1_micromechanicsIntroduction
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Transcript of 1_micromechanicsIntroduction
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First Order Concerns"Scalar properties"
Definitions
Some "General" EquationsNote: There are lots of ways to express these relationships. (For example, two forms are given for .
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Micromechanics and MacromechanicsContinuous fiber composites
Goal: To determine the composite behavior from the properties of the constituents. (i.e. to determine the "effective" or "homogenized" properties) Typical requirement: periodicity of the material structure. (unit cell) Examples
How are effective properties equivalent to the original heterogeneous configuration?Consider the following comparisons Simulated experiment of composite vs. equivalent material Volume averaged stress vs. volume averaged strain of composite vs. that for
equivalent material Strain energy of composite vs. equivalent material
What are the constituents?
Micromechanics: constituents = fibers + matrixResult = effective engineering properties
Macromechanics: constituents = lamina of different types and orientationsResult varies: effective engineering properties, ABD matrices
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Approximations: geometric and otherwise
What stresses are assumed to exist? What variations of stresses are assumed?…Ditto for strains
Approximate analyses for determining effective properties violate reality in one or more ways: Distribution of constituents Shape of constituents Compatibility of displacements Equilibrium Dimensionality (e.g. 1D or 2D analysis)
What is the source of error for the fiber-direction effective modulus?
What are the sources of error for the effective modulus perpendicular to the fiber direction?
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Strength of Materials Micromechanics
Assume = “fiber direction”
Property Assumptions Idealized model
* Matrix and fibers loaded in parallel* Same axial strain* Determine average stress
* Matrix and fiber loaded in series* Same axial stress* Determine average strain
* Matrix and fiber loaded in parallel* Same axial strain* Determine from net contraction
Because the constitutive matrix is symmetric,
* Matrix and fiber loaded in series* Same shear stress* Determine average strain
* In each case we assume only one and exist.* In each case we assume either constant or . Also, within a single constituent both
the stress and strain are constant.
Notation for solutions
superscript “c” composite value … either same for all constituents or is the
“effective” value
subscript f fiber value
subscript m matrix value
Vf, Vm = volume fractions of fiber and matrix, respectively
Effective Longitudinal Modulus,
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Use these in definition
Result:
General:
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Effective Transverse Modulus,
Combine these with definition
Result
General
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Effective Shear Modulus,
Since for this 1-D analysis
Combine these with the definition
Result:
Same form as transverse modulus not considered very accurate
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Load in x1 direction
and consequence of symmetry of constitutive matrix
Challenge: Calculate directly using SOM.
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Some Simple Formulas for Transport PropertiesTransport Properties
p.105
heat conduction
electrical conduction
moisture diffusion
transport of electrical and magnetic fields
Estimate
Longitudinal: (i.e. rule of mixtures)
Transverse: Use Halpin-Tsai equations
and
natural log?
Moisture Absorption (mass transport … diffusion)
p.106
Fickian … like heat conduction
Non-Fickian
Text gives solution for 1-D Fickian (I think).
Some “useful” equations for estimating moisture content of plate vs. time (how long
for saturation)
We will return to this if time permits.
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Thermal Expansion
p. 101 Schapery
(3.67)
(3.68)
For
We may return to derivation later.
How would you calculate thermal expansion coefficients using finite elements?
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Moisture Induced Expansion
p. 104
Usually assume (since fibers prevent expansion)
densities
Compare with thermal expansion equation