8- Mechanics of Composites - A Historical Review

8/16/2019 8- Mechanics of Composites - A Historical Review

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2 C.T. Herakovich / Mechanics Research Communications41 (2012) 1–20

“More important than any one new application is the new

‘materials’ concept itself. It marks a shift from concern with

substances to concern with structures, a shift from artisan to

scientist as man’s artificer, a shift from chemistry to physics as

the basic discipline, and a shift, above all, from the concrete

experience of the workshop to abstract mathematics, a shift

from starting with what nature provides to what man wants

to accomplish”.

Peter F. Drucker, The Age of Discontinuity, 1969

1. Since the beginning

A historical review of the mechanics of composites must first

consider the question “what initiated the study of these hetero-

geneous, anisotropic materials”? Composite materials have been

present since the beginning of time. Many objects such as plants

and animals are fibrous composite systems. This is very evident in

trees and their leaves, in the wings of birds and the fins of fish. The

human body is the most complex fibrouscomposite system. Onone

level it consists of a musculoskeletal system of bones, muscles and

tendons. On a microscopic level these objects themselves are com-

posite systems consisting of a variety of components that give rise

to heterogeneous, anisotropic materials.The first production and man-made use of a fibrous com-

posite material appears to be the papyrus paper made by the

Fig. 1. New testament on papyrus.

Egyptianscirca4000 B.C. They laid up strips from the fibrous

papyrus plant in two layers with one layer at right angles to the

other. In present day mechanics terminology, such a fabric would

be called an unsymmetric, cross-ply laminate. While it is possible

that the Egyptians used a symmetric laminate to eliminate curva-

ture, this author has found no indication that such was the case.

The development of papyrus paper was so important to the Egyp-

tians that they guarded the secret of how it was produced, thereby

creating a monopoly. Papyrus paper revolutionized the way peo-

ple saved valuable information. As a result, it was ancient Egypt’s

greatest export for many centuries.

Cuttings from the papyrus plant also were used in bundles by

early Egyptiansto make boats,sails, baskets and ropes. Fig.1 shows

an image of a passage from the NewTestament written on papyrus

around the beginning of the 3rd century, some 1800 years ago. It

is referred to as the Bodmer Papyrus XIV-XV in the Vatican (2007).

Fig. 2 is an example of present-day Egyptian artwork on papyrus

paper.

Another early, but totally different, application of a man-made

fibrous composite was the use of straw to strengthen bricks made

from mud. According to the Book of Exodus (“do not give them

straw for their bricks, make them find their own straw”), this prac-

tice was used as early as1300 B.C.; it is still in use today. Fig. 3 is a

picture of such a brick that was taken inthe Middle East inthe mid-20thcentury. Present-day mechanics would classify these bricks as

randomly reinforced, short fiber composites.

According to Hartman et al.(1996), ancient Egyptians also made

containers of coarse fibers drawn from heat softened glass, and the

French scientist Reaumur considered the potential of forming fine

glass fibers as early as the 18th century. It was not until 1939 that

continuous glass fibers were produced commercially (Knox, 1982).

These glass fibers were produced mainly for high temperature

electrical applications. Two more decades passed before the “so-

called” advancedfibers wereproduced, boron(Talley, 1959) carbon

Fig. 2. Artwork on papyrus.


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C.T. Herakovich / Mechanics Research Communications41 (2012) 1–20 3

Table 1

Early activities, contributions and accomplishments in mechanics of composites.

Year Activity People Country

4000 BC Papyrus paper developed Egyptians Egypt

1660 Hooke’s Law Robert Hooke Great Britain

1780s Young’s modulus defined Thomas Young Great Britain

1821 Formulation of general equations of elasticity Claude-Louis Navier France

1822 Anisotropic equations of elasticity Augustin-Louis Cauchy France

1837 Strain energy density defined – 21 elastic constants George Green Great Britain

1887/1889 Uniform strain modulus prediction W. Voigt Germany1892 Treatise on theMathematical Theoryof Elasticity A.E.H. Love Great Britain

1929 Uniform stress modulus prediction A. Reuss Germany

1935 Papers on anisotropic bodies S.G. Lekhnitshkii Russia

1938 Owens-Corning developed fiberglass Owens-Corning USA

1941 Air Force Materials Lab initiated composites activity Robert T. Schwartz USA

1941 Fiberglass fabrics available to market Owns-Corning and H. Goldsmith USA

1946 Mathematical Theory of Elasticity I.S. Sokolnikoff USA

1947 Anisotropic Plates S.G. Lekhnitshkii Russia

1950 Theory of Elasticity of an Anisotropic Elastic Body S.G. Lekhnitshkii Russia

1954 Japan Society of Reinforced Plastics formed Tsuyoshi Hayashi Japan

1954 Fabricated glass reinforced plastic glider Tsuyoshi Hayashi Japan

1961 Theory of Anisotropic Shells S.A. Ambartsumyan Russia

1965 Strength of Unidirectional Lamina A. Kelly and G.J. Davies Great Britain

1967 ModernCompositeMaterials L.J. Broutman and R.H. Krock USA

1967 Journal of Composite Materials Vol. 1 No. 1 Stephen W. Tsai USA

1968 CompositeMaterialsWorkshop Tsai, Halpin and Pagano USA

1969 The Analysis of LaminatedComposite Structures Lee Calcote USA

1969 Primer on CompositeMaterials:Analysis Ashton, Halpin and Petit USA

1970 Theoryof LaminatedPlates J.E. Ashton and J.M. Whitney USA

1970 Theory of Anisotropic Plates S.A. Ambartsumyan Russia

1972 Theory of Fiber ReinforcedMaterials Zvi Hashin USA

1975 Mechanics of CompositeMaterials R.M. Jones USA

1979 Mechanicsof CompositeMaterials R.M. Christensen USA

(Soltes, 1961) and aramid (Kwolek, 1964). The development of the

advanced fibers in the late 1950s and early 1960s spurred great

interest in the development of theoretical and applied mechanics

for applications to fibrous composite materials and structures.

Fromthe earliest applicationsof fibrous composites by theEgyp-

tians to theintroduction of advanced composites in thesecond half

of the 20th Century, roughly 6000 years have passed. The progress

in the use of fibrous composites in the most recent fifty yearswas much greater than that during the preceding nearly six thou-

sand years. The Egyptians were artisans in that they undoubtedly

developed their products through trial and error. During the past

fifty years, theoretical and applied mechanics has been employed

in order to exploit the vast potential of man-made fibrous com-

posites. These advancements are exemplified dramatically by the

application of advancedfibrouscomposites in SpaceShipOneand its

launch vehicle White Knight, (Fig. 4, see http://www.scaled.com/).

SpaceShipOne is an all-composite, suborbital spaceplane launched

in 2003 by Scaled Composites.

Fig. 3. Brick with straw fibers.

2. The early years

Table 1 summarizes the early activities, contributions and

accomplishments related to advances in the mechanics of fibrous

composites. The remainder of this paper is organized according

to subject matter. Topics covered include constitutive equations,

micromechanics, laminates, thermal and moisture effects, damage

and failure, experimental methods, interlaminar stresses, tubes,plates, nanocomposites, and university and government programs.

This leaves many related subjects still to be reviewed.

3. Anisotropic, elastic constitutive equations

Discussions on the advances in the development of constitu-

tive equations for elastic materials can be found in Love’s work

(1892–1927), Sokolnikoff (1946/1956) and Timoshenko’sHistory of

StrengthofMaterials (1953). The development of constitutive equa-

tions for homogeneous, elastic materials began with the work of

Hooke (1678) who stated that for an elastic body there is propor-

tionality between stress and strain. Navier (1821) generalized this

Fig. 4. White knight and SpaceShipOne.


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idea to arrive at differential equations describing elastic response;

however, his equations included only one elastic constant. Cauchy

(1822), building on the work of Navier, developed equations of

elasticity for isotropic materials with two elastic constants.1

Cauchy (1828) generalized his results for anisotropic (or

aeolotropic) materials and found that there are, at most, twenty-

one independent constants. However, he believed that only fifteen

of these constants were “elastics constants”. Like Cauchy, Poisson

(1829) believed that there were only fifteen elastic constants.

These beliefs were based upon a molecular structure of solids and

intermolecular forces. During this period, there were two com-

peting theories as to the number of elastic constants, either 21

or 15. The question was resolved when Green (1839) introduced

the concept of strain energy and arrived at equations of elastic-

ity from the principal of virtual work. The equations developed

using Green’s approach have 21 independent, elastic constants.

Lord Kelvin (Thomson, 1855) used the First and Second Laws

of Thermodynamics to argue for the existence of Green’s strain

energy function. The existence of the strain energy functionand the

presence of 21 independent elastic constants in the most general

anisotropic case is now the accepted theory.

The most general form of the anisotropic constitutive equations

for homogeneous, elastic, composite materials can be written2:

1 2 3 4 5 6

≡

xx yy zz yz zx xy

=

C 11 C 12 C 13 C 14 C 15 C 16C 12 C 22 C 23 C 24 C 25 C 26C 13 C 23 C 33 C 34 C 35 C 36C 14 C 24 C 34 C 44 C 45 C 46C 15 C 25 C 35 C 45 C 55 C 56C 16 C 26 C 36 C 46 C 56 C 66

ε xxε yyε zz yz zx xy

≡

ε1ε2ε3ε4ε5ε6

(1)

where ij and ij are normal and shear components of stress,respectively, εij and ij are the normal and shear components of strain, respectively, and C ij is the symmetric stiffness matrix with

21independent, elastic constants (orstiffnesscoefficients).(The sin-

gle subscriptnotationof stress and strainin (1) is common practice

for analysis of composite materials.) These constitutive equations

appear in the same form in Lekhnitskii (1947) Anisotropic Plates

book (p.10, Eq.(2.1)). It is noted that thepreface to the 1947 editionactually was written in May 1944.

Lekhnitskii also shows that a monoclinic material (one plane of

symmetry) has 13 independent constants, an orthotropic material

(three planes of symmetry) has 9 independent constants, a trans-

versely isotropic material (isotropic properties in one of the planes

of symmetry) has five independent constants, and an isotropic

material (properties independent of direction) has two indepen-

dent constants. He also discussed the case of a material with

cylindrical anisotropy.

Inversion of (1) gives expressions for the strains in terms of

stresses and compliance coefficients S ij:

ε xxε yyε zz yz zx xy

=

S11 S12 S13 S14 S15 S16S12 S22 S23 S24 S25 S26S13 S23 S33 S34 S35 S36S14 S24 S34 S44 S45 S46S15 S25 S35 S45 S55 S56S16 S26 S36 S46 S56 S66

xx yy zz yz zx xy

(2)

The constitutive Eq. (2) can be written in terms of the engi-neering constants, elastic moduli E i, Poisson ratios ij and shearmoduliGij. As an example,for an orthotropic material withprincipal

1 Dates given forNavier andCauchycorrespond to when they read their paper to

theParis Academy. Publication,if any, was at a later date.2 Commonly usednotations for compositemechanicsand developments of many

of the results presented in this article are described in detail in the author’s text

(Herakovich, 1998).

directions 1, 2, 3, Eq. (2) takes the form (3) when written in termsof engineering constants and reduced notation:

ε1ε2ε3ε4ε5ε6

=

1 ⁄ E 1 −21 ⁄ E 2

−31 ⁄ E 3 0 0 0−12 ⁄ E 1

1 ⁄ E 2 −32 ⁄ E 3 0 0 0

−13 ⁄ E 1 −23 ⁄ E 2

1 ⁄ E 3 0 0 0

0 0 0 1 ⁄ G23 0 0

0 0 0 0 1 ⁄ G31 0

0 0 0 0 0 1 ⁄ G12

1 2 3 4 5 6

(3)

The symmetry of the compliance matrix provides additionalrelationships between the moduli and Poisson ratios.

The earliest publications employing anisotropic constitutive

equations for the solution of real problems appear to be those of

Lekhnitskii. During the years 1935–1942, he published papers on

plane problems, cylindrical anisotropy, torsion and bending. Wood

was the primary material considered in his work. See the bibli-

ography in Lekhnitskii’s Theory of Elasticity of an Anisotropic Body

(1950) for additional references. It is possible that Navier and co-

workers used the equations that they had developed in their work

on wood structures, but no references have been found by this

author.

4. Micromechanics

The study of composite materials at the fiber and matrix level

is referred to asmicromechanics . It is desired to predict the overall

effective (or average) elastic properties and inelastic response of

the composite based upon the known properties, arrangement andvolume fraction of the constituent phases. Examples of compos-

ites at the fiber and matrix level are shown in Fig. 5. Fig. 5a shows

carbon fibers in an epoxy matrix and Fig. 5b is a photomicrograph

of ceramic fiber (silicon carbide) in a titanium matrix. The silicon

carbide fiber has a tungsten core that is clearly visible in the figure.

The carbon fibers are actually a collection (called tows) of numer-

ous carbon filaments (2000–30,000 or more). As indicated in these

figures, ceramic fibers typically have a much larger diameter than

carbon fibers. The distribution of fibers is quite uniform in metal

matrix composites, but is variable in resin matrix composites. This

significant difference in the distribution of fibers requires that a

larger region (number of fibers) be considered as the representative

volume element (RVE) for micromechanics studies when the fiber

distribution is nonuniform. Whenthe fibers areuniformly spaced asin Fig. 5b, it is reasonable to consider a singlefiber andsurrounding

matrixmaterialas the RVE. In this lattercase,symmetryarguments

often can be used to reduce the region under consideration even

further.

A wide variety of methods for predicting the effective thermo-

elastic properties of composites have been offered (Table 2). The

earliest works are those of Voigt (1889) and Reuss (1929). While

theseearly studies were concerned primarilywith polycrystals,the

theories can be applied to fibrous composites. Voigt assumed that

the strains were constant throughout the material under load. In

contrast, Reuss assumed that the stresses were constant through-

out the material. Hill (1952) showed that the Voigt assumption

results in upperboundson effectiveelasticproperties andthe Reuss

assumption results in lower bounds.


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Table 2

Developments in micromechanics.

Year Development Author(s) Country

1837 Strain energy density defined – 21 elastic constants George Green Great Britain

1887/1889 Uniform strain modulus prediction W. Voigt Germany

1929 Uniform stress modulus prediction A. Reuss Germany

1957 Determination of the elastic field of an ellipsoidal inclusion and related problems Eshelby Great Britain

1960 Prediction of Elastic Constants of Multiphase Materials Paul USA

1962 The Elastic Moduli of Heterogeneous Materials Hashin USA

1963 Elastic Properties of Reinforced solids: some theoretical principles Hill Great Britain1963 Variati onal Approach to the Theory of The Elastic Behavior of Multiphase Materials Hashin and Shtrikman USA and Israel

1964 Theory of Mechanical Behavior of Heterogeneous Media Hashin USA

1964 Theory of Mechanical Properties of Fiber-Strengthened Materials – I. Elastic Behaviour Hill USA

1964 The Elastic Moduli of Fiber-Reinforced Materials Hashin and Rosen USA

1965 The Principle of the Fiber Reinforcement of Metals Kelly and Davies Great Britain

1967 Modern Composite Materials Broutman and Krock USA

1968 Composite Materials Workshop Tsai, Halpin and Pagano USA

1972 Theory of Fiber Reinforced Materials Hashin USA

1972 On t he E ffective M oduli o f C omposite M aterials: S lender R igid I nclusions a t D ilute C oncentrations Russel and Acrivos USA

1975 A Theory of Elasticity with Microstructure for Directionally Reinforced Composites Achenbach USA

1979 Analysis of Properties of Fiber Composites with Anisotropic Constituents Hashin Israel

1991 Mechanics of Composite Materials: A Unified Micromechanical Approach Aboudi Israel

1993 Micromechanics: overall properties of heterogeneous materials Nemat-Nasser and Hori USA

The development of micromechanics models for predicting the

effective properties of composites experienced a flurry of activity

beginning in the late 1950s and extending through the 1960s and

into the 1970s. The earlier works were concerned with the pre-

diction of effective properties for materials (both solids and fluids)

consisting of inclusions in a carrier material (see Hashin, 1964).

Paul (1960) and Hill (1963) used energy approaches to obtain

upper and lower bounds on elastic moduli of heterogeneous mate-

rials consisting of inclusions in a matrix. In general, the inclusions

were of arbitrary shape, but both authors did make reference to

fiber-like inclusions. It is noteworthy that they both showed that

the Voigt and Reuss approximations are upper and lower bounds

on moduli. Hashin and Shtrikman (1963) presented a variational

approach to derive upper and lower bounds for the effective elas-

tic moduli of quasi-isotropic and quasi-homogeneous multiphasematerials of arbitrary phase geometry. Hill (1964) addressed the

elastic mechanical properties of fiber-strengthened materials. In

(1972) Russeland Acrivos considered theeffective modulus of com-

posites with slender rigid inclusions at dilute concentrations.

Hashin andRosen (1964) employeda concentriccylinder assem-

blage (CCA) model to develop upper and lower bounds as well

as specific expressions for (some of) the effective elastic mod-

uli of transversely isotropic composites. Their model consists of

an assemblage of concentric cylinders, each cylinder consisting

of a fiber core surrounded by a matrix annulus, such that the

size of the cylinders varies as needed to fill the entire volume of

material.The ratio of fiber radius to cylinder radius is held constant

throughout, thereby maintaining a constant fiber volume fraction

in each cylinder. Four of thenecessary five effectiveproperties fora

transversely isotropic composite can be determined using the CCA

model.

In the following from Christensen (1979), with subscripts f and

m indicating fiber and matrix respectively, V fiber volume fraction,

E axialmodulus,Poisson’s ratio, shearmodulus, kbulkmodulus,and the fiber and matrix are taken to be transversely isotropic, the

CCA model provides expressions for four of the effective properties

in terms of the phase properties and fiber volume fraction.Effective axial modulus:

E ∗1 = V f E f + (1 − V f )E m +4V f (1 − V f )(v f − v m)

2m

((1 − V f )m)/(k f + f /3) + V f m(4)

Effective axial Poisson’s ratio:

12 = (1 − V f )m + V f f

+V f (1 − V f )( f − m)

m/(km + m/3) −m/(k f + f /3)

((1− V f )m)/(k f + f /3) + V f m/(km + m/3) + 1

(5)

Fig. 5. Fiber and matrix photomicrographs.


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Effective plane strainbulk modulus:

K ∗23 = km +m

3

+V f

1/[k f − km + ( f −m)/3] + (1 − V f )/(km + 4m/3) (6)

Effective axial shearmodulus:

∗12m

= f (1 + V f ) +m(1 − V f )

f (1 − V f ) +m(1 + V f ) (7)

It is evident from Eqs. (4) and (5) that the first two terms cor-

respond to a rule of mixtures. The last term is typically very small

for most composites in use today. Thus, the rule of mixtures (i.e.

Voigt upper bound) is a very good predictor for the effective axial

modulus and effective axial Poisson’s ratio. This cannot be said for

the other two properties.

Chamis and Sendeckyj (1968) presented an extensive critique

of the theories known at the time for predicting the thermoelastic

properties of fibrous composites. The theories reviewed were clas-

sified as: netting analysis, mechanics of materials, self-consistent

model, variational, exact, statistical, discrete element, semi empir-

ical methods, and theories accounting for microstructure. Theyincluded comparisons of predictions by different theories for uni-

directional glass-epoxy, boron-epoxy and graphite-epoxy.

Hashin (1972) gave an extensive theoretical treatment of

micromechanics. He considered effective elastic, viscoelastic and

thermoelastic properties, thermal and electrical conduction, and

electrostatics and magnetostatics behavior.

Achenbach (1974) and Achenbach (1975) considered wave

propagation in fiber-reinforced composites with microstructure.

The composite with microstructure is distinguished from a com-

posite that is modeled as a homogeneous, anisotropic continuum

using effective properties. The point is made that for dynamic

response such as wave propagation, the characteristic lengths of

the deformations may be small and the effective modulus theory

may not suffice. The proposed theory showed good comparisonwith ultrasonic data for fibrous composites and finite element pre-

dictions. Lectures on this subject were given at the International

Centre for Mechanical Sciences (CISM) in Udine, Italy, in July 1973

with publication of the (expanded) monograph in 1975.

Aboudi (1991) presented micromechanical analysis methods

for composite materials and provided an in-depth analysis of the

Method of Cells for thermo-elastic, viscoelastic, nonlinear behav-

ior of resin matrix composites, initial yield surfaces and inelastic

behavior of metal matrix composites, and composites with imper-

fect bonding. The method of cells consists of a periodic square

array of rectangular subcells, one representing the fiber and three

similar subcells representing the matrix. This model provides a

computationally efficient method for predicting inelastic response

of composites.The effects of different types of fiber orthotropy on the effective

propertiesof composites were considered by Knott and Herakovich

(1991a). Nemat-Nasser and Hori (1993) presented a treatise on the

mechanics of solids with microdefects such as cavities, cracks, and

inclusions, including elastic composites.

Discrete element methods such as the finite element method

have been used to predict effective properties of unidirectional

composites. The earliest work using finite elements appears to

be that of Foye who studied effective elastic properties, inelastic

response, and stress distributions in unidirectional boron/epoxy.

Finite element studies can be valuable when the fiber distribution

is very regular as shown for the ceramic fiber in a titanium matrix

of Fig. 5b, but less so for random fiber distributions such as the

carbon/epoxy of Fig. 5a.

Fig. 6. Axial modulus predictions for carbon/epoxy.

4.1. Micromechanicsmodel comparisons

Figs. 6–9 show comparisons of micromechanics predictions for

the effective properties E 1, E 2, 12, and G12 of unidirectional car-bon/epoxy (Lissenden and Herakovich, 1992) as a function of thefibervolume fraction. The methods comparedinclude: Voigt, Reuss,

concentric cylinder assemblage, self-consistent, method of cells,

Mori-Tanaka and strength of materials.

Several important features are evident from these comparisons.

For the effectiveaxial modulus,E ∗1 (Fig.6) essentially all models givethe same prediction, with the lower bound Reuse model being the

exception. Thus, a simple rule of mixtures (the Voigt upper bound),

provides excellent predictions for the effective axial modulus.

Schapery (1967) has shown that the results for linear elastic

materials can be extended to linear viscoelastic materials in a sim-

ple and accurate manner.

5. Lamination theory

Possibly the most fundamental result for the application of

fibrous composites in structural and devices is Classical Lamination

Theory. The theory follows the original works of Pister and Dong

(1959), Reissner and Stavsky (1961) and Dong et al. (1962)

The theory considers an assemblage of layers bonded together

to form a laminate. The individual layers are taken to be homoge-

neous with properties that can range from isotropic to anisotropic.

Typically, the layers are unidirectional fibrous composites with the

Fig. 7. Transverse modulus predictions for carbon/epoxy.


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Fig. 8. Shear modulus predictions for carbon/epoxy.

fibers in the kth layer oriented at an angle k from a global x-axisas depicted in Fig. 10.

Analysis results in the fundamental equation relating theinplane forces {N } and moments {M } acting on the laminate to

the midplane strains {ε◦} and curvatures {} through coefficients[ A], [B] and [D] that are functions of the material properties, layers

thickness and stacking sequence of the layers.N M

=

A BB D

ε◦

(8)

The effectiveengineeringproperties of symmetriclaminates can

be predicted from Eq. (8) through a series of thought experiments

where the laminate is subjected to a series of specified loadings.

With the laminate compliance defined:

[a∗] ≡ 2H [ A]−1 (9)

The results of these thought experiments provide expressions

for the engineering properties of the laminate. Examples are:

Axialmodulus:

E x =̄ xε◦ x

=1

a∗11(10)

Fig. 9. Poisson’s ratio predictions for carbon/epoxy.

Fig. 10. Composite laminate.

Poisson ratio:

xy = −ε◦ yε◦ x

= −a∗12a∗11

(11)

Shearmodulus:

G xy =̄ xy ◦

xy

=1

a∗

66

(12)

Coefficient ofmutual influence:

xy,x = ◦ xyε◦ x

=a∗

16

a∗11(13)

The coefficient of mutual influence (13) quantifies the shear

strain associated with normal strain; it is non-zero when the lam-

inate compliance term a∗16 is non-zero.Specific examplesof therange of engineeringproperties thatcan

be affected through the choice of material and stacking sequence

are presented in Figs. 11–13. These figures show the variation in

axial modulus, Poisson ratio and shear modulus for T300/5208car-

bon/epoxy.

These three figures show that the effective engineering prop-

erties of angle-ply laminates are higher than those of thecorresponding laminae. Further, Poisson’s ratio of angle-play lam-

inates can exhibit values greater than 1.0, and the shear modulus

of angle-ply laminates is largest at 45◦.

Another most interesting result for laminated composites

(Fig. 14) is the fact that the through-the-thickness Poisson’s ratio

Fig. 11. Axial modulus – unidirectional and angle-ply laminates.


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Fig. 12. Poisson’s ratio – unidirectional and angle-ply laminates.

xz is negative over a significant range of fiber orientations forsomeangle-ply laminates (Herakovich, 1984).

Another interesting feature of laminates is that, depending

on the stacking sequence of the layers, they can exhibit cou-

pling between inplane and bending effects. Laminates that are

unsymmetric about the laminate midplane have a non-zero [B]

matrix resulting in coupling between inplane and out-of-plane

responses (see Eq. (8)). Unsymmetric laminates exhibit curvature

when subjected to pure inplane loading. Likewise, unsymmetric

laminates exhibit inplane strains when subjected to pure bending

moments. More on unsymmetric laminates is provided in a later

section.

6. Environmental effects

6.1. Thermal effects

Environmental effects often play a critical role in the choice

of material for many applications in devices and structures.

Fig. 13. Shear modulus – unidirectional and angle-ply laminates.

Fig. 14. Through-the-thickness Poisson’s ratio.

Composites often arethe material of choice where thermal stressesor thermal expansion are important. The coefficient of thermal

expansion in the fiber direction of unidirectional composites is

often near zero and can be slightly negative. This has huge

consequences whendesigninglaminates for low,or matching, coef-

ficients of thermal expansion. Thermal stresses can be extremely

important for the application of fibrous composite materials as

essentially all composite materials are fabricated at an elevated

temperature. The constituentphases become bonded at an elevated

temperature resulting in residual thermal stresses in the composite

after it has cooled to room temperature.

Fundamental problems at the micromechanics level are predic-

tion of the residual stresses and the effective thermal properties of

unidirectional composites. At the laminate level, it is necessary to

predict the residual stresses and the laminate effective coefficientof thermal expansion (CTE). This latter property is very important

as it is one of the unique aspects of laminated composite materi-

als: composite laminates can exhibit CTE values over a wide range

including zero, positive and negative.

The earliest papers dealing with thermal effects in anisotropic

materialsappear to be those by Ambartsumyan(1952) who consid-

ered thermal stresses in anisotropic, laminated plates, and Hayashi

(1956) who considered thermal stresses in orthotropic plates.

The earliest works at the micromechanics level appears to be

that of Van Fo Fy (1965) who considered thermal effects in com-

posites consisting of periodic arrays of continuous, circular glass

fibers. He used stress analysis to determine exact thermal coef-

ficients for specific phase geometries. Levin (1967) presented an

approach for determining the effective coefficients of thermalexpansion for two phase composites with isotropic phases. The

work used an extension Hill’s approach and included bounds on

the expansion coefficients of transversely isotropic, unidirectional,

fiber-reinforced composites. Rosen (1968) investigated thermal

expansion coefficients for composite materials. Much of this work

is incorporated in the later paper by Rosen and Hashin (1970) on

expansion coefficients.

Schapery (1968) derived upper and lower bounds as well as

specific approximations for thermal expansion coefficients of lin-

ear elastic and viscoelastic composite materials. He extended the

previous work of Levin and Van Fo Fy for an arbitrary number

of constituents and phase geometries, for isotropic phases. The

approach provided upper and lower bounds using the principles

of complementary and potential energy. Approximate expressions


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Fig. 15. CTE – unidirectional and angle-ply laminates.

for the axial and transverse thermal coefficients of expansion for

unidirectional, fiber-reinforcedcomposites were presented.Hashin(1979) extended Schapery’s elastic results for composites with

transversely isotropic phases. The final forms of the predictions for

the axial and transverse thermo-elastic coefficients of expansion

(as presented by Daniel and Ishai, 1994) are:

˛1 =E f ̨f V f + E m˛mV m

E f V f + E mV m=

(E˛)1E 1

(14)

for the coefficient in the fiber direction, and

˛2 = ˛2 f V f

1 + 12 f

˛1 f ˛2 f

+ ˛2mV m

1 + 12m

˛1m˛2m

− (12 f V f + 12mV m) (E˛)1E 1(15)

for the coefficient in the transverse direction. In the above, f and

m refer to fiber and matrix, respectively, V is volume fraction, E

is modulus, ˛ is coefficient of thermal expansion and is Pois-son’s ratio, (E ̨ )1 =E f ̨ f V f +E m˛mV m and E 1 is the rule of mixturecomposite modulus in the fiber direction.

Additional works on thermal effects in composites include the

review article by Tauchert (1986) and that by Herakovich and

Aboudi (1999).

The first presentation of the thermal-elastic formulation for

composite laminates was by Tsai (1968). An early textbook presen-

tationof the formulationis thatby Calcote (1969). A mostimportant

result of the formulation is an expression for the effective coeffi-cient of thermal expansion { ¯̨ } for a symmetricN -layered laminate,

namely:

{ ¯̨ } = [ A]−1N

k=1

[ Q̄ ]k{˛}kt k (16)

Ashton et al., 1969 presented results for the varia-

tion of thermal strains as a function of fiber orientation

for unidirectional and angle play laminates. Fig. 15

shows that rather large, negative coefficients of ther-

mal expansion are possible for a typical carbon/epoxy

material (T300/5208 in Fig. 15) over a range of fiber orientations

for angel-ply laminates.

Fig. 16. Finite width couponunder axial load.

6.2. Moisture effects

The analysis of moisture effects in organic matrix composites is

analogous to that for thermal effects at both the micromechanicsand laminate levels. Much of this work is detailed in three volumes

edited by Springer (1981), Springer (1984), and Springer (1988a).

Volume 3, Chapter 1 (Springer, 1988b) provides a broad review of

the effects of temperature and moisture on organic matrix com-

posites. In general, moisture effects are not nearly as significant as

thermal effects.

7. Interlaminar stresses

The first publication concerned with interlaminar stresses in

laminated composites appears to be that of Hayashi (1967) who

investigated interlaminar shear stresses in an idealized laminate

consisting of orthotropic layers separated by isotropic shear lay-

ers. Other important early works include those by Bogy (1968)who investigated the singular behavior of stresses at the inter-

section of a boundary and bonded dissimilar isotropic materials,

and the first three-dimensional (numerical) analysis of inter-

laminar stresses in laminated composites by Pipes and Pagano

(1970).

Pipes and Pagano provided the first complete analysis of the

problem of an axially loaded, laminated coupon with free edges

(Fig. 16). They formulated a reduced system of elasticity equa-

tions governing the laminate behavior by assuming independence

of the stress and strain state on the axial coordinate and then

solved the system of equations using the finite difference method.

Their results showed the existence of all three interlaminar stress

components in the boundary layer regions along the free edges

of finite width laminated coupons under inplane tensile load-ing. They presented results for a variety of fiber orientations and

laminate stacking sequences and showed that the width of the

boundary layer is approximately equal to the thickness of the lam-

inate, that the interlaminar normal stress z and the interlaminarshear stress zx can exhibit singular behavior as the free edge isapproached, and that the sign and magnitude of the interlami-

nar stresses are functions of the laminate configuration including

material type, fiber orientations, layer thicknesses and stacking

sequence.

The free edge problem has been studied on a continuing basis

ever since the original work in the late 1960s. The finite dif-

ference solution of Pipes and Pagano was followed quickly by a

three-dimensional finite element solution by Rybicki (1971). Later,

it was recognized that the tensile coupon problem also could be


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Fig. 17. Free edge deformations – quasi-isotropic laminates.

formulated as a two-dimensional finite element problem because

of the independence of thestress andstrainstates on the axial coor-

dinate. The finite element formulation for cross-ply laminates as a

two-dimensional problem waspresentedby Foyeand Baker(1971),and the two-dimensional finite element formulation for laminates

including off-axis layers, thermal stresses and non-linear response

was presented by Herakovich et al. (1976).

Noteworthy approximate analytical solutions include a pertur-

bation solution by Tang (1975), a variational approach by Pagano

(1978), a solution employing complex stress potentials and eigen-

function series by Wang andChoi (1982), and solutions based upon

statically admissible stress states (Kassapoglou and Lagace, 1986;

Rose and Herakovich, 1993). The free edge problem has also been

investigatedexperimentally, e.g., Pipes and Daniel (1971), Oplinger

et al. (1974), and Herakovich et al. (1984). The experimental inves-

tigations provided physical evidence of a boundary layer withlarge

strain gradientsnear free edges. Allof theabove studies haveclearly

shown interlaminar stresses are the result of the mismatch in Pois-son’s ratios and coefficients of mutual influence and the presence

of a stress free boundary. The laminate stacking sequence plays

an important role in the magnitude and sign of the interlaminar

stresses.

Fig. 17 shows examples of the deformations of two-dimensional

finite element grids of the generic cross-section near the free edge

of carbon/epoxy laminatesunder axialloading(Buczeket al., 1983).

Results are presentedfor two differentstackingsequences of quasi-

isotropic laminates with the±45◦ always adjacent to one another.

It is evident from this figure that the displacements (and related

stresses and strains) are a strong function of the stacking sequence

with the interlaminar normal stresses being positive or negative

depending upon the stacking sequence. Figs. 17 and 18 combined

provide a complete picture of thepossible edge effects on the threegeneric planes (top face, free edge and transverse cross-section) of

a finite width coupon under axial load.

Fig. 18 (Herakovich et al., 1984) shows Moiré fringe patterns for

the axial displacements on the face and free edge of angle-ply, car-

bon/epoxylaminates subjectedto axial loading. On the coupon face

it is evident that thewidth of the edge effectis approximately equal

to the thickness of the laminate. On the edge, the shear strains zxare proportional to the gradient of the fringe lines and are max-

ima at theinterfaces between the layers; the 10◦ and 30◦ laminates

exhibit much higher shear strain (and stress) than the 45◦ lami-

nate. Analytical studies are in agreement with the fiber orientation

dependence which is directly related to themismatchin layerprop-

erties. It is also evident from the figure that the displacements on

the face and on the edge are uniform along the length

Fig. 18. Moiré fringe patterns – angle-ply laminates.

8. Unsymmetric laminates3

As noted previously in the section on lamination theory,

unsymmetric laminates exhibit coupling between inplane and

out-of-plane responses. Hyer (1988) reviewed many features of

3 This section waswritten by Mike Hyer, with a very fewmodifications provided

by the author.


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Fig. 19. Possible shapes of anti-symmetric cross-ply laminates.

unsymmetric laminates including curing and out-of-plane shapes,

stability considerations, and effects of materials properties, lami-

nate thickness, laminate aspect ratio, and stacking sequence.When a flat, unsymmetric, fiber-reinforced laminate layup is

cured in a press or autoclave at an elevated temperature, it will

develop curvature when cooled to room temperature and released

from all constraints of the curing process. The laminates will often

have two shapes at room temperature, and the laminate can be

changed from one shape to the other by a simple snap-through

actionin the form of a momentappliedto the edges of thelaminate.

The developed curvatures are due to a mismatch in the thermal

expansion behavior of the layers within the laminate. The exact

shape of the cooled laminate, and whether it has multiple shapes

or just a single shape, depends on layer material properties, the

geometry of the laminate, and the temperature change from the

cure temperature to room temperature.

Considering a square unsymmetrical cross-ply laminate, forexample, and considering a simplified description, when two

shapes occur, one shape is near-cylindricalwith the cylindergener-

ator parallel toone setof oppositeedgesof thelaminate. Thesecond

shape is also near-cylindrical, but with a curvature of opposite sign

and with the cylinder generator direction perpendicular to that of

the first shape, namely, parallel to the other set of opposite edges.

In terms of principal curvatures,the principle curvatures of the two

shapesare of opposite sign and the major principal curvature direc-

tionof oneshape is perpendicularto themajor principal directionof

the other shape. Furthermore, the major principal curvature direc-

tion in each case is parallel to an edge of the laminate. When a

single shape occurs, that shape is a saddle with its principal curva-

tures aligned with the laminate edges.When this behaviorwas first

observed it was nothing morethan a curiosity.However, the behav-ior has attracted continuing attention since the initial work by Hyer

(1981a) to empirically study the phenomenon. Representations of

these different shapes are shown in Fig. 19.

Continuing with a cross-ply laminate as an example, classical

lamination theory, with its linear strain-displacement relations,

predicts the shape of all unsymmetric laminates to be a saddle,

and since classical lamination theory is a linear theory, that is

the only shape predicted. However, when cooling a thin lami-

nate of any reasonable size, e.g., 4-, 8-, or 16-layers, 0.5 m square,

from its cure temperature to room temperature, the out-of-plane

deformations are many times the laminate thickness. Deforma-

tions of this magnitude bring into question the validity of using

the linear strain-displacement relations. Therefore, considering

cross-ply laminates, Hyer (1981b) and Hyer (1982) and Hamamoto

and Hyer (1987) employed the geometrically nonlinear von Kár-

mán strain-displacement relations, an energy approach, and the

Rayleigh–Ritz technique wherein approximate displacement func-

tions were used to develop a semi-closed form solution to the

problem thatexplained howthe existenceof twocylindricalshapes

or a single saddle shape depended on the laminate geometry and

the temperature change relative to the cure temperature. Obvi-

ously, the thermo-elastic properties of the laminate material were

important too.

Interestingly, the developed theory predicted a saddle shape

for all laminate geometries, but for laminates with large side-

lengths, the saddle shape was predicted to be statically unstable,

and therefore never to be observed, while the two cylindri-

cal shapes predicted were predicted to be stable. For laminates

with small side-lengths, the saddle shape was predicted to be

stable and the only shape predicted to exist, i.e., the same as

the prediction of classical lamination theory based on the lin-

ear strain-displacement relations. Data from a limited number of

laminates correlated reasonably well with the predictions of the

theory.

In later work Jun and Hong (1990) questioned the assumption

made by Hyer (1981a) of ignoringany in-planeresidual shearstrain

caused by cooling the laminate. As was shown in their extension

of the semi-closed solution form developed by Hyer, the shapepredictions and the dependence on laminate geometry are influ-

enced by the assumption andthe influence was quantified. Jun and

Hong (1992) were also the first to consider laminates with fiber

angles other than 0 or 90◦. A further extension of their semi-closed

form earlier work (1990) was developed to include the possibility

of twist curvature, while takingintoconsideration theneed to have

a tractable set of equations.

Using the Rayleigh–Ritz approach, Peeters et al. (1996) also

investigated laminates with fiber orientations other than 0 and

90◦. They erroneously assumed that for all unsymmetric lami-

nates the principal curvature direction was oriented at 45◦ relative

to the laminate edge and they treated the laminate as if it was

square in the principal curvature coordinate system. Experimental

results from only a single+ 30◦ angle-ply laminate were presentedto compare with the theoretical model. The comparison between

experiment and predictions was reasonable, but the authors went

on to explain how manufacturing problems, material property

uncertainties, and material inhomogeneity could have influenced

their experimental results.

To explore the predictive capabilities of the Rayleigh–Ritz

approach beyond shape predictions, Dano and Hyer (1996) inves-

tigated the snap-through event of a cross-ply laminate. Using a

force-control set-up, based on dead weights, the magnitude of

a concentrated force necessary to cause the laminate to snap

from one cylindrical configuration to the other was measured.

The motivation for this experiment was as follows: With unsym-

metric laminates having multiple shapes, there was the potential

for developing morphing structures, in this case, structures thatchanged shape on command. This required knowledge of the level

of actuation necessary to achieve the snap through. So, the origi-

nally developedenergy-based Rayleigh–Ritz analysis was modified

to include an applied concentrated force, the simplest form of an

actuation force. The deformations of the laminate as the force was

applied resulted in complex expressions for the work done by the

force, but the snapping force magnitude measured in the exper-

iments correlated well with the predicted level. Later, as will be

discussed, considerable work was done by a number of investiga-

tors in the area of morphing.

While the initial work based on developing semi-closed

form solutions was very helpful in explaining the fundamental

mechanics of the multiple-shape phenomenon, the application of

finite-element approaches opened new opportunities to study the


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problem, particularly for very general unsymmetric laminates. One

of the difficulties of using finite-element analysis was the existence

of multiple solutions at room temperature. Finite-element analysis

hadto be coaxed into converging to thevarious solutionsby know-

ing somethingabout eachof the multiple shapes the laminatecould

attain, using a small force or moment to ‘move’ the convergence

of the finite element solution toward a particular shape, and then

removing the small force or moment so only thermal effects were

causing the computed deformations.

Somefinite-element codes would provideinformationabout the

stabilityof various solutionscomputed. Schlechtet al. (1995) useda

finite element approach to compare with the earlier Rayleigh–Ritz

results of Hyer (1981a) for cross-ply laminates. The comparisons

between the results for the cross-ply laminates from the two dif-

ferent approaches were good. Laminates with 45◦ ply orientation

were also considered, but there was no comparison with results

from other approaches. Schlecht and Schulte (1999) considered

the effect of temperature-dependent properties and the finite ele-

ment approach to study the cooled shape and snap-through forces

and displacements for square cross-ply and [02/2]T laminates.The principal curvature directions of the latter laminates were also

computed as a function of , as wasthe response of circularunsym-

metric laminates.

Dano and Hyer (1998) extended the Rayleigh–Ritz approachto study much more general laminates by using more complex

approximate displacement functions. Shape predictions for more

general laminates using this extended Rayleigh–Ritz approach

were compared with predictions from finite element analyses and

experiments. The correlations among the two predictedshapes and

the experimentallymeasuredones were quite good. Going one step

further, Dano and Hyer (2002) used the extended Rayleigh–Ritz

approach to predict the forces required for snap through of general

unsymmetric laminates and conducted force-control experiments

to measure these forces. The measured and predictedsnap-through

forces were in good agreement.

Hufenbach and Gude (2002) used a genetic algorithm and a

Rayleigh–Ritz solution to find families of laminates with desired

curvature characteristics. Cross-ply and angle-ply laminates wereconsidered, though details of the approximate in-plane displace-

ment response assumed in the analysis were not given. Hufenbach

et al. (2002) repeated some of the work discussed in (2002) and

presented additional empirical results. A concept for producing

snap through on command was also outlined. Ren et al. (2003) pre-

sented an interesting alternative to the concept of multiple cooled

shapes by presenting predicted results for unsymmetric laminates

following the original approach by Hyer (1981b), but assuming

the laminate was cured on a cylindrical tool rather than in a flat

state before it was cooled. The initial cylindrical shape had a sig-

nificant effect on the role of laminate dimensions on the stability

characteristics.

A number of investigatorscontinued to study unsymmetric lam-

inates. The influence of laminate aspect ratio, defined as the ratioof laminate length to laminate width, on the stability character-

istics of cross-ply laminates, and therefore the existence or lack

thereof of cylindrical shapes, was investigated by Gigliotti et al.

(2004) using the original formulation of Hyer (1981a) and finite

element analysis. The conclusion was that long, narrow cross-ply

laminates did not exhibit multiple shapes, rather they cooled to

a unique shape with a dominate curvature in the long direction.

The lack of multiple shapes was associated with the loss of bifur-

cation behavior, i.e., the solution for laminate shape did not yield

two solutions, rather just one. It was also concluded that the orig-

inal formulation of Hyer (1981a) was incapable of predicting the

behavior of long, narrow laminates. Potter et al. (2007) measured

the load vs. displacement relations for a two-layer cross-ply lami-

nate and observed the deformation behavior. The load was applied

at the center of the plate and normal to the surface of the curved

laminate. The laminate exhibited complex local snap-through

behavior that was not predicted.

Tawfik et al. (2007) used finite element analysis to study square

cross-ply laminates. The shape predictions were compared with

results using the original theory of Hyer (1981a) and the finite ele-

ment analysis was extended to compute the force, applied at the

centerof thelaminateand normal to the curvedsurface,to produce

snap through. Also, calculations were made to determine the crit-

ical laminate aspect ratio and it was concluded that snap through

will not occur if the laminate is too narrow. Pirrera et al. (2010)

used more sophisticated approximate displacement functions and

the Rayleigh–Ritz approach, considered cross-ply laminates, and

provided detailed predictions of shapes and the forces to cause

snap through, where again, the force was applied in the center

of the laminate and normal to the curved surface. Mattioni et al.

(2008) considered unsymmetric laminates with piecewise varia-

tions of the fiber angle within a layer and concluded that multiple

shapes were still possible. Betts et al. (2010) used the approach

developed by Dano and Hyer (1998), relabeling the nomenclature,

and a three-camera system with markers on the laminate to mea-

sure laminate shape. This measurement arrangement was unique

and provided high-resolution data. The comparison between the

predicted and measured shapes was good.A potential advantage of using the multiple shapes of unsym-

metric laminates for morphingstructures was thatactuators would

only have to provide theforces to snap the laminate from oneshape

tothe other.Actuator energywouldnot berequiredto holdthe lam-

inate in a particular shape. Actuator energy would only be required

for transforming the shape. This could be a significant advantage,

assumingthe shapes developedby the unsymmetric laminatewere

desirable shapes, and environmental or other influences did not

impact the shape, or the transforming of the shape. Dano and Hyer

(2003) investigated use of shape memory alloy wires, stretched

between short struts, or posts, that were mounted normal to the

surface of an unsymmetric laminate. When electric current was

passed through the shape memory alloy wires, their temperature

would increase andthroughthe martensitic-austenitictransforma-tion, the wires would contract and produce a tensile force between

the struts. The tensile force acting through the length of the strut

produced a moment on the laminate that, if sufficient, would cause

the laminate to snap through. A similar shape memory alloy wire

and strut arrangement on the opposite side of the laminate would

reverse snap the laminate. The laminate, strut, and shape mem-

ory alloy wire arrangement was modeled using the Rayleigh–Ritz

approach, and the wire temperature to produced snapping was

predicted. Of course, strut length and the number and diameter

of shape memory alloy wires were design variables. Experiments

were developed, with care taken to electrically insulate all com-

ponents, and snap through measurements made. The correlation

between the predicted and measured snap-though temperature

was good.Using an alternative concept, Schultz and Hyer (2003) used

piezoelectric actuators, bonded to the surface of an unsymmetric

laminate, to produce snap through. The design variables were the

area of the actuator, the strain-voltage capabilities of the actuator,

the maximum voltage that could be applied to theactuator, and the

method of bonding the actuator to the laminate. Interestingly, for

thin laminates the stiffness of the actuator could alter the cooled

shape of the unsymmetric laminate, if not globally, then certainly

in the neighborhood of the actuator. A model was developed to

consider all these issues, the model being based on a multi-step

Rayleigh–Ritz approach, where the steps were (1) laminate cur-

ing, (2)bonding the actuator to the laminate, and (3) application of

voltage to the actuator. Reasonably good correlation between the

predictedand measuredvoltageto produce snappingwas achieved.


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Fig. 20. Laminated tube – axial independent loading.

Other investigators considered the same basic approachof using

piezoceramic actuators. Gude and Hufenbach (2006) used a finite

element approachto model theactuators andlaminate. Bowenetal.

(2007) considered using mechanical loads to augment the effect of

the actuators.Portela et al.(2008)concludedthat theactuator prop-

erties must be matched to the laminate.Failure to do this matching

results in either the multiple shape characteristics of the laminate

being eliminated, or the actuator not being powerful enough to

snap the laminate.

9. Tubes

The laminated circulartube (Fig.20) isoneofaveryfewcompos-

ite structural configurations for which an exact elasticity solution

is available. The analytical solution for axisymmetric, mechanical

loading is based upon the works of Lekhnitskii (1950a), Scherrer

(1967), Reissner (1970), Pagano (1971), and Reissner and Tsai

(1974). Thermal stresses were included by Hyer and Cooper (1986)

and several follow-on works by Hyer and co-workers Wilson and

Orgill (1986) included material and geometric nonlinearities. The

availability of an analytical solution permits direct and efficient

in-depth study of the problem parameters. For an N-layer compos-

ite tube under x-independent, axisymmetric, thermo-mechanical

loading, the displacements in thekth layer can be written:

u(k)( x, r ) = ε◦ x x (17)

v (k)( x, r ) = ◦ xr (18)

w(k)(r ) = A(k)1 r + A

(k)2 r − + (k)ε◦ xr +˝

(k) ◦r 2 + (k)r T (19)

In the above,ε◦ x and ◦ are the axial strain and angle of twist per

unit length, (k), ˝(k) and (k) are known functions of the layer

material properties, T is the uniform temperature change, A(k)1

and A(k)2 are unknown layer constants to be determined from inter-

facial displacement and stress continuity equations and external

boundary conditions.

There are two interesting facets for the response of laminated,

composites tubes. Axisymmetric tubes of a [±] configuration

exhibit coupling between axial and shear response. This is due

to the fact that the individual layers are not at the same radius

from the axis. In addition, the response of tubes is a function of

the tube aspect ratio RI /h where RI is the inner radius and h is

the laminate thickness. Fig. 21 demonstrates the effect of aspect

ratio for the radial coefficient of thermal expansion (CTE) of a car-

bon/epoxy. Results are presented for unidirectional and angle-ply

laminates. The unidirectional laminae exhibit negative CTE only

over a very small range of fiber orientation near 90◦. In contrast,

the angle-ply laminates exhibit negative CTE over a large range of fiber orientation; the range of negative values and magnitude are

very dependent on the tube aspect ratio. For angle-ply tubes, the

inner radius of the tube actually shrinks when subjected to heat-

ing; this effect is more pronounced as the aspect ratio decreases

(i.e., the tube is thicker).

The solution for layered tubes can be extended to the case of

a layered, solid cylinder by using the fact that singular displace-

ments arenot permissible atr = 0.Thisconditionis used toeliminate

Fig. 21. Radial CTE fortubes.


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Fig. 22. Plate in cylindrical bending.

one of the unknowns in the solution. Hashin and Rosen used the

solid cylinder approach in their (1964) paper on the concentric

cylinderassemblage model for micromechanics of composites with

isotropic fiber and matrix. Avery and Herakovich (1986) used the

solid cylinder approach to study the effect of fiber anisotropy on

thermal stresses, andKnott andHerakovich (1991b)used itto study

the effects of various fiber morphologies on effective composite

properties.

10. Plates

The first publication concerned with anisotropic plates appears

to be that of Huber (in Polish) in (1921). The English translation of

the title to his work is “The theory of anisotropic rectangular plates,

with special consideration of reinforced concrete slabs, iron, etc.”

Another early publication on anisotropic plates is that by Hayashi

(1941). See theAnniversary Volumeof collectedPapers of Tsuyoshi

Hayashi (1973) for this publication and other related works by

Hayashi, including his 1947 doctoral thesis that contains his work

dating from pre-World War II. Lekhnitskii’s Anisotropic Plates first

appeared in 1944 with a secondedition in (1957). Ambartsumyan’s

Theory of Anisotropic Plates appeared in (1967) and the transla-

tion from Russian in 1970. Whitney’s book on Structural Analysisof Laminated Anisotropic Plates was published in (1987). Reddy’s

book entitled Mechanics of laminated composite plates: theory and

analysis was published in (1997) with a second edition in (2004).

Most solutions for laminated plate problems involve some level

of approximation. Finite element and other approximate solutions

are common. However, there is one problem for which an analytic,

elasticity solution is available. It is the cylindrical bending problem

(Fig. 22) presented in a series of papers by Pagano (1969), Pagano

(1970), and Pagano and Wang (1971). The plate is composed of N

orthotropic or angle-ply layers, simply supported along its edges

x= 0 and x= L, and subjected to transverse loading q( x) over the

entire the top surface, z =H . For cylindrical bending, all displace-

ments, stresses and strains are independent of axial coordinate y.

Thus the solution to the problem corresponds to the solution fora generic x-z plane; it shows that, in contrast to classical plate

theory, the results are a function of the plate length to thickness

aspect ratio. The classical plate theory predictions converge to the

elasticity solution for large aspect ratios.

11. Failure

11.1. Unidirectional lamina

The axial tensile strength of a unidirectional lamina is typically

controlled by the fiber ultimate strain (or stress). Kelly and Davies

(1965) provided a method for predicting axial tensile strength as a

function of fiber and matrix strengths, and the constituent volume

fractions. Fora compositethat fails when thefibersattains thefiber

Fig. 23. Off-axis tensile coupon failure comparisons.

ultimate stress ult f

, the composite failure stress ult c can be written

in terms of the stress in the matrix at this strain level m and the

fiber and matrix volume fractions as:

ult c = ult f V f +

m(1 − V f ) (20)

They also presented a relationship specifying the critical fiber

volume fraction required for the composite strength to be greater

than the bulk matrix strength.

11.2. Quadratic failure criteria

Several quadratic failure criteria have been presented to predict

failure of a unidirectional composite in a state of combined load-

ing. They represent attempts to provide better correlation between

theory and experiment by inclusion of all components of stress in

an equation representing the failure criterion. The quadratic crite-ria are based upon the mathematical premise that a second order

curve has more parameters with which to fit experimental data

thandoes a straightline.These criteriagenerally arenotbasedupon

the physics of thefailure mechanisms.While they may provide bet-

ter correlation between theory and experiment in some situations

they are limited in that the sign of the normal stress components

must be known a priori if the positive and negative strengths are

different (which is often the case with composites).

Failure criteria based upon polynomials of strength tensors

represent an attempt to mathematically overcome one of the

shortcomings of the quadratic criteria, namely, to account for the

differences in tensile and compressive strengths. They have the

additional advantage of being scalar equations of tensor quantities.

Hence they are invariant and transformations between coordinate

systems can be affected using the tensor transformation laws.

Tensor polynomial failure criteria were first considered by

Gol’denblat and Kopnov (1965) and Ashkenazi(1965). The criterion

most oftenemployedovertheyearsis thesecondordertensor poly-

nomial criterion proposed by Tsai and Wu (1971). It is a complete

quadratic, tensor polynomial with linear terms included. The crite-

rion assumes that there exists a scalar function f ( i) that describesthe failure surface in stress space. For failure to occur, f ( i) mustsatisfy the condition:

f ( i) = F i i + F ij i j ≥ 1 (21)

In (21), F i and F ij are tensor quantities of strength parameters

that can, at least in theory, be determined from a series of tests on

the composite.


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Fig. 24. Failed angle-ply laminates.

Fig. 23 shows a comparison of strength predictions using the

tensor polynomial criterionand three maximum stresscriteria with

experimental results for two resin matrix, unidirectional, off-axis

tensile tests. The tensor polynomial gives the best predictions over

the full range of fiber orientations.

Thefiber orientation and stackingsequence of a laminatedcom-

posite has a strong influence on the strength and mode of failure.

This is particularly true if free edges are present. Fig. 24 shows the

significant effects of stacking sequence for tensile tests on angle-

ply laminates with different stacking sequences. When the + and− layers are stacked in an alternating fashion, the mode of fail-

ure is primarily due to fiber breakage. In contrast, when the +

and − layers are grouped together, edge effects and interlaminarshear stresses dominate; the specimen fails due to delamination

at a much lower axial stress. Alternating the layers of the [±30]S specimen resulted in a 48% increase in strength.

12. Damagemechanics

Failure is often an ill-defined term when referring to com-

posite materials and composite structures. These heterogeneous,

laminated materials typically exhibit many local failures prior to

rupture into two or more distinct pieces. Local damage in the form

of matrix cracks, fiber breakage or fiber buckling, and fiber/matrix

debonding may initiate and grow (accumulate) throughout the

structure prior to rupture. The local failures are referred to as“damage” and the development of additional local failures with

increasing load or time is called “damage evolution or damage

accumulation”. The term “damage mechanics” has been coined to

describe the study of the initiation and evolution of damage up to

and including rupture.

The earliest published works on damage mechanics appear

to be those by Kachanov (1958) and Robotnov (1968) on the

application of a continuous damage variable for creep failure of

metals. The concept was generalized, within the framework of

irreversible thermodynamics, for isotropic materials under mul-

tiaxial loads by Lemaitre and Chaboche (1974), Hult (1974), Leckie

(1978) and Murakami (1983). Books on damage mechanics include

those by Lemaitre andChaboche (1985), Bazant andCedolin (1991),

Krajcinovic (1996) and Voyiadjis and Kattan (1999).

A damage theory for composites that has received considerable

attention is the mesoscale composite damage theory proposed by

Ladeveze (1983, 1986). This theory has been shown to be robust

for predicting the damaged response of composite materials and

structures under a wide variety of conditions. It is based upon

the method of local state expressed in terms of state variables

and the associated thermodynamic forces. The theory is called themesoscale composite damage theory because it is based upon the

assumption that the damage is uniform through the thickness of

individual layers of the composite. Mesoscale is a term indicating

that the scale of the analysis is between micromechanics (i.e., the

level of the fiber and the matrix) and laminate analysis. The theory

is based upon the mean value of the stress in each layer and allows

the damage state to vary from layer to layer in a laminate. When

delamination is of interest, damage between layers is introduced

through consideration of damage to the idealized interfacial layer

(Ladeveze et al., 1990; Allix and Ladeveze, 1992).

Additional developments of the model were concerned withthe

computation of the intensities of the different damage mechanisms

up to ultimate fracture (Ladeveze, 1992) and bridging the model to

the micromechanics scale, Ladeveze and Lubineau (2002). Appli-

cation of the model to the development of damage in a stiffened

composite panel was given by Flesher and Herakovich (2006).

The foundation of the model is the expression for the strainenergy density of a damaged layer written in terms of the

mean, effective layer stresses.

E D = 1

2

11

2+

E ◦1

(1−d1) +

112−

E ◦1

−2◦

12E ◦

1 11 22 +

222+

E ◦2

(1−d2) +

222−

E ◦2

+ 212

G◦12(1 − d6)

(22)

The di in (22) are damage variables and the ± notation indi-

cates a non-zerocontribution dependingupon the signof the stress.

The associated thermodynamic forces are then:

Y 1 =

211+

2E ◦1(1 − d1)2

Y 2 = 222+

2E ◦21 − d22

Y 6 = 212

2G◦12(1 − d6)2

(23)

The model has demonstrated good comparison between theory

and experiment for a variety of materials and loading conditions.

13. Experimentalmethods

Testing composite materials has proven to be much more diffi-

cult than testing homogeneous, isotropic materials. The difficulties

are associated primarily with load introduction, development of aspecified,uniformstateof stressin a desired regionof thetest spec-

imen, obtaining stress–strain results into the nonlinear range, and

controlling the type and location of failure. In addition, the funda-

mental testsof tension,compression andshear mustbe extended to

include consideration of material anisotropy. Unidirectional, con-

tinuous fibers in a polymeric matrix present the most difficult

case.

An early discussion of test methods for fibrous composites was

given by Waddoups (1968). His work was largely based on two Air

Force Contractor Reports by Rogers (1965, 1966). The experimental

work reported on had been conducted at three organizations, Gen-

eral Dynamics Fort Worth Division, Illinois Institute of Technology

Research Institute (IITRI), and Texaco Experiment Inc. Waddoups

discussed the advantages and disadvantages of methods for


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Fig. 25. IITRI style tensile coupon.

tension, compression and shear testing of unidirectional and lam-

inated boron/epoxy. Three tensile specimens were considered:

dogbone, straight sided coupon and sandwich beam. Two com-

pression specimens were considered: short column coupon and

sandwich beam. For shear tests, a sandwich cross beam configu-

ration was used. Linear and nonlinear response, failure, and cyclic

loading and unloading were considered.

TheAmerican Society forTestingand Materialsplayed a keyrole

in the development andacceptance of suitable test methods. ASTM

Subcommittee on Composites, D-30, formedin 1964, ledthis work.

Between 1966 and 2005 ASTM published sixty-six Special techni-

cal Publications (STPs) related to testing and design of composite

materials. More recently, e.g., Adams (2011) has reviewed testsmethods for composites in a series of articles in high Performance

Composites.

13.1. Tension

As a result of the above noted work and many later investiga-

tions, the standard specimen for tensile testing of unidirectional

and laminated fibrous composites is the IITRI tensile coupon. An

example is shown in Fig. 25.

The specimen is a straight-side coupon with fiber-glass tabs

bonded at the ends for load introduction.

13.2. Compression

Compression testing proved to be more difficult than tensile

testing because of issues associatedwith specimen alignment, fiber

kinking, and local splitting and crushing of the specimen at the

ends where load is introduced. The standard compression speci-

men is a short coupon under combined axial and shear loading.

While a short gage length is required in order to negate the pos-

sibility of fiber kinking, the specimen must be sufficiently long to

eliminate load introduction end effects. The IITRI Compression Fix-

ture was introduced in 1977. A sketch from Adams (2011) is shown

in Fig. 26. Numerous versions of this fixture have been introduced

over the years. In (1988), Lin and Pindera presented an IITRI-like

fixture that can be used for cyclic, tension-compression loading of

flat specimens.

13.3. Shear

A wide variety of specimens have been employed to obtain

the shear stress-strain response of fibrous composites. Early work

included sandwich cross beam, picture frame and rail shear. Three

specimens that are now accepted as providing good results for

shear properties are: the [±45]S tensile coupon, the Iosipescu V-

notched specimen and the off-axis tensile specimen.

The [±45]S tensile coupon was proposed by Rosen (1972) as a

simple test for determining the in-plane shear modulusG12 of uni-

directional lamina. He showed that for linear elastic response, the

shear modulus canbe expressessimply intermsof theaverageaxial

Fig. 26. IITRI compression fixture.

stress ̄ x , and the axial and transverse strains, ε x and ε y, respec-

tively, as:

G12 =̄ x

2(ε x − ε y) (24)

The Iosipescu specimen (1967) was originally designed as a

round specimen with a V-notch groove for shear tests on metals. It

was proposed for flat composites by Herakovich and Bergner et al.

(1977). Advantages of the Iosipescu specimen are that it is not lim-

ited to linear elastic response (as is the [±45]S tensile coupon) andcan be used for out-of-plane response (i.e., G23) as well as inplane

response. The specimen can provide shear response well into the

nonlinear range for some laminate configurations. Fig. 27 shows

a typical test fixture arrangement used for Iosipescu tests of flat

laminates.

Fig. 27. Iosipescu shear specimen and fixture.


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13.4. Off-axis tensile test

Tensile tests on unidirectional off-axis lamina have been used

to measure the shear modulus G12. The results are dependent on

the fiber orientation angle and the specimen length/width aspect

ratio. It has been shown that for polymeric matric composites, an

angle of 45◦ and aspect ratio of 10–20 are recommended (Pindera

and Herakovich, 1986).

14. Nanocomposites

The term nanotechnology first appeared in (1974) in a paper

presented by Taniguchi in Tokyo, Japan. Nano denotes a scale of

10−9 and nanotechnology refers to the properties of atoms and

molecules measuring roughly 0.1–1000 nanometers. The first arti-

cle on materials at the nano scale was by Kroto et al. (1985) who

published the results of their work on the development of a sta-

ble, cage-like molecule of 60 carbon atoms arranged in the shape

of a soccer-ball or geodesic dome. They called the molecules buck-

erminsterfullerenes or buckyballsafter Richard Buckminster Fuller

of the geodesic dome fame. Richard Smalleyat Rice University was

actually the one who discovered the arrangement of atoms that

provided the correct stable shape (see Smalley, 1991). The devel-opment of buckyballs was followed by the development of carbon

nanotubesby Iijima (1991). Carbon nanotubes(CNTs) are cage-like,

cylindrical fullerenes, roughly 1.5 nanometers in diameter and a

few hundred nanometers to microns long. They exhibit exceptional

specific stiffness and strength and are highly elastic. They can be

formed coaxially, one nanotube inside another, to form multiwall

CNTs.

Because of their high specific modulus and strength combined

with their exceptional elasticity, carbon nanotubes are of great

interest to the composites community. The goal is to disperse car-

bon nanotubes in the matrix of a fibrous composite with the hope

of improving the mechanical properties of the composite. The work

to date has been largely associated with fabrication issues. Much

of this work is reviewed in four articles published by Chou andco-authors with the most recent by Chou et al. (2010). Sun and co-

workers reported improved compressive properties of glass and

carbon fiber composites using nanoparticle infused matrix materi-

als. These works are summarized by Uddin and Sun (2009).

All things considered, it may be said that, as of this writing,

additional research is required before major advancements in the

mechanical properties of fibrous composites are realized through

the use of nanoparticle enhanced materials.

15. University andgovernment programs

The first significant government research activity on fibrous

composites in the United States appears to be that at the Air Force

MaterialsLaboratory at Wright-Patterson Air Force Basein theearly1960s. The history of the activity at AFML was detailed by Tsai

in (2005). Soon after the initiation of work at AFML, NASA Lang-

ley Research Center joined the activity in composites promoting

their use in commercial aircraft. Virtually all government labora-

tories now have very active programs in mechanics of composites.

And, of course, the National Science Foundation has played a very

strong role in the funding of research in mechanics of compos-

ites.

Educational and research activities in mechanics of compos-

ites first appeared at universities in the United States in the late

1960s. The first course in composites was taught by Jack Vinson

at the University of Delaware in 1969. That same year, a Depart-

ment of Defense research grant was funded at Virginia Tech under

the leadership of Dan Frederick. Formal programs were established

at both of these universities in 1974. At Delaware, a Center for

Composite Materials was formed with Jack Vinson, Tsu-Wei Chou,

Roy McCullough and Byron Pipes providing the leadership. The

Delaware Center has maintained a strong relationship with indus-

try. At Virginia Tech, the NASA-Virginia Tech Composites Program

was established under the leadership of the author and John Davis

at NASA Langley Research Center. Under this cooperative pro-

gram, students completed a research residency of six to twelve

months at NASA’s Langley working with NASA engineers. The first

course in mechanics of composites at Virginia Tech was taught by

the author in 1974. Courses in mechanics of composites are now

taught in many universities in the United States and around the

world.

16. Closure

Duringthe process of writing this invited article it became clear

that there was too much material to be discussed, at the depth

desired by the author, and in the time frame and space available,

if all aspects of composites mechanics were to be covered. Thus,

the time came when it was necessary to say that’s enough for now.

Hopefully, there will be time in the future to add additional top-

ics. Apologies to those who have made significant contributions inareas not covered. You are many!

17. Books onmechanics of composites

Following is a partial listing of books on mechanics of compos-

ites.

• Aboudi, J., 1991. Mechanics of Composite Materials: A Unified

Micromechanical Approach. Elsevier.• Achenbach, J.D., 1975. A Theory of Elasticity with Microstruc-

ture for Directionally Reinforced Composites, CISM Courses and

Lectures No. 167. Springer-Verlag, New York.• Ambartsumyan, S.A., 1970. Theory of Anisotropic Plates. Tech-

nomic Publishing Co. Inc. Stamford, CT.• Ashton, J.E., Halpin, J.C., Petit, P.H., 1969. Primer on Composite

Materials: Analysis. Technomic.• Ashton, J.E., Whitney, J.M., 1970. Theory of Laminated Plates.

Technomic.• Agarwal, B.D., Broutman, L.J., 1990. Analysis and Performance of

Fiber Composites, second edition. John Wiley & Sons.• Broutman, L.J., Krock, R.H. (Eds.), 1974. Composite Materials, vol.

1–8. Academic Press.• Buryachenko, V., 2007. Micromechanics of Heterogeneous Mate-

rials. Springer.• Calcote, L.R., 1969. The Analysis of Laminated Composite Struc-

tures. Van Nostrand Reinhold.• Chawla, K.K., 1987. Composite Materials – Science and Engineer-

ing. Springer-Verlag.• Christensen, R.M., 1979. Mechanics of Composite Materials. J.

Wiley & Sons.• Daniel, I.M., Ishai, O., 1994. Engineering Mechanics of Composite

Materials. Oxford University Press.• Gibson, R.F., 1994. Principles of Composite Material Mechanics.

McGraw-Hill, Inc.• Halpin, J.C., 1992. Revised Primer on Composite Materials: Anal-

ysis, second edition. Technomic.• Hashin, Z., 1972. Theory of Fiber Reinforced Materials. NASA CR-

1974.• Hearmon, R.F.S., 1961. An Introduction of Applied Anisotropic

Elasticity. Oxford University Press, Oxford.• Herakovich, C.T., 1998. Mechanics of Fibrous Composites. John

Wiley & Sons, Inc., New York.


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• Holaway, L., 1993. Polymer Composites for Civil and Structural

Engineering. Blackie Academic & Professional, Glasgow.• Hyer, M.W., 1998. Stress Analysis of Fiber-Reinforced Composite

Materials. McGraw-Hill, Boston.• Jones, R.M., 1975. Mechanics of Composite Materials. McGraw

Hill.• Kelly, A., 1994. Concise Encyclopedia of Composite Materials,

revised edition. Elsevier.• Kelly, A., Rabotnov, Y.N. (Eds.), 1986. Handbook of Composite,

vols. 1–4. North-Holland.• Lee, S.J. (Ed.), 1991. International Encyclopedia of Composites,

vols. 1–5. VCH Publishers, Inc., New York.• Lekhnitskii, S.G., 1963. Theory of Elasticityof an Anisotropic Body.

Holden-Day.• Lubin, G., (Ed.), 1982. Handbook of Composites. Van Nostrand

Reinhold Co.• Obraztsov, I.F., Vasil’ev, V.V., (Eds.), 1982. Mechanics

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