3.1.1.1 Infinitesimal Strains
The French school considered the displacement of the points of a body as a continuous function meaningful only at the places occupied by molecules; the deformation was defined, before considering geometric intuition, then analysis [30, 77]. Betti devi- ated from this approach; he ignored the corpuscular nature of the bodies, modeling them as continua and followed a purely analytical approach. He made instrumen- tal use, because of their convenience, of the infinitesimals, quietly abandoning the mathematical rigor of the Italian school carried out by Piola and Bordoni.13
13Betti knew that, if desired, he could rewrite all the ‘less rigorous’ steps developed with the use of infinitesimals with a strict mathematics.
Deformation was defined as the change in length of the linear element:
ds2=dx2+dy2+dz2, (3.1)
where(x,y,z)are the present coordinates of the point P of the continuum, a function of the coordinatesξ,η,ζthat P has in the reference configuration. Betti admitted that “the variation of length of the linear elements and the element themselves are so small quantities that one can ignore the power of higher order with respect to that of lower order” [20].14The variation that Betti made ofds2operated then on functions ofξ,η,ζ:
dsδds=dxδdx+dyδdy+dzδdz=dxd(δx)+dyd(δy)+dzd(δz), (3.2) which is possible for the exchangeability of the operatorsd andδ. The variations δx,δy,δzcoincide with the components of the vector of displacement[u(ξ,η,ζ), v(ξ,η,ζ),w(ξ,η,ζ)]and the (3.2), divided byds2, appears in the extended form as:
δds ds = du
dx dx
ds 2
+du dy
dy ds
2
+dw dz
dw ds
2
+ dv
dz +dw dy
dy ds
dz ds +
dw dx +du
dz dz
ds dx ds +
du dy +dv
dx dx
ds dy ds.
(3.3) For small strains the variationδdsapproximates the differencedsand consequently the ratioδds/dsapproximates the relative variation of length of the elementdsin the direction ofdx,dy,dz. To individuate the components of the deformation it is enough to assume suitable values for dx,dy,dz. For instance one assumesdy = dz = 0, and thusds =dx, to obtain the deformation alongx:δds/ds = du/dx. The same considerations apply for the other directions. With some mathematics Betti also obtained the expressions for the angular distortions, that is the variation of the angle between orthogonal segments.
The strains, infinitesimal because they have a physical meaning for small displacements only, are indicated by Betti on the footprints of William Thomson [88]15as:
14p. 3. From now on the quotations from theTeoria della elasticitàby Betti refer to the offprint by Soldaini of 1874 [20].
15p. 391.
(1 + c)dz
(1 + a)dx (1 + b)dy
dz
dx dy
y
x z
-2f π-2h
2 π-2g
2
P π
2 (a)
(b)
Fig. 3.1 Geometrical meaning of the coefficients of strains according to Betti.aUndeformed state.
bDeformed state Betti du
dx = a dv
dz +dw dy = 2f dv
dy = b dw
dx +du dz = 2g dw
dz = c du dy +dv
dx = 2h
Thomson du
dx = f dv
dz +dw dy = a dv
dy = g dw
dx +du dz = b dw
dz = h du
dy +dv dx = c.
Figure3.1illustrates the geometrical meaning of the coefficientsa,b,c,f,g,h.
Betti defined the angular distortions focusing more on the mathematical aspects than the physical ones: in fact 2f,2g,2hrepresent the variation of the right angle, while f,g,h are those that today are called the components of the strain tensor (along witha,b,c). The use off,g,his generally convenient in the mathematical treatments which by their nature require the essential use of the strain tensor.
Finally Betti showed that the componentsa,b,c,f,g,hof strain uniquely define the displacement field apart from a rigid motion.
3.1.1.2 Potential of the Elastic Forces
The concept of potential was an integral part of mathematical physics since Betti’s first works. As already mentioned, in the early works [16] Betti introduced the poten- tial as a primitive function of the forces without attributing to it a particular sta- tus of physical magnitude. He soon changed approach and ‘force’ began to take on an ambiguous meaning, indicating both the Newtonian force and the (thermo-) mechanical magnetic or electrical potential energy.
Betti published only articles of thermology and heat propagation, but still showed good knowledge of thermodynamics, which came to mathematical physics thanks to the work of William Thomson. In theTeoria della elasticità, by means of the first and
second principle of thermodynamics, he gave a physical meaning to his potential, today included under the name of potential energy.
The thermodynamical theory was developed for homogeneous thermal processes, although there was awareness that in a real body the processes are generally hetero- geneous. Betti, following with William Thomson the current approach to thermody- namics with the aid of the differential calculus, considered the continuumSdivided into infinitesimal elements, each of which is treated as homogeneous. The potential of the elastic forces is thus given by the sum of the potential of the elastic forces of all the infinitesimals, and then by an integral. More precisely, ifPexpresses the potential of an infinitesimal element, the potentialof the whole continuum is:
=
S
P dS. (3.4)
Betti proposed that the potential of the elastic forces be a function of the infinitesimal strains, in the footprints of Green [54]. He assumed thenatural state, as the reference stable configuration from which to measure the strains; thus in the development in series ofPhe could neglect the first-order terms. He also neglected the terms of the order higher than the second, obtaining so the quadratic form:
P= 6
i=1
6
j=1
Aijxixj. (3.5)
wherexα, α=1,2, . . . ,6 represent the generic components of the strain. For the stability of the equilibrium the quadratic form should be negative definite (remember that the potential is potential energy with sign reversed). Similarly to Green [54] for an isotropic body Betti came to the expression [20]16:
P=A2+B2, (3.6)
where
=a+b+c, 2=a2+b2+c2+2f2+2g2+2h2. (3.7) Notwithstanding the evident reference to Green in the expression of the potential, the constantsAandBare not those that Green used for the elasticity of isotropic bodies, but are connected to the Lamé constantsλ˜ andμ˜ [54]17:
A= −λ˜
2 B= − ˜μ. (3.8)
16p. 18. LetEbe the tensor of deformation,andare respectively the trace ofEandE2.
17p. 253. The tilde distinguishes the two Lamé constants from Betti’sλandμ.
3.1.1.3 The Principle of Virtual Work
The third chapter of theTeoria della elasticitàopens with the quotation:
To determine the relationships that must exist between the forces acting on a homogeneous elastic solid body, and the deformations of the elements of the same, for there is equilibrium, we will use the following the principle byLagrange: for a system, whose virtual motions are reversible, to be in equilibrium it is necessary and sufficient that the mechanical work done by the forces in a whatever virtual motion, be equal to zero [20].18(A.3.4)
Betti therefore did not consider the equilibrium equations as relations between external and internal forces, but between external forces and strain-displacements.
He used the principle of virtual work (“the principle of Lagrange”), because in such a way he could express the virtual work of the internal forces without making them intervene directly. It is worth noting the way in which he stated the principle of Lagrange: there is no physical obstacle to its validity, there is only the interest of the mathematician who wants to clarify whether the work is negative or zero. Assuming bilateral constraints, the work can be equated to zero.
The equation obtained by Betti is:
δ+
S
ρ(Xδu+Yδv+Zδw)dS+
σ(Lδu+Mδv+Nδw)dσ = 0. (3.9) In it,δis the virtual work of the internal forces,ρis the mass per unit of volume, (X,Y,Z), the components of the force per unit of mass (accelerating force) in the volumeSand(L,M,N) the force per unit of surface on the surfaceσ, the boundary ofS. Passing from the variational Eq. (3.9) to the equations of equilibrium is simple for Betti; indeed similar elaborations had already been carried out by Navier, Green, William Thomson, and Clebsch.
In this point of theTeoria della elasticitàBetti, without specifying the form of the potentialP, limited to obtain the local and boundary equations of equilibrium that were written as [20]19:
ρX = d dx
dP da + d
dy dP 2dh+ d
dz dP 2dg ρY = d
dx dP 2dh+ d
dy dP db + d
dz dP
2df (3.10)
ρZ = d dx
dP 2dg+ d
dy dP 2df + d
dz dP dc,
18p. 20. Our translation.
19p. 22.
L=dP daα+ dP
2dhβ+ dP 2dgγ M = dP
2dhα+dP dbβ+ dP
2dfγ (3.11)
N = dP
2dgα+ dP 2dfβ+dP
dcγ.
The sign of the second members of these equations is contrary to the one usually found in modern textbooks because Betti orientated the normaln, of whichα,β,γ are the components, to the surfaceσtoward the interior instead of toward the exterior as is done today.