where, 0E is elastic modulus at 0 x = ， β is non-homogeneous coefficients. Z–axis is along the thickness direction, the radius is R and the thickness is h as shown in Fig.1. The basic equations for Reissner’s spherical shell are 2 2 2 2 3 3 3 2 2 2 2 2 3 3 2 2 3 2 1 ( ) (1 )( ) (1 )( 1)( ) 2 5(1 )( 1) ( ) 0 1 1 ( 1 ( ) (1 )( 1 ( ) (1 )( 2 2 ) 5(1 ) )( ) y y y x x x x y y x x x y h x h x h y x x y x y x y w x h x h x h x y x x y x x y h x y ϕ ϕ ϕ ϕ ϕ ϕ β μ β μ β μ β μ ϕ ϕ ϕ ϕ ϕ ϕ β μ β μ β μ ϕ β μ ∂ ∂ ∂ ∂ ∂ ∂ + + + + − + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − + − − = ∂ ∂ ∂ ∂ ∂ ∂ − − + − + − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + − + ∂ 2 0 0 2 2 2 2 4 3 2 2 2 0 2 2 2 1 ( ) 0 5 (1 )( ) 12 (1 ) 5 ( 0 ( ) (1 ) (1 ) 2 (1 2 2 ) ) 0 ) y y x x w h y w hE x E h x y x x E h x w x x x w y ϕ ϕ ϕ β κ μ ψ β ϕ ψ ψ ψ β ψ κ β β β β μ β ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ⎪ − − = ⎪ ∂ ⎪ ∂ ∂ ∂ ⎪ + − − + + ∇ + − = ⎪ ∂ ∂ ∂ ⎪ ⎪ ∂ ∇ ∂ ∂ + ∇ + + ∇ − + − + = ⎪ ∂ ∂ ⎩ ∇ ∂ (2) where, xϕ， yϕ are the angle displacement, w is the deflection, κ is the curvature, ψ is the stress function. 3. The boundary conditions As the crack surface is free, the boundary conditions are 0, 0, 0 0, 0 y xy y y xy M M Q N N = = = ⎧ ⎨ = = ⎩ (3) Further, they can be expressed as Fig.1 The functionally graded spherical shell h/2 z y x ( ) E x
RkJQdWJsaXNoZXIy MjM0NDE=