Presentation of the Variance Method

Fatigue failure is caused by the normal stress σ n ( t ) and the tangential stress τ n ( t ) which acts in the tangential direction s → of the fracture plane with the normal n → (Figure 1). Based on this knowledge, Macha, E. and Niesłony [1] developed a method for determining the position of the critical plane based on the variance of the equivalent stress.

Figure 1.Direction of vector, n → and s → and of stress, σ n ( t ) and τ n s ( t ) in the case of any position of the fatigue fracture plane in relation to the system of axes x, y, z [2][3].

With:

The components of the stress vector have the following expressions:

The general formulation of the criterion is:

where:

F, B are the criterion constants;

σ − 1 and τ − 1 are the fatigue limits under tension-compression and torsion loading, respectively.

The average of the director cosines must satisfy the following orthogonality conditions:

General Formalism

The proposal consists of introducing the influence of the stress gradient into the criterion of the statistical approach. The formalism of the modified EWALD Macha criterion is defined by the following expression:

With:

F and B the criterion constants; K = ( σ − 1 τ − 1 − σ − 1 ) 2 − 1 .

σ − 1 : Tensile-compression fatigue limit and τ − 1 : Torsional fatigue limit;

τ η s ( t ) et τ η s ( t ) , are functions of the constraint components σ i j ( t ) ( i , j = x , y , z ) .

-G(t), geometric gradient of stress normal σ η ( t ) to the physical plane with normal h, is defined at time t by [4]:

Case c) criterion for combining maximum tangential and normal stresses in a fracture plane[3].

In this particular case of the criterion (B = 1), the expression for the equivalent stress is

σ i j ( t ) ( i , j = x , y , z ) are the stress components.

The variance of the equivalent stress is written as:

With μ x s t the variance-covariance matrix of the variable x k :

Generally, the analytical resolution of this method poses a problem. To eliminate this difficulty, the cosine directors l ^ n , m ^ n , n ^ n are replaced by the trigonometric functions of the three Euler angles ψ, θ, φ (Figure 2).

Figure 2.The trigonometric functions of Euler’s three angles [3].

Considering that the problem is plane and that the facets concerned are those with the normal in the plane ( i → , j → ) , we have the condition k → ⋅ h → = n ^ 3 = 0 .

This consideration allows us to obtain the conditions on the Euler angles with θ = π / 2 .

With θ = π / 2 , the condition, and the cases of rotations (Figure 2), we have the following matrix of cosine directions:

Statistical Method for Estimating Lifespan

There is a point of convergence with the previous method, which is the determination of the counting variable, but the difference lies in the formulation of the fatigue criterion and also the damage law. We would like to note that the Miner rule does not consider stresses below the endurance limit of the material defined experimentally. For the effect of fatigue is cyclic (i.e. For any stress above or below the endurance limit, cyclic application could influence fatigue performance).

However, the statistical approach takes this notion into consideration by introducing a coefficient 0 ⊲ α ≤ 1 . The simplified algorithm of this method is presented in Figure 3.

Figure 3.Algorithme de calcul de durée de vie par la méthode statistique.

In practice, the analysis of random loads is done using counting cycle algorithms. Random loading is composed of different mean and alternating values. Nowadays, the Rainflow algorithm is the most widely used in calculating the fatigue life of materials, which allows us to obtain a certain number of loading cycles and half-cycles with different mean values, and applied stress amplitudes during damage accumulation; it is applied to the uniaxial stress state. In random loading, the most commonly used assumption is a linear Palmgren-Miner hypothesis, where the sum of damages is proportional to the number of loading cycles. In this accumulation process, only the stress amplitudes above the endurance limit are taken into account. However, in the definition of the fatigue phenomenon, it is observed that even amplitudes below the fatigue limit can influence the material’s fatigue performance. To address this problem, Ewal Macha and Adam Nieslony (2012) [1] introduced a coefficient, which leads to:

Avec:

T 0 —calculation time for the equivalent stress,

n i —the number of cycles with stress amplitudes to—the number of cycles with stress amplitudes to determined by the rain-flow method,

S ( T 0 ) —degree of damage to,

m—exponent of the S-N curve (Wöhler),

σ a f —fatigue limit,

N f —number of cycles corresponding to the fatigue limit,

a P M coefficient taking into account the influence of stresses below the endurance limit σ a f (in this work, we consider a P M = 0.5 ).

After determining the lifespan, we proceed to calculate the fatigue lifespan defined by:

The tests were conducted on cruciform specimens made of 10HNAP steel, the chemical composition of which is summarized in the following table:

The dimensions and shapes of the specimens are specified inFigure 4 [5].

The diagram of the test bench is shown in Figure 5. Two pairs of orthogonal cylinders allow for biaxial tensile-compressive loading of the specimens. The specimens are tapered in their central part. The deformation measurements are carried out using 45˚ rosettes (loading directions, which are the two axs of symmetry of the test specimens).

Figure 4.Geometry of the steel specimens used by E. Macha.

Figure 5.Test bench for cruciform specimens.

For the variable amplitude biaxial loading: HS: hydraulic unit, SM: servomotor, SV: servovalve, F: rigid frame, CS: cruciform specimen, uv: displacements, Ix Iy: control εij: relative deformations, MS: acquisition system, MFDC: acquisition software, GRS: function generator, Cont.: Control system.

The transition from deformations to stresses is carried out using the behavior laws of the elastic domain. The presence of a three-gauge rosette instead of a rosette containing only two, which is strictly necessary since the principal stress directions are fixed and known (these are the two axs of the loads), is simply intended to verify the absence of material distortion and thus tangential stress. This can also help verify the correct orientation of the strain gage directions (during their bonding) according to the axs of the specimens. The static characteristics of the 10HNAP steel are summarized in the following table.

Table 1 presents the results obtained in terms of sequential life at the crack initiation of GP9310 and GP9313 from Macha tests made available to us [8].

Table 1. Fatigue life comparaison.

The fatigue life obtained by the “statistic + gradient” approach is close to reality with a ratio of 1.14 and 1.21 respectively for the GP9310 and GP9313 sequences, compared to that obtained by [8] and the simple statistical approach.

In a company, the maintenance department is at the heart of the financial budget. However, the higher the maintenance cost, the more the financial aspect of the company will be affected. That being said, we found that the lifespan obtained by [8] is conservative because it is shorter compared to the experimental one, but in terms of maintenance cost, it is not good because according to the reports, we will perform maintenance two times before reaching the experimental lifespan and we will discard the part that could still be useful. On the other hand, our determined lifespan is closer to the experimental one. This favors the maintenance cost. Finally, we advise engineers to consider our proposal in their sizing.