# Multiaxial Fatigue Analysis

Multiaxial Fatigue Analysis, using S-N (stress-life), E-N (strain-life), and Dang Van Criterion (Factor of Safety) approaches for predicting the life (number of loading cycles) of a structure under cyclical loading may be performed by using OptiStruct.

In Uniaxial Fatigue Analysis, OptiStruct converts the stress tensor to a scalar
value using user-defined combined stress method (von Mises,
Maximum Principal Stress, and so on). In Multiaxial Fatigue
Analysis, OptiStruct uses the stress tensor
directly to calculate damage. Multiaxial Fatigue Analysis
theories discussed in the following sections are based on
the assumption that stress is in the plane-stress state. In
other words, only free surfaces of structures are of
interest in Multiaxial Fatigue Analysis in OptiStruct. For solid elements, a shell
skin is automatically generated by OptiStruct, shell elements are
used as-is. Multiaxial Fatigue
Analysis features are activated by setting
`MAXLFAT`=YES on
the FATPARM Bulk Data
Entry.

The stress-life method works well in predicting fatigue life when the stress level in the structure falls mostly in the elastic range. Under such cyclical loading conditions, the structure typically can withstand a large number of loading cycles; this is known as high-cycle fatigue. When the cyclical strains extend into plastic strain range, the fatigue endurance of the structure typically decreases significantly; this is characterized as low-cycle fatigue. The generally accepted transition point between high-cycle and low-cycle fatigue is around 10,000 loading cycles. For low-cycle fatigue prediction, the strain-life (E-N) method is applied, with plastic strains being considered as an important factor in the damage calculation.

Sections of a model on which Fatigue Analysis is to be performed must be identified on a FATDEF Bulk Data Entry. The appropriate FATDEF Bulk Data Entry may be referenced from a fatigue subcase definition through the FATDEF Subcase Information Entry.

The Dang Van criterion is used to predict if a component will fail in its entire load history. The conventional fatigue result that specifies the minimum fatigue cycles to failure is not applicable in such cases. It is necessary to consider if any fatigue damage will occur during the entire load history of the component. If damage does occur, the component cannot experience infinite life.

## Uniaxial Load

## Proportional Biaxial Load

In models with non-proportional biaxial loads, principal stresses can vary non-proportionally, and/or with changes in direction. Typically, out-of-phase loading of stress components is known as non-proportional multiaxial load.

## Non-Proportional Multiaxial Load

In Multiaxial Fatigue Analysis, proportional biaxial loads and non-proportional multiaxial loads are considered in OptiStruct.

## Non-proportional Cyclic Loading and Non-Proportional Hardening

The plasticity model used in Multiaxial Fatigue Analysis will take care of non-proportional hardening, if applied load is non-proportional.

## Critical Plane Approach

Experiments show that cracks nucleate and grow on specific planes known as critical planes. The Critical Plane Approach captures the physical nature of damage in its damage assessment process. It deals with stresses and strains on the critical planes.

OptiStruct searches for the most damaging plane by 10 degrees of $\theta $. On each plane, OptiStruct assesses damage using tensile crack damage model and shear crack damage model. At the end of a search, OptiStruct reports damage at the most damaging plane which is a critical plane.

## Stress-Life (S-N) Approach

The Stress-Life Approach for the Multiaxial Fatigue Analysis is similar to Uniaxial Fatigue Analysis. See the S-N Curve and Cycle Counting sections of Uniaxial Fatigue Analysis for introductory information for Stress-Life approach in Multiaxial Fatigue Analysis.

### 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 mean stress correction methods 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 method
should be expected to correlate test data for all materials in all life regimes.
There is no consensus yet as to the best method to use for multiaxial fatigue life
estimates. Multiple methods are available and can be used concurrently in OptiStruct Multiaxial Fatigue Analysis. For stress-based mean stress
correction method, Goodman model is used for tensile crack. Findley model is used
for shear crack. You can define mean stress correction methods after the `MCORRECT` field on the FATPARM
Bulk Data Entry.
If multiple models are defined, OptiStruct selects the model which
leads to maximum damage from all the available damage values.

### Goodman Model

The Goodman model in Uniaxial Fatigue Analysis is used at critical plane to assess damage caused by tensile crack growth.

### Findley Model

The constant $$k$$ is determined experimentally by performing fatigue tests involving two or more stress states. For ductile materials, $$k$$ typically varies between 0.2 and 0.3. Its default value in OptiStruct is 0.3.

### FKM

FKM mean stress correction is available for both Uniaxial and Multiaxial S-N Fatigue. For more information, see Mean Stress Correction under Uniaxial S-N Fatigue in the User Guide.

`CHK`is set to YES in FATPARM for the proportional loadcase, the above equation is calculated on maximum shear stress plane. For more details, see Proportional Biaxial Load.

## Strain-Life (E-N) Approach

If the applied load is non-proportional multiaxial cyclic, OptiStruct runs the Jiang-Sehitoglu plasticity model to calculate the total strain and elasto-plastic stress.

^{1}In OptiStruct, isotropic hardening part is removed from Jiang-Sehitoglu's original model, which is best described in deviatoric stress space.

### Yield Function

The notation $$\tilde{S}$$ is used for the deviatoric stress tensor, $$\tilde{a}$$ is the backstress tensor and $$k$$ is the yield stress is simple shear.

### Flow Rule

### Hardening Rule

There are four material parameters in this model, $$c$$, $$r$$, $$X$$, and $$k$$. All of these constants are computed from the cyclic stress strain curve of the material (Equation 23). A simple power function is fit to this curve to obtain three material properties; cyclic strength coefficient, $$K\text{'}$$, cyclic strain hardening exponent, $$n\text{'}$$, and elastic modulus, $$E$$.

The shear yield strength, $$k$$, is obtained by setting the plastic strain to 0.0002 (0.02%) and dividing by $\sqrt{3}$. Both $$c$$ and $$r$$ are obtained by selecting a series of stress strain pairs along the material cyclic stress strain curve and describe the shape of the curve. Ratcheting rate is controlled by $$X$$ which is set at a fixed value of 5. The number of components ($$M$$) is set to 10.

### Non-Proportional Hardening

Non-proportional hardening is the term used to describe loading paths where the principal strain axes rotate during cyclic loading. The simplest example is a bar subjected to alternating cycles of tension and torsion loading. Between the tension and torsion cycles the principal axis rotates 45 degrees. Out-of-phase loading is a special case of non-proportional loading and is used to denote cyclic loading histories with sinusoidal or triangular waveforms and a phase difference between the loads. Materials show additional cyclic hardening during this type of loading that is not found in uniaxial or any proportional loading path.

You will need to input the `coefkp90` and
`coefnp90`. The default value of `coefkp90` is
1.2. Default value of `coefnp90` is 1.0.

^{2}

The rate of hardening is controlled by the constant $$b$$. Since you are interested in the stable solution for fatigue calculations, $$b$$= 5 is selected for numerical stability. These equations are now sufficient to solve for the stresses and strains under any arbitrary loading path. The solution now proceeds by incrementally solving these equations, which can be solved in either stress, or strain control. The initial conditions are that the stress and backstress start at. All material constants start at their initial values as well and are updated after each loading increment.

### Notch Correction

In uniaxial loading approximate solutions such as Neuber's rule are used to compute
the stresses and strains during plastic deformation. Unfortunately, Neuber's rule
cannot be directly extended to multiaxial loading because there are six unknowns and
only five equations. To overcome this problem Koettgen ^{3} proposed a structural yield surface to obtain local
elastic-plastic stresses and strains. Having defined a "yield surface," standard
cyclic plasticity methods can be used to solve the unknown stresses and strains at
the notch. The material memory effects are built directly into standard cyclic
plasticity calculations. The method is based on the same concept at the analytical
or experimental nominal stress - notch strain curves used in uniaxial fatigue
analysis such as the one shown below.

Where, the constants $$K*$$ and $$n*$$ represent the behavior of the structure rather than the material. Uniaxial Neuber's rule is employed to establish the constants $$E*$$, $$K*$$ and $$n*$$.

- Apply uniaxial Neuber's rule to get $$K*$$ and $$n*$$ (Equation 22).
- Run Jiang-Sehitoglu plasticity model in pseudo stress control with pseudo
constants $$K*$$ and $$n*$$ to obtain the pseudo stress-local strain
response. Now total strain is available.
(24) $\u25b3\epsilon =\left[f\left({K}^{*},{n}^{*},{E}^{*}\right)\right]\u25b3{\sigma}^{e}$ - Run Jiang-Sehitoglu plasticity model in strain control with material
constants $$K$$ and $$n$$ to obtain the local strain - local stress
response. Finally, stress is also available.
(25) $\u25b3\sigma =\left[f\left(K,n,E\right)\right]\u25b3\epsilon $

### Proportional Biaxial Load

If applied load is proportional biaxial load, you do not have to calculate incremental plastic stress and strain using Jiang-Sehitoglu plasticity model. Because direction of principal stresses do not change, you know where the maximum shear stress and the maximum tensile stress are developed and what their values are with Mohr's circle. What you need to do is convert pseudo stress range to stress range using notch correction method such as Neuber's rule or Glinka's rule. OptiStruct applies Hoffmann-Seeger method for this situation.

If only one static subcase is referenced in FATEVNT, the cyclic
load is always proportional biaxial load. In this case, if `CHK` is
set to YES in FATPARM, OptiStruct runs Hoffmann-Seeger method to calculate stress and strain from pseudo stress. If
multiple static subcases are referenced in FATEVNT, currently
OptiStruct runs Jiang-Sehitoglu plasticity model.

- Plane stress states. Out-of-plane principal stress is zero.
- The principal stress and strain axes are fixed in orientation.
- The ratio of the in-plane principal stress is constant.
- The uniaxial stress-strain curve such as Ramberg-Osgood curve can be extended for use with suitable equivalent stress and strain parameters such as the von Mises parameter
- Hencky's flow rule is applicable.

- The elastic values of the signed von Mises stress and strain from nominal
stress:
(26) $\begin{array}{l}{\sigma}_{eq}^{e}=\frac{{\sigma}_{1}^{e}}{\left|{\sigma}_{1}^{e}\right|}\sqrt{{({\sigma}_{1}^{e})}^{2}+{({\sigma}_{2}^{e})}^{2}-{\sigma}_{1}^{e}{\sigma}_{2}^{e}}\\ {\epsilon}_{eq}^{e}=\frac{{\sigma}_{eq}^{e}}{E}\end{array}$ - The equivalent strain and stress ${\sigma}_{eq}^{}$ and ${\epsilon}_{eq}^{}$ is obtained by solving the following
equations:
(27) $$\begin{array}{l}{\sigma}_{eq}^{}{\epsilon}_{eq}^{}={\sigma}_{eq}^{e}{\epsilon}_{eq}^{e}=\frac{{\sigma}_{eq}^{e}}{E}\\ {\epsilon}_{eq}^{}=\frac{{\sigma}_{eq}^{}}{E}+{\left(\frac{{\sigma}_{eq}^{}}{{K}^{\text{'}}}\right)}^{\frac{1}{{n}^{\text{'}}}}\end{array}$$ - Calculate plastic modulus, $$H$$.
(28) $${\epsilon}_{p\text{}eq}^{}={\left(\frac{{\sigma}_{eq}^{}}{{K}^{\text{'}}}\right)}^{\frac{1}{{n}^{\text{'}}}}$$ Plastic portion of the equivalent strain:(29) $$H=\left(\frac{{\sigma}_{eq}^{}}{{\epsilon}_{p\text{eq}}}\right)$$ - The principal stresses are calculated by:
(30) $\begin{array}{l}a=\frac{{\sigma}_{2}^{e}}{{\sigma}_{1}^{e}}=\frac{{\sigma}_{2}^{}}{{\sigma}_{1}^{}}\\ {\sigma}_{1}^{}=\frac{{\sigma}_{eq}^{}}{\sqrt{1-a+{a}^{2}}}\\ {\sigma}_{2}=a{\sigma}_{1}\end{array}$ - The principal total strains are calculated by:
(31) $\begin{array}{l}\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\end{array}\right\}=\left[\begin{array}{cc}\frac{1}{E}+\frac{1}{H}& -\frac{v}{E}-\frac{1}{2H}\\ -\frac{v}{E}-\frac{1}{2H}& \frac{1}{E}+\frac{1}{H}\end{array}\right]\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\end{array}\right\}\\ {\epsilon}_{3}=-\left(\frac{v}{E}+\frac{1}{2H}\right)\left({\sigma}_{1}+{\sigma}_{2}\right)\end{array}$

The same procedure can be applied to calculate the amplitude of each quantity assuming Massing assumption is valid for equivalent stress and strain.

Range of each quantity can be obtained by doubled amplitude.

### 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 OptiStruct 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 and the Morrow model are
available. You can define damage models after the
`MCORRECT` field on the FATPARM Bulk
Data Entry. If
multiple models are defined, OptiStruct selects the model which
leads to maximum damage from all the available damage values.

#### Smith-Watson-Topper Model

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.

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

`CHK`is set to YES in FATPARM for proportional loadcase, (Equation 35) is calculated on the maximum shear strain plane. For more details, see Proportional Biaxial Load. If ${\tau}_{f}^{\text{'}}$ or ${\gamma}_{f}^{\text{'}}$ are not available, OptiStruct 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.

#### Brown-Miller Model

#### Morrow

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

Where,

$$A=1.3+0.7S$$

$$B=1.5+0.5S$$

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.

If `CHK` is set to YES in the
FATPARM Bulk Data Entry for proportional load cases, the
above equation is calculated on the maximum shear strain plane. For more details,
see Proportional Biaxial Load.

In case of proportional load, you do not have to search critical plane to find maximum damage plane, because direction of principal stress and principal strain do not change, you know the direction of plane where the maximum shear stress amplitude and the maximum normal stress amplitude. At the plane, you can easily calculate required information using Mohr's circle.

If `CHK` is set to YES in
FATPARM for proportional load case, damage in Fatemi-Socie
model is calculated on maximum shear strain plane, which is 45 degrees away from a
principal stress plane.

Damage in a SWT model is calculated in the maximum principal stress plane.

#### References

## Dang Van Criterion (Factor of Safety)

Used to predict if a component will fail in its entire load history. In certain physical systems, components may be required to last infinitely long.

For example, automobile components which undergo multiaxial cyclic loading at high rotational velocities (like propeller shafts) reach their high cycle fatigue limit within a short operating life. The conventional fatigue result that specifies the minimum fatigue cycles to failure is not applicable in such cases. It is not necessary to quantify the amount of fatigue damage, but just to consider if any fatigue damage will occur during the entire load history of the component. If damage does occur, the component cannot experience infinite life. Fatigue analysis based on the Dang Van criterion is designed for this purpose.

Fatigue crack initiation usually occurs at zones of stress concentration such as geometric discontinuities, fillets, notches and so on. This phenomenon takes place in the microscopic level and is localized to certain regions like grains which have undergone local plastic deformation in characteristic intra-crystalline bands. The Dang Van approach postulates a fatigue criterion using microscopic variables in the apparent stabilization state; this is a state of elastic shakedown if no damage occurs. The main principle of the criterion is that the usual characterization of the fatigue cycle is replaced by the local loading path and so damaging loads can be distinguished.

- Evaluate the macroscopic stresses $${\sigma}_{ij}\left(t\right)$$, for each location at a different point in time.
- Split the macroscopic stress $${\sigma}_{ij}\left(t\right)$$ into a hydrostatic part $p(t)$ and a deviatoric part $${S}_{ij}\left(t\right)$$.
- Calculate the stabilized microscopic residual stress $$dev\text{\hspace{0.17em}}p*$$ based on the following equation:
(49) $$dev\text{\hspace{0.17em}}{p}^{*}=Min\left(Max\left({J}_{2}\left({S}_{ij}\left(t\right)-dev\text{\hspace{0.17em}}p\right)\right)\right)$$ The expression is minimized with respect to $\rho $ and maximized with respect to $$t$$.

- Calculate the deviatoric part of microscopic stress.
(50) $${s}_{ij}\left(t\right)={S}_{ij}\left(t\right)+dev\text{\hspace{0.17em}}{p}^{*}$$ - Calculate factor of safety (FOS):
(51) $$FOS=Min\left(\frac{b}{\tau \left(t\right)+ap\left(t\right)}\right)$$ (52) $$\tau \left(t\right)=0.5Tresca\left({s}_{ij}\left(t\right)\right)$$ Where, $$b$$ and $$a$$ are material constants.

If

`FOS`is less than 1, the component cannot experience infinite life.

### OptiStruct Factor of Safety Setup

- The torsion fatigue limit and hydrostatic stress sensitivity values required for an FOS
analysis can be set in the optional
`FOS`continuation line on the MATFAT Bulk Data Entry. - The Dang Van criterion type can be selected on the FATPARM Bulk Data Entry.
- Factor of Safety output can be requested using the FOS I/O Options Entry.

^{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)