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Free-Edge Stresses in a Laminate

Material Properties

Layer material stiffness properties are entered in the same format used by MaterialMinds`laminate. The difference is that 3-dimensional properties must be used.

The following property list has the order {E_x, E_y, E_z, G_xy, G_xz, G_yz, ν_xy, ν_xz, ν_yz].
ThermProp is a list of thermal properties {alpha1, alpha2}.  
Note: A hybrid laminate can be defined by nesting multiple materials together (note the double bracket, even for a single material). These properties have been used in several studies of the edge stress phenomena.


prop = {{20 10^6, 2.1 10^6, 2.1 10^6, .85 10^6, .85 10^6, .85 10^6, .21 , .21 , .21}} ;

thermProp = {{.22 10^-6, 15.2 10^-6}} ;

σ_33 of [0/90] _S Laminate

Create a 0/90 laminate.

As with the laminate package, the format is {{angle1, matl1, h1},{angle2, matl2, h2},...}. Angles must be in radians. Unlike other packages, edgeStress assumes that the laminate is symmetric. Therefore, only half the laminate is entered. The first layer is on the outside surface, the last is adjacent to the centerline.

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

Because only a single case is needed, edgeSolve is placed directly in the case function. Twelve terms are assumed. The applied mechanical strain is 10^-6.

soltn = edgeSolve[prop, thermProp, lam, 12, 10^-6, 0] ;

Plot the 33 component of stress along the centerline (z=0).
The plot goes from y=0 to y=2, with 50 evaluation points. The y coordinate is normalized by the total thickness.
Note: Most references normalize the y coordinate by 2h.

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σ_23 of [0/90] _S Laminate

The same solution vector is used to evaluate σ_23(or σ_4using contracted notations). The number of evaluation points is increased because stress gradient is very large near the edge (y=0).
The stress evaluation is at the 0/90 interface (z=0.25). σ_23would be zero at the centerline.

                                                                                               ...                                                                                                 23



σ_33at Free-Edge of [0/90] _S Laminate

It is visually appealing to plot the z coordinate on the vertical axis, and stress as the horizontal axis. verticalPlot is a convenient method to reverse the normal axes for a list plot.

Using the function to recreate one of the curves from Figure 4.

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Average σ_33 in [45/-45/0/90] _ (2S) Laminate Taken Over Interval y/h=0 to y/h=16

The interlaminar stresses in a laminate can be singular at the free edge. The method used in this package cannot represent a singularity. The peak edge stress will continue to change as the number of terms in the approximation series is increased. However, the average stress, taken over a small distance near the edge will converge with a finite number of terms. The average stress has also been suggested as a failure prediction method.

For thicker laminates, it is convenient to use the utility homoStackDeg from the laminate package. This allows one to enter a list of angles in degrees.
The function returns a stack definition list in standard form. All of the layers of 0.1 in. thick.

lam = homoStackDeg[{45, -45, 0, 90, 45, -45, 0, 90}, .1] ;

Perform the solve with 20 terms.Evaluate for an applied axial strain of 10^-6.

soltn = edgeSolve[prop, thermProp, lam, 20, 10^-6, 0] ;

Plot the average stress. To compute the average, the stress is integrated from y=0 to y=1/16 (one ply thickness in from the edge).

                                                                                               ...                                                                                                 33



Average σ_33 in [45/-45/90/0] _ (2S) Laminate resulting from thermal stress, assuming ΔT=-200F.

The code can also handle the effects of residual thermal stress. The thermal property list included thermal expansion coefficients.

Using the stored solution, create a case for zero mechanical strain, and a -200F temperature change from the stress-free condition.

soltn = edgeSolve[prop, thermProp, lam, 8, 0, -200] ;

Plot the average 33 stress.

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This is different than the reference paper, but I now believe the original calculation was incorrect.

Created by Mathematica  (March 7, 2004)