Page 223 - Biaxial Multiaxial Fatigue and Fracture

Critical Plane-Energy Based Approachfor Assessment of Biaxial Fatigue Damage whem ... 201

In equation (1) Gij is the stress tensor (both i and j are equal to1,2,3), and the stresses 01 1,
622, and q3 are principal stresses. Since the thin-walled tubular specimens are in plane stress
condition, 92=0.
Principal stresses can be calculated from applied stress amplitudes:

o, =om, +[oI sin (e)] (34
I

where orn1 and crn2 are the longitudinal and the transverse mean stresses, respectively. The
angle 8 is the angle during GI cycles of stressing in a block loading history at which the Mohr's
circles are the largest and has the maximum value of shear strain. Angle 8 varies from 0 to
2an. Angle I$ corresponds to the phase delay between loads on the longitudinal and transverse
axes.
2
In Eq (2) G~ is the strain tensor (both i and j are equal to1,2,3) where the strains &ll, ~ ~ and ,
~33 are principal strains calculated using the elastic-plastic strain constitutive relation (Eq(4)).
In Eq (4), the first square bracket presents the elastic and the second square presents the plastic
components of strains:

where the deviatoric stress Sij is defined as the difference between the tensorial stress oij and
1
hydrostatic stress ( -okk ):
3
1
s, = 6, --BJ,
3
where &,=1 if i=j
S,=O if i#j
E is the elastic modulus, ve=O.3 is the elastic Poisson's ratio for the material used in this study
and Gkk is the summation of principal stresses. The stress amplitude of the hysteresis loop at
half-life cycle was associated with the stabilized cyclic stress-strain loop. The cyclic stress-
strain curve can be described with the same mathematical expression as for the monotonic
stress-strain curve. The relationship between the equivalent cyclic plastic strain E& and the
equivalent cyclic stress 0% obtained from uniaxial stabilized cyclic stress-strain data is:

The coefficient and the exponent in (6) are the cyclic strength coefficient, K*=1227 MPa,
and the cyclic strain hardening exponent, n*=0.36, respectively.
Eq (4) presents elastic-plastic deformation and correlates the tensorial cyclic stress and
strain components for 3D state of stress and strain. A simple form of this equation for the