The bar system shown in Fig. P12.1a consists of two long “planks” of length L and cross-sectional dimensions b × h that are securely bonded together at their interface. The materials have moduli of...

The bar system shown in Fig. P12.1a consists of two long “planks” of length L and cross-sectional dimensions b × h that are securely bonded together at their interface. The materials have moduli of elasticity E1 and E2 and mass densities ?1 and ?2, respectively. (a) By carrying out a complete derivation similar to the one in Section 12.1, show that axial deformation (the axis remains straight; cross sections remain plane and remain perpendicular to the axis) is possible if E1/E2 = ?1/?2. Note: You will need to replace Eq. 12.3 with the two equations P1 = A1s1, P2 = A2s2 Figure P12.1 (a) Two-material bar undergoing axial deformation; (b) cross section of the bar; (c) free-body diagram of the bar segment from x to (x + x). In addition to the F equation in Eq. 12.5, you will need to use the free-body diagram in Fig. P12.1b and write an equation for M. In these equations, include the inertia terms from the upper block and the lower block as two separate terms. (b) Give the equation of motion for axial deformation of this system. For homogeneous bars, your answer should reduce to Eq. 12.8.