PSI - Issue 12

Valerio G. Belardi et al. / Procedia Structural Integrity 12 (2018) 281–295 V.G. Belardi et al. / Structural Integrity Procedia 00 (2018) 000–000

284

4

The terms of the matrix are no more constants, depending on the angular coordinate θ , as the layer exhibits a di ff erent elastic response according to the specific direction of the external load, they are:

( k ) 66 2 c ( k ) 22 − 2 Q ( k ) 66 ) c

2 4 cs + Q ( k ) 26 − Q

4 + Q

2 s 2 + Q ( k ) 12 + 2 Q

Q ( k ) Q ( k ) Q ( k ) Q ( k ) Q ( k ) Q ( k )

( k ) 11 c

( k ) 12 + 2 Q

( k ) 16 c

( k ) 26 s

( k ) 22 s

2 + Q

4

11 ( θ ) = Q

2 s 2 + Q

2 2 cs

( k ) 66 c

( k ) 16 c

( k ) 26 s

( k ) 12 + Q

2 + Q

( k ) 11 + Q

( k ) 16 − Q

12 ( θ ) = Q 16 ( θ ) = Q 22 ( θ ) = Q 26 ( θ ) = Q 66 ( θ ) = Q

2 cs + Q 2 cs + Q

2 + Q 2 + Q

2 4 c 2 − 1 2 4 c 2 − 1

( k ) 66 s ( k ) 66 s

2 1 − 4 s 2 + Q 2 1 − 4 s 2 + Q

( k ) 12 − Q

( k ) 11 + 2 Q

( k ) 22 − Q

( k ) 12 − 2 Q

( k ) 16 c

( k ) 26 s

(4)

4 + Q

2 s 2 − Q

( k ) 66 2 c

2 4 cs + Q

( k ) 22 c

( k ) 12 + 2 Q

( k ) 16 s

( k ) 26 c

( k ) 11 s

2 + Q

4

( k ) 66 c

( k ) 22 − Q

( k ) 12 − 2 Q

( k ) 12 − Q

( k ) 11 + 2 Q

( k ) 26 c

( k ) 16 s

( k ) 66 + Q

( k ) 22 − 2 Q

2 s 2 + Q

( k ) 66 c

( k ) 16 c

2 − s 2 2 cs

( k ) 11 + Q

( k ) 12 + 2 Q

( k ) 26 − Q

being c = cos( θ ), s = sin( θ ). The relations for the whole rectilinear orthotropic composite circular plate are derived from the stress-strain relation in Eq. (1). As stated by the classical plate theory (Jones (1975)), the displacement and strain fields are continuous along the plate thickness since the layers must be bonded together to be part of the same laminate, without the occurrence of relative slip between the layers which behave as a comprehensive elastic body. Consequently, the total laminate stresses are the sum of the homologous stress terms of all the N layers composing the rectilinear orthotropic composite circular plate:

   ε r ε θ γ r θ

   σ r σ θ τ r θ

   =

   ε r ε θ γ r θ

  

  

   σ r σ θ τ r θ

   k =

N k = 1

N k = 1

Q ( θ )

= Q ( θ )

(5)

k

in which the terms of the laminate reduced sti ff ness matrix are obtained as Q i j ( θ ) = N k = 1 Q ( k ) i j ( θ ). Moreover, the radial, circumferential and orthogonal displacement components of rectilinear orthotropic composite circular plate, evaluated along the coordinate directions r , θ and z are derived in accordance with the Kirchho ff -Love hypotheses for thin-plates:     u r ( r , θ, z ) = u ( r , θ ) − z ∂ w ∂ r u θ ( r , θ, z ) = v ( r , θ ) − z 1 r ∂ w ∂θ u z ( r , θ, z ) = w ( r , θ ) (6)

where u , v and w represent the radial, circumferential and orthogonal displacements of plate mid-surface ( z = 0).