Experimental validation of a fracture plane model for multiaxial cyclic loading A. Carpinteri, R. Brighenti, A. Spagnoli Dipartimento di Ingegneria Civile, Universita di Parma Viale delle Scienze, 431 Parma, Italy Abstract Several approaches have been conceived in order to analyse the high-cycle fatigue behaviour of metallic structures under multiaxial cyclic loading. According to the critical plane approach, for example, the fatigue fracture plane is determined on the basis of the amplitudes or values of certain stress components or a combination of them. By applying the weight function method, the present authors have recently proposed to correlate the orientation of the critical fracture plane with the averaged principal stress directions. In this paper, experimental data collected from the relevant literature, concerning different types of metals under out-of-phase sinusoidal normal and shear stress states, are examined to assess the predictive capabilities of some critical plane criteria, also including the model developed by the authors. 1 Introduction Many metallic structural components are subjected to cyclic loadings (e.g. Carpinteri [1]), which often provoke multiaxial stress states. Several multiaxial high-cycle fatigue criteria can be found in the literature to evaluate whether fatigue failure may occur or not (for example, see You and Lee [2] for a review on the subject) : empirical formulae, stress invariant-based criteria, stress average-based criteria, energy-based criteria and the critical plane approach. According to the last approach mentioned, the fatigue fracture occurs in a plane where the amplitudes or values of certain stress components (or a combination of them) attain their maximum. Alternatively, the position of the critical plane can be correlated with that of the principal stress directions (Ohnami et al. [3]). However, the principal stress directions under cyclic loading are generally time-varying and, therefore, averaged principal stress directions should be considered.
52 Boundary Element Technology In the present paper, a critical fatigue fracture plane model recently proposed by Carpinteri et al. [4,5] and based on the weight function method is outlined. Applications of some critical plane criteria to series of experimental tests under out-of-phase sinusoidal normal and shear stress states are discussed together with the predictions of the model presented in Refs [4,5]. 2 A Critical Fatigue Fracture Plane Model In the following we analyse the correlation between the orientation of the experimental fatigue fracture plane and the principal stress directions. Since such directions are in general time-varying under cyclic loading, an appropriate averaging procedure is required to obtain the mean principal directions. This procedure can be performed, for example, by employing suitable weight functions which account for the main factors influencing the fatigue fracture process, as has recently been proposed by Carpinteri et al. [4,5]. The main aspects of such a procedure are presented hereafter. 2.1 Principal Stress Directions The matrix A of the principal direction cosines /%, m%, %%, with n = 1,2,3, relative to the the principal stresses a (with o\ > *2 ^ *3 ), consists of nine elements, but only three of them are independent because of six orthonormality conditions. On the other hand, the orthogonal coordinate system PI with origin at point P and axes coincident with the principal stress directions (Fig.l) can also be defined through the Euler angles,, V Therefore, analogously to the case of the direction cosines, we only need three independent parameters to determine the principal stress directions. The procedure to obtain the principal Euler angles from the components of matrix A is rather lengthy, although very simple (see Ref. [4] for details). It is worth noting that, conversely to the notation adopted in Ref. [4], in the present paper the maximum principal direction 1 is described by the two angles and 9. Hence, the orientation of the 1-axis is now defined by two parameters only. If the Euler angles are known, the principal direction cosines can be written as follows : A = C C C\y Sty S\y C< S*CflCnf + CA^I,/ 5", (1) where s and c correspond to sin and cos, respectively, while the subscripts represent the arguments of such trigonometric functions. The calculation of the principal Euler angles at each time instant t from the matrix A(t) consists of two stages described in Ref. [4]. Note that the ranges of
Boundary Element Technology 521 Figure 1 Principal stress directions 3 described through the Euler angles such Euler angles at the end of this two-stage procedure are as follows : and -n 1 2<y(t)<n 1 2. 2.2 Averaging Procedure In order to determine the mean position (,,i/? ) of the principal stress axes 1, 2 and 3 under cyclic loading, it seems realistic to average the Euler angles by employing suitable weight functions, W( f& ), which take into account the main factors influencing the fatigue fracture behaviour : _1_ w "I if with W = (2) Consider the following weight function : = 1. (3) In this case, the summation W is equal to the number N of time instants being considered and, consequently, the weighted mean values of the Euler angles, obtained from eqns (2) for W( t^ ) = W\( t^ ), coincide with the simple arithmetic averages. Now let us examine another possible weight function :
522 Boundary Element Technology if if <c< 1 (4) It only includes into the averaging procedure those positions of the principal axes for which the maximum principal stress <7j is greater than or equal to the product of the constant coefficient c, with < c < 1, and the normal stress fatigue limit, o ^, deduced from the S-N curve for fully reversed axial loading (i.e. the loading ratio, /?, is equal to -1). The weight of such positions exponentially depends on the coefficient /% = - (1 / m), where m is the negative slope of the S-N curve considered. Then the following weight function is analysed : < c < 1 (5) where ^,nax^k ) ^ the maximum shear stress at time instant f&. Furthermore, the parameter T^ represents the shear stress fatigue limit determined from the S-N curve for fully reversed torsion (R = -1), whereas m^ = - (1 / m *) depends on the slope m * of this S-N curve. Finally, a linear combination, called U^, of the weight functions W^ and Ws s considered : tf &l(*k Onf and?mnv( tlf ) < C Tf,f ( V ) / \/Mgy C7j ( t if ) U<v J if O i ( t^ and and (6) ( \ (T \a\(tk)\, \C af } " max\ *k ' T, The foregoing weight functions are plotted in Fig.2 in the case of a generic nonproportional sinusoidal loading. In particular, Figure 2(b) displays the constant function W/, whereas Fig.2(c) represents the functions W^ to W+ Final-
Boundary Element Technology 5 TIME Figure 2 Modifications of the time history of angle f t) by means of the weight functions W(t) to ly, the function * W,- (with / = 1 to 4), which has to be averaged to determine the mean Euler angle according to eqns (3) to (6), is shown in Figs 2(d) and 2(e). 3 Experimental Validation Few papers in which the orientation of the experimental fatigue fracture plane is reported can be found in the literature. In the following, experimental data related to a synchronous out-of-phase sinusoidal loading for round bars under
524 Boundary Element Technology combined bending and torsion are analysed (Nishihara and Kawamoto [6]). The plane stress condition of normal and shear stresses at the generic point P of a round specimen (Fig.3) can be expressed as follows : 'xy (where / stands for longitudinal), while the other components of the stress tensor are equal to zero. The phase angle (3 is equal to (in-phase), 6 or 9 (out-ofphase). Different values of the amplitude ratios (X /d/, J have been examined (Tables 1 and 2), whereas the mean stresses a/^, and %,% are equal to zero. In Nishihara's tests [6], round bars of mild steel with.15% C content (Table 1) and grey cast iron with 3.7% C content (Table 2) were employed. The mechanical characteristics of the mild steel are: E = 2 GPa, v =.3, cjy = 277.6 MPa, o*f= 5.4 MPa, t*f = 214.7 MPa, m* =, m, = 2.. For the grey cast iron, the mechanical characteristics are: E = 11 GPa, v =.25, c^f = 96.1 MPa, Taf = 93.7MPa,m<,= 19.4, m,= 19.9. The orientation of the experimental fatigue fracture plane can be compared with that of the theoretical critical plane determined according to some of the current criteria (e.g. Findley [7], Matake [] and McDiarmid [9]). In particular, the results obtained on the basis of the maximum amplitude of the shear stress (Matake and McDiarmid criteria) are considered. Note that the direction of the shear stress vector C, acting on the generic plane A whose normal w is defined by the angles and (Fig.3b), is generally time-varying under cyclic loading. In order to determine the amplitude and mean l,m (7) X N : normal stress C : shear stress (b) Figure 3 (a) PXYZ coordinate system relative to the generic surface point P of a round specimen; (b) Puvw coordinate system, with the w-axis normal to the material plane A.
Boundary Element Technology 525 Table 1 Experimental fatigue fracture plane in mild steel specimens (%xp) [6] and theoretical critical plane orientation (r cai) for normal and shear stress state. Test No. 1 2 3 4 5 6 7 9 1 11 13 14 15 16 17 [MP a] [MPa] 194. 3 2. 3 5. 6 17. 3 11. 3.. 21. 1 194. 2 15. 2 1. 9 244. 5. 6 5. 2. 1 1. 6 116. 4.. 4.9 916 166. 142.3 1.6 97.1 6. 131.5 5.7 4.9 117.9 14.1 136. 14.5 *'"...2.5 oo oo.5.5.2.2.5.5 Theoretical results for which (n^, - * The critical plane is undetermined. ft n 6 6 6 6 9 9 9 9 9 9 Exp. n 22 3 22 Pre. sent study (' Heal) W, W, o 34 1 1 ) < ± 1 are underlined. W3 w. 34 1 1 Matake McDiarmid 33 11 * * value of this vector, Papadopoulos [1] has recently proposed to analyse the components of C along the u- and v-axis (Fig.3b), where the direction u is obtained as the intersection between A and the plane described by the w- and the Z-axis, and the direction v is chosen to form a right-hand orthogonal coordinate system Puvw : Cu = u C C =VC: + m () By employing the direction cosines of the u- and v-axis and recalling eqns (7), eqns () can be written in this form : ^ - f sin (cor) + g cos(cot) + ^ = psin(cot) + qcos(cot) + u,m v,m (9)
526 Boundary Element Technology Table 2 Experimental fatigue fracture plane in grey cast iron specimens (r^p) [6] and theoretical critical plane orientation (T^i) for normal and shear stress state. Test Ref. 1 2 3 4 5 6 7 9 1 11 13 14 [MPa] 13. 93.2 95.2 34 56.3.. 93.7 67.6 99.6 14.2 97.1 75.1 71.3 [MPa].. 19.7 41.6 6. 9.1 94.2 46.9 1.6 2.6 21.6 4.6 9.6 6.1 *"*'...2.5 oo.5.2.2.5 Theoretical results for which * The critical plane is undetermined. n 6 6 9 9 9 9 9 Exp. (riexp ) n 25 34 49 16 33 3 37 Pre. sent study (Veal) V W, W2 W4 34 1 ± 1 are underlined. 44 34 1 Matake McDiairmid (Veal) n 33 * where C%,m and C^m are the mean values of the components of C along the u- and v-axis, whereas the functions /, g, p and q depend on the stress amplitudes o/,a and T«and the phase angle /3. Equations (9) represent the ellipse described by the tip of the shear stress vector C on the plane A during a loading cycle. This ellipse is centred at point (Cun\ m), and its semi-axes are given by : (1) The amplitude of the shear stress C coincides with the major semi-axis C«of the above ellipse (Papadopoulos [1]). Then, the orientation of the critical plane according to Matake and McDiarmid criteria is obtained by maximizing C«with respect to the angles and. The orientation of the experimental fatigue fracture plane is expressed by the angle r}^, between the normal to the cracked plane and the longitudinal axis
Boundary Element Technology 527 (Y-axis) of the specimen. This experimental value is compared with T ^/ = arcos (ra j) for the criterion proposed by the present authors (that is, in this case r\^i is the angle between the Y-axis and the weighted mean direction 1 of the maximum principal stress), and with r ^/ = arcos (w^) for the other criteria being considered (in this case r ^/ is the angle between the Y-axis and the w-axis, Fig.3b). Note that, when calculating the weighted mean principal stress directions (Section 2.2), the coefficient c is assumed to be equal to.5. The results of such a comparison are presented in Tables 1 and 2. It can be seen that the predictions of the present study are overall fairly satisfactory for the different types of loading conditions and materials examined, especially in the case of low values of the phase angle. In particular, the results obtained from the weight functions V/2 and W^ exhibit the best correlation with the experimental data. Insofar as the criteria by other authors are concerned, the agreement with the experimental results is not so good. In conclusion, the normal to the fracture plane seems to closely agree with the expected direction 1 of the maximum principal stress, by employing the weight functions VJ^ or H^. 4 Conclusions A critical fatigue fracture plane model developed by Carpinteri et al. [4,5] has been outlined. According to this model, the principal stress directions appear to be fundamental in determining the orientation of the plane where the fatigue fracture is expected to initiate. However, since the principal stress directions may change at each time instant, an averaging procedure which accounts for the main factors influencing the fatigue fracture process is carried out by employing the weight function method. Experimental data collected from the literature have been compared with the results deduced by applying different critical plane criteria, also including the method proposed by the authors. The experimental data considered concern tests on round bars under combined bending and torsion, in the case of a synchronous out-of-phase sinusoidal loading [6]. The results derived through the present authors' model are in good agreement with the experimental data, especially when particular weight functions are employed. More in detail, for the examined tests, the normal to the experimental fracture plane agrees with the weighted mean direction 1 of the maximum principal stress by adopting the functions ^ or V/4, especially in the case of low values of the phase angle between the applied loads. Acknowledgements The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MURST) and the Italian National Research Council (CNR).
52 Boundary Element Technology References 1. Carpinteri, A. (ed.), Handbook Of Fatigue Crack Propagation In Metallic Structures, Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 1994. 2. You, B.-R. & Lee, S.-B., A Critical Review on Multiaxial Fatigue Assessments of Metals, Int. J. Fatigue, 1,pp. 5-244, 1996. 3. Ohnami, M., Sakane, M. & Hamada, N., Effect of Changing Principal Stress Axes on Low-cycle Fatigue Life in Various Strain Wave Shapes at Elevated Temperature, Multiaxial Fatigue ASTM STP 53, pp.622-634, 195. 4. Carpinteri, A., Macha, E., Brighenti, R. & Spagnoli, A., Expected Principal Stress Directions for Multiaxial Random Loading - Part I: Theoretical Aspects of the Weight Function Method, Int.J. Fatigue, 21, pp.3-, 1999. 5. Carpinteri, A., Brighenti, R., Macha, E. & Spagnoli, A., Expected Principal Stress Directions for Multiaxial Random Loading - Part II: Numerical Simulation and Experimental Assessment through the Weight Function Method, Int. J. Fatigue, 21, pp. 9-96, 1999. 6. Nishihara, T. & Kawamoto, M., The Strength of Metals under Combined Alternating Bending and Torsion with Phase Difference, Memories of the College of Engineering, Kyoto Imperial University, 11, pp.5-1, 19. 7. Findley, W.N., A Theory for the Effect of Mean Stress on Fatigue of Metals under Combined Torsion and Axial Load or Bending, J. Engng Industry, Trans. ASME, 1, pp.31-36, 1959.. Matake, T., An Explanation on Fatigue Limit under Combined Stress, Bull. JSME, 2, pp. 257-263, 1977. 9. McDiarmid, D.L., A General Criterion for High Cycle Multiaxial Fatigue Failure, Fatigue and Fract. Engng Mater. Struct., 14, pp. 429-3, 1991. 1. Papadopoulos, I.V., Critical plane approaches in high-cycle fatigue: on the definition of the amplitude and mean value of the shear stress acting on the critical plane, Fatigue and Fract. Engng Mater. Struct, 21, pp. 269-25, 199.