Journal of Solid Mechanics Vol. 6, No. 3 (2014). 310-321 Imlementing the New First and Second Differentiation of a General Yield Surface in Exlicit and Imlicit Rate- Indeendent Plasticity F. Moayyedian, M. Kadkhodayan * Mechanical Engineering Deartment, Ferdowsi University of Mashhad, Iran Received 18 Aril 2014; acceted 20 June 2014 ABSRAC In the current research with novel first and second differentiations of a yield function, Euler forward along with Euler backward with its consistent elastic-lastic modulus are newly imlemented in finite element rogram in rate-indeendent lasticity. An elasticlastic internally ressurized thick walled cylinder is analyzed with four famous criteria including both ressure deendent and indeendent. he obtained results are in good agreement with exerimental results. he consistent/continuum elastic-lastic moduli for Euler backward method are also investigated..all rights reserved. Keywords: Rate-indeendent Euler backward/forward methods; Consistent elastic-lastic modulus; Internally ressurized thick walled cylinder 1 INRODUCION P LASICIY describes the deformation of a material undergoing non-reversible changes of shae in resonse to alied forces. he main imortant of coding an elastic-lastic roblem in a finite element rogram is how to udate the state of stresses from tn, tn 1, where t n is the first time and tn 1 is the end time of any load ste. In general, there are four integration algorithm methods in udating stress, i.e. 1-exact integration algorithm methods, 2-exlicit integration algorithm methods such as Euler forward method, 3-exlicit/imlicit integration algorithm methods and 4-imlicit integration algorithm methods such as Euler backward method. he first one is more accurate which solves a first order lasticity differential equation because of the difficulty of the comlicated roblems the other three are in much attention esecially the fourth one. he main oint in associated lasticity is the direction of the incremental lastic strain with resect to the yield surface. If one accets that direction to be normal to the yield surface at the beginning of the load ste then the exlicit integration algorithm methods is used, and the intersection oint of the trial stress with the yield surface must be obtained. On the other hand, if the direction of the incremental lastic strain is normal to the yield surface at the end of the interval of the load ste, the imlicit integration algorithm methods are emloyed. In these algorithms, obtaining the intersection of the trials stresses with initial yield surface is unnecessary and it needs some trial and error iterations to reach normality at the end of load ste. During the iterations, obtaining the second differentiation of the yield surface is necessary. Moreover, using an elastic-lastic modulus in global finite element rocedure which is consistent with the linearized imlicit algorithm is inevitable to reserve the quadratic rate of asymtotic convergence. o overcome these * Corresonding author. el.: +98 9153111869; Fax: +98 511 8763304. E-mail address: kadkhoda@um.ac.ir (M. Kadkhodayan).. All rights reserved.
311 F. Moayyedian and M. Kadkhodayan difficulties, obtaining the second differentiation of a general yield surface can hel to accomlish the numerical rocedures. In exlicit/imlicit integration algorithm method, the direction of the incremental lastic strain from the values of the beginning and the end of the interval of the load ste with the aid of different criteria can be evaluated. Krieg and Krieg [1] investigated the accuracy of the integration of the constitutive equations for an isotroic elastic-erfectly lastic von Mises material. Schreyer et al. [2] examined tangent redictor-radial return aroach and the elastic redictor radial corrector algorithm with von Mises, isotroic hardening model under lane stress condition. Simo and aylor [3] demonstrated that the concet of consistency between the tangent oerator and the integration algorithm emloyed in the solution of the incremental roblem lays a crucial role in reserving the quadratic rate of asymtotic convergence of iterative solution scheme based on Newton's method. Ortiz and Poov [4] investigated the accuracy and stability for integration of elastic-lastic constitutive relations. Potts and Gens [5] introduced five methods to treat this draft and recommended the 'correct' method in finite element calculations. Ortiz and Simo [6] roosed a new class of exlicit integration algorithms for rate-indeendent and rate-deendent behaviors, with lastic hardening or softening, associated or non-associated flow rules and non-linear elastic resonse. Simo and aylor [7] develoed an unconditionally stable algorithm for lane stress elastolasticity condition, based uon the concet of elastic redictor-return maing (lastic corrector). Dodds [8] derived an extension of elastic-lastic-radial return algorithm and a consistent tangent oerator which satisfy the requirements for stable, accurate and efficient numerical rocedure for lane stress condition with mixed hardening. Ortiz and Martin [9] characterized the conditions under which an elastic-lastic stress udate algorithm reserves the symmetries inherent to material resonse. Gratacos et al. [10] investigated the generalized midoint rule for the time integration of elastic-lastic constitutive equations for ressure-indeendent yield criteria and the existence and uniqueness of the incremental solution was discussed. Ristinmaa and ryding [11] introduced a unified aroach to establish an exact integration of the constitutive equation in elastolasticity, assuming the total strain-rate direction to be constant. Kadkhodayan and Zhang [12] roosed a new efficient method, the consistent DXDR method, to analyze general elastic-lastic roblem. In recent years, Foster et al. [13] modified sectral decomosition technique in imlicit algorithms for models that include kinematic hardening for geomaterials. Kim and Gao [14] resented a general method to formulate the consistent tangent stiffness for lasticity. Hu and Wang [15] roosed a yield criterion with three indeendent invariants of the stress tensor and the deviatoric stress tensor to fulfill the exerimental observations. Ding et al. [16] develoed a stress integration scheme to analyze a three-dimensional sheet metal forming roblems and imlemented into ABAQUS/Exlicit via User Material Subroutine (VUMA) interface roblem. Nicot and Darve [17] showed that emloying regular flow rule for granular materials disaeared as soon as more general loading condition was alied. heir model highlighted the great influence of the loading history on the shae of the lastic Gudehus resonse-enveloe. Oliver et al. [18] roosed an imlicit/exlicit scheme for non-linear constitutive scheme which rovided an additional comutability to those solid mechanics roblems where robustness is an imortant issue. Ragione et al. [19] described the inelastic behavior of a granular material in the range of deformation that recedes localization by idealizing the material as a random array of identical, elastic, frictional sheres and assuming that articles moved with the average strain. Kossa and Szabo [20] resented semi-analytical solutions for the von Mises elastolasticity by combined linear isotroic-kinematic hardening. u et al. [21] resented a semi-imlicit return maing algorithms for integrating generic non-smooth elastic-lastic models. Valoroso and Rosati [22] resented a general rojected algorithm for general isotroic three-invariant lasticity method under lane stress conditions. Gao et al. [23] using exerimental and numerical studies demonstrated that stress state had strong effects on both lastic resonse and ductile fracture behavior of an aluminum 5083 alloy. Cardoso and Yoon [24] derived a stress integration rocedure based on the Euler backward method for a non-linear combined isotroic/kinematic hardening model based on the two-yield surfaces aroach. Ghaei and Green [25] develoed an algorithm for numerical imlementation of Yoshida-Uemori two-surface lasticity model into finite element rogram with the aid of the return maing rocedure. Ghaei et al. [26] introduced a semi-imlicit scheme to imlement Yoshida-Uemori two-surface model into finite element method. Kossa and Szabo [27] roosed a new integration scheme for the von Mises elastic-lastic model with combined linear isotroic-kinematic hardening. Rezaiee-Pajand et al. [28] roosed two new exonential-based aroximate formulations for associative Drucker- Prager criterion lasticity and could obtain the accurate solution of the roblem. Becker [29] roosed a new lasticity algorithm based uon observation from the closed form integration of a generalized quadratic yield function over a single ste. Moayyedian and Kadkhodayan [30] resented a new first and second differentiation of a general yield surface and imlemented it for different time steing schemes including the exlicit, traezoidal imlicit and fully imlicit time steing schemes in rate-deendant lasticity. Moayyedian and Kadkhodayan [31] resented a new non-
LOAD INCREMEN IERAION LOOP Imlementing the New First and Second Differentiation of a General Yield Surface 312 associated viscolastic flow rule (NAVFR) with combining von Mises and resca loci in lace of yield and lastic otential functions and vice verse with the aid of fully imlicit time steing scheme. wo imortant factors in integration algorithms are accuracy and stability. In exlicit algorithms the size of load ste lays a crucial role in stability and accuracy, i.e. a very large load ste cannot guarantee the stability and accuracy of the algorithm. On the other hand, in the full imlicit algorithms, even by choosing a large load ste, the stability is guaranteed. In fact, this is the main advantage of imlicit algorithms to exlicit algorithms. However, choosing a large load ste may not give a roer accuracy. DIMEN Presets the variables associated with the dynamic dimensioning rocess INPU Inuts data defining geometry, boundary conditions and material roerties LOADPS Evaluates the equivalent nodal forces for ressure loading, gravity loading, etc. ZERO Sets to zero arrays required for accumulation of data INCREM Increments the alied loads according to secific loads factors. ALGOR Sets indicator to identify the tye of solution algorithm e.g. initial stiffness, tangential stiffness, etc. SIFFP Calculates the element stiffness for elastic and elastic-lastic material behavior FRON Solves the simultaneous equation system by the frontal method RESIDU Calculates the Residual force vector, INVAR Evaluates the effective stress level. YIELD and FLOWPL Determines the flow vector, a and also d D CONVER Checks to see if the solution rocess has converged OUPU Print the results for this load increment END Fig. 1 Program flow chart for two-dimensional elastic-lastic alications [32]. In this aer, with the aid of obtained exressions for the first and second differentiation of a general yield surface in the revious work of the authors in [30] an internally ressurized thick walled cylinder is considered with erfectly lastic and linear-isotroic hardening behavior of material and coded in Comaq Visual Fortran
313 F. Moayyedian and M. Kadkhodayan Professional Edition 6.5.0 with the aid of references [32-36] in finite element and [37-40] in lasticity theories. Euler forward method with sub-increment and Euler backward method along with the continuum and consistent elastic-lastic moduli with different criteria are investigated and the obtained results are comared to each other and exerimental results. Fig. 1 shows the flow chart of the rogram for two-dimensional elastic-lastic alications for a global finite element. 2 EULER FORWARD (EXPLICI) - MARIX FORMULAION he yield F can be written as: F, 0, (1) where is the stress vector and is the hardening d d for the work hardening. Defining F F F F F F x y z yz xz xy a,,,,,, (2) We have d D d, e (3) where dddd ' D e D, dd D a. H d b D (4) hus H is obtained to be the local sloe of the uniaxial stress/lastic strain curve and can be determined exerimentally [30]. 3 EULER BACKWARD (IMPLICI), (GENERAL CLOSES PON PROJECION) - MARIX FORMULAION For a linear-isotroic hardening in any load ste we have ' n1 Y H n n1. (5) he yield function can take the following form, F F f k n1 n1 n1 n1 n1 ' f n1 Y H n n1 0. (6) he following stes have to be followed in a fully imlicit algorithm.
Imlementing the New First and Second Differentiation of a General Yield Surface 314 trial 1. Assume lastic loading, that is Fn 1 0, the lastic flow residual Rn 1 and yield condition can be defined as: R a Fn 1 Fn1, n1, n1 n1 n n1 n1 where, n D n n 1 1 1. 2. Linearized the above equation. Because n 1 and d n stage, then k 1 k, n1 D n1 and we have: (7) are assumed to be fixed during return-maing n1 n1 n1 n1 0, k 1 k k R n1 n1 n1 k 2 k a n1 n1 0 k k k ' 2 k F a H (8) and the Hessian matrix is: 1 1 an1 n1 D n1. (9) 3. Solve the linearized roblem to obtain 2 k and n 1 k n 1 d as: k k k 2 k Fn 1 an1 n1 Rn1 n1 k k k a n1 n1 an1 H k k k n 1 n 1 R n1, k 2 k a n1 n1 k 1 k 1 : 1 n D n. 4. Udate the lastic strain k k n 1 ', and the consistency arameter k as: n 1 (10) k 1 k 1 k n n n (11) 1 1 1, and k1 k 2 k n1 n1 n1. (12) he rocedure summarized above is simly a systematic alication of Newton s method to the system of Eq. (10) that results in the comutation of the closest oint rojection from trial state onto the yield surface. Note that in
315 F. Moayyedian and M. Kadkhodayan the Euler backward method normality is enforced at the final (unknown) iteration. he above rocedure converged when R R and Fn 1 in Eq. (7) reach to rescribed tolerances. n 1 n 1 4 CONSISEN ELASIC-PLASIC MODULUS - MARIX FORMULAION By differentiating the elastic stress-strain relationshis and the discrete flow rule, we obtain d n1 D dn1 dn1 d n1an1. an1 dn1 n1 d n1, (13) he following algorithmic relationshi is obtained: d d d a n1 n1 n1 n1 n1. (14) On the other hand, differentiating the discrete consistency condition yields ' a d H d n1 n1 n 1 0. (15) hus, from Eqs. (14) and (15) we have d n1 e a ed dn1 ' H e a D D n1 n1 n1., (16) Finally, substituting Eq. (16) into Eq. (14) yields the exression for elastic-lastic relation d D d n1 e n1 n1, (17) where the consistent elastic-lastic modulus is defined as below e a eded D H e a e n1 n1 ' D D n1 n1. n1, (18) Note that the structure of Eq. (18) is analogous to Eq. (4). he elastic modulus D as the continuum modulus in Eq. (4) is substituted by the algorithmic modulus n 1 defined by Eq. (9). It should be noted that to comute n1 in Euler backward method to udate stress state and construct the consistent elastic-lastic modulus, the derivation of the second differentiation of the yield surface is inevitable and for a comlicated criteria is cumbersome where a comrehensive aroach to overcome this difficulty was resented in the earlier work of the authors is emloyed in current research [30].
Imlementing the New First and Second Differentiation of a General Yield Surface 316 5 RESULS AND DISCUSSIONS In the following, different investigations and comarisons are achieved contained with the new exressions for differentiation of a general yield surface as in [30] for rate-indeendent lasticity: 1. Comaring between Euler forward and backward methods for four different criteria containing indeendent and deendant criteria to hydrostatic ressure. 2. Comaring the results obtained from Euler forward and backward methods with those of exerimental data. 3. Comaring the continuum and consistent elastic-lastic moduli for Euler backward method. Figs. 1 and 2 show the variation of circumferential strain and stress at the inner surface with the internal ressure while the von Mises criterion, erfect-lastic material and also linear isotroic hardening behavior such that ' E H 10 are used. he geometry of the cylinder is assumed as b/a=2. he variations of circumferential strain of the inner surface (r/a=1) with ressure and also circumferential stress distributions in the interval of (1 r/ a 2) for Euler forward and backward methods are considered here. Fig. 2 shows that with increasing the ressure and develoment of lastic zone, the differences between the results of Euler forward and backward methods increase. Moreover, in the same ressure, the hoo strain redicted by the imlicit integration algorithm is larger than that of exlicit one. In addition, the hoo stress redicted by Euler backward method is smaller than that of Euler forward method, Fig. 3. Furthermore, the results show that the material hardening increases the difference between two integration algorithm methods. Figs. 4, 5, 6, 7 and 8, 9 show the same arameters based on resca, Mohr-Coulomb and Drucker-Prager criteria, resectively. In using Mohr-Coulomb criterion it is assumed that tan0.879 and for the linear isotroic hardening material ' H E / 20 and in Drucker-Prager criteria it is assumed that ' H E / (10 3). Fig. 2 Circumferential strain of the inner surface with increasing ressure using von Mises criterion. Fig. 3 Circumferential stress distributions using von Mises criterion.
317 F. Moayyedian and M. Kadkhodayan Fig. 4 Circumferential strain of the inner surface with increasing ressure using resca criterion. Fig. 5 Circumferential stress distributions using resca criterion. Fig. 6 Circumferential strain of the inner surface with increasing ressure using Mohr-Coulomb criterion. Fig. 7 Circumferential stress distributions using Mohr-Coulomb criterion.
Imlementing the New First and Second Differentiation of a General Yield Surface 318 Fig. 8 Circumferential strain of the inner surface with increasing ressure using Drucker-Prager criterion. Fig. 9 Circumferential stress distributions using Drucker-Prager criterion. he results show the same trend as exlained for Figs. 2 and 3 reviously. he results show the differences between the exlicit and imlicit integration algorithm methods in Mohr-Coulomb and Drucker-Prager criteria are larger than those in resca and von Mises criteria, resectively. In other words, when the yielding criteria are deendent to the hydrostatic ressure, the differences between the results obtained from the Euler forward and backward methods are more significant than those of the hydrostatic ressure indeendent yielding criteria. Furthermore, it can be shown that in non-linear isotroic hardening materials these differences increase comared with erfect-lastic materials. able 1. shows the required iteration number for converged results when the consistent and continuum elastic-lastic moduli in Euler backward method are used. As it is seen, the required iterations in each increment for consistent elastic-lastic modulus is less than that of the continuum one, i.e. the former reserves the asymtotic rate of quadratic convergence in imlicit integration algorithm. Moreover, the Mohr-Coulomb and von Mises criteria have the highest and the lowest required number of iterations, resectively. Furthermore, the number of required iterations increases when lastic zone is develoed. In erfect-lastic materials, in addition, the number of iterations in last increments increases which can be attributed to the initiation of some kinds of instability in this tye of material. It is also observed that emloying the consistent elastic-lastic modulus instead of the continuum one causes more decreasing in required iteration number in hydrostatic ressure deendent yielding criteria comared to indeendent ones. Finally, emloying the consistent elastic-lastic modulus with Euler backward method for non-circular shae criteria in deviatoric lane needs more iteration in each load ste. Fig.10 show a comarison between the results obtained by Euler forward and backward methods and exerimental results. he figure demonstrates the variations of internal ressure in over strain 100% ( E / Y 1) at external surface of the vessel with the ratio of external radius/internal radius, (b/a), for erfect-lastic material and von Mises criterion. It is seen that using Euler backward method could redict the exerimental result more accurately. It is also observed that differences between the numerical simulations and exerimental data are maximum in b/a=1.6 and minimum in b/a=2.4. It should be noted the results agree with the results in [30] too.
319 F. Moayyedian and M. Kadkhodayan able 1 he number of required iteration in each load ste using, (a) von Mises criterion, (b) resca criterion, (c) Mohr-Coulomb criterion, (d) Drucker-Prager criterion. (a) von Mises criterion and erfect-lastic material Ste 1 2 3 4 5 6 State el el-el el-l l-l l-l l-l Continuum 1 1 2 3 5 8 Consistent 1 1 2 2 3 5 von Mises criterion and linear isotroic hardening material Ste 1 2 3 4 5 6 State el el-el el-l l-l l-l l-l Continuum 1 1 3 4 5 7 Consistent 1 1 2 2 3 4 (b) resca criterion and erfect-lastic material Ste 1 2 3 4 5 State el el-l l-l l-l l-l Continuum 1 3 3 6 9 Consistent 1 2 2 3 5 resca criterion and linear isotroic hardening material Ste 1 2 3 4 5 State el el-l l-l l-l l-l Continuum 1 4 4 5 9 Consistent 1 2 3 3 4 (c) Mohr-Coulomb criterion and erfect-lastic material Ste 1 2 3 4 5 State el el-l l-l l-l l-l Continuum 1 4 6 8 11 Consistent 1 2 3 4 6 Mohr-Coulomb criterion and linear isotroic hardening material Ste 1 2 3 4 5 State el el-l l-l l-l l-l Continuum 1 4 7 8 10 Consistent 1 2 3 3 5 (d) Drucker-Prager criterion and erfect-lastic material Ste 1 2 3 4 5 6 State el el-el el-l l-l l-l l-l Continuum 1 1 4 5 7 10 Consistent 1 1 2 2 4 5 Drucker-Prager criterion and linear isotroic hardening material Ste 1 2 3 4 5 6 State el el-el el-l l-l l-l l-l Continuum 1 1 4 6 6 9 Consistent 1 1 3 3 3 4
Imlementing the New First and Second Differentiation of a General Yield Surface 320 Fig. 10 Comarison between the Euler forward/backward methods and exerimental data. 6 CONCLUSIONS A new rocedure with using newly obtained first and second differentiation of a yield function [30] is imlemented for Euler forward and backward method in rate-indeendent lasticity. In the following, some more obtained results are exlained: 1. Alying more loads and develoment of lastic zone make the differences between the obtained results from Euler forward and backward and also continuum and consistent oerators more obvious. 2. It was demonstrated that the hydrostatic ressure deendent yield criteria continuum and consistent elasticlastic moduli were more sensitive in emloying Euler forward and backward methods. 3. Comaring the results obtained from von Mises and resca and also Drucker-Prager and Mohr-Coulomb criteria show that emloying consistent elastic-lastic modulus in imlicit integration algorithm for noncircular surfaces in deviatoric lane needs more number of iterations to converge. REFERENCES [1] Krieg R.D., Krieg D.B., 1977, Accuracies of numerical solution methods for the elastic-erfectly lastic model, Journal of Pressure Vessel echnology -ransactions of the ASME 99:510-515. [2] Schreyer H.L., Kulak R.F., Kramer J.M., 1979, Accurate numerical solutions for elastic-lastic models, ransactions of the ASME 101: 226-234. [3] Simo J.C., aylor R.L., 1985, Consistent tangent oerators for rate-indeendent elastolasticity, Comuter Methods in Alied Mechanics and Engineering 48:101-118. [4] Ortiz M., Poov E.P., 1985, Accuracy and stability of integration algorithms for elastolastic constitutive relations, International Journal for Numerical Methods in Engineering 21:1561-1576. [5] Potts D.M., Gens A., 1985, A critical assessment of methods of correcting for draft from the yield surface in elastolastic finite element analysis, International Journal for Numerical and Analytical Methods in Geomechanics 9:149-159. [6] Ortiz M., Simo J.C., 1986, An analysis of a new class of integration algorithms for elastolastic constitutive relations, International Journal for Numerical Methods in Engineering 23:353-366. [7] Simo J.C., aylor R.L., 1986, A return maing algorithm for lane stress elastolasticity, International Journal for Numerical Methods in Engineering 38:649-670. [8] Dodds R.H., 1987, Numerical techniques for lasticity comutations in finite element analysis, Comuters & Structures 26:767-779. [9] Ortiz M., Martin J.B., 1989, Symmetry-reserving return maing algorithms and incrementally external aths: A unification of concets, International Journal for Numerical Methods in Engineering 28:1839-1853. [10] Gratacos P., Montmitonnet P., Chenot J.L., 1992, An integration scheme for Prandtl-Reuss elastolastic constitutive equations, International Journal for Numerical Methods in Engineering 33:943-961. [11] Ristinmma M., ryding J., 1993, Exact integration of constitutive equations in elasto-lasticity, International Journal for Numerical Methods in Engineering 36:2525-2544. [12] Kadkhodayan M., Zhang L.C., 1995, A consistent DXDR method for elastic-lastic roblems, International Journal for Numerical Methods in Engineering 38:2413-2431. [13] Foster C.D., Regueiro R.A., Fossum A.F., Borja R.I., 2005, Imlicit numerical integration of a three-invariant, isotroic/kinematic hardening ca lasticity model for geomaterials, Comuter Methods in Alied Mechanics and Engineering 194:5109-5138.
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