Issue 25

D. A. Hills et aliii, Frattura ed Integrità Strutturale, 25 (2013) 27-35; DOI: 10.3221/IGF-ESIS.25.05

1

1 II G K K       0 II II I I

I 



1

I II I K d K   

 

II

  

    

,

(17)

I

II

0

For this case, the stresses, instead of being given by Eq. (12), are instead given by       1  1  0 0 0 , I II ij I II ij ij r r r f f G d d                       

(18)

so that the direct   p x , and shearing   q x , are given by     1  1  , I II r p x           

x

x



int

I

II



 



f

f

(19)

0   d  

0   d  





G

G

0

0

I 







1 

1 





 

r 



r

0   x     d

0   d    

,

q x

II x f f



int

I

II



 



(20)





r

r

G G

0

0

When II K is positive, closure is implied through the asymptotic region. However, depending on the punch angle,  , and the coefficient of friction, f , various slip regions are implied at the edge and/or interior of the contact. To compute the implied slip extents we substitute Eq. (19) and (20) into the slip condition     q x fp x   , and solve for 0 / x d , again, denoting any boundary between stick and slip as s x , which gives 1 I K is negative and

II

II

s x f   

  

   

f f f f

  

I

II





r

(21)

I

I

d

f



0





r

f

0.2 0.4 0.6 0.8 1.0 1.2 1.4

60° Punch K I ,  K II

stick

- slip

x s d 0

0

1

2

3

4

f

f

0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.2 0.4 0.6 0.8 1.0 1.2 1.4

90° Punch

120° Punch  K I ,  K II

 K I ,  K II

stick

- slip

- slip

stick

+ slip

x s d 0

x s d 0

0

1

2

3

4

5

0

1

2

3

4

5

6

K is negative and

II K is positive, for punch angles of

Figure 3 : Plots of the implied regions of slip and stick, when I   60 , 90 ,120      , where the red line denotes the position at which the  

  q x fp x   condition is met, and the blue line the

position at which the  

  q x fp x   condition is met.

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