A RECURSIVE CALCULATION OF THE CONSISTENT TANGENT OPERATOR IN ELASTO-PLASTICITY

Transcription

1 Recursive calculatio of cosistet taget operator i elasto-plasticity XIII Iteratioal Coferece o Computatioal Plasticity. Fudametals ad Applicatios COMPLAS XIII E. Oñate, D.R.J. Owe, D. Peric ad M. Chiumeti (Eds) A RECURSIVE CALCULAION OF HE CONSISEN ANGEN OPERAOR IN ELASO-PLASICI BOJAN SARMAN, MIROSLAV HALILOVIČ, MARKO VRH, BORIS ŠOK Laboratory for Numerical Modellig & Simulatio (LNMS) Faculty of Mechaical Egieerig, Uiversity of Ljubljaa, SI-1000 Ljubljaa, Sloveia boris.stok@fs.ui-lj.si, Keywords: Computatioal Plasticity, Numerical Itegratio, Cosistet aget Operator, ABAQUS/Stadard, UMA, LD2004. Abstract. he paper presets a derivatio of the cosistet taget operator (CO) for the class of iterative costitutive itegratio algorithms. he derivatio is based o a variatioal aalysis of algorithmic equatios with respect to the strai depedecy of state variables i each iteratio. Such treatmet results i a recursive formulatio, where the value of CO is updated based o its old value. he proposed formulatio is demostratively applied to the cuttig-plae algorithm (CPA), for which it is kow that a closed form solutio for CO does ot exist. 1 INRODUCION A oliear boudary value problem i cotiuum solid mechaics is, owadays, efficietly solved with the fiite elemet method, ad a appropriate itegratio scheme is adopted to umerically tackle the give material costitutive behaviour. With referece to the latter, a wide class of retur-mappig algorithms based o a operator-splittig methodology prevails, particularly, i the computatioal rate-idepedet plasticity. Basically, the operator-splittig methodology follows the additive split of the costitutive equatios ito the elastic predictor part ad, whe required, the subsequet plastic corrector part with the elastically predicted state variables iteratively projected back oto the yield surface. Such methods, usually also referred to as elastic predictor-plastic corrector algorithms, are classically performed accordig to the backward-euler approach. Because of the chose backward-euler approach, a implicit system of equatios is obtaied, which is classically solved iteratively with the Newto-Raphso algorithm. Alteratively, if the operator-splittig methodology is performed accordig to the forward-euler approach, the corrector part directio is evaluated accordig to the curret state. Such approach results i a explicit system of equatios which is solved without matrix iversio. Such methods are usually referred as the cuttig-plae algorithms ad are, accordig to their explicitess, simpler to implemet ito a FEM code ad computatioally more efficiet. If summarizig i mathematically more geeral way, whe a ewly developed costitutive model is implemeted ito a fiite elemet method program, a time itegratio scheme for solvig the 886

2 model s equatios must be chose. his gives a system of equatios which is commoly solved iteratively with the Newto-Raphso algorithm. Fially, it ca be cocluded the solutio strategy roughly remais uchaged regardless of the form of costitutive model. Although a geeral solutio method is available i priciple, the last challege due to implicit FEM remais the derivatio of cosistet taget operator (CO). he cosistet taget operator is eeded to esure the quadratic covergece of global Newto-Raphso algorithm which govers the fulfilmet of equilibrium equatios. Classically the derivatio of cosistet taget operator is model depedet ad whe developig ew costitutive model special attetio must be addressed. I this cotributio the recursive approach to calculatio of the cosistet taget operator is preseted, where the iterative solutio procedure (such as Newto-Raphso algorithm) is applied to itegrate the costitutive equatios. hus the cosistet taget operator becomes a part of iterative solutio procedure without additioal care eeded for its derivatio. he derivatio is based o variatioal aalysis of algorithmic equatios with respect to the strai depedecy of state variables i each iteratio. I cosequece, the proposed approach results i aalytically derived recursive formulatio of the CO for a geeral class of plasticity models. A extesio to other costitutive models, e.g. viscoplasticity, represets o additioal effort. he approach is applied to a case, where the itegratio of costitutive equatios is doe iteratively with the cuttig-plae algorithm (CPA) [1]. With the proposed approach the CO for CPA was derived for the first time. 2 CONSIUIVE MODELLING OF PLASICI RELAED PHENOMENA Firstly, let us briefly review goverig equatios of a icremetal theory of plasticity, which ca be optioally exteded or modified to embrace also other pheomea (path depedet hardeig, damage, path depedecy of fracture, etc.) that are observed i solids. Such material s behaviour is mathematically described as a system of differetial-algebraic equatios (DAE): he quatities 0 d Ckl dkl d kl (1) d d s d s d, s 1, 2,, p,,,, represet respectively the yield fuctio, plastic potetial, stress tesor, strai tesor ad yield stress. For cosideratio of other icluded pheomea, p additioal state variables are itroduced. Variable d deotes the plastic multiplier. It should be emphasized, that the preseted system of equatios serves for illustratio oly. I fact, the methodology i the sequel is completely geeral; the oly coditio is that the model is described with DAEs where the drive variable is the strai icremet. 887

3 3 ALGORIMIC REAMEN OF INIIAL VALUE PROBLEM he task of a itegratio scheme applied to the system (1) is to calculate the fial stress state ( ) ad the other state variables 1 (, 1,, s) 1 that result from the applicatio of the prescribed total strai icremet. he mechaical state ( ), (, 1,, s) at the begiig of the icremet is assumed to be kow. As the problem is oliear ad requires, i cosequece, a iterative solutio procedure a appropriate iitial guess must be chose. A majority of the applied itegratio methods assume that the algebraic coditio i the system (1) is fulfilled at the ed of the cosidered icremet ad the differetial equatios derivatives are approximated accordig to a chose scheme (e.g. backward-euler scheme, forward-euler scheme): 0 (k 1) 1 1 (k 1) (k 1) 2 1 : : σ (k 1) (k 1) 3 0 (k 1) s (k 1) s3 s s ψ σ σ C ε C 0 0, 1, 2,, p Withi the system of equatios (2) the Voigt otatio is used for a descriptio of tesor variables. Itroducig a vector of ukow state variables Σ σ,, 1,, s,, the (k 1) system (2) ca be rewritte i a form Ψ f ( Σ 1 ) C ε 0, where C represets the exteded stiffess matrix: C1111 C1122 C1133 C1112 C1123 C1131 C2211 C2222 C2233 C2212 C2223 C 2231 C3311 C3322 C3333 C3312 C3323 C 3331 C C1211 C1222 C1233 C1212 C1223 C C2311 C2322 C2333 C2312 C2323 C 2331 C (3) 0 C3111 C3122 C3133 C3112 C3123 C he solutio of a oliear system of equatios is classically obtaied with the applicatio of Newto-Raphso root fidig algorithm which arises from liearizatio Ψ 0 : (4) ΛΨ f ( Σ ) C ε f Σ 0 1 Σ 1 1 (0) With the iitial state Σ 1 kow, the update is i each iteratio determied ( k1) ( k1) as Σ 1 Σ 1 Σ 1, where the correctio Σ 1 is extracted from (4). he iterative procedure is repeated util the give covergece criteria are met. (2) 888

4 5 A RECURSIVE CALCULAION OF HE CONSISEN ANGEN OPERAOR Σ1 o derive the CO, a aalytical expressio for ε must be foud. Cosiderig the strai update 1 ad cosequetly d 1 d, it ca also be expressed as Σ1 ε. he CO will be calculated cosecutively over all iteratios with a recursive expressio. For a derivatio of the CO we cosider, i order to properly establish fuctioal relatios for a differetiatio with respect to ε, that i the system of equatios (4) the expressios Λ deped o Σ 1, Σ 1 ad ε, whereas the value C is idepedet of ε ; thus, Λ ΛΣ 1, Σ 1, ε. Cosiderig that dλ 0 the followig recursive expressio is obtaied: (5) (k 1) Σ 1 1 (k 1) Σ 1 f J 1 C J 1 J 1, J ε ε Σ Observatio of the fial result reveals some favourable properties. From the structure it ca be oticed, that there is o presece of secod order derivatives, i fact, all quatities have already bee used i the solutio procedure determied with (4). Cosequetly, oce the itegratio scheme is implemeted, the CO implemetatio is straightforward. I the ext sectio the derived approach will be applied to the cuttig-plae algorithm for which it is kow, that a closed form solutio of CO caot be derived [2-5]. For this reaso the proposed approach offers a attractive alterative solutio [1]. 6 NUMERICAL VALIDAION WIH APPLICAION O CUING-PLANE ALGORIHM 6.1 Cuttig-plae algorithm Plasticity models, defied as a system of differetial-algebraic equatios (1), are, due to structure, split ito two phases the elastic trial ad the plastic correctio. he elastic trial is made firstly (6) Σ C ε where, accordig to the structure of C, a projectio of the stress state σ ito the trial stress tr state σ 1 σ σ is made. All other variables retai their values. Because at the ed of the elastic trial a step beyod the yield surface is made, a projectio back oto the yield surface must be foud. As the etire strai icremet ε was already applied, it remais for the plastic correctio phase to covert a portio of the elastic strai ito the plastic oe, sice i e p the correctio phase the relatio ε ε 0 must hold. he startig poit for the plastic correctio phase is the trial poit, which is reached at the ed of the elastic trial. Accordig to the basic cocept of CPA, returig to the yield surface 1 0 is performed iteratively over several cuts. he described geometrical iterpretatio arises from the fact, that the algebraic coditio i (1) is liearized aroud the curret state Σ i each -th iteratio

5 ( k1) ( k) ( k) 1 = 1 Σ Σ 1 0 Because a explicit solutio ca be obtaied as the fial result ad matrix iversio i equatio (4) avoided, the algebraic coditio i (7) is treated separately from the remaider of the differetial evolutio equatios. Itroducig for the state variables the otatio,,,,,, Σ ς σ, ad discretizig the differetial equatios accordig to the forward-euler scheme yields 1 s ς R R C ( k) ( k) κ 1, : σ,, Cosiderig, that equatio (7) is idepedet of ( k1) ( k) ( k) 1 1 ς ς 1 (7) (8), it ca be also expressed as = 0. From this equatio, whe cosiderig (8), the correctio of the plastic multiplier ca be explicitly extracted 1 ς R 1 1 ad the state is correspodigly updated i each iteratio ( ( ( ( ( (k ς ς R, Σ ς, Equatios (9) ad (10) are iteratively repeated util the covergece criteria are met. 6.2 Cosistet taget operator for cuttig-plae algorithm For the CPA it is kow, that it is simple to implemet due to its explicit ature. O the other had the literature [2-5] idicates a lack of derivatio of the cosistet taget as its major shortcomig. I this sectio the methodology derived i sectio 5 will be applied. Itegratio with CPA is withi a curret -th icremet subdivided ito two steps, elastic trial ad iterative plastic correctio. Because the elastic trial state is reached i the first step, it is obvious, that the startig value for CO will be (0) Σ1 ε I the secod step iterative plastic correctio defied with equatios (7) ad (8) is applied. For the purpose of derivatio we defie: ( k) ( k) 1 1 ς ς 1 ( k1) ( k) ( k) (9) (10) (11) C. = + 0 (12) Λ ς ς R 0 which ca be equivaletly expressed as, recogized, that quatities ( 1) ( ) ( ) ( ) Λ Λ 0. From the algorithm it is 1 2 ( 1) ( ) ( ) ( ) ς, ς,, R deped o ε, thus, Λ Λ ς, ς,, R, ε. By cosiderig that dλ 0, the followig recursive expressio is obtaied: 890

6 ( k1) ( k) ( k 1) ς 1 ( k) R ς 1 ( k) I R 1 ε ς 1 ε ε (13) ( k 1) ( k1) ( k) ( k) 1 ( k) ( k) 1 = 1 1 R ς ς R ε ς ς ς ε which yields the recursive expressio of the cosistet taget operator i the stacked form: ( k 1) ς1 Σ 1 ε ε ε (k 1) (14) 6.3 Numerical example combied torsioal ad uiaxial loadig of a circular otched specime I order to show efficiecy of the proposed recursive approach, a combied torsioal ad uiaxial loadig of a circular otched specime is cosidered. A 5mm-thick sheet metal cylider with otch detail, as show i Figure 1a, is o both eds loaded i such a way that a combiatio of torsio ad uiaxial loadig coditios is imposed. he rotatio () t ad displacemet ut () are applied at the odes coected through the kiematic couplig o each ed of the cylider. o take the ifluece of give loadig coditios ad resultig loadig path o the covergece of global equilibrium solutio scheme ito accout, the loadigs are applied as show i Figure 1b. he elastic-plastic material respose is modelled with the simple vo Mises model ad rather complex LD p model [7,8]. As show i Figures 1c ad 1d the equivalet plastic strai of 1.30 is achieved at the ed of the loadig process i the case of the vo Mises model, ad approximately 1.20 i case of the LD p model. Figure 1: Combied loadig of otched cylider a) schematic view of model; b) prescribed loadig; c) equivalet plastic strai distributio for vo Mises model; d) equivalet plastic strai distributio for LD p model. 891

7 For the purpose of compariso of the effectiveess of the proposed approach, the derived recursive formulatio of cosistet operator applied to CPA is compared with the default ABAQUS/Stadard. he umber of iteratios eeded to achieve the global equilibrium i each load icremet is preseted i Figure 2. Figure 2a correspods to the ABAQUS/Stadard default algorithm, Figure 2b to the CPA with the derived CO. I the case whe the CO is ot available for the implemeted umerical algorithm, the cotiuum taget ca be used. Figure 2c presets such a case, whe cotiuum taget operator is utilized i cojuctio with the CPA. Figure 2: Covergece behaviour for simulatio of loadig of otched cylider (vo Mises model) a) ABAQUS/Stadard default; b) CPA with CO; c) CPA with cotiuum taget operator. ABAQUS/Stadard default itegratio scheme ad CPA with CO exhibit similar computatioal behaviour regardig the umber both of load icremets ad iteratios. Similarities ca be also observed i the covergece of global equilibrium solutio scheme i each load icremet eve though load icremets chage its size or loadig directio. O the other side, CPA with cotiuum taget operator exhibits much smaller efficiecy ot oly from the viewpoit of iteratios eeded to achieve global equilibrium i each icremet, but also from the viewpoit of icremet size as a idex of simulatio s progressio. he latter is a cosequece of the poor covergece behaviour which disables equilibrium solutio procedure to icrease the icremet size. he umber of all iteratios eeded durig the problem aalysis, whe the CPA with the CO is used, is approximately oly oe third of the umber of iteratios exhibited whe the CPA with cotiuum taget operator is used. A larger differece i the computatioal efficiecy of the CPA with CO ad CPA with cotiuum taget operator ca be observed o a more otrivial costitutive model, as for example LD p model. As show i Figure 3, the umber of all equilibrium iteratios is agai decreased for more tha 70%. Figure 3: Covergece behaviour for simulatio of loadig of otched cylider (LD p model) a) CPA with CO; b) CPA with cotiuum taget operator. 892

8 7 CONCLUSION I this cotributio, where the iterative solutio procedure is applied to itegrate the costitutive equatios, the recursive approach to calculatio of the cosistet taget operator is preseted. he derivatio is based o variatioal aalysis of algorithmic equatios with respect to the strai depedecy of state variables i each iteratio. I cosequece, the proposed approach results i the aalytically derived recursive formulatio of the CO. As a illustrative example the methodology has bee applied to the CPA. For the CPA the literature reports that a closed form solutio caot be obtaied, ad from that reaso a cotiuum taget operator is used, which leads however to the loss of quadratic covergece i a global Newto-Raphso s iterative loop. Motivated by this apparet obstacle we preset a aalytical derivatio of the CO which leads to a explicit recursive expressio. I each iteratio of the CPA procedure, the curret value of the CO must be updated, yieldig at the ed the fial value of the CO whe the CPA is coverged. he approach has bee verified by cosiderig the mechaical respose of combied loadig of a otched cylider. Correspodig computer simulatios have bee performed usig material behaviour that obeys the vo Mises ad LD p models, respectively. he cosidered material models have bee implemeted ito ABAQUS/Stadard via the UMA material subroutie. he correctess ad computatioal efficiecy of the developed CO algorithm is proved by accuracy ad computatioal performace compariso with ABAQUS/Stadard default, where possible, ad by a compariso with the simulatio results obtaied whe usig the cotiuum taget operator. he low umber of iteratios i the global iteratio loop idirectly cofirms the correctess of the preseted CO derivatio ad its implemetatio ito the FEM code. REFERENCES [1] B. Starma, M. Halilovič, M. Vrh, B. Štok, Cosistet taget operator for cuttig-plae algorithm of elasto-plasticity. Comput. Methods Appl. Mech. Egrg, 272: (2014). [2] A. Aadarajah, Computatioal methods i elasticity ad plasticity: solids ad porous media, Spriger, New ork, [3] V. Vavourakis, D. Loukidis, D.C. Charmpis, P. Papaastasiou, Assessmet of Remeshig ad Remappig Strategies for Large Deformatio Elastoplastic Fiite Elemet Aalysis, Computers & Structures, 114: (2013). [4] Huag, J., Griffiths, D.V., Observatios o Retur Mappig Algorithms for Piecewise Liear ield Criteria. Iteratioal Joural of Geomechaics 8 (2008). [5] J.C. Simo,.J.R. Hughes, Computatioal Ielasticity, Spriger-Verlag, New ork, [6] E.A. Neto de Souza, D. Perić, D.R.J. Owe, Computatioal Methods for Plasticity: heory ad Applicatios, Wiley, Chichester, West Sussex, UK, [7] Barlat, F., Aretz, H., oo, J.W., Karabi, M.E., Brem, J.C., Dick, R.E. Liear trasformatiobased aisotropic yield fuctio. It. J. Plasticity 21: [8] oo, J.W., Barlat, F., Dick, R.E., Karabi, M.E. Predictio of six or eight ears i a draw cup based o ew aisotropic yield fuctio. It. J. Plasticity 22: