Materials Background

Materials Chemistry

Composite Theory

Composites are made from a combination of reinforcing fibers and matrix components to create a material with specific optimal properties. The two components control different aspects of the composite - reinforcing fibers influence the strength and stiffness properties while the matrix acts to transfer loads between fibers and protects against environmental conditions. The matrix binds the fibers together and is usually composed of metals, polymers, or ceramics.

Reinforcing fibers are usually much stronger than the bulk material because they have fewer defects. However, the fibers are only effective uniaxially and not in transverse directions. Different types and lengths of fibers are available to suit specific needs. Continuous fibers are used in applications that require high strength and stiffness in a single direction. Randomly oriented short fibers offer less strength, but improve the isotropic properties of the composite. Fabric fibers may also be woven in specific orientations to improve strength and prevent torsion over a large area.

The strength and stiffness of a composite depends on the orientation of the reinforcing fibers. Both the longitudinal and transverse modulus of elasticity influences the total stiffness of a composite. The longitudinal stiffness in the direction of the fiber orientation can be calculated by the rule of mixtures, which is a volume average summation of the elasticity in the fibers and in the matrix.

El = Efvf + Em(1-vf)

where Ef is the modulus of the fibers, Em is the modulus of the matrix and vf is the volume fraction of fibers in the composite.

The transverse modulus of elasticity is

1/Et = (1-vf)/ Em + vf/Ef

This equation indicates that the fibers do not greatly contribute to the stiffness in transverse directions unless their volume fraction is high. Both the longitudinal and transverse modulus of elasticity assumes that the stresses are equal in the fibers and matrix. While thermal and moisture expansion are also factors that affect stiffness, they do not contribute greatly in this application.

The strength of the composite takes into account the longitudinal and transverse tensile and compressive strengths. The longitudinal tensile strength is the only property controlled predominantly by the fiber strength. The longitudinal compressive strength is usually lower than the tensile strength by a factor of one-half or greater. The transverse tensile strength is mostly controlled by the matrix strength, defect concentration within the matrix, and the adhesion between the matrix and the fibers.

Failure caused by buckling is governed by the shear properties of the composite as well as fiber misalignment. The misalignment is especially due to the curing process; the fibers become misoriented when the polymer component shrinks between 3 to 9% of its volume. Depending on the processing, there may be fewer defects present in the matrix of a composite than in the bulk matrix material. However, the fibers may add stress concentrations. The same factors are considered in predicting the transverse compressive strength, and because of the complex nature of these factors, the strengths cannot be calculated to a great degree of accuracy.

The fiber orientation must be correctly chosen to achieve the appropriate strength and stiffness for the board. Because loose fibers are very hard to control and process, we are looking at either fibers woven into fabrics, or fibers pre-impregnated with epoxy. Woven fabrics allow easy processing because they come in sheets that can be cut and wrapped to shape, requiring only the addition of epoxy. However, the directions of the fibers are specified in advance, thereby constricting the fiber orientation and strength. Fibers in a weave are also not as strong as those oriented in a single direction within one layer because they contain a slight bend from being entwined together. Fibers can also be packaged as prepreg layers, in which the fibers are oriented in the same direction and are already held together by epoxy. Prepreg layers are easy to process and allow for maximum manipulation of strength in all directions because the orientation of fibers per layer can be independently arranged to achieve the desired overall strength.

A composite laminate can be made up of several prepreg layers with varying fiber orientation. The main advantage of these laminates is the ability to compensate for weakness in the transverse directions. The properties of a laminate depend on the tractions, N, the bending moments, M, and the stiffness of each layer. As a laminate is bent, each layer has a certain strain, e, and concequently a curvature, k. The traction must balance the stresses determine from the strain and curvature in each ply. The resulting equation is the sum of two matrices, A and B.

N = Ae + Bk

A is the extensionsl stiffness matrix which accounts for the strains. B is the coupling stiffness matrix which accounts for the curvature.
The moments of the ply stresses must also balance the moments of external forces.

M = Be + Dk

where D is the bending stiffness matrix. These equations combine to form the single equation that can be used to determine the stiffness of the entire laminate as well as the maximum allowable stresses that may be applied:

|e| = |A B| |N|
|k| |B D| |M|

There are various ways in which to orient the fibers of consecutive layers, but the two under consideration for this project are the symmetric and cross-ply laminates. In a symmetric laminate, the angles of orientation are symmetric to a central layer (e.g. 45, 0, 0, 45 degrees). These laminates do not experience bending or warping due to thermal effects. Cross-ply laminates consist of fiber orientations of 0 and 90 degrees, which simplifies the design and construction.

Materials Selection

Because the board must be very strong and moderately stiff, some of the materials we first considered for the fibers were carbon, glass and Kevlar. The tensile strengths of each are comparable at 1100 MPa, 1800 MPa, and 2000 MPa, respectively. However, the Young's modulus of carbon in the fiber direction of 230 GPa is much different compared to 55 GPa and 80 GPa for glass and Kevlar respectively. We chose to use carbon prepreg for the interior layers because it is easy to cut and layup. A Kevlar/carbon weave will be used for the outer layer because it is very strong and is easy to wrap around the board.

The core material is very important to the design because it must be lightweight as well as strong in compression. Some of the materials we originally considered were honeycomb, wood and several types of foam. Wood is much too heavy to use for the core and is also very hard to shape. Foam is light and relatively easy to shape, but is not as strong in compression and will lose its shape after several uses. Therefore we chose to use nomex honeycomb because it has all of the properties we are looking for.


Materials Properties

Properties Carbon Prepreg Kevlar/carbon Weave Nomex Honeycomb
E (fiber direction MPa) 131,000 75,800 192.5
E (transverse direction MPa) 6,200 131,000 192.5
G (MPa) 4,830 3,450 63
Poisson's Ratio 0.25 0.30 0.05
Ply Thickness (m) 0.0001 0.0005 0.005