CBEMS155/MAE156FAMQuiz 3 SolutionGiven:• Non isotropic cubic crystal• Bulk modulus 1/K = −∆V/VP• ∆V/V = exx+ eyy+ ezzRequired:Prove: K =C11+2C123Assumption:Hydrostatic pressureSolution:The structure is under hydrostatic pressure soσxx= σyy= σzz= −P (1)For a non-isotropic cubic crystal we haveσxx= C11exx+ C12(eyy+ ezz)σyy= C11eyy+ C12(exx+ ezz)σzz= C11ezz+ C12(exx+ eyy)(2)Adding the three stress terms together we getσxx+ σyy+ σzz= C11(exx+ eyy+ ezz) + 2C12(exx+ eyy+ ezz)−3P = C11(∆V /V ) + 2C12(∆V /V )−3P = (C11+ 2C12)(∆V /V )P = −(C11+ 2C12)(∆V /V )3(3)Substitute the above expression into the bulk modulus definition we get11K= −∆V/VPK = −P∆V/VK =(C11+ 2C12)(∆V/V )3(∆V/V )(4)so we getK =(C11+