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# CTE calculation

## CTE calculation

(OP)
I am doing a similar task as WK95 is, learning to design with composites by hand and have a related question that I think warranted its own posting. I'm using Barbero's book and am in micromechanics having just read the thermal expansion section-you calculate the CTE properties of the laminate using properties of the fiber and matrix, which makes sense. I was surprised to see however that this calculation uses the tensile modulus of both constituents as part of the calculation. What physical mechanism of the modulus comes into play to affect CTE? With metal design I have not seen this, and doubt that there is a connection there but the laminates are a different animal of course. Can some one explain how the modulus affects thermal expansion, or point me to a source?
Thanks,
Bob

### RE: CTE calculation

Considering the longitudinal fiber direction, the ply must deform such that strain of the fibers is the same as the matrix (strain compatibility). If the fiber and matrix have different CTE values (usual), then one material is going to *want* to expand more than the other. The elastic modulus affects the influence of how effective the constituent is at getting to its "natural" position, which is not the final position of the ply (combined fiber/matrix).

In other words, the ply's CTE is going to be closer the CTE of the constituent with the higher elastic modulus. Thinks about it this way. If the matrix had nearly zero stiffness, then the ply CTE would be the same as the fiber's. The matrix would just be along for the ride. And vice versa.

Brian
www.espcomposites.com

### RE: CTE calculation

When I took composites, the prof first took it back to a CTE problem everyone had seen in strength of materials (which was in turn taught by HIS old prof) say a steel tube around an aluminum tube, both ends securely fastened at room temp, find the CTE as the temp changes. As Brian said, a tension will develop between the two materials. In the context of this problem that everyone was more comfortable with, he solved it as fiber/matrix 60/40 or whatever the area fraction was. This is as isotropic as the composite. I think the prof developed a couple special cases like a 0 CTE composite, and a negative poison ratio composite.

### RE: CTE calculation

(OP)
Excellent answers, that makes perfect sense now. Barbero does mention in several places that there is an assumed perfect bond between the two materials and this is one of them. Thanks Brian and moon.

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