Random non-proportional fatigue tests with planar triaxial fatigue testing machine

ABSTRACT. Complex stresses, which occur on the mechanical surfaces of transport machinery in service, bring a drastic degradation in fatigue life. However, it is hard to reproduce such complex stress states for evaluating the fatigue life with conventional multiaxial fatigue machines. We have developed a fatigue testing machine that enables reproduction of such complex stresses. The testing machine can reproduce arbitrary in-plane stress states by applying three independent loads to the test specimen using actuators which apply loads in the 0, 45, and 90 degree directions. The reproduction was tested with complex stress data obtained from the actual operation of transport machinery. As a result, it was found that the reproduced stress corresponded to the measured stress with an error range of less than 10 %. Then, we made a comparison between measured fatigue lives under random non-proportional loading conditions and predicted fatigue lives. It was found that predicted fatigue lives with cr, stress on critical plane, were over a factor of 10 against measured fatigue lives. On the other hand, predicted fatigue lives with ma, stress in consideration of a non-proportional level evaluated by using amplitude and direction of principal stress, were within a factor of 3 against measured fatigue lives.


Focussed on Multiaxial Fatigue and Fracture
Random non-proportional fatigue tests with planar tri-axial fatigue testing machine INTRODUCTION he main objective of this study is to develop a fatigue testing machine that enables reproduction of complex stresses occurring on the mechanical surfaces of transport machinery in service such as automobiles and railway vehicles.It is well-known that non-proportional loading generating such complex stresses brings a drastic degradation in fatigue life [1,2].Methods for prediction of fatigue life in consideration of life degradation have been proposed using fatigue testing results with two types of fatigue testing machines, a planar cross axial testing machine and an axial and torsion testing T machine [3,4].However, it is hard to reproduce such complex stress states with conventional multiaxial fatigue testing machines.In this study, we have developed a fatigue testing machine that enables generation of arbitrary in-plane stress states which occur on a mechanical surface where fatigue crack initiates.Also, fatigue tests under random non-proportional loading conditions were performed with the advanced tasting machine.

Testing Machine
he advanced testing machine has three structural features.First, the advanced testing machine consists of mechanical structures for generating three independent loads in the 0, 45, and 90 degree directions using actuators which apply loads.For generating arbitrary in-plane stress states, namely, x, y and xy in the xy plane, on the test specimen, three independent loads in the same plane are required.Stress states generated by conventional multiaxial fatigue testing machines are shown in Fig. 1.  1 ,  2 , and  1 in Fig. 1 are the maximum principal stress, the minimum principal stress, and the maximum principal stress direction in the xy plane, respectively.Since conventional machines consist of mechanical structures for applying two independent loads to the test specimen, stress states to be generated are limited. 1 generated by the planar cross axial testing machine is limited to 0 or 90 degrees as shown in Fig. 1 (a).When the axial stress direction is x,  y generated by the axial and torsion testing machine is limited to 0 as shown in Fig. 1 (b).On the other hand, in the advanced testing machine, the structure for applying the load in the 45 degree direction is added to the planar cross axial fatigue testing machine. 1 ,  2 , and  1 generated by the advanced testing machine are therefore described in the following expression: where 0, 90, and 45 are stresses generated by loads in the 0, 90, and 45 degree directions, respectively.The limit of 1 is eliminated for the advanced testing machine.In other words, this testing machine could generate arbitrary in-plane stress states.T Second, the advanced testing machine generates loads with mechanical structures each of which includes an actuator on one side and a reaction wall on the other side.These structures enable simplification of actuator control.Third, the advanced testing machine consists of a fixed frame which forms a 45 degree test axis and two movable frames which form 0 and 90 degree test axes.These structures enable prevention of a misalignment between the center of the specimen being shifted by loading to one direction and each loading axis.As a result, the prevention of misalignment can avoid a bending stress and a fracture of the test specimen by that bending stress.Fig. 2 and Tab. 1 show the advanced testing machine structures and specifications.Two frames, each of which has an actuator and a reaction wall, are installed on the fixed frame trough linear motion guides.An electro-hydraulic control system is installed in the testing machine.This testing machine was manufactured by Shimadzu Corporation.

Test Specimen
Two types of specimens made of SS400 were used as shown in Fig. 3.One was designed as a cylindrical projection on the center of the specimen as shown in Fig. 3

Method for Reproducing Complex Stress
When reproducing complex stresses or strains with the advanced testing machine, a stress or strain matrix is composed of stress or strain states that are generated by applying a unit load to each axis.In the elastic region, a stress in i direction (i) is described in the following expression using a force in j direction (L j ) and a stress in i direction generated by applying a unit load in j direction (Si,j): The inverse matrix in Eq. ( 4) is calculated by either measurement results using a rosette gauge by applying a unit load to each axis or numerical results using a FE model with the constraint conditions of the testing machine as shown in Fig. 4.

Reproducibility of Complex Stress
omplex stress states were reproduced in the center of the flat type test specimen as shown in Fig 3 (c) by the advanced testing machine.The stress states to be reproduced had been measured using a rosette gauge on the mechanical surface of actual transport machinery in operation.The reproduced stress was compared with the measured stress in Fig. 5.It was found that the reproduced stress history was in good agreement with the measured stress.The errors in each peak stress were within 10 %.

Fatigue testing results under non-proportional loading conditions
We made a comparison between measured fatigue lives and predicted fatigue lives under random non-proportional loading conditions.The fatigue tests were conducted by applying one random waveform for 240 seconds repeatedly.To execute the nonproportional test, the waveform having different amplitude and phase was generated in each axis.
For predicting fatigue lives, two types of stresses,  cr (Eq.( 5)) calculated by critical plane method [5] and  ma (Eq.( 6)) in consideration of the non-proportional level calculated by amplitude and direction of principal stress, were used.
where  cr is the angle between the direction perpendicular to the critical plane and x axis, and  is the material constant expressing the influence of non-proportional loading.f ma is the parameter expressing the intensity of non-proportional loading and described as: where  I(t) is the maximum absolute value of principal stress at time t, and  I(t) is the direction of  I(t) .Each stress is calculated by numerical results using the FE model shown in Fig. 4. Predicted fatigue lives were calculated using stress cycle counting, i.e.Rainflow method, and linear cumulative damage rule based on the modified Miner's rule.
It was found that predicted fatigue lives with  cr were over a factor of 10 against measured fatigue lives as shown in Fig. 6.On the other hand, predicted fatigue lives with  ma were within a factor of 3 against measured fatigue lives.It can be noticed that the errors of predicted fatigue lives using ma are caused by the relationship between maximum principal stress and minimum principal stress.The minimum principal stress in condition A (the most conservative as shown in Fig. 6) has the opposite sign as the maximum principal stress.On the other hand, the minimum principal stress in condition B (the most non-conservative as shown in Fig. 6) has the same sign.When the maximum principal stress is tensile, the compressive minimum principal stress has a promoting effect on fatigue crack initiation and propagation.On the other hand, the tensile minimum principal stress has a suppressing effect.The stress states in condition B can only be reproduced in the advanced testing machine; therefore, the advanced testing machine is effective in developing the method for high-accuracy prediction of fatigue life in consideration of the minimum principal stress effect.
a) Planar cross axial testing machine.(b) Axial and torsion testing machine.

Figure 1 :
Figure 1: Stress states under Mohr's stress circle generated with each testing machine.
Magnification view of A.
(b).The other was flat shaped as shown in Fig.3 (c).The chemical composition and mechanical properties of SS400 are shown in Tabs. 2 and 3. A-A' cross section view of projection type.(c)A-A' cross section view of flat type.

Figure 4 :
Figure 4: FE model with constraint conditions of testing machine.

Figure 5 :
Figure 5: Comparison between reproduced and measured stresses.

Table 1 :
Specifications of planar tri-axial fatigue testing machine.
Accordingly, loading conditions ([L j ]) for generating arbitrary in-plane stress states ([ i ]) generated at the evaluation point of the test specimen are stated as: