2. Foreplay

In this part we deal with the concepts of endurance limits and the accumulation of damage. We also present the general form of fatigue criteria.

2.1. Endurance limit and accumulation of damage, uniaxial case

In the uniaxial case, the rigorous definition of the endurance threshold is the half-amplitude (half of the variation) of load defined under stress below which the lifespan is infinite. However, since in practice the lifespan can never be infinite, endurance limits are defined at \({10}^{7}\), \({10}^{8}\), etc. load cycles. There is another way of looking at it: since in practice there is no such thing as an infinite lifespan, we use the concept of accumulation of damage. The cumulative damage approach consists in defining a limit in the number of cycles clear of which the cumulative damage is equal to one. So the limit at \({10}^{7}\) means that after \({10}^{7}\) cycles the cumulative damage is equal to 1.

2.2. Fatigue criterion, multiaxial case

In the literature, a number of criteria have been proposed to define the endurance threshold under multiaxial cyclic loading. The general form of these criteria is:

\({\text{VAR}}_{\text{amplitude}}+a\mathrm{\times }{\text{VAR}}_{\text{moyenne}}<b\) eq 2.2-1

where \(b\) is the endurance threshold under simple shear, \(a\) is a positive dimensionless constant. \({\text{VAR}}_{\text{amplitude}}\) is a certain definition of the half-amplitude (half of the variation) of the cycle and \({\text{VAR}}_{\text{moyenne}}\) is a variable in relation to the average stress (or sometimes the deformation) or the stresses (or sometimes the deformations). Different models are distinguished by different definitions of \({\text{VAR}}_{\text{amplitude}}\) and \({\text{VAR}}_{\text{moyenne}}\).

To go from endurance to accumulation of damage, we can define an equivalent stress (or deformation):

\({\sigma }_{\text{eq}}\mathrm{=}{\text{VAR}}_{\text{amplitude}}+a\mathrm{\times }{\text{VAR}}_{\text{moyenne}}\) eq 2.2-2

This equivalent stress gives us a unit damage on the fatigue curve. Since the second member of the inequality [éq 2.2-1] corresponds to the shear threshold, a shear fatigue curve is needed. But shear fatigue curves are rare because they are difficult to obtain, so an attempt is made to use fatigue curves in alternating tensile compression. To do this, it is necessary to be consistent at least in terms of the endurance threshold, i.e. multiply \({\sigma }_{\text{eq}}\) by a constant of the order of \(\sqrt{3}\) in order to be able to use the fatigue curve in traction. The value \(\sqrt{3}\) is the exact value for a Mises criterion, experimentally this coefficient is smaller than \(\sqrt{3}\).

2.3. Definition of a loading amplitude in the multiaxial case

In Code_Aster, there are two definitions of loading amplitude in the multiaxial case:

\(A\): radius (half diameter) of the sphere circumscribed to the path of the load;

\(B\): half of the maximum distance between any two points on the trip.

It is clear that in the case of a load defined on a sphere, \(A\) and \(B\) give the same amplitude. On the other hand, if we take a (two-dimensional) path in the form of an equilateral triangle with side \(l\), the definition \(A\) gives us \(l/\sqrt{3}\), while the definition \(B\) gives us \(l/2\). To work in a conservative framework we take as a definition of the amplitude (half-variation) of a loading path the radius of the circumscribed sphere (or circle for the 2D case).

2.4. Definition of the shear plane

At a point \(M\) of a continuous medium we express the stress tensor \(\sigma\) in an orthonormal coordinate system \((O,x,y,z)\). To the unit normal \(n\) of components \(({n}_{x},{n}_{y},{n}_{z})\) in the orthonormal coordinate system, we associate the stress vector \(F=\sigma \text{.}n\) with components \(({F}_{x},{F}_{y},{F}_{z})\). This vector \(F\) can be decomposed into a vector normal to \(n\) and a scalar carried by \(n\), i.e.:

\(F=Nn+\tau\) eq 2.4-1

where \(N\) represents normal stress and the vector \(\tau\) represents shear stress. In coordinate system \((O,x,y,z)\), the components of the vector \(\tau\) are noted: \(({\tau }_{x},{\tau }_{y},{\tau }_{z})\). The vector \(\tau\) can be deduced directly from [éq 2.4-1] and from the normal constraint:

\(N=F\text{.}n\) where \(\tau =F-F\text{.}nn\text{.}\) eq 2.4-2

Figure 2.4-a: Representation of stress vectors \(F\) and shear stress vectors \(\tau\)