Nonlinear Saint-Venant compatibility conditions for nonlinearly elastic plates

2011 ◽  
Vol 349 (23-24) ◽  
pp. 1297-1302 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Sorin Mardare
Keyword(s):  
Elastic Plates ◽  
Compatibility Conditions ◽  
Nonlinearly Elastic

NONLINEAR SAINT–VENANT COMPATIBILITY CONDITIONS AND THE INTRINSIC APPROACH FOR NONLINEARLY ELASTIC PLATES

2013 ◽  
Vol 23 (12) ◽  
pp. 2293-2321 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
SORIN MARDARE
Keyword(s):  
Symmetric Matrix ◽  
Elastic Plates ◽  
Compatibility Conditions ◽  
Korn’S Inequality ◽  
The Neumann Problem ◽  
Nonlinearly Elastic Plate ◽  
Matrix Fields ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Nonlinearly Elastic

Let ω be a simply connected planar domain. First, we give necessary and sufficient nonlinear compatibility conditions of Saint–Venant type guaranteeing that, given two 2 × 2 symmetric matrix fields (Eαβ) and (Fαβ) with components in L2(ω), there exists a vector field (ηi) with components η1, η2 ∈ H1(ω) and η3 ∈ H2(ω) such that ½(∂αηβ + ∂βηα + ∂αη3∂βη3) = Eαβ and ∂αβη3 = Fαβ in ω for α, β = 1, 2. Second, we show that the classical approach to the Neumann problem for a nonlinearly elastic plate can be recast as a minimization problem in terms of the new unknowns Eαβ = ½(∂αηβ + ∂βηα + ∂αη3∂βη3) ∈ L2(ω) and Fαβ = ∂αβη3 ∈ L2(ω) and that this problem has a solution in a manifold of symmetric matrix fields (Eαβ) and (Fαβ) whose components Eαβ ∈ L2(ω) and Fαβ ∈ L2(ω) satisfy the nonlinear Saint–Venant compatibility conditions mentioned above. We also show that the analysis of such an "intrinsic approach" naturally leads to a new nonlinear Korn's inequality.

Radially Symmetric Steady States of Nonlinearly Elastic Plates and Shells

2016 ◽  
Vol 124 (2) ◽  
pp. 243-278 ◽  
Author(s):  
Alexey B. Stepanov ◽  
Stuart S. Antman
Keyword(s):  
Steady States ◽  
Elastic Plates ◽  
Plates And Shells ◽  
Nonlinearly Elastic
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