to the unique solution of a two-dimensional model consisting of a set of variational inequalities that are posed over the abstract completion of a non-empty, closed and convex set. We recall that the two-dimensional limit model obtained departing from Koiter's model is characterised by the same variational inequalities as the two-dimensional model obtained departing from the variational formulation of a three-dimensional linearly elastic generalised membrane shell subjected to remaining confined in a prescribed half-space, with the remarkable difference that the sets where solutions for these two-dimensional limit models are sought do not coincide, in general. In order to complete the justification of Koiter's model for linearly elastic generalised membrane shells of the first kind subjected to remaining confined in a half-space, we give sufficient conditions ensuring that the sets where solutions for these two-dimensional limit models are sought coincide.

Figures (5)

• Figure 1: An example where the condition \ref{['continuity']} holds. The figure here represented is a cross section of a portion of a cylinder, whose extension by reflection along one of the straight edges of its boundary is denoted by the dotted pattern. The unit-vector $\bm{q}$ identifying the orthogonal complement to the half-space where the shell has to remain confined is given and it is equal to $(0,0,1)$. The vector $\bm{a}^3(y)$, denoted by a solid line, is reflected (in the sense of Section 9.2 of Brez11) onto the corresponding vector $\hat{\bm{a}}^3(\hat{y})$, denoted by a dotted line, where $\hat{y}\in\overline{\hat{\omega}}\setminus\omega$ is the unique antipodal point with respect to the straight edge of the boundary $\gamma$ that corresponds to an appropriate point $y \in\omega$. We observe that $\bm{a}^3(y)\cdot\bm{q}=\hat{\bm{a}}^3(\hat{y}) \cdot\bm{q}$ even though in general $\bm{a}^3(y) \neq \hat{\bm{a}}^3(\hat{y})$. The remaining vectors of the contravariant basis under consideration can be extended likewise. Other examples of parametrisation of surfaces $\bm{\theta}$ and unit-vectors $\bm{q}$ for which this extension machinery is possible are described in Figure \ref{['fig:3']}.
• Figure 2:
• Figure 3: The set $\hat{\omega}$ is an extension by reflection of the original domain $\omega$. First, we reflect the strips of width $\delta(\bm{\eta})$ across the edge of $\gamma_0$ it is adjacent to. Second, we reflect the strips of width $\delta(\bm{\eta})$ across the edges of the intermediary extension that contain the portion of boundary $\gamma\setminus\gamma_0$. The hatched patterns denote the portion of $\omega$ where the given function $\bm{\eta}\in\bm{U}_M^{(2)}(\omega)$ vanishes. The dotted pattern denotes the points in the domain extension where the extension of $\bm{\eta}$ by reflection vanishes (cf., e.g., Theorem 9.7 in Brez11). The point $\hat{y}$, which is represented by a black dot, is the antipodal point corresponding to $y\in \omega$ with respect to $y_0\in\gamma$, denoted by a hallow dot. This means that there exists a number $c>0$ such that $\hat{y}=y+2c\nu(y_0)$. In the case illustrated in (a), where the set $\gamma_0$ is the union of two parallel edges, the extension vanishes near the corners of $\omega$ and the extension by reflection of the contravariant basis with respect to the remaining edges is performed in the manner described at the beginning of section \ref{['Sec:5']}. If the set $\gamma_0$ consists of one edge only, then the extension by reflection of the contravariant basis is performed in the usual manner when $y_0$ is not a corner point (b). Otherwise, the extension by reflection of the contravariant basis is carried out with respect to two points $y_1$ and $\hat{y}_2$ that are antipodal with respect to the corner point $y_0$ under consideration (c). The latter is equivalent to considering the extension by reflection of the contravariant basis at the point $y_1$ to the point $\hat{y}_1$ first, and then considering the extension by reflection of the contravariant basis at the point $\hat{y}_1$ to the point $\hat{y}_2$.
• Figure 4: Three examples of surfaces defined over a rectangular domain $\omega \subset\mathbb{R}^2$ to which Theorem \ref{['th:density']} can be applied. (a) A portion of a spherical cap such that the contour where the boundary conditions of place are imposed lies on a plane parallel to the orthogonal complement of $\bm{q}=(0,0,1)$; (b) A portion of a cylinder such that the image of the edges of $\gamma$ that are orthogonal to $\gamma_0$ lies on a plane that is parallel to the orthogonal complement of $\bm{q}=(0,0,1)$; (c) A portion of a hyperboloid of revolution such that the symmetry axis director is parallel to $\bm{q}=(0,1,0)$. In order to apply Theorem \ref{['asymptotics']}, we need to require the validity of condition \ref{['dpcmp']}. Note that this is in line with the conclusion obtained for linearly elastic elliptic membrane shells CiaMarPie2018. Assuming the validity of \ref{['dpcmp']} certainly affects the choices for the middle surfaces for which the asymptotic analysis presented in Theorem \ref{['asymptotics']} holds.
• Figure :

Theorems & Definitions (16)

• Theorem 3.1
• Lemma 3.1
• Theorem 4.1
• proof
• Lemma 4.1
• Lemma 4.2
• proof
• Theorem 4.2
• Lemma 5.1
• proof