Stress concentrations at holes of general form in a nonlinearly elastic plate

1970 ◽  
Vol 6 (5) ◽  
pp. 558-561 ◽  
Author(s):  
L. M. Vardanyan
Keyword(s):  
Elastic Plate ◽  
Stress Concentrations ◽  
Nonlinearly Elastic Plate ◽  
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.

On the Existence of Solutions to the Generalized Marguerre-von Kármán Equations

2006 ◽  
Vol 11 (1) ◽  
pp. 83-100 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Liliana Gratie
Keyword(s):  
Operator Equation ◽  
Shallow Shell ◽  
Existence Of Solutions ◽  
Vertical Displacement ◽  
Existence Of A Solution ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Nonlinearly Elastic Plate ◽  
Von Kármán ◽  
Nonlinearly Elastic

Using techniques from asymptotic analysis, the second author has recently identified equations that generalize the classical Marguerre-von Kármán equations for a nonlinearly elastic shallow shell by allowing more realistic boundary conditions, which may change their type along the lateral face of the shell. We first reduce these more general equations to a single “cubic” operator equation, whose sole unknown is the vertical displacement of the shell. This equation generalizes a cubic operator equation introduced by M. S. Berger and P. Fife for analyzing the von Kármán equations for a nonlinearly elastic plate. We then establish the existence of a solution to this operator equation by means of a compactness method due to J. L. Lions.

Dynamical Stress Concentration in an Elastic Plate

1962 ◽  
Vol 29 (2) ◽  
pp. 299-305 ◽  
Author(s):  
Yih-Hsing Pao
Keyword(s):  
Stress Concentration ◽  
Elastic Plate ◽  
Wave Length ◽  
Thin Elastic Plate ◽  
Circular Cavity ◽  
Stress Concentrations ◽  
Compressional Waves ◽  
Concentration Factors ◽  
Stress Concentration Factors ◽  
Dynamical Stress

Stress concentrations around a circular cavity in an infinitely extended, thin elastic plate, during passage of plane compressional waves, are discussed. The dynamical stress concentration factors are found to be dependent on the incident wave length and Poisson’s ratio for the plate, and, at certain wave lengths, they are larger than those encountered under statical loading.

ON THE GENERALIZED VON KÁRMÁN EQUATIONS AND THEIR APPROXIMATION

2007 ◽  
Vol 17 (04) ◽  
pp. 617-633 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
LILIANA GRATIE ◽  
SRINIVASAN KESAVAN
Keyword(s):  
Original Problem ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Von Karman ◽  
Strict Positivity ◽  
Von Kármán Equations ◽  
Conforming Finite Element ◽  
Nonlinearly Elastic Plate ◽  
Von Kármán ◽  
Nonlinearly Elastic

We consider here the "generalized von Kármán equations", which constitute a mathematical model for a nonlinearly elastic plate subjected to boundary conditions "of von Kármán type" only on a portion of its lateral face, the remaining portion being free. As already shown elsewhere, solving these equations amounts to solving a "cubic" operator equation, which generalizes an equation introduced by Berger and Fife. Two noticeable features of this equation, which are not encountered in the "classical" von Kármán equations are the lack of strict positivity of its cubic part and the lack of an associated functional. We establish here the convergence of a conforming finite element approximation to these equations. The proof relies in particular on a compactness method due to J. L. Lions and on Brouwer's fixed point theorem. This convergence proof provides in addition an existence proof for the original problem.

Sign in / Sign up

Export Citation Format

Share Document