A Multiaxial Variable Amplitude Fatigue Life Prediction Method Based on a Plane Per Plane Damage Assessment

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1 American Journal of Mecanical and Industrial Engineering 28; 3(4): ttp:// doi:.648/j.ajmie ISSN: (Print); ISSN: (Online) A Multiaxial Variable Amplitude Fatigue Life Prediction Metod Based on a Plane Per Plane Damage Assessment Bianzeube Tikri, *, Fabienne Fennec 2, Bastien Weber 3, Jean-Louis Robert 2 Department of Mecanical Engineering, Polytecnic University of Mongo, Mongo, Cad 2 Department of Mecanical Engineering, Clermont Auvergne University, Clermont-Ferrand, France 3 Department of Metallurgical and Materials Engineering, ARCELORMITTAL Maizieres Researc, Maizieres-Les-Metz, France address: * Corresponding autor To cite tis article: Bianzeube Tikri, Fabienne Fennec, Bastien Weber, Jean-Louis Robert. A Multiaxial Variable Amplitude Fatigue Life Prediction Metod Based on a Plane Per Plane Damage Assessment. American Journal of Mecanical and Industrial Engineering. Vol. 3, No. 4, 28, pp doi:.648/j.ajmie Received: June 23, 28; Accepted: July 9, 28; Publised: August 7, 28 Abstract: A multiaxial variable amplitude fatigue life prediction metod is proposed in tis paper. Tree main steps are distinguised. Te first one concerns te counting of multiaxial cycles and uses te normal stress to a pysical plane as te counting parameter. Ten a multiaxial finite fatigue life criterion allows one to assess te material life corresponding to eac cycle on any pysical plane. A damage law and its cumulation rule describe te damage induced by eac cycle plane per plane. By tis way te critical plane for a given multiaxial stress istory is found out. It is assumed to be te fracture plane and te fatigue life of te material is traduced as te number of repetitions of te sequence up to crack initiation. At tis stage, material fatigue criteria and linear and nonlinear damage laws assume tat te material is damaged. One distinguises among tese criteria critical plan type wose formalism can identify te crack initiation plan. An application is given for eac load. In te context of multiaxial solicitations of variable amplitude, a validation of te estimation of te orientations of te priming planes is carried out based on experimental results on cruciform test pieces; te estimated orientations are close to tose observed experimentally. Keywords: Multiaxial Fatigue, Variable Amplitude, Fatigue Life, Damage Law, Cycle Counting. Introduction Most mecanical structures or macine elements are nowadays fatigue-prone components. As bot economical and environmental constraints lead nowadays to a decrease of material weigt for some given work conditions, te materials are tus often submitted to ig stress levels. In te case of variable amplitude loading, stress amplitudes are consequently larger and induce fatigue damage of materials. te durability assessment as become by tis way a ceck point for lots of mecanical components tat engineers ave to design. An accurate assessment of te level of safety of a structure requires a toroug examination of te wole components to identify and ten verify teir critical areas. despite te fact tat for uniaxial variable amplitude or multiaxial constant amplitude loadings, some efficient tools are developed concerning cycles counting and multiaxial fatigue criteria, few adequate metods are proposed for te most general case of solicitations, i.e. te case of multiaxial variable amplitude stress states istories. Te purpose of tis paper is to describe a stress based approac tat allows one to assess te fatigue life of materials submitted to suc a kind of loading. Tis work as been developed witin te framework of a collaboration between SOLLAC (steel manufacturer), te researc department of EDF (electric power generation) and te laboratory of Solid Mecanics of INSA-Lyon.

2 48 Bianzeube Tikri et al.: A Multiaxial Variable Amplitude Fatigue Life Prediction Metod Based on a Plane Per Plane Damage Assessment 2. Principle of te Multiaxial Variable Amplitude Fatigue Life Prediction Metod Designers ad for a long time to assess te reliability of mecanical structures wic are submitted to variable amplitude stress states. In te case of uniaxial variable amplitude stress istories, usual fatigue life prediction metods use cycles counting, as level cross counting or more often Rain flow counting, and damage and cumulation rules tat are devoted to traduce te fatigue damage penomenon. For multiaxial variable amplitude stress states, few metods owever are proposed today even if a more and more accurate assessment of fatigue prone components is required. Carpinteri [] and Wang [2] proposed a metod derived from low cycle fatigue tat assumes te material to be more sensitive to te sear stress effect or to te normal stress effect. Zeng-Yong [3] as built a continuum damage model tat defines a damaging process independently from any cycle counting. Nicolas [4] recently proposed a fatigue life assessment based on statistical parameters of te stress istory. Te presented metod is derived from classical uniaxial stress states fatigue life assessment metods. It is based bot upon te extension of a multiaxial fatigue life criterion to finite lives and te definition of a counting variable in order to identify cycles troug multiaxial variable amplitude sequences. Te two main points of te metod are detailed in te following sections. Te counting variable tat is used as to be good representative of te all components of te stress tensor and also of teir real evolution versus time. Te fatigue life is assessed at te point of te mecanical component were te stress states istory is known by considering all te possible material planes troug tat point. As te crack initiation is developing on one particular plane of te material, all te possible orientations of te considered plane are examined and te damage is assessed troug te stresses tat te loading induces on tat plane. Tis concept is te origin of te plane per plane damage and fatigue life valuation. Two linear and non linear damage law and cumulation rules ave been adapted to multiaxial stress states to allow fatigue lives assessments. Te metod is general, i.e. suitable for any kind of loading istory. Te different steps of te procedure are described on te following flow cart (figure ). Plane per plane investigation Multiaxial variable amplitude stress states istory Counting variable sequence on a given plane Definition of te counting variable Rainflow counting Identification and extraction of multiaxial cycles Multiaxial fatigue criterion Fatigue life assessment of te material plane for eac extracted cycle Damage law and cumulation rule Damage and fatigue life assessment of te considered plane for te wole sequence Critical plane determination (most damaged plane) and fatigue life of te material Figure. Flow cart of te plane per plane fatigue life assessment metod. Te assumption is made tat te critical plane of te material, i.e. te plane were te damage reaces te igest value, is te fracture plane of te material and consequently implies its fatigue life. Te major points of te proposed metod are first te counting of multiaxial cycles from a six time series istory (i.e. te six components of te symmetrical stress tensor) and also te use of a multiaxial critical plane fatigue criterion to assess te damage of any multiaxial cycle. Te following sections give details of tese steps. 3. Multiaxial Cycle Counting Te Rain flow counting procedure is te emergent tecnique tat is used today to identify stress cycles. It is based upon te appearance of a closed loop in te stressstrain material response [5]. Te present purpose is not to develop te usual Rain flow tecnique even if te practical application may vary rater widely, despite te fact tat te

3 American Journal of Mecanical and Industrial Engineering 28; 3(4): basic principles are well known. Tis point as led to an industrial frenc consensus about a standard procedure. Actually te Rain flow counting can be applied only to a uniaxial sequence. In te case of multiaxial constant amplitude stress states, te cycles counting can be correctly obtained wen considering any non constant component of te stress tensor because all te components ave te same frequency and as a consequence te same time period (figure 2). Wen a cycle occurs on one cannel, all te oter cannels experience also a cycle. te oters, a cycle may occur on one cannel but tere is rarely a corresponding cycle at te same time period on anoter cannel (figure 3) cycle Time 2 Time Figure 2. Identification of a cycle for a multiaxial constant amplitude stress states istory. Cycles counting are really more complex for istories of multiaxial random stress states. As a matter of fact, wen all te components of te stress tensor are independent one from Figure 3. Problem of cycle s identification for a multiaxial variable amplitude stress states istoty. Te counting and identification of cycles need consequently te requirement of a counting variable wic must be a correct representation of te stress states and of teir evolution as a function of time t. J 2 J 2 I I time 2 time (a) Figure 4. Evolution of invariants J 2 and I : (a) during a pure torsion-compression cycle, (b) during a pure torsion cycle. (b) Stress or deviatory stress invariants are not representative of te evolution of stress states as a function of time enoug to be considered as te counting variable. Figure 4 gives, for a pure tension-compression stress states cycle (respectively a pure torsion stress states cycle), te corresponding evolution of te second invariant J 2 of te deviatory stress tensor and te first invariant I of te stress tensor. Te first invariant I is not suitable to count cycles for a pure torsion stress states as it remains equal to zero during te wole cycle (te ydrostatic pressure is equal to zero for suc stress states). Te second invariant J 2 of te deviatory stress tensor is not usable too as in eiter tension or torsion cases; its frequency is twice te real frequency of te stress states. Tis variable would not allow one to identify te real cycles. As te fatigue crack initiation on a given plane is induced by stresses tat are acting on it, te counting variable as to be closely related to tese stresses. Te sear stress acting on a given plane may rotate in tis plane during te stress istory. Consequently two components are necessary to describe it properly and tus make te Rain flow counting be inapplicable from tis point of view. Te normal stress acting on a considered pysical plane tat is defined by its unit normal vector (figure 5) is proposed as te counting variable. Two angles ϕ and γ define te orientation of te unit vector tat is expressed by: sin sin P γ γ cos ϕ 3 γ cos sin γ ϕ ϕ Figure 5. Definition of te unit normal vector to a pysical plane. 2

4 5 Bianzeube Tikri et al.: A Multiaxial Variable Amplitude Fatigue Life Prediction Metod Based on a Plane Per Plane Damage Assessment Te identification of a cycle is governed by te fact tat te counting variable does not remain constant wen te stress tensor is canging it is to say wen any of its six components is varying wit respect to time. Oterwise no stress evolution is detected and te counting procedure can not identify any stress cycle. From tis point of view, te identification of cycles is very similar to te uniaxial stress states cycle occurrence conditions. Wen a cycle occurs witin te counting variable istory, te corresponding multiaxial stress cycle is extracted from te multiaxial stress states sequences (figure 6). Te fatigue life of suc a cycle is ten determined by te use of a multiaxial stress criterion. (t) Figure 6. Identification of a multiaxial stress states cycle. 4. Fatigue Life Prediction Metod 4.. Multiaxial Critical Plane Criterion A multiaxial fatigue criterion based on te critical plane concept is employed to assess te life of any multiaxial stress cycles [6]. For tat purpose it as been extended from infinite fatigue lives (endurance limit) to finite ones. Te formulation of te critical plane endurance criterion is given by equation ().[ ] represents te stress states cycle., and τ are te so-called fatigue limits of te material for a reversed tensile test (stress ratio R equal to -), a reversed R and a zero-to-maximum tensile test torsion test ( ) R respectively. E is te fatigue function of te criterion. ([ ( t )],, τ ) E ij (), Te formulation of te criterion adapted to finite lives is: ([ ( t )], ( N ), ( N ), τ ( N ) E ij (2) were, and t t 2 time τ are te fatigue strengts of te material corresponding to N cycles for te same cyclic stress states as, and τ. Equation (3) indicates tat te fatigue life of te material for te multiaxial cycle [ ]is N cycles. For te endurance criterion, te fatigue life corresponds to te tresold of te endurance domain. Te fatigue function E of te criterion is obtained from searcing te critical material plane. A time dependent fatigue indicator E ( t) is defined as a linear combination of te different mean and alternate stress components of te stress vector φ tat is acting on te material plane wic unit normal vector is denoted. E ( t) τ a τ a were ( t) and ( t) ( t) + α a ( t) + β θ a m (3) are te alternate sear stress vector and te alternate normal stress tat are acting on te considered plane at time t. m is te normal stress acting on tat plane during te cycle. τ a ( t) is obtained by building te surrounding circle to te so-called loading pat wic is constituted by te tip of te sear stress vector τ ( t) during te cycle (figure 7). Te loading pat is generally a closed loop, especially for constant amplitude solicitations. Te center O of te surrounding circle defines te mean sear stress vector τ m ( t) acting on te material plane during te cycle (it is generally assumed it as no contribution to te fatigue damage of te material tus it is not involved in te fatigue indicator E ( t) ). Te alternate sear stress vector is stated as: v O τ τ (t) a ( t) τ( t) τm τ m τ a (t) Figure 7. Definition of te mean and alternate components of te sear stress vector. Finally te fatigue indicator E of te criterion during a cycle for a given pysical plane is obtained by: ( t) (4) E Max E (5) t Te critical plane criterion concept requires searcing for te most damaged plane (denoted as te critical plane) during O u

5 American Journal of Mecanical and Industrial Engineering 28; 3(4): te considered stress cycle, it is to say te plane were E reaces te igest value: Te parameters α, β and E Max E (6) θ are used to establis te damage indicator of te criterion. Tey are obtained by τ ( 2 2 N θ τ α + ; β τ τ N N N α α, β and θ describe in fact te sensitivity of te material to te alternate sear stress, to te alternate normal stress and to te mean normal stress. Tey determine te respective contributions of tese stress components to te fatigue damage of te material. In te case were te tree fatigue strengts vary wit te same factor versus te number of cycles, tese respective sensitivities of te material remain te same. Oterwise some of tem may become predominant for te fatigue damage process depending on te microstructure of te material [7]. Te fatigue life of te material for a given multiaxial cycle is determined by solving equation (2). An implicit algoritm is used to calculate te life N. It is based upon te meaning of te difference between te criterion fatigue function E and te teoretical value ( E ) wen te fatigue strengt of te material submitted to te given multiaxial cycle is reaced:. If E is greater tan unity ( E > ), te cycle as larger stress amplitude tat wat te material is able to endure N times. Te real fatigue life is less tan N cycles (reference life) tat were used to make te first calculations of E. Parameters α, β and θ are recalculated for a smaller life N and a new fatigue stating tat te multiaxial critical plane criterion is verified for te tree basic fatigue tests up to N cycles tat give te tree fatigue strengts, and τ. For eac one of tese fatigue tests one as: E. Finally te parameters are expressed as: θ 8θ 2 α function E is assessed. 2. In te oter case ( E < ), te real fatigue life is greater tan te reference one. Parameters α, β, θ and te fatigue function E are calculated once again for a greater number of cycles. It is to notice tat tis concept allows one to assess te real fatigue life of any material plane. As a matter of fact, all te possible material plane are not submitted to te same levels of sear and normal stresses. Tis induces various values of te damage indicator depending on te orientation of te considered plane as sown for te two following examples. Figure 8-a gives te distribution of te damage indicator E for a fixed principal stress directions cycle. Figure 8-b gives te same representation for a cycle wic principal stress directions rotate. Te damage indicators were calculated in bot cases for te number of cycles N D corresponding to te endurance tresold of te material. In oter words α, β and θ were establised by using te fatigue limits, and τ. (7) E angle ϕ (a) angle γ Figure 8. Distribution of te fatigue indicator E : (a) for a constant principal stress directions cycle, b) for a rotating principal stress directions cycle. E angle ϕ (b) angle γ

6 52 Bianzeube Tikri et al.: A Multiaxial Variable Amplitude Fatigue Life Prediction Metod Based on a Plane Per Plane Damage Assessment Tese two distributions were obtained wit material data given in table : Table. Material and stress states cycles data. Material Ref. N D (cycles) (Mpa) (MPa) Case (a) XC 48 [8] Case (b) 35 Cr 4 [9] τ (MPa) ( t ) ij sin( ωt) sinωt sin( ωt) π 2 95 sin ωt + ϕ and γ are te angles tat define te orientation of te unit normal vector to te considered plane. Te distribution of te damage indicator all over te possible material planes sows te more severe aspect of te multiaxial rotating principal stress directions cycle. Wen principal stress directions are fixed, a limited number of planes are critical. In te case of rotating principal stress directions, an infinite number of planes are equally critical. In tis case of course te microstructure defects or weakness induce te site of te crack initiation among all tese possible planes Plane Per Plane Damage Cumulation Te fatigue life of a material plane is derived by te metod described above for all te cycles tis material plane experiences. A damage law is used to assess te damage induced by eac cycle and a cumulation rule allows one to obtain te amount of damage corresponding to te wole sequence. Two damage rules may be used for tis step of te metod. Te first one is te well known linear Miner s rule [8], te second one is te non linear law proposed by Lemaitre and Zi Yong Huang [9]. Tis law gives te damage increase δ D due to δ N identical uniaxial stress cycles defined by teir amplitude a and teir mean value m [7] as: δd ( D) β + α M a ( b )( D) m β δn α If < (small amplitude cycle), A m Is te material fatigue limit as a function of te mean stress. It is given by te endurance constant life diagram of te material and is expressed as: ( ) A b m a, b, β and M are material parameters, R m is te material ultimate tensile strengt. Tis non linear damage law allows small amplitude cycles to contribute to te material damage and takes into account te occurrence order of te cycles. 5. First Validation of te Fatigue Life Prediction Metod Te first validation of te proposed metod against experiments is realised wit biaxial random stress states istories. Te tests were carried out by Jan Papuga [] in te laboratory of professor Maca in Opole (Poland), on cruciform specimen (figure 9) made of low carbon steel denoted HNAP. Te tables 2 and 3 give respectively te cemical composition of te material and its mecanical static properties. Were: a A ( m ) α a if > ( ) R m a a m A m (large amplitude cycle), Table 2. Cemical composition of te HNAP steel. Figure 9. Cruciform specimen used for biaxial stress states fatigue tests. Elements C Mn Si P S Cr Cu Ni Contents (%) Table 3. Material mecanical static properties. e [MPa] R m [MPa] ν E [MPa] Tree material crack initiation S-N curves, and τ for respectively reversed tension-compression tests ( R ), zero to maximum tensile tests ( R ) and reversed torsion tests ( R ), are required. Tese material data are plotted on figures, and 2.

7 American Journal of Mecanical and Industrial Engineering 28; 3(4): Log( - ) τ - Log(252.) - (N) 82. τ - (N) Figure. Fully reversed tension-compression S-N curve ( N ) Log(28) (N) Log(26988) Log(N) Log(N) Figure. Zero to maximum tensile S-N curve ( N ). Log(26988) Log(N) Figure 2. Fully reversed torsion S-N curve τ ( N ). Fourteen sequences of biaxial stress states (figure 3) were tested. Experimental lives are compared against predicted ones, wic were calculated by using bot linear and non linear damage laws. Lives are expressed as te number of repetitions of sequences up to crack initiation. Predicted lives are plotted against experimental ones in figure 4. Figure 3. Biaxial stress states sequences (sample). Lemaitre & Caboce Miner Predicted Fatigue Life Experimental Fatigue Life Figure 4. Experimental results against predicted assessments. Predicted fatigue lives are generally conservative, especially wen te non linear damage law is used. Tis comes from te fact tat according to tat non linear law small amplitude cycles bring teir own contribution to te damage and make decrease te fatigue lives. Te average ratio between experimental lives and predicted ones are equal to 2.2 and 3.4 for Miner s rule and Lemaitre and Caboce non linear damage law. Te determination of te critical plane gives te fatigue life of te material up to crack initiation and also te orientation of te expected fracture plane corresponding to eac sequence. Te distribution of te fatigue damage of pysical planes can be plotted versus te orientation of teir normal vector defined by te two angles ϕ and γ. Te figure 5 gives an example of suc a distribution obtained for one treated sequence. A double symmetry wit respect to

8 54 Bianzeube Tikri et al.: A Multiaxial Variable Amplitude Fatigue Life Prediction Metod Based on a Plane Per Plane Damage Assessment ϕ 9 and γ 9 is observed because of te pure biaxial stress states of te sequences. Te distributions ave identical sapes for Miner s rule (rigt side) and Lemaitre and Caboce s rule (left side) Figure 5. Distribution of te damage versus pysical planes orientations according to te non linear / linear damage laws. A life assessment requires te examination of all te possible material planes. Practically, a selected number of pysical planes are examined and te calculation procedure is repeated for eac of tem. An optimization of te orientations of te investigated planes as been realized [] so tat teir normal vectors are equally distributed in a tree dimensional space and as allowed a strong reduction of time calculations. Suitable algoritms were also developed for te determination of te alternate sear stress vector τ a ( t) by building te smallest surrounding circle to te loading pat. Te exact geometrical solution is now rapidly found for any kind of loading pat. Te calculation times are now admissible wit respect to current design request from an industrial point of view. 6. Conclusion Lemaitre & Caboce Miner. ϕ critical 9 ϕ critical 8 A new fatigue life prediction metod suitable for any kind of multiaxial variable amplitude stress states istory is proposed. It is a stress approac based upon a plane per plane damage distribution, directly related to te stresses experienced by tese material planes. Te identification of multiaxial cycles is processed for any plane by considering te normal stress to tis plane as te counting parameter. A multiaxial fatigue criterion extended from endurance to finite lives allows one to assess te life of te cycle wit accounting for te six components of te stress states tensor. Linear or non linear damage laws are usable to express and 9 8 γ critical make te cumulation of damage versus time. Te procedure allows te assessment bot of te crack initiation plane and te fatigue life of te material. A first validation of te life prediction metod is realized by te way of fourteen biaxial random stress states istories issued from tests carried out on cruciform steel specimens. Te average values of te ratio between experimental lives and expected ones indicate conservative assessments of 2.2 by using Miner s damage rule and 3.4 wit Lemaitre and Caboce damage law. References [] Carpinteri A., Spagnoli A., Critical plane criterion for fatigue life calculation: time and frequency domain formulations, 3rd International Conference on Material and Component Performance under Variable Amplitude Loading, VAL25, 25 pp [2] Wang C. H., Brown M. W., Life Prediction Tecliniques for Variable Amplitude Multiaxial Fatigue, Journal of Engineering Materials and Tecnology, 996, Vol. 8/367. [3] Zeng-Yong Yu, Sun-Peng Zu, A New Energy-Critical Plane Damage Parameter for Multiaxial Fatigue Life Prediction of Turbine Blades, [4] Nicolas R. Gates, Ali Fatemi, Multiaxial variable amplitude fatigue life analysis using te critical plane approac, Part I: Un-notced specimen experiments and life estimations, International Journal of Fatigue, 27 pp [5] Yingyu Wang, Critical Plane Approac to Multiaxial Variable Amplitude Fatigue Loading, Frattura ed Integrità Strutturale, 25 pp [6] Wang, Y. and Susmel, L., Te Modified Manson-Coffin Curve Metod to estimate fatigue lifetime under complex constant and variable amplitude multiaxial fatigue loading. International Journal of Fatigue, 26, pp [7] Bannantine, J. A. and Socie, D. F., A variable amplitude multiaxial fatigue life prediction metod. In Fatigue under Biaxial and Multiaxial Loading, ESIS, ed. K. Kussmaul, D. McDiarmid and D. F. Socie. Mecanical Engineering Publications, London, 99, pp [8] Miner, M. A., Cumulative damage in fatigue, Journal of Applied Mecanics, 945, pp [9] Zi Yong Huang, Danièle Wagner, Claude Batias, Jean Louis Caboce, Cumulative fatigue damage in low cycle fatigue and gigacycle fatigue for low carbon-manganese steel, 28. [] Jan Papuga, Quest for fatigue limit prediction under multiaxial loading, Procedia Engineering, 5t Fatigue Design Conference, Fatigue Design 23. [] Robert, J. L., Fogue, M. and Bauaud, J., Fatigue life prediction under periodical or random multiaxial stress states. In Automation in Fatigue and Fracture: Testing and Analysis, ASTM STP 23, ed. C. Amzallag. ASTM, Piladelpia, 994, pp