Astm stp 546 1974

Compton 395 Materials 396 Specimen Preparation for Test 397 Laboratory Testing 399 Data Analysis and Test Results 400 Discussion of Results 403 Conclusions 407 Interlaminar Shear Fatigu

COMPOSITE MATERIALS:

TESTING AND DESIGN

(THIRD CONFERENCE)

A conference sponsored by the AMERICAN SOCIETY FOR TESTING AND MATERIALS Williamsburg, Va., 21-22 March 1973

ASTM SPECIAL TECHNICAL PUBLICATION 546

C A Berg, F J McGarry, and S Y Elliott, coordinators

List price $39.75 04-546000-33

AMERICAN SOCIETY FOR TESTING AND MATERIALS

1916 Race Street, Philadelphia, Pa 19103

Library of Congress Catalog Card Number: 70-185534

NOTE The Society is not responsible, as a body, for the statements and opinions advanced in this publication

Printed in Baltimore, Md

June 1974

Foreword

The Third Conference on Composite Materials: Testing and Design was held 21-22 March 1973 in Williamsburg, Va Committee D-30 on High Modulus Fibers and Their Composites of the American Society for Testing and Materials sponsored the conference in conjunction with the Metallurgical Society of the American Institute of Mining, Metallurgical, and Petroleum Engineers and the American Society of Mechanical Engineers C A Berg, University of Pittsburgh,

F J McGarry, Massachusetts Institute of Technology, and S Y EUiott, Douglas Aircraft Company, served as coordinators Most of the papers presented at the eight sessions are included in the volume which complements the first and

second conference publications, ASTMSTP 460 and ASTMSTP 497, Composite Materials: Testing and Design

Analysis of the Flexure Test for Laminated Composite Materials—/ M

Whitney, C E Browning, and A Mair 30

Methods for Determining the Elastic and Viscoelastic Response of

Com-posite Materials—Z) F Sims and J C Hatpin 46

Validity of ±45 deg Laminate Test and Rail Shear Test 48

Experimental Results 56

Conclusions 60

Analysis, Testing, and Design of Filament Wound, Carbon-Carbon Burst

Tubes-i? C Renter, Jr and T R Guess 67

Analysis 69

Numerical Results and Discussion 75

Conclusions 82

Elastic Torsional Buckling of Thin-Walled Composite Cylinders—Z) E

Marlowe, G F Sushinsky, and H B Dexter 84

A Correlation Study of Finite-Element Modeling for Vibrations of

Com-posite Material Panels—£ A Thornton and R R Clary 111

Description of Correlation Study 112

Discussion of Results 118

Concluding Remarks 128

Bundles and Unidirectional Fiber/Matrix Composites—5 L

Phoenix 130

Single-Fiber Tensile Strength Model 131

Generalized Fiber Bundle Strength Analysis 136

Probabilistic Tensile Failure Theories for Unidirectional Composites 143

Summary 149

Debonding of Rigid Inclusions in Plane Elastostatics—G P Sendeckyf 152

Basic Equations 154

Partially Bonded Elliptic Inclusion 155

Analysis and Discussion of Results 159

On the Determination of Physical Properties of Composite Materials by a

Three-Dimensional Finite-Element Procedure—S / Kang and

G M Rentzepis 166

The Finite-Element Method 167

Computation of the Elastic Constants 172

Applications 174

The Unidirectional Fiber Composite 174

The Lamellar Composite 180

The Short-Fiber Composite 180

Discussion 186

Laminate Strength—A Direct Characterization Procedure—£" M Wu and

J K Scheublein 188

Basic Procedures for Establishing Laminate Failure Criterion 189

Experimental Results and Discussion 202

Summary and Conclusion 206

Experimental Studies

Stress-Rupture Behavior of Strands of an Organic Fiber/Epoxy Matrix—

T T Chiao, J E Wells, R L Moore, and M A Hamstad 209

Experimental 210

Results 213

Analysis and Discussion 214

Conclusions 223

Effect of Temperature and Strain Rate on the Tensile Properties of

Boron-Aluminum and Boron-Epoxy Composites—Z) A Meyn 225

Methods of Fiber and Void Measurement in Graphite/Epoxy

Com-posites—£" alley, D Roylance, and N Schneider 237

Materials 238

Experimental Methods 239

Discussion of Results 244

Conclusions 248

Evaluation of Experimental Methods for Determining Dynamic Stiffness

and Damping of Composite Materials—C W Bert and R R Clary 250

Experimental Techniques 251

Free Vibration 252 Pulse Propagation 254 Forced Vibration Response 257

Application to Composite-Material Structures 260

Conclusions 263

Environmental Effects

Effect of Salt Water and High-Temperature Exposure on Boron-Aluminum

Composites—// E Dardi and K G Kreider 269

Salt Exposure 271 High-Temperature Exposure 275

Conclusions 280

Effects of Moisture on the Properties of High-Performance Structural

Resins and Composites—C E Browning and J T Hartness 284

Materials and Experimental Procedures 285

Results and Discussion 288

Conclusions 302

Materials Parameters that Govern the Erosion Behavior of Polymeric

Composites in Subsonic Rain Environments—G F Schmitt, Jr 303

Apparatus Description 304

Polymeric Composites Erosion Behavior 306

Discussion and Conclusions 322

Lightning Protection for Composites—// T Clark 324

Evaluation of Candidate Coatings 325

Evaluation of Coatings on Boron/Epoxy Substrates 328

Evaluation of Coating Schemes on an F-4 Rudder 330

Evaluation of Coatings on a Simulated Boron/Epoxy Empennage

Structure 330 Conclusions 339

Fatigue and Fracture Behavior

Torsional Fatigue Behavior of Unidirectional Resin Matrix Composites—

R C Novak 345

Experimental Procedure 346

Results and Discussion 349

Conclusions 358

Cutouts-iJ E Rowlands, I M Daniel, and J B Whiteside 361

Orthotropic Stress Analysis 362

Laminates 363 Geometric and Loading Effects 364

Summary and Conclusions 373

Crack-Tip Deformation Measurements Accompanying Fracture in Fibrous

and Laminar Composites—/ H Underwood 376

Procedures 377 Discussion of Results 380

Summary 391

Uniaxial, Biaxial, and Fatigue Properties of Polyester Fiber Glass—L H

Irwin, R A Dunlap, and P V Compton 395

Materials 396 Specimen Preparation for Test 397

Laboratory Testing 399

Data Analysis and Test Results 400

Discussion of Results 403

Conclusions 407

Interlaminar Shear Fatigue Characteristics of Fiber-Reinforced Composite

}A2Atx\a\s—R Byron Pipes 419

Interlaminar Shear Fatigue Results 423

Fiber Tension and Interlaminar Shear Fatigue 427

Conclusions 430

Wear Properties

Mechanical and Physical Properties of Advanced Composites—PV T

Freeman and G C Kuebeler 435

Evaluation of Graphite Fiber Reinforced Plastic Composites for Use in

Unlubricated Sliding Bearings—7? D Brown and W R Blackstone 457

Test Equipment 459 Bearing Specimens 460

Test Results 461 Discussion 468 Conclusions 474

Wear of Glass Fiber Reinforced Composite Material—^ L Ward 477

Experimental Procedure 478

Experimental Results 480

Discussion 492

Metallic Composites

Effect of Filament-Matrix Interdiffusion on the Fatigue Resistance of

Boron-Aluminum Composites—/ R Hancock and G G Shaw 497

Materials 498 Experimental 498 Results 500 Discussion 504 Conclusions 505

The Notched Tensile Behavior of Metal Matrix Composites—Ậ M Prewo 507

Experimental 508 Results and Discussion 509

Conclusions 520

Reinforcement of Metals with Advanced Filamentary Composites—C T

Herakovich, J G Davis, and H B Dexter 523

Applications of Composite Reinforced Metals 524

Tensile Behavior of Boron/Epoxy Reinforced Metals 528

Concluding Remarks 541

Comparison of the Mechanical Behavior of Filamentary Reinforced

Aluminum and Titanium Alloys—/ / Toth 542

Experimental Procedures 544

Results and Discussion 546

Summary and Conclusions 558

Plastic Deformation Processing and Compressive Failure Mechanism in

Aluminum Composite Materials—il/ Chang and Ẹ Scala 561

Analytical Studies 563

Analytical Models and Calculated Results 564

Comparison with Experiments 571

' Test Results 587 Analytical Investigation 593

Yielding of Bladder 616

Totally Plastic Bladder 619

Short-Term Failure Predictions 620

Numerical Evaluation 623

Conclusions 630

Development of a Unique Graphite/Epoxy Antenna Subreflector—£ Y

Robinson, R A Stonier, and C L Lofgren 632

Low-Weight, Impact-Resistant Helicopter Drive Shafts-/ E Figge,

J Henshaw, P A Roy, and E F Olster 651

Background and Philosophy 652

Drive Shaft Design Criteria 653

Drive Shaft Design 654

Fabrication 657

BalUstic and Mechanical Tests 657

Conclusions 660

Experimental Stress Intensity Factor Measurements in Orthotropic

Com-posites—/ Tirosh and C A Berg 663

Path Independent Integral 664

Numerical and Experimental Evaluations 666

Evaluation of the Method 668

Conclusions 672

STP546-EB/Jun 1974

Introduction

The American Society For Testing and Materials in conjunction with the

Metallurgical Society of the American Institute of Mining, Metallurgical, and

Petroleum Engineers and the American Society of Mechanical Engineers, held

the Third Conference on Composite Materials: Testing and Design on 21-22

March 1973 in Williamsburg, Va The objectives of this conference were the

same as those of its organizing group (ASTM Committee D-30), that is, to

stimulate research and promote the understanding of the behavior of fibers and

their composites, including reexamination of conventional tests in the light of

structural design data requirements and composite properties Because the field

of advanced composites is so rapidly growing and changing, it is possible to meet

such broad objectives only insofar as the progress to date is concerned—and this

the Third Conference has done successfully The previous two similar

confer-ences were held in New Orleans in 1969 (STP 460) and Anaheim in 1971 (STP

497), and succeeding conferences will continue to be held as long as

encouragement of new developments continues to be needed This volume

includes most of the papers given at the conference—and because its eight

sessions (on testing methods, analytical treatments, experimental studies,

environmental effects, fatigue and fracture behavior, wear properties, metallic

composites, and aircraft applications) covered the total subject adequately in

both depth and breadth, it will make a valuable addition to the library of the

scientist, analyst, designer, or testing engineer who is seeking a better

understanding of the mechanical behavior of composites

The papers in this volume provide a focus on the design of engineering

applications of composite materials in industrial, military, and consumer uses;

the testing of such materials and products; and the service behavior and

performance of the products Contributions dealing with design and test

methods, fatigue and fracture phenomena, effects of environmental conditions,

friction and wear behavior, impact response, manufacturing technology, and

related topics are emphasized Both theoretical and experimental approaches are

described, and useful data are presented It is hoped that the dissemination of

the results of recent advancements and successes will encourage similar fallouts

to other commercial and military areas

Special thanks for the preliminary organization and scheduling of the

conference papers are due to the original chairmen Professor C A Berg,

University of Pittsburgh and Professor F J McGarry, of the Massachusetts

L B Greszczuk ^

Microbuckling of Lamina-Reinforced

Composites

REFERENCE: Greszczuk, L B., "Microbuckling of Lamina-Reinforced

Com-posites," Composite Materials: Testing and Design (Third Conference), ASTM

STP546, American Society for Testing and Materials, 1974, pp 5-29

ABSTRACT: Compressive strength and failure modes of unidirectionally

lamina-reinforced composites were studied both theoretically and

experi-mentally The failure modes investigated included microbuckling and

com-pressive strength failure of the reinforcement The influence of the following

parameters on compressive strength was investigated experimentally: lamina

volume fraction, lamina end fixity, lamina thickness, specimen geometry, and

matrix properties The two-dimensional microbuckling theory showed good

correlation with test data from specimens designed to fail by microbuckling

As the Young's modulus of the matrix increased, the failure mode changed

from microbuckUng to compressive failure of the reinforcement

KEY WORDS: Composite materials, compression, microbuckling, reinforced

plastics, theories, microinstability, composite structures, failure, stresses

Nomenclature

2c Thickness of matrix between laminas (see Fig 1)

Ef Young's modulus of the reinforcement or laminas

Er Young's modulus of the matrix

Gr Shear modulus of the matrix

h Thickness of the reinforcement or laminas

k Volume fraction of reinforcement

L Length of composite specimen

m Number of buckle waves

P Compressive load

R Radius on the end of the laminas

T Thickness of composite specimen

' Staff engineer Advance Structures and Mechanical Department, McDonnell Douglas

Astronautics Company, Huntington Beach, Calif 92647

W Width of composite specimen

X Parameter defining end fixity of reinforcement

acE Critical composite stress for microbuckling in the extension mode

acs Critical composite stress for microbuckling in the shear mode

Oe Euler column buckUng stress

Vf Poisson's ratio of the matrix

Reinforcement microbuckling as a plausible failure mechanism for

lamina-reinforced and fiber-lamina-reinforced composites subjected to compressive loading has

been studied by several investigators, including Biot[i],^ Rosen[2],

Schuerch[J], Hayashi[4], Chung and Testa [5], and others Most of the studies

are theoretical and are based on a mathematical model consisting of alternate

layers or laminas of reinforcement and matrix material, loaded in compression

parallel to the reinforcement direction A review of the theoretical work on the

problem is presented by Greszczuk[6,7]

To verify the validity and limitations of the microbuckling theory, extensive

experimental studies have been conducted on nearly perfect lamina-reinforced

model composites, the results of which are presented herein

Theoretical Considerations

For a two-dimensional composite model consisting of alternate layers of

laminar reinforcement and resin, two types of microbuckling failure can take

place if such a model is subjected to compressive loading parallel to the direction

of the reinforcement: microbuckling in the extension mode and microbuckling

in the shear mode The two failure modes are illustrated in Fig 1 Microbuckling

in the extension mode is governed by the following equation [2]:

•n^Efh^k 12L'

whereas microbuckling in the shear mode is predicted by

The various terms appearing in Eqs 1 and 2 are defined in the Nomenclature

and in Fig 1 For a composite made of a given combination of materials, the

composite stresses OCE and acs have to be minimized with respect to the

number of buckle waves, m, to give the minimum values of acs and acs- Por

^ The italic numbers in brackets refer to the list of references appended to this paper

any given k, the smaller of the two values, OCE or (^CS> is the governing stress In

general, Eq 1 gives a lower microbuckling stress for A: < 0.10, whereas for

k > 0.10 the minimum stress is given by Eq 2 Inasmuch as in actual composites

the fiber volume fractions are usually greater than «0.10, only Eq 2 is of

practical interest

In deriving Eqs 1 and 2, it was assumed that the ends of the individual

laminas are simply supported If the laminas are not simply supported, then Eq 2

will be of the form [8]:

Gr , Ti^Efk /mh\

where X is dependent on the end fixity of the individual laminas Table 1 gives

the values of X as a function of LjTR, where L is the specimen length andR is

the radius of the end of the lamina For a lamina with square ends, R =°° and

therefore X = 4, which corresponds to a fixed-end condition For laminas with

rounded ends, 1 < X < 4, depending on the value of JR Fori? = 0, X = 1, which

corresponds to laminas with simply supported ends

50 1.04

CXI

1

It can be readily seen that if the composite is long {L ^^h), the second term

in Eq 3 has a negligible influence on 0^5, and therefore the influence of lamina

end restraints is unimportant If, however, the composite test specimen is of

such geometry that the second term in Eq 2 is not negligible, then end

conditions of the laminas will influence ffc5 and have to be taken into account

Most of the specimens tested were of such geometry that the second term in

Eq 2 was not negligible, and end conditions of the laminas did influence

acs-Experimental Studies on Microbuckling of Lamina-Reinforced Composites

To establish the validity of the microbuckling theory, the influence of the

following parameters on the compressive microbuckling strength of

lamina-reinforced composites was investigated; lamina end fixity, lamina volume

fraction, specimen geometry, lamina thickness, and matrix properties In all

models the lamina reinforcement was 6061-T6 aluminum strips The matrix

materials used were a urethane resin with a Young's modulus of ~2.5 ksi and

two different epoxy resin formulations with Young's moduli of approximately

23 and 62 ksi, respectively

The dimensions of the test specimens were selected such that the specimens

would fail by microbuckling rather than by Euler column buckling With the

lamina-reinforced composites that were tested, the relationship between the

critical Euler buckling stress, a^, and the critical microbuckling stress, acs, for

microbucking in the shear mode is [8]:

id

\k(l -k)n^Ef — (-)

2

For a typical specimen with L = 2.5 in., h =0.0195 in., Ef= 10.1 X 10* psi,

Gf = 904 p&i,k = 0.50, and T = 1.0 in., evaluation of Eq 4 gives

ffe = 322acs Even for composites with a low-volume fraction of laminas (k = 0.16), the term

in front of acs was greater than 100 Further discussion of the design of test

specimens is presented in [8]

Influence of Lamina End Restraints

To verify experimentally the influence of lamina end restraints on the

compressive strength of composites, four composites were fabricated and tested

Two contained laminas with square ends, and two contained laminas with

rounded ends Figure 2 shows the two types of composites The laminas were

GRESZCZUK ON MICROBUCKLING

FIG 2-Lamina-reinforced composites with lamina ends corresponding to fixed and

simply supported conditions

made of 0.040-in.-thick 6061-T6 aluminum strips, while the matrix was a

urethane resin with average Young's modulus of «*2385 psi The laminas were

spaced to give a reinforcement volume fraction of «>0.50 The rounded ends of

the laminas had a radius of 0.025 in., and the specimen length, L, was 2.5 in

Consequently, for composites reinforced with round-ended laminas, X= 1.04,

which is close to the condition of simply supported ends (See Table 1.)

The test results as well as the test-theory comparison are given in Table 2 It is

seen that excellent agreement exists between test and theory Moreover, the

results show that, for the models used, the influence of lamina end restraint is

not negligible and has to be taken into account

Influence of Reinforcement Volume Fraction

To estabUsh the influence of reinforcement volume fraction on compressive

strength, lamina-reinforced composites with the following average reinforcement

volume fractions were fabricated and tested in compression: k = 16.2, 24.6,

32.1, 48.4, 65.0, and 75.3 percent The reinforcing material consisted of

»'0.0195-in.-thick 6061-T6 aluminum strips; the matrix material was urethane

TABLE 1-Influence of end restraint on microbuckling of

Square Round Square Round

Lamina Thickness,

in

0.0399 0.0399 0.0422 0.0420

Reinforcement Volume, % 48.7 48.7 51.4 50.0

Composite Stress at Failure Test, psi

2490

2490

2280

2280

" Specimens consisted of a!0.040-in,-thick aluminum laminas embedded in soft urethane

resin The nominal dimensions of composite specimens IR and IF were: 0.465 in

(thickness) by 2.00 in (width) by 2.50 in (length); the corresponding dimensions for

composite specimens 2F and 2R were: 1.00 in by 2.05 in by 2.50 in Specimens were

tested in the direction of a 2.50-in dimension

* The theoretical values were calculated from Eq 3 For composites made with round-ended

laminas, X was taken as 1.04; for composites made with square-ended laminas, \ was taken

as 4

resin Figure 3 shows several lamina-reinforced composites with various

reinforcement volume fractions The nominal dimensions of the test specimens

were 1.0 by 2.0 by 2.5 in Typical failures of the specimens are shown in Figs 4

and 5 Figure 4a shows the failure of a composite containing 24.4 percent

reinforcement by volume, while Fig Ab shows the failure of a composite with

64.7 percent of reinforcement by volume The dimensions of the specimens

shown in Fig 4a were, approximately, 2.5-in height by 2.0-in width by 1.0-in

thickness To ensure that the failure modes of the specimens tested were

representative of composites, a wide specimen (3.71 in wide, 0.10 in thick, and

2.50 in long) was also fabricated and tested The failure mode of this specimen

(Fig 5) was similar to those shown in Fig 4 Typical load-deflection curves for

composites with various volume fractions of laminas are shown in Fig 6

The comparison of measured compressive strength values with the results

predicted by the two-dimensional microbuckling theory is shown in Fig 7

Inasmuch as the composites shown in Fig 3 were made with square-ended

laminas, it was assumed that X = 4, which corresponds to the fixed-end condition

of individual laminas With this value of \ Eq 3 for the critical composite stress

for microbuckling in the shear mode becomes

,a^^'JMl

"«-i^*^(|-) <«

The curve segment for microbuckling in the extension mode was calculated

from Eq 1 The curve segment corresponding to microbuckling in the shear

mode was calculated from Eq 5 using the average properties (E,- and v^) of the

matrix The average properties of the matrix were obtained from the same resin

batches as those used to make the various composites The resin castings were

co-cured with the various composite models and were tested for mechanical

properties at approximately the same time as the various composites Resin

properties shown in Fig 7 are an average from all the resin castings that were

prepared and tested The average values for Young's modulus and Poisson's ratio

were Ef = 2650 psi and Vf = 0.466

The fact that theory (which assumes microbuckling failure) and test results

show good agreement leads to the conclusion that the failure of the specimens

for which data are presented in Fig 7 was by microbuckling Moreover, the

results shown in Fig 7 verify the accuracy of the two-dimensional microbuckling

theory for predicting the microbuckling compressive strength of

lamina-rein-forced composites As demonstrated in[S], the test-theory correlation can be

improved from that shown in Fig 7 by normalizing the test data with respect to

the average properties of the matrix By normalizing the test data, the variations

of matrix properties from specimen to specimen can be accounted for

That numerous buckle waves were not observed in the specimens described

here (Figs 4 and 5) is apparent from Eqs 1 and 2 Numerous buckle waves will

% T - 0 3 9 9 IN

h - 0J)19S IN

0.03 DEFLECTION (IN.I

( s ) MACHINE STOPPED AND

^ LOAD REVERSED

0.02 DEFLECTION (IN.I

DEFLECTIONS WERE OBTAINED FROM MACHINE HEAD TRAVEL

LOAD RATE WAS 0.0S IN./MIN

FIG 6-Load-deflection curves for lamina-reinforced composites containing various

volume fractions of reinforcement

MICROBUCKLING IN THE SHEAR MODE, (EQUATION (5) IN TEXT)

•MICROBUCKLING IN EXTENSION MODE (EQUATION (1) )

form in the specimen only if microbuckling is in the extension mode, which

requires that the reinforcement volume fraction be less than ^sS percent[J,5] ^

All the specimens tested here contained reinforcement volume fraction equal to

or greater than k= \6 percent Thus these specimens buckled in one buckle

wave, m = \, which minimizes acs given by Eq 2 and 5 By making

lamina-reinforced composites wdth k<\0 percent, several buckle waves were

here

^ For composites with E^lEy ratio of «>4000, which applies to the specimens discussed

observed in Ref J, as one would expect from examination of Eq 1 To make

ocE given by Eq 1 a minimum requires that m> \ Because composites with

low fiber volume fraction are of little practical interest, no experiments for

microbuckling in the extension mode were performed

Influence of Reinforcement Thickness

Compression tests were performed on composites made with urethane resin

and reinforced with lamina of three different nominal thicknesses: 0.0118,

0.0195, and 0.0415 in The specimens contained approximately 49 percent

reinforcement by volume and were similar to those shown in Fig 3 The

specimen dimensions were selected so that the length-to-lamina thickness ratio

remained approximately constant A detailed description of the test specimens

as well as the test data is presented in [8] The effects of lamina thickness on

compressive strength of composites are summarized in Table 3

Inasmuch as there were variations in Young's moduli of the resins used in

various specimens, as well as variations in L/h and k, the test data have been

normalized with respect to average properties, as shown in Table 3 The results

given in Table 3 do not show any apparent influence of the absolute lamina

thickness on the compressive strength of lamina-reinforced composites This

conclusion can also be drawn from the results presented in the following section

The failure modes of composites made with laminas of various thicknesses were

simOar to those shown in Figs 4 and 5

Influence of Specimen Geometry

To establish the relative influence of the first and second terms in Eq 5 on the

microbuckling compressive strength of lamina-reinforced composites, the

TABLE 3-^Effect of lamina thickness on compressive strength of composites

GRESZCZUK ON MICROBUCKLING 17

geometry of the test specimens was varied The primary variables were specimen

length and lamina thickness (while specunen width and thickness remained

constant) Several tests were also performed on specimens in which the specimen

length and width remained constant and the specimen thickness varied Figure 8

shows several of the test specimens, while the compressive failure of one of the

specimens is shown in Fig 9

The test data on the influence of specimen geometry on compressive strength

of composites are presented in Figs 10, 11, and 12 Figure 10 shows the effect

of specimen length on the compressive strength of composites made with

urethane resin and reinforced with «s0.0193-in.-thick aluminum strips, while

Fig 11 shows similar data for composites made with '*'0.0411-in.-thick

aluminum laminas The test data shown in the latter two figures were normaUzed

for average resin properties and a reinforcement content of 50 percent [8] The

solid curves in Figs 10 and 11 were calculated from Eq 5 using average resin

properties and a reinforcement content of 50 percent By plotting the

compressive stress in composites as a function of (-7-) , the theoretical curves in

Figs 10 and 11 can be made to converge Figure 12 shows such a plot It also

contains test data from previous sections for composites with various L/h ratios

As shown in Figs 10, 11, and 12, as Z, and L/h increase, the compressive

stress, Oc, of the composite asymptotically approaches a constant value given by

the first term of Eq 5, that is, by

r-V (6)

which is only a function of the shear modulus of the matrix and the volume

fraction of the reinforcement Conversely, as L and L/h decrease, the second

term in Eq 5 becomes dominant For the latter case (L/h,L-> O), the

compres-sive microbuckling strength is governed by the properties of the reinforcement

and is relatively insensitive to the properties of the matrix For composites with

anall values of Z,, the compressive microbuckling strength is also strongly

affected by the lamina end restraints, as discussed previously

As shown in Fig 12, the test data show a good correlation with the

theoretical values predicted from Eq 5 These results further confirm the vaUdity

of the two-dimensional microbuckling theory for lamina-reinforced composites

It is shown that, by varying L and h but keeping L/h approximately constant,

the test data for composites made with laminas of various thicknesses fall close

together This behavior further shows the insensitivity of the compressive

microbuckling strength of lamina-reinforced composites to the absolute

thick-ness of the laminas

GRESZC2UK ON MICROBUCKLING 19

•;ili:\;i'rlTli

nil-•• " • • i » • • : ; - i l - k ! - 1 '-Ill - 1- nil-••nil-•• 111

* • » • • ii' ' i'.i

FIG 9-Compressive failure of lamina-reinforced composite made with urethane resin

and 0.0195-in.-thick aluminum laminas in the amount ofk = 48.8 percent (L = 4.75 in.)

THEORY BASED ON THE FOLLOWING AVERAGE PROPERTIES:

MICROBUCKLING FAILURE PREDICTED FROM EQ (5)

837 PSI 0.0193 IN

0 5 0

1 0 1 X 10^ PSI

-o-J 2"

SPECIMEN LENGTH, L , inches

FIG lO-Effect of specimen length on compressive microbuckling strength of

composites made with 0.0193-in.-thick Utminas

Influence of Resin Properties

To establish the influence of resin properties on the microbuckling

compressive strength of lamina-reinforced composites, additional composites

made with epoxy resin formulations, denoted here as Resins B and C, v/ere

prepared The average Young's modulus of Resin B was approximately 23 ksi,

while for Resin C the Young's modulus was ~62 ksi The reinforcement for

these composites consisted of 0.0195 in.-thick 6061-T6 aluminum strips

Composites with three different reinforcement volume fractions were fabricated

and tested A complete description of the test specimens and the test data is

r'

• THEORY BASED ON THE FOLLOWING AVERAGE PROPERTIES:

837 PSI O.Olill IN

0.50 10.1 X 10" PSI

It 6 8 SPECIMEN LENGTH, L, inches

10

FIG 11—Effect of specimen length on compressive microbuckling strength of oosites made with 0.041 l-in.-thick laminas

com-^resented in [5] Figure 13 shows the compressive strength of lamina-reinforced

:omposites as a function of reinforcement volume fraction and Young's modulus

'or the resin For comparison, previously discussed results on composites made

vith urethane resin are also shovm therein

The composites made with urethane resin (E «* 2500 psi), failed by

micro-juckling The composites made with Resin B (Ef ~ 22 990 psi) failed by

nicrobuckling; however, the buckle wavelength was shorter than that in

;omposites made with urethane resin, as can be seen by comparing Figs 4 and

14 Composites made with Resin C (Er «* 62 380 psi) failed in a complex maimer,

IS shown in Fig 15 The failure was by transverse tension, resin shear, overall

FIG 13-Test results on effect of matrix properties on compressive microbuckling strength of lamina-reinforced composites

composite shear, reinforcement yielding, and possibly by inelastic

micro-buckling The reinforcement stresses at failure ranged from 50 100 to 57 700

psi, depending on the volume fraction of the reinforcement These stresses are

significantly higher than the compressive yield strength or the compressive

ultimate strength of 6061-T6 aluminum sheet, which are 35 000 and '^=45 000

psi, respectively (Ref/O) A comparison of experimental results for

lamina-reinforced composites made with different resins with theoretical results

predicted from E q 5 is shown in Figs 16 and 17 Figure 16 shows the

compressive failure stress in the composite as a function of shear modulus of the

resin The horizontal portion of the curve was calculated assuming compressive

failure of the reinforcement given by the following approximate equation"*

[^^|(1-^)]

(Oc)u = Ofu\k+-^(l-k)\ (7)

where (0^)1/ is the ultimate compressive strength of the composite, and Ofi/ is

the ultimate compressive strength of the reinforcement Figure 17 shows the

test-theory comparison for the reinforcement stress in the composite at failure

of the composite as a function of the shear modulus of the resin By plotting

reinforcement stress, Of, rather than composite stress Oc, test data for

composites with ^ « 32 percent and k^ 65 percent can be plotted on the same

graph, inasmuch as the term k(l — k) is approximately the same for the two

cases

Conclusions

On the basis of results presented herein, it can be concluded that the

two-dimensional theory, corrected for the influence of end restraints, accurately

predicts microbuckling compressive failure of lamina-reinforced composites It

can be further concluded that the mode of failure is influenced by the properties

of the reinforcement, the properties of the matrix, and the volume fraction of

the reinforcement For composites made with low-modulus resins, failure is by

microbuckling, but, as the modulus of the resin increases, the composite failure

stresses are governed by the compressive strength of the reinforcement

' A better approximation for (CTc)f/is

where, in addition to the terms defined in the text, Oy* is the stress in the resin

corresponding to the strain in the reinforcement at ofi/ This equation was not used because

complete compressive stress-strain curves were not available for laminas The assumptions

made in using Eq 7 are obvious

isd j o T X aamivJ xv ssaats wiawaoacwKiaH aAissaH<)woo

GRES2CZUK ON MICROBUCKLING 29

Acknowledgment

The work described herein was sponsored by the Air Force Materials

Laboratory, Air Force Systems Command, United States Air Force, Wright

Patterson AFB, Ohio, under contract F33615-71-C-1399

References

[1] Biot, M A., Mechanics of Incremental Deformation, Wiley, New York, 1965

[2\ Rosen, B W in Fiber Composite Materials, American Society for Metals, Metals Park,

Ohio, 1965, Chapter 3

[3] Schuerch, H., Journal, American Institute of Aeronautics and Astronautics, Vol 4,

No 1, Jan 1966, pp 102-106

[4] Hayashi, T., "On the Shear Instability of Structures Caused by Compressive Loads,"

AIAA Paper No 65-770, presented at the Joint Meeting of The American Institute of

Aeronautics and Astronautics, the Royal Aeronautical Society, and The Japan Society

for Aeronautics and Space Science, Aircraft Design and Technology Meeting, Los

Angeles, CaUf., Nov 1965

[5\ Chung, Wen-Yi and Testa, R B., Journal of Composite Materials, Vol 3, Jan 1969,

pp 58-80

[6] Greszczuk, L B in Analysis of the Test Methods for High-Modulus Fibers and

Composites, ASTM STP 521, American Society for Testing and Materials, 1973,

pp 192-217

[7] Greszczuk, L B., "Microbuckling of Unidirectional Composites," Air Force Materials

Laboratory Report AFML-TR-71-231, Jan 1972

[8] Greszczuk, L B., "Failure Mechanics of Composites Subjected to Compressive

Loading," Air Force Materials Laboratory Report AFML-TR-72-107, Aug 1973

[9] Timoshenko, S P and Gere, J M., Theory of Elastic Stability, McGraw-Hill, New

York, 1961

[10] "Metallic Materials and Elements for Aerospace Vehicle Structures," MIL-HDBK-5A,

Department of Defense, Waslyngton, D C , Feb 1966

Analysis of the Flexure Test for

Laminated Composite Materials*

REFERENCE: Whitney, J M., Browning, C E., and Mair, A., "Analysis of the

Flexure Test for Laminated Composite Materials," Composite Materials:

Testing and Design (Third Conference), ASTMSTP546, American Society for

Testing and Materials, 1974, pp 30-45

ABSTRACT: Equations applicable to a general class of symmetrically

lami-nated beams ate derived by considering a beam as a special case of a lamilami-nated

plate The beam bending stiffness thus becomes a function of all the bending

stiffness coefficients of a laminated plate The validity of this approach is

verified by comparing theoretical results to flexure data on graphite/epoxy

angle-ply and quasi-isotropic laminates In addition, it is shown that flex

strength on general composite laminates is extremely difficult to interpret,

even though the stresses can be calculated from the modified beam theory

Discontinuities in the in-plane stresses at layer interfaces lead to a state of

stress which is difficult to compare to standard laminate tensile coupons

KEY WORDS: composite materials, laminates, beam theory, bend tests,

anisotropy, angle ply composites

Bending properties of composite materials are often characterized by using

simply supported beams under concentrated loads at the center or

quarter-points Results from such tests are commonly based on homogeneous isotropic

beam equations For laminated materials these formulas must be modified to

account for the stacking sequence of the individual plies It has been shown by

Hoff[7]^ and Pagano[2] that layered beams in which the plies are oriented

symmetrically about the mid-plane and the orthotropic axes of material

symmetry in each ply are parallel to the beam edges can be analyzed by classical

* This work is based on both an in-house research project sponsored by the Mechanics

and Surface Interactions Branch, Nonmetallic Materials Division, Air Force Materials

Laboratory and a thesis submitted to the Air Force Institute of Technology by A Mair for

the Master of Science degree

' Materials research engineer, research chemist, and graduate student, respectively, Air

Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio 45433

^The italic numbers in brackets refer to the list of references appended to this paper

WHITNEY ETAL ON THE FLEXURE TEST 31

beam theory if the bending stiffness EI is replaced by the equivalent stiffness

El 1 ^/ defined in the following manner:

^ 1 1 / = £ Ei^kik (1)

where £"11 ^^ is the effective bending modulus of the beam, £"11^ is the modulus

of the kt\\ layer relative to the beam axis, / is the moment of inertia of the beam

relative to the midplane, I^ is the moment of inertia of the kih layer relative to

the midplane, and n is the number of layers in the laminate

In addition to being used as a quality control and materials acceptance test on

unidirectional materials, the flexure specimen has been used to characterize the

modulus and strength of laminates constructed of angle-ply orientations [5,¥,5]

Equations which are applicable to beams constructed of angle-ply layers are

nonexistent

In the present paper equations which are applicable to a general class of

symmetric laminates are derived by considering a beam as a special case of a

laminated plate In light of the actual specimen dimensions employed in

composite flexure experiments, such an assumption seems justified The validity

of this approach is verified by comparing theoretical results to flexure data on

graphite/epoxy angle-ply and quasi-isotropic laminates Results show that flex

strength on general composite laminates is extremely difficult to interpret, even

though the stresses can be calculated from a modified beam theory

Disconti-nuities in the in-plane stresses at layer interfaces lead to a state of stress which is

difficult to compare to standard laminate tensile coupons

Analysis

The constitutive relations for the bending of a laminated anisotropic plate of

thickness h are of the form [6]

(2)

where Mx, My, Mxy are the resultant bending and twisting moments per unit

length, Kx, Ky, Kxy are the bending and twisting curvatures of the plate, and Djf

are the plate bending stiffnesses In terms of the anisotrojjic reduced stiffnesses

for plane stress, Qij, the bending stiffnesses are of the form

•h/2

•>-h/2

(3)