The expression over the brace is the variation of the strain energy density 6 aUo Using the properties of calculus of variations8/0=50 SUodv=8/ Uodv=8u=8(/ tiu;ds+/fu;dv)=8(-v where v is the potential of the external loads. Therefore 6=6(U+V which is known as the Principle of minimum potential energy(PMPE). In fact this expression only says that II is stationary with respect to variations in the displacement field when the body is in equilibrium We can prove that it is indeed a minimum in the case of a linear elastic material: Uo=2Ci]kIEN. We want to show I(v)≥∏(u),v I(v)=I(u)分v=t Consider i=u+ du t4(u2+6u)ds-/t(2+6)dV I()+2/cnn6 The second, fourth and fifth term disappear after invoking the PVD and we are left with II(u+ Su)=II()+/Ciju dei SEudV� � � � The expression over the brace is the variation of the strain energy density δU0: ∂U0 δU0 = δ�ij ∂�ij Using the properties of calculus of variations δ () = δ(): � � �� � � δU0dV = δ U0dV = δU = δ tiuidS + fiuidV = δ(−V ) S V where V is the potential of the external loads. Therefore: δΠ = δ(U + V ) = 0 which is known as the Principle of minimum potential energy (PMPE). In fact this expression only says that Π is stationary with respect to variations in the displacement field when the body is in equilibrium. We can prove that it is indeed a minimum in the case of a linear elastic mat 1 erial: U0 = 2Cijkl�kl. We want to show: Π(v) ≥ Π(u), ∀v Π(v) = Π(u) ⇔ v = u Consider u¯ = u + δu: � �1 � Π(u + δu) = Cijkl(�ij + δ�ij )(�kl + δ�kl) dV V 2 � � − ti(ui + δui)dS − ti(ui + δui)dV S � V � 1 1 =Π(u)+ �2 V �2 C � ijkl�ijδ�kldV + V 2 Cijklδ�ijδ�kldV − tiδuidS − fiδuidV S V The second, fourth and fifth term disappear after invoking the PVD and we are left with: 1 Π(u + δu) = Π(u) + Cijklδ�ijδ�kldV V 2 2