Calculate the modulus of resilience for the material having the stress-strain behavior shown in the Animated Figure 6.12. Tensile strength: 450 MPa (65,000 psi) Strain=0, Stress=0 MPA, Stress=0 psi 500 70 Strain=0, Stress=0 MPA, Stress=0 psi 60 400 10 psi 40 50 . MPa 300 Stress (MPa) 200 Yield strength: 250 MPa (36,000 psi) 40 200 30 30 Stress (103 psi) 20 20 100 10 100 0 10 0.005 0 0 0.10 0.20 0.30 0.40 Strain
The Correct Answer and Explanation is:
0.469 MJ/m³
The modulus of resilience (Ur) is a mechanical property of a material that quantifies the amount of strain energy it can absorb per unit volume without undergoing permanent, or plastic, deformation. Graphically, it is represented by the area under the linear elastic portion of the stress-strain curve, up to the yield point. For materials that obey Hooke’s Law in their elastic region, this area forms a triangle, and its value can be calculated using the formula:
Ur = ½ * σy * εy
Here, σy is the yield strength, and εy is the strain at the yield point.
First, we must identify these two values from the provided stress-strain diagram. The yield strength (σy) is explicitly stated on the graph and indicated by an arrow pointing to the curve. The value is given as 250 MPa.
Next, we need to determine the corresponding strain at this yield point (εy). This value is not directly stated, so we must calculate it. We can do this by first finding the modulus of elasticity (E), which is the slope of the initial, straight-line portion of the curve. Using the detailed inset graph, we can pick a point in the linear region, for instance, a stress of 200 MPa corresponds to a strain of approximately 0.003.
Modulus of Elasticity (E) = Stress / Strain = 200 MPa / 0.003 ≈ 66,700 MPa or 66.7 GPa.
Now, we can use this modulus and the known yield strength to find the yield strain using Hooke’s Law (σy = E * εy):
εy = σy / E = 250 MPa / 66,700 MPa ≈ 0.00375
Finally, we can substitute the yield strength and yield strain into the formula for the modulus of resilience:
Ur = ½ * (250 MPa) * (0.00375)
Ur = ½ * (250 x 10⁶ N/m²) * (0.00375)
Ur = 468,750 N·m/m³ = 468,750 J/m³
Converting this to a more common unit, we get 0.469 MJ/m³.
