A multiaxial stress-strain correction scheme Holger Lang 1, Klaus Dressler 2, Rene Pinnau 3

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1 1 ntroucton 1 A multaxal stress-stran correcton scheme Holger Lang 1, Klaus Dressler 2, Rene Pnnau 3 1, 2 Fraunhofer Insttut für Techno- un Wrtschaftsmathematk, Fraunhofer Platz 1, Kaserslautern, Germany holger.lang@twm.fraunhofer.e, klaus.ressler@twm.fraunhofer.e 3 Technsche Unverstät Kaserslautern, Erwn Schrönger Strasse, Geb. 48, Kaserslautern, Germany pnnau@mathematk.un-kaserslautern.e Abstract A metho to correct the elastc stress tensor over tme at a fxe pont of an elastoplastc boy, whch s subect to exteror loas, s presente an mathematcally analyse. In contrast to unaxal correctons Neuber or ESED, our metho takes multaxalty phenomena lke cyclc harenng/softenng, ratchettng an non-masng behavour nto account usng Jang s moel of nonlnear elastoplastcty. Our numercal algorthm s esgne for the case that the scalar loas are pecewse lnear an can be use e.g. n connecton wth crtcal plane/multaxal ranflow methos n hgh cycle fatgue analyss. In aton, a local exstence an unqueness result for a generalze soluton of Jang s system of scontnuous ornary fferental equatons s gven. Keywors. Jang s moel, elastoplastcty, multaxal hgh cycle fatgue analyss, urablty, fferental nclusons, Flppov theory, scontnuous ornary fferental equatons MSC classfcaton: 74C15, 74D10, 65L06 1 Introucton 1.1 Motvaton. Let U R 3 be a oman, representng a homogeneous elastoplastc metallc boy, that s subect to loas, an whose ponts x Ū unergo a nonlnear multaxal consttutve materal law of the form σt, x = t 0 C ep ετ, x τ, 1 where the elastoplastc tensors C ep R epen on the whole hstory/memory of the materal an ts ponts. In our artcle we are especally concerne wth the moel of Jang an Sehtoglu, as t s qute successful n practce an wely accepte n the engneerng communty. It s rate-nepenent an reprouces many multaxal stress-stran phenomena lke nonlnear sotropc/knematc harenng or ratchettng, see [14, 15]. We brefly name t Jang s moel. For an explct representaton of C ep, we refer to 28. Let us asume, we know the elastc stress tensor e σt, x va superposton of fntely many lnear elastc, statc PDE results,.e. e σt, x = n L nt e σ n x, e σ n x = C e ε n x 2 wth gven scalar loa functons L n : [0, T ] R. C R s Hooke s tensor of elastcty c kl = Eν 1 + ν1 2ν δ δ kl + E 1 + ν δ kδ l = λ δ δ kl + 2µ δ k δ l 3

2 1 ntroucton 2 E = Young s moulus, ν = Posson s rato, λ, µ = Lamé s constants. Due to the ecouplng of tme t an space x n 2 an ue to the lnear stress-stran relatonshp, the latter approach s numercally very cheap compare to the costs when solvng the full transent problem wth the exact nonlnear stress-stran relatonshp 1 wth hysteress. The general case see K n 10: For x Ū, we have e σ n x an thus e σt, x n R 3 3 s = {A R 3 3 : A = A T } R 6. 4 The specal notch case see L n 11: If x U, we know from the theory of lnear elastcty, that an approprate orthonormal bass exsts such that all e σ n x an thus e σt, x are n R 3 3 sp = {A = a R 3 3 s : a 13 = a 23 = a 33 = 0} R 2 2 s R 3. 5 If x les n an area, where the real stress s hgh, e.g. n a notche regon, plastcty comes nto play, so the elastc stress by far overestmates the real stress. Typcally these regons are small, an the overall materal behavour s elastc. Our goal s to rectfy e σt = e σt, x at fxe x n a postprocessng step to a better stress σt R 3 3 s resp. R 3 3 sp from now on referre to as the real stress by an approprate mofcaton of Jang s moel, base on a pseuo parameter approach of Koettgen et al. [16]. Eventually, ths s the pont on whch fatgue analyss may be set up. Instea of solvng numercally expensve nonlnear PDE, we are focussng on a much cheaper nonlnear ODE moel. In ths paper we wll 1. gve a proper efnton of the correcton moel n [16], whch s base on a strct couplng we call t smultanety between elastc an real stress space. 2. prove local exstence an unqueness for our correcton moel. Almost by the way, we wll get local exstence an unqueness for Jang s consttutve moel as well. 3. present an algorthm, whch s more stable an faster than exstng ones e.g. [12, 16] because of a the strct smultanety couplng n both stress spaces, whch makes t possble to avo teratve soluton algorthms. b a scontnuty etecton for pecewse lnear elastc stress nputs. Remark 1.1 fels of applcaton In engneerng HCF-urablty, e.g. n the the automotve nustry, loa functons L n are typcally sample wth a frequency of approx. 400 sec 1. One half hour on a test track woul consequently yel about ata ponts. So t s an wll be n the next years mpossble to solve the full transent nonlnear PDE problem n an acceptable computatonal tme wth suffcently fne fnte element scretsaton of the oman Ū. Nowaays, a wesprea practce to fn a remey s to correct scalar equvalent sgne norms/proectons of e σt va the ESED or Neuber s metho, see e.g. [19].

3 1 ntroucton 3 elastc stress space V Se M e n e e s R e α e ρ e α 0 Se Y real stress space V S M R e n s 0 α 3 α ρ e R e ρ S Y R ρ Fgure 1: Movements n both stress spaces 1.2 Defnton of the correcton moel Frst, usng the well known orthogonal ecomposton R 3 3 = R 3 3 R 3 3 h = {A R 3 3 : tr A = 0} RI R 8 R 1, R 3 3 R 3 3 h wth the assocate lnear orthogonal proectons ev : R 3 3 R 3 3 an sph : R 3 3 R 3 3 h we efne R 3 3 s = ev R 3 3 s = {A R 3 3 R 3 3 sp s : a 33 = a 11 a 22 } R 5, 6 = ev R3 3 sp = {A R 3 3 s : a 13 = a 23 = 0, a 33 = a 11 a 22 } R 3, 7 whch are evatorc subspaces of R 3 3 s, corresponng to 4 an 5. Our nces s, p, an h mean symmetrc, plane, evatorc an hyrostatc.e. sphercal. Notaton. Let : an enote the stanar Euklan scalar prouct resp. norm on R 3 3 R 9, B r σ = {τ R 3 3 s : τ σ < r} an { π : R 3 3 σ/ σ f σ 0 s B 1 0, σ 0 f σ = 0. The unerlyng functon spaces are C k [0, T ], = space of k-tmes countnously fferentable functons, C k pw[0, T ], = space of pecewse C k functons, AC[0, T ], = space of absolutely contnuous functons where the respectve mage epens on the context. Note that AC = W 1,1. A. We assume that symmetrc elastc stresses e σ Cpw 1 [0, T ], V, where V s ether R3 3 s R 3 3, are gven. Ther evatorc parts sp or e st = ev e σt, eṡt = t e st = ev e σt 8

4 1 ntroucton 4 are external nput/force nto ε pl, ξ, e α 1,..., e α m, e Ṙ, α 1,..., α m, Ṙ = Je s t, ξ, e α 1,..., e α m, e R, α 1,..., α m, R 9 a system of frst orer scontnuous ornary fferental equatons. That way, J e s s explctly tme epenent, the so calle elastc stress control. B. After solvng ODE 9, the real stress evators st are efne smultaneously.e. parallel an rescale to the elastc stress evators e st at the corresponng pont n real stress space σ, ε, ε el, s, e α, e ρ, α, ρ = K e s, ε pl, e α 1,..., e α m, e R, α 1,..., α m, R, σ h 10 resp. σ, ε, ε el, s, e α, e ρ, α, ρ = L e s, ε pl, e α 1,..., e α m, e R, α 1,..., α m, R, 11 each a system of smple explct algebrac equatons. For a better geometrc unerstanng, we refer to fgure 1. There are a lot of parameters appearng n the efnton of Je s, L, K, whch are all lste n 20 below. A. Defnton of Je s on Ω = [0, T ] R V m m 0, V 0, where V = ev V from 6,.e. V = R 3 3 s, or 7,.e. V = R 3 3 sp. For t, ξ, e α 1,..., e α m, e R, α 1,..., α m, R Ω we set an e ρ = e ρ e a ρ e e b ρ e R, e α = m =1 e α, α = m =1 α. 12 Se Y = Be ρ e α the elastc yel surface, 13a Se M = Be R0 the elastc memory surface, 13b S M = B R 0 the real memory surface. 13c Now f e st Be ρ e α no loang,.e. elastc, efne for = 1,..., m Otherwse we eclare ε pl = 0, ξ = 0, e α = 0, e n = e nt, e α 1,..., e α m = π e st e α, e Ṙ = 0, α = 0, Ṙ = eṡ n = e ṡt : e n. If e ṡ n 0 neutral loang, let 14 hol agan, else plastc loang set for = 1,..., m e χ = e Q 2 e n : π e α 1 + e a χ e e b e χ R 15a e c = e c 1 + e a 1 e e b 1 ξ + e a 2 e e b 2 ξ 15b e e ξ α = e c e e eχ α +1π r n e e α 15c r e ξ α = m ξ e α 15 =1 { e e c R 1 e α / e R f e α ξ R = Be R0 π e α : e ξ α + f e α Be R0 c 15e

5 1 ntroucton 5 n the elastc stress space, an n the real stress space. Further, we eclare χ = Q 2 e n : πα 1 + a χ e bχr 16a c = c 1 + a 1 e b1 ξ + a 2 e b2 ξ 16b e α χ +1 ξ α = c r n πα 16c r ξ α = m ξα 16 =1 { cr 1 α /R f α B R 0 ξ R = πα : ξ α + f α B R 0 c 16e ξ e ρ = e ρ 0 e a ρ e b ρ e e b ρ er ξ e R, e h = e n : ξ e α + ξ e ρ. 17 The values of Je s are fnally gven by e Ṙ = ξ ξ e R, e α = ξ ξ e α, ξ = eṡ n e h, εpl = ξ e n, Ṙ = ξ ξ R, α = ξ ξ α. 18 Of course, e α, α, ε pl reman n V, as V s a lnear space. We remark that the rght han-se of 9 oes not epen on ε pl. The ntal values for 9. We assume that they satsfy the followng contons: 0 < e α 0 < e m r, 0 < e α 0 e R0, =1 m 0 < α 0 < r, 0 < α 0 R0, =1 ε pl 0 arbtrary, ξ0 0, e s0 m e α 0 e ρ e a ρ e e b e ρ R0. =1 19a 19b 19c 19 For vrgn materal we shoul atonally have e α 0 = α 0, e R0 = R0 an e α 0 1, e R0 1 an e σ0 = 0. The parameters for 9. We nee elastc parameters e p l for the elastc stress space, an real parameters p l for the real stress space, namely e a, e a χ R; e b χ 0 e Q ; 1 < e a ρ < 0; e b ρ < 0 < e b, e c R, e r, e c, e ρ 0 20a a, a χ R; b χ 0 Q ; 1 < a ρ < 0; b ρ < 0 < b, c R, r, c, ρ 0 20b l = 1,..., 7m + 6, = 1,..., m, = 1, 2. The e a way that nf e a ξ 0 e e b ξ > 1, nf ξ 0 a, e b e b, a, b have to be chosen n such a ξ > 1 = 1,..., m 21 The p l are physcal parameters Jang parameters, epenng solely on the materal, whereas the e p l are fctve an epen atonally on the consere pont x, n partcular on the

6 1 ntroucton 6 geometry of U. For each x we nee an approprate set of e p l. The affects of p l resp. e p l on the materal/part propertes as ratchettng or cyclc harenng/softenng are explane n [14, 15]. In any case, we shoul have e ρ e a ρ ρ a ρ, as R e R for vrgn materal, see 29 an [12], chap. 4. B. Defnton of L the specal notch case, V = R 3 3 sp on V [0,T ] V [0,T ] V [0,T ] m 0, [0,T ] For all 0 t T we set e ρt, e αt, αt as n 12 an V [0,T ] m 0, [0,T ]. ρt = ρ aρ e bρrt, st = αt + ρt e e st e αt. 22 ρt The real yel surface s efne as S Y t = B ρt αt. Fnally the real stress an strans are σt = D 1 st, ε el t = C 1 σt, εt = ε el t + ε pl t. 23 where D s the lnear somorphsm D = R 3 3 ev sp R 3 3 : R 3 3 sp sp R 3 3 sp. If e σ an σ are not plane, our correcton s applcable too, but only f the hyrostatc component σ h s avalable as atonal nformaton. Defnton of K the general case, V = R 3 3 s on V [0,T ] V [0,T ] V [0,T ] m 0, [0,T ] V [0,T ] m 0, [0,T ] R [0,T ]. Here the same equatons as for L hol, except that we replace the frst equaton n 23 by σt = st + σ h ti. The parameters for 10 an 11. Here were ust have E > 0 an 0 < ν < 1/2 n 3. But note that they on t have any affect on σ. Remark 1.2 smultanety By efnton 22 we enforce parallelty of e s e α an s α. Note that n 15 an 16 the same yel surface normal e n s use both for e S Y an S Y. That s the way, the movements n both stress spaces are couple. The movement of the e α an α an thus e α an α an the evelopment of e R an R ffer n general, unless e p l are p l equal. For llustraton see agan fgure 1, where m = 5. Remark 1.3 former work Defnng Je s, L an K, we clearly follow the ea of Koettgen et al. [16], whch they call the e σ approach. They evelope ths kn of scheme for the consttutve moel of Mroz [18]. Later Hertel [12] aapte ths approach to Jang s moel. But compare to our formulaton, there are some fferences n [12, 16]:

7 1 ntroucton 7 The ntegraton of ṡ = he eṡ, h = e n : ξ α + ξ ρ 24 h propose n [16] cannot be realze n a consstent manner wth the fact that Jang s moel comprses expanson/contracton of the yel surface. So, urng ntegraton of 24, we have two yel surfaces wth fferent ra ρ an e ρ an thus two yel contons to be satsfe at the same tme, whch s nconsstent wth the smultanety ea. In contrast to our case, a purely knematc verson of Mroz s moel wth only one yel conton e ρ = ρ s refere to n [16]. An atonal avantage of our formulaton s that we get r of equaton 24, so the number of fferental equatons to be ntegrate s reuce. As the physcal correcton ea s per efnton smultaneous, we are ntegratng the equatons n elastc an real stress space parallelly couple an not teratvely n seres e σt 25,e p l ε pl t Iter.,p l σt, as t s one n [12, 16]. So, as we on t have a stress-controlle ntegraton followe by completely separate stran-controlle ntegratons, ths yels more accuracy an much more spee, as we have to hanle the problems wth scontnutes only once. We have the garuantee that plastc yelng starts/stops exactly at the same pont n tme. That way, our algorthm has turne out to ecrease the number of expensve functon evaluaton by a factor more than 21 compare to [12] yelng a spee factor of more than 10. Remark 1.4 the orgnal stress controlle Jang moel Snce e α, e Ṙ, ξ, ε pl o not epen on α, R, you rescover the orgnal stress-controlle Jang moel [14] as a subsystem ε pl, ξ, α1,..., α m, Ṙ = J s t, ξ, α1,..., α m, R 25 on Ω = [0, T ] R V m 0,, where s = evσ Cpw 1 [0, T ], V s the external nput an ε = C 1 σ + ε pl. Just rop the atonal equatons 16 an the e at each elastc varable an parameter. Here the elastc stress an the real stress space conce, uner the assumpton that the nput 8 s the real stress, 13c oes not exst. The ntal values/contons analogeous to 19 are 0 < α 0 < r, 0 < m α 0 R0 26a =1 ε pl 0 arbtrary, ξ0 0 26b s0 m α 0 ρ aρ e bρr0. 26c =1 Conton 30 becomes ɛ > 0 small m r α ρ 0 a ρ b ρ c R + ɛ. 27 =1 c It can be straght forwarly shown that n 1 C ep = C I 2µ 2µ + h n n. 28

8 2 local exstence an unqueness of soluton 8 2 Local exstence an unqueness of soluton The elastc stress controlle functons Je s n 9 an J s n 25 are scontnuous. So unfortunately, the stanar theory of systems of ornary fferental equatons, where usually at least local Lpschtz contnuty s requre, oes not apply. We nee a generalze soluton concept to guarantee one an only one soluton of the respectve systems of equatons. For other elastoplastc consttutve laws comprsng nonlnear knematc harenng e.g. the moels of Melan-Prager, Armstrong-Freerck or Chaboche wellposeness results are alreay avalable. The reaer s referre to the work of Brokate/Krec [2, 3, 4]. The mathematcal technques there rely on the theory of varatonal evoluton nequaltes an movng convex sets n Hlbert spaces. However, ths s not easly applcable here, as Jang s moel atonally comprses nonlnear sotropc harenng,.e. the yel surface the convex set n Brokate s theory may contract an expan. So n ths paper, we wll stuy a completely fferent approach. We wll conser 9 resp. 25 as fferental nclusons n orer to apply the theory of Flppov [9]. 2.1 Frst analytcal facts Lemma 2.1 The yel an memory ra are couple n a one-to-one manner. The mappngs 0, e R e f e ρ 0, n 12, 0, R f ρ 0, n 22 are strctly ncreasng. Further e ρ e R e ρ 0, ρ R ρ 0, 29a Especally e ρ e R 0 e ρ e a ρ = e σ Y, The nverse mappngs are gven by ρ R 0 ρ a ρ = σ Y. 0 < e σ Y e ρ e ρ 0 <, 0 < σ Y ρ ρ 0 <. e ρ e f 1 1 e ρ/ e ρ e ln 0 1 b e, ρ f 1 ρ a ρ Proof: Clear, because a ρ, b ρ, e a ρ, e b ρ < 0 < ρ 0, e ρ 0. 1 ρ/ρ0 1 ln. b ρ a ρ 29b So t oesn t matter whch of e R or e ρ to choose as the governng varable. We ece to take e R. Lemma 2.2 If e α e r, then ξ e α : e n 0. If e α e r for all = 1,..., m an at least one of the nequaltes s strct, then ξ e α : e n > 0. The analogeous assertons hol for ξ α an e n. Proof: For each, multplcaton of 15c wth e n yels usng the Cauchy-Schwarz nequalty e ξ α : e n = e c e r 1 e eχ α +1 e e n : π e α 0 r

9 2 local exstence an unqueness of soluton 9 note that e χ 0 wth at least one strct nequalty, an therefore the assertons follow after summng over = 1,..., m. Durng elastc/neutral loang, ξ = ε pl = 0 ue to 14. Durng plastc loang, we conclue from 18 the next lemma wll guarantee e h > e h mn > 0 that 0 < ξ = ε pl ξ max. Therefore ξt = t 0 εpl τ τ s the accumulate plastc stran: constant urng elastc/neutral loang an strctly ncreasng urng plastc loang, n partcular non-negatve ue to the ntal conton 19c. Lemma 2.3 If e α e r for = 1,..., m, a suffcent conton for e h e h mn > 0 or equvalently 0 < ξ ξ max s wth an ɛ > 0. m =1 Proof: In both cases n 15e, we have e c e r e α e ρ 0 e a ρ e b ρ e c R + ɛ 30 ξ e R e c R 1 e α / e R e c R 31 Hence e h 15c,15,17 = 31 e c e e eχ α +1 r 1 e e n : π e α + e ρ e r 0 a e ρ b ρ e e b er ρ e ξ R e c e r 1 e α / e r e ρ 0 e a ρ e b ρ e c R Note here that e χ 0 > e a ρ, e b ρ. As e ṡ s boune over any nterval [t n 1, t n ], where e s C 1 eṡ, C 0, t s n fact globally boune on [0, T ], thus ξ ξ max <. The next lemma proves that the e α reman n ther lmtng ra e r as alreay asserte but not prove n [15] n the case of proportonal monotonous loang. Lemma 2.4 Let e n C 1 pw [ξ0, ξ T ], V B 1 0, where 0 ξ 0 < ξ T <, be gven as nput nto the fferental equatons for the partal backstresses 15a-15c an 16a-16c. Then each e α C 1 pw [ξ0, ξ T ], V. If e α ξ 0 < e r, then e α ξ < e r hols n the whole nterval [ξ 0, ξ T ]. The analogeous asserton hols for α an r nstea of e α an e r. Proof: Obvously, snce all e b 0 < 1 + e I < e c 15b e c > 0 an ξ ξ 0 0, we have 1 + e a 1 + e a 2 =: e C. 32 where e I enotes the nfmum n 21. Let us frst conser the nterval [ξ 0, ξ 1 ], where e n s C 1. Here we have the polar ecomposton e α = e α π e α. Dfferentaton wth respect to ξ yels e ξ e α π e α + e α ξ π e α = e 15c ξ α = e c e e eχ α +1π r n e e α. 33 r

10 2 local exstence an unqueness of soluton 10 Snce 1 = π e α 2, we have 0 = ξ π e α 2 = 2π e α : ξ π e α, an therefore after scalar multplcaton of 33 wth π e α e ξ e α = e c e r n : π e e eχ α α e e C e r 1 e eχ α +1. r e r As e b χ 0, we have e χ e Q 1+ e a χ =: e K, an wth the substtuton e γ = e α / e r we obtan the scalar raal fferental nequaltty e ξ γ e C 1 e γ e K. In both cases e K > 1 an e K = 1, separatng the varables leas us to e γ < 1. e α n the worst case each may assymptotcally reach e r an the frst summan n 30 may become too small an the whole moel bursts. Ths usually occurs, f the nputs e st are too large. Anyway the moel 9 s ust able to get along wth nputs that are small enough. So far, we have not been able to prove a suffcent a-pror-estmate epenng on the W 1,1 norm e s 1,1 of the nput lke for example Brokate n [2, 3] for elastoplastcty moels of lower complexty. A way to fn a remey n practce s to use more backstresses. Unfortunately, a larger m yels more fferental equatons an more parameters. 2.2 Proof of the man theorem Let us frst conser the Jang correcton moel. For our theoretcal purposes, we rewrte 9 equvalently n the form ẏ = Je st, y, y = ε pl, ξ, e α 1,..., e α m, e R, α 1,..., α m, R or ẋ = Ie sx = 1, Je sx, x = t, y. 34 Through ntroucton of tme as an artfcal space varable x 0 = t, the system 34 s autonomeous an we have pushe our tme scontnuty nto space. We prove exstence an unqueness of a Flppov-soluton t xt of the fferental ncluson { } k ẋ Ie sx = conv v : x k k N Ω s.t. x k x an Ie sx k k v,.e. a functon x AC[0, T ], Ω, Ω = [0, T ] V R V m m 0, V 0, satsfyng 34 for almost every t. Of course, f Ie s s contnuous n x, we have Ie sx = {Ie sx}. The set of scontnuty ponts of Ie s has measure zero. More precsely speakng, t s gven by S Ωe Y SΩe M SΩ M wth the smooth submonfols cylners S Ωe Y = {x Ω : ϕe Y x = 0}, ϕe Y x = e st e α e ρ e a ρ e e b e ρ R, 35a S Ωe M = {x Ω : ϕe Mx = 0}, ϕe Mx = e α e R, 35b SM Ω = {x Ω : ϕ Mx = 0}, ϕ M x = R, 35c α

11 2 local exstence an unqueness of soluton 11 separatng Ω nto the omans Ω e Y = {x : ϕe Y < 0}, Ω e M = {x : ϕe M < 0}, Ω M = {x : ϕ M < 0}, 36a Ω +e Y = {x : ϕe Y > 0}, Ω +e M = {x : ϕe M > 0}, Ω + M = {x : ϕ M > 0}, 36b where Ie s s Lpschtz an the soluton x therefore wll be smooth. Remark 2.5 γ-conton All of the omans 36 an all ther mutual ntersectons Ω + e Y e M = Ω e Y Ω+e M,..., Ω ++ e Y e MM = Ω e Y Ω+e M Ω+ M,... are locally connecte, as a consequence they satsfy Flppov s γ-conton,.e. Ω e Y {t = const} = Ω e Y {t = const} for almost all t,... see [9], 6 where the bounary at the rght han se s to be taken n the relatve topology of the hyperplane {t = const}. The whole theory [9], 7, 9, 10 s eveloppe for scontnutes n the space varable x. But n [9], 6 an 9, t s shown that uner the γ-conton the ntroucton of ummy tme x 0 above oes not nfluence soluton propertes,.e. exstence an unqueness. In the sequel, we assume that e ṡ n = e n eṡ > 0, where the scontnutes at the surfaces S Ωe Y, S Ωe M an SΩ M occur. In the case e ṡ n 0, ue to 14, the whole movements,.e. the functons ε pl, ξ, e α, e R, α an R, reman constant. 1. We frst conser the actve loang of S Ωe Y,.e. the the case, where the soluton reaches a pont x S Ω e M, S = SΩe Y = Ω e Y = Ω+e Y. Here we have Ie s = conv{i, I + } wth functons I x = lm Ie sx = 1, 0, 0, 0,..., 0, 0, 0,..., 0, 0 x Ω e Y,x x I + x = lm Ie sx x Ω + e Y,x x = 1,,, e c e e eχ α 1 1 e 1 n e e α 1 ξ,..., e c m n r 1 e c R 1 e α ξ, e,...,, R e α m e r m eχ m e α m ξ, whch are smooth up to Ω e Y an Ω +e Y respectvely. an enote some scalar resp. tensoral quanttes. To have a unque element I 0 = λ I + λ + I + Ie s T x S, t s necessary an suffcent that the system of lnear equatons 1 1 λ 1 ϕe Y, I ϕe Y, I + = 0 has one an only one soluton. Vector calculus leas us to ϕe Y = eṡ n, 0, 0, e n,..., e n, e ρ 0 e a ρ e b ρ e e b ρ er, 0,..., 0, 0 λ +

12 2 local exstence an unqueness of soluton 12 The proectons of the functon value lmts on the surface normal are ϕe Y, I = e ṡ n > 0 an ϕe Y, I + = 15e = eṡ n e n : e c e e eχ α n e e α ξ + e ρ e r 0 a e ρ b ρ e e b er ρ c R 1 e α e R { eṡ n e n : e c e e eχ α } n e e α + e ρ e r 0 a e ρ b ρ e e b er ρ e ξ R ξ ξ 15c,15,17 = eṡ { } n e n : e ξ α + e 17,15 ξ ρ ξ = e ṡ n e 18 h ξ = 0. Hence λ = 0, λ + = 1, Ie sx T x S = {I + x} an theorem 2 n [9] 10 yels unqueness. We see that the soluton curves reach S an a slng moton occurs as long as e ṡ n > 0. The soluton x cannot enter Ω +e Y. As I+ an S are smooth, x wll be smooth urng the slng moton along S. 2. We conser the actve loang of S Ωe Y an SΩe M. Here the surface of scontnuty s gven by S = S Ωe Y SΩe M, where the normals ϕe Y an ϕe M are lnearly nepenent obvously ue to the ecouplng nto the elastc an real stress space. So S Ωe Y an SΩe M locally ve Ω at x S nto four quarants, so we have Ie s = conv{i, I+, I +, I+ + } wth four lmtng values of Ie s. It s straght forwar to see that ϕe Y, I ϕe Y, I + ϕe Y, I+ ϕe Y, I + + ϕe M, I ϕe M, I + ϕe M, I+ ϕe M, I + + λ λ + λ + λ + + = has exactly one soluton, thus Ie sx T x S = {I + + x}. Here as well a slng moton occurs, the soluton can nether enter Ω +e Y nor Ω+e M. As S s smooth an I + + s Lpschtz, x wll be smooth urng the slng moton along S. 3. It s clear that the cases, where the soluton x meets a pont on S Ωe M Ω e Y or SΩ M Ω e Y cannot occur. The actve loang of S Ωe Y an SΩ M s treate exactly n the same way as n the secon step, an the actve loang of S Ωe Y, SΩe M an SΩ M the octant s as well treate straght forwarly. Theorem 2.6 Local exstence an unqueness of soluton Let V = R 3 3 s or V = R 3 3 sp. a Jang correcton moel. As long as conton 30 s satsfe an none of e α an the e α passes through the orgn, there exsts one unquely etermne, absolutely contnuous Flppov-soluton of 9 uner the ntal values/contons 19, f the elastc stress evator e s s of class Cpw 1 [0, T ], V. b Jang moel. As long as conton 27 s satsfe an none of α an the α passes through the orgn, there exsts one unquely etermne absolutely contnuous Flppov-soluton of 25 uner the ntal values/contons 26, f the stress evator s s of class Cpw 1 [0, T ], V Proof: a 1. Assume frst e σ C 1 [0, t 1 ]. Unqueness: See steps above. To ensure local exstence by applyng the frst part of theorem 1 7 n [9], we remark that the functon t, x Ie sx s set value an upper semcontnuous n the sense of [9], sect. 3, 5 n t, x. In orer to verfy ths, see [9], lemma 3, 6 an the corollary after lemma 4.

13 3 numercal mplementaton 13 for all t, x the set Ie sx s nonempty, boune, close an convex. 2. Inucton: e σ C 1 [t n 1, t n ], 0 = t 0 < t 1 <... < t N = T, n = 1,..., N. b Clear, as Jang s constutve moel 25 s a subsystem of 9. Just rop each e. 2.3 A specal case If for any {1,..., m} the ratchettng parameter e Q s equal to zero, the fferental equaton for the partal backstress e α smplfes to e α 15c = e c e r e n e α ξ 18 = e c e r ε pl ξ e α the rght-han se beng a generalzaton of the backstress evoluton equatons of Armstrong/Freerck [1] resp. Chaboche, see [4, 5, 14], snce e c = e c ξ s non-constant. If we nsert ε pl = ξ e s e α/ e ρ, equaton 37 becomes e α = e e e st e αt c ξt ξt r e e α 38 ρt a system of ornary fferental equatons of frst orer. If the functons e α, e s, e ρ, ξ are of class AC an e st e αt e ρt, ξ 0 for almost every t [0, T ], we can solve 38 by varaton of constants, yelng the explct formula e α t = 1 {e t e α 0 + e e sτ e } ατ r e W t 0 e Ẇ τ τ 39 ρτ for almost every t [0, T ], where t e W t = exp 0 e c ξτ ξτ τ = exp e c ξ t 0 An analogeous equaton to 38 an soluton formula 39 hols of course n the real stress space too. Usng 39, we can gve an alternatve proof of lemma Numercal Implementaton Throughout ths secton, we are concerne wth the notch case 9, 11. Clearly, our algorthm s applcable to 25 as well. For mplementaton of Je s an L, we conser ẏ = Je st, y n the non-reunant form, where y R 32m+1+3 where the scalar prouct on V = R 3 3 sp R3, A a 11, a 22, a 12 T, consere as subspace of R 3 3, s gven by x, y M = x T My, M = , x 2 M = x, x M x, y R We conser a screte fnte sequence of gven elastc stress tensors e σ 0,..., e σ N t 0 = 0,..., t N = T, N N an assume e σ to be pecewse lnear: at tmes e σt = e σ n + t n + 1 e σ n t n, e σ n = e σ n e σ n 1, e σt = e σ n t n 40

14 3 numercal mplementaton e σ σ Fgure 2: Elastc stress e σ an correcte stress σ for n = 1,..., N an t [t n 1, t n ]. Ths s the usual case n engneerng practce, where only such sample ata or sgnals are avalable. Theorem 2.6 guarantees a unque soluton. Because of rate nepenency we may assume t n = t n t n 1 = 1 an thus T = N. Applcaton of ev to 40 mples e st = e s n + t n + 1 e s n, e s n = e s n e s n 1, eṡt = e s n 41 Dscontnuty etecton at Se Y. The ea s smply to reparametrze τ n = t n + 1 an to calculate the exact ntersecton pont e s n of the straght arc { τ n e s n 1 + τ n e s n : τ n [0, } wth Se Y, whose poston remans constant on [0, τ n ]. Settng e β n 1 = e s n 1 e α n 1 we have e s n = e s n 1 + τ n e s n wth τ n = f e s n 0 an e β n 1, e s n M + e β n 1, e s n 2 M e s n 2 M e β n 1 2 M e ρ 2 1/2 n e s n 2 M 42 τ n = 43 otherwse. If τ n [0, 1, t follows from the slng moton reflectons n secton 2 that e sτ Se Y τ for τ [τ n, 1], otherwse e sτ Be ρτ e ατ = Be ρ0 e α0 for τ [0, 1]. Ths kn of front checkng leas to a sgnfcant reucton of reecte steps, when usng a stanar ODE metho wth error control, see Harer et al. [11], ch. II.6. The scontnutes at Se M an S M are left to the step sze control. Usng a constant step sze woul yel too large errors when crossng these scontnutes.

15 3 numercal mplementaton The algorthm. It turns the nto Input e σ n, n = 0,..., N, e p l, p l l = 1,..., 7m + 6 Output σ n, s n, ε n, ε el n, εpl n, e α n, e α n, e R n, e ρ n, α n, α n, R n, ρ n, ξ n n = 0,..., N Lower nces n are loop nces whereas the upper nces enumerate the backstresses. It reas as follows A ε pl 0, ξ 0, e α 0, e R 0, α 0, R 0 := acc. to 19 B e α 0, e ρ 0, α 0, ρ 0, s 0 := as n 12, 22, e s 0 := D e σ 0 as n 41 C for n := 1,..., N D e s n := D e σ n, e s n := e s n e s n 1 as n 41 E τ n := as n 42, 43 F f 0 τ n < 1 G H I J else e s n := e s n 1 + τ n e s n, e s n := e s n e s n ε pl n, ξ n, e α n, e R n, α n, R n:= DoPr5-soluton at τ = 1 of ẏ = Je sτ, y over τ [0, 1] wth ntal values ε pl n 1, ξ n 1, e α n 1, e R n 1, α n 1, R n 1 an external nput e sτ = e s n + τ e s n, e ṡτ = e s n ε pl n, ξ n, e α n, e R n, α n, R n := ε pl n 1, ξ n 1, e α n 1, e R n 1, α n 1, R n 1 acc. to 14 K e α n, e ρ n, α n, ρ n, s n := as n 12, 22 L σ n := D 1 s n, ε el n := C 1 σ n, ε n := ε el n + Ď 1 ε pl n as n 23 Here D = 2/3 1/3 0 1/3 2/ , C 1 = 1/E ν/e 0 ν/e 1/E 0 ν/e ν/e ν/e, Ď 1 = That way, we push the trval ntegraton 14 nto an outer loop stepsze t n = 1, ratenepenency!. For the nner loop step H, varable step sze h we have teste all methos wth step sze control contane n Shampne/Rechelt [20], among whch the Runge-Kutta metho of Dorman/Prnce see RK457m n [6], DoPr5 n [11], sect. II.5 or [20], sect. 5 has turne out to be the fastest. The ntal step sze an the steps sze control has been performe as n [11], sect. II.4..

16 4 concluson The butterfly test. We conser e σt = L 11 t e σ 11 + L 22 t e σ 22 + L 12 t e σ 12 as n 2, n whch e σ 11 = 1, 0, 0 T, e σ 22 = 0, 1, 0 T, e σ 12 = 0, 0, 1 T. All parameters for the steel S460N an an approprate notch of an axs are taken from [12] an [13]. e a 1 e b 1 e a 2 e b 2 e Q e ρ e +2 e r e c 1 9.5e e 3 4.3e e e 2 7.0e 3 1.4e +0 e a ρ 2.88e e e 2 6.8e e e 1 1.5e 2 1.4e +0 e b ρ 6.16e e e 2 7.1e e e 2 2.0e 2 1.4e +0 e a χ 2.85e e e 1 8.0e e e 1 2.0e 2 1.4e +0 e b χ 6.45e e e 1 7.0e e e 2 3.5e 2 1.4e +0 e c R 1.00e +2 r c a 1 b 1 a 2 b 2 Q ρ e e e 3 2.4e e e 1 1.7e e 2 a ρ 3.20e e e 2 4.0e e e 1 3.0e e 2 b ρ 7.60e e e 2 2.0e e e 1 2.0e 2 1.0e 2 a χ 5.00e e e 1 3.0e e e 1 2.0e 2 1.0e 2 b χ 2.60e e e 1 2.5e e e 1 3.0e 2 1.0e 2 c R 1.00e +2 an E = 2.085e 5, ν = 3.0e 1. Here m = 5, so altogether, the number of nepenent fferental equatons s 36, the number of parameters 84. All quanttes of physcal menson stress are measure n MPa. You shoul montor, that suffcent conton 30 s not volate urng ntegraton. The crtcal value at ts rght han se s e ρ e 0 a e ρ b e ρ c R Lkewse, you shoul montor that e Rt, Rt les n the correcton regon { e R, R 0, 2 : e f e R = e ρ > ρ = fr}. As propose n [12], we conser the smooth peroc nput L 11 t = sn2ωt, L 22 t = 0, L 12 t = snωt, ω = 2π/8 44 over the tme nterval [0, T ] wth T = 80,.e. 10 cycles. 44 s obtane from the orgnal one by a rotaton of the coornate system. The results are epcte n fgure 2. We compare the exact soluton DoPr5 algorthm wth smooth nput an relatve accuracy δ = 10 9 wth our soluton wth sample nput N = an relatve accuracy δ = The relatve error s epcte n fgure 3. The largest peaks occur, where σ s very close to zero. But the reucton of runnng tme s remarkable, as seen n fgure 4, where we compare DoPr5-Algorthm wth smooth nput an varous relatve accuraces δ wth the soluton of our algorthm wth sample nput ata: k {0,..., 5}, t n = 4 k, N = 80 4 k sample ntervals. Of course, the falsfcaton of the nput loang path s of orer O t n. All loang paths n [12] have been teste as benchmark tests wth smlar results concernng spee an accuracy. 4 Concluson We have proven local exstence an unqueness of soluton Jang consttutve moel of elastoplastcty. A proper formulaton for a stress space stress-correcton moel has been gven.

17 4 concluson t t Fgure 3: Relatve error 10 0 runnng tme 10 1 δ = 10 3 no scont. etect. δ = 10 4 no scont. etect. δ = 10 5 no scont. etect. δ = 10 3 scont. etect. δ = 10 4 scont. etect N δ = 10 5 scont. etect. Fgure 4: Runnng tmes wthout scontnuty etecton

18 references 18 Through the smultanety couplng nherent n the moel, we have been able to mplement a soluton algorthm whch s much faster than exstng ones. Through a scontnuty etecton metho for sample nput ata further mprovement of performance has been acheve. Parameter entfcaton. The qualty of results of our moel s comparable wth approaches usng an ncremental form of Neuber s formula, where no elastc parameters are neee, see Glnka et al. [10]. For practce, fnng approprate e p l n our moel may be a problem. But the avantage of our correcton scheme s that we have the freeom of austng the e p l or even the p l n orer to optmally ft measurements or nonlnear PDE results wth Jang s consttutve materal law. A ervaton of elastc parameters e p l from unaxal equvalent loa-notch stran curves can be foun n Hertel [12], sect. 4.4, automatc optmzaton of the p l, e p l n Lang et al. [17]. Future tasks. Numercal challenges for the future are a further reucton of reecte steps, maybe by etectng the scontnutes at Se M an S M wth Runge-Kutta nterpolaton/ense output methos of Enrght/Jackson [7, 8]. An open queston s, f t s possble to generalze our theoretcal result to nputs σ resp. ε of certan subsets S, E AC [0, T ], R 3 3 s n orer to have proper Jang -operators F : S E, σ ε = F[σ], F 1 : E S, ε σ = F 1 [ ε] turnng stress nto stran an vce versa. Acknowlegements. Thanks to the DFG an the GKMP Kaserslautern for the fnancal support an to Olaf Hertel TU Darmstat, ept. of materal scence for hs kn provson of ata. References [1] Armstrong P. J., Freerck C. O.: A mathematcal representaton of the multaxal Bauschnger effect. CEGB, report RD/B/N 731, [2] Brokate M.: Elastoplastc consttutve laws of nonlnear knematc harenng type. Ptman research notes n mathematcs seres, Vol. 377, pp , [3] Brokate M., Krec P.: On the wellposeness of the Chaboche moel. Internatonal seres of numercal mathematcs, Vol. 126, pp , [4] Brokate M., Krec P.: Wellposeness of knematc harenng moels n elastoplastcty. Mathematcal moellng an numercal analyss, Vol. 32, No. 32, pp , [5] Chaboche J.-L.: On some mofcatons of knematc harenng to mprove the escrpton of ratchettng. Internatonal ournal of plastcty, Vol. 7, pp , [6] Dorman J. R., Prnce P.J.: A famly of embee Runge-Kutta formulae. Journal of computatonal an apple mathematcs, Vol. 6, No. 1, pp , [7] Enrght W. H., Jackson K. R.: Interpolants for Runge-Kutta formulas. ACM transactons on mathematcal software, Vol. 12, No. 3, pp , [8] Enrght W. H., Jackson K. R.: Effectve soluton of scontnuous IVPs usng a Runge-Kutta forumla par wth nterpolants. Apple mathematcs an computaton, Vol. 27, pp , 1988.

19 references 19 [9] Flppov A. F.: Dfferental equatons wth scontnuous rghthan ses. Kluwer Acaemc Publshers, [10] Glnka G., Bucynsk A., Rugger A.: Elastc-plastc stress-stran analyss of notches uner non-proportnal loang paths. Archves of mechancs, Vol. 52, No. 4-5, pp , [11] Harer E., Noerset S.P., Wanner G.: Solvng ornary fferental equatons I. Sprnger- Verlag Berln Heelberg, [12] Hertel O.: Weterentwcklung un Verfzerung enes Näherungsverfahrens zur Berechnung von Kerbbeanspruchungen be mehrachsger nchtproportonaler Schwngbelastung. Dplomarbet, Bauhaus-Unverstät Wemar, [13] Hoffmeyer J., Dörng R., Schlebner R., Vormwal M., Seeger T.: Lebensauervorhersage für mehrachsg nchtproportonal schwngbeanspruchte Werkstoffe mt Hlfe es Kurzrssfortschrttkonzeptes. FD-2/2000, FG Werkstoffmechank, Technsche Unverstät Darmstat, [14] Jang Y., Sehtoglu H.: Moelng of cyclc ratchettng plastcty, part I: Development of consttutve relatons. Transactons of the ASME, Vol. 63, pp , [15] Jang Y., Sehtoglu H.: Moelng of cyclc ratchettng plastcty, part II: Comparson of moel smulatons wth experments. Transactons of the ASME, Vol. 63, pp , [16] Köttgen V. B., Barkey M. E., Soce D. F.: Pseuo stress an pseuo stran base approaches to multaxal notch analyss. Fatgue an fracture of engneerng materals an structures, Vol. 18, No. 9, pp , [17] Lang H., Pnnau R., Dressler K.: Parameter optmzaton for a stress-stran correcton scheme. AGTM report, TU Kaserslautern, No. 264, 2005; [18] Mroz Z.: An attempt to escrbe the behavour of metals uner cyclc loas usng a more general workharenng moel. Acta mechanca, Vol. 7, pp , [19] Neuber H.: Theory of stress concentraton for shear-strane prsmatcal boes wth arbtrary nonlnear stress-stran law. Journal of apple mechancs, Transactons of the ASME, Vol. E28, pp , [20] Shampne L. F., Rechelt M. W.: The Matlab ODE sute. SIAM ournal on scentfc computng, Vol. 18, No. 1, 1997.