PSI - Issue 2_A

J. Felger et al. / Procedia Structural Integrity 2 (2016) 2504–2511

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J. Felger, W. Becker / Structural Integrity Procedia 00 (2016) 000–000

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where ψ x and ψ y are the inclination angles and w represents the mid-plane deflection. Within the analysis of plates, it is convenient to introduce the stress resultants M x = h / 2 − h / 2 σ x zdz , M y = h / 2 − h / 2 σ y zdz , M xy = h / 2 − h / 2 τ xy zdz , Q x = h / 2 − h / 2 τ xz dz , Q y = h / 2 − h / 2 τ yz dz . (2) In the present work, we focus on orthotropic material behaviour. If the three planes of symmetry coincide with the coordinate planes of the defined rectangular coordinate system, the constitutive relations can be established in the form   M x M y M xy   =    D 11 D 12 0 D 12 D 22 0 0 0 D 66      ψ x , x ψ y , y ψ x , y + ψ y , x   , Q y Q x = κ A 44 0 0 A 55 w , y + ψ y w , x + ψ x . (3) The displacement field according to Eq. (1) and the constitutive relations in Eq. (3) together with the equilibrium equations lead to a system of partial di ff erential equations (PDE-system) with the mid-plane deflection w and the inclination angles ψ x and ψ y as unknown functions. If the plate is only subjected to transverse forces and moments along the boundary, cf. Fig. 1, the governing PDE-system can be written as A 44 ( w , yy + ψ y , y ) + A 55 ( w , xx + ψ x , x ) = 0 , Applying the principle of virtual work (Reddy, 1997), the boundary conditions corresponding to the local { n , s } - coordinate system can be expressed as w = w ∗ or Q n = Q ∗ n , ψ n = ψ ∗ n or M n = M ∗ n , ψ s = ψ ∗ s or M ns = M ∗ ns . (5) The FSDT allows for prescribing three physically natural boundary conditions along the plate edges in contrast to the classical plate theory where artificial Kirchho ff shear forces have to be defined. In fracture mechanics, the local near-tip field in the vicinity of the notch tip is of special interest. From the gen eral theory of elliptic boundary value problems containing singularities at conical or angular points (Grisvard, 1980; Kondrat’ev, 1967) it is known that local solutions in the vicinity of the singularity can be written in the form w ∼ i K i r λ i W i ( φ ) , ψ x ∼ i K i r λ i Ψ i x ( φ ) , ψ y ∼ i K i r λ i Ψ i y ( φ ) , (6) where λ i are the singularity exponents ( λ i ≤ λ i + 1 ) with their associated intensity factors K i , W i ( φ ) , Ψ i x ( φ ) and Ψ i y ( φ ) are smooth functions depending on the circumferential coordinate. Eq. (6) yields   w ψ x ψ y   = O ( r λ i ) ,   w , x , w , y ψ x , x , ψ x , y ψ y , x , ψ y , y   = O ( r λ i − 1 ) ,   w , xx , w , xy , w , yy ψ x , xx , ψ x , xy , ψ x , yy ψ y , xx , ψ y , xy , ψ y , yy   = O ( r λ i − 2 ) (7) (8) Therefore, the leading order terms for r → 0 are associated with the second-order derivatives. Substituting the expan sion of Eq. (6) into the PDE-system (4), collecting terms of equal order in r and considering only the leading order terms yields the reduced PDE-system A 44 w , yy + A 55 w , xx = 0 , D 11 ψ x , xx + D 12 ψ y , xy + D 66 ( ψ x , yy + ψ y , xy ) − κ A 55 ( w , x + ψ x ) = 0 , D 12 ψ x , xy + D 22 ψ y , yy + D 66 ( ψ x , xy + ψ y , xx ) − κ A 44 ( w , y + ψ y ) = 0 . (4) 2.2. Asymptotic solution with r λ i − 2 r λ i − 1 r λ i for r → 0 .

D 11 ψ x , xx + D 12 ψ y , xy + D 66 ( ψ x , yy + ψ y , xy ) = 0 , D 12 ψ x , xy + D 22 ψ y , yy + D 66 ( ψ x , xy + ψ y , xx ) = 0 ,

(9)