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Nonlinear Kirchhoff-Love shell models derived from the Ciarlet-Geymonat energy: modelling and well-posedness

Ionel-Dumitrel Ghiba, Trung Hieu Giang, Catalina Ureche

Abstract

Starting from a three-dimensional model based on the Ciarlet-Geymonat energy, we derive nonlinear shell models within the classical elasticity theory of compressible isotropic materials. The Neo-Hookean term involving the norm of the deformation gradient leads to an energy depending on the first, the second, and the third fundamental forms of the deformed midsurface. The coefficients appearing in the resulting shell models depend on the classical Lamé coefficients of the three-dimensional material, on the thickness of the shell, and on the mean and Gaussian curvatures of the reference configuration. This shows that the behavior of the shell is influenced not only by the elastic coefficients but also by the initial geometry of the three-dimensional thin body. The purely volumetric Ciarlet-Geymonat contribution of the three-dimensional energy leads to two-dimensional energies depending on the mean and Gaussian curvatures of both configurations, namely the undeformed and the deformed midsurfaces. Since a purely asymptotic derivation may lead to nonlinear terms for which the lower semicontinuity of the resulting functionals is not clear, we combine the asymptotic reduction through the thickness with Simpson's quadrature rule applied to the purely volumetric energy terms, ensuring that the lower semicontinuity is inherited from the three-dimensional model. After deriving the model, we establish the well-posedness of the proposed shell energies. More precisely, we prove coercivity and lower semicontinuity property of the resulting functional and show the existence of minimizers in appropriate Sobolev spaces. A key ingredient in the proofs is a polyconvexity concept in the shell theory, together with some results concerning the weak convergence of terms involving the mean curvature of the deformed midsurface.

Nonlinear Kirchhoff-Love shell models derived from the Ciarlet-Geymonat energy: modelling and well-posedness

Abstract

Starting from a three-dimensional model based on the Ciarlet-Geymonat energy, we derive nonlinear shell models within the classical elasticity theory of compressible isotropic materials. The Neo-Hookean term involving the norm of the deformation gradient leads to an energy depending on the first, the second, and the third fundamental forms of the deformed midsurface. The coefficients appearing in the resulting shell models depend on the classical Lamé coefficients of the three-dimensional material, on the thickness of the shell, and on the mean and Gaussian curvatures of the reference configuration. This shows that the behavior of the shell is influenced not only by the elastic coefficients but also by the initial geometry of the three-dimensional thin body. The purely volumetric Ciarlet-Geymonat contribution of the three-dimensional energy leads to two-dimensional energies depending on the mean and Gaussian curvatures of both configurations, namely the undeformed and the deformed midsurfaces. Since a purely asymptotic derivation may lead to nonlinear terms for which the lower semicontinuity of the resulting functionals is not clear, we combine the asymptotic reduction through the thickness with Simpson's quadrature rule applied to the purely volumetric energy terms, ensuring that the lower semicontinuity is inherited from the three-dimensional model. After deriving the model, we establish the well-posedness of the proposed shell energies. More precisely, we prove coercivity and lower semicontinuity property of the resulting functional and show the existence of minimizers in appropriate Sobolev spaces. A key ingredient in the proofs is a polyconvexity concept in the shell theory, together with some results concerning the weak convergence of terms involving the mean curvature of the deformed midsurface.
Paper Structure (13 sections, 9 theorems, 123 equations)

This paper contains 13 sections, 9 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. The dimensional reduction of the three-dimensional model
  3. The three-dimensional parental problem and geometry of the referential configuration
  4. The ansatz and the approximations
  5. The proposed models
  6. Model I up to order ${\rm O}(h^5)$
  7. Model II up to order ${\rm O}(h^3)$
  8. Model III up to order ${\rm O}(h^5)$
  9. The existence results
  10. A general existence theorem
  11. The convexity of energetic terms
  12. The existence results for the proposed models
  13. Conclusions and final remarks

Key Result

Lemma 3.1

Let $\omega \subset \mathbb{R}^2$ be an open bounded domain with Lipschitz boundary. Let $y_0 \in C^2(\bar{\omega}, \mathbb{R}^3)$ be such that $\partial_{x_1} y_0(x')$ and $\partial_{x_2} y_0(x')$ are linearly independent for every $x' \in \bar{\omega}$ and let $\gamma _{y_0}$ be a nonempty relativ where $R_1(m)$ and $R_2(m)$ denote the principal radii of curvatureThe condition $\dfrac{h}{2}\max\

Theorems & Definitions (20)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 10 more