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

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along the loaded radius, as it should be expected due to isotropy. In addition, ω ο and P c for the three materials were obtained (using Eqs.(45)), respectively, equal to: (3.79 o , 453.70 MPa), (4.24 o , 405.92 MPa), (3.38 o , 508.66 MPa). For the same data, the displacements along the radius normal to the loaded one (for θ=ϕ o +π/2), are plotted in Fig.4.

3.0E-05

(b) v r

30 o o

(a) v r

2.0E-05

(c) v r

1.0E-05

(b) v θ

(a) v θ

(c) v θ =0

0.0E+00

0.00 Polar displacement components [m] 0.01 0.02

0.03

0.04

0.05

r [m]

Fig. 4: The polar components of the displacement field along the radius normal to the loaded one for transtropic discs made of the same materials as the discs considered in Fig.3.

It can be seen from both Figs.3 and 4 that the degree of anisotropy (defined as the modulus of elasticity along the strong- vs. that along the weak-anisotropy direction) influences the variation of the displacements according to a strongly non-linear manner. For example for r=R=0.05 m the radial displacement along the loaded radius is equal to 6.5x10 -5 m for the isotropic material, 7.7x10 -5 for the serpentinous schist (δ=2.15) and 10.5x10 -5 m for the material with δ=4. Similarly, along the radius normal to the loaded one, the respective numerical data are equal to 1.7x10 -5 m, 1.9x10 -5 m and 2.9x10 -5 m. What is, also, worth mentioning is that the solution, although given in series form, is proven capable of describing the displacement field in the limiting case of isotropic materials with excellent accuracy as it can be concluded by comparing the present data with the ones obtained from the solution by Kourkoulis et al. (2012). In order to further enlighten quantitatively the role of the anisotropy ratio, the deformed shape of the disc is plotted in Fig.5 (red continuous lines) in juxtaposition to the initial circular shape of the discs (black dotted lines). Only the

0.06

0.06

ϕ o +ω o

ϕ o +π/2

0.04

0.04

ϕ o +π/2

0.02

0.02

ϕ o =45

o

ϕ o =45

o

0.05

0

0

-0.06 -0.04 -0.02

0

0.02 0.04 0.06

-0.06 -0.04 -0.02

0

0.02 0.04 0.06

ω o

y [m]

y [m]

ω o

0.05

-0.02

-0.02

ω o

ω o

-0.04

-0.04

Deformed

Deformed

Undeformed

Undeformed

-0.06

-0.06

x [m]

x [m]

(a) (b) Fig. 5: The deformed (red lines) and undeformed (black dotted lines) shapes of an isotropic (a) and a transtropic disc with δ=4 (b).