Statement of the Problem and Model Examples

1.1 Governing Equations and Basic Definitions

We consider a linearly elastic isotropic body of uniform thickness. For mechanical and geometrical parameters, we introduce the following notation: ρ is the mass density of the body, Ε is Young’s modulus, ν is Poisson’s ratio, 2 h is the thickness of the body.

In this book we use the orthogonal coordinate system specified by the vector equality

α1,α2,α3=Mα1α2+α3n,

where α1, α2 are parameters of the lines of curvature on the midsurface of the body Γ (α3 = 0); n is the unit midsurface normal; α3 is the distance from Γ along the normal (see Fig. 1.1).

In the coordinates α1, α2, α3 the 3D dynamic equations of elasticity can be written as*

Hi∂σii∂αi+1Hj∂σji∂αj+∂σ3i∂α3+1HiHj∂Hj∂αiσii−σjj+1HiHj∂Hi∂αjσij+σji+1HiHj∂HiHj∂α3σ3i+1Hi∂Hi∂α3σi3−ρ∂2υi∂t2=0,

Hi∂σi3∂αi+1Hj∂σj3∂αj+∂σ33∂α3−1Hi∂Hi∂α3σii+1Hj∂Hj∂α3σjj+1HiHj∂HiHj∂α3σ33+1HiHj∂Hj∂αiσi3+1HiHj∂Hi∂ajσj3−ρ∂2υ3∂t2=0,

and

ii=E21+υx2υ1−υ∂υ3∂α3+1Hj∂υj∂αj+1HiHj∂Hj∂αiυi+1Hj∂Hj∂α3υ3+1Hi∂υi∂αi+1HiHj∂Hi∂αjυj+1Hi∂Hi∂α3υ3,σ33=E21+υx2υ1−υ1Hi∂υi∂αi+1Hj∂υj∂αj+1HiHj∂Hi∂αjυj+1HiHj∂Hj∂αiυi+1Hi∂Hi∂α3υ3+1Hj∂Hj∂α3υ3+∂υ3∂α3,σ3i=E21+υ1Hi∂υ3∂αi+∂υi∂α3−1Hi∂Hi∂α3υi,σij=E21+υ1Hi∂υi∂αj+1Hi∂υj∂αi−1HiHj∂Hi∂αjυi−1HiHj∂Hj∂αiυj,

where i ≠ j = 1, 2 and

=c2c1=1−2v2(1−v)

with

1=E1−υ1+υ1−2υρ,c2=E21+υρ.

Here σkl (k, l = 1, 2, 3) are the stresses, vm (m = 1, 2, 3) are the displacements, t is time, Hi are Lame’s coefficients, c1 is the dilatation wave speed, c2 is the distortion wave speed.

In coordinates (1.1.1) Lame’s coefficients Hi (i = 1, 2) are defined by the formula

i=Ai1+α3Ri,

where Ai are the coefficients of the first quadratic form 12dα12+A22dα22 of the midsurface Γ, Ri are the principal curvature radii of Γ. The derivatives of Lame’s coefficients in equations (1.1.2), (1.1.3) can be written as

Hi∂αj=∂Ai∂αj1+α3Rj,∂Hi∂α3=AiRii≠j=1,2.

The first formula (1.1.7) is well known in differential geometry; the second one is evident.

Throughout this book, with the exception of Chapter 8, we impose the following boundary conditions on the faces of the body

3k=Qk±α1α2atα3=±hk=1,2,3,

where k± are given loads. These boundary conditions are traditions are traditional in the theory of shells. The case of fixed faces

k=0atα3=±hk=1,2,3

will be investigated in Chapter 8.

Further we always assume that the geometrical parameter η = h/R (R is a typical radius of curvature of the midsurface Γ) is small, i.e. we consider only thin walled bodies.

Apart from the geometrical parameter η, dynamic processes in thin walled bodies are characterized by the following two physical parameters: l, the ratio of the wavelength to R, and T, the ratio of the time scale to c2−1. It is convenient to express them in terms of the small parameters η

=ηq,T=ηa.

The indices q and a are the basic concepts of the book. We shall call them the variability and dynamicity indices, respectively.

In stationary dynamics, the parameters l and Τ (and consequently q and a), as a rule, are constant for each harmonic wave in the region occupied by the body. In non-stationary dynamics, they depend on coordinates and time, i.e. the indices q and a are local.

Now we dilate the scale of the independent variables, setting

i=Rηqξi,α3=Rης,t=Rc2−1ηari=1,2

and assume that differentiation with respect to the dimensionless variables ξi and τ does not change the asymptotic order of unknown quantities. Dependence of the unknown quantities on the thickness variable ζ(–1 ≤ ζ ≤ 1) will often be defined explicitly. Otherwise we shall assume that differentiation with respect to ζ does not change their asymptotic order.

In variables (1.1.11) equations (1.1.2) and (1.1.3) become

−q1Hi∂σii∂ξi+1Hj∂σji∂ξj+η−1∂σ3i∂ς+RHiHj∂Hj∂αiσii−σjj+RHiHj∂Hi∂αjσij+σji+RHiHj∂HiHj∂α3σ3i+RHi∂Hi∂α3σi3−η−2aE21+vR∂2υi∂τ2=0,η−q1Hi∂σi3∂ξi+1Hj∂σj3∂ξj+η−1∂σ33∂ς−RHi∂Hi∂α3σii−RHj∂Hj∂α3σjj+RHiHj∂HiHj∂α3σ33+RHiHj∂Hj∂αiσi3+RHiHj∂Hi∂αjσj3−η−2aE21+vR∂2v3∂r2=0,

and

ii=E21+vx2Rv1−vη−1∂υ3∂ς+η−q1Hj∂υj∂ξj+RHiHj∂Hj∂αiυi+RHj∂Hj∂α3υ3+η−q1Hi∂υi∂ξi+RHiHj∂Hi∂αjυj+RHi∂Hi∂α3υ3,σ33=E21+vx2Rv1−vη−q1Hi∂υi∂ξi+η−q1Hj∂υj∂ξj+RHiHj∂Hi∂αjυj+RHiHj∂Hj∂αiυi+RHi∂Hi∂α3υ3+RHj∂Hj∂α3υ3+η−1∂υ3∂ς,σ3i=E21+vRη−q1Hi∂υ3∂ξi+η−1∂υi∂ς−RHi∂Hi∂α3υi,σij=E21+vRη−q1Hj∂υi∂ξj+η−q1Hi∂υj∂ξi−RHiHj∂Hi∂αjυi−RHiHj∂Hj∂αiυj,

Where i ≠ j = 1, 2 and

i=Ai1+ηςRi*

With

i*=Ri/R.

In the consequent transformations it is sometimes convenient to present relations (1.1.13) in the...