Page 362 - Introduction to Continuum Mechanics

346 The Elastic Solid

5.88. Verify the relations between C,y and the engineering constants given in Eqs. (5.29.2a)

5.89. Obtain Eq. (5.29.3) from Eq. (5.29.2)
5.90. Derive the inequalities expressed in Eq. (5.30.4)
5.91. Write down all the restrictions for the engineering constants for a monoclinic elastic solic
5.92. Show that if a tensor is objective, then its inverse is also objective.
1 T
5.93. Show that the rate of deformation tensor D = :r[(Vv) + (Vv) ] is objective

5.94. Show that in a change of frame, the spin tensor W transforms in accordance with th
r
equation W * = QWQ + QQ r
5.95. Show that the material derivative of an objective tensor T is in general non-objective
5.96. The second Rivlin-Ericksen tensor is defined by

where Aj = 2D [See Prob. 5.93]. Show that A2 is objective.
5.97. The Jaumann derivative of a second order tensor T is

where W is the spin tensor [see Prob. 5.94]. Show that the Jaumann derivative of T is objective.
5.98. In a change of frame, how does the first Piola-Kirchhoff stress tensor transform ?
5.99. In a change of frame, how does the second Piola-Kirchhoff tensor transform?
5.100. (a) Starting from the assumption that

and

show that in order that the constitutive equation be independent of observers, we must have

r
(b) Choose Q=R to obtain

where R is the rotation tensor associated with the deformation gradient F and U is the right
stretch tensor.
(c) Show that