The formulation of the theorem of reciprocal work is perhaps the most notable contribution of Betti to the theory of elasticity:
If, in a homogeneous elastic body, two systems of displacements are respectively equilibrated to two systems of forces, the sum of the products of the components of the forces of the first system by the corresponding components of the displacements of the same points in the second system is equal to the sum of the products of components of the forces of the second system by the components of the displacements at the same points of the first [20].20(A.3.5) This theorem is presented and demonstrated in the absence of volume forces only, in theTeoria della elasticità.
The demonstration, relatively simple, started from the equilibrium equations writ- ten for the two equilibrated systems of forces and displacements. Betti retraced in reverse the steps by which he had obtained the equilibrium equations using the prin- ciple of virtual work and obtained the expression:
σ
Lu+Mv+Nw dσ =
σ
Lu+Mv+Nw
dσ, (3.12)
where(u,v,w)is the vector field of the displacements associated to the active surface forces (L,M,N), solution of the elastic problem. The apices indicate forces and displacements of two distinct elastic problems, still for the same continuum.
Betti came back to the theorem of reciprocal work in 1874 [19]21 by extending the theorem to the case of volume forces of componentsX,Y,Z, reaching thus the expression with which it is known today:
20p. 40. Our translation.
21p. 381.
σ
Lu+Mv+Nw
dσ+ρ
S
Xu+Yv+Zw dS=
σ
Lu+Mv+Nw
dσ+ρ
S
Xu+Yv+Zw dS.
(3.13)
He also indicated the role attributed to his theorem:
In this paper I show a theorem that, in the theory of the elastic forces of solids, takes the place of Green’s theorem in the theory of the forces acting according to the law of Newton, and for what the applications is concerned I just deduce formulas similar to that of Green’s functions for the potential [19].22(A.3.6)
Green’s theorem recalled by Betti has the expression:
σv∂u
∂ndσ=
σu∂v
∂ndσ, (3.14)
whereuandvare harmonic functions and represent the potentials of central forces in a portionSof a homogeneous and isotropic space void of sources, delimitated by the surfaceσwith normaln.23To obtain Eq. (3.14) Green started from Dirichlet’s elliptic problem, defined by the harmonic equation of the potential and by the boundary conditions:
v=0 inS, v=v onσ. (3.15)
Hereis Laplace’s operator andvan assigned function onσ.
Betti in theTeoria della elasticitàstarted instead from the field equations of the elastic problem for an elastic and homogeneous continuum [20]24:
ρX+(2λ+μ)d
dx +μ2u=0 ρY+(2λ+μ)d
dy +μ2v=0 ρZ+(2λ+μ)d
dz +μ2w=0,
(3.16)
22p. 379 Our translation.
23In [53] Green enounced a more general theorem than (3.14), today known as the second Green’s
identity:
σv∂u
∂ndσ+
S
uv dS=
σu∂v
∂ndσ+
S
vu dS.
Functionsuandvare (whatever their form) endowed with the necessary conditions of regularity ([53], p. 23, par. 3, not numbered equation); by imposing thatuandvbe harmonic functions the Eq. (3.14) is obtained, not made explicit by Green.
24Equations 32–33, pp. 33–34.
with2= dxd22 +dyd22 +dzd22, and the boundary equations:
L+2
λ+μdu dx
α+μ du
dy +dv dx
β+μ du
dz +dw dx
γ=0 M+μ
du dy+dv
dx
α+2
λ+μdv dy
β+μ dv
dz +dw dy
γ=0 N+μ
du dz +dw
dx
α+μ dv
dz +dw dy
β+2
λ+μdw dz
γ=0.
(3.17)
In the relations (3.16), (3.17)λandμare not Lamé’s constants, usually denoted by the same symbols, but the constantsAandBof the relation (3.6) with sign reversed.
The analogy between (3.12) and (3.14) starts from the way both are obtained: the field equations are multiplied by arbitrary displacement fields and integrated by parts, so to reduce the maximum order of the derivatives. The aim is to relate the solution of differential equations to a quadrature formula by means of special functions (now calledGreen functions).25
Betti’s reciprocal work theorem is often used in educational presentations of the theory of elasticity to derive the reciprocity theorem of Maxwell. Furthermore Betti’s theorem is reinterpreted for concentrated forces. By assuming only two forcesfiand fj, applied respectively to the pointsiandjof an elastic body, and ifuij anduji are respectively the displacement inidue to the forcefjand the displacement injdue to the forcefi, Betti’s reciprocal theorem gives:
fiuij =fjuji, (3.18)
that assumingfi =fj =1, furnishesuji =uij. This is the very Maxwell’s theorem, as formulated in [48].26
The above considerations are only intended to motivate the association between the theorems of Maxwell and Betti. This association was not, and there was no reason it should be, evident to the two scholars who moved driven by different purposes. Betti wanted to find a possible method of solution of his differential equations; Maxwell was moved by considerations of a more physical character, to shed light on certain properties of the elastic relationships.
25Green’s integral formula provides the functionv, solution of (3.14), at a pointPinternal toS starting from the knowledge ofvonσ. On the basis of the (3.15) it is given by:
v= 1 4π
σv∂
∂n
v+1 r
dσ.
Hereris the distance ofPfrom the pointsQofσ. The functionv(P,Q), sometimes called aGreen function, satisfies Laplace’s equation and is such thatu=(v+1/r)=0 onσ[53], p. 29. More frequently, one calls the whole expressionua Green function. Among the authors which individuate invGreen’s function to signal Betti, Rudolf Otto Sigmund Lipschitz, and Carl Neumann. Green seems to prefer the use of the functionu[53], p. 31, § 5, Eq. 5.
26p. 297.