Crack Paths 2012

From Eq. (1) it appears that the equivalent stress (Seq (t) is linearly dependent on the

stress state components 6X (t) and txy (t), so it can be expressed as

11

(Seq : Z a j X j : a 1 X 1 + 3 2 x ’2 J21

where: al I sin (20¢), a; I 2cos (20L), x1 I 6X, x2 I txy.

From theory of probability [7] it results that the variance of random variable being a

linear function of some randomvariables is expressed by the following formula

I1

l'loeq : zajgl'lxj + 2zajakl'lxj:k al2l'LxI +a2llx2+ 2a1a2l'LxIx2 7 j=1 j

where: uasq - variance of equivalent stress osq,

uxl - variance of normal stress 6X,

uxg - variance of shear stress txy,

uxlxg - covariance of normal6X and shear stress txy stresses.

Under biaxial random stationary and ergodic stress state, the variances uxl, [1X2 and the

covariance [ixlxg in Eq. (3) are constant.

In the method of variance for determination of the critical plane position the m a x i m u m

function of Eq. (3) is searched in relation of the angle 0t occurring in coefficients a1 and

a2. After reduction, the variance of equivalent stress ugeq versus the angle 0t can be

written as

uqeq : sin2(2ot)p.X1 + 400s2 (20c)uX2 + 2sin(4ot)uX1X2 .

(4)

In the method of damage accumulation the damage degree SPM(TO) during observation

time T0 was calculated (Fig. 1) with use of the Palmgren-Minerhypothesis

k

.

l . Z— for

oeqaai 2 a-oaf ,

eq,ai < a ' G a f

where:

6 m m— amplitude of the equivalent stress,

m — coefficient of Wohler’s curve slope,

Gaf— fatigue limit,

N0— numberof cycles correspondingto the fatigue limit Gaf,

nI — numberof cycles with amplitude osqm,

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