Mathcad Exercise 3. A simple strength 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.

In this section i'll show you how to implement a composite materials design tool based on the desired strength of the composite material. First we need to ensure that all the relevant materials properties are defined, including the elastic strains to failure of the fibre and matrix.

If we recall the previous section describing the strength of composites with aligned fibres, then we now know that the onset of failure occurs at the lesser of the failure strain of either the matrix or the fibres - assuming that the two materials are linear elastic to failure, i.e., BRITTLE! Thus we can define a critical strain using the MIN() function in MathCAD.

We can now define 3 different stresses, (a) the stress in the composite parallel to the fibres at the critical strain, (b) the stress in the composite when only the fibres remain to support the load and (c) the stress in the composite when only the matrix remains to support the load. Each of these 3 stresses depends on the volume fraction of fibres so we make them all functions of f, the fibre volume fraction since we want the design tool to be able to find a desired volume fraction.

The ultimate tensile strength of the composite depends on which is the larger of (a) or ((b) if the the failure strain of the fibres is greater than that of the matrix, or (c) if the failure strain of the matrix is greater than that of the fibres). This logical statement can be written using the IF() and MAX()functions in MathCAD and, since all 3 stresses are functions of f, we shall also make the UTS a function of f.

We can now plot out each of the 4 stresses to see how the working strength and UTS vary with volume fraction. 

We can also create a simple calculator to determine both the working and ultimate tensile strengths as well as implement the ROOT() function to enable use to find the volume fraction of fibres that would enable us to produce a composite with either a desired strength or strength to density ratio parallel to the fibres.

If you look closely at the MathCAD design tool you should notice that we have been re-using almost all the functions that we derived for the stiffness design tool in the strength design tool. In the next exercise we'll plot out the load deflection curves as you would measure in a simple tensile test.