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 HyperWorks 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, HyperWorks selects the model which leads to maximum damage from all the available damage values.

Smith-Watson-Topper Model

This model is for tensile crack growth. In very high strength steels or 304 stainless steel under some local histories, cracks nucleate in shear, but fatigue life is controlled by crack growth on planes perpendicular to the maximum principal stress and strain. Both strain range and maximum stress determine the amount of fatigue damage.

Figure 1. SWT Damage 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,

ε_{1 a}

and maximum stress on the principal strain range plane,

σ_{n, \max}

.(1)

ε_{1} σ_{n, \max} = σ_{a} ε_{a} = σ_{a} (\frac{σ_{f}^{'}}{E} {(2 N_{f})}^{b} + ε_{f}^{'} {(2 N_{f})}^{c})

The stress term in this model makes it suitable for describing mean stresses during multiaxial loading and non-proportional hardening effects.

Fatemi-Socie Model

This model is for shear crack growth. During shear loading, the irregularly shaped crack surface results in frictional forces that will reduce crack tip stresses, thus hindering crack growth and increasing the fatigue life. Tensile stresses and strains will separate the crack surfaces and reduce frictional forces. Fractographic evidence for this behavior has been obtained. Fractographs from objects that have failed by pure torsion show extensive rubbing and are relatively featureless in contrast to tension test fractographs where individual slip bands are observed on the fracture surface.

Figure 2. 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.(2)

\frac{Δ γ}{2} (1 + k \frac{σ_{n, \max}}{σ_{y}}) = \frac{τ_{f}^{'}}{G} {(2 N_{f})}^{b_{γ}} + γ_{f}^{'} {(2 N_{f})}^{c_{γ}}

The sensitivity of a material to normal stress is reflected in the value $k / σ_{y}$ . Where, $σ_{y}$ is stress where a significant total strain of 0.002 is used in HyperWorks. 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.

τ_{f}^{'}

γ_{f}^{'}

are not available, HyperWorks calculates them using the following relationship. The transition fatigue life,

2 N_{f}

, is selected because the elastic and plastic strains contribute equally to the fatigue damage. It can be obtained from the uniaxial fatigue constants.(3)

2 N_{f} = {(\frac{E ε_{f}^{'}}{σ_{f}^{'}})}^{(\frac{1}{b - c})}

The Fatemi-Socie model can be employed to determine the shear strain constants.(4)

\frac{Δ γ}{2} (1 + k \frac{σ_{n, \max}}{σ_{y}}) = \frac{τ_{f}^{'}}{G} {(2 N_{f})}^{b_{γ}} + γ_{f}^{'} {(2 N_{f})}^{c_{γ}}

First, note that the exponents should be the same for shear and tension.(5)

\begin{array}{l} b_{γ} = b \\ c_{γ} = c \end{array}

Shear modulus is directly computed from the tensile modulus.(6)

G = \frac{E}{2 (1 + ν)}

Yield strength can be estimated from the uniaxial cyclic stress strain curve.(7)

σ_{y} = K^{'} {(0.002)}^{n^{'}} = \frac{σ_{f}^{'}}{ε_{f}^{' \frac{b}{c}}} {(0.002)}^{\frac{b}{c}}

Normal stresses and strains are computed from the transition fatigue life and uniaxial properties.(8)

\begin{array}{l} \frac{Δ ε_{p}}{2} = ε_{f}^{'} {(2 N_{t})}^{c} \\ \frac{Δ ε_{p}}{2} = \frac{σ_{f}^{'}}{E} {(2 N_{t})}^{b} \\ σ_{n, \max} = \frac{Δ σ}{4} = \frac{E Δ ε_{e}}{4} \end{array}

Substituting the appropriate the value of elastic and plastic Poisson's ratio gives:(9)

\begin{array}{l} \frac{Δ γ_{e}}{2} = 1.3 \frac{Δ ε_{e}}{2} \\ \frac{Δ γ_{p}}{2} = 1.5 \frac{Δ ε_{p}}{2} \end{array}

Separating the elastic and plastic parts of the total strain results in these expressions for the shear strain life constants:(10)

\begin{array}{l} {τ^{'}}_{f} = \frac{1.3 Δ ε_{e}}{2} (1 + k \frac{σ_{n, \max}}{σ_{y}}) \frac{G}{{(2 N_{t})}^{b_{γ}}} \\ {γ^{'}}_{f} = \frac{1.5 Δ ε_{p}}{2} (1 + k \frac{σ_{n, \max}}{σ_{y}}) \frac{1}{{(2 N_{t})}^{c_{γ}}} \end{array}

Brown-Miller Model

This model is for shear crack growth. Brown and Miller conducted combined tension and torsion tests with a constant shear strain range. The normal strain range on the plane of maximum shear strain will change with the ratio of applied tension and torsion strains. Based on the data shown below for a constant shear strain amplitude, Brown and Miller concluded that two strain parameters are needed to describe the fatigue process because the combined action of shear and normal strain reduces fatigue life.

Figure 3. Fatigue Life vs Normal Strain Amplitude

Morrow

Morrow is the first to consider the effect of mean stress through introducing the mean stress

σ_{0}

in fatigue strength coefficient by:(11)

ε_{a}^{e} = \frac{(σ'_{f} - σ_{0})}{E} {(2 N_{f})}^{b}

Thus, the entire fatigue life formula becomes:(12)

ε_{a} = \frac{(σ'_{f} - σ_{0})}{E} {(2 N_{f})}^{b} + ε_{f}^{'} {(2 N_{f})}^{c}

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:(13)

\frac{Δ \hat{γ}}{2} = \frac{Δ γ_{\max}}{2} + S Δ ε_{n}

Where,

Δ \hat{γ}

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.

Δ γ_{\max}

is the maximum shear strain range and

Δ ε_{n}

is the normal strain range on the plane experiencing the shear strain range

Δ γ_{\max}

. Considering elastic and plastic strains separately with the appropriate values of Poisson's ratio results in:(14)

\frac{Δ γ_{\max}}{2} + S Δ ε_{n} = A \frac{σ_{f}^{'}}{E} {(2 N_{f})}^{b} + B ε_{f}^{'} {(2 N_{f})}^{c}

Where,

$A = 1.3 + 0.7 S$

$B = 1.5 + 0.5 S$

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,

σ_{n}

is one half of the axial mean stress leading to:(15)

\frac{Δ γ_{\max}}{2} + S Δ ε_{n} = A \frac{σ_{f}^{'} - 2 σ_{n, m e a n}}{E} {(2 N_{f})}^{b} + B ε_{f}^{'} {(2 N_{f})}^{c}

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.

References

¹ Jiang and Sehitoglu "Modeling of Cyclic Ratcheting Plasticity, Part I: Development of Constitutive Equations," Journal of Applied Mechanics, Vol. 63, 1996, 720-725

² Tanaka, E., "A Non-proportionality Parameter and a Cyclic Viscoplastic Constitutive Model Taking into Account Amplitude Dependencies and Memory Effects of Isotropic Hardening," European Journal of Mechanics, A/Solids, Vol. 13, 1994, 155-173)

³ Koettgen V.B., Barkey M.E., and Socie, D.F. "Pseudo Stress and Pseudo Strain Based Approaches to Multiaxial Notch Analysis" Fatigue and Fracture of Engineering Materials and Structures, Vol. 18, No. 9, 1995, 981-1006)