CompositesWorld
Issue link: https://cw.epubxp.com/i/697353
JULY 2016 28 CompositesWorld Trace: A simple, must-have tool This recently discovered constant in laminate calculations can put designing with CFRP materials on par with designing with metals. » Trace is the sum of the three elastic constants of all materials — the longitudinal, transverse and shear moduli that are their diagonal compo- nents in the mathematical matrix of all carbon composites (see Fig. 1). In the fgure, two diferent carbon composite materials are shown, each in two diferent laminate confgurations. But for both, if we add the diagonals (multiplying the third diagonal by 2) in each of the laminates, we get the same numbers for diferent laminates, for each material. Tis number is the trace of that material. In Fig. 2 (p. 29), we discover something more: We divide those same diagonal values by that material's trace. What happens? In all four cases, we get 1.00. Regardless of the material or laminate, the result is 1. Each laminate, regardless of the material, has the same fraction of trace values, e.g., 0.37 for the frst laminate. In my work, this has proven true with all laminates for all carbon fber-reinforced plastics (CFRP) materials. Te implication is clear: Trace is a universal constant, and in CFRP, the fractions of longitudinal, transverse and shear moduli in trace are, likewise, constant. For example, the longitudinal stifness is 88% of trace for all CFRP materials, and is so within 1.3% accuracy. Tis accidental discovery has made composites much easier to under- stand, and for practicality's sake, the use of trace can be made even easier by focusing primarily on the measured longi- tudinal stifness, that is, trace divided by 0.88. Why? Because the transverse and shear, together, share only 12% of trace. Tey are small enough that they need not be measured for our purpose here. Tis makes sense because we all know that longitudinal stif- ness is fber-dominated, and the transverse and shear stifness are matrix-dominated and, therefore, are not needed for the determina- tion of laminate stifness. Trace, it turns out, is the only constant that clearly defnes the stifness of the material, and we've found it is the only constant that we need with CFRP. Tus, the design of CFRP composites can be as simple as that for metals because, as is true with the latter, we need to test for only one constant. (Trace, by the way, is equally appli- cable to glass-fber and aramid-fber reinforced polymers, but with less accuracy than with carbon; i.e., with variation greater than 1.3%.) Why not just use the longitudinal stif- ness, and forget about trace? Tere is a funda- mental diference between the two. Trace is an invariant. It does not change when the coordi- nate system rotates. Longitudinal stifness, on the other hand, changes as we rotate it. It does By Stephen Tsai / Contributing Writer Fig. 1 Finding a material's trace Here, two diferent CFRP materials are shown, each in two diferent laminate confgurations. But for both, if we add the diagonals (multiplying the third diagonal by 2) in each of the laminates, we get the same numbers for diferent laminates, for each material. This number is the trace of that material.