by the s a m e c o n c e n t r a t i o n of s t r a i n , w h i c h c a n be calculated using Hookas law.
Thus K o = Ke and from equation 2.1 Ko = K t as shown in
figure 2.1. The validity of N e u b e r 's expression is not necessarily obvious when yielding takes place at the notch root. H o w ever, it s u g g e s t s that as K o d e c r e a s e s , K e
increases in non-linear deformation, so their product could be constant.
For fatigue application equation 2.1 is usually altered by replacing Kt with Kf and defining K o and Ke as the ratio
of the ranges of stress and strain which result in:- Aa A s
K f = ( — . — ) - 2.2
A S A e
W h e r e :-
act = Stress range at notch root
AS = Nominal stress range
a g = Strain range at notch root
Ae = Nominal strain range
Kf = Fatigue concentration factor Equation 2.2 may be transposed giving
% £
All terms on the left side are determinable for each reversal from the load history and cyclic stress-strain curves, and all terms on the right side represent the local stress/strain behaviour of the material at the notch r o o t .
If the nominal conditions away from the notch are elastic then:- A S Ae = --- - 2.4 . E G i v i n g :- i’ Kf . AS = (Act. A g .E) - 2.5 * T h e r i g h t h a n d s i d e of the a b o v e e q u a t i o n is a determinable constant for each half-cycle of load. The equation is of the form, X.Y = C, which is a rectangular hyperbola [39, 43].
For simulations of the material at the notch root, the smooth specimen must be strained until the product of the stress and the strain equal the constant C, where C = (Kf.AS)2 /E as shown in figure 2.2.
2.3 The Cyclic Stress-Strain Curve
The cyclic stress/strain curve which provides a measure of the s t e a d y - s t a t e c y c l i c d e f o r m a t i o n r e s i s t a n c e of a material can be different from the monotonic stress/strain curve. The cyclic stress/strain curve, is the locus of tips of the stable hysteresis loops from several companion tests at different completely reversed constant strain amplitudes [43]. Such a steady-state stress- a m p l i t u d e strain-amplitude curve can be compared directly with the monotonic stress/strain curve. Cyclically induced changes in d e f o r m a t i o n r e s i s t a n c e t h e n b e c o m e i m m e d i a t e l y apparent. If the cyclic stress/strain curve is above the monotonic curve the material is said to cyclically harden;
if the cyclic curve is below the monotonic curve the material is said to cyclically soften [44].
Cyclic stress/strain properties are determined by testing s m o o t h p o l i s h e d s p e c i m e n s u n d e r ax i a l c y c l i c s t r a i n control. The cyclic stress/strain curve is defined as the locus of tips of stable "true stress/strain h y s t e r e s i s loops" obtained from companion test specimens. A typical stable hysteresis loop with a cyclic stress/strain curve drawn through the loop tips is shown in figure 2.3.
As shown in figure 2.4 the height of the loop from tip-to- tip is defined as the stress range (Act). For completely
reversed testing one half of the-stress range is generally equal to the- stress amplitude. While one half of the width from tip-to-tip is defined as the strain amplitude ( A 6 T /2). The p l a s t i c s t r a i n a m p l i t u d e is f o u n d by
subtracting the elastic strain amplitude (Ae4 /2 ) from the total strain amplitude.
A Gp A GT
2 2
For elastic conditions:
a s e Act = — - 2.7 2 2E Where; E = modulus of elasticity T h e n : A Gp A 6T A <j = - - 2.8 2 2 2E
The relation between cyclic stress and plastic strain can be described mathematically by a power function similar to that used for the monotonic curve [44, 45].
Act A Gp n 1 — = k' ( ) - 2 . 9 2 2 A G. 2.6 28
w h e r e : A a — = S t a b l e S t r e s s A m p l i t u d e 2