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 .


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



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?


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,


Use these in definition

Result:

General:


Effective Transverse Modulus,

Combine these with definition

Result

General


Effective Shear Modulus,

Since for this 1-D analysis

Combine these with the definition

Result:

Same form as transverse modulus not considered very accurate


Load in x1 direction

and consequence of symmetry of constitutive matrix

Challenge: Calculate directly using SOM.


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.


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?


Moisture Induced Expansion

p. 104

Usually assume (since fibers prevent expansion)

densities

Compare with thermal expansion equation

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1_micromechanicsIntroduction