Acceleration Waves in the von Karman Plate Theory

tinuous“ von K´arm´an plate theory so as to cover situations when some physical quantities
suffer jump discontinuities at a certain curve.
4 Acceleration waves
2 Conservation laws
In the recent paper [2], all Lie point symmetries of system (1) are shown to be variational symmetries of the functional (2), and all corresponding (via Noether’s theorem) conservation laws admitted by the smooth solutions of the von K´arm´an equations are established. Each such conservation law is a linear combination of the basic linearly independent conservation laws
arXiv:math-ph/0006016v2 27 Jun 2000
1 Introduction
The von K´arm´an plate theory is governed by two coupled nonlinear fourth-order partial differential equations in three independent variables (Cartesian coordinates on the plate middle-plane x1, x2 and the time x3) and two dependent variables (the transversal displacement function w and Airy’s stress function Φ), namely
(Latin) indices range over 1, 2 (1, 2, 3), unless explicitly stated otherwise; the usual
summation convention over a repeated index is used and subscripts after a comma at a certain function f denote its partial derivatives, that is f,i = ∂f /∂xi, f,ij = ∂f /∂xi∂xj, etc.
bending tensor Kαβ being given in terms of w and Φ through the following expressions:
N αβ = εαµεβν Φ,µν , M αβ = −D (1 − ν)δαµδβν + νδαβδµν w,µν , Qα = M,αµµ + N αµw,µ, Eαβ = (1/Eh) (1 + ν)εαµεβν − vδαβδµν Φ,µν, Kαβ = w,αβ.
The theory under consideration allows an exact variational formulation, the von
K´arm´an equations being the Euler-Lagrange equations [1] associated with the action
Gαβ = (1/Eh) (1 + ν)δαµδβν − νδαβδµν Φ,µν − (1/2) εαµεβνw,µw,ν, F α = Gα,νν.
Table 1 Conservation laws
w - translations
X1
=
∂ ∂w
Φ - translations
X14
=
∂ ∂Φ
time - translations
energy P(α4) = −w,3Qα − Φ,3F α + w,3βM αβ + Φ,3βGαβ Ψ(4) = T + Π
wave momentum P(α2) = δα1L + w,1Qα + Φ,1F α − w,1βM αβ − Φ,1βGαβ Ψ(2) = −ρ w,1w,3 P(α3) = δα2L + w,2Qα + Φ,2F α − w,2βM αβ − Φ,2βGαβ Ψ(3) = −ρ w,2w,3
∂Ψ(j) ∂x3
+
∂P(µj) ∂xµ
=0
(j = 1, 2, . . . , 14) ,
whose densities Ψ(j) and fluxes P(µj) are presented (together with the generators of the respective symmetries) on the Table 1 below in terms of Qα, M αβ, Gαβ and F α,
The von K´arm´an equations (1) describe entirely the motion of a plate, the membrane stress tensor N αβ, moment tensor M αβ, shear-force vector Qα, strain tensor Eαβ and
is the strain energy per unit area of the plate middle-plane and T = (ρ/2) (w,3)2 ,
is the kinetic energy per unit area of the plate middle-plane.
1Published in: Integral Methods in Science and Engineering, Chapman & Hall / CRC Research Notes in Mathematics 418, Boca Raton, FL, pp. 131-136 (2000).
the bending rigidity, E is Young’s modulus, ν is Poisson’s ratio, h is the thickness of the plate, ρ is the mass per unit area of the plate middle-plane, δαβ is the Kronecker delta symbol and εαβ is the alternating symbol. Here and throughout the work: Greek
rotations
X6
=
x2
∂
∂ x1
−
x1
∂
∂ x2
moment of the wave momentum P(α6) = x2P(α2) − x1P(α3) + ενµw,µM αν + ενµΦ,µGαν Ψ(6) = x2Ψ(2) − x1Ψ(3)
rigid body rotations
X7
functional
I [w, T − Π
(2)
where
Π = (D/2) (∆w)2 − (1 − ν) εαµεβνw,αβw,µν − (1/2Eh) (∆Φ)2 − (1 + ν) εαµεβνΦ,αβΦ,µν + (1/2)εαµεβνΦ,αβw,µw,ν,
∂ ∂Φ
center-of-mass theorem P(α9) = −x3Qα, Ψ(9) = ρ (x3w,3 − w) P(α10) = x3P(α7), Ψ(10) = x1Ψ(9) P(α11) = x3P(α8), Ψ(11) = x2Ψ(9) P(α12) = x1F α − Gα1, Ψ(12) = 0 P(α13) = x2F α − Gα2, Ψ(13) = 0
D∆2w − εαµεβν w,αβΦ,µν + ρw,33 = 0, (1/Eh) ∆2Φ + (1/2) εαµεβν w,αβw,µν = 0,
(1)
where ∆ is the Laplace operator with respect to x1 and x2, D = Eh3/12(1 − ν2) is
=
x1
∂ ∂w
X8
=
x2
∂ ∂w
scaling
X5
=
xµ
∂
∂ xµ
+
2x3
∂
∂ x3
angular momentum P(α7) = M α1 − x1Qα + wεαν Φ,ν2, Ψ(7) = ρx1w,3 P(α8) = M α2 − x2Qα + wεναΦ,ν1, Ψ(8) = ρx2w,3
(3)
Ω
Σ
corresponds to each of the conservation laws listed in Table 1. It holds, just as the respective conservation law, for every smooth solution of the von K´arm´an equations.
The balance laws are applicable even if Ω is intersected by a discontinuity (singular) manifold (on which the corresponding densities Ψ(j) and fluxes P(αj) may suffer jump discontinuities) provided that the integrals exist. We are ready now to extend the “con-