Page 405 - Biaxial Multiaxial Fatigue and Fracture

A Multiaxial Fatigue Life Criterion for Non-Symmetrical and Non-Proportional Elasto-Plastic ... 389

For an elastic-plastic axial strain, E, = E: + E!', one has:

where vel and vp, are respectively the elastic and plastic Poisson's ratio. This would imply
that a material model should be used to separate the elastic and plastic strains once the total
strain is given. Instead, a simplified approach will be adopted here. In order to determine the
material function K in the general case, it is found useful to introduce an effective Poisson
ratio v , defined in the elastic-plastic range as following:

where E is the Young's modulus and E,, is the secant modulus. With this definition
one always can write E,. = E, = -vex. The equivalent stress amplitude, beq, may be determined
by setting the equivalent strain amplitude, Eq. (2), in the uniaxial cyclic stress-strain curve,
that is described in mathematical form by the Ramberg-Osgood equation:

For numerically determining the value of the material function K for each value of the
effective shear strain yeri, Eq. (lo), a convergent iterative procedure has to be employed. First,
an initial guess value of v = vel is set, by which the transverse strains may be evaluated as
E, = E, = -vE~, for a given value of longitudinal strain. Then, the equivalent strain is calculated
and, by employing the cyclic stress-strain curve, Eq. (15), the equivalent stress and the secant
modulus E,? are determined. Finally, the new estimated value of the effective Poisson ratio is
calculated, Eq. (14). The procedure is repeated, by employing the calculated value as initial
guess, until the difference between the guess and the obtained value is sufficiently small.
Values of K calculated for a range of effective shear strain amplitudes are shown as hollow
circles in Fig. 3-a, b and c, for the Inconel 718 alloy, the Mild Steel and the SAE 1045 steel,
respectively. Until the material response is elastic, a constant value of K, i.e. K,, , Eq. (13), is
achieved while in the plastic region K is a decreasing function of the strain.
In order to speed up the evaluation of the parameter K when predicting the fatigue life, the
following interpolating expression has been adopted for defining the material function K: