A bar of a steel alloy that exhibits the stress-strain behavior shown in Figure \(6.21\) is subjected to a tensile load; the specimen is 300 mm (12 in.) long and has a square cross section \(4.5 \mathrm{~mm}(0.175 \mathrm{in}\).) on a side. (a) Compute the magnitude of the load necessary to produce an elongation of \(0.45 \mathrm{~mm}\) \((0.018\) in.). (b) What will be the deformation after the load has been released?

Hooke's Law defines the relationship between stress (σ) and strain (ε) as σ = Eε, where E is the Modulus of Elasticity. We need to determine the stress at which the elongation of 0.45 mm occurs. First, we need to calculate the strain from the elongation: ε = (elongation) / (original_length) = (0.45 mm) / (300 mm) = 0.0015 Now, we can find the stress by using the relationship between stress and strain given in Figure 6.21, which is approximately linear: σ = Eε Let's say the Modulus of Elasticity of the steel alloy (E) as given in Figure 6.21 is 200 GPa. So, converting GPa to N/mm², E = 200 x 10^3 N/mm². σ = 200 x 10^3 N/mm² x 0.0015 = 300 N/mm² Now, we can calculate the force: Area of the cross-section (A) = Side^2 = (4.5 mm)^2 = 20.25 mm² Force (F) = Area (A) x stress (σ) = 20.25 mm² x 300 N/mm² = 6075 N So, the magnitude of the load necessary to produce an elongation of 0.45 mm is 6075 N (approximately).

When the load is released, the steel bar will partially recover its shape due to its elastic properties. However, it may not return to its original length completely. To determine the permanent deformation, we need to find the strain after unloading. This can again be using the data provided in Figure 6.21 regarding the relationship between stress and strain. Let's say that after unloading, the stress (σ') is reduced to 0 N/mm² since the load is removed. The relationship between the σ' and ε' can also be used from Figure 6.21, and assuming it's linear: σ' = Eε' Now, as σ' = 0, ε' will also be 0. This implies that the deformation after the load has been released is also 0. In reality, there might be a little permanent deformation, and finding its value would require more information than provided in the exercise. However, for answering this part, the deformation after the load has been released is approximately 0, implying that the steel bar would regain its original length after unloading.