PSI - Issue 3

Christos F. Markides et al. / Procedia Structural Integrity 3 (2017) 334–345 Christos F. Markides, Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2017) 000–000

336

3

P frame

P frame

σ r = – P(θ)

σ r = – P(θ)

2ω ο

2ω ο

x

x

y

y

ϕ ο

ϕ ο

θ

θ

L

L

z=re iθ

z=re iθ

R

R

d

d

r

r

O

O

z

z

2ω ο

2ω ο

Orthotropic Transtropic E x = E z = = E E y = = E΄ G xy = G yz = = G΄ G xz = = G ν xz = = ν ν yx = ν yz = = ν΄ Orthotropic disc Transtropic disc

(a)

(b)

– P frame

– P frame

x

G΄

G xy

y

ν΄

x

ν, G Planes of isotropy

ν yx

y

ν zx

Ε z

G xz

Ε

ν yz

G yz

z

Ε

ν΄

Ε x

Ε

z

y

Ε΄

Three planes of elastic symmetry

One plane of elastic symmetry

Fig. 2: (a) The configuration of the problem for the orthotropic Brazilian disc and definition of symbols. (b) The transtropic Brazilian disc as a particular case of the orthotropic one. x-axis is parallel to the traces of planes of elastic symmetry that subtend an angle ϕ o with P frame ’s axis (Fig.2a). P frame is parabolically distributed along two finite antisymmetric arcs of the disc’s perimeter L, each one of length 2Rω ο , as:           2 2 r c o o c max P P 1 sin sin , P 0, P P                    (1) In addition, d is considered comparable to the disc’s radius R, and therefore plane strain conditions are assumed. Then, following Lekhnitskii’s formalism (1981) for orthotropic “cylindrical” bodies in plane strain state in the absence of body forces, the generalized Hooke’s law, the equations of equilibrium, the (single) condition of compatibility in terms of stresses and the boundary conditions for stresses, are written for the present problem, respectively, as:

x                             11 x 12 y y 12 x 22 y xy u v y x , ,   u x v y

(2)

66 xy









(3)

xy  

xy

0,

0

x  

x

x y

x y

 

 