Evaluation of Stresses and Strains in Aligned Fibre Composites
This section contains a pair of worked examples showing how to determine the stresses (given imposed strains) and strains (given imposed stresses) applied at an arbitrary angle, q, to the fibres in a composite made from aligned continuous fibres with a volume fraction of fibres, f. We shall make extensive use of the tensor notation to solve these problems in conjunction with the composite stiffness Q(f) and compliance S(f) matrices derived in the previous class for properties measured parallel/perpendicular to the fibre axis and the oriented composite stiffness Qc(f,q) and compliance, Sc(f,q), for properties measured at an arbtrary angleq to the fibre axis.
Calculation of stress state given an applied set of strains
First, define a system of engineering strains, applied to a composite with a volume fraction of fibres, f, and an orientation with respect to the fibre axis of q°.
The applied strain is rotated in order that the strains parallel and perpendicular to the fibre axis can be determined. It should be noted that the rotation of strain requires first the conversion of engineering strain to tensor strain, rotation through the required angle q then conversion back to engineering strain using the R matrix.
The stresses in the composite is simply obtained by mutiplying the stiffness matrix of the composite at 0° by the rotated strain
We can now determine whether or not this particular set of strains applied to the ply/composite will result in failure. For the sake of illustration, the equations have been written out in full so that all the individual steps involved in the calculation can be seen.
where Fail(s,f) is the Tsai-Hill maximum strain energy failure criterion in which X(f) and Y(f) are the failure stress of the composite with a volume fraction of fibres f measured parallel and perpendicular to the fibres, respectively.
Thus in the example shown, the imposed strains are insufficient to fracture the composite since the substitution of the stresses into the TSai-Hill strain energy criterion results in an answer that is < 1. This concept can be explored in more detail by associated MathCad file.
Calculation of strain given stress
First, define a system of engineering stresses, applied to a composite with a volume fraction of fibres, f, and an orientation with respect to the fibre axis of q°.
Given the imposed stress, s, the strains are calculated using the compliance of the composite, Sc(f,q) calculated at the test angle for this fibre orientation q.
Now that we have the strains calculated in the testing direction, we repeat the process described above for determining whether or not failure occurs.
In this case the imposed stresses were sufficient to fracture the composite as the result is > 1. Again this concept canbe explored by studying and using the associated MathCad file.
The analysis above has developed two functions that can be used in any composite laminate to determine whether or not failure occurs. The functions have been written in terms of fibre fraction, f, and orientation, q, so that they can be used in the analysis of multiply laminates.
