A clamp is required for processing of electronic components. The clamp has a square cross-section of width \(x\) and given depth \(b\). It is essential that the clamp have low thermal inertia so that it reaches temperature quickly. The time \(t\) it takes a component of thickness \(x\) to reach thermal equilibrium when the temperature is suddenly changed (a transient heat flow problem) is \[t \approx \frac{x^{2}}{2 a}\]

where the thermal diffusivity \(a=\lambda / ho C_{p}\) and \(\lambda\) is the thermal conductivity, \(ho\) the density and \(C_{p}\) the specific heat per unit mass.

The time to reach thermal equilibrium is reduced by making the section \(x\) thinner, but it must not be so thin that it fails in service. Use this constraint to eliminate \(x\) in the equation above, thereby deriving a material index for the clamp. Use the fact that the clamping force \(F\) creates a bending moment on the body of the clamp of \(M=F L\), and that the peak stress in the body is given by (Appendix B, Section B15):

\[\sigma=\frac{x}{2} \frac{M}{I}\]

where \(I=b x^{3} / 12\) is the second moment of area of the body. The table below summarizes the requirements.

Data From Appendix B.15

Function Constraints Objective C-clamp of low thermal inertia Depth b specified I Must carry clamping load F without failure Minimize time to reach thermal equilibrium Free variables Width of clamp body, x Choice of material