Page 202 - Biaxial Multiaxial Fatigue and Fracture

186 T LAGODA ET AL.

For distinguishing the positive and negative work in a fatigue cycle, we introduced functions
sgn[e(t)] and sgn[a(t)J in Eq. (3). Then, during a tension half-period, the sign of work
0.25 &(t)o(t) sgn[e(t)] is determined by the strain sign and during a compression half-cycle the
sign of work 0.25 &(t)o(t) sgn[o(t)] is influenced by the stress sign.
Let us introduce the two-argument logical function sgn(x, y), sensitive to signs of variables
x and y, defined as follows
‘1 when sgn(x) = sgn(y) = 1
0.5 when x=O and sgn(y)=l or y=O and sgn(x)=l
sgn(x, y) =. 0 when sgn(x) =-sgn(y) (5a)
-0.5 when x=O and sgn(y)=-1 or y=O and sgn(x)=-l
,- 1 when sgn(x) = sgn(y) = -1

When sgn(x, y) = 0.5 or -0.5 according to Eq. (5a), then always W(t) = 0 according to Eq. (4).
Thus, Eq. (5) can be written in a simplified way
‘1 when sgn(x) = sgn(y) = 1

sgn(x, y) = 0 when

<- 1 when sgn(x) = sgn(y) = -1
(5b)

Then Eq. (4) can be written as follows

Let us notice that Eq. (6) expresses the positive and negative parameter of strain energy den-
sity in a fatigue cycle and it allows to distinguish energy (specific work) under tension and un-
der compression. Expression (6) has another advantage: the parameter of strain energy history
has the zero mean value, while cyclic stress and strain change symmetrically in relation to the
zero levels. If we do not introduce the sign of stress and strain under cyclic loading, the fre-
quency of the strain energy history W(t) would increase twice, the amplitudes would decrease
twice and the mean value would be different from zero. It is not acceptable because it leads to
counting of a greater number of fatigue cycles with smaller amplitudes [20].
Under cyclic loading, if stress and strain reach their maximum values Ga and &, the maxi-
mum parameter of strain energy density, Eq. (6) and its amplitude are equal to

W, =0.5~,&, . (7)

Figure 1 shows interpretations of amplitude of the strain energy density parameter. Assum-
ing W(t) according to Eq. (6) as the damage parameter, we can replace the standard character-
istics of cyclic fatigue (0, - Nf) and (& - Nf) by a new one, (W, - Nf). In the case of the high-
cycle fatigue, when the characteristic (0, - Nf) is used, the axis 0, should be replaced by W,,
where