Handbook for the Engineering Structural Analysis of Solid Propellants

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FROM Distribution authorized to U.S. Gov't. agencies only; Test and Evaluation; 01 APR 1971. Other requests shall be referred to Office of Naval Research, Arlington, VA 22203-1995.

AUTHORITY USNASC ltr,

11 Apr 1972

THIS PAGE IS UNCLASSIFIED

UTEC CE 71-O89

SPU3VUCATION 214

1

Handbook for the Engineering Structural Analysis of Solid Propellants "MAY 1971

Prepve~d by

J. Edmund Fitzgerald William L Hufferd

With the Universily of UtahsU

J =5(•

~Sposored

by

OFFICE OF NAVAL RESEARCH AND

•--• I

NAVAL WEAPONS CENTER CHI4A LAKE

W

CHEMICAL PROPULSION INFORMATION AGENCY Distr•bution limited t. U. S. G*v't agecies only; test a3d e~kluA; 1 Aoril 1!7'1. Other requee for this docuea', mwut be referred tv the Office

ef iiaral Rxsearch, Codt 439:, Irlinota, Virginia 22217.

3ost Av:'"", Copy

•"

C.

Best Avai~lable Copy

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The folJ'owing pages are purposely left blank:

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ODOC-TCA. Sep 71

*,

CPIA PUULICATION 214

(kTEC CE 71-089

andbook for the Engineering Structural Anailysis, of Solid Propellants MAY 1971

Prepared by

iJ.Edmund Fizgerald

William L. Hufferd Under Contract NOOO14-67-A-0325-0001 With, the UIfersity of Utah

Sponsored by OFFICE OF NAVAL RESEARCH AND NAVAL WEAPONS CENTER CHINA LAKE

CHEMICAL PROPULSION INFORMATION AGENCY

$@GM-N " "%A U#V"*f ." " P'"WCS-• • 0"T1- ""4"-"at t

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D~~OCUMENT CONTROL DATA- RC&i~ Srcurily ciausdfieatton,of title. body of abstract and Indexing winoto.twp no,.t be *niered w.h~nUe-

IOIGiNA TONG AC IlVITYý(Couporsj* author)

2a. RE "ORt

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report 1z ri-',:1iied1

o# ,.i

TY C L',SIvICA IS1CTO TION

SECCRI

Unlversity-Nof Utah 3RPORT

TI(LE

Handbook for the Engineering Structural Analysis of Solid Propellants 4

OEC RIP TIVO

NOTCA (Ty'pe Of report Andi1.10usiv. dotes)

Handbook II AU THORIS) (First nap*. nmiddf

nit Il last nam~e)

*J. Edmund Fitzj'erald William L. Hufferd s

RpctptrT

DATE

70.

May 1971 C44MC

ORGAJ

O

4.

TOTAL NO

ORIGIt4ATOWS REPORT

Cont~ract N00014-67-A-0325-000l1 C.

Ib.

OF PAGES

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(Any other numbers that may 5* eaatdnod

this report)

CPIA Publication No. 214

d.____________________ 4

0

11-

OISTRIS.JTION STATEMIEPT

SUPPLEMENTARY.NOTES1

12. SPONSORING

MILITARY ACTIVITY

(1)Office of Naval Research, Arlington (2)Naval Weapons Center, China Lake ACTA

TbA Aandbook presents a~review and discussion of some of the refinements ýind improvements that have been incorporate'd int9 the methods of determining the grain st~ructural, integr'ity of soylid rocket mnotors during the past fifteen years. The Harnibook has been written primarily for the solid propellant grain designer/ [No items on this DD Form 1473 are subjec;t to 'the document distribution statement.]

D~ "0v.17 D S/N 0101 -807-6811

Uncl ass-i fled

(AE1 "

t Clsiicto

Ak-31404

I

Security Classification

I

:KIEV

WORD

-

-T

W-

,,-0 t .

_

-'

T

Solid Propellants Solid Propellant Grain Design Solid Propellant Rocket Motors Structuiral Integrity Vi scoel asti ci ty Nonlinear ViscoelasticityI

Aging Stress-Strain Analysis

I.

)

DD ,Nv.,1473 5/ki

Clol-so0-6ez

(BA)

Unclassified

Security Claosifi stion

40

Ir

FOREWORD This timely and comprehensive volume was prepared under the joint spolisorship of the Structural Mechanics Program of the Office of Naval Research Ran,, of t,,he ,alai Weapons Center, China Lake, and the Polyme.. S.r.c.. through ONR Contract N00014-67-A-0325-O001 with the University of Utah, the below indicated reasons.

for

In the structural integrity analysis of solid rocket propellant motors, intrinsic material and geometric complexities, such as interrelated thme, temperature and shape dependent grain response, have dictated many simplifying assw-aptions which are becoming increasingly incompatible with rising performance requirements. While over the years propellant mechanics research b]as resolved many of these obstacles, the current trend toward higher solid filler content is greatly aggravating the seriousness of those remaining. Accordingly, there is now a rapidly growing need for a sound and comprehensive survey and collation of inadequately exploited research gains, and ?or their proper integration into grain design technology. The purpose of this present bold but sound effort to achieve these formidable goals is twofold. The first is to provide in fact Just such a needed guide for inmediate design utilization. The second is to provide an authoritative advance document to serve as the basis for a scheduled joint Air Force-Navy sponsored engineering review and, where appropriate, a supplemental expansion of its contents, by the various specialists in the Govern ent-lndustry solid pr1ellant community who serve on the Joint Army-4avy-NASA-Air Force .Structural Integrity Committee for solid rocket motors.

~

The willingness of the authors, Professors J. Edmund Fitzgerald and W. L. Htfferd, to undertake such P dual purpose mission is ciomiendable, The fact that the effort had the concurrence of all the prospective reviewers on this very effective Committee is indicative of the professional respect and acknowledged leadership they enjoy among their colleagues in this technically aggressive and highly competitive cvinunity. An initial review of the draft of this "basic" document by the undersigned reveals that both objectives of the undertaking are admirably met and that the confidence placed in these authors is lindeed well founded. Assuredly, this. integration of present knowledge with its planned supplamentetion will greatly assist our solid rocket engineers to provide the requi,,itjstructural integrity in critical propellarniigrains of our solid rocket molrs. In view of the viable, dynamic nature of defense technology, however, our continuing adequacy in this vital area of solid rocketry can be insured on even a minimum basis only by the effective implementation of -the further research and development requirements identified in this and future a•endcients to this timely and authoritative handbook. March, 1971

!li

John M. Crowley Structural Mechanics Program Office of Naval Research

MR~

PREFACE This Handbook re-presents an attempt to present an accurate report and evaluation of the cupr~nt state-of-the-arit of solid propellant grain structural integeity .Majorý emphasis is given to the requirements for meaningful *ýiateriai ch 4r,.tor4.at~ioo, structural analysis dnd-failure analysis of solid analyst. In preparing this 'Handbook, we have relied heavily on the published and unpublished works of many of our colleagues from~ the solid rocket community. in addi-tion, conversations with technical personnel of various solid rocket motor companies were carried out for the purpose of obtai'iing accurat? information on current structural integrity practices. 'For their generous cooperation, we are partic~larly grateful to:

*

Aerojet Solid Pr(ýpjlsion Company Atlantic Research Corporation Hercules Incorpor~ated Lockheed Propulsion Company Mathematical Science Cornoration RocketdyneT, Thiokol C.,'1unca! Corp-)ration Un~ited Technology Center

We wish to acknowledge the StructurzI Mechanics Program of the Office of~ Neval Researcnl which join~tly sponsored this effort under Contract NQO0l4-67-A-.%'3?5-O0G1 w~th the Naval W4eapons Center. -Special thanks are due to Mr. John Crowley for h~s inception of'the handbook concept as well as hils constant efforts toward bringing-this-Haahdbcok to fruition. Tn this regard, the effecti'ie assistance and support of Dr. Arnold Aficeff nf the Naval Weapons Center and Mr. Irving Silver of the'N%,al Air Systems Commn~nd is also appreciated. We'are 0Iýo indE~bted to 'Irs. Merle Bryner, Mrs. Marilyn Harris, Mrs. Kaye Bowen and Mrs. Johanna Broadbent for their devotion and skill in the task of typing this matiuscript. J. Edmund Fitzgerald William L. Hufferd

February, '1971

V

,ii TABLE of CONTENTS

I./

1.L General ..... ............................ 1.2 Loading Environment ....... .................. 1.3

Methods.

Preliminary De.ian ,lis

.1.1 1.3 !.4

..........

1.4 Final Design Analysis Methods ..... ............. 1.5 Special Design Considerations ..... ............. 1.6 Experimental Stress Analysis Methods .......... ............... 1.7 Failure Analysis Methr.s ...... 1.8 Material Characterization Methods .............. 1.9 Linear Viscoelasticity ..... ............... "1.10 Thermoviscoelasticity ...................... 1.11 Nonlinear Viscoelasticity ..... ............... 1.12 Appendices ....... ......................

1.5 1.6 1.7 1.7 1.8 ... 1.9 .1.9 1.9 1.10

II. LOADING AND ENVIRONMENTAL CWNSIDERATIONS .................... 2.1 Introduction 2.2 Specified Loads;2.1 2.2.1 Thermal Loads ..... ................ . "2.2.2 Acteleration Loads ..... ............... Axial Acceleration ..... ........... .... Transverse Accelev:ation ..... ............ 2.2.3 Dynamic Loads . . ...... ................ Vibration ....... ................... Shock ...... .............. Transportation and Handling. ....... .... . 2.2.4 Special Loads and Environments ...... Loads ....... . .. . and Design .. Induced 2.3 Propellant .................. 2,,.3.1 Cure Shrinkage 2.3.2 Pressurization Loads ..... .............. S2.3.3-\Flight and Combined Loads ..... ........... ........................ Humidity 2.4 AgIng and 2.4.1 Relative Humidity ...... ............... 2.4.2 Aging' ...................... Minimization of Adverse Aging EfFects ....... Life Predictions ,Service ... .. . . . . . .. Programs . ................ \iurveillance

III.

2.1 2.1, 2.6 2.7 2.8 2.9

2.9 2.15 2.16 2.J7 2.18 2.18 2.22 2.23 2.25 2.28 233 2.35 2.47

Non-Destructive Test Techniques ....... ..... 2.4.3 Closure ......... .................... 2.5 Manufacturing and Processing Consid,.-at~ins ........ .................... 2.6 Nomenclature .......... °.. ....................... 2.7 References .

2.49 2.52 2.52 2.59 2.60

PRELIMINARY DESIGN ANALYSIS ..................... 3.1 Introduction ........ 3.2 Temperature Loadipgs ...... ................. .. 3.2.1 Shrinkage During Cure ... ............. 3.2.2 ThermalVCooling and Temperature Cycling . ... SHollow Cylinder .... ................ ... ..... .......... Solid Cylinder.... .. . End-Bonded Hollow Cylinder......... ....

3.1 3.3 3.5 3.6 3.9 3,15 3.16

vii

3.3

3.4 3.5 3.6 3.7

"3.8 3.9 3.10 3.11 3.12 IV.

V.

3.2.3 Aerodynamic Heating ...... .............. .................... Dynamic Loads ......... . . 3.3.1 Shock Loads ................... . 3.3,2 Vibration . . . . . . . *ý.. . . . . . ... Lateral Vibration of a Starpoint .... ....... Lateral Vibration of a Circular Port Grafn. . .... ..... Axial Vibration .... ......... Thermomechanical rnonlinn And ................ Generation .... Heat Vibration Dpsign Analysis Suramary .... .......

3.17 3.19 3.19 3.21 3.25 3.32 3.33 3.35 3.38

3.40 Acceleration Loads ....... .................. .. 3.40 3.4.1 Axial Acceleration .................... ... 3.45 3.4.2 Lateral Acceleration .......... 3.46 . ... ........... Pressurization Loads ......... 3.46-3.5.1 Hollow, Cylinder .................... 3.48 ............. 3.5.2 Star Perforated Grain 3.50 ......... Finite Length End Correction Factor. Stress/Strain Concentration Factors in Star 3.53 _.,.2 .... ... ... ... Perforated Grains 3.56 .......... 3.7.1 Simple Slot Grain Geometry ..... 3.61 "3.72 Slot V•41th Effect . .......... .......... 3.7.3 Positive Wedge Angle Geometry . . ........ 3.7.4 Negative Wedge Angle Geometry.......... ......... 3.69 3.7.5 Elliptical Slot Tip Geometry ..... 3.72 3.7.6 Extension of Photoelastic Test Results . ... 3.76 .......... Design Analysis Procedure Summary ..... . ..... .3.79 Equivalent Hollow Cylinder .......... 3.84 ....................... Closure 3.89 ............ .......... Nomenclature. 3.91 ............... References ........

FINAL DESIGN ANALYSIS 4.J Introduction ............ . ... ............. ... 4.2 Numerical Techniques .... ................. 4.3 Outline of the Flnite Element Method .... ......... ...... .. Incompressible'Reformulation ........... .... Development of, Bagic 'Equations ...... ...... . ............. Solution of Equations. 4.4 Industry Practices . ............... ....... 4.5 Future Developments . . ..... ... ..... ... ... ... 4.6 Nomenclature ..................... o... ... ........................... 4.7 References SPECIAL DESIGN CONSIDERATiONS ..................... 5.1 Introduction .................. 5.2 Transition Regions .. . . .............. 5.3 Grain Terminations .... 5,3.1 Relief Flaps ..... ................. .. ... Simple Prellminary Design Procedure Singular Behavior of Case-Grain Terminations. Energy Balance Approach ..... ...........

"viii

4.1 4.2 4.6 4.7 4.14 4%22 4.27 4.28

4.31 4.33 5.1 5.1 5.4 5.6 5.9 5.11 5.19

5.3.1

Relief Fl~ps .. . . . . . 5..6 Simple Preliminary Design Procedure . , .. . 5.9. Singular Behavior. of Case-Grain Tpminations . 5.11 Energy Balance 'Approach ...... ....... ..... 5,19" 5.3.2 -Grain End Contouring................. . . 5.32 5,3.3 Summary and Design Guidelines ..... . 5.85 5*.4. Nomenclature ........... .... ................ 5.87 . . . . . . . 5.5 References

VI.

VII.

EXPERIMENTAL ANALYSiS MftHiQpS . 6.1 'Introduction ... .... ........ .. ...... .. 6.1 6.2 'Photoelasticity ..... ..... ......•. ........ 6-.1 6.2.1l Three Dimensional Photoelasticfty ............ 6.3 Stress Freezing Technique .. ,........ 6.4, Scattered Light Technique ..... . . . .... . .. 6.5 "6..2.2 Limitations of PhQtoelasMcity .......... 6.7 6.2.3 Applications of Photoelasticity to Problems ' of Grain Structural Integrity .. . ..6 6.3 Instrumentation.... ' ** '. . 6........611 6.4 Structural' Test V6hicles . . .. . .......... ..... 6.18 6.5, Nomenclature. . ......... .............. .. ..... 6.216.6 References ..................... . . ..... ... .. 6.23 ' FAILURE,-ANALYSIS 7.1 Introduction ...... .......................... 7 7.1 7.ll Acceptable Margin of Safety .. . . . . 7.1. 7.2 Discrete Failure Analysis . . ..... ...... 7. 7.3.1 Failure Surface Criterion . .... 7.6 7.3.2 Fracture .............. -....... 7.9 7.3.3 CumulatiVe Damage . ......... . 7......07.4 References . .. . . . . . . . . . . . . . . . . . . . . 7.13,

. ,

.,

..

VIII.

IX.

MATERIAL CHARACTERIZATION 8.1 Introduction ......... ..... 8.2. Mechanical Property Tests .................. 8,.3 Thermal Property Tests ' /. 8.4 Nonlinear Characterization .......... * .......... 8.4.1 Physical Nonlinearities ....... ..... 8.4.2 Geometrical Nonlinearities . ............ 8.4.3 Irreversible Micro-Structural Changes . . ..'. 8,4.4 Closure ...... ........... ......... 8.5 Nomenclature ..... ....................... .. 8.6 References . .... ... ... . .. . .. ... ..

.,

8.1 8.3 8.9 8.11 8.12 8.17 8.18 8;248.25 8.27

LINEAR VISCOELASTICITY '. . . ... .,9.1. . . . .............. 9.1 Introduction ... 9.2 General Considerations." .... ..... -. .. .. .9.1 9.3 Description of a LinearVy, Viscoelastic Material .... 9.2 9.4 Constitutive Equations ... .................. .... 9.3 9.5 Stress Analysis. . ........ *.... . . . . .... 9.8 9.6 Eigenvalues qf Relaxation Operators Applied, to Linear an6 Nqnlinear Solid Propellant Predictions 9.17.

1

ix

-j

"19.6.1" Development of Eig~nvalue Procedures, ......

9,18

9.19

. Constant Strain Rate Input for Cons~tant Strain Rat-..... ue6ston " - -- I~pu s . . . . . . . . . . .. ... . .. . . . . . .. "... .. . .... S•.cond Order. N~put .... !C ',

X.

re

97

Nomencla I

9-.8

Reference

.

.

. . . . . . ..

....

9 2 _• 9.12 6

. . . .

o '

"

-9.31 . .. 9 33

. -..,-."

THERMOVISCOELA• ICITY .10.1

Lttr6duction-.10'..................

10.1

,...10.1 10t2' Thermorheologibfily Simple Materials ...... 10.3 U0.3 Calculation -of Ti.me-Temperature Shift racor .* 10.3 l0.3ol Uniaxia] Tensile Stress.Re!axation Test Shift Factor..I0.6 tI.3:2 Determination of, 10.13 '."."."...".".. -10.4 Material Characte'rizatiod .... . ............. Transient Problems .... ...... Solution-MetFhods- . . . ... ........ .10.7 .Conclusio s . ............... "10.8 Nomenclature ...

I0j5 '10.6

10.9 .,XI.

.

.. ..

10.18' 10.19

. . ... .... ........... ...

References . .............

..

10.21 10.22

10.23

........

NONLINEAR VISCOELASTICITY 1.l .... ............ ..... .. ,l1.1 Introduction . . .... .. .11.3 , 1l.1.1 Linear Versus Nonlinear Analysis ..... . ll'.7 S11.2 ' Mechanisms of Noblinear Viscoelastic Behavior . . 11.14 I11.3- Classification of Viscoelastic Constitutive Theories 11.19 11'.4 Recent Devel-opments in Constitutive Equations . . 11.27 . . ... ..... ....... •11.5 .Equation Devalopment ... 11.30, 11.6 -A General Theory of Fading Memory ,11.6.1 Classification of Fading Memory .

.

-Characteristics

.....................

.

.. .

11.31

11.38 Position of Previous Fading Memory Theories.. 11.......11i39 . Green-Rivli n Theory .... Coleman-,ioll Theory ..... ..... .......... 146, 11.51 . . ............ Wang 4§ Theorie5 ." 11.57 . ..... .......... Wang-Boweni•e eory . .. .. 11.60 -...... jblemay-Miz~l-Theory ....... 11.6.6 ° Dev~lopmentof A Memory Function Norm . . . . 11.62" 1.63 1.. . ..... Fading Memory 'Hypothesis ...... Developmbnt of a Memory Function Norm ,, . . . 11.65 11.7 Further Development of, A•Thermodynamic Constitutive 72 Theory '1.'....................... 113.2 , 11.7.1 Thermodynamic Prelimmifaries. ............ .844.. I1..7,.2 Constitutive Assumptions . .9. 11.7.3, ThTarmodynamjc Restriction-n Constitutive :-l.ý, ..... .... Functionals . . ..... ." "? 11.7.4ý Discussion of Results and Comparison with ' .. 11.103 ..... ' .% Previous Theories' 11.3 Approximate Theories and Dispussion of. Applications .11.106 .. 'm,6 11.8.,3 Prel 4 imiharies . .... ............. 11.8.2 Representation of Constitutive Functionals .110 ............ and Approximate Theories 11.6.2

."-

.

I

X

41

lo",

IN,

11.8.3 11.8.4

-11.9 1•0.lO

11.11 11.12

•

XII.

,pecial Theories 6f Material BehaVior.iI.I15--.117 . .. . Closure and-Conclusions ........ IdealizedThermoviscoelastic Cylinder Problem ..... . . . 11.120 Comparisoa ofxVarious Noplinear Methods of Analysis 11.13. Nomenclature . . . .' .,. ... . . . . . . . . . . . . . . . 11.133 References .. ....... . ............ ...1. . . Appendix 11A - "On Mathematical Forms for .the "Material Functions in Nonlinear Viscoela~ticity," by R. 0. Stafford....... ...... 11.143.. 1....1.....

RESEARCH NEEDS

12.1 12.2 APPENDICES

Introduction . ........... Specific.Research Needsi. ......... ........ ...... •

.

A.

SUMMWRY OF LINEAR ELASTICITY EQUATIONS

B.

FINITE ELEMENT ANALYSIS

.

PARAMETRIC'DESIGN CUR•VES

.

... --

""_D.. •--P

F"E

E.

O-ELASTIC STRESS/STRAIN .CONCENTRATION FACTORS FOR . STAR SIAP& PRORELLANT GRAINS SAMPLE MOTOR STRUCTURAL INTEGRITY ANALYSIS

F.

'MOTOR EXPERIENCE

'..

--

/

.

S•

xi A!

-.

12.1 12.2

....

ftt,

M1'

LL5i7:zo

!.11

• ,/.

'era~rrl"

INTRODUCTION

in&

In the

"•tn

""

-Thl Handbook Presents a revfew and discussion of some of the refine mens .and improvements that have been incorporated into the methodl of determining the grain. structurl 'integrity of solid. propellant rockit motors "durinq-the past fiftwn years.

Major emphasis is given Io discussions of

the /current state-of-the-art practices along with a critical, appraisal of the accu,'aty and •range of applicability pf structural analilis methods.-

experimenral verificatio dnd importeance.s-of Y " The necessity -. 4 ' ' 4

'

-

of -grain struc-

tural ititegrity, is similarly stressed. A grain structural integrity analysis is an evaluation of the ability of a solid propellant roqket motoW to perform satlsfactorily throughout a .specified environment; and i's comprised of two parts: 'agrai6 struttural analysis and a; failure. analysis." A grain structural analysfs is the de-

-.4,

.

j

termination of the ýtresses, strains, deflection s,and deformati ns. a solid propellant.'grain may. be .subjected to during its lifetime. 'Astr•.• .analys.is, when coupled with.appropriate failure datp of the.component materials through a failure analysis, de-fines the''imiting environment i.n which a solid rocket ractor may be expected to perform satisfactorily. Most often, however, the environment of a rocket.motor, is specified by the prir, contractor or the sponsoring government agency to the propul.-

sion system subcontractor.

Thus, a coupled structural and failure analysis,

or.equivalentlya grain structur-1 integrity analysis, instead of defining

*l~l

S•

.,

-

-

-

I>

/

San operational environment, serves to dete/Mine a minimum margin of safety for satisfactory motor operation throughout the specified service life of the motor. In determining a minimum margin of safety, consideration must be-"given

not only to. the statistical variations inherent in the experimental

determination of material property data, but also to the loads encountered by the motor (e~g.,

vibration, acceleration, pressuriz~tioý, etc.),

the physical environment of the motor (e.g., aging conditions, humidity, tehiperature, etc.), and the inaccuracies inhererlt in the analysis methods or artificially introduced through simplifying asgumpti.ons.

The margin

of safety determined through proper consideration of these factors I .an indication of the overall system reliability. 7

If-these factors were pre-

cisely known, there would only be required a margin of safety greater than zero. -Inasmuch as this is not the case, and quite often assumptions orapproximations -must be gSade regarding specific information which is un-

;

avaiiafle, arbitrary' restrictions are placed on an acceptable'minimum margin of safety.

These restrictions reflectan ignorance fortar.._asociat.ed,

iwi-th the structul-al analysis, the loading conditions., propellant behavior and failure analysis as well as the physical environment and mis-siqn.rea qutreme.nts.

The degree of arbitrariness- of these restrictions has. been

somewhat lessened by _the motor experience gained throughout the industry during the'-past decade.

This experience has been gained in all aspects

of grain structural integrity and has resulted in generally 4cceptable ' englaserin~approximations and assumptions regarding failure data, aging behavior, structural analysis.and minimum margin of safety which, are valid fort'he most part for engineering Aaly•lis purposes,

Within this context,

1.2

VIt'

Sapproximations and simplifying assumptions are normally introduced for, say, material property data for which data is available for a similar material, or for a motor design for whieh experience exists with a.similar configuration.

The validity and usefulness of these assumptions and ap-

proximations are discussed herein, cautioning that in reality there is no satisfactory substitute for the actual information required.

In the case

of vastly different materials or design configurations', the required information must, of necessity, be obtained through more sophisticated analyses,

Smore

extensive laboratory testing;.a'nd if necessary, determination of grain structural integrity may ultimately require-verificati'\n throigh full-scale motor tests.

In any casd, simplifying assumptions and

pproximations re-

quire substantiation through pa~t experience or appropriate analyses and experimentation. 1.2

Loading Environment This chapter serves as an introduction to the subject by considering

the loading conditions to which a solid rocket motor is li-kely t9 be sub-

"jected. Temperature, dynamic, acceleration, pressurization and combined loadings are discussed in terms of their origin and relative severity in conventional motor designs.

These loadings are discussed in some detail

in this chapter so that future chapters on analysis methods will not be unduly repetitive in describing the loading environments.

This chapter also considers the influence of the physical environment which a solid propellant rocket motbr experiences during its lifetime.

This Is

a most difficult and significant problem facing the structural integrity engineer.

Of the factors producing adverse effects which serve to reduce

1.37

/

the operational service life of a solid rocket motor, the normal aging of propellants and liners, and relative humidity level during storage are qenerally accepted as being the most critical.

Proper evaluation of the ef-

fect of these storage conditions on the structural integrity of a solid rocket motor requires consideration of the physical,

chemical and physlo-

chemical changes of all age-sensitive system and sub-system components. 1.3

Preliminary Design Analysis Methods This chapter is the first of two chapters dealing specifically with

the methods of performing a grain structural analysis. Preliminary design configuration analysis methods are discussed here and final design analysis methods are dealt-with in the following chapter. A preliminary design analysis of a prospective candidate motor configuration determines if a given grain design has ierit and possibly gives qualitative or semi-quantitative indications of how the design may be structurally improved.

At this state in the analysis of a'solid propel-

lant grain approximations and simplifyihg assumptions in the analysis methods are warranted.

Design data sheets and approximate engineering

formulas are reconier'ded for the analysis of conventional motor designs. Extensive numerical analyses at this level are- unwarranted.

The addition-

al accuracy gained from using a computer analysts is often unjustified in viei 6f possible approximations made regarding, say,material properties or failure data, and also, the uacqrtainty of the final design configuration does not justify the expense of computeranalyses. may exist

n the case of new or novel grain designs,

analyses may be required.

An exception,

in which case computer

In these cases, and particularly In the caqe of

radically new grain designs, development of new analyses coupled with 1.4

(--

U

"experimental subscale motor tests is recommended in place of relying on computer analyses of questbnable appltcability. In this chapter, approximate engineering analysis methods are given fbr the 'loading conditions' discussed in Chapter 2. The approximate methods consist ot formulas for calculating stresses, strains and deflections for thick-walled hollow cylinders.

Empirically derived relationships for

determining s~tress concentratidn factors for slotted and star configurations, anA cuLrves of fiitk

length end correction factors are included-

The ptesentation of. this material has been parameterized in terms of web fractions and length to ditaeter ratios. "Becaese of the approximate and preliminary nature of the analysis methods discussed in this chapter, only the expressions used for determining maxinmum values of stress, strain and deflection are given. sign-analysis.

These values are sufficient for a preliiminary de-

Profiles of stress, strain and deformation as a function

of length for finite length hollow cylinderý have been obtained by means of

rical solutions to the equations of elasticity.

AMP,.•nalysis

methods presentedare based on infinitesimal linear

elasticity theory.

Indicati,.,i

of how time and temperature ýffects may

be incorporated are also discussed.

For the most Part, because of the

pre-liminary natbre of a preliminary design analysis, these modifications are not called for at the preliminary design stage. 1.4

Final Design Analys s Methods This chapter considers the

ses.

methods of conducting final design analy-

Preliminary design analysis methods were 8i;c,ussed In the preceding

chapter. The -final stage of a grain structural analysis is periormed after a preliminary design analysis has indic~ted the potential adouiacy of a given 1..'

k

-

-

grain configuration.

Whereas a preliminary desIgn analysis normally in-

vestigates loads and regions of a propellant grain generally thought to represent critical structural integrity parameters, under simplifying assumptions and approximations, a final design analysis usually encompasses the total lodding ewivironnent and the entire propellant grain, normally under- less restrictive assumptions and approximations. The level of sophistication required in the final analysis and design -tage

is'determined by the complexity of the grain configuration and the

severity of the loading environnent.

Presently, the final stage of a grain

structural analysis involves extensive use of approximate numerical tdchniques.

Few closed-form analytical solutions are obtained during the final

analysis stage because of the complexities O•the problems involved and the Moative ease of developing numerical analysis methods for obtaining, approximate solutions.

A brief description of the numerical techniques

"commonly.6ued'throughoot the solid propellant industry is presented here along with ia discussion of current industry.practices. 1.5 Soeclal Design Consideratigns

Several areas of grain structural integrity analyses require special consideration.

Particular theoretical and experimental investigations

have been carried out for " TRANSITION REGIONS "• GRAIN TERMINATIONS The results of some of these studies are sunmmarized in this chapter.

In

some cases the results are quite qualitative and, at best, are only suited for preliminary design analysis efforts when coupled with competent,

1.6

A

engineering judgment.

Considerable detail is presented to illustrate

design and analysis procedures. 1.6 Experimental Stress Analysis Methods Experimental stress analyses may serve as the primary analysis tool or confirmatory experimentation of other analysis techniques.

Experimental

methods are frequently used as the main analysis tool for complex grain configurations when the validity of the results of analytical and numerical analyses is seriously questioned. 'Experimental

methods are also em-

ployed for confirmation of analysi§ and failure predictions with subscale and prototype motor tests used as ultimate verification of graIn structur-" al integrity.

Properly used, experimental stress analyses represent power-

ful tools for the designer/analyst,

complementing analytical ahd approxirTate

numerical analysis techniques. Presently, experimental methods make considerable use of photoelasticity, displacement measuring devices and Structural Test Vehicles (STV's), which model the essential features of production delivery motors.

These

topics are discussed ih this chapter. 1.7

Failure Analysis Methods A failure or strength analysis comprise$ the final stage of a grain

SsItrtfctural integrity analysis.

The results of a strength analysis are ex-

pressed as a factor of safety or margin of safety.

Determining a minimum

safety factor requires consideration of the statistical variations inherent in the experimental determination of material property data, the loads encountered by the motor (e.g., vibration, acceleration, pressurization, etc.), the physical environment of the mot6r (e.g,

1.7

aging c6nditions,

/

II

humidity, temperature, etc.), and the inaccuracies inherent in the analysis methods or artificially introduced through simplifying ascumptions.

The

determined through proper consideration of these factors

margin of safei

is an intc•iation 6f the overall system reliability.

If these factors were

preýiseiy known, there would be no real requirement for a margin of safety greater than zero.

InasmUch as this is not the case, and quite often

assumptions or approximations must be made regardilg specific informatibrn which is unavailable,.Arbitrary restrictions are placed on an acceptable miimum margin of 'safety.' These restrictions reflect an ignorance factor associated with the structural analysis,-the loading environment, propellant behavior and filure criteria as well as the physical environment and mission requirements. 1.8 Material Characterization MethOds 'The material characterization-of highly filled solid propellants consttutes one of the' major problems. to be resolved before proper structural integrity analyses co be made. There exist at present several, essentially standard, test specimen geometries for the above purpose.

These are generically

. Uniaxial specimens of several ,varieties -

Biaxial.strip specimens

.

Torsion, single and doublý-lap shear

. Triaxial (poker-chip) s pimenis Diametral specimens.

. The

pl

o. ,,l -of

preparation are well covered In the ICRPG

Sol.id Propellant Mechanical Behavior Manual, CPIA Publication No. 21, I.I

1.8

6(

.t

I-

.

September 1963,

including

its

various addttlons and revisions.•

This sample

preparation aspect is, therefore, not discussed herein. In addition to methods and specifications for sample preparation, the COIA Manual presents specific test'procedures to be followed when using the above specimens. chapter.

This area occupies th!e discussions of this

The publ'shed procedures are based uponi the-use of a linear visco-

elastic constitutive equation' and the MoWeland-Lee reduced time integral, using ah experimentally determined time-temp~rature shift factor.' The deficiencies of the test procedures center primarily upon.the validity of the above assumption•. l.q

Linear Visboelastfcl. This chapter summarizes the equations of-linear viscoelasticity

and also contains a.•discussion on the eigenvalue interpretation of linear viscoelasticity,'which provides a rapid'means of conducting "pseudo"' nonlinear analysts. .

,,

1.10 Themovlscoelasticity

The analysis associated with problems in thermoviscoelasticity, ranges. from the simplicity of linear viscoelastic-analysis t

complexity of,

nonlinear viscoelasticity. The reason for this wide range of complexit

lies in the physeca•

assumptions relative to-the effect of temperature upon the materi4l be,;. •

havior.. A discussion of these assumptions and present methods of thermo "Vscoelastic analysis are presented in this chapter.1.11

Nonlinear Viscoelasticity The ability to predict an-lytically the mechanical response of a

structure requtres as a prerequisite the characterization or mathemati-,

1.9

-

"

.

the mechanica)" pesponse of.each of the materials'in

cal dcriptiof. Sthe struct

e.re

Theehematheati'cal descriptors of the'constituent ma-

terial .46tse,-or consfitutiie equations asrthey areocalled, together 'With a kqi'owldgd of the applied

rrace oads and displacements and the

field equations of engineering. nI anics, comprise a system of equations whose •oiuticn yields the•,&tte of stress ,ndstratn for every point within-' the body.

To predict-'the success or faidure of A grain design req;ires

corpat'inr The. calUlated -stress or strain states within the body to some failure

tehion. One therefore finds that qn aplysis of a •structure is

only as good as the constitutive equations defining material response of little consequence if.e and also that. a&failure analysis is'

-

.

(-

j"

'

predict-

ed state of-stress is-largely'in error., Also, the determinatioi: of geaeral, failure criteriýfor thret-dimefiional stortes of stressgenerally requirses the calculation of the stress state jn l multiaxial loading'conditions.

rboratory samples subjected to

Thu s,tthe determination of appropriate fail-

ure criteria ir.also dependent on the constitutive equations defining material resýponse.

'

While the sequence'coNstituti ve equation, loads~definitiong strUctural analysis and failure.definition are obviotuFTy totally interrelated, and the final usable.jIswer t6 4 performance prediction is equally dependenft upon the accuracy of each of the aboye elements in a design, the discussion of this chapter 'iscoticerned mainly with the development of acceptable constitutive equations,,.. 1.12, Appendicet

-

A number o{ appendices ore included -to sUpplement the text materAal, Appendix A summariles the equations of linear elasticity.

1.*T0 -

•

1 .I0.

If

'

"

Appendix B presents a further discuss-ion of finite element computer programs fromn the user's point of-vieew." -Atypical 'computer program is 'listed along with descrip~tions of repaitred input and output data

Three

sample problemý,ar& solved to illustra~te the use of finite element computer ptograms. Appendix C contains parametric design curves for prelimin~ary design ai~alysis of cylindrical grains with vdrious end conditions and .subjected to thermal, pressureand'acceleration loads, Nm

-Appeddix 0 presents phiotoelastic test, dpta for evaluating star' -ialley stress/strain concentration factors. Appendix E prdsents a sample motor struictuiral- integrity~ anajygis to i'llustrate the design and analysis procedure for the novice *designe~r/analyst. elmn gainedcompute through~prsetsa •rterdipsio Appndx compendium of motoroffiitexperience a Appendix'F presents

out thp soli'd~prop~liant industry, during the~past decade.

Motor failures

-and subsequent corrective action talk n are 1discussed in the hope that such information migit~benefit the-entire industry in preventing similar typejfailures'in the futurg.' Inasmuch as the greatest effort has been *made

to avoid c6mpromtisng tae-proprietorsfiip of the-vartous oompanies Qr~causing'any embarrassment to'any cbmpany or presenting:materl~al *of a classified nature, these discussions' take on a rather general form -In which the specific details relating to motor progr~ams, mission Objectives and propellant type are for the most part opiitted. It is still felt, how.ever, thAt 'this miaterial will benefit thp new engineer entering the solid propellant industry.

ST.11.

II. LOADING AND ENVIRONMENTAL CONDITIONS

. 2.1

INTRODUCTION' The'loads encbuntered by af'solid rocket motor are nonially classified

as two types: sp*.ified loads and induced or derived loads. •ecified "loads arn fixed by mission'requirements demanded by the prime contractor ..or sponsori ng government agency in 8FP or nitor specification documents. These loWds are typically the.operational temperature environment, acceleration, vibration, shock, trarosporotation and handling loads, and 'the physical e~vironr*nt (e.g;, aging 'condi'ions, humidity, etc.).

Induced'

loads arise from'a particular'-selectfon of the propellant, proe'ssing. techniques and grain confIguration satisfying the mission obj the 1mot6r.

Induced loads are"typically cure shrinkageqp'r

tives of ure, flight

and certain coimbn6d ,oads. The_'origln and severity of the•k in this chapter.

loads and e"vironments are ditcussed

Sugges-tions for minimizing adverse effechs of these Tbads

through variations-In material ,properties or design configurationsare given-' A section dealing with manufacturing andprocessing considerations is also inc,luded. 2,2

SPECIFIED LOADS 2.2.1

.

THEIM4A1 LMOS.

The most seypre temperature loading isomost often. l cycling.

teUPverature

The critical areas of analysis are typica-lly the inrir bore

in internally R.c'forated grains and the case-grain termination points (i.e., gr6in ends).

.

,

Thermal stresses-and strains arise because of the difference between the coefficient of thermal expansion of the propellant and the motor case. 2.1

'A

The' coefficient of thermal exjansion of propellants, liners, and insulation materials is typically an order of magnitude larger than that of motor case materials.

Thus, upon cooling to temperatures lower than the

motor cure temperature, thermal stresses and straIns are induced in a propellant grain due to.the restrained shrinkage of the propellant, liner and insulation buffer materials.

The difference in linear coefficients

of expansion for composite propellants with a steel or fiberglas typically-5 X 104 X

-5 (OF). .

(OF)I;

case is

with an aluminumcase this difference is about

This difference typically ranges between 8 X 10-5 and

10-4 (OF)-l for double-base propellants and a steel or fiberglas case. the magnitude of thermally induced loads depends upon the propellant and grain design selected to satisfy the motor requirements, inasmuch as this

-. 'sei1tiondetermines the propellant cure shrinjage which is equivalent to ,a,-pre•Ibed temperature-loading.

Cure shrinkage stresses and strains

are discussed further in §2.3.1.

The operational temperature range is

normally specified i'n an RFP or motor specification document.. In performing thermal stress ane strain analyses, the calculations ma•-be referred to the propellant cure temperature and cure shrinkage, stresses ard strains superposed, or the calculations may be referred to

/

)he zero stress/strain temperature of the propellant, T1 . Thiis tdmperature, is defined to be the temperature'at which thermally induced stresses and strains vanish.

As noted in §2.3.1,

because of propellant shrinkage

during cure, the'stress free temperature will usually be higher than the cure temperature.

The cbret process for convention'I>tast double-bas'e

prQpeilants, described in §2.3.1,

tsually results in a stress free state

"-at ambient temperatures, however..

I

4.2

/

p

I

The temperature T, may be conveniently determined by several techniques.

(

Oro. method is to subtract the eqtevale-nt ietierature

decrease associated with the cure-shrinkage from the propellant cure temperature, T= T-

=T

1-p

C ap

V0C t÷

(a.

+

3

where Tc is the propellant cure temperature, (p if the propellant linear coefficient of thermal expansion and a is the net volumetriccure shrinkage.

U

The net volumetric shrinkage

of polybutadiene propellants is ty.pically

.0.2% and that of slurry cast double-base propellants 0.5%.

The shrink'age

of conventional cast double-Pase propell\ants is substantially lower. Alternately, the zero stress-/strain temperature may be determined from analog or subscale motor tests.

In these tests the temperature of the

cured motoris slowly raised above its cure tenperature.and measurements of the internal' configurations versus temperature are'recorded.

The

t-mperature- at which the internal geometry of the motor coincides with the original,.mandrel configuration is thedn defined to be the stress'free temperature.

r

Measuremennts made in this manner indicate that T, is typically

150Fhighcr than the propellant cure temperature for polybutadien'e prqpellants and 22 0 F hiqhIerfor slurry cast double-base propellants [1]. Thermally induced stresses and strains may be minimized by reduction of'the zero stress-strain temperature or through design optimization. Minimization of the zero stress-strain temperature is accomplished by -

(

reducing the propellant cure temperature or introducing a complicated

cure

ýcTe as discussed in §2.3.1.

Design optimization procedures are

\currently based on engineering intuition of the analyst gaihed through pasý

Smotoi

experience and on parameterized comiputer analyses. 2.3

11 At grain termination points, bond stresses are reduced by introdizing intentionally relejased areas called flazpp or boots,

=-,

~stress-rplief grooves, fillets, wedges, 'etc.

•I

fcomposed

or by the use of

Stress relief flaps are.

of materials which-havestrength and elongation characteristics

~greater than the Propellant throughout the anticipated temperature range

""

the motor.

•Iof

They are also selec ted to have insulative capabilities/

consistent withth

loca-otitc

tiunts

asbestos filled buna-n ruLbers typically fulfill these requirements. Inner bore hoop siresses and strains are minimized through design considerations which will become evident in subsequent discussions of grain analys>s methods in Chapters 3 and I/

effects have become important in recent years Aerodymami cheating 1

Swith the development of supersonic aircraft and sophisticated air-launched attack missiles.

The structural problem which results from aerodynamic

heatingarises when a solid rocket motor, which has been in storage at a low temperatulre, or externally attached to a high-flying aircraft, for a length of time sufficient to allow the major portion of the propelant grain to reach equilibrium, is then subjected to the thermal barrier which results froma supersonic

dashiof the aircfaft.

The temperature of the

boundary layer and that of the missile skin is raised appreciably because of.the dissipation of energy generated in the boundary layer at high speeds and the shearing work done on the flu'id by the 'viscous stresses within the'boundary layer at high velocittes.

The result, is

that

temperature

gradients will exist within the Motor case and the propellant. Since the major pOrtion of the propellant grain doesn't have time to react to the temperature gradient caused by aetrodynamic heating, bond

2.4

stresses in addition to those already present due to thermal cooling are induced. As a general rule, the magnitude of this additional stress is/ small, on the order of a few p'i. This component of thermally induced stress normally acts for only a short period of time before expansion of the Insulation or liner buffer materials (whAch expand an order of magnicase) Stude-more-t&7l induces a compressive stress component which results in an overall net reduction of bond stresses.

One sees, then,

that the structural problem associated with aerodynamic-heating is a decline in the

strdhgtb properties of the propellarit-liner-insulation-

case bond system due to the rapidly rising temperature field.

This

problem reaches catastrophic proportions when the teiperature rite at the propellant-case interface is such that the bond stress capabilities at this interface decrease ,oe rapidly than the bond stresses decrease. This behavior is described scheitacaily in Detail I below.

Propellant/li nen/case bond stress capabilities Aero-heating bond stress Thermal cooling

S--k, U

Then=_

i

col n5

,

bond stress

u*-IStart

of aero-heating

S4 .Failure iDetail I.

time

SCHEMATIC REPRE-SENTATION OF AEPRDYNAMIC HEATING EFFECTS 2.5

_-.=A-

Y Upon aerodynamic heating, the net bond stresses initially increase,

in i-•ignitude-s1 ightly,- afn then- decre.mse-with time.---The'propellant-.-tasE interfacial bond 3tress capabilities do-crease mopnctnbi-ily with time during aero-heating.

If at some time during this process, the

interfacial bond stress capabilities decay t6,the point wherethey,,are less than the induced bond stresses,

Ahen

failure will ensue.

A normal procedure for minimizing adverse effects of aaro-heating is to insulate the external surface of the motor case with cork or some other good light-weight insulation material which has a heat transfer coefficient comparable to that of t.'e propellant. •This insulation inhibits the magnitude of the temperature differential between the case and the propellant, and decreases the temperature flux.

Another method for de-

creasing adverse effects, which is currently being researched by a number ,-

of companies throughout the country, involves the use of high temperature curing adhesives, elastomiF- and propellants. ml S

The use of these materials

izes degradation throu h imporvement of the high temperature physical

Sprbpert~ies of the compo-nent nterial.

These materials have not been

developed sufficilntly to be c6psidered state-of-the-art as yet, however. 2.2.2

ACCELERATION LOADS

This section presents-"rtef-dtscussion of two classes of acceleration loading cnditions: (i) 0(i)

storage slump launch and manuvering

Transportation of a solid rocket motor normaliy induces acceleration "loads of the order of + 3 g's or less.

Because of the load reversal during

transportation, this type of acceleration loading Is better described under dynamic loadings and will be considered in a subsequent subsection.

2.6

C)

AXIAL ACCELFRATION Axial;• or longitudinal accel ration loads occur under vertical storage, transportati'

and launch conditions.

During vertical storage of a solid

rocket nmtor, tOe propellant grain is subjected 'to a one gravity body force.

Normally a

I. g

Icad is not sufficient to produce a critical shear

stress along.the case-propellant bond interface.

An exception to this occurs.

-in the case of large solid rocketrmotors in which inadequate-orain-terminations are provided, since bon". shear stresses are proportional to the motor dtameter. to occur.

In this. situation it is not unusual for grainend unbonding

The major problem associated with vertical storage, however, is

the occurence of large propellant deformations, or slump.

Slump can be a

critical design factor for storage above ambient temperatures (i.e., 70'F). At lower temperatures, the stiffness of the propellant usually lessens the magnitude of

fdrmations.

Gral.n deformations under axial storage conditions are prdporti

al

to the square of the motor diameter; thusý slump of a propellant grain is particularly pronounced in the case of long term vertical storage of relatively large- solid motors,-as for example,'in a'silo.

This condition may

,also become critical in smaller motors when propellant slump provides gas flow restrictions not accou. ;ed for i6 the ballistic design of the motor. An example of this is a motor with a submerged nozzle or radial slots. Slump characteristic' of a solid propellant are controlled by the creep properties of.the propellarit.

Thus, .increasing propellant stiffness

will-reduce the magnitude of slump.defdnnat onsý

however, the-adverse

-

effects associated with increasing the propellant stiffness, in particular,the reduction in propellant elongation capabilities, usually overshadow any benefi $

itms, axial slump is most often-daTtwith- W-des 2.7

Ii-

-

I procedures which allow for large deformations.

Stress relief flaps, or

boots, provided at grain terminations to prevent grain end unbonding during temperature cycling, are usually also adequate for preventing grain end unbonding during axial storage. During launch of a solid rocket motor, high shear stresses are induced at the case propellant bond interface.

These stresses are a maxim='

at the forward end grain teri~inations, and are directly proportional to the acceleratiqn magnitude and morur diameter. The maxiumni shear stresses normally occur near the time of maximum acceleration, rather than immediately, upon launch.

Axial acceleration stresses are more spvere in an unprezsurized

motor, such as in a second stage vehicle, than one which is internally pressurized.

PreSsurization induces a hydrostatic compressive field which

tends to enhance propellant strength capabilities and lessen somewhat the effects of body forces.

High temperature acceleration is more severe than

low temperature acceleration because of reduced propellant case bond strength capabilities. -'-

Axial launch and infl-ight accelerations are the more important

acceleration loads for high acceleration motors. Axial acceleration stresses may be minimized by maximiziong the extent of bonded area, in particular grain end support,-and through design of grain terminations in such a manner as to minimize stress concentrations at grain case singularities. TRANSVERSE ACCELERATION Transverse, or lteral accelerations occur under horizontal storage, transportation, and maneuvering during free flight. at moderate temperatures, propellant deformation.

Horizontal storage

like vertical storage, can produce significant In addition to providing gas flow constrictions--

/,

-.........

•

from star poiits or longitudinal slots closing together, there is 'the more- severe probl sm of case'dvality in flight

4eight motors.

•

This latter

pro bils corrected by providing stiffening rings to prevent the case All of-the above effects are mini-

from becoming oval during storage.

mized, or at least compensatedifor, by periodically rotating the motor ninety degrees.

tn the case'of long, thin st'arpoints, start~p deflections

are often limited y providing consumable styrofoam- supports tor the " individual starpoints.

i' )

iiflight maneuvering normaily does not induce significant acceleration loads.

For very high accelerition motors bending of the motor case due',

to sharp maneuvers may occasionally result in graii cracking -or grain epd unbonding.

Normal design practice, for other Toading'conditions,

-howeyer,

generally result in adequate structural capabilities during free flight

.

maneuvering. 2.2.3

DYNAMIC LOADS

..

VIBRATION Vibration'of s6lid rocket motors is generally recogntied as a ootential structural integrity problem for applications in which severe or sustalned vibration enviroments arc incoi damping..

d because of significanrtpropellant

.

Vibration effects are most severe during ground and •inflight

transportation.

Free flight vibration is not normally damaging •o the

propellant grain because of the relatively short burn times of solid rocket motors.

Resonant burhing, however, may lead to significant

structural problems, particularly in the case of thin unsupported grain The high iluminum content webs (e.g., thin starpoints, wagon-wheels, etc.). of-most modern solid propellants tends to reduce -omewhat the problem of

"2.9

t

combustion instability.

For certain. ombustionconditions, qas flow

conditions and iiiternal graio.ponfigu:•atfons,

acoustic instability may

result in.pressure waves of,sufficient magnitude to cause propellant fracture and subsequent catastrophic mutor failure. Jr critical problem associated with vibration is'.that of generating local temiperature increases sufficient to cause either spontaneous ignition'of the propellant

severe

ur

- mechanical degradaion..

TMe,

rate of'anargy di'ssipation int&heat for a linear v/;sco&stic material is pro

rtional to the frequency of.vibration,,material stiffness and

the square of the magnitude of the deformation. problem is typicall'y post

Thus,'the vibrtjpn

severe for conventional motors under th

"frequency, first resonant~mode, at high temperat res.

low

In this siation

the propellant stiffness is a minimum for Vibration.conditions,.4nd the motion of, free-surfaces (e.g., a starpointl maximum energy dissipition into heat., cated by the chatacteristically lant mechanical properties.

is greateýtYresulting in

The probylem is further compl'i-

-strong temperaturedependence of propel-

This temperature dependence makes the

energy dissipation, very ,sensitive to temperature variations so that a -continuing periodic forced motioh gives'use to substantial temperature fincreases. At low temperat-ur

A

the propellant behavesi more nearly elastictlly

so that motion and'energy " t\dissipztion i psplln are ethat~ reduced.- Also,' tbe heat generated is more readily

nducted away from sources of heat,:generaztion

at low temperatures than at hightemperatures.

-

Another probleu occurs for sustained Vibration of very high fraction motors.

mass

During sustained vibeation, the' temperature increase 2

S2.'•0

-

(.

(

/

Q1

associated with energy dissipatidn causes the propellant grain to expand 2

•

to f/i

a'ailable fjeq volume.

If insuffizienft free volume is provided,

propellant--propellant or propellant-case

contact may be' made.

This,"

contact may result in.local temperature increases an order o• magnitude; or mord, higher than average temperature increases; or it

wy'cau'se

jtructural jfailure due' to degraded propellant mechanical proherties.. In the event that temperature increases are not sufficient to caase autoignitio-,

there, st-ill exi~sts the pqssibility of inducing sufficient

degradation of material properties, as a result.of. cyclic loading, to, cause -.hemomechanical breakdown of thl propellant or propellant-liner-casebond. Eamples of prol proellant depolymerizati d extended-cycl'ic loadtng have-been presented by Tormey and Britton [2j, Thd vibrational capabilities of solid rocket mofprs a~e currently determined from full scale motbr tests.

The specific nature of the test

.'envjronment isdictated by t~ie applicable m'i].itary specification.

'A

typic-I.specification will require vibration testin5 in each of three "mutually or thgonal directions (transverse , vertfcal,

7

and longitudinal)

at input amplitudes of 0.100 iich double amplitude disp

Cement or 5 g's

peak ac.celeratio6 intensity "t frequency rangs.between 2 and*500 cps ýnd 5,to 2000 cps. In.addition, a certain porti-on of the test is.carried under sinusoidal oscillation while 'Continuously varyihg the frequency

Sout

or under broad band randomýexcitation.'

Most military specifications now-

require 30 minutes-dwell time at

Ahe resonant frequencies bietweeh

Z and 500 cps. oevels

ach

Table I presents a comparison of some measuredvib-

.

Aion

aistic artecvifation specifications.

In the past, lnumerou~s inquiries,have been fmad•c regarding the question of how realistic are the vibration specificatiJans.

S•L,

•

•2.11.

Th-ese inq~irles

"

7

-

TABLE

i

I

4

I

CMPARISOM OF SOMEMEASURED VIPATIOM LEVELS WITH VIBRATION SPECIFICATION

.

IaDVIBRATION LEvEL

or 0PRATIO.KMU3 1.

[3

TQA.NSPORTATION AftO HAX ING

Sbipoent by Coi-on Larrier

/A.

/to k m 0.35g pea.kt••x 1.7 1. giptax aPart

"

c. 2.

*

cross-ccuntr (I t1I0 mph)

S 3.

at 10 cp-.ý)

1;-3 g(peak) less. tk'r cps regionjokiurred 1%Iof tir-

;

RaII •oad' a.

-

"•€

"/• b.

/

"

over the road (50 to 70 mph)' -

-

1.1 to S.0 g(l-ak) for 4Z mn 15 mnin. in each axis (sweep rate for each 5-500-S cps cycle)

0•

3.7 g(p.ak) max i; 240-350

Tructor-Trailer / ,

Sinusoidal resonance search: 500 cp! resonance dwell: Part 2. Sinusoidal 1.3 to 5.0 2(peak) for 30 pin. at each resonance (2 to 500 cp. ) 3 Sinusoidal updor sweep f equency: Part I.

.

. Truck od q•0-25 mph) e roads a.b rough p

switoJing shack e (tmnsient) .

.

0.8 g(pzak)(max at 1000'cps) 2.2 g(peok)(max value noted 93% of vibration-was less than 0.75g) 2,0 g(peak)(predominant 're-qeAcies in 2.5 to 7.5 and 50 to 62cps regions) 35 g(ptea)

-

with 8 mph impact

-

-I

• 4:

V-.

Air;taft,, propllIer- . dri~en' with reoiprocating' or turbo engine

N

2.8 9trm)(Max at B80 cps)-

Alrcraft, jet engiine

"6. Heli apter

S

7.

,

.B

-7.0 g(,--s)(Wax At 4u0 cps)

t~o g(•eak)(3..0 to 35 cps)

Sips

O.Cg( ak)Lmax's al 1.5 '

a.

c41M seas

b.

rough seas.

c.

iergereny manurvers

15 cps) O- 9(peak)dax's at 0.1 and Sand

1

Minuteawn Stze !U/Boing ranporir

"

.

3.0 g(peak)(m•a's at 2.5, 12 45 c-s)"

*

/ -3.

w 0.6 9(pehI10 tiatek

pak

noted ini he 0.25 to 0.5 g range than in the 0.i to 0.75 range)

-

l.GOg{pe&)--- lonlt.di nal 1.5 g(peak)..-,Lteral 3.0 g (peak--vertical'

2. Mike I (storlage-tai1 uncher)

II.

, ,

Shi Peont by Special Transporters

1,

-

,

" " 5.0 g( ms)(max atnO0 cý&ps). / 5.0 g(peak)(l to 250 cps)

Titan III Segrent/tractor Trailer Transporter

-'

"

1.3 g(peak'--fore-aft 1.0 g9peak)--lateral and vertical

A'

AIR LAUftE4i. ROCKETSs

-,A.

Captive FlightI (tactical aircraft)

.S

4.3 q(rmi--lonqitrudinal '9.3 g(iwsl--Tateral 11,8 g(rms)--vertical cover; 20 to 2000 ops wit.h a .axium In the SO to 1000 cps rgion)

1. carried in bay (doors ooen) (q - 1480 psf)

"-tspectrum 2.

Part 1. Sinusoidal Resonancejearch; Sto 0 cps or S to 2000 cpv. Par~t 2.' Sinusoidal ;Zesonece Dwell' 5 or 10 g(oeak) for 30 minute / at ech resonanr. , 'aPrt 3. "Sinusoidal Swee?ýFrequercy:-/. 6r 1(0 g(pcai) 4for 2 hoUrs in each axis (sweep rate -'20 inin. ..- per 5-2000-5 cps cycle) .

4.7 g( M ) for a0chI.; at -. f

Externally Carried _33,000

6.3 to 9.0 g(rmsJ for 600 koots it 5,000 "ftý altit de 3.0 g(peak)( -to 180 cps 8.

Powered,

Free-Flight

.

-

0.86 to 2.8

" Part 1.

(M)

.

Part 2.

-2.2

*

,

Siriusoidil Sweep Frequency.. Sto 20 q(peak) for 2>paie Sin each axis (sweep r cps cycle). min. for eacl 5.2000-5 2 Racdoc: 0.04 g lcPS P7.4 g-rms) . "f-30 min. 4n each a'Yis. (100 to ,7•0 O cps with 6 d0/octave rill b

each end to 50 and 2000 cps

4,

EUTABLE

1. (ccntinuea) COKPAfiIS0~

OF SGR t4ZASURED

MODE OF OPERATION Il1.

YEL ,,,Tý' lu Y10-FATpw Syr TFICAT!ONS (3]

MEASURED VIBRATION LEVELS

GROUND LAUNCHED RVCkETS A.

Lauichetd frog Stationary Site

B.

Launched from Mobile launcher 1.

SPECIFIED-ENVIROMENT Part 1.

Sinusoidal with Sweep frequency: 5 to-50 g(peak) for 39•min. ih eith axix (sweep rate. 20 min. each 5- 1.Y-'5 cps cycle)

Parlt 2.

Random:

4.t g(nras)(max PSD noted , 2 0,007 g /pps at j2c-Ocps)

Captive Transportation, Tracked Vehicle (30 to

1.6S g(peak) at 76 cps-longltudinal 1.04 g(peak) at 80 cps-

Is mph)

lateral

.02 g /cps (5.30 g-ras)

to15'/c~ 5.-res, s (46.3 for 30 min. in each -axis (W0D to 1000 cps with 6 db/octsve roll off each *nd to SO and 2000 cps respectU~ely)

1.95 g(peak) at 80 cptvertical 2.75 g(ras)(PS0 spectrtm peaks

at 10 cps) IVY.SHIP LAUHCHED RDOCYES A., aptive Transportation Ships)

"(Tact'cal 1.

5e.rye 3(peak)ý17 to 170 cps)

Z. PT Boat 3. Submarne

6.0 9(peeak)(10 to~ 140 c-ps)

8, "owerad S.

2.0 Q(pegk)(15 to 160 ;ps) •""-

Free Fltigit

8.3 3(rm)(max. PSI) noted was 0 .03 /lcps in-700 t'•~•

2.1

-

S9

9.'9

. ,.

.Il

2.13

C

resulted in a recent study being conducted to compare available motor

-

data wi~th current vibration ipecifications [8].. The major objection to current specifikations is that the resonace dwell requirements are an unrealistic facsnýnile of the actual motor vibration environment.

Wagner

[31 recommended that resonant frequency tests be.abandoned and repiaced by random sweep vibration Lests since these are more representative'of the actual environment of a ýolid rocket motor. It is easily demot.strated that the maximum internal heat generation in a solid propellant grain' occurs during vibration at resonance; and also that as th: temperatute of the-propellant increases, the resonant freqsency of'the motor initially decreases.

Most current specifizations

require that during resonance dwell tests the input motor vibration frequency be varied to follow changes in the resonant frequency of the motor wh'tch results from the temperature increasing.

This environment is more sever)e

than the vibration environment a solid rocket motor actually encounters/ durinq its l.ifetime. Questions have also been raised regarding the requirement that input acceleration intensities (usually +.5 g's) be monitored at fixed motor/ shaker

attachmdnt points.

Amplifi

tion factors frequently. result in

t+10 g's acceleration occurring at ntin6des,

This loading situation is ih-

tended to simrulate aircraft vibration&l loading of an externally mounted rocket motor.

In practice, however, the vibrational loading is due mainly to

air and wind buffeting loads.' The result is that the motor, in fact,; it a nain'source of vibration loading for the aircraft. -ated in present vibration tests.

This condition is not simu-

/

P

Sore improvement in pte sent vibratiQn test',ng and specifications are . expected to result from the captfve flight tests of instfumented bomb dummy units in a program sponsored by the Air Force-Rocket Propulsion L,aboratory, and from Condor Motors, presently instrumented and being monitored during captive flight by RockeBtdyne.

2.14 -/

/I

SHOCK Shock loads normally occ> when a solid rocket motor is dropped or subjected to d'seve're mechanical jolt during handling or transportation. The various ,blast waves of .nuclear explosions are also an important source of shock.loads.

The time averaged magnitude

waves from nuclear explosions,

of these loads, excluding shock

is nqrmally in the range of one to five g's.

Peak local intensities range between 25 and 75 g's; Peak shock load intensities.act for a very short time during which propellant normally behaves essentialy as an elastic material with a glassy modulus. undamaged.

As a consequence,

The major damage that Dccurs.is usually bending or warping

flight weight rocket motor cases. *

the .propellant grain itself is itsually.,

lant grain as a-result of sýh

Possible damage inflicted on a propel-

loads may be either propellant-liner-case

unbonding, or the developmentoofoa sh6ck-wave with sufficient epergy to causp detonation. Case-grain unbonding is most likely to occur at intermeddate,to low teMperatures either in the immediate area of the-point of application of the shock load, or at ctse-grain terminations.

At high temperatures the

propellant is more complianf.and can better withstand large deformationS.

"The compliant, high-strength, elastomeric end-rilease flaps employed at critical grain-terminations interfaces to relieve thermal stresses also serve .to reduce the probability of peel failure at these locations during shock. loading of a Aktor. tii! not

Detonation of propellants. is a complicated process which is completely vnderstood.

Some of the possible Initiation mechanisms are

adbabatic heating oA compare

gases~within void areas,

-

- _..2:15

friction between

.

...

-

I

the solid particles in the propellant, fracture of energetic solid oxidizer particles and viscous heating oftheý propellant binder' [4-10]. There are indications that theenerny release due to viscous damping is not suflicient to initiate detonation of propellants [10,11]; however, this qtestion has nnt yet been completely resolved. In the 4sence of quantitative analyses, laboratory and full scale motor tests are routinelyiconduct*d to determine the shock sensitivity of peopellants and loaded motors.

Laboratory tests generally consist of

(

impact sensitivity tests In which physical impact is caused by a falling weight such as in the ERL test; or shock sensitivity tests, in which the initiating rhock is gener tsd by a donor explosive, ascn the gap test; by a projectile.

or by high velocity impact,

References5, 6 and 8ts

contain further discussio s of laboratory methods of determining and evaluating shock sensitiv ty of propellants.

An analytical model which

qualitatively relates the experiental conditions of shock pressure, shock impulse and accepter dia

ters in regard to initiatJon of detonation has

been discussed by Pratt .7]. TRANSPORTATION AND HANDý ING Some of the loads ýrising from transportation .iid handling of rocket motors haveI been discu sed briefly in the previous sections. tude and vibrational

4

The magni-

equency of transportation loaaý vary somewhat

depending upon 'the ca~rier._ Some measured vibration levels for transportations are shown in Table I.

Specifications--,typically require 30

minute resonance dwell -tests at each resonance between 2 and 500 cps at 'ak

input accelerations between 1.3 and 5 g's, and sinusoidal s~eep tests

from 5 to 500 cps at 1.3 to 5 g's peak input acceleration. 2.16

1

During handling of a solid rocket niotor datnage is most likely to' be inflicted upon the motor case. -This damage may consist of bending .of the motor case or nozzle, dents in flight weight cases, bending or dentinig attachment or handling lugs, etc.

Grain unbonding'in the area

of the dent-is alto-likely to occur. tuý'

scale motor tests usually consist of- drpping motors, unpackaqed

aVd packaged in protective shipping containers, from various heights in various attitudes

onto concrete or steel slabs.

Visual inspectioi•

and

non-destructive tests, such as X-ray inspection and ultrasonic inspection of critical bondiregions are used todetermine the severity of the damage. Some specifica

ons require simply that:theimotor not detonate,whereas

others require that it operate satisfactorily aftpr drop tests. tests--for shock sensitivity have been mentioned above.

C6rmnon

Usually dropping

a flight weight motor which i-s not encasOedin a protective container will result in irreparable demage t•_ the motor case. 2.2.4- SPECIAL -OADS

AND .ENV IRON

S

Occasionally a solid rocket motor wi fully in a special enviro.nment.

be required to operate success-

T,;,e spin/ernvirorment has -probably had the

greatest attention, although little has beere

publisl)o. in the open litera-

---- ture on ithe structur6l behavior of a spinning ,.ropelant grain,

It .is

possible that the spin rate may induce appreciab&e-Pnertial stresses and deformation within the grain.

Star-shaped gr~in geometries are--usually

a-voided, since the deformations of starpoints wili, norma

y~e excessive.

The spin env.ironmentfiirequetly -le.ds to eratic burning characteristics; the most noteworthy of Which is errosive burning.

Strong coupling between

the burning characteristics and structural behavior is to be expected. -

*

2.17

Other loads and'environments, which will not be discussed in detail in this handbook, rezult from requireme-iits for heat sterilizable propellants for sp;ce application, resistance to radiation and special environments pec-. "ar to nuclear explosives. Generally,

particular propellants are

developed and tailored'specifically to survive in these enviromnents.. "

PROPELLANT AND DESIGN INDUCED LOADS

!!2.3

The induced loads, as mentioned in Sec. '2.1,

arise from a .particular

selection of the propellant, processing techniques and grain configuration satisfying the notor mission objec~tives.

These loads typically consist of

cure shrinkage, pressure, flight and. certoin combined loads. 2.3,1

CURE SHRINMAGE

Cure'shrinkage stresses and strains are induce41during propellant cure w,;en the propellant is transformed from a highly viscous fluid-like material into. a solid.

tie majority of propellant cure shrinkage takes placq prior

to the initial- point of propellant solidlf~ication (i.e.,,gel); however, ,4,

cure shrinkage strsses devieop only, after gel, in'asmuch as propellants display fiuid-properties prior to gelaton and cannot support substantial propellant gel and shear stresses. The shrinkage that takes place between coiplation of cure is restrairind by the motor case. As a result,^shrinkage or residual s~res'es exist in a sGil rocket motor upon completion of cure. For propellants wMch gel at an ea

stage of-the cure process, the

possibility of developing shrinkage stresses which are greater than the stress capabilities .of the partially cured propellant exist3.

The result

is that a motor may exhibit cracks or unbona regions- mme.diately upon "removal from cure.., These effects are most evidertiin high mass friction, 2.18

I

/g=

/

-g

composite propellanrt motors and slurry cast high energy double base propellants which are cured under conditions leading to indesirahle nonuniform temperature fields. curing

Under certain condi.ions, however, as it shall be seen,

in a nonuniform temperature field leadsto a reduct'ri in cure shrink-

age stresses and strains. The amount of shrinkage that takes place during cure of a solid rocket motor is a consequence of the particular choice of propellant and processing techniques selected to satisfy a given motor s temparature environmental requirements. \

Conventional composite propellants are cured by a polymLrization

-process which is generally accompanied by a volume shrinkage which is proportioal

to the binder volume present.

Because of propellant shrinkage during

cure, the zero stress/strain temperatui• will be higher than the cure temperature.

Composite propellant rocket motors are typically cured between

130 °F-and 145 0 F.

0

The cure'shrinkage that takes place is equivalent to a

150 F tgnperature drop [1].

Thus, the zero strtss/strain temperature, which

isnormally taken to be the reference te,,:perawure for thermal stress and straln calculations, ranges between 145 0 F and'60OF for most conventional composite propellants.

The stress

f;ree temperature for slurry cast

double base propellants is.typically 22°F higher-than the cure temperature

[1]..These propellants are typically cured between 1ll5°*and

40°F.

The cure of conventional cast double-base propellants is basically a mutual diffusion'process between casting powder and casting solvent with no chemical reaction occurring until the final stages 6f the cureprocess [12].

Casting powder granules, consisting primarily ofvnitrocellulose

with metal fuel, oxidizer, ballistic modifiers and stabilizers incorporated to improve motor performance, are loaded 'into an empty motor chamber by a 2.19

p.:

pneumatic conveyor.

The casting solvent which is a high energy liquid

containing nitroglycerin with diluents to increase plasticization arcd reduce sensitivity, is then added under pressure. the liqoid -toflow into interstices

Thic pressure causes

in the powider granules until voids The solvent is absorbed by the

'have been replaced by casting solvevit.

casting powder %hich in turn swells the powder into interstices 'ýonwrly' Mechanical displacements are often

occupied'by the absorbed liquid.

applHed to the propellant by means of

rams

to aid grain consolidation.

After the casting powder and solvent have combined to form a single phase material,the temperature is raised to about 120'F and the propellant Propellant containment by the

,cure completed. f20OF

cure r'mperature.

Thus,

rams

is maintained at the

cure shrinkage in cast double-base

propellants results-essenti'allyfrom bed'

settling and collapse of micro-

voids in the casting powder gz 4nules during ambient-temperature mixingof solvent and powderand is not chemical- in nature.

Some' chlemicSi

shrinkage does occur d-&ring completion of cure at the elevated, cure temperature; however, because of the partial cure at ambient temperature a decreated net shrinkage relative to composite propellants results. because of the compressive loading on the propellant grain

Furthermore,

during the entire cure process,

an essentialll

stress free condition

.e6xists in the grain when it is returned to a~bient tenrature.

"The volumetric cure shrinkage of most composite polybutadiene propellants can be adequately described by the relation

AV

-

{1

-

(2.2)

exp(-1t)}

V0 where a and ý are experimentally determined constants.

The constant a

represents the net voluietric shrinkage, and the product ac represents' 2.20

the initial rate of shrinkage. ,The net volumetric shr~nkage a is usually not explicitly temperature dependent, hgwever, the shrinkage rate aC is ttroagly temperature dependent since polymerization, in general, .thermally

is a

activated process which'is usually adequately 4escribed by

-first order kinetic theory.

Procedures for performing cure shrinkage

tests are not readily available in the 'pen

literature, however, a con•on

technique makes use of a mercury dilatometer submerged-in a constant tempera-' -j

•ture

bath.

The'rise of the-mercury-column in a capilliay is mootored

using'a-cathetIometer.

More direct techniques for measuring th//e stress free

temperature were discpssed in §2.2.f., Determination of cure shrinkage stresses in a solid rocket motor represents a diffucult'task which is not easily carried out.. Thacher [1]

hIs

analyzed recently the d6uble-base manufacturing process, and Cost [13] intro" ducedan analytical approach wnich may prove t6obe worthwhile for determining .shrinkage stresses in composite propellant;. ouring polymers 'under and

formulated

isothermal

the basic

differential operators.-

as

functi'ons

cations

of space,

of 'how

material

and steady-state

governing

of

The

time,

In'this report Cost treated

differential

conditions,

equations

in terms

material "properties

temperature and

behav,

thermal

may

were treated

degree of cure.

be related 'to

molecular

Indiparaa-

meters were given,.based on Bueche's theory of molecdiar viscoelastiyity. Such a theory may serve as a guideline-for future developments in this area; however, lacking,experimental verification, these deveiopments should be 6onsidered tc be of a preliminary nature,

Development of analysis

techniques for cure shrinkage stresses will likely be necessary in the future with the.evolution of more complicated and more highly constrained -

grain configurations.

f2.?

./),

Several techniques have been successfully used forminimizing cure shrinkage stresses and reducing the stress free temperatures.

One method

is to use propellant binders and ingredients whlch undergo minimum shrinkage during polymerization. 'Unfortunately, seldom meet motor perf6drmance requirements.

the resultai" pi'pellants Another effective means of

reducing shrinkage stresses is to cure the propellant at a lower temperaturE for a longer period of time.

Still another alternatIVe is tp step

cure the motor starting et an intermediate,temperatu.e •pd gradually increasing the cure temperature in steps at various stages of the cure process.

This process produces a nonuniorm temperature- field wbich in

Ssome.situations results in a more favorable strength distr'bution during Step.cure cycles are usually determined-empirically by performiing•pro-

ure.

pellant cure shrinkage tests at a number of different temperatures.

.-

A step

cure has been successfully used by Lockheed Propu)s-ion for polysulfide, Dolycarbutene and slurry cast double base propeliay'-s.,

Poube-.base propel-

lants a-e often cured in a nonuniform temperature,field. Auxjili:ry benefits are derived frow botn the stepeure and the longer term low temperatuýE cure.

These methods of propellant curing reduce

the strejs free temperature and thereby reduce the stresses and btrc.ins resulting frum temperature cycling.

Thes4'curing methods .also tend to

produce a more fu1lly cured propeiAant and thul. reduce aging degradation due to propellant post curing. 2.3.2

PRE'SUPIZATI'N LOADS

Pressurization loads arise duriq ignition if g solid r.-opeiiant rocket

o-ocor and, act until motor burnout.

Ignition -ressurization induces

a compressive hydrostatit pressure thIroughout the grain with superimposed tenfsile ; hoop components of stress ar~d. .-train at the imner bore. ..

-

4 .

'

'The

/

mp

pressurization loads imposed on a solid rccket motor are. determined by the propellant'properties

(e.g.', bulk,compressibility, burn rate, pressure

exponent, temperature sensitivity, etc.), the grain configuration and the stiffness of thQ motor case.

The ballistic properties of the propel-

lant deternine the grain configpration and the motor operati'ng pressure. Low modulus case materials such as fiberglas cases typically g'

rise

to more severe pressurization loads because of their lower st;Tfness. .The hoop strain

dt

the Tiner bore and the stresses and strains at

grain terminations are usually the critical design parametersfor pressurization lmadinge particularly'for low temperature firings where the propellant has less elongation capabilities than at high temperatures.

Design

considerations which lead to a reduction of thermally induced stresses and strains -also tend to reduce pressurization stresses and strains. 2.3.3-

FLIGHT AND'COMBIN[D LOADS

Certain combined loads are of nmrp significance in determining the structural integrity of a solid rocket motor than the individual loads applied separately.

The most signific-ant combined loading is normally

that of low temperature firin*g.

A 'iow temperature firing of a solid

rocket %mtor superimposes the stress state resulting from ignition pressurization-upon the existing stress state due t- thermal cooling.

As in

the individu'al lqadings, the critical strain field occurs at the inner bore. The bunG Atresses at grain terminations are also of major importance. Thermal strains are usually higher than pressurization strains with the possible exception of fiberglas case motors in which pressurization strains if 20% are not uncommon.

Thermal strains. are also induced at a -low

loading ra- ,.the motor cooling rate, whereas pressurization strains are 2.23

inducec at a higherloading rate,

Propellant strength and elongatio'i'.. *

capabil ties are higher der a hydrostatic pressu're so that thermal.. cycling will'usually dominate de ign considerations. There is some

,

evidence.which indicates that if a motor has not failed during low temperature cyc~lrg it is not likeiy~to fail upon firing [14]. addi'tional 4

ý1

evidence

that

There is

suggests existing flaws or cracks may_-.

-

enlarge and propagate leading to catastrophic f'ilure during firing [15].

gI

Launch of a solid rocket motor superpose, pressurization,

the stress states due to

thermal cooling or cure shrinkage and axial acceleration.

The addition of launch acceleration- loads in considering low temperature firirgs is usually not required iasmuch as the propel1nt stiffness and strength capabilities are substantially'highier at low temperatures than at high temperatures.

At high temoeratures, on the other hand, 'thermal

stresses and strains dre'inconsequential-and launch acceleration stresses dominate design considerations because of the, reduced propellant strength and stiffness. The above, combined loading situations, and others which may occur, require considerations of cumulative !jamage effects iii assessing motor structural integrity.

2.4 AGING AND HUMIDITY The influence of the physical envirbmnent which a solid propellant rocket motor experiences during its lifetime is e most difficult and significant problem facing the structura\ integrity engineer,

Of the

factors producing adversc effects which serve to reduce the ep-rational service life of a solid rocket motor, the normal aging of propellants and liners, and relative humidity level during storage are geherally 2.24

(J

"-.-

'.

/

I',

accepted as being the mnost critical." M6tors are 61so frequently reluired

to be i•.erv

us to salt spray resulting from shipboard storage and •\

'

"trans.0ortation, a d biological attack -(e.g.,, fiingus qrowth in tropical climates).

These !atter-envi'ronments are not discussed herein, however.

Some of the factors. which 'infTuence aging behavior are described in Table II.

Prbper evaluation 'of the effect of these storage conditions on

the structural ittegrity of a solid rQýKet motor requires consideration of the physical, chemical.and physiochemical ch•anget of all age and euviro'nmetit-sensitive system and subsystem components.

Some of tihe more'

impqrtint factors influencing operational 'servJce life as well as current practices of establishing4 cuss•,

nd verifying ser~vice life predictions are dis-

in the following subsections.

For convenience,

the effects of,

relative humidity level are di!5;ussed separately from other aging degradation mechanisms. General surveys of environmentai by

Fishman [16],

and aging effects have been presented

Bills, Fishman and Myers [17] (97.0,

Mechaical Property-Testing for-Surveillance"),

'Appiicationsof

and Kelley [18].

articles contain rather extensive reference up to about 1966.

These

Hence,

the references cited herein refer, for the most part, to more recent literature on this subject.

Of the recent literat.ure, references 19 tnroU~gh

29-pretty well, reflect the current state-of-the-art. 2.4.1

-RELATIVE HUMIDI1Y

The presence -of moisture severely degrades the mechanical and chemical properties o.f mostý-solid propellarts [16, 17,

19-25, 30-323.

This degrod-d

tion is typically manifest as swelling of the binder matrfx, rediction ih

Smodflus

and retardation'of propellant ignition,_often leachi-ig of sbrface

2.25 y"

^

IJ

TABLE 11.

FACTORS INFLUENCING PROPELLANT DEGRADATION OUR•flG AGING [18]

FACTOR

K1ANIFESTAT ION

Change of Chemical State

FAILUME MODE

Ho-dening. embrittlewent. gassing, i¢ccumulation of degradation products . viscous flow enhancement, change of adhesivity.

A.

Chemical reactipt*ty of pro-Ilant components singularly or in combination

S.

,hemcat Interaction with

S

enironment

ity of propellant at surfaces tnd within bulk. -

Inc-eased tendency to ci'ack durIng storage, ignition, or temperature cycling; possible burning rate change, impulse loss, ignitiion problems, and linear

separation.

1.

s A; in

nonhcogne-

Sidmte•, S as A

Atmosphere

Moisture

.1a.

b.

c. 2. C.

0.

r E.

Gaseous or solid aecomposition products (aiutocatalys is) Air (oxygen. ozone, contamiinants in air)

Ot**r materials in motor (liner, mt-atIs, etc.)

Factors whi:, mayinfluence rate of cnange 1.

Temperature

2.

Stress state

Irradiation

Same as A

Same as A

-U

Time scale of degradation

Polymer crosslinking or degradation

Same as A

Surface changes

ULmnown

Phase changes which and

Hysteresis of temperature-dependent physicl properties

Increased tendency-to crack during storage ignitidn, or tempera"tnqcling.

2.

Recoverable strains

Probably minor

Probably minor

3.

Diffusion of mterials

Monhomogeneity of propellant droperties; oxidizeo-poor surfaces, porosity, shrinkage .

Crack development, increased tedency to crack du'-ing toraq, ignition, or temperature cycling.

Cracks at fillets, line'

Incre'asedafnting areas and rates.

1.

Background

2.

Induced

Bacteriological action

Change "n Ptyslc&A State A.

Reversiblc phy:ical thanges 1.

depend on time ~tee~raturm

°

a.

Gases

b. " Plasticizer c. hoisture 2.

Irreversible physical changes 1.

Strain beyond reversible IImit caused by: a. b.

Gravity A-crzrst"on (duning transpcrt)

"c. Thormal to.

separation,

b'vjseus deforation, deweLting (blanaiig)

gradients

Environxental tempc-atu~e.

2'26

oxidizer particles is observed.

The mechanism of this degradation is

primarily a reversion process in which chemical scission of polymer network cross-links and consequent reduction in modulus is caused by hydrolytic attack at cross-linik sites.

Epoxide cured propellants are

relativeiy insensitive to hydrolytic attack: varying deqees of susceptibility.

Imine cured_-prope-flants have

Double-base propellants are usually

less influenced by'moisture than, composite propellants. Moisture enters a solid propellant or liner-propellant interface through a diffusion process.

The depth of penetration appears to be con-

trolled by the relatiye humidity level,

the ratio of volume to surface

area exposed and the duration of e)Dosure.

Inasmuch as moisture is stored

to a large,'extent in the binder matrix,reduced strength and increased elongation capabilities of .the propelllant are-observed at ambient temperatures and above; wh1ereas 'at low temperatures significantly reduced elonga.tions dre observed, probably due to moisture embrittlement or freezing-in water which restricts polymer chain motion.

The strength of the liner-

propellant bond is reduced at all temperatures due to moisture diffusion at high relatiwe humidity levels. suggests- that Available information

the

effects of ',i-h moisture

content are more severe for propellant in a strained state than in an unstrained state.

This attribute may lead to particularly acute problems

during temperature cycling of a solid rocket motor due to breathing of the p.,opeliant or rocket motor interior with the external ,envwronment.

In this

situation frost and water may condense on the propellant sur'face and accumulate in subsequent temperature cycles, eventually leading to structural ,failure of the propellant-in areas of high stress concentration. 2 ?7

.1

(.

Inasmuch as moisture induces drastic and rapid degradation of propellant and propellant-liner bonds, exposure of a solid propellant

grain to high

humidity levels must be avoided since it is not always possible or practical, to select propellant poljkmers in which water is not sol'uble.

"p

I

the-humidJty level in

Fortunately

a solid propellant grain may be maintained at

an

acceptable-low level-relatively easily through proper implementation of hermetic seals and-dessication. -

Past experience has indicated that short

nd probably even long'ter exposureto relatively-low hu 1dity leveels produces no effect directly attributable to the level or duration of exposure.

It is generally "accepto that at levels below about 30% RH no

effect is observed". The effects of relative humidity have also been observed to be reversible to a certain extent.

The original properties of a propellant grain

which hastbeen inadvertently exposed to a high humidity level, but whitd

has

not yet structurally failed, jre substantially recovered by dessication of the grain.

As a general rule, the drying recovery time is the same as the

exposure time to moisture.

Dessication vf un-aged propellants also tends

to remove moisture introduced during mixing, casting and curing operations. 2.4.2

AGING

In addition to being sensitive to moisture level's,

olid propellants

experience changes due to normal aging during long term storage.

These

changes-ire reflected'in changes in the chemical and physical properties of th- propellant and liner-propellant bond.

Unlike the effects of moisture,

however, the effects of aging are irrerversible.

Most propellants typically

exhibit between 25 and 50 pe'rceit degradation during aging. here is restricted-to chemorheological aging.

S2.8

The' discussion

Mechanical aging degradation

II

results from sustained or, cyclic application of the loads discussed previously and is normall'

handled.through cur,•ulative damage considderations.

Several factors are•{nown to influence the aging characteristics uf propellants which inturn affect the shelf-life of a solid rocket motor. Theýminant aging mechanisms affecting propellant behaviw&r-w-,weh-nor-ma ly occur simultaneously'

are continued post-curing, oxidative cross-linking and

polymer'chain scission [16-18,20-23,30,31].

Additional

consideration must

also be given to surface versus bulk aging characteristics [17,25-27,30-38] tj

and migration effects [17.20.23-25,33,35,37,39,40].

The infiunce

of

these factors is dependent to a greater or lesreer extent upon the propellant polymer and cure system,curd cycle, cure catalysts, ballistic modifiers and agirg te'mperature. Rost-cure curative reactions result from the slow continuation of reactions not driven to completion during the normal cure cycle.

These

reactions result in an increase in the propellant modulus due to the formation of additional network cross-links. Oxidative cross-linking is primarily a surface pheonoenon which results from free-radical attack dt double bonds in the polymer chain backbone. This mechanism also results in an increase in stress ard decrease in the strain properties of solid propellants. Chain scission is-largely determined by the cure system in softening oy the propellant.

This phenomenon,

accentuated by the presence of moisture; however, and Carboxy-terminated polybutadient (CTPB)

and results

as mentioned before, is Hydroxy-terminated (HTPB)

prooellants frequently display

this reversion process during high temperature

aging.

Polyurethane pro-

pellants also undergo chain scission during aging due to splitting of functional linkage. 2.29

L

__Vr

.Distinct differ ences between' the surface and bulk aging chara'teristics of propellants have'been noted jrimarily due to surface oxidation of the propellant.

This surface\ox.idative cross-linking results in a

considerably stiffer propellant surface.

Surface skin effects,

-

notably.

hardening of the grain inner bore, has been observed to a depth of onehalf inch in some cases [34,36,38,40]. -

Significanft variations in the aging

bqhayior of propellant aging in sample cartonsand propellant aged in rocket motors'has also been observ

,,

,36,38].

These variations have

been attributed in part to the fact that motors are characteristically

(

cured at a higher temperature than the oven temperature because of internal exothermic reactions.

Cartons, on the other hand, are cured at a tempera-

ture more nearly equal to the oven temperature. Migration of soluble species is

", najor Loncern at the propellant-

liner-insulation bond interfaces [17,20-23,25,37,39,40].

Soluble species

such as low ,molecular weight polymer, bu~'ning rate catalysts, plasticizers, moisture and degradation products may milrate acrdss bond interfaces causing both chemical and physical changes.

An e'act relation between

ingredient migration and the physical and chemical changes is not presently known.

'Such a relation is influenced in'a ýomplicated manner by timee,

temperature, cOncentrationf and relative solubility of the migrating. species. The predominant physical effcct of all migratory species is degradation the adhesive bond between the liner 0'of tion or case.

In addition,

arid propellant or liner and insula-

the propellant and the liner of insulation may

harden or soften eithey separately or jointly.

Typically, migrating species

from t~e propellant tnto the liner or insulatiori'act as plasticizers causing the -iner or insulatioW to soften and swell and the propellant to harden and shrink resulting in high localized stresses -nd straThs at the bond 2,39

"w

a-

interface as well as a weakened adhesive bond.

Plasticizer migration

from certain elastomeric insulations into the propellant, hand, normally softens the propellant.

on thp other

In ot~ier situations,

suchas

I.\

a.curative imbalance between the liner and the piopellant, Ak hardening of either or both the propellant and the liner

Cross migration

may result.

of Oeher ingredients may have similar results depending on the particular ingredients and concentrations involved.

It suffices to observe that

migration invariably degrades the adh4sive bond system.

-

-

Migration in composite propellants ha, been observed -to be particu,larly critical, in CTP3 propella;ts, which have a notoriouslhistory of difficult bonding problem- to begin with; and for liquid alkylferrocene ballistic modifiers.

rellants employing

Migratory behavior has

.

also been' observed of diioctyl aze-late ,DOZ) and circo light oil.- The primary migrating species of double-base Oropellhrmts is the energetic plastirizer nitroglycerine.

o

Diethyleire gl-yc6l dinitrate .(DEGDN),

triethylene glycol dinitrote (TEGDN), (TMETN),

tr lmethylethylene trinitrate

dibutyl phtnalate (DBP)"and triacetii

are also known Co migrate

when used in CMDB propellartts-. Isodecyl Pelargonate (IDP)

denmnstrates

mi-ratory behavior in both doublt-base and composite propellants.,.

An

additional coocern with double-base prqpeiiants is the decomposition of certain ingredients to form products which increase the sensitivity or stability of the propellant. The storage or aging temperature in-luences the rate at which the above processes occur, the relative severity of degradation, and, certain extent, if a given aging mechani fm will occur.

to a

In general,

increasing the aging temperature accaierates the rate at which degradation takes place.

It is also nctad that the degradati

2

in prQpellant and

..

IP liner-propellant properties observed during high temperatkure aging is"

I,,

significantly greater .than the degradation observed during amoient temperature aging, even for prolonged p~eriods.pf time [17,18,25,27-35,41-44]. Post-cure curative reaction rates are accelerated by increasing the. storage temperature.

K

In this situation, high temperature agiitg conpletes-

the normal cure process. Migration rates and the relative degradation .. of the liner-propfellat adhesi~ve;bond due to migration are significantly inhceased at high temnpýeratures. Surface hardening due to oxidative cross-linktng also appe~ars to beaccentuated at elevated temperatures.

-On the other hand, propellant

softening due to excess chain scission over-cqntirued post-cure crosslinking, npticeably absent under ambient temperature storage- conditions, fias been observed im-ýCTPB, temperature-.aging..

HTPB and polyurethane proppilants during high

For the most part,Ach~e processes discussed i~n the previous paragrai are de-emphasized under l'ow temperature storage coridi~tions;

(

However,

"an alternate problem may be introduced for composite propellAnts. containing liquid'ýaikylferrocenes which may crystall}ize during low' temperature storage,

Crystallization has been observed to be most

sevdrefor propeliants containing n-butyl ferrocene liquid burn rate

\

catalyst at temperatures below about -4 0 °F [25]. T-his •hen••on

has, been,

attributed to the ekisterfce of n-butyl ferrocene in a.siellcu•oed state;" -which uhnderg oes a change of phase to a crystalline statL duc to shock c6nditions ,na~ced by temperature .excursionsor mechanicalloading:.'

•

,Substantially, reduc4.dpropellant elongations are obtained as a result

2.32

-'

S..

!I -.

! Ak

E.

o-

of ci'ystallizatlon.,

Of equal importahice is the Dossibility of propel-

Since n-butyl ferrocee is a highly •ehergetic plast-

lan6t detonation.

cizer any severe mechanical jol-t,such as a low tamperature firing, may fractuie'n-butyl ferrocene crystallites thereby releasing sufficient ehergy-to initiate and propagate detonation mechanisms resultiqg in catastrophic

motor failure.

In addition to the primarily physical effects discussed pre~iously, aging also affe'cts the ballistic properties of so-lid propellant grains. The normal ballistic changes are changes in burn rate. p'ressbre and .,temperature sensitivity of burn rate and igniteab4 lity caused primarily by hardening of the propellant and evaporation and migration of volatile catalysts [20,21,23,25,35,37,43]. MINIMIZATION OF ADVERSE AGING EFFECTS The probiem of controlling and minimizing aging effects has only been partially solved by the propellant chemist.

The problem facing.

the propellant chemist is that of formulating completely stable solid propellants which u6ndergo insignificant changes in all aging environments.

This goal has-effectively been attained only for polybutadiept-

acrylnitrile acrylic acid terpolymer (PBAI'P propellants [34j, Which typically 'undergo-about a .25% decrease in strain properties 'during the ------

first few • mQnths of aging and then remain sta•le thereaftere. -C, .. )

.

',-,.

.

Cont ' nued-post-cure cross-linking is exhjoited by Ail composite'

.

°rope, Ilants tb a greater~or lesser extent. One effective means of c ntrollin p.ost-cUre reactions 'has been to extend th e cure cycle to

.

'. .,,

-

II IM

I

MM"

I

nminimized by sealing the rocket; motor interior in an inert gas environment..

A positive internal pressure is often maintained to circumvent

breathing of the motor during environmental changes.

This procedure has

e~fectiveiy reducea the promlem associated with oxidative cross-linking to an inconsequential level. Migration effecti may be reduced to tolerable levels through consideration of the equilibrium concentrations of migrating species and ÷ha

:

nf minrntinn harko2 ri,

Drimrill ____rmfA

6ffnr+ hac hoon

÷nwn"A

the elimination or reduction of the degradation of adhesive bonds attributed

to plasticizer migration. znd

liners

For composite propellant applications,

inz~aiHtinns hac- hben deveTopnead whirh arp resistant

aigratlng plasticizers [25,45].

f

tM abhsrntion

Non-functional plasticizers have also

been introduced into clastow.ric insulations intended for low temperature applications.

Conventional powder-embeddment case-bond ;ystems used with

double-base propellants are reasonably resistant to plasticizer -rigratioh; however, the high glass transition temperature of the epoxy in which the

Li

casting p.rowder granules are =hp,•._dded ini prer!,ides low temperatures [20,46-48].

tse

of rhis system at

Adequate low temperature behavior has been

demonstrated-using a double-layer bonding concept [473. to resist plasticizing and plasticizer migration,

One layer, selected

is used as a coupling

layer to bond a propellant bonding layer tc the case or i'nsulation.

The

propellant bonding layer is attached to the propellant through mutual diffusion and chemical interactions. SERViCE-LIFE PRED'CTiWNS Presently, the predictioti of aging degradation is largely based on past experience with similar propellants, and on fairly extensive surveillance 2.3b

•

.

~*M,

.'

-

-

J

-tE

,iM

As noted in a subsequent subsection, however, predictions surveillance s orvice-life of making initial means is an inadequ~ate

test progr'amns. -t~stina

-

a-time

and is best suitod for revising service-life predictions during~the lifeof thpi6(tor and as a termi.,al measure of ultimate'service-life.

I

;

In an effort to olftain 'an indicatin of the aging degradation of pew propellant formulations or new liper-propellant bond systems due to long term storage at dmbient temperatures short term accelerated aging tosts are conducted at elevated temperatures.

Accelct•ated aging test data are

I.

then reiatV to a~ibient temperature aging conditions assuming that the ratc of aging degradation obeys the classical Arrhenius

.•'

.

, k'

A exp(-E0AkT)

equation

•

I

(2.3)

_

I

where of reaction at temperature.T

Srate

I

EA - activation energyv assumed constant A

ccnstant

k

Boltzmann' s constant

(2.3) is used to predict propeillant..physieftl property degrada-

_""Equation

..

'tion by treating the. rate ofreaction k' as an average rate associated with soime change, A, in a pertinent mechanical property,

Thus,

(2.4)

, -

Defining A has the critical change signifyinng the end of useful service "_

A%_ j ^ IJ

-c-

I.-

"t A' exp(EA/kT) 2.36 2.36

(2.5)

a

to CuR.

otw4La' eqUdl

IM

Equationl (25 inirflC1Ce

This behavior has

that a plot of log tý ve~, US l/T should be linear.

I 'data -

for polyurethane propellar~ts where

WMY

2 which presents

-bedn observed for some Oi-ope'll-ants as shown iý Detail

A was-taken to be doubling of

the initial strain at maximum stress- or the decrease to one-half of the IrIILidi

(AIt should t'Ie reccallled that poyue~an

maximumi nominal stress.

propellants frequently underjo a reversion, e.g., sfeigudrhg temperature aging. Most- poolybutadiene propellahts, on the other handý, will harden under high temperature aging). 100,000

/000

S

-o15' 1,000

10

FProol0

_

.

I

o

endof .. ,efui IWe

220'

I_

n

_

_

_

_

_

i1 ofic the deraaion o ai ven O i

_

_

Al

_

sofamterning*ina

Aih notrersn strahen tempe raturos Thas beenauedior inPwhich

athe de radtioh tion7

faqgive--!n pI-sica paamtr

snome i t

f _

ropetallt 3 o3 liea ufw

temiperatires may be cross plotted as a fictRon of aging temperature asdi

shown in Detail 4.

A curve of this type may

used for interpohting 4e

aging degradatioh at other, storage temperatures.

UU

a

i.r Iin o the l rae

U,__

:::LI

know

11~. 01

l

ction in Strain thataTtten"Capablity

Detail

-

asu

I

..

-

-

-.

---.

. .-.

i

Detail

4.

•

The validity of the above kinetic approach is basedon o the aassumption o t thattlie same processes-.ccur .over the range of storage temperatures and

-•

Ueeecsdwe •)

.

~

times; the only effect of the increased temperature, IS assumed to be an sn h tagtowr A rrhe ius e Styan D~rel ationsa tmincrease in the rae that of-reaction. ThisIncrease assumption leads to theapproxinmately well rule-of-thw'ib a ten-degree in temj~ratire

i~~known

doubles the rate of reaction. ~~~be • ._

Arrhenl'us type relat''........ ......

"

t

2.3':

~to make service-life predictions inasmuch as the assui•Ftion of a cr~stant, temperature Indepe-rndeint activation energy is ncet valid, and prccesses .which do not occur, or are at least of little consequerle at ar~blcnt • :

.i•" i"

This behavior has been observed to a

exercised when using the straightforward certain extent for sone propefl . "l, I

• I

-a..

.... m

reit

3.

This

!

0*-.--

S4

--

-

2.38

Ions•• I

!

Ul temperatures are initiated at elevated,.temperatures.

The activation

energies for competing processes (e.g.., hý,drclytic chain scission and oxidative cross-linking) are most-probably different. Typically the derdto

*

predicted fromni

c lrtdaging tests is substantiaily

larger than that observed at idwer storage tem'peratures.

Thus, the

conmmon rule-of:-thumb that 5 and l0, weeks aging a't 160'F is equivalent, respectively, to 5.and 10 years ambient storage can be seriously misleading, and may result in eliminating propellants from consideration which aer ade'quate for the actual storage environment. Despite the inherent Oefici~ncies associated with accelerated aging tests such tests are required on new prc

ilant foi'mulatlons

lin

order to

Obtain even some indication of the aging characteristics of the propella nt; even though such information is difficult to interpret at this time. Improytn~mnts in the method~ of analyzing and interpreting accelerated

aging test~data are required, ifthis type of data isto be success-

0fully used fcr quantitative service-life predictions in the future.

Some Indic'ations of possible extensions of present'atnalysls methods are delu Ue'inthe following paragraphs.

Further improvements in service,-

life predict~ixe capabilities will-be forthcoming from the extensive chemical aging program being sponsors,"-J z~y the Air Force Roclet Propulsion Laboratory.

This study is aimed specifically at relating chemical changes

during aging,.to mnecharical propery chanages. Temperaturea dz-peniierice of the activation~ energy EA iiiay be introauced in a straightforward manner by considering the Van't Hoff equation f'rnm which

/Arrhenius obtained (2.3) [491,

2.39

'

4,

¾

I]og kW,

;

dlok

-

E.

•(2.6)

kT'

dT

As an initial point .of departure, there is evidence (e.g.,.[CO]) that tne activation'energy for fracture of viscoelastie materials is linear There are also indications that the activation energy

in temperature. for •

reactions is also linear in temperature [49].

c.e,,,,,

a fi-st,approximation one may take E tion energy at absolute zero.

E-

Thus, as

ET, where Eo is the act va-

Substituting this expression in (2,6)

SK.R

and carrying out the implied integration, k,'

A(T)

(2.7)

exp(-Eo/kT)

The analogue of (2.5) is then (2.8)

t = A' T' expjE /kT) -

2

'•

where n - k/E.

Equation (2.8).incorporates temperature depend'-nce into

the activation energy EA; owever, it is still based on the assumption that either only one process is occurring or that all processes have the same activation energy. undoubtedly happens,

If multiple aging nechanisms are os-trring, which

then onne maw wri.

(2.8) a4ti th

suggestion of

Ree and Eyring [51) in the form t -£ where n= k/EI.

I A (T)n

(2.9)

exp(Ei/kT)

Equation (2.9) may lead to fruitful

cMslUlts if one

considers a separate activation energy for ea-ch qf thes conc-nr-.zt pifocesses of chain slssio-n, o-ieat';ve z-,,s-i1nkUn9 These actlvatiu

and contipued

post curing.

eriergies may be related to changes in, say, cross-link

2.40

'

-OR-U

densi,ty in an eff~ort to relate molecular parameters to obsOfrved mechanical behavior. Mechanical properties (2,9) enter only through the damage index Ai the coefficients A.

Itwill be recalled that A 'is defin~ed as 'the critical

change in a particular parameter which signifies the end of useful service lift. Inasmuch as the stress state dur'ing aging has a strck'?g inflvence on aging behavior, as no..:J pr-eviously,,it may be desirouý to introduce a more direct dependence on the stress state. This may be accomplished by rewriti~ng (2.9) as t =E

A1 (T) exp[(Eo

)k](.0

where W is the work done in the stes state] ineet

]i

(2

usual context, of the kinetic theory of the strength of solids W is taken to' be 1/2 cc, ('i.e., 'the.strain energy associated with uniaxi~a) tens )na on molecular bonds). However, inasmich as we are not attempting, or Ilk&

suggesting, th'at this~a~proach be interpreted molecularly in the context of

kinetic

thie classical

theory of.

solids,

a more general

expression may be considered from a phenomenologicall point of view.

From

this point of view, W, for example, may be taken to be the energy of a linear viscoelastic material as roted In [52] f t W .0 ij (-0 s. (T)dT

j

d vol

'

(2.11)

corresponding to the loading history experienced by an aging propellant grain. F-

t eiiratwure dependence into W in an

tb"'

ad hoc manner througb th-e assumption of thermorheologically simple material C

behiavior.

2.41

While the above approach is probably simewhat mor~e geneeal than present methods of analyzing accelerated-aging test data, it &,oes not account for the fact that Certain processes occur at elevated tempera'tures which dre 6f no impoi'tance under am'bient temperature storage

condltiofls.

This behavinr m~ay he aircoiinted for'nncsibly by intrnmiring

a threshold temperatureefor a process in the expression for the activa-

I*

tion energy for the proc-est. "rhis temperature may also be incorporated intd the expressfon.-of'a W and a shift factor rela~~ng storage-temperature and time with degradat~ion dedu~ced.

I

Clearly much more work is.

~required to quantitatively define the relationshipi between ambient

and elevated temperature storage. Our intention here ~is only'to pointout a possible approach and hope that it will stimulate further study on thils-complex problem.

cntttv

qain

o

Diretdvelpmet o costiutie euatonsforaging viscoelastic matearjals' is another aporoach~to the problem of predhjlng the aging be~avibr of solid propallants.

For anwaging, linear Vis~ceelastic material

a general differential constitutive equation may

K-,n+

++

P_ (t)

Ldtndtn 1

...

+ P t a

.[qo (t

ni%

written in the form-i

-

*..

+q

(t Ie

dO

2.2

Lubliner [53] has poin~ted out, however, that selection of'a model for a.--

representing a tigme varying material requires mo're- ca~re than for a time invariant one.

It -Is also noted that Vt.s solution- of (2.12) becomes

complicated except for the nw-st simple -Adels.

Pis

to introduce a time depr-dent kernel (i.e., relaxation function) into 2.42

0

&

4

I

k tthi common Boltzmann integral of linear viscoela.caty

4 -

t

d

k (t"

.

(T)d-

time-irvariant linear viscoelasticity,

/-In

ax~ton f.i.n..t.n kerneT-jt,T). O

S.,ar

V-1,

Rabotnov has

fcn,,

(2.d k(t,T) is simply the re-

dy be chosen tor the aging,

iiscussed two-forcis of k(t,T),

k(t,-E) :h(T) dp(t T)

(2.14)

..

'-Equation (2.14) has been used to descrIbe creep in conkrete, and (2.15) : is_-.queotly, encountered in solving problems in the theory of heredltAry elasticity with boun-•aryconditiin/s given.on a vari-able bounda'-y. ,-

Su'b-

tituting (2.14) into (2.13) gives

A

fc

"whereas the func.tion g(t) in (2.15) can be extracted from the integral to give

' (t) ,

g(t)

b(t-')

s(T)d.

(2,17)

•.

"Tiie functions h(T) and g(t) characterize the aging behavior of the material and may be determined experimentally from tests-of aged sp1c•,. Aging effects may also be treated within the-framework of'reduced

variable concepts in th- sa_'u w-an'-r tha tCnpcAt~re is under the il-lr1assumption of thermorheologically gimple materidl behavior.

Fulmer

[54,55] has successfully,.treated environment, as a redaced variable 2.43 4,

2.

3

.

J1

q creep failure of bolymers.

i inivestigations of,

Stauffer and. Wineman

56] recently suggested a mole formal approach which represents a~extension to the tnelmorheologi,,eal lytsimple theoty.

a(t). ,

for

t

an a iTng\fcoustitutiveoquatio•

((s

Et

Their studysuggests

.

d:

i

12.18a)

where t~je functional P(s) is tfie reduced v'riable characterizing the pffect

Q

of environment~on the relaxatjon modului. It is clar that the samei I consi&rations :an be introduced'into (2.16) or ('2.q) and also into the previous kitneticý approach if a technique, such as the idea of a threshold temperatuire•, on be successfui1y derived ror'

eparating mech'anism. which

occur at elevated temperatures but niot- at lower temperatures,

*

* The reT•xation function" E is determined in the presirnce of the environmental h~stor Y for the entire tine •niterval (--,t2k Stupposing, welaxation rmodulus ,is determined for

for diScUssion purposes, that te

a -series of aing'times at constant temperature.

Then., se'ries of ,curves

as depicted in Detail 5 would result.

A

tim

Detail 5.

pro4pertles

....

S.

'""-'+-

",time I--

-

DEPENDENCE OF RELAXTION FUNCTION ON AGING TIME AT CONSTANtT

STEMPERATUIRE.,.,,

•

A,+ •

•.

..,

-

-

9

.,,-.

N

.. ,.

.

.

.-...

..

.

.

.

Changes in glassy and rubbery response may be accounted for simply ,

through scale changes.

If the sl^pes through the transition regions are

identical for all aging

times then a shift function may be determioed'

relating aging timieto.relaxation response. elevated and ambient temperature

aging

If a relation between

can be developed, than a masteý

I-

,temperature-time curve for aging may be constructed in the identical way

Ir

,

that a master relaxation modulus curve is constructed.

If the slopes

"through .the transltion region vary with aging time or temperature, which they will do, then a more complicated reduced variable must be introduced., In principle,. It-should be poss-,ble to do this in a manner analogous to th. construction of strain and temperature dependent shift factor (i.te., by verticjal as weil as'horizontaltranslations)"

Rotations

;nmy also be 'required; however. The reduced variable approach for Sfudying -appealing for several reasons.

the effectsof aging is

First, the approach is quite general..

It is equally well suited for studying the influence of other environmerits, such as humdldity, irradation,• "bioliglcal conIuamination, etc. Seccndly, the approach is also suiteo for. assessing damage inflicted by

*

mechanlc~l aging.

By introducing a slightly different nbtation and a

more general interprctation, (2.18) may be'cast in the'form siflir

to

the equations discussed in CI,apter 11 for describing the permanent r eory behavior if. solid propellants.

Thus, by combining various as"-ýcs of

the reduced variable appruach to aging and enviromental effects discussed here with the developments of Chapter 11, it is not too difficult to see now a general ncnllnear constitutive theory for solid propellants including" the effects of aging and enyironment may be 2.45

S"" S4,

I'

t

developed.

lI

r

.The reduced variable approach to'aglng may also afford a more dieect opportunity for relating chemical chanAges durig.aging to observed rechanical behavior from the molecular viewpoint of..Kellev and Williams [57,581.

Chemical changes may'be dharacttrized 1'y thefr influence on ;

reduced variable.

For example, the equilibrium modulus Eel frcm rubber-

elasticity theory, is related to the crosslinP densit'yv;

e Eea 3-

f

kT

('2o19)

Thus, the vertical scale changes in ', shown in Detail 5 O bote, are related to changes in crosslink density.

Observing changes in crosslink

density through changes in the modulus, in swelling with aging time then supplies the information for determining a vettical scaling shift factor for'Ee. .

Aging temperature effects can be Incorporated as discussed'-

previously. ,

r(hariges in other molecular parameters with aging time and

temperature can-be ased in a zimlfiarmanner.

Chain sclssicn may be

determined fr m ch.-nges in 'blndcr molecular weight.

lThe c6bined effects

iof conctrrent chain scissitn aad cross-linking can be separated to,a certain extqnt and measured by continuous stress relaxation\ard intermitc tent stress, masuremeht techniques.

Dielectrii, and dynamic properties

can also Me Osed to measure changes in internal structure (e.g., molecul.&r Sweight, -

degree of crystaliinity, dewetting. crosslink density, conforma-*

tional changes, etc.).

f

OxidMt

4tn

-may--be measured-by tecb-

niques ;uhich determine awounts of free anS bound oxygef,ipA De

pro•r•llart.

nmination of these and other kinds of chemical chp.nges during aging

and',establishment of fheir relationship to mechanical behavior should lead to a better understarlding of-aging degradation and a More realistic.appro'I to service-life predictions.

A, nluportant consideration -in studles, of th-4 2,46

"0,] •

-•

r

..

.

.

*

.

• -

".

."

--

•-

•

.

. -

=__ -1.,

.

!

•,

-

:

•

*-

.

,

.

_J..-*'__.

.

•

•

.•*-

,• .

• :

-

r.r(.., _

.

:[

_____ • *.

'

-. ,,_

• _ •t---- _

'yoe isthat only chanqes in'mechanicai oronarties whirh xro "eleaw-+t a motor ItructuraI- inteiri'ty determination shbuld be sought; extrapo-4 lations of proo~i'ant propert'ies which do not enter directly into a sr-uctura-lintegrity analysis can be'seriously mi~sleading. The lenjth and-~breadth of the discussion here on's'ervice-11fe, prediction Isindicative of-the lmportacce and severity of this nrobleni. While this pre:;entatlon has not consjoered present methods of ,_,i-.ing

0service-l-ife predicti~ns'jivfdetail, several a-prbaches have bein described which nemcie certain deficieficies from current service-life predk-ctio') me'hos.

Soui coinbination of the approaches describeid above, coupled'I-

with the informatioriforthcoming from the Air Force sponsored COerical Aging program, should lead. to improved se,

-if predictive capabilities.

Current techniques for maki"ng service-life predl0~~ons are adequately treated

in Riferences0 17, 28,. 33, 42, 44 and 59 through 67;

O

* SURVEML MICE PRDGOGAS, In recdent years, due to a general lack of confidence in servi~ce-life' -predictive-capabilities,

Mssive surveillance programus 'supposedly aimed at

maximum-cost. effectiveness are inltiatid with each new mnotor program r17, 26-ý9, 38, 4g-44, 611, WhAle -it is not our- intention to fully revie'w past surveillance prograI!B-

sowe of the nvre ivwcrtan~t factors which contribute

to a meaningful surveilianct prog ra-mwlll be briefly discussed. .*it-will be seen that surveililance program~ achieve maxim'um effecttveness when they function-as a terminal meA~urement of ultimate service life. FOllowinj anv initial service-life prediction based'on analysis of accelerated aging test data by some means such as those mentioned previously,

propellant samples, subscale-prototype motors and full-scake motors 'are

placed in storage under service cond tions 12 to 18 months before the first delivery of-production motors.

Periodically, subscale .and full-scale motors ý

are-test-fl red to 'determine ballistic changes.

gJhysical property tests are

also conducted to evaluate degradation of particular paraweters which are determined to Weimportant to the determination of grain structural integrity. These tests should typically involve selected uniaxial., biaxial' and triaxial bond adhesion. t~sts at various temperatures under loading environments carefully designed to simulate the motor loading envi ronment.

Rec~ntly determiha-

tion of the cohesive fr~acture energy has been added to the list of relevant parameters [68]. An assessment of damage accumulation may be4 provide'd by conducting combined and repeeted loading tests making use of =Wmuative damage concepts [17, 27, 28, 59, 60, 66. 67, 6gb1

Because of the sfIgnifi-

cance of the dl fferences observed beb~een surface/ ýnd blagncarton and motor aging, arid aging in a strained versus/h~f stress free-state prototyp so!scale motors should be aged and.period'ically tested te failure under' a critical loading environarnt, such'as, for,exampl1e,

low tenmperature cycling.

Odcasional dissection Of full-scale motors allows correlation of the behavior observed from car't~ storage with that observed fromn motor storý'ge. evaluation of-aging in acstressed`state, and eiv'aluatiot, of surface and 1itkerface effects i n the full-scale wotor [24, 33, 34, X6- '49,48, 69:! Miniature test speci mi-is, have been developed for determining-propelLant properties neav~ the

inner bore surface and the case._graininter~face tL3,39, 40, 70, 7413. As 'these tests are evaluated ter-vice life predi~ctions may be, continuously tipdated. *program

Contingency souples; should be provided in eni agling surveillance such as this to substantiate that a first failure does indeeit, 2.48

I

represent a deterioration of motor service life and to obtain 5tatistical data for predictions of mean service life and probable distribution of1 fail'ures. failure.

Extreme value statistics may be used for predictions of a first By allowing 12 to 16 i,&nthq lead time on controlled acting of

motors and propellant saiples, sufficient time is provided from the first indication of motoryage-cut to determine an appropri.•te course of action tor the motors remaining in the field. A comprehensive surveillance program, such as that outlined in the V>Yprevious paragraph,

is effective in evaluating the ultimate service-life

of'a solid rocket motor; however, the greater usefulness of such a comprehensive aging program is the rat 4 onal basis it provides for extending the useful service-life of a solid rocket motor beyond contractor requirements or objectives.

The value of this capabil.ity is readily recognized

when it is recalled that in" Southeast Asia, as in Korea, it has been necessary to use weaponry which has gone substantially beyond the predicted storage life. ENN-DESTRUCTIVE TEST TECHNIQUES. The romprehensive aging program outlined in the previous section has one serious drawback; namely,, the high Costs involved.

Testing of propel-

laht samples aged in cartons is not too expensive; however, structural testing of aged'STV's and full-scale motors, and dissection of full-scale motors 'for propellant samples is quite expensive.

Furthermore, when discrepancies.

exist between the controlled aging enviromnent and field service conditions, or when uncertainties exist about actual field conditions,

it is frequently

necessary to remove motors f,-om field storage for structural testing if a meaningful evwduation of motor service life is to be made. 2.49, .

........

Removing motors

si

Ii

~~zn 2..O bn nait

crnu

h

rrid

I

a etetion

has been given to the development of Mon-.Destrtuc!'Ave Test (NOT) techniques which are ap~plicable to sclid propellant surveillance and which are adaptable to field use. Soine of the NDT techniques i-n use throughout the i~iustry are swnimrjzed in Table 111.

Not all of these t;echniques are readily usable for field

inspection of in-service motor~s, howev~er.

Also, these NOT inspection tech--

niques are usually adequate for determining iffailure has occ~urred inthe form of crackingi, ulribonding,, etc.; but they are, for the mos~t part, not directly applicable to an assessment of proppliant or c-ue/grain bond degra* dation due-to aging.

For these reasons, recent efforts have been directed

toward development of NDT-mettods for recording changes in physical properties which are relevant to structural integrity detemiui~hation.

Hardness

relaxation measurements have been suggested as a mieans of evaluating surface agin~g[72J. Emb'edded gages have 4lso ;been suggested for monitoring changes in propellant properties. The instru~mentation i'equire4. for monitor1ing internal stresses and strains in a propellaint gri is not avial -sithin tepresent state-of-the-art technology, however, This instrtumentation problem is dif~figult and complex.

Satisfactory solution requires a knowledge of the

stress-state existing in the motor and complete characterization of the propellant response. The Air Force Rocket Propulsion Lbboratcry is sponsoring the development of an enbedded gage for nm6nitoring changes in propellant properties due to aging in a solid rocket motor. This instrumentation isvitally needed; however, progress on this difficult problsn will most likely be slow and the development of reliable instrumtentation will probably take several more years' research., 2.50

IWA

0

COMMONM

OCTECTION TECIW1NJ

FOR DETECTION OF

OTER COKSIDERATIOnS

*rUtica MnflCUtaic

(AmR)

wuantitstive cpowsitional changes

ZaSrered ?%flectamc Ultraviolet spectroscopy

viLSnT

Soft is above San as above cytli4di-d

c

Poorer then ATh rqec fcytliepo

.aisreocaise

CandtiM

Freetiocts; onentrs;ationla prod mecoranis efecs occ.rtoroo onin tsabise coorretaatio

Gasdlroatnlftphync IRtre tnois. o FilSetonel(isbe

oaw

£1

urae ast dsu--tc

Phoicoe Stat missiona

Catiodbsraieon

Infratire ds

roelantHepe a

mehais

Inaubperatmn Ml. W

eqie

r-sfalchngs

.ca .

3~~~~~bgieauk~n Stairit Prlf Ilmnata C.

f urac or seb-sif t ion efewiattloa roiat. if heat trar isi tin ora~loa

f-racition Traailse fadonty X-M iary io

u-n

r*uilte

Poros ty edfectt. voictds cr yk

scintillatton read out reflectione Propertiesaiie n

-01osinlSal C.FrplatHogn-

on

Scatter

i tFroff ce

atiss rate chanas causaof b) ormca thanges: oraaffect loc-al u-

So.1nrspst"

Radilatin Tasmoles in Perrr-

A physi fll al

~t-tias)

trin o rpelnteadt &s ce cit narae r-. quired

eo~l r

ln

Coreaioiototalrh

sofllte oouttbino readalitt Low Frequency Wt Trns msion orTranmtWCak turefledtiosn Prpeted&ia*&ton

Ntc r

voiss

inrspaain ae hne causedsio doomcal hars orcompsiton-

Develaped Vaey hoingho lnane re ndiredofn &arvace gre intsa rao

Scattetranamosln o

Arelectie

X-5,

separaton

Sape*nP

M-rimMcwae

o-

Alpyia

rsalnt

roota

m

wocroeA

orlto

totbil

atto sta:ia

2.4.3

''LOSURE

rvosdsusosi

Frr ;

~

are~~~

~

~

wihaigad-h

4

~lxpolm

espnta

xoueo

asoiae

rp-tgant

h

enlrnmt Thzslto tIhs rte sol atl opeea tact wit the renvirormnt

Tiheussonei cs he sn seen

Pex, howiever. Thle uoluti'mtsolutheeironto~ corse,

issociate dwith ging ndt~o poellants

wth atgingx are ore com

irbes asoncyiated wilte agtno anprope~llant- nra

ton sythems

whic presnot aghe. Advese objective~ repreent aificl task evrnieqirtingy t controlleoy w~rtno

and balane ofproseat curant reatiosewthy oe hnmcon-

s~ct isstionvro*n. undertaingssas These coy urse, is

:

1

prviusthav ated aonly aren partcia-

thus farln.o Sevrapllastuis are proelsent-lyindera

:z

1

Zowve, evra o !Z 0eye ::o

nensiv researchl

whicd systel

ask requirend for

'In the aufthrs' opinio" p~ost motor failures do not result from a design 4afWciency, but rather,~',rr the "result of some. obvious defect in prbcesing ttrat could hive been avoided through thoughtful- consideration of *

the prncesslng merodt and controls Mqieiured to mi~antain the structural integrity of a solid propellant grain.

To overcew the occurrence of

motor failures attributtatle to prbces's.ng errors it is oecessary that the structural integrity enginter be famillvir with the actual m~ethods of motor

2.52

,

-

/ ,

/ manufa4ure, the-processing &ý.rtrls that are feasible In a plant operation and the controlF that are necessary to maintain motor structural integrity. Thus, the purpose of the dlscus:o.%n

herein Is to acquaint the practicing

engineer wi6 some of the more Important manufacturing and processing consikarations which are rWiated to motor structural integrity. emphasis here, is given to the .

The

pdsystems existing in roicket motors

bond failures repvesent the n.ajority of all motor failures.

sli

The Integrity of any bond i-n a solid rocket motor is directly related to the processlng_. mthcds. Good precessing, methods Tead to reliable bond systems, whereas bad prmcesing methods lead to poor bond systems which *i

i~weiably resuit'.i!ý costly repairs, 4nd in s.one instances, even in motor rejection.

In fabricating a stress relf'f flap, caution must be exercised

to ensure the bond integrity between the flap ard the ,notor case, within

the flap, and between the flap and the propellant. The bond between the flap and the motor-case is the strongest of all interfaca-l bond systems and will represent an area of coqw ern only if Impi'oper bonding techniquez are used. The best"bonding procedure is to vulc.-nize the flap in place using ancured or only partially cured rubber stock.

An alternate, equally acceptable "-thod is to use a high tempera-

ture (3OOWF) adhesive system to secondarily bond a cured rubber flr, to the case.

In this case, it is nccessary to determine any detrimental effects

the high temperature oost cure of the rubber flap may have orn the strength and compliance characteristics of the flap or on the prooellant - liner flap bond-capability.

-

In the event that the high temperature cure does

result in Z'rious degradati6n of the flap material, acceptable results may be frequently obtained with a low temperature cure (160°F) secondary bond.

In this case, as in the above cases, proper prucessing methods and 2,53

WIN

-~r

-P

,

-

controls vith regard to cleaning and su~fce preptration~of theý case and

the flap, and 1"he aging conditions of' Ce adhesive are- a prerequisite for obtaining acceptable bonds. 7he quality of this bondA s norma~lly determirned fromi single lap or double lap~ shaar tests.

The integrity of the flap is asstireed only through vulcanization of uncured rubber into a continuious, orhe-piece fl ap. Inthis process, it is 't

desirable to vulcanize the flap to the motor i~ase simuitaneouslyo if possible, to avoid the use of secondary bonds.. Seconda-'y bonds wi thin the flap i tsel f are usual ly r.)t recommnended.

The potential savi ngs', 'when compared

with costs in time and money of extensiia repairs, are not sufficient to justify secondary bonds in this critical region,. The quality of this bond

[is

[

determined through small angle arvi lerge angie tl8fi"F) peel tests. The small angle tests are probably more reprenentative of actual motor coaditions. ~The interpretation of the results of either -test is, at-best-,--nl-y qual-i-ta.-

Ialso,,

The integrity of the-propeliant-liner-insuletior. (i.e., flap) bond is

to reiterate, related to pr~ocessing methods. Tks quality of this bond

MU

is normally determined through bond-in-tension and peel tests of' tfie propellant-liner-insulptiofl interface. The acceptance criteria for bond-intension tests is that the interfacial bon4 strengyti in tension be at least as great as the propellant sttength, and that the observed failure mode be Acohes mb failure on the propell1ant.

The question of when a failure is or is not a cohesive failure is still unresolved. As a general rule-of-thw~b, however, one may interpret a cohesive

failure as one in which at-least 1/16 inch of propellant remai~s on the liner surface. The primary requirement of a ,good bond system Is that theI strength of the bond be as great as or greater than the strength of the

weakest component of the system, however. 2.54

A

-

peel tests..

Tn these tests, as in the bond.in-tension test, ft~would,be

Sdesirable to have a" cohesive failure in the propellant, liner, or ev--n in ,%the

insulation.

Unfortunately;, the -failure mode, and of course, the

feilure load, i~s greatly influenced by the angle of peel, type of peel specimen b)eing, Lv-ed,' an& the rat

ZII

that peel test dat~a are extremely, difficult .t

--

the partir~ula~r

6••'e test.

TKO resu'.t is

'trpret quantitatively.'

"•Mostpresent peel analyses do not~indicate what the peel capability of a SdpartiCular bond sys-tem under a'gi'ven load environment should-be, nor for that matter, do they assess"ithe ewrespondence between )e'Ll" ang'le ip a motor and angle ef~peel duringFa ,laborat~ory peel teit.

TAu,., these tests"

are only~qualitative and are of 6oz~t use in comparison of aaJhesive systems or evaluation of p.rocessing studi-es. .•

appliled -directly to Assess

peelntess. yet.

Iond tseytest

The result: of thi's -test may" be-

,týhe. i ntegri ty

of a-givyer, propel Ilant/liner/i nfula-

s,- aser, these tests are net conducted

Also, with theslefilure

:

mayv bý obtalnd

More quantitative results of adhe'sive 4ond,.sy~st~

-, from the bli~ster peel test [73•,74].

bass

•

in th

ppe

ntrouid

e

blla t, liner tevin i,

.there are no analyses available. In the previous paragraphs, major emphas•i's was placed on ob'taining

cohesive propellant failures.

Tihe justification for reqiring s th

ype

*Mr uniaierslso deietody-esmybiotie of failure is based on the heuristic argument tho past .experience-with toasest

ei~

solid rocket motors has indicated that grain unbonding'at grain ternmira-"

tions is much less likely to occur when laboratory test.sof the-bonresystem have resulted in cohesive propellant failures.

In the event, Wal laboratory

tests predomienantly result dn adhesive failures between the propellant and pliner or the liner and insulation, correctnve measures should be,.taken prior to mneufacturing a motor, almost irrespeecive t 2.55

f the faiutre loadst -

-V

Ther'e ar~e several means olf improving~ a b'rid ýysen

tv

f aiUres.. Usually only minior modifications ave~requi red.

avoid adhes Ive

*

It is~a ger~erj,>ý

ally accepted fact 'that~the rnaj6'rity ,& all adhesive fa IIur~s in laboratory cedu'resg,

ThMs

the mephods o-F manufacturinq~the laboratory speci mins

should-1irst be revie'wed to ensure that OroRpA c~ie~i a 'S¶$elt 'su~fce:prqPtration of all~bortding'sq1rfaces), and that adhes'ives, 'ir4ers n.ued the ovcmanr Thag stae COndlttioS Of l-in&Poplat'n aidhesive ingrediefits; O~ould also beadpyiopel.UI~h wer

evaluate.4,snems

=

and pdesives are greatly influenced by

**.

-these factors, anid they have a defintai, rtstricted shelf-life. knother 4t

--

ý,factor worthy of serious ccnsideration is the storage environment of "tabgratory test specimens. prior f.o their testIilg.

It is well known t9.hat

the storage envi roevuent is a ma.jor influence of bond strength an in~dei., ence, early consideration should be given to rsic n

hwettvNimdity lee to, testing and.',urfgtestng

lurb o the

tshvedring specim'en hindl-ing prior

h

hsregard, the use of solvents -in

i

sto6rage anid test arias sh-puld be avoideed. Ini the-4'nt that ad1hesive-failures~or low strongth bo.1 s still predominate aft" considering the factor-S inthA,. previous para~raph, extensive proces~ftfig studies inay be employed 'For ft proving a: bond sys ten. Liner-propellant failure strengths Increase Viet~ liner tliickneS.4 isiAn-, 'creased. Wh failure mode also tends V5ýard a cohesive Prop~aiat -fai1Lre as the liner thickness increases.- Thu's, one 091nificant processing study isto deteri~ne the iiner-propellkfjt strerigth ard failure~m'm of liner thickness.

Liner cure or prerure prior to propellafIt castng also 2.

At£

as i functlo~n

45

A

Inf1 u Sn

s len liner-propellant &ond strength and failure mode.

Hence, bond

studieS mayhoe performed to detetmihe the optimum liner cure or precue.e; Various diffe'ent surface preparation techniques are elso worth considering. Typlically,-the best bonds are obtained with rubber surfaces which have

Len.

sandblasted and loose .particles i'emoved by dry inert gas flOw, or by acid etching.

Ultraionid cleaning i

dethed ppear to be a reliable means of

cleaning hard surfa~es quch as metalz and plastlcs.• In tht'ase of sand-. :4

bla5te•'surfaces,

care should be exercised in choosing the grltsize to.

avoid damage to the insulation and in maintaining a dry gaS flow for %l.-a

removing loose particles.

Vapor degreasing after sandblasting'is 'not

recommeoded un-less the ptrt has been ;.xpos-ed to the atnosphere-•fr some ,ime, in which case, care must again be exercised to make sure only clean 3

.

*

solvents are used and no solvent remairts on the paft.

Acid etclhing

generally results in excellefit bonding surfaces.

The precaution that must

be takin here .is to control the depth of cheiic•il

4ttack.. hcid etching is

u•desirable ,or'small rubber parts as it is veiry e~sy to sariously degrade "7,

(

the bu'lk rubber pr6perties.,

,

It"is worth rnentioning that slmilIar precautions shoul.d be t~ker, with regard to obtaining good Insulation flap-to-case bonds and flap4o-flap.> bonds.

In these instanc'es, processing studies similar to~th.se mentioned

"above may be undertaken to i0prove a particular bond system;

An additional

.nsderation here' is that of the 'pressure environment durtng cure of the I-Mainta.! uniform.pressure of sufficient magnftdde is of sv ri•ticuWhr impptmnft6- when dealing with pressure-sensitfve adhesive f 4 lms.

n all Taborato,7 processing studies, only those processing ne~ods' which ca.be carripcover.into plantmanufacturing bperatit.s, wittf due. regard for-. Gsts-inWolyed, should be considered.

2.517 ..

-A!I

,

--

,4

*

2.6 NO"ELATURE AA.t

Censtant

IC*

Activation anarg,

constant

EA

It Act'ivation anetry at temporrture T

to

ActivAtion energy at absolute zero Wliriu moftlus

E4a

*CC~leration of gravity

9 *

u-Aoinv

h(-:)

-

Funct on

Aging function

Cuostst.

k

W. *

-l

T

rtmlucto

-

zwo strossistraln tieratuare.".

at

Propeallant 0"r t".ewbtwr

Vlin T *V

a

u

Sconstant&

Stain

WIT*

#(S)

-

* a

~* ~.~

~

o

edocad vartabl* f*or

mrti

Stress

rossink density

2159 ..... . .......

-- I-

%

1 414

7-7

RFFFRFNCFS

1. Lockheed' Prop~lsion CaW&-y, *Engineering Heoods fai Integilty-Analysis," Combined Final Report Contrcts No. AFt,4(&ll )-8013' and Il-04-495-40RD-aW, 144y 1963. 2. Tormey, J. F. and Briltton, S. C., 'Effect of C~yclic LoWadis mn S.&Al Propellant Grain Structures;* AIAA Journal, Vol, I Mo. 8j,.Augut 1%63. 3. Wagnqr, F. R.. *Solid Rocket Load Defini1tion. Stui4y.. Thembibratior Envlronment,". Solid Rocket Structural Integrity Informiation Center, College of R1'4).Contrat No. F04011.Engineering, Lhjfvrsity df Utah, AFWTR-6-Lj 67-C-004), Noevember '1969.,1 _Lg*osjons in~ 4. Bcwdenr F.- P. and Yvffev, A. D). i Ini ti ati a~GG Liggids and Solids. Cpmr3irdge Unierity resso 9.12.Q 5. Cook, M.A., The Science of High Explosives,. Reinhold Publishing Corpora-toNew o. .6.

rwin, 0. R,, Salriwa, P. K. ElwellI I S. -,andValo~rH. -H, "Shock inSth Seminar on the Semsitlvity of' Solid C~wipo-11te.PropeaU. Sunsitivtky.of ?ew Hateftals. MI.) 01A Pu '1ication go. 1509 July 1967.

7. 'Pratt, -T.,H, "Initiation of iWtnation iro Solid Prwellhts," Tchn~ical Report -1?77, Pohm*Woi Rfas Co~i*, U6e' 98 8. ftapaweiky, R.- S ; "Sensitivity of Ekplosiye Sy~steý tq-A toatfon nod' Sudttoni1cn Reactions,* ITT Rese'arch Ins-titute. fChlca oIIlois, 1964S.

..

T. -A. arnd-Tulis, &;J.'; Shck-ti"i 15el ttI vity Testing5, Sth SEr'sons, 4e,-Sfinar on the Sensiti-vi-t of lkw Kaftlhisl 1%0)0 CPI PtbllcatIoln. 'ft.

150, J3uly'967-.

..

-

-

,

X-and William, H. L. , '1Erivtc_*svrnwiit of~ Shock Intensity Secor, G,~ A ill viscotlastlc M~aterials," Final Rkpoft-tao tISAW, Redst~ -Arsenal,, Alabama for eontract No. DAAN 0'1-67-C-1441. Uniersty of UtaoA October 1969. Fracture of Simul&ted Solid Prwpllant," in Ti. -Secor, Ca. A., H nrwi, "The Chemistry and. Mchanics of Comba&stion vith Application to Rocket, Engine 5y'stemis,".. LTEC TH 7046~4, Univarsity of Utah, Hfovew~er 1970. 10.

12.

Thacher; J'. H., "'Structural Analysi~s of a porto f-teCsDube 91se- Mnufac~tri"n Process ,~ ultno te5hMeiqq~ of the 1QQPfi vwrking eroup on W.chancalt Gehpvor,* CiK'A Ptblication*1W. 119,1_ Vol. 1, pp. 457-471, October '1966.-

13.

Cos-% T. 1., Analyti cal Methods for Qeft~riinlng the Shrinkaoa $tresses "Itioyu wr-cT KAterlals During Ciure," Technical Report S-72, (Contracts No.1' fAH01-67-6-0947 and i)AAiHOl-68-C089) tDcewbfrr 1968. -.-- k 2.60

-..

--.

1.':4;g'

(3.

.

.4Xl9,

Sold ocetkoor,"AFRPLiR7-1 UE DO 71-041),, College of Engneeing Unverityof Utah,, (Contract No. F04611-70-10-0006), ~ cGati '.i ns ofi~Cumulative Damage in the o. Paaetr& Prearaionof Parmetic rai DeignCuresa~nd the Prediction ofGainFaiui~~asPresurzaton, Reort1341-26F, AerojetGenbral Coprto,(otatN.N00017-69-C-4423), 'August 1970. 16. ishanN.,"Eniromenal ffets n SlidPropellants," Feature Rcke St~turl Itegi~yAbstracts, Vol. 3, No. 1, ArtileSold R.,Th Crack Criiclit.i 1-22,f

PiI s. .W..arato

SL

~(

*17.

Annmu,*IRGSldPropellant Mechanical Behavior Manual," CP'IA

)18.

Kelley, F. Me, *MSolid Propellant Mechanical Properties Te~sting, Failure, Crteria nd gin,"Advancei in Chemistry Series, Number 88, HauacueFzards and Testing," -pp. 186-243, American Chemical Soit,1969. OPropellants,

19. 'CO1obw, P. C. 'AndtKthm G.-F., *Hmfidtidty and Temperatume Effects on tw-Creep Behavior ofSldPropellants." Bulletin of the 4th Meeting .'lteICRPG Working rg on Mechanical Behavior, CPIA Publication pp J-40, October 1965. -ft -M ,Vo=., 20.

Dickinson, L. A.., Atschuleo, M.H., McClay, R. E.,'and Tice, H. B., "Case Banding Technology -A Review of Selected Applicatlop,ns(U)," Proc. -f -the - nuainand Cae-Bonc:n S osium (U), CPIA Publicatio-n

21.

Stensen,, R., '$Che"~~ and Physical Factors Governfihg the Storage Life of- Solid Prpel lant Itacet Nftors,TM AIMA Paper No. 68-526, ICRPGI AIM 3rd SolidAu Conference, Atlantic6 City, N&~ Jersey, June 4-6, 1958,-

,4.Kim,.C. S., "Mechano-Chemical.Effects in Propellant Binder Aging -

(U),"

BullIeti n Lf the 7th -1.CRPG Methanical Behavior Working Groop Meeting (U), CPIA ubilcation No. 177. pp. 303-314, October, 968.

iii of23. M~osher, 1. P., ýgin Mechanisms in CTPBPropellants (IJ),M the 7th ICRPGQMc hcal , Behavior Workfng Grogp Meeting. (U), CPIA

-

24.

Beavior

No.PIA7

Puliaio97M13,Vl.I

Hart, W. D.,Briggs, W. E.an dF ran-z; W'K. , Experi 46 til Investigations Wrig groMeing, anlTBPoelnsmulletin of the 8th JANNAF NechanclBhvo No.90 I.p.'9-5,Mrc eia9CPIA 193, Vol. Bhvo okn rM Publication 2-3, ad 9-.61 2p

2.61

.............

26,

Butt-in, S.:-C., et al, "~Engineering Analysis of Systemi Surveillance Pro-% grams," Bulletin of the 8th JANNAF Mechanical BehaviorlWorking Group Meeting71CPIA P1cation No. 193, Vol. Itpp. 297-318-t Morch.1970.

27. Thadher, J. H.,*et al, "Structural Evaluation and Characterization of Materials, Stibsystems, and System5 in a Solid Rocket Surveillance-, Program," Bulletin of the 8th JANNAF Mechanical Behavior' Wo,41nj

Group %e Q.

28.

PAubllcation-, Na. 193t Viol. 89-pp. 319-33u,

Leeminiq, H.,, et al,.. "Service Life Prediction and Verification," Bulletin ,)f the 8th J-ANNAF r~echanical Behavior Working Group Meeting CPI UDii~Icationr No. 193, o61&t p. 331-M42. March 1970.

.29. Vqers, J. Lo., et al, "Organization and ftnagemint of Aging and Surveillance Programs," bulletin of the 8th JANNAF Mechanical Behavior Workint Group neetingI cvA -~~o~nMiu1939 ve"N nn 343-3, fa'ch 1970.. 30.*- Lefming, No, etls f"S-olid Propellant Structural Test',Vehlccle, Cuiml ative Damage and Systems Analysis," L-*I Report No. AFRPL-TR-68-13'j, Contract F0461 l-67-C-O0O0), Lockhed Propuisjuai C.:Manv. Redlands, Cal ifornl a, _,ctober 1968. 31.

Leeming, No, et al, *Solid Propellant Structural Test Vehicle and Systems W ysis." Final Report No. AFRPL-oTR-70-lO, Contract No. F04611-69-C002, Lockheejd Propu~lson Company, -Redlands, California, March 1970.

32.

Oberth, A. E. and Bruenmer, R. S's $."The Cause of Mois-ture Embrititlement in Solid Pr.*ellants,," Bulletin of the 51.h ICRPS.Madhanical Behavior Workng rowMeetng.CrA Publication flo. -113t VoL l--1, pp. 43-52~,

33. Olds, R. N. and Tho~son,, A. R.,-"Comparison of Operational Motor and~ AcceleratedAjth2g -Saple Propellant Properties," Builletin of the 5jth F~IA PublicaICRPG Mechanical bifivlor Workfng Group Naeetliý- (D tion No. ,J59, "ol 1, pp. 1-14, Octoer 1967. 34.

Bennett,, 5.3. and Lay~ton, Lo Not, -Agi ng Effe~t o . d-Pi4 pe1l-antin Laboratory Samples and Full Scale Motors ('J)," Sulletinf of the 7th ICRPG Mechanical Behavior Working Group Meeting (UT;M1X Public-ati-on Wo 117, pp. 339.130, Octobr%16. -

35.

Pi ckett, N. F., "Chbracterlzation of C5APropellant," NAYWEPS Report 901.3 (INdS TP 3997), Q.. S. Naval Ori~ance Test Station, China Lake, Cali fornta, 'April, 1966. 1.1.

Hart, W. Do and Brigq4, W., E., Ekperlmeotal- and -Theoeti~Correlation of Aging Effects Between Laboratory 'and Analog Motorst AIMK Paper No. 68-527,.ICRPG/AIAA 3rd Solid Propulsion Conference, Atlantic City, New Jersey, June 4-6, 1968.. 37. DeWitt, 1.L., o~etermlnation of Concentration Gradlenf of-Ingredients in Aged Propellanti," Prssented-at-the 25th ICRPG Meeting. of the Working Group on Analytical Chemistry, Picatiniiy Arsenal , iw Jersey, June 1968. 2.62

36.

C

.

*.

A

38. Anonymous, "Service Lifi Improvemmnt Program." 'Final Report TASK 5, TWR-2984, Contract NO. AF 04(694)-926, Thiokol Chemical Corporation, -Wasatch Division, Brigham City, Utah, 20 August 1968.

*

319.

Miller,-W. H.and Fulbright, J.L., "Variation of Propellant Mechanical Properties Near PrplatLnr Restrictor, Insulation Interfaces," (U), Proc. of the Insulation and Case'onding S-ymposium (U), CPIA Publicaffin W. 159, pp. 153-M7, November 1967.

_40.

FulbrightC J. Li-and Miller 8~. -H.. "Failure Analysis of Sol'id Propellant Graihs Based on Dissected Motor Properties,! Bulletin of the 6th ICRPG Mechanical Behavior. Working Group Meeting, CPIA Publication ~'16. la T1 pp. 377_392, October '1967.

.41.

Lohr, J.3., Wilson, 0.E., Hamaker,-F. M.,, Stewart, W. J., "Accelerated Testing of th~e Mechanical and Thermal Iptegrity of Polymeric Materials," .J.Spacecraft Rockets, Vol. 5,No. 1,-pp.' 68-74, 1968. Moon, E.L.., "Samsonov,-A.,.and-Hyers,,J. L.~, "Revised Minutemnan Service Life Estimating Procedure," TRW 12138-6D002-RO-O0, Contract 404701-68-C-0327, TRW Systems Group, San Bernardino,, California

42.

31 March 169M

__-----

43.

tj'ancis,, E.C.and Carlton, H., uA PWAA/A Propellant'Surveillance Report (U)," bulletin of the 7th ICRPG Mechanical Behavior Working Group Meeting (U),"*CPIA Publication No. 177, pp. 327-338, October 1968.

44.

Vilts P. M., "Surviillance Prograui~for Stage 11 Minuteman," Bulletin of the 8th 3ANNWJ Mechanical BehaVior dr-king Group Meetina CTI Frubl~icatolE 193, Vo51. T1, 7pp. 255,,266, March 1970. :., Fife, W.B. and Wel-I, R.-D., Iproved Insulation for Advanced W~eapon Systems (U),' Proc.. of the Insulation and Case Bonding §,iim CPIA Publication N.59, pp. 21-42,, Noveer 1967. Whelan, W.P., J1%, Van Duskuk, P. R. and Kiley, L Y "New Materials for Solid Propulsion Combustion Chamber Insulation IU)," Proc. of -- the Insulation and Case-Bonding Sympsium (U), CPIA Pubiica-tio-nNo. 159, pp.. 3-20, KNoemer 1957.

*

45. *46.

*41.

Greever, W.L., "An Extended"Temperature Rafnge Case Bond S stem CHOB Propellants (U),"l Proc. of the Insulation an aeBndnM im CPIA Publication No. 159,, pp. 117-129, Nove 1967.

48.

Vriesen, C.W.and Schloss, H. R., "Adhesive Study - Case Bonding Sol-id Rocket Motors (U)," Proc. of the Insulation and Case Bon-ding 12 vme 97 SyMposium, CPIA Pulcto o 5 Eyring, H. and Eyring, E. M.,- Modern Chemical Kinetics," Reinhold Publishing Corp., 'Tc9Y.,

49. 50.

Bartehev, G. M. and Zuyev, Yu. S,., "Stren'.ith arnd Failure of Viscoelastic Materials," (translated b-T.-7a .iiiP. Jaray), erain Press, New York 1-968.

51.

Ree, T.and Eyring, H.,, "T heory on Non-Nextonian Flow, 1. Solid Plastic System," J.-Appl_~.*._Py Vol. 26, ~pp 793-809x 1955. 4

52. Fitzgerald, J. E., "Theimoec1hncs of Nonl1inear PoTymers under Nonequilibrium Processes," Presented-at the Winter Meeting, Soc. .Rheology, February 11971 (submitted for publicatfon-w-iPm'. SOC.RheologyY*.

IiMolecular 53.

Lubliner, J., "Rheological Models for Tim~e-Variable Materials," Nuclear Engineer'ing and Qesic2n, Vol. 4, pp. 287-291, 1966.

54.

Fulmer, G. E.,

55.

II I Ft

"tivvirowment, inAddition-to Stress, Temperatwre and HWeight as a R~educed Variable,.in Environmental-. Stress Cracking," Pol. 5M~. -Sd... pp. M-0194,' 19067. Fulmer, G.E., "Kinetics of Environmental Stress Cracking," Prociedings of the Fifth International Congress on Rheology, Kyoto, Japan, 1968.

56.

Stouffer, D.C.aO~ Wineman, A.S., "Constitutive Representation for

57.

Kelley, F. N.adWilliams, 'Ij.L., "The Engineering of.Polymers for Mechanical'Behavior," Rubber Chem. and Tech., Vol. 42, pp. 1175-1185, 1969.

'Lindar Aging Enviromiental-Dependent Viscoelastic Materials," Paper presented at the Winter Meeting of the Society of Rheology, Un~ve),s~ity- of Utah, Salt Lake City, 1-3 February 1971.

68. Wli M.L.end Kelley,, F.N., "Application of the Interaction Matr ixMthod to6 Solid Rock~et Design," Bulletin of 'the8th MeetIng of tho JANNA Mechanical Behavior Vorking, Gruk, PTublication No.

1_

_

_

_

_

_

__._,_

_

_

_

_

_

_

_

_

5. Briar, H. P.'s 'Raleatiohships Between Propellant Failure Times in Vaiuia s; aodeso Failure," Bulletin ofthe 6th Meeting of the ICRPG

q

etDt ()"Bletno 61.:.W

-1 I

17ndT.R

137-_

h'lhIRGMechanical o UCI Workin ubiainn.18 Behavior __48____Marc_

Brar0... Wiegand, T. A., n "Ard%-to StatistoLicealp prait FailurefrmLbaty Criteriata Bulletin of th~ r e'7tb f ICRPGMehnclBavo h Workin ru GoupMehang, (0)v~o, V PIA NuBlication No. 6177, p 31-2$ ol. o 1968. 48,Oto~l~4 61. lyir. J.L.,

nd Mon,

th Poplio Mnuem Sstm, 3ulet fh8ý AIA 2.6L,"evc4 ~

rdi~inPorm

o

L' A,, Morrill, L. G. and 3ersche, C. V., "Predicting Propellant Storage Life by.Svperposition," Bulletin of the 5th ICRPG Mechanical Behavior Working Group, CPIA PublIcation No. 19, IFT-T, pp-. Oc~toer16.

Maitit,

63. • 64. ,.j

65. --'

"66.

Planck, II.W.,((PAnother Look at Predictions of the Service Life of Propellant Grains from Laboratory Test Data on Aged Specimens," Bulletin of the 6th ICRPG Mechanical Behavior Working Group Meeting CPIA Publication T,-14, octo-befr 77. 967. {'J Chappell, R., N.', Jensen, F. R. and Burton, R. W., "Statistical Service Life Prediction: Minutemzri Third-Stage Propellant Grain," J. Spacecraft Rockets, Vol. 5, No. 1, pp. 42-44, 1969. Bfils, K. W., Jr. and Steele, R.- D., "Effects of Chemical Change and Time-Dependent Response and Failure Properties on Grain Storage Life (U)," Bulletin of the 7th ICRPG Mechanical Behavior Working Group Meeting (U), CPIA Publication No. 177, pp. 351-358, October -1968.

67.

Majerus, J. N.', Bual, 'qF. and Wiegano, J. H., "Behavior and Varia, bility ',\Solid Prd)lants and Criteria for Failure and for Rejection,"'\,. Spacecraft Rockets,' Vol. 2, pp. 883-845, 1965.

68.

Layton, H. L. an6\Bennett, S. J., "A Fracture Mechanics Approach to Surveillance," Bull1tin of the 8th JANNAF Mechanical Behavior Working Group' Meetin, PA"Publication No. 193, Voi. ,p 20926,rch 1970.

-.69.

White, B. B.4 "Case Cutting'Techniques for Rocket Motor Dissection," Bulletin of the 6th ICRPG Mechanical Behavior Working Group Meeting, --PA Publicatlon No. 158, Vol. I, pp.307-528, Octo-br 1967.

70.

Briggs, W. E. and Hart, W. D., "A Special Miniature Specii=en for evaluating Propellant," Bulletin of the 8th JANNAF Mechanical Behavior Workin• Group Meeting, CIA P'ublication No. 193, Vol. 1, pp. 5 4 -, March 1970.

71.

Robinson, C. N., Graham, P. H. and Sturms, C. E., "A Microtest Specimen for Evaluation of Propellant Tensile Properties," Bulletin' of the 5th ItRPG Fibhanical Behavior Working Group Meeting, tPFT7o'.'115, Vol. 1, pp. 65-82, O'ctober 1966.

72.

Leeming, H. and Anderson, G., "Nondestructive Relaxation Modulus Measurement,' Bulletin of the 8th Meeting of the JANNAF Mechanical Behavior Working Groyu, CPIA Publication No. '9, M.Vo7, March 1970.

73.

Jones, W. R., Jr., "Cohesive c.aid Adhesive Polymer Fracture Investigation," Ph.D. Thesis, University of Utah, June 1970.

74.

Williams, M. L., "The Continuum Interpretation for Fracture and Adhesion," J. Appl. Pol. Sci., Vol. 13, pp. 29-40, 1969. 2.65 --

III. 3.11

PRELIMINARY DESIG! ANALYSIS

INTRODUCTION

A preliminary design analysis of a prospective randidate motor configuration determines if a given grain design has merit and possibl-y

I.the

gives qualitative or se~i-quantitative indications of hiow the design may'be structy laly improved. At this state in the analysis of a solid propell~nt grain appr~oxlmations and simplifying assumlptidns in

analysis methods are war'a-ptWd.

Design data sheets and approximate

engineerihng formulas ara rae-otmer-ded for the analy';is Of convefltiQnalap motor designs. Extensive num~erical analyses at'this level are unwarranted. The add~itional accuracy galned from using a computer analysis is d'ften unjustified in view of possible approximations made regarding, s~ai material properties or 4faildre data, and also, the uncertainty 'ft f1inal design configuration does not jtmtify the expefise of conputer, analyses.

An excep tion may exist ini the case of new or novel grain

designs, in which case, comruter an~alyses may be re~uired.

In these

cases, and particularly in the case of radically new graiin desiýKs, development of new analyses coupl~d with experimental su:)sr.le motor

tests is recommended in place 6f relying on computer analyses of questionable applicability.

.3.

Iqtte, Iol~lo~ing sections, -approxim~ate-erigineering tnays~is pethodi' are coithe. oiad~ ing, rdthis c ns

>r

methods cosiird

es strai

.a'nd

n~hi

cat~

ihcvssed in Chap'ter 2, The. api~roximnateof formulas for calculating-stress

deflections' for thick-walle~ holipw cylinders.

Empirical ly~

derived.re'latidrships~fordetemnining stress concentration factors for slot,ted and star .c~bnfigurations, and 4 ctifes o~f fAni'te lengtW~end correction facorý,are incladed. The presentAtion of this material has beerparmtr ized in.ter~s -ofj web fractions and length to diarnet& i'aýaios7.

Beca~se of the

approximate

Iprelimin-ary ,naturo of the 'analysis mieth',dq discussadl4,ný-is

chapter only

,eexpressiont used for determining maxinibm yalues of stress,

strairp and deflectitil are 91ven., These valu~s are sufficient for a pr'eliminary -design-';analy s7. ~ rofiles of stress, strain ana deformation as a functfon of le'ngth for fini~te ktngt -hollow cylinders have been obtained by me~s~~ mtedifference~sOuTtions *.0 the equations of'elasticity formulated in termns of- 5Sruthwell stress functions. Jhese resultsý -re contained in the fonnof pavitnetric cu~rves in ref-trences 1, 2 and 3.

I

*Extesv param~etrit curves-*whfch :,.ifcinl ~

sipiyeiiay

design analyses are presented- inAppend x C.

*

The pl~opallaat is,assuned to be in~omipressible. Ifhe influence'of ratio, on. stresG and stranrsos

/Poisson~s.

is discussed in a subsequent"ý

ch-Aier

SI

The analys~is meth6ds. ptesented here are based on infihitesimal linear, tlstciy hery

hepetien

e~aons of elasticity are sunimarized,

in,Appendixk '~to this handbook.. Indicotions of'how time~,and temperature effecti may'be incorporated are also discussed.

For the-most part, because

of the\oreliminary nature of a pteiirninary design analysis, threse mouifilr cattons areapnt called for-et the preliminary design stage. ,

3.2

--

A

V'

3.2

TfPWi•TURA LOADINGS Therm.l stresses and strains, as'discussed in Chapter 2, result

from a difference in linear and bulk coefficients of thermall-expaions between the propellant and thie motor case.

-o

Typical values of the

bredticed" coefficient of (linear) thermal expansibn for pol1.Sutadiene .

!listed below I aPrJ double base propellants for various case materials are PUKEDED COEFFICIENT OF THERMAL

XPA • SGN,

Pr"pel i ant

Case' -Material"

Base

tDouble

'

. 25

Steel

.

Alw.i num.

4 x 10,

Fierlas"

5xO1-5

" ylon 66

,

7.

.,.

n7"'

-F

X" ".

95 x

5

3.7 x 10-.

x.

These values of a. are represntative of manhy propellant* in these two classes of'propellants.

*The use of tne above values is reccmnended

.'when thermal expansion datV-kn the particular propellant being considered Most-prdpel!Ants will not have .reducqd coefficients of

,isunavailable:

thenmal expansion, which vary more than.± ten percentSfrom the above value3. In performing themal' strs-ss aqd strain analyses, the calculations

"may b rerfeered to the pjopel lant' cure'temperature and cure shrinkage stresses and strains superposed, or the calculations may Se referred to the zero stress/strain temPerature of the propellant, T.

This tempera-

ture is deflncd to be the terioerature at which thermally induced stresses

"The 'reduced"

coefficient of thermal expansion occurs frequently it, thermal stress: and strain analyges and is defined by the relation 0R =

p

[(T+

-

Vc)i(l tV/

•c. 3.3

i

*

.

'

.

'

I

-

a,

Iand

strains va~nisji. 'Beau~e of, pr~opa ant shrihnkag'a during cure, the

z~ro st~ress/strain temp&4ýure will 'De hither than the citire tarperature. Iti~ofeniire -convenieiit t o,

keh '&r0

tsiT~r~ tr

re

as the reference for thermdl. stre.ýs'ahalysis,

'

The tlio eratu~e TVmay be Convenlentiy determiined by several tech~niqties.

Qme met~od is to' subtract the Oquivalent.'temfperature'dacrease,

asso~ciated with the cure shrinkage from the propell ant pure taperature'

'C

p

where T i's the propellant cure tam~peraiure-, a Us the propiefllnt linear Scoeffi ci.ept of thermitl expansfon and a -is-the net volvmevric cure shrinkage.

The net volumetric shrinkige of polybutadiene pr~opellants istypically

* .'002 and that of .slurrisy cast doub1l basi propel ;ants O.095. The shrinkse. of *

conventional c~ast double base prcpeilanta.1s considerably leissas iridi~cated in ~tle Oreviouq chajitrx.1the-se Values are suffi-,cietLtlY ýr'presen~tative to be valid n preimina7.r si-y-ss Alternatively, the zero's~tress/straiin tenperature may be, detevnidned

from~ analogue.or subscale motor tests. Inthe~e testi-the twemre.at~re of the cure4 motdr is-flowlyraised above ~scu're temerature and tes~~ents of-the internal coiffigurat~dh versus tompdrature are recorded. The temperature at whichf the internal gmtwnitry of the miotpr. coinocides wi~th the ori~ina1,z mandrel ctrtfiguratfon is then 4efined to be., the zero stress/strain' teperature. iReasurienetts m~ade in this mannir -Cir'icate that

isZypically ISOF higher than the liropellant cure ton-

Penipture for pollybutadi ene propellIants and -?2"F highar yeT-'double -base~ j4.5*

propellants -

These temperature increa~es are Inciose agreement ~~3.4

'x

'wi tb those calculated u~ing equation'(3.1) anid the values givern aboveP for typical volumetric,cure shrinkage of these propellants. These values

can be safely Uised in preli~minary design ana~lyses since there is little vafriation for a large number of propellants.*3.2.1

SHRINKAGE DUPING C1uRE The volutnetrlr-cure shrink~age of most polybuain

rpiat

n

sci~,Joub~e' base p~ropel Iants can be adequately described by the relat-ion,.

wihere a and'o are ,experimentally determi ned constants.

The cotistanit

represents the n~et vouerc'~rnk~age, and the prodUct ag" represents the initial rate of shrlprkage. The rnet volumetric shrinkage &tis usually g

mnot 49plicitly temperatuire.dapende Z,however, the shrinkage I-ate Qs. is strongly temperature dependent sinc plyuierization, in general, is a

-

themallIy Aat+vatel process .whicthi, i2dai ly adequaieTy described by first qrder.1kinetic theory.

Procedures for. performing cure shrinkage

tests* are- not.readiiy available in the open literature, however, a cQ'Pn

*technlqui~

ma~s. use of amercu~y dil tometer submerged' ina constant --temperature bath. -The rise of-the mercury column-ifll a capillary is

monitored using a cathetometer. Detr~nnaion~-fsue srinag stresses in. a-solid rocket-motor -Xc

repretents a difficult task which is beyond the current state-of-the-art ca~pabilities. ..Reccntly, ~howev~er, Cost [6) due n nlyia approach which may .prpve to bee worthwhile in the -futurf for determining [6) shrinkage stresses. In this report, Cost treated curifig polymers

3,5ý

-

6

unider 1sothefrml and sted~y-'state thermO& conditionfs, 'and formiulated ~~the basic govertiing--idifferential- eqmtivs -in -ems-f4-frn4l--

[..

nrperators'. *The mtt.erial properties wer'e treated. as functions of space, -time, tempeiaturpi and degree of cuire.

imdicattibhs -of how m~aterial be-

havior may be related to molecuarl paftmeters. were giv&n, b~ased on Buectie's theo'q of wlcul4ar viscoelastic-ity. Such a thieory may seerve as ,a-gui delitne for futuzre developwnts in this area, however, lacking experimren~tal ve~rification, these developmwnt~s should be considered to0 k, of .apreliminary nature.

Develhcnent of analy~1s techniques for

cure-shrinktge stressts will quite 14kelv be necessary in the fuiture with the evolution of izord ci~plicated and more highly coostrained fgrain ,Configuratlons.

pr thte present state-of-th-r #r~ ztor configurations, adverse *

.effects

of cluret shrinkagi cans -for-the-Most-part, bel-Wided in preimiý-

nary andfinral 'des ign, analysis Ohves.through experiance anPd'egineez1P9 jUdgent bASed 06~ tht gu0-delindt OU~-nd ifl 1 hater 2.-

typicallymoiielwtprtueccig

Fo

tvnibowtr

configir'otions, the. critical ar'eas of analysis are the. i nner bore and the.

case

grain

tatdnain-bf,

POT~ts

In the case of ver-y high m~ass fraction m~otors, tht radial compcnent of theý case-grain interfacial stress at thefotorlndplane may be the ltjo~t2 ing destgn parawiter.

.3.6

-A

urni~~~i

w en d1VjI w

1WViiiu

cyc~ling effects~, the stresses-

6nd strains'iVdetem,ined onily for 3 11ott t~aiperature soak of the grain.Gling v effects ai-e then accounted- for Ur the failure analysis using

some cumulative-dmage-~rifie.

This is the procedure adopted here for

preliia ry desIgiI dnalyiyý~. in. p'rforfngitheruWs-traln orthermal~ straes~ analyses, a nu.mober of s~itopiify~rig asswivptions are noniiaily Introduced. First, the grain geomietry is-idealized to-in-nflinite length hollow cylinder. Correction *for

81it24* fattors.

geometry--and finite length are irtroducea as multiplicative It is also typtcally assumned that the case is infinitel~y rigid.

Physi~aallYothis assumrption is equivailent to assuani ng -that the ratio U AE,/E is n~ejlioble compared_ to one, where b denotes -the outer grain radius, h the-case thickness and _E--wWE~ are Young's Moduli of the propel Tarit. and the case reipectively.-..On.- also nobnnally- assuaneta .uhifonn temperature distributicn throughout tbe grain.

This is equivalent to

S assv: n~,- that tht graiii -is very slewly =,ped. Th-last assumiption typi Cal ly -11a&d1 -tha.t

p.p e"it

is incompressit4I-.--(i~e.,v

1/2).

The assumption of,-cm.aaicail incom~pressibility norma~ly leads to an inconsistený:y wheai Poisscn's-'ratio, v, the elastic modulus E and line,ar .t co~efficient of expansion a (or equivalently the bu'lk- modulio K(and bitlR coafficitint of-,exoansion 3ci) 1%re. treated as independent quzantities., For Incompressible materials, then-rMedynamic rest~rictions ~require that vanish.

Thus, the developm~ent of thermal stresses and strains ft icm

presiible material is ex.-1uded.. In this cise one t~ypi~cally maka the ad hoc assumptiorithat. as but the quantity E

;I.½the bulk modulus K-becoomes infinite,

U3(l,- V) rem~ains finite and cr is also assumed to be 3.7

-

nonzero.

can bs ,'efOved most easily byr*eformu,

These inconsistencip

at4on -vtlthead •c Introduction of

"-I,•ttiigthe-bets•r-thi•oel-stk rlatfon.

a Gri•nisenf

Spec-flcally, the limit m

i .3 1

)1 V

K

-3

----

i- assume-4 to approach a fi.•ite value as

v4ga M. ct-#,3

co•u•re

i

or,

cryst~l i ttice strtctas, the quantl;{• - 1htmnt) is known.as the Gr:uinisen ýorts

existence is demonstrated throagh consideration

the nonliar vle-depenece of the -fequihqy ofa lattice vibration

....

of specified wave'vector;

C81 .. '

The existence of-t~he llmit-(3.3) for

amorpheus matrials Is based on thero

The ustful-

myraca ag uments

ess bf the .relation (3.3).n performing themal stress analyses has been

4.suggstel bY

Freudenthal

and Fitzgerald

It is worth notin§ thai'th'

forwatitio:.d inconsistencies do not

exist in current firnite element computer programs which incorporate reformulation[ 12J.

Herrmann's

This -is due to-the-fact that PoissWst"

ratlo is not taken to be It' in these programs.

-

Instead, the assouptionr

>2• whir X and-i are the Lame constants, and .mean as additional uraknoun at pressure function f =u3ej/2M(1+v) -ýs introduced is made that

eac

element, wheree, is bhe first stress invariant.

V-01. Hermanti's refomulatlon can be shOcw

in the limit as

to be equivalent to (3.31)

the relation (3.3l, in peeio tmng- thermal stress analyses The use o0f a later section of solid ,-ocket ftors is discussed in greater detail tw% of this handbook.

In the fdllowing s'ctlons, for the most part, the

3.8 S..

... ,,.•

... .•-

.

-

•

I.m

results will be based on the asumptibns of Incompr~essibility and a non. ze.coef c,4ont¢-f exuparsion-a.

Althugh inconsistent, these assmtptiOns

-- ,------have led to rfsults which have been proyea-measn-b~ly adequate in the -d6Ig-npurposes.

past for.prelii4na

•-•1•~t

CYLINDER

.

S°JdeP the assumptions stated above, the maximum inner bore hoop strain is independent of material ph'sical *_peties

loge{l

c(a)

and is given by [4,5,13,14,15]

(3.4)

+ 'c'(a))-, e

e~

where (3Y2) aR X2 AT,

- "(a)

(3.5)

and Q!R

'

p

-

2/3(1+v)

a ac

propellant reduced,

coefficient'of linear expansion x

b/a - Ratio of graln outer radius to innesr bore radius

rATT1

-

T -Temperature

decrement from zero stressistrain

from zero stress/strain temperature T,. The natural logarithm of the hoop strain has been introduced-in equation (3.4) II for calculation of the actual hoop strains since measurements.on coled analogue m•tors have indlcateý that the hoop strain is better described in terms of natural strain

45

.

The relationship

between c (a) and e.(a) is shown in figure 1, whereit-is seen that the

-

3.9 . ..

4 17! ýT 7-

44 t

:7, 14 ±t±t

tt-4--

-4 t+

.4 f fT ;4 ,_ ý_

I

I-

t-i

I

1 .1

411

1

I

11

4-T T TIT

t

UIT T*

41'.

. ý__,+

1-n

_Z:4

Ar

Tiff' H, I

j! txz TMW

7ýqr 7 _T

Tt

4 +10ý1

5-

T

.....-----

aT, ý4+

tit

I ®r

+r-7 V-Z 7=7

.4--

fl:,=

4- 14. -4ý

-4-

14

7. u L 147W

1.1

.........

Ttt

ýfll

O il-

..

_T -T"

.4t

tt

?"

ir

t4.

deviation associated with using equation (3.5) reaches about 10 percent .......... at- a hoop s racent ; For -small strains, the difference betveen (3.4) and (3,5) is negligible. In arriving "at equation (3.5) a c'•dition-of Pl?.ne strun has been assumed. If a condition of so-called "generSlized" plane straini • aseneralized" plan strain ;assunfd equation (3.5) becomes (3/2) (aRA2-Q)

AT

(3.6)

.

The maximum hoop stra n_.at---lootemperatures .•i-lcted for co!Iventional grain geometries by (3,-6)---tyT-6caly on i6i4 order of ten percent less than that given by (4.5). Equations (3.4)--nd (3.5) areiico ended for preliminar, analyses, -Since UW tend to produce conservall..ve- res

i and the condition ofp

generalized" plane strain is unrealistic

'-FEqtiation (3.5, is* applI cable foijon9 relation between x and gra'

cfircular port grains.

length-to-diameter rato-fL/D

equation (3,5) is valid, bised on nwrerical analyses [lg i-n figure 2.

The

for which -

is shown-

All poinnts, to.the Might of the 6Eecorrespond to geometries

for which (3.5) is applicable. Values of x and L•D corresponding to points to týe -_left of 'the curve represent geometrtiesf.-r which flnite-__

,

grain length correction factors mus; be Applled,

Fihite length coffee-

tlon factors are discussed in a subsequent-section obf--this chapter.' In addition to finite length correctionAn •

td__cor-rec-ti-ons-for

concentration factors for star perforated graifis must also be applied "By ",generaze axial3-strai n.

pare strain is meant a condition of constant (nonzero) --

3.1-

?

-.

.

. _ _ . -. .- . - . . . .

. .. .

.

Tri-77.. TF

72 -

:1,;;.

fl.4

-.1

id ý1-4;ý4il

-4!

P4:

I

lt

IT J;

j.4ý1,Z

7 ta

Ar.

4r'

T-

ir4

lit'

,44

tit Doi

t 1a

.,t.. 41 ,

,

,

IT

ýtz- L

I A. 4T.

4

4.4+

4

I;tl

ýt4 1

if a, mm

44_

MI. LU

to equation" (3.5). V

This correcti'on Is also discussed In a later section I

-of this report.. "

the maximum

Under-the-assumption's leadingto equation (3.5),

"radial component of the stress is the case-grain interfacia' bond stress given by

.

j)

.-.

-E A-);2 4 )

(3.7

The inner-bore- hoop- s'iress is given by

•

.

I&

AT . o~)2ý,2 ,4EP

*

P)

*-(3.

SOLID CYLINDER

.

t

for;.hollow cylinders The prevlous equatlornr have teen de•Y'e'pel under a condition of plane strain. To-dalculate thermal. stresses and.. stra-ins arising in an incomprs'slbie, infinite'length'ssolld cylinder, rigidly resLralned,,the basic theioelastic equations must be modIfled. if meaningful results are to be obtai.ed, since mathenatically 'infinite _s-.-rgidl

es are developed fort-eve•-an-nfitntesimal

temperature change in a

Yestrained,.infinite length h6llow'cylinder.

ThiU result is- due

to the aforementioned inconslstent assumptioni of. the *independence of the bulk coeffilcent of thermal expansion iný th.e bulk compresibillty. There are two methods by %rich an incomipressible solid cylinder can be -,rst, as in th•e •ae of the i ',finite

handled in, a prelfiinary analysis.

le.,gth hollow cylinder. Poisson's ratio may be taken to be.one-hWlf and'. and nonzero

"linear coeffirient

of

thermal

expansi•n

with the rjgld case replaced by a flexible thin case.

assumed

..

Under ,hese con-•"

dltions ,th maximm.. -radial bon(A stress is given by

AhE 01AT r(b)

c3.9) 2(i-V2 )b -

3.13

I-"k

~_er

For a steel case 4h a qra~n,.radius to case thickness of 100 and a "typ 'e.al vilue of aR taken f!bm the. table on page 2, this equation repre. tress of 25 psiV0 F.

sent.5 a 6

The radial strain for this condition

is constant independent of ýaterial physical properties and' radial Dosition and is given by 0 .0y"

"(r)=. (ý/2) a AT r. pr

'An alternate approach to this problem is.-tueformulate the basichoc intrio-,uctign of, the Gruneisen

,thermlastic equations wth the, a

relatlor

6.3);:

In tthis.case a rigid case may be treated with the

maximum radial ,bond stress given by (3.11)

or(b)

whe;.e .6 is' treated as a material "constant. TIn 'the case of a flexible thin case-the maxinnmir

'-A

bond stress ,is ,given by

.

+

"

_R~

The radial stroan is again indeoendent of-,the.radial co•rdinate physical properties, and is given by

, and mataZ I

.r

\\E

C•(r)

f' = (3/2)(ai p-aR,)

(.-3.13)

T

a typilal composite propellaoit with itulk modulus K : 500 ksi

v i,6 x ,l0 _ Of expansion o.p= and linear coeof-Cn. constanta is determ-Nned tn be.841)si/.F.

N.t

3..14

0oV

(0 .F) , the Gruneisen.

Using this value for 'a

-

/

eation (3.1l)g4ve's a'bond stress of 75 psi/*F for a-rigid case, and "equatlol, (3.12) gives a stress of 23, psi/ 0 F for a flexible thin steel

case with b'i" = iO0.

It is seen that equations (3.9) and (3.12). give

equivalent results.for 'the case of a flexible thin tase, as they sfiould. The..re.selts for a rigid, or very thick case, are seen to be aboutithree 'Lime:s greater than thatfor' z thin case. The strain represented by.equation (3.13) is about one-ýenth that "given,by equation (3.10). This is to be expected, however, since the

asstiqption of a rigid case severely limits def lctions! From the above..discuss4ons, it, is appareh-t that for geometries* which can be analyzed using either classital techni-qdes or through introduction of.a GrUneisenrrelation; the results will not be significantly -,different.

The introduction of a 8runeisenoconstraint, howevir, places

the equations ol'.thermoelasticity-6n an -admits-blr•thermodynaMic basis. Ti,.a greater use of a Grimneisen constraint will be for highly. oonfineOi "geomeiries that is, very high. mass fIraction moto~rs with relatively

stiff, or tlick cases. Before extersiveuse can be made of a Grujneisen constanti however, further investigation is required to verify that 0 is indeed a constant.

In 'particular more accurate dete,minations of the

bulk modulus of propellants ad .Vebehavior of tho_ cVefFicient of themeal expansion "inmultiaxial stress statet must be obtained. The analysis described'here for solid ckmers is useful for the. preliminary analysis of, say,(,x_'tridge-loaded end-burn-ing grains.

3.15

f

\

•~

NEND-BONDED HOLLOW CYLINDERf ,The 'enaining grain geometry that is- readliy handled in -a preiimi'nary analysis is the finite lengthhholLaw cylinder erth both ends bonded sUbjected to a u.lfom' temperatuA decrease.

'

•As- in the previow.

cases a nAmber of simplifyfng asstnp6tions can -.e intvvduced. ,First, unpf im end 4ef4ct-at'e assured With no bendiflg. The caseis _a]o t;Nated as belrig rigid, and Poisson'. satio" is t'a ento be 1/2 with a nonzero coefficient of expansica.

Oa.hed on these assw-np-

tions+,I.e pertinent stresses ani strains are 4jiven bY.L1

Ti S%(a} .

?),2

•

(.)E

AT,

-a(r)2.

E

(3.1~4)

,3..

+3X2 1 6T/2

(3.17)

-

£r

c -

PC

,

) 1-1-31 2 ) IT/2 ,

("

(a )

•

- 2(3.9)

.-.

These equations can 0 so be wdified- to inclUdea

rinelsen constraint,

however the resulting 'rations, to a lipear aýproxinmatlon, vield -results whlcboagree buite well. wits the stresses and str&ins Preicted by ssand, •trains s r -, IhteuJsby -quations (3. 14) through (3.19). jor by less than one percent. ..

most ýpplictons,

Sdiffer

results. will

3

3.16

""

// S... .....

.......

.. _-_

.

. .

. . . . . -

.

...

.

.

.

3.2.3

AERODYNAMIC HFATING Aerodynamic., heating stresses and strains can be hzadled in a

,',Tatively easy m.anner, in .a prpliminary design analysis.

The conservative

approaph is to.asstme that the'propellant grain .undergoe's a' step radial displacement corresponding to a -tep tempercture increase at the motor caso.

The resultant stresses and strain9 induced 1n the propellant grain

a?- then ssiperposed with theimal coolin

stresses dnd strains.

Tempjera-

"ture gradients thk ugh the case an'propellant-case interface apd expansion of insulation or lner mater1als are neglected "In th~s approximation. SUnder these loading conditions and asst6ning plane strain cohditions the. ,-tnaxiinu 4

.rdial bond tress due to only aerodynamic heatingris' given by

~(b)

-

u2(,N2-1)

t

E AT /(3+X.)2 )

and the inner bore hoop straim is given by r4,k7

0

'a)

aC A7Tc/(3+'2)

where .AT denote. tho stL, tempereture increase at the motor case. C

?

Equations (32)Ad32)may 4e adde'd to equations (3.,6'

and (3.3)

respectively to Potain the bond stress and oore hocp strai-n in

cooled

pr4eiiant grain subjected to aerodynamic heating: (X2-1)

r( ..

_

.

Ep {aR AT'+ 2ac ATc/(3+X2')}

eo(), = 3aR X AT/2 + 4"• •Tc/(3+x

2

)

(3.23)

.

"I"t,can be slen, from'a comparison of equations (3.20) and (3.21)

/

with (3.,22) and (3.23), that the magnitude of the strestes and strains

I':

3.17 .1°

f,

1:

,7

_

_

_

____

___

.

A

-

f

)

induced by aerodynamic heating will usually notfbe. sigificant. critical factor under

The'J

this loading is the rapid decline p the propel-

lant-case in.te?-factal bond stress capabilities caused by the rapid temperature increase at the propellant cas "nd the time scale of the ,mperatur

interface.. The magnit~de

increase at the propell~nt case

ir~terface is usually minimized through the use of external insulatton, such as'cQrk, on tneenotor case&which ablates and transfers the energy

0

absorbed as teat back'.into the air flov. 1. In'obtaining the above results, the heat transfer problen was'ne*

glected b imposing constant time and spatial variation of the temperature.

Thls.Prob.em is coplicated by 'the fact that for'high velocity

flow6 the acrodynamt heat~hg of 66 boundary layer affects the heat transfer and the frictito)'appreciably. duced when or ionizes

:,high

Further complications ere intro-

temperaturestecome so high that the gas

dissociates

(an unlikely prospect for solid rocket motdrs), or for -very

altitude high velocity flight where themeap fiee path of the mole:-

culb becomes of the o or of the~boundary la&br thickness and the ,

cýiitinuum-treatment is no longer valid (aIso unlikely for solid rocket motors).

At subsonic ovelocities aeodynaic heating is. usually• negli-

gible, .wh.eres at high °speeds, on the other hand, the rate of heat flow to the missile skin increases rougIly lii proportion o the flight velocity (if the surface is Oia-ntained ai consan 't empe-rature). The zpproximations

hnt oduced Above ariiundbubtedly sufficient for RefinknePtiwhich include lntroductlon of

preliminary design purposes.

time and spatial distri butirop Qotemperature are discussed in th

low-

Ing chapter on final design analysis techniques.' S~3,18 f7s

U

--

m

m•

"

* 3.ý3

DVN AMIC LOADS Dynamic loads arp not treated extensively at: the preliminary design

stage since determining the dynamic response'of a solid rd~ket motor involves solution, of a coupled thetynomecharical problei., which is not easily. solved anal~tipall.y.' Simple probleains, such as stan point deflections, and Ini ,o-me cases axis.ymmetric geometries can be dealt w th

S

-through the introduction of. slrpl ifying assuwpti'ons.

3.3.1 SHOCK LOADS Shock loads occur when a solid r-ocket motor is subjected to a severe mechanical Jolt such as dropping during handling or 'shipping. Since these * loads'act over a short . period of time, the propellant normally responds as-an eiasticjnterial with a glassy modulus.

The most severe da'iiage

likely to. occur is unacceptable inelastic deformation of the case. Damage to the propellant grain is usually msinimal, althoug'h the p'Ossibility does exist for grain unbonding at low teniperatv~rs caused by large deflectionsik of the case.

-Stresses and strains in'the propellant

grain-.can be estimated through an approximate conversion of the shock loads to an equivalent gravity'loading, and tubsequently treating shock loading as an, acceleration loading. Por a motor subjected to an axial *shock, the shear stress at the propellant-case interfaqe is given by £ ~

=p

n-g b (X2 -1)/2X2

.

w~iere inertial effacts due to straining have been, neglect.ed aid, p -%Propellant Density

* n-g ;-.Equivalent Acceleration in Gravities. 3.19

(3.24)

-o-

1ypic'a1"-.a

multiplicatve. factor of 3 is appl ie tp iquation (3.24)

in an attempt to account for stress concentrations at grain termination points1[3] Becaus' of the short loading duration, grain defor-matlos. 'will be small ,nd can be neglected under axial shock loadings, latnral, or transverse shock, deformaticns.

A

however, can produce significant grain.

The-maxitfw inner bore hoop strain foý a rigidly itcesed,

incompressible cylindrical grain subjected to a lateral shock is given b(a) ; 3/4 a-

--

(3.25)

urder plane strain conditions. Equations (-. 241) and (3. 25) can be used foro geametriet other than cylindrically •3erforated grains7

In usirg (3.24) an equlyalent circu-

lar port radius can be estimated, or rwre simply, the product of the total propellalt weight and the equivalent acceleration loading can be divided-bythe total-bonded area to give an average shear stress.

In

the case of star perforated grains a multlplicative strain concentration factor may be applied to equation (3.25) for estimating bore strains. ?lo~fications, Introduced due to partial head ýnd bonding are "discussed in a subsequent section. of this chapter. Lateral shock loading of star perforated grains can be treated in similr approximate manner. sa

Assw-ning inccimpressibiliiy and treating

the star point as a catilevered plate; of uniform thicknes- the stress at the point of support is given by-, a

3 n-g p Z2 /h,

(3.26)

"3.20 S.

iN

." I-'+

, :

*

1

where I a I ngth of starpoint, b - averale starpolnt thickness "Normally a factor of 2 Is #ýltiplied into (3. 2C) to account for the stress

zoncontration 'at the point of• support.

The deflection of the innermNt

point of the star tip may-be estimated from the relatlon v'I-

,3 n, QP •

-)

)Ep

h2

(3.27) *

The'results of this section may be used to estimate the gross bd---. havior of zolid rocket motors subjected to shock loads, however, these results must Le regarded as approximate since the true dynamic problem

j

has not been considered. 3.3.2

VIBMATION

Vibration of solid rocket motors is generally recognized as a , .potential structural fnt~eqritdm for applications in which severe

or sustained Sibration enviroents are encountered.

For example,

possible effects of cyclic -loading of solid propellants have been, vividly shown by TQrmey and Britton E17].

They reported on an extreme

amount of disslpative heating and gralnr'damage which occurred in vibration tests of solid propellant rocket motors-. Vibration is not dealt with extensivily at the preliminary design stage, an'J in fact, it has only been recently that fini-te elemertt, cornputer programs have been available for considering vibration in final design analyses.

Simple geometries, however, such as star points

3.21

,

"

----

-

_

-jk

7.j'--

subjected axkl

-

toL

late.ral vibrations and circular pdrt grains sub.jected~to _Sa tgrinsa dealt wi ch In a1n approximate M~anner. I

-thejolldwlng paraq-rap~hs- approxiihate fonwdas are giyen for estimating ampli tude ieat-ios and di ssipati664.>-A -bref genera discussion is also

-anot -as yet-.been completely solved

-- hi le -the -0WlyYsribet.

the comnplex- grrairr geometri-es of -practi-cOl moior syt

-for

ksee refer-

encur-3-F an~d-A 9-for-trevt-ts-end-en-*ttnd~edbitrpvIs zajýalyti~al-solutions- have been-btaiaed 'for simple" systCems whic'i seem to ~ ytre Vwe-.-1 -thexperi ment-,qta (reiferences 20 through 24). Suc6&iiifor.r~aton~WI~flenotcap~le~~ pov~~ngquantitative inforitiatibn foirmor design arad analysis purposes,,can provide valuable insight infowtora property and grain design. charact~eristics which are advantageous fbr

In___ii_-n--t

t

thermoviscoelastic analýyses

al~ difficult eS encountered in unc6upl~d

.i c

rottitically stron§ tempcrature

ernfteol' prop&lfaint mechanical proper'i~ makestvao -wheat gni~a4~n~esu t-itig -frontýyjclic loading very 'sensltv ýe --

tur term.

* -*

Or er-

genie7---tie-t

-nonl inear heat source &-oasr -adhnce h tcouplhd-it sy Moreover, in'rkckei niotor-vl-tration, eng

propellant grain frord the vlbivating case by the ihert~al-t-_ction of the mass of the -orope-1Tant--Vaii4--e--the---accelerating-case: Disregarding Tertila anid--matrial property degradation, large temperature increases aft-ecountý!red whenever the applied stress or strain exce'eds a certain crlticai value. The coiiibination of temperf'ure-dependent properties and

inertia leadis to the poss~thl I'ity, Qf temperature and displacement jump instabi-lities which are simil~ar to the p~ipoouella obs~rved in a nonlinear 3;2

-

-

s pri ng-ibass sy tei in -whWtch.

pn-

twti4

4 -ilaent

Irn considering the vibratioil response of solid rocket motors, two propellant phystcal property parameters are of particuliar significance. hese are the log-log slope of-the reduced relaxation modulus in the ti'me-temperature region of Interest (sl6pe of theccurve log' modulus ve..Aus 'log reduiced tinie) anod the slope of the iogý shift factor, al`ý versus the Vtemperature range of interest,

The ampl-ication facto't' at resonance

(with or witi~out jum~p instability eff~ects) which, along with the ajceleration level, detem~ines ths peak str'ains imposed- -6nf t--prope-1Iant A-i-- a fu~nctioni of -. he relaxaiton modulus slope only. As this slope Jecreases' -(I.e., as the propellant becomies more elastic and lss-v il&isi) dampifig dc-crezses and the propell]ant strain asip~ltude at- resor'an-e 16-;-

creases. .Convarsiiy,

as-

tfts slope increases, cpr~espondinq. to an increase

in viscous: responso, the strains at resonance decrease arnd the reson~nce ---broadens..: Cleirly,-wvith all other factors equal, a propellant witth a

0 *

~

large slope- would be sub~ecte'd to s~aller deformations at "esoi~ance. ~ The slo~pe of the shif~t factor~versus teiip~rature curve is a measure '

of the temperature sensitivity df-th~e viscdelastic properties of a pro-~--~.

pellant.

Since the mechanical Oroperty temperature sensitivity along with

i nerti a-l oadil9 'condl ti ons praduce6 the nonlinear Jump instability 'effect, it follows that for otherwise identicaRl conditions, th4jump stability' effect will be most predominant for propellants which have a large shlft~factor ý'ersus' temperature slope. The above discuss~ion has been concerned onliy with what mlijht be called>k'eversible"' termomechanical effedt's resulting fomi the pr'opellant

*

3.23

ther-ioviscoelastic properties and cycl1o loading conditions. -

"j rreer,-

sible" effects 4.ich include fracture, 4degradation' or decomposition er-" .is and autoignition ,which can be,'theý restilIt of the combined high ,temperature and cyclic strain conditions of the vibration environment have not been accounted for.

"

,

The.e phengmena ame; of course., also of

-paramount impor::a-:e, however the establis'hmnent of a failure criterion for the thermal-v';Dration enviiooment is a difficult task. has shown that for the conditions

Experience

sually encolnftered in solid rocket

motor vibration, fracture-pr severe degradition will usually precede end prevent te "I•j.K

eraire rises to leVels at which autfignitio'n will occur.

Propellant suscaptfbility to fracture, under prescribed Oibration conditi.ons is a significant,,fActor in practical sl•tuations, however, '

and is found to vary s1gizficantly frot proqellant "to propellant as well as for various transient loading-ZlOtlons. The proNb-sns of physical or chemical degradition'forl'the combined the io.,echankca -environment is undoubtedly the most difficult and least understood of the failure mechanisms known to be signiffcant. for vibration of solid propellent. Degiradatloti of'CT$PB popel1ant under sustained vibratiorn has been shown by Tormey and Britton [17.Degradation of other composite propellantcfornnulatiOns has also been shown [24 perience indicates that the|ri

Ex-

is not a .gross difference in the vibration

behavior of double base and conposi-te'propeilants. As mentioned above, the dynamic behavior of simple geometries

can

be approximated in a relatively easy manner using lumped parameter single degree of freedom models.

In the fo'ilowing paragr~aphs the isothermal

-teady-state sO'insoidol vibration behavior of starpoints

and the

-

3.24

--

*--

I

'

case-grain interface under lateral vibration modes, and the propellant grain under an axial vibration miode are discussed.

The presentatign"

of this material essentially parallel's the developmern. in reference 15'.

LATERAL VIBRATION OF A STARPOINT The star point under a lateralsmode of vibration is idealizeid as a massiess cantilevered plate with a concentrated, equivalent mass at ,its free end.

The mbdel is shown in figure 3A.

The dimengi6n in the/z

direction is assumed suffic..ly ldrge so that end-effects-may be mreg-

/ected-

The moel' is excited at the fixee end.

SThe ,quat.oI

pf motion, governing "Pte elastic response of 'the model

is (3.28

kv

".. ,

.

.

.

-

'

Swhere effective mass'of the 1,it.,,ed mass systems, lateai- dplace" nt of the mass, m v . •=latev

displaceein,t of the base support,

k = spring'constdnt of the model' and the superior dots indicate differentiation with respect to. time

t.

'The viscoelastic responstof theimodel is obtained in a straightforWard manner from. the c6cresponding elastic sol tion by replacing' the elastic s prirg constant of the model, k, by a colex spring constant k*'

k. k* -- %(k'1k") ,,

(3.29)

3.25 *

1_

'

--

b

,e

-~

rmre

~~~~

*+V

m.

Vb

*

I

o,~

Y, Vb elt

-

--

VV

X.e

,~b A.

b

LUMPED MASS MODEL OF SLENDER STAR POINT UNDER LATEIAL EXCITATION.

4

v•

-

Y

Y, •s

~B.

Vm

LU,*PZD MASS MODrL. OF CASE AND G AIN LINDEP.-LATEP.AL EXCITATION.

10,1E

-,

SLAB

0

y4 .

-

C.

LIMPED MASS MODE' O /iNFINITE SLA

Vb....t •L

FIGUjRE 3

____________L_

41

ADC:EBNDGRI.,

w i Vibra,,-'ur Modelsm ~3.26

H '~~ort•@

/

ll•

i•ql

H

i

R

where W

,

circular excitation frequency

U = slope of relaxation modulus versus reduced time curve,

assumed constant, and a powevc.law representation of the complex spring constent of the ,material h,,s beer, assumed. vibration, the excitation

For the case of steady state sinusoidal vb And the response, vm are ass.rned to be of

the form' (3.30)

vb`- Vb expiwt) Vm

(3.31)

Vm exp[(wt -a)]

where Vb =" iagnitude of excitaion Vm

magnitude of response

u= phase ang'e Substituting (3.29), (3.30) and (3.31) into (3.28), the equation of motion governing the viscoelastic behavior of the model is obtained; Meu'2V m + (k' + ik)

nvm 2 (k1 + ik")

INvb e la

(3.32)

From this evur_ )n the amplitude ratio of the responseto the excitatioon is readily determined to be

I

=V b !

_-..e(

3.27

3.331,

II

2-A where the )Loss tar.gent 8 is define~d through the relation

8 "

9Ik'

(3ot3) cap be -ampitud isobaind

-Equation

•

'

(3.34)

rati

simplified through introduction of a nat ral

V~+

frequency is

[,,

m

and a reduced frequency ratio

:

w

Substituti'ng {3.35) and (3.36) Into (3.33; a simpler expression for the" -. iplitude ratio is obtained;

a

=

(V',,n)'2n. ,(3.38)

•The behavior of the a•ilitude ratio as a function of-the frequency ratio n is shown in figure 4 for several values of p. .learly, the amplitude.

Imax

ratio is a maximu'• when the excitation frequency is equal to the natural frequenicy of the nodel; that is, when n =, ,

-

3

3

The phase angle hy wnich the response lags. the ecCtation may also be determined

from (3.32);

"tan

(339)'

13.28

,,•

2' - " _•... • ,.;'•'•'._.t 4,-

:•S

44

0

(A

::7

NIt

c~

0

C70

,

PA"A NU

/01' 3.291

)1

w

II It is seeý

at at maximum amplificatioh, the phase angle is simply

related to t

loss tangent; viz.,

:•

=tan"'

.

(1/0)

(3.40

From (3.38) and (3.40) it is seen that the maximum a'rplifIcatibn factor and the phase angle lag',of the response can .be estimated-knowing only one piopellant physical Mroperty,'nameiy the slope n of the lbgrelaxation modulus versus reduced time curve.

Knowing n,-the loss

tangent+*8 is 4etermined using the'Pelatlon S=tan(nw/2)

I

.

This relatiQn existt theoretically for linearly -viscopi1astj; materials,. however for nonlinear, highly solids.loaded propellants this i-elatlor, has been -shown-_o- be in error

.

Nevertheless, 'eqati W(3'.41 is

an acceptable approximation for preliminary desfgn purposes when the dynamic behavior of the propellant is unknown.Although the complex stiffness of the model, or propellant,-is not required for estimating the response for an excitation frequency equal to the natural frequencyof the model, this Irformation is needed for 'determining the response for any other, frequeiicy.

The effective mas.M

and the spring constantk-* are readily dete!lmntd tle

(b.e

Me - (33/140) (.w•'t 1,.. and

k* - (1/3)

S.

(h/i)3 E*

42(1/3) (h/t) (E'- + iE",V)j

Ifiear den'si ty of cartilevered 0 atnv,

z,-= length of cantilevered'piate, ,3,30

,

(3.)

IPA"

g

acceleratioo of gIravity ,, h ý'pjatd thicknes's =averg tckess

of starpoint,

*=comp~lex niodulus, !

4.

.•

°

V-Ew

= storage'modulu-,

'V

= loss SE"•n modulus.

,

13= E"/E',-= loss tan~gent Before, considlering trie lateral vibration of a cylindrical port propellant .grain 'it should be noted that the maximum amplification ratio 0 (.36) is also valid for acceleration forced vibration. Also, although it appears that a high loss tangent 'impVyingra large amount of internal

damping would serve to decxrase the ampliflcatior factor,-an increase in internal damping significantly incraeases'Intirnal hcay generation, anp Shence

temperature rise, at frequencies away,from resonance since the resonance is broadened. -It should also be noted that the simple theory presented here applies only to Euler-'BernoJilli beams. of th beam i

When the thickness (i.e., starpoint) is such that shearing deformations becomes dignpiicant. (roughly when the height approfches the length), then

Timoshenko beam theory must be used to determine the response s2nc the Finally, it should also be remembered • 'only a the single degrey of freedom model hasbeen empnoyed in the idealization of the starpoint geometry.

Any real continuum, of course, possesses an infinite ,uxnber

of degrees of freedom,

however this approximation should be valid for the lower modes of vibration,

3.31

..- .

LATERAL VIBRATION OF A CIRCULAR PORT GRAIN The lateral vibration of an in01nite length cylindrical port grain is idealized by the pin-ended m9del showr in figure 3B. ratio (3.33) is also applicaole for' this model, values of B, k* and Me are required.

The amplitude

however, new and different

For the geome.try of this model, the

effective mass is-given-by Me,= (29/70) (wX.e9) ,

(3.4)

where SWe

=

wc bhh + ppU(t)2-a

2)

(3.45)

witff We = linear denr'Ity of the.beam, Z

4

p

iength of be=in, = density,

h = motor case thickness, . 2b = grMn O.D., 2a = grain I.D.,

/

and the subscripts c and p refer to case and propellant properties

C

respectively. -re••

I

For this model, the'frequency terms are absorbed in the

nd imaginary parts of the complex spring constant; k* - k(

+ ik"(,)

(3.46)

where k'

{E='ni(b4-a4)/4 + Ecb 3 h) k" 'T4 Ell wn7 (b•_4)44]

48 Ecb•1

,

,

(3.47) 3.48)

* and the loss tangent is given by • = ký" 3 /48 Ecbbh.

(3.49)

3.32

5,'-,.

0

Assuming B

< 1, the approximate expression for the amplitude ratio

at a = 0 4s obtained;

max= bL

It can{L

(3.50)

nb

Vm4Ech/E"

seen here that high amplification factors may be observed for

"lateral vibration be.cause of the strong influence of the case stiffness. "0

The natural frequency at resonance' is approximAtely given by 2-~n • (70) 48}Ecb 3 h4 "(29) 14 w n I

(3. l)

c1

AXIAL VI3RATION In a manner similar to that above, the axial vibrdtion of a casebonded cylindrical grain cay be treated by lumping its mass at the renter as shown in figure 3C.

The slab which is also shown in figure 3C

is

mathematically equivalent to the cylinder and is used in the following section in dis~cssing thermomechanical coupling and heat generation. before,

As

the grain is assumed sufficiently long-so that an9 effects nay

be neglected. As before, the amplitude ratio is also given by (3.37) with a new definition of the natural frequency wn"

For this geometry it is more ne'

convenient to work with a nondimensional effective mass M * defined by

e

g M Me e .-

-X4. {(X2-1)/X21og

x + 2(logex-l) '

The effective mass versus X is shown in Figure 5.

LI.

3,33 7

(3.52)

Ima i

4 44,

UtI -t

-4-

7Pi I. t

114

44.

41 ;eHi

t-7t

rt

............

4+4+' aT

, ;

, ++++ .

_M4 -44-

1"

+ +1

11

;4, at

I........... t ý71.-k

t 4M

Zile

f4

t tv

P4

i

t:

Wit

r

r #4

-444 ++

+f+

+

+

t4 ljý

14

IT

H-U

4+

T!

44 _T!

4i it

"The soring constant is also medified from that given above; k*= 2irG*/loge

=

( 2 ,,/log A)(,

+I

G")Xn. ,

assuming a power I'aw represetltat-on for the shear modulus.

(3.53) The loss

tangent 6 is now giv~n by... •

B =G"!G'"

•(3.54)

Finally, the natural frequency is redefined to be [

2GI•"ge.

-

,

35*

4

so that the amplitude ratio is indeed given by (3.37) with the maximum, value given by (3,3)., eI

THERMOMECHANICAL COUPLING AND HEAT GENERATION As mentioned before, sustained yi•ration of a solid rocket motor can lea,' to suv-•tantial internafl dissipati1ve heatirq, particuIIarIy in the vi:inity of regions of high "loial strains.

The resulting high, local

temperatures can produce signific4nt mechanical or chemical degradation of the propellant, and conceivabI, even cause iutoignition of the propel-lant., The nature of the the;,momechanical coupling pr•5b•4.-as been"studie.d for slabs under lateral vibrations and for cyl.inders under axial shear 7

vibratiors, (see references (15, 20 throdgb 24, and 28). , The slab shown in figure 3C is dynamically equivalent to the long solid rocket motor under axial vibration,- which is 11so shown in figure 3C.. This slab. ,

geometry has been treated for one-dmer.;i

"

dimensional [22, 28] heat transfer condtions.

1

2

' 2

2-

23

d two-

,,The specific details of

these studies are not presented here however, theogeneral results are

discussed. 3.35

_=.0

%a

i

FcrI Phear iodg of 4,Wbration the rate of .mechartical dissipation

D is

given ty

where y' •s

the shear strain.

Xyy

Thus, dissipation and hence, heat

generalion is seer to be troportional to the saiare of the.strain magniThis ooints out the sIiIgptcance of the aforementioned high local

tude.

-

The, strain amplitude is dettrmined using\the ratio of output

strains.

I

-©

mrotion amplitude to input amplitude and the phase Oelation.

A relation

similar ito (3.56) also exltts for stresses indidati~ng high local stresses It qan a,•so be teen from

alzo significantly irnfluence heat generation.

9-

0(3.56) that the maximum disslpation occyrs at the natural frequency since G" is a maxinum at this frequency.

-.

The nature of the dynamic response of the viscoelastic slab under steady state th -C---aT-vibraon_conditionsis show•i in figures 6 and 7 where the amplitude ratio and mpximw~n temperature rise in the slab are In figure 7 the

l(/n. presented as a function of- the frequency ratio

actual temperature ris-e-is- 56F times the.-reduced temperaturO, *-

The

normalizing factor'H in these figures is a thickness parameter which has the dimension of length

H .4

and is deft-ned by

w

Žk3K. 'IS7

na

_

n'

where, K = Propellant C6nectivity , issumed constant Propellant los! tangent

8

i

,

-

slope of re'laxation modulus curve

=

natural fraqmubncy,

n

,

,

.

-

-

n

3.36 -

.2

Sim-

1 >

1

It o

>

icrJ -

0~

*P-4

7

7

and k1 and a

are constants defined by J* = ('iJ") = (k3ik2 )

t

"

(3.58)

"The relation (3.58) has haen found to wpll approximate the complex compliance of some composite propellants over a wide range of frequencies"and temperatures [-1,

In (3.58)

28].

difference above a reference temperature. S.

e is the temperature

The constant a can be deter-

mined from the shift factor versus temperature curve. The aforementioned studies have also included random-loading processes,

The equations for thermomechanical response to stationary random-

loading proceises have been shown to be similar to those for harmonic loading [23, 28]

VIBRATION DESIGN ANALYSIS SUMMARY S-,Before

closing this discussion on vibration, some of &e pertinent

results will be summarized for easy access. The amplitude ratio for forced displa'cmient and forced load

vibration is given by (3.37) for the laTtra~ly vibrating starpoint and slab and axial shearing vibrations of the circular cylinder;

-

(3.37)

[(l

It is also observed that for steady state (3nd adibatic) conditions the maximum amplitude ratic depends on only the loss tangent; viz.

-

and occurs at the frequency w :n

(/

where

)2• n

.l (3.39) 3.38

)

-

and the phase anale (3.40)

:tan • can bc astiiated from (3. 41),

The loss tangent

(3.41)

tan (nT/2)

=

in,the eventý that actual propellant data is unavailable, although errors on the order of 20 percent are common when using (3.4i) for highly solids ,loaded

propellants.

This value of 0 gives a maximum amplification factor of

propellants.

The slope

2.25.

A typical value of s is about 0.5 for composite

n

of the Atress relaxation modulus curve typically

ranges between 0.2 and 0.3 for composite propellants, and is somewhat ,l•ower for double base propellants. (

The',maximum dissipation is given by (3.56);

(3.56)

I-I ixyli,

which points up the importance of minimizing-'strair (or stress) concentrations.

Significant temperature rises werc seen to occur at frequencies

equal to about one-half the natural frequency., that an increase in slabt thickness (i.e.,

Results have also shown

an increase in grain web

thickness) will increase the steady state temperature, assuming strain s ....... ,n•cu. g

I,

The equilibrium temperature is related to

UG" h2

Axy 12 /2K,

Disregarding the temperature dependence

where

hi is the slab thickness.

of G",

doubling the t!ab thirknocse

ifolid •irease in the qteadv 5tate tmp

wVb ..... ness) results in a four-

~r~t•."-.

The e'atio of tne output

amplitude along with the phase relation determines the strain amplitude needed for estimating dissipation and the equilibrium temperature. 3.39

3.4

ACCELERATION LOADS Acceleration loads can be treated in the same maniier as shock loads.

In most cases normal acceleration loads will•,,poduce negligible stresses and strains. The exceptions to this rule are the large diameter solid rocket motors undergoing 1 g vertical or horizontal storage slump, and some tactical missiles which are subjected t6 very high launch ýccelerfitionis. Storage slumD of large solid motors in which inadqquate grain terminatitns have been provided can be a critical design faCtor for-storage above ambient temperatures. at the grain ends.

The result is that grain unbonding may occur

In large diameter motors in whlbh adequate grain

terminations have heqr, orovided. 'large deoat'n

f

h

y•i

a

serve to constrict •he gas fl ow resulting in errosive burning, part-icularly in the area of a submerged no'zzle or radial slots.

At Inw temperatures,-0

the propelTant stiffness signoificantly reduces $lump deformations.

Para-

metric curves for determining storoge slump deformations are presented in Appendix C.

3.4.1 .AXIAL ACýELERAT!ON The shear stress at the prope'llant-case interface, is'given by

,

eqvation (3.24) for axial acceleration;

i I

0

•

/ ,

T

rz =

ng b (x2-!)/•2

(3.59)

,

wh .. ere a stress cor,centration factor of 3 has beer, ,Int..oduucedto account for stress concentrations at the forward arain termination point, and inertia effects due to straining have again been neglected. lar configurations,

For irregu-

the shear stress may be caiculated from the simple

forbnula T

rz

3 n.g W/A

(3.60)

,

where

W '=

total propellant weight,

A

total bonded a"ea.

=

Head end bonding of the propellant grain to thd case serves to reduce acceleration, stresses by lessening th# propellant weight supported In shear by the motor case.

The ratio of load carried-by a full head end

bond to the total load for Poisson's ratio v.= 1/2, is shown in Figure 8 as a function of length-to-diameter ratio. and in Figure 9 as a function of the grain radius ratio,

x-..

For a value of Poisson's ratio less thark

1/2, the ratio of the load carried by tie bhead end bond is decreased.

This

influence of Poisson's ratio is discussed in a following chiapter.

"Although full head end bonding serves to significantly reduce accelera,tion stresses and slump deformations, as indicated in Figures 8 and 9, the low temperature storage and firing capibilities of the motor may be severe t iy compromised. of thep-ropeiiant grain

Head end bonding effectively doubles the length - ;.'-ermal stress and strain calculations.

he axial deflection at the port of an axial acceler-aLing propellant grain is given by [14 3.41

Cli

-

..-

OR

,

0

ID

D

IIx.

It Cy

C~

T..

--

x

> I-'-

3.42

0Ivc

[4

*:'',.

.

. ..

"

-

4I.I

--

'..

,:

I

.

, ...

.

i

.-

. ...

"

.....

"

' I+ ;' 1

. "

-- •

-

--I

:

.

"

"I

*,

".
-

S"Ith h

h '-%:dt a time-independent failure

envelop may be 4ons ructed for uniaxial failiu•'data of polymers. The Simitlý f3A, ure enveIope is shown below;

One of the difficulties in applying the Smith faihire envelope to propellants is~that It does not reflect the path dependent charac-teristics of propellant failure.

Another difficulty is that the

eetension to mulftiaial failure is not well understood. Andersor and Bennett [9,1O] have proposed an energy-failure &nd second str•so-invarlants, and 'ter~oni.- t_• o -ý fIrst have successfully correlated anlaxial, biaxial and hollow ellipsoid failure da-a.

The h.llow v••Lri.

test a-pear to be ideally suited

for development of a failure criterion for solid propellants since failure data may be obtaii;ad in the four oztants of stress space required *or the ccnplete deicripti'Lq

of failure of initially isotropic

materi als.

7.2.2

FRACTURE The preceed,-ing dUssio,, has beel concern.d win:

ateralt ..

.ic...ot

Zx,,,bt

flows

or detects.

faiiure of

When pre-existing

flaws exist Willianrs [121 has propose, a viscoelac;tic extension of 7.9

L

~/

-

r,.•

Griffitlh'ý .•rltt -l'(etacre toecry whereby =pe•••:

function of crack length and tIme exceeds a cerwrain critical energy required for creation of a new burface.

This approach has Ogen.

successfully applied to the analysis and design of case-grain termination relief flaps dnd inner bore failure by port cracking durirng thermal cooling and pressurization. CUMULATIVE DAMAGE

7.2.3

Several approaches have been proposed, in recent years for asessing'

/

,•

-damage accumulation in solid propell-ants based on extension of Miner's linear cumulative danage laW. ,These studies have considered accumuWOO, -v, ener-y, .strss and straIn and have bee6 mainly concerned with repeated temperature cycling and combined temperature and pressure -loads. The iws,t exens*-,* cc. , tulative damage work has bee by Bills [12-15] at Aerojet General Corporation.

ca!V-ied out

Bills has considered

statistical 'Implications in evaluating cumulative damage and failUre using a maximum principal stress approach.

These.studies have

resulted in development of criteria for iolid propellant screeningand preliminary engineering'designs.

Shift a~tors have been intro-

duced for pressure and environment41 facto•i

wv;ch are employed in a

manner similar to that of the time-temperature shift factor.

Recently,

it has been concluded that motor firngs generally ignore previous damage.

In particular, a set oIf motors will fire successfully after

temperature cycling if h~one of the motors failed during temperature cycling.

7.10

N SIn

addition to the studies•conducted at Aerojet several other

-indeitgdcicns hdve recently beer, canpl'eed.

II .

Roýketidyne ZTh6j

apprOdched cuni-ulati've damage using fracture mechahics cor;.iderations.

.

Atlantic Research Corporation [17] considered appl1.cat*.s of the

S--

principles of absolute reaction rate tkeory as derii.......n

1

6 olsky-Eyring-exp~ressions. "To

. ,l

Lockheed Propulsion Eompony [18] has

proposed using vclumetric response as a damage index.

Thd general

results of th.se investlga~tlons have demonstratef that enviro"nmental fa'•'Fis can produce very large change.s -*n the time-to-Failure "data, and that the•,statistical aspects of cumulative dg'ae testing are significant and complex; thus requiring the use of extreme value statistics for evaluation of grain reliability. •

Hercules [19] has. re.en.y completed a cumulative damage study of CMDB propellants. •They studied the use of stress cumulat vei damage, strain failure index, totai energy fF

r

index and sess-

strain failure envelope, and found that clthoug"roh

of the four

criteria accurately-accounted for propellant behavior, each of the _iI

four were at, least qualitatively valid for- correlating various aspects of propellant behavior.

A nonlinear madification 1.

The "stres softenin!

effect incrases as ýM

*square of the strain value, Ie., the deviation from linearity is greater at the higher strain

Si

values and, equivalently, longer times. Comparison of (8-13) and (8-14) with the latter rewrtten as

a~ke 1(t)] qi ka(st(t)J

-

k-kl

a(8-17)

shows that tte deviation frm hoogeneity, for k >-I, I.e., increased rate of strain, i*

*-

,

. proportional to k(1-k) . -proportional to the basic nonlinearviscosity

coefficient, in,andC * proportional to the square of the basic strain rate, R1 . Quite obviously, both the deviations of (8-16) with respect to '

the elastic modulus and of (-17) with respect to the viscous components< will only be detectable in a test to the extent that the deviations exceed the d~ta scatter. Both of the types of deviation becme more marked as the ratio of the strain rate of tS' two tests increases (as k becomes larger).

8.16

S. . C..

.

.

. • • .

8.4-.2 -GEOMETIC NOK-LINEARITIES i

Assum nalmateril] that is fundamentally linear, one can nevertheless observe; "Parent

.nonlirearit~ies in the stress-strain relation.

This phenomena occurs and is observable at 13rge strains, say above 10%: in typical solid propellants. The reasons are purely; gwmetric (or more properly, kinematical) in that the expression for stress is linear in strain and as usually used is referenced to. the infinitesimal strain (tensor). As is demonstrated

il

in any text on large defamation elasticity, the'use of this infinitesimal strain measure is only an approximation to the correct stressstrain expression.

Thus, the large strain nonlinearities result.

As a matter of fact, .tf onhe observes actual -!nearity in the relation of, say, uniaxial .stress .to the infinitesimal measure of strain, c, for very .large strains,, say 20% to 50%, then the material is basically'nonlinear. S•equations

That is, since a basically (physically) linea.-

material should show nonlirearities at large strain with respect to the of infinitesimal elasticitty the occurrence of linearity in, say, a uniaxial tension, test at these large strain levels will be &A indication of basic nonlinear behavior.

Io

One would then expect to

•

obserI-fairly aueisol large nonlinearities in, say, tha biaxiAorc or other multinprxmto tes

si•1sri

these same larie stra is tannnlerte 'letlls. It shousldbe r dded that much of the lietarite

stanepeso.Tutelrg axial test at

"

eut

ret orted in tests

above the 10% strain level is because a of rrepe ductbailty of the test !"samples as well as careless control and/or observation of temeratui e 7-

and humitety levels duing f

stigo reasulticn, thn sch large data scatter

8.17

that a linear -elation may ar, well b a proper pr~o~dure.

Sli'ce

tto the Naserved data. This is-

the data

have

a lAge error, one may as

well a&dpt the simplest constitutive equation, i,liAear ore•' and obtain qiestionable analysis results readily

And

inexpensively,

It makes

little sense to emloy barely tractable nolinewr theories with data of low relability.

The validity of any analysis based upor. poor data can

only be assured with the concomittant use of large safety factors ,c;;*ver, say above 3.0.

IRREVERSIBLE MICRO-STRUCPJURL CHANGES

8.4.3

The third ,aJor type of nonlineariti is caused Ly essentially-

lrreversibla microstructural changes such as . polymer bnnd breakage .

vacuole formtion In the binder

. dmtting vacuole fomulation between the binder and solid filler particles. Unfortunately the single constant straen-vate tests di:.cussed previously in this chapter are not sufficient to detect th~se irreversible chaisges... -I

test does not reverse nov, repeat the straining pattern.

Thus, a constant strain kete test will 'show micro-struc¢iural changes as physical nonlinearities which were described earlier. The previously referenced paper by Farris and Fitzgerald [46] has been followed up by extensive work on irreversible changes or permanent ueryr

effects in the doctoral thesis of Farris [47] and the essence of

the work with applications has been reported by Fitzgerald and Farris in reference [48]. Currently, further effort along these lines is being 8.18

pursued by Farris, now at Aerojet-General Corporation, Sdcraniento, Califo)rnia,

and at the University of Utah.

The essential test procedure for the detection of these microstructural changes is based upon the additivity portion of the linearity law.

Two distinct types of tests are generally recommended, loading; unloading; reloading, and ramp-stra",. rest, additional ramp-strain. For example, Fig. 6

shuws the relaxation modulus calculated

from tests run at straiAJl eiel&,iffering by a factor of two.

The

results obey the homogeneity rule of linearity in that the resultant modulus is independent of strain. Thus, if one were to infer from the

vbcve that linearity. (linear

viscoelasticity) held, the predicted results of Fig. 7

wouid hold.

It is observed, however, that quite a different experimental curye results.\ A triply repeated ramp strain teqts proý,jced the results-of Fig. 8

wt-ere the calculated curvg-is based upon a constitutive equation

utilizing the ratio of the miaximum strain, average

Ij1ciiII,

to o weighted

11c,11121, the so-called Lebesgue-21 norm, times the present

value of strain, eiz(t). For cormarison, Figure 9

snows the results of a linear visco-

elastic prediction versus a "pemranent

wn.emry" prediction for a triple

ramp strain. it is thus clear that a different set of nonlinearity is observed with reputed ramp tests.

It is thought that the preknt type of

nonlinearity is caused by bond breakage in the'propella't binder, Reference [48] discusses this point at leangth. 8.19

011~

0

00 ..... . ... 4J

0o

V)

4n.

U).A LU

0

0

La

0 E

4-P

-4-0

-1U

0

CL

-4)

b-

1

040

ci

E4J

6c Co

o8.2

AA

0

0 • "4.'IJt• CC LASa

4-

•

I...a*c-

Gus

4-'4J

-

I-a.

4-a -)

LaJO•

V-)

.CA.

C

0C

-o-

40

''-

4.

C 0

I

f

CA

I-

V)C *

-hA Ci

4'\

sda'-4

F

pdc

____

5ssauJ

'4...

ItIL 44

c~J4J

LUJ

LnL CV,~

.

CU

Cl.

UW 4.),

LIM

-I-

8-22.

*

"

i

-4'-

-

,

...

;30

It

100~

I

Sexperia-ental rate strain constant experimental

/

3"

",

/

*

if linear viscoelist

"-"predcted

"

4

A2

"

4N.

'

**

/

IN

"*

e '

/

F.

i.08

,, "•

,'

' S/r.

- "'"

-

/

-

¢

//

/

S.

-

u/

/

,0i

,

V 0

R

__

8.330

f

15

•.,

.,.

ur

*

".04in o

02

•

e, t ~Inu e-

.. "•,

.1-0

.08

CONSTANT STRAIN RATE TEST [48].);'*;; INTERR}UITED FC2 OUTPUT 9.STR~ESS ' Figure ,_,

8.23

..

x• ',0

*0

i

tO

\

CI.A

CLOSUR

,M

..tn test results hMve Won pointed

Seieral caus'es-'af ronltnea• ,out ,and °discussed.

•; 1procedures,

I:staple V.

,,",

It. has been s

' th~ato one

(straining histories) i nV'l vi

ust,., pectfy2 test

more pxttfnstve .•ests ,than'..

cotant-stran and single raw-strain in order to- detett anU

• }

":

d~~istinguish-the various ronltearities..-,

•.

To- the"extnt"thittthe obser-)ed nontl Uear tiei ,are within acceptabe.

.,.

•'•tolerance

*

dd use, it is racamended that a ltnear

levels for'the tn "haracterztion be used.-"

"W ere the degree of nonlinearity is large for the-ftended use,pnt

recourse to adnonlsnear donstitutive 1

*

-procet

es delst wain•in

relation wuistlVbe requ red

hoapter 1m.os • "I* ,ona" *mr

Tetst 4tt

,l-r~a

siiWlhere lineatrty is andicaged foruse, .itnear elattocty and4lnear vtscoelaesltc-ty

isy be Applied. 1

.

Oler addntconal Wnt her ted. If tests

-

s s,,

.

onaipeemaritn

.repeated loading sh

iy

ar erfor the-ite

" rteversible chanoslare condusted aftereletio•nw sorained to the.vwtI nonlWneare tie

ri, if

anticipated working strain, then thi. remaining inydictedfort uhe, lncategore of stbic phyticar .

_

szation if

d s

r,,be been, Ths ,

l

or, large strain thduced non'rynearties premayously db.ossed. a. tshakiedwn test"

be-,

on aoften be,.a in order prior. t6

Thus,

et characteri

-

themaximw anthe iattrital also inolves repeate

i

-loadings and nloadi

(

ylfS.- It is thts latter s pointpq which proAcdes hope

for li near cr pseudo-linear char~cterization and'analysis of "per•anent mary" type materl.als.

-" *°

~

- -

.

-"

8.24.

.

-

......

________

I oi

0

_

Constant

"Ea

Tensile o[dulus

-

SE

k

Constant

Stainrate

R

'

"

"

+ -

9.

=

Constant Principal Strain*

.-

EC/Ei.

Principal Strain' - -R ".

s Straenrate

r VicipaflStrns"

° "-•

Prltcipal Stress

8.25

"'*

i""

8.6 REFERENCES

(Th

1.

Brlttofi, S. C., NCharacterization of Solid Propellants as Structural Materials," Solid Rocket Structural Integrity Abstracts, Vol. 2, No. 4, pp.I-7", October 1,955.

2.

Williams, M.J...( /Stmural Analysis of Viscoelastic Materials," AIMA Jovkýhal, Vol. 2, pp. 785-808, 1964.

3.

K(ruse, R. B., ¶Laboratory Characterization of So~id Propellant• Mechanical Properties,* A.IAA Paper No. 65-147, 6th Solid Propellant Rocket Confirence, Wkashington, D. C., February 1965.

4.

Bollard R.'J. Mtp. et al.,

.'5.

"Stiructura1 Integrity An&lysts of Large Solid ;Opeflant Moto"'Gra.nS,"NMSC Report No. 65-21-2, Mathematical Sciences Corporation (Contract N0. NAS *7-242), July 1965.

Dill, F H., -"Tb*eConrq.n*, o~f Constitutive Relations and-failure

Criteria and thir-Essent•al Role in Design," Paper presented -at the ICRPG/A1AA 2nd Solid Propulsion Conference, Anaheim, July 1967,

6.

Physical Characterizetion," Poper presented Kruse, R.I ,."Propellant at the 7th Annual Meeting of-the ,CRPG Macha .ical Behavior Working -r-oup, Orlando, November 1968.

/•'

.~.at alT~., 7. Dill, No. E. 69-%0-1,

8.

anical Frperties Testing, .. '7?RQ-Prope'alant; KelIey,, F.J-, Failure Cr1ite..a. and A'ingr•' Advancis in Chemistry, No. 88,' ="Prop11ants, hfa.c.tt e, Hazards and Testing," pp. 188-;43, ne taner ~nCh-lca'- Sd.ety, 1969.

10 Report "St~ruct~al-jnt! ity tudies,-NS )Ithemtical/Scteees.te, Inc. (Coptract HAS 7-464), WzrmWer J069.

-

1lant Failure Criteria," S9. Jonest- J. Wandauis., W. 0., -r AI'A:Paps. No. 65-•57, AIM 6th Solid Propellant Rocket Conference, • shingtdn, D.C., February 1965. 10.

yopellant. Failure Mechanisms,* Bulletin of the 3rd Jones, J. W Meetji# W-'. ICRPG Working Group on Mechanical Behavior, Yo CPAjqica~ton. • ,p ' 371:g 1%641.~ Ofl__-;.:, No'.

11.

Cost,

12.

COst

'Analysis of the Biaxial Strip and

A.L,-And Parr, C. H.,

llant Characterization'" Repoift Shear Laf Tests for Solid No; S-73, Rohm & Haas Compa& (Contract DA-O1-021-ANC-11536(Z)), May 1967. 1. andParr, O H., LT

"Analysis of the Blaxial Strip Test

"for-Polýertit Materials," J. Materials, J MLSA, Vol. 4, pp. 312-323, --1969. M.~

8.27 -.-..-

7-

•

.•

L

t

•

13. wiI lisms. W4

m.

L M'

~

.

t~i

Fitzqerld, a.E.6 vA Bjaxial Test Vor Solid relat,3iltn

*15.

J. '0 StMs O)ISt$rtb1imfm ie Eli ipttce Olis -with 1oncetra9tg toads Acthng Along the Axes 1of '~atj~qattl tra, l C., Ugtr 1.11 Report on Engi~roartne

Ortsbaie

16.

Boisbane, J .Jftfrth~v- 0s JAW Pear.)c

m n~ok 1-M7 S tessYtiftorVnit~ra]s Irgele a of SolidO lMseTi

17. t~i~ds~yO S. H.* ot aI, *The T-Kaxial jTenslon Failure of Viscoslastic MUM &Ias,* AML 63-1529, Arwpace Research Laboratories -_(Cantract)Io.O M'.31(6161-8399). SUptuier 1963. 18. Messiur, A. IL,6Stress Disiributionx in -Poker-Chip Tv. 1,oSpac1ims 19.Bri' sins J J.9 Streses, Strains and Displacmmsnts lin -thiwoker Ci Nadianical Odavg, orA.,

___

179V

37

20.

Harbert, S. C 'railTensile Failure of Solid Propellants,* Sul eImU. o-f the 3rd PHeti t ofAqlm ZCP tim P±M on *hchahi cal

21.

LindsaW .G., "Strets DNstributim n-/W a Poker Chip Spectm sub~ject CN61 m4 Loads,' Sul U%1i dofs 39*'d'Umti"4ef 00OseA Vwt asr~ ~~dWncaj Reaw ---IOTA POlicatiom -No. - 1p. 573, I4 -;.to

t~ia~rbert, l a.C., "Triaxi A Tlestlx -of Selid pelhuItfts;* 4th fteting of ti-b-CRPGWori myoon *d&wiZAa

fti A of the vl~or. T

Lindscw 9.( IL. *'%tyrostatlc Tins'll Frctur of a-0t'u-O.yAtne Elastomer, ARt. 66-0029"Aerospace Research Laboratories-. (Caotra~N. Fw3-l) 2217)o, February 196. 2.Lindsay. G. H., 'TriaxisA Fracture Studies,* J.AI,1 389 Ar o..4843.-485Z 1067. 23.

__

,25. SweetW., K.H. end Sills,, KM., "Poisson's Ratios of the JRMF Pawel o-n P

26.

DetevnAtJOi,*Blei

~ical Prortie -of Solid Prdollants,'!1Fl

Bisdel ,, L., Mmeasuemikt of Poissom.'s Ratio inTent -

17th JM4AF 'sw)ane Pihcation no.

Mi taitcal'!

"1l,, p. 203', 1953.

8.28

-o

o1

-

k-Bul1letin of the oer~E

,U,. Rinbird, R.: W. and Vernon, j. H. C., "An Instrument for the Measurement of Volume Changes Occurring in Tensile Testing," Bulletin of the 19th Meeting .of the JA-AF Panel on Physical PropeR•ies' of Solid Pro int, SPIA Publication No. PP13, p. 39, 1960. 28.

Kruse, R. L., "Dilatometric Behavior of Composite Solid Propellants tnder Uniaxial Tension", Bulletin of the 20th Meeting of the JANAF Panel on Physical PropertTes 0?olid Publication No. 14U. p. 307, 1961

29.

Wosland, N. .C. "An App4ratus for Neasuring i-ulk Modulus of Solid Propellants", Bulletin of the 20th Mee of the JANAF Panel on Physical Propertie$ of Sold aIA Publication No. 14U, p. 317, 1961

130.

Svob, G. J., ete at... "Volume Changes in Polyurethane Propellants -Subjected to- Small Strains",. Bulletin of the 20th Meeting of the 4ANAF Panel on Physical Properties of Soliod Propellants, Vol.

PAu

()

cM on No.. 1

T,

29S, 1.

31. ...

Fthn N. and Rtine, J. A., "Dilatation of Composite Propellants", Bulletin of the 2eu Meeting of the !CRPG Workin, Goup on MechanicalBeh•avlor, cPIA Publication No. 77, p. 349, 1963

32.

Ferris, '. -. , "Strain Dilatation in Granular Filled Elastomers", 'Supplement to the Bulletin of the 2nd ReetirM of the ICRPG Working. Group on ftchancal Pehaor , ication tIPuNo. Z7A7 p. 55, 1954

33.

Fishmmis, . and Rinde, J. A., "Development. of a Dilatational Equationof-State", Bulletin of the 3rd Meetl of the ICRPG Working Group on Mechanical lavio,. Vol. 1, CPIA Publication No. 6t0, 1964'

34. Farris, R. J., "Dilatation of Granular Filled Elastomers under High Rates of Strain%, .. Appl. Pol. Sct., Vol. 8, p.,23, 1964

L7'7i

"

35.

Surland, C. C., "Compressibility of Elattomers with Crystalline Fillers and ticrovoid Inhoiageneittes Related to Various Empirical Equations of State for Liquids and Solids", J. Appl. Pol. Sc., Vol. 11, pp. 1227-1?49, 1967.

36.

Surland,*C. C., "Compressibility and Other Thermodnamic Properties of Polymers", J. Appl. Pol. Sdcj, Vol. 12, pp. 1423-1437, 1968.

37.

Farr.is, R. 3.. "The Influence of Vacuole Formation*'on the Response and J.ilure of Filled Elastomars", Transactions of Soc. Rheolo2y, Vol. 112, pp. 315-334, 1968.

38.

Bennett,-S."S.4. and Anderson, G. P., "Mechanical and Failure Properties of Propellant in a Multiaxial Stress Field", Bulletin of the 4th MeettnQ-of the_1RPG Workip2 Group on MechanicaR , 1r, CPflA Pbblication oi. 94U, Uctober 1965.

39.

Anderson, G. P. and Bennett, S. J., "Mechanical and Failure Properties of an B6 Percent Solids PBAN Propellant in Multiaxial Stress Fields', .Exp.lNech., Vol. 8, pp. 411-418, 1968 8,29-

,z

.,1 ,

4L•--

--

-

--

-

-

40. Andersoi, G. P. and Bennett, S. J., "Fracture of Polymneric Matirials in Multiaxial Stress Fields%.AFRPL.-TR-70-36, Thiokol Chemical Corp., (Contract F04611-69-C-0036), March l?7Th. 41.

Jones, J, W. and Cantey, 6. E.. "Inwestigatior~s of Propellant Dynamic Responses, Viscoelastic Linearity and Ther~rheologica1 Behavior", Bulletin of the 3rd Meeting of tte ICRPG MirirgGroup zin jechani-. Cal Behavi or, CPIA Publicatior, No6U

42.

Schapery, R. A. and Cantey, D.., "Ther mecharvW;al Response Studies of Solid Plopellants Subject'o4 to Cyclic and Random Loading", AIAA Paper No. 65-160, AIMA 6th, Solid-Pro4pellant flucket Conference, Washington, Di. C.,, Fet4ruary 1?065

43. Allen, E. L. and Willouqjhtby Di. A., "A Siniple, Accurat-i Method for Determining Thermil Conductivity of Solid Propellants", Report S-160, Rohmn and Haas Compaty (Contracts DMAHOI.-67-C-0655 and

DAAHO1-68-C-0632'), June 1968

.44. Allen, E,. Lý and Willough~by. 0. A., mA Simple, Awirate Method for -Determining Themal Coihductivity of So~id Propellarats", Bu'11~tin of tbk 7th Meetiw. of the ICRPG Workii:9 Grow~ on Mechania Be-(;I ol--te No. 177, pp. 4743,,, October 1968 45. Joneg, J., Fitzgerahld, J. E. and Francis, E. C.,, "Thermal Stress Investigation of Solic4 Propellant Grains: Volume i - Theory 'and Ex* ~periuent', LPC Repoý,t Noi. 670-F-1~. Lock-Need Propolsion Coqmpan (Cnrc HF04(6fl)-4O13),, May 1963 46.

Farris, R.:and Fitzgerald, J. E., *D~eficiencies of Viscoelastic Tbeories as Apilied to Solid Prope~lantih, Bulletlii of the 8th JANNAF Nechinical Behavior Working Grou , OTA-METT-ation No.

47. Farris, R..J., *Hwogereous Constitutive Equations fn-' Materials oith Permainent Memory"m, UTEC TH 70-083 Project THEmi'S Report AFOS&, 70-1962-T7P, University of Utah, July 197U 48. Fitzgerald, J. E, and Farris, R. J., "Characterization and Analysis Methods for Nonlinear Vi~scoelastic Mate~rials", Prcjert, THEMIS Report, UTEC TH 70-204,-U~n iv~rt ~ty of Utah, M!ovember 1970.

8.30

(

SIX.

LINEAR VISCOELASTICITY

INTRODUCTION

L9.1

"---The literature on linear viscoelasticity is quite externsive; rather than attempt a decidedly incomplete listing, we cite only a-few references [1-12], which in themselves contain extensive references to additional works as well as to the original papers. Certain portions of this chapter have been prepared from lecture notes of classes taught by Dr. W. G. Knauss j at the California Institute-of Technology, and Dr. Mi. L. Williams

2

at the University of Utah.

We appreciate

their permission to include this miaterial. It is also observed that this discussion is quite--abbreviated since "* the intention here is merely to provide an illustrative introddictitn to viscoelasticit,. 9.2

-

GENERAL CONSIDERATIONS

)When

-

-.

considering the stress-strain relation of an elastic material, it is evident

that for a particular value of stress there is associ

a particular value of strain, and regardless of the length of time the stress

" that

acts upon the body, or what path was followed in appot•y1Wit-,

the strain ,'emains cQnstant.

In viscoelastic materials, howevei; when_

-

stress ir applied to the body, the strain state depends upon the-manner

..a

in which 'the stress is applied; that is, whether the loadi-is-4pplied Lecture notes on Theory of Viscoelasticity, California Institute of Technology, Pasadene, California, 1965-1966. "Engineering Analysis of Viscoelastic Media," Lake City, Utah, March 20-24., 1967. ••'

9.1

University of Utah, Sal-t

-

rapidly-or _s_1ow1-y--Thus, the-+tstory of loading must be considered as

In, addition, a viscoelastic body will,

well as tli-f-figitude of_ the load.

not maintain a constant deformation under a constant stress,-regardless of the loading pattern, rather it will deform or creep with time. Also, .if such a body is constrained at constant deformation, the stress necessary gradually diminishes, or relaxes. ftohold it Without •bming-u~nnecessarily ilvolved in semantics, one may coaI'Aer this time dependent-effect to--be the underlying d

ition between visco-

elastic materials and elastic materials. OF A LINEARLY VISCQELASTIC MATERIAL

-3-*DE"SRIPTION

al. linearity is defined i-terms-ofsuperposition (additivity) (homogeneity) of action and reactfiis. ---For----"and scaTar multiplication

- -Mater

p-: -am xial tensile-bar-wM__ke considered

i-I_ 1uStati-e prp0g-

rb-ra-yI--aditig-§coi-tiOns.

-here rather than a general- sol-id-unO-we

Many

c--oncepts of viscoelas-tic_ material behavior can be demonstrated using this -

simple .Stress -state, and In most cases, the extension to-general

three-

-dimensional-considerations is straightforwar&. ""Letthe force displacement relation be gilven In t--gerneL_ fo---

u(t) -

where [tIs

Wt

-

-

a time -operator ch-aracteristic of-tm

- of- the tensile bar. -

F(t)

becomes a constant, unit dimensions).

-,(9-1)

la1 -properties

(For a linearly elastic naterial, the operator r E being Young~s modulus, assuming the bar to have Equation (1) describes a linear force-displacement

relation if the operator [

has the properties -that 9.2

------

0

[*I]t [kF( t197=

k~jPItF(t)]

and " /]t [~Fj(t) + FA•)) I,

=

[,J

F 1 (t) + [Wt F2 (t)

(9-3)

-

where k is a constant'

Equation (2) is an expressicn of the homogeneity

requirement (scalar mgltiplication) and (3) expresses the addicvity or superposition requir

(i,)

ent.

These are the only requlr,3ments for lineari.ty,

and thus

.

k u(t)

E[*It

Ek F(t)]

u.(t) =u

1 (t)

+ u2 (t)]

4

and [i]t W [F1 (t) + F2 (t)]

(g-5)."

9.4:'ONSTITUTIVE EQUATIONS Consider the special case of an applied time-varying force, a step function of~nagnitude AF- al t,, -- t unit step finction

0. Let H(t) denote, che Heiviside

and let c(t) = [*It H(t).

Then the corresponding'

displacement Au is given by

Au(t) - c(t) AF,

(9-6)

Now let a second force AF2 be applied-in a step-like manner at a time' -r after the first l-oad,-i.e.,

-F2 (t) = AF2 H(t-Ti.

.

'

,9-, - .

4/4

e.M2

~~It Z:.4

If thk operator L' has not chainged in the time-interval 0 < t

10,-

*

J.

(V

't

--..

,00(0

..

M

,

.

IV.4.lC

S.

b

.,H

the poymr maeia.A huhte.1Ids de mrehanichaly

soweffjto

unconV

otebne

omk '

the cthemical reinkages arfoimng tepolye

natrally seewok, thelstrupossible eer

onigualython sam

stehe orc i reurdtomhl ae themsr intat themialet. I ii

hnd athey do so, Tex e 'time

toe

remiac-.tthisei strindener the whe~d-in stjse to he'araceestraifuctonditof. 'ths I plyuel ea' material nigreIthoattheseladsadddtion propinellant hae ome ffact. Sponc the chlermial -

Ve rosink nepatwork the-lasjetru

isysatera is

times aminder the smae

reacls imng prpollym th aer

athce snerameihnath binder it

ths bn

suadecftm theatR initialthelod'chalnpt w sexce th i uatm toe

these sois.This ~ o supolye ihfs, lenfo.a propiguellan

whi bepsomewhts

urethose ir Thes bisnderweanosubece tha the sam strahinbonition.e deps eadily seaacen i-i Figun1ta the saelxto

time

bu binder f'a k*sld and propellant.rSince the

f

btaiong ar

loads'"dii te soid the too

s arem oyer thenot

aileein aroesllat waeis l are

derparticles t gintthe rfaci

rexthe oadThis rwlthation IPsnatual ton insec tat parof dat a pufisede

byt fromdetha plymer chaing

samerbl

lalond mtit . icus due toni

snodiuqmchlorid fblle

aloy-

uar.rthanepojmae. Thcuis isimntariant ousay tatd theysoeidsexhibint tom e depenrennt deiraeeistc butl t

in thena sametioi

eim s

thilner elpolymerihoconditions~

sold ue hv reinocdsult.dradbodfiur *

'.

'

ilreuti

II.j

-h-

0' 4J

0f4-)

raCwL

4J.

_______c)

U'l SU6 Pj'

C .

S.

a

!

if

'

4 {s required to be a positive monotonedecreasing, continuous function of s which goes to zero rapid•- is s

+

Tf the first'-two concditions hold theo. a sufficient con-

-.

dition for the last condition to hold is that h(s) decas to zero in such a way that the limit relation

ZrM srh(s) = 0

"monotonically for large s.

it(s)

(11-14)

For example, the function

(l

(sU+1)-P

.

isan influence function of order r for r < p, and the exponential

h(5) = exp(-ls)

,

>0

(11-16)

is aA influence function of arti.trary orde-. The r•vZllection of a particular history f(s) was first defined' analogously to the L given

p;

H f()')Ih

V

norms for a given influence function h and a

=:f

IP(s)j

h(s)]p ds]

11.47

if 1 < p
1/2 "such that the

constittutive f nctional G of (13)

is Frechet.-differentiable at the

zero history in the Hilbert space of histories.

The nth-order

. stronger principle of fading memory is the requirement that there exist an influence function of order constitutive functional be n-times

r > n + 1/2

such t; :t the By

Frechet differentiable,

sufficiently retarding- any given motion one can justify approximating the constitutive functional of the retarded motion by a multilinear function of time derivatives of the given motion A the present time.

This theorem gives the result that the nth-order ,a-,!Tn-_

Ezrickoen material may be interpreted as an asymptotic appriximation

Q

to the theory of general simple materials obeying Coleman and Noll's strong principle of fading memory for suificient.,y slow motions. Several deficiencies maky be found in a theory of materials which uses (17)

and (18)

or (19)

to characterize the effects of memory.

For one thing, the norm is weighted with a decaying obliviator so that finite an- permanent memory characteristics are excluded.

It

is also noted that there is no unique way of choosing the influence function although the existence of 4.njis) is a material property.

of the required type in C7)

p

FurtbAr, the arbitrary numbers

are extremely difficult, if not impossible, to determine ex-zerlmentally.

11.50

11.5

-

-.

--

~

---

r-•

WANG'S THEORIES has proposed two fading memory theories.

WaLg [70,7i]

Wang's

first theory generalize,; the res-01ts of C16ieman and Noil discussed '

In place of the influence function, utilized by

immediately above.

Wang formulates a principle of fading memory by

Coleman and Noll,

means of an obliviatiirg measure.

An obiiviEating measure p on the

real time axis [0,-) is the Lebesgue-Stielties me'sure associated I with a non-decreasing Vower semi-continuous real function a(s) with the properties:

i) l

a(s)

0 for s < 0 ;le

vi) (s) is bounded, i.e., lrm a(s)

From condition (ii)

p ([0,c)) = M.

=

M< •.

In terms of the ob!iviating

measure 1, the recollection of a history f(s) is defined by

If a(s) is continuously differentiable, the function

h(s) =

(11-23)

.

serves the purpose of Coleman and Noll's influence function if

11.51

/.

/ p,,+"i~l'l llll+'+IIIII "Iii •i!11-1+I I

It

nm"oncltoiicaliy decreases

when

Alternatively, if h(s)

is large.

s

is an influence function as defined above and we set ,'9

Jh(sY/

ds J)

for all, Bore! sets I efO,),* then P is an obliviating measure and '(22)

Although the greater generality of an

reduces to (15).

obliviating measure includes a broader class of materials as being

9

endowed with fading memory, Wang's theory suffers from the sam.=

-*

deficiencies as Coltman and Noll's theory discussed previously. Permanent and finite merly characteristics are excluded ard there is no unique way of determining the functions

d(s).

In this same work Wang [70) remiarks upon the possibility of extending his formulation of a principle of-fading memory to a much bigger class of deformation histories. *

-

Considering the class of

all deformation histoiies measurable with te~pect to a preassigned d

obliviattng measure, Wang introduces-a metric function makes this/"class a Frechet spacei

The metric function d, associated

with a preassigned obliviating measure,

id(D,E)

defined by

sl-Ey

o

which

(1-25).

•~

where D and E are two histories, car oe shown to giye the desired Namely, a

topology on the space uf all measurabic historigs. sequemi~x

tk

IIlv?

l

11 .52

.

[

-----

II'i

,-

tll~r 1i1111i1111rllfi

i

liIT-

'

i|9

i~w f and only if d(D ,E)

n

as n

-*0

Wlang's [71] second the~ory is based upon the topology. oýf un i._rm corf~rgence on compact Sets rather thdn the Hilbert-s'ace topcl ly of, and liol - discussed previously.

I,,.,appftximation ColiiednrJ

Most oficonclusions ,of the

theoremq pronved byCeerr n NO Ii a26

remain un-

changed;however, the definition of fading memory is~diffErerft.

Wang's

formul-ation is motivatedb the-desirt to satisfy the stress relaxa;tion theorem trivially "and to include finite rreipory as a special -q~b of fading memory.. Before discussing Wang's results for weak arid strong 'Fading memory some introdu~ctory notions will firs~t be given.

Consider the

mechanical coristitutive-relation,for'ýml material

cf

2 ~..

T(t)=

'~

the fo

T (C(t-s)V,(lk2} s=O

where C(t-s) is the history of the right Cauchy-Green tensor.. The order' of a simple material is defined to be the. smallestI nteler p (o

to.be a functional evervwhere contiruous tth"respect to the topolog,, on Dfi(T).7

With the, above definitions.a simple material is said to obey the weak p Mincipal' of fading memory i,f its canstitutivq. functional 1,. ,continuous at every rest\history (i.e.,'C(ý-s) with" respect"to the topology defined tn'D(i). defined and ,used in"•l

The &-retardation is

anal6gous manner to that.of Coleman and N9ll

tional that hap a linear ,s~uth

C(t) for all s > 0)

Wang obsee'res that any linear constituiive func-

discussed-aboVe.

t =-

.,

dependence on the -defofmation history near

as the linearly viscoelastic material, cannot satisfy

his-weak principle of fading memory. The:above definition*impl ies thet a simple materidl

ibeing the.

weak principle of fading memory must have t~d material memory para-&

meters:

the time of sentience" n and the grade of sentience 6 " 0.0'

Briefly the idea bahind the time, of sentience 6f a mater{ii

(for a

certa{n -fixed rest history C) is that if a material obeys Wanng's principle

of fading memory,,then for every

6 >0 and 1 -0 'suCh that if wh',en e[t-n,t], ten Ir(D(v)

(T) -

F

G, therý exist nupbers 6for all 0 f' 0

and

the influence measure

k i,

is the injfZuene fnetion.associated with which is actually the Radon-Nikcdym 11.60

ge

0

mI |'1

derivative of P; i.e., "b

dv

(s)ds(1-)

for every interval (a,b) c (0,-).

The function k also satisfies

the conditions

and as s

decays to zero essentially as o(l/s).

-+

(11-30)

k(s) ds 0. The function norm

p

is0

said to have the a*,uentiaZ Fatou-pro9perty when the following prop-

ert. holds: (v• ."if -fo' fz1 S. ..

f

f2

• L and if fn

"" ,

'•

o

i

fo

"pointwise v - a. e. , -then p(f n) I P~fo

ýalld a ;ion-Ytmrit "If all five of these properties hold, then p *isfunbtion norm, retative :o p, with the sequent -1

Z

Fatau property.

For'the wst part, this discussion will consider only functions satisfying properties (i) - (v)enumerated above. The set of all f & L+ satisfyig p(f) L = P.


-. 0

" *

~~(11-117)

*Thx.procedure followed from this point on by previous workers

has been to claim that r may be arbitrarily selected and thn to require the generalized stress functionalh to be independent of g. This procedure, as we shall note subsequently, -creates difficulties for certain processes.

An alternate, equally acceptable and possibly

more useful approach in describing real material behavior is outlined here which removes some of these difficultles.

A detailed discussion

11.96

4°

•

,,..

.

and intrapretation of the assumption Introduced and the results obtained here, as wsll as a corparison with previous thennodynamic theories is• delayed until the following section. It is first notWd thdt the generalized stress functi6nal may be deco•rposed In the following manner [31]:

(11-118)

-(rt; g) - =o(rt) + .=(rt; g)

"where =-,*is a functional of rtj independent of'g, and

is a funci

tional of rt and a function of g. Next, in analogy with previous thermodynamic theories the association

4.E=Drpo, is asAowned.

Introducing

(l 1-119)

(118)

along with (119) into (116)

it

is conduded that

o-E(r ;.g)"" - 6rp(rt•rt)

e-I

Q(rt;

g)'g

_0.

"(11-120) "The quantity 6 aefined by ""t 6 = [°_(r; g)r -

)](l-ll

6t(rrr

is called the internaZ dsaoipation. Thus, (120)

(11-121)

simplifies to

11.97

R"

11.7.4

I

(11-122)

pe6 >_q-g

I

DISCUSSION OF RESULTS AND COMPARISON WITH PREiVOUS THEORIES

The restrictions imposed on the constitutive functionals and admissible processes by the requirement that the lo~cal rate of entropy0 -,,

production be.non-negative are.embodied in the assumption (1.19) the dissipation inequality

and

(I22).-

In order to demonstrate the mo~tivation for the development of the last paragraph in the previous section and to demonstrate .thne limitted applitcabili ty of -0revtous thermooynamt c, theort es we return to (117)

W•pd explore in greater detail the restrictions imposed on.

admissible processes and the resulting constitutitve functtonals,. Equation (117) is Intehepreted as imposing restrictions on admissible proceses astwell as constitutive functionals. The space of historias o0 (processes) for whichb 17) reis ostulated to be valid rincludes historios that are smooth.at the.cudent time as well as historiesn

that exhibit Jump discontinuities

d at the prisent time.

For histories

t hat r may be jumps at the present time it is argued possess that specified independently of rot and g. One then concludes that thet generalized stress fun•tional from she mined

of rintg and ipsdeter-

free energy functional through functienal dt.fferentaa-,

ialp lsoninlte tht xhbi

tion;

dis independ

(.e., "(poseo wh(r1)i

a te reen tme11r.isore beal,.

11.9a hm

time

(11-tvd)o

as wellmas histories

.nlude

The internal dissipation 6 is then-defined by

s

6'-rp(rt l r

(1-124)

and the dissipation inequality Spe6

>_ q.g

(11-125)

results.. Since 6 is independent of g, we may set g =0 and obtain

k"'

>_ 0 .(11-126)

o ,~

*

Thit is, the internal dissipation is non-negative.

This latter

ineqva ity I$ interpreted as a statement that the Clausius•Duhem inequality-implies.the Clausius-Planck inequality

[31].

From (124).it is noted that histories wf~h' Jumps at'•the pretent time -are incapable of exhibiting instantaneolis dissipation. The above argument is supposedly valid for all histories.

How-

ever, it is-rot applicable for- histories which are smooth at the present time.

For histories which are smooth at the current time r"

-may not be selected independently of rt,

In particular'it is required

that ý be the limit of the ttestriction ft as s -+0.

In this case one

cannot conclude that =_is independent of g, andJ hence the subsequent arguments are not valid.

•

11 .99

Returning now to (118) leading to- (122)

we explore the validity of the arguments

for histories which are smooth at the current time.

The decomposition (118)

is motivated by experience with'simpler

theories of materipls of- the differential type. bn

These materials have

,studied by several workers including a recent in-depth study

by the luthors such as (llQ)

£79].In this work the author notedthat a decomposition

may be obtained for the stress for general nth-order

Rivlin-Ericksen materia.s with one comnponent of the stress derivable from a (non-equilibrium) potenLial, while the other compnnent. "Called. tthe extra stres8, may be associated with higher order dissipative mechanisms.

In the situatior, here, the generalized stress functional

has one, component ;-o wh'ich is derivable from thebon-eq~ilibrjium free energy functional p, while the remaining component to instantaneous dissipation.

=-,,gives rise

The assumption embodied in (119)

motivated by two sources; first, (119)

Is.

is a result of application, of

the ClausiAus-Duhem inequality for histeries with jumps at the present time, and secondly, a similar expression is obtained in studies of' materials of the differential type.

The associati'

the deIfinition of the internal dissipation gi contrast to (124)

f

(119') (121)

leads to In

it is' observdd.that histories which are smooth at"

the present time give rise to instantaneous dissipation.

This is

clearly a desirable property for viscous materials and general inelastic deformations such as, for ixample, plasticity. the dissipation inequality

(122)

It is noted that

is formally identical to (125)

the exception that s is given by (121)

11 .100

in place of (124)

It is

with

also observed that for smooth histories at the presint time 6, given by (122),

is not independent of g and hence (126)

cannot be, concluded.

Thus, the internal dissi'pation may be negative in histories smc. 'h at

the

present

time.

internal

The

dissipation is the amcunt by which internal working exceeds the rate of gowth of internal, energy less heat storage, or alternatively 6/8 is the amount by which'the entropy growth pn exceeds the quotient of non-mechanical power by temperature.... Roughly speaking, a negative internal dissipation then mnears that energy can be added to a body at rate faster thAn it can be dissipated by internal stress working' or heat storage.

-This result is not physically unreasonable.

Thus,"

the requirement of strictly non-negative dissipation is not implied by the Clausius-Duhem inequa.lity (except in a homothermal field) for histories continuous a- the current time.. This requirement may bie met.'if.the:

less restrictive and less general Clausius-Planck inequality

is assumed to hold (see [31)

.

)

From the above discussion,

through. (122)

give reasonable

r~strlctions on the constitutive functionals for histories smooth at the current time, whereas .the approach and arguments of Coleman! and others are inappl~icable.

It is easily shown that these results.

also reduce to previous results (123) with jumps at the. present time.

through (126)

for histories

For histories with a jump at the

r is independent of r r in (123) presenttime ' ,

and the inequality. (123)

.holds only if

l.10

7

•

Hence, 020) reduces t-o (1241

i-

(11-127)

oV- 0. and (1ii9)

follows at a-result, as-.-

well as (126). In conclusion, it is apparent that the development.presented generalizes

here, which -is based oil the reasonable assumption (119)

Coleman's results to include siioO'th nistories at the presen.t time,, prese-_ nd a~lsoqcontains his results for histories'with Jumps at "time..

his developmefit leads also to the physically desfrable result

that th3tinstantaneocs dissipation is allowed. •fivially,.it is noted in practlcal phys. •situati ons ju'np dlscontinuities do not occur. Be-fore clr,4ing this disC,,zsion, the results of (118) •

(121)

S,

through

will be assimilated'in4terms of the more f4miliar quantities b,

and q:

=0 n AptF

" (11-7128)

*x',

S

00

(F(F,) ;o.

)(k) S=O S0C

:

"

oI ,r)=(rt,e;;g)

3

S=O.

"(11Y

00

,

S=O

Truesdell 18o] has •hidependenrfiy arrived at similar concl-usions.4-,102

I

••

In addition, the stress So and the entropy n are deriwvble from the free energy functional p;

.- {)

-

p00) DJo

(II-132)"

'

(F

,(t) = Pe p(Ftet)

11.7.5

(11-133)

EQUILIBRIUM THERMODYNAMICS

The results of the previous section may be easily shown/to reduce to the proper results for equilibrium processes., Lt

The iothem.nal staticcontinuatic- of r by anount a is defined by

rt•(s•

Spt(s-O

,

=(11-i34)J

• ,' ' ? +-

+

. •

•l•.•-,+•+ +,++:

z

-+

+'+

,..,

•

•..

:

jj•ql

J

++

.

,.

..

+!

• •

"-"L-,++•!.,•1

#VC.

S&>W >~146D > UtO1

FO4M

tamvn

SP>TN

____________________________

hstrslA

14233

0!5 -

-.

j

1

ft0(piu

It

TimeM hatVA

11 15

DLXUA

fAA.I

4S~r ptootoa )

Oa

wi UmuatkAl fowun Ow the materWi tnanUom In tvoaim visco.l.kity

349

OhW-pidctloiw for the FOAImnhterid are shovn 'mi Fig. 1, and clearly none of the em-sitiutive eqjmtions follow the data in a satisfactory manner. From inequoities (4.18) the supeWition theory is superior to the others, and this is

zti OVbJ6 4er , The Unear form is not •iown as it was ahnost uniformly 98 per cet otthe *oduct form. The reduced-time equation (4-.12) gave a good prediction of tee LIN > SUP

(4.16)

Thus. we know a priori that only the superposition form can improve on the cprodiet fttowever, the third-order term de nihates, particularly at high Sstrcsse, and the superposition prediction is observed to have a significant error.

]!~ .157

IL-.O.SrwwouD

852

5. UM~A=CAL DMTA: STAXSSý-JRJAXATIOX

FORMUTATIOX,

Hcre, the thW-dorder integrAl polynomWa tthcoyX a6rn~pred witluiý6 sets of uni&xial data., which wene previously analysed wvith two ,quite, Aifferent threedimeasional constitative equations for, ir comprrssible, -waterir s's )l~Guanr =nd TýANI (1907) used a eimiplifcati~m of Coleýinn emd Xofls fiti~ Hia' i viscoelastitity theory to anays, styrene butadikn. rubber-, ZuAP" ad Cw~r'INS90), used the 35(Z elastic fluid theory td analys6 poliobqtylene, Broth dealt With enim*#wer Limited tv onevery large stvTins (80 t* 100 per Cent); liowe'em, * wieemt wns obtained atimmegh predietions tended or ýwo.-stcp tetts where good~a Ummn~

a0

dieAgS in~ theXac zeo of aniLnxompressiblke material, let

where a~4t1

Is - 2A + I/9

-,

The DIV. elaztia Mauid theory becorpes

AAms bw h tatig A'

vy ifeea standpo nt3 boffa iwvetiatons ended f(A) + 2 r

+2

(+

I(k,0~ MQ (W- 1) +

I

(12 -) 81

P8 Q~tIA; + 710).

A5.5))

56

Both invesigator~s revealed that three materia functions wene zueniont to describe the one-step meponse, and all can. be well represente by elexnentary functions like tP or IriS An equivalent third-*rder in~tegrl polynomial theory cain be develoyed ffr~m the R-foxrnetion of (2.0). By subtracting j tr a (to elkniiuate. the hydrostatic pres. srme) anid txpreming M in terms of A,where At A(1.g), one obtains

On mathematical fornms for the matermial ifuroons in norJlcar vivcel'idtlcity

+

44

#-±

(At' - I

'

_(A

I

22

353

1)- -2 ý)-~z

1

+1)• 0( W-. finite stia~ins, we write Since (8.8) is restricted to smill Ta l~

d .)

57

and expmad the terms i;u the eqi~aions in pow.-rs of e. Eqatuaion (5.8) e& be made to coinkide with (5.5) andt (e, to third order min by choosing thre fe',inwing definitions for the #'5 *

s *ad Car

a v. mwdL3AYs A

+

t

4

+8-

-4 A

~The errur indueed by ign oring fourth and higher teams ean be readity •timatcd Fbr ole-stcp tersts, (6.7) r(

to

)

i

US7 + omit

of loas is an open one. Neither of rae papers presented TSe q5e.t8on isti-step pplion5.,dlu c c.Eqatis of ep datea, non ot weio was in the range dein (.5 an) d (a.10)9 myeued to determine which approximatin aoino (57).wquatiotsh

"Adre xandt•

'a

11.159

I

a-I0

J93

x

.

1.

1

1.

0

00~i

/

90~cn

.

2

a,

Ex-d

Lin33

rd Prrg

Lo i n

i oroa 170

920

I0

1

_1 Pi.5

'30.ac O

2 0 tts-elaino

S U

ItGITRl

/o

eod U

S

4

A'

eps

Z. t

10

A

e

-

S-m

1o

0

10

10

0

40

0.i Sttqn-!sat.o of PLU

50

so

-zoo,-

.S

(fint and, 4vr~

4I

---

Z.Lc.

,

I

V

N

wilpiediet -t4larger response. F1k, ireashig stepsa bo~tW raterhi~s yield th~e refilts t

SU4' > JARN > P'ROD)

LIN> PRtOD> SUT

SUri-> ftQD,/,LIN .

By Pwaluating relative inagnitudes, one finds; *

Ar

S.

.

.

.

.

e

--

A

,

7

-

APPENDIX A

-

StfARY OF LINEAR ELASTICITY EQUATIONS

i.....

A.V- CYLINDRICAL EQUATIONS: EQUATIONS OF MOTION:

e + L@z VP+ aT3,

1

-

+

ToT z

(~z

+ R

2Tre

3 z 4,-

a lfrz

r

+

-

P -tT

2

+

+ Irr ,

• 2W

r

r

.. . STRESS-STRAIN:

Ea

II•+.

tZ

• r.z

r

au 4v r '

oz)j + a(T v(a,+ 8

a+a)

(

TO)

O

.3w

=

Fr"Fe rr

--

To

TZ•

=-o r e .

r

"Irz =Tz"

ar

-"+-yoz

'T Te

5z,

L

+

4

COMPATIBILITY:

V4. = 0

=' stress function

,

Ia

22

i

a2

a'

A.2 AXIALLY SYM44ETRIC E.QUATIONS EQUTIONS OF Wrl".2r:

ar

az

r"U

r +No dependence on O 'sre -ez

-F+ '3z E-Z+ Tr-z•Z 0 "arrz 5z

=p =

r

STRA I '-DISPLACEMENT: au

$

wa

u

"C

w "R

=

=

-,'($tress-strain •.

relation qnchanged from above)

Ii

STRESS FURC-TIONAL APPROACH V

= 0

V2 =

+

(NO BODY FORCES; EQUILIBRIUM)

? •_+

a-

rZ +.7--T--7

or=i 9z

C~vV2+-•r] TZz

2

= -V

SA,2

9

--

0

r

r

E

r a•rz

E

r W3z

(1+v 1 1

6

az

a2

(1+-.[(-2V)V

=

U

4++•-

+4

E ~rDz

W

[12\)V2ý I) E

A.3 AXIALLY

2$ +

_

rl

SYMMETRIC EQUATIONS

Dar

-

TWO-DIMENSIONAL:

EQUATIONS OF MOTION:

or "e

"' aa'

+R +R

p

a2 u

~STRAI N-DI2',o~ SPLACEM•ENT:

•

o

-

aru,

ar-

r

-6

a

uw

PLANE STRESS: a,7

.

0

-(a•

v a-

v or

'L

-

. Ez

+ a+

+ ac(T

-

To,

(T-TO)

•(T aB

o A.3

S!j

*PLANE ýS!kAIN:

+.

a

*

Hence, ~to change Olarie stresý in~to plane str~in, substitute:(

L.E

STR

fa Vt a-r.d -~-y-forv,

'~FUNqrOR.ALAOPROACH'(NO

BODY.FORCES A~D EQUILIBRIUK):*

V40-

II

A + 2cx~ r

r ar In-r

~

C

r A,

d

A.4 ENERGY THEOREMS '4INIMUMW'OTENTIAI. ENE!kGY THiEORN4: Of a'Ll -displAcelen1ts sý!t1.Afyinhg ccnP-at*'bflitj and the di!ýpltacemeitt

-

*bounda-.y c.onditions, :those- which 'satisfy teequi 11bri um, equati ons and thd boumda'y Lot.di tibns make tthe 'potdfital energy$,. V,- an 3bsa'iirtE nVI!n. xu.1

-

'4

C

F-

T

/ w (e.,j dt-J-T U do

= 2 + 21i)e

w

Uc

(30X + 2p)a(T

-4,e2

-

0 To)e

/

(r+ =W(T-To)

.(eij e

=

u (T,)

-(eij)

ail + e 2 z+ e33

NE

E

If the potential energy 's an absolute minimwu

thep the displace-

merits are such ds to setisfy eovu'librium and stress coiditions.

MINIMUM CfNPLWaiENUhRY ENERGY THFOREM: Of all stress tensor fields rij that :-atisfy tfe equations of equilibriw; and the boundary :onditions whee stresses are ptes~ribed

Hmake

those whiO satisfy compatibility and displacement botndary ccditiwls

the co

anV~ abso'Llte Minfirp.ll

o1exnlta '-

A.5

I

-

-

*

.

CONVERSE'THIOREM:' If the complcemrtaryUenergy V* i~s irn ab501JtS Minnimkm then stfjisfy' Cwpatjbijity and%dis~placemeft. the str~s~es .are-su~ck-As

REISSNEWS NIXED PRIHýIPLE:

valuo of Elastic equilibriui I's 4jitejngished-by r,stationadry providid the fur tiopal JR when T~nd u1 smr va~'ed indepandenitly, s symwetric. t~hat the t.ensor T

TUi d6 T

A.5 REAIO$ITWE4

-

-jtjk.~juj do'.

L~ICCjTA-4T5 2

Ev

,

3vK

3K(-

____

rt

K

-S:

vTO~~-

E

-- 2

-.-

vT

*--

3 K-ANj½

S3K-FE

E

___

r)

r dlr-

15,1 2

_____u

ýK

2'4

r I~~

..

'A

___ A-7

APPENDIX B FINITE ELDEME•T ANALYSIS

(o

a

{I

•,,

I

-

-

.

.

......

-

... .. "_ ..

.

'

--U ___ --

iL

___

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. _

...

...

___

_

_

_

_ _

-•____- _.:'

_ .

__ •

_

-__

-

Il

A

JAPPENDIX B FINITE ELENTA,ý ALYSIS

B.-A

INTRODUCTION Id chapter four of this handbook the fundamental '(elasticity)

equations of the finite element method wers developed, and their rpplication to grain structural analyses discussed briefly,

In this

"appendix, the-finite element method of analytis is further discussed., The emphasis -of th,i_-sscuss49n- is. on the use -of a finite element computer program instead of upon the theory of the technique or the mechanics of computer programs.

Thus, this appendix -reflects, the attitude of the user

rather than the producer.

Those- persons c.?siring to pursue the theory in

greater deta.l than presented herein are referred to the many text books and survey articles. ,. In particular, reference I is an excellent introduction

4

to the subject and C2-4]contain particularly lucid, thorough descriptions Can:aysls .f7Thiepp1ication-of the finite element method to orain struc,-ra qroblems.

Sk),A

-

sample computer program listing is included here for illustrative purposes.

Listiags of computer programs for-particular analyses are also

contained in the reports rfll-erenced in [5] through [18]. B.2 SUMMARY.OF THEORY As mentioned

iin

chapter four, finite element solutions normally begin

with the staCment of a variational principle.

A

functional is defined which

has as it! arguments the reveva:7t physical variables of the problem. then ssown that the particular functions (arg

It is

cfrtain admissible types)

which minimize the funqt.onal are in fact the ones that-satisfy the governing differential equations of the problrm.

For example, the Theorem of Minimum

A') NWIS

Poteritial Energy statai that the ptnilenergy assv~rs an absolute minimum

for thase displacements satisfying the equilibriumi equation, provided thatI classes of admissible functions are limited td thost! satisfying the boundary conditions.,--

It is obvious, then, that one could (I) guess an approximate solution tZoI aproblem.,2 clutete approximate solution~. -as

the better.

otnileery,

and (3) compare to another

Of the two, the one with t~he minimum enerqy Is chosen

We then search for anothe~r approximation to comare.

By sUch

a process, the solution will eventually be reached, if it iz p'ossible to reach a solution.

A nore systematic process. a~in to the Rayleigh-Ritz:

method,* Is to assume an approxilute solution, with unkndwn parameters-

We

then detemilne the unknown paraiat~ers in stich a manner as to 'minimize the,,. ftictional.

Then, we know that thea solution we have is the best of the typeI

Iwith which we stý;rted.

The^ finite eltemsnt methed is1sessenitially suc! a

Process

-

-----

-----

---

-

The origin of the name of the "'finite elenvnt" arises from, the fact *.at, to perform the process ot minimizat'c

Q cussed abev,

the body hein§t

anelyzed is divided into small sbegfons, over which expressions for the displacement are assuird.

For exan~1e, suppose We wisn, to analyse the brody

shown Jrn Figure 1. This body which is axs~mtric, is subjected to axisynwr-etric loads, e.g., axial acceleration. W~e could suppose that this body may be repre~sented by a collcction of rirjs of trlo.,ular cross-section as in Figumm 2. This is;, in fict, thie. type elemen't used for such problems.

However,

tie triangles usually are conb1ned intZo qua~rilaterils within the program, and we need on~ly con-.eri ourselves with the quadrilateral.

T---

Thub, we might

(,

A J

C.

-

-Y

il.

I Figure

i.

I

Rpreentaeive

AdBysetric

Solid

4

I

_ _

I

Lily N

I S

I

III

.1 FiFure 3,

1

Quadrilateral Idea1izaeio

II

-1

-

of Solid

I -

______

1;

break up the body into the subregions such, es shown for simplicity o00Y in Figure 3.

The right hahd section, of the6b)qy i.-

rhown

subdivided.

In the axisymetric case, thKse elemnents ire each actually rings.

!If

this ware a plane probiem, they vould be prisims with axis perpendiccular to the plane of the paper. rhe displacement field is no assumed in some form for each element. This In.roduces several unknown paramrters,

Thege parameters can be solved

ftfr in te-,m of the 'dlspfacements at~particular points in tha elements, usually the comers which are known as the nodal points. "Hence,

we have a

set ot subregions, for' each of which we have an assumed displacementt spape depending upon the unknoýin nodal point displacements.

The functional for

each region now can be calculated and minimized with respect to the unknown displacements.

This leads to a system of algebraic equations in the nodal

f8e~d which minimizes the functional and from which the stresses capn be "0

•,alCul ated..

In actual practice, the ebove is not carried out as easily-a3 it.is described.

The handlitig of the large amount of data and solution of the proe__•b]L__ms W

resu•ting system of equations a effitient mannsre.

must be solved-ii-n

-.

However, this capabilit+y has been built into the .comput~r

program and as such is not the user's problem. One point which is significwnt to a user is the form of the displacemont assumtion.

if the function is lineiBr.in each element, then the dis-

Splacements along tuhe edges of each elewnt will coincide. des-, reble,; and ig frequently used,

A

-BA

This appears

Higher order assumptions will lead to

.Te nus linear assumption also lends itself to dpproximatic-l

diLfferant

of..arbitrary fields as the e1iments becomi, smaller.

The point to 5a nioted

~that in areas ofý bodies. where the actual displacement field is linear, a

linear approximation is- adeqtiate 'with large el~ements. In areas where the displacemenit is more coMpi. 1nswlsaie be requi reAd to approximate t he actual displacements.

In par,,-14cu ar, in areas where the dispiacd~ents

iare progressively mm-sinoniinear, the alemeiits -must-be prpgressi-vely sma-11ler..-

-it additton, errors in tbe di-splacamawr. approxima~tion produce even larger errors In stress calculations because stress is related tp the derivatives of disonkcenlentS. 8.'3 APPLICATION, OF THE FJNITt ELEM~ENT MEETHOD 'Y'hts and the next subse~tilott wi i Ishow 'in detail the steps involved in using a finite element pýroqram and will include some numerical examnples. This is undoubtedly the simrplest way to discuss- the practical use.,of the--fn ii~eeeen

ehd

Because the actual use of a program is dependent

on the program, one fisust sacrilfice somm generality by rteferencii-to, a partlcular program, to te one

Hovevev-, we shall assuumeste anvaiiabls'~"Iyograi~simrilar -te~nSectio-n a.S Specific- details 'may change fiom progr-am

torrograidbut'the general iddas remain the sane. 6,3.1

INPUT DATA-' Broadly speakinig, the data that must be input to.a computer,programi can

be seDarated out into several categories:

control information, nodal point

data, element data., material properties, -ýe load data.

These categories are

not uniaue 'and son-6 progran's may.. com~bined one or more into a single category. For example, it is quite natural to iunpý the last three tor3ther because for any eletvnt we must associalte the material -rf which it ;aonslsts an~d the

0-z

-

loads towhic

it is subjected.

W shall discuss Ve type of data ii each

cate•ory as though they were separate, and Inditate the conventions wherever possible.

--

I Cont~rol information- -----, (a) Title: usuaUy the first item of information required is the title, etc., or *u:ayid46nifitation the user desires (b) Number of nodal points (c) Number cf rows of .,cdal points (d) Numbe~ of mnate~rials (e) AxisyrnrZetric or plane problem (f) Number of cardi-of same type to be read Data of this type refers to the problem in ger,-ral and/or-to the proc.ess of inat to the 6ompriuter. qd i rd _ omthtn. . problem may also be input to this category. In t*is case, either item (b) or (c) information will be required: not h,-h fl- - -!......

S~~Nodia

which, Also, Item (e) will not be needed for a progriam that iW specifically either plane or axisymmetric. , point data

(a)

Nodal point number a single number or two (I. 3) .oordinates if the nodal point array is associated with a ma; :'Lc (b) Coordinates of the nodal point (c) Indicatein of whether forcjes or displacements, or neither, are specified in each coordinzate direction (d)-" Specified forcer and/or displacement, if any (e) Any other relevaX.i'iiormation, e..g. , .temperature, if it-iP associated with nodal points, or special types' of control -either

KEWor r'

i ism7

.

°.

Element datP-

(a) E10-menat number ()Nodal pointsý to which element is connected i(,) Mate rial, identification -All tis, inform~ation may be asa~rbed into other areas. The element fuermay be associated with a particular nodal point to which the elemnent Is con~nected; this eliminates ite-na (a) and Item (b); Item (c) can be handled a request to read the data while reading nodal point information. bei,ýJproprties (a) Mechanical properties - modulus, 'etc. (b) Thermal properties expansion coefficient ýc) Physical properties etc. -dencity,

(d) Identification, Load data (a) Element to which 'Load is applied (b) Magnitude of load - pr-eseure: temperature change, etc. T1h6Se data and material properties may be associated with element data,

hýC

-and

t

specilj~c noaL"

corresponen~ce

Itf is not necessary to do so, however.' Final comr.-!-ts on the imiut data Ir. general, it is necessary to define the jposition of every nodal point, Oernentg connecited to that nodal pq'intthe mate_*ia-' propbrties for eacb element, ab well as nodal point and elemi-ent lqtds. Becau,., a given probldm mifght involve close to 1006 ~~a points, with sorriewhat fewer elements, the am~ount of datz. 'jould, be. pr6hibitattve if- each nodal point and elemeneit was treated separately. This js not done, of course, A progran. -ill incorporate SO or fself -gener~ati 1 points freature. F eapiit6nodal are sepaftt ed by s~everWa: far~ which no datýa is a- ecified, the pvog~r'm mifgIA generate the. reldiriield ifitermedia'te poixits along a etraIight line jo'irtiIg the,

0-the

*

gi ven 'P.,nts; withx u.iiAermT -sacig SK

r,

'th- elevi Atsi left, out woui4d

~~~be gefjr-ated, By a6tis

are 6

0

ratingprpcess.' the number of input data is co;nsidei-

The onlyi nodal Points a nd Ce4z Yents tha' require Operification

*Z'rd~~ 5oe

&l-ang 'the boundary of the object, as wkell as inte~rior Points for 83

-

II which chan•." from previous data occur.

For example, when two different

materials are adjacent, the transition from the fi-ýst to the second must be V specified. B.3.2

I

OUTPUT DATA

At the conclusion of an analysis, a great deal of information has been created - stresses, displacements,

strains. etc.

All this information is At this point, there is no convenient way to reduce the volume unless the analyst knows that only certain results are needed. In general, the output..

analysis will yield hundreds of pages of results..

14

Provisicn can be made,

however, to write the results on a tape, and then to have this information presented graphically in terms of stress and strain contours, etc.

The user must be cautioned against placing total confidence in the re------- nf-------

__analyst

%----, l,

T.h

;

a..

. -

-ac

th.od :s

a u....u..

t.oo.

1

uI i. & .

.st

o

not a panacea for the analyst. Results will be dependent on the ability of the to cantiire tlis essential fsoelris of the prnh).-M hir proper layout of

'

the nodal points and elements describing the object of concern. There is no substitute for intelligent engineering consideration of the problem. B. 4

EXAtPLE PROBLEMS This subsection makes mnre specific the comments of the preceding sub-

section by presenting a group of examples of finite element analyses, including the whole process of solving the problem. the reader is

referred to the references

For additional examples,

listed at the end of the appendix.

The program used to solve the examples is

thie Rohm and Haas AMGO32A,

suitably modified for use on a Univac 1108 computer. The program listed in a folloing section [7] is due to Wilson and is similar in operation the Rohin and Haas Prooram used for these examples. B.4.1

UNIAXIAL COMPRESSION OF A RIGHT CIRCULAR CYLINDER The first example is

the simplest problem that one can formulate

uniaxial coqpression of a circular cylindrical sample, B.8

-

The setup of the

to

-

I

•

I problem is

i

gu

4.ho,ývnFigure i

*th -. the necesbary defining quantities. y

I

WN

4()

shows the physical problem

As shown in Figure 4(B),

it is

possible to reduce the problem to one of smaller proportions by taking into account symmetry.

This is in a form that can be analyzed by an axisym-

mctric program, i. e. , a body of revolution loaded by axisymnmetric forces. We note that points on the axis of the *bodycan only deform in the z-direction, •hile points on the r-axis can only move in the r-direction, this from the ,smmetry of the problem.

Thus, we have established displacement boundary

along the zSonditions and r-axes. -nown, while the side is free. Figure

4(C).

On the top surface, the pressure is

The grid is added and nodal points defined in

The grid is the simple, obvious one - an array of squares

0.5 inch x 0.5 inch coveriz~g the section of the cylinder, a total of 50 elements •ard 66 nodal poina. The nodal points now have (I, J) coordinates associated with them as shown. I The displacement conditions are indicated by rollers. The nodal points on the two ?ixes can roll along those axes, except that the (1. 1) nodal point cannot move at all. This figure defines the problem in enough detail to generate the input data, which is shown in Figure

5.

For convenience in discussing th2

"data, we have numbered the columns in groups of 10,

1-9 and blank.

The first line of data is simply a title in the.required format. --

The second line of data gives the number of nodal poin: cards to be read 34), and number of rows (11). The next 34 cards are nodal point cards, followed by IZ cards giving flement propertie•, and loads, and finally a card ending the data.

Note that I

columnn number, 3

row number.

B,9

V31

- 11

;Ii

PRSSRE

J 3I

DIAMETER: 25 N

41

I!

t'

F

PS

100

,.LJ1

!

t

•

I"'•

HEIGHT:I1OIN. -r ~PRESSURE: 1000 PSI x106 PSI, ~~~~E ~ =30

'

•.

0

tt tt.I

SYMMETRY

'

((A)

S-

-

-

i-

Figur

4.St

i. i

-1

,B.1_

ofp SimpeCopesinPrbe

El IftI AMLIVA

S1"I'LI.

"m

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CL

CfP41-Ilk2,L

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,

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I Ul) 'Iti1*.

~~ fi| u~l

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u il3 olu I9

(Jiwli'l b1111111

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)

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i

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31IM11 . ( Ill. 11I I IUI I~ |

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liUo1i1li I Ul I

9 I

Lllll 11 5

.3.

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nn

. t)ilil

I nno0,0

tLNL 'W1 .)A I A

SFigure

5.

Input for Sirnpl,, Compression Problem

Otto

a

*

I

I In the system'used in the program here, the number of an element is

associated with theXridal point of smallest (I, J) connected to it. poinl.s are input, we then specify whether an elerntr not, and what to do abou -the 'element properties. easily seen by refererAceto the data.

iq to

As nodal

ssociated orbe

e

This is probably most

As the computer reads the nodalpoint

cards, the following data is found: I, J. element typei four boundary codition codes, - coordinate, 'or vairple, Card,.A 1 -. ,

z - coordinate, four loads and/or displacements. Is

X-1 I

SColumn

Data

5

1

I=l

10

1

J=l

15

1

read element data

16 17

1 1

ur specified u specified

18

0

no moment or rotation specified

19

0

no slope specified

2 1-30

0.0

r - ctwrdinate

31-40

0.0

z - coordinate

41-50

0.0

u

51-60

0.0

uz

61-70

0.0

moment/rotation

71-80

0.0

slope

"

M

I

"itthe number in Column 15 were 0, the program assumes the same data as for the previous element; i- it were 5, no element is associated with this point; 2, 3, and 4 also have specific meaninga,

Tf the numo ti br

Cc'-mnnP

16-18 is zero, no data is specified; if 1, displacement/rotation data; if 2, force/moment data. otherwise. r,

Column 19 has a 1 if displacement is along a slope, 0

Column 20 is blank.

The remaining columns, by 10's, contain

z, u r# uz, moment of rotation, and displacement slope.

If the data in a

given field is zero, it may be left blank. Thus, the first six cards refer to nodal points (1, 1) to (6, 1),

for all of

which the z-displacement is zero, the r-displacement specified (aq zero) for only (1, 1). (6, 1).

Each nodal point has an element associated with it,

The element data is read for (1, 1),

slope or moment is specified.

except

and the same for-the rest.

No

Coordinates are given and applied forces or

B.12I

U

53

displacements are zero.

Note that all thu ,iudai points in Row I are

specified since it is part of the boundary. Fornodai poiun

(1, i), elernenL data is t6 be read, and the same data

holds until nodal point (1, 10) because only 1 in Column 15 requests data. The first element card is 27, on which is read, by 10's, the following: E" ,-' na.1

.--

o- ,

adil; radial body force; axial body force; pressure; shear,

a number to indicate where the pressure and/or

*a.and

Rhe-

r acts.

Cagd Z8 is

Alsoiread, and indicates what information is new by a I in he appropriate I

For all the elements qp to (1, 10), E = 30 x 106 psi, v = i/3

column.

Note that all the boundary nodal points are entered as data; and on the left, an element is entered with each nodal point. Starting with

(1,

10),

the lower left corner of the left element, we must

indicate the pressure load on the top of this element.

Trus, we request

element data to be read as Cards 29 and 30, where p = 1000.0 psi and a 3 indicates the pressure and face dn which it acts (according to a standard scheme).

Since we don't chexnge E or Y, these values carry on.

For ele-

ment (Z, 10) we again read the pressure, since the program does not carry

on the pressure from element to ele

ant.

This process continues Wcnodal

(6i 10), with wh'ich -o element is assocated. Nodal points (1, 11) through (6, 11) finish the regions boundary.

No ele-

ment is associated with these, and only (1, 11) has a specified force or dis-

C)

placement (ur = 0).

While the above description is somewhat tedious, it gives a detasled illustration of data requirements, both in Idiid and arrangement.

I

-

.%.,•tptLir this problem is shown -n Figure 6.

Not all the output is shown, for it is too voluminous.

It consists of the

following: (1)

Botndary conditioi; itiformation for verifying the boundary conditions

(2) '.-oordinates of all nodal points-includes all tlhe interior points for which no coordihxates were specified (3)

Displacements oi all nodal points

(4)

Stresses and strains in each element - radial, hoop, axial. and shear stress and strain; and maximum and roinimunn stresses and strains

B.13

*

41

4

1

.4)0

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'01 0.0 100Ca 0a N w 0

1)0 0 NO %17 NO 1)

0

0

0

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=0

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.

8 ONU

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00

0

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00

00

0 0

00

00

00

00

0

0

0

0

0

0

0

B 0-0 0

4N.

A0

01

8

0

00

0

0

s

0-0 .

4

1

IJ

I'I Only data for elements (11) through (4,4) is shown.

There is no

need to snow more, as the values are the same for each row. changing nl y r- a phenomenon we should have anticipated, due to symmetry. (We

J:in

Notice that ar = -1000 psi, /3 X1

'6;

problem,

'•J

B,.-4.2

fr

= to = -vt

a

-0.

0, as is correct.

= 0.111 x 10-,

as is correct.

Also

t =

=

In this very simple

the dis -acernent field is linear, with consequent good results. THERMAL EXPANSION OF A RIGHT CIRCULAR CYLINDER

This example uses the same shape as Example 1, - circular cylinder, but the loading and boundary conditions are different. In this example we' subject the cylinder to a temperature rise of 100°F so that adT= 6.5 x 10-4, and restrain the expansion which would normally occ,'r. This requires removal of the input cards specifying the applied pressure, and addition of a

U

I to Column 17 of each card for Row II to indicate that u z = 0 for these points. Note that-both the nodal point cards for the interior nodes of Row 10, and the material cards corresponding to those nodal points are removed. On the first material card the thermal load is then added. The input is shown in Figure 7 . The results are shown in Figure 8 in the form of the output for Elements (1, 1) - (4, 4). Again the displacement field is linear, and the resulting stresses are exactly correct. B .4.3

CENTRIFUGAL LOADING OF A RIGHT CIRCULAR CYLINDER

Example 3 uses the same data as Example 2, except that the thermal load is replaced by a radial body force, rwa = 1000. This gives the stresses set up by rotation with angular velocity w. The results for this case are s ummarized in Figure 9. The stresse's a0, Vz and displacement u r are shown. The exact solutions are given as solid lines and the results froin the finite element program as points.

exceilelt. J.15

Again, the agreement is

II i

II

,

;

,

1

5

1 .2

5

I

~

h 1

'I

, h

""b 1

(•

211,

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6 I7

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2,5

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50 0')O0

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O1000 U

3 9

b500i) 1Olh0

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.o

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4

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12149;6789

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,f STPAINED THE,:NAL [X,'),TOW

IAV,.0A

A

7111

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• ]oi53:3,5 .•uJII()rJ(WI(W'

0.O

0o.n-

n.u

O.)

t•.0

t.fU (Mh UAIA Figure

7

Input for Restrained Thermal Expansion Problem

B. 16

0.0

!! vI~

it z.

00

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ova .aV)

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owa

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o1. an 00

an

6

0N

6

~'aan n~

in 000 00

06 .

6

6

j 00

6

b)6 3tý6 0

6

n 00O

6

06 00 i

00 6

a )

i

n I0I 6a 0

6

00

6

an

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0

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6

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I..

3.0

. .. .

2.0

----

_____ _

-1. 1.\

"CLC

STRESSES AND DISPLACE;E MS FOR ROTATING SOLID CYLINDER

_'r

0

Co

Q

0

-

,.

0.51.0

1.5

2.0

RADIUS,-r (in.)" /$

Fig-Are

9.

Stresuea and Lispllactments for Rotating Solid Cyli'nder

8.18 i

Q°

0.5

2.5

.

B.4.4

INTERNAL PRESSURIZATION AND'ROTATION OF A HOLL3W CYLINDER

In the next two examples the geometry is changed to a hollow cylinder. I.e inner radius is 0. 5 inch ant, the outer 3.0 inches so that the wall thickesss is 2.5 inches, as was the r'adius of the original cylinder.

In both

exAmples. the longitudinal deformation is restrained, creating a problem in 0 strain. For Ixample 4, an internal pressure of 1000 psi is applied; and for Example 5, radial body force is applied, with r w9 1000. The resul ts of the analysei art-shown in Figures

8.4.5

10 and 11.

INIERNAL PRESSURIZATION OF AVCOMPOSITE CYLINDER

'To illustrate the ability of the program to handle problems with multiple materials, consider the problem of a composite cylinder subjected to inter-

nai pressure."the loading is again an internal pressure of 1000 psi magni:;de. The cylinder is now composed of two maArials: an inner portion (r: 0.5 inch to r = 1.5 inches) of copper, 4na an outer portion (r = 1.5 inches :3 r = 3.0 inches) of eteel. The result of the finite element analysis and exact solution are shown in Figure 12.

Note that discontinuities exist in

V.€ displacement gradient and gradient of w., as well as in a' and a . In ",hsparticular example an analysis with a mesh of one-half the original size (o.z5 inch compared to 0.5 inch) was performed.

The solid points for a0

are from the second analysis to demonstrate the accuracy with which the discontinuity is captured.

B,19 *,

.

L

LL

Ur

0

3.0

-

1 U1,

O

•, 10

--

.D

-

CLU

t4.0

0.5

Figure

1.0

10.

1.5 2;0 RADIUS, r (in.)

,

2.5

Displacerne'nts and Stresses for Hollow Cylinde-" with Internal Pressure

B.20

3,0

t j.L 4.0

8.0

.1 U 3,0

6.0

r

W\ 6

4.0

b -

j

wý

\' .41.0

40.0

"Ix

0

00.5

1.0

1.5

2.0

2.5

RADIUS, r (in.) Figure

11.

Displacements and Stresses in Rotating Hollow Cylinder B, 21

3.0

5

1.01

.

4

-....

DISPLACEMENTS AND STRESSES INHOLLOW CYUNDER OF TWO MATERIALS UNDER INTERNAL PRESSURE

;•-,Ur

"3

0

0.6

•

-

•

=

O (FINER MES)

I

Z2,-"4

1 0.2

'\

0.5

••...

101.5

2.0

qz---.

r (in.)

S~RADIUS,

Figure

j

12

.

2.5

Dieuplacementc and Stresses in Hiollow Cylinder of Two Materials Under Internal Pressure

B,22

3.0

SAMPLE FINITE ELE1MENT COM1LITER PROGRM

G.5

PART A

PROGRAM LISTING (UNIVAC 1108, FORTRAN IV)

-

ON F'-Nn~O T?,; AFAITQAPY AXIcSYlMrrTp Tf' C,(-%g C CCVMCtI

ltI'4MNP,":LWCL.thtJv'~AT t'iu'At'C

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ion WRITE (6#2009) H3 CALL EXIT IIll C(%14T I IJUE C PEA" AND PRTINT OF ELVmEf'.? POPEOr'ic'5 WRITr (b#?00i) N=O 13tC READ

IF

(5.1003)

.X(4J,.)

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B. 23

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READ loir PpzpT

OF PRErsuv~r .Om*'oA'vY CO01¶!TTON,J IF (?tt;i.pc) ?&ifl.3j0ppan 20A' %PITr 16,2MO) DC 300 L=IPNUVIPC RFAO (501004)3 IFC(L.vJRCfL30P(L) 30q WPITF (6#2007) I0CfL3..J~f~L).PP(L) Ai. CONTIP:uE

c

OETEPPINE "AmIV VwflTo 00 34.0 N=1.N1P4(L DO 34.0 1:1,. 00 325 L=1,4~ IF

II~W.J)

O

325,3250!20

32M~ J:KK 323 CtNTINUE

340 COtITINUE c

SOLVE t'ON-LINFAP STWUCTVRE AY SUCCVMI'E PIOPROXIMATIONS DC' 3'S0 N=IozNUMEL 350' EPS W) =0. n 001500 NNN=!;pwc FOP*M r-YIFNESS MATRIX CALL STIFF c SOfLVE: FOR DISPLAC!'*FNTS CALL QANSOL WAITE U#Z.006) c COC.PUTE* SrPF.SSES CALL STRESS 50(l Ce,,TINUtE GO TV' 50 1001 FORMAT 1002 FC!F..g 1003 FORMAT 1004~ FORMAT A00ý FORMAT

(2!5s2vI0.0 (I5*F5.0tSPj0 *q) (61S) f215#FI0*0l (SF10.f)

2000 FI)RMAT (1IN! 12A6/ 1 30r4C KUNSER 2 30140 UILMIER ' 30140 t1JUNRER 'I 30140 KA"$4CR 5 30140 AX!Ai

MF OF OF CIF

NOVAL POMTS ------- 13 ELEMENTS-------~01FF. WATERIALSS...PRE5SSJrE CARDS---- 13/ ACCFLEPATIO-------E2/

&30140 ANGUJLAR VELOCITY--------------

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B.24

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PART A

C

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-

LISTING (Continued)

ADD ELEMIrNT STZFVNIESS TO TOTAL STIFFN5SS 00~ 200 1=1,4% nm 2flh K=102

00 2r'O L1.P2 ),J=Lv(J)*L- at.I1I-KSWIV-

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IV (PItUIPC ?60f3J0.?60 261 DO 300 L=.:1NUdPC PP:PPfL1./l6

I)42.'(j) ZX=2QRZ IF MIPP) 262#2*6'..62

26? RX:3.0 Z)=3.0 2611 JI:2?*IKSPSFT JJ=2*J.-KSHIFT 26% IF (IT-tJO) 27fl.270*28e1 270 SINAZO.0 COSAZ1.r) IF frOr)EM)~ 2712t72.'??P 271 Slt;A=SIN fCODE f I C~SADcQCOQfc II)) 8 fl 3I I )=R()-RYO fSTP:A.$)-COSA OW I 260 IF fJJf 300#!00#28c 289 IF C4JJ.ND) 2V0*?91l,30f, 2901 SINA=0.O COSA21.00

IF ICOCCIji) ?x291202OA2 191 SN*:SINdCOM(W) I

B*.26

PART A

(,;,I

LISTING (Con~tinued)

-PROGRAM

P)

SIN.(Aor 7C(SIIs

=0(JJ)LjJ;-7y

30)(1 CONT INUE Z. nt;PLicr"!iT P.C. C 31AP, nt) unn0 f NL-f," N:?.V-l-KSHIFT 31'. IF

(v0i)E(M4)-i.)

3170,3r#'1-

IF

(C7 r'E0A)-3.)

'Q'a,30q3t1a

31',

SC TON O00 380~ CALL ~0IY(..0

~Rt~~,'

4.00 C CPNTJI'JE

C

-

I01TE FILOCK OP FtQTI-'?.T.S ONt TAPE Ame SwIFT UP LOWER BLO'CK

6fl'.n.ca0

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FILE 11vr AONsT0V) =A (K v

42n~ ZALI. FYIT FOR E~ 'nr C 50

0

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200tF)Pl

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OLaro5v) SURPOUT111 C1wor (IJI 3)-T NPNie ',IU A~lI'4~~r~iir^MAnTMPTPP pyr~laT!12)$Tpql0 7 fro( 1T2)S!pk~ 4FC ;,

1 CrF(60)#TODO ) ICI nn

79l

P00

PPPOl)PAOB.6

V10)

i

..

PART A

P ROGRAM LISTING (Continued)

MF*It~' 1,.YrF) -Trp,,p) 101A

)0-4,10410i1O4

PTI^=n.0

prr'N~r (M- 1 WMTYPE )-EF(N¶-1 I VTYP-) Ir ([P ) 70. 71 ?1) 7e% PATlr:=(TEak'-- (m-1.1.VTYPrF)/r)Er

71

Dc ir'l.

KK = -

TEMPAPTFP4P-0 EPl;R:EE(7) I/f7(1) IF

1"SRM-F4".(Kj) )

I,,lr..

10A RAl :=(FE(7)/(EPS(N)1r1f tr(1))t.O-YYI;N~fmTyoE3),XYNI(p4TvPE) Erni -- EW'-Afio 10 A CONJTINUE IF W"PP) I0-8h. k

C( I -2) =C(W'4EFI2) C (2.' #2

C(3.

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C(.30).-,r)'/EEE3)

T(2*.1):((C(" C1f '1.2)CY2

c

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FORK' C~AlAfRILATERAL

ST!FFtIF4SS 14ATRYX

DO 94 91= I.', RAlZ IF c("N)) IF

~.1Q

(C')PEMIA"))

6)TMP

93.q?091

9! RPRwv)=Rl",

DO 921 JJ:1@6

B. 28

PART A

Jj=j

-PROGýjAM

LISTING (Continued)

If,*

i CALL

j)

Gt( )(1 '30

~~~CALL

TIT~,,

CALL TPISTViIs2,5)~ CALL TflISTFI;?P,5) CALL

7 T'ISTFL'.4N5)

Dr 11411111,P6

De. 140

jj=103

13M' ~Rf Tijr

S!,RPetT I

tIf) !~STF

; I,jsJJwit

0,F'l 12)t P 2 roo~,r j00 f ( OO

I-

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ZZ (2 :ZzC I! I Do ar "1,1 on tenf.1=1.6 lop, O(Itjvo.o c 3. F0064 CALL rriTEPfXIR;'sZ7,

8.29

)F (1

),7(0 1fj

9 0 .sA U7(900),

~ PART A

-

LJSTIN,ý (Cniud

-PROGWA

IF (NPP) 104#106.10k~ CID

4 1CV 0

~TO

108

1,;10)= 115)s

3-3

DO 110 JS.1u6 C

4.

FORm

C07FCETDSLCMN

TRANSFORMATIOm MATRIX

*or i i 3,.~tOz

)#/COMb

DNv3*3)=(FP(t2)-.PR(1 1t DO 12n ~. H(2fJ):0Dt(2. I HC..Jl)Z00(3#1) 12

O. 2 , XIF (AGLtEl))

122o125r125

COSA:C OS IANGL~F:I)) IJ=2e1 HIfe1J-1)=TEPV.COS#4HtIK.IJ)*S1NA C

i2A CONT104UE ~ FORMa ELEMENT, c;T1PFIES DO 130 .jzit1O 00 1:ýO KZ:t6 IF (Will4)) 2,0.p 1a2A 00 129 1=116

VA'TpIY 41)'.Wri*f)

13"I COWI7NUE

IF I0$(X.I))

138014001."ie

B. 30

PART A

C

LISTING (Continued)

-PROGRAM

6. FORM THEOMJAL LOS!' MAtTPIX IF ,ripp) 14so1bO,1ws 1845 TTc3)ZO.G I~~1(1 , EE 4) 15A

COmM=rO('ilTYPE )*AJGFG*S2 TP(1).ZC0MlAXI(7) + IXI(21*TT(3) TP(2)-C0MfA.XI(91 + yI(1)*(TT(j)4TT(3~))

T,"(i)=roAk!; 1i~)4 x (Iis)*TTW~ Cl(%VM-kO (klTYPE)*SACFL t

DO 166C 'i12010 Oe. 16nC J:1Nf6

4n00Do 41

J1=1#6A

411) HWC I vJ)=A(

P.J).(ItP.A'i.J

PRETU~n ON FQR SYVTNV SUBRPOUTINE 5INYEPlPD.ZZ) DOMEVr4N /PA(E4#4)

C0M A(C,)M-A/iJ4.' Ir(2:1Rr)ans-.

ONFOR1:14T 2) SUBPOUZI-E 3)

-t

ER

9P

-DIMEN.ION COMMO

"DT

'XI#Il60*.#33q

/PLAE4

N.

-_

PkART A

-PROGR(AM

LISTING (Cotinui~ed)I

1.10

10 Ot pr

Sm Y I I )tu.O

00

;,. I:1i6

lFRX)$.LT.IF-jO) r0 To 1')0

XI(.VX1(4)*Ym(I)*ZCI)/A(T) .10ft Cr~sT IN1A NETUPN ON FM '.-oDIFY StPOOLITINE %fvrlFY(I.I%.'EQe'48A'4O#fU)

Of) 250 M=Z#.4eANI) 235*235#e230

VIXU)

IF (1EO-K) 2509240924.0 A(N#M)t0.0 250l COPiTINVE

Enjo

ON Fnfl PANSOL 5Skl~QAUT1NPF

WIPSOL

REiWWO It REWIP.~) 12 Nr~:0

I

S GOK

OAIOS'YBOK tDC C,

9.TTROKOFEUT, 1,

ion

N--W4

PART A'.

r~c

I

Son, N~1et4;J

XFA

)A(N4j) *2i#V)

rO D5t

C

rFfWT1)

RFADPNEXT/A(Nf

Z2.

PROGRAM U1STING (Continued)'

3,

(11)~,~P~

,rF L~:CED VTOIJTIN

OUITCI PLOCK

Do 3-qN100 GOf

1-0f'

22A f#4)

;

/

f~

DO 2cD K=L.M4Y

(lLmL-P

IF

1-OC

374a0 fl

GO

(12)

375' ýlO47S ST 6)Aus)P:.QeN

"C 00

SACXelýTTTO DC 40lDC 450r M=I*NN DC4:!9 K=JN M

451A 83A(NM N)ietNR)

BCSPAC0U!'E 12~5 CCMTON 40

c O~fIEP N9Nnl~vsIN PB.33

bV =

O

AP

PART A

II,Ef8##

I

-PROGRAM

1?)IJiC( 12) #VYXU

LISTING (Continued)

s(90fl) UP490OO) #1J7(900) l?) @R(13AC')

2 CODEC 1030)~*TRP(l v I tC7() *.SC10neI) I

H~1-NfeO).QR('.)

#PP(210).t(4 ANLE ('4) oSI"G( If))

ZZ(4).C(4.t4) .H(6.tln) D(6.6)eF(6.1O)#TP(6),XiC(1O)

CC'vfW"I /pLANL/ rgpp CCu.PlITE ELEMIE14T StrcySVS

c

DC Ain0 M=I1ýhu"EL

CALL 111AD(N#VOL) I)(Nplt)=VTYPE

12n P:1.T=q(J)

ki

DO

153

K1=11I

16~GTc 165 11P(q)=0.O P(M10) 0. 16% 00' 170 1:1.6

Oft

-

00' On K=1#100

FRP(2I):TP(?) TP(J )=TP(1)4HH(T.'t) P(V)

174 DC IAn 1v1.3 SIGI I)=-TT( I D0 IAO K=103 Ianl SlG( I):Sl;IG )4CC1.K)*PR(K)SIGI Ia)L (I4ý'.) .Rp ( CALCULATE EnEDGY TERuc; C Do' 250 1:1.lo Ccm~ftC.O0 00 200 x=1010 20A C0"=PCOIMMS(1.K)*Pfx 25"i XpF=XpE+COmm'l I) c

CALCLOLATE EFFECTIVr STa"AN~ IF (NP) 251P252#2Si

B.34

PART A

c

0

OUTMIT

-C

LIS-LING (Continued)

-PROGRAM

S.'TcEsF-s

CALCI'LP.TE P'PPICTPAL

Sc'rrrt_;c

CR:SCQT((SJ6t2)-.S1G(I))/?.O),. S1G(C,) CC.cp c

2

+STGL'i*.2

SREcsýrs PkkALLFL Tr LY!,F T.-J I:1 ripiv 1 (1

1

SIG (0)CX*CnS?A4CTc, (4) tcPPA *CC

IQZF

(1,1P) OO,0uin

IO I ¶~(VPI) 11Os32 k.3t 6 WRIOF (6,20001

10qi

IF !Y.T-TRE '3 K

p00

X 10'Diý.£,

2

2 37H ANGLE IJ-S.TpFSS jK.-5TRESS, SHrAP 2001 FORmAT (7Zo,216I.,P9.,PEGp 2006 FOqRaaT (L56#OAPPPOYrMATF FUNV14DPNTAL FREOWJNCY ENO)

B.35

1341NCTR5

£1205)

______ _-

,

I

- [

,

-

aV

I' PART B

-INPUT

DATA

The following is a description of the input data used to describe the problem to the computer. A.

IDENTIFICATION CARD - (72H) Columns I to 72 of this cvard contain information to be printed with

results.

"B. CONTROL CARD - (415, 3F10.2, 215) Columns 1 - 5 Number of nodal points (900 maximum) 6 11

10 Number of elements (800 maximum)

- 15 Number of different materials (12 maximum)

16 - 20 Number of boundary pressure cards (200 maximum) 21 31

-

40 Angular velocity

41

50 Reference temperature (stress free temperature)

51-

55 Number of approximations

56

C.

30 Axial acceleration in the Z-direction

-

60 = 0 Axisymmetric analysis = I Plane stress analysis

MATEFUAL PROPERTY INFORMATION The following group of cards must be supplied for each diffe rent

mate riak First Caý.rd--(215, Columns

1

-

6 -

2F10.0)

5 Materials identification - any number from I to 12. 10 Number of different temperatures for which properties are given - 8 maximum

11

-

20 Mass density of material

21

-

30 Ratio of plastic modulus to zlastic modulus

B.36

0

I Following Cards - (8F10,0) Oae card for each temperature Columns

1

-

10 TamperaZure

11

-

20 Modulus cf elasticity - Er and Ez 30 Poisson's ratioV

21

rz

D.

31

-

40 Modulus oi elasticity - E0

41

-

50 PoA-kson's ratio - P0 r and vOz

51

-

60 Coefficient of thermal expansion - c r and az

61

-

70 Coefficient of thermal expansion - c0

71

-

80 Yield stress - (y

NODAL POINT CARDS - (215, F5.0, 5F10.0) One of the first steps in the structural analysis of a two-di,-nensiona:

solid is to select a finite elem'ent representation of the cross- section of the body. Elements and nodal points are then numbered in two numerical sequer.c! each starting with one.

The following group of punched cards numerically de. fine the two-dimensional structure to be analyzed. There i13 a card for each nodal point and each card contains the following information: Columns

0

I

-

S11

10 Number which indicates if displacements or forcee are to be specified - 20 R- ordinate

21

-

30 z - ordinate

31

-

40 XR

5 Nodal point number

41

50 XZ

51

60 Temperature

If the number .. , column 10 is

0 - XR is the specified R-load and XZ is the specified Z-load. I - XR is the specified R-displacement and XZ is the specified Z-load. 2

- XR is the specified R-load and XZ is the specified Z-displaccment.

3 - XR is the specified R-displacement and XZ is the specified Z-displacement. B.37

___Wes_______V90_W__

W!_____

All loads are considered to be total forces acting on a one radian segment (or unit thickness in the case of plant stress analysis). Nodal point cards must be in numerical sequence. If cards are omitted, the omitted nodal points are generated at equal intervals along a straight line between the defined nodal points; the necessary temperatures are determined by linesr interpolation; tho boundary code (column 10). XR and XZ are set equal to se ro. E.

ELEMENT CARDS - (615) One card for each element Columns

1 -

5 Elemeat number

O Nodal Point I 11 - 15 Nodal Point 3 S6

16 21 -

20 Nodal Point K 25 Nodal Point L

26 -

30 Material Identi-

1.

Order nodal points counter-clockwise around element.

16 -. Maximum diffe rence between nodal point 1. D. mukt be 'ese than 27

fication Element cards must be in element number sequence. If element cards are omitted, the program automatically generates the omitted information by incrementing by one the preceding I, J, K and L. The material identification code for the generated cards is set equal to the value given on the last card. The last element card must always 'o supplied. Triangular ,lements are. also permissible, they are identified by repeating the last nodal poLAr number (i. e. , 1, J. K. L). F.

PRESSURE CARDS - (215, IFI0.0) One card for each boundary element wbich is subjected to a normal

pressure. Columax

5 Nodal Point 1 6 - 10 Nodal Point3 iI - ZO Normal Pressure

1 -

B.38

___

____________

________

PRESSURE

•~NORMA•L

I As shown above, ihe boundary element must be on the left as one progresses from I to J.

Surface tensile force is input as a negative pre.ssure.

B..39

PART C

A.

-

ADDITIONAL REMARKS AND OUTPUT DATA

MATERIAL PROPERTIES

Material properties vs. temperature are input for each material in tabular form. The properties for each element in the system are then evaluated by interpolation.

The mass deiisity of the material is required

only if acceleration loads are specified or if the approxirnkate frequency is desired.

Listing of the coefficients of thermal expansion are necessary

only for thermal stress analysis.

The plastic modulus ratio and the yield

stress are specified only if nonlinear materials are used. B.

SKEW BOUNDARIES If the number in columns 5-10 of the nodal point cards is other

than 0,

1, 2 or 3, it, is interpreted as the magnitude of an angle in degrees.

This angle is shown below.

0

S

B. 40

The terms in columns 31-50 of the nod~l point card are then interpreted as follows: XR is the specified load in the s-direction XZ is the specified displacement in the n-direction The angle 0 must always be input as a negative angle and may range from .0,001 to -180 degrees.- Hence, + 1.0 degree is the sa:-ie aR -179.0 degrees. The displacements o! these nodal points which are printed by the program ire:

)

the displacement in the 3-direction uz

the displacement in the n-direction

C.

USE OF THE PLANE STRESS OPTION

/

A one punch in column 60 of the control card indicates the body is a plane stress structure of unit thickness.

In the case of plane stress analysis,

the material property cards are interpreted as follows: Columns 11

-

20 Modulus of elasticity - Er

21

-

30 Poissonl3 ratio

31

-

40 Modulus of elasticity - E z

V

-

The corresponding stress-strain relatioaship used in the analysis

Or Vz

_ 0 P 12I)1_

0

r

2(+VjrzJ 2(#+ P).

E E

1).

z &

I'r.#

[rzjLwhere

"'1g -r

0 -00

Er

z

EI

OUTPUT DATA Th*. following information is developed and printed by the program: 1. 2.

Reprint of input data Nodal point displacements

B.41

It

*

-=I

.•

.Z

"I

~ 7,N

44

U#

-

3. 4.

Stresses & the cent.r of each elerrient An approirimate fundamentai frequency. (The'displacements ,--Lor thegiven load condition are ýIzed a; an approximate mode shape-in the calculation of a freqiuency by RFaleigh's procedure.,

-

E.-

A considerable amount o, enginee•ring Judqernent~must be used in• irtterptietation qf this frequency.)

PLOT OF FINITE ELEMENT MESH

f

The program automatically devrelops'a plot of the outhne of each eldment in the, system.

This serves as an excellent"heAc on the input data.

In' order to obtain the plot from AGC'a computer operation, an additional Oharge card must be aubrgtted with the job. If only a plot of the mesh is '&de

the calculation of disPk.ements and strqsses may be eliminated by specifying more pressure cards than actially exist. The first 30. columns

~fthe identification card are used a~a &/t~itle for the plot.

-'.4-

S..

-

---

-"N

B42) B

"REFERENCES 1.

Zienkiewic'z, O., C.,.The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hi 11 Fbok Company, New York, 1967.

2.

Anderson, J. M. ,"A Review of the Finite Element Stiffness Method as Applied to Propellant Grain Stress Analysis Feature Article, SRSIA, (AFRPL-TR69-2?0), Vol. 6, No. 4, pp. 1-54, October 1969.

3.

Pau, C. , The Applicatiun of Numerical Methods to the Soution of Structural1Integrity Problems of Solid Propellant Rockets,""SRSIA, No. 2, October 1964. Pau, C. H., "The Application of Numerical Methods. to the Solution of

S4.

Vol.

1, •

"Structural Integrity Problems of Solid Propellant Rockets II", SRSIA, Vol. J, No. 1, January 1967'. 5k Becker, E. B..and Brisbane, J. J., "Application of the.Finite Element Method to Stress Analysis of 4old Propellant Rocket Graitis," Technical ReportS-76, Rohm Haas Company,. Redstone ResearCh .abmrator~s, November '1965. •,~~ý , ,S ° .6.

Pister, K. S., Taylor, R. L. and Dill, E. H., "A'ComnJtzr Program for Axially Symmetric El asti city Problems ," Mathemati cal Sci enceS Corpora ti on Report No., 65-21-3,, December 1965.,

7.

Wilson, E. L., "A Digital tonuter Program for the Fin.te Element Analysis of Solids with Nonlinear Material Properties," Aerojet-General Corporation Report TM-23," July 1965.

8.

Wilson, 6. L., "A Computer Program for the Dynamic Stress Analysis of Under-lund Structures," Structural Engireering-Laboratovy Report 67-3, 'Univer-sity of California, Berkeley, February 1967.

-q)

9. Hermann, L. R., Taylor, R. L. and Green, D. R., "Finite Element Analys;s for Solid Rocket Motor Cases," Report No. 67-4, Structural Engineering Laboratory, University of California, Berkeley, March 1967. 10.

Brisbane, j. J. and Becker. E. B., "Stress Apal~yis' of Solid Propellant Grains Under Tr'ansverse Accel-er,.ion Loads," Technical-Report S-116, Rc,.m anI Haas Company, lFstone Research Laboratories, March 1967.

11.

Dunham, R. S. and Taylor, R. Lv, "'Finite Element Analysis of Axisymmetric Solids withArbitrary Loading," Structures Report 67-5, University of Cal it'ornia, B#'keley-, JLne 1967..

12.

Nickell, R. E., '"Stress-Wave Analysis jn Layered Thermoviscoelastic Materials by the Extended Ritz Method," Technical Report S-175, Rohm'and Haas Company, Redstone Research-,Laboratories, Oct3ber 1968.

B.43

S.

. .. .. -' ." ..

.

...

.

=

••

-

l- .--• .... ..

.. . .... . . ..

13.

BecKer, E. B. and Pau, C. H., "Application of the Finite Ele-nt Method to Hekt, Conduction in Solids," Technical Report S-4l, Rohin and Haas Compary, Redstone Resear'ch Laboratories, 46ovember 1968.

14.

Leeaiing, H., et al, "Solid Propellant Structural Test Vehicle Cumulative' S•atrage, and Syst.ms Andlyses," Appendix, Final Technical Report AFRPL-TR-68-130, October 1968.

15.

Cost, T. L., "Thermomechanilcal Coupling Phenomena in Non-Isothermal •Viscoelastic Solids, Technical Report S-226, Rohm and Haas Company, Redstone Research Laboratories, August 1969.

16.

'Becker' E. B., Brisbane

J. J. and Schkade, 4A. F. , Jr. , "Investigation of Techniques of Three-&imensionai'Finite Element Stress Analysis," Tzchnical Report- S-250, Rohm and HaastCompar.y; Redstone Research Laboratories, March 1970.,

17.

DL.,ham, R. S., "Dynamic Stress Analysis of One-Dimensional Thermorheologically Simple Viscoelastic Solids with Nonlinear Heat Conduction 4nalys~is," Report No. RK-JR-70-13, U. S. Army Missile Command, Redstcne Research Laboratories, July 1970.

18.

Taylor, R. L., Goudreau, G. L., and Pister, K. S., "Thermomechanical Analysis of Viscoelastic Solidso Structural Engineering Laboratory Report 68-7, University of California, Berkeley, June 1968.

CI

B.44

B.44

N

N

....... . . . . . .

.

.

. '-.

% . .

...

F APPENDIX C

-

PARAMETRIC DESIGN CURVES

I ~o.

p

4

ic _

_

_

_

[Tz

FOREWORD

The parametric desi~n data presented in this Appendix has been compiled from the following tovo references:

1).. LOCKHEED PROPULSION COMPANY STRUCTURES MANUAL, December 1969 El4.-2)

Messnier, A. M. and Schiessmann, D.:

"Barameter Calculation

of Simple Propelldnt~urains for Teeperature Cycling,

j

Pressurization, aný Acce'seration", Appendix D,. Study of Mechanical P'4vties of So'kid Rocket Propellants, Aerojet-General Report No. 0411-10F, March 1962.

Permission to publish, this material is greatly appreciated.

WI

.3.

General

Curves are presented to aid in making preliminary stress ani strr.in analyses of propellant grains. Inner bore hoop strain, interface radial stress and interface shear stress values may be obtained for a wide rangip of b/a and L/b ratios. Linear interpolation may bs used as required fo," problems in which intermediate values of the governing parameters exist. The primary loading ,conditions associated with propellant grain structural problems are a ftuiotion of pressure,, temperature and &ooeleration. Rosulits of a stress analysis for each of the three loading conditions may be ouperimposed to obtain total stresses and strains resulting from a combined loading condition. Analysis was performed based on the irntinitesimal theory of elasticity and subject to all of the associated limitati,.ns and assumptions, In particular, the assumptions are made that the propellant in a homogeneouss isotropic, elastic solid and that the magnitudes of tha strainas rotations and displacements are small.

Actual grain configurations consist of a concentric hollow cylinder with %quarG ends, The analytical solutions are thus valid only for the ! isyvmntr.o1n. cae. Mechanical properties end load levels usad in the analy3ea are as f•llows: Linear

of ,oefient Therma2

&-pansion

Linear Coelff"ciant of Thermai E-X

,nicn

Case

in .089 x 10.-6 in*F

ao

Prropellant 'a a 63 x 10 "° p

Poisson's Ratio

Case

--. 0.3

Poissones Ratio

Propellant

VP =0,5

Modulus of Elasticity Modulus of Elasticity

-Case •

S

-30X 106 psi

Propellant

Thickness of Case to-Diameter Ratio

t/D - 0.001.95

Preesure Loading

P a 500 pai

Axial Acceleration

A a 10

CA0

S

p

- 1000 psi

•.2

Recommnded Analysis ?rcedvr*ý

All of tht desim cu-veb are baaed on aneletical solutions of a Consequently, a star bore concentric hallow cylinier witch squars snd,

coni'uration must be converted to an oquiwilent ho*iow cylinder. Absolute, wriitm.ds of th•e stresses &a strains may be obtained f-a As

the design -uryes no follow*% Detarvine considaerd the dimensions and loading conditions to be

i.

Outar rmdiua of propllan,

b

2.

!inner radius of propcllani.

a

3.

Lcngth of propel)

L

4. 5*

Pressure Cure etmperature

6.

Acoeleration in number of g's

7.

q.

nt

-

Poissoi's Ratio of propellant is

0.5

p AtTras. AT

n Vp

Modulus of propellant.

B.

Find appropriate soluitions for specified geoetrj Linear interpolation and end bonding condition. may be used as required.

C,

M'21tiply solutions found in B by the applicable -1. factors lined in Table

D,

Determine" strain concentration factor at the star tip and multiply .- n*er bore hoop strain by this frictor. For original grain dtsign, consideration should be given ta the following recommoonded design practices.

C.2

I 'U1 I

m

60

;1

I

I

4

42

43

1 -ZU

jI•I.u

,

34) 0

. ._V _-l! ..i-i-... ........... L4

C;.3

0

-.

0

°

V)iO

"E

.3

Fecovriended Design Practices

In addition to ensuring that strain levela are kept dowa A. to a minimun it is necessary to achieve a high strength bond between the In gemeral it is good practice to keep the bond propellant and tne case. of the propellant modulus. 10 percea.t atress lower than radius A compromise must be made between star tip fillet B. Structurally, a star tip that subtends the maximum and sliver loss. angle is the best and the ideal tip geometry is a 2:1 ellpse. * Structural reliability decreases as grain constr•aints C, Highly bonded grains with srall in*er bores are poor structurally. increase. Free surfaces and unbonded regionri tend to relieve thi stresses and strains in the system. There are several basic characteristics of the propellant D. grain systems that can be assumed. While defornati'mns of the motor case have a direct 1. influence on the propellant-grain, ths grain is incapable of causing any Mase deformation effects. All strains and stresses generated by independent 2. loads can be added to obtaiu results for combined loads.

3.

Steady state thermal conditions are alwuyb more severe than the transients between the steady state condition. If a motor is to be placed in a low temperature otorage box until steady state conditions prevail and then mored to a high temperature storage. the most severe structural conditions will occur 1-n the low temperature envircminent.

within the limitations of elastic theory. SThis statement is true however, for the case of viscoelastic materials there is evidence to the fact that actual tip geometry is not important

C.4

(,

UA

I

.A

Sample Problem

problem. V lpae Use of the design curves is beet illustrated by a combination following the for solved ir I Figure The problem shown in of loads and miterial properties LOADING CONDITIONS

I

1.

Temperature Cycling to +350?.,tT a -S*?

2.

Pressure Loadings p

3.

Axial Accelerations A a 31) g

12%90 psi

MATERIAL PROPERTIE S Yw0.5

11

Poisson's Ratio of Propellant

29

Poissonts Ratio of Case

v

3.

Modulus of Elasticity

Ep a3500 psi

0

*0,3

of Propellant

h.

iO6 e

Modulus of Slastic.ity

psi

do30x.L0

of Case

-6

5.

of Linear Cc-fficient for Thermal ' .9anvion PropellaA

6.

Linear Coerficiont of Thermal Expeunsion for Case

7.

Density of Propellant

8.

Cure Temperature of Propellant (Streas Free Condition)

63 x

aP

C.5

in-OF

0

5.9 x I0Cin-O - 0.063

p T

-

cur•

12C0F

in

-

in

III

Iii I! ]I

I3 I

""

.

IC

4)6r

IMF

CALMUTIONS

L

35 in.

b

110 in.

L/b .- 350 b/i - 3.33

t/ vbero

-o0.00195 1 1

370 and

92 145* max. -

3.31

2r

o gbore'

One End is Bonded Presuriiation Interfaoe Radial Stresa Actual Pressure

, 8tre=-- ,'-I"=- :,500

psi

-:1'

Interpolate between F-1gu-es l0,11i a•• Streas

a

r -492

[]=.

1268

j-

3 14

psi

LWoaion near free end Innor Bore Hoop Strain Strain aa Value ahown x Strain Concentration Factor Interpolate between Yigxner lO,lland

13,14

C.7

,

__

'

--

w 14.20

Strain Strain

x 3.316

I

w.13.90%

a

Location Near Bonded End ,ni.-rface Shear Stress Stress - Value shown x P

Mouu -0ln

Lntvrpolate between Figires

17 and

WOEs,,-

63.8 psi

- 18.2x ,•x

.

f-15

Ne6.r Inner Bore Surface

Location: Te~e ra~turo •ycling

Interface Radial Stream Actual Temp

ropellant

sho•i x Modulus

Strss -Valv.e

"

VOk

x Difftronce

-190F

"InLerpolete\betwoen Figure&3Q2,33 and _37,38 Stress ,'

--

x .8 2 j~

B 8tress :

r.

x

v 310 psi

-Ation at

'ree*Fmd

Inner Bore Hoey Strain Strain

Actual Temp.

Strain - Viue shovn x Cone-ntrett~on x Difference Fact or -796F Intetpoi=.tAbetweer. Flurea 34,3YJnd 6.1x3.31.

Strain Strain

37,38

za

*a 21.6%

"L,-;ation Near Bonded W4

C.8 -'/

!

1•

K

\

I!

Actual Temp. Difference

Intorface Shear Strees

Stress v

alue shown x

P~ro~ellant Modulus

-79"F

psii 1o lO00

Intvarpolate between Fipzres 34,35and Stress

u

23.5

Siress -

1O00,

39

-79

- 89,5 psi

Location near* irmer Bore -\

Accoleration.

/

•~nterface Radial Stress

/"

Stress

*a

Actuial .show x Acceleration

.u.

intorpolate between Figures

o

Stress

0.4 3f.8 Psi

St~ress

i98x

60/asnd

4 xb.555 Actual

Stress -Value' show.- x Acealeration x twe.n ~Interpolate Exd Firured 60 Oree Aoc~ticm Nat be

I

g tress

-4-.8 xX x 0x 5

Strs "

a-8.0 P-1i

n

b 63

-b63

\I

iLoxatic'n Near Freo 3hd

17

1.-

C,9

1 /-

I~mer Dore Hoop Strain

*1

Actual Strain Strain uValt'. shown x Concontrati~• x Accelera'tion 10 Factor Ici

1000 xPropel1a Modulus

S,

x

b

"

interpolatA between ?igux"4' Strain Strain.

0.58 x 3.31 0a

-a x

60 and

63

x 1 x 00

.0.92% Cole

Location Roar Bonded End-

"

----o - 4-- -

-. -!--

-

-

.!5

Parametric Curves

Structural problems commonly encountered in solid propellant grains can best be described by illustrations which show thA Z-6in deformations under load.

The propellant solidifies at its cure temperature and is then strese free as shown in Figure 2A When the motor is cooled to an operating , or storage temperature level, the propellant trios f.o assume the configuration

shown in Figure 2B Figure 2C shows the low temperature configuration that results because the propellant is bonded to the case and cainot assume the unbonded geometry of Figure 2B Iner bore hoop strain is a measure of the percent increase in inner bore radius from Figure to Figure 2C It is important to note that strains are measured from the stress free configuration 2B and not from initial dimensions shown in Figure " 2A A plot of inner bore hoop strain, since it represents percent change in radiu:s will have the satwe shape as the inner bore of the grain. The defcrmed grain shapes obtained by pressuriration and acceleration loLding conditions are shown by Figures 21) and 2E respectively. Mechanical properties, constants, and loading conditions which wore used throughout the analyses, to prepare the design curves, are presented in sectieh .1.

Configurati-1s of geometries analyzed are shown by Figure

3F

and a list of solutions are presented by Tabia -2. Reference to the list of solutions will facilitate use of the design curves and permit the rapid solving of problems. The design curves are grouped by type of loading and classified within each group by geometry, type of stress or strain and definitive bimensionless ratios. Each design curve thus catalogued is referenced to the a-propriate figure and page number.

C.11

'N---

z-,

II

0

ou1

4-2

04

C-0

10

t. IC

12J

4

t.

P4>

CL

mL

/b

II

1'*

No Ind Botudia

b

One End Bonded

|L

b

Both Znds Banmded

S.....

•1

' Direction of" Acceleration

S Configaration of Geometries Analyz"ed

:

,

Figure C.13

3

t-

"TABLE -2 LIST OF SOLUTIONS M./AD CONDITION Prezaurization

TEOMEIY L/b No end bonding No end bonding No end bonding One end bonded

O

-

One end bonded Both ends bonded

t

Pressurization

FIGURE

•

Interface

4

4

2,3,94,6

4)5 6,7 71,9 8,9 10,11 1.2

8a

bd

e

b/a

STRESSE

Poth ends bonded

2

Interface

2

End

/4

Interface

84

8

rnd Interface End-^

12,4" 13,14 16,17

2

Interlace

2 4

End

18 19,20

84

Interface End Interface

8

End

21 22,23 2,3s4,6

24 -25.227-7

Temp. Cycling .No

No end bonding end bonding No end bonding One end bonded

2 h 8 2 2

Interface

.

Bo

e• s bonded

14

14

Fnd

:37,38

8

p

40,41

End

4

Interface

45

14 "8

End Interface

46$47 48

43,44

End

Acceleration

No end bonding

2

Interface

end bonding No end bonding

14 2 2 ""4 a 8

One end bonded

K \

C. 14

2,3,,h,6

49,50

2,0,,9,6

52,53

954 456,57

8

14 Acce•lertion

42

2

8

end bonded

3

Interface

2

Both ends bondeid

IOne

36

Interface

Temp. Cycling

*No

28,29 3013L 32,33 36

1 1 Interface End Interface

8

One, end bonded

2,3o,4,6

itrface

End >

Interface End Tnterface •nd

I58.59

I60

61,62 63 64,65 2,3•14•

64

L-t of Solutions (Cont'd)

LOAD CONWITION Acceleration

GEOMETRY Both ends bonded

-- "

L/b

S¶7RESEs

2

Interface

2 2

Fwd end AFt end Interface Fwd end Aft Interface ld end Aft end

i4

V Acceleration

Both zndi. bonded

C.1

C;.15

4 4 Ii i 8 8

b/a

FIGtME

2,3,4,6

67,68

2,3,o,6

69 70 71,72 73 74 75,76 77 78

7I -TT Shear Stress Equal to Value i

-i--

7-,-

-ý4t-

ýShiown X ModulusH

-

i.4 4.

1001~

0

UP

-

b/LA

-

500 ps 2

-5

[PrcetofToal Stess mcnlLnth ov

-4640~~~~~~ 3 0

50

- bo6 b/ 20 6

Shontec StPresssCudbyPesure

6/, 2ý

b/

16

31ý'

0

80

9

500

-J

I

AI5

b.2

0

Hoop Strain Equol

2C00

tob71

Sluo:Sow

60

100

80

Percent of Total Segment Length MIoop Strain Caused by Pressure No End Bonding //

Fiure

5

C.17 4\

20~

.

-IShear Stress Equal to Value

17i~

SShownX 10~~10

.--

-

-

....

4'4

-too

-~

tb/a2

b/a.3

b/o -

/-

90

NEn;boBon,

Fhwnxgu

6-esre

C.18

10

J6

I7

Hoop Stvoiri Equal to Value Shown

U

co

b/a 2

b/c 3

/

0•,

6

.

L,,

• • 'T" , .,

ki10

0

b/o4

6

20

30

.- •

' ' .• •• •

40,0

• • • •

60

.70

Parcei oi Total Segment Lengthj

~Hoop Strain Caused by Pressure Wo End Bondiing,

F!.gure

C119

7

.s-

•• ,

80

90.

100.

30

Stress Eauol to Vl

1TShear

*010

b1/.1-4

*

500 psi

/P

I

~diaI a~ Stress Equal to Valu~e -'

-52o

Shown

-460

HH+:40

I

20o

*Percent

0

~

6

#3.0 7

of Total Segment Lang-h

liiteifcce Stresses Cauased by Pressu~re No Ee'd Bonding

C. 29

tt~I 0

9

0

IE *0I

Hoop Strain Equal to Value Shown b /a.

bA-

-

ED

0 0

10

__Percent

-i

20

6/c

-

-. b/o 6

4

-

-T

30

40

s0

60

70

of, Total Segment Length Hoop Stvain Caused by Pressure No End Bonding

80

70

1001 P

500 psi

L/b 8

30

4

20

-

0

6

b/

-TT

-t5t'

-490

I7 4

b/

3

0708

0405

Ln Pe~~~~cen''. csdb rsue~~nt~~~rfoc~~L/ UCLndB~i.

Pigu~e-1 -47.2

Nl 00

CA2 PSItdS~nn 2t~.

~I

1

Is

31.

11

74

1 CL,

A

-~Perceont

EfToal Seto VLuengtho

On* End Bonded

C.23

-

100j

'Shea

rt

-

-

-

60-

0

4 Q

b

-61-4,

20-

-

,t00

.~~~80P

4Z

- r'0

.2

ll

_7

0- 40

C.2

5,

4

psi

144 1T7

-

7

P500

0

9

200

T-L-t iI T -r,-,:I b/o quolto A'Axil aiI Sr,.ts

Shear Stress Equal to Value 25

o uu

111111iIF

I- HI

20

b/o

10 10

0.

b/a-2 ~p -460~-

-430

10

2

T130!' T0I

0

6

0

80

9

LHbffII

Rojiolt Stes Equal tome; Value L Hoop byPrePressLadie IL traj~~o~bed

psi -

L/

~tif

-L0

50

0

Hoop Strain Equal to Value Shown

I

10

QLL-j

IIIIII

CL

0

0~

/

6/

~p

.W

500psi

. 1 1 1

0

10

20

50 60 70 Percent of Total Segment Length

30

40

Hoop Strom Caused by Pressure Loading One Enid BS-,nded

I_____Figure C.26

14

L,/b -4

T-rr 80

90

100

90

tt ~ 0.*

c

Shear Stress Equal to Value Shown X 2 I U

-4.-

t

100

UC

60

40-b40

Interface Shear Stress. psi

-T..i: 90

4-

::L:p-SO psi

41

-44

=I U

-'-f

70-

bAbA

¶

io0

f b/ ~

-1

2

-

Jr7

IO

~ 4o

t

,

_

: 1" 4i0

in/oac

0

Axial psaltoVau Stress,

End Stresses Caused by Pressure Loading On* End Bonded

Fi

ýr

.

15

2-7~

_

_

__IF

SherS-'s, Equal to Value

S20 I..o

U10 .4--

pae500

86

-480

C4o2

-46

-400

0

Stress Equal to Value Pesr

*vIrRadial

*~-420

10

14 REiHft±L++J 20

70 60G 30 40 50 Percent of Total Segment Length

Stresses Caused by Pressuie Loading One End Bonded Figure

16 C.28

80

90

100

psi

PY

V-P.~r-'T1CV4-3 0 ,0 V-6-3

TRIM TRIM

-

.

.'Grain Gross-Section (Segment III)

S0

0

0.,

0.8

i-.2

t.6

Raaial Dist.9nce to Specimen Center from.Liner-G&rai Interface, d, in.

•IGU F

"'----_•

F-5. GRAIN ALLOWABLLE•-tUdAX1

.VARIATION

WITH DISTANCE

iii F.7

AI. s.AIN

RMM

2.0 2.0

4.2

I

A

--

=

F

110[1

10Rangre

of VA11100 Prnm 0 t' IIo .

Pan Samples of Mix Grain

""

O ,

Cross Section (Seg III)

4.-X

+

00

Spe cimen Series ©

Location •o

"§ s.

80 m

SI

•

60

L

'6

JANAF SPECIMEN'TESF CONDITIONS ~~X-4ead Rate = 2.0 i-n./min

Se e

Ss a ' Ide._.

"X

111-7 11-13

/

--

0o

S"

"Temperature = 77-F-

Tri-

-

"0

V-4-,2

v _-' ,LV-0

0.4

0.8

,

136

1.2

2.0

Radial Distance to Specimen Center From Liner-Grain Interface,- d, in. FIGURE F-6,'GRAIW ALLOwABLE--UNIAXIAL STRESS VARIATION WITH DISTANCE FROM LINER-GRAIN INTERFACE[C]

*

i

,S-

-

F.8

Trim Trim

Trim,

LLL

Grid System

__________a.

Caae '6 ,~E 29 x 16e

-

a~~~~~~J .iU

s

'-

4ner.ial

~

'

i= 120

loduu±j Di'stribu'tion

FIGURE F-7 MATHEMATICAL MODEL FOR BOOSý-SUSTAIN GRAJN MOTOR[P)

F.9 --

-

-

-

= ~ m r -'L

F......

•2

-- •2 -- -3F500 psi

'~~ •"

L_

30 99

7_______•

Sustain

,•'Psi

'-"Boost Grain

/

ra

18 S: 1 ,•

OS =-17%

Analytical Prediction Considering Dissected Motor Propellant Mechanical

" I•• -

Properties

Prediction Considering /?Non-Aged-Homogenous Propellrnt

S-Anfalytical'

F0

Poperties

*o

S.I -

--6

MS94%

I........ ..

-2

Station Number, in.,

FIGUR F-8

0

2

4

(See sketch atove)

m KuUTED MAJ(-MU m p sm -TRskvAL LiA-s i a, io N, G~RAIN ADJACINIT TO LINWR IN CERM11 R4OIN!OF

cA~ TN

HOTOFI,

,

F.10

Pthe

ccentral

properties

reio of this riiii~nr~o~p'~e cas'dterminired firom unz-ed control saj,,ples,

a' dflO

em~oyovin

the heteroqo~eous, propellant modulus disti'1bution indic'ted at the top of fiaure F-8 and in fi~gureF-I.'

Using the data _from the dissected

motor, a neg'ative margin of safety of 17% was calculated at the prooella~'t/liner bond. - Considerirtg tI'e conventional liaboratory co~rtroi data a positiye~margi6 o'f safety of 94% was predicted'. The c~rcuraferenltial crack. schematically illustrated 4". figure F-8, was observed during rad'ographic Inspection lof the temperature cycled motor and coý-fi med by visual inspection of th'e dissected grain.' 'Figure F-9 Mlustrate!' the associated *significance of heterogeneity

ElThe

at the inneor bore of the restrilctor -portion of this- slotted grain design. upiaxiza strain at failure of specimeps takean across %,he disscpted grain web give a relptive assessment of the local-immediatt vicinity of res-ri c~tr-reducii n in pr~pellant allowable."elongation, -The failure in this demo~nstration motor was attributed to a

Q

Icirative imbalance between'the propellant and linier which caused a

'3hardeningof the propel~lant and reduction of elongati~an at;zth6 propeliant liner Interface.,

gThis

failure and subsequent structural integrity reasstssirent program-'lllustrate~t the Importance and usefulness of motor dissection as a means of assessing grain structural in~tegrity. The curves of mehanical property data obtained from the dissected motor su~erOoses Cwt-ul ati Ys damiage effects resulting from aging ard temperature cycling. Thus,*compj~rison, of such data wieth un-aged control sample data should -Pruvide the gra-In structqra'l integ'rity ehqineer wi-th a higiher confidence

F.11

4

I."

V

.--4

4

1

-co CIA

c-.:.