The Design of Composite Materials and Structres

Composite Materals Design - A Simple Tensile Test Simulator

You should download the MATHCAD document that contains this design exercise, if you haven't done so already, 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 build a simple tensile test simulator using MathCAD that builds on the equations developed previously in the strength calculator, but first we must recognize that there are two modes in which a tensile test can be carried out, one where a deflection is imposed and the resultant load measured and the second where a load is applied and the resulting deflection measured.

The first case, where a deflection is imposed and the resulting force of resistance by the materials is measured, is typical of the type of tensile test carried out in a screw driven machine. Thus load is a function of deflection, though it is actually easier to write down stress as a function of strain. For composite materials we have seen that the yield and ultimate tensile strength depend on the volume fraction of fibres, so the function that we define for stress will need to have two variables, volume fraction f, and strain e.

Now for the logic used to create a single function....

When the strain is less than the lesser of the matrix or fibre failure strain the composite will behave as a linear elastic solid and the stress is simply the elastic modulus of the composite parallel to the fibres multiplied by the strain (Hooke's Law), E_x(f).e.
In MathCAD this is best implemented using the IF() function where the logical test is based on strain being less than the critical strain.
Once this critical strain is reached we then need to consider two further possible scenarios, (a) that the fibres remain intact or (b) that the matrix remains intact.
Again, the selection of the appropriate case can be implemented in MathCAD using the IF() function where the logical condition is based on which is the lesser strain to failure, e_m or e_f.
In both cases, the composite will continue to behave as a linear elastic solid - we assume that the matrix and fibre are both linear elastic to fracture! - but the elastic modulus will be reduced, for case (a) the stress will simply be the product of the elastic modulus of the fibre and strain scaled by the volume fraction of fibres, f, while for case (b) the stress will be the product of the elastic modulus of the matrix and the imposed strain scaled by the volume fraction of matrix, (1-f).
Finally we need to account for the fact that when the composite reaches a strain where it actually breaks, the measured load drops to zero.
Again, an IF() statement can catch this condition in each of cases (a) and (b) above with the logical test being if the strain is greater than the respective failure strain.

To plot load as a function of deflection we can vary the imposed strain from zero to the larger of the matrix or fibre failure strain, - or any desired value for that matter - then for deflection, plot strain multiplied by the gauge length, while for load, plot the product of stress and the original cross-sectional area, noting that stress is a function of volume fraction, f, and strain, e.

Question.

Answer.

How would you use the same logic to define strain as a function of an imposed stress? Use your answer to plot the load - deflection response for a tensile test carried out under load control using a servo-hydraulic testing machine.

A suitable logical function would define strain in terms of the imposed stress and the volume fraction of fibres, , testing first for an imposed stress less than the "yield stress" and then handling the elastic behaviour of the remaining, unbroken, component of the composite. Recall that for solids which are linear elastic to failure, strain is just stress divided by the appropriate stiffness (Hooke's Law)

	Review this section on strength of aligned fibre composites.
	Move on to examine the effect of orientation on strength of aligned continuous fibre reinforced composites.