# 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 HyperLife.

In Uniaxial Fatigue Analysis, HyperLife 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, HyperLife 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 HyperLife. For solid elements, a shell skin can be generated in the FEA.

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.

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 HyperLife Multiaxial Fatigue Analysis, non-proportional multiaxial loads are considered.

## 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.

HyperLife searches for the most damaging plane by 10 degrees of $\theta $ . On each plane, HyperLife assesses damage using tensile crack damage model and shear crack damage model. At the end of a search, HyperLife 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 HyperLife 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 in the Multiaxial Fatigue dialog. If multiple models are defined, HyperLife 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 Findley criterion is often applied for finite long-life fatigue.

Where, ${\tau}_{f}^{*}$ is computed from the torsional fatigue strength coefficient, ${\tau}_{f}^{\text{'}}$ using:

The correction factor $\sqrt{1+{k}^{2}}$ typically has a value of about 1.04. Note that ${\tau}_{f}^{*}$ has to be defined based on amplitude. If ${\tau}_{f}^{\text{'}}$ is not defined by you, HyperLife automatically calculates it using the following equation.

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 HyperLife 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.

## Strain-Life (E-N) Approach

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

The Jiang-Sehitoglu model
is one of the more successful incremental multiaxial cyclic plasticity models. ^{1} In HyperLife, isotropic hardening
part is removed from Jiang-Sehitoglu's original model, which is best described in
deviatoric stress space.

### Yield Function

The Mises yield function, $F$ , is expressed as:

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

The normality rule is given by:

Where, ${\tilde{\epsilon}}^{p}$ is the exterior unit normal to the yield surface at the loading point, as:

### Hardening Rule

The total backstress $\tilde{\alpha}$ is divided into $M$ parts.

The evolution of the backstress (translation of the yield surface) for each of the parts is given by:

Where, $dp$ is the equivalent plastic strain increment and ${c}^{i}$ , ${r}^{i}$ , and ${X}^{i}$ are three sets of non-negative and single valued scalar functions:

Material memory is contained in the $a$ terms. The plastic modulus function, $H$ , is derived from the consistency condition which requires that the stress state be on the yield surface when plastic deformation is occurring.

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 13). 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.

The 90 degrees out-of-phase loading path has been found to produce the largest degree of non-proportional hardening. The magnitude of the additional hardening observed for this loading path as compared to that observed in uniaxial or proportional loading is highly dependent on the micro-structure and the ease with which slip systems develop in a material. A non-proportional hardening coefficient, $a$ , can be introduced which is defined as the ratio of equivalent stress under 90 degrees out-of-phase loading/equivalent stress under proportional loading at high plastic strains in the flat portion of the stress-strain curve. This term reflects the maximum degree of additional hardening that might occur for a given material. HyperLife measures the non-proportional hardening coefficient, $a$ , at 0.003 strain amplitude using:

Where,

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

The degree of non-proportionality, $C$ , is a fourth ranked tensor which is a function of the plastic strain and the direction of the plastic strain increment.

The amount of non-proportional hardening is computed from Tanaka's parameter. ^{2}

For any proportional loading $A=0$ and for 90 degrees out-of-phase loading A = 1/sqrt(2). The material constants, ${r}^{i}$ and the yield stress $k$ are related to the non-proportional hardening parameter as:

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.

Cyclically, nominal stress - notch root strain response has all the features associated with stress-strain response such as hysteresis, memory, cyclic hardening and softening. The concept of pseudo stress, ${\sigma}^{e}$ , or strain, ${\epsilon}^{e}$ , is defined as theoretical elastic stress or strain computed using elastic assumptions. Neuber's rule may be written in terms of pseudo stress as:

These concepts can be generalized to three-dimensional stress and strain states with a structural yield surface at a notch. Having defined a structural yield surface, standard cyclic plasticity methods can be used to solve the unknown stresses and strains at the notch. For multiaxial loading, the total stress or strain is obtained by superposition of the individual load cases. A relationship between pseudo stress and notch strain is described with a power function that has the same form as a stress-strain curve.

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 $n*$ .

For plane stress on the surface, ${\sigma}_{z}^{e}$ , ${\tau}_{yz}^{e}$ and ${\tau}_{xz}^{e}$ are all zero and this yield function can be written as:

This equation defines a structural yield function, ${F}_{o}$ , which has the same form as the yield functions used in the plasticity models for stress-strain calculations, that is, a Mises yield function in terms of pseudo stress.

- 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.$$\u25b3\epsilon =\left[f\left({K}^{*},{n}^{*},{E}^{*}\right)\right]\u25b3{\sigma}^{e}$$
- 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.$$\u25b3\sigma =\left[f\left(K,n,E\right)\right]\u25b3\epsilon $$

### 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 HyperLife 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 from the Fatigue Module dialog. If multiple models are defined, HyperLife 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, ${\epsilon}_{1a}$ and maximum stress on the principal strain range plane, ${\sigma}_{n,\mathrm{max}}$ .

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 $k/{\sigma}_{y}$ . Where, ${\sigma}_{y}$ is stress where a significant total strain of 0.002 is used in HyperLife. 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 ${\tau}_{f}^{\text{'}}$ or ${\gamma}_{f}^{\text{'}}$ are not available, HyperLife 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.

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 ${\sigma}_{0}$ 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, $\text{\Delta}\widehat{\gamma}$ 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. $\text{\Delta}{\gamma}_{\mathrm{max}}$ is the maximum shear strain range and $\text{\Delta}{\epsilon}_{n}$ is the normal strain range on the plane experiencing the shear strain range $\text{\Delta}{\gamma}_{\mathrm{max}}$ . 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, ${\sigma}_{n}$ 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.

#### 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:$$dev\text{\hspace{0.17em}}{p}^{*}=Min\left(Max\left({J}_{2}\left({S}_{ij}\left(t\right)-dev\text{\hspace{0.17em}}p\right)\right)\right)$$
The expression is minimized with respect to $\rho $ and maximized with respect to $t$ .

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

If

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

### HyperLife Factor of Safety Setup

- Select the FOS tool from the fatigue modules.
- The torsion fatigue limit and hydrostatic stress sensitivity values required for an FOS analysis can be assigned in the Assign Material module.
- Assign load histories and proceed to evaluate.

^{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)