Page 416 - Biaxial Multiaxial Fatigue and Fracture

400 M. FILIPPINI ET AL.

v, v,, , vpr Effective, elastic and plastic Poisson’s ratio
6, v, Spherical coordinates
0; Uniaxial fatigue strength coefficient
w Angular frequency

APPENDIX

On an interference plane of normal n (q, 6) a shear strain vector is acting denoted as 1/2y, :

1
-y,(v,,6,r)=~(t).n-[n.~(t)-n]n
2

In the most general case where all the six components of the strain tensor ~(t) are present,
the squared intensity of 1/2y, , i.e. 1/4(y,)2 =(1/2y,,).(1/2yn) is given by, [21]:

1 2
;(Y,,) = [(E:+E:~+E~) sin2q cos2 ~+(E:~+E:+E~~) sin’y? sin2 8+

+(E: +€:; +E:) COS’v,+
+2 (E,E, + + E~E~) p sin 6 cos 6+
sin2
~
~
E
+2 ( E ~ + ~E,E~ + q~,) sin q cos q sin B+
~
+2 (ex&, + E ~ + ~E,E,) sin v, cos v, cos tp] -
E
-[(E:) sin4q cos4 6+(&:) sin4q sin4 a+(&:) cos4 p+
+4$)
+(~E~E~ sin4q cos2 sin2 tp+
+(25eZ +4%) sin2q cos2 v, sin2 8+
+(~E=E, +4~:) sin2q cos2 v, cos2 a+
+ ~ E ~ E ~ ~ X Y ) sin4p cos3 6 sin B+ ( ~E,E, ) sin3v, cos q cos3 6+
(
+(~E,,E~~) sin4v, sin’ B c0s8+(4&,&,) sin3v, cosq sin3 6+

+(~E:E~) COS^^ sinv, sin0+(4~,~,) COS^^ sin9 cost^+
+(~E~E~~ sin3q COSY, cos2 0 sintp+
+sE,E,,,)
+
+( ~E,E~ SE~~E~,,~) cosp sinZ 8 cos 6+
sin’q
+( ~E~E, SE,E=) sin2v, cosz 9 sin 6 cos 01
+
where for the off-diagonal components of the strain tensor the standard equality cy = yq/2
between the mathematical and engineering notation holds.