13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Figure 1. Sandwich plate model 2.1. Governing equations Based on the non-classical theory, and by adopting the Власов assumption, the governing equations for dynamic analysis considering the membrane forces can be derived as [6]: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + +2 0 1 1 0 2 2 1 1 2 2 y x z x y xy y x x x z x y y x z y G h w w w w w h N N N q k x y t x y x y G h w D J x y x y k x t G h w D y x x y k y τ τ τ β β ρ β β β β ν ν β ρ β β β ν ν β ∂ ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ ∇ + + − + = ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎛ ∂ ⎞ ∂ ∂ ∂ − + ⎛ ∂ ⎞ + + − + − = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ∂ ∂ ⎛ ⎞ ∂ − + ∂ + + − + ⎜ ⎟ ⎜ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ∂ ⎠ ⎝ ⎠ 2 2 0 y J t β ρ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ∂ ⎪ − = ⎟ ⎪ ∂ ⎩ (1) where 1 2 2 h h h = + , xN , yN and xy N are membrane forces, for this paper, they are generated by thermal stresses. zG , D, ν, hρ and Jρ are equivalent compositional parameters defined as: 3 3 2 2 1 2 2 2 3 3 1 2 1 3 3 2 6 2(1 ) 2 6 2(1 ) z h h E h h E G h h h h ν ν ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = − − + − ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ + + ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ (2) 3 3 3 3 1 2 2 2 2 2 2 1 2 2 3(1 ) 2 2 3(1 ) 2 2 E h E h h h D ν ν ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − + − − ⎢ ⎥ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − − ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ (3) 3 3 3 3 1 1 2 2 2 2 2 2 2 1 2 1 2 3 1 2 2 1 2 2 E h E h h h D ν ν ν ν ν ⎧ ⎫ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − + − − ⎢ ⎥ ⎢ ⎥ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − − ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎣ ⎦ ⎣ ⎦ ⎩ ⎭ (4) 1 1 2 2 2 h h h ρ ρ ρ = + (5) 3 3 3 3 2 2 2 1 2 2 1 3 2 2 3 2 2 h h h h Jρ ρ ρ ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − + − − ⎢ ⎥ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ (6)
RkJQdWJsaXNoZXIy MjM0NDE=