Problem $3$

Based on the stress$-$strain model in Figure P$1\mathrm{.}15,$ graphically determine:

a$)$ Modulus of resilience.

b$)$ Toughness. Hint: the toughness $(u_t)$ can be determined by calculating the area under the stressstrain curve.

$u_t={\displaystyle \underset{0}{\overset{{}_f}{}}}d$

where ${}_f$ is the strain at fracture. The preceding integral can be approximated numerically by using a trapezoidal integration technique:

$u_t={\displaystyle \underset{i=1}{\overset{n}{}}}{u}_i={\displaystyle \underset{i=1}{\overset{n}{}}}\frac{1}{2}({}_i+{}_{i-1})({}_i-{}_{i-1})$

c$)$ If the specimen is loaded to $40$ksi only and the lateral strain was found to be $-0\mathrm{.}00057in.\frac{?}{in}.,$ what is the Poisson's ratio of this metal?

d$)$ If the specimen is loaded to $70$ksi only and then unloaded, what is the permanent strain?