<P> For the tension performance equation P C R = σ / ρ (\ displaystyle P_ (CR) = \ sigma / \ rho), the first step is to take the log of both sides . The resulting equation can be rearranged to give l o g (σ) = l o g (ρ) + l o g (P C R) (\ displaystyle log (\ sigma) = log (\ rho) + log (P_ (CR))). Note that this follows the format of y = x + b (\ displaystyle y = x + b), making it linear on a log - log graph . Similarly, the y - intercept is the log of P C R (\ displaystyle P_ (CR)). Thus, the fixed value of P C R (\ displaystyle P_ (CR)) for tension in Figure 3 is 0.1 . </P> <P> The bending performance equation P C R = σ / ρ (\ displaystyle P_ (CR) = (\ sqrt (\ sigma)) / \ rho) can be treated similarly . Using the power property of logarithms it can be derived that l o g (σ) = 2 ∗ (l o g (ρ) + l o g (P C R)) (\ displaystyle log (\ sigma) = 2 * (log (\ rho) + log (P_ (CR)))). The value for P C R (\ displaystyle P_ (CR)) for bending is ≈ 0.0316 in Figure 3 . Finally, both lines are plotted on the Ashby chart . </P> <P> First, the best bending materials can be found by examining which regions are higher on the graph than the σ / ρ (\ displaystyle (\ sqrt (\ sigma)) / \ rho) bending line . In this case, some of the foams (blue) and technical ceramics (pink) are higher than the line . Therefore those would be the best bending materials . In contrast, materials which are far below the line (like metals in the bottom - right of the gray region) would be the worst materials . </P> <P> Lastly, the σ / ρ (\ displaystyle \ sigma / \ rho) tension line can be used to "break the tie" between foams and technical ceramics . Since technical ceramics are the only material which is located higher than the tension line, then the best - performing tension materials are technical ceramics . Therefore, the overall best material is a technical ceramics in the top - left of the pink region such as boron carbide . </P>

Briefly explain the selection of each variable on the x and y axes and why