Classical Lamination Theory (CLT) predicts the properties of laminated composite materials made up of many individual fiber-reinforced layers, including the stress and strain on a composite resulting from a given load. For my class in composite materials, I wrote a series of MATLAB functions to perform these calculations. This undertaking was entirely extracurricular, and taught me a lot about implementing models in MATLAB.
Introduction
Overview of laminated composites
Models for lamina properties
Classical Lamination Theory
Implementing CLT in MATLAB
The code
References
A fiber-reinforced composite is made up of multiple layers known as laminas, each of which is composed of long thin fibers embedded in a matrix of supporting material. The determination of a lamina's properties, and even more so those of the composite made up of various individual laminas, is not straightforward. There are no derivations from first principles of the properties of such a lamina. Composites are non-isotropic, meaning their properties are not identical in every direction (unlike metals, for instance). Numerous empirical models have been developed to predict the elastic modulus, shear modulus, and Poisson's ratio of a lamina from the properties of its constituent fiber and matrix, giving results that more or less match the measured properties of the actual material (some properties can be predicted more accurately than others). The rotation of the lamina introduces another complication, as the properties must then be transformed by a relation involving the angle of rotation. The properties of the entire composite depend not only on those of its individual layers and the orientation of those layers with respect to the composite, but also the thickness of each layer and its position in the composite stack. Classical Lamination Theory was developed to deal with these considerations. Its implementation in computer code is straightforward (if tedious), as it depends only on linear relations and the sine and cosine functions.
Lamina model: fibers, matrix, [void] - diagram
Coordinates (1-2 plane)
Fiber and matrix properties
Rotation convention
Laminate: stacking convention and coordinate system
E1, E2, G12, ν12
Transformation under rotation (brief overview; main discussion in CLT section)
Stiffness matrix ABD, composed of individual stiffnesses A, B, and D
[NM] = [ABD][ek], [ek] = {[ABD]^-1}[NM]
Calculation of [ABD] from layer properties and rotations (Q, Qbar)
Calculating stress from strain, in the x-y and 1-2 directions
manual P, Qbar, NMek for simple calx or user input
main calculator: matrix of fiber/matrix properties + angles [P] -> [Qbar] (Qbar)
[Qbar] -> A, B, D (ABD)
[ABD] + ([NM] or [ek]) -> [ek] or [NM] (NMekAuto)
[ek] + [Qbar] + [z] -> [Sigma xy] (Sigma_xy)
[Sigma xy] + [P] -> [Sigma 12] (Sigma_12)
[Sigma 12] + [F12] -> [FS12] (FS12, for project)