Composite Materials Design - Effect of Orientation on Stiffness and Strength | You should download the MATHCAD document that contains this design exercise and open it using version 11 or higher of MATHCAD in order to follow along.
There are some exercises for you to try at the end of this section. |
In this section I'll show you how to use MathCAD to calculate the elastic constants of a simple fibre reinforced composite at any angle to the fibres. We will then use the functions developed to calculate the stresses and strains that result in the composite when a given set of deflections or loads are applied to that composite material. Finally, we will develop a function that will predict the uniaxial strength of a composite in which all the fibres are aligned parallel to each other for any fibre orientation relative to the applied stress.
As in all the previous examples we will need to define the mechanical properties of the matrix and fibre materials, remembering that the fibre may be anisotropic.
Next the micromechanical equations that define the elastic properties of the aligned fibre composite are defined in terms of the fraction of fibres f, since ultimately, we will shall want to calculate the properties of the material for any volume fraction of fibres.
We do the same for the strength parallel, X(f), and perpendicular, Y(f), to the fibres and implement the Tsai-Hill maximum strain energy criteria of failure for multi-axial loading, recalling the the stress is actually a vector of three stress, the stress parallel and perpendicular to the testing direction and the shear stress: we'll use the built-in min() function to pick the appropriate minimum strain parallel to the fibres and minimum stress perpendicular to the fibres.
We now set up the compliance matrix for the composite parallel to the fibres using the simple models set out above such that the compliance is solely a function of the volume fraction of fibres, f. The stiffness tensor Q is just the inverse of the compliance tensor S!
We can define a rotation matrix, T, where q is the rotation in the x-y plane from the SPECIALLY ORTHOTROPIC '1-2' lamina (where the principal stresses are aligned parallel and perpendicular to the fibre axes) to the GENERALLY ORTHOTROPIC 'x-y' lamina (where the principal stress are not aligned with the fibre axes). The R matrix is required to convert tensor strain to engineering strain. Finally, we define the compliance (Sc) and stiffness matrices (Qc) of the composite at any angle such that the compliance and stiffness matrices are functions of both the fibre volume fraction, f, and the angle between the applied stress and the fibre axis.
The 3 elastic modulii are then
which can be plotted out as
So we can now see that the elastic modulus of a composite in which all the fibres are aligned in a single direction drops off rapidly after about 3 or 4° of misalignment. Notice how MathCAD combines and hides all the matrix algebra when defining the 3 main elastic constants in terms of volume fraction and angle.
 | Return to review the effect of orientation on elastic properties in aligned fibre composites.
|  | Move on to examine the effect of fibre orientation on the measured strength of aligned continuous fibre reinforced composites.
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