Multiaxial Fatigue

Multiaxial fatigue can be observed within many structures which are used in everyday life.  The effects of fatigue have to be evaluated using adapted models which consider some specific mechanisms.  Engineers are often surprised to see the large number of criteria.  There are so many different models available in this area.  These models are not only different because of the different types of equations they present, but also because of their critical criteria.

Mutiaxial fatigue tests can be performed allowing various stress states, such as biaxial traction states, torsion traction states, to be tested.  Tests simultaneously involving several loading modes are said to be in-phase or proportional if the main components of the stress or strain tensor simultaneously and respectively reach their maximum and minimum and their directions remain constant.  If it is not the case, they are said to be off-phase or non-proportional.

Many researchers have attempted to reduce multiaxial stress/strain state to uniaxial one, which is used in fatigue life calculation.  Such uniaxial parameter is often called ‘equivalent’ and it means that the same fatigue life is obtained under uniaxial (‘equivalent’) and multiaxial stress/strain state.  The reduction is based on the multiaxial fatigue failure criterion. Numerous multiaxial fatigue failure criteria have been proposed in recent decades.

Critical Plane Approach

The critical plane approach assumes that the fatigue failure of the material is due to some stress or/and strain components acting on the critical plane.  It is based upon the experimental observation that in metallic materials fatigue cracks initiate and grow on certain planes.  In this approach, the aspect of microdamage or even short crack propagation are not considered.  The critical plane approach concerns the crack initiation process that is usually related to fatigue failure at high cycle fatigue regime. However, it was successfully used also at low cycle fatigue regime.

Critical plane criteria reduce the multiaxial state of stress/strain to the equivalent-uniaxial state;  this single damage parameter is used to calculate fatigue life or damage degree on a plane using the standard S-N curves.  Any phenomena regarding to the crack propagation are not considered.  Fatigue life to crack initiation is estimated.


Findley Criterion

max(τa,n + kσn) = f

where τa,n is the shear amplitude on a plane with a unit normal n, i.e. τa,n = ∆τn/2 and σn is the normal stress on lthe same plane.  The two material coefficients k and f can be determined from fatigue loading tests.

The critical plane orientation coincides with the plane orientation where the maximum value of this linear combination occurs. It depends on the material coefficient k. Findley noticed that k value was small for ductile materials and the position of the critical plane for these materials approached to the direction of maximum shear stress.  A high k value is characteristic for brittle materials like cast iron, and the critical plane position is then compatible with the position of maximum principal stress direction σ1. Findley did not define a mathematical formula for the material coefficient f.  Some researchers assume that it can be determined from the shear-mode cracking.

McDiarmid Criterion

In the McDiarmid criterion, the critical plane is a plane with the maximum shear stress amplitude τmax.  To calculate the limit state, besides τmax, the effect of normal stress in the same plane σmax is considered. The mathematical criterion can take the following form

τmax / (τaf A,B) + σmax /(2σu) = 1

where τaf A and τaf B are torsion fatigue limits, for the case of an increase in cracking type A or B, and σu is a monotonic tensile strength. By transforming the formula above, we obtain a relationship defining the equivalent stress can be obtained:

σMD = τmax + kσmax ≤ τaf A,B


k = τaf A,B /(2σu)

Matake Criterion

According to Matake, the critical plane is one of two planes of maximum shear stress τns,a with a higher value of normal stress σn,max

τns,a + kσn,max ≤ τaf

Under this approach, there is only one coefficient in the criterion, which is determined using the following formula

k = 2 τaf / σaf − 1

Dang Van Criterion

Dang Van assumes that fatigue damage occurs with some grains which are not correctly oriented within the representative volume, which corresponds to some grains going through a plasticity phenomenon whereas the macroscopic stress does not reaches elasticity limit and mainly remains in a elastic strain state.  He proposes a criterion which is based on a meso-macro switch, considering the stress distribution within the grains which get plasticized.  Dang Van’s criterion is based on a two-scale approach.  In metals the fatigue crack initiates at the grain level in the persistent slip bands (PSB) due to alternating shear stresses.  The tensile micro-scale hydrostatic stress will open the crack and accelerate its growth along the PSB.  The macroscale stresses which appear in the macroscale equilibrium equations are denoted as S and the micro-scale stresses appearing in the grains are denoted as σ.  The criterion for fatigue is postulated in the micro-scale as

τ (t) + aσh(t) = b

where τ (t) and σh(t) are the instantaneous shear and hydrostatic stresses on the slip band.  If τ (t) + aσh(t) < b, the structure will elastically shake-down to the applied loading.

The macroscale stress and microscale stress are related as

σ(t) = S(t) + dev ρ

where ρ is the stabilized residual stress tensor; dev = deviatoric part of.  During the elastic shakedown the microscale yield surface will expand and move according to isotropic and kinematic hardening laws and the stabilized residual stress tensor ρ designates the distance from the center of the yield surface from the origin. The state of shakedown is found if the stress path is completely inside the expanded and displaced yield surface.


Dang Van Fatigue Failure Criterion in Micro-stress Meridian Plane

Kandil-Brown-Miller Criterion with Wang-Brown Modification

In 1973 Brown and Miller observed that fatigue cracks started and propagated on the maximum shearing plane while being helped by the stress normal to the plane.  In 1982 the Kandil-Brown-Miller criterion has been developed adopting a critical plane approach, justified by experimental observations of the nucleation and growth of cracks during loading: depending on the material, stress state, environment, and strain amplitude, fatigue life is usually dominated by crack growth along either shear planes or tensile planes.  Kandil, Brown and Miller proposed that both the cyclic shear and normal strain on the plane of maximum shear must be considered due to the fact that cyclic shear strains help to nucleate cracks, while normal strains contribute in their growth.

Referring to the Mohr circles, the maximum shear strain amplitude Δϒmax and the normal strain rate Δεn are

Δϒmax /2 = (ε1 – ε3 )/2


Δεn = (ε1 + ε3 )/2

where ε1 and ε3 are respectively the maximum and the minimum principal strain. Since ε3 = – νε1, the maximum shear strain amplitude and the normal strain rate can be also written as

Δϒmax /2 = (ε1 – ε3 )/2 = ε1 (1+ ν) /2


Δεn = (ε1 + ε3 )/2 = ε1 (1 – ν) /2

Applying Basquin-Manson-Coffin criterion,

εa,eq = C1σ’f (2Nf)b/E + C2ε’f (2Nf)c

For most of the common metallic materials in the elastic field, the Poisson coefficient can be considered equal to 0.3, so the strain components under investigation assume the form of

Δϒmax = ε1 (1+ ν) = 1.3 ε1


Δεn = ε1 (1 – ν) /2 = 0.35 ε1

From these considerations it follows that C1 = 1.3 + 0.35 = 1.65 and, through the same methodology adopted for the elastic field, the parameter associated to the plastic field (ν = 0.5) becomes C2 = 1.75. The final equation to be used in for life estimation according to Kandil-Brown-Miller is then

Δϒmax /2 + Δεn /2 = 1.65 σ’f (2Nf)b/E + 1.75 ε’f (2Nf)c

The equation is modified by Wang-Brown as

Δϒmax /2 + Δε*n /2 = 1.65 σ’f (2Nf)b/E + 1.75 ε’f (2Nf)c

where the normal strain excursion

Δε*n = max (εn (t))  – min(εn (t)) = εn,max – εn,min

tA < t < tB

The critical plane is the plane with maximum fatigue damage.

Socie-Fatemi Criterion

Socie et al. observing fatigue fractures came into conclusion similar to those by Brown and Miller, that is, the normal strain εn in the plane of maximum shear strain accelerates the fatigue damage process through crack opening.  Crack opening (by maximum normal stress) decreases the friction force between slip planes.

This model was born as a modification of the Brown-Miller’s critical plane model, considering mainly the maximum shear strain amplitude and the maximum normal stress on the maximum shear strain amplitude plane. Considering the normal stress also the crack closure effects which may derived from the application of cyclic loadings can be taken into account. The fatigue parameter to be used in life prediction is then defined by

a,eq = Δϒmax (1 + n σn,max / σy )

where σn,max is the maximum normal strain on the maximum shear strain amplitude plane, σy is the yield stress, n is a material constant which can be found by fitting the uniaxial experimental data against the pure torsion data. In order to use this fatigue parameters, the shear strain formulation of the Basquin-Manson-Coffin equation has to be used, in which it can be applied the axial and shear properties τ’f = σ’f /30.5,  bϒ = b, ϒ’f = ε’f /30.5,  and cϒ = c,

Δϒmax (1 + n σn,maxy )/2 = τ’f (2Nf)b/G + ϒ’f (2Nf)c

The  critical plane is the plane experiencing the maximum value of shear strain amplitude ϒmax.