design composite chambers

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Design of Composite Material Chambers Aug. 18for Solid Propellant Missile Motors Technical Parer, Aug. 1985[. PERFOPRMING ORG. REPOR-T N'j)BSER

7. AAJTMON(U) 6. "ONTRACT OR GRANT NUMBER(a)

James LeRoy SzatkowskiS. PCRFORMING ORGANIZATION NAME ANO AOORESS to. PROGRAM ELEI.ENT. PROJECT. TASK ~ EllA& OAK UNIT NUMBERS

. Naval Plant Branch Representatives OfficeP.O. Box 157, Ma.-ana, UT 34O44-O157

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y O -~' Composite material, missile c-m%.trs, c.. nosite pressure vcssels,

material properties, lzminatc ur.,3Lsis,

220. ASSTIIIACT Con.w= X O*o* .4g. If 004*0@WV d"$1" b? wEe' f 06A review of the technical research on composite materialsused in the manufacture of solid propellant missiles has beenconducted. Current design methods were compared to the methodsgiven in the literature. Most of the designers of compositestrutures use empirical relationships based on data compiledfrom previous designs, to. piedic the structure's characteris-

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tics. No universal approach to characterization of materialproperties of composties has been found, because no one hasbeen able to adequately characterize the material properties ofthe constituent materials and their interrelationships. Recentanalyses have shown that only finite element analysis (FEA) canadequately represent the material loading within the compositematrix. Accurate material properties, obtained only by detailedanalysis of experimental work, are required to be used inthe calculations. The properties are deduced from the experimen-tal results by computer analysis by finite elements. Accurateanalysis of the total structure of a composite material chamberwil l also require the use of 3-dimensional FEA. The use of 3-DFE has been limited because of canputer time and costs. However,new ccmputers will now handle reasonably sized 3-dimensionalproblems within acceptable limits of time and cost.

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DESIGN OF COMPOSITE MATERIAL CHAMBERSFOR SOLID PROPELLANI MISSILE MOTORS

byJames L. Szatkowski, PE

B.S.E.E. , University of Utah, 1976H.5.N.E., Naval Pcltgraduate School, 1984

8 09 03 .092


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TABLE-OF CONTENTSAbstract iTable of C-ntents iiNomenclature iiiIntroducticonTheoret icalDevelopment/Li terature Review 2Current Applications 7Conclusions/Recommendations 10Appendix 1 12Appendix 2 14,List of Reicrenzes 18

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NOMENCLATURE

E Young's modulusG Shear ModulusK stress'concentration factorM momentP pressurex,y,z principle directions, cartesion coordinatesr,O principle directions, polar coordinatesV volumeW weight stress, normalT stress, shearC strainO angular displacement, in degreesv Poisson's ratio

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INTRODUCTIONComposite materials have been used or solid rocket motor

applicat ions for almost three decades. The do'elopment of thesematerials progressed at a rate significantly ahead of the basictheoretical analysis, so a trial and error development programwas the foundation for years of manufacturing. The designparameters of isotropic materials are highly developed, and dataon the material properties, including most failure properties,are readily available. However, no similar information isavailable for composite materials. Designers must rely on themeager collection of data that has been compiled from experi-mental programs with similar designs and composite structures.

Significant work has been performed in the developmentprograms of rocket chambers, to test and evaluate sub-scalepressui-e vessels of similar design and composite structure.These sub-scale results are then scaled-up to the design require-ments. Unfortunately, the scaling factors are not well under-stood and failures often occur: this is pr-.marily the result ofthe, lack of fundamental knowledge of the composite structuresmaterial interactions.Study has been devoted to this fundamental deficiency.Researchers have attempted to apply basic isotropic materialrelationships to specific composite structures and have hadlimited succesa in the determination of the characteristics.Isotropic materials such as metals, crystals and some plastics,are not purely isotropic; their fai lure properties account fo rthe irregularities and characterize the failure mode(s) thatreault according to the size, location and orientation of theseirregularities. Since composite materials are, by their verynature, anisotropic, the attempt to characterize their materialproperties with relationships that relate the defects in theirstructure to their strength and failure modes is misguided.In this paper I will survey the theoretical development of

composite structures and contrast the theory with the currentmanufacturing design practices. I will discussthe proofingmethods and those methods that are used to analyze damage topressure vessels. Conclusions and recommendations will beoffered to improve the design and analysis methods used incurrent chamber design and structural analyses.

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THEORETICAL DEVELOPMENT/LITERATURE REVIEW

Laminate Theory (LT)The general laminate theory is a mathematical means ofcalculat ing the material 'propert ies of a composite from theangles of fiber orientation and the propert ies of the layers ofthe fibers. The matrix propert ies are included in this cmlcula-t ion and the matrix is asaumed to be homogeneous and perfectlybonded to the fibers. The elastic propert ies of the layers aredetermined from the Young's moduli Ell. E12, the shear modulus

G12, and the Poisson's ratio v12 using Hooke's-law relationship:a =EE, Eqn.1

where strain is proportional to stress.The laminate propert ies are calculated by using the angle ofthe laminate direction and the angle of the material , which arethen added vectorially (see Appendix 1).Theory implies that the stacking sequence and boundarieswill have no effect on laminate strength. Pagano and Pipes Illhave shown that this is not true. , Their work illustratedsignificant stacking sequence effects, as well as significantboundary effects. Tolbert E2] also reported that the laminatetheory failed in the analysis of thick-walled cylinders, but wasable to represent~thin-walled vessels without serious error.. Theprimary fault of this theory is in its assumption of zeroboundary stress fields. It seems to accurately account for th estress fields remote from the boundary layers.Nany researchers have attempted to confirm this theoret icaldevelopment; however., there has been'little success in correlat-ing the experimental results to the theory. The assumptions madeto simplify the calculations do not allow for the physicaldefects -in a manufactured laminate structure. of a thicknessgreatur than one-tenth of an inch (33.,For rough calculations,,. the general laminate theory can beused to obtain the approximate laminate strength; the result canbe refined 'by experimental trial and error procedures. Chamis(43 published a procedure, based on laminate theory, that can beadapted for use on small computers or hand-held calculators.The relationships that' apply to material propert ies are no taffected by thickness and are discussed in the theory of appliedmechanics. Fracture mechanics 'change with apeciven thickness.Fracture mechanics theory is highly developed for isotropicmaterials, but not for anisotropic laminate structures. Further,

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the relationships between the fracture characteristics of thickand thin fiber-reinforced laminates have not been established.

Linear Elastic Fracture Mechanics (LEF_Several investigators have modified LEFM for a homogeneousanisotropic material (see Appendix 2). They have developedexpressions for the czack tip stress field, using Hooke's law,for a homogeneous linear aniaotropic material in the case ofplane strain and pure shear. The results were analogous to the

isotropic m~terial case. The results for non-homogeneousanisotropic laminates were different, but were characterized bythe "inherant flaw model . In this model, a "high intensityenergy region" adjacent to the holes and at the tip of thenotches was postulated. This region wa's assigned a dimensionwhich was then used in the plastic zone currection factor fo risotropic materials to modify the stress. intensity factors andthe stress distribution. The investigators found that singlecracks do not nucleate and grow, as in metals, under fatigueloading. This l_ ads to the question of how applicable LEFM isfor composite mat.Rrials.While trying to explain this phenomenon and the, characteris-tic problem, the investigctors postulated a non-fracture mech-anics stress fracture criteria. A "point stress model postula-ted that failure would occur when the average stress at somedistance from the discontinuity reached the critical stress.This model worked well with some laminates, and not so well withothers.Using the LEFM and the R-curve concept, which relates th erising fracture toughness of the material to the increasing crackextension, the s.ze of the plastic zone for 'conventional mat-erials of thin sheets may be considered. For fiber-reinforcedthin sheets, the point and average stress models explain th eincrease in toughness as the notch length increases, as a resultof the stress distribution near the notch t ip. These approaches

consider the deviation of -the load/crack opening displacementcurve from linearity because of the damage zone formationand seLf-similar crack extension. This does not apply in thicklaminates or in laminates with delaminations (5].Shih and Logsdon 62 found during their study of thick-section graphite/epoxy composi tes that the LEFN did not applydirectly to the composi tes ' which 'had cracks perpendicularto the fiber ori1rtation. They also found that in three-pointbending the failure mode was interpiy failure. This mode offailure faults the basic assumptions in LEFM.Self-simi)-- creck-propegatio, another onaumpt ion of the

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homogeneous anisotropic model ,' is not assured because of theanisotropy and heterogea.i.:.y o f the laminate microstructure.Only for fractures that :.e' confined to interlaminar planes,where the matrix phase domina tes the fracture response andwhere self-simitar propagation is assured, can LEFM be usedeffectively [7]. In these cases, tests can determine the stressintensity factors, KI, KII, and KIII for the opening, shearingand tearing modes, respectively, as well as the mixed mode offracture.

Other researchers found that the values of toughness reachedasymptotic values and nonhomogeneities because of fiber sizeand ply thickness were not significant. Simple macromechanicalmodels based on LEFM, such as the boundary integral equation(BIE)' method, could be employed to predict fractures.Further models, such as the "shear-lag model-V were develop-ed to predict the behavior of unidirectional fiber-reinforcedcomposites that were suasceptible to matrix splitting ahead ofthe crack t ip vs. fiber. This two-dimensional model assumes thatthe fibers support all the axial force (because of their higherelastic modulus) and the matrix supports the shear and trans-verse normal forces. This method can be modified for angle plylaminates also. The micromechanical models, such as the BIE andthe "shear-lag'. model, were generally more accurate in thefracture predictions than in the macromechanical models.The macromechanical models which treat the composite as ahomogeneous anisotropic medium do not fully explain the physicalfailure mechanism, even when they agree with the experimentalresults. The microvechanical models have the potential toaccount for the -effects of material nonhomogeneity. These

nonhomogeneitioee canbe addreas*4 by careful analysis with finitee lement methods C83.

2-Dimensional in~t.Elemeant

The fin tte element method o f failure characterizationassumes that when an Individual finite element exhausts all ofits capacity for strain nnergy, it fails, forming a crack. Theelement is then removed from the matrix; the rectione loads ithad are transferred to the adjacent elementsa its stiffnessproperty is reduced to zero. t the reaction loads exceed thecapacity of ad3acent elements , they too fail. Should the appliedstress on the composi te remain constant during this process,catastrophic failure occurs. This model can be used to study th4,basic fa lute,processes that take plaoe in matrix material. Th emodel. con study poat-debond elidwng between fiber and matrix,fiIer fracture, and fiber pullout, to aentioa' a few of ,the manypossibilities.

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The full characterization of the failure of an inhomogen-eous anisotropic laminate material is exceptionally complex. Theplanar characteristics have been formulated by Wang [9] whosomodel considers a paztially closed delaminntion having friction-less crack-surface contact. The result is the classical inversesquare-root singularity with in algebraic multiplicity of two.With frictional contact, the results depend on tCa frictionalcoefficients and the fiber orientations of the adjacent dis-similar plies. if the delamination has a very small area ofcrack closure, the model in simplified by taking the limit of thepartially closed delamination. Wang used Leknitskii 's complex-variable stress potentials in conjunction with an eigenfunctionexpansion method, which leads to a standard eigenvalue problem.From this eigenvalue problem, stress singularities and thegeneral solution structures of deformation and stress fieldsassociated with composite delamination are determined. Thecomplexities of the composite delamination phenomenon include th einherent crack-tip singularity, the effect of anisotropy of eachconstituexit fiber lamina, and the abrupt change of stiffness orply orientation through the laminate thickness direccion. inaddition, the 3-dimensional state of stress and deformation thatexists at the delamination gives rise to all modes of fracture,vhich renders the problem mathematically intractable.

For pressure vessels and large olanar structures it is notonly canvenient, but accurate to model these structures as2-dimensional. This simplification allows accurate mathematicalformulation and solution with finite element analysis and methods,as advanced rec3ntl'y by Wang E10) and many other authors.

,These analyses require the material properties to beaccurate for the fracture conditions.. This requirement willforce the testing methods to accurately characterize the cbnditions in the composite, including the fiber-resin interaction andthe resin propertie* in the laminate Eli]. The resin propertiesin the laminate may differ significantly from test couponsbecause of processing variations and conditions that cannot beduplicated in a test coupon. There may be a significantly higher'concentrat ion of trapped air bubbles and airborne particles. Theenvironmental conditionsand windIng variables, such as windingspeed and tension, will affect the resin properties and, hence,the laminate properties.

. 3-dimensionsl F1nite Elmaent

In the structures where the stress state ii not planer, anda, true 3-dimensional stress state exista, the only accurateanalysis of the fracture behavior 'is by 3-dimensional finitee lement . Although this method has been formulated for 7ears, themathenatical diificulties, because of the sheer number of

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calculations required, have precluded its effective uss untilr'ecently. The structures of principal concern include aircraftstructural members and pressure vessell that are noL perfectspheres. Cylindrical pr4"sure vessels have ktresa states thatapproach planar in -he cylinder section, but have complicated3-dimensional stress states in the dome end sections.Wang 412] has solved the problem of composite dslaminationand failure using the theory of anisotropic laminate elasticityand incerlaminar fracture rechanics. These solutions require th e

use of 3-dimensional finite element analysis to accommodate the3-dimensional stress -state and interactions of the lnminrtematerials. This problem is very complex because of the geometricand material discontinuities and the inherently coupled modes offracture. Wang presents the effects of the geometry, thelamination and the crack variables. The basic governing equa-tions are lengthy and complex and are well presented in E12], soI omit them here. His results do show that the stress intensityfactor for the tearing mode is one or two orders of magnitudehigher than the other modes. .Wang also confirmed that the fiberorientations, ply thickness, stacking sequences &nd otherlamination variables all effect the delamination behaviorsignificantly.A more efficient, cost-effective finite element methodwas developed based on the concept of the mode I.II magnitude byNishioka and Altluri (133. This wasa one-step, smaller meshrefinement of their 'hybrid 3-dimensional method that used atwo-step solution with a very fine mesh.

ajsicn Effects

The sequence of the stacking of the fiber layers has asignificant effect on the strength of the composite E1,10].Herakovich discussed the theoretical inplLastiona of the stacking,sequence through the theories analy;ed by finite element methods(14].. The use of adjacent 9 layers leads to high interlaminarshear stresses and that interlaminar normal stresses are verydependent on the stacking sequence. Therefore, ad3acent Gilayers should generall.y be avoided. Interspersed to layers areprnferred to reduce the extremes of interlaxinar snear and normalstresses. Thiyia '.-caune of "mismatch" of the engineering.propert ies bstveen. adjacent layers. The interlam inar shearstresses are a fienotion of the coeftictents of mutual influencethat cun be ten LiAes greater'then the mismatch of Poisson'sratio. This mismatch reaches a peak at 100 le 116fiber range fo rtG lamina* combination*. It can be reduced by a factor of-twowhen the layers are Interspersed betv*em 01savd 90' ayers4

Layer thickneas both total and essin/f ibe ply, affect& the


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laminate structural characteristics. Apart from the obviouslinear effect of multiple plies of apecific strength. thecomposite's fracture Cnd failure characteristics are altered withvariations in thickness, The stacking sequence of the plies alsoeffects the ovarall thickness; a clustered aequence of plies canbe twice as thick as an alternated sequence as ohown in Figure1. Although the in-plane elastic properties of such laminatesare independent of stacking sequence, the strength and toughnesscan vary, ignificantly (15].

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Stacking Sequence and Fiber OrientationSFigure 1.

Resin systems are formulated tc provide specific matrixproperties, processing characteri,;.ics and fiber'resin bonds.These resins must have chemical characteristics that are compati-ble'with the fiber(s) being used in the composite. Damagetolerance of a composite structure is a function of the resinvolume content and the plastic flow characteristics of theresin. This resin also effects the compressive strength of thecomposite. The failure m6de frequently observed is a delamina-tion fracture in the through thickness direction. Therefore,improving the compressive strength of the resin/fiber systemshould improve the damage tolerance of the composite. Resinsystems with elastomeric additives,, thermoplastic additives andviryl modifiers have shown improved tolerance to damage (16.,The interlaminar shear strength of the composite is dependenton the fiber-rasin bond; fiber surface treatment may improve the=omposite's characterist ics C171.

CURRENT APPLICATZONSThe designers of the chambers for modern solid rocket motorsuse many methods. The designer is given the parameters andallowables for a feasible design based on previous data andexpected gains through new materials and/or approaches. The

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structural members of the design are chosen to balance reli-ablility and performance. The reliablility is calculated bythe expected number of failures under normal operating condi-tions. The performance of missile chaLbers is calculated fromthe product of the pressure inside the chamber and the volume ofthe chamber divided by the weight of the chamber and is repre-sented by:

* Performance = PrV/W Eqn.2 The theories that are applied to the requirements are in twocategories: thin-wall and thick-wall. The typical missilechamber is in the thin-wall category, whereas the segments fo rthe filament-wound chambers used in the solid rocket motors fo rthe Space Shuttle Program are in the thick-wall category. Theexact transi-tion point between the two criteria has not beenfully characterized.With these limitations, the desi ier calculates th eloads the chamber will experience and, after -nlying appropriatemargins of safety for the reliabili ty cri` e.ion, designs th erequired composite structure. He typically predicts the expectedfiber stresses by extrapolating those that were experienced inprevious, designs similar to the one in development.From the data that he has access to on other similardesigns, the designer derives material properties. The structuraltesting of design prototypes during development is crucial. An

initial design 'is then formulated by combining netting theory,. awinding model, and dome-contour empirical equations. These contour equations are varied through several dome designs toobtain c design that meets the design criteria. Two-dimensionalfinite element analysis is. applied to locate the hoop. reversalpoints, to size port reinforcements, 'and to determine th ehardware geometries. Theae finite alement codes may be separatedforthe forward and after dome regions, if they are different byvirtue of the s mmetry of the chamber. This separation maysignificantly reduce the computer ' time required to process th emodel. The separ tion is made at a point on the chamber where he loading is pur ly, or nearly so, membrane load. The 2-Dfinite element analysis provides the designer with the displace--m eats,strains, and stresses for the loading Of the fiber matrix ~analysis.Safety factors and margins of safety are applied in accord- ancewith the MIL Handbook V Interaction Equation based on the

worst-caseoet of peak loads. The skirts are designed for thechamber after t o initial design of the pressure vessel. Theskirts provide for the structural transmission of the boost* energy, as well as providing. the handling and attaching fixtureV for the pressure vessel. The skirts are designed to resistbuckling in all mo ds including: bending, differential pressure,il


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exial loading and shear.Tests are required, to adjust empirical and semi empirical

formulas and relationships which will account for designs & testresults that differ from the previously used values.These adjustments usually account for the differences in

material testing properties from stsndard tensile and coupontests and allow for the composite material interaction. AlthoughSthesedjustments would allow for the back-calculation ofmaterial propert ies as they apply in the composite structure, therelationships are usually adjusted instea' The adjustedrelationships become very effective for the desijn in considera-t ion and can result in actual performance at failure 3oing within

a few perceat of predict ions.This empir ic ism is an acceptable and prudent method ofdesign for those applications that closely resemble in structure,or composition, a previous design with a given data base. Forradical designs or composite structures, the empiricism would beas costly as developing the theory to the application.Three-dimensional analysis is used when other methods failto account for an anomalous condit ion or a failure that cannot beexplained with exist ing analysis. The software and engineeringis available to make this a routine evolution and the basis formost design evaluations. Current testing procedures include the instrumentation ofchambers, real-tim e and standard radiography, structural loading,

and the hydro-proofing and hydro-bursting of chambers. The prooftesting of composite materials was studied by Hahn and HwAng['38]. The lifetit. and strength of most materials usuallyshow more scatter than the other anterial properties, since theyare related to local defects in the homogeneous structure. Inthe composite structures, lifetime is related to staticstrength. The effect of proof testing on subsequent strength hasbeen found to be negligible. The actual amount of proof stressthat can be endured haa not been exactly characterized, althoughno apparent reduction in strength hak% been noted in vesselstested through the design loads

Materials for the composite structures are constantly inreview. Current experimental work in this field is attempting toincrease the rerformance of composi te pressure chambers on theorder of 15-25%. Laboratory teating of nes fibers, resins andSiber/resain systeas has had promising results (193. Theseresults represent tests with smal l pressure bottles which havebeen standardized in dimensions for compilation of data. Theresults would then be scaled to alfuture'design using existingmethods.

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I

CONCLUSIONS AND RECONKENDPATIONSThe current level of theoretical development of composit

chambers is well beyond the practical application level used ithe design and manufacture of these chambers. The most promisinarea of fundamental growth that can be applied to the designinand manufacturing areas is that of accurate 3-dimensional finiteelement analysis (3-DFE). The limiting factors precluding thitransition include:1) The reluctance of industry to adapt hardware ansoftware to perform those analyses.2) The lack of expertise at the industry level to ef-ectively employ the procedures of this method, including thselection of the regions where 3-DFE analysis is required.3) The lack of testing methods and systems to determine *hmaterial properties within the composite structure by deductiothrough 3-DFE methods; e.g. the deduction of the materiaproperties from test results instead of vice versa.The longitudinal material property values that are compilein the l i terature are not very useful in the designs; actuadesigns include many property-reducing factors such as stresconcentrations, fiber wash, fiber slipping, resin-rich anresin-lean regions. delaminations and voidb. These factorsalthough not, part of the design, exist in all manufacturedchambers to some degree. The analyses of the current designs arbased on the data from previous teats and have empiricall

included these effects on a macroscopic scale. This then doeinclude the 'unknowns . The problem often arises where somdamage or a manufacturing defect beyond those that are inherenoccurs and an onalysis-is required to determine the suitabilityof the chamber for use. This analysis, if limited to twodimensions, may mais a critical stress condition its the triaxialcondition that actually exists.Bec4use of the multiplicity of possible failure mode*, thprediction of the ultimate failure strength is very difficulfrom the theory. The original designs have built-in factors ocafety and empirical factor4, regarding the actual test data osimilar designs. These facto"s cannot be directly applied to ththeoretical optimization calculb;:ions. Fundamental. data will

have to be obtained in order to rmach an optimum design condition. Crit -nl design optimizations can utilize. successivapproximations to create the proper angles of windings, domcontours and fiber layers, as well as resin content. Throutines to accomplish this can be creat%, with significanpay-back on the investment by providing designs th,: approach thmaximum possible performance and rellablil4ty wita, given materals.10

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I' recommend that current designers adapt the testing anddesign methods that utilize 3-DFE as their foundation. Thehardware and software required to accomplish this exists. Thecosts are justifiable. Software to do limited 3-DFE is currentlyavailable for PC's that can be on the designer's desktop. Noreadvanced CAD routines incorporate 3-dimensional graphics that candisplay the grids and visually display the stresses that a designis experiencing to the designer/engineer. These tools for designexist and should be incorporated into our current design/evalua-tion techniques.

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APPENDIX IGeneral Notes on Laminate Theory from "Composite MatarialsHandbocx -hel M. Schwartz, McGraw-Hill

Laminate analysis is a mathematical means of determiningelastic properties of the complete laminate from the angles offiber orientation and elastic moduli of the individual layers.Conversely, the method is also used to determine stress andstrains in each layer produced by stretching and bending forceson the complete laminate.The elastic properties of a laminate (Ex,Ey,Gxy,Yxy) aredetermined in three steps:1) The elastic propert ies of each layer are determined fromthe moduli Ell. El'2. and G12 and Poisson ratio v 1 2 of'the layer

material . These propert ies are determined from the Hooke's-lawrelationship between stress and strain in the principal materialdirections (parallel and perpendicular to the fiber direction).

2) The elastic properties of each layer in the principallaminate directions (parallel and perpendicular to the axis ofthe laminate structure) are determined from the material-direction properties and the angle between the laminate and thematerial directions.3) A summation of the laminate-diredtion elastic propertiesof all layers, taking into account their relative positions withrespect to *.he midplare of the complete laminate, yields th e

elastic properties of the complete laminate (Fig. 2).Laminate analysis is also used to determine stresses andstrains in each layer due to forces on the laminate. Layerstrains are determined form laminate strains and rotations,taking into a-count the Layer position rslati* to the middlesurface of the complete laminate. Stresses in each layer are'then determined from layer strains and 'Lhei Hooke's- law relation---ship. Thus, the stresses and strains at a point in the. structurecan be calculated in everydirection, that is, 0 xe73 . TXy and G 11.022, T12 in every pl.y.

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A o AFA0 ARE LAvMATEY --G, -EOME TlIC

ATAPOINTEACHAYER O H E C T N S . I ANO 2FIBERS ARE PRIN4CIPAL

OIRECTIOI SOFPLY FIBERS.

IStIED d IEOMETRIC * ANDi DIRECTIONS FOR OOfA iLAMENATL

-IECFICIN '22 .22 01 1 LAYER 2

- 1 1LAYER 4

:. STVISSICtO IN POINCIPIK I AND I2 DIRECTIONS FOR EACHLAYER

4 f 1 1 LAYER 3

General Laminate Structure, with LoadsFigure 2.

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APPENDIX 2General Notes on Linear Eleastic Fracture Mechanics (LEFM) Theoryfrom "Fracture Toughness Testing and Its Applications-Symposiumon Fracture Toughness Testing and Its Applications, ASTM SpecialTechnical Publication No. 381, American Society For Testingand Materials. 1964

The introduction of stresses in a body, caused by a crack,is a feature of the strength 'of the material. The stresses nearthe crack tip may result in the growth, or perpetration of th ecr-ack or f law. The nonlinear and plastic effects are neglectedin the gross features discussed in this presentation, but may besignificant in a given scenario. The stress-field expansions forthe crack tips are applicable for the general case of an isotro-pic elastic body. The three modes, opening, edge:-sliding andtearing, (1,II and III respectively) are shown in the figure (3).

Y YX X

YThe Basic Modes of Crack Surface Displacements.Figure (3).

The surfaces of a crack dominate the distribution ofstresses in the vicinity of the crack. Other boundaries areremote and affect only the intensity of the uc'rl stress field,as does the loading forces applied to tW' body. The openingmode, I, is associated with the loca.l disp.acmernt where th ecrack surfaces move directly apart. The edge sliding mode, II,is where the displacements move over one another, perpendicularto the leading edge of the crack. The tearing mode, XII, iswhere the crack surfaces slide parallel to the leading edge ofthe crack.

Modes I and II can be treated as plone-extensionel problemsin the theory of elasticity, symmetric and skew-symmetric,respectively. Mode III is a pure torsion or shear problem.Using the notation in figure (4), the resulting stress anddisplacement fiOelds are derived:

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Mode I.ax (KI/(2nr)l/2 )coaO/2EI-ein(8/2)sin(30/2)Jay= (KI/(27irr)" 2 )cos0/2[1+s~in(0/2)ain(30/2)I

T xy = (KI/(2nrr)1' 2 )esin(0/2)cos(0/2)cos(30/2)OZ2= V(OFX +Oy), T =T' Z .0u = (KI/G)Cr/(2ir>Ji'2 cos (0/2)

[1-2 v *sin 2 (0/2)J-(K'I/G)(r/ (2 T)]11 2 sin (0 2)

[2-2 v -cos 2 (O)/2)]w,= 0.

Mode II: x -CKII/(2iir)1' 2Jsin( WY2.) E2cos( 0/2)*coa(3Qt2))

=y fKII/(2lTr)i 13s n(0/2)cos(EG/2)cos(3O/2)* xyu (KII/(2irr)1/ 2 ]coa(G/2)t2.-ain( ()'2)cos(30/2)I

z V x + y) T xz uT y . 0u (KII/G)Cr/(21t)]l' 2 sin(e/2)

C2-2 vg.co& 2 (0)/2)3* v (.KIZ/G)(r/(2Wr)3l12 cons 3/2)

C-14.2 v +Sin 2(8)/2)3W 0.

Mtode III:Txz -(KIII/(2lTr)1/ 2 )sin(0/2)

w a (KIxr/G)C(2r)/ tl,1/2 sin (0/2)U 0v aO.


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/ axz

--- leading edgeof the crack

zCoordinates of the Stress Components in the Crack Tip.Figure (4).The parameters, KI,, KII, and KrII in the equations arestress-intensity factors for the corresponding three types ofstress and displacement fields. These factors must containthe

loading force magnitudes linearly (for linear elastic bodies) anddepend on the configuration of the body including the cracksize. They, therefore, reflect the redistribution of stress in abody due to the introduction of a crack and the force transmis-sion through the crack t ip.It has been shown that, for the general homogeneous aniso-tropic case, the cracJ%-tip stress fields and their intensityfactors, the complete analogy with the isotropic case is preserv-ed. By judicious definition of the anisotropic stress-intensity

factors, they are identical to those for the isotropic came.However, for the non-homogeneous anisotropy, the reaultant stressfields and intensity factors, as well as stress distributions,may well be different and result in different singularities thanthose observed in the homogeneous case.The introduction of vipcoelasticity into the problem resultsin similar intensity factors and stress distributions in thevicinity of cracks; however, this causes these factors to nowbecoame variables as a function of time.The complete discussion and developaent, as wel as exten-

sions to other and general case* is found in the above"creditedtext and this presentation is merely a condensation of the1S


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principle governing equations. It is given for completeness andthe readers reference.


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REFERENCES

1. Pagano, N.J. and Pipes, R.B., ' The Influence of StackingSequence on Laminate t'2-ength," Journal of Composite Materials,vol. 5. Jan. 1971. pp. -- 572. Tolbert, R.N., "5tiffness Properties for Dynamic Modeling ofComposite Graphite-Epoxy Cylindrical Orthotropic Shells," NASACR-170942, Dec. 19833. Harris, C.E. and Morris, D.H., "Fracture Behavior of Thick,Laminated Graphite/Epoxy Composites," NASA CR-3784, 19844. Chamis, C.C., "Prediction of Fiber Composite MechanicalBehavior Made Simple," NASA TM-81404, Feb. 19805. Peters, P.M.W., On the increasing fracture toughness atincreasing notch length of 0/90 and 0/+45/0 graphite/epoxylaminates," Composites, vol. 14, Oct. 1983, pp. 365-3696. Shih, T.T. and Logsdon, W.A., "Fracture Behavior of a Thick-Section Graphite/Epoxy Composite, Fracture Mechanics: ThixrteenathConference, ASTM STP-743. 1981, pp. 316-3377. Jurf, R.A. and Pipes, R.B., "Interlamin&r Fracture ofComposite Materials," Journal of Comoosite Materials. vol. 16,Sept. 1982, pp. 386-3948 . Mahishi, J.M. and Adams, D.F., "Fracture Behaviour of aSinglefibre Graphite/Epoxy Model Composite Containing a BrokenFibre or Cracked Matrix, "Journal of Haterjj e , vol. 18.1983, pp. 447-4569. Wang, S.S., The Mechanics of Delamination in Fiber-Reinforc-ed Composite Materials, Part I - Stress Singularities andSolution Structure," Mechanics of Comoqsite Materials-1983.Proc. of the Winter Annual Meeting of Mechanical Engineers, ASMEAND-Vol. 58, pp. 83-10410. Wang, S.S., The Mechanics of Delamination inFiber-Reinfort.od Composite Materials, Part II,- The DelaminationBehavior and Fracture Mechanics Parameters," Mechanicr__qpoaosite Materials 1983. Proc. of the Winter Annual Meeting ofM4chanical Engineers, ASME AMD-Vol. 58, pp. 105-13311. Marloff, R.H., "Finite Element Analys is o Biaxial StressTest Specimen for Graphite/Epoxy and Glass Fabric/Epoxy. Composit-eo," CompositeMaterialsa Testina and D9iaQn (Sixth Conference).ASTL 2TP 77Zj 1982, pp. 34-49

S~18


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12. Wang, S.S., "Fractire Mechanics for Delamination Problems inComposite Materials," Journal ,f Composite Materials, vol. 17,May 1983, pp. 210-22313. Nishioka, T. and Atluri, S.N., A Simple Estimation ofStress Intensity Factors for Tt :ough-Cracks in Angle-Ply Laminat-es," Engineeriig Fracture Mechanics, vol. 16, No. 4, 1982pp. 573-58314. Herakovich, C.T., On the Relationship Between EngineeringProperties and Delamination of Composite Materials," Journal ofComposite Materials vol. 15, July 1981, pp. 336-34815. Herakovich, C.T., "Influence of Layer Thickness on theStrength of Angle-Ply Laminates," Journal of Composite Materials,vol. 16, May 1982, pp. 216-22716. Williams, J.G. and Rhodes, M.D., "Effect of Resi.n o: ImpactDamage Tolerance of Graphite/Epox , Laminates," CompositeMateri.als: Testing and Design (jixth Conference). ASTM STP 787.1982, pp. 34-4917. Varma, D.S., et. al. "Intorlaminar Shear Properties ofGraphite Fiber, High-performance Resin Composites, PolymerComposites, vol. 4, no. 2, Apr. 1983, pp. 98-10318. Hahn, H.T. and Hwang, D.G., "Failure Characterization of aGraphite/Epoxy Laminate Through Proof Testing," CompositeM aterials: Testing and Dgsain (Sixth Conference). ASTM STP 787.1982, pp. 34-4919. Mumford, N.A. and Sasso, Se., Advanced Pfateriels for SolidRocket Cmposite Cases , presented at the 1985 JANAF PropulsionMeeting; 9-12 April 1985, San Diego, CA (Lknited Distr ibut iop-U.S. Government Agencies Only)


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FIL MED

design composite chambers

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