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LA-8204-MS

Informal ReportWum 1

c“ 3 ClC-l 4 REPORT COLLECTION

m.-C

8.—

REPRODUCTIONPnDv

Developmental Studies of Constitutive Models

for Plastic-Bonded Explosives

Lsil!lKLOS ALAMOS SCIENTIFIC LABORATORY—Post Office Box 1663 Los Alamos. New Mexico 87545

An AffumativcAction/EqualOpportunityEmployer

Photocomposition by Chris West

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by the Uni!ed SIates Government. Neither the Umtcd

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state or reflect tho~ of the Uni!cd States Government

or any agency thereof.

).

.

UNITED STATES

DEPARTMENT OF ENERGY

CONTRACT W-740 B-ENG. 36

LA-8204-MSInformal Report

UC-45Issued: June 1980

Developmental Studies of Constitutive Models

for Plastic-Bonded Explosives

Robert M. Hackett*Randall L. Peeters

William R. Oakes, Jr.

l--

-.‘

—

“Aerospace Engineer, Propulsion Directorate Technology Laboratory, US Army Missile CommandRedstone Arsenal, AL 35809.

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .,,,..

II. MODELINGTECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,A. Creep Model Formation . . . . .... . . , . , . . ,-, . . . . . , . ‘. , . . , . . . . , . . . , . . . . . . . . . . , . . ,B. Viscoelastic Model Formation ‘.. . . . .. ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III. SUMMARYANDCONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..0,

1

1

225

8

8

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b

iv

‘

DEVELOPMENTAL STUDIES OF CONSTITUTIVE MODELSFOR PLASTIC-BONDED EXPLOSIVES

by

Robert M. Hackett, Randall L. Peeters, andWilliam R. Oakes, Jr.

ABSTRACT

Typically, elastic and elastic-plastic theory are used in structuralanalysis computer programs to model the mechanical behavior of high ex-plosives; these models, however, do not fit the observed behavior of plastic-bonded explosives. This report discusses the development of an equation-of-state creep model and a linear viscoelastic model for the analysis of thesematerial systems and shows comparisons between experimental results andanalytical model predictions.

————— ————————— ————— —

I. INTRODUCTION

We require accurate constitutive models of the in-dividual weapon components for the development ofmodern weapon systems. These models ale neces-sarily complicated because of irregular geometriesand the nonlinear nature of the different materials.Highly developed stress analysis computer codescapable of modeling three-dimensional geometriesand loading conditions are available and are beingused at the Los Alamos Scientific Laboratory(LASL). However, the results obtained with thecodes cannot be relied upon unless the constitutivebehavior of the component materials is well under-stood and modeled.

In predicting the response of systems utilizingplastic-bonded high explosives (HE), thecharacterization of the HE material is of paramountimportance. To date, little information concerningthe mechanical response of this class of materials isavailable. Experimental programs have shown thematerial to be highly complex and somewhat unique

in comparison to other materials that exhibit thesame general physical characteristics. For example,the high-solids loading with a plastic binder resultsin a stiff polymeric material characterized by smallstrain capability, combined with a creep-likeresponse. Therefore, the task of developing suitableconstitutive models to describe the response of theHE materials, and to subsequently be used in com-puter codes to predict overall mechanical-thermalsystem response, is a formidable one. This reportdescribes some of the initial efforts at LASL todefine the critical modeling parameters. Thesemodels are mathematically sound and yield resultsthat correlate with experimental observations, Theanalytical developments cited in the report arebased primarily on material testing already con-ducted at LASL. The described analytical work canalso help define the direction and role of furthertesting. A continued analytical/experimental in-teractive development program of this nature isclearly warranted, based upon the results of the in-itial work reported herein.

1

II. MODELING TECHNIQUES

In the development described, the initial model-ing is based upon a postulated elastic-plastic ther-mal creep response. The models to be described donot portray this complete phenomenon, but aredeveloped in such a manner to be contained in thisdomain. For example, although the modelspresented are formulated without a consideration ofthermal effect, such a consideration can be incor-porated subsequently without reconsideration of thecomplete basis of the formulation. On the otherhand, a consideration of “damage,” which goesbeyond the conception of a “deformable” continuum,would, in all likelihood, require a significant refor-mulation of the considered models. In fact, as will beseen, reasons to consider a theroy that includesdamage in the model formulation may exist.

The present analytical formulation is based upona characterization of the time-dependent materialbehavior. Three accepted methods of modeling thisbehavior are: (1) the phenomenologicaI or equation-of-state creep theory, (2) the memory or hereditarytheory, and (3) the nonlinear viscoelastic aproach.The analytical studies described here are basedupon methods 1 and 3. All of the data used in theformulations are based upontesting of an inert plasticdesignated 900-10.

A. Creep Model Formulation

room temperaturebonded material,

InitiaI studies centered on the equation-of-statecreep theory. Creep models of this type are used innuclear reactor technology to predict the deforma-tion, under sustained loading, of metaI structuressubjected to high temperatures. Such models existin numerous nonlinear finite element computercodes, including ADINA2 (Automatic DynamicIncremental Nonlinear Analysis), which is presentlyemployed at LASL, The two one-dimensional creepmodels in ADINAforms.

have the following mathematical

(1)

where ~Cis the uniaxial creep strain, rJ is the uniaxialstress at a sustained load, t is the time since the ap-plication of the sustained load, and a. through az are

2

material parameters obtained from a series of uniax-ial creep tests conducted at different stress levels;and

= F [1-exp(-Rt)I + ct.,cc (2)

where

F = aoexp(alu),(3a)

R = a2 (~)a4, (3b)

and

~ = a5exp(a6U)$ (3C)

where aO through ae are material parameters. TheADINA code also provides for the inclusion of a“user-developed” third creep model.

By using creep data obtained from a series of fourcreep tests, shown in Figs. 1 and 2, an attempt wasmade to model the response of the inert material us-ing Eq. (1) and Eq. (2).* Early results indicated thatEq. (1) could not be employed to model the inertmaterial. The parameter az from Eq, (1), as deter-mined from the recorded creep data, has a value lessthan unity that is not admissible in the creep strainrate calculation step in ADINA. It was also found,from early investigation, that Eq. (2), when

provided with values of the constants a. through a,from the creep test data, could not reproduce load-deflection histories. Thus, it was concluded that Eq.(1) and Eq. (2) were inadequate as creep models forthe characterization of the inert material.

Two new creep models were then developed for theADINA code. Both of these models were based uponthe creep test results displayed in Figs. 1 and 2,

They were coded into the Hewlett-Packard finiteelement package, and an attempt was made toanalytically predict the three constant-strain rate-to-failure curves for the inert material shown in Fig.4,——_—_____*[norder to quickly perform the necessary parameter studies, anefficient single-element finite element code was written for theHewlett-Packard 9825A calculator. The program utilizes thefour-node isoparametric axisymmetric quadrilateral element andthe same elastic-plastic thermal-creep analysis found in theADINA code. Experimental creep data is read from a flexibledisc; the experimental and predicted response curves are plottedon a multicolor x-y recorder. Figure 3 shows the various compo-nents in the system.

r

d’

m.m

900-lE STRAIN VS TIME

room t.mp.mt.uf-e Oc.fmtant “OOmpre.. ion stress

1.s HP.

c a,0S4

GL

Z3W.

:

; 1.8 w.

<< m.S02

{&sw.

“k

~t S9W

4,,”.

awm

Time (hre)

Fig, 1,Total compressive strain us time for sutainedconstant load.

am

s.W4

m.M2

mono

-

,/”..- ~,---------------

900-;9 STRAIN k TIME I.Onst.nt

room temperature .t.re..eampr.a.aioo

/

/

(i/ < 5 ~ >

as w.<,,S9,..

4,s-

1.0w.

---n.0 I@.

——-—— a..wmd initial re. pon.e

7 7 ? -i

Log Tim- (hrd

Fig, 2.Total compressive strainsustained constant load.

us log time for

The firat newly developed creep law was basedupon a Prony series fit to the experimental creepdata and has the form

cc = A+B1exp(-Rlt)+B2 exp(-R2t)

+B3exp(-R3t )+llbexp(-R4t)+Ct, (4)

where

A = -( R1+B2+B3+BQ), (5a)

B1 = A1UA2,

AqB2 = A30 ,

R3 = @Xp(&

ABR~ = A7a ,

and

(5b)

(5C)

3 (5d)

(5e)

(5f)r = AgexP(AIOU), -

with the units ofu and t being pounds per squareinch and hours, respectively. By choosing an ap-propriate functional form foreachof theconstantsinEq. (4), asdemonstrated inEq, (5), each constantinEq. (5) was evaluated by using a least squares curvefit. The four creep tests provided four data points.The values of R, through R, were preset to 0.01,0.1,1.0 and 10.0, based upon earlier curve fittinganalyses. The resulting values of the constants inEq. (5) are:

Al = -3.2517 X10-5 A5 = 1.8759 X10-3

A2 = 5.8234 x10-1 AT = -!5.8980X1 0-6

A3 = 2.6708 x10-6 A8 = ?3.2636x10-1

Aq = 9.0099x1 0-1 A9 = -1 .0179X1 0-5

A5 = 2.4385x1 0-” AIO = -5.4725 x10-3.

This model reproduces the constant load creepcurves from which it was derived and predicts creepresponse for uniaxial creep specimens subjected tovarying sustained load levels, but because of thenature of the formulation, it could not be employedin the Hewlett-Packard finite element model toreproduce the constant-strain rate-to-failure curvesof Fig. 4. Attempts at doing so resulted in overflowerrors associated with the nature of the loading.Therefore, we concluded that Eq. (4) provided amodel of limited application in its present state.

We then developed a creep model of a somewhatdifferent nature. Again, based upon the test resultsshown in Fig. 1 and 2, an expression of the form

= aoual[l-exp(-a~t)] + a3t‘c (6)

3

G!E.!2 f

SINGLE ELEMENT

D

++FINITE ELEMENT CODE

‘ADINA MODEL ++

●E-P + CREEp

●ARBITRARY LOAD HISTORY

YISCQ

VISCOELASTIC FORMULATION

I

I

, f

k

QROJ2L71sc

UNIAXIAL TESTDATA

(/’HP 9825DESKTOP COMPUTER

MULTICOLOR HARDCOPY GRAPHICS

Q EXPERIMENTAL DATA

● ANALYTICAL COMPUTATIONS

. CORRELATION

Pig. 3.Hewlett-Packardsystem medtodigplaytest resultsanddevelop finiteelementmodeh.

4

#9470-10 STRESS VS STRAIN

room k.mperat.ur./ ~ \

0.1

/ / —0.al

-train rat e (l/rein)

tZzc..

oompress ion

-!!7

<l*c”

a 001 ,Ux”

ton= i0“

Fig. 4.Constant-strain rate-to-failure experimentaltest results.

was proposed, with aO through as being materialparameters, and with u and t having unita of poundsper square inch and hours, respectively. When weused this model to reproduce the stress-strain curvesof Fig, 4 with the Hewlett-Packard finite elementprogram, we found that values of 5.658 x 10-’, 0.7,and 7.5 x 10-5 for a., a,, and ao, respectively, aresatisfactory, but the parameter az is rate sensitive.This parameter has the form

az ‘a- 6 0. (7)

The rate dependence of the a and /3 terms is shownin Table I. These values were used to obtain thestress-strain curves shown in Fig. 5. In the Hewlett-Packard finite element program, the values of theinitial modulus of elasticity E. and Poisson’s ratio ~were 600,000 psi and 0,36, respectively. Both valueswere obtained from prior testing and analysis. The

TABLE I

RATE-DEPENDENT PARAMETRIC VALUES

Strain Rate (rein-’)

0.001 0.01 0.1LY 14 80 400

/3 0.008 0.032 0.132

10

CREEP MOOEL

-treat! Ve et.rain I/<’-’12 P “

900-10 inork

room t..mper.t.ra

cxmpre..i.nn /

~train rate (llmin

4 /’

model prediction

A A experimental data

m I

Fig. 5,Constant-strain rate-to-failure predictions us-ing Eq. (6).

analysis also was based upon an assumption of in-compressible creep strains, which is consistent withthe ADINA theory that it duplicates. For simplicity,these analyses were limited to elastic, rather thanelastic-plastic, material behavior.

The excellent agreement between correspondingcurves in Figs. 4 and 5 supports the formulation ofEq. (6) as being representative of material behaviorover a restricted range. The limitation of theprogrammed material behavior to the elastic rangewas stated earlier. This model was aiso extrapolatedto the case of constant sustained loading (-zerostrain rate) with results that appear to correlate withmeasured creep data; however, this correlation hasnot been completely checked at this time. Westrongly recommended that this model be furtherstudied and developed. The Hewlett-Packard finiteelement program is undergoing an extension fromthe initial elastic range limitation to the generalelastic-plastic modeling capability. It provides thedeveloper with an extremely efficient means of exer-cising in-depth any proposed creep model, such asthe one of Eq. (6), relative to time-step sensitivity,load function, etc.

B. Viscoelastic Model Formulation

Additional studies were focused on the develop-ment of viscoelastic material behavior models. Thiswas partly based upon shortcomings associated with

5

equation-of-state creep theory models, e.g., lack of acapability of recovery behavior prediction. Becausethe degree of viscoelastic linearity of the HE was anunknown, the general overall approach was con-sidered to be nonlinear. Nonlinear viscoelastictheory is highly formulated,’ but applied with dif-ficulty. We felt that a sensible approach to thedevelopment of a viscoelastic model is to initiallyformulate a linear model that, as will be seen,provides the basis for a nonlinear analysis. Afterdetermining how well this model portrays thematerial response, the direction of further develop-ment will become more clear.

The nonlinear viscoelaatic constitutive theorydeveloped by Schapery4 was derived by the use ofprinciples of the thermodynamics of irreversibleprocesses. In this formulation, the nonlinearity iscontained in a “reduced time, ” where the reducedtime is an implicit function of stress. It is based, inpart, on the observation that nonlinear stress relaxa-tion of various materials can be described in terms ofthe same time-dependent properties found in thelinear range. The corresponding creep expression is:

c(t) = go Jo u(t) + gl$$(@u)W dT. (8)

This form is employed by Schapery under conditionsof constant temperature and uniaxial stress IJ,where

and

(9a)

(9b)

are the reduced times and au is a shift factor. Theterms g., gl, gl, and an are functions of stress at cons-tant temperature; JO and @ (t) are the time-independent compliance and the time-dependentcreep function, respectively, and r ands are generictimes.

None of the parameters in Eq. (8) are fundamen-tal physical or thermodynamic constants and,therefore, all must be determined from creep, relax-ation, or other mechanical tests of the material be-ing considered.

For linear viscoelastic behavior, with & = g, = g,= a. = 1, Eq. (8) reduces to

aCJ(T) &,c(t) = Jo u(t) + & (t-l) ~ (lo)

or the equivalent form

c(t) = f: J(t-T)- dl. (11)

In developing an expression for J(t), the creepcompliance, the creep test data shown in Figs. 1 and2 were combined, and produced the somewhat sur-prising narrow data band shown in Fig, 6. By using aPTony series curve fit, the following expression wasobtained.

J(t) = o+Plexp(-81t)+p2 exp(-t52t)+P3exp( -63t)

+pqeXP(-6qt )+p5exP(-6~t)+p6t3 (12)

with the unita of t being houra, and where

P . .0000211956 ok = -.0000009812

01 = -.0000111843 05 = -.0000035347

P2 = -.0000004538‘6 =

-.0000000246,

03 = -.0000023488

900-10 CiEEP COMPLIANCE vs TIME

I

?00. t.mp.eat.u..

ctmpve*. ionI lJ#

I I I I—

Ip---+= I I L_-’. d.....d L. I

Lag Time (ht-.)

Fig. 6.Creep compliance us log time for four tests.

6

with the values of al through & preset to 0.01, 0.1,1,0, 10.0 and 100.0.

A linear viscoelastic program, using the expressionfor J(t) from Eq. (12) and the loading from Fig. 7,was written for the Hewlett-Packard calculator dis-cussed earlier, and used to predict the experimentalstress-strain behavior shown in Fig. 8, The results ofthe program are shown in Fig. 9. The similaritybetween the two curves (Figs. 8 and 9) is interesting.Notably, the slopes of the two curves are approx-imately the same, after two pulses. However, theviscoelastic model predicts complete strain recoverybetween pulses, whereas the actual test data showthis not to be the case, Comparison of the results ob-tained with the linear viscoelastic model to the ex-perimental data might lead one to postulate a new“damage theory.” However, refinement of this modelmight be expected to yield a damage-like response.

8

4

2

m

: : g. g 5d d d d ❑“

Axial S&rain

Fig. 8.Stress-strain diagram corresponding to ex-perimental puked pressure loading of Fig. 7.

CYCL”IC LOADING

str.3ss vs time fao01.

$

* ,,

Time

CYCLIC LOAD ING

drain v- time

#

Time

Fig. 7.Pulsed pressure loading with extendedrecovery time between pulses.

B

LI NEAR VI SCOELASTI C MOOEL

c,trem. v. train

I

Axial Strain

Fig. 9.Stress-strain diagram predicted using Eq. (11).

III. SUMMARY AND CONCLUSIONS

Three time-dependent constitutive behaviormodels have been formulated, Clearly, all arelimited and exploratory. Equation (4) predicts thetime-dependent behavior under sustained constantload. Equation (6) correlates extremely well withtest data based on constant-strain rate-to-failuretesting in the “elastic range, ” but is, in ita presentform, strain-rate dependent. Equation (6) also ap-pears to accurately predict time-dependent behaviorunder conditions of constant load. Equation (11), alinear viscoelastic model, provides a reasonablecomparison with experimental data on cyclicloading and poses some very interesting questionsrelated to damage.

These simple analytical models, together with theHewlett-Packard finite element program, can beutilized to gain more valuable insight into the con-stitutive behavior of this class of materials.

ACKNOWLEDGMENT

REI?ERENCES

1. W. E. Haisler and D. R. Sanders, “Elastic-Plastic-Creep-Large Strain Analysis at ElevatedTemperature by the Finite Element Method,”Comput. Struct., 10, No. 2, 375-381 (April1979).

2. K. J. Bathe, “A Finite Element Program forAutomatic Dynamic Incremental NonlinearAnalysis, ” Acoustics and Vibration Laboratory,Mechanical Engineering Department, Mass-achusetts Institute of Technology, report 82448-1 (September 1975).

3. W. N. Findley, J. S. Lai, and K. Onaran, Creepand Relaxation of Nonlinear Viscoela8ticMateriaZs (North-Hollard Publishing Company,Amsterdam, New York, Oxford, 1976).

4. R. A. Schapery, “On the Characterization ofNonlinear Viscoelastic Materials,” Polym. Eng.Sci., 9, No. 4,295-310 (July 1969).

The authors gratefully acknowledge the efforts ofHarry Duhamel who helped conduct the experimen-tal studies.

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