ADINA analysis of large deflections and stresses in bending fatigue specimens

Computers dr .Wuctures Vol. 17. No. Printed in Gnat Britain.

Sd,

pp. 86S870,

0045-7949/83 Pcrgamon

1983

s3.00+ .m Press Ltd.

ADINA ANALYSIS OF LARGE DEFLECTIONS AND STRESSES IN BENDING FATIGUE SPECIMENS JAN-OLOF NILS~ON Steel Division Research and Development Centre, S-81 1 81 Sandviken, Sweden

and MICHAEL HEHENBERGER Sandvik Coromant Research Centre, S-126 12 Stockholm, Sweden Abstract-Laboratory tests of flapper valve steel bending fatigue specimens are compared to large deflection ADINA analyses based on the nine-node shell element. Three standard thicknesses, l/l00 in. (0.254 mm), 3/200 in. (0.381 mm) and l/50 in. (0.508 mm), are investigated. Both the longitudinal and the anticlastic curvatures could be brought into good agreement with experiment through proper modelling of the grip section by bakelite brick elements in sandwich form. The calculated stress maxima are observed in the specimen edge region. This result is consistent with observations that fracture initiation usually occurs in that region.

1. INTRODUCTION A basic requirement on flapper valve steel is a high fatigue strength. In order to study the cause and position of fracture initiation, test specimens are blanked out of hardened and tempered steel strip and subjected to bending in a specially designed fatigue test apparatus[l]. Figure 1 shows the specimen geometry which has emerged from years of empirical studies and was designed to yield an even stress distribution in the longitudinal direction. In a previous study[2] the effect of defects like pits, gouges and oxide inclusions on bending fatigue frac-

tures were investigated. Since such defects can be assumed to be distributed uniformly over the specimen surface, it was proposed that the high frequency of initiation points in the edge region (see Fig. 2) is caused by stress enhancements due to edge deflections. It was also pointed out that due to the nature of the bending fatigue test procedure it is necessary to use advanced non-linear finite element methods for the analysis of deflections and accompanying stresses. For thicknesses of 0.254, 0.381 and 0.508 mm and deflections of up to about 10 mm, it is impossible to obtain reliable results from linear analysis. A recent analysis[3] showed that the FEMprogram ADINA[4] was a suitable tool for the present study due to its capability to handle geometric non-linearities in shell and plate problems. By employing this program we hoped to provide a quantitative assessment of both longitudinal and transverse curvatures[5] as well as associated stresses in the critical crack initiation region. Since these theoretical results could also be checked by means of available experimental data, they could be used to firmly support earlier suspicions[2] and further to produce reliable S-N (stress vs number of fatigue cycles to fracture) curves for the steel material considered. 2.

FINITE

ELEMENT

MODELLING

In order to make the computer simulation as realistic as possible, an elastic boundary condition

was used corresponding to bakelite plates of 0.2 mm thickness on either side of the specimen in the grip section. These plates are used in all bending fatigue tests to avoid stress concentrations and a resulting premature failure at the grip edge. Values of 0.35 for Poisson’s ratio and 5. lo3 MPa for the elastic modulus were assumed for bakelite.

I

I

I/

v

Y

I

-I

Fig. 1. Drawing of specimen geometry. The force is applied at section I. Transverse deflections and stresses are computed for section II.

Fig. 2. Fatigue fracture of the bending fatigue specimen considered. The crack initiated at a small indentation at the edge. 865

J.-O.

866

NIL.WN

and M.

The specimen itself was treated as an elastic continuum with the corresponding values of 0.3 and 2.1 . 10’ MPa respectively. An implication of this is that no effects of e.g. residual stresses are accounted for. The finite element mesh consisting of 27 brick elements, 3 shell transition elements and 33 nine-node isoparametric shell elements (6) is shown in Fig. 3. Here we have taken advantage of the lateral symmetry of the specimen and confined the calculations to the right half. This gives a correct description of the situation provided the symmetry line is subjected to the boundary condition that ~-displacements are all zero. According to experimental practice the force is applied along a line corresponding to nodes 195-201 (indicated in Fig. 3). Furthermore the force is always perpendicular to the specimen. This was accomplished in the calculations by choosing appropriate values of the x- and z-components of the force. The force and the corresponding deflection angle were measured in situ in the testing device. The boundary condition corresponding to the situation in the grip was represented by 27 brick elements, 9 of which represent the grip section of the specimen and 18 representing the bakelite plates. Initially a linear analysis of the problem was attempted, but it soon became clear that this led to erroneous results. For instance, the radius of curvature of the specimen close to the grip became too small, and the concomitant stresses in this part beceme unrealistically large. Linear calculations can only be expected to yield reasonable results for small deformations and strains. However, in order to describe the displacement and accompanying considerable axial and transverse curvatures encountered during our fatigue tests, we had

HEHENBERGER

to include geometrically non-linear effects. This can be done in ADINA by using the total Lagrangian formalism[6] in which all static and kinematic variables are referred to the initial configuration, Significant deviations from the linear results appeared due to the up-dating of the stiffness matrix with equilibrium iterations after each of the 10 discrete steps in which the force was applied. To illustrate the computational complexity of the problem we might mention that 24 min. of IBM 3033 CPUtime were necessary to analyse the 0.381 mm fatigue specimen with element geometry as shown in Fig. 3. with 229 nodes and 843 degrees of freedom. Obviously, the thicker specimen (0.508 mm) required less, and the thinner specimen (0.254mm) more computer time. In fact, convergence could only be reached for the last by dividing the step size by a factor of ten. 3. RESULTS

Using the ADINA-PLOT program [7] together with PLOT79[8j it was possible to present the results in a rather convenient form. The ilIustrations in Fig. 4 are examples of the use of ADINA-PLOT, showing the deflection at various time steps. Here, as well as in all subsequent examples, the force is applied perpendicular to the specimen surface along the line defined by nodes 195201 (see Fig. 3), referred to below as “section I”. On the other hand, since crack initiation usually occurs in the edge area defined by elements 9, 12, 15, 18, 21 we define a second line, “section II”, along nodes 139-145 (see Fig. 3). As mentioned above, three different thicknesses (t = 0.254,0.381, 0.508 mm) are considered. The case t = 0.381, however, is the one investigated most

SECTION I

i,

SECTION II

Fig.

3.

Finite element mesh in our model. Nodes 195-201 refer to section I and nodes 139-145 refer to section II.

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ADINA analysis of large deflections and stresses in bending fatigue specimens

Fig. 4. An example of the use of ADINA-plot showing deflections at various time steps. The force is applied in 10 steps and increases to a maximum value of 24.3 N at time step 10.

in our experiments, and will therefore also be treated as the standard case below. In order to be able to trust the stress predictions, we have chosen two criteria for examining the accuracy of the calculations: (1) The longitudinal curvature can be checked by calculating the deflection angle at node 195 (section I) and comparing it to the angle at which the force is applied along section I. In other words, if the structure responds properly, the force should act normal to the surface. (2) The anticlastic curvature can be checked experimentally by means of a device termed Talysurf, originally designed for surface roughness measurements. For a given deflection at node 195 (we have chosen two values, 2.5 and 5 mm), the experimentally measured and the theoretically calculated transverse deflections of section II ought to agree. Figure 5 illustrates how well the first criterion is satisfied. In view of the fact that the experimental boundary conditions in the grip region (rounded bakelite edges, imperfect fit, etc.) can hardly be modelled in full detail, we find the discrepancy of roughly 10% at maximal deflection within reasonable limits. Figures 6 and 7, showing the anticlastic deflection for t = 0.381 mm at displacements of 2.5 and 5 mm respectively, illustrate the similarity between the experimental and theoretical curves. Considering the inaccuracies involved in evaluating Talysurf-results the agreement must be regarded as good. During the calculations, it was observed that the edge deflection was more pronounced in thick specimens. This effect is shown in Figs. 8 and 9 for displacements at node 195 of 2.5 mm and 5 mm respectively. It may be observed in Fig. 8 that the thickness dependence in this case is rather small, the edge deflections being approx. 40 pm. Increasing the displacement from 2.5 mm to 5.0 mm yields larger

DEFLECTION

thoroughly

4 2

ANGLE

AT

NODE

195

I

Fig. 5. Angle of deflection at node 195 for t = 0.381 mm. Dashed curve refers to experimental points and continuous curve refers to ADINA-calculations. The force is applied at section I in 10 steps and increases to a maximum value of 24.3 N at time step 10.

ANTICLASTIC

-451 00

05

’ 10

I IS

I

eo Transverse

DEFLECTION

’ 25

30

35

40

15

50

J

55

ctmtancc (mm)

Fig. 6. Anticlastic deflection for displacement 2.5 mm at section II. Dashed curve refers to experiments and continuous curve refers to ADINA-results.

J.-O. Nt~sso~ and M. HEHENBERGER ANTICLASTIC

DEFLECTION

TRANSVERSE

1

t

STRESS

350

VARIATION , _-P 6--

i

Fig. 7. Same as Fig. 6 for displacement S.Omm.

Fig. 10. Transverse stress variations for displacement at section

ANTICLASTIC

I

of 2.5 mm for thickness 0.254mm(*), 0.381 mm( + ) and 0.508 mm(O).

DEFLECTION

TRANSVERSE 800

STRESS ,

VARIATION ,

, .. ,Q

700

Fig. 8. Anticlastlc deflectlon usmg ADINA of thicknesses 0.254mm(*). 0.381 mm( +) and 0.508 mm(O) for a displacement at section I of 2.5mm.

ANTICLASTIC

DEFLECTION

Fig. 9 Same as Fig. 8 for displacement 5.0 mm.

deflections in the edge region. Moreover, the thickness effect now becomes more pronounced as shown in Fig. 9. The maximum edge deflection, 75pm, is assumed for t = 0.508 whereas the thinnest specimen only deflects 52 pm. It should be pointed out that the shape of the anticlastic bending is parabolic for all cases considered and no evidence of a gull-wing shape is observed[5]. The anticlastic curvature is accompanied by a corresponding increase of stresses towards the edge, examples of which are shown in Figs. IO and 1I when the displacement of node 195 is 2.5 and 5.0mm,

Fig. 11. Same as Fig. 10 for displacement 5.0 mm.

respectively. It is clear from these figures that the stress level increases for increasing thickness. Furthermore, the stress increase from the centre to the edge is more pronounced in thick specimens. This may in particular be inferred from Fig. 1I, showing a stress increase of 270 MPa for r = 0.508 and 160MPa for t =0.254. By way of contrast there are comparatively small longitudinal stress variations. Let us consider the example shown in Fig. 12, where node 195 is displaced 6.9 mm, corresponding to the last time step used. If the stresses in elements 9-24 are calculated, using two gaussian points in the outermost region in each element (see Fig. 13) the stress curve in Fig. I4 is obtained. According to this figure, longitudinal stress variations are much smaller than variations in the transverse direction. However, there appears to be a stress maximum in element 21 rather close to the curved region in the narrow part of the specimen. For the case considered in Fig. I4 when t = 0.381 mm, the von Mises stress varies between 800 and 880 MPa, the maximum being assumed in element 21. These values should be compared with the value predicted by the standard formula cr = [6P,/(1*)], where P is the applied force and t is specimen thickness. When the maximum value of 24.3 N is inserted a stress of 1004 MPa is obtained. This particular case illustrates a general observation made in this investigation, namely the tendency for the standard formula to

ADINA analysis of large deflections and stresses m bending fatigue specimens CENTER

LINE DEFLECTION

r

i

30

Fig. 12. Deflection of centerline at an applied force of 24.3 N for t = 0.381 mm.

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(Figs. 6 and 7). This fact indicates that the boundary conditions chosen in our model give a true description of the physical situation. Therefore, there is reason to believe that also the stress values computed-which are not experimentally measureable-are reliable. Experience from tests performed in our own laboratory has led to the conclusion that the stress distribution in our bending fatigue specimen is not uniform. This is primarily due to the observation that fracture initiation predominantly occurs in the edge region[2]. A detailed fractographic study performed recently showed that more than 50% of all fractures had nucleated less than 1 mm from the edge[9]. This is consistent with the FEM-calculations presented in this paper, showing a significant stress increase towards the edge (see Figs. 10 and 11). Although there are comparatively small longitudinal stress variations there appears to be an enhancement of stresses in elements 18 and 21. If this is so, there ought to be a high fraction of initiation points in this region. In order to check this we examined 100 fractured specimens of thickness 0.381 mm. 58 of these cracks had started in a region corresponding to elements 18 and 21. Therefore, the stress maximum in elements 18 and 21 is justified from an experimental point of view. However, these results imply that slight modifications of our fatigue specimen geometry ought to be made if stress enhancements be removed. If the stress concentration in elements 18 and 21 are due to notch effects from the curved edge, an improvement could probably be achieved by increasing the radius of curvature (r = 4 mm in Fig. 3). As a result, the effective volume tested would become larger. If lon~tudinal stress variations reflect the specimen geometry--and therefore may be influenced by design-transverse stress variations are unavoidable since they depend upon the Poisson contraction[lO], which is an inherent property of all elastic materials. The large effect of stress enhancements in the edge region is elucidated by the experiment performed by Nilsson and Persson[2]. They showed that extremely small hardness indentations positioned in the edge region inevitably caused initiation (see Fig. 2) whereas large indentations in the central parts of the specimen were unable to initiate fracture. The results, explained in terms of a stress increase towards the edge, are thus consistent with the present calculations (see Figs. 10 and 11). However, an effect which cannot be explained in our model is the observation that virtually all edge cracks initiate at some finite distance from the edge. A tentative explanation of this effect is that crack initiation is retarded at the edge due to residual compressive stresses from the polishing procedure. An additional effect may come from the rounded edges which is not accounted for in the calculations. The explanation given is only tentative and it should be pointed out in this context that the discrete nature of the numerical solution procedure does not allow a detailed analysis very close to the edge, unless a much finer element mesh is used.

B 2

1

1

12

9

Fig. 13. Positlon of elements 9-24, corresponding to the region in which edge stresses are calculated (see Fig. 14). STRESS

VARIATION ALONG

EDGE

Fig, 14. Longitudinal stress variations in the oute~ost gaussian points in elements 9-24. A significant stress enhancement occurs in elements 18 and 21. overestimate stresses. Moreover, the standard formula does not have the ability to predict stress variations. 4. DISCUSSION

The present investigation shows an excellent agreement with experimental results both in terms of deflection angle (Fig. 5) and anticlastic deflection

5. CONCLUSIONS Excellent agreement in terms of deflections and stresses have been obtained between the ADINAcalculations and bending fatigue experiments. (1)

870

J.-O. NIL~~~Nand M.

(2) The calculations clearly show that the stress level increases towards the specimen edges, particularly in elements 18 and 21. This result is consistent with the experimental observations of a very high fraction of initiation points in this region. (3) Standard formulae available in the literature can neither predict correct stress levels nor stress distributions. This is due to geometrically non-linear effects which elementary theory cannot account for. (4) The geometry of our bending fatigue specimens is close to optimal since longitudinal stress gradients are rather small. However the ADINA-calculations indicate where improvements can be made. Acknowledgements-This paper is published by permission of SANDVIK AB. The authors would like to thank Mr. Gerhard Persson at Sandvik Steel R&D Centre, and Prof. K.-J. Bathe for valuable discussions. REFERENCES 1. R. Johansson and G. Persson, Influence of testing and material factors on the fatigue strength of valve steel. Purdue Compressor Technology Conference (Edited by

HEHENBERGER

W. Soedel). Purdue University, West Lafayette Indiana, U.S.A. (1976). 2. J.-O. Nilsson and G. Persson, Bending fatigue failures in valve steel, Purdue Compressor Tech. Conf. (1980). 3. M. Hehenberger and J.-O. Nilsson, A stress analysis of bending fatigue specimens using the finite element method, Purdue Compressor Tech. ConJ (1982). 4. K. J. Bathe. ADINA. a FEM-nroeram for automatic dynamic incremental non-linear-aniysis. Rep. 82448- 1, Mechanical Engineering Department, MIT (1978). 5. D. G. Ashwell, The anticlastic curvature of rectangular beams and plates. J. Aeronaut. Sot. 54, 708 (1950). 6. K. J. Bathe, A geometric and material non-linear plate and shell element. Comput. Structures 11. 2348 (1980). 7. ADINA-PLOT-A program for display of input and output data using ADINA. Rep. AE82-3. ADINA Engineering (March 1982). 8. N. H. F. Beebe, PLOT 79, Departments of Physics and Chemistry, University of Utah, Salt Lake City. 9. G. Persson, Influence of surface finishing on fatigue properties of valve steel, Purdue Compressor Tech. Conf (Edited by R. Cohen). Purdue University, West Lafayette, Indiana, U.S.A. (1982). 10. S. P. Timoshenko, Theory of Elasticity, p. 287. McGraw-Hill, New York (1970).