Modeling of the stress-strain relationship for specimens made of S 355 J 0 steel subjected to bending block loading with mean load

The paper presents results of calculation for modelling of the stressstrain relationship in the case of block loads with mean load value. A model, based on the stable hysteresis loops, was assumed and modified to use for block loading analysis. For stress history calculation, the proposed model and two other constitutive models were used. Results of fatigue test of specimens made of S355J0 steel subjected to bending block loading with mean load value are presented and used to verify the proposed model. In the tests, the mean load was increased and decreased in subsequent blocks. The changes of strain recorded during the tests shown in the paper indicate a different behavior of the material. Damage accumulation degree for block sequence was used to compare the results of calculations. It was shown, that stress history parameters (stress amplitude and mean stress value in this case) are similar for all investigated models.


Focused on Mechanical Fatigue of Metals
Modeling of the stress-strain relationship for specimens made of S355J0 steel subjected to bending block loading with mean load INTRODUCTION bservation of the relation between strains of the structural materials and loading applied to the structure allow assessing the phenomena which occurs during fatigue of materials [1,2].For given material the value of load is a parameter which decides about strain-stress relationship.The area of the stress-strain hysteresis loop is identified with the plastic strain energy density.This energy parameter is widely used in so called energy based criteria to describe and predict the fatigue life of material, eg.[3,4] as well for unnotched and notched components eg.[5].In the case of complex load, during operation of the structures, some changes of the shape and location of the hysteresis loop are observed.It has to be considered in the algorithm of the fatigue life calculation, where the influence of the different load parameters on the strain-stress relationship is important.Using a proper combination of cyclic loading blocks it is possible to realize complex load histories as laboratory tests.Also the mean stress influence can be investigated -in this case, a constant component should be added to the generated blocks of variable amplitude loading.The load sequence can change the final fatigue life of materials and components, because damage accumulation is changed.The most important load sequences are these, which are strongly different with their parameters -amplitude and mean stress.Usually the Hi-Lo or Lo-Hi sequences are used [1].Studying papers on that problem, we can find that there is no standard procedure for fatigue life calculation and some researchers obtained completely opposite results [6].Most of those considerations were O based on experimental results which were obtained for different materials, loading conditions, applied stress levels.It is possible to describe a proper damage accumulation rule for such cases only.Fatemi and Yang [7] present a wide number of different cumulative fatigue damage and life prediction theories.They conclude that none of them is widely accepted.Each damage model can only account for one or several phenomenological factors, such as load dependence, multiple damage stages, nonlinear damage evolution, load sequence and interaction effects, over load effects, small amplitude cycles below the fatigue limit and mean stress.
The tests presented by Chiou and Yip [8] shows the effect of the mean load based on the example of research on AISI 316 alloy steel subjected to uniaxial loads of constant amplitude and the saw-toothed waveform.Curve parameters of cyclic strengthening were determined for a fixed value of the mean strain of εm= 0.1%; 0; 0.2%; 0.4% resulting in the following values: K'=745.594;722.571; 693.080; 587.699MPa and n'=0.1572;0.1507; 0.1424; 0.1170 [8], respectively.In the case of block loads involving different mean values in each load blocks, distinct changes of fatigue life are observed.The authors Memon at al. [9] shows the results of fatigue tests under conditions of two-stage block loads in Lo-Hi and Hi-Lo sequences (Fig. 1) for different sequences of amplitude and the mean stressess.The algorithm of fatigue life calculation, proposed by the authors, uses Lemaitre's model of kinematics hardening and also additional stress-strain analysis based on MES method.Final degree of damage accumulation was calculated with the use of linear Palmgren-Miner rule.The results present good agreement between calculations and tests in the cases, where amplitudes of the load in analyzed sequences were on similar level and maxi-mum stresses were near the yield limit.Bigger difference be-tween amplitudes of the loading in the block sequence strongly influences on the level of the degree of damage accumulation for Hi Lo and Lo-Hi cases.The effect of block loads on the fatigue life is shown by Pawliczek and Lachowicz [10], which have involved bending of specimens from S355J0 steel to certain maximum stress levels.In the case of loads at yield point, a difference in fatigue life has been observed depending on the direction of increase of the mean load, and fatigue life was twice lower if the highest mean load value occurred in the beginning of the load sequence.Similar tests were performed by Pawliczek [11].The paper describes the research on the relationship between stress and strain under bending block loads where the mean value of the load varied for each block segment.A significant impact of such a load on strains generated in the material, and consequently on the fatigue life, is shown.It can be seen, that analyze of the stress-strain relation is an important part of block loading fatigue tests where mean load is applied.Such investigations allow considering that effect and verifying the influence of the mean load on it.The aim of this paper is to create an algorithm for stress history parameters definition and its application for calculations of accumulation damage degree.

MODEL FOR STRESS HISTORY PARAMETERS DEFINITION
he model for stress amplitude and mean stress value calculation bases on the model of the hysteresis loop presented by Chiou and Yip [8].I was assumed that the shape of the stable hysteresis loop remains unchanged under the same strain amplitude and different mean strain level conditions.Additionally, the stable hysteresis loop shifts in accordance with the value of the mean strain.This results in symmetry of the hysteresis loop according to the point described by mean strain ε m and mean stress σ m (Fig. 2).
T Figure 2: Description of the stable hysteresis loop [7].
The parameters on the Fig. 2 are as follow: εmax, σmax -maximum strain and stress at point A respectively, εmin, σmin -minimum strain and stress at point B respectively, ε m , σ m -mean strain and stress in the point respectively, ε a , σ a -amplitude of strain and stress in the point respectively, ε Apl , ε Ael -plastic strain and elastic strain at point A respectively.
For situation presented in Fig. 2 the most conservative case exists.All calculations are performed for known strain history ε(t) registered during the test, so all strain parameters in Fig. 2 are available.Take into account geometry of the stress-strain relation based on the presented assumptions and using standard Ramberg-Osgood formula for its mathematical description, the strain amplitude ε a can be expressed as: where: σ max -maximum stress at point A, E -Young modulus, K' = K'(ε m ) -cyclic strength coefficient, n' = n'(ε m ) -cyclic fatigue exponent.Assuming additionally, that cyclic strength coefficient and cyclic fatigue exponent values depend on the actual value of the mean strain, it should be noted, that coefficients K' and n' are defined for actual mean strain ε m [8].Eq. ( 1) allows calculating stress at point A.
From hysteresis loop and Ramberg-Osgood relationship, we can use the amplitudes of elastic ε ael and plastic ε apl strains: Substituting elastic part in Eq.(2) using Eq.( 3), after transformation, the mean stress σm can be calculated from the following formula: This form of equation (Young modulus) suggests that mean stress is responsible for elastic part of deformation.However, the parameter ε apl in Eq.( 4) is still undefined.Also to solve this problem, the Ramberg-Osgood equation can be used.Bearing in mind, that the shape of the stable hysteresis loop does not change for the same strain amplitude and different mean strains we can calculate stress amplitude σ a using relation It is very important to use coefficients K' = K'(εm=0) and n' = n'(εm=0) obtained for standard, symmetric loading (εm=0 and σm=0) in the case of Eq.( 4).All the equations presented above are adequate for cyclic load with the mean load, where load conditions are constant during the test.However, it was mentioned before, for block loading location of the hysteresis loop can be different for each segment of block.Some researches of this problem [11] allows observing, that changing the men load during test a new, stable location of the hysteresis loop is registered.Strong flow of the mean strains along strain axis usually arises at the end of life of the specimen, so it can be assumed, that each, new position of the hysteresis loop stabilises at new levels of mean strain and stress (Fig. 3).After first block, where strain εm1 and stress σm1 exists a new position of the hysteresis loop is observed and described by parameters ε m2(measured) and σ m2 .Notation "measured" means mean strain of the measured strain history.It should be noted, that after first sequence some value of permanent deformation ε Apl1 exists in material after unloading (Fig. 3).This part of strains participates as the initial state of the material for next stage of loads.In the case, for which the second block will have zero mean load value it leads to situation, that presented algorithm will resulting with non-zero mean stress, what is not corresponds to the real stress conditions.The strain ε Apl1 as the "starting" point for each sequence must be taken into account in calculations.

THE ALGORITHM FOR STRESS HISTORY CALCULATION
aving regard to the considerations presented above the following algorithm for stress history calculation can be defined, where all the steps must be repeated for each sequence in the block load for whole registered time history of the strain: Input data: εmax εmin εa εm 2.
Calculation of σmax where K'(εm) and n'(εm) are defined for actual mean strain value:

7.
Calculation of the damage accumulation degree for given stresses (σa, σ m ) and the fatigue life can be predicted.
Additionally, in order to define the stress history parameters, some kinematic models of cyclic deformation were used.
Usually, those criteria base on the von Mises's plasticity surface as the yield criterion.For such calculation then the flow rule for a plastic strain increment calculation must be specified.Additionally, during computation with use of the constitutive equations it is important to define the way the plasticity surface is translated.Translation can be described by some hardening rules.During numerical tests it was found that the best results of calculations were obtained for Garud-Mroz [12] and Chu [13] models of the constitutive equations of the cyclic plasticity description.These cyclic plasticity models are used very often for energy based criteria for fatigue life estimation, eg.Lachowicz [14], to describe stress-strain state and to calculate energy dissipated in material.

FATIGUE TESTS
The fatigue tests conducted included subjecting the samples made of S355J0 steel to fatigue loads with including the mean load values.The Tab. 1 presents properties of the tested materials.The tests were performed under bending loads applied in blocks with different mean load values.Two paths of the change in the mean load value in blocks were applied (Fig. 4): a) mean load was increased from zero to maximum value for the following sequences in blocks of loading, b) mean load was decreased from maximum to zero value in successive blocks of loading.
The mean value of the load was as follow: 1: σ m = 0, 2: σ m = ⅓σ a , 3: σ m = σ a .The whole number of cycles N b in block was divided to three equal parts n 1 =n 2 =n 3 =⅓N b .The block applied in the tests had a length N b =15000 cycles.The view of the fatigue tests stand MZGS100 is presented in Fig. 5a.The stand allows realizing tests under cyclic bending with mean load value, which is applied by spring actuator.The applied loads and observed deformations (strains) of the specimens were measured during the tests.The strain bridge NI USB9237 and the author's software (Fig. 5b) were used.It was possible to observe on-line the time courses of loading (amplitude and mean value of the bending moment) and strains and additionally moment-strain curve was displayed.An example time histories of bending moment and corresponding strains are shown in Fig. 6.Then, the registered signals were used to define the changes of the moment-strain relation for both of applied block load paths.The results of fatigue test are presented in Fig. 7 and Fig. 8 for the load paths shown in Fig. 4a and Fig. 4b.
Figure 7: Hysteresis loops at the increasing mean load value.
In the case of Fig. 7 higher value of the moment amplitude in the first phase of fatigue testing resulted with strain amplitude ε a =0.3% after 2000 cycles.The increase in the mean value of load in each sequence resulted in a shift of the loop towards the average value of ε m =0.35%.When M a =M m =10N·m, the increase of the mean strain value to ε m =0.45% was observed but the variable strains are in the elastic range (curves 5 and 6 in Fig. 7).The maximum strain of εmax=0.6% and the mean strain ε m(max) =0.47% were recorded.For the case shown in Fig. 8 in the initial phase of the test the highest value of the mean moment was applied.In the case where Ma=15N·m and Mm=5 N·m (curves 3 and 4 in Fig. 8), there is no increase plastic strain as compared to the corresponding case (curves 3 and 4 in Fig. 7) with the increase of the mean load value.This may indicate the strengthening of the material caused by the mean load at the beginning of the block.The reduction in the mean moment value to zero in the last block enlarged the area of the hysteresis loop and a little shift of the hysteresis loop after 3000 load cycles is observed.At the same time, the mean load ε m =0.2% was recorded that remained after the strain (curves 5 and 6 in Fig. 5).The maximum strain of εmax=0.5% and the mean strain εm(max)=0.15%were recorded.

RESULTS OF CALCULATIONS
or registered histories of the strains the proposed algorithm and two kinematic models (Garud-Mroz and Chu) of cyclic deformation were used to calculate the stress history parameters: stress amplitude and mean stress value in this case.Figs. 9 and 10 presents hysteresis loops as stress-strain relationship calculated by the use of Garud-Mroz model.Form of the graphs corresponds to the measurement results presented in Fig. 8 and Fig. 9, respectively.

F
It can be seen, that for curves 5 and 6 of Fig. 9 some compression stresses reordered.The same effect is visible for curves 1 and 2 on the diagram for the case of decreased mean load (Fig. 10).The aim of performed calculations is to establish a parameters of the stress history (stress amplitude σa and mean stress σm in this case) for each sequence of the applied block loads respecting both paths of changes of the mean load using proposed model, Garud-Mroz proposal and Chu also.Results of these investigations are presented in the Fig. 11 for increased mean load value and in the Fig. 12 for decreased mean load value.Comparing all analyzed models it can be seen, that stress amplitude are very close to each other for all cases, however for mean stress the proposed model gives results with higher values: about 40% for 7000 and 10000 cycles (second sequence in block load) and 80% for 12000 and 15000 cycles (third sequence in block load).It will generate higher level of damages -then lower, predicted fatigue life will be calculated for such load sequence.In general, such observations are indicated in the research.

Figure 3 :
Figure 3: Location of the hysteresis loops for the next sequence in the block load.

Figure 4 :
Figure 4: Block loading applied in fatigue tests: a) for increasing mean load value, b) for decreasing mean load value.

Figure 5 :
Figure 5: Fatigue tests: a) the view of the fatigue tests stand, b) software's user interface.

Figure 6 :
Figure 6: Registered moment and strains history.

Figure 8 :
Figure 8: Hysteresis loops at the decreasing mean load value.

Figure 9 :
Figure 9: Stress-strain loops at the increasing mean load value calculated with the use of Garud-Mroz model.

Figure 10 :
Figure 10: Stress-strain loops at the decreasing mean load value calculated with the use of Garud-Mroz model.

Figure 11 :
Figure 11: Comparison of the stress history parameters calculated by proposed model and models of Garud-Mroz and Chu for increased mean load value.

Figure 12 :
Figure 12: Comparison of the stress history parameters calculated by proposed model and models of Garud-Mroz and Chu for decreased mean load value  a : proposed model  m : proposed model