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Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 1
Composite materials
and structures:
Professor C Bhaskaran
NIET
Syllabus
Objective:
To understand the design and
fabrication of composite materials
and structures.
Unit 1. Stress-strain Relation:
Introduction - Advantages and
applications of composite materials,
reinforcements and matrices –
Generalized Hooke’s law –
Elastic constants for anisotropic,
orthotropic and isotropic materials.
Unit 2. Methods of analysis:
Micro mechanics – mechanics of materials
approach, elasticity approach to determine
material properties.
Macromechanics - stress-strain relations,
with respect to natural axis, arbitrary axis
- determination of material properties.
Experimental characterization of lamina.
Unit 3. Laminated Plates:
Governing differential equation for a
general laminate,
angle ply and cross ply laminates.
Failure criteria for composites.
Unit 4. Sandwich Constructions:
Basic design concepts of sandwich
construction
-Materials used for sandwich
construction
- Failure modes of sandwich panels.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 2
Unit 5. Fabrication process:
Various open and closed mould
processes.
Manufacture of fibres .
Types of resins and properties and
applications
- Netting analysis.
Books:1. Calcote L.R., “The analysis of laminated composite structures”
Von Nostrand Reinhold company, New York.
2. Jones R M., “Mechanics of composite materials” McGraw-Hill.
3. M. Mukopadhyay “ Mechanics of composite materials and
structures” Universities Press.
4. Isaac M Daniel & Ori Ishai “Engineering Mechanics of
composite materials” Oxford University Press.
5. Avtar K Kaw. “Mechanics of composite materials“ crc press.
6. Krishan K Chawla “Composite materials: Science and Engg”.
References:
1. Agarwal B D & Broutman L J “The analysis and performance
of Fibre composites”. John Wiley and sons.
2. Lubin G. “Handbook of Advanced Plastics and Fibre glass”
- Von Nostrand Reinhold Co., New York.
3. Dowling “Mechanical Behaviour of Materials”.
Blank page
Composite materials and structures:
Unit 1. Stress-strain Relation:
Introduction- Advantages and application
of composite materials, reinforcements
and matrices – Generallized Hooke’s law-
Elastic constants for anisotropic,
orthotropic and isotropic materials.
Definitions:
A composite material is a material system
composed of two or more physically
distinct phases whose combination
produce aggregate properties that are
different from those of its constituents.
A composite material may be defined as one which
satisfies the following conditions:
1. It is manufactured.
2. It consists of two or more physically and/or
chemically distinct, suitably arranged or distributed
phases with an interface separating them.
3.It has characteristics that are not depicted by
any of the components in isolation.
[Ref: J F Schier and R F Juergens (sept. 1983)
Astronautics and Aeronautics]
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 3
Natural composites:
1. Coconut palm leaf : concept of fibre
reinforcement
2. Wood : Cellulose fibres in a lignin matrix
3. Bone : short and soft collagen fibres in a
mineral matrix called apatite
What is a composite?
A composite is a structural material which consists of
two or more constituents. The constituents are
combined at a macroscopic level and are not soluble in
each other. One constituent is called the reinforcing
phase and the other in which it is embedded is called
the matrix. The reinforcing phase material may be in
the form of fibres, particles or flakes. The matrix phase
materials are generally continuous.
Example:
Concrete reinforced with steel,
Epoxy reinforced with graphite fibres.
Difference between an alloy and a composite
material?
Composite material is, in effect, a mixture of
two or more materials(e.g., concrete, a mixture
of cement and aggregate Whereas an alloy is a
solid solution of alloying elements in t he host
metal. The atoms of the alloying elements take
positions in the crystal structure, as impurities,
whereby the metal gets strengthened, known
as, solid solution strengthening .
metals
ceramics polymers
Ceramic-metal composites Metal- polymer
composites
ceramic -polymer composites
structural Composite materials
The Two phases:
matrix - plastics(polymers),
metals, or ceramics
and
reinforcements - Fibre or particles
Composite materials
examples:
Wood, plywood, concrete
reinforced rubber
fibre reinforced plastics
fibre reinforced metals
and so on
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 4
Classification
Based on
i) the Matrix Material
ii) the Shape of reinforcement
Based on matrix material
Composite materials
PMC MMC CMC
PMC - Polymer Matrix Composites
MMC - Metal Matrix Composites
CMC - Ceramic Matrix Composites
Based on reinforcement shape.
Fibre composites
and particulate composites.
Particulate composites can have
small particles or flakes as
reinforcements.
Reinforcements:
(metal, polymer or ceramic)
fibre
particle flake
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 5
Types of fibre-reinforced composites
Type Fibre Matrix
Polymer Glass, Carbon,
Aramid (Kevlar),
Boron
Epoxy, Polyester,
Polyimide
Polysulfones,
Metal Boron, Carbon,
Borosic,
Silicon carbide,
Alumina
Aluminium,
Magnesium,
Copper, Titanium.
Ceramic Silicon carbide,
Alumina,
Silicon nitride.
Silicon carbide,
Alumina,
Glass-ceramic,
silicon nitride
Carbon Carbon Carbon
Glass fibres come in several varieties.
Designated S-, A-, C-, or E-glass.
Each variety has special characteristics.
S-glass is exceptionally strong.
C-glass is extremely resistant to
corrosion and chemical attack.
A-glass has good resistance to chemicals.
E-glass is a non-conductor of electricity.
Though economical, glass fibre is
relatively heavy. Of the common synthetic
reinforcements, it has the least efficient
strength-to-weight ratio.
Aramid Fibre resists impact. It is used
extensively in bulletproof vests and body
armor. Racing drivers wear aramid suits that
help protect them from burns in fiery, high-
speed crashes. Aramid is commonly known as
Kevlar, produced by DuPont. Aramid fibre’s
cost is between glass and carbon. Aramid is
more difficult to work with than glass and has
a tendency to absorb moisture.
Carbon Fiber is a very strong fiber and
extremely stiff. It is lighter in weight than
glass fiber. Carbon fibers come in several
varieties and strengths and are the most
expensive kind of fiber reinforcements. They
are typically used in airplanes and spacecraft.
Carbon fiber reinforced composites are also
used in products such as bicycle frames,
tennis rackets, skis, and golf club shafts.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 6
Boron is an extremely hard natural element
and
ceramics are hard materials that can
withstand high heat and harsh chemicals.
Ceramic material is a compound containing metallic and
non-metallic elements.
Oxygen, nitrogen, carbon are the major
non-metallic elements.
Examples:
-clay : Traditional ceramics,
consisting of fine particles of hydrous
aluminum silicates and other minerals.
-silica : SiO2 , basic for all glassy materials.
-Alumina : Al2O3. -silicon carbide: SiC.
-carbides and nitrides: such as Tungsten carbide (WC),
Titanium carbide (TiC),
Titanium nitride, Boron nitride.
Different fibers can be combined to make a
composite to cost less or perform better.
Composites that are made of more than one
fibre/resins are called hybrid composites.
Fibers with special characteristics are used
when a composite must be exceptionally
strong or heat-resistant;
--for high-performance military aircraft and
aerospace applications.
What are advanced composites?
Advanced composites are composite materials
traditionally used in aerospace industry. These have
high performance reinforcements of a thin diameter in
a matrix material such as epoxy and Aluminum.
Example: graphite/epoxy, kevlar /epoxy, and
boron/ aluminum composites.
What are the advantages of composite materials ?
--have high specific strength and specific stiffness.
-- Fatigue properties are better than common
engineering materials
- Tailorability of physical properties to suit specific applications.
-Low maintenance
-Corrosion resistance
-Self lubrication (specialized composites)
-Long life (if UV protected)
-Low weight (compared to the alternatives)
-better appearance and surface finish.
-Radar-invisible
-Low thermal signature
Disadvantages:
Disadvantages:
- poor reliability and repeatability.
- Anisotropic
- PMC’s are liable to be attacked by chemicals and
solvents
- generally expensive and man intensive .
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 7
Materials
solid, liquid, gas
Metallic and non-metallic materials
Organic and inorganic materials
Metals – Ferrous and Non-ferrous metals
Ferrous metals:
Iron, Steel, steel alloys, cast iron.
Non-ferrous metals:
Aluminium (Al), Copper (Cu),
Magnesium (Mg), Manganese (Mn),
Gold (Au), Silver (Ag),
Tin (Sn), Titanium (Ti), Zinc (Zn), etc
Non-metallic materials:
Boron, Carbon, Silicon, Sulfur,
phosphorus, oxygen, nitrogen
Polymer:Is a compound formed of repeating
structural units called ‘mers’, whose
atoms share electrons to form very
large molecules.
Polymers usually consist of carbon plus
one or more other elements such as
hydrogen, oxygen, nitrogen, and
chlorine. Theses are organic polymers.
Inorganic polymers are those
without carbon atom. Eg., glass ,
silicon rubber.
Cotton, silk, wool and rubber are natural
polymers.
Polyethylene, PVC (Poly Vinyl Chloride),
Nylon, Terylene are synthetic polymers,
synthesised from low molecular weight
compounds.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 8
Polymers:can be divided in to
Thermosetting polymers
and Thermoplastic polymers
(and also elastomers).
Thermoplastic polymers:
These materials gets softened when
heated and can be formed in to various
shapes, which will be retained on cooling.
Can be subjected to several cycles of
heating and cooling.
Polyethylene, PVC, Nylon, Polystyrene, etc.
[ C H2-CH2 ]n
Polyethylene monomer
0000000000000000 chain formation in making a polymer
Thermosetting Polymers:
some of the polymers undergo some
chemical change on heating and convert
them in to a rigid structure. They are like
the yolk of the egg, which on heating sets
into a mass, and , once set can not be
reshaped. Such polymers are called
thermosetting polymers.
Eg., Phenolics, Amino resins, Epoxies.
Elastomers: Polymers that exhibit significant elastic
behaviour is termed as elastomers.
Eg., Natural rubber, Neoprene, Polyurethane.
Plastics, elastomers, Fibres and Liquid Resins:
Depending on its ultimate form and use, a polymer
can be classified as plastic, elastomer, fibre or liquid
resin.
When a polymer is shaped in to hard and tough
utility articles by the application of heat and pressure,
it is used as a ‘plastic’. Typical examples are
:Polystyrene, PVC, Polymethyl methacrylate.
When vulcanised in to rubbery products exhibiting
good strength and elongation, polymers are used as
‘elastomers’. Eg., Natural rubber, Synthetic rubber,
Silicone rubber.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 9
If drawn into long filament like materials,
whose length is at least 100 times its
diameter, polymers are said to have been
converted in to ‘fibres’. Typical examples
are Nylon and Terylene.
Polymers are used as adhesives, potting
compounds, sealants etc., in a liquid form
are described as liquid resins. Typical
examples are: commercially available
epoxy adhesives and polysulphide sealants.
Ceramics- a compound containing metallic
and non-metallic elements.
Oxygen, nitrogen, carbon are the major non-
metallic elements.
Examples:
-clay : Traditional ceramics, consisting of fine
particles of hydrous aluminum silicates
and other minerals.
-silica : SiO2 , basic for all glassy materials.
-Alumina : Al2O3. -silicon carbide: SiC.
-carbides and nitrides: such as Tungsten
carbide (WC), Titanium carbide (TiC),
Titanium nitride, Boron nitride.
Carbon fibre cloth used to make
composites
Hybrid composites:
Different fibers can be combined to
make a composite to cost less or
perform better. Composites that are
made of more than one type of fibre
or resin are called hybrid composites.
Applications:
1. Aircraft, Aerospace & Military
2. Marine field
3. Automotive
4. Sporting Goods
5. others.
Aircraft and Aerospace and military
applications:
Optimally an Aircraft, missile, satellite launch
vehicle and satellite requires a minimum weight
design to attain greater speeds and increased
payload and fibre - reinforced composites have
been found to be ideal for this purpose. Carbon
fibre or kevlar fibre reinforced polymer (FRP)
composites are being extensively used in the
production of aircrafts, nowadays. Civil and
military aircraft wings, fuselage, empennage
components are being designed and built using
these composites. -contd-
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 10
-Continued-
For Missiles and Launch vehicles, interstage
structures, solid motor cases, liquid storage
tanks etc., uses composite materials very
successfully. Satellites also use composite
materials in the form of sandwich structures for
main body and other structural components.
Carbon fibre composites are vey useful in the
design of dish antennas, especially for
temperature stability.
-continued-
-cotinued-
Strength and stiffness of composites are major
considerations for the aircraft, missile and
launch vehicles, while stiffness and low (or zero)
coefficient of thermal expansion are the major
requirements for space (satellite) applications.
FRP is used for the construction of antennas,
booms, support trusses and struts.
Marine field applications:
Potential applications in the marine field range
from small components such as radar domes,
masts and piping to large-scale structures,
submersibles and off-shore structure modules.
Glass reinforced plastics (GFRP) are used in the
construction of boat hulls, including yachts, life
boats, dinghies, canoes, speed boats, fishing
boats, and passenger boats.
The popularity of GFRP with boat builders
lies in its competitive low cost ( in
comparison to wooden hulls), a trouble-
free performance, low maintenance cost
and aesthetics.
Military and commercial hovercrafts also
uses GFRP. Fast patrol boats are made of
hybrid glass/carbon laminates in place of
steel and aluminum.
Automotive field:
FRP’s are being used to make many parts
of cars. Exterior parts of the cars, such as
canopy, door etc are made of composites.
Chassis components leaf spring is made of
FRP.
Sporting goods:
Because of the reduction in weight
many sports goods are being made
using composite materials.
Tennis rackets, fishing rods, archery
bows, bicycle frames, sail boats and
kayaks, oars, paddles, canoe hulls,
javelins, helmets, golf clubs, hockey
sticks, athletic shoes surf boards etc.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 11
The Soldier’s Load and Life Expectancy:
The humble soldier’s life expectancy is increased
not only by Kevlar vests but by composite
helmets, advanced composite goggles, gloves
and more effective, lighter weaponry using
composites in their construction. The general
weight reduction of standard equipment has not
lessened his (or her) load though – one just has
to carry additional equipment.
However what has changed is that there are
many soldiers alive today who would be dead
were it not for advanced composite armor and
flak jackets.
Hooke’s law:
States that the deformation is directly
proportional to the load or strain is
directly proportional to the stress up
to elastic limit, for a linearly elastic
material.
{ε} = [1/E] {σ},
where E is the proportionality
constant, called Young’s modulus.
Anisotropic, orthotropic, isotropic
materials:
Definitions:
Isotropy: properties at a point are
same in all directions.
Orthotropy: Three planes of symmetry
exists. Equality of property
in each plane, in one direction.
Anisotropy: properties are different
everywhere, any direction.
STRESS STRAIN
STRENGTH
STIFFNESS
TOUGHNESS
Stress is the load which we apply and strain is
the effect. Stress can be normal stress( tensile or
compressive) and/or shear stress. And the strain
can be normal strain and/or shear strain.
Stress and strain are related through Hooke’s law.
Stiffness is the load required to produce unit
deformation, in the direction of the load.
Toughness is the energy absorbed by the
material before it breaks.
The area under the stress-strain diagram gives
the energy per unit volume .
Generalized Hooke’s Law
Hooke’s law gives the stress-strain relations
for engineering materials.
The connecting parameters are the elastic
constants through engineering constants
for the materials.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 12
In the most general case the stress –
strain components are related by the
generalized Hooke’s law, as given in the
following equations, in the matrix form.
There are 81 elastic constants
( figure showing the stresses, on face 1)
2 σ12
σ11
σ13
3 1
Identification of stress, strain components:
σ11, σ22, σ33 are normal stresses.
σ12 etc are shear stresses.
σ1, σ2, σ3 are normal stresses.
σ4 = τ23, σ5 = τ31
and σ6 = τ12, are shear stresses.
Similarly strains also. Replace σ with ε
Stress – strain relation for an anisotropic material
the above can be written in indicial form as:
σij = Cijkl εkl ,
where, (i,j,k,l = 1,2,3)
and Cijkl are known as Stiffness coefficients.
Or
we can also have the strain-stress relation as,
εij = Sijkl σkl
where, Sijkl are compliance
or flexibility coefficients.
Stress and strain tensors are required to
be symmetric, that is,
σij =σji, εij = εji .
This reduces the number of elastic
constants to 36.
we can write the stress- strain relations or
the constitutive relations as follows:
stress-strain relations for an anisotropic material 3-D,
considering stress symmetry
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 13
σi = Ʃ Cijεj
similarly,
εi = Ʃ Sijσj
i=j=1 to 6
It can be seen that, as per the requirement
of energy considerations,
Cij=Cji and Sij=Sji.
And hence the number of independent
elastic constants for an anisotropic
material reduces to 21.
stiffness coefficients for anisotropic materials
(symmetric about diagonal) (3-D)
Similarly compliance matrix [Sij] also.
C11 C12 C13 C14 C15 C16
C12 C22 C23 C24 C25 C26
C13 C23 C33 C34 C35 C36
C14 C24 C34 C44 C45 C46
C15 C25 C35 C45 C55 C56
C16 C26 C36 C46 C56 C66
Compliance or flexibility matrix
3- Dimensional
S11 S12 S13 S14 S15 S16
S12 S22 S23 S24 S25 S26
S13 S 23 S33 S34 S35 S36
S14 S24 S34 S44 S45 S46
S15 S25 S35 S45 S55 S56
S16 S26 S36 S46 S56 S66
We can write Hooke’s law, in the expanded
form, for strain, as,
ε1 = s11σ1 + s12σ2 + s13σ3+s14σ4+s15σ5+s16σ6
ε2 = s12σ1+s22σ2+s23σ3+s24σ4+s25σ5+s26σ6
-------- etc.
S11 to S66 and C11 to C66, the elements of the
compliance and stiffness matrices respectively
are the elastic constants.
So we need 21 elastic constants to study an
anisotropic material, at the minimum.
Orthotrpy is that the material has three
planes of symmetry.
If 1, 2, 3 are the coordinate axes
normal to these planes and the load is
acting along these directions, then we can
write the relations :
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 14
Stiffness matrix for orthotropic materials 3-D
C11 C12 C13 0 0 0
C12 C22 C23 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 0 0 C66
ε1 S11 S12 S13 0 0 0 σ1
ε2 S12 S22 S23 0 0 0 σ2
ε3 = S13 S23 S33 0 0 0 σ3
γ23 0 0 0 S44 0 0 τ23
γ31 0 0 0 0 S55 0 τ31
γ12 0 0 0 0 0 S66 τ12
Hooke’s law for 3-D orthotropic material
(Strain-stress relation)
Stiffness matrix for orthotropic materials 2-D
Needs FOUR (4) elastic constants.
C11 C12 0
C12 C22 0
0 0 C66
Constants for Isotropic materials:
S11= S22= S33= 1/E,
and S12 = -ν/E = S13 = S23
S44 = S55 = S66 = 1/G = 2(1+ν)/E
Needs only two elastic constants, S11 and S12.
Needs two engineering constants to be
evaluated, given by E and G or ν.
Elastic constants for various epoxy matrix
composites:(Fibres along x-axis)
Material Exx
GPa
Eyy
GPa
Gxy
GPa
νxy νyx VfSp.
Gravity
Graphite 181 10.3 7.17 0.28 .016 0.7 1.6
Boron 204 18.5 5.59 0.23 0.5 2.0
Glass 38.6 8.27 4.14 0.26 0.45 1.8
Kevlar 76 5.5 2.3 0.34 0.6 1.46
Now, the elements of the compliance
and flexibility matrices are formed by
engineering constants or elastic properties
of the material, such as,
Young’s modulus E,
shear modulus G
and Poisson’s ratio ν.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 15
Determination of elastic constants for
orthotropic materials:
Most often the material properties are
determined in the laboratories in terms of the
engineering constants, E, G, ν. These are
measured using simple tests like uniaxial tension
test, pure shear, torsion test etc.
The engineering constants for an
orthotropic materials are, E1, E2, E3,
ν12, ν21, ν23, ν32, ν31, ν13,
G12, G23, and G31.
But six Poisson’s ratios are not
independent, only three are required.
The strain-stress relation for an orthotropic
material is:
ε1 s11 S12 S13 0 0 0 σ1
ε2 S12 S22 S23 0 0 0 σ2
ε3 S13 S23 S33 0 0 0 σ3
γγγγ23 = 0 0 0 S44 0 0 ττττ23
γγγγ31 0 0 0 0 S55 0 ττττ31
γγγγ12 0 0 0 0 0 S66 ττττ12
Initially, apply σ1, keeping other forces zero, then
we have all the elements in the stress matrix as
zeroes, except σ1.
ε1 = S11σ1 ----(1)
ε2 = S12σ1 ------(2)
ε3 = s13σ1 ------(3)
γ23 = γ31 = γ12 = 0
The young’s modulus in direction (1) is E1 and is
defined as E1=σ1/ε1 = 1/S11.
Poisson’s ratio:
In general terms Poisson’s ratio, νij,
is defined as the ratio of the negative of
the normal strain in the direction ‘j’ to the
normal strain in the direction ‘i’, when the
only normal load applied is in direction ‘i’.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 16
The Poisson's ratio ν12 is defined as,
ν12 =-ε2/ε1 = -S12/S11.
The Poisson’s ratio ν13 is defined as
ν13 = -ε3/ε1 = -S13/S11
Similarly apply load in the direction 2 only, and
get,
E2 = 1/S22,
ν21 = -S12/S22
ν23 = -S23/S22
Similarly apply load in the direction 3 only,
and get,
E3 = 1/S33,
ν31 = -S13/S33
ν32 = -S32/S33
Now apply,τ23, keeping others zero.
Then, γ23 = S44 τ23.
Shear modulus in plane 2-3 is defined as,
G23= τ23/γ23 = 1/S44.
similarly, G31 = 1/S55
and G12 = 1/S66.
In the above expressions, twelve
engineering constants have been defined
as follows:
E1, E2, E3 in each material axes,
Six Poisson’s ratios ν12, ν21,ν23,ν32,ν13, and ν31,two for each plane, and three shear modulii for
each plane, G23, G31 and G12.
However, Six Poisson’s ratios are not
independent of each other.
From the above expressions, we have,
E1 = σ1/ε1 = 1/S11
E2 = σ2/ε2 = 1/S22
ν12 = -ε2/ε1 = -S12/S11
ν21 = -ε1/ε2 = -S12/S22
ν12/ν21 = (-S12/S11) / (-S12/S22)
= (S22/S11) = E1/E2
or, we may write,
(ν12/E1) = (ν21/E2)-------(i)
(ν13/E1) = (ν31/E3)-------(ii)
(ν23/E2) = (ν32/E3)-------(iii)
These are known as
Reciprocal Poisson’s ratio equations.
Thus there are Nine (9) elastic constants to be
evaluated through Nine(9) material properties.
Composite Materials & Structures 9/24/2013
Prepared By Prof. C Bhaskaran 17
The stiffness matrix elements can be
obtained by inverting the flexibility or the
compliance matrix, as ,
C11 = (S22S33-S232)/S
C22 = (S11S33-S132)/S
C33 = (S11S22-S122)/S
C12 = (S13S23-S12S33)/S
C23 = (S11S13-S23S11)/S
C13 = (S12S23-S13S22)/S
C44 = 1/S44, C55 = 1/S55, C66 = 1/S66
Where, s =
End of unit 1
S11 S12 S13
S21 S22 S23
S31 S32 S33
Assignment-1:
1. Write down the generalized Hooke’s law.
2. Write the stress- strain relation for an orthotropic
material (3-D) and state the flexibility and stiffness
matrices in terms of the engineering constants, E’s,
ν’s and G’s.
3. Explain what are anisotropic, orthotropic,
monoclinic, transversely isotropic and isotropic
materials.Date of submission: 12/8/13
-------0000--------
Macromechanics: Macromechanics deals with the establishment
of the stress- strain relationship and the
strength and stiffness of the composite material
applying the average properties established
through micromechanics analysis.
Stress and strain relations for uni-directional
and bi-directional lamina are developed using
the lamina properties. Methods for the
evaluation of the properties of the angle lamina
are established and the strength and stiffness of
the laminate are arrived at.
Note:
Crystalline and amorphous nature of
materials,
lattice structure,
grains and grain boundaries.
Long order and short order arrangement of
atoms and molecules.
Solid, Liquid, Gas.
Slurry ( mixture of liquid and solid)