.--Bulletin 41

(Part 7 of 7 Parts)

REPRINTED FROM

THESHOCK AND VIBRATION

BULLETIN

DECEMBER 1970

A Publication ofTHE SHOCK AND VIBRATION

INFORMATION CENTERNaval Research Laboratory, Washington, D.C.

Office ofThe Director of Defense

Research and Engineering

DYNAMIC ELASTOPLASTIC RESPONSE OF GEOMETRICALLY NONLINEAR ARBITRARY SHELLS

OF REVOLUTION UNDER IMPUSLIVE MECHANICAL AND THERMAL LOADINGS

T. J. Chung, J. T. Oden, R. L. Eidson, J. F. Jenkins, and A. E. MastersResearch Institute, The University of Alabama in Huntsville

Huntsville, Alabama

This paper discusses the application of the finite element method todynamic. geometrically and materially nonlinear behavior of complex,thermoplastic structures subjected to impulsive mechanical and thermalloading. As a byproduct, the capability of predicting dynamic stabilityand postbuckling behavior of thermoelasto-plastic shells is alsoobtained. Numerical integration of the equations of motion and heatconduction for the discrete model is performed by the use of parabolicacceleration and cubic displacement methods. A new technique forevaluating impulsively-introduced buckling loads is proposed. Numericalexamples include the dynamic response of an impulsively loaded elasto-plastic spherical cap. For an evaluation of various effects, a varietyof special cases are considered.

INTRODUCTION

Despite the preponderance of literatureon static or quasi-static structural behavior,in the physical world the application of loadsto a structure in most engineering problemstakes place dynamically. Traditionally, mostanalysts turn their back on this fact, chieflybecause of the mathematical difficultiesinvolved in predicting dynamic structuralresponse. When one adds to th~se difficultiesthe obvious complications involved in depictingelastoplastic behavior and in includingtransient thermomechanical effects induced byimpulsive loads. he encounters a class ofnonlinear problems which are likely to betreated successfully only through the efficientuse of electronic computation and modernmethods of approximate analysis. It is to thisclass of problems that the present investiga-tion is directed; specifically, this paper isconcerned with the application of the finiteelement method to a general class of problemsinvolving the thermomechanical behavior ofelasto-plastic continua.

Preliminary studies of impulsively-loaded beams, rings, and plates, includinglarge deformations and rigid-plastic behavior,are available in the literature (1,2,3],although thermal effects and transient elasto-plastic response appear to have been ignored.Leech et al (4] studied large-deflections androtations of cylindrical panels of elastic,elasto-plastic, and strain-hardening materials.They used finite difference equations with

81

step-by-step numerical integration as a methodof solution. More recently, Stricklin et al(5] presented extensive results on nonlineardynamic response of shells of revolution inwhich geometrical nonlinearities were included.This study, which employed the finite-elementtechnique combined with various schemes ofnumerical integration, concluded that theHoubolt method (6] was superior to a numberof other methods of numerical integration(e.g. the Chan. Cox, and Benfield method [7],the method of "approximation of loads matrix,"and the Runge-Kutta method). In theiranalysis, material nonlinearities were nottaken into account. Oden [8,9] and Oden andPoe [10] studied applications of the finiteelement method to a class of problems indynamic cou pled thermoelasticity and solvedone-and two-dLmensional problems; however,the possibility of yielding was not considered.

The present study concerns the develop-ment of a general finite-element formulationof dynamic problems of large, thermoplasticdeformation and heat conduction in bodiessubjected to impulsive mechanical and thermalloads, and the application of these formula-tions to a class of nonlinear dynamic problemsinvolving thin, elastoplastic shells ofrevolution. Following this introduction, weexamine the basic equations governing themechanics and thermodynamic a of a class ofthermoelastic-plastic solids. We then examinediscrete analogues of thesl! equations obtainedby application of the finite-element concept.So as to obtain specific forms of the equations

governing the motion and heat conduction oftypical finite elements, we consider specificforms of the constitutive equations for thefree energy and the heat flux. The resultingequations are sufficiently general to applyto a wide range of problems. We then directour attention to certain special cases ofparticular importance which involve the pre-diction of the dynamic response of arbitraryshells of revolution subjected to impulsiveloads. The analysis accounts for largesymmetric or asymmetric deflections and elasto-plastic behavior andj therefore, includes thework of Stricklin [5 as a special case.

The computing technique involvesdegeneration of the equations of motion andheat conduction equations into a step-by-steptime integration. The parabolic accelerationmethod (11] and/or cubic displacement method(6] are used to integrate the equations ofmotion. For the case of heat conduction,special matrix manipulations are performed tocombine time rates of change of temperaturesand displacements such that a linear variationwithin a time increment is obtained. Thisleads to a suitable recurrence formula forstep-by-step time integration. For each timeincrement, a complete elasto-plastic analysisbased on Huber-Mises yield criteria and theassociated flow rule of Prandtle-Reuss isperformed. To this end, the shell thickness isdivided into a number of layers to examinethe yield condition. All nonlinear terms areplaced on the right-hand side of the equationsto serve as incremental loads dependent on theprevious histories. Finally, a new method ofpredicting impulsive buckling loads is pro-posed, which requires far less operations thanthe so-called "threshold load method" [12,13].

tangent to the deformed material lines Xl inthe current configuration, we may also expressVI J as (GI J - 61 J) /2, where GI J - Qt . fJ isthe metric tensor (deformation tensor) ofthe material frame in the current configuration.

In addition to the purely kinematicalrelations, we also introduce three physicallaws that, assuming certain smoothnessproperties hold, are valid at each point inthe continuum:

(aIJXJ ,I) ,I + PoF. = Poll. (2)

poe = alJYIJ + q:1 + Poh (3)

poe~ - q;1 - Ph + ~qle,l ~ 0 (4)

Here alJ=aJl is the symmetric Kirchhoff stresstensor, Po the mass density in Co. F. thecartesian components of body force, £ is theinternal energy, superimposed dots (') indicatetime rates of change, ql the components ofheat flux, h the heat supply. e the absolutetemperature, and ~ the specific entropy.Equation (2) is Cauchy's first law of motionwhich insures that linear momentum is balancedover B; (3) is a local form of the law ofconservation of energy and (4) is the localClausius-Duhem inequality. In obtaining(2) ,(3) ,(4) we assume, of course, that massis conserved; i.e. P ;-G=Po where G=det GIJ.

In many caaes, it is convenient tointroduce the free energy, ~ • £ - ~e. Then'(3) can be written in either of the twoalternate forms,

BASIC EQUATIONS

For future reference, we review brieflycertain basic relations pertaining to thethermomechanical behavior of continuous bodies.

Po~ = OIJYIJ - Po~i - 0

poe~ - q:1 + Ph + a

where a is the internal dissipation.

(5)

Consider a continuous body B in a ref-erence configuration Co in which ~e establisha fixed, rectangular cartesian frame ofreference XI and an intrinsic, material frameXl. The particle labels Xl coincide with XIin Co, but thereafter differ in accordancewith the motion XI=XI(~,t) of B. IfUI (!'t)S XI (1t, t) - Xl are the rectangularcomponents of displacement. it is easilyshown C14] that the components VIJ of theGreen-Saint Venant strsin tensor are

where commas denote partial differentiationwith respect to the material coordinates(e.g. UI,J = Oul/OXJ). If £1 = (61J+UI,JHJdenotes the transformation of ao orthonormalbasis iJ corresponding to XI into vectors fJ

82

THER.lofOPLASTIC ITY

To describe the behavior of a specificmaterial, we must add to (1)-(5) constitutiveequations characterizing the material.Typically, these depict the free energy,stress, entropy, and heat flux as functionalsof the histories of YIJ and e. In thepresent study, however, we shall considermaterials for which ~,alJ ,~, and q arefunctions of the current values of thedeformation and temperature and possiblytheir rates and gradients. This is sufficiep.tfor the development of theories of thermo-viscoelasticity and thermoelasticity. It isour intention, however, to develop a for-mulation for the analysis of dynamic thermo-elastic-plastic problems. For such problems,it appears to be necessary [15] to expressthe free energy as a function of e, the

where k1J is the thermal conductivity tensor.Finally, to depict yielding in the body wemake use of the Prandtl-Reuss flow rule

where f=f(crIJ ,T) is the yield function and~ the normality parameter. and ~p is theequivalent yield strsin. Following well-established arguments (16], we may show that

(9)

(10)

( lIb)

(lIa)EIJ""_ ZpqZreEIJHErt ..E(p) + Z,"E kt'"

where

Unfortunately, insufficient experimentaldata is available to suggest plausible formsof tpe constitutive equations for either ~or Ylr), and for quantitative results weare forced to treat thermoplastic phenomenain an approximate way; that is, we shall takeadvantage of existing incremental theoriesof plasticity to predict yielding of structuressubjected to impulsive loads (indeed, thisis convenient since the numerical techniquesto be used describe the behavior incrementallyin time), and we shall only approximate thedissipation terms in the coupled equationsof heat conduction.

"inelast~~" strain yg), and the elasticstrain YIJ) and to have ?n additional con-stitutive equaf~~n for Y1r). While theinclusion of YIJ in ~ may possibly have littleeffect on the equations of motion, it has apronounced effect on the form of the heatconduction equation for the material as itcontributes directly to the dissipation cr.

Within this framework, suppose that weconfine our attention to infinitesimal strains,still leaving open the possibility of largedisplacements and rotations, and assume thatthe free energy is of the form

and E (p) = dO/d"r.(p) is the equivalent"plastic modulus". Other details are similarto those discussed by Hill [17J.

FINITE ELEMENT FORMULATION

cp = .! E I) •• yC e )y< e ) + B I J yC. )T -E-....22 IJ.. 1J + 2T,-1'

+ FIJ "yn)y~~) (6)

We now decompose the body into acollection of finite elements and, followingthe usual procedure, approximate the localdisplacement and temperature over the elementaccording to

where E1Jkt,BIJ ,'" ,FIJ" are arrays ofmaterial constants and T = e - T. is thechange in temperature. Introducing (6) into(4) and making the usual arguments concerningthe satisfaction of (4) for arbitrary strainand temperature rates, we find that

(12)TUI

where .N(~ and ~(~ are local interpolationfunctions and u~ and TN are the components ofdisplacement and temperature at node N of theelement; N being summed from 1 to N•. thetotal number of nodes of the element.Introducing (12) into (I), we obtain

(7)- P. 1\

a = _F1J.nYI~ y~~) (13)

Obviously, the last term in (6) does noteffect the stress, entropy, or the equationsof motion; it is introduced only to providean approximate nonzero dissipation termwhich depends upon the rates of plasticstrain. While we do not propose equationsfor yff) or yFf), our numerical scheme willallow these rates to be approximated in theanalysis.

and, if u~ takes on an increment bu~ ;:C~ ,this results in increments

To (7), we add the constitutive equationfor heat flux which, in the present study,is taken to be the classical Fourier law,

OS)

(8) The general equations of motion of the

83

( 16)

where mNM is the mass matrix and PNk are thecomponents of generalized force at node N. Byconsidering incremental changes ~~ 60lJ

6(OYIJ/OY~), and 6p~ of terms in (i6) w~ easilyderive the following incremental equations

the element. Beginning with the generalequation of heat conduction for finite elements[18,10,17], and introducing (7), (10), and(12), we arc able to show that

-fTo+A.4RM)AN[B1J(,VIJ - -vd ) + (c!To)I\t-f'lJdvv

(17)

The first integral represents the "geometric"stiffness of the element. To specify theremaining integral, note from (7) and (10) that

Here kHII is the thermal conductivity matrix[10], qN is the generalized heat flux atnode N, [10], ON is the generalized dissipationat node N which, in our case, is given by

(IJ ••. P",ON = JF Yl J Van All dv (24)

v

By incrementing ~l r ,YlJ. and TN and Iineariz-ing in the increments, we can recast (23)into the equivalent matrix form

(18)(25)

For computational reasons, it is sometimes(as in the following sections) convenient torewrite (17) in a form in which a "plastic"and thermal generalized force appear on theright-hand side. Then we rewrite (17) in theform

where ~ is the specific heat matrix, 6~ thevector of incremental generalized temperatures,Q and ~ are thermal conductivity sndmechanical coupling matrix,.the elements ofwhich are functions of YIJ ,YIJ. and T of thestate prior to supplying the increments,6[is a vector of generalized heat fluxes,anq 6£ an incremental dissipation vector.

(19) SHELLS OF REVOLUTION

where the integral represents the usual linearstiffness matrix and

The foregoing theories can be appliedto a shell of revolution upon inserting theinterpolation functions (12) to the shellsurface strain tensor of the form (20)

~ j[IJ.n. ArM C MuNNk = E A'UnCNR IJUk '0. + NMI JUkv

(26)

A similar procedure is followed to obtainthe incremental heat conduction equation for

CNII1JT,ndv(21) The application of the finite element

method to a shell of revolution is welldocumented [21,22,23,5]. In the presentstudy, a curved ring element with four genera-lized displacements (meridional displacement,tangential displacement, transverse displace-ment, and the meridional rotation) are chosen.The interpolation functions consist of alinear variation of meridional and tangentialdisplacements and a cubic variation oftransverse displacement. The configurationof the element is shown in Figure I.

where e ~ = middle surface strains; K ~ =bendingastrains a,~=1,2; µ = shell th~cknesscoordinates.

(22)

(lla) .For simplicity(20) to more

f IJ[ k II ~= B (ANII + C"MIJUk)uT +v

Here J1J.n is the second term inNote that (20) is linear in ,~.in notation, we convert (19) andconventional matrix notation; viz

84

x3

Fig. 1 - Shell element configuration

The equations of motion and heat con-duction applied to a shell of revolution areidentical in form to (22) and (25), respec-tively.

In order to account for the asymmetricloadings, we make use of a Fourier seriesexpansion of all displacements, strains, andstresses. For example, a given function f(displacement, strain, or stress) may bewritten as

integration procedures discussed in thefollowing section and summed through all orsignificant harmonics.

HETHODS OF SOLUTION

The analysis of the response of nonlinearcomplex structures by methods other thandirect numerical time step integration isdifficult, if not impossible. The two mostpromising techniques to determine the dynamicelasto-plastic response of geometricallynonlinear shells of revolution under impulsiveloading aPeear to be the parabolic accelerationmethod [llJ and the cubic displacement method(12,SJ. The former assumes that theaccelerations within a small time incrementvary quadratically. Accelerations. velocities,and displacements are then derived by directintegration and are substituted into theequations of motion. The latter, on the otherhand, assumes that thedisplacements are cubicwithin the time increment. Velocities andaccelerations are then derived by use offinite differenees and are substituted intothe equations of motion. Both methods providerecurrence formulas from which the responsesmay be calculated for each incremental timestep.

We note that both the equations of motionand heat conduction contain all nonlinearterms in the right-hand side of the equations.These terms are treated as displacement-dependent or stress-dependent nonlinear loadvectors. The "ith time step" re~urrenceformulas for the elasto-plastic. geometricallynonlinear response, are as follows:

where n is the harmonic number, P is the anglebetween points around the tangential direction,and the subscripts A and B denote the symmetricpart and the antisymmetric part of f res-pectively. Nonlinearity, however, leads to acoupling of certain A and B termS so that thegoverning equations are

sin np (27)For the parabolic acceleration method:

A~ = :£.(ll1'.1 + l.i~ + 6f; - IS. IV, . t.t '""'~ = ,l!. + "3 ill!)

6t2 ..ll~..l = lz. + 12 6);1.1

where

. 2 .. llt3 ...!t = 6~-1 +"3 tit 6l!!-1 + 6ti~-1

till!!: •• + IS'· ~u i = ll.fr n + lib n + li~:(28)

~0n + ~n ll~r. = llf/l n + lib' + llie n

l4 n 6Ta· + ~ n llI;l r. + ~ n llus n = 6Fj; n + l:.1!~•(29)

where right- side contain 2n(A+ B) unknowns.

The responses and temperatures arecalculated at each tangential e station foreach harmonic based on the step-by-step

85

!1. = ll!.!J-l +l.lt lli!,!-l+ f2 lit2 llJh-l.... 2,.. 2 , ...U\U = At "'.l!.l - At "'~-l

llt2 _X. = (tl. + 12 10 1

with the initial conditions:

llijp =t!,-l(liJ.'.p - ~!!p)

uu: = ~-l (llt. - ~ 1!,p). 1

llPo = llt (ll~ - ll~)

culated and summed as shown in (28) and (29)for each process of nonlinear load-updatingiterations.

As a byproduct of thcse numerical pro-cedurcs, it is possible to determine adynamic buckling load applied in any mannerto an arbitrary shell of revolution. Themethod consists of solving an eigenvalueproblem of the form

For the cubic displacement method:

(2li + llta10 A~ = llt2 (lll) + lit + At)+ t!.(5Al!.1-1 - 4A.\!.I_a + AW-s)

with

ll~ : 6~t(llA.\!.I - 18A~_1 + 9AJLt-a - 2AJ.!.1-s)

IA~ : Ata (2~ - SA1!.1-1 + 4Al,!,l-a - A~_3)

o (30)Obviously, there are fewer operations in

the cubic displacement method than in theparabolic acceleration method. A comparison ofthe solution stability and computing timeis given in the following section.

For the solution of the transient heatconduction equations with effects of inertiaand thermoplastic coupling included, anefficient m~thod of direct integration bycombining AI" and ll.i!,sat each node of thefinite element model has been proposed. Apaper describing complete details is forth-coming.

Impulsive loads may be treated easilyby removing the loads at the end of any initialportion of a time step and allowing the furtherstep integrations to continue as far asdesired under only the influence of previoushistories. The analysis of elasto-plasticresponse consists of calculating the equivalentyield stresses at each time increment todetermine incremental plastic loads satisfyingthe yield criteria and assocaited flow rule.Since the current stresses vary through theshell thickness, it is necessary to dividethe thickness into s number of layers andcheck the status of yielding for each layerand for each element of the shell. Allelements in which plastic "unloading" takesplace are determined at each step and areexcluded from the contribution to plasticloads. For a given aoplied load and giventime increment, an updating of the plasticload vector continues with each trial process,dependent on the previous histories, untilsatisfactory convergence is reached. Onceconvergence is obtained, the current historiesare carried over to the next time increment,and the updating of plastic loads is onceagain repeated until acceptable values areachieved for that time increment.

The geometric stiffness matrix and othernonlinear matrices are treateq also as loadvectors dependent on previous 'responsehistories. Unlike the elasto-plastic response,these nonlinear incremental load vectors areupdated with simple iterations within eachtime increment irrespective of plastic yieldconditions. For asymmetric loadings orirregular heat flux, harmonically coupledFourier response coefficients must be cal-

86

where ~ is the geometric stiffness matrix,~ is the plastic stiffness matrix, and wdenotes the natural frequency. Thesenonlinear matrices are obtained from alternateforms of nonlinear load vectors updated ateach time increment under trial dynamicbuckling loads. The integration time stepsare carried to the maximum peak response from

'which ~~ are calculated. The lowest frequencyresulting from an eigenvalue solution of (30)is recorded. The next higher trial dynamicbuckling loads are applied and the identicalprocedure repeated to obtain the next lowerv&ue of frequency. With sufficiently largetrial dynamic loads applied, a negativefrequency may result. A plot of the lowestfrequency versus the trial dynamic bucklingloads will determine the zero frequencycorresponding to the critical load. Pre-li~inary calculations have produced promisingresults. A paper devoted to this subjectwith complete results is forthcoming.

EXAMPLE PROBLEMS

In order to substantiate the theory andproposed numerical procedures. a computerprogram was written and applied to a sphericalcap shown in Figure 2. This example wasconsidered by many authors and extensivelystudied by Stricklin, et al [5]. Theysuccessfully obtained the geometricallynonlinear dynamic responses of this sructure.

The results of the present study coincidewith those of Stricklin. et al for the specialcase of elastic response. Our main interest,however, lies in the elasto-plastic response,and we wish to consider sufficiently largeloads to induce yielding in the shell.

The spherical cap is subjected to asuddenly applied load of 1000 psi, of infinite-duration, transverse to the middle surface.A yield stress of 24,000 psi and a plasticmodulus of 21,000 psi are used. A total offive ring elements was used in this analysis.A timc integration increment of 10-6 sec.produced a stable solution by both the cubicdisplacement method or thc parabolic accelera-tion method. It should be mentioned thatthe present example retains the geometricnonlinearity only through those nonlinear

'.

- - --Linear

Elasto-Plastic

.3

- - -Geometric nonlinear

---..---'Static solution

6t = 1.0 10-ssec•2 ~ I \ R = 22.27 in .t D 0.41 in.

P = 1,000 psiC E = 10.5 lOspSi.... v • 0.3......x .I P = 0.10066 10-3lW Lb. sec"/in3p..< a = 26.67°~ Ep = 21,000 psiE-<

0°yield = 24.000 psiz

~lWUgtI)HQ

~~

-.3static solution

Fig. 2 - Elasto-plastic dynamic response

strains necessary for moderate rotations.

As seen from the results in Figure 2,amplitudes of the linear elastic response aregenerally lower than the nonlinear response,the elasto-plastic response peaking at almosttwice the magnitude of the linear elasticresponse. The geometrically nonlinearresponse remains approximately halfway betweenthe two. Frequencies of each case appear tobe equal, as expected. Unfortunately, theauthors were not able to locate literaturedealing with similar analysis concerned withthe dynamic elasto-plastic response of anaxisymmetric shell and consequently. no meansof comparison were available. However, thepresent study offers a promising tool forthe analysis of extremely complicatednonlinear structural behavior under arbitrarydynamic loads. It has been observed that,in general, the cubic displacement method ofintegration requires less computing time;

although solution stability for the parabolicacceleration method occur s slightly at alarger time increment. A more detailed ex-ploitation of the present computation capa-bility must await future studies.

ACKNOWLEDGEMENT

The support of this research by theAir Force Office of Scientific Researchthrough Contract F4462-69-C-0124 is gratefullyacknowledged. The assistance of Mr. J. K. Leein certain aspects of the programming workis also appreciated.

REFERENCES

1. E. H. Lee and P. S. Symonds. "LargePlastic Deformations of Beams UnderTransverse Impact," Journal of Applied

87

Mechanics, Vol. 19, No.3, pp. 308-314,Sep. 1952

2. R. H. Owen and P. S. Symonds, "PlasticDeformations of a Free Ring Under Con-centrated Dynamic Loading," Journal ofApplied Mechanics, Vol. 22, No.4, pp.523-529, Dec. 19S5

3. N. Jones, "Finite Deflection of a Rigid-Viscoplastic Strain Hardening AnnularPlate Loaded Impulsively," Journal ofApplied Mechanics, Vol. 35, No.2,pp. 349-356, 1968

4. J. W. Leech, E. A. Witmer, T. H. H. Pian,"Numerical Calculation Technique forLarge Elastic-Plastic Transient Deforma-tion of Thin Shells." AIAA Journal,Vol. 6, No. 12, pp. 2352-2359, Dec. 1968

S. J. A. Stricklin, J. E. Martinez,J. R. Tillerson, J. H. Hong, andW. E. Haisler, "Nonlinear DynamicAnalysis of Shells of Revolution byMatrix Displacement Method," AIM/ASME11th Structures, Structural Dynamics,and Materials Conference, Denver,Colorado, April 22-24 1970

6. J. C. Houbolt, "A Recurrence MatrixSolution for the Dynamic. Response ofElastic Aircraft," Journal of AeronauticalScience, Vol. 17, pp. 540-550, Sep. 1950

7. S. P. Chan, H. L. Cox, and W. A. Benfield."Transient Analysis of Forced Vibrationsof Complex Structural-Mechanical Systems,"Journal of Royal Aeronautical Society,Vol. 6, pp. 457-460, Jul. 1962

8. J. T. Oden, "Finite Element Analysisof Nonlinear problems in the DynamicalTheory of Coupled Thermoelasticity,"Nuclear Engineering and Design,Amsterdam, pp. 465-475, July

9. J. T. Oden, Finite Elements of NonlinearContinua, McGraw-Hill. New York, inpress.

10. J. T. Oden and J. \.1. Poe, "On the NumericalSolution of a Class of Problems inDynamic Coupled Therriloelasticity,"Developments in Theoretical and AppliedMechanics, Vol. V, Pergamon Press, Vol. V.

11. E. L. Wilson and R. W. Clough, "DynamicResponse by Step-by-Step Matrix Analysis,"Symposium on Use of Computers in CivilEngineering, Laboratorio Nacional deEngenharia Civil, Lisbon, Oct. 1962

12. J. S. Humphreys, "On Dynamic SnapBuckling of Shallow Arches." AIAA Journal.Vol. 4, No.5, May 1966

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13. W. B. Stephens and R. E. Fulton,"Axisymmetric Static and Dynsmic Bucklingof Spherical Caps due to CentrallyDistributed Pressures," AIM Journal.Vol. 7, No. II. Nov. 1969

14. A. E. Green and W. Zerna, TheoreticalElasticity, Oxford at the ClarendonPress, 1968

15. A. E. Green and P. Naghdi, "A GeneralTheory of Elastic-Plastic Continua,"Archives for Rotational Mechanics andAnalysis, Vol. 18. pp. 251-281, 1965

16. W. Prager, "The Stress-Strain Laws ofthe Mathematical Theory of Plasticity--A Survey of Recent Progress, Journalof Applied Mechanics, Vol. 26. pp.101-106, 19S9

17. R. Hill, "The Mathematical Theory ofPlssticity," Oxford at the ClarendonPress, Ely House, London

18. .1. T. Oden, "Finite Element Formulationof problems of Finite Deformation andIrreversible Thermodynamics of NonlinearContinua--A Survey and Extension ofRecent Developments," Japan-U.S. Seminaron Matrix Methods of Structural Analysisand Design, August 25-30, 1969,Tokyo, Japan

19. J. T. Oden and G. Aguirre-Ramirez,"Formulation of General Discrete Modelsof Thermomechanical Behavior of Materialswith Memory," International Journal ofSolids and Structures, Vol. 5, No. 10.pp. 1077-1093, 1969

20. V. V. Novozhilov, Thin Shell Theory,P. Noordhoff Ltd. - Groningen -The Netherlands

21. O. C. Zienkiewicz and Y. K. Cheung.The Finite Element Method in Structuraland Continuum Mechanics, McGraw-Hill.1967

22. J. A. Stricklin, W. E. Haisler, H. R.Mac Dougall, and F. J. Stebbins,"Nonlinear Analysis of Shells ofRevolution by the Matrix DisplacementMethod," AIAA Journal, Vol. 6, No. 12,pp. 2306-2312

23. E. A. Witmer, T.H.H. Pian, E. W. Mack,and B. A. Berg, "An Improved DiscreteElement Analysis," ASRL TR 146-4. Part I,March 1968

page1titles. Bulletin 41 THE SHOCK AND VIBRATION DECEMBER 1970

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page4titlesE(p) + Z,"E kt'" + FIJ "yn)y~~) (6) 83

page5titles-fTo+A.4RM)AN[B1J(,VIJ - -vd ) + (c!To)I\t-f'lJdv (IJ ••. P", 84

page6titlesx3 , . t.t '" !t = 6~-1 +"3 tit 6l!!-1 + 6ti~-1 85

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page7titles+ t!.(5Al!.1-1 - 4A.\!.I_a + AW-s) A~ : Ata (2~ - SA1!.1-1 + 4Al,!,l-a - A~_3) o 86

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