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Composite Micromechanics

Spherical Particles

Modulus

In[1]:=

Needs["MaterialMinds`micromechanics`"]

Use a case from [4], pg 57. Glass Microspheres in a polyester matrix.

In[2]:=

matrix={1,.45};

In[3]:=

particle={40.8,.21};

In[5]:=

? CompositeSpheres

CompositeSpheres[{Em,num},{Ei,nui},vi,opts] returns the effective properties of a composite wi ... r,SphereUpperBoundShear, and SphereLowerBoundShear. Returns list for composite properties {Ec,nuc}

In[6]:=

CompositeSpheres[matrix, particle, .5, ShearModelCSAUpperBoundShear]

Out[6]=

{5.35916, 0.353278}

Plot the Young's modulus, E using various assumptions for the shear modulus model. This may be compared to Fig. 2.4 of Christensen, [4].

Build a function that is simply a shortcut for plotting the desired quantities. The last [[1]] means take the first element of the list returned by CompositeSpheres, which will be the modulus.

In[8]:=

func[x_,c_]:=CompositeSpheres[matrix,particle,c,ShearModel->x][[1]];

Use a built in capability to draw legends on plots

In[12]:=

<<Graphics`legend`

Plot[{
func[SphereLowerBoundShear,c],
func[SphereUpperBoundShear,c],
func[SphereThreePhaseShear,c],
func[CSAUpperBoundShear,c]
},{c,0,1},
PlotStyle->{Hue[.2],Hue[.4],Hue[.8],Hue[1]},
AxesLabel->{"vol frac","Modulus"},
PlotLegend->{"Upper Bound","Lower Bound","Three-Phase Model","Hashin"},
LegendPosition -> {.8, 0},
LegendTextSpace -> 4]

[Graphics:HTMLFiles/index_7.gif]

Out[22]=

⁃Graphics⁃

Phase Average Stress

Compute effective properties for a particular volume fraction (0.4)

effec=CompositeSpheres[matrix,particle,.4]

{3.82251, 0.378821}

Phase average stresses in matrix and particles for a unit axial average stress

PhaseAverageStress[effec,matrix,particle,.4,{1,0,0,0,0,0}]

{{0.577525, 0.189058, 0.189058, 0., 0., 0.}, {1.63371, -0.283586, -0.283586, 0., 0., 0.}}

Fiber Reinforced Composite

Match Table 1 of  [5].

fiber={345,9.66,2.07,.2,.3};
epoxy={3.45,.35};

TableForm[Table[Flatten[{c,YoungsToBulk@CompositeCylinders[epoxy,fiber,c]}],{c,0,1,.2}],
TableHeadings->{None,{"v","EA","k","GT+","GA","nuA-"}}]

v EA k GT+ GA nuA-
0 3.45 4.25926 1.27778 1.27778 0.35
0.2 71.7752 4.64386 1.5317 1.40474 0.317707
0.4 140.092 5.08589 1.86497 1.54497 0.286624
0.6 208.402 5.59923 2.30424 1.70067 0.256686
0.8 276.704 6.20265 2.88398 1.87456 0.227831
1. 345. 6.92215 3.71538 2.07 0.2

Thermal Expansion Coefficient

Match the figure in [9]

In[23]:=

fiber = {7 ,.2};
matrix = {.315 ,.382};

In[28]:=

Plot[{Levin[CompositeCylinders[matrix,fiber,c],matrix,fiber,6,.49][[1]],Levin[CompositeCylinders[matrix,fiber,c],matrix,fiber,6,.49][[2]]},{c,0,.8},
PlotStyle->{Hue[.1],Hue[.5]},
AxesLabel->{"vol frac","α"},
PlotLegend->{"Long.","Trans"},
LegendPosition -> {.8, 0},
LegendTextSpace -> 2]

[Graphics:HTMLFiles/index_11.gif]

Out[28]=

⁃Graphics⁃

Created by Mathematica  (March 7, 2004)