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Lamination Theory Package Examples

Needs["MaterialMinds`laminate`"]

Layer properties for T300/208 in form {E_x, E_y, G_xy, ν_xy}, using metric units

prop = {181 10^9, 10.3 10^9, 7.17 10^9, .28} ;

myStack = {0, 0, 0, 0, 90, 90, 90, 90}//PlyThickness[125 10^-6]//Sym//Mat[1] ;

We need to relate strains to applied load, so invert the inplane stiffness matrix

a = Inverse[Amat[prop, myStack]]

{{5.16783*10^-9, -8.7852*10^-10, 8.33089*10^-10}, {-8.7852*10^-10, 1.65644*10^-8, 2.19053*10^-8}, {8.33089*10^-10, 2.19053*10^-8, 5.49711*10^-8}}

MatrixForm[a]

( 5.167834285594072`*^-9 -8.78519953075865`*^-10 8.330887670364936`*^-10  ... 0692666`*^-8 8.330887670364936`*^-10 2.190529001069267`*^-8 5.497106590250056`*^-8

Compute the laminate strain for a unidirectional load of N_1=1 MN/m.

strain = a . {10^6, 0, 0}

{0.00516783, -0.00087852, 0.000833089}

Tabulate the layer stress in the layer coordinates

TableForm[plyStress[prop, myStack, strain], TableHeadings {Automatic, {"Sigma x", "Sigma y", "Tau xy"}}]

Sigma x Sigma y Tau xy
1 9.37025*10^8 5.88152*10^6 5.97325*10^6
2 9.37025*10^8 5.88152*10^6 5.97325*10^6
3 9.37025*10^8 5.88152*10^6 5.97325*10^6
4 9.37025*10^8 5.88152*10^6 5.97325*10^6
5 1.27283*10^7 4.43653*10^7 3.11571*10^7
6 1.27283*10^7 4.43653*10^7 3.11571*10^7
7 1.27283*10^7 4.43653*10^7 3.11571*10^7
8 1.27283*10^7 4.43653*10^7 3.11571*10^7
9 1.27283*10^7 4.43653*10^7 3.11571*10^7
10 1.27283*10^7 4.43653*10^7 3.11571*10^7
11 1.27283*10^7 4.43653*10^7 3.11571*10^7
12 1.27283*10^7 4.43653*10^7 3.11571*10^7
13 9.37025*10^8 5.88152*10^6 5.97325*10^6
14 9.37025*10^8 5.88152*10^6 5.97325*10^6
15 9.37025*10^8 5.88152*10^6 5.97325*10^6
16 9.37025*10^8 5.88152*10^6 5.97325*10^6

Ref:  [8], page 157, Example 1

thermProp = {-0.3, 28.1} 10^-6 ;

Set up a symmetric, quasi-isotropic stack with layer thickness of 2 10^(-3)m.

myStack = {0, 90, 45, -45}//PlyThickness[2 10^-3]//Sym//Mat[1] ;

The laminate thermal expansion coefficients can be calculated from the layer properties.
beta is a thermal curvature coefficient that is non-zero if the stacking sequence is non-symmetric.

TableForm[plyStress[prop, thermProp, myStack, {0, 0, 0}, 1]//Chop, TableHeadings {Auto ... x y xy

σ x σ y τ xy
1 406925. -269943. 12814.5
2 121059. -258041. -1330.82
3 406925. -269943. -12814.5
4 121059. -258041. 1330.82
5 121059. -258041. 1330.82
6 406925. -269943. -12814.5
7 121059. -258041. -1330.82
8 406925. -269943. 12814.5

Properties {Ex, Ey, Gxy, νxy} for a typical carbon/epoxy material. Units are MSI.

prop = {18 , 1.2 , .85 , .35} ;

carpetPlot[youngsModulusCarpet[prop], YAxis->"Young's Modulus"]

[Graphics:HTMLFiles/index_22.gif]

Layer strengths for T300/5208. Units are MPa. Form is {F_XT, F_YT, F_XC, F_YC, F_S}

strength = {1500, 40, 1500, 246, 68} ;

Cross-ply stack. The plot is for average stress, therefore the total laminate thickness is not important.

myStack = {{0, 1, 1}, {Pi/2, 1, 1}} ;

Layer failure envelopes using the maximum stress criterion

envelopePlotter[{prop}, myStack, maxStress[strength]]

[Graphics:HTMLFiles/index_28.gif]

⁃Graphics⁃

By using innerEnvelopePlotter, only the first-ply failure envelope is shown.

innerEnvelopePlotter[{prop}, myStack, maxStress[strength]]

[Graphics:HTMLFiles/index_31.gif]

⁃Graphics⁃

Created by Mathematica  (March 7, 2004)