Introduction to Multiaxial Fatigue

7/30/2019 Introduction to Multiaxial Fatigue

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Introduction to Multiaxial fatigue


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The essential elements of any fatigue analysis are:

Identification of the loading environment

Establishment of the relationship between applied loads and local stresses and/or strains

The relationship between local stresses/strains and fatigue damage

For multiaxial analysis, these three elements can be analyzed at various levels of detail, and

with the increasing detail comes a cost of analysis time and interpretation of results.Loads are applied simultaneously in several directions, producing stresses with no bias to a

particular direction. In 3D geometry, these stresses are called multiaxial.

For accurate fatigue damage calculation, one must identify multiaxial stresses and use

appropriate algorithms.

Stresses from loads applied at multiple locations and how they combine at particular or

critical locations need to be investigated carefully.

If we assume that fatigue damage initiates at the surface of a structure then any direct or

shear stresses normal to the surface are zero, and there are only two principal stresses

which may be non-zero.

Multiaxial fatigue


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Hence there are three basic cases:

Case 1 Uniaxial stress state there is one principal stress which is significantly larger than

the second for the whole of the load history and whose angle does not change.

Case 2 Proportional biaxial stress state the ratio of the two principal stresses is nonzero,

but remains constant for the duration of loading. Angle also remains constant.

Case 3 Non-proportional stress state either the biaxiality ratio or the angle of the

maximum principal changes significantly through the time history.

For case 1,

No special algorithm corrections need to be applied to convert elastic stresses and strains to

elastic-plastic. The Neuber correction, or similar methods, are sufficient.

For case 2,

Procedures should be used to take into account the fact that the loading is non uniaxial. Two

such procedures are due to Hoffmann & Seeger and Klann Tipton - Cordes.

For case 3,

A full multiaxial notch correction procedure should be used. So, results from highly multiaxial

elements with high levels of strain should be treated with caution.

Multiaxial fatigue


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Stress-based approaches,

Strain-based & Energy-based approaches,

Critical plane models,

Fracture Mechanics approach for crack growth.

Multiaxial fatigue life estimation methods


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Equivalent stress approach

Equivalent stress approaches are extensions of static yield criteria to fatigue.

The most commonly used equivalent stress approaches for fatigue are the maximum

principal stress theory, the maximum shear stress theory (or Tresca), and the octahedral

shear stress theory (or von Mises).

Since fatigue crack invariably occur at the surface of the material where the normal stress is

usually zero, the practical problem can be simplified to a biaxial system of stresses.

In general, both a mean stress and an alternating stress are applied and the problem

concerns fatigue failure under a biaxial system of constant principal stresses (1m , 2m ) and

a biaxial system of alternating principal stresses (1a , 2a ).

In most general case the mean and alternating principal stresses are not collinear so the

directions 1m, 2m do not coincide with the directions 1a, 2a.

Three of the most common criteria that have been postulated are shown below, applied to a

biaxial stress system of principal alternating nominal stresses 1a , 2a. Taking the uniaxial

stress (or equivalent nominal stress amplitude) at a particular life N as faN, the criteria may

be expressed in dimensionless form and are listed below:

Stress based approaches


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Maximum principal stress criteria:

Failure occurs at the life N when,

1a = faN (or) 2a = faN

Minimum shear stress criteria:


1a - 2a = faN

Octahedral shear stress criterion (von mises criterion):


(1a)2 + (2a)

2 - 1a 2a = (faN )2

where, 1a , 2a are principal alternating nominal stresses with 1a > 2a

After computing equivalent nominal stress faN, S N approach discussed earlier topics can be

used for fatigue life calculations by setting faN equal to the appropriate fatigue strength Sf

It was studied that from the above theories of failure, that the non-shear stress and the

octahedral shear stress agree more accurate (never differ by more than 16%)



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Sines method

Accepting that fatigue damage is essentially due to local slip reversals, the shear stress is a

logical criterion for failure under alternating stresses but the same reasoning does not apply

to the mean stress.

The theory proposed by the Sines is considered the best for fatigue failure under multi

axial alternating and mean stresses acting together.

Sines suggested that the effect of mean stress is determined by the normal forces on the

hear planes and is such that the life is reduced when the forces are tensile and increased

when they are compressive.

Under this criterion the failure life under the stress system (1a , 2a ), (1m , 2m )

corresponds to that under the uniaxial fatigue stress fm faN where,

fm = 1m + 2m

faN = (1a2 + 2a

2 - 1a 2a)1/2



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Fatigue material properties are typically based on uniaxial stresses

Real world stress states are usually multiaxial

This fatigue result gives the user some indication of the stress state over the model and

how to interpret the results

Biaxiality indication is defined as the smaller in magnitude principal stress divided by the

larger principal stress with the principal stress nearest zero ignored.

Stress Biaxiality indication values :

Biaxiality of zero corresponds to uniaxial stress

Biaxiality of 1 corresponds to pure shear

Biaxiality of 1 corresponds to a pure biaxial state

Stress biaxiality


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Strain-based approaches are used with the strain-life curve in situations where significant

plastic deformation can exist such as in low cycle fatigue or at notches.

Analogous to equivalent stress approaches, equivalent strain approaches have been used as

strain-based multiaxial fatigue criteria.

Similar to the equivalent stress approaches, equivalent strain approaches are also not

suitable for non - proportional multiaxial loading situations.

The most commonly used equivalent strain approaches are strain versions of the equivalent

stress models as follows:

Maximum principal strain theory:


1a = aN (or) 2a = aN

Minimum shear strain theory:


( 1a - 2a )/(1+) = aN

Strain based approaches


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Energy-based approaches use products of stress and strain to quantify fatigue damage.

Several energy quantities have been proposed for multiaxial fatigue, such as:

Plastic work per cycle as the parameter for life to crack nucleation.

Plastic work is calculated by integrating the product of stress times plastic strain

increment for each of the six components of stress.

The sum of the six integrals is the plastic work per cycle.

Application of this method to high cycle fatigue situations is difficult since plastic strains are

small.

Total strain energy density per cycle, consisting of both elastic and plastic energy density

terms.

Energy, however, is a scalar quantity and, therefore, does not reflect fatigue damage

nucleation and growth observed on specific planes.

Energy based approaches


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Experimental observations indicate that cracks nucleate and grow on specific planes (also

called critical planes).

Depending on the material and loading conditions, these planes are either maximum shear

planes or maximum tensile stress planes.

Multiaxial fatigue models relating fatigue damage to stresses and/or strains on these planes

are called critical plane models.

These models, therefore, not only can predict the fatigue life, but also the orientation of the

crack or failure plane.

Different damage parameters using stress, strain, or energy quantities have been used to

evaluate damage on the critical plane.

Critical plane model

Assumption:

Only stress or/and strain components acting on the critical plane are responsible for the

material fatigue failure.


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Fatigue calculations are post processing functions.

They are preceded by a linear or nonlinear finite-element analysis.

The fatigue solver imports FEA results involving multiaxial repeated loads, fluctuating loads,

and rapidly applied loads.

Cracks usually begin on a component's surface unless there is a flaw in the material. So,

fatigue solver uses only the surface elements and nodes from the results.

FEA Software must calculate surface-resolved stresses (plane stresses on a structure's

surface) to correctly calculate multiaxial relationships and assessments.

Shell models produce surface stresses by default. However, many solid models produce

stress results in coordinate systems that must be transformed into surface resolved

stresses. And it takes surface stresses to correctly calculate biaxiality ratios and perform

multiaxial assessments.

Rigid elements and bonded contacts are used at precise locations, to transfer loads as

realistic as possible.

Results from the fatigue analysis are stored in a database. They show that greatest damage

or shortest life appears.

Finite Element based Multiaxial fatigue

Introduction to Multiaxial Fatigue

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