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2 /0 X~'A 5S1 CIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 558 CIlPV, UILU-ENG ISSN: A LARGE STRAIN PLASTICITY MODEL FOR IMPLICIT FINITE ELEMENT ANALYSES By ROBERT H. DODDS, JR. and BRIAN E. HEALY A Report on a Researh Projet Sponsored by the DAVID TAYLOR RESEARCH CENTER METALS AND WELDING DIVISION ANNAPOLIS, MARYLAND Researh Contrat: N C-0035 DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS JANUARY 1991

3 REPORT DOCUMENTATION PAGE 4. lltje and SUbtItle 1. REPORT NO. UILU-ENG A Large Strain Plastiity Model for Impliit Finite Element Analyses 7. Author{s) R.H. Dodds, Jr. and B.E. Healy 9. PerfOl'l11k1g OrganIzatIon Name.net Addreaa University of Illinois at Urbana-Champaign Department of Civil Engineering 205 N. Mathews Avenue Urbana, Illinois Sponaorlng OrganIzation Name.net Addreaa David Taylor Researh Center Metal and Welding Division, Code 281 Annapolis, Maryland Reipient'. Aeaalon No. 5. Report Date January 199' Perfor'rNng OrganIzation Report No. SRS ProfetfTaak/Wortt Unit No. 11. ContrIlCt(C) or GnInt(Q) No. N C Type of Report & Pertod Covered Annual: to SUpplementary Note. 18.AbstrKt (limit: 200 words) The theoretial basis and numerial implementation of a plastiity model suitable for finite strains and rotations are desribed. The onstitutive equations governing J2 flow the0!y' are formulated using strainsstresses and their rates defined on the unrotated frame of referene. Unhke models based on the lassial J aumann (or orotational) stress rate, the present model predits physially aeptable responses for homogeneous deformations of exeedingly large magnitude. The assoiated numerial algorithms aommodate the large strain inrements that arise in finite-element formulations employing an impliit solution of the global equilibrium equations. The resulting omputational framework divores the finite rotation effets on =-,....._-,...,... -_... +_"'..."... :_+rt...._n+~"... 1"'"tI..f +)...0 "'1"\+.0", +"... '1...An+o +h.o ~f"'\+o...;nl...0""....,...,...,("1.0.i"\.'1t.o.,.- n 1.1"' :1 (+';,...0\ C"I+a~ :Sll(:1111-:SUt:::s:s Id.Lt:::s HU1l1 ll1lt::t;1a.l.1ull U~ L.llt:: ~al.t::.., L.U ut'ual.~ 1.11\.; 111al.t::~lal ~\';"''pull'''~ uv\.;~ a luau \Ull1~J..,I.t::p. Consequently, all of the numerial refinements developed previously for small-strain plastiity (radial return with subinrementation, plane stress modifiations, kinemati hardening, onsistent tangent operators) are utilized without modifiation. Details of the numerial algorithms are provided inluding the neessary transformation matries and additional tehniques re9.uired for finite deformations in plane stress. Several numerial examples are presented to illustrate the realisti responses predited by the model and the robustness of the numerial proedures. 17. Doument AnatyaIa Desripton Large Strains, Large Rotations, Plastiity, Finite Elements, Numerial Methods, Impliit, Total Lagrangian b. IdentIftera/Open-Ended Terms. COSATI field/group 18. Avallabllty Statement Release Unlimited 19.5eurtty tass (ThIs Report) UNCLASSIFIED 20. Seurtty Class (ThIs Page) UNCLASSIFIED 21. No. of Pages Pr1e (See ANSI-Z39.18) OPTIONAL FORM 272 (4-77) Department of Commere

4 A LARGE STRAIN PLASTICITY MODEL FOR IMPLICIT FINITE ELEMENT ANALYSES By Robert H. Dodds, Jr. and Brian E. Healy Department of Civil Engineering University of Illinois A Report on a Researh Projet Sponsored by the: DAVID TAYLOR RESEARCH CENTER METALS AND WELDING DIVISION Annapolis, Maryland University of Illinois Urbana, Illinois January 1991

5 ABSTRACT The theoretial basis and numerial implementation of a plastiity model suitable for finite strains and rotations are desribed. The onstitutive equations governing J2 flow theory are formulated using strains-stresses and their rates defined on the unrotated frame of referene. Unlike models based on the lassial Jaumann (or orotational) stress rate, the present model predits physially aeptable responses for homogeneous deformations of exeedingly large magnitude. The assoiated numerial algorithms aommodate the large strain inrements that arise in finite-element formulations employing an impliit solution of the global equilibrium equations. The resulting omputational framework divores the finite rotation effets on strain-stress rates from integration of the rates to update the material response over a load (time) step. Consequently, all of the numerial refinements developed previously for small-strain plastiity (radial return with subinrementation, plane stress modifiations, kinemati hardening, onsistent tangent operators) are utilized without modifiation. Details of the numerial algorithms are provided inluding the neessary transformation matries and additional tehniques required for finite deformations in plane stress. Several numerial examples are presented to illustrate the realisti responses predited by the model and the robustness of the numerial proedures. - ii -

6 ACKNOWLEDGMENTS The first author (R.H. Dodds) was supported by the David Taylor Researh Center Metals and Welding Division (Code 281), Annapolis, Maryland under ontrat N C-0035 to the University of Illinois. The seond author (B.E. Healy) aknowledges support provided by the Center for Superomputing Researh and Development under grant SCCA from the illinois Dept. of Commere. Computations reported here were performed on an Apollo DN workstation operated by the Department of Civil Engineering. Aquisition of this omputer was made possible by a grant from Hewlett-Pakard, In. - iii -

7 TABLE OF CONTENTS Setion No. Page 1. INTRODUCTION KINEMATICS, STRAIN-STRESS MEASURES AND THEIR RATES ELASTO-PLASTIC CONSTITUTIVE FRAMEWORK Seletion of Stress-Strain Rate Plastiity Rate Equations Stress Updating on the Unrotated Configuration NUMERICAL PROCEDURES FOR FINITE STRAINS Stress Updating Proedure Tangent Modulus for Stiffness Updating Plane-Stress Idealization Polar Deomposition NUMERICAL EXAMPLES Homogeneous Finite Extension Homogeneous Finite Shear Crak-Tip Blunting In Small-Sale Yielding SUMMARY AND CONCLUSIONS REFERENCES Appendix A iv -

8 LIST OF TABLES Table No. Page 1 Relative Computational Effort Required for Polar Deomposition v -

9 LIST OF FIGURES Figure No. Page 1 Initial and deformed onfigurations illustrating two methods of deomposition Homogeneous finite extension. Comparison of finite-element results with analytial solutions. Inremental, linear-elasti material: E= 1, v= a Homogeneous finite simple shear. Comparison of finite-element and analytial solutions for the Cauhy normalstress. Inremental, linear-elasti material: E = 1, v = b Homogeneous finite simple shear. Comparison of finite-element and analytial solutions for the Cauhy shear stress. Inremental, linear-elasti material: E = 1, v = _ Finite-element model (plane-strain) for boundary layer idealization of the small-sale yielding problem Crak-tip deformations for inreasing applied load in the SSY model 26 6 Comparison of finite-element results for initially blunt tip with asymptoti HRR fields and with sharp-tip models (r and f) are oordinates in initial onfiguration) vi-

10 A LARGE STRAIN PLASTICITY MODEL FOR IMPLICIT FINITE ELEMENT ANALYSES 1. INTRODUCTION High-end workstations and mini-superomputers are making feasible the routine onsideration of plastiity with large strain and rotation effets in finite-element analyses. Suh diverse phenomenon as post-bukling deformations, metal forming, ontat/indentation and the miromehanis of dutile frature may be realistially modeled. Finite rotations of material axes (those attahed to material points that rotate with the ontinuum in a loal sense) ompliate the definition of strain-stress rates and their numerial integration to advane the material response over a load ( or time) step. Traditionally, onstitutive models for large strain plastiity in finite-element odes are ast in a spatial setting whih mandates use of an objetive stress rate to remove that part of the total stress rate due to simple rigid rotation of the material. In a spatial setting, the omponents of Cauhy (true) stress are defined relative to a fixed, Cartesian system; thus rigid rotati-on alone alters the stress omponents. While numerous objetive stress rates may be onstruted [2] with eah leading to a potentially different material response, the Jaumann stress rate has been implemented universally in both expliit and impliit odes given its apparent simpliity, for example [17] and [1]. Over the past ten years, serious objetions to onstitutive models employing the Jaumann rate have developed as more omplex material behavior is onsidered (e.g., kinemati hardening and visoplastiity) and as the magnitude of deformations experiened in the appliations has inreased (plasti strains exeeding %). The first objetion addresses the inreased omplexity of numerial algorithms to aommodate the spatial setting; tensorial state variables within the plastiity model, for example, the bak-stress in kinemati hardening, must also be expressed using an objetive rate and modified to reflet finite rotations. Proessing of the purely kinemati effets due to finite rotations is thus interwoven with integration of evolution equations for the internal state variables. Consequently, development of eah new material model requires potentially individual treatment of finite rotations. The seond objetion to use of the Jaumann rate onerns the physially unaeptable stresses predited at large strains under ertain onditions. The problem of simple finite shear illustrates the defiieny [5]. An inremental, linearelasti material law is used to relate the J aumann stress rate to the rate of deformation expressed in a fixed Cartesian system. The predited Cauhy stresses osillate in an unrealisti manner (a12 atually reverses sign). Nagtegaal and de Jong [22] noted suh stress 1

11 osillations with kinemati hardening in elasto-plastiityfor a material whih strain hardens monotonially in tension. Atluri [2] later showed that similar osillations exist for isotropi hardening unless the elasti strains are vanishingly small. The osillatory response derives from the onstant spin rate tensor harateristi of simple shear while the atual rigid-body rotation diminishes with inreasing deformation, approahing Tf/2 in the limit. The Cauhy stress obtained using the onstant spin tensor beomes erroneous one the logarithmi shear strain, Y12, exeeds 100%. Atluri [2] demonstrated that removal of the osillatory response in simple shear may be aomplished through definition of alternate stress rates or through a more general onstrution of the hypo-elasti material law. In a desire to retain the simplest hypo-elasti material law as a diret generalization of the onventional small-strain forms, Green and Naghdi [9] introdued an objetive stress rate that has been disussed extensively by Dienes [5], Johnson and Bammann [15] and Atluri [2]. A Cauhy stress measure and its objetive rate are defined on an unrotated orthogonal referene frame established through polar deomposition of the total deformation gradient at eah material point. This onstitutive model predits monotonially inreasing stresses in simple shear for inremental, linear-elastiity. Using this onept of an unrotated referene frame for onstitutive modeling, Flanagan and Thylor [8] developed the PRONTO 2-D and 3-D [32] odes for transient dynami analysis with expliit time integration. An impressive olletion of material behaviors and ontat algorithms are inluded in these odes. Constitutive omputations are performed using strains, stresses and their objetive rates defined on the unrotated referene frame. Effets of finite rotations are thus transparent to integration algorithms for stresses and the material state variables. The numerial arhiteture of existing small-strain plastiity models is fully retained. Flanagan and Taylor note their omputational hallenge was development of an exeptionally effiient algorithm for evolution of the polar deomposition with time in the globally expliit solution. The present paper desribes the implementation and performane of the unrotated referene frame onept for finite-element solutions that use impliit methods to resolve the global equilibrium equations and as suh represents an extension of Flanagan and Taylor's work. While effiient methods for polar deomposition remain an issue, two additional hallenges fae the developer of an impliit ode: 1) effiient and aurate shemes to integrate the plastiity rate equations over the very large strain inrements harateristi of impliit methods, and 2) onsistent tangent operators to maintain quadrati rates of onvergene for global equilibrium iterations. Aordingly, the ontents of the paper are as follows: (i) a desription of the kinematis of finite deformation and development of the strain-stress rates, (ii) a brief development of the rate independent plastiity theory for finite strains and a disussion of numerial tehniques to integrate the plastiity rate equations, (iii) details of the omputational steps to proess finite rotations at a material point, development of a onsistent tangent operator, and ompliations arising for the plane-stress idealization. The paper onludes with the solution of several example prob- 2

12 lems that illustrate the physially aeptable responses predited by the material model and the robustness of the numerial implementation. Finite simple extension and shear are examined with omparisons made between analytial and numerial solutions. Severe blunting at a rak-tip is analyzed in the final example with speifi attention given to the global onvergene rate. 2. KINEMATICS, STRAIN-STRESS MEASURES AND THEIR RATES Development of the finite strain plastiity model begins with onsideration of the deformation gradient F = ax/ax, det(f) = J > 0 (1) where X denotes the Cartesian position vetors for material points defined on the referene (undeformed) onfiguration. Position vetors for material points at time t are denoted x (onfiguration B in Fig. 1, after Flanagan and Taylor [8]). The displaements of material points are thus given by u = x-x. In stati analyses we assoiate the time-like parameter t with a speified level of loading imposed on the model. Stress and deformation rates are thus defined with respet to the applied loading rather than with time. The polar deomposition of F yields F = VR= RU (2) where Vand U are the left- and right-symmetri, positive definite streth tensors, respetively; R is a orthogonal rotation tensor. The prinipal values of V and U are the streth ratios, 'N, of the deformation. These two methods for deomposing the motion of a material point are illustrated in Fig. 1. In the initial onfiguration, B o, we define an orthogonal referene frame at eah material point suh that the motion relative to these axes is only deformation throughout the loading history. With the RU deomposition, for example, these axes are "spatial" during the motion from Bo to Bu; they are not altered by deformation of the material. However, during the motion from Bu to B these axes are "material"; they rotate with the body in a loal average sense at eah material point. Strain-stress tensors and their rates referred to these axes are said to be defined in the unrotated onfiguration. The material derivative of displaement with respet to an applied loading parameter is written as v = x (i.e., the material point veloity in dynami analyses). The spatial gradient of this material derivative with respet to the urrent onfiguration is given by L = ~ = av ax = FF- 1 ax ax ax (3) 3

13 Axes Are Material: Follow the Rotation r Rigid Axes Attahed To A Material Point Axes Are Spatial: Do Not Follow Deformation Fixed, Global Axes +-- F Axes Are Spatial: Do Not Follow Deformation Axes Are Material: Follow the Rotation Fig. 1. Initial and deformed onfigurations illustrating two methods of deomposition.

14 The symmetri part of L is the spatial rate of the deformation tensor, denoted D; the skewsymmetri part, denoted W, is the spin rate or the vortiity tensor. Thus, where L=D+W (4) W represents the rate of rotation of the prinipal axes of the spatial rate of deformation D. When integrated over the loading history, the prinipal values of D are reognized as the logarithmi (true) strains of infinitesimal fibers oriented in the prinipal diretions if the prinipal diretions do not rotate. It is important to note that D and W have no sense of the deformation history; they are instantaneous rates. Using the RU deomposition of F, the spatial gradient L may be also written in the form in whih the following relations are used (5) (6) :if = RV +RV (7) and The first term in egn (6) is the rate of rigid-body rotation at a material point and is denoted Q. The spin rate Wand Q are idential when the prinipal axes of D oinide with the prinipal axes of the urrent streth V. Simple extension and pure rotation satisfy this ondition. The symmetri part of the seond term in eqn (6) is alled the unrotated deformation rate tensor and is denoted d (8) d = 1- ( V-IV). 2 (9) The unrotated rate of deformation defines a material strain rate relative to the orthogonal referene frame indiated on onfiguration B in Fig. l. Using the orthogonality property of R that d(r TR)/dt = 0 the unrotated deformation rate may be expressed in the simpler form as (10) 5

15 d = RTDR. (11) The priniple of virtual displaements applied in the urrent (B) onfiguration readily demonstrates the work onjugay of the the spatial rate of deformation, D, and the symmetri Cauhy (true) stress, T. Sine omponents of both D and T are defined relative to the fixed, global axes, the onjugate stress measure for d on the unrotated onfiguration is given simply by where t is termed the unrotated Cauhy stress, i.e., T is the tensor t expressed on the fixed global axes. In subsequent setions, finite-element solutions are onsidered whih employ a Total Lagrangian (T.L.) desription of the motion [4]. Constitutive quantities d and t omputed on the unrotated onfiguration must be transformed into stress-strain measures required within the T.L. framework. The T.L. deformation measure is the Green strain given by whih has the rate E = ~ CFTF + FTy) = FTDF (14)... By using the RU deomposition for F in eqn (14) and the transformation of eqn (11), the deformation rates are related by Upon equating the stress work rates per unit volume in the urrent(b) and referene (Bo) onfigurations. the stress measure onjugate to the Green strain rate is determined to be the seond Piola-Kirhoff stress, denoted S, By substituting the RU deomposition of F into eqn (16) and using the transformation in eqn (12), we find Finally, we need the relationship between the rates of t and S. The derivative of S yields s = m-liv- I + tr(d)s - V-IUS - SUV- I (18) (12) (13) (15) (16) (17) 6

16 where tr(... ) denotes the trae of the symmetri tensor D. 3. ELASTO-PLASTIC CONSTITUTIVE FRAMEWORK 3.1 Seletion of Strain-Stress Rate The simplest form of a hypo-elasti onstitutive relation is adopted to ouple a materially objetive stress rate with a work onjugate deformation rate. The Jaumann and Green Naghdi objetive stress rates are T J = T- Wf+TW = : D (19a) (19b) where the modulus tensor C may depend linearly on the urrent stress tensor and on history dependent state variables. One the objetive stress rate is evaluated using C : D, the needed spatial rate of Cauhy stress, T, is found by omputing W or Q and transposing the above equations. In a finite-element setting, these rate expressions are numerially integrated to provide inremental values of the Cauhy stress orresponding to load (time) steps. When D vanishes both the Jaumann and Green-Naghdi rates predited by the onstitutive models also vanish; however, the two stress rates lead to different spatial rates of Cauhv stress sine Wand Q are Qenerallv not idential. Use of the soin tensor W in eon ' '-'" J... (19a) auses the physially unreasonable response predited for the finite shear problem; the Green-Naghdi rate leads to a realisti response (see numerial examples below). The Jaumann rate is adopted extensively in finite-element odes -- the quantity W is readily available as a by-produt of omputing D whereas omputation ofq requires polar deompositions of F. Hughes and Winget [12] reognized that a onstant spin rate W (or rotation rate Q) limits the aeptable step sizes for impliit odes. They developed a numerial integration sheme for eqn (19a) that retains objetivity of the Jaumann rate for rotation inrements exeeding 30. Suh refinements, however, do not remove the fundamental ause (W) of the osillatory response in simple shear. Roy, et al. [27] reently implemented a 2-D, impliit finite-element ode based on the Green-Naghdi rate as expressed in eqn (19b). They employed the Hughes-Winget proedure to integrate T GN using Q omputed from polar deompositions of F at the start and end of eah load inrement. The Green-Naghdi rate may be written alternatively as the rate of unrotated Cauhy stress, i, expressed on the fixed, Cartesian axes (20) 7

17 Transformation of the spatial deformation rate D in this expression to the unrotated deformation rate d yields Constitutive omputations, equivalent to the Green-Naghdi rate in eqn (19b), therefore an be performed using stress-strain rates defined on the unrotated onfiguration. Updated values of t are rotated via R to obtain the updated Cauhy stress at the end of a load inrement. The numerial problems of integrating the rotation rates in eqns (19a) and (19b) are thus avoided. Moreover, internal state variables of the plastiity model, e.g., the bak-stress for kinemati hardening, are also defined and maintained on the unrotated onfiguration and thus never require orretion for finite rotation effets. The simpliity derived from this onstitutive framework is very appealing and it is adopted in subsequent developments of the finite strain plastiity model. The potential disadvantage of this onstitutive framework is the numerial effort to ompute R for use in eqns (11,12) from the polar deomposition F = RU at thousands of material points for eah of many load steps. This issue is disussed in the setion on numerial proedures. 3.2 Plastiity Rate Equations The inremental plastiity theory onsidered here assumes initial isotropy of the material and neglets strain-rate effets. A von Mises yield surfae and assoiated flow rule are adopted. A mixed isotropi-kinemati hardening model defines subsequent yield surfaes. The Mises yield surfae is given by where ts' is the deviatori part of the shifted stress vetor t s, R is the radius of the yield surfae in deviatori stress spae, and Ep is the effetive plasti strain. R is related to the effetive tensile stress Y by The shifted stress ts is given by where t is the urrent Cauhy stress on the unrotated onfiguration and tb is the bakstress on the unrotated onfiguration whih loates the enter of the yield surfae (for isotropi hardening, tb = 0). Further developments require kinemati deomposition of the total strain rate d into elasti and plasti omponents. The multipliative deomposition of the deformation gradient (21) (22) (23) (24) 8

18 (25) appears most ompatible with the physial basis of elasti-plasti deformation in rystalline metals (see, for example, [3]). FPrepresents plasti flow (disloations) while Ferepresents lattie distortion; rigid rotation of the material struture may be onsidered in either term. Substitution of this deomposition into the spatial rate of the displaement gradient eqn (3) yields We now impose the restrition that elasti strains remain vanishingly small ompared to the unreoverable plasti strains; a behavior losely followed by dutile metals having an elasti modulus orders of magnitude greater than the flow stress. Consequently, FP and Fe are uniquely determined by unloading from a plasti state. This onsiderably simplifies the above expression and permits separate treatment of material elastiity and plastiity. U sing the left polar deomposition and writing the streth as the produt of elasti and plasti parts yields Identifying the elasti deformation as and using the small elasti strain assumption, we have Consequently, the expression for L is approximated by L = L e + II. (30) As in eqn (5), the symmetri part of this approximation for L is taken as D with the result that (26) (27) (28) (29) D = De + I)P. (31) Given the restrition of vanishingly small elasti strains, the multipliative deomposition of the deformation gradient in eqn (25) leads to the familiar additive deomposition of the spatial deformation rate D into elasti and plasti omponents. The onversion of D to the unrotated onfiguration using eqn (11) provides the deomposition sheme needed for d as 9

19 (32) One the above transformation of elasti and plasti strain rates onto the unrotated onfiguration is aomplished, remaining steps in development of the finite-strain plastiity theory are idential to those for lassial small-strain theory. If the elasti strains are not vanishingly small, the inrementally linear form of this hypo-elasti material model predits hystereti dissipation and residual stresses for some losed loading paths, for example, the path defined by finite extension -+ finite shear -+ tension unloading -+ shear unloading [18]. Unoupled loading-unloading for extension and shear produes no residual stresses. For finite-strain plastiity of dutile metals having large modulus-to-yield stress ratios this situation is not a serious onern sine plasti strains are ommonly times greater than the elasti strains. 3.3 Stress Updating on the Unrotated Configuration The plastiity rate equations are numerially integrated over a finite time (load) inrementusing the elasti preditor-radial return algorithm [7,16,19,29,30]. Beause integration of the stress rate ours with all quantities ast onto the unrotated onfiguration, algorithmi details of the integration proedure are idential to those developed for onventional small-strain plastiity models. The elasti preditor-radial return method provides the most auray for both single step and subinrementation shemes (the strain inrement is divided into m subinrements with the plastiity integration proedure applied suessively over eah subinrement). Moreover, the proedure is unonditionally stable and mixed isotropi-kinemati hardening is easily inluded. The plane-stress idealization introdues additional omplexities at two levels. First, the (u, v) nodal displaements do not provide a means to ompute the through-thikness strain inrement, tl.d 33. The updated stress t33 must be zero yet non-zero values of the bak stress, t33(b), and the shifted stress, t33(s), are required to math the Baushinger effet predited by a orresponding 3-D model defined with plane-stress boundary onditions. The elasti preditor-radial return algorithm, for example, an be exeuted iteratively in a 3-D setting to ompute simultaneously tl.d 33 and the updated stresses under the onstraint that t Simo and Taylor [31] and Keppel and Dodds [16] provide details of two suh shemes. The seond ompliation introdued by plane-stress involves the F33 term of the deformation gradient whih aounts for finite hanges of material thikness due to loading. This term must be onstruted from inrements of tl.d 33 determined by the stress update proedure. Computation of F33 is desribed in the next setion. 4. NUMERICAL PROCEDURES FOR FINITE STRAINS The numerial algorithms in this setion are developed for a Total Lagrangian setting. Only minor modifiations are required for use of these same algorithms in an Updated 10

20 Lagrangian setting. The global solution is advaned from time (load) tn to tn+l using an inremental-iterative Newton method. Iterations at tn+l to remove unbalaned nodal fores are onduted under fixed external loading and no hange in the presribed displaements for displaement ontrolled loading. Eah suh iteration, denoted i, provides a revised estimate for the total displaements at t n +l, denoted U~?l. Fully onverged displaements at tn are denoted Un. Following Pinsky, et al. [23] a mid-inrement sheme is adopted in whih deformation rates are evaluated on the intermediate onfiguration at 1/z(u n + U~?l) = U~i~lj2 The hoie of 0.5 represents a speifi form of the generalized trapezoidal rule that is unonditionally stable and seond-order aurate. Key and Krieg [17] have demonstrated the optimality of the mid-point onfiguration for integrating the rate of deformation and the resulting orrespondene with logarithmi strain. The following setions desribe the omputational proesses performed at eah material (Gauss) point to: 1) update stresses, 2) provide a onsistent tangent matrix for updating the global stiffness matrix and 3) resolve ompliations arising from the plane-stress idealization. A brief disussion of the proedure to ompute the polar deomposition of the deformation gradient is also provided. The organization of a partiular finite-element ode ditates whih operations are performed in the element dependent routines and whih are performed in the material models; thus no partiular distintion is made here. Standard finite-element proedures to ompute deformation gradients at Gauss points in a T.L. setting are also omitted. 4.1 Stress Updating Proedure The omputational steps are: Step 1. Compute the deformation gradients at n + V2 and n + 1!lex (i) ) F(i) _ U + U n +l n+l -!lex ax (i) F(i) _ (1 + U n +112 (i) (i) I n + 1 = det(f n + 1 ) (33) n+ll2 - ax (34) Step 2. Compute polar deompositions at n + V2 and n + 1 (35) Step 3. Compute the i th estimate for the spatial gradient of the displaement inrement over the step where (36) (37) 11

21 ( (i) ) a(~ 0») ~F(i) = a U n+ 1 - Un = U ax ax Step 4. Compute the i th estimate for the spatial deformation inrement over the step Step 5. Rotate the inrement of spatial deformation to the unrotated onfiguration AdO) - RO)T. A DO). RO) il - n+lh il n+lh (40) Step 6. The terms of ~d (i) define the strain inrements for use in a onventional small-strain plastiity model. Invoke the small-strain plastiity model to provide the i th estimate for the unrotated Cauhy stress at n + 1 (38) (39) (41) where C denotes the small-strain integration proess using the elasti-preditor, radial return algorithm. The integration proess requires state variables at n: the unrotated Cauhy stress, the equivalent plasti strain, and the bakstresses on the unrotated onfiguration. Step 7. The unrotated Cauhy stress at n + lis transformed to the 2nd Piola-Kirhoff stress at n + 1 as required for the T.L. setting - S U) - JU). U(i)-l. t(i). U O)-l n+1 - n+l n+l n+1 n+l (42) Key advantages of the above steps are the absene of half-angle rotations applied to stresses (and bak-stresses) found in o-rotational rate formulations, eqn (19), and most importantly, the ability to use an existing small-strain plastiity model for Step 6 without modifiation sine all quantities are referred to the unrotated onfiguration. The disadvantage is the need to perform two polar deompositions for eah stress update. Finally, onverged deformation inrements ~D are summed over k load steps to define the logarithmi strains for output n=k eij = I!::J)ij n=l 4.2 Tangent Modulus for Stiffness Updating A tangent modulus matrix, denoted [C ep ], is needed to form the element-struture stiffness matrix in impliit odes. The moduli ouple inrements of Green strain with inrements of 2nd Piola-Kirhoff stress required by the IL. formulation. To maintain a quadrati onvergene rate of the global Newton iterations, the tangent operator must be onsistent with the numerial algorithm employed to integrate the stress rate just desribed. Consisteny implies that the finite stress inrement predited by the tangent oper- (43) 12

22 ator ating on a strain inrement mathes, to first order, the stress inrement predited by the integration proedure. The small-strain plastiity model provides the onsistent tangent modulus [30] that relates the unrotated stress inrements and unrotated deformation inrements (in matrixvetor form) (44) The needed form of the above relation for the global T.L. approah is (45) where {~EG} is the inremental Green strain. To transform [C;p] ~ [C ep], the inremental forms of the rate transformations in eqns (15) and (18) are employed (46) where tr(.. ) denotes the trae of a tensor and ~d = U- I ~EG U- I. (47) Attempts to ombine eqns (46) and (47) into a transformation operator yield a nonsymmetri [C tp ] even though [C;p] is symmetri. Moreover, the resulting expression is unneessarily omplex and very diffiult to express in the matrix form of eqn (45). To preserve the symmetry of [C ep], two assumptions are made to develop an approximate transformation operator: (1) the material is inompressible suh that tr(~d) ~ 0 and (2) the term ~U may be negleted in omparison to U and S. With these two assumptions, the approximate transformation of tangent moduli may be written in matrix form as (48) Terms of the 6x6 matrix [T] (for 3-D) are derived from the symmetri, positive definite matrix U omputed from the polar deomposition F = RU at the urrent onfiguration. The 3-D form of [T] is given below. Axisymmetri and 2-D speializations are derived by omitting the appropriate rows and olumns. The row-olumn ordering of [T] is: x, y, z, xy, yz, xz. To shorten the notation, we introdue the following terms: U rr-l. U rr1; U rr-l. U U-I. U rr-l. U rr-1 1 = U11, 2 = u21 3 = u22, 4 = 31, 5 = u32, 6 = u33. (49) 13

23 With this notation, [T] is given by u 2 1 u 2 2 u~ 2u1U2 2u2U4 2ulU4 u 2 2 u3 2 us 2 2u2U3 2u3US 2u2Us [T] = u4 2 us 2 U~ 2u4US 2uSU6 2u4U6 (50) UIU2 U2U3 U4US 2 UIU3 + U2 U4U3 + U2US UIUS + U2U4 U2U4 U3US USU6 U2US + U4U3 U3U6 + U~ U2U6 + U4US UIU4 U2US U4U6 UIUS + U2U4 U2U6 + U4US UIU6 + U~ Numerial tests demonstrate that this approximate transformation of tangent moduli maintains the onvergene rate of the global Newton iterations (subsequently disussed example problems show this). Use of the onsistent moduli for the unrotated onfiguration in eqn (44) appears more important for good onvergene rates than the purely geometri transformation approximated by eqn (48). 4.3 Plane-Stress Idealization The F 31, F 32, F 13, and F23 terms off vanish for motion restrited to the 1-2 plane (planestress, plane-strain, and axisymmetri idealizations). The FIb F 12, F 21, and F22 terms are determined from the in-plane displaements. The F33 term is neessary for omputation of J = det(f); J appears in the stress and tangent moduli transformations, eqns (42) and (43). For plane-strain analyses F33 = 1; for axisymmetri analyses F33 = (Ro + u )/Ro where Ro is the undeformed radius of the material point. For plane-stress onditions, F33 is simply the urrent thikness, T, divided by the undeformed thikness, To, F33 = ~ = To + ~T To To The hange in thikness is obtained by integrating the unrotated deformation rate over the loading history to define the through-thikness logarithmi strain InCl + ~~) = e33 = I I d33 = (51) D33 = I tj.d33 (52) where equivalene of the (3,3) spatial deformation and (3,3) unrotated deformation terms is noted for motion in the 1-2 plane. The solution of the above expression for ilt and the substitution into eqn (51) provides the needed expression for F33 as The term I ~d33 is maintained as a history dependent quantity at eah Gauss point in the same manner as the aumulated plasti strain fp. (53) 14

24 4.4 Polar Deomposition The polar deomposition F = RU is a key step in the stress-updating algorithm and must be performed twie for eah Gauss point for eah stress update, i.e., at n + 1;1 and n + 1. The omputational effort required for the polar deomposition should be insignifiant relative to the element stiffness omputation and the equation solving effort. For their expliit ode, Flanagan and Taylor [8] developed an algorithm for the integration of R = QR that maintains orthogonality of R for the very small displaement inrements harateristi of expliit solutions. Numerial tests readily show their proedure fails for large displaement inrements experiened with impliit global solutions. The following algorithm removes suh approximations and yet remains omputationally very effiient with the framework of an impliit solution. Step 1. Compute the right Cauhy-Green tensor and its square (54) (55) where only the upper-triangular form of the symmetri produts (6 terms) are atually omputed and stored. Step 2. Compute the eigenvaluesl!,l~ andl~ of C. A Jaobi transformation proedure speifially designed for 3x3 matries is used to extrat the eigenvalues. Do-loops are eliminated by expliitly oding eah off-diagonal rotation form. Two or, at most, three sweeps are needed to obtained eigenvalues onverged to a 10-6 tolerane. Step 3. Compute invariants of U and the det(f) (56a) (56b) (56) Step 4. Form the upper triangle of the symmetri, right streth, U, and it's symmetri inverse, V-I (see [11]) where I denotes a unit tensor with the f3 oeffiients defined by (57a) f31 = l/(/ullv - IIIv), {32 = Iulllu, {33 = I~ - IIu (57b) 15

25 Similarly, the inverse of U may be formed diretly as (57) where the y oeffiients defined by Yl = l/iiiu(/ullu-illu), Y2 = Iu1f2u- 111u(/t;+IIu), (57d) Y3 = - IIIu - IuCPu - 2IIu), Y4 = Iu Step 5. Form R as the produt R= FU- 1 (58) The FORTRAN ode listing for the above proedure is given in the appendix. Table 1 summarizes the relative omputational effort required for (1) generation of the element tangent stiffness matrix, (2) stress updating at all Gauss points of the element (inluding two polar deompositions at eah Gauss point), and (3) the relative time required for a single polar deomposition. Results are given for an axisymmetri, 8-node isoparametri element and a 3-D, 20-node isoparametri brik element. Both elements employ redued integration rules; 2x2 for the axisymmetri element and 2x2x2 for the 3~D element. Computations were performed on a Unix workstation. The CPU time required for generation of the element tangent stiffness is assigned a unit value for eah ase. The results learly demonstrate that polar deompositions are not a omputational issue in an impliit ode. Table 1. Relative Computational Effort Required for Polar Deompositions Computation 8-Node Axisymmetri 20-Node 3-D (2x2 Gauss Rule) (2x2x2 Gauss Rule) Element [K T ] Element Stress Updating Single Polar Deomposition NUMERICAL EXAMPLES Numerial results for three example problems are presented in this setion. The examples demonstrate the exellent performane of the finite strain model for both 2-D and 3-D 16

26 onfigurations. The first two examples onsider finite, homogeneous deformation in unoupled extension and simple shear. By adopting an inrementally-linear material, analytial solutions may be onstruted for these two problems to assess the auray of the finite-element solutions. The third example onsiders the plane-strain, Mode I smallsale yielding problem for a material that follows the rate independent, inremental plastiity theory. An initially blunt noth tip is opened to several times the initial width in a boundary layer model that approximates the onditions at a rak tip in an infinite body The finite-strain model is implemented as a pre- and post- proessor for the existing small-strain plastiity model in our researh finite-element system (POLO-FINITE [6]). The small~strain model required no hanges; we onsider this to be a signifiant advantage of adopting the unrotated onfiguration to perform onstitutive omputations. 5.1 Homogeneous Finite Extension Consider a unit blok of material aligned with edges parallel to the oordinate axes (see Fig. 2). The blok is onstrained and loaded onsistent with uniaxial tension in the Xl diretion. The displaement field is given by where a, k are onstants and t is a time-like loading parameter that inreases monotonially from zero. For a unit ube, a may be taken as unity. The oordinate streth ratios are then In the absene of rotation, the unrotated Cauhy, Jaumann, and Green-Naghdi stress rates are idential, i.e., R = I, F = U, d = D, and W = O. The hypo-elasti relations all have the form (59) (60) t = AItr( d) + 2Jld (61) where A and Jl are the Lame onstants. Diret integration of these relations yields (62) for the Cauhy stresses. The streth ratios are related by from whih the orresponding axial fore is found to be (for a ube with unit initial edge lengths) (63) (64) 17

27 ~ O.B... Axial Fore, Finite-Element (3-~ & Plane-Stress) A Axial Strain, Finite-Element :::l 2.0 Exat ~ 1 ~.~ d 1.5 ~ ~ 0.7 rrj f-' (X) ~ ~ ) U ~ ~ X ~ 1 2 U.~ Deformed ~ 0.4 S ~ I ~ ~ ~.~ 0.3 ~ ~ bi) / / 0 L 0.1 J-/' Thikness = 1 b= 1 ---I Al J 0.0, I d ~ Streth Ratio, A 1 Fig. 2. Homogeneous finite extension. Comparison of finite-element results with analytial solutions. Inremental, linear-elasti material: E = 1, v = 0.3.

28 where E and v are Young's modulus and Poisson's ratio, respetively. This load-streth response, plotted in Fig. 2, exhibits a maximum load effet ap 1/ aai = 0 at a ritial streth ratio of Al = Plane-stress and 3-D finite-element models are analyzed for this finite extension problem. The plane-stress model ontains four, linear quadrilateral elements; the 3-D model ontains eight, linear brik elements. The integration order for the plane-stress elements is 2x2 with a 2x2x2 order used for the 3-D elements. Constitutive omputations are performed by the inremental plastiity model with a yield stress large enough to prevent plasti deformation. Numerial results for these two models should be idential provided the through-thikness strain omputations desribed in eqns (51) and (53) are performed during solution of the plane-stress model. Both models are loaded by displaements imposed on the fae Xl = 1. Thirty (30) equal size inrements are imposed to reah the deformation Al = 7. Iterations at eah load step are performed until the onvergene test given by II R II ~ II p II * 10-4 (65) is satisfied, where II II denotes the Eulidean norm, R is the residual fore vetor, and P is the vetor of total reations at the onstrained nodes. For both the plane-stress and 3-D solutions, a total of 60 iterations are performed for the 30 load steps; two iterations are needed for onvergene at eah load step. The plane-stress and 3-D solutions are idential. Figure 2 ompares the analytial and omputed axial fores as a funtion of the streth ratio. The figure also ompares the exat axial strain, In AI, with the finite-element approximation obtained by summing deformation inrements as in eqn (43). The maximum error in predited axial fore is 0.3 % while the maximum error in the predited logarithmi strain is 0.1 %. 5.2 Homogeneous Finite Shear In the finite simple shear problem disussed by Dienes [5], material undergoes simultaneous strething and large rotation. Analytial solutions for eah stress rate are now available for assessment of numerial implementations. A unit ube of material is again employed as shown in Fig. 3. The displaement field is given by where a is a onstant and t is a time-like loading parameter that inreases monotonially from zero. The deformation gradient and material displaement derivative are simply (66) 1 F = 0 o at o 1 L= o a o 0 0 (67) 19

29 T I 1 L 1 X 2 l.-1--.i ~ Deformed Shape / ;# / / // / / / / / / // X 1 Unrotated Cauhy Rate, Eqn (73) Finite-Element 8 Inrements N o /, /, /, /, /, /, /... /... / Jaumann Rate, Eqn (71 ~)y -- A 16 Inrements -1.0 o Shear Strain, Yxy Fig. 3a. Homogeneous finite simple shear. Comparison of finite-element and analytial solutions for the Cauhy shear stress. Inremental, linear-elasti material: E = 1, v = 0.3.

30 F shows that the deformation is isohoriwithl = 1. Sine motion is restrited to thex 1 -X 2 plane, R must have the form os f3 sin f3 0 R = - sinf3 osf3 0 (68) o o 1 where the angle f3 is given by f3 = tan- 1 (at/2). (69) The rate of deformation in the fixed Cartesian system, eqn (5), and the unrotated onfiguration, eqn (11), are o 1 0 a D = o sin2fj os 2f3 0 a d = - os2f3 sin 2f3 0 2 o 0 0 Dienes [5] adopted the inrementally-linear, eqn (61), for the Jaumann and unrotated stress rates. The onstitutive models are thus (70) T J = Altr(D) + 2. ld (71a) i = Altr( d) + 2. ld (71b) The integration of eqn (71a) yields the following solution for the Cauhy stresses TIl = - T22 =. l(1 - os at), T12 =. l sin at (72a) while the integration of eqn (71b) yields the unrotated Cauhy stresses as t11 = - t22 = 4. l1n(osfj), t12 = 2. l(2fj - tanfj) (72b) These stresses are rotated to the fixed Cartesian system using R from eqn (68) to yield TIl = - T22 = 4. l[os2fjln(osfj) +fjsin2fj- sin 2 fj], T12 = 2. l os 2fJ[2fJ - 2 tan 2fJ In (osfj) - tanfj] (73) Figure 3 ompares the analytial solutions for the Cauhy stresses obtained using the Jaumann rate and the unrotated Cauhy rate. The solution for the Jaumann rate exhibits a physially unaeptable harmoni osillation while the solution for the unrotated 21

31 Fig. 3b. Homogeneous finite simple shear. Comparison of finite-element and analytial solutions for the Cauhy normal stress. Inremental, linear-elasti material: E = 1, v = 0.3.

32 Cauhy rate inreases monotonially with inreasing deformation. The Green-Naghdi rate yields the same solution for the stresses as eqn (73). The finite-element model for the finite shear problem ontains four, linear quadrilateral elements with an integration order of 2x2. To maintain the isohori deformation desribed by eqn (66), the model must be loaded by presribing the displaements at all nodes. Thus no iterations are neessary. The logarithmi shear strain, Yxy, is inreased to a magnitude of 8 in separate analyses using 8 and 16 equal size displaement inrements. For a unit ube, the shear strain equals the imposed displaement along the edge X 2 = l. The omputed Cauhy stresses for these analyses are ompared with the exat solutions in Fig. 4. The finite-element stresses very losely math the exat solution and show very minor dependene on the load-step magnitude. In the first of 8 inrements, the error is 3%; as Yxy ~ 8 the error dereases to less than 0.1 %. 5.3 Crak-Tip Blunting In Small-Sale Yielding Small-sale yielding (SSY) in Mode I plane-strain haraterizes the deformation near a rak tip in an infinite body. Rie and Traey [26] and MMeeking [20,21] developed a boundary-layer approximation for the infinite body model that is suitable for finite-element analysis. The SSY model onsists of an annular region ontaining either a sharp or smoothly blunt rak tip whih is subjeted to inreasing displaements of the elasti (Mode I) singular field on the outer irular boundary. SSY mod~ls are ommonly employed in frature mehanis studies to investigate ontinuum based, miromehanis parameters that desribe the initiation of dutile rak growth. Very effiient finite-element models that inlude the effets of large strains and large material rotation are essential for studies that investigate suh parameters. Figure 4 shows the inner portion of the plane-strain finite-element model developed to solve the SSY problem. The rak is modeled as a noth of initial width b o having a semi-irular tip. The mesh extends to a radius R = 500Ob o and ontains 2328 nodes, node isoparametri elements. The smallest element at the noth tip has length b o /12. The use of redued (2x2) integration eliminates loking due to the inompressible plasti deformation. The uniaxial, true stress-logarithmi strain urve follows a power-law form where the material onstants seleted for analysis are: fo = 0.002, a o = 60, n = 10, a = 1, and Poisson's ratio v = 0.3. Symmetrial boundary onditions are applied on the rak plane (Xl ~ b o, X 2 = 0). The noth surfae remains tration free. Displaement inrements of the elasti K]-field for Mode I are imposed on the outer irular boundary. The boundary displaements are inreased monotonially in 40 equal inrements to the level KJ / (TaR) = 1.11 at whih point the plasti zone extends = RI10. Iterations at fixed KJ are performed until the onvergene test, eqn (65), with a tolerane of 5 x 10-4 is satisfied. This tolerane insures 0.1 % onvergene of the strains for this problem. (74) 23

33 2328 Nodes 737 Elements R = SOOOb o X t 2 KI Displaement Field Imposed on Remote Boundary N +:-- '-Symmetry Conditions.. X 1 Fig. 4. Finite-element model (plane-strain) for boundary layer idealization of the small-sale yielding problem.

34 Figure 5 shows the deformed near-tip region at inreasing levels of the noth opening. Equivalent plasti strains in the noth-tip element exeed 2 at the maximum load. Figure 6 shows the opening mode Cauhy stress, aee, and equivalent plasti strain ahead of the noth tip on the rak plane. The radial distane is normalized by J / a o where J is the value of Rie'sJ-integral [24]. A domain integral method [28] is used to extrat J from the numerial solution; for SSY onditions J = K1(1 - v 2 ) / E. The noth opening b an be estimated as 0.5J/a o ; the horizontal axis thus spans 0 ~ lob. Outside the 'blunting' zone of size r = 3b, the present stresses ahieve a steady-state ondition whih sales with J/a o ' One b/b o > 3, exellent agreement is observed between the present stresses and those of a onventional small-strain model ontaining a sharp rak tip (modeled by singularity elements). The plasti strain distribution, shown for maximum load in Fig. 6, reveals that the zone of finite strains extends to r = 3b beyond whih the strains are a only a few multiples of the yield strain Eo. The small-strain, asymptoti stresses (HRR) of Huthinson [13] and Rie and Rosengren [25] are shown for omparison. The present finite strain results for an initially blunt noth and those for a small-strain model with an initially sharp noth both fall below the HRR solution at inreasing distanes from the tip, i.e., the HRR solution ontains only the leading term of the full SSY field and is orret only for r ~ 0. Eah of the 40 loading inrements required an average of3 iterations for onvergene. Elements at the noth-tip sustain strain inrements of about 4.3 % or 21 x Eo. To gauge the onvergene rate, this problem was analyzed using a onventional small-strain plastiity model-the same number of iterations were required for onvergene. The finitestrain solution required 10% more CPU time than the orresponding small-strain analysis. A solution using 200 inrements produed essentially no differene in the stresses and a maximum 1.5 % differene in noth-tip strains. A solution with 20 inrements onverged without diffiulty but with a loss of auray in noth-tip strains. 6. SUMMARY AND CONCLUSIONS The numerial implementation of a flow theory plastiity model suitable for large strains and large rotations has been presented. The onstitutive omputations are formulated using strains. stresses. and their rates ast on the unrotated referene frame thereby removing many of the ompliating details of previous finite strain models. Polar deomposition of the deformation gradients is employed to establish the unrotated referene frame; the resulting onstitutive model is equivalent to one based on the Green-Naghdi stress rate but muh simpler to implement numerially. The stress updating proess is developed for an impliit, Total Lagrangian formulation of the finite-element method. Details of the various transformations and the onsistent tangent modulus are given inluding a very effiient algorithm for 3-D polar deomposition. This finite-strain material model retains 25

35 (a) (b) N ~ Note: Deformations to Sale b/b o Noth-Tip b/21 wtt1l (C) \... Noth-Tip (a) (b) I 3.36 I 1.58 () I 4.96 I 2.08 Fig. 5. Crak-tip deformations for inreasing applied load in the SSY model.

36 5 0.5 ',I HRR 4- " o Small-Strain Solution (Sharp-Tip Model) 0.4 N '-.J a()() a O 1 I. )II J r X7 I JTl 1') I- ~. 4 '-J ~~.. no u- au~ -I 0.3 b/b o KJ /(ej2or) -I r 3.36 I ~ 4.96 I 1.11 Results for ()= 0 ~ 0.1 Epl Fig. 6 1 o 1 2 J 4 5 rj(j ja o ) Comparison of finite-element results for initially blunt tip with asymptoti HRR fields and with sharp-tip models (r and 0 are oordinates in initial onfiguration). 0.0

37 the full numerial arhiteture of a onventional small-strain plastiity model with isotopi-kinemati hardening and has been implemented as a pre- and post- proessor for suh a model in our finite-element ode. Numerial tests demonstrate that for an impliit global solution, the omputational effort required for the polar deompositions is insignifiant relative to the effort required for updating element stiffnesses. Three numerial examples illustrate the aeptable responses predited by the material model for simple homogeneous deformation of an inremental, linear-elasti material and for dutile frature analyses of a material following inremental plastiity. Large step sizes are aommodated without undue loss of solution auray or onvergene rate of the global equilibrium iterations. 7. REFERENCES 1. ABAQUS User's Manual, Version 4.8, Hibbitt, Karlsson & Sorensen, In., Providene R.I., Atluri, S. N., "On onstitutive relations at finite strain: hypo-elastiity and elasto-plastiity with isotropi and kinemati hardening," Computer Methods in Applied Mehanis and Engineering, Vol. 43, 1984, pp Asaro, R. J., "Crystal Plastiity," Journal of Applied Mehanis, 50, 1984, pp Bathe, K. J., "Finite element proedures in engineering analysis," Prentie-Hall, Dienes, J. K., "On the analysis of rotation and stress rate in deforming bodies," Ata Mehania, Vol. 32, 1979, pp Dodds, R. H., and Lopez, L. A., "Software virtual mahines for development of finite-element systems," International J. for Engineering with Computers, Vol. 13, 1985, pp Dodds, R. H., "Numerial tehniques for plastiity omputations in finite-element analysis," Computers & Strutures, Vol. 26, No.5, 1987, pp Flanagan, D. P., and Taylor, L. M., ':.\.n aurate numerial algorithm for stress integration with finite rotations," Computer Methods in Applied Mehanis and Engineering, Vol. 62, 1987, pp Green, A. E., and Naghdi, P. M., '~general theory of an elasti-plasti ontinuum," Arh. Rat. Meh. Anal., Vol. 18, No. 1965, pp Hibbitt, H. D., Maral, P. V, and Rie, J. R., ':.\. finite element formulation for problems of large strain and displaement," International Journal of Solids and Strutures, Vol. 6, 1970, pp Hoger, A., and Carlson, D. E.,"Determination of the streth and rotation in the polar deomposition of the deformation gradient," Quarterly of Applied Mathematis, Vol. 10,

38 12. Hughes, T. J. and Winget, J., "Finite rotation effets in numerial integration of rate onstitutive equations arising in large-deformation analysis," International 1. for Numerial Methods in Engineering, Vol. 15 (12), 1980, pp Huthinson, J. W, "Singular behavior at the end of a tensile rak in a hardening material," Journal of the Mehanis and Physis of Solids, Vol. 16, 1968, pp Jaumann, G., "Geshlossenes system physikalisher und hemisher differentialgesefze," Sitz Zer. Akad. Wiss. Wein, (IIa), Vol. 120, 1911, pp Johnson, G. C. and Bammann, D. J., '~disussion of stress rates in finite deformation problems," International 1. for Solids and Strutures, Vol. 20 (8), 1984, pp Keppel, M. and Dodds, R. H., " Improved numerial tehniques for plastiity omputations in finite-element analysis," Computers & Strutures, Vol. 36, No.1, 1990, pp Key, S. Wand Krieg, R. D., "On the numerial implementation of inelasti time dependent and time independent, finite strain onstitutive equations in strutural mehanis," Computer Methods in Applied Mehanis and Engineering, Vol. 33, 1982, pp Koji, M., and Bathe, K. 1., "Studies of finite-element proedures: stress solution of a losed elasti strain path with strething and shearing using the Updated Lagrangian J aumann formulation," Computer and Strutures, Vol. 26, No. 1/2, 1987, pp Krieg, R. D., and Key, S. W "Implementation of a time independent plastiity theory into strutural omputer programs. In Constitutive Equations in Visoplastiity: Computational and Engineering Aspets, AMD-20 (Edited by J. A. Striklin and K. J. Sazalski), ASME, New York, 1976, pp MMeeking, R. M., and Rie, 1. R., "Finite-element formulations for problems of large elasti-plasti deformation," International Journal of Solids and Strutures, Vol. 11, 1975, pp MMeeking, R. M., "Finite deformation analysis of rak-tip opening in elasti-plasti materials and impliations for frature/' Journal of the Mehanis and Physis of Solids, Vol. 25, 1977, pp Nagtegaal, 1. C., and de long, 1. E., "Some omputational aspets of elasti-plasti, large strain analysis," International J. for Numerial Methods in Engineering, Vol. 12, 1981, pp Pinsky, PM., Ortiz, M., and Pister, K. S., "Numerial integration of rate onstitutive equations in finite deformation analysis," Computer Methods in Applied Mehanis and Engineering, Vol. 40, 1983, pp Rie, J. R.,'~ path independent integral and the approximate analysis of strain onentrations by nothes and raks," Journal of Applied Mehanis, Vol. 35, 1968, pp Rie, J. R., and Rosengren, G. E, "Plane strain deformation near a rak tip in a power law hardening material," Journal o/the Mehanis and Physis of Solids, Vol. 16, 1968, pp Rie, J. R., and Traey, D. M., "Computational frature mehanis," in Numerial and Computer Methods in Strutural Mehanis, eds. SJ. Fenves, et al., Aademi Press, New York, 1968, pp

39 27. Roy, S., Fossum, A. F., and Dexter, R. J., "On the use of polar deomposition in the integration of hypo--elasti onstitutive laws," submitted for publiation. 28. Shih, C. F., Moran, B., and Nakamura, T., "Energy release rate along a three-dimensional rak front in a thermally stressed body," Internationall of Frature, Vol. 30, 1986, pp Simo, 1. C., and Ortiz, M., ''A unified approah to finite deformation elasto-plasti analysis based on the use of hyper-elasti onstitutive equations," Computer Methods in Applied Mehanis and Engineering, Vol. 49, 1985, pp Simo, J. C., and Taylor, R. S., "Consistent tangent operators for rate-independent elastoplastiity," Computer Methods in Applied Mehanis and Engineering, Vol. 48, 1985, pp Simo, J. C., and Taylor, R. S., ''A return mapping algorithm for plane stress elasto-plastiity," Internationall of Numerial Methods in Engineering, Vol. 22,1986, pp Taylor, L. M. and Flanagan, D. P., "PRONTO 2D, a two-dimensional transient solid dynamis program," SAND , Sandia National Laboratories, Albuquerque, NM,

40 APPENDIX A FORTRAN CODE FOR POLAR DECOMPOSITION 31

41 W N ****************************************************** * * * mtmplr -- polar deomposition of the deformation * * gradient into the rotation tensor and * * the right streth tensor. (3x3) * * * ****************************************************** subroutine mtmplr( f. r, u, ui, datf ) impliit intarer (a-l) real (3,3). r(3,3), u(e), ui(6), e(6), ee(6), & ev(3), et(e), iu, iiu, iiiu, a!. bl, l, a2, & b2, e2, d2, date (l) (2) (3) (4) (5) (6) t(l) t(2) t(3) t(4) t(5) t(6) (l) (2) (3) (4) (5) (6) C,CC,u and ui are in symmetri upper triangular form. ompute the metri tensor. f(l,l)*f(l,l)+f(2,l)*f(2,l)+f(3,l)*f(3,l) f(l,l)*f(l,2)+f(2,l)*f(2,2)+f(3,l)*f(3,2) f(l,2)*f(l,2)+f(2,2)*f(2,2)+f(3,2)*f(3,2) f(1,1)*f(l,3)+f(2,l)*f(2,3)+f(3,1)*f(3,3) f(1,2)*f(l,3)+f(2,2)*f(2,3)+f(3,2)*f(3,3) f(1,3)*f(1,3)+f(2,3)*f(2,3)+f(3,3)*f(3,3) (l) (3) (6) (2) (5) (4) ompute the square of the metri tensor. (1)*C(1)+(2)*(2)+C(4)*(4) (1)*(2)+(2)*(3)+(4)*(5) (2)*C(2)+(3)*(3)+C(5)*(5) (1)*C(4)+(2)*(5)+(4)*(6) (2)*C(4)+(3)*(5)+(5)*(6) (4)*C(4)+(5)*(5)+(6)*(6) ompute the prinipal values of the metri tensor. all mtmevd( t, ev ev(l) ev (2) ompute the invariants of the right streth tensor. the determinant of deformation tensor is produt of right streth eigenvalues. sqrt(ev(l» sqrt(ev(2» ev(3) iu iiu iiiu detf a1 b1 1 u (1) u(2) u(3) u(4) u(5) u (6) a2 b2 2 d2 ui(l) ui (2) ui(3) ui(4) ui(5) ui(6) sqrt(ev(3» ev(1)+ev(2)+ev(3) ev(1)*ev(2)+ev(2)*ev(3)+ev(1)*ev(3) ev(1)*ev(2)*ev(3) iiiu ompute the right streth tensor. 1.0/(iu*iiu-iiiu) iu*iiiu iu*iu-iiu a1 * a1 * al * a1 * a1 * a1 * b1 + 1*(1) 1*(2) bl + 1*(3) l*c(4) 1*(5) b1 + 1*(6) - (l) - (2) - (3) - (4) - (5) - CC(6) ompute the inverse of the right streth tensor. 1.0/(iiiu*(iu*iiu-ii u» iu*iiu*iiu-iiiu*(iu* u+iiu) -iiiu-iu*(iu*iu-2.0* iu) iu a2 * a2 * a2 * a2 * a2 * a2 * b2 + 2*(1) + d2*c(1) 2*(2) + d2*c(2) b2 + 2*(3) + d2*(3) 2*C(4) + d2*(4) 2*C(5) + d2*(5) b2 + 2*(6) + d2*(b) ompute the rotation tensor. r(l,l) f(1,1)*ui(1)+f(1,2)*ui(2)+f(1,3)*ui(4) r (1,2) f(1,1)*ui(2)+f(1,2)*ui(3)+f(1,3)*ui(5) r(1,3) f(1,1)*ui(1)+f(1,2)*ui(5)+f(1,3)*ui(b) r(2,1) f(2,1)*ui(1)+f(2,2)*ui(2)+f(2,3)*ui(4) r(2,2) f(2,1)*ui(2)+f(2,2)*ui(3)+f(2,3)*ui(5) r(2,3) f(2,l)*ui(4)+f(2,2)*ui(5)+f(2,3)*ui(6) r(3,1) f(3,1)*ui(1)+f(3,2)*ui(2)+f(3,3)*ui(4) r (3,2) f(3,1)*ui(2)+f(3,2)*ui(3)+f(3,3)*ui(5) r(3,3) f(3,1)*ui(4)+f(3,2)*ui(5)+f(3,3)*ui(b) return end ****************************************************** * * * mtmevd -- ompute eigenvalues of 3x3 symmetri matrix stored in paked format * ****************************************************** *

42 w subroutine mtmevd( k, lamda ) impliit integer (a-z) real k(l),lamda(l),kbari,kbarj,kbar,ki, & kj,mi,mj,sale,alpha,gamma,x,xsign,jatol, & thold,sqtol,ratiok,rad,errork,swap, & m1,m2,m3,k1,k2,k3,k4,ks,k6 logial vgtst data maxswp/15/.jatol/1.0e-041 m1 1.0 m2 1.0 m3 1.0 k1 k(l) k2 k(2) k3 k(3). k4 ks k(4) k(5) k6 k (6) lamda(l) k1 lamda(2) k2 lamda(3) k3 swpnum 0 kj kj kj ki ki ki mj mi iexp sale m1 m2 m3 k1 k4 k2 k6 k5 k3 initialize lamda. m, sweep parameters. sale [k] to avoid potential problems with exponential overflow and underflow. k1 mine k2,kj mine k3,kj k1 max( k2,ki max( k3,ki ompute the sale fator and do the saling int ( ( 10g10(kj)+log10(ki) ) * 0.2S 1. 0 I ( 10.0 ** iexp ) m1 * sale m2 * sale m3 * sale k1 * sale k4 * sale k2 * sale ka * sale k5 * sale k3 * sale begin a new sweep 10 swpnum = swpnum + 1 thold = ** swpnum sqtol = jatol * jatol if ( thold.it. sqtol ) thold sqtol enter sweep loop -- work on lower triangle only. rows are done from top to bottom olumns are done from left to right. skip when already within tolerane. ratiok = (k4*k4) I ( k2*k1 ) if ( ratiok.it. thold ) go to 15 kbari -m2 * k4 kbarj -m1 * k4 kbar k2 * m1 - k1 * m2 rad ( kbar * kbar I 4.0 ) + kbari * kbarj xsign 1. 0 x kbar I sign(xsign, kbar) * sqrt(rad) if ( (abs (x).it. j atol*abs (kbarj». or. & (abs(x).lt.jatol*abs(kbari») then alpha 0.0 gamma = -k4 I k2 else alpha gamma end if kbarj I x -kbari I x ki k5 kj k6 k5 ki + gamma * kj ka kj + alpha * ki kj k1 mj m1 ki k2 mi m2 k1 kj + alpha * alpha * ki * alpha * k4 m1 mj + alpha * alpha * mi k2 ki + gamma * gamma kj * gamma * k4 m2 mi + gamma * gamma * mj k4 0.0 row 3 and olumn ratiok = ( k6*k6 ) I ( k3*k1 ) if ( ratiok.it. thold ) go to 20 kbari -m3 * ka kbarj -m1 * ka kbar k3 * m1 - k1 * m3 rad ( kbar * kbar I 4.0 ) + kbari * kbarj xsign 1.0 x kbar I sign(xsign, kbar) * sqrt(rad) if ( (abs(x).lt.jatol*abs(kbarj».or. & (abs(x).it.jatol*abs(kbari)) ) then alpha = 0.0

43 w ~ gamma else alpha gamma end if -k6 / k3 kbarj / x -kbari I x ki k5 kj k4 k5 ki + gamma kj k4 kj kj + alpha ki kl mj ml ki k3 mi m3 k1 kj + alpha alpha ki alpha * k6 ml mj + alpha * alpha * mi k3 ki + gamma * gamma * kj * gamma * k6 m3 mi + gamma * gamma * mj k6 0.0 row 3 and olumn 2 20 ratiok = ( k5*k5 ) / ( k3*k2 ) if ( ratiok.it. thold ) go to 25 kbari -m3 * k5 kbarj -m2 * k5 kbar k3 * m2 - k2 * m3 rad xsign ( kbar * kbar / 4.0 ) + kbari * kbarj 1.0 x kbar / sign(xsign, kbar) * sqrt(rad) if ( (abs(x).lt.jatol*abs(kbarj».or. & (abs(x).it.jatol*abs(kbari)) ) then alpha 0.0 gamma -k5 / k3 else alpha kbarj / x gamma end if -kbari / x ki k6 kj k4 k6 ki + gamma * kj k4 kj + alpha * ki kj k2 mj m2 ki k3 mi m3 k2 kj + alpha * alpha * ki * alpha * k5 m2 mj + alpha * alpha * mi k3 m3 ki + gamma * gamma * kj * gamma * k5 mi + gamma * gamma * mj k lamda(l) larnda(2) larnda(3) end of sweep loop update eigenvalue vetor -- lamda kl / ml k2 / m2 k3 / rn3 vgtst =.true. hek off-diagonal elements for onvergene errork = k4 * k4 / ( k2 * kl ) if ( errork.gt. sqtol ) vgtst errork = k6 * k6 / ( k3 * kl ) if ( errork.gt. sqtol ) vgtst errork = k5 * k5 / ( k3 * k2 ) if ( errork.gt. sqtol ) vgtst if vgtst) go to 30 if swpnum.it. maxswp ) go to 10.false..false..false. eigenvalues have onverged. reorder and exit. if (lamda(2).it.lamda(i)) then swap lamda(l) lamda(l) lamda(2) lamda(2) swap end if if ( lamda(3) swap lamda(l) lamda(3) end if if ( lamda(3) swap lamda(2) lamda(3) end if return end.it. lamda(l) lamda(l) lamda(3) swap.it. lamda(2) lamda(2) lamda(3) swap ) then ) then