 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
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| M. C. Delfour, "Intrinsic differential geometric methods in the asymptotic analysis of linear thin shells," Mech. Solids. 41 (5), 66-129 (2006) |
| Year |
2006 |
Volume |
41 |
Number |
5 |
Pages |
66-129 |
| Title |
Intrinsic differential geometric methods in the asymptotic analysis of linear thin shells |
| Author(s) |
M. C. Delfour (Montréal, Canada) |
| Abstract |
In earlier papers a completely intrinsic differential calculus on C1,1 submanifolds of
codimension one in ℝN has been developed by a marriage of tangential derivatives and the
oriented distance function. Its potential has been illustrated by investigating some linear
models of thin shells based on truncated series expansions with respect to the variable
normal to the midsurface. In this paper we analyze the asymptotic behavior of three models
for an arbitrary constitutive law. Given a midsurface with Lipschitzian boundary in a
C1,1 submanifold of ℝN, we show that solutions of the intrinsic linear P(1,1),
P(2n,1) and P(2,1) models of thin shells converge to solutions of asymptotic shell models which consist of a coupled system of two variational equations. The first is
the asymptotic P(1,0) model which yields the generally accepted classical membrane shell
equation and the Love-Kirchhoff terms. The second is a generalized bending equation. In the
bending dominated case and for the special constitutive law specified by two Lamé
constants, the quadratic term of the second equation of the asymptotic P(2n,1) and
P(2,1) models is the classical bending dominated equation. Also Naghdi's model is an
approximation of the reduced P(2,1) model and Koiter's is a projection of Naghdi's. A
detailed analysis of the three asymptotic models is given: existence and spaces of
solutions, decomposition of the equations, and construction of the associated effective
constitutive laws. Strong/weak convergence is established in the natural spaces and norms
under an assumption on the asymptotic behavior of the constant of continuity of the
right-hand side for shells without boundary or shells with homogeneous Neumann boundary
conditions (quotient space) or homogeneous Dirichlet boundary conditions on a part of the
boundary. |
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|
| Received |
15 January 2005 |
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