A bar of a steel alloy that exhibits the stressstrain behavior shown in Figure \(6.22\) is subjected to a tensile load; the specimen is \(375 \mathrm{~mm}(14.8 \mathrm{in}\).) long and has a square cross section \(5.5 \mathrm{~mm}(0.22 \mathrm{in}\).) on a side.(a) Compute the magnitude of the load necessary to produce an elongation of \(2.25 \mathrm{~mm}(0.088\) in.). (b) What will be the deformation after the load has been released?

First, we need to determine the initial cross-sectional area and engineering strain of the steel bar. The cross-sectional area can be calculated as: \(A_0 = L^2\) where \(L\) is the side of the square cross section (\(5.5 \mathrm{~mm}\)). For the engineering strain (\(\epsilon\)), we can use: \(\epsilon = \frac{\Delta L}{L_0}\) where \(\Delta L\) is the elongation (\(2.25 \mathrm{~mm}\)) and \(L_0\) is the initial length (\(375 \mathrm{~mm}\)).

In the elastic region, Hooke's Law states: \(\sigma = E \cdot \epsilon\) where \(\sigma\) is the engineering stress, \(E\) is the elastic modulus (also known as Young's modulus), and \(\epsilon\) is the engineering strain. We can rearrange the equation to find the tensile load (\(P\)): \(P = \sigma A_0 = E \cdot \epsilon A_0\) The elastic modulus \(E\) of the steel alloy must be provided. Let's assume \(E = 200 \mathrm{GPa}\) for this example. Now we can calculate the tensile load to produce the desired elongation: \(P = E \cdot \epsilon A_0\)

The deformation after releasing the load will depend on whether the bar was subjected to only elastic deformation or if it also went into the plastic region. If the load required to produce the given elongation keeps the stress within the elastic limit, we can assume that the bar will spring back to its original length (no permanent deformation). However, if the load goes beyond the elastic limit, we will have to consider the plastic region. In this case, the stress-strain curve must be provided, letting us find the plastic deformation after unloading the bar. This can be done by finding the corresponding strain value on the curve after unloading, and then using the initial length and engineering strain to find the actual post-unload elongation. Without the actual stress-strain curve, it's impossible to calculate the exact deformation. So our answer depends on whether or not the material remains in the elastic region when subjected to the calculated load.