Composite training

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Simulation Of Composite
Material
By Indranil Bhattacharyya
6th August 2016

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Introduction of
Composite Material

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What is a composite Material?

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Composites – Polymer Matrix
Polymer matrix composites (PMC) and fiber reinforced plastics (FRP) are
referred to as Reinforced Plastics. Common fibers used are glass (GFRP),
graphite (CFRP), boron, and aramids (Kevlar). These fibers have high specific
strength (strength-to-weight ratio) and specific stiffness (stiffness-to-weight
ratio)
Matrix materials are usually thermoplastics or thermosets; polyester, epoxy
(80% of reinforced plastics), fluorocarbon, silicon, phenolic.

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Composites
Fibers
Matrix materials

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Composites – Polymer Matrix
Reinforcing fibers
Glass – most common and the least expensive, high strength, low
stiffness and high density. GFRP consists 30-60% glass fibers by volume.
– highest specific strength, toughest fiber, undergoes plastic
deformation before fracture, but absorbs moisture, and is expensive.
Aramids (Kevlar)
– boron fibers consist of boron deposited on tungsten fibers, high
strength and stiffness in tension and compression, resistance to high
temperature, but they are heavy and expensive.
Boron
– more expensive than glass fibers, but lower density and higher stiffness with
high strength. The composite is called carbon-fiber reinforced plastic (CFRP).
Graphite (99% carbon) or Carbon (80-95% carbon)
The average diameter of fibers used is usually less than .0004 inch (.01 mm). The
tensile strength of a glass fiber could be as high as 650 ksi (bulk glass Su = 5-150
ksi)

Ken Youssefi
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Properties of Reinforced Plastics
The mechanical properties of reinforced plastics vary with the kind, shape,
relative volume, and orientation of the reinforcing material, and the length of
the fibers.
Effect of type, length, % volume, and orientation of fibers in a fiber reinforced
plastic (nylon)

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Applications of Composites
Phenolic as a matrix with asbestos fibers was the first reinforced plastic developed. It
was used to build an acid-resistant tank. In 1920s it was Formica, commonly used as
counter top., in 1940s boats were made of fiberglass. More advanced developments
started in 1970s.
Typically, although not always, consumer composites involve products that require
a cosmetic finish, such as boats, recreational vehicles, bathwear, and sporting
goods. In many cases, the cosmetic finish is an in-mold coating known as gel coat.
Consumer Composites
A wide variety of composites products are used in industrial applications, where
corrosion resistance and performance in adverse environments is critical. Generally,
premium resins such as isophthalic and vinyl ester formulations are required to meet
corrosion resistance specifications, and fiberglass is almost always used as the
reinforcing fiber. Industrial composite products include underground storage tanks,
scrubbers, piping, fume hoods, water treatment components, pressure vessels, and a
host of other products.
Industrial Composites

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Composites – Metal Matrix
The metal matrix composites offer higher modulus of elasticity, ductility, and
resistance to elevated temperature than polymer matrix composites. But, they are
heavier and more difficult to process.

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Composites – Ceramic Matrix
Ceramic matrix composites (CMC) are used in applications where resistance to
high temperature and corrosive environment is desired. CMCs are strong and stiff
but they lack toughness (ductility)
Matrix materials are usually silicon carbide, silicon nitride and aluminum oxide, and
mullite (compound of aluminum, silicon and oxygen). They retain their strength up
to 3000 o
F.
Fiber materials used commonly are carbon and aluminum oxide.
Applications are in jet and automobile engines, deep-see mining, cutting tools,
dies and pressure vessels.

Ken Youssefi
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Composites
Fibers
Matrix materials

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Composites – Metal Matrix
The metal matrix composites offer higher modulus of elasticity, ductility, and
resistance to elevated temperature than polymer matrix composites. But,
they are heavier and more difficult to process.

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Composites – Ceramic Matrix
Ceramic matrix composites (CMC) are used in applications where resistance
to high temperature and corrosive environment is desired. CMCs are strong
and stiff but they lack toughness (ductility)
Matrix materials are usually silicon carbide, silicon nitride and aluminum oxide,
and mullite (compound of aluminum, silicon and oxygen). They retain their
strength up to 3000 oF.
Fiber materials used commonly are carbon and aluminum oxide.
Applications are in jet and automobile engines, deep-see mining, cutting
tools, dies and pressure vessels.

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Material Model

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Mathematical formulation of Composite Material
Analysis of the composite material requires a knowledge of anisotropic elasticity and specific failure criteria
Isotropic : All material properties are independent of the direction
Anisotropic : Direction dependent
Homogeneous : Composition saem
Heterogeneous : Composition varies
Elastic Coefficient of the material
Stress on an Element

 ,0
 0,


 ,
0U
0C
 = Uo + Co

Uo= Uo(xx , yy , zz , xy , yz , zx , x, y, z, T)
 
 
 
xx 1 xx yy zz xy xz yz
yy 2 xx yy zz xy xz yz
yz 6 xx yy zz xy xz yz
f , , , , ,
f , , , , ,
f , , , , ,
       
       
       
M-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Expressed in compliance matrix form
 = S·

Expressed in stiffness matrix form
 = C· 

In general, stress-strain relationships such
as these are known as constitutive
relations
Note that the stiffness matrix is traditionally
represented by the symbol C, while S is
reserved for the compliance matrix!

Internal Energy
0 0 xx xx yy yy zz zz
xy xy xz xz yz yz
C U σ ε σ ε σ ε
2σ ε 2σ ε 2σ ε
    
  

yy0 0 xx 0 0 zz
xx xx xx yy xx zz xx
xy yz0 0 xz 0
xy xx xz xx yz zz
U U U U
U U U
     
  
      
    
  
     

Strains
0 0 0
xx yy zz
xx yy zz
0 0 0
xy xz yz
xy xz yz
C C C
ε ε ε
σ σ σ
1 C 1 C 1 C
ε ε ε
2 σ 2 σ 2 σ
  
  
  
  
  
  
xy = 2xy yz = 2yz zx =
2zx

Hooke’s Law (Anisotropic)
xx 11 xx 12 yy 13 zz 14 xy 15 xz 16 yz
yy 21 xx 22 yy 23 zz 24 xy 25 xz 26 yz
zz 31 xx 32 yy 33 zz 34 xy 35 xz 36 yz
xy 41 xx 42 yy 43 zz 44 xy 45 xz 46 yz
xz 51 xx 52 yy 5
σ C ε C ε C ε C γ C γ C γ
σ C ε C ε C
     
     
     
     
   3 zz 54 xy 55 xz 56 yz
yz 61 xx 62 yy 63 zz 64 xy 65 xz 66 yz
ε C γ C γ C γ
  
     
The 36 coefficients C11 to C66 are
called elastic coefficients

ij jiC C
21 independent constants

Hooke’s Law

Hooke’s Law
The generalized Hooke’s law is an
assumption, which is reasonably accurate
for many material subjected to small strain,
for a given temperature, time and location

Strain Energy Density
21 1 1 1 1 1
0 11 xx 12 xx yy 13 xx zz 14 xx xy 15 xx xz 16 xx yz2 2 2 2 2 2
21 1 1 1 1 1
21 yy xx 22 yy 23 yy zz 24 yy xy 25 yy xz 26 yy yz2 2 2 2 2 2
21 1 1 1 1
31 zz xx 32 zz yy 33 zz 34 zz xy 35 zz x2 2 2 2 2
U C ε C ε ε C ε ε C ε γ C ε γ C ε γ
C ε ε C ε C ε ε C ε γ C ε γ C ε γ
C ε ε C ε ε C ε C ε γ C ε γ
      
     
     1
z 36 zz yz2
21 1 1 1 1 1
41 xy xx 42 xy yy 43 xy zz 44 xy 45 xy xz 46 xy yz2 2 2 2 2 2
21 1 1 1 1 1
51 xz xx 52 xz yy 53 xz zz 54 xz xy 55 xz 56 xz yz2 2 2 2 2 2
1 1 1 1
61 yz xx 62 yz yy 63 yz zz 64 yz x2 2 2 2
C ε γ
C γ ε C γ ε C γ ε C γ C γ γ C γ γ
C γ ε C γ ε C γ ε C γ γ C γ C γ γ
C γ ε C γ ε C γ ε C γ γ

     
     
    21 1
y 65 yz xz 66 yz2 2
C γ γ C γ 

Isotropic materials have only 2
independent variables (i.e. elastic
constants) in their stiffness and compliance
matrices, as opposed to the 21 elastic
constants in the general anisotropic case.
Isotropic material
Eg: Metallic alloys and thermo-set polymers

The two elastic constants are usually
expressed as the Young's modulus E and the
Poisson's ratio n.
Alternatively, elastic constants K (bulk modulus)
and/or G (shear modulus) can also be used. For
isotropic materials
G and K can be found from E and n by a set of
equations, and vice-versa.

Hooke's Law in Compliance Form

Hooke's Law in Stiffness Form

An isotropic material subjected to uniaxial tension in x
direction, xx is the only non-zero stress. The strains in
the specimen are
Youngs Modulus from Uniaxial Tension

The modulus of elasticity in tension, Young's
modulus E, is the ratio of stress to strain on the
loading plane along the loading direction.
2nd Law of Thermodynamics and understanding that
under uniaxial tension, material must elongate in
length implies:
E > 0

Shear Modulus for Pure Shear
Isotropic material subjected to pure shear, for instance, a
cylindrical bar under torsion in the xy sense, xy is the
only non-zero stress. The strains in the specimen are

Shear modulus G:Ratio of shear stress to
engineering shear strain on the loading plane

Since both G and E are required to be positive,
the quantity in the denominator of G must also be
positive. This requirement places a lower bound
restriction on the range for Poisson's ratio,
n > -1
G=E/2(1+n)

Bulk Modulus for Hydrostatic stress
For an isotropic material subjected to hydrostatic
pressure , all shear stress will be zero and the
normal stress will be uniform

Also note:
K > 0
Under hydrostatic load, material will change its
volume. Its resistance to do so is termed as bulk
modulus K, or modulus of compression.
hydrostatic pressure
K =
relative volume change

The fact that both bulk modulus K and the elastic
modulus E are required to be positive, it sets an
upper bound of Poisson's ratio
n < 1/2
K=E/ 3(1-2n)

Orthotropic material has at least 2 orthogonal planes of
symmetry, where material properties are independent of
direction within each plane.
Eg: Certain engineering materials, 2-ply fiber-reinforced
composites, piezoelectric materials (e.g.Rochelle salt)
Orthotropic material require 9 independent variables (i.e.
elastic constants) in their constitutive matrices.
Orthotropic material

The 9 elastic constants in orthotropic
constitutive equations are comprised of
3 Young's modulii Ex, Ey, Ez,
3 Poisson's ratios nyz, nzx, nxy,
3 shear modulii Gyz, Gzx, Gxy.
Note that, in orthotropic materials, there are no
interaction between the normal stresses x, y,
z and the shear strains yz, zx, xy

Hooke’s law in compliance matrix form

Hooke’s law in stiffness matrix form

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Governing stiffness Equation
Assumptions :
σ3=σ4= σ5= 0 ;
Considering the plate is very thin
G13 = G12 ; E1 ; E2 ;G12 & v12 to be entered
Inversion

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Stiffness Matrix
for
Composite Material

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Mechanics of Orthotropic Material
A composite has a fibber orientation of 30 deg construct [a]
matrix by calculating the direction cosines ;
0.866 0.5 0
-0.5 0.866 0
0 0 1

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Classical Laminated Plate Theory
Assumptions for CLPT
1. The laminate consists of perfectly bonded layers. There is no slip
between the adjacent
layers. In other words, it is equivalent to saying that the displacement
components are
continuous through the thickness.
2. Each lamina is considered to be a homogeneous layer such that its
effective properties are
known.
3. Each lamina is in a state of plane stress.
4. The individual lamina can be isotropic, orthotropic or transversely
isotropic.
5. The laminate deforms according to the Kirchhoff - Love assumptions
for bending and stretching of thin plates (as assumed in classical plate
theory). The assumptions are:
a. The normals to the mid-plane remain straight and normal to the
midplane even after
deformation.
b. The normals to the mid-plane do not change their lengths.

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Stacking
Forces and Moments on a plate

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Deformation of a plate

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Force and Moment Results

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Force and Moment Results

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Composite Layups

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Composite Layups

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ORIENTATION

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Failure of Composite

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Types of Failure in Composite
Common types of failures are :
Fiber Breaking / Matrix crazing – cracking / disbanding and delamination
can be arranged as per below categories .
• Maximum Stress Criteria
• Maximum Strain Criteria
• Tsai- Hill failure Criteria
• Truncated maximum-strain Criteria
Failure criteria are complemented with Laminate failure to predict single layer data.

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Types of Failure in Composite
The strength of single layer composite material described by the following strength
values :
• Tensile Strength in fiber direction FiT
• Compressive Strength in the fiber direction F1c
• Tensile Strength in transverse direction F2T
• Compressive Strength in transverse direction F2c
• In plane shear Strength F6
• Inter laminar shear strength F4 F5
• Biaxial interaction coefficient f12
Strength Ratio R = UTS/ Actual Stress

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Maximum Stress Criterion

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Maximum Stress Criterion

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Maximum Strain Criterion

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Tsai- Hill Criterion
The criteria is similar to Von Misses criterion for Metals and modified for orthotropic materials
“f indicates failure stress

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Tsai- Wu Criterion
The criteria uses a complete quadratic expression to draw the failure envelop

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Fiber-Matrix Failure Criteria

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Composites Model for Explicit

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Appendix

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