Volume 2, Issue 1
The Obstacle Problem for Shallow Shells: Curvilinear Approach.

Alain Léger & Bernadette Miara

Int. J. Numer. Anal. Mod. B, 2 (2011), pp. 1-26.

Published online: 2011-02

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  • Abstract
We start with a three-dimensional equilibrium problem involving a linearly elastic solid at small strains subjected to unilateral contact conditions. The reference configuration of the solid is assumed to be a thin shallow shell with a uniform thickness. We focus on the limit when the thickness tends to zero, i.e. when the three-dimensional domain tends to a two-dimensional one. In the generic case, this means that the initial Signorini problem, where the contact conditons hold on the boundary, tends to an obstacle problem, where the contact conditions hold in the domain. When the problem is stated in terms of curvilinear coordinates, the unilateral contact conditions involve a non penetrability inequality which couples the three covariant components of the displacement. We show that nevertheless we can uncouple these components and the contact conditions involve only the transverse covariant component of the displacement at the limit.
  • AMS Subject Headings

74B99 74K25 74M15

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COPYRIGHT: © Global Science Press

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@Article{IJNAMB-2-1, author = {Alain Léger and Bernadette Miara}, title = {The Obstacle Problem for Shallow Shells: Curvilinear Approach.}, journal = {International Journal of Numerical Analysis Modeling Series B}, year = {2011}, volume = {2}, number = {1}, pages = {1--26}, abstract = {We start with a three-dimensional equilibrium problem involving a linearly elastic solid at small strains subjected to unilateral contact conditions. The reference configuration of the solid is assumed to be a thin shallow shell with a uniform thickness. We focus on the limit when the thickness tends to zero, i.e. when the three-dimensional domain tends to a two-dimensional one. In the generic case, this means that the initial Signorini problem, where the contact conditons hold on the boundary, tends to an obstacle problem, where the contact conditions hold in the domain. When the problem is stated in terms of curvilinear coordinates, the unilateral contact conditions involve a non penetrability inequality which couples the three covariant components of the displacement. We show that nevertheless we can uncouple these components and the contact conditions involve only the transverse covariant component of the displacement at the limit.}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnamb/296.html} }
TY - JOUR T1 - The Obstacle Problem for Shallow Shells: Curvilinear Approach. AU - Alain Léger & Bernadette Miara JO - International Journal of Numerical Analysis Modeling Series B VL - 1 SP - 1 EP - 26 PY - 2011 DA - 2011/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnamb/296.html KW - asymptotic analysis KW - differential geometry KW - obstacle problem KW - shallow shells KW - Signorini conditions AB - We start with a three-dimensional equilibrium problem involving a linearly elastic solid at small strains subjected to unilateral contact conditions. The reference configuration of the solid is assumed to be a thin shallow shell with a uniform thickness. We focus on the limit when the thickness tends to zero, i.e. when the three-dimensional domain tends to a two-dimensional one. In the generic case, this means that the initial Signorini problem, where the contact conditons hold on the boundary, tends to an obstacle problem, where the contact conditions hold in the domain. When the problem is stated in terms of curvilinear coordinates, the unilateral contact conditions involve a non penetrability inequality which couples the three covariant components of the displacement. We show that nevertheless we can uncouple these components and the contact conditions involve only the transverse covariant component of the displacement at the limit.
Alain Léger and Bernadette Miara. (2011). The Obstacle Problem for Shallow Shells: Curvilinear Approach.. International Journal of Numerical Analysis Modeling Series B. 2 (1). 1-26. doi:
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