PSI - Issue 42

Christos F. Markides et al. / Procedia Structural Integrity 42 (2022) 202–209 Stavros K. Kourkoulis and Christos F. Markides et al. / Structural Integrity Procedia 00 (2022) 000 – 000

204

3

2. Theoretical analysis 2.1 The contact stresses and the relation between R 1 and R 2 (for a fixed h value)

According to the general formulae for the contact problem of two cylindrical, linear elastic, isotropic and homo geneous bodies, compressed against each other by an overall load P frame , and under the additional condition that R 2 → R 1 , the contact angles ω 2 , ω 1 and the contact pressure P ( ϕ ), defined in Fig.2, for the cavity (denoted by index 1) and the indenter (denited by index 2), are given by Markides & Kourkoulis (2016) as:

y

y

U formed indenter

frame P

Undeformed indenter

frame P

Deformed indenter Undeformed indenter

Undeformed indenter

1 

j 

1 

2 R

j  2 

2 R

L − Deformed indenter

2 

2 R

2 

2 R

2 

L

L −

L

j t

j t

j 

j  2 

( j) t

( j) t

L −

2 

L −

2 

L

2 

L

O 

j t

O 

j t

1 R

1 R

Deformed CTBD Undeformed CTBD

Deformed CTBD Undeformed CTBD

j t

j t

L

L

j ( ) P  c P

frame P

c P

j ( ) P 

frame

h

h

1 R

1 R

R

R

L

L −

L

L −





j j t Z 

j j t Z 

Undeformed CTBD

Undeformed CTBD

x

x

O

O

(a) (b) Fig. 2. The contact problem and the contact pressure for the CTBD’s cavity (a) and for the indenter (Newton’s 3 rd law) (b).

 

1  + +

2

1  

K P

R K

1

(1)

sin

,

,

−

2 



=

=

1 = +

frame

2

2

 

(1 ) 

4

4  

2 R t

R

− 



1

1

2

2 ) R h R R R h R + + − + 2 1 1 2 ( 1 ( )

2

(2)

1 1

cos

−

=

1 

2

   

   

cos

j 

3 (1 )   −

( ) P P

1 = −

,

j  ( )

sin

P P =

=

j 

2 

(3)

max

c

c

2

sin

8

K

2 

where κ =3 – 4 ν or (3 – ν )/(1+ ν ) for plane strain or generalized plane stress, with ν the Poisson’s ratio, and μ the shear modulus. Assuming that h and R 1 are given, then from Eqs.(1, 2) (and assuming that sin ω 2 →sin ω 1 , since R 2 → R 1 ), the magnitude for R 2 required so that the whole cavity comes in contact with the indenter is obtained as:

2 1 = + − 1 2 4 R R

2

frame 1 sin K P R t   2

(4)

R

2

1

2.2 The 1 st fundamental problem for the isolated CTBD In order, now, to solve the problem of an isolated CTBD, the parabolic pressure that is quantified in Eq.(3) is transformed into a statically equivalent parabolic distribution of point forces F j ( X j , Y j ), which act on the points Z j = x j +i y j (centers of infinitesimally small holes) arranged along the arc – LL of the intact BD, standing, thus, as the cavity of the CTBD (see Fig.3).