Issue 41

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 41 (2017) 396-411; DOI: 10.3221/IGF-ESIS.41.51

normally intersected by the planes of isotropy, is a plane of elastic symmetry. For the specific purely plane strain configur ation, the present study is concerned with a thorough parametric investigation of the displacement and strain fields developed all over the disc cross-section.

P frame

Half-ball bearing

Upper jaw

= – P ( θ )

σ r

x

y

2 ω ο

ϕ ο

Guide pin

d

θ

L

R

z = r e i θ

r

O

xz -planes of isotropy Ε , v , G

2 ω ο

z

xy -plane of elastic symmetry E ΄, ν ΄, G ΄

Lower jaw

– P frame

Figure 1 : A disc made of a transversely isotropic material compressed between metallic jaws of radius ranging from 1.5 R to infinity: Configuration of the analytic problem and definition of symbols. In this context, a Cartesian co-ordinate system { O ; x ; y ; z } is considered at the disc center so that the xz -plane to be parallel to the planes of isotropy, the so-called strong direction, and the xy -plane to be parallel to the disc cross-section, known as the week direction. The strong direction is characterized by a Young modulus E , a Poisson ratio ν and a shear modulus G = E /(2(1+ ν )); for the week direction the respective engineering constants are E ΄, ν ΄ and G ΄(actually, ν ΄ char acterizes dilatation/contraction in xz -planes of isotropy for compression/tension along y -direction). The force P frame exerted by the loading frame to the disc through the metallic jaws forms an arbitrary angle ϕ o with respect to x -axis, i.e., the strong direction (Fig.1). In addition, and taking advantage of the isotropic disc-jaw contact problem, it is here assumed, in a first approximation, that P frame is parabolically distributed along two finite arcs of the disc periphery L , according to the law [16]:     2 2 1 sin sin c o o P P            (1) In Eq.(1) it is assumed that P ( θ )>0 so that the pressure on the loaded rim will be σ r = – P ( θ ). Moreover, it holds that P c =max{ P ( θ )} and angle ω o corresponds to the half loaded arc. These two quantities may equally well be defined in two different ways at one convenience. Namely, assuming without loss of generality that the jaw radius is equal to 1.5 R and re sorting to the isotropic disc-jaw contact problem, one may postulate [14]:

6 ( ) 



4  J



K P

P

3

1

( ) 1   

o frame

frame

  o 

o

o o  





( ) o 





Arc

P

K

( )

sin

,

,

(2)

c



32 ( ) 

G

Rd

K Rd

G΄

4

o

J

where, assuming plane strain conditions, it holds that [14]:

398