Analysis of composite materials used in aircrafts, it’s complete details and failures. Contains the fabrication, methodology, components used

1. ANALYSIS OF
COMPOSITE
MATERIALS (UNIT-
2)
Introduction and Mechanics
and Material Approach
(Strength of Material
Approach)

2. Basic Definitions
Micromechanics and Macro-mechanics
 Micromechanics is the study of the behavior of fibers and
filaments, matrices, interfaces, and interphases in a
composite upon the application of stress and strain.
 Micromechanics (or, more precisely, micromechanics of
materials) is the analysis of composite or heterogeneous
materials on the level of the individual constituents that
constitute these materials.
 Macro-mechanics is the study of the behavior of the lamina or
laminate when stresses or strains are applied.
 Macro-mechanics is the study of composite material behavior
wherein the material is presumed to be homogeneous, and
the effects of the constituent materials are detected only as
averaged apparent macroscopic properties of the composite
materials.

3. Lamina
 A lamina (also called a ply or layer) is a single flat layer of
unidirectional fibers or woven fibers arranged in a matrix.
Laminate
 A laminate is a stack of plies of composites. Each layer can
be laid at various orientations and can be made up of
different material systems.

4. Micromechanical Analysis
1. Mechanics of materials (simplifying assumptions unnecessary to
specify the details of the stress and strain distribution at the
micromechanical level, geometry of the microstructure is
arbitrary)
2. Elasticity theory (elasticity theory models involves the solution of
actual local stress and strain fields and actual geometry of the
microstructure is taken into account)
3. Empirical solutions (curve fitting to elasticity solutions or
experiments)

5. Analysis Assumptions
1. The fibers are: (i) continuous, (ii) straight, (iii) infinitely long in
the X1 direction, (iv) perfectly aligned with X1 axis, (v)
circular in cross section, and (vi) arranged in a periodic
square array.
2. The fiber and matrix materials are (i) homogeneous, (ii)
isotropic, and (iii) linearly elastic.
3. The fiber and matrix are perfectly bonded at their interface.
4. Mechanical loads are applied at infinity.
5. Loads and material properties do not vary along the X1
direction.

6. Void Content
 During the manufacture of a composite, voids are introduced in
the composite This causes the theoretical density of the composite
to be higher than the actual density. Also, the void content of a
composite is detrimental to its mechanical properties. These
detriments include lower
• Shear stiffness and strength
• Compressive strengths
• Transverse tensile strengths
• Fatigue resistance
• Moisture resistance
The volume of void is given by

7.  The volume fraction of the voids is
Where, Vv = volume of voids, v_c = Total volume for
composite, ρ_ce= experimental density, ρ_ct=theoretical
density, wc = weight of composite

8. Elastic Moduli
 The Elastic Modulus is the measure of the stiffness of a material.
In other words, it is a measure of how easily any material can be
bend or stretch. It is the slope of stress and strain diagram up to
the limit of proportionality.
 There are four elastic moduli of a unidirectional lamina:
• Longitudinal Young’s modulus, E1
• Transverse Young’s modulus, E2
(Young's modulus is a measure of the ability of a material to
withstand changes in length when under lengthwise tension or
compression)
• Major Poisson’s ratio, ν12
• In-plane shear modulus, G12
Shear modulus, also known as Modulus of rigidity, is the measure
of the rigidity of the body, given by the ratio of shear stress to
shear strain.

10. Mechanics of Material Approach
 Strength of Materials Approach
 Elasticity Approach
 Semi-Empirical Models

11. STRENGTH OF MATERIALS
APPROACH
The following assumptions are made in the strength of materials
approach model:
• The bond between fibers and matrix is perfect.
• The elastic moduli, diameters, and space between fibers are
uniform.
• The fibers are continuous and parallel.
• The fibers and matrix follow Hooke’s law (linearly elastic).
• The fibers possess uniform strength.
• The composite is free of voids.

13.  From a unidirectional lamina, take a representative volume
element that consists of the fiber surrounded by the matrix.
 The fiber, matrix, and the composite are assumed to be of the
same width, h, but of thicknesses t_f , t_m, and t_c,
respectively. The area of the fiber is given by

14.  The two areas are chosen in the proportion of their volume
fractions so that the fiber volume fraction is defined as
 And the matrix fiber volume fraction V_m is

15. Longitudinal Young’s Modulus
 From the given figure under a uniaxial load Fc on the
composite VE, the load is shared by the fiber F_f and the
matrix F_m so that

16.  The loads taken by the fiber, the matrix, and the composite
can be written in terms of the stresses in these components
and cross-sectional areas of these components as
( σ = F/A )

17.  Assuming that the fibers, matrix, and composite follow
Hooke’s law and that the fibers and the matrix are isotropic,
the stress–strain relationship for each component and the
composite is

18.  The strains in the composite, fiber, and matrix are equal (εc =
εf = εm); then, from Equation
 Substitute above equation into volume fraction
-------A

19.  Equation A gives the longitudinal Young’s modulus as a
weighted mean of the fiber and matrix modulus. It is also
called the rule of mixtures.
 The ratio of the load taken by the fibers F_f to the load taken
by the composite F_c is a measure of the load shared by the
fibers.

20. Problem:
 Determine the longitudinal elastic modulus of
a unidirectional glass/epoxy lamina with a
70% fiber volume fraction. Use the properties
of glass and epoxy from E_f =85 GPA and
E_m = 3.4 GPA, V_f=0.7 and V_m =0.3
respectively. Also, find the ratio of the load
taken by the fibers to that of the composite.

21. Solution:
Given
Longitudinal elastic modulus of the unidirectional lamina is

22. The ratio of the load taken by the fibers to that of the composite
is

23. Transverse Young’s Modulus
 The composite is stressed in the transverse direction. The
fibers and matrix are again represented by rectangular blocks
as shown.

24. The fiber, the matrix, and composite stresses
are equal. Thus,
-----A

26. By using Hooke’s law for the fiber, matrix, and composite, the
normal strains in the composite, fiber, and matrix are
--------C

27. Substituting Equation (B) and Equation (C)
in Equation (A) and using Equation (A) gives
-----(D)

28. Major Poisson’s Ratio
 The major Poisson’s ratio is defined as the negative of the
ratio of the normal strain in the transverse direction to the
normal strain in the longitudinal direction, when a normal load
is applied in the longitudinal direction.
 The deformations in the transverse direction of the composite
is the sum of the transverse deformations of the fiber and the
matrix

29. Because the thickness fractions are the same as the volume fractions,
per Equation

30. Problem:
Determine the major and minor Poisson’s ratio
of a glass/epoxy lamina with a 70% fiber volume
fraction. Use the properties of glass and epoxy
from V_f =0.7, V_m = 0.3, v_f =0.2 v_m=0.3,
E_1 =60.52 and E_2 = 10.37, respectively.
Solution:
Given

31.  The major Poisson’s ratio is:
The longitudinal Young’s modulus is
The transverse Young’s modulus is
The minor Poisson’s ratio