Crack Paths 2006

The Matake Criterion

Matake [7] introduced a linear combination of the shear and normal stresses on the

critical plane, similar to the Findley proposal

(8)

N

¸¹·

W

1

m

W

k

¨©§

a n

ans ,

,

af

W V

where Vn,a

is the normal stress amplitude on the critical plane. However, the critical ,

W

N

f

plane orientation coincides with the maximumshear stress amplitude. For such

orientation of the critical plane, it is possible to determine the material coefficient k. For

uniaxial torsion loading, the plane of the maximumshear stress amplitude does not

experience the normal stress. Therefore, the material coefficient k is determined from

Vn,a

= 0.5Vaf; where Vaf

uniaxial push-pull tests. For fatigue limit,

= 0.5Vaf,

is the

ns,a

fatigue limit under push-pull tests (R=-1). Eq. (8) takes the following form

af af k W V V 12 12 . af

(9)

From Eq. (9), the k coefficient is as follows

1 2 af a f k V W .

(10)

Under random loading, the equivalent shear stress history has the same mathematical

form as Eq. (5). However, for the Matake criterion the maximumshear stress range

determines the critical plane orientation.

^ ` ^ `)( min ) ( m a x 0 0 t t n s T t n s T t W W

:),( s n

W

G G '

ns

,

(11)

where T is the time of observation. Damagedegree D(i) is computed on the critical plane

for each i-th stress level according to the general Eq. (6), where F=W.

The MaximumNormalStress Criterion on the Critical Plane (max{Vn})

This failure criterion comes from the static hypothesis of material strength. According

to this criterion, the maximumnormal stress range is responsible for the fatigue of

materials. For the cyclic loading it leads to the following equation

V V V

NN

m

,

(12)

¸¸¹·¨¨©§

V

,

a n

f a f 1