Multiaxial fatigue

Multiaxial fatigue occurs when the local stress state is multiaxial. The stress components may vary in proportion to each other (proportional stresses) or may vary independently of each other (non-proportional stresses). The manner in which these stresses and the corresponding strains interact influences the fatigue damage development at the location in the structure at which the multiaxial stress state occurs.

Cracks typically initiate on the shear planes from the free surface of a material (stage I initiation and growth) and re-orient to an opening mode tensile crack for continued growth (stage II growth). Stage II cracks grow by an extension mechanism produced by shear deformation at the crack tip (see the following figure).

Models and Stages of Crack Growth

Given these models of crack initiation and growth, both shear and tensile parameters can correlate with fatigue life depending on the stage of crack growth.

For multiaxial strain states, Miller has proposed a classification of crack growth into Case A and Case B cracks (see the following figure). These classifications can be used to determine on which plane the crack will grow, as well as provide physical parameters to relate to fatigue life. Case A cracks tend to be shallow elliptical cracks. Case B cracks intersect the surface at a 45° angle and so are also influenced by the shear stresses on these planes.

For Principal Strains Ordered as ε1 ≥ ε2 ≥ ε3, Cracks First Initiate on Planes of Maximum Shear Strain (Stage I Growth) and Then Grow Tensile Planes (Stage II)

Brown and Miller proposed separate damage parameters for case A and case B cracks. See the illustration for contours of constant fatigue life in the figure below. Note that the axes of the gamma plot are one-half the maximum shearing strain for the horizontal axis and the average of the maximum and minimum principal strain (that is, the location of center of the largest Mohr’s circle of strain) on the vertical axis.

Curves of Constant Fatigue Life on the Γ-Plane

Based on these observations that have been made of fatigue crack initiation and growth, it is clear that loading modes and the crack growth stage in addition to the type of material have a direct relation to the fatigue life of a component or structure. This influences the choice of damage parameter to use for an analysis and whether or not the crack growth will be confined to a certain set of planes for the entire time, or if it will grow on different planes at different times.

Equivalent stress and strain approaches

Equivalent stress and strain approaches reduce the multiaxial stress or strain states to a single scalar value that can be used to calculate fatigue damage just as for a uniaxial state of stress. These approaches are exact if the state of stress is uniaxial, and are generally very good if the state of stress is proportional or nearly proportional. If the state of stress is non-proportional, then the critical plane approach is recommended, because the fatigue cracks will grow on different planes, as described previously.

There are various ways to compute equivalent stresses or strains. The methods that are implemented in Specialist Durability are described below.

The signed von Mises stress is computed by using the formula

where the sign, *SGN,*is the same as the sign of the principal stress with the largest magnitude.

The SGN-term is introduced since von Mises stresses are positive by definition. Hence for simple fully reversed uniaxial loading the von Mises stress amplitudes are just half the real stress amplitude. For plane stress, this formula reduces to

Additionally, the time histories of the maximum principal stress and the maximum shear stress can also be used to calculate fatigue damage.

For the stress-life approach and the pseudo-stress based strain-life approach (for the signed von Mises equivalent stress only), the stress quantities used are calculated based on the theory of elasticity.

Approach	Parameter Name	Equivalent Stress Quantity
Stress-lifeStrain-life	Signed von Mises	Signed von Mises plane stress
Stress-lifeStrain-life	Maximum Principle	Maximum principle plane stress
Stress-lifeStrain-life	Maximum Shear	Maximum shear stress plane stress
Stress-lifeStrain-life	Full 3D Signed von Mises	Signed von Mises full stress tensor
Stress-lifeStrain-life	Full 3D Maximum Principle	Maximum principle full stress tensor
Stress-life Strain-life	Full 3D Maximum Shear	Maximum shear full stress tensor

The critical plane approach

Findley observed that for fatigue crack initiation, damage accumulates independently in different directions (planes) in a material. For multiaxial stress states, more damage is caused on certain planes than on other planes. The planes that have the most damage are termed critical planes.

The basic steps involved critical plane approaches are similar, although the method for determining the amount of damage on each plane in the material depends on the choice of damage parameter.

To determine the critical plane, the multiaxial stress and strain histories at a particular point must be known or estimated. Then, this multiaxial stress state is resolved to particular planes (orientations) in the material to compute the normal and shear stresses and strains acting on the plane as in the figure below. Fatigue damage for these resolved stress and strain histories is computed based on a rainflow counting and an appropriate damage parameter. When all of the calculations for the planes are complete, the plane with the highest calculated damage is the critical plane, and determines the fatigue life of the structure at that location.

The Strain Tensor (Illustrated) and the Stress Tensor Can be Resolved to Candidate Critical Planes in the Material

Although there are an infinite number of planes at a particular point, candidate critical planes are commonly taken to be ten degrees apart, because it has been observed that fatigue damage does not vary much for planes less than ten degrees apart. Furthermore, if the damage is to be calculated on a traction free surface of a structure, the number of candidate planes can be greatly reduced.

A local coordinate system (Ρ, Ψ, ϴ) is used to transform the stresses and strains from the component coordinate directions (x, y, z). These coordinate systems are defined in the figure below, where the z-direction is traction free and in the direction normal to the plane of the component.

There are three critical plane damage parameters available in Specialist Durability, the crack opening parameter and the shear parameter for plane stresses and a full 3D critical plane (normal stress) for full 3D stress tensor histories (surface under pressure, calculation below the surface). These use normal stresses and shear stresses, respectively.

Definition of Local Coordinate System Used in the Critical Plane Analysis

The former two rely on plane stresses (that is, sz, txz, and tyz are zero). Since cracks start from the surface of a specimen this assumption is valid. The projection on the plane with given angle j is given by:

For shear stresses, you also have to take into account plane shearing. Therefore, the shear stresses in the 45°-planes (ϴ = 45°) are also calculated by

The stress quantities used are calculated based on the theory of elasticity.

Approach	Parameter	Mode
Stress-lifeStrain-life	Critical Plane, Open Mode (I)	I: normal stress
Stress-lifeStrain-life	Critical Plane, Maximum Shear (II + III)	II + III: max shear
Stress-lifeStrain-life	Full 3D Critical Plane (Normal)	I: normal stress

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