Mean Stress Correction
Depending on the material, stress state, environment, and strain amplitude, fatigue life will usually be dominated either by microcrack growth along shear planes or along tensile planes.
Critical plane models incorporate the dominant parameters governing either type of crack growth. Due to the different possible failure modes, shear or tensile dominant, no single mean stress correction model should be expected to correlate test data for all materials in all life regimes. There is no consensus yet as to the best mean stress correction model to use for multiaxial fatigue life estimates. Multiple models are used in HWTK GUI Toolkit Multiaxial Fatigue Analysis. For strain-based mean stress correction, one model for tensile crack growth, Smith-Watson-Topper is used and two models for shear crack growth, Fatemi-Socie model and Brown-Miller model are available. You can define damage models . If multiple models are defined, HWTK GUI Toolkit selects the model which leads to maximum damage from all the available damage values.
Smith-Watson-Topper Model
This model, commonly referred to as the SWT parameter, was originally developed and continues to be used as a correction for mean stresses in uniaxial loading situations. The SWT parameter is used in the analysis of both proportionally and non-proportionally loaded components for materials that fail primarily due to tensile cracking. The SWT parameter for multiaxial loading is based on the principal strain range, ε1a and maximum stress on the principal strain range plane, σn,max .
The stress term in this model makes it suitable for describing mean stresses during multiaxial loading and non-proportional hardening effects.
Fatemi-Socie Model
To demonstrate the effect of maximum stress, tests with the six tension-torsion loading histories were conducted, that were designed to have the same maximum shear strain amplitudes. The cyclic normal strain is also constant for the six loading histories. The experiments resulted in nearly the same maximum shear strain amplitudes, equivalent stress and strain amplitudes and plastic work. The major difference between the loading histories is the normal stress across the plane of maximum shear strain.
The loading history and normal stress are shown in the figure at the top of each crack growth curve. Higher maximum stresses lead to faster growth rates and lower fatigue lives. The maximum stress has a lesser influence on the initiation of a crack, if crack initiation is defined on the order of 10 mm, which is the size of the smaller grains in this material.
These observations lead to the following model that can be interpreted as the cyclic shear strain modified by the normal stress to include the crack closure effects.
The sensitivity of a material to normal stress is reflected in the value k/σy . Where, σy is stress where a significant total strain of 0.002 is used in HWTK GUI Toolkit. If test data from multiple stress states is not available, k = 0.3. This model not only explains the difference between tension and torsion loading, but also can be used to describe mean stress and non-proportional hardening effects. Critical plane models that include only strain terms cannot reflect the effect of mean stress or strain path dependent on hardening.
If τ'f or γ'f are not available, HWTK GUI Toolkit calculates them using the following relationship. The transition fatigue life, 2Nf , is selected because the elastic and plastic strains contribute equally to the fatigue damage. It can be obtained from the uniaxial fatigue constants.
The Fatemi-Socie model can be employed to determine the shear strain constants.
First, note that the exponents should be the same for shear and tension.
Shear modulus is directly computed from the tensile modulus.
Yield strength can be estimated from the uniaxial cyclic stress strain curve.
Normal stresses and strains are computed from the transition fatigue life and uniaxial properties.
Substituting the appropriate the value of elastic and plastic Poisson's ratio gives:
Separating the elastic and plastic parts of the total strain results in these expressions for the shear strain life constants:
Brown-Miller Model
Morrow
Morrow is the first to consider the effect of mean stress through introducing the mean stress σ0 in fatigue strength coefficient by:
Thus, the entire fatigue life formula becomes:
Morrow's equation is consistent with the observation that mean stress effects are significant at low value of plastic strain and of little effect at high plastic strain.
Influence of Normal Strain Amplitude
Analogous to the shear and normal stress proposed by Findley for high cycle fatigue, they proposed that both the cyclic shear and normal strain on the plane of maximum shear must be considered. Cyclic shear strains will help to nucleate cracks and the normal strain will assist in their growth. They proposed a simple formulation of the theory:
Where, Δˆγ is the equivalent shear strain range and S is a material dependent parameter that represents the influence of the normal strain on material microcrack growth and is determined by correlating axial and torsion data. Δγmax is the maximum shear strain range and Δεn is the normal strain range on the plane experiencing the shear strain range Δγmax . Considering elastic and plastic strains separately with the appropriate values of Poisson's ratio results in:
Where,
A=1.3+0.7S
B=1.5+0.5S
Mean stress effects are included using Morrow's mean stress approach of subtracting the mean stress from the fatigue strength coefficient. The mean stress on the maximum shear strain amplitude plane, σn is one half of the axial mean stress leading to:
Select either the Fatemi-Socie model or the Miller-Brown model for shear crack growth mode. The SWT model is always used for tensile crack growth. Morrow method is also available.
Damage in a SWT model is calculated in the maximum principal stress plane.
Likewise, damages in Brown-Miller and Findley models are calculate on the maximum shear strain plane and maximum shear tress plane, respectively.