Betti reciprocal theorem

A theorem relating two equilibrium states in the theory of small deformations of an elastic body [a1] (cf. also Elasticity, mathematical theory of). In physical terms, the theorem equates the work which would be done by the surface tractions and body force of one state acting through the displacements of the other state to the work of the tractions and body force of the second state acting through the displacements of the first.

Small displacements from the unstressed state referred to a rectangular Cartesian system $ x _ {i} $ are denoted by $ u _ {i} ( x ) $. The stress components $ t _ {ij } $ are symmetric and depend only on the strains, the symmetric part of the deformation gradient $ { {\partial u _ {i} } / {\partial x _ {j} } } $. The elastic coefficients of the linearized stress-strain relation $ t _ {ij } = c _ {ijkl } u _ {k,l } $ then possess the symmetries $ c _ {ijkl } = c _ {jikl } = c _ {ijlk } $. The equilibrium equations are:

$$ { \frac \partial {\partial x _ {j} } } ( c _ {ijkl } u _ {k,l } ) + F _ {i} = 0 $$

throughout the region $ V $ occupied by the body, where $ F _ {i} $ are the body force components. On the surface $ S $ of the body, the components of traction are given by

$$ T _ {i} = c _ {ijkl } u _ {k,l } n _ {j} , $$

where $ n _ {i} $ is the unit surface normal.

For an elastic body, the work required to deform an element must depend only on the final state of strain, so that a strain energy density exists and the elastic coefficients possess the additional symmetry $ c _ {ijkl } = c _ {klij } $. It follows that the differential operator in the equilibrium equations is self-adjoint (cf. also Self-adjoint operator). For two equilibrium states $ u _ {i} $, $ u _ {i} ^ \prime $ with body forces $ F _ {i} $, $ F _ {i} ^ \prime $ and tractions $ T _ {i} $, $ T _ {i} ^ \prime $, the divergence theorem and the equilibrium equations then lead to the reciprocal theorem

$$ \int\limits _ { S } {T _ {i} u _ {i} ^ \prime } {d S } + \int\limits _ { V } {F _ {i} u _ {i} ^ \prime } {d V } = \int\limits _ { V } {c _ {ijkl } u _ {i,j } u _ {k,l } ^ \prime } {d V } = $$

$$ = \int\limits _ { S } {T _ {i} ^ \prime u _ {i} } {d S } + \int\limits _ { V } {F _ {i} ^ \prime u _ {i} } {d V } , $$

under suitable smoothness conditions.

E. Betti used this theorem to provide a formula for the average strain produced in a body by given forces and to apply the method of singularities to the solution of elastic problems (for a discussion, see [a2]). The theorem has also been used to specify the deformation of least strain energy among a class of deformations [a3], and to determine the resultant force and moment over part of a surface where displacements are known by means of certain auxiliary solutions [a4]. The latter approach has been extended to determine load-displacement relations in problems in second-order elasticity theory [a5].

References

[a1]	E. Betti, Nuovo Cim. (2) , 7–8 (1872)
[a2]	A.E.H. Love, "A treatise on the mathematical theory of elasticity" , Cambridge Univ. Press (1927) (Reprint: Dover, 1944) MR0010851 Zbl 53.0752.01
[a3]	R.T. Shield, C.A. Anderson, "Some least work principles for elastic bodies" Z. Angew. Math. Phys. , 17 (1966) pp. 663–676
[a4]	R. T. Shield, "Load-displacement relations for elastic bodies" Z. Angew. Math. Phys. , 18 (1967) pp. 682–693 Zbl 0149.42901
[a5]	Z. Bai, R.T. Shield, "Load-deformation relations in second order elasticity" Z. Angew. Math. Phys. , 46 (1995) pp. 479–506 MR1345808 Zbl 0830.73011