Composite Materials

Composite Material
• Composites or Advanced composite materials – a material consisting of tiny diameter,
high strength, high modulus fibers embedded in an essentially homogenous matrix
• The orientation of the fibers can be different in different layers
• This results in a material that is anisotropic ie. Having mechanical and physical
properties dependent on fiber direction
• This also results in heterogeneous property ie. It consists of dissimilar constituents that
are separately identifiable.
Composite Material
• Composites differ from metals
• Properties are not uniform in all directions
• Strength and stiffness can be tailored to meet loads
• Possess a greater variety of mechanical properties
• Poor through the thickness strength, short transverse
• Composites are usually laid up in two dimensional form, while metal may be used in billet,
forged, bar forms
• Greater sensitivity to environmental heat and moisture
• Propagation of damage will be through delamination rather than through thickness cracks

Condition Composite Behavior relative to metals

Load Strain relationship More linear strain to failure
Static Greater sensitivity
Notch-sensitivity
Fatigue Less sensitivity
Transverse Properties Weaker
Mechanical Properties variability Higher
Sensitivity to Hydrothermal environment Greater
In-plane delamination instead of through
Damage growth mechanics
thickness cracks
composites
• Advantages of composites over metals
• Light weight
• Resistance to corrosion
• High resistance to fatigue damage
• Reduced machining
• Tapered sections and compound contours easily accomplished
• Can orient fibers in direction of strength or stiffness
• Reduced number of assemblies and hence reduced number of fasteners when
cocured or cobonded
• Absorb radar microwaves
• Thermal expansion close to zero reduces thermal problems in outer space
applications
• Disadvantages of composites over metals
• Material is expansive
• Lack of established design allowables
• Corrosion problems from improper coupling with metals, especially when carbon
or graphite is used
• Degradation of structural properties under temperature extremes and wet
conditions
• Poor energy absorption and impact damage
• May require lightning strike protection
• Expensive and complicated inspection methods
• Reliable detection of substandard bonds is difficult
• Defects can be known to exist but precise location cannot be determined.
 Composites

Organic Matrix Metal Matrix

• High strength fibers such as boron, • Boron/Aluminum

graphite which provide basic strength • Graphite/Aluminum
• Matrix such as epoxy, polyamide or • Titanium and nickel are under
any other thermo plastic material research
stabilizing these thin fibers in • High temperature application
compression • Metal matrix provides better
• Matrix also acts to redistribute load resistance against buckling for
in shear between fibers in case of filaments in compression and hence
individual fiber failure or laminate provide much higher compression
transition capability especially at higher
temperatures
composites
• Most commonly used fibers in aerospace industry are
• Carbon
• Graphite
• Kevlar
• Boron
• Available as unidirectional tapes or woven cloth
• Collimated
• Preimpregnated
• Matrix
• Resins- most commonly used in aerospace applications
• Metals
• Ceramic materials
– The matrix is cured by means of time, pressure and temperature
– The most important element in determining the material behaviour
– The selected matrix formulation determines the cure cycle and affects the properties such as

• Creep
• Compressive strength
• Shear strength
• Thermal resistance
• Moisture resistance
• Ultraviolet resistance

All these above properties affect the long term stability of the composite structure
Composites

• Basically there are two types of matrix materials

• Thermoplastic and thermosetting
• Advantages of thermoplastic over thermoset are

• High service temperature

• Shorter fabrication cycle
• No refrigeration required for storage
• Increased toughness
• Low moisture sensitivity
• No need for chemical cure
• The matrix is susceptible for moisture absorption
• This degrades its mechanical properties
• For airframe structures this loss of mechanical properties should be taken
care in design stage
Composite Material

Relative Advantage
Mechanical
Properties Thermoplastics Thermosets Metal

Corrosion Resistance xxx xxx x

Creep xxx xxx x
Damage Resistance xx x xxx
Design flexibility xxx xxx x
Fabrication time xxx xx x
Final part cost xxx xx x
Finished part cost xxx xx x
Moisture resistant xx x Xxx
Physical properties xxx xxx Xxx
Processing cost xxx xx X
Raw material cost x xx xxx
Reusable scrap xx - Xxx
Shelf life xxx x xxx
Solvent resistance xxx xx X
Specific strength xxx xxx X
Strength xxx xxx X
Weight Saving xxx xx NA

Best Good fair

 composites
Matrix Properties
• most widely used
• Max. temp 930 C
Epoxy • best structural properties
• easy to process
• toughened versions available now
• Max. temp 1800 C
Bismaleide • easy to process
• toughened versions available now
• variety of matrix types
• Max. temp 3200 C
Polyimide
•difficult to process
• expensive
• relatively poor structural characteristics and hence limited usage in structural
polyester components
• easy to process
• same as polyester
• more difficult to process
Phenolic
• can be used at higher temperature than polyesters and expoxies
•Low smoke generation
• greater improved toughness
Thermoplastics •Unique capabilities
• have processing difficulties
 composites

Property Desired properties

• Increases fiber strain
Impact • Good Matrix shear strength
• Moderate interlaminar fracture toughness
• Moderate interlaminar fracture toughness
Delamination • Moderate matrix fracture toughness
• Good fiber/matrix interfacial adhesion
• Good fiber/matrix interfacial adhesion
• high compressive strength in 00 direction
Compression • high post impact compressive strength
• high matrix modulus
• good time-dependent properties: fatigue, creep etc,
• Low viscosity prepregging matrices
• Melt processible matrices
Processing
• Solvent-resistant matrices
• Thermo-oxidative stability
Composite Materials

STIFFNESS ANALYSIS
 Composite Materials
Definition: Composite material is defined
as a material having atleast two different
phases.

Load carrying phase – Fibers

Eg. Carbon, Glass, Kevlar, Boron etc.

Load distribution phase – Matrix

Eg. Phenolic, Epoxy etc
 Composite Materials
• A necessary prerequisite for the design and safe
operation of composite structures is an accurate
knowledge or their strength and stiffness
properties
• An assessment of these properties solely by
testing of critical components or of prototype
structures is feasible but is usually encumbered
by punitive cost and time requirements.
 Composite Materials
• Therefore, analytical methods must be employed with
the aim of

– Facilitating trade-off studies in the early design

phases
– Providing insight into the overall structural response
– Identifying critical parts of the composite structure.

• The ability to compute the responses of composite

materials and to estimate strength levels provides a
powerful tool for the designer who is faced with the need
to optimize the materials and the structural geometry
simultaneously.
 Composite Materials
• Composite materials used in structural
applications are most effective in the form of
plates or shells.
• In these structures, principal stresses develop in
the plane of the laminates while out-of-plane
stresses are secondary except near holes,
cutouts or free edges.
• The classical lamination theory is normally used
to analyze stress fields that are free of stress
concentrations.
 Composite Materials
• Solution methods based on this theory can provide
only in-plane stresses in each of the layers.
• In the cases of structures with complex shapes or
under complex loadings, detailed local stress
analyses are often preceded by a finite element
analysis based on a physical model of the structure
• The model consists of composite elements
assumed to be globally homogeneous and
anisotropic and provides reasonably accurate force
and displacement data at the element junctions .
 Composite Materials
• With these data and the assumptions of laminate
theory, a detailed analysis of the in-plane stresses
and strains in each of the layers of the element can
then be performed.
• Analysis based on laminate theory can be of two
types
• Stiffness analyses aiming to calculate the laminate
stiffness from known elasticity data of the
constituent layers
• Strength analyses in which the strength of the
laminate is calculated from known strength
properties of the layers.
Composite Materials

SIGN CONVENTION
AND NOTATION
 Composite Materials
Y 1
2

X –Y – Laminate directions
1 – 2 – Lamina directions
E1 – Modulus of elasticity in fiber direction, ν12 – Major Poisson’s ratio
E2 – Modulus of elasticity in transverse direction,
G12 – In-plane Shear Modulus
 Composite Materials
Formulation of stiffness Matrix
Stress = Stiffness coefficient * Strain
Consider, δ = (PL/AE)
Therefore, P = (AE/L)(δ) δ

Where (AE/L) – Stiffness Coefficient

• Stiffness coefficient in a 2D state of stress
is defined as Cij – stress introduced at the
location ‘i’ caused by a unit strain at
location ‘j’.
 Composite Materials
In the matrix form the
relationship can be
written as
C11
{σ} = [C] {ε} where C21 C22
{σ} = {σx, σy,, σz, ςxy, ςyz, ςzx} C31 C32 C33
{ε} = {ε x, ε y,, ε z, γxy, γyz, C41 C42 C43 C44
γzx} C51 C52 C53 C54 C55
[C] = is as shown, Upper C61 C62 C63 C64 C65 C66
half is symmetric
For 2D state of stress,
 Composite Materials
{σ} = {σx, σy,, ςxy}, {ε} =
{ε x, ε y,, γxy,}
In the abbreviated form, {σ} C11 C12 0
= [C] {ε}
C11 = Ex/(1-νxyνyx) C21 C 22 0
C22 = Ey/(1-νxyνyx)
C12 = νyxC11
0 0 C33
C21 = νxyC22
C33 = Gxy
For reasons of sym. C12 =
C21
 Composite Materials
• The laminate theory is based on the premise that
the mechanical response of laminate can be
determined by the superposition of the
contributions from its constituent layers.
• The stiffness coefficients of a unidirectional 00
laminate, consequently, are the summation of the
stiffness coefficients in 00 for all layers.
• In the more general case of multidirectional
laminates, the stiffness coefficients of the non 0 0
layers, prior to their summation, must be
transformed into components that correspond to
the direction of the laminate principal axes.
 Composite Materials
In mathematical terms, the known coefficients
Cij of a layer with principal axes different from
the laminate axes is given as

 x '  C11 C12 0   x ' 

     
 y '  C21 C22 0   y ' 
   0 C33   xy '
 '
 xy   0
 Composite Materials
Must be transformed into coefficients Cij’ which
satisfy the equation

 x  C11 ' C12 ' C13 '   x 

     
 y   C12 ' C22 ' C23 '   y 
  C ' C ' C '  

 xy   13 23 33   xy 
 Composite Materials
TRANSFORMATION OF CO-ORDINATES AND
DISPLACEMENTS
 x'  cos  sin    x 
    
 y '  sin  cos    y 

U '  cos  sin   U 

    
V '   sin  cos   V 
Composite Materials
 Composite Materials
TRANSFORMATION OF STRAINS

Strain Transformation matrix is given by

  x   cos 2  sin 2   cos sin     x ' 

    

 y   sin 2
 cos 2  cos sin     y ' 
  2 cos sin   2 cos sin  cos 2   sin 2    xy '
 xy  

This is abbreviated as {ε} = [Tε] {ε’}

 Composite Materials
TRANSFORMATION OF STIFFNESS CO-EFFICIENTS
Strain Energy stored in the typical layer of a laminate is

U  t / 2  ( x x   y y   xy xy )dA
Is an invariant
A property and it is same in either co-
ordinate systems. Hence,
[σ]T {ε} = [σ’]T {ε’} but, [σ] = [C’] {ε} and [σ’] = [C] {ε’}
Therefore,
{ε}T [C’] {ε} = {ε’}T [C] {ε’}
Put {ε’} = [Tε] {ε}
{ε}T [C’] {ε} = {ε}T [Tε]T [C] [Tε] {ε} = {ε}T [C’] {ε}
 Composite Materials
TRANSFORMATION OF STIFFNESS CO-EFFICIENTS
Its elements are

 C11 '   m 4 2m 2 n 2 n4 4m 2 n 2 
C '   2 2 
 12  m n (m 4  n 4 ) m2n2  4m 2 n 2  C11 
C13 '   m 3n  
 mn(m 2  n 2 )  mn 3  2mn(m 2  n 2 )  C12 
  4  
C22 '  n 2m 2 n 2 m4 4m 2 n 2  C22 
C23 '  mn 3 mn(m 2  n 2 )  m 3n 2mn(m 2  n 2 )  C33 
   
C33 ' m 2 n 2  2m 2 n 2 m2n2 (m 2  n 2 ) 2 

TRANSFORMATION OF STRESSES
Stresses are transformed from global to local axes
through similar transformation matrices.
 Composite Materials
Types of laminates: Loading:
• Unidirectional(0/0/0) •In-Plane

• Cross ply (0/90/0/90/0/90) •Out of Plane

• Angle ply (45/-45)

State of Stress in Laminates:

• Membrane state of stress
• Bending state of stress
• Coupling between membrane
and bending
 Composite Materials
Membrane state of stress:
• Laminates with a balanced and symmetrical
stacking order will develop under in-plane loading,
a membrane state of stress
• With the constraint of symmetry, the individual
layers of such laminates may be of dissimilar
materials and may have different thickness
• The determination of the stress distribution is
facilitated by the replacement of laminate stress by
stress resultants N = ∫σi dh. This is mathematically
represented as
 Composite Materials
 N x   x h   A11 A12 0   x 
       
 N y    y h    A12 A22 0   y 
 N xy   xy h   0 0 A33   xy 
   
Where h is the thickness of the laminate.
•The coefficients Aij are the sums of the proportionate
contributions from the individual layers to the
stiffness of the laminate
n
Aij   Cij ' (hk  hk 1 )
(k )

k 1
 Composite Materials
Membrane state of stress:
• The term hk denotes the distance from the plane of
symmetry to the outside face of layer k, so that (hk – hk-1)
denotes the thickness of the layer.
• In the simple case of a unidirectional laminate with θ=0 it
follows that
n
Ex
A11   C 11 ' (hk  hk 1 )  h
k 1 1  xy yx
•Generally the stress resultants Nx, Ny and Nxy are either
given as external loads or can be calculated as internal
loads by analytical means
 Composite Materials
• The actual problem is the determination of the stress and
strain distributions in the individual layers as functions of
the laminate strains.
• The laminate strains are readily obtained by the inversion
of the stiffness matrix.
• From the laminate strains the strain components in a
typical layer k are calculated by means of the
transformation

x' x 
   
  y '    T   y 
 '  
 xy   xy 
 Composite Materials
• The corresponding stresses follow from the stress
transformation equations as the product of stiffness matrix
and the strain components of layer

 x '  C11 C12 0   x ' 

     ' 
 y '  C21 C22 0   y 
   0   
C33   xy '
 '
 xy   0
• After establishing the state of stress and strain in all of the
layers, safety margins can be assessed by correlation to
appropriate failure criteria
 Composite Materials
POPULATION OF STIFFNESS MATRICES
• The population of a stiffness matrix depends on the stacking
order of the laminate.
• Balanced and symmetrical stacking orders lead to laminates
with orthotropic properties in which, under normal loads, no
coupling of normal and bending stresses occurs.
• In below mentioned stacking orders the stresses due to
normal loads and bending moments are coupled, for
example in
 Balanced and non-symmetrical laminates
 Non-balanced and non-symmetrical laminate
• The following examples display typical laminate stacking
orders, their displacement patterns due to normal loads and
bending moments, and the population of their stiffness
matrices
 Composite Materials
POPULATION OF
STIFFNESS
MATRICES
• In classically orthotropic
laminates both the normal
and shear stresses and the
bending and torsional
stresses are decoupled.
 Composite Materials
POPULATION OF
STIFFNESS
MATRICES

• In pseudo-orthotropic
laminates the normal
and shear stresses are
decoupled while the
bending and torsional
stresses are coupled
 Composite Materials
POPULATION OF
STIFFNESS MATRICES
• Laminates with symmetrical but
unbalanced stacking orders are
classically anisotropic; the stresses
due to normal loads and bending
moments are decoupled, but the
normal and shear stresses as well
as the bending and torsional
stresses are coupled
 Composite Materials
GENERAL STATE OF STRESS
• In laminates with balanced and symmetrical stacking orders
and subject to in-plane loading, the resulting membrane
stresses depend only on the in-plane strains

 N x   x h   A11 A12 0   x 
       
 N y    y h    A12 A22 0   y 
 N xy   xy h  0 0 A33   xy 
   
• Laminates under bending loads experience curvatures and display a
very different stress distribution
• The relationship between applied bending moments and the
conjugated curvatures of the neutral plane for orthotropic laminates is
given by the equation
 Composite Materials
 M x   D11 D12 0  x 
     
 M y    D12 D22 0  y 
 M xy   0 0 D33   xy 
 

• The coefficients of the stiffness matrix have the form

n
Dij  (1 / 3) Ci j ' (hk  h
3 3
k 1 )
k 1
• They are readily recognized as the contributions of the
constituent layers to the overall bending stiffness of the
laminate.
• In the special case of a unidirectional laminate with θ = 0
one obtains
 Composite Materials
3
Ex h Ex I x
D11  
12(1   xy yx ) (1   xy yx )
• Under combined in-plane and bending loads, the stiffness
matrix incorporates contributions from the previous
expressions for A and D
• These contributions are not coupled i.e., the curvatures do
not affect the membrane strains, and the membrane strains
do not affect curvatures
• In laminates with unsymmetrical stacking orders, the in-plane
stresses depend on the membrane strains as well as on the
curvatures, and the bending stresses on the curvatures as
well as on the membrane strains
 Composite Materials
 N   A B   
    
M   B D   
• In the extreme case of unsymmetrical and unbalanced
laminates, the stiffness matrix may be fully populated.

 N x   A11 A12 A13 B11 B12 B13    x 

N    
 y   A12 A22 A23 B12 B22 B23    y 
 N xy   A13 A23 A33 B13 B23 B 33   xy 
   
 M x   B11 B12 B13 D11 D12 D13    x 
 M y   B12 B22 B23 D12 D22 D23    y 
    
M xy   B13 B23 B33 D13 D23 D33   xy 
 Composite Materials
• B is called the coupling matrix. It’s coefficients are given as
n
Bij  (1 / 2) C 'ij
(k ) 2 2
(hk  hk 1 )
k 1

• After the compilation of the stiffness matrix, the unknown

strains and curvatures as functions of the known in-plane
loads and bending moments can be calculated by inversion
of the stiffness matrix
• The calculations of stresses in the individual layers
corresponds, in principle, to that of the membrane stresses.
• It must be observed, however, that the laminate strains are
composed of two contributions. Membrane strain
components, which are constant through the thickness of
the laminate.
 Composite Materials
• Strain components from the curvatures of the laminate
which vary linearly through the thickness of the laminate.
• The intersection of εx and κx,z lies in the reference plane of
the laminate which does not necessarily have to be a plane
of symmetry.
• The resulting strains in the kth layer in the direction of the
laminate principal axes can be expressed as

  xk    x   x 
 k    (hk  hk 1 )  
 k     y     y    y 
 k    2  
 xy   xy   xy 
 Composite Materials
• Their transformation into the principal directions of the layers
{ε’}k = [Tε]k{ε}k allows the calculation of the layer stresses by
means of the equation {σ’}k = [C]k {ε’}k
Composite Materials

STRENGTH ANALYSIS
 Composite Materials
• The strength determination of a laminate can be approached
in two different ways
•On the basis of the overall laminate properties, or
•ON the basis of the properties of the individual layers.
• First approach requires the experimental construction of a
failure surface for a given laminate, covering all possible
failure modes
• The strength of the laminate is satisfactory if the stress tensor
for a critical loading condition lies inside the failure surface
• Generally this procedure is considered impractical because
firstly, it is applicable only for a specific laminate stacking
order and secondly, it does not provide insight into the
fracture process
 Composite Materials
• The second approach recognizes that, under increasing
loads, the eventual failure of the most highly loaded layer will
cause a redistribution of stresses but will not necessarily
precipitate failure of the laminate
• The stress redistribution as such, or a further load increase,
can lead to the failure of a second layer which will induce
another redistribution of stresses.
• This process repeats itself until the residual strength of the
remaining laminate layers is exceeded and the laminate as a
whole
• The strength analysis, therefore, requires for each load step
a calculation of the prevailing stresses in the layers of the
laminate.
 Composite Materials
• Failure of a layer occurs when a combination of σx, σy and
Tau-xy exceeds a critical value which must be defined by a
failure criterion.
• The formulation of a failure criterion presupposes the
experimental determination of the strength properties of a
typical layer of the laminate.
 Composite Materials
• For isotropic materials, subject to a plane state of stress, a
relationship .  v  f ( x ,  y , xy )
• The formulation of a failure criterion presupposes the
experimental determination of the strength properties of a
typical layer of the laminate.
 Composite Materials
FAILURE CRITERIA
• For isotropic materials, subject to a plane state of stress, a
relationship
 v  f ( x ,  y , xy )
Can be established as a measure for the stress intensity.
• The function f(σx, σy, ςxy) reduces the plane state of stress to
an equivalent uniaxial stress σv
• Failure occurs when σv exceeds the ultimate strength of the
material under uniaxial loading.
• The best known failure criterion for metals VonMises-Hencky
σv = SQRT(σx2 + σy2 3q2 - σx σy) is based on this premise.
• This definition of a simple equivalent stress is not possible
for orthotropic materials because of the directional
dependence of their strengths
 Composite Materials
• Fiber reinforced resins have the additional problem of
moisture and temperature effects on their strength properties.
• Suitable failure criteria for such materials have the form f(σx,
σy, ςxy, X,X’,Y,Y’,S) = 1.0 where X and X’ represent the tensile
strengths parallel and normal to the fiber direction of a layer,
Y and Y’ the compressive strengths and S the shear strength.
• Accordingly, failure occurs when the above function reaches
or exceeds the value 1.0
• Several failure criteria in the stress and strain domain have
been developed for fiber-reinforced materials which attempt
to accommodate their material properties.
 Composite Materials
• The probability is low, however , that one particular criterion
can satisfy all requirements.
• The commonly used failure criteria include the maximum
stress, maximum strain and interaction criteria
INTERACTION CRITERIA
• The maximum stress and maximum strain criteria are
pragmatically defined failure criteria of great simplicity
• The first better founded failure criteria for orthotropic
materials were derived from the Von Mises/Hencky
hypothesis and contained quadratic and mixed linear terms
• Continued improvement have subsequently led to the
formulations of at least a dozen new criteria of which but two
will be addressed
 Composite Materials
DISTORTIONAL ENERGY CRITERION (NORRIS)
• The distortional energy criterion is predicated on the
anisotropic plasticity theory and was introduced by Norris.

2 2
 x  x y  y  xy
2

2
  2  2  1 .0
X XY Y S
• When σx and σy are compressive stresses, the ultimate
strengths terms X and Y are replaced by X’ and Y’.
• The mixed linear term is sometimes deleted to produce the
simpler criterion
 Composite Materials
QUADRATIC INTERACTION CRITERION
• The quadratic interaction criteion can be formulated in
tensor notation in terms of stress components for 2D state
of stress as
• Fxx σx2 + Fyy σy2 + Fss qxy2 + Fxσx + Fyσy + 2Fxyσxσy = 1
• Five of the six strength parameters Fij in this equation are
derived from the ultimate strengths X, Y and S.
• Fxx = (1/XX’), Fyy = (1/YY’), Fx = {(1/x) – (1/X’)},
Fy = {(1/Y) –(1/Y’)} and Fss = (1/S2)
• Fxy = (-1/2)√(FxxFyy)
 Composite Materials
SAFETY FACTORS
• Safety factors impose constraints on the design of
structures
• Cardinal rules for the choice of safety factors are no failure
at ultimate loads and no yield at limit loads
• In composite construction the rational for selecting safety
factors is different from that for metal construction because
the quality of a composite structure is influenced by many
factors
 condition and handling of the raw materials
 Peculiarities of the curing process
 volume and distribution of fibers
 environmental conditions
 Composite Materials
MARGINS OF SAFETY
• A reserve of load carrying capacity exists when the stress
point lies within the failure surface
• It’s magnitude can be determined by considering that the
prevailing state of stress leads to a certain value, H of the
failure criterion
2
x
2
 y  xy
2

2
 2  2 H
X Y S
• Failure occurs when upon multiplication of the stresses with
a reserve factor, RF, the value of the failure criterion is unity
i.e.,
2

2
  xy
2
RF 2 ( x2  y2  2 )  H
X Y S
 Composite Materials
• In short, RF2 * H = 1.0, or RF = SQRT(1/H)
• The implication is that
• If RF > 1 Failure limit is not yet reached
• If RF = 1 Failure limit is reached
• If RF < 1 Failure limit is exceeded
• And finally, Margins of safety is defined as MS = RF - 1
 Composite Materials
References:
1. J.C. Halpin, “Primer on Composite Materials: Analysis”,
TECHNOMIC Publishing company, 1984.
2. Tsai, S.W., “Mechanics of composite Materials”, Part II,
Technical report AFML-TR 66-149, Nov 1966.
3. Tsai S.W., “Strength characteristics of composite
materials”, NASA CR-224, Apr 1965.
4. Tsai S.W., ‘Think Composites’
5. Bergmann H.W., “Course on the mechanics of composite
materials and structures”, Nov 1991