7A -A128287 AN IMPROVED NTAIN'HARDENINGMCHARACTERIZAION IN THE I//IADINA CODE USING RHE..A ARMY ARMAMENT RESEARCH ANDIDEVELOPMENT COMMAND ABERDEENPNOVI, A D GUPTA ET AL

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UNCLASS siEDSECURITY CLASSIFICATION OF THIS PAGE (35.. Doe Raed)

REPORT DOCUMENTATION PAGE BREDISRCINREPORT NUMBER )2._GOVT ACCESSION NO. 3. RAECIPIIENTS CATALOG MUMMER

TECHNICAL REPORT ARRRL-TR-02484 -' -14. TITLE (and Sabana) S. TYPE OF REPORT & PERIOD COVERED

AN IMPROVED STRAIN HARDENING CHARACTERIZATION INTHE ADINA CODE USING THE MECHANICAL SUBLAYER il

MODELS. PERFORMING ORG. REPORT NUMBER

AUTH~t~s)S, CONTRACT OR GRANT MNMER~a)

Aaron D. GuptaJoseph M. SantiagoHenry L. Wisnievaki ______________

9. PERFORMING ORGANIZATION NAMIE AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASKUS Army Ballistic Research Laboratory AE OKUI UBRATTN: DRDAR-BLT RDT&E IL161102AH43Aberdeen Proving Ground, MD 21005It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

USA Armament Research & Development Command May 1983US Army Ballistic Research Laboratory (DRDAR-BLA-S) IS3. HNMER OF PAGES

Aberdeen Proving Ground, MD 21005 -4414. MONITORING AGENCY NAME 6 ADDRESS(I( dIft.,.iut from Controlling Offes) IS. SECURITY CLASS. (of this AMP.c)

01 UNCLASSIFIEDIS. DECLASSIFICATION/DOWNGRAOING

SCHEDULE

IS. DISTRIBUTION STATEMENT (of this Repeat)

Approved for public release; distribution unlimited.

M7 DISTRIBUTION STATEMENT (of the abstract entered In block 20fI different hrem Report)j

IS. SUPPLEMENTARY NOTES

IS. gEy WORDS (Coninue an "Dee aide itgnocomfoend d ealir by block nasb.,)

Finite Element Method Besseling-White Hardening ModelStructural Response Modeling Mechanical Sublayer Model

TheBeselig-Witesublayer model for simulating the nonlinear strainharenig bhavorof.plastically deforming solids has been implemented in theADIN fiiteeleentcomputer program as an additional material option. TheDeselig-Witemodl, lsocalled the mechanical sublayer model, simulates

perfectly plastic elements, called sublayers. Sublayer properties are deter-mined from a polygonal fit to the stress-strain curve. This new material option

89cumTY CLASSIFICATION OP THIS PAGE (11INmi Diut. 880twe04

a,

UnclassifiedsuCUmTY CL.ASSIFICATION OF THIlS PAS(3w Dft W

Item 20. Continuation

1has been validated by comparing predictions using this model and the two linearhardening models (kinematic and isotropic) presently in the standard version ofADINA. Comparison of predictions for the transverse deflection history of animpulsively loaded clamped-edge circular plate shows a close correlation betweenthe sublayer model and the kinematic hardening model and a somewhat poorer, butstill reasonable, correlation between the sublayer model and the isotropichardening model.

Unlasiie

SEUIYCASFCTO FTI ACb o at*

-- &

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS ........................ 5

I. INTPODUCTION .. .. ................ ........ 7

II. FORMUL4TTON OF TME MECHANICAL SIBLAYPR MODEl .. .. .. ....... 9

A. Derivation of the Sublaver Equations .. .. .......... 12

B. Computational Procedure. .. ................. 13

r. Program Implementation .. .. ................. 15

III. DYNAMIC RESPONSE ANALYSIS OF PLATE .. .. ............. 16

A. Plastic Comparison. .. .. .................. 18

R. Elastic-Perfectlv Plastic Ccmparison .. .. ..........18

C. Elastic-Linear Strain Hardening Comparison. ..... .. ..... 20

IV. CONCLUJSIONS. .. .. ........................ 20

R EFE.RENCES .. .... ....................... 23

APPENDIX A - PROGRAM~ LISTING OF SUBLAYER MODEL SUBROUTINE. . . . 27

LIST OF SYMBOLS. .......................... 35

DISTRIBUTION LIST. .. .. ..................... 37

Accession For

DTIC TABUnannounced 0Justification

Distributi~on/__

Availability CodesjAvaii and/or-QDist Special

*1IW3

LIST OF ILLISTRATTONS

Figure Page

I. Mechanical sublayer modeling of the uniaxial stress-strain

curve upon initial loadine and reversal of loading . ...... .. 11

2. Tensile uniaxial stress-strain data for 2024-0 aluminum takenfrom Reference 26 approximated by a number of material

response representations ....... .................... .. 17

3. Finite element modeline of half the plate showing a typical

element and the direction of polar deflection ........... .. 19

4. Comparison of the deflection histories at the pole for purely

elastic responses and an elastic-perfectly plastic resnonse . . 21

5. Comparison of the deflection histories at the Pole for an

elastic-linear hardening response using the isotrovichardeninq, kinematic hardening and sublaver models .......... 22

I5i: i

.1

5

I. INTRODUCTION

This investigation is part of an ongoing research program to improvethe structural response modeling capabilities at the Terminal BallisticsDivision of the Ballistic Research Laboratory. Among the computer programsused to model structural response at the BRL, the ADINA finite element codehas been used intensively since its acquisition in 1978. It is a welldocumentedI ,2,3 and widely used4 ,, 6 general purpose program forcalculating the static and dynamic responses of complex structures, with thecapability of modeling both geometric and material nonlinearities. Inparticular, concerning the elastic-plastic behavior of materials, the codeemploys the Prandtl-Reuss theory of plastic flow7 with either the vonMises yield condition or the Drucker-Prager yield condition. Using the vonMises condition, linear isotropic hardening or kinematic hardening can bemodeled, while only elastic-perfectly plastic behavior can be modeled withthe Drucker-Prager condition. Although linear hardening is adequate forsituations where small amounts of plastic strain are involved, its use inpredicting the large strain response of materials exhibiting nonlinearplastic behavior can result in considerable error. Hence, in order toimprove the modeling of strain hardening in the ADINA code, the task ofincorporating the Besseling-White sublayer model into the ADINA plasticityformulation was undertaken.

1. K.-J. Bathe, "ADINA - A Finite Element Program for Automatic DynamicIncremental Nonlinear Analysis," Report 82448-1, Acoustic and VibrationLab, MIT, Dept of Mechanical Engineering, Sep 75 (rev. Nov 79).

2. K.-J. Bathe, "Static and Dynamic Geometric and Material Nonlinear

Analysis Using ADINA," Report 82448-2, Acoustic and Vibration Lab, MIT,

Dept of Mechanical Engineering, May 76 (rev. May 77).

3. M.D. Snyder and K.-J. Bathe, "Formulation and Numerical Solution ofThermo-Elastic-Plastic and Creep Problems," Report 82448-3, Acoustic andVibration Lab, MIT, Dept of Mechanical Engineering, Jun 77.

4. K.-J. Bathe (ed.), "Applications Using ADINA," Proceedings of the ADIRAConference August 1977, Report 82448-6, Acoustic and Vibration Lab,

'I MIT, Dept of Mechanical Engineering, Aug 77.

5. K.-J. Bathe (ed.), "Nonlinear Finite Element Analysis and ADINA," Pro-ceedings of the ADINA Conference August 1979, Report 82448-9, Acousticand Vibration Lab, MIT, Dept of Mechanical Engineering, Aug 79.

6. K.-J. Bathe (ed.), "Nonlinear Finite Element Analysis and ADINA," Pro-ceedings of the 3rd ADINA Conference, MIT, 10-12 Jun 81 or ComputersA Structures, 13, No. 5-6, 1981.

7. R. Hill, The Mathematical Theory of Plasticity, Oxford University Press,London, 1950, pp. 38-45.

7

PRUviOUS PAGEF B

"f

The Besseling-White model, also called the mechanical sublayer model,has been employed for some time in a number of structural responsecodes 8- '1 and in particular in the PETROS series of shell responsecodes 12- 15 developed for the BRL. This model simulates nonlinearstrain hardening by summing the stresses from an array of elastic-perfectly plastic elements, called sublayers, for the material stress at apoint. Nonlinear hardening is modeled in a piece-wise linear manner by eachsublayer yielding at a different stress intensity, with the sublayer yieldstrength being determined by a polygonal fit to the uniaxial stress-straincurve.

8. H.A. Balmer and E.A. Witmer, "Theoretical-Experimental Correlation ofLarge Dynamic and Permanent Deformations of Impulsively-Loaded SimpleStructures," Tech Docu Rpt No. FDL-TDR-64-108, Jul 64, Air Force FlightDynamics Lab, Wright-Patterson AFB, Ohio.

9. J.M. Santiago, H.L. Wisniewski and N.J. Huffington, Jr., "A User'sManual for the REPSIL Code," BRL Rpt No. 1744, Oct 74, USA BallisticRes Lab, APG, MD (AD A003176).

10. B. Hunsaker, Jr., D.K. Vaughn and J.A. Stricklin, "A Comparison of theCapability of Four Hardening Rules to Predict a Material's PlasticBehavior," Second National Congress on Pressure Vessels and Piping,ASME, San Francisco, CA, 23-27 Jun 75.

11. R.W.H. Wu and E.A. Witmer, "Analytical and Experimental Studies of Non-linear Transient Responses of Stiffened Cylindrical Panels," AIAA

Journal, Vol. 13, No. 9, Sep 75, pp. 1171-1178.

12. L. Morino, J.W. Leech and E.A. Witmer, "PETROS 2: A Finite-DifferenceMethod and Program for the Calculation of Large Elastic-PlasticDynamically-Induced Deformation of Multilayer Variable-Thickness Shells"BRL Contract Rpt No.12, Dec 69, USA Ballistic Res Lab, APG, MD (AD 708774).

13. S. Atluri, E.A. Witmer, J.W. Leech and L. Morino, "PETROS 3: A Finite-Difference Method and Program for the Calculation of Large Elastic-Plastic Dynamically-Induced Deformations of Multilayer Variable-Thickness Shells," BRL Contract Rpt No. 60, Nov 71, USA BallisticRes Lab, APG, MD (AD 890200L).

14. S.D. Pirotin, B.A. Berg and L.A. Witmer, "PETROS 3.5: New Developmentsand Program Manual for the Finite-Difference Calculation of LargeElastic-Plastic Transient Deformations of Multilayer Variable-ThicknessShells," BRL Contract Rpt No. 211, Feb 75, USA Ballistic Res Lab,APG, MD (AD A007215).

15. S.D. Pirotin, B.A. Berg and B.A. Witmer, "PETROS 4: New Developments*and Program Manual for the Finite-Difference Calculation of Large

Elastic-Plastic, and/or Viscoelastic Transient Deformations ofMultilayer Variable-Thickness (I) Thin Hard-Bonded, (2) Moderately-ThickHard-Bonded, or (3) Thin Soft-Bonded Shells," BRL Contract Rpt No. 316,Sep 76, USA Ballistic Res Lab, APG, MD (AD B014253L).

8

*1#

The sublayer model has been programnmed in the ADINA code as anadditional material option (model 12). Presently, it is available for useonly with the explicit (central difference) time integration method. Inorder to insure that the sublayer model is correctly implemented in the code,parallel calculations have been performed using the sublayer model, thekinematic hardening model, and the isotropic hardening model to represent alinear strain hardening material. The correlation of deflections using thesublayer model and the kinematic hardening model is excellent. Thecorrelation between the sublayer model and the isotropic hardening modelalthough somewhat poor, as is to be expected, is still reasonable.

This report describes the formulation of the sublayer model and itsimplementation in the ADINA code. It also gives the results of thecomparison between the sublayer model, the ADINA kinematic hardening model,

and the ADINA isotropic hardening model. The material in this report hasbeen published in preliminary form in the Proceedings of the First Chautauquaon Finite Element Modeling.

1

II. FORMULATION OF THE MECHANICAL SUBLAYER MODEL

The Besseling-White or mechanical sublayer model originated with theconcept by Duwezl7, 18 of modeling nonlinear plastic hardening andunloading behavior by superposing the responses from a continuous aggregateof elastic-perfectly plastic elements. First, Whitel9 and thenBesseling 20 in a more general setting extended Duwez's one-dimensionalmodel to three dimensions and laid the basis for using the mechanicalsublayer model as a computational procedure by proposing a discrete array ofelements or sublayers. The computational use of the mechanical sublayermodel was first carried out at MIT by Balmer and Witmer8 in analyzing theresponse of structures using the finite difference method.

16. A.D. Gupta, J.H. Santiago and H.L. Wisniewski, "An Improved StrainHardening Characterization in the ADINA Code Using the Mechanical Sub-layer Concept," First Chautauqua on Finite Element Modeling, Harwichport,MA, 15-17 Sep 80, pp. 335-351.

17. P. Duwez, "On the Plasticity of Crystals," Physical Review, Vol. 47,.- ri 1935, pp. 494-501.

18. H.F. Bohnenblust and P. Duwez, "Some Properties of a Mechanical Modelof Plasticity," J. Appl. Mech., Vol 15, 1948, pp. 222-225.

19. G.N. White, Jr., "Application of the Theory of Perfectly Plastic Solidsto Stress Analysis of Strain Hardening Solids," Tech Report 51, GraduateDiv of Applied Math., Brown University, Aug 50.

20. J.F. Besseling, "A Theory of Plastic Flow for Anisotropic Hardening inPlastic Deformation of an Initially Isotropic Material," Report S. 410,Nat. Aero. Rich. Inst., Amsterdam (NLL), 1953.

9

The mechanical sublayer model assumes that the physical stress C.. at a1j

point in the material can be decomposed into a weighted sum of N sublayer

stresses o..13

N

oILji= 1 j(l

with W a the sublayer weighting factors. Sublayers are assumed to be

elastic-perfectly plastic and to experience the same physical strain E.. •1J

They share identical elastic constants, but have distinct values for theiryield strengths. The values of sublayer yield strengths and weightingfactors are determined from a polygonal fit to the uniaxial stress-straincurve (for example, see Figure 2), with the total number of sublayers N equalto the number of nonzero slope straight-line segments employed.

We can briefly describe how the sublayer model simulates nonlinearhardening by examining the case of uniaxial stress. Upon initial loading,all the sublayers respond elastically, but as loading proceeds sublayers willsequentially become plastic as they each reach their yield limit. Overall,this results in progressively diminishing values of the stress-strain modulusas the total stress increases, as illustrated in Figure I. Upon unloadingand reversal of loading, all the sublayers initially become elastic andremain elastic until the sublayer with the lowest yield stress again becomesplastic, whereupon the process repeats itself in the reverse direction, butwith one significant difference: Straight-line segments are now twice thelength of the corresponding segments on the initial loading curve due to eachsublayer stress having to change from a given value of the yield stress toits negative value rather than from zero to the given value. Thus, the

sublayer model exhibits a type of kinematic hardening behavior first proposedi by Masin'21 that has subsequently achieved acceptance in modeling

cyclic loading.22 ,2 3

21. G. Masing, "Eigenspannungen und Verfestigung beim Messing," Proceedings

2nd Intern. Congress for Appl. Mech., Zurich, Sep 1926.

22. Z. Mroz, "On the Description of Anisotropic Workhardening," J. Mech Phys

Solids, Vol. 15, 1967, pp. 163-175.

23. Z. Mroz and N.C. Lind, "Simplified Theories of Cyclic Plasticity," Acts

Mech., Vol. 22, 1975, pp. 131-152.

10

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A. Derivation of the Sublaver Equations

An incremental nlasticitv law is used to calculate the sublaver

stresses. For a Riven sublayer a, the sublayer stress increment AO . isea ij

related to the elastic strain increment AC.. through the isotropic Hooke's

law for the material,

j= E Aa + -aV (2)

where 9 is Youne's modulus and V is Poisson's ratio. The strain incrementea

ACij is the sum of the elastic sublaver strain increment Ac.j and the

plastic sublaver strain increment Apj

ea PaAC.. = AC.. + AC..

1j 1j 1j

where, as already mentioned, the total strain increment Ac.. is the same for1j

all sublayers at the noint. Each sublaver is assume-d to obey the von Misesyield condition:

s? Sa 2 aC ) 2(4)ii ij f

where 0 is the sublayer yield stress andy

ij j - 0kk ij

is the sublaver deviatoric stress. Should the material exhibit strain ratesensitivity, the yield stress for each sublaver can be made rate dependentthrough the relation

0a = Oa [1 + (e/Da)l/N](6yO o/C (6)

we 0

where a is the static sublaver yield stress, 2 is the second invariant of

deviatoric strain rate tensor, and Da, Na are empiricall! determined

constants2 4 which generally are assumed to be identical for all sub-layers. To comolete the set of equations, the associative flow rule isapoended in order to determine the plastic strain increment:

24. N. Perrone, "Response of Rate-Sensitive Frames to Impulsive Load," J.Engr. Mech. Div., ASCE Vo1. 97, No. EMI, Proceedings Paper No. 7890Feb 71.

12

_ * -

..... =AA-S.

-- A( S (7)

where AXa is the usual plastic flow parameter that is adjusted to keep thestress state at the yield value. Combining equations (2), (1), and (6) weobtain the incremental equation used by the sublaver model to determine thesublayer stresses:

A --E [Ac +VA 6 ] a Sa. (8)ij 1+v + s-v.kk ii] ij

""where for convenience we replace AX bv

Xa =E A a

Eouations (1), (4), (5), (8), and, when treating a rate sensitivematerial, equation (6) are the basis for the mechanical sublayer plasticitycalculation. When the suhlayer stress rate lies inside the yield surface, as

determined by (4), a in (8) is set equal to zero, otherwise Aa is deter,

mined by the requirement that the resulting stress state satisfy the yield

condition. The details of determining the sublaver weighting factors Wa and

the static sublaver vield stresses a a from the polygonal fit to the uniaxial0

stress.strain curve are described in References 8 and 12.

R. Computational Procedure

The computational procedure assumes that at a Riven time sten or cycle n

in the code's computational algorithm the strain increments AC.. from the1J

previous to the current cycle and the sublaver stresses O. (n-i) at the11

previous cycle are known for all Gaussian integration points. Using this

information the sublayer model computes first the current sublayer stresses"a .(n) and from these the current physical stresses aF. j (n) by means of the

algorithm2 5 which we now outline.

25. J.M. .antiago, "Formulation of the Large neflection Shell Equations forUse in FiniteDifference Structural Response Codes," RL Rpt No. 1571,Feb 72, VISA Rallistic Res Lab, APG, MD (AD 740742).

13

i _ _ _ _ _ _ - . -4

- . ... ..-~ - - - - ... . - .. . . . . . . -.. . ,, "

Starting with the sublaver having the smallest value of the yield stress.

a trial elastic stress increment ASij is computed by assuming that no plastic

flow occurs during the increment so that AA - 0 and equation (8) becomes:e E V

AaC~ + .. C (10)Lij =+ I Aij 2 Lkk 6ij (0

Adding the elastic stress increment to the previous sublaver stress, a trial

sublayer stress for the current cycle is obtained

Ta a eo.. = a. .(n-l) + Aa.. (11)ij 13 i1

Writing the von Mises yield function for the sublaver as

a a oa a 1 a 2 2 a (12)(ij) -- 'i j - (Kk) 3 y o)(2

the trial sublayer stress is checked by substitution in the von Mises yield.a Ta e

If * (a.) <0, then the addition of the elastic stress increment AO..13 13

results in a stress state inside or on the yield surface, and hence the trial

sublayer stress has Riven the correct sublaver stress for the current cycle:

aj = Ta (13)

3a Ta

If, however, a(a ij) > O,then Yielding occurs, and according to (8) the

trial sublayer stress needs to be corrected to obtain the current sublayer

stress:

a.) Ta a aj(n) = aj A i(n-1) (14)

"" To determine the flow parameter Aa , equation (14) is inserted in the von

Mises yield function (12) and, in order 'to insure that the resulting sublayer

stress state is on the yield surface, the function is set equal to zero; i.e.a a-*5 (a. i (n)) 0 0. This determines a quadratic equation in La (Reference

25),which is solved for the smallest positive root:

14

_ 1 •.

Bx B B 2 C] (15)

where

toS(n-i) S~. n-i)

i 2

B = a.. S..(n-l)1j 1j

Ta .C = * (aij)

In some instances when the strain increment is too large, the discrimi-

nant under the radical in eauation (15) can become negative due to the trialsublaver stress causinR a state too far outside the yield surface. In suchinstances a subincrevental iterative procedure described in Reference 9 isemployed to reduce the strain increment to a number of subincrements in order

to obtain a real solution for AP

Once the Plastic flow parameter AP is determined, the sublayer stress

a . (n) is computed from (14) and is stored for the next computational cycle.

This Procedure is repeated until all the current sublaver stresses at the

given integration point are found, and then these are summed with the

appropriate weighting factors to determine the physical stress at that Point

for the current cycle

N

a. (n) = _ ac. (n) (16)

C. Pro2ram Implementation

The ADINA code is structured to accept nonlinear material models notcurrently included in the material librarv. No changes to the program arenecessary except for inserting the user supplied material model subroutine interma of the anpropriate source program call variables describedin Reference 1.

15

i . .. . ' °....................... .

The sublayer model has been inserted as material model 12 (subroutineOEL3D2), which is intended for triaxial states of stress. As presently

programmed the sublayer model can only be used for explicit time integration.The subroutine calculates the current stresses from the strain increments andthe previous stresses as explained earlier, but has not as yet beenprogrammed to calculate the stress-strain matrix (subroutine MIDEP3) neededto determine the stiffness matrix required for implicit time integration.Because the sublayer stresses at each cycle must be stored in order toperform the stress calculation for the subsequent cycle, provision forstorage must be made in accordance with the number of sublayers used. Thismeans that when more than one sublayer is employed the sublayer model willrequire more storage than the ADINA kinematic hardening model.

Ill. DYNAMIC RESPONSE ANALYSIS OF PLATE

In order to evaluate the ability of the mechanical sublayer model asimplemented in the ADINA code to reproduce the dynamic response of structureswith nonlinear material characteristics, calculations have been undertaken tocompare code predictions with known experimental results. Some preliminaryresults from this investigation are now available concerning a comparisonbetween the sublayer model and the existing ADINA hardening models.Employing each plasticity model, the code was used to solve a probleminvolving the large transient deflection of an impulsively loaded,clamped-edge, circular flat plate.

The problem is taken from an experimental report26 and concerns aplate 6.35mm thick with a radius of 63.5mm, made of 2024-0 aluminum. Thismaterial exhibits a considerable degree of nonlinear hardening which can beclosely approximated by the sublayer model as illustrated in Figure 2.

However, for purposes of comparing the sublayer model (model 12) to theisotropic hardening model (model 8) and the kinematic hardening model (model9), we need only consider the following material responses: purely elastic,elastic-perfectly plastic, and elastic-linear strain hardening. Hence, forthis purpose the following values of the material properties were used:

a (yield stress 85.5 MPayE (Young's modulus) 73.7 GPa

ET (strain hardening modulus) 6.85 GPa

v (Poisson's ratio) 1/3

p (mass density) 2775 Kg/m3

The deformation was initiated by subjecting the plate to a uniform impulsevelocity of 53.09 m/s.

26. E.A. Witmer, F. Merlis and S.D. Pirotin, "Experimental Studies ofExplosively-Induced Large Deformations of Flat Circular 2024-0 AluminumPlates with Clamped Edges and of Free Thin Cylindrical 6061-T6 Shells,"BRL Contract Rpt No. 13A , Jan 74, USA Ballistic Res Lab, APG, MD(AD 917518L).

16

... J - __

240"

220f C 32

K 2001I 28

-62 _8

:* 16024-

In 4 SUBLAYERS22PURELY ELASTIC 2

120 PERFECTLY PLASTIC -18- -LINEAR. HARDENING 1

10 0 14

80 12

10

0 02 .04 .06 .8 1STRAIN

Figure 2. Tensile uniaxial stress-strain data for 2024-0 aluminumtaken from Reference 26 approximated liv a numbier of

material response representations.

1 17

IAUi

By passing a symmetry plane through a diameter, only half the plate had

to be treated. Twenty 3-dimensional brick elements were used to model half

the plate, as illustrated in Figure 3. Each element used 16 nodes except for

those bounding the pole which used only 12 nodes, giving a total of 142 nodesfor the problem. Two Gaussian integration points were employed along eachcoordinate direction, making a total of eight points per element. All nodesalong the circular boundary were assumed to be fixed. Explicit timeintegration was used for all calculations and the total Lagrangianformulation was used for the nonlinear geometric analysis.

A. Elastic Comparison

The purely elastic response was modeled by increasing the value of the

yield stress a to a level insuring elastic behavior only.(See Figure 2.) The

elastic calculations were performed using first a linear and then a nonlineargeometric analysis.

For the linear analysis, the elastic response of the sublayer model(model 12) was compared with the response of the ADINA linear analysis model(model 1) and the elastic response of the ADINA strain hardening model(either model 8 or 9, since the isotropic and kinematic models coincide forelastic response). The numerical results from the three models were found tocoincide exactly. This is illustrated in Figure 4 for the history of thedeflection at the pole.

For the geometric nonlinear analysis the elastic responses of the

sublayer model and the strain hardening model were compared. Again, it wasfound that the models gave coincident numerical results, as shown in Figure 4.

Hence for both linear and nonlinear geometric analyses, the sublayermodel and the ADINA models are elastically consistent. Moreover, it is

observed that for the given problem, nonlinear geometric effects are alreadysubstantial at levels of maximum deflection in the order of 60% of the platethickness.

B. Elastic-Perfectly Plastic Comparison

For the elastic-perfectly plastic comparison, the yield stress was

returned to the value of a = 85.5 MPa and the strain hardening modulus ET~ywas set equal to zero, as shown in Figure 2. The sublayer model (with onlyone sublayer), the isotropic hardening model, and the kinematic hardeningmodel were compared using the nonlinear geometric analysis.

*Excellent correlation between the results of the three models was foundalthough they no longer coincided numerically. As portrayed in Figure 4,the deflection history curves for the models overlap. Hence, for elastic-perfectly plastic behavior, the sublayer model and the ADINA plasticity modelsgive the same results physically. Also, comparing the elastic-perfectlyplastic response with the purely elastic response, we see the dissipative

effects of plastic flow in the reduced and smoothed deflection peaks at late

times.

18

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C. Elastic-Linear Strain Hardening Comparison

To model strain hardening behavior, the strain hardening modulus wasreset at the value of XT - 6.85 GPa. Two sublayers were used with theyield stress of the second sublayer raised to a value high enough to insureits elastic behavior, so that only the first sublayer could become plastic.This insured that the sublayer model would exhibit linear hardening behavior,so that a meaningful comparison could be made with the standard ADINA models.The nonlinear geometric analysis option was used for the comparison.

The correlation between deflections from the sublayer model and thekinematic hardening model is still very good, as illustrated in Figure 5, butthe curves no longer overlap as they did for the perfectly plastic case. The

agreement between the values of the stress components at Gaussian integrationpoints ranged in the order of one to three significant figures.

On the other hand, the deflection using the isotropic hardening modelwas only in fair agreement with the deflections from the other tvo models,although still physically acceptable. However, this is hardly surprisingsince the isotropic model bases its plasticity calculation on the concept ofan expanding yield surface rather than a shifting yield surface as do theother models. Hence,whenever a significant amount of plastic flow reversaloccurs, the models can be expected to predict different responses.

Calculations were also performed using four sublayers to model thestress-strain curve as shown in Figure 2. However, for the level of loadingimposed on the plate, most of the plastic flow was experienced by the firstsublayer, so that the results were graphically indistinguishable from thoseof the two-sublayer model.

IV. CONCLUSIONS

The mechanical sublayer model has been implemented in the ADINA finiteelement structural response code, extending its capability to modelelastoplastic constitutive behavior to nonlinear kinematically hardeningmaterials. The model is formulated to treat triaxial states of stress. Itis currently programed for use in both the linear and nonlinear geometricanalyses of transient response employing explicit time integration. Thesublayer model subroutine has been exercised, and calculations using thesublayer model and the standard ADINA hardening models show that results arein close agreement and appear physically reasonable.

A sublayer model for biaxial states of stress is currently being devisedto treat plane stress problems. When ready, this model will also beimplemented in the code. It now appears that the sublayer model can beeasily reformulated to permit implicit time integration. Further effort isunderway to validate the model with available experimental results and withpredictions from reliable finite difference response codes, such as REPSIL9

and PETroS4,15 Eventually, the sublayer model can be expected toreplace the existing kinematic hardening model, thereby considerablyextending the material modeling capabilities of the ADINA code.

20

L f..,

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LPLE

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a~ <0

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44

r4,4

22

I___ g

REFERENCES

1. K.-J. Bathe, "ADINA - A Finite Element Program for Automatic DynamicIncremental Nonlinear Analysis," Report 82448-1, Acoustic and VibrationLab, MIT, Dept of Mechanical Engineering, Sep 75 (rev. Nov 79).

2. K.-J. Bathe, "Static and Dynamic Geometric and Material NonlinearAnalysis Using ADINA," Report 82448-2, Acoustic and Vibration Lab, MIT,Dept of Mechanical Engineering, May 76 (rev. May 77).

3. M.D. Snyder and K.-J. Bathe, "Formulation and Numerical Solution ofThermo-Elastic-Plastic and Creep Problems," Report 82448-3, Acoustic and

d Vibration Lab, MIT, Dept of Mechanical Engineering, Jun 77.

4. K.-J. Bathe (ed.), "Applications Using ADINA," Proceedings of the ADINAConference August 1977, Report 82448-6, Acoustic and Vibration Lab,MIT, Dept of Mechanical Engineering, Aug 77.

5. K.-J. Bathe (ed.), "Nonlinear Finite Element Analysis and ADINA," Pro-ceedings of the ADINA Conference August 1979, Report 82448-9, Acousticand Vibration Lab, MIT, Dept of Mechanical Engineering, Aug 79.

6. K.-J. Bathe (ed.), "Nonlinear Finite Element Analysis and ADINA," Pro-ceedings of the 3rd ADINA Conference, MIT, 10-12 Jun 81 or Computers& Structures, 13, No. 5-6, 1981.

7. R. Hill, The Mathematical Theory of Plasticity, Oxford University Press,: i London, 1950, pp'. 38-45.

8. H.A. Balmer and E.A. Witmer, "Theoretical-Experimental Correlation ofLarge Dynamic and Permanent Deformations of Impulsively-Loaded SimpleStructures," Tech Docu Rpt No. FDL-TDR-64-108, Jul 64, Air Force FlightDynamics Lab, Wright-Patterson AFB, Ohio.

9. J.M. Santiago, H.L. Wisniewski and N.J. Huffington, Jr., "A User'sManual for the REPSIL Code," BRL Rpt No. 1744, Oct 74, USA BallisticResearch Lab, APG, MD (AD A003176).

t

10. B. Hunsaker, Jr., D.K. Vaughn and J.A. Stricklin, "A Comparison of theCapability of Four Hardening Rules to Predict a Material's PlasticBehavior," Second National Congress on Pressure Vessels and Piping,ASME, San Francisco, CA, 23-27 Jun 75.

11. R.W.H. Wu and E.A. Witmer, "Analytical and Experimental Studies of Non-linear Transient Responses of Stiffened Cylindrical Panels," AIAA

Journal, Vol. 13, No. 9, Sep 75, pp. 1171-1178.

12. L. Morino, J.W. Leech and E.A. Witmer, "PETROS 2: A Finite-DifferenceMethod and Program for the Calculation of Large Elastic-PlasticDynamically-Induced Deformation of Multilayer Variable-Thickness Shells,"BRL Contract Rpt No.12, Dec 69, USA Ballistic Res Lab, APG, HD (AD 708774).

23

REFERENCES (Continued)

13. S. Atluri, S.A. Witmer, J.W. Leech and L. Morino, "PETROS 3: A Finite-Difference Method and Program for the Calculation of Large Elastic-Plastic Dynamically-Induced Deformations of Multilayer Variable-Thickness Shells," BRL Contract Rpt No. 60, Nov 71, USA BallisticRes Lab, APG, MD (AD 890200L).

14. S.D. Pirotin, B.A. Berg and E.A. Witmer, "PETROS 3.5: New Developments

and Program Manual for the Finite-Difference Calculation of LargeElastic-Plastic Transient Deformations of Multilayer Variable-ThicknessShells," BRL Contract Rpt No. 211, Feb 75, USA Ballistic Res Lab, APG,lD (AD A007215).

I,

15. S.D. Pirotin, B.A. Berg and E.A. Witmer, "PETROS 4: New Developmentsand Program Manual for the Finite-Difference Calculation of LargeElastic-Plastic, and/or Viscoelastic Transient Deformations of

Multilayer Variable-Thickness (1) Thin Hard-Bonded, (2) Moderately-ThickHard-Bonded, or (3) Thin Soft-Bonded Shells," BRL Contract Rpt No. 316,Sep 76, USA Ballistic Res Lab, APG, MD (AD B014253L).

16. A.D. Gupta, J.H. Santiago and H.L. Wisniewski, "An Improved StrainHardening Characterization in the ADINA Code Using the Mechanical Sub-layer Concept," First Chautauqua on Finite Element Modeling, HarwichportMA, 15-17 Sep 80, pp. 335-351.

17. P. Duwez, "On the Plasticity of Crystals," Physical Review, Vol. 47,1935, pp. 494-501.

18. H.F. Bohnenblust and P. Duwez, "Some Properties of a Mechanical Model

1 of Plasticity," J. Appl. Mech., Vol 15, 1948, pp. 222-225.

19. G.N. White, Jr., "Application of the Theory of Perfectly Plastic Solidsj I to Stress Analysis of Strain Hardening Solids," Tech Report 51, Graduate

Div of Applied Math., Brown University, Aug 50.

20. J.1. Besseling, "A Theory of Plastic Flow for Anisotropic Hardening inPlastic Deformation of an Initially Isotropic Material," Report S. 410,Nat. Aero. Rsch. Inst., Amsterdam (NLL), 1953.

21. G. Masing, "Kigenspannungen und Verfestigung beim Messing," Proceedings

2nd Intern. Congress for Appl. Mech., Zurich, Sep 1926.

22. Z. Mroz, "On the Description of Anisotropic Workhardening,",J. Mech PhysSolids, Vol. 15, 1967, pp. 163-175.

23. Z. Mroz and N.C. Lind, "Simplified Theories of Cyclic Plasticity," ActsMech., Vol. 22, 1975, pp. 131-152.

24

REFERENCES (Continued)

24. N. Perrone, "Response of Rate-Sensitive Frames to Impulsive Load," J.Engr. Mech. Div., ASCE Vol. 97, No. EML, Proceedings Paper No. 7890,Feb 71.

25. J.M. Santiago, "Formulation of the Large Deflection Shell Equations forUse in Finite-Difference Structural Response Codes," BRL Rpt No. 1571,Feb 72, USA Ballistic Res Lab, APG, MD (AD 740742).

26. E.A. Witmer, F. Merlis and S.D. Pirotin, "Experimental Studies ofExplosively-Induced Large Deformations of Flat Circular 2024-0Aluminum Plates with Clamped Edges and of Free Thin Cylindrical 6061-T6Shells," BRL Contract Rpt No. 134, Jan 74, USA Ballistic Res Lab,APG, MD (AD 917518L).

I

25

APPENDIX A

PROGRAM LISTING OF STI1BLAYER MODEL SUBROUTINE

27 I.VOs ,AGE

1. Element Group Control Card (2014) Page X.5 (Ref. 1)

Columns Variable

Model 57-60 NPAR(15) 12, for sublayer model

NCON 65-68 NPAR(17) 2, for sublayer model

IDW 77-80 NPAR(20) (Number of sublayers x 6) + 7

2. Material Property Data Cards Paste X.11 (Ref. 1)

a. Material number card

No change

b. Material Property Card Page X.11 (Ref. 1)

Columns Variable

1-10 PRnP(1,N) Young's modulus, F

11-20 PROP(2,N) Poisson's ratio, V

3. Sublayer Data

a. (NSI)AL(ILAY), TLAY - 1,NIJMMAT) (1615)

where

j NSU1RL(LAY) - the number of sublayers in laver ILAY.

NUMMAT - NPAROO6 Number of different sets of material properties.

b. (SIC,I.Z(ILAY,ISR), ISD - 1,NRBL) (5EI5.6)

ISG,D. - the magnitude of the ordinate of the 1-dimensional stress-strain curve.

NSHL - Number of sublavers in the laver (material) in question.

ILAY - Material in qluestion.

EPS,DZ - the magnitude of the abscissa of the 1-dimensional stress-strain curve.

NSRL - Number of sublavers in the laver (material) in question.

29

MCJEL P"AG LhIK-l F1

C EALUATE STR~ESS INCREMENTS AND STRESSES

c THE FOLLOWING VARIABLES ARE USED IN THIS SUNROUTINE

c SIG PREVIOUS STRFSSES

C EP PREIOUSSTRiAINS

c STRESS CURPENT STWESSES (TO HE CALCULATED)cSTRAIN CRETSTRAINS (6 1 V E N)cIPEL a sMATEPIAL ELASTIC (INITIAL VALUE)

C a 2. MATEkIAL PLASTICC DELEPS INCREMENTAL SWkAINSC CELSIG INCREMENTAL STRESSES, CALCULATED ON THE ASSUMPTIONC OF ELASTIC oEHAVION DURING STRAIN INCI4EPFt'T (DEOPFS)C L NWe OF SUF-INCIEMENTSC LC CWFCK ON SUB-INCREMENTSC PROPMI YOUNGS MODULOSC PROP(2 POISSONS RATIOC YR TRIAL STRESSESC TC CORP.ECTOR S7HFSSESC To STRESS PER/SUHIAVER

DIMENSION PkOP~l)EPS,(11*S(,(NSL,)COMMON /EL/ IN~oICOUIi~NA40)NUEGNEGLNEGLIMASSIDAMP*,1 ISTATNOOFKLINIEIG.IMASSNIDAM~PNCOMMON /VAR4/ NGMOfFXIttvrKSTFPITEMAIEQREF.ITEKPRi,I IIWEF.IE(UITIPR1,KPLOTNKPLOTECOMIAON /MTmn3D/ O(3E),STRESS(6).STRAIN(6),IPTN4EL.IPSCOMVON /MATER/ COEFF(6JSI(,MA(6)9EkET(6)COMMON /MATfPI/ NSURL(4,.SIGIOZ4.S).EPS)VZ(4,5)COMMON /oisflR/ DZsDi9)COMMON /ELSTP/ TIPE*IOTkFDIMENSION DFLSIGuA).DELEPS(b),TR(6),TC(6),TM(6),STATE (2)I EQUIVALENCE (NPAR(3) .INbNU)DATA STATE /PH' E92I*P/

IPELDxIPELIF(IPT *NE. I)GOTO 30

C of*** CALCULATION OF MATERIAL CONSTANTS

PVUPR0P(?)CALL MATPPO IMAIPoPIIOP)SIAMNSIGMA (1)

CF ACTORal.A)=YM/(1..PV)

AI=Al/(19-?o*PV)Ed mAl1PVAluAl-'

C 1. CALCULATE INCKEMENTAL STRAINSC

30 DO 32 IwI.6STRESS(I)=O~n

32 CONTINUE

30

IF(INDNL JoEQ 3)6070 36C TOTAL LA(6RANGIAN

DO 34 Iz1,634 DFLEPS (I)aSTPAIN (I)-EPS (I)

C bT40 UPDATFU LAGPAN(bIAN36 00 38 Ix1.638 (ELEPS(I)*STPAIN(I)

C40 IPLAST=O

DO 250 ISP=19NSPLC STRESS PEP SUBLAYE.R 516(1589I)

DO 43 I1,E,

43 CONTINUEC KPRI=O SKIP CALCULATIONt JUST PRINT STRESS

IF(KPRI 9EQ. O)GOTO 228Lul

SIGPS~izSIGMA (ISH) OSIGMA (ISs) *FACTORt*FACTORCC 29 CALCULATE TRIAL ELASTIC STRESS 1INCREPAENTSC

DELSIG(1)=Ai*rEfLEPS(l) * Hl*(DELEPS()tELFPS(3))DELSIG(2)uAl*CFrLFPS(2) + [41*(DkLFPS(1)*UELEPS(3))fESIG(3)uAl*LFLFPS(3) * Eb*(DFLEPS(1)+DELEPS(2))DELSIb(4) .C1*DELFPS (4)DELSI6 (5)=cl.DELEPS(S,)DELSIG (b)mCI*DELEPS (6)

45 LCzl50 SURINCzl@/ELI C 3. CALCULATL. TRIAL ELASTIC STRESSES

CDO 60 IZl.A

60 CONTINUETAURzTR (1) TP (2).TR (3)

CC 4. CHECK THROUGH VON kISES-HENCKY YIELD CRITERIAC YIELfl PHI

1 (TAUR*Oi*2*0OSIGMSJ)/3*0

C

MLAN'DA=O0IF(CZ*LE.0. *ANO. L .ekgo)(OTO 215IF(CZ oLE* no)(OTO 110

CC 5. COM4PUTE NECESSARY CORRECTORC

TC(1)uTM(l)-S'

31

TC (3) &TH(3) -SATC(4)=TM(4)TC(S)NTM(5)TC46)aTM(6)TAUCaTC (1)*TC (2)TC(3)

C 6. CALCULATE PLASTICITY PARAMETER FROM 04UFFINGTON MODEL

1 -(TAUC*2/3*0)E42uTC (1)*TR(l )*TC(2)*TRC2l*TC(3)*Tk(3)*2.0*(TP(4)*TC(4[.I TY()*tC(t;).TR(6)*TC(6))-TAUC*TAUR/3o0LIsCpuBZ*N4z-A7OCZ

CC TEST A7

C.IF AZ IS NE~GATIVE - PRINT FRRUk MESSAGEC IF AZ IS ZERO - SVP.-INCREMENTC IF AZ IS POSITIVE - CONJTINUIEC

IF(AZ) 6091SO9100AO WR ITE t+ 14)90 FORVAT(@3'.4X9'AZ NEGATIVE A791')

60TO 180

C TtST r)ISCRI41NAhNTC IF DISCR IS NEGATIVE SUH-IPNCPEP'ENTC

100 IF(DISCA *LT, Oe0)6GT0 150

C TEST RZC IF HZ IS NEEATIVE Ok zEPO Suo-ICwEtMLNT

IF(1Z .LF. O.0)GOTO ISOIC COOPUTF MLA96DAhLAPDAwCZ/(RZ*SQPT(OISCP))

C 7. EVALUATE SUBLAYER RKLIEVED ELASTIC STWESSESC

110 00 120 19,10TM(I)aTR(I)-HLAMDA*TC(I)? CONTINUE

CCC CH4ECK THE SUB-"INCREMENT NUOREPC

IF(LCoEQ.L) GOTO 210LC*LC.I

C WSAXF SUPR-INCREMFNjTS SPALLEPC150 LaL*1

EL uL

C CHECK MAXIMUM NUMBER OF ALLOWANLE SUb-IteCREMENTS

32

IF(L *Lk. 100)60TO 4SbRITE (6.170)

170 FOPI'AT(93STPESS CALCULATIUN UtNSATISFACTORY(100 SUR-INCREMENTS)1 /)

180 WRITE4e.3000) ISIBLsLC9NEL#IPT9TI0E3000 FORPAT19 SURLAVERSO914,S Stib-INtCPMENT L8191499 LC=19I49

It ELEI"LNT NUMIBE~afeI4ts IP'OINTu'.Ii,,' TIMF*I*E16EolWRITE(6930) CZoA?oPZ

3005 FORM'AT(@ CZu'.E'15.a.' A?.',E15.e.' Hl*El5eH)

3010 FORMiATI' i. Sl.6ESR1) 9Fb

WRITE (93021) ITC (1)91=196)3021 FORMAT($ TC 996E15$)

bkITE(693022) (T"i(1)*Iulqb)302 FOR#'DT(' TI' 996f1.8s)

3023 FOkI'*T( DEL9IG 996E15.6)STOP 'STRESS CALCULATION' UNbATISFACTOPYO

c REACHED PLASTIC SOLUTION210 IPLAST=IPLAST.1

GOTO k~bC ELASTIC TI' E(UALS TRIAL TR PER~ SUE4LAYER

?IS' D0 220 Tub*ATv' (IuTR )

220 CONTINUESTRESS POTATIOP IS APPLIED IN LARG~E DISPLACEI'E4T/STRAIN

M2 IF(IhVNL9Nf*3)60T0 ?26OMEGAjmj~lS(4).DISVI(6)OME6A2uUiISD (S)-OISO (9')OEA3sDISO (T)-DISD(V)

I OPEGiA3SIG6IS~.6)-OIiGA3*SIG(IS8,4))

TM(A).TM4(6)..b.(0ME0A30(SIG(1SS,3)-SIG(ISS,2))-I OIE6A2*SIG(ISb*4)eONE6A1*SIG(1$Bob))

P6IF(IUPDT *NE, 0)GOTV 22A

C

S STORE CALULAE TTA STESSOM PRI SU(SAYE FTRES TE

26DO 230 Iw1.6STREIsbq(I)wTREm T'(I) CtF(I)

230 CONTINUE

C3

250 CON~TINUILECC eses CALCUILATION' OF ELASTIC-PLAS~TIC STRESSESI *. (E N D

IF(KPR! A0O 0GOTO TOOIPFJL~w1IF(IPLAST *GTe 0)IPtLVx?

CC U P 0AT IN 6C

IF (IUPLIT.NF..0)COTO 6~15IPFLxIIPLfL110 610 IZ1,6

C615 P~ETURN

CC PRINTING OF STPESSLSC

700 IF (IPkI*IF(4.0 oAND9 IPT.H.C.1) wwITF (692100) tELc 9, CALCUJLATE HYDROSTATIC ANDo nFVIATOkIC STRESSESC

S94(STESS(1).STRFSS(2),STh-ESS(31)/3,uSXzSTkESS (1) -SPASY=STIRFSS (?)-S0S?=STlkSS(3)-S'

f FTAz.5 * (Sx**2?.SY**2*SZ**2)1 5TkfSS(4',**2 0 STIIESS(b)**? +*TtS()*

FTAxS(RT (3o*FT6)

IF (IfUNLoNE.P) 60O To $no

C IN TOTAL LA(PA~bIAN FO.H'ULATION CALCULATE CAUJCHY STPkSSESC

IF (IPk1cI.Qoo)WRITE (602P00) IPT.STATEIXlPELD).STIRESSFTAL.LC

CALL CAUCI43C

POO IF (IPk~I.EQeQ)I WRITE (ho?200) IPTSTATF(II'ELD),STR~ESSFTA.LLC14ETUNN

?100 FOkPAT (/16H4 ELEMENT STkES6,'Y0XIlHQUIVALkNT/I 131h NUM/IPT STATE STRESb-XX STRFbS-YY2S-ZZ STPFSS-XY STOESSmA? STI4ESS-YZ SURESSXL LC /141~

2200 FOPMAT (SX .J?,IX.A2,6ILASTIC16E14.6,1A*r1..6,93,6*X.I3)

34

U.ST OP S'. . .O1.

Da, Na empirically determined constants in the strain rate equation.

E Young's modulus (GPa).

E T tangent strain hardening modulus (GPa).

S. deviatoric stress tensor at the (th sublaver.

6 ij Kronecker delta.

2 second invariant of the deviatoric strain rate tensor.

E physical strain at a point.ij

C kk hydrostatic comnonents of strain.

v Poisson's ratio.

P mass density (Ke/m3 ).

a.. physical stress at a point (MPa).z13

C kk hydrostatic components of the sublayer stress (WPa).

a. . (n) sublayer stress at the ath suhlaver at the current cycle (HP&).

13d3. (n-1) sublayer stress at the ath xuhlaver at the vrevious cycle (Mea).lIj

TaC .. trial sublayer stress (MPa).

! O sublaver yield stress (MPa).Y0a static sublayer yield stress (MPa).

0

'* von Mises yield function.

w sublayer weighting factor.

AC strain increment.

nlastic flow parameter.

AO aublayer stress increment (MPa).

, 13

A. elastic stress increment (MPa).

ij

35

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