Flexure of a circular nonlinearly elastic plate by a concentrated force

1973 ◽  
Vol 9 (2) ◽  
pp. 177-181
Author(s):  
L. G. Odnoral
Keyword(s):  
Elastic Plate ◽  
Concentrated Force ◽  
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.

Propagation of Elastic Pulses and Acoustic Emission in a Plate—Part 2: Epicentral Responses

1981 ◽  
Vol 48 (1) ◽  
pp. 133-138 ◽  
Author(s):  
A. N. Ceranoglu ◽  
Yih-Hsing Pao
Keyword(s):  
Acoustic Emission ◽  
Transit Time ◽  
Transient Response ◽  
Elastic Plate ◽  
Numerical Results ◽  
P Wave ◽  
Concentrated Force ◽  
Center Of Rotation ◽  
Double Couple ◽  
Time Required

In the first part of this paper expressions for Green’s dyadics in terms of the generalized ray integrals for both unbounded and bounded media were given. In this part Cagniard’s method is applied to obtain the transient response along the epicentral points of an elastic plate. Numerical results are shown for a concentrated force, a single-couple, a double force, a double-couple without moment and a center of rotation up to 10 transit time required for the longitudinal (P)-wave to cross the thickness of the plate.

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.

Propagation of Elastic Pulses and Acoustic Emission in a Plate—Part 1: Theory

1981 ◽  
Vol 48 (1) ◽  
pp. 125-132 ◽  
Author(s):  
A. N. Ceranoglu ◽  
Yih-Hsing Pao
Keyword(s):  
Acoustic Emission ◽  
Elastic Plate ◽  
Dynamic Process ◽  
Ray Theory ◽  
Transient Waves ◽  
Concentrated Force ◽  
Center Of Rotation ◽  
Material Defects ◽  
Double Couple

Transient waves generated by a variety of dynamic nuclei of strains including a concentrated force, a single-couple, a double-force, a double-couple without moment, a center of rotation, and a center of explosion in an elastic plate are analyzed. Some of these sources, or a combination of them, could be used to model the dynamic process of material defects. The analysis is based on the generalized ray theory and Cagniard’s method and the solutions are presented in terms of Green’s dyadics for a plate.

THEORY OF BRILLOUIN SCATTERING BY AN ISOTROPIC ELASTIC PLATE

1984 ◽  
Vol 45 (C5) ◽  
pp. C5-103-C5-107
Author(s):  
D. R. Tilley ◽  
E. L. Albuquerque ◽  
M. C. Oliveros
Keyword(s):  
Elastic Plate ◽  
Brillouin Scattering
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