Composite Materials Design - Effect of Orientation on Stiffness and Strength
We showed how to use MathCAD to plot out the variation of the elastic properties with orientation in a previous lesson. In this exercise we will develop an approach which will enable the strength to be evaluated as a function of orientation and to determine whather or not a simple laminate fails when subjected toan arbitrary set of imposed stresses. We have already defined two simple functions that express the strength of the aligned fibre composite parallel and perpendicular to the fibres as a function of the fibre fraction and incorporated these into the Tsai-Hill maximum strain energy failure criterion.  Does the laminate break under a set of imposed strains/displacements?In order to determine when failure occurs we simply need to establish what the imposed stresses are when aligned with the fibre axis. We shall start by considering a set of imposed strains ex,ex gxy applied at an angle q to the fibre axis.  The imposed engineering strains are first converted to tensor strains, then the tensor strains are rotated through the appropriate angle using the T(q) matrix and then converted back to engineering strains.  Since the strains are now aligned parallel and perpendicular to the fibres we simply multiply the stiffness tensor of the laminate by the strains to determine the stresses parallel and perpendicular to the fibres.  The stresses can now be substituted into the Tsai-Hill criterion and we can determine if the imposed strains were sufficient to fracture the laminate or not, depending on whether or not the result was greater than or less than 1.  Thus we have now defined a function PLYFAILSTRAIN(volume fraction, angle, imposed strain) that will enable us to determine, given an imposed set of strains, whether a unidirectional laminate with a volume fraction of fibres, f, will fail or not. Does the laminate break under a set of imposed stresses/loads?As in the previoous example we shall define the imposed stress tensor and its orientation with respect to the fibre axis  The simplest way to proceed, since we already know what the compliance of the laminate is at this orientation is to determine the strains that result from the imposed stress by multiplying the compliance tensor (itself a function of the volume fraction of fibres and the angle) with the imposed stress tensor. i.e.,  We can now substitute these strains into the PLYFAILSTRAIN() function that we defined in the previous section to determine if the laminate fractured.  Hence we can define a similar function, PLYFAILSTRESS(volume fraction, angle, imposed stress tensor) that will tell us whether or not the simple laminate fractures using the Tsai-Hill maximum strain energy criterion.
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