The Design of Composite Materials and Structres

Composite Materials Design - A Simple Stiffness Calculator

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.

The design tool we shall create will enable us to calculate the elastic properties of an aligned continuous fibre composite in the two principle directions, i.e. parallel and perpendicular to the fibres. We will then add the ability to determine the volume fraction or weight fraction of fibres needed to create a composite with specified elastic properties in the principle directions. Finally i’ll show how to extend the design tool to determine the principle elastic properties of composites made from mixtures of more than one type of fibre.

In the first part of the design tool we must write down the elastic constants of both the fibre and matrix materials - that way, at a later date, we can enter the values for any fibre/matrix combinations we choose. A table of values is provided and remember that the fibre might be anisotropic! Any units can be used and Mathcad automatically converts them to their SI equivalent. (Units such as ksi, Msi, MPa, GPa - may need to be defined in terms of basic units such as Pa and psi e.g., ksi = 10³*psi)

The only variable in the design of these composites, once the fibre and matrix materials have been defined, is the volume fraction of fibres, f. So we make all the properties of the composite material functions of f. That way we do not need to explicitly define a value for f. For completeness we have also included the two functions necessary to convert back and forth between weight and volume fractions.

Now let’s see how the various elastic properties of the composite vary as the volume fraction of fibres is increased from 0 to 1.

Question.	Answer.
The variation of those same properties with weight fraction can also be plotted. How?	Simply change E_x(f) to E_x(f_v(f)) - that way when we substitute a fraction from 0 to 1, it is first converted from weight to volume fraction and that newly calculated value is then substituted in the equation that describes the modulus parallel to the fibres.

To use the properties calculator, simply enter a value betwee 0 and 1 for the volume fraction and the elastic and physical constant of the composite are displayed.

Question.	Answer.
How would you use the calculator to show the properties for a given weight fraction of fibre?	change f:=0.4 to f:=f_v(0.4) , you can then enter any number you want, between 0 and 1, and the fraction will first be converted from weight to volume before being used in the property calculator.

Say I wanted to design a fibreglass composite with an elastic modulus parallel to the fibres the same as that of aluminum, 69 GPa. Is it possible? To do this we must first go back to the top of the mathcad document and change the properties of the fibre to reflect those of E-glass. A table of mechanical and physical properties of a number of common fibres and matrices is provided.

There are two ways to solve for an unknown in MathCAD - first using a solve block as shown below

and secondly using the root function

When using either approach, an initial guess for the unknown is required. Also you should note that when using the root function, it is better to write the equality as a ratio E_x(f)/E-1 rather than E_x(f)-E. This is because Mathcad uses numerical methods to iterate to a solution and very large or very small numbers can cause problems.

Question.	Answer.
Is it possible? Fibreglass with the same elastic modulus as aluminum?	Answer. NO! the fraction is too high to be made - remember, the practical limit of 60% by volume. So what material would we choose for the fibre? If we look at the equation describing E_x, then basically any fibre whose modulus multiplied by 0.6 (the maximum amount of fibre)is greater than that of the desired value will work . So put in the values for AS4 graphite and look at the new answer!

Question.	Answer.
How about a fibreglass with the same stiffness per unit weight as aluminum?	Re-enter the properties for E-glass in the fibre section. Now re-write both the solve block and the root equation such that E_x(f)/r(f) = E/r where r should have been defined as 2.7 gm.cm^-3 for aluminum. (use gm for grams not g which is the acceleration due to gravity!) In this case YES!, it is possible to make a fibreglass composite with the same stiffness per unit weight as aluminum.

	Return to review the previous section on stiffness of aligned fibre composites.
	Move on to examine the strength of aligned fibre composites.
	Some composite materials are constructed using more than one type of fibre, for example, mixing carbon and kevlar fibres is common, where the high modulus of carbon is combined with the high toughness of the much lower stiffness kevlar. In the following section I'll demonstrate how to modify the simple design tool to accommodate hybrid or mixed fibre reinforcements. This section will show you some new concepts in MathCAD such as How to use vectors How to use summations when the number of terms being summed may be redefined How to "Vectorize" calculations How to use a simple programming block to define a function