Minimum norm ellipsoids as a measure in high-cycle fatigue criteria

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1 6 th World Congress on Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 005, Brazil Minimum norm ellipsoids as a measure in high-cycle fatigue criteria Nestor Zouain Mechanical Engineering Department, COPPE, EE - Federal University of Rio de Janeiro, C.P , Rio de Janeiro, RJ, Brazil. nestor@ufrj.br 1. Abstract This paper focuses on the use of minimum norm ellipsoids in stress-based fatigue criteria. We consider both the theoretical issues involved in constructing mathematical models for high cycle fatigue endurance limits and the computational features concerning the application of the resulting fatigue criteria. In the present approach, the macroscopic level of continuum is enhanced with internal variables corresponding to the accommodated state at the mesoscopic level of a representative volume. That is, elastic shakedown in the representative volume is used in order to explain the mechanisms of fatigue endurance of a body undergoing elastic strain variations at the macroscopic level. We propose a scalar measure for the amplitude of shear stress variations based on the definition of an enclosing hyper-ellipsoid of minimum Frobenius norm. This equivalent shear amplitude is combined with the maximal mean stress, in both linear and quadratic functions, leading to two alternative fatigue criteria. The predictions of these criteria are compared with experimental data. The distinction between minimum norm enclosing ellipsoid and minimum volume enclosing ellipsoid is discussed. We argue in favor of using the Frobenius norm rather than the volume of the hyper-ellipsoid. The computation of the minimum norm ellipsoid is discussed and a practical algorithm is proposed.. Keywords: High cycle fatigue, fatigue criteria, shakedown, minimum norm ellipsoid. 3. Introduction Stress-based criteria for multiaxial high cycle fatigue have widespread application in mechanical design [1], within the framework of total-life approaches. This paper focuses on the use of minimum norm ellipsoids in stress-based fatigue criteria. We consider both the theoretical issues involved in constructing mathematical models for high cycle fatigue endurance limits and the computational features concerning the application of the resulting fatigue criteria. The models for high cycle fatigue of metals considered here are based on the description of the infinite life endurance as an elastic accommodation (i.e. shakedown) process at the mesoscopic level [, 3, 4, 5]. In the present approach, the macroscopic level of continuum is enhanced with internal variables corresponding to the accommodated state at the mesoscopic level of a representative volume. Elastic shakedown in the representative volume, with the size of some grains, is used in order to explain the mechanisms of fatigue endurance of a body undergoing elastic strain variations at the macroscopic level. We consider a scalar measure for the amplitude of shear stress variations based on the definition of an enclosing hyper-ellipsoid of minimum Frobenius norm. This equivalent shear amplitude is combined with the maximal mean stress, in both linear and quadratic functions, leading to two alternative fatigue criteria. The predictions of these criteria are compared with experimental data in order to validate the proposed shear amplitude measure. In summary, the damaging potential of a cyclic loading at each point of the body is basically measured by the size of the minimum norm enclosing ellipsoid in the deviatoric space. This measure admits a variational formulation in terms of macroscopic internal variables. The distinction between minimum norm enclosing ellipsoid and minimum volume enclosing ellipsoid is discussed. We argue in favor of using the Frobenius norm rather than the volume of the hyper-ellipsoid. The crucial issue concerning the feasibility of the computation of the minimum norm ellipsoid is discussed and a practical algorithm is proposed and demonstrated. The proposed fatigue criteria are also compared to some of the criteria found in the literature [6, 7, 8]. 4. Fatigue Criteria The notation adopted here is as follows. The mean stress and deviatoric part of a stress tensor are denoted σ m = 1 3 trσ S = σ σ m1 (1) 1

2 where 1 is the identity. The principal stresses are {σ i, i = 1,, 3}. Fatigue analysis is based on the data defined either as a cyclic loading program or as a set of bounding constraints on the considered feasible loading programs. In general, we represent this information in the form of a prescribed stress variation domain, given as a set of stress tensor values {σ k, k = 1 : m} at each point of the body under variable loadings. These stress values come from the discretization of a given cyclic stress variation or from the vertices of the assumed feasible domain of loading mapped in the stress field space. The projection of the prescribed stress variation domain = {σ k, k = 1 : m} onto the deviatoric subspace is denoted dev, while σ k m and S k are the mean and deviatoric parts of σ k. Then, fatigue endurance for is ensured if the following inequality is satisfied F ( ) 0 () where F represents a scalar function, of set-valued argument, defining a particular criterion. We consider some examples in the sequel. It is well known that shear stresses and hydrostatic pressure produce different effects on high-cycle fatigue endurance [1, 9]. Accordingly, several equivalent scalar measures for variable shear and mean stresses are proposed in the literature. It is also commonly recognized that, for complex loadings, the issue of evaluating an equivalent shear amplitude is far more demanding than computing an equivalent mean stress. τa equiv 1 S a J a (3) The symbol τa equiv denotes the equivalent shear stress amplitude, while S a is the modulus of the associated simple shear tensor (subscript a stands for amplitude). The same shear measure is also written above in terms of the second stress deviator invariant, J, for the purpose of comparing to many references. The goal of this work is to propose a criterion capable of assessing fatigue endurance under conditions of very general variable multiaxial loadings. The main feature of our model is the definition of the equivalent shear stress modulus amplitude, S a, as the Frobenius norm of a linear operator, L, associated with the minimum norm ellipsoid enclosing the stress path in the deviatoric space. That is, our proposal is S a = L (4) The definition of the linear mapping L = L( ) and the computation of L are main issues in this work, considered in Section 5. In addition to the definition of the shear amplitude, another important choice in tayloring a fatigue criterion is the mathematical representation of the interaction between the critical mean stress and the adopted deviatoric stress amplitude. Likewise in (3), we denote the chosen critical mean stress by σ mc I 1c /3, where I 1 is the trace of a stress tensor. Two admissibility conditions are considered in the following subsections Linear interaction (Drucker-Prager) 1 S a + 3ασ mc β (5) 1 The material parameter β represents the limit of S a under pure shear fluctuation; thus, it is equal to the experimentally observed fatigue limit under alternating torsion usually denoted t 1. The material constant α is the nondimensional interaction parameter. The factor η that amplifies the prescribed stress variation so as to attain a critical domain, that is corresponding to equality in (5), is obtained as η = β 1 S a + 3ασ mc (6) When η < 1 the given loading conditions are safe, while if η > 1 there exists at least one feasible cycle in leading to failure by high-cycle fatigue. In this special sense, η may be called a safety factor with respect to fatigue failure.

3 4.. Quadratic interaction 1 S a + 3ασ mc (3γασ mc + β) β (7) The factor η that amplifies the prescribed stress variation so as to achieve equality in (7) is obtained as β η = (8) 1 S a + 9(1 + γ)α σmc + 3ασ mc This format of quadratic interaction, given by (7), preserves the meaning of the material constants α and β the same as in the linear case. The constant γ determines the contribution of the quadratic term in the interaction function. 5. The minimum Frobenius-norm ellipsoid We consider now some basic elements concerning hyper-ellipsoids in order to find a variational formulation for the ellipsoid with minimum Frobenius norm enclosing the deviatoric stress path dev. An ellipsoid in the deviatoric space IR n (n = 5 in general) is the image of the unit ball by a linear mapping L, shifted by A, that is E = {S IR n S = A + LY, Y IR n Y, Y 1} (9) Then: (i) the center is A, (ii) the lengths of the semi-axes (or principal radii) of the ellipsoid are the singular values {r i, i = 1 : n} of L, that is the non-negative square roots of the eigenvalues of LL T and (iii) these semi-axes are oriented along the unit eigenvectors e i of LL T [10, p. 71]. Further, the sum of the squared radii is obtained by computing the Frobenius norm of L. However, for the purpose of finding enclosing ellipsoids, we may restrict our attention only to canonical forms of ellipsoids where n Y = n and L is symmetric and positive semi-definite. Indeed, for any linear mapping L, eventually non-symmetric, the polar decomposition theorem [11, p. 3] ensures that there exist a symmetric positive semi-definite tensor V and a proper orthogonal tensor R such that L = V R and V = LL T. Then, V represents the same ellipsoid. Consequently, for the canonical representation of an ellipsoid, the lengths of the semi-axes are the eigenvalues {r i, i = 1 : n} of L, which are oriented along the unit eigenvectors e i of L. Moreover, for the present application we must include the cases when L has some null eigenvalues and thus the hyper-ellipsoid is contained in an affine variety (a translation of a subspace [1]) with dimension strictly lower than the deviatoric space dimension (five, in general). This is because the stress variation dev may be dimension defective in the deviatoric space, thus leading to one or more null semi-axes, that is imposing zero eigenvalues to L. Then, aiming to eliminate the dummy variable Y from the representation (9), we consider a pseudo-inverse for L as defined in the following. First, we order the proper values of L so that r 1... r nr > r nr +1 = = r n = 0 (10) where n r is the rank of L. Then, denoting by {M i ; i = 1 : n} the set of eigenvectors of L, we write L = ( ri M i M i) = Q diag(r 1,..., r nr ) Q T (11) i=1:n r where diag(r 1,..., r nr ) IR nr nr is the diagonal matrix whose entries are {r i ; i = 1 : n r } and Q IR n n r is the matrix whose columns are the first n r eigenvectors. This is written in the present notation as col j Q = M j for j = 1 : n r. That is, L transforms the unit ball in IR n into a full dimension ellipsoid of IR n r. Whenever n r is strictly less than n the tensor L has no inverse in IR n. Then, we use the following Moore-Penrose pseudo-inverse (Horn and Johnson 13, p. 41; Golub and Van Loan 10, p. 57) L = ( r 1 i M i M i) = Q diag(r1 1,..., r 1 n r ) Q T (1) i=1:n r 3

4 We can write now the following equivalent representation for the ellipsoid given in (9). E = {S IR n L (S A) 1, S A + R(L)} (13) where R(L) is the image of IR n by L. In the present notation the Frobenius norm of L verifies L = trl = ri = i=1,n i=1:n r r i (14) We are now prepared to present the definition of the minimum Frobenius-norm enclosing ellipsoid in a convenient variational formulation. The tensor L IR n n and the center A IR n defining the minimum ellipsoid enclosing dev constitute the unique solution of the following problem. min L (S A) 1 S dev } (15) (L,A) IR n n IR n{ L 6. Computing the minimum Frobenius-norm ellipsoid enclosing m points in IR n This section is devoted to the analysis and solution of the abstract problem formulated in (15). A simple and effective algorithm is proposed, which is totally autonomous in the sense that can be programmed, and included within a fatigue post-processor, just following the step-by-step instructions given here. We consider now the solution of the minimization problem (15), rewritten below in terms of the set of m sampling points defining dev ρ = min { L L (S k A) 1, k = 1 : m} (16) (L,A) IR n n IR n The Lagrangian of (16) is L(L, A, λ) := L + k=1:m λ [ k L (S k A) 1 ] where λ IR m is the vector of dual variables. Then, the optimality conditions are: L L [ λ k L (S k A) (S k A) ] L = 0 (17) k=1:m k=1:m λ k L L (S k A) = 0 (18) λ k [ L (S k A) 1 ] = 0 L (S k A) 1 λ k 0 (19) Moreover, we find that the following equations are consequence of the above optimality conditions. L 4 = k=1:m A = λ ks k k=1:m λ (0) k [ λk (S k A) (S k A) ] (1) k=1:m The algorithm proposed in the next Subsection computes all the elements of the minimum Frobenius norm ellipsoid. It is based on the optimality conditions given by (19), (0) and (1). It is assumed here that in the implementation of the algorithm each symmetric second order tensor S (a deviatoric stress tensor) is represented by a column vector collecting the components referred to an orthonormal basis in the deviator space. For instance, if σ x, σ y, σ z, S x, S y and S z denote direct total and deviatoric stresses and S xy, S xz and S yz denote shear components, then 3 S (1) = S x = 1 (σ x σ y σ z ) () 6 S () = 1 (S x + S y ) = 1 σ y (3) S (3) = S xy S (4) = S xz S (5) = S yz (4) 4

5 is a compatible set of coordinates corresponding to an orthonormal basis. This allows to present the algorithm in intrinsic notation, and also facilitates its implementation. At each iteration, a new approximation of the matrix L 4 is computed, by using (1), and its eigen-pairs are then found. The nested iterative procedure to solve this five dimensional (or smaller) eigen-problem may be any classical procedure or any embedded system routine. In the application presented herein we have programmed the cyclic-by-row Jacobi algorithm, exactly as described by Golub and Van Loan [10, p. 430] and complemented by ordering the eigenvalues in a non-increasing sequence. The numerical procedure proposed here is a dual algorithm, where the Lagrange parameters are focused in the first place. In step (35) of the algorithm, we compute the pre-image Y k of S k A. This pre-image should be inside the unitary ball when the sample point is feasible. In the subsequent step, (36), the present approximation of λ k is reduced if the pre-image is strictly inside this ball. For the definition of the computational rank used in the algorithm, we refer to Golub and Van Loan [10, p. 61]. 7. An algorithm Set (typically): tol = 10 3, tol r = 10 1, a = 0.9 and iter max = 100 (1) Initial values. λ k = 1 m k = 1 : m (5) A = 1 m S k (6) k=1:m do while ( ρ old ρ tol ρ and iter < iter max ) n r = n (7) Q = 1 (8) M i = col i 1 i = 1 : n (9) r i = 1 i = 1 : n (30) ρ old = 0 (31) ρ = n (3) iter = 0 (33) iter iter + 1 (34) Y k = Q diag ( r 1 1,..., r 1 n r ) Q T (S k A) k = 1 : m (35) λ k ( Y k ) a λk k = 1 : m (36) A = L 4 = k k=1:m λ ks k k=1:m λ k (37) [ λk (S k A) (S k A) ] (38) Compute all the eigen-pairs {(ri 4, M i ), i = 1 : n} of L 4 (39) Find n r, the numerical rank of L 4 for tol r (40) col j Q = M j j = 1 : n r (41) ρ old = ρ (4) ρ = (43) i=1:n r r i end do (3) Final computation. L = Q diag(r 1,..., r nr ) Q T (44) 5

6 8. Fatigue criteria based on enclosing ellipsoids We propose in the sequel two fatigue criteria, with linear or quadratic shear-pressure interaction, based on the definition of the minimum Frobenius-norm ellipsoid enclosing the shear domain. The definitions below apply to both criteria. σ mc := max{σ m σ } (45) Sa := ri = L (46) where L = i=1,n min L + (S A) 1 S dev } (47) (L,A) IR n n IR n{ L Adopting the latter definition produce fatigue criteria that predict independence of the phase angle in combined torsion and bending, as it is experimentally observed. In fact, consider the combined torsion and bending loading σ x = σ x + σ xa sin Ωt (48) σ xy = σ xy + σ xya sin(ωt ϕ) (49) where σ x and σ xy are normal and shear stress components and (σ x, σ xy, σ xa, σ xya, ϕ, Ω) are the loading parameters defining mean stresses, amplitudes, phase, and frequency. In this case the loading program is itself an ellipsoid, hence this loading domain dev coincides with its enclosing minimum F-norm ellipsoid. So, we only need to compute the corresponding linear mapping L and its Forbenius norm. This gives L = r 1 + r = 3 ( σ xa + 3σ xya) (50) This shows that any criteria using this measure for the shear amplitude will predict independence of the fatigue endurance limit with respect to the phase angle Minimum F-norm ellipsoid with linear interaction (MFEL) Combining the definitions (45), (46) with (5) we choose a simple fatigue criterion, here denoted MFEL, essentially based on the concept of minimum Frobenius-norm ellipsoid of the deviatoric stress path. The particular form of this criterion for the case of combined shear and uniaxial traction, is obtained as the explicit condition σ xa 3 + σ xya + α(σ x + σ xa ) β (51) by taking into account (45), (46), (5) and (50). The above equation exactly coincides with the analogous particularization of Papadopoulos criterion given in Papadopoulos et al. [7, p. 30]. This is a very simple and successful formula that is proven to be very effective in the comparison of several criteria with respect to combined torsion and bending experiments, as reported in the cited reference. The prediction of this model for the influence of the mean tensile stress in pure traction fatigue tests is given by the condition ( ) α σ xa + ασ x β (5) This linear approximation for the experimental data in traction-compression fatigue tests is sometimes poor. This is one of the reasons to prefer a quadratic interaction for shear and mean stress in fatigue modelling, although an additional material constant must be introduced; see e.g. Cruz and Zouain [5]. As mentioned above the material constant β equals the fatigue limit in the alternating torsion test. The material constant α is usually identified so as to exactly reproduce the alternating traction limit f 1. This approach, which is not always the best option, leads to the following formulae. β = t 1 α = t 1 f (53) 6

7 Since α is positive this fatigue model can only be applied to hard and brittle materials, those having r = t 1 /f 1 > 1/ Minimum F-norm ellipsoid with quadratic interaction (MFEQ) Combining the definitions (45), (46) with (7) we choose a quadratic fatigue criterion, here denoted MFEQ, essentially based on the concept of minimum Frobenius-norm ellipsoid of the deviatoric stress path. The particular form of this criterion for the case of combined shear and uniaxial traction is written as σ xa 3 + α γ(σ x + σ xa ) + αβ(σ x + σ xa ) + σ xya β (54) by taking into account (45), (46), (7) and (50). The critical amplification factor for combined shear and traction-compression is 3β η = 3σxa + 9α (1 + γ)(σ x + σ xa ) + 9σxya + 3α(σ x + σ xa ) Thus, the predicted alternating bending limit is (55) f 1 = 3β 3 + 9(1 + γ)α + 3α (56) while the repeated bending limit is f 0 = 6β (1 + γ)α + 6α (57) We consider now the identification of the material constants α and γ in view of the above results. First, we notice that this fatigue criterion is able to represent the complete range of metals, from ductile to brittle behavior, provided a suitable value for γ is selected. Let us examine now a common strategy to identify α and γ that consists of exactly reproducing the data corresponding to alternating traction-compression f 1 and repeated tensile stress f 0 (besides setting β = t 1 ). So, we can solve the system of equations consisting of (56) and (57) to compute the material constants as where α = 4r m 1 4(r m) r := t 1 f 1 γ = 3r 6rα 1 3α (58) m := t 1 f 0 (59) However, the fatigue limit in repeated bending is seldom reported in experimental studies and sometimes corresponds to a peak stress that is close to the yield limit for the material (or even beyond this boundary). We adopt here a different parameter identification strategy. First we set β = t 1 and then compute α and γ by using (58) and (59) with two fictitious limits f 0 and f 1 that are varied until we attain a good overall fitting to the whole batch of experimental data available for the material. 9. Applications In this section the MFEL and MFEQ fatigue criteria, proposed in previous sections, are assessed by its application to fatigue strength experiments encountered in the literature. We compare to the results obtained by the Papadopoulos criterion, which is one of the leading present fatigue strength criteria [7]. For being compatible, in our context, with the performance index adopted in the just cited reference, we proceed as follows: For each experiment, for which a critical domain is obtained, we compute the value i := 1 η 100 (%) (60) η 7

8 where η is given by equation (6) or (8), respectively if the MFEL or MFEQ criterion is been evaluated. This index i measures, for a given experiment, the relative difference between the left and right hand sides of inequality (5). just like the index adopted in [7]. In particular, when η = 1 we get i = 0 and this means that the experimentally determined domain perfectly agrees with the critically amplified domain predicted by the criteria. In the following, the proposed criteria are applied to bending and torsion tests conducted on hard metal specimens collected in the comparative study of [7]. Table 1 shows the experimental results, reported by [14], for a series of plane bending and torsion tests on specimens made of a 30NCD16 steel, from sample, neck diameter of 10mm and ultimate strength σ u = 1880 MPa; the proposed models performances are evaluated and compared with Papadopoulos criterion. In Table 1 we demonstrate the proposed criteria performance according to index (60). Material parameters are as follows: MFEL with α L = and β = 410 MPa, and MFEQ with α Q = 0.006, γ = 190 and β = 410 MPa. Present results are compared to Papadopoulos criterion [7], with α P = and β = 410 MPa. Table 1: Fatigue limits for torsion-bending tests on a 30NCD16 steel; data from Froustey [14] σ x ( MPa) σ xa ( MPa) σ xya ( MPa) ϕ xy (deg) P. et al. (%) MFEL (%) MFEQ (%) Mean value Standard deviation Maximum deviation In Table, for conciseness, we report the basic statistical information corresponding to the evaluation performed on the following experiments: 1. The Hard Steel tested by Nishihara and Kawamoto [15], as reported in Liu s fatigue limits collection [16].. The 4CrMo4V steel tested by Lempp [17], as reported in Liu s fatigue limits collection [16], including the fatigue limit in repeated tension. 3. The 34Cr4 steel tested in [18], as reported in Liu s fatigue limits collection [16], including the fatigue limit in repeated tension. Table display the performance index (60) of the proposed models and Papadopoulos criterion [7] for the above three collections of bending and torsion experiments on hard metals [15, 17, 18]. The repeated columns in Table, for the Hard Steel and 4CrMo4V steel, are due to the fact that for such materials the visual inspection of the interaction diagrams for the available set of experimental data does not show any tendency for a quadratic sensitivity with the mean stress. 8

9 Table : Models performance in bending and torsion experiments on hard metals Hard Steel 4CrMo4V 34Cr4 MFEL MFEQ MFEL MFEQ MFEL MFEQ & P. et al. & P. et al. & P. et al. Number of tests Parameter identification α L α Q β ( MPa) γ Performance Mean value (%) Standard deviation (%) Maximum deviation (%) Conclusions The minimum Frobenius-norm enclosing ellipsoid is proposed in this paper as a tool for measuring shear amplitude of variable stresses in fatigue life assessment. Two fatigue criteria are then proposed, both using the Frobenius norm of the minimum-norm ellipsoid enclosing the deviator range. One of these criteria assumes linear interaction between shear to mean stress, and the other one adopts quadratic interaction. The criteria are able to represent a set of collections of experimental data already used in the fatigue literature as benchmark problems. The performance of the proposed assessment procedure, in correctly modeling these experimental results, compares well to known fatigue criteria. The rigorous definition of the minimum Frobenius-norm ellipsoid is accompanied by an effective algorithm to compute the corresponding semi-axes, which was demonstrated in some benchmark problems. 11. References [1] S. Suresh. Fatigue of Materials. Cambridge University Press, UK, second edition, [] K. Dang Van, B. Griveau, and O. Message. On a new multiaxial fatigue limit criterion: Theory and application. In M. W. Brown and K. J. Miller, editors, Biaxial and Multiaxial Fatigue, EGF 3, pages Mechanical Engineering Publications, London, [3] P. Ballard, K. Dang Van, A. Deperrois, and Y. V. Papadopoulos. The effects of mean stress and stress concentration on fatigue under combined bending and twisting. Fatigue Fract. Eng. Mater. Struct., 18(3): , [4] N. Zouain and I. Cruz. A high-cycle fatigue criterion with internal variables. Eur. J. Mech. A/Solids, 1(4): , 00. [5] I. Cruz and N. Zouain. A shakedown model for high-cycle fatigue. Fatigue Fract. Eng. Mater. Struct., 6():13 135, 003. [6] B. Crossland. Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. In Proc. Int. Conf. on Fatigue of Metals, pages Institution of Mechanical Engineers, London, [7] I. V. Papadopoulos, P. Davoli, C. Gorla, M. Filippini, and A. Bernasconi. A comparative study of multiaxial high-cycle fatigue criteria for metals. Int. J. Fatigue, 19(3):19 35,

10 [8] Bin Li, J. L. T. Santos, and M. de Freitas. A computarized procedure for long-life fatigue assessment under multiaxial loading. Fatigue Fract. Eng. Mater. Struct., 4: , 001. [9] K. Dang Van and I. Papadopoulos, editors. High-Cycle Metal Fatigue, From Theory to Applications, volume 39 of CISM Courses and Lectures. CISM - International Centre for Mechanical Sciences, [10] G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press, London, third edition, [11] M. Šilhavý. The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin, [1] J-B. Hiriart-Urruty and C. Lemarchal. Convex Analysis and Minimization Algorithms I. Springer- Verlag, Berlin, [13] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, [14] C. Froustey. Fatigue Multiaxiale en Endurance de l Acier 30NCD16. Docteur de l E.N.S.A.M., L École Nationale Supérieure d Arts et Métiers, [15] T. Nishihara and M. Kawamoto. The strength of metals under combined alternating bending and torsion with phase difference. In Kyoto Imperial University, editor, Memoirs of the College of Engineering, vol. XI, no. 5, pages Kyoto Imperial University, [16] J. Liu. Beitrag zur Verbesserung der Dauerfestigkeitsberechnung bei mehrachsiger Beanspruchung. Diss, TU Clausthal, [17] W. Lempp. Festigkeitsverhalten von Sthlen bei mehrachsiger Dauerschwingbeanspruchung durch Normalspannungen mit berlagerten phasengleichen und phasenverschobenen Schubspannungen. Diss., Uni. Stuttgart, [18] R. Heidenreich, H. Zenner, and I. Richter. Dauerschwingfestigkeit bei mehrachsiger Beanspruchung. Forschungshefte FKM, Heft