Page 287 - Biaxial Multiaxial Fatigue and Fracture

An Analysis of Elasto-Plastic Strains and Stresses in Notched Bodies Subjected to Cyclic ... 27 1

+
Ae:, = - SpdA
ASP,
2G
+
Aep, = - SzdA
a;2
2G
AS;
Ae" - - SGdA
f
13- 2G
de" - s+S;2dA
12- 2G
+
Ae' - - S;,dA
23- 2G
Aea - -+ AS;3 Sp,dA
3'- 2G
where:

The specific form of the stress-strain function, f(o,,), must be obtained experimentally from
an uniaxial cyclic test.

EQUIVALENCE OF INCREMENTS OF THE TOTAL DISTORTIONAL STRAIN ENERGY
DENSITY

It is proposed, analogously to the original Neuber rule, to use the equivalence of increments of
the total distortional strain energy density contributed by each pair of corresponding stress and
strain components.

[SF, Ae:, + e:,ASF, = S;,Ae:, + .PIAS:
Sf2 Aef2 + ef2 ASf2 = SP, AeP2 + er2 ASP,
Sf3Aef3 + ef3ASf3 = SP,AeP, + ey3ASP,
S12Ae;, = S;2Ae:2 +e;2AS:2
Si3Ae;, +e,',AS;, = SJ3Ae,q +e,",AS;,
S:3Ae:3 + e:3AS:3 = SZAe:, +e,"jAS?,

The equalities of strain energy increments for each set of corresponding hypothetical elastic and
actual elastic-plastic strain and stress increments at the notch tip can be shown graphically (Fig.
4) as the equality of areas of the two pairs of rectangles representing the increments of total
strain energy density (including the complementary one). The area of dotted rectangles
represents the total strain energy increment of the hypothetical (fictitious) elastic notch tip input