PSI - Issue 8

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

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

is found that for this stacking sequence the neglected circumferential variability does not influence significantly the accuracy of the evaluation of the displacement field and of the plate sti ff ness.

2. Constitutive equations

A circular portion of a laminate composite rectangular plate can be treated as composite circular plate with rectilin ear orthotropic material properties, i.e. the material features a unique set of principal directions, meanwhile the plate geometry is axisymmetric. Unlike the case of circular isotropic plate where the mid-surface deflection depends only on the radial coordinate r , the bending analysis of a rectilinear orthotropic composite circular plate is a bidimensional problem because of the circumferential variability of the bending sti ff ness matrix. Rectilinear orthotropic layer stresses and strains relations in the cylindrical coordinate system ( r , θ, z ) are obtainable applying the proper rotation matrix to the stress and strains vectors defined in the global Cartesian reference frame:   σ r σ θ τ r θ   k = T ( θ )  σ x σ y γ xy   k   ε r ε θ γ r θ   k = T ( θ ) − T   ε x ε y γ xy   k (1) where T ( θ ) is the transformation matrix used to transfer a vector from the global Cartesian coordinate system to the cylindrical one and θ is the angle delimited by the x -axis of the Cartesian coordinate system and by the r -axis of the cylindrical coordinate system. Stress-strain relations, under plane stress and strain conditions, for the k th rectilinear orthotropic layer composing a laminate plate, expressed in the global Cartesian coordinate system ( x , y , z ), (Jones (1975)):   σ x σ y τ xy   k = Q k   ε x ε y γ xy   k =    Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 Q 16 Q 26 Q 66    k   ε x ε y γ xy   k (2) ff ness matrix, keeping in mind that the principal material coordinate system and the global Cartesian coordinate one does not coincide for a finite rotation around the z − axis that is common for the two reference systems. Moreover, the combination of the layer stress-strain relation in the Cartesian coordinate system (2) and the equa tions for the coordinates transformations (1), returns the stress-strain relation for the k th layer in the cylindrical coor dinate system:   σ r σ θ τ r θ   k = Q ( θ ) k   ε r ε θ γ r θ   k =    Q 11 ( θ ) Q 12 ( θ ) Q 16 ( θ ) Q 12 ( θ ) Q 22 ( θ ) Q 26 ( θ ) Q 16 ( θ ) Q 26 ( θ ) Q 66 ( θ )    k   ε r ε θ γ r θ   k (3) being Q k the layer transformed reduced sti

in which:

Q ( θ )

= T ( θ ) Q

k

T ( θ )

T

(4)

k

is the transformed reduced sti ff ness matrix in the cylindrical coordinate system.