Transformation of stress components

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Transformation of stress components and of
elastic constants
Principal material coordinates are natural
coordinates
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Sign convention
It is often necessary to know the stress-strain
relationships in non-principal coordinates
(off-axis) such as x and y. Therefore How do
we transform stress and strain? How do we
transform the elastic constants?
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Transformation of stress components between
coordinate axes
This is obtained by writing a force balance
equation in a given direction. For example, in
the x direction
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By repeating this, the complete set of stress
transformations in xy coordinates can be obtained
()
where c cosq and s sinq
And in the 12 system we have
with
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Similarly, we have
()
Now, remember that for a 2-dimensional lamina in
its principal coordinates we showed that
()
(remember only 4 independent constants).
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Then, substituting () into (), and then into
(), we find
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and the components of the transformed reduced
stiffness matrix are as follows
(How many independent constants among these
transformed components?)
In terms of engineering constants, we have
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EXAMPLES
Nylon-reinforced elastomer composite
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Zinc (hexagonal symmetry)
Carbon AS4/epoxy composite
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Anisotropy of bone (J. Biomechanics 1992)
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Therefore, from the relationships
it is clear that we must know (and find ways to
measure and/or predict) the principal elastic
constants E1, E2, G12, n12 ! This is the topic
of micromechanics models
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Micromechanics models for elastic constants

• Microstructural aspects
• The elastic constants can be measured through
  careful experiments. However, it is desirable to
  be able to reliably predict lamina properties as
  a function of constituent properties and
  geometric characteristics (such as fiber volume
  fraction and geometric packing characteristics).
• Thus we will be looking for a functional
  relationship of the following type

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Typical transverse cross-sections of
unidirectional compositesComposites with low
volume fractions tend to have a random fiber
distribution, whereas with high volume fractions
the fibers tend to pack hexagonally.
Silicon carbide/glass ceramic,
fiber diameter 15 mm, Vf 0.40
Carbon/epoxy, fiber
diameter 8 mm, Vf 0.70
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Expected volume fractions in composites using
representative area elements
NOTE - It is assumed that the fiber spacing (s)
and the fiber diameter (d) do not change along
the fiber length, so that area fractions volume
fractions
Max(Vf) _at_ sd Vf 0.907
Max(Vf) _at_ sd Vf 0.785
In practice, max(Vf) 0.5 0.8
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Fiber content, matrix content, void content

• For n constituent materials we must have for
  volume fractions vi Vi/Vtot

In many cases this simply reduces to
For weight fractions wi Wi/Wtot, we have
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(No Transcript)
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Since weight (density)(volume), we obtain
immediately
thus, a Rule of Mixtures, which for 2-phase
composites becomes
And in terms of weight fractions instead of
volume fractions
The void fraction can be calculated from measured
weights (not fractions) and densities
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Other structural aspects to be quantified
The fiber length distribution
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The fiber orientation and the fiber orientation
distribution
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The same formula may be used to calculate the
real depth of a hole from a (SEM) picture at an
angle
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2. Elementary mechanics models
Longitudinal Youngs modulus - Assume an
isostrain situation (Voigt 1910) implicitely
perfect interfacial adhesion
Assuming linear elasticity for the fiber and
matrix
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Assume Ac, Af, Am are the cross-sectional areas
of the composite, the fiber and the matrix. From
balancing the forces in the fiber direction, we
have
Therefore
and
Thus
(Voigt)
Also
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Transverse Youngs modulus - Assume an isostress
situation (Reuss 1929)
In this case it is easily shown that
(Reuss)
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Voigt bound
Reuss bound
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A few remarks

• The longitudinal modulus is always greater than
  the transverse modulus
• The ratio E11/E22 may be considered as a measure
  of the degree of anisotropy (or orthotropy) of
  the lamina
• Poisson ratio n12 is also predicted by a rule of
  mixtures
• The shear modulus G12 is predicted by an inverse
  rule of mixtures
• Improved (tighter) bounds were developed by
  various authors (Hashin Rosen)

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3. Semiempirical models

• The most successful semiempirical model is that
  of Halpin Tsai (1967)

where p represents composite moduli, for example
E11, E22, G12 or G23 pf and pm are the
corresponding fiber and matrix moduli. x is the
measure of the degree of reinforcement of the
matrix by the fibers, which depends on fiber
geometry and distribution, loading conditions,
etc. It is essentially a fitting factor.
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These equations are quite accurate at low volume
fractions, a bit less so at higher volume
fractions. They have been modified to include the
maximum packing fraction. Specific values include
the following
Note finally that when x?0, the inverse rule of
mixtures is obtained, and when x??, the direct
rule of mixtures is obtained.