4. If the formula of sigma =500varepsilon ^0.5 exists between the true stress and true strain for a steel. Obtain amount of elastic strain at a strain equivalent to 0.25 ? (E=200GPa) 5.During a tensile test on a specimen of 1cm^2 cross-section,maximum load of 80 kN and area of cross-section at neck was 0.5cm^2 Calculate ultimate tensile strength UTS) of specimen.

4. To find the elastic strain at a strain equivalent to 0.25, we can use the given formula $\sigma = 500\varepsilon ^{0.5}$. We need to rearrange the formula to solve for $\varepsilon$.<br /><br />Given:<br />$\sigma = 500\vRearranging the formula to solve for $\varepsilon$:<br />$\varepsilon ^{0.5} = \frac{\sigma}{500}$<br /><br />Substituting the given value of $\sigma$:<br />$\varepsilon ^{0.5} = \frac{E \cdot \varepsilon}{500}$<br /><br />Simplifying the equation:<br />$\varepsilon ^{0.5} = \frac{E \cdot \varepsilon}{500}$<br /><br />Dividing both sides by $E$:<br />$\varepsilon ^{0.5} = \frac{\varepsilon}{500/E}$<br /><br />Taking the square of both sides:<br />$\varepsilon = \left(\frac{\varepsilon}{500/E}\right)^2$<br /><br />Simplifying further:<br />$\varepsilon = \frac{\varepsilon^2}{500^2/E^2}$<br /><br />Multiplying both sides by $E^2$:<br />$E^2 \cdot \varepsilon = \frac{\varepsilon^2}{500^2}$<br /><br />Dividing both sides by $\varepsilon$:<br />$E \cdot \varepsilon = \frac{\varepsilon}{500^2}$<br /><br />Simplifying further:<br />$E \cdot \varepsilon = \frac{\varepsilon}{250000}$<br /><br />Dividing both sides by $E$:<br />$\varepsilon = \frac{\varepsilon}{250000E}$<br /><br />Simplifying further:<br />$\varepsilon = \frac{1}{250000}$<br /><br />Therefore, the elastic strain at a strain equivalent to 0.25 is $\frac{1}{250000}$.<br /><br />5. To calculate the ultimate tensile strength (UTS) of the specimen, we can use the formula:<br /><br />$UTS = \frac{Force}{Area}$<br /><br />Given:<br />Force = 80 kN = 80000 N<br />Area = 0.5 cm^2 = 0.5 x 10^-4 m^2<br /><br />Substituting the given values into the formula:<br />$UTS = \frac{80000}{0.5 \times 10^{-4}}$<br /><br />Simplifying the equation:<br />$UTS = 80000 \times 10^4$<br /><br />$UTS = 800000000$<br /><br />Therefore, the ultimate tensile strength (UTS) of the specimen is 800 MPa.