Chapter 3: Mechanical Properties of Materials

1. Chapter 3: Mechanical Properties of Materials • Section 3.4: Hooke’s Law • Section 3.5: Strain Energy

2. Hooke’s Law • Robert Hooke in 1676 discovered linear relation between stress and strain • Young’s modulus E is constant of proportionality (slope in linear region) • Linear elastic region is terminated with proportional limit • Can you find the elastic modulus for mild steel?

3. How does carbon content affect mechanical properties? • The modulus for alloy steels is independent of carbon content • Only proportional limit increases, but modulus remains constant • Spring steel has a modulus of 29,000 ksi. What is the stress at a strain of 0.002? What is the stress at a strain of 0.007? • Note Hooke’s Law only applies to linear region of stress strain diagram

4. Can you increase strength of a material? • When a material is loaded beyond the yield strength, permanent deformation sets • After external load is removed elastic deformation is recovered • Recovery is along same slope for linear region (same modulus) • Deformed material has permanent strain at O’, but higher yield point at A’; It strain hardens ! • Does the ductility of the material increase or decrease after the deformation?

5. Strain Energy • When an external load is applied in tensile test, external work is essentially applied to deform the material • An increase in incremental load results in incremental deformation • The work is average of force times distance traveled • Where does this work go? • It is stored in the material as strain energy

6. Strain Energy • External work = Strain energy due to deformation • Strain energy per unit volume: strain energy density • For linear elastic behavior

7. Energy related material properties Modulus of resilience Modulus of toughness • Strain energy up to fracture of material • Area under stress strain curve up to the proportional limit (elastic strain energy) • Represent ability to absorb external energy without permanent change in shape

8. Toughness favorable for design of accidentally loaded members Controlled by alloying element Nylon exhibits high toughness

9. Example 1: The stress-strain diagram for steel alloy having an original diameter of 0.5 in. and a gauge length of 2 in. is given in the figure. Determine the following: • The modulus of elasticity • The load on the specimen that will cause yielding, • The ultimate load the specimen will support • The modulus of resilience

10. Solution: • The modulus of elasticity • The load on the specimen that will cause yielding, • The ultimate load the specimen will support • The modulus of resilience

11. Example 2: The figure shown is the stress-strain diagram for an aluminum alloy. If a specimen of this material is stressed to 600 MPa, determine: • The permanent strain that remains when the load is released. • The modulus of resilience before and after application of load.

12. Solution: At 600 MPa. • The permanent strain that remains when the load is released. • Strain recovers from D to C after removal of load. Can you find strain CD? • Permanent strain OC is then found as

13. Solution: At 600 MPa. • Modulus of resilience before and after load application. • What is relation for modulus of resilience? • Before load application Did the toughness increase or decrease after removal of load • After load application