A Modified Neuber Method Avoiding Artefacts Under Random Loads

Transcription

1 A Modfed Neuber Method Avodng Artefacts Under Random Loads T. Herbland a,b, G. Calletaud a, J. L. Chaboche c, S. Qulc a, F. Gallerneau c a Mnes Pars Pars Tech, CNRS UMR 7633, P 87, vry cedex, France b CTIM, Pôle DSL, Senls, France c ONRA, DMS, P72, Châtllon cedex, France Abstract Surprsngly, Neuber's method s n error when appled to non regular cyclc loads. Some examples are frst shown, n whch the predcted result by the classcal formulaton produces a non physcal local ratchettng n the notch, for a global load made of a large symmetrc cycle and of a small cycle nsde the large one. A modfcaton of the orgnal method s then propopsed to avod the occurence of ths artefact. It s successfully tested aganst fnte element computatons on several notched specmens. 1 INTRODUCTION Many engneerng components subjected to cyclc loadng contan notches lke grooves, holes, keyways, welds, etc. When such a component s loaded, a stress concentraton appears at the notch root. Fatgue can then be reduced snce, even f the component remans globally elastc, a plastc zone develops at the notch root, and an early crack ntaton can be observed. In order to predct fatgue lfe of such components, engneers have to compute the local stresses and strans. Such type of calculaton may be qute long, especally when non-lnear consttutve equatons are used. Smplfed methods are then appled, as an alternatve to FM computaton. Although many solutons have been nvestgated, lke Molsk-Glnka [1], Glnka [2,3], llyn and Kujawsk [4] Neuber s rule [5] remans the most currently used for ndustral applcatons. In the case of unaxal constant ampltude loadngs, t appears to be a good approxmaton; however, a ratchettng phenomenon may appear when the method s appled to varable-ampltude loadngs. In ths paper, a modfcaton to Neuber s rule s then proposed, and tested for several loadngs spectra. 2 PRSNTATION OF NUR S MTHOD Ths method makes t possble to get an approxmaton of the elasto-plastc stress and stran at a notch root, from an elastc computaton, as represented n Fgure 1. σ nom σ nom (σ e = K t σ nom, ε e ) (σ N,ε N ) Fgure 1 : Illustraton of Neuber s rule Result of an elastoplastc computaton ; local stress and stran accordng to equaton (1) If ths notched component, loaded by a nomnal tensle stress σ nom, would reman purely elastc (Fgure 1 ), ths load would nduce a local stress σ e and stran ε e at the notch root. Defnng K t as the stress concentraton factor, σ e can be expressed as σ e = K t σ nom. σ e and ε e can be reached through a smple elastc FM

2 computaton. To calculate the elasto-plastc stress σ N and stran ε N, Neuber [5] postulates that, for a local plastcty, the product of stress and stran at the notch root does not depend on the plastc flow : N N σ ε = e e σ ε (1) Fgure 2 s a graphcal representaton of Neuber s rule n a stress-stran space. quaton (1) mples that the area of the rectangle defned by σ e and ε e s equal to the one defned by σ N and ε N. One can also remark that the elastc pont (σ e, ε e ) and the elasto-plastc pont belong to a hyperbola defned by σ ε = σ e ε e Fgure 2 : Classcal Neuber's method Classcal extenson to cyclc loadng Ths method can be extended to cyclc loadngs. The classcal approach conssts n transferrng the classcal constructon on a cyclc dagram, as llustrated n Fgure 2, usng stress and stran ranges nstead of stress and stran n equaton (1). The subsequent pont s found on the cyclc curves, followng equaton (2) : N σ ε N e e = σ ε (2) Ths soluton s not satsfactory, snce t cannot take nto account mean stresses that would be present n global loadngs. Ths s why more precse strateges have been proposed [6,7] (Fgure 3), based on an update of the stress-stran curves for each reversal. For a gven loadng branch (), the current pont always refers to the last couple (σ N -1, ε N -1) reached n the prevous loadng step. The resultng computaton scheme s gven by equaton (3). For nstance, Fgure 3 llustrates the case of the frst reversal. σ ε 1 ε 2 σ Fgure 3 : xtenson of Neuber s rule to cyclc loadngs wth an updatng strategy

3 N N N N e e e e ( σ σ 1 )( ε ε 1 ) = ( σ σ 1 )( ε ε 1) (3) Once determned the frst maxmum stress pont 1 by the classcal method, one can smply compute the stress and stran for the second extremum 2 by nvertng the (σ,ε) axes and choosng pont 1 as the new orgn. In some case, the local stress tensor s not unaxal. A seres of possble extensons can also be found n the lterature, for nstance, usng Von Mses nvarants nstead of components (equaton (4)) N N J ( σ ) J ( ε ) = J ( σ ) J ( ε ) (4) Ths s stll an open problem, wth no unque soluton for plane stran or axsymmetrc problems. It wll not be treated n ths paper, whch s restrcted to plane-stress cases. N N J ( σ N σ N 1 ) J ( ε ε 1) = J ( σ σ 1) J ( ε ε 1 ) (5) 3 RATCHTTING FFCT FOR VARIAL-AMPLITUD LOADING A 2D plane-stress notched test specmen s chosen, so that the normal stress s null at the notch root, and an unaxal stress state appears wth only σ When applyng Neuber s method to varable ampltude loadngs, we observed a ratchettng phenomenon, even for a lnear knematc hardenng. Ratchettng was not observed when computng the same notch subjected to the same loadng n smulatons wth the FM code ZeuLoN. Moreover, such a localzed ratchet s unrealstc n plastc confnement condtons. Fgure 4 llustrates the loadng geometry. Only one quarter of the test specmen s modelled, to respect symmetry condtons, dsplacements are fxed n drecton 1 on the specmen axs, and fxed n drecton 2 on the bottom. A tensle stress σ nom s appled at the top of the specmen. Fgure 5 presents the response at the notch root under the loadng of Fgure 4. The local elastoplastc stress σ N s drawn wth respect to the local total elastoplastc stran ε N. On Fgure 5 these values are calculated followng the classcal Neuber algorthm, on Fgure 5 they are calculated followng the new Neuber algorthm, that wll be explaned n the next part. σ nom σ Ε (MPa) T t Fgure 4 : Mesh and boundary condtons appled to the specmen ; Loadng hstory, σ e = K t σ nom

4 Table 1 : Materal propertes (lnear knematc hardenng) R 0 (MPa) C (MPa) d (GPa) ν ,3 Fgure 5 : Ratchettng response under the loadng defned n Fgure 4 Intal algorthm Modfed algorthm 4 NUR S MTHOD MODIFICATION Fgure 6 shows an elastc loadng that wll be used to explan the prncple of the new strategy. The problem s that the cyclc rule must take nto account the pont where a small loop reaches the boundary left by the last large loop. The constructon s llustrated n Fgure 7. Steps A and are made accordng to the prevous defned algorthm. The proposed modfcaton s made durng step 3: accordng to classcal plastcty, when the reloadng branch reaches pont C, the subsequent curve must follow the ntal branch, wth ts orgn at pont O, whch produces the evoluton after pont C shown n step 4. Keepng as the orgn of the reloadng branch would produce an extra-hardenng, as shown n step 4. Fgure 6 : response of an elastc computaton σ 1 ε 2 A σ 3 C A O ε 1 σ 2 ε 3 Step 1 Step 2 Step 3

5 σ 4 D σ 4 D Step 4 ε 4 O ε 4 Step 4 Fgure 7 : Neuber's classcal (Steps 1, 2, 3, 4) and modfed (Steps 1, 2, 3, 4 ) algorthm applcaton to loadng of Fgure 6 Ths algorthm can be compared to the ranflow technque: the classcal Neuber method s appled untl pont C, where σ c = σ a, Once the elasto-plastc loop s closed, t s erased from the loadng hstory, and the elasto-plastc soluton for next ponts lke D s calculated by choosng O as orgn. The algorthm s fully detaled n Table 2. It s executed at each stress ncrement. n s the number of reversals to be erased at the begnnng of the algorthm, before computng the soluton (σ Ν, ε Ν ). The value of n used for ncrement s determned at the end of the algorthm for ncrement k-1. Pe represents the ampltude of the reversal whch ends at pont, accordng to Neuber s formula. The resultng expresson s then: Pe J ( σ σ 1 ) J ( ε ε 1). = ( Table 2 : The modfed algorthm (*) For each ncrement k endng at a pont of the loadng hstory: rase the n reversals prevous to the current ncrement from the loadng hstory (n s determned at ncrement k-1) The new orgn for Neuber s calculaton s now the latest peak before to the erased reversals In case of unloadng: The new orgn for Neuber s calculaton s now pont Re-ntalzaton: n = 0 (**) If loadng hstory contans less than 3 reversals: go to (***) If Pe > Pe 1 n n = n + 2 : the two last reversals wll be suppressed at the next ncrement go to (**) (***) Calculaton of σ N, ε Ν accordng to Neuber s rule and go to (*) As shown n Fgure 5, the present method allows the cycle to reman symmetrc for the loadng defned n Fgure 4. Fgure 8 shows a complex hstory computed wth the present algorthm. Reversal suppressons occur at ponts 1, 2 and 3. The result obtaned wth a knematc hardenng rule s shown n Fgure 9. The materal has the same propertes than n Table 1. A non lnear knematc hardenng rule p p X = C ε dx ε, materal propertes n Table 3) gves the result presented n Fgure 10.

6 σ e (MPa) A O C D 1 F G H 3 I 2 K J Tme Fgure 8 : Test loadng hstory showng asymmetrc reversals Fgure 9 : lastoplastc response to loadng of Fgure 8 wth a lnear knematc hardenng followng : Classcal Neuber s algorthm New Neuber algorthm Table 3 : Materal propertes (non-lnear knematc hardenng) R 0 (MPa) C (MPa) d (GPa) ν ,3 Fgure 10 : lastoplastc response to loadng of Fgure 8 wth a non-lnear knematc hardenng followng: Classcal Neuber s algorthm New Neuber algorthm

7 5 CONCLUSIONS Local stress and stran hstory was determned followng classcal Neuber method at the notch root of a notched component under varable-ampltude loadng. The elastoplastc soluton was compared to a soluton reached by FM computaton. A ratchettng effect that dd not appear by usng FM, was observed by usng Neuber s rule. Consderng that ths ratchet was due to asymmetrc cycles, a modfed Neuber method was proposed. Lke the classcal one, ths new algorthm can only be appled to unaxal stress states. It s also based on Neuber equvalence rule, but a change of orgn at certan moments of the hstory makes t possble to avod the shft due to asymmetrc cycles. Ths new algorthm was tested successfully under several complex load spectra, for lnear and non-lnear knematc hardenng rules. 6 RFRNCS [1] K. Molsk and Glnka G, Materal and Scence ngrg, 50, 1981, [2] G. Glnka, ngrg. Fract. Mech., 22, 1985, [3] G. Glnka, ngrg. Fract. Mech., 22, 1985, [4] F. llyn, D. Kujawsk, ngrg. Fract. Mech., 1989, 32, [5] H. Neuber, J. Appl. Mech. (Trans. ASM), 1961, 28, [6] J. Lematre, J.L. Chaboche, Mécanque des matéraux soldes, nd edton. [7] M. Chaudonneret, J.P. Culé, La Rech. Aérosp., nglsh Transl., 1985, , 33-40