PSI - Issue 8

V.G. Belardi et al. / Procedia Structural Integrity 8 (2018) 368–378

372

V.G. Belardi et al. / Structural Integrity Procedia 00 (2017) 000–000

5

As remarked by the dependence on the angular coordinate θ of the reduced sti ff ness matrix considered in the cylin drical coordinate system Q ( θ ) k , the k th rectilinear orthotropic layer features a di ff erent elastic response according to the specific radial direction along which the bending sti ff nesses are evaluated. As stated by the classical lamination theory (Jones (1975)) the layers are perfectly bonded together, i.e. no relative slip between them can take place. Consequently, the displacement and strain fields are continuous along the composite plate thickness. Therefore, the overall laminate stresses can be determined summing up the homologous stress terms of all the N layers composing the rectilinear orthotropic composite circular plate:   σ r σ θ τ r θ   = N k = 1   σ r σ θ τ r θ   k = N k = 1 Q ( θ ) k   ε r ε θ γ r θ   = Q ( θ )  ε r ε θ γ r θ   (5) keeping in mind the definition of the laminate reduced sti ff ness matrix terms: Q i j ( θ ) = N k = 1 Q ( k ) i j ( θ ). Because of the quasi-isotropic stacking sequence examined, the dependence of the displacement and strain fields on the angular coordinate θ can be neglected as the circumferential variability of these quantities is not practically relevant. Therefore, in the frame of Kirchho ff -Love theory for thin plates, the components of displacement and strain fields in the cylindrical coordinate system, along the directions r , θ and z , are:

∂ u r ∂ r

∂ w ∂ r

ε r =

u r ( r , z ) = u ( r ) − z

u r r

(6)

ε θ =

u θ ( r , z ) = v ( r ) u z ( r , z ) = w ( r )

∂ u θ

u θ r

∂ r −

γ r θ =

in which: u , v and w represents the plate’s mid-surface ( z = 0) radial, circumferential and deflection displacements. In addition, each strain is composed of two contributions: the first one represents the mid-surface strains and the second one is related to the contribute in strains of the mid-surface curvatures:

   − ∂ 2 w ∂ 2 r − 1 r ∂ w ∂ r 0

  

  

=    ε 0 r ε 0 θ γ 0 r θ

   + z

  

+ z    κ r κ θ κ r θ

  

=    ∂ u ∂ r u r ∂ v ∂ r −

   ε r ε θ γ r θ

(7)

v r

Moreover, the integration over the N -layers in the laminate thickness t of the elemental moments leads to the stress resultants per unit width; the resulting bending moments and the torque moment are expressed by:

   =

Q ( θ )

  

  

z +    κ r κ θ κ r θ

  

   σ r σ θ τ r θ

   M r M θ M r θ

  

  

   κ r κ θ κ r θ

   ε 0 r ε 0 θ γ 0 r θ

   ε 0 r ε 0 θ γ 0 r θ

z 2    

   

t / 2

z k

N k = 1

+ D ( θ )

dz = B ( θ )

zdz =

(8)

− t / 2

z k − 1