Suggestions for Further Study 23 2.7.Calculate the strain hardening expo- 3 X 104 N.Assume no volume change nent for a material whose true stress during plastic deformation. and true strain values are 450 MPa 2.11.A metal plate needs to be reduced to and 15%,respectively.Take K=700 a thickness of 4 cm by involving a MPa. rolling mill.After rolling,the elastic 2.8.In Eq.(2.6),the relation properties of the material cause the plate to regain some thickness.Cal- culate the needed separation be- tween the two rollers when the yield is given.Show in mathematical strength of the material after plastic terms for what condition(pertaining deformation is 60 MPa and the mod- to a possible change in volume)this ulus of elasticity is 124 GPa. relation is true. 2.12.A cylindrical rod of metal whose ini- 2.9.Show that the true and engineering tial diameter and length are 20 mm stress and strain are related by and 1.5 m,respectively,is subjected to a tensile load of 8 x 104 N.What =1+e) is the final length of the rod?Is the and load stressing the rod beyond its elas- e=In (1+e) tic range when the yield strength is 300 MPa and the elastic modulus is for the case when no volume change 180 GPa? occurs during deformation,that is,2.13.Calculate the Poisson ratio of a cylin- before the onset of necking. drical rod that was subjected to a ten- 2.10.Compare engineering strain with true sile load of 3500 N and whose initial strain and engineering stress with diameter was 8 mm.The modulus of true stress for a material whose ini- elasticity is 65 GPa,and the change tial diameter was 2 cm and whose fi- in diameter is 2.5 um.Assume that nal diameter at fracture is 1.9 cm.The the deformation is entirely elastic. initial length before plastic deforma- 2.14.Calculate the Poisson ratio for the case tion was 10 cm.The applied force was where no volume change takes place. Suggestions for Further Study See the end of Chapter 3.Further,most textbooks of materials science cover mechanical properties.2.7. Calculate the strain hardening expo￾nent for a material whose true stress and true strain values are 450 MPa and 15%, respectively. Take K
700 MPa. 2.8. In Eq. (2.6), the relation t
 li l 0 d l l
ln l l 0 i 
ln A A 0 i  is given. Show in mathematical terms for what condition (pertaining to a possible change in volume) this relation is true. 2.9. Show that the true and engineering stress and strain are related by t
(1 ) and t
ln (1 ) for the case when no volume change occurs during deformation, that is, before the onset of necking. 2.10. Compare engineering strain with true strain and engineering stress with true stress for a material whose ini￾tial diameter was 2 cm and whose fi￾nal diameter at fracture is 1.9 cm. The initial length before plastic deforma￾tion was 10 cm. The applied force was Suggestions for Further Study 23 3 104 N. Assume no volume change during plastic deformation. 2.11. A metal plate needs to be reduced to a thickness of 4 cm by involving a rolling mill. After rolling, the elastic properties of the material cause the plate to regain some thickness. Cal￾culate the needed separation be￾tween the two rollers when the yield strength of the material after plastic deformation is 60 MPa and the mod￾ulus of elasticity is 124 GPa. 2.12. A cylindrical rod of metal whose ini￾tial diameter and length are 20 mm and 1.5 m, respectively, is subjected to a tensile load of 8 104 N. What is the final length of the rod? Is the load stressing the rod beyond its elas￾tic range when the yield strength is 300 MPa and the elastic modulus is 180 GPa? 2.13. Calculate the Poisson ratio of a cylin￾drical rod that was subjected to a ten￾sile load of 3500 N and whose initial diameter was 8 mm. The modulus of elasticity is 65 GPa, and the change in diameter is 2.5 m. Assume that the deformation is entirely elastic. 2.14. Calculate the Poisson ratio for the case where no volume change takes place. Suggestions for Further Study See the end of Chapter 3. Further, most textbooks of materials science cover mechanical properties.