Including the normal to shear stresses ratio in fatigue life estimation for cyclic loadings

The paper presents the estimation of the fatigue life under multiaxial cyclic loading of two construction materials. The main aim of this paper is to present a new method which allows evaluation of fatigue life during the design and construction phase of machine elements. In paper three well known multiaxial fatigue criteria based on the critical plane approach verified. This paper contains a proposition to define a new way of determining an orientation angle of the critical plane. The comparison between experimental and theoretical results varying the critical plane orientation appears to be satisfactory.


Introduction
The practical aspect of more than 170 years' history of fatigue phenomenon research is the proposed methods of estimating the fatigue life of materials or structures. Their main objective is to predict the time of the reliable and safe operation of a machine or device already at the construction or design stage. The structural elements of machines and devices are subject to loads varied in terms of their properties. These are generally multi-axis load states. The complex nature of the occurring fatigue processes resulted in a large group of fatigue criteria constituting the basic tool in the estimation algorithm for fatigue life. These hypotheses reduce the spatial tension state to the equivalent single-axis tension state. The multi-axial fatigue criteria arising from the above hypotheses may be divided due to the physical nature of a parameter deciding on destruction into: stress, strain and energy (stressstrain). Stress criteria constitute one of the largest groups of fatigue criteria taken from the classic theories of damage. During their adaptation, a set of criteria was established based on the critical plain concept. This group of criteria will be the subject of considerations in this study because criteria based on tensions are preferred by engineers when calculating the fatigue life of structural elements.
The main aim of this paper is present a new method which allows evaluation of fatigue life during the design and construction phase of machine elements.
In paper three well known multiaxial fatigue criteria based on the critical plane approach are verified: Carpinteri-Spagnoli criterion [1], modified C-S criterion [2] and one proposed by the author [3]. The proposed model has been verified based on the results of own and literature experimental studies, obtaining satisfying results.

Fatigue strength evaluation
Generally, the estimation of fatigue strength consists of several stages (Fig.1). The first step includes measurement, generation or calculation of the stress tensor components according to the following equations, in the case of biaxial fatigue (for example, cyclic bending and torsion): (2) where σxx(t) refers to stress induced by bending, and τxy(t) refers to torsion-induced stress. Further: σa -amplitude of normal stress induced by bending; τa -amplitude of shear stress induced by torsion; ω -pulsation; φ -phase shift; t-time. The input data within the model presented are tension values. This study uses normal tension amplitudes σa for calculations, from bending and the shear stress amplitude τa from torsion.

Fig. 1. A scheme for estimating fatigue life under cyclic loading
The critical plain concept is strictly related to the reduction of the multi-axis state to the single-axis equivalent [4]. The course of normal stress turned at angle α with respect to σxx is given by the formula Whereas, the course of tangential stress was written as: where: αη -maximum angle determined by normal stresses In this study, three expressions determining the angle β were used for considerations. First is the angle proposed by Carpinteri (βC), which is defined in relation to the direction determined by the maximum in the normal direction [1] where B2 is the fatigue limit ratio expressed by means of the formula It has been assumed that the ratio of the of fatigue limits corresponds to the stresses for Nf = 2,000,000 cycles. The second expression (βmod) is a modified form of equation (6), namely the calculation of the critical plane orientation angle the ratio of normal to shear stress amplitude is used instead of the fatigue limits.
The third expression (see Eq. 9) is a new expression for angle β, which is used for determining the critical plain orientation angle. The pre-assumptions concerned the following: -the consideration of the ratio value of normal to shear stress amplitude in the function of cycle number instead of the ratio of fatigue limits; -including values lower than 1 and higher than √3; -the use of ctg function, which, in the first calculations, was almost a perfect match for the analysed data.
The reduction of the multi-axis load state to the single-axis equivalent state is another essential step in estimating the fatigue life. The calculations involved the modified criterion in the plain of maximum normal and shear stresses. This is a new criterion based on the assumptions considering pure load states.

Experimental research and analysis of literature data
This publication presents a description of experimental studies along with an analysis of their results for 16Mo3 steel. Experimental tests for simple load states, as well as for a combination of proportional bending and torsion, were carried out on the MZGS-100 fatigue machine by the Achtelik project (Fig. 2). The MZGS-100 test stand consists of a drive system, head, loading system and control-measurement system. The parameter controlled during tests was the total moment course Mc loading the sample. The required sample load was obtained through the proper balancing of rotating discs and setting a lever turning angle.

Fig. 2. MZGS-100 fatigue machine
Steel 16Mo3 is boiler steel, these are creep-resistance grades, maintaining good ductility and malleability -in hot and in cold conditions. They are perfect in treatment and they guarantee welding simplicity with the use of conventional methods and welding materials. Owing to the increased presence of molybdenum (Mo), they are applied in the conditions of higher temperature. Whereas, the presence of chrome (Cr) makes steel resistant to corrosion in the environment of water steam. The chemical composition of steel 16Mo3 is presented in Table  1, and strength properties are shown in  Smooth samples with the round "diabolo" type section (Fig. 3) without a geometrical notch were used in fatigue tests.

Fig. 3. The geometry of samples used in tests
The range of fatigue tests included: bending τa = 0 ,torsion σa = 0 , bending with torsion τa = 0.5 σa. The results of fatigue tests using the S-N fatigue diagram in a double logarithmic system was presented in fig. 4 and 5. The second material analyzed in this work is GTS45 cast iron, the description and results of the research were taken from the ref. [7]. Cast iron is a casting alloy of iron with carbon and other components containing from 2 to 3.6% of carbon in the form of iron carbide or graphite. Cast irons, GTS45 was selected for the analysis of fatigue tests results. The geometries of samples are presented in Fig. 6. Chemical composition is presented in Table 3, and strength properties are specified in Table 4. Regression equation coefficients according to ASTM norm for the respective load variants are presented in Table 5.   Fig. 6. The geometry of samples used in tests [7]

Verification of the proposed multiaxial criterion
The

Summary
On the basis of the literature overview, the analysis of the experiment results and literature results as well as the performed calculations, the following conclusions were drawn: 1. A new proposal of angle β made by the author is based on the ratio of normal and shear stresses and it allows a wider range of that ratio than <1;√3>. It is crucial with the analysis of materials with non-perpendicular characteristics. 2. Carpinteri's proposal was modified by the author through the replacement of the ratio of fatigue life limits with the value of the ratio of normal to shear stress for a given number of cycles. 3. After the analysis of the results of comparing the calculated fatigue life values with the experimental ones, it may be stated that the proposed algorithm of estimating fatigue life provides satisfactory outcomes for multi-axis cyclic loads what is shown by the average dispersion equalling 3.00. 4. In the future, further analysis of the proposed model for subsequent construction materials will be carried out