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

342

9

characterized by a Poisson’s ratio ν΄. In Fig.2, the transition from the orthotropic disc (Fig.2a) to the transtropic one (Fig.2b) is shown; shear moduli along the discs’ thicknesses are also shown, where, in addition to Eqs.(46), it holds that G yz ≡G΄ and G xz ≡G=E/2(1+ν), not counting, however, for the plane strain analysis considered here. Then, inserting Eqs.(47) for β ij in all previous formulae (and of course in Eqs.(42) and (43)), the displacement field for the as above described transtropic disc is automatically obtained. As a next step, the way anisotropy influences ω o and P c is investigated. In this context, consider Eqs.(45) and let first ϕ o =0; then, compression in x-direction causes dilatation in y-direction characterized by a Poisson’s ratio ν xy which, due to the symmetry of the strain tensor, is equal to Eν΄/E΄. Secondly, for ϕ o =π/2, compression in y-direction causes dilatation in x-direction characterized by a Poisson’s ratio ν΄. In the general case where P frame forms an arbitrary angle 0<ϕ o <π/2 with respect to the planes of isotropy ν(ϕ o )-value for Poisson’s ratio in ϕ o ±π/2-direction and Muskhelishvili’s constant κ(ϕ o ) may be obtained, in a first approximation, as follows:

2

2

E

E

                 2 ΄cos ΄sin  

(48)

2

2    sin

΄

cos

 

 





2

E΄

E΄

2

E

   3 4          3 4 ΄   

(49)

2

2    sin

cos





2

E΄

Substitution in Eqs.(45) from Eq.(49) provides the respective ω ο (ϕ o )- and P c (ϕ o )-values for the transtropic disc in the case of a non-arbitrarily predefined angle ω ο (one can always skip this procedure using an arbitrary value for ω ο and Eq.(44) for P c ). In addition, in the examples following, use is made of the Lekhnitskii’s (1981) approximating formula:

EE΄

(50)

G΄







E 1 2 ΄ E΄   

(c) v θ =0

0.0E+00

(a) v θ

-3.0E-05

(b) v θ

(c) v r

-6.0E-05

(a) v r

30 o

-9.0E-05

(b) v r

-1.2E-04

0.00 Polar displacement components [m] 0.01 0.02

0.03

0.04

0.05

r [m]

Fig. 3: The polar components of displacement along the loaded radius for transtropic discs made of three different materials.

4. Applications For the potentials of the solution introduced to become evident, some typical applications are described here for the displacements developed in transtropic discs with various degrees of anisotropy, subjected to parabolic distribution of radial stresses. In all cases the discs’ geometry is constant (R=0.05 m, d=0.01 m). In Fig.3, the polar components of displacement, v r =u(x,y)cosθ +v(x,y)sinθ and v θ = – u(x,y)sinθ+v(x,y)cosθ, are plotted along the loaded radius, i.e. for θ=ϕ o , in the case ϕ o =30 o and P frame =20 kN, for three discs made of different materials, namely: (a) a serpentinous schist with E=58 GPa, ν=0.34, E΄=27 GPa, ν΄=0.12 (Barla & Innaurato 1973), (b) a fictitious material of increased anisotropy ratio equal to δ=4:1, with E=60 GPa, ν=0.30, E΄=15 GPa, ν΄=0.10 and (c) a fictitious isotropic material with E=E΄=30 GPa and ν=ν΄=0.30. In all three cases, the same formulae obtained here for displacements were used (for m=15 additional terms). Actually, in that latter case (c) it turns to be μ 1 =μ 2 =i (see Eqs.(10), (11)) and v θ =0 all