Crack Paths 2009

C O L D - D R A W RIENSGI D U ASLTRESSES

= ∞ d ρ ), determined by

The longitudinal residual stress data for an unnotched bar (

Elices [13] through the neutron and X-ray diffraction technique, are shown by solid

symbols in Fig. 2a, where the radial coordinate r is normalized with respect to the bar

radius

0 R (i.e.

0 R/= r*) andrthe residual stress with respect to the residual stress value

). By interpolating such data through

at the bar centre (i.e. resI σ σ best fitting polynomial, a continuous cu v is obtained (Fig. 2a). ) 0 ( = σ * ) ( ) ( = resI

) ( rresI

Such an axisymmetrical stress distribution has been used to numerically assess the

residual stress distribution in a notched bar (

= 009.0 d ρ ). The dimensionless residual

stress profile, in correspondence to the reduced section S-S, is reported in Fig. 2b (in

such a case, the dimensionless radial coordinate is given by R/ * r= ). r

A generic axysimmetrical residual stress distribution given by

)(

) ( * ) 0 (

(1)

σ

r

=

σ

r

=

⋅

σ

r

) (

) (

) (

resI

resI

resI

can be approximated through a power series expansion as follows:

B r A

dr

r

r

≅

i )(

resI

( ) r

i n

i resI

i

i n

resi

i

i n

resi

⎜ ⎜ ⎝ ⎛

⎟ ⎟ ⎠ ⎞

i d 1 σ

) (

i

() r

σ

⋅ =

∑ ∑ = = ⋅ = 0

⋅

∑ =

*)(

(2)

) (

!

) (

0

)(

0

0 ) (

r =

where *r is given by Rr/ (R is equal to

0 R for unnotched bar and to R for notched

are expressed by the following equation:

bar), and the coefficients

)(resiB

(3)

σ

B

i Ri d

dr

dr

B

resi

resi

i

i

⎢⎣⎡

⎟ ⎟ ⎞

⎤

/ *

⎜ ⎜ ⎝ ⎛

σ

σ

B

R i d

r

=

⋅

[

]

=

*

=

!

) (

* ) 0 (

!1

i

resI

r

=

with

)( ) ( resI

) (

=

)(

)(

0

⎥

) (

i

r

r e s i r e s I

⎥ ⎦

⎠

) ( ) (

)(

0

01.82

3.2

(a)

(b)

0.0 0.2 0.4 0.6 0.8 1.0

ρd =

ρd = 0.009

2.4

0.4

1.6

0.0

0.8

-0.4

0.0

-0.8

-0.8

-1.6

0.0 0.2 0.4 0.6 0.8 1.0 D I M E N S I O N L E S S R A D I A L C O O R D I N A T E , r* -1.2

D I M E N S I O N L E S S

R A D I A L C O O R D I N A T E , r*

Figure. 2. Dimensionless residual stress distribution

against dimensionless radial

σ

*

resI

) (

coordinate r * : (a) unnotched round bar; (b) notched round bar.

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