Ellipticity of Elasticity Problems Strain Energy and Extenal Work

13 For the isotropic Hooke's law, we saw that there always exist a strain energy function W which is positive-definite, homogeneous quadratic function of the strains such that, Eq. 7.20

hence it follows that

14 The external work done by a body in equilibrium under body forces bi and surface traction ti is equal to / pbiuidQ+ tiuidr. Substituting ti — Tjnj and applying Gauss theorem, the second term Jo J r becomes

/ Tij nj uidr— (Tij ui),j dl — / (Tj,j ui + Tij ui,j )dil (9.16)

J r Jo Jo but Tijui,j — Tij(Ej + lij) — TijEij and from equilibrium Tij,j — -pbi, thus

/pbiui dll+ ti uidF— pbiuidl+ (Tij Eij — pbiui)dl (9.17)

pbiuidl + ti uidr Jo J r

— 2 f Tij Eij di J o 2

External Work

Internal Strain Energy

that is For an elastic system, the total strain energy is one half the work done by the external forces acting through their displacements ui.

0 0

Post a comment