2.4 Piola’s Stress Tensors and Theorem
2.4.2 The Piola-Kirchhoff Stress Tensors
The introduction of Piola’s name to qualify the stress tensors pulled back to the reference configuration is due to Truesdell and Richard Toupin [73] who often refer to the works we have examined in this paper. Frequently, Kirchhoff is mentioned in the same breath as Piola, and this attribution is also due to Truesdell and Toupin; we shall clarify why. Even though we are focused on Piola’s contributions, we will also summarize Kirchhoff’s contribution for a more complete study of the subject. In fact, unlike Piola, Kirchhoff was conscious of introducing a new idea, the stress pulled back in the reference state to study finite deformations. Unfortunately, Kirchhoff’s mathematical treatment was not as good as Piola’s: so the complementarity of under- standing and misunderstanding of mathematical and physical concepts by the two scientists justifies Truesdell’s juxtaposition of Piola’s and Kirchhoff’s names.
In 1852, Kirchhoff [35] published a paper in which he studied the problem of elastic equilibrium in presence of finite displacements. Kirchhoff maintained that he
88pp. 23–24; p. 620.
89pp. 596–597; 246–248; 185.
90p. 620. Our translation.
was inspired by Saint Venant [65], who had formulated a clear definition of a finite measure of strain (which is now indeed called Green-Saint Venant strain tensor) and had given some hints on how to obtain equilibrium equations for non-infinitesimal displacements, claiming that
When tensions are considered over the slightly inclined planes into which the three material planes initially rectangular and parallel to the coordinates have changed, we have, for the six components, the same expressions, as functions of the dilatations and the distortions [the components of the Green-Saint Venant strain tensor], that we have when displacements are very small [65].91(A.2.34)
The conclusion drawn by Saint-Venant in this passage does not seem so clear to a modern reader, and is probably the cause of Kirchhoff’s uncertainties in the con- sidered paper. Quite surprisingly, in fact, Kirchhoff’s article was somewhat obscure and presented incorrect expressions according to modern standards. It is not clear from the text whether Kirchhoff intended to follow an approximated reasoning, or if he made genuine errors. According to Todhunter and Pearson [71]92 Kirchhoff himself later realized the weakness of this paper and did not want to re-publish it in hisGesammelte Abhandlungen[36].
These are Kirchhoff’s words on how he claimed to derive local equilibrium equa- tions in the case of finite displacements (some evident typographical errors have been amended):
I will denote byξ,η,ζthe coordinates of a point after deformation, byx,y,zthe coordinate of the same point before it. I imagine that in the natural state of the body there are three planes, parallel to the coordinate planes, through the point(x,y,z); the parts of these planes, which lay infinitely near the mentioned point, are transformed by the deformation in planes which form non-square, finite angles with the coordinate planes, but infinitely smaller than 90owith each other. I imagine to project the pressure underwent by these planes after the deformation on the coordinate axes, and denote these components: Xx,Yx,Zx,Xy,Yy,Zy, Xz,Yz,Zz, in such a way that for instance Yx is the component alongyof the pressure to which is subjected the plane which was orthogonal to thexaxis before the deformation. These nine pressures are in general non-orthogonal with respect to the planes on which they act, and there are not three equal to other three, like in the case of infinitely small displacement.
Once established the conditions for the equilibrium of a part of the body which before the deformation is an infinitely small parallelepiped with sides parallel to the coordinate axes of lengthdx,dy,dz, one obtains the equations:
ρX= ∂Xx
∂x +∂Xy
∂y +∂Xz
∂z ρY= ∂Yx
∂x +∂Yy
∂y +∂Yz
∂z ρZ= ∂Zx
∂x +∂Zy
∂y +∂Zz
∂z
⎫⎪
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎭
. . . (1)
if we denote byρthe density of the body and by X,Y,Z the components of the accelerating forces acting on the body at the point(ξ,η,ζ). One obtains these equations by considering that the sides and the angles have changed infinitely little, and so one can use the same
91p. 261. Our translation.
92art. 1244, p. 50.
considerations introduced in the equilibrium in presence of infinitely small displacements [35]93(A.2.35)
Thus, Kirchhoff focused on three infinitesimal faces which are parallel to fixed coor- dinate planes and pass through a generic point which undergoes a finite displacement.
He then projected the stresses arising after the deformation on those faces on the fixed coordinate axes and wrote the local equilibrium equations with respect to the same axes. Kirchhoff’s equations (1) above seem inconsistent when what has been said in the previous section is considered. Indeed, they have a similar form of Eqs. (2.11) and (2.19), but do not coincide with them for two reasons:
1. It is not clear how the components Xx,Yx, . . .may coincide with those of Piola’s first stress tensor. Indeed, no information is provided either on how the area affected by the stress changes during deformation, or on the change of metric between the present and the reference configuration.
2. It is not clear where ρis measured. If ρis the mass per unit volume in the present configuration, as it seems to follow from Kirchhoff’s words, this is again inconsistent with the Eq. (2.11), since the mass density is required to be measured in the reference configuration.
It is strange that a sharp expert in physics and a well-educated mathematician like Kirchhoff wrote such inconsistencies. This may perhaps be explained by the fact that Kirchhoff was studying a problem of finite displacements with infinitesimal strain, as explicitly stated in the above quotation, and as conjectured by Saint-Venant:
[…] the mutual distances of points very close vary only in a small ratio […] [65].94(A.2.36) One may then suppose that Kirchhoff considered the body as almost undistorted so that areas and volumes do not vary. In this case, it is still possible to derive local equilibrium equations for the stress components in the present configuration, projected on the fixed coordinated axes, by means of standard procedures. This should be represented by Kirchhoff’s equations (1), ifρis taken as the density in the reference configuration.
It is remarkable how the developments by Piola and Kirchhoff are in a way each other’s mirror images. In the second derivation of the local equilibrium equations which Piola presented inMeccanica de’ corpi naturalmente estesi, he first introduced what we now call Piola’s second stress tensor: its components are the Lagrange multipliers of his variational problem. Then, he introduced what we now call Piola’s first stress tensor simply as a mathematical stratagem with which to write the local equilibrium equations in the present configuration; no mechanical meaning is given to its components. On the other hand, Kirchhoff began by considering from a physical point of view the quantities that we now call the components of Piola’s first stress tensor. Later, he introduced the components of what we now call Piola’s second stress tensor only to obtain a constitutive relation for the components of the first.
93pp. 762–763. Our translation.
94p. 261. Our translation.
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The Mathematicians of the Risorgimento
Abstract The constituting phase of the Kingdom of Italy was a time of recovery of mathematical studies. The political unity facilitated the inclusion of Italian math- ematicians in the context of European research, in particular the German one. The internationalization of Italian mathematics is customarily associated with a trip taken in 1858 by some young mathematicians including Francesco Brioschi, Enrico Betti and Felice Casorati in Europe. In a few years we assist in the development of some schools that will maintain their role even in the 20th century. Among them, those promoted by Enrico Betti and Eugenio Beltrami were undoubtedly the most impor- tant. In this chapter we present briefly the contribution of two of the leading pioneers and their students.