Beltrami’s first thorough paper on elasticity theory is dated in 1882 [5], and is about the formulation of elastic equilibrium equations in a space with constant curvature where a continuum with volume S and boundaryσis placed. He got the moves from the elasticity equations obtained by Lamé [60]58 in curvilinear coordinates, and from later works by Carl Neumann and Carl Borchardt [23,70]. These last sim- plified Lamé’s calculations by adopting elastic potential in curvilinear coordinates, and obtaining equilibrium equations by the variation of its integral over the vol- ume occupied by the elastic body. According to Beltrami, however, their approach, though leading to correct results, could be improved and it was possible to put into evidence some aspects of a certain importance. Lamé, Carl Neumann, Borchardt, started either from direct equilibrium of forces (Lamé) or from the elastic potential (Carl Neumann, Borchardt) with respect to Cartesian coordinates, which implied that space was supposed Euclidean. Beltrami derived elastic equilibrium equations directly, without preliminary hypotheses on the nature of space.
Like Carl Neumann and Borchardt, Beltrami used a purely analytical approach, starting from the equality of virtual works of inner and outer forces. The key idea lies in the definitions of metrics, and, from this, of strain, which reduces to the usual one when the metrics is Euclidean:
58p. 290.
Letq1,q2,q3be the curvilinear coordinates of any point in a three-dimensional space, and let59:
ds2=Q21dq21+Q22dq22+Q23dq23
be the expression of the square of any line element, in such a space[. . .]then, by posing δθ1= ∂δq1
∂q1 +δQ1
Q1, δω1=Q2 Q3
∂δq2
∂q3 +Q3 Q2
∂δq3 q2 , δθ2= ∂δq2
∂q2 +δQ2
Q2, δω2=Q3 Q1
∂δq3
∂q1 +Q1 Q3
∂δq1 q3 , δθ3= ∂δq3
∂q3 +δQ3
Q3, δω3=Q1 Q2
∂δq1
∂q2 +Q2 Q1
∂δq2 q1 we may write
δds
ds =λ21δθ1+λ22δθ2+λ23δθ3+2λ3δω1+λ3λ1δω2+λ1λ2δω3 where the three quantitiesλ1, λ2, λ3, defined by
λi=QidQi ds
are the direction cosines of the angles that the line elementdsforms with the three coordinate [lines]q1,q2,q3[5].60(A.3.9)
In essence, Beltrami expressed the length of the infinitesimal elementdsas a function of the quantitiesδθ1,δθ2,δθ3,δω1,δω2,δω3, which he chose as candidates for strain.
Remark that the curvilinear coordinatesq1,q2,q3are implicitly assumed orthogonal, since in the expression fordsthe contribution of the productsdq1dq3,dq2dq3,dq1dq2
is missing. In Euclidean space one has Q1 = Q2 = Q3 = 1, and the quantities Beltrami chose in order to characterize strain coincide with the components of the tensor of the infinitesimal strain.
Beltrami then defined the virtual work of inner forces:
(1δθ1+2δθ2+3δθ3+ 1δω1+ 2δω2+ 3ω3). (3.33) Here1, 2, 3, 1, 2, 3are coefficients of undefined nature, “since the vari- ation of the line element depends on the six quantitiesδθ1,δθ2,δθ3,δω1,δω2,δω3. The six multipliers(1, 2, 3, 1, 2, 3)are functions ofq1,q2,q3, “the mean- ing of which is not necessary to investigate” [5],61but which later will become inner forces. Beltrami expressed the equality of the virtual works done by the external bulk and boundary forces,Fiandφirespectively, and by inner forces:
59Remark thatQ1,Q2,Q3in general depend onq1,q2,q3, even if Beltrami did not state it explicitly.
60pp. 384–385. Our translation.
61p. 386.
(F1Q1δq1+F2Q2δq2+F3Q3δq3)dS+
(φ1Q1δq1+φ2Q2δq2+φ3Q3δq3)dσ+
(1δθ1+2δθ2+3δθ3+ 1δω1+ 2δω2+ 3ω3)dS=0.
(3.34)
After having laboriously developed the integral containing inner forces in function of the displacementsq1,q2,q3, he gave a geometrical and mechanical meaning to these quantities, and recognized, as already remarked, the components of the infini- tesimal strain of curved space in the quantitiesδθ1,δθ2,δθ3,δω1,δω2,δω3, and the components of the stress in the coefficients1, 2, 3, 1, 2, 3. He obtained three local equilibrium equations plus three boundary equations which “coincide with those that Lamé obtained by the transformation of the analogous equations in curvilinear coordinates”.62
In any case, the result that Beltrami considered important, and that represents the main contribution of his paper, is to have shown the independence of the equations he obtained on Euclid’s fifth postulate:
What is more worth remarking, and that appears evident from the process kept here to obtain those equations, is that the space to which they are referred is not defined by other than the expression (1) of the line element, without any condition for the functionsQ1,Q2,Q3. Then equations (4), (4a) have a much greater generality than the analogous ones in Cartesian coordinates and, in particular, it is immediately worth remarking them to be independent on Euclid’s postulate [5].63(A.3.10)
Until now, Beltrami did not advance any hypothesis on the nature of inner forces, that is on constitutive relations. In the following steps, he first assumed conservative inner forces with potential function of the strain components, then considered isotropic bodies, for which the potential is:
= −1 2
Aθ2+Bω ,
θ=θ1+θ2+θ3, ω=ω21+ω22+ω23−4(θ1θ2+θ1θ3+θ2θ3).
(3.35)
Beltrami examined curved spaces with constant curvature, where isotropy is defined by two coefficientsA,Bindependent ofq1,q2,q3. Under this condition he obtained relatively simple local equilibrium equations:
A Q1
∂θ
∂q1+ B Q2Q3
∂(Q2θ2)
∂q3 −∂(Q3θ3)
∂q2
+4αBQ1x1+F1=0, A
Q2
∂θ
∂q2 + B Q3Q1
∂(Q3θ3)
∂q1 −∂(Q1θ1)
∂q3
+4αBQ2x2+F2=0, (3.36) A
Q3
∂θ
∂q3+ B Q1Q2
∂(Q1θ1)
∂q2 −∂(Q2θ2)
∂q1
+4αBQ3x3+F3=0.
62Note by Beltrami:Leỗons sur les coordonnộes curvilignes, Paris, 1859, p. 272.
63p. 389. Our translation.
Here α is the curvature, and xi = δqi(i = 1,2,3) are the displacement components [5].64 Whenα = 0,Q1=Q2=Q3=1, one recovers the equations by Navier, Cauchy and Poisson.
If the elastic (micro-)rotation and cubic dilatation vanish, in a space with positive, uniform curvature Beltrami observed an elastic deformation which has a certain
‘analogy’ with that provided by Maxwell’s theory [47]:
We then obtain a deformation, free from both rotation and dilatation, in which force and displacement have in each point the same (or opposite) direction and constantly proportional magnitudes. Such a result, that has no counterpart in Euclidean space, presents a remarkable analogy with certain modern concepts on the action of dielectric means [5].65(A.3.11) In spaces with uniform curvature, Beltrami interpreted electro-magnetic actions (stresses in ether around a current, spherical waves) by means of the contact action in ether particles. Quite pragmatically, he supposed the space with positive, negative, or vanishing curvature, depending on the phenomenon; calculations on the stress to which ether is subjected select the correct curvature.