10.1.1.127.441(1)

ORAL REFERENCE: FT436 AN EFFICIENT CRITICAL PLANE METHOD FOR DUCTILE, SEMI-

DUCTILE AND BRITTLE MATERIALS

C. Gaier and H. Dannbauer

Magna Powertrain, Engineering Center Steyr, Steyrer Strasse 32, A-4300 St. Valentin, Austria

ABSTRACT In mechanical engineering fatigue assessment of components, where complex non-proportional loads are acting on, plays an important role to identify critical spots and to predict the lifetime till crack initiation. Today’s fatigue life prediction software has to deal with stress results of huge finite element models consisting of several hundred thousands or even millions of nodes and elements. Therefore, efficient algorithms for fatigue assessment are necessary to reduce development time and costs. Nevertheless, such methods should be able to deal with randomly and non-proportionally loaded components made of ductile, semi-ductile or brittle materials. Both cyclic tension/compression (or bending) and torsion loading and its combinations should be predicted correctly. In this contribution a new method for high cycle fatigue is presented, which covers both infinite and finite life region, and which is based on the critical plane approach combined with rainflow counting of load cycles and linear damage accumulation. Comparisons with test results of combined in-phase bending and torsion loaded specimens show an excellent agreement. A practical example demonstrates the applicability of the proposed method for daily engineering purposes. KEYWORDS Fatigue life prediction, Non-proportional loads, Multiaxial stresses, Critical plane method INTRODUCTION In mechanical engineering, computational simulations of stress and strain distributions on mechanical components with the Finite Element Method (FEM) are widely used. Nevertheless, much experience and know-how are still necessary to obtain good results of lifetime simulations, which are based on stress/strain FEM results. A lot of fatigue life calculation methods exist (e.g. a large variety of damage parameters, which are difficult to survey; stress or strain-based methods for HCF/LCF, different methods for proportional/non-proportional loading and brittle/ductile material), and it is not easy to select a method, which is most suitable for a specific problem. Typical complex shaped and dynamically loaded components in automotive engineering are engine blocks, connecting rods, crankshafts, gearboxes and body in white structures, where the location of the critical areas are not known in advance. Appropriate and efficient methods are needed for fatigue analysis of such complicated geometries. They should fulfill several requirements:

Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.

• They should be able to predict both cyclic tension/compression (or bending) and torsion loading and

its combinations correctly for any kind of material (ductile, semi-ductile or brittle). As basic input material data, a tension/compression S-N curve, defined by its slope, cycle limit and endurance limit σfl, and the fatigue limit τfl for shear should be enough.

• Application of classical cycle counting procedures as e.g. rainflow counting [1]. • Assessment of tri-axial stress states, which occur on surfaces under pressure. Cracking may also start

below the surface, if the surface is hardened or if there are pores inside. • Assessment of randomly and non-proportionally loaded components. • Independency from coordinate system. This can be achieved e.g. by application of critical plane

criterions with sufficient small angle between planes (5 to 10 degree for practical needs). • From a practical point of view the calculation time for big finite element models consisting of more

than one million nodes and elements, where several hundred load channels with several ten thousand samples are simultaneously acting on, should not exceed one night.

In this contribution a new method will be presented, which fulfill all above requirements. MATERIAL BEHAVIOR FOR COMBINED IN-PHASE BENDING-TORSION LOADING For ductile material (e.g. steel) it was found by a lot of test results ([3-8], Figure 1), that the fatigue limit forms an ellipse in the normal/shear stress diagram for combined in-phase bending-torsion loading:

122

=

+

flfl ττ

σσ (1)

This relation represents the well-known Gough ellipse. For brittle material (e.g. cast iron) the fatigue limit forms a parabola in the normal/shear stress diagram:

12

=

+

flfl ττ

σσ (2)

Eqn. 1 and 2 can be combined to the empirical formula [2]:

12

=

+

fl

k

fl ττ

σσ (3)

For the exponent k it is rather intuitive to take the ratio tensile to shear fatigue limit:

fl

flkτσ

= (4)

The k-value ranges between 1, which represents a brittle material, and 2, which represents a ductile one. By a linear combination of Eqn. 1 and 2 another widely used empirical expression can be obtained [9]:

( ) ( ) 12122

=

+

−+

−

flflfl

kkττ

σσ

σσ (5)


For k=2 Eqn. 3 as well as Eqn. 5 become to Eqn. 1, whereas for k=1 Eqn. 2 is obtained. Also for other k-values Eqn. 3 and Eqn. 5 deliver almost the same results as shown in Figure 2. These formulas will be used in the following to check the result quality of the proposed lifetime prediction method.

Figure 1: Fatigue limit of different steels and cast iron for combined in-phase bending and torsion (this figure taken from Ref. [3] is based on test results from [4-8])

Figure 2: Empirical representations of the fatigue limit for different k-values and combined in-phase bending and torsion by different formulas (see Eqn. 3-5)


PROBLEMS OF CLASSICAL APPROACHES For non-proportional loadings and multiaxial stress states with rotating principal stress directions, the critical plane criterion is a widely applied and commonly accepted concept. The principle is rather simple: For all material planes in a certain angle distance (e.g. 10 degree) a damage analysis will be performed. The plane with maximum damage is assumed to be critical for fatigue failure. The critical plane approach guarantees results independent from the coordinate system, if the angle distance is sufficiently small. For free surfaces, where the stress state is biaxial, it is enough to consider all planes perpendicular to the surface (e.g. 18 planes with 10 degree distance). For tri-axial stress states, which occur on surfaces under pressure (e.g. bearings) or inside of components, much more planes have to be considered, i.e. all planes, whose normal vectors form a hemisphere (e.g. about 200 planes with about 10 degree distance). Damage is produced by closed load cycles in the stress-strain path. According Palmgren-Miner, the damages of each load cycle can be linearly accumulated to the total damage. Although linear damage accumulation has been criticized sometimes, it is still the most common way for fatigue analysis because of its simplicity and effectiveness, and the accuracy is sufficient in technical practice. For random loads, closed cycles must be detected by a suitable counting algorithm as e.g. rainflow counting [1]. The number of cycles can be economically stored in a rainflow matrix. The question arises, for which quantity a rainflow counting should be performed in combination with the critical plane criterion. A stress vector is acting on material planes, but cycle counting procedures can be applied only to scalar quantities. The easiest way is applying rainflow counting to the normal stress. Considering normal stress is good for brittle failure, but shear stress is responsible for ductile failure. Nevertheless, for tri-axial stress states rainflow counting cannot be applied to shear stress because of its vector nature, i.e. shear stress may rotate or even form random curves in the material plane. THE METHOD OF SCALED NORMAL STRESS The main idea of the proposed method to solve above problems is damage equivalent scaling of the normal stress in dependence on the stress state (tension – torsion – hydrostatic, and its combinations). A suitable quantity, denoted as V is needed to characterize the stress state. For a best fit to test results as presented previously (Figure 1 and 2), the following procedure is proposed:

a) Calculation of the local principal stresses 321 σσσ >> at each time step.

b) Calculation of the ratio minimum to maximum principal stress at each time step:

1

3

σσ=V for 31 σσ > (6)

3

1

σσ

=V for 13 σσ > (7)

The ratio V is a value between –1 and +1 with special meanings:

o V = –1: Dominating shear or torsion load o V = 0: Dominating tension/compression or bending load o V = +1: Hydrostatic stress state


c) In dependence on V the normal stress of the considered material plane will be scaled by the following time dependent scaling factor:

Vkf )1(1 −+= (8)

The quantity k is defined by Eqn. 4. It can be easily recognized, that for brittle materials with k = 1 the stress state will not be modified at all (f = 1). Therefore, the normal stress will be used directly, which is in accordance with the normal stress hypothesis for brittle materials. For ductile materials f depends on the load situation: For tension/compression loading (V = 0) the stress still remains unchanged, whereas for shear loading (V = –1) the normal stress will be increased by the factor k. In this way, the stronger damaging effect of shear stress is taken into consideration in a quantitatively correct manner. For hydrostatic stress states (V = 1) the factor f will be linearly extrapolated, i.e. the normal stress will be decreased. This behavior is in good accordance with the distortion energy criterion for ductile materials, where the Von Mises stress becomes zero. According to Eqn. 8, the factor f becomes to zero for very ductile materials with k = 2. From a computational point of view, it is much more efficient, to scale the stress tensor just before transforming the tensor into the planes instead of scaling the normal stress directly in every plane.

d) Next the critical plane criterion is applied: In each plane a rainflow counting of the normal stress

history is performed. The resulting stress amplitudes can be used together with a tension/compression S-N curve for the calculation and linear summation of partial damages according to Palmgren/Miner. Before damage analysis the S-N curve should be locally modified according to influences like mean stress (which can be quantified by means of an Haigh-diagram), notch support effect (which can be considered by taking into account stress gradient [10-13]), temperature, surface roughness, surface treatments, etc. The plane with maximum damage is assumed to be critical. The ductility of the material has already been taken into account by scaling of the normal stress or stress tensor.

VERIFICATION Here the proposed method of scaled normal stress will be verified for combined in-phase bending/torsion loading of cylindrical un-notched specimens, for which a lot of test data (Figure 1 and 2) and commonly accepted empirical formulas exist (Eqn. 3-5). For this purpose the fatigue limit in the normal/shear stress diagram for combined in-phase bending/torsion loading will be mathematically derived and compared with the empirical formulas. On the specimen’s surface the plane stress state can be written as tensor :

=

000000

ττσ

σ (9)

Here the stress quantities are already stress amplitudes.


Principal stresses can by found by solving the equation:

000

00

det =

−−

−

i

i

i

σσττσσ

(10)

The following principal stresses will be obtained:

( )221 4

21 τσσσ ++= (11)

02 =σ (12)

( )223 4

21 τσσσ +−= (13)

Now the critical plane criterion has to be applied. For proportional loading without mean stress effects the plane with maximum normal stress is critical, i.e. the principal stress according Eqn. 11 acts in the critical plane. Following point c) of the proposed method of scaled normal stress, 1σ according Eqn. 11 will be multiplied by the scaling factor f according Eqn. 8. To obtain the fatigue limit for combined in-phase bending/torsion loading, the scaled normal stress must be equated with the known fatigue limit for tension/compression:

flf σσ =1 (14)

Some simple manipulations are necessary, to obtain the fatigue limit as a function of the normal and shear stress amplitudes. Putting Eqn. 6, 8, 11 and 13 into 14, the following relationship can be derived:

( ) ( ) 12122

=

+

−+

−

flflfl

kkττ

σσ

σσ (15)

Comparing Eqn. 15 with Eqn. 5 it can be seen, that these expressions are completely identical! This excellent agreement with empirical formulas confirms the postulated relationship for the damage equivalent scaling factor f for stresses according to Eqn. 8. PRACTICAL APPLICATION: SIMULATION OF A DIESEL ENGINE CRANKSHAFT The proposed method of scaled normal stress was implemented in the commercial fatigue software tool FEMFAT, developed at the Engineering Center Steyr [12, 14, 15]. FEMFAT is established in automotive engineering for about fifteen years now and continuously improved. Finite element stress results are taken as input data for fatigue assessment. FEMFAT can take into consideration different influences on fatigue life as stress gradient (to take into account the notch support effect), mean stress, surface roughness, surface treatment, temperature and much more. In the following fatigue analysis of a diesel engine crankshaft is presented. In order to perform an accurate fatigue life prediction of a crankshaft it is necessary to take a couple of different effects into account. The crankshaft itself is a linear reacting structure that undergoes large nonlinear displacements. Additionally, an elasto-hydrodynamic (EHD) oil film model is required, being capable to consider the stiffness and damping properties of the oil film inside the journal bearings. An efficient computation of all these effects requires different algorithms (e.g.: FEM for linear reacting structure, MBS for large nonlinear movements and EHD software for the oil film dynamics). The FE,


MBS and EHD software contribute to this integrated simulation process with their particular advantages. The modal representation of the crankshaft (component modes, Craig Bampton Theory [16]) is imported into an MBS Software (e.g. ADAMS, RecurDyn). The reaction forces and moments of the elasto-hydrodynamic oil film are computed in a co-simulation process using a user-written subroutine within the MBS software. The result of the time integration of the MBS solver are, among many other result sets, the modal coordinates of the crankshaft, representing the amplitude of each mode shape at each time step. A superposition of all mode shapes weighted by the corresponding modal coordinates gives the total deformation of the flexible structure. Each deformation of a FE structure is related to a clearly defined stress distribution. Consequently, each component mode shape (deformation) corresponds to a clearly assigned stress distribution (modal stress), which is an output of a subsequent FE analysis. The resulting stress state of the FE structure is again computed by a linear combination of the modal stresses. The single modal stress shape’s modal coordinate is the same as the modal coordinate of the corresponding component mode and a result of the MBS (Figure 3).

Figure 3: Proceeding of the durability analysis based on modal stresses Figure 3 outlines the procedure of the multiaxial and channel-based fatigue lifetime prediction. The modal stresses and modal coordinates are the input for the durability analysis. Each channel consists of a modal stress and the time history of the corresponding modal coordinate. FEMFAT then computes the resulting stress states for each time step (stress history). It has to be mentioned, that the modal based approach provides the technique for fatigue life prediction of any vibration dominated problems or dynamic loaded parts, like components of an engine. Figure 4 and 5 left shows the distribution of safety factors against endurance limit on the finite element mesh, calculated by FEMFAT with the scaled normal stress approach. The minimum safety factor can be found in notches of the main bearings. The safety factor also changes with the rotational speed of the crankshaft as shown in Figure 5 right. The minimum can be found at 7000 rpm, where the safety factor still do not fall below the critical limit of one. Therefore the crankshaft is safe, and also in tests no cracks have been observed. Figure 5 also shows safety factors obtained with pure normal stress (without scaling). The difference in the safety factors between pure normal stress und scaled normal stress is rather small, which indicates, that loading of the crankshaft is dominated by bending. Therefore, torsion loading is more a small part of the total loading, which acts on the crankshaft in daily working.


Figure 4: Finite element mesh of the crankshaft together with distribution of safety factor against endurance limit; whole crankshaft (left), detail at main bearing (right)

Figure 5: Endurance safety factor distribution at notch (left), endurance safety factors against rotational speed of crankshaft, comparison of pure normal stress and scaled normal stress

Nevertheless, the question arises, why the safety factors obtained by the scaled normal stress are slightly higher than the safety factors obtained by pure normal stress, although the difference is quite small. In theory, on the surface a biaxial stress state is acting, therefore V ≤ 0 must be valid (according Eqn. 6 and 7). It follows that f ≥ 1 (according Eqn. 8) should be true. This discrepancy to the obtained results of the crankshaft can be explained by the limited accuracy of the finite element method. The surface at the notches of the main bearings cannot be absolute exactly modeled by finite elements, because there remains folds between the faces of finite elements. Therefore the normal stress perpendicular to the surface is not exactly zero. A further refinement of the FE mesh at the notches of the main bearings would reduce the differences between pure normal stress and scaled normal stress almost to zero.


CONCLUSIONS A fast and reliable stress based fatigue life prediction method has been presented for both high cycle finite and infinite life applications. The method uses an extended critical plane criterion and can be applied to all kind of metals (ductile, semi-ductile, brittle). For random loading classical cycle counting procedures like rainflow can be used. For combined in-phase bending and torsion loading exact results compared to empirical formulas are obtained. It has been shown, that the method can be easily combined with the finite element method for fatigue analysis of large and complicated shaped structures (e.g. automotive parts like crankshafts, engine blocks, cylinder heads etc.). The critical areas need not to be known in advance. Future efforts take aim to investigate the result quality for combined out of phase bending and torsion loading. REFERENCES

1. Matsuishi M. and Endo T. (1968), Fatigue of Metals Subjected to Varying Stress, Proc. Kyushi

Branch JSME, pp. 37–40. 2. Dietmann H. (1973), Werkstoffverhalten unter mehrachsiger schwingender Beanspruchung, Teil 1:

Berechnungsmöglichkeiten, J. of Materials Technology, Nr. 5, pp. 255-263. 3. Dietmann H. (1973), Werkstoffverhalten unter mehrachsiger schwingender Beanspruchung, Teil 2:

Experimentelle Untersuchungen, J. of Materials Technology, Nr. 6, pp. 322-333. 4. Gough H.J. and Pollard H.V. (1935), The Strength of Metals Under Combined Alternating Stresses,

Proceedings of the Institute of Mechanical Engineers, Vol. 131, pp. 3-103. 5. Gough H.J. and Pollard H.V. (1935), The Effect of Specimen Form on the Resistence of Metals to

Combined Alternating Stresses, Proceedings of the Institute of Mech. Eng., Vol. 132, pp. 549-573. 6. Gough H.J. and Pollard H.V. (1937), Properties of some Materials for Cast Crankshafts, with Special

Reference to Combined Stresses, Proc. Inst. Autom. Eng. 31, pp. 821-893. 7. Gough H.J. (1949), Engineering Steels under Combined Cyclic and Static Stresses, The Engineer, pp.

497-499, 510-514, 540-543, 570-573. 8. Gough H.J. (1950), Engineering Steels under Combined Cyclic and Static Stresses, Journal of

Applied Mechanics, Vol. 50, pp. 113-125. 9. Socie D.F. and Marquis G.B. (2000), Multiaxial Fatigue, SAE, Warrendale, U.S.A. 10. Hueck M., Thrainer L. and Schuetz W. (1983), Berechnung von Woehlerlinien fuer Bauteile aus

Stahl, Stahlguss und Grauguss – Synthetische Woehlerlinien, VBFEh, Bericht Nr. ABF 11, Duesseldorf.

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13. FKM-Guideline (2003), Analytical Strength Assessment, Forschungskuratorium Maschinenbau, VDMA Verlag, 5th Edition, Frankfurt am Main, Germany, 2003.

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16. Prandstötter M., Riener H. and Steinbatz M., Simulation of an Engine Speed-UP Run: Integration of MBS – FE – EHD - Fatigue, ADAMS User Conference 2002 - Europe.


NOMENCLATURE

σ Normal stress amplitude

σ1 Maximum principal stress

σ2 Middle principal stress

σ3 Minimum principal stress

σfl Alternating tension/compression fatigue limit

σi Principal stresses

τ Shear stress amplitude

τfl Alternating shear fatigue limit

k Ratio tensile to shear fatigue limit

f Damage equivalent scaling factor for normal stress or stress tensor

V Ratio minimum to maximum principal stress


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