Transcript Ch3 Micromechanics

CH3
MICROMECHANICS
Assist.Prof.Dr. Ahmet Erklig
Objectives
o Find the nine mechanical:
o four elastic moduli,
o five strength parameters
o Four hygrothermal constants:
o two coefficients of thermal expansion, and
o two coefficients of moisture expansion
of a unidirectional lamina
Micromechanics
Determining unknown properties of the composite
based on known properties of the fiber and matrix
Micromechanics
Uses of Micromechanics
 Predict composite properties from fiber and matrix data
 Extrapolate existing composite property data to different
fiber volume fraction or void content
 Check experimental data for errors
 Determine required fiber and matrix properties to produce
a desired composite material .
Limitations of Micromechanics
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Predicted composite properties are only as good as fiber
and matrix properties used
Simple theories assume isotropic fibers many fiber
reinforcements are orthotropic
Some properties are not predicted well by simple
theories


more accurate analyses are time consuming and expensive
Predicted strengths are upper bounds
Notations
Subscript f, m, c refer to fiber, matrix, composite
ply, respectively
v
volume
V
volume fraction
w
weight
W
weigth fractions
ρ
density
Terminology Used in Micromechanics
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•
•
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Ef, Em – Young’s modulus of fiber and matrix
Gf, Gm – Shear modulus of fiber and matrix
υf, υm – Poisson’s ratio of fiber and matrix
Vf, Vm – Volume fraction of fiber and matrix
Micromechanics and Assumptions
 Approaches:
 Mechanics of materials approach,
 Semi-empirical approach; Involves rigorous mathematical
solutions.
 Assumption: the lamina is looked at as a material whose
properties are different in various directions, but not different
from one location to another.
Volume Fractions

Fiber Volume Fraction
𝑣𝑓
𝑉𝑓 =
𝑣𝑐

Matrix Volume Fraction
𝑣𝑚
𝑉𝑚 =
𝑣𝑐
𝑉𝑓 + 𝑉𝑚 = 1
Mass Fractions

Fiber Mass Fraction

𝑊𝑓 = 𝜌
𝑉𝑓
𝑓
𝑉𝑓 + 𝑉𝑚
𝜌𝑚
Matrix Mass Fraction
1
𝑊𝑚 = 𝜌
𝑉𝑚
𝑓
(1 − 𝑉𝑚 ) + 𝑉𝑚
𝜌𝑚
𝜌𝑓
𝜌𝑚
𝑊𝑓 + 𝑊𝑚 = 1
Density
Total composite weigth:
wc = wf + wm
Substituting for weights in terms of volumes and
densities
𝜌𝑐 𝑣𝑐 = 𝜌𝑓 𝑣𝑓 + 𝜌𝑚 𝑣𝑚
Dividing through by vc gives,
𝑣𝑓
𝑣𝑚
𝜌𝑐 = 𝜌𝑓 + 𝜌𝑚
𝑣𝑐
𝑣𝑐
𝜌𝑐 = 𝜌𝑓 𝑉𝑓 + 𝜌𝑚 𝑉𝑚
𝑊𝑓 𝑊𝑚
1
=
+
𝜌𝑐 𝜌𝑓 𝜌𝑚
Density
When more than two constituents enter in the
composition of the composite material
where n is the number of constituent.
Void Content
Effects of Voids on Mechanical Properties
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†ower stiffness and strength
L
Lower compressive strengths
†
Lower transverse tensile strengths
†
Lower fatigue resistance
†
Lower moisture resistance
†
A decrease of 2-10% in the preceding matrix†
dominated properties generally takes place with
every 1% increase in void content .
Void Content
Evaluation of Four Elastic Moduli
There are four elastic moduli of a unidirectional lamina:

Longitudinal Young’s modulus, E1

Transverse Young’s modulus, E2

Major Poisson’s ratio, υ12

In-plane shear modulus, G12
Strength of Materials Approach
Assumptions are made in the strength of materials approach

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 fiber and matrix follow Hooke’s law (linearly elastic).
The fibers possess uniform strength.
The composites is free of voids.
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


Representative Volume Element (RVE)
This is the smallest ply region over which the stresses and
strains behave in a macroscopically homogeneous
behavior. Microscopically, RVE is of a heterogeneous
behavior. Generally, single force is considered in the RVE.
RVE
RVE
matrix
fibre
Longitudinal Modulus, E1
Total force is shared by fiber and matrix
Longitudinal Modulus, E1
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
The strains in the composite, fiber, and matrix are equal (εc = εf = εm);
Longitudinal Modulus, E1
The ratio of the load taken by the fibers to the load taken by the composite
is a measure of the load shared by the fibers.
Longitudinal Modulus, E1
Predictions agree well with experimental data
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
The fiber, the matrix, and composite stresses are equal.
σc = σf = σm
the transverse extension in the composite Δc is the sum of the
transverse extension in the fiber Δf , and that is the matrix, Δm.
Δc = Δf + Δm
Δc = tc εc
Δf = tf εf
Δm = tm εm
tc,f,m = thickness of the composite, fiber and matrix, respectively
εc,f,m = normal transverse strain in the composite, fiber, and matrix,
respectively
Transverse Young’s Modulus, E2
By using Hooke’s law for the fiber, matrix, and composite, the normal
strains in the composite, fiber, and matrix are
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
Major Poisson’s Ratio, ν12
Major Poisson’s Ratio, ν12
Major Poisson’s Ratio, ν12
Major Poisson’s Ratio, ν12
In-Plane Shear Modulus, G12
Apply a pure shear stress τc to a lamina
In-Plane Shear Modulus, G12
In-Plane Shear Modulus, G12
In-Plane Shear Modulus, G12
FIGURE 3.13
Theoretical values of in-plane
shear modulus as a function of
fiber volume fraction and comparison with experimental
values for a unidirectional
glass/epoxy lamina
Halphin-Tsai Equation


Longitudinal Young’s Modulus
Major Poisson’s Ratio
Transverse Young’s Modulus, E2
For a fiber geometry of circular fibers in a packing geometry of a square
array, ξ = 2. For a rectangular fiber cross-section of length a and width b
in a hexagonal array, ξ = 2(a/b), where b is in the direction of loading.
Transverse Young’s Modulus, E2
In-Plane Shear Modulus, G12
For circular fibers in a square array, ξ = 1. For a rectangular fiber crosssectional area of length a and width b in a hexagonal array,
ξ = 3 𝑙𝑜𝑔𝑒 (𝑎 𝑏), where a is the direction of loading.
Hewitt and Malherbe suggested choosing a function
In-Plane Shear Modulus, G12
Elasticity Approach
Elasticity accounts for equilibrium of forces,
compatibility, and Hooke’s law relationships in
three dimensions.
The elasticity models described here are called
composite cylinder assemblage (CCA) models. In a
CCA model, one assumes the fibers are circular in
cross-section, spread in a periodic arrangement, and
continuous.
Composite Cylinder Assemblage
(CCA) Model
CCA Model
Longitudinal Young’s Modulus, E1
Major Poisson’s Ratio
Transverse Young’s Modulus, E2
The CCA model only gives lower and upper bounds of the
transverse Young’s modulus of the composite.
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
FIGURE 3.21
Theoretical values
of transverse
Young’s modulus as
a function of fiber
volume fraction and
comparison with
experimental values
for boron/epoxy
unidirectional
lamina
In-Plane Shear Modulus, G12
In-Plane Shear Modulus, G12