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Effective Properties of a 2D Weave

The effective properties of the weave can be calculated from the 3-dimensional model.
Matrix properties for AS4/977.

matrixProps = {0.65 * 10^6, 0.37} ;

Generate plain weave model using parameters previously defined. Output is hidden here, however weaveRVE must be executed prior to using showRVE.
Note that np1 = 3 and np2 = 3.  In this case, there are 3 piece-wise sections to be used when approximating the sinusoidal parts of the bundle cross-section and the bundle path.

pweave = weaveRVE[plainWeave, weaveInputForm1[packingDensity, FilamentDiameter, numberOfFilamentsInBundle, bundleWidth, VolumeFraction], {3, 3}] ;

Show the representative volume element (RVE) of the defined plain weave.  By setting surfaceColors → True, we can distinguish between the warp and fill fiber bundles.

showRVE[pweave, surfaceColorsTrue]

[Graphics:HTMLFiles/index_4.gif]

⁃Graphics3D⁃

Generate 5 harness satin weave model

sweave = weaveRVE[satin5harness, weaveInputForm1[packingDensity, FilamentDiameter, numberOfFilamentsInBundle, bundleWidth, VolumeFraction], {3, 3}] ;

Show the representative volume element of the 5-harness satin weave defined for the warp group of bundles only (set groups → {1}).
surfaceColors is defaulted to False here since we are only dealing with one group.

showRVE[sweave, groups {1}]

[Graphics:HTMLFiles/index_8.gif]

⁃Graphics3D⁃

Show the representative volume element of the 5-harness satin weave for all groups using surface colors to distinguish between warp and fill bundles.

showRVE[sweave, surfaceColorsTrue]

[Graphics:HTMLFiles/index_11.gif]

⁃Graphics3D⁃

Compute the effective properties of each weave using the default upper bound estimate. The Mathematica function MatrixForm is used here for ease in viewing the Cij matrix.

MatrixForm[effectiveProperties[pweave, matrix, bundleProps]]

( 6.559992663970444`*^6    685897.1081950658`       1.359636532414121`*^6    0         ...      0                        0                        0                        582795.1403424414`

MatrixForm[effectiveProperties[sweave, matrix, bundleProps]]

( 7.542200120743876`*^6    689306.8674801969`       951701.1048950515`       0         ...      0                        0                        0                        599864.9004348294`

Since no coupling is evident from the above calculations, we can compute the effective engineering constants.
In this step, the effective engineering constants for each weave are calculated. The relevant properties are in PSI.
    First, consider the plain weave. Here only mechanical properties are calculated because thermal properties are not entered.  
    Note that the function defaults to an upper volume average estimate.

pprops = effectiveEngConstants[pweave, matrix, bundleProps]

{5.5939*10^6, 5.5939*10^6, 1.42885*10^6, 582795., 1.21433*10^6, 1.21433*10^6, -0.0477001, 0.734615, 0.734615}

Next calculate the mechanical properties for the 5-harness satin weave.

sprops = effectiveEngConstants[sweave, matrix, bundleProps]

{7.00337*10^6, 7.00337*10^6, 1.47147*10^6, 599865., 802510., 802510., 0.0219581, 0.550272, 0.550272}

Calculate all effective properties, including thermal, using 2 different estimates using upper bound estimates.

ppropsUpper = effectiveEngConstants[pweave, matrix, bundleProps, matrixThermal, bundleThermal] ...  spropsUpper = effectiveEngConstants[sweave, matrix, bundleProps, matrixThermal, bundleThermal] ;

The order of the effective properties is listed below.

standard3DOrder = {E , E , E , G  , G  , G  , ν  , ν  , ν  , α , α ,  ... y   xz   yz        xy        xz        yz        x        y        z        xy        xz        yz

The increased number of undulations in the plain weave reduce its in-plane stiffness.
The properties can be viewed in table form and compared. In order to view the properties in this manner, these 2 nested lists are flattened (converted to 1 list).

TableForm[Transpose[{Flatten[ppropsUpper], Flatten[spropsUpper]}], TableHeadings {standard3DOrder, {"Plain Weave", "5-Harness Satin Weave"}}]

Plain Weave 5-Harness Satin Weave
E  x 5.5939*10^6 7.00337*10^6
E  y 5.5939*10^6 7.00337*10^6
E  z 1.42885*10^6 1.47147*10^6
G  xy 582795. 599865.
G  xz 1.21433*10^6 802510.
G  yz 1.21433*10^6 802510.
ν       xy -0.0477001 0.0219581
ν       xz 0.734615 0.550272
ν       yz 0.734615 0.550272
α       x 4.04681*10^-6 3.48853*10^-6
α       y 4.04681*10^-6 3.48853*10^-6
α       z 0.0000264403 0.0000312226
α       xy 0 0
α       xz 0 0
α       yz 0 0

Created by Mathematica  (March 7, 2004)