1 Premise
Most theoretical presentations of Classical Mechanics are still developed in the Euclid 3D spatial context with the time playing the role of evolution parameter.
This approach is also undertaken in treatments dealing with equilibrium formulations in terms of a reference placement.
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A non-exhaustive but significant list of these treatments, dating from the fifties of the past century up to the present, is collected hereafter to put in evidence the relevance in Continuum Mechanics of the matter under investigation.
Toupin [1, Eq. (10.24–10.25)], Truesdell and Toupin [2, \(\mathsection \)210], Truesdell and Noll [3, Eq. (43 A.6)], Fung [4, \(\mathsection \)16.3], Malvern [5, Eq. (5.5.18)], Gurtin [6, Ch.IX.27], Oden and Reddy [7, Ch.5.8], Marsden and Hughes [8, Ch.5.4], Ogden [9, \(\mathsection \)(3.4.2], Crisfield [10, Ch.(10.4)], Podio-Guidugli [11, Ch.II.10], Nguyen Quoc Son [12, \(\mathsection \)1.2.4], Holzapfel [13, Eq. (8.42)], Lubarda [14, Ch.6], Asaro and Lubarda [15, Ch.5.7], Oden [16, Ch.4.4], Gurtin, Fried, Anand [17, Ch.24], Oden [18, Ch.4.3], La Carbonara [19, Eq. (4.88)], Mariano and Galano [20, Ch. 3.6], Salençon [21, 22], Taroco [23, \(\mathsection \)3.7], Merodio and Ogden [24, Eq. (13)], Corigliano et al. [25, Eq. (54)].
However, when developing a geometric treatment, the manifold \(\,\mathcal {E}\,\) of spacetime events, with \(\,\dim (\mathcal {E})=3+1\,\), comes readily to be regarded as the natural mathematical setting, even in the realm of classical mechanics.
Fundamental topics such as motion, velocity and the description of change of observer, can be properly investigated only in the spacetime context wherein the basic notions of time-projection \(\,t_\mathcal {E}:\mathcal {E}\mapsto \mathcal {Z}\,\) onto the time-line \(\,\mathcal {Z}\,\), time-arrows \(\,\textbf{Z}\,\) and dynamical trajectory \(\,{\mathcal {T}}\subset \mathcal {E}\,\) can be introduced, as described in the next Sect. 2.
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2 Spacetime context
Let us here summarise basic elements of the geometric spacetime theory.
A mathematical observer defines a projection from the event manifold \(\,\mathcal {E}\,\) onto the time-line \(\,\mathcal {Z}\,\), that is a surjective submersion1\(\,t_\mathcal {E}:\mathcal {E}\mapsto \mathcal {Z}\,\) which assigns a time instant \(\,t_\mathcal {E}(\textbf{e})\,\) to each spacetime event \(\,\textbf{e}\in \mathcal {E}\).
Tangent vector fields (sections in geometric language) of the tangent bundle \(\,\varvec{\pi }:{T\mathcal {E}}\mapsto \mathcal {E}\,\), are maps \(\,\textbf{v}:\mathcal {E}\mapsto {T\mathcal {E}}\,\) such that the composition \(\,\varvec{\pi }\circ \textbf{v}:\mathcal {E}\mapsto \mathcal {E}\,\) is the identity in \(\,\mathcal {E}\).
The spacetime manifold \(\,\mathcal {E}\,\) is fibred into spatial slices \(\,S\,\) which are integral manifolds of the kernel of the differential:2Integrability is assured by Deahna-Frobenius3 condition \(\,\texttt{d}\texttt{d}t_\mathcal {E}=0\,\) which is trivially fulfilled due to nilpotency of exterior differentials [27].
$$\begin{aligned} \texttt{d}t_\mathcal {E}:\mathcal {E}\mapsto {({T\mathcal {E}})^*}. \end{aligned}$$
(1)
The time-vertical (or spatial) sub-bundle \(\,\varvec{\pi }:{V\mathcal {E}}\mapsto \mathcal {E}\,\) of the tangent bundle \(\,\varvec{\pi }:{T\mathcal {E}}\mapsto \mathcal {E}\,\) is made of those vector fields \(\,\textbf{v}:\mathcal {E}\mapsto {V\mathcal {E}}\,\) such that:Each observer also defines in the spacetime manifold of events a congruence of time-lines, generated by a vector field of transversal time-arrows \(\,\textbf{Z}:\mathcal {E}\mapsto {T\mathcal {E}}\). Time-lines do intersect each space-slice just at a single event.
$$\begin{aligned} {\langle }\texttt{d}t_\mathcal {E},\textbf{v}{\rangle }=0. \end{aligned}$$
(2)
A direct sum decomposition \(\,{T\mathcal {E}}={V\mathcal {E}}\oplus \text {Span}(\textbf{Z})\,\) holds and tangent vectors \(\,\textbf{V}\in {T\mathcal {E}}\,\) may be uniquely split as sum \(\,\textbf{V}=\textbf{v}+\lambda \cdot \textbf{Z}\,\) with \(\,\textbf{v}\in {V\mathcal {E}}\,\) and \(\,\lambda \in \Re \).
The time-vertical bundle \(\,{V\mathcal {E}}\,\) with \(\,\dim ({V\mathcal {E}})=3\,\) is endowed with a field of metric tensors which are symmetric and positive covariant tensors \(\,\textbf{g}\in {\textsc {Cov}}({V\mathcal {E}})\,\), so that each spatial slice is a Riemann manifold.4
The dual bundle \(\,{({V\mathcal {E}})^*}=(\text {Span}(\textbf{Z}))^\circ \,\) is made of those covectors in \(\,{({T\mathcal {E}})^*}\,\) which vanish on time-arrows \(\,\textbf{Z}\in {T\mathcal {E}}\).5 The bundle \(\,{({V\mathcal {E}})^*}\,\) may also be identified with the factor bundle \(\,{({T\mathcal {E}})^*}/({V\mathcal {E}})^\circ \).
Without loss of generality the following tuning property can be assumed:The notion of time-arrow field, referred to as observer rigging in [28] and observer field in [29, 30], is a discriminant which puts into evidence the superiority of a proper 4D spacetime context when compared with the standard 3D spatial context where the time plays just the role of ordering parameter.6
$$\begin{aligned} {\langle }\texttt{d}t_\mathcal {E},\textbf{Z}{\rangle }=1\circ t_\mathcal {E}:\mathcal {E}\mapsto \Re . \end{aligned}$$
(3)
Given a pair of vectors and covector frames:the associated the Haar matrix is defined by:The determinant \(\,\det (\textsc {HAAR}(\{\textbf{a}^*\},\{\textbf{a}\}))\,\) is multilinear and alternating in both bases. An (exterior) form is a field of multilinear alternating tensors.
$$\begin{aligned} \begin{aligned}&\,\{\textbf{a}\}:=\{\textbf{a}_1,\textbf{a}_2,\textbf{a}_3\}\,\subset {V\mathcal {E}}, \\&\,\{\textbf{a}^*\}:=\{\textbf{a}^*_1,\textbf{a}^*_2,\textbf{a}^*_3\}\,\subset {({V\mathcal {E}})^*}, \end{aligned} \end{aligned}$$
(4)
$$\begin{aligned} \textsc {HAAR}(\{\textbf{a}^*\},\{\textbf{a}\})= \left[ \begin{array}{ccc} {\langle }\textbf{a}^*_1,\textbf{a}_1{\rangle }\,\,&{}\,\,{\langle }\textbf{a}^*_1,\textbf{a}_2{\rangle }\,\,&{}\,\,{\langle }\textbf{a}^*_1,\textbf{a}_3{\rangle } \\[4pt] {\langle }\textbf{a}_2^*,\textbf{a}_1{\rangle }\,\,&{}\,\,{\langle }\textbf{a}_2^*,\textbf{a}_2{\rangle }\,\,&{}\,\,{\langle }\textbf{a}_2^*,\textbf{a}_3{\rangle } \\[4pt] {\langle }\textbf{a}_3^*,\textbf{a}_1{\rangle }\,\,&{}\,\,{\langle }\textbf{a}_3^*,\textbf{a}_2{\rangle }\,\,&{}\,\,{\langle }\textbf{a}_3^*,\textbf{a}_3{\rangle } \end{array}\right] \end{aligned}$$
(5)
Since maximal multilinear alternating tensors are all proportional, dual volume forms \(\,\varvec{\mu }\in {\textsc {Max}}({V\mathcal {E}})\,\) and \(\,\varvec{\mu }^*\in {\textsc {Max}}({({V\mathcal {E}})^*})\,\) are defined by:7In the time-vertical bundle \(\,{V\mathcal {E}}\,\) a metric field can be introduced as symmetric positive definite covariant tensor \(\,\textbf{g}\in {\textsc {Cov}}({V\mathcal {E}})\,\) or equivalently as a positive definite \(\,\textbf{g}:{V\mathcal {E}}\mapsto ({V\mathcal {E}})^*\,\) fulfilling the self-duality property \(\,\textbf{g}^*=\textbf{g}\).
$$\begin{aligned} \varvec{\mu }^*{(\{\textbf{a}^*\})}\cdot \varvec{\mu }(\{\textbf{a}\})=\det \Big (\textsc {HAAR}(\{\textbf{a}^*\},\{\textbf{a}\})\Big ). \end{aligned}$$
(6)
Setting \(\,\{\textbf{a}^*\}=\{\textbf{g}\textbf{a}\}\,\) in Eq. (5), we get the associated Gram matrix:With the notion of Gram matrix a volume form \(\,\varvec{\mu }_\textbf{g}\in {\textsc {Max}}({V\mathcal {E}})\,\) may be introduced, to within an orientation-dependent sign, by the rule:According to the definition in Eq. (8) the volume form \(\,\varvec{\mu }_\textbf{g}\in {\textsc {Max}}({V\mathcal {E}})\,\), compatible with the spatial metric \(\,\textbf{g}\in {\textsc {Cov}}({V\mathcal {E}})\,\),8 is uniquely defined, once a positive space orientation is fixed.
$$\begin{aligned} \textsc {GRAM}(\{\textbf{a}\})= \left[ \begin{array}{ccc} \textbf{g}(\textbf{a}_1\,,\textbf{a}_1)\,\,&{}\,\,\textbf{g}(\textbf{a}_1\,,\textbf{a}_2)\,\,&{}\,\,\textbf{g}(\textbf{a}_1\,,\textbf{a}_3) \\[4pt] \textbf{g}(\textbf{a}_2\,,\textbf{a}_1)\,\,&{}\,\,\textbf{g}(\textbf{a}_2\,,\textbf{a}_2)\,\,&{}\,\,\textbf{g}(\textbf{a}_2\,,\textbf{a}_3) \\[4pt] \textbf{g}(\textbf{a}_3\,,\textbf{a}_1)\,\,&{}\,\,\textbf{g}(\textbf{a}_3\,,\textbf{a}_2)\,\,&{}\,\,\textbf{g}(\textbf{a}_3\,,\textbf{a}_3) \end{array}\right] \end{aligned}$$
(7)
$$\begin{aligned} \varvec{\mu }_\textbf{g}^2(\{\textbf{a}\}):=\det \Big (\textsc {GRAM}(\{\textbf{a}\})\Big ). \end{aligned}$$
(8)
2.1 Motion along the trajectory
A primary example of powerfulness of the four-dimensional spacetime representation is given by the notions of body trajectory and motion.
A motion \(\,\varvec{\varphi }:\mathcal {E}\mapsto \mathcal {E}\,\) of a body in the spacetime manifold \(\,\mathcal {E}\,\) of events is described by an observer as a one-parameter group of movements:which are automorphisms of the dynamical trajectory submanifold \(\,{\mathcal {T}}\subset \mathcal {E}\,\) fulfilling \(\,\forall \;\alpha ,t\in \mathcal {Z}\,\) the property of simultaneity conservation, according to the commutative diagram:
Movements form a commutative group under the composition rule:The spacetime velocity of motion is defined by:Taking the derivative \(\,\partial _{\alpha =0}\,\,\) of Eq. (10)\(_1\), we have:Comparing Eq. (13) with the tuning property Eq. (3), we get the decomposition in space and time components:with the spatial component qualified by time-verticality:Moreover, by definition of push:taking the derivative \(\,\partial _{\alpha =0}\,\,\) in Eq. (11), we infer the spacetime velocity is pushed by the motion:
$$\begin{aligned} \varvec{\varphi }_\alpha :{\mathcal {T}}\mapsto {\mathcal {T}}, \end{aligned}$$
(9)
(10)
$$\begin{aligned} \varvec{\varphi }_{(\alpha +\beta )}= \varvec{\varphi }_\alpha \circ \varvec{\varphi }_\beta =\varvec{\varphi }_\beta \circ \varvec{\varphi }_\alpha ,\quad \forall \,\alpha ,\beta \in \mathcal {Z}. \end{aligned}$$
(11)
$$\begin{aligned} \textbf{V}_{\varvec{\varphi }}:=\partial _{\alpha =0}\,\varvec{\varphi }_\alpha :{\mathcal {T}}\mapsto T{\mathcal {T}}. \end{aligned}$$
(12)
$$\begin{aligned} {\langle }dt_\mathcal {E},\textbf{V}_{\varvec{\varphi }}{\rangle }=1\circ t_\mathcal {E}. \end{aligned}$$
(13)
$$\begin{aligned} \textbf{V}_{\varvec{\varphi }}=\textbf{v}_{\varvec{\varphi }}+\textbf{Z}, \end{aligned}$$
(14)
$$\begin{aligned} {\langle }dt_\mathcal {E},\textbf{v}_{\varvec{\varphi }}{\rangle }=0. \end{aligned}$$
(15)
$$\begin{aligned} (\varvec{\varphi }{\uparrow }\textbf{V}_{\varvec{\varphi }})\circ \varvec{\varphi }=T\varvec{\varphi }\cdot \textbf{V}_{\varvec{\varphi }}, \end{aligned}$$
(16)
$$\begin{aligned} \varvec{\varphi }{\uparrow }\textbf{V}_{\varvec{\varphi }}=\textbf{V}_{\varvec{\varphi }}. \end{aligned}$$
(17)
2.2 Change of observer
Definition 1
(Change of frame) In mechanics a change of frame is an automorphism \(\,{\varvec{\zeta }}:\mathcal {E}\mapsto \mathcal {E}\,\) of time-bundles \(\,t_\mathcal {E}:\mathcal {E}\mapsto \mathcal {Z}\,\) and \(\,t_{\varvec{\zeta }}:\mathcal {E}\mapsto \mathcal {Z}\,\) over the identity \(\,\textbf{id}_{\mathcal {Z}}:\mathcal {Z}\mapsto \mathcal {Z}\,\), according to the commutative diagram:
(18)
From Eq. (18), commutativity of push and exterior differential gives:Newton frame changes are characterised by no change of clock rates:The spacetime motion \(\,\varvec{\varphi }_\alpha :\mathcal {T}\mapsto \mathcal {T}\,\) is pushed by the change of frame to a motion \(\,({\varvec{\zeta }}{\uparrow }\varvec{\varphi })_\alpha :{\mathcal {T}_{\varvec{\zeta }}}\mapsto {\mathcal {T}_{\varvec{\zeta }}}\,\), with \(\,{\mathcal {T}_{\varvec{\zeta }}}={\varvec{\zeta }}(\mathcal {T})\,\), as depicted by the commutative diagram:
and the relevant velocities are related by push:Moreover, comparing the splitting of the spacetime velocity \(\,{\textbf{V}_{{\varvec{\zeta }}{\uparrow }\varvec{\varphi }}}\,\) performed by the privileged observer and pushing the splitting of the spacetime velocity \(\,\textbf{V}_{\varvec{\varphi }}\,\) in Eq. (14), we get:The computations in Eqs. (22) and (23) clarify why in the spacetime context covariance (commonly referred to as objectivity) of the spacetime velocity \(\,\textbf{V}_{\varvec{\varphi }}\,\) holds true under any change of observer.
$$\begin{aligned} \texttt{d}t_{\varvec{\zeta }}={\varvec{\zeta }}{\uparrow }\texttt{d}t_\mathcal {E}. \end{aligned}$$
(19)
$$\begin{aligned} \texttt{d}t_{\varvec{\zeta }}=\texttt{d}t_\mathcal {E}. \end{aligned}$$
(20)
(21)
$$\begin{aligned} \begin{aligned} {\textbf{V}_{{\varvec{\zeta }}{\uparrow }\varvec{\varphi }}}&\,=\partial _{\alpha =0}\,({\varvec{\zeta }}{\uparrow }\varvec{\varphi })_\alpha \\&\,=T{\varvec{\zeta }}\circ \,\partial _{\alpha =0}\,\varvec{\varphi }_\alpha \circ {\varvec{\zeta }}^{-1}\\&\,={\varvec{\zeta }}{\uparrow }\textbf{V}_{\varvec{\varphi }}. \end{aligned} \end{aligned}$$
(22)
$$\begin{aligned} \left\{ \begin{aligned}&\,{\textbf{V}_{{\varvec{\zeta }}{\uparrow }\varvec{\varphi }}}={\textbf{v}_{{\varvec{\zeta }}{\uparrow }\varvec{\varphi }}}+\textbf{Z}, \\&\,{\varvec{\zeta }}{\uparrow }\textbf{V}_{\varvec{\varphi }}={\varvec{\zeta }}{\uparrow }\textbf{v}_{\varvec{\varphi }}+{\varvec{\zeta }}{\uparrow }\textbf{Z}. \end{aligned} \right. \end{aligned}$$
(23)
This conclusion is not in contrast with the lack of objectivity concerning the spatial component \(\,\textbf{v}_{\varvec{\varphi }}\,\) of the velocity, declared in [2, p. 555].
Indeed, the transformation rule:expressing objectivity of the spatial component \(\,\textbf{v}_{\varvec{\varphi }}\,\) of the velocity, according to Truesdell & Noll [3] and followers, implies by Eq. (23) that \(\,{\varvec{\zeta }}{\uparrow }\textbf{Z}=\textbf{Z}\,\) which means there is no spatial relative velocity between the involved frames.9
$$\begin{aligned} {\textbf{v}_{{\varvec{\zeta }}{\uparrow }\varvec{\varphi }}}={\varvec{\zeta }}{\uparrow }\textbf{v}_{\varvec{\varphi }}, \end{aligned}$$
(24)
In classic treatments [3], where time is a just an ordering parameter, the time-arrow field \(\,\textbf{Z}\,\) is not even defined and the discussion above is simply out of reach.
2.3 Material, spatial and referential fields
Material, spatial and referential fields are defined as follows [31]:After [3, 6] referential fields are often improperly referred to as material fields in the literature. No one-to-one correspondence may exist between material and spatial fields, according to the clear distinction made in [31] and recalled in above.
-
Material vector fields are based on the trajectory and take values on the time-vertical tangent bundle to the trajectory manifold. Material tensor fields are linear maps on material vector and covector fields.
-
Spatial vector fields are based on the trajectory and take values on the tangent bundle to a space-slice manifold. Spatial tensor fields are linear maps involving spatial vector and covector fields.
-
Referential vector fields are based on a chosen local reference shape and take values on the relevant tangent bundle.
Moreover:Parallel and convective derivatives of any tensor field are defined by Leibniz rule, due to invariance of their scalar values.
-
Spatial vectors can be parallel transported along any line \(\,\textbf{c}:\Re \mapsto \mathcal {E}\,\) drawn in spacetime endowed with a connection. The forward parallel transport is denoted by \(\,{\Uparrow }\,\) with inverse \(\,{\Downarrow }\). Invariance under parallel transport results in vanishing of the covariant derivative \(\,\nabla \,\) along the vector \(\,\textbf{t}=\partial _{\lambda =0}\,\textbf{c}(\lambda )\,\) tangent to the line:Parallel transport to vectors based on alien manifolds outside the ambient spacetime is not feasible due to lack of a connection.$$\begin{aligned} \nabla _\textbf{t}(\textbf{v})= \lim _{\lambda \rightarrow 0}\frac{1}{\lambda }\Big ({\Downarrow }_{-\lambda }(\textbf{v}\circ \textbf{c})(\lambda )-\textbf{v}(0)\Big ) \end{aligned}$$(25)
-
Material vectors can be transformed along the motion by push \(\,{\uparrow }\,\) (with its inverse \(\,{\downarrow }\,\)) to get other material vectors, or along a placement map to get referential vectors. Invariance under push or pull results in vanishing of the Lie10 (or convective) derivative along the motion:$$\begin{aligned} \mathcal {L}_{\textbf{V}_{\varvec{\varphi }}}(\textbf{v}):=\partial _{\alpha =0}\,(\varvec{\varphi }_\alpha {\downarrow }\textbf{v}) =\partial _{\alpha =0}\, \Big (T\varvec{\varphi }_{-\alpha }\cdot (\textbf{v}\circ \varvec{\varphi }_{\alpha })\Big ). \end{aligned}$$(26)
3 Genesis of Nanson’s formula
A configuration \(\,{\varvec{\varOmega }}\,\) of a body is detected by an observer as a spatial slice \(\,S\cap {\mathcal {T}}\,\) of the trajectory manifold \(\,{\mathcal {T}}\).
The formula contributed by Nanson in [33] deals with an outer oriented material surface \(\,\varSigma \,\) with \(\,\dim (\varSigma )=2\,\), drawn in a configuration \(\,{\varvec{\varOmega }}\,\) of geometric dimension \(\,\dim ({\varvec{\varOmega }})=3\).
Nanson investigated about the changes of area and unit normal versor of the material surface \(\,\varSigma \,\) by effect of the motion \(\,\varvec{\varphi }:{\mathcal {T}}\mapsto {\mathcal {T}}\,\) along the trajectory \(\,{\mathcal {T}}\,\) of dimension \(\,\dim ({\mathcal {T}})=\dim (\mathcal {E})\).
Normal lines the material surface \(\,\varSigma \,\) embedded in \(\,{\varvec{\varOmega }}\,\) are generated by those vector fields \(\,\textbf{n}_{\varSigma }:\varSigma \mapsto V_{\varSigma }\mathcal {E}\,\) whose associated one-forms \(\,\textbf{g}\cdot \textbf{n}_{\varSigma }:\varSigma \mapsto (V_{\varSigma }\mathcal {E})^*\,\) annihilate the tangent bundle \(\,T\varSigma \,\):Then \(\,\dim \big (\text {Span}(\textbf{n}_{\varSigma })\big )=3-2=1\).11
$$\begin{aligned} \textbf{g}\cdot \textbf{n}_{\varSigma }\in (T\varSigma )^\circ , \end{aligned}$$
(27)
Spatial vector fields \(\,\textbf{n}_{\varSigma }:\varSigma \mapsto V_{\varSigma }\mathcal {E}\,\) normal to \(\,\varSigma \,\) are assumed to be \(\,\textbf{g}\)-unitary, i.e. such that \(\,\textbf{g}(\textbf{n}_{\varSigma },\textbf{n}_{\varSigma })=1\,\) and the sign of \(\,\textbf{n}_{\varSigma }\,\) is conforming to the outer orientation of the material surface \(\,\varSigma \).12
Similarly, the unit spatial vector field \(\,\textbf{n}_{\varvec{\varphi }(\varSigma )}:\varvec{\varphi }(\varSigma )\mapsto V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\) normal to the moved material surface \(\,\varvec{\varphi }(\varSigma )\,\), is such that \(\,\textbf{g}(\textbf{n}_{\varvec{\varphi }(\varSigma )},\textbf{n}_{\varvec{\varphi }(\varSigma )})=1\,\) with the corresponding covector field \(\,\textbf{g}\cdot \textbf{n}_{\varvec{\varphi }(\varSigma )}:\varvec{\varphi }(\varSigma )\mapsto (V_{\varvec{\varphi }(\varSigma )}\mathcal {E})^*\,\) fulfilling the annihilation property:The area two-forms \(\,\varvec{\mu }_\varSigma \in {\textsc {Max}}(T\varSigma )\,\) and \(\,\varvec{\mu }_{\varvec{\varphi }(\varSigma )}\in {\textsc {Max}}(T(\varvec{\varphi }(\varSigma )))\,\) induced on \(\,\varSigma \,\) and on \(\,\varvec{\varphi }(\varSigma )\,\) by the volume three-form \(\,\varvec{\mu }_\textbf{g}\in {\textsc {Max}}({V\mathcal {E}})\,\) are defined by contractions with the pertinent unit normal field:
To simplify, we often omit the dot symbol \(\,(\cdot )\,\) of linear dependence and adopt, for the time-vertical restricted tangent movement, the familiar notation:13The determinant \(\,J_{\varvec{\varphi }}:=\det (\textbf{F}_{\varvec{\varphi }})\,\) of this tangent movement is defined by Euler–Jacobi identity [34, 35]:14
In a frame \(\,\{\textbf{a}\}\,\) on \(\,{V\mathcal {E}}\,\) as in Eq. (4), the pullback is given by:
$$\begin{aligned} \textbf{g}\cdot \textbf{n}_{\varvec{\varphi }(\varSigma )}\in (T\varvec{\varphi }(\varSigma ))^\circ . \end{aligned}$$
(28)
(29)
$$\begin{aligned} \textbf{F}_{\varvec{\varphi }}:=T\varvec{\varphi }:{{V\mathcal {T}}}\mapsto {{V\mathcal {T}}}. \end{aligned}$$
(30)
(31)
$$\begin{aligned} (\varvec{\varphi }{\downarrow }\varvec{\mu }_\textbf{g})(\{\textbf{a}\}):=\varvec{\mu }_\textbf{g}(\{\textbf{F}_{\varvec{\varphi }}\textbf{a}\}). \end{aligned}$$
(32)
Proposition 1
(Nanson) Area-form \(\,\varvec{\mu }_\varSigma \,\) and unit normal \(\,\textbf{n}_{\varSigma }\,\) to an outer oriented surface \(\,\varSigma \subset \mathcal {T}\,\) and area-form \(\,\varvec{\mu }_{\varvec{\varphi }(\varSigma )}\,\) and unit normal \(\,\textbf{n}_{\varvec{\varphi }(\varSigma )}\,\) to the pushed surface \(\,\varvec{\varphi }(\varSigma )\,\) fulfil on \(\,\varSigma \,\) the relation:
(33)
Proof
Let \(\,\{\textbf{a},\textbf{b}\}\,\) be a frame at \(\,T\varSigma \).
Then the triplet \(\,\{\textbf{n}_{\varSigma },\textbf{a},\textbf{b}\}\,\) is a frame on \(\,V_{\varSigma }\mathcal {E}\,\) and the tripletis a frame on \(\,V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\).
$$\begin{aligned} \{\textbf{n}_{\varvec{\varphi }(\varSigma )},\textbf{F}_{\varvec{\varphi }}\textbf{a},\textbf{F}_{\varvec{\varphi }}\textbf{b}\}, \end{aligned}$$
(34)
The tensor product \(\,\textbf{g}\textbf{n}_{\varSigma }\otimes \textbf{n}_{\varSigma }\,\) defined by:is the \(\,\textbf{g}\)-orthogonal projector on \(\,\text {Span}(\textbf{n}_{\varSigma })\,\) and \(\,\textbf{I}-\textbf{g}\textbf{n}_{\varSigma }\otimes \textbf{n}_{\varSigma }\,\) is the complementary projector on \(\,T\varSigma \). Similarly for \(\,\textbf{n}_{\varvec{\varphi }(\varSigma )}\).
$$\begin{aligned} (\textbf{g}\textbf{n}_{\varSigma }\otimes \textbf{n}_{\varSigma })(\textbf{v}) :=\textbf{g}(\textbf{n}_{\varSigma }\,,\textbf{v})\cdot \textbf{n}_{\varSigma },\quad \textbf{v}\in V_\varSigma \mathcal {E}, \end{aligned}$$
(35)
Hence, for any \(\,\textbf{v}\in V_\varSigma \mathcal {E}\,\) and \(\,\textbf{F}_{\varvec{\varphi }}\textbf{v}\in V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\):Recalling Eq. (29), rewritten in terms of tensor products, Eq. (36) yields:
Each one of Eqs. (37) states no more than:
$$\begin{aligned} \left\{ \; \begin{aligned}&\,\varvec{\mu }_\textbf{g}(\textbf{v},\textbf{a},\textbf{b}) ={\langle }\textbf{g}\cdot \textbf{n}_{\varSigma },\textbf{v}{\rangle }\cdot \varvec{\mu }_\textbf{g}(\textbf{n}_{\varSigma },\textbf{a},\textbf{b}), \\&\,\varvec{\mu }_\textbf{g}(\textbf{F}_{\varvec{\varphi }}\textbf{v},\textbf{F}_{\varvec{\varphi }}\textbf{a},\textbf{F}_{\varvec{\varphi }}\textbf{b}) ={\langle }\textbf{g}\cdot \textbf{n}_{\varvec{\varphi }(\varSigma )},\textbf{F}_{\varvec{\varphi }}\textbf{v}{\rangle } \cdot \varvec{\mu }_\textbf{g}(\textbf{n}_{\varvec{\varphi }(\varSigma )},\textbf{F}_{\varvec{\varphi }}\textbf{a},\textbf{F}_{\varvec{\varphi }}\textbf{b}). \end{aligned} \right. \end{aligned}$$
(36)
(37)
The volume of a parallelepiped is height times base-area.
3.1 Role of the cofactor map
Central in the investigation about Nanson’s formula is the notion of cofactor map, as defined e.g. in [11, Eq. (1.17)] and [17, \(\mathsection \)2.11]:
The tangent \(\,\textbf{F}_{\varvec{\varphi }}:V_\varSigma \mathcal {E}\mapsto V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\) and its \(\textbf{g}\)-adjoint \(\,\textbf{F}_{\varvec{\varphi }}^{A}:V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\mapsto V_{\varSigma }\mathcal {E}\,\) according to the metric \(\,\textbf{g}\in {\textsc {Cov}}({V\mathcal {E}})\,\) are related by the defining identity, for any \(\,\textbf{v}\in V_{\varSigma }\mathcal {E}\,\) and \(\,\textbf{u}_{\varvec{\varphi }}\in V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\):Equation (39) is equivalent, in terms of dual maps, to:Symmetry of \(\,\textbf{g}\,\) means \(\,\textbf{g}^*=\textbf{g}\,\) and Eq. (40) gives:It is easy to see that:stating inverse and adjoint are commuting operations.
(38)
$$\begin{aligned} \textbf{g}(\textbf{F}_{\varvec{\varphi }}\textbf{v}\,,\textbf{u}_{\varvec{\varphi }}) =\textbf{g}(\textbf{F}_{\varvec{\varphi }}^{A}\textbf{u}_{\varvec{\varphi }}\,,\textbf{v}). \end{aligned}$$
(39)
$$\begin{aligned} {\langle }\textbf{g}\textbf{F}_{\varvec{\varphi }}\textbf{v},\textbf{u}_{\varvec{\varphi }}{\rangle } ={\langle }\textbf{g}\textbf{F}_{\varvec{\varphi }}^A\textbf{u}_{\varvec{\varphi }},\textbf{v}{\rangle } ={\langle }(\textbf{g}\textbf{F}_{\varvec{\varphi }})^*\textbf{u}_{\varvec{\varphi }},\textbf{v}{\rangle } ={\langle }(\textbf{F}_{\varvec{\varphi }}^*\textbf{g}^*)\textbf{u}_{\varvec{\varphi }},\textbf{v}{\rangle }, \end{aligned}$$
(40)
$$\begin{aligned} \textbf{g}\textbf{F}_{\varvec{\varphi }}^A=\textbf{F}_{\varvec{\varphi }}^*\textbf{g}:V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\mapsto (V_{\varSigma }\mathcal {E})^*. \end{aligned}$$
(41)
$$\begin{aligned} \textbf{F}_{\varvec{\varphi }}^{-A}=\textbf{F}_{\varvec{\varphi }}^{A-}:V_\varSigma \mathcal {E}\mapsto V_{\varvec{\varphi }(\varSigma )}\mathcal {E}, \end{aligned}$$
(42)
To clarify the role of the cofactor maplet us observe that the image \(\,\textbf{F}_{\varvec{\varphi }}\textbf{n}_{\varSigma }\in V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\) of a normal versor \(\,\textbf{n}_{\varSigma }\in V_\varSigma \mathcal {E}\,\) transformed by a motion \(\,\varvec{\varphi }:\mathcal {T}\mapsto \mathcal {T}\,\), is a vector not necessarily still normal to the moved surface \(\,\varvec{\varphi }(\varSigma )\).15
$$\begin{aligned} \textrm{cof}(\textbf{F}_{\varvec{\varphi }}):\,V_\varSigma \mathcal {E}\mapsto V_{\varvec{\varphi }(\varSigma )}\mathcal {E}, \end{aligned}$$
(43)
On the other hand, by definition of cross product \(\,\textbf{a}\times \textbf{b}\in V_\varSigma \mathcal {E}\,\):the vecor \(\,\textbf{a}\times \textbf{b}\in V_\varSigma \mathcal {E}\,\) is orthogonal to the surface \(\,\varSigma \,\) and hence proportional to \(\,\textbf{n}_{\varSigma }\in V_\varSigma \mathcal {E}\).
$$\begin{aligned} \textbf{g}\cdot (\textbf{a}\times \textbf{b}):=\varvec{\mu }_\textbf{g}(\textbf{a},\textbf{b}), \quad \forall \,\textbf{a},\textbf{b}\in V_\varSigma \mathcal {E}, \end{aligned}$$
(44)
Moreover, for any \(\,\textbf{c}_{\varvec{\varphi }}\in V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\) we get the chain of equalities:Since the expression at the l.h.s. of Eq. (45) vanishes for any \(\,\textbf{c}_{\varvec{\varphi }}\in V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\) tangent to \(\,\varvec{\varphi }(\varSigma )\,\), it follows the vector field:is orthogonal to the moved surface \(\,\varvec{\varphi }(\varSigma )\,\), hence proportional to the unit normal \(\,\textbf{n}_{\varvec{\varphi }(\varSigma )}\,\), as quoted in [11, Eq. (1.18)].
$$\begin{aligned} \begin{aligned} \varvec{\mu }_\textbf{g}(\textbf{F}_{\varvec{\varphi }}\textbf{a},\textbf{F}_{\varvec{\varphi }}\textbf{b},\textbf{c}_{\varvec{\varphi }})&\,=\varvec{\mu }_\textbf{g}(\textbf{F}_{\varvec{\varphi }}\textbf{a},\textbf{F}_{\varvec{\varphi }}\textbf{b},\textbf{F}_{\varvec{\varphi }}\textbf{F}_{\varvec{\varphi }}^{-1}\textbf{c}_{\varvec{\varphi }}) \\&\,=\det (\textbf{F}_{\varvec{\varphi }})\cdot \textbf{g}(\textbf{a}\times \textbf{b},\textbf{F}_{\varvec{\varphi }}^{-1}\textbf{c}_{\varvec{\varphi }}) \\&\,=J_{\varvec{\varphi }}\cdot \textbf{g}(\textbf{F}_{\varvec{\varphi }}^{-A}(\textbf{a}\times \textbf{b}),\textbf{c}_{\varvec{\varphi }}) \\&\,=\textbf{g}(\textrm{cof}(\textbf{F}_{\varvec{\varphi }})\cdot (\textbf{a}\times \textbf{b}),\textbf{c}_{\varvec{\varphi }}). \end{aligned}\nonumber \\ \end{aligned}$$
(45)
$$\begin{aligned} \textrm{cof}(\textbf{F}_{\varvec{\varphi }})\cdot {\textbf{n}_{\varSigma }}:\varSigma \mapsto V_{\varvec{\varphi }(\varSigma )}\mathcal {E}, \end{aligned}$$
(46)
We may then state, as equivalent to Nanson’s formula Eq. (33) on \(\,\varSigma \,\), the expression on \(\,\varvec{\varphi }(\varSigma )\,\):
Equivalence between Eqs. (33) and (47) is inferred from the equality:To prove Eq. (48) it suffices to observe that for any \(\,\textbf{u}_{\varvec{\varphi }}\in V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\):Hence, by arbitraryness of \(\,\textbf{u}_{\varvec{\varphi }}\in V_{\varvec{\varphi }(\varSigma )}\mathcal {E}\,\):
From Eq. (50) and the definition of cofactor in Eq. (38), we infer Eq. (48) and hence equivalence between Eqs. (33) and (47).
(47)
$$\begin{aligned} \varvec{\varphi }{\uparrow }\big (J_{\varvec{\varphi }}\cdot \textbf{g}\cdot \textbf{n}_{\varSigma }\big ) =\textbf{g}\cdot \textrm{cof}(\textbf{F}_{\varvec{\varphi }})\textbf{n}_{\varSigma }. \end{aligned}$$
(48)
$$\begin{aligned} \left\{ \begin{aligned} \,{\langle }\varvec{\varphi }{\uparrow }(\textbf{g}\cdot \textbf{n}_{\varSigma }),\textbf{u}_{\varvec{\varphi }}{\rangle }&\,=\varvec{\varphi }{\uparrow }{\langle }\textbf{g}\cdot \textbf{n}_{\varSigma },\varvec{\varphi }{\downarrow }\textbf{u}_{\varvec{\varphi }}{\rangle } \\&\,=\varvec{\varphi }{\uparrow }{\langle }\textbf{g}\cdot \textbf{n}_{\varSigma },\textbf{F}_{\varvec{\varphi }}^{-1}\textbf{u}_{\varvec{\varphi }}{\rangle } \\&\,={\langle }\textbf{g}\cdot \textbf{F}_{\varvec{\varphi }}^{-A}\textbf{n}_{\varSigma },\textbf{u}_{\varvec{\varphi }}{\rangle }. \end{aligned} \right. \end{aligned}$$
(49)
(50)
A further equivalence is proven by considering, for any pair of tangent vectors \(\,\textbf{a},\textbf{b}\in T_\textbf{x}\varSigma \,\), the generated parallelogram tangent to \(\,\varSigma \,\) at \(\,\textbf{x}\in \varSigma \,\) with area \(\,{\mathcal {A}}_{\varSigma }\,\) and the pushed parallelogram with sides \(\,\textbf{F}_{\varvec{\varphi }}\textbf{a},\textbf{F}_{\varvec{\varphi }}\textbf{b}\in T_{\varvec{\varphi }(\textbf{x})}\varvec{\varphi }(\varSigma )\,\) tangent at \(\,\varvec{\varphi }(\textbf{x})\in \varvec{\varphi }(\varSigma )\,\) with area \(\,{\mathcal {A}}_{\varvec{\varphi }(\varSigma )}\,\), defined by:The original expression of Nanson formula, see [2, Eq. (20.8)] and [9, Eq. (22.2.18)], may be expressed in geometric terms as follows:
The pushed area is given by:Let us prove equivalence of Eqs. (47) and (52).
$$\begin{aligned} \begin{aligned} {\mathcal {A}}_{\varSigma }&\,:=\varvec{\mu }_\varSigma (\textbf{a}\,,\textbf{b}), \\ {\mathcal {A}}_{\varvec{\varphi }(\varSigma )}&\,:=\varvec{\mu }_{\varvec{\varphi }(\varSigma )}(\textbf{F}_{\varvec{\varphi }}\textbf{a}\,,\textbf{F}_{\varvec{\varphi }}\textbf{b}). \end{aligned} \end{aligned}$$
(51)
(52)
$$\begin{aligned} \begin{aligned} \varvec{\varphi }{\uparrow }{A}_{\varSigma } ={A}_{\varSigma }\circ \varvec{\varphi }^{-1}:=&\, \varvec{\varphi }{\uparrow }\Big (\varvec{\mu }_\varSigma (\textbf{a}\,,\textbf{b})\Big ) \\ =&\,(\varvec{\varphi }{\uparrow }\varvec{\mu }_\varSigma )(\textbf{F}_{\varvec{\varphi }}\textbf{a}\,,\textbf{F}_{\varvec{\varphi }}\textbf{b}). \end{aligned} \end{aligned}$$
(53)
Application of Eq. (47) to the triplet \(\{\textbf{u}_{\varvec{\varphi }},\textbf{F}_{\varvec{\varphi }}\textbf{a},\textbf{F}_{\varvec{\varphi }}\textbf{b}\}\) with \(\,\textbf{u}_{\varvec{\varphi }}\in V_{\varvec{\varphi }(\textbf{x})}\mathcal {E}\,\) and \(\,\{\textbf{a},\textbf{b}\}\,\) vectors in \(\,T_{\textbf{x}}\varSigma \,\), by Eq. (53) leads on \(\,\varvec{\varphi }(\varSigma )\,\) to the expression:Arbitrariness of \(\,\textbf{u}_{\varvec{\varphi }}\in V_{\varvec{\varphi }(\textbf{x})}\mathcal {E}\,\) and non-degeneracy of the metric \(\,\textbf{g}\,\) in Eq. (54) yield Eq. (52).
$$\begin{aligned} {\mathcal {A}}_{\varvec{\varphi }(\varSigma )}\cdot \textbf{g}(\textbf{n}_{\varvec{\varphi }(\varSigma )}\,,\textbf{u}_{\varvec{\varphi }}) =(\varvec{\varphi }{\uparrow }{\mathcal {A}}_{\varSigma })\cdot \textbf{g}\big (\textrm{cof}(\textbf{F}_{\varvec{\varphi }})\cdot \textbf{n}_{\varSigma },\textbf{u}_{\varvec{\varphi }}\big ). \end{aligned}$$
(54)
In particular, setting \(\,\textbf{u}_{\varvec{\varphi }}=\textbf{n}_{\varvec{\varphi }(\varSigma )}\,\) in Eq. (54), we get:This means that the factor of proportionality between the image of \(\,\textbf{n}_{\varSigma }\,\) through the cofactor map, given by \(\,\textrm{cof}(\textbf{F}_{\varvec{\varphi }})\cdot \textbf{n}_{\varSigma }\,\), and the normal unit vector \(\,\textbf{n}_{\varvec{\varphi }(\varSigma )}\,\) to \(\,\varvec{\varphi }(\varSigma )\,\), which are parallel vectors, is equal to the area-ratio:The equivalence between the Euler–Jacobi volumetric identity Eq. (31) and Nanson formula Eq. (33) leads to the conclusion that this formula is of a purely kinematical nature, dealing with a surface \(\,\varSigma \,\) embedded in a 3D configuration \(\,{\varvec{\varOmega }}\,\) of a body in motion along the dynamical trajectory.
$$\begin{aligned} {\mathcal {A}}_{\varvec{\varphi }(\varSigma )} =(\varvec{\varphi }{\uparrow }{\mathcal {A}}_{\varSigma })\cdot \textbf{g}\big (\textrm{cof}(\textbf{F}_{\varvec{\varphi }})\cdot \textbf{n}_{\varSigma },\textbf{n}_{\varvec{\varphi }(\varSigma )}\big ). \end{aligned}$$
(55)
$$\begin{aligned} {\mathcal {A}}_{\varvec{\varphi }(\varSigma )}/(\varvec{\varphi }{\uparrow }{\mathcal {A}}_\varSigma ). \end{aligned}$$
(56)
The unit normals to a surface pushed by the motion are spatial vectors which are not dragged by the motion.
3.2 Nanson’s formula in the literature
In the literature Nanson’s formula has been variously quoted.
A flawed expression was exposed in Rodney Hill early paper [36, Eq. (31)]. With the present notations, the statement therein was:with \(\,\rho \,\) and \(\,\rho _{\varvec{\varphi }}\,\) volumetric mass densities in \(\,{\varvec{\varOmega }}\,\) and \(\,\varvec{\varphi }({\varvec{\varOmega }})\).
$$\begin{aligned} \rho _{\varvec{\varphi }}\cdot ( {\mathcal {A}}_{\varvec{\varphi }(\varSigma )}\cdot \textbf{n}_{\varvec{\varphi }(\varSigma )}) =\varvec{\varphi }{\uparrow }\big (\rho \cdot ( {\mathcal {A}}_{\varSigma }\cdot \textbf{n}_{\varSigma })\big ) , \end{aligned}$$
(57)
By fixing a frame \(\,\{\textbf{a}\}\,\) in \(\,{T{\varvec{\varOmega }}}\,\), Euler–Jacobi volumetric identity Eqs. (31)–(32) yield for the volumes:Hence, by conservation of mass (spuriously implicit in Eq. (57)):Equation (57) exposed in [36] consists then of the flawed geometric statement:In Eq. (60) area weighted unit normals are improperly related by multiplication according to volumetric ratio and push forward.
$$\begin{aligned} \begin{aligned} \mathcal {V}_{{\varvec{\varOmega }}}&\,:=\varvec{\mu }_\textbf{g}(\{\textbf{a}\}), \\ \mathcal {V}_{\varvec{\varphi }({\varvec{\varOmega }})}&\,:=\varvec{\mu }_\textbf{g}(\{\textbf{F}_{\varvec{\varphi }}\textbf{a}\}) =\varvec{\varphi }{\uparrow }(J_{\varvec{\varphi }}\cdot {\mathcal {V}}_{{\varvec{\varOmega }}}). \end{aligned} \end{aligned}$$
(58)
$$\begin{aligned} \rho _{\varvec{\varphi }}\cdot {\mathcal {V}}_{\varvec{\varphi }({\varvec{\varOmega }})} =\varvec{\varphi }{\uparrow }(\rho \cdot {\mathcal {V}}_{{\varvec{\varOmega }}}) =\rho _{\varvec{\varphi }}\cdot \varvec{\varphi }{\uparrow }(J_{\varvec{\varphi }}\cdot {\mathcal {V}}_{{\varvec{\varOmega }}}) \;\Longleftrightarrow \;\varvec{\varphi }{\uparrow }\rho =\rho _{\varvec{\varphi }}\cdot (\varvec{\varphi }{\uparrow }J_{\varvec{\varphi }}). \end{aligned}$$
(59)
$$\begin{aligned} {\mathcal {A}}_{\varvec{\varphi }(\varSigma )}\cdot \textbf{n}_{\varvec{\varphi }(\varSigma )}=\varvec{\varphi }{\uparrow }(J_{\varvec{\varphi }}\cdot \mathcal {A}_{\varSigma }\cdot \textbf{n}_{\varSigma }). \end{aligned}$$
(60)
This wrong expression has to be corrected to Eq. (52) where area weighted unit normals are related by push according to the cofactor map \(\,\textrm{cof}(\textbf{F}_{\varvec{\varphi }}):=J_{\varvec{\varphi }}\cdot \textbf{F}_{\varvec{\varphi }}^{-A}\).
As a matter of fact, the flaw in Eqs. (57)–(60) is evident since in general the vectors \(\,\varvec{\varphi }{\uparrow }\textbf{n}_{\varSigma }\,\) and \(\,\textbf{n}_{\varvec{\varphi }(\varSigma )}\,\) are not parallel one another and hence not proportional.
A correct expression of Nanson’s formula in cartesian coordinates was reported by Carl Erik Pearson in [37, Eq. (31–32)] and a little later by Clifford Ambrose Truesdell III and Richard Toupin in [2, Eq. (20.8)].
A similar treatment was later exposed by Rodney Hill in [38, Eq. (1.32)] and by Raymond Ogden in [9, Eq. (22.2.18)].
A trivial typo, resulting however in a serious flaw, is found in more recent lecture notes by Ragnar Larsson [39] with \(\,\textbf{F}_{\varvec{\varphi }}^{-1}:T(\varvec{\varphi }({\varvec{\varOmega }}))\mapsto {T{\varvec{\varOmega }}}\,\) in place of \(\,\textbf{F}_{\varvec{\varphi }}^{-A}:{T{\varvec{\varOmega }}}\mapsto T(\varvec{\varphi }({\varvec{\varOmega }}))\).
This mistake is there hidden by an over-simplified notation with domain and range manifolds not evidenced.
3.3 Material surfaces
Let us now consider the case of a material surface \(\,\varSigma \,\), with \(\, \dim (\varSigma )=2\,\), moving along its own trajectory \(\,\mathcal {T}_\varSigma \subset \mathcal {E}\,\) with \(\, \dim (\mathcal {T}_\varSigma )=2+1\).
In this case, the surface motion \(\,\varvec{\varphi }_\varSigma :\mathcal {T}_\varSigma \mapsto \mathcal {T}_\varSigma \,\) is defined only on this material trajectory, and the time-vertical restricted map:acts only on the time-vertical \((2+1)\)D tangent bundle \(\,V\mathcal {T}_\varSigma \).
$$\begin{aligned} \textbf{F}_{\varvec{\varphi }_\varSigma }=T\varvec{\varphi }_\varSigma :V\mathcal {T}_\varSigma \mapsto V\mathcal {T}_\varSigma , \end{aligned}$$
(61)
Therefore, the Euler–Jacobi determinant:is of order two and evaluable just on the bundle \(\,V\mathcal {T}_\varSigma \,\) by:By Eq. (51), this expression is equivalent to:Unlike the previous cofactor map defined in Eq. (38), the new one:doesn’t act on \(\,\textbf{n}_{\varSigma }\in V_\varSigma \mathcal {E}\,\) to \(\,\varSigma \,\), which are out of its domain of definition \(\,V\mathcal {T}_\varSigma \).
$$\begin{aligned} J_{\varvec{\varphi }_\varSigma }:=\det (\textbf{F}_{\varvec{\varphi }_\varSigma }), \end{aligned}$$
(62)
$$\begin{aligned} \varvec{\varphi }_\varSigma {\downarrow }\varvec{\mu }_{\varvec{\varphi }_\varSigma (\varSigma )}=J_{\varvec{\varphi }_\varSigma }\cdot \varvec{\mu }_\varSigma . \end{aligned}$$
(63)
$$\begin{aligned} \varvec{\varphi }_\varSigma {\downarrow }\mathcal {A}_{\varvec{\varphi }_\varSigma (\varSigma )}=J_{\varvec{\varphi }_\varSigma }\cdot \mathcal {A}_{\varSigma }. \end{aligned}$$
(64)
$$\begin{aligned} \textrm{cof}(\textbf{F}_{\varvec{\varphi }_\varSigma }):=(\varvec{\varphi }_\varSigma {\uparrow }J_{\varvec{\varphi }_\varSigma })\cdot \textbf{F}_{\varvec{\varphi }_\varSigma }^{-A} :V\mathcal {T}_\varSigma \mapsto V\mathcal {T}_\varSigma , \end{aligned}$$
(65)
The investigation can anyway still be carried out by considering just frames on \(\,\varSigma \,\) and their push by the motion \(\,\varvec{\varphi }_\varSigma :\mathcal {T}_\varSigma \mapsto \mathcal {T}_\varSigma \).
However, Nanson’s formula is no longer applicable.
In this respect the remark after Eq. (1.19) in [11] is worth to be carefully read and mentioned.
4 Equilibrium and stress fields revisited
The formidable theoretical construction, which stands as monumental basis of Continuum Mechanics (CM), was mainly set out in the course of the XVIII century with basic contributions by Jacob, Johann and Daniel Bernoulli, Leonhard Euler, Jean-Baptiste Le Rond d’Alembert, Joseph-Louis Lagrange, and at the beginning of the XIX century with the decisive completion due to Augustin-Louis Cauchy.
The whole edifice of Continuum Mechanics is based on stating equilibrium as a variational condition concerning a force system \(\,\textbf{f}_{\varvec{\varOmega }}\,\) composed of bulk actions per unit volume on the elements of a regularity partition \(\,\mathcal {P}({\varvec{\varOmega }})\,\) of the body configuration \({\varvec{\varOmega }}\,\) and surficial actions per unit area on its boundary \(\,\partial \mathcal {P}({\varvec{\varOmega }})\).
The linear space \(\,\mathcal {V}_{\varvec{\varOmega }}\,\) is made of kinematic fields \(\,\textbf{v}\in \mathcal {V}_{\varvec{\varOmega }}\,\) each with square integrable gradient in the elements of the regularity partition \(\,\mathcal {P}({\varvec{\varOmega }})\).
Firm and bilateral boundary constraints, acting between adjacent elements of the partition \(\,\mathcal {P}({\varvec{\varOmega }})\,\), consist of the assignment of a linear subspace \(\,\mathcal {L}_{\varvec{\varOmega }}\subset \mathcal {V}_{\varvec{\varOmega }}\,\) of conforming kinematic fields.
In boundary value problems the conformity space includes all kinematic fields vanishing in a boundary layer of each element of the partition \(\,\mathcal {P}({\varvec{\varOmega }})\).
Frictionless constraints are characterised by vanishing of virtual power performed by their interaction field for any conforming kinematic field.
The property of equilibrium means the force system doesn’t perform virtual power for any kinematic field \(\,\delta \textbf{v}:{\varvec{\varOmega }}\mapsto T_{\varvec{\varOmega }}S\,\) which is rigid, i.e. such that any line drawn in the body does not tend to change its length.
“ The state of equilibrium of a deformable body is not altered if any part of it is replaced with a rigid body of the same geometry.”
Peculiar to \(\,3D\,\) Continuum Mechanics is the implicit description of the fields of virtual infinitesimal isometries \(\,\delta \textbf{v}:{\varvec{\varOmega }}\mapsto T_{\varvec{\varOmega }}S\).
This notion is based on conceiving a virtual motion \(\,\delta \varvec{\varphi }_\lambda :\delta \mathcal {T}\mapsto \delta \mathcal {T}\,\) along a virtual trajectory \(\,\delta \mathcal {T}\subset S_{\varvec{\varOmega }}\,\) drawn in the spatial slice \(\,S_{\varvec{\varOmega }}\,\) across the configuration \(\,{\varvec{\varOmega }}\,\), with \(\,\delta \varvec{\varphi }_0\,\) identity in \(\,\delta \mathcal {T}\).
Rigidity is imposed by vanishing of the Lie derivative of the metric tensor field \(\,\textbf{g}:{T{\varvec{\varOmega }}}\mapsto ({T{\varvec{\varOmega }}})^*\,\) along virtual movements \(\,\delta \varvec{\varphi }_\lambda :{\varvec{\varOmega }}\mapsto S_{\varvec{\varOmega }}\,\), a condition which in terms of virtual velocity fields \(\,\delta \textbf{v}=\partial _{\lambda =0}\,\delta \varvec{\varphi }_\lambda \,\) writes:Virtual infinitesimal isometries are equivalently expressed by vanishing of the associated mixed Euler stretching tensor [41, 42]:which provides the local rate of elongation of any material line drawn in \(\,S_{\varvec{\varOmega }}\,\) through a point of evaluation in \(\,{\varvec{\varOmega }}\,\), in the sense that:being \(\,{\dot{\textbf{h}}}:=\partial _{\lambda =0}\,(\delta \varvec{\varphi }_{\lambda }{\Downarrow }(\delta \varvec{\varphi }_{\lambda }{\uparrow }\textbf{h}))\).
$$\begin{aligned} \mathcal {L}_{\delta \textbf{v}}(\textbf{g}):=\partial _{\lambda =0}\,(\delta \varvec{\varphi }_\lambda {\downarrow }\textbf{g}) =\textbf{0}. \end{aligned}$$
(66)
$$\begin{aligned} \textbf{D}(\delta \textbf{v})=\textbf{g}^{-1}\cdot \tfrac{1}{2}\,\mathcal {L}_{\delta \textbf{v}}(\textbf{g}):=\text {sym}_\textbf{g}\nabla (\delta \textbf{v}):{T{\varvec{\varOmega }}}\mapsto {T{\varvec{\varOmega }}}, \end{aligned}$$
(67)
$$\begin{aligned} \tfrac{1}{2}\,\mathcal {L}_{\delta \textbf{v}}(\textbf{g})(\textbf{h},\textbf{h}) =\textbf{g}(\dot{\textbf{h}},\textbf{h}) \end{aligned}$$
(68)
In Eq. (67) defining the Euler stretching tensor, \(\,\nabla \,\) is the covariant derivative associated with the spatial metric \(\,\textbf{g}\,\) in the Euclid space \(\,S_{\varvec{\varOmega }}\,\) [43].
The relevant Levi-Civita connection is metric preserving, flat and symmetric. The corresponding (path independent) foreword parallel transport by translation is \(\,{\Uparrow }\,\), with inverse \(\,{\Downarrow }\).
The linear Euler operator \(\,\textbf{D}:\mathcal {V}_{\varvec{\varOmega }}\mapsto \mathcal {H}_{\varvec{\varOmega }}\,\) maps David Hilbert space \(\,\mathcal {V}_{\varvec{\varOmega }}\,\) of regular kinematic fields into the Hilbert space \(\,\mathcal {H}_{\varvec{\varOmega }}\,\) of symmetric \(\textbf{g}\)-mixed tensor fields which are square integrable in the elements of the partition \(\,\mathcal {P}({\varvec{\varOmega }})\).
The \(\textbf{g}\)-adjoint \(\,\nabla (\delta \textbf{v})^A\,\) of \(\,\nabla (\delta \textbf{v})\,\) is defined by the condition:Symmetric and antisymmetric parts are defined by:The inner product between mixed tensors is computed, in terms of the linear invariant \(\,J_1\,\) and in cartesian coordinates by:According to this inner product, \(\textbf{g}\)-symmetric and \(\textbf{g}\)-antisymmetric mixed tensors are orthogonal one another.
$$\begin{aligned} \textbf{g}(\textbf{a}\,,\nabla (\delta \textbf{v})^A\cdot \textbf{b}) =\textbf{g}(\nabla (\delta \textbf{v})\cdot \textbf{a}\,,\textbf{b}), \quad \forall \,\textbf{a},\textbf{b}:{\varvec{\varOmega }}\mapsto T_{\varvec{\varOmega }}S. \end{aligned}$$
(69)
$$\begin{aligned} \left\{ \begin{aligned}&\,2\,\text {sym}_\textbf{g}\nabla (\delta \textbf{v}):=\nabla (\delta \textbf{v})+\nabla (\delta \textbf{v})^A, \\&\,2\,\text {skew}_\textbf{g}\nabla (\delta \textbf{v}):=\nabla (\delta \textbf{v})-\nabla (\delta \textbf{v})^A. \end{aligned} \right. \end{aligned}$$
(70)
$$\begin{aligned} (\textbf{T},\nabla (\delta \textbf{v}))_\textbf{g}:=J_1(\textbf{T}^A\cdot \nabla (\delta \textbf{v})) =\textbf{T}_{ij}\nabla (\delta \textbf{v})_{ij}. \end{aligned}$$
(71)
The stretching operator: \(\,\textbf{D}:\mathcal {V}_{\varvec{\varOmega }}\mapsto \mathcal {H}_{\varvec{\varOmega }}\,\) is linear and continuous from the Hilbert space \(\,\mathcal {V}_{\varvec{\varOmega }}\,\) of the vector fields in \(\,{\varvec{\varOmega }}\,\) to the Hilbert space \(\,\mathcal {H}_{\varvec{\varOmega }}\).
The notion of equilibrium is expressed by vanishing of the virtual power expended by the force system \(\,\textbf{f}\in \mathcal {F}_{\varvec{\varOmega }}\,\) for any virtual rigid kinematic field:16In Eq. (72) \(\,\{\mathcal {F}_{\varvec{\varOmega }},\mathcal {V}_{\varvec{\varOmega }}\}\,\) is a pair of Hilbert spaces in duality.17
$$\begin{aligned} {\langle }\textbf{f},\delta \textbf{v}{\rangle }_{\varvec{\varOmega }}=0,\quad \textbf{f}\in \mathcal {F}_{\varvec{\varOmega }}, \quad \forall \,\delta \textbf{v}\in \textrm{Ker}(\textbf{D})\subset \mathcal {V}_{\varvec{\varOmega }}. \end{aligned}$$
(72)
The implicit form of the rigidity constraint allows us to introduce a field of Lagrange multipliers with the mechanical meaning of Cauchy stress field \(\,\textbf{T}\,\), a basic observation attributed in [2, fn,4, p. 595] to Gabrio Piola [45].
Let \(\,\varvec{\mu }_\textbf{g}\,\) and \(\,\partial \varvec{\mu }_\textbf{g}:=\varvec{\mu }_\textbf{g}\cdot \textbf{n}\,\) be the metric volume and area-forms on \(\,{\varvec{\varOmega }}\,\) and on its boundary \(\,\partial {\varvec{\varOmega }}\,\), with \(\,\textbf{n}\,\) outward unit normal vector to \(\,\partial {\varvec{\varOmega }}\).
Volterra integral transformation in terms of the exterior differential \(\,\texttt{d}\,\), yields, for 3D integral domains, the expression of Gauss-Ostrogradskij divergence theorem of a smooth vector field \(\,\textbf{u}:{\varvec{\varOmega }}\mapsto {T{\varvec{\varOmega }}}\,\):with \(\,\partial {\varvec{\varOmega }}\,\), boundary of \(\,{\varvec{\varOmega }}\,\), oriented consistently with the orientation of \(\,{\varvec{\varOmega }}\).
$$\begin{aligned} \oint _{\partial {\varvec{\varOmega }}}^{}\varvec{\mu }\cdot \textbf{u}=\int _{{\varvec{\varOmega }}}^{}\texttt{d}\Big (\varvec{\mu }\cdot \textbf{u}\Big ) =\int _{{\varvec{\varOmega }}}^{}\textrm{div}(\textbf{u})\cdot \varvec{\mu }, \end{aligned}$$
(73)
Let \(\,\varSigma _{\varvec{\varOmega }}\,\) be the linear space of mixed tensor fields \(\,\textbf{T}\in {\textsc {Mix}}({T{\varvec{\varOmega }}})\,\) which are \(\textbf{g}\)-square integrable in each element of \(\,\mathcal {P}({\varvec{\varOmega }})\,\) with their divergence, and the linear space \(\,\mathcal {V}_{\varvec{\varOmega }}\,\) of vector fields \(\,\delta \textbf{v}\in {\textsc {Mix}}({T{\varvec{\varOmega }}})\,\) which are \(\textbf{g}\)-square integrable in each element of \(\,\mathcal {P}({\varvec{\varOmega }})\,\) with their gradients.
Setting \(\,\textbf{u}=\textbf{T}^A\delta \textbf{v}\,\) in Eq. (73) with \(\,\textbf{T}\in \varSigma _{\varvec{\varOmega }}\,\) and \(\,\delta \textbf{v}\in \mathcal {V}_{\varvec{\varOmega }}\,\) we may define the divergence \(\,\textrm{Div}\,\) of a mixed tensor field \(\,\textbf{T}\,\) by a formal application of Gottfried Wilhelm Leibniz rule of calculus:This yields George Green integral formula [46, 47]:The boundary integral on the l.h.s. of Eq. (75) may be rewritten as:The gradient can be split in \(\textbf{g}\)-symmetric and \(\textbf{g}\)-antisymmetric parts:with:having respectively the meaning of local stretching and spin.
$$\begin{aligned} \textrm{div}(\textbf{T}^A\delta \textbf{v}) =\textbf{g}(\textrm{Div}(\textbf{T})\,,\delta \textbf{v}) +(\textbf{T},\nabla (\delta \textbf{v}))_\textbf{g}, \end{aligned}$$
(74)
$$\begin{aligned} \begin{aligned} \oint _{\partial {\varvec{\varOmega }}}^{}\varvec{\mu }\cdot \textbf{T}^A\delta \textbf{v}&\,=\int _{{\varvec{\varOmega }}}^{}\texttt{d}\Big (\varvec{\mu }\cdot \textbf{T}^A\delta \textbf{v}\Big ) =\int _{{\varvec{\varOmega }}}^{}\textrm{div}(\textbf{T}^A\delta \textbf{v})\cdot \varvec{\mu }, \\&\,=\int _{{\varvec{\varOmega }}}^{}\Big ( \textbf{g}(\textrm{Div}(\textbf{T})\,,\delta \textbf{v}) +(\textbf{T},\nabla (\delta \textbf{v}))_\textbf{g}\Big )\cdot \varvec{\mu }. \end{aligned} \end{aligned}$$
(75)
$$\begin{aligned} \oint _{\partial {\varvec{\varOmega }}}^{}\varvec{\mu }\cdot \textbf{T}^A\delta \textbf{v}=\oint _{\partial {\varvec{\varOmega }}}^{}\textbf{g}(\textbf{T}^A\delta \textbf{v}\,,\textbf{n})\cdot \partial \varvec{\mu }=\oint _{\partial {\varvec{\varOmega }}}^{}\textbf{g}(\textbf{T}\textbf{n}\,,\delta \textbf{v})\cdot \partial \varvec{\mu }. \end{aligned}$$
(76)
$$\begin{aligned} \nabla (\delta \textbf{v})=\textbf{D}(\delta \textbf{v})+\textbf{W}(\delta \textbf{v}), \end{aligned}$$
(77)
$$\begin{aligned} \left\{ \begin{aligned}&\,\textbf{D}(\delta \textbf{v})=\text {sym}_\textbf{g}\nabla (\delta \textbf{v}), \\&\,\textbf{W}(\delta \textbf{v})=\text {skew}_\textbf{g}\nabla (\delta \textbf{v}), \end{aligned} \right. \end{aligned}$$
(78)
By virtue of George Green formula Eq. (75), the equilibrium operator \(\,\textbf{D}':\varSigma _{\varvec{\varOmega }}\mapsto \mathcal {F}_{\varvec{\varOmega }}\,\) on the Hilbert space \(\,\varSigma _{\varvec{\varOmega }}\,\), in duality with the stretching operator \(\,\textbf{D}:\mathcal {V}_{\varvec{\varOmega }}\mapsto \mathcal {H}_{\varvec{\varOmega }}\,\) defined by Eq. (67), is given by:According to Eq. (71), this equation provides the representation of a force system \(\,\textbf{f}_{\varvec{\varOmega }}\in \textbf{D}'(\textbf{T})\,\) in equilibrium on \(\,{\varvec{\varOmega }}\,\) in terms of:In Eq. (79), we may assume \(\,\textbf{T}=\text {sym}_\textbf{g}(\textbf{T})\,\) without loss of generality, due to orthogonality of \(\textbf{g}\)-symmetric and \(\textbf{g}\)-antisymmetric mixed tensors. Then:and the inner product \(\,(\textbf{T},\textbf{D}(\delta \textbf{v}))_\textbf{g}\,\) of Eq. (75) may be replaced with the product \(\,(\text {sym}_\textbf{g}\textbf{T},\textbf{D}(\delta \textbf{v}))_\textbf{g}\).
$$\begin{aligned} \begin{aligned}&\,{\langle }\textbf{D}'(\textbf{T}),\delta \textbf{v}{\rangle }_{\varvec{\varOmega }}={\langle }\textbf{T},\textbf{D}(\delta \textbf{v}){\rangle }_{\varvec{\varOmega }}:=\int _{{\varvec{\varOmega }}}^{}(\textbf{T},\textbf{D}(\delta \textbf{v}))_\textbf{g}\cdot \varvec{\mu }\\&\quad =\int _{{\varvec{\varOmega }}}^{}(\textbf{T},\nabla (\delta \textbf{v}))_\textbf{g}\cdot \varvec{\mu }-\int _{{\varvec{\varOmega }}}^{}(\textbf{T},\textbf{W}(\delta \textbf{v}))_\textbf{g}\cdot \varvec{\mu }\\&\quad =-\int _{{\varvec{\varOmega }}}^{} \Big (\textbf{g}(\textrm{Div}(\textbf{T})\,,\delta \textbf{v})+(\textbf{T},\textbf{W}(\delta \textbf{v}))_\textbf{g}\Big )\cdot \varvec{\mu }+\oint _{\partial {\varvec{\varOmega }}}^{}\textbf{g}(\textbf{T}\textbf{n}\,,\delta \textbf{v})\cdot \partial \varvec{\mu }. \end{aligned}\nonumber \\ \end{aligned}$$
(79)
-
volumetric bulk force vector field \(\,-\textrm{Div}(\textbf{T})\,\) in \(\,{\varvec{\varOmega }}\,\),
-
volumetric bulk torque antisymmetric tensor field \(\,\text {skew}_\textbf{g}(\textbf{T})\,\) in \(\,{\varvec{\varOmega }}\,\),
-
surficial force vector field \(\,\textbf{T}\textbf{n}\,\) on \(\,\partial {\varvec{\varOmega }}\).
$$\begin{aligned} (\textbf{T},\textbf{W}(\delta \textbf{v}))_\textbf{g}=(\text {sym}_\textbf{g}\textbf{T},\textbf{W}(\delta \textbf{v}))_\textbf{g}=0, \end{aligned}$$
(80)
The representation of the equilibrium system of forces is thus replaced with an equivalent one made of the sole:Symmetry of \(\,\textbf{T}\,\) is a convenient choice18 and not a consequence of rotatory equilibrium, as usually stated and still recently repeated [24] in the literature.
-
volumetric bulk force vector field \(\,-\textrm{Div}(\text {sym}_\textbf{g}\textbf{T})\,\),
-
surficial force vector field \(\,(\text {sym}_\textbf{g}\textbf{T})\textbf{n}\,\) on \(\,\partial {\varvec{\varOmega }}\).
On the other hand, the assumption of a non-symmetric stress tensor is legitimate although non-convenient for description purposes and leads to representative bulk torques per unit volume.
What is not allowable, in the context of Euler implicit representation of rigidity constraint Eq. (67), is the presence of surface torques per unit boundary area.
In the regard, most attempts of modifying Euler implicit representation have clashed against redundancy of the implicit formulations, as pointed out in [49] and recently discussed in [50].
The duality Eq. (79) trivially entails the following polarity properties:Taking the polars in Eq. (81), we get:For 3D bodies, the polarity Eq. (82) can be stated in the context of Functional Analysis by means of Banach closed range theorem and Korn inequality [44].
$$\begin{aligned} \left\{ \begin{aligned}&\,\textrm{Ker}(\textbf{D})=(\textrm{Im}(\textbf{D}'))^\circ , \\&\,\textrm{Ker}(\textbf{D}')=(\textrm{Im}(\textbf{D}))^\circ . \end{aligned} \right. \end{aligned}$$
(81)
$$\begin{aligned} \left\{ \begin{aligned}&\,\textrm{Im}(\textbf{D})=(\textrm{Im}(\textbf{D}))^{\circ \circ } =(\textrm{Ker}(\textbf{D}'))^\circ , \\&\,\textrm{Im}(\textbf{D}')=(\textrm{Im}(\textbf{D}'))^{\circ \circ }=(\textrm{Ker}(\textbf{D}))^\circ . \end{aligned} \right. \end{aligned}$$
(82)
The former polarity in Eq. (82) provides the variational condition of kinematic compatibility for stretching fields \(\,\varvec{\varepsilon }\in \big (\textrm{Ker}(\textbf{D}')\big )^\circ \subset \mathcal {H}_{\varvec{\varOmega }}\).
Existence and uniqueness (to within additive rigid kinematic fields in \(\,\textrm{Ker}(\textbf{D})\,\)) of a kinematic field \(\,\textbf{v}\in \mathcal {V}_{\varvec{\varOmega }}\,\) fulfilling this variational compatibility condition is proven by Eq. (82)\(_1\) which gives:The latter polarity in Eq. (82) provides the variational condition of static equilibrium for force fields.
$$\begin{aligned} \varvec{\varepsilon }\in \textrm{Im}(\textbf{D})\;\Longleftrightarrow \;\exists \,\textbf{v}\in \mathcal {V}_{\varvec{\varOmega }}\, : \,\varvec{\varepsilon }=\textbf{D}(\textbf{v}). \end{aligned}$$
(83)
Existence and uniqueness (to within additional self-equilibrated fields)19 of the stress field \(\,\textbf{T}\,\) fulfilling the equilibrium condition \(\,\textbf{f}\in \big (\textrm{Ker}(\textbf{D})\big )^\circ \,\) follows form Eq. (82)\(_2\):This proves existence of a Cauchy stress field \(\,\textbf{T}\in {\textsc {Mix}}({T{\varvec{\varOmega }}})\,\) in equilibrium with the force system \(\,\textbf{f}\in \mathcal {F}_{\varvec{\varOmega }}\).
$$\begin{aligned} \textbf{f}\in \textrm{Im}(\textbf{D}')\;\Longleftrightarrow \;\exists \,\textbf{T}\in {\textsc {Mix}}({T{\varvec{\varOmega }}})\, : \,\textbf{f}=\textbf{D}'(\textbf{T}). \end{aligned}$$
(84)
By virtue of the duality in Eq. (79), from Eq. (84) we infer the virtual power principle (VPP) [50]:In terms of mass-form \(\,\textbf{m}=\rho \,\varvec{\mu }\,\) and of natural stress \(\,\varvec{\varSigma }=\rho ^{-1}\,\textbf{T}\,\), with \(\,\rho \,\) scalar mass density, the VPP writes:In classical Continuum Dynamics the bulk force system:includes, á la d’Alembert [51], both the external force system \(\,\textbf{b}_{\textsc {ext}}\,\) and the inertia term \(\,-\rho \,\textbf{a}\,\), with \(\,\textbf{a}\,\) spatial acceleration field in inertial frames and \(\,\rho \,\) scalar mass density.
$$\begin{aligned} {\langle }\textbf{f},\delta \textbf{v}{\rangle }_{\varvec{\varOmega }}=\int _{{\varvec{\varOmega }}}^{}(\textbf{T},\textbf{D}(\delta \textbf{v}))_\textbf{g}\cdot \varvec{\mu },\quad \forall \,\delta \textbf{v}\in \mathcal {V}_{\varvec{\varOmega }}. \end{aligned}$$
(85)
$$\begin{aligned} {\langle }\textbf{f},\delta \textbf{v}{\rangle }_{\varvec{\varOmega }}=\int _{{\varvec{\varOmega }}}^{}(\varvec{\varSigma },\textbf{D}(\delta \textbf{v}))_\textbf{g}\cdot \textbf{m},\quad \forall \,\delta \textbf{v}\in \mathcal {V}_{\varvec{\varOmega }}. \end{aligned}$$
(86)
$$\begin{aligned} \textbf{b}=\textbf{b}_{\textsc {ext}}-\rho \,\textbf{a}, \end{aligned}$$
(87)
To eliminate constraint reactions \(\,\mathcal {L}_{\varvec{\varOmega }}^\circ \subset \mathcal {F}_{\varvec{\varOmega }}\,\) from the equilibrium condition, it suffices to restrict the space of regular kinematic fields to the conformity subspace \(\,\mathcal {L}_{\varvec{\varOmega }}\subset \mathcal {V}_{\varvec{\varOmega }}\). The restricted stretching operator \(\,\textbf{D}:\mathcal {L}_{\varvec{\varOmega }}\mapsto \mathcal {H}_{\varvec{\varOmega }}\,\) admits the dual equilibrium operator:defined by the variational condition:In Eq. (88) the Hilbert space \(\,\mathcal {L}'_{\varvec{\varOmega }}\,\), dual of \(\,\mathcal {L}_{\varvec{\varOmega }}\,\), are made of active force systems, quotient of force space \(\,\mathcal {F}_{\varvec{\varOmega }}\,\) modulo reactive systems in \(\,\mathcal {L}_{\varvec{\varOmega }}^\circ \).
$$\begin{aligned} \textbf{D}':\varSigma _{\varvec{\varOmega }}\mapsto \mathcal {L}'_{\varvec{\varOmega }}=\mathcal {F}_{\varvec{\varOmega }}/\mathcal {L}_{\varvec{\varOmega }}^\circ , \end{aligned}$$
(88)
$$\begin{aligned} {\langle }\textbf{D}'(\textbf{T}),\delta \textbf{v}{\rangle }_{\varvec{\varOmega }}={\langle }\textbf{T},\textbf{D}(\delta \textbf{v}){\rangle }_{\varvec{\varOmega }}, \quad \forall \,\delta \textbf{v}\in \mathcal {L}_{\varvec{\varOmega }}. \end{aligned}$$
(89)
Remark 1
Truesdell and Toupin [2, \(\mathsection \)232 p. 595], in the detailed historical notes of Ch.V “Variational principles” do attribute to Gabrio Piola [45, 52‐54] an early proposal of adopting the Lagrange multiplier method to introduce the notion of stress field by duality with the rigidity constraint, as in Eqs. (72) and (84) above. According to the notes of [2, \(\mathsection \)210, p. 553] “The equations of motion expressed in terms of a reference state”, Piola was also the first to conceive referential formulations of equilibrium.
Moreover, according to notes in the next [2, \(\mathsection \)210, p. 554] “Virtual work and the Lagrange-d’Alembert Principle”, Piola was also the first in proposing redundant conditions of rigidity.
5 Evolutive equilibrium
Let us consider at time \(\,t\in \mathcal {Z}\,\) a body in the spatial configuration \(\,{\varvec{\varOmega }}\,\) whose kinematics is defined by a linear space \(\,\mathcal {V}_{\varvec{\varOmega }}\,\) of piecewise regular spatial vector fields \(\,\textbf{v}_{\varvec{\varOmega }}:{\varvec{\varOmega }}\mapsto T_{\varvec{\varOmega }}S\).
The body is assumed to be subject to affine kinematic constraints described by a linear subspace \(\,L_{\varvec{\varOmega }}\subset \mathcal {V}_{\varvec{\varOmega }}\,\) of conforming kinematic fields and by an imposed kinematic field \(\,\textbf{w}_{\varvec{\varOmega }}\in \mathcal {V}_{\varvec{\varOmega }}\,\), under the action of a force system \(\,\textbf{f}_{\varvec{\varOmega }}\in \mathcal {F}_{\varvec{\varOmega }}=(\mathcal {V}_{\varvec{\varOmega }})'\,\) and of the reactive system exerted by the affine constraints.
Admissible kinematic fields at \(\,{\varvec{\varOmega }}\,\) must accordingly belong to the affine set \(\,A_{\varvec{\varOmega }}:=\textbf{w}_{\varvec{\varOmega }}+L_{\varvec{\varOmega }}\).
In a time lapse \(\,\alpha \in \mathcal {Z}\,\) the movement \(\,\varvec{\varphi }_\alpha :{\mathcal {T}}\mapsto {\mathcal {T}}\,\) along the trajectory \(\,{\mathcal {T}}\subset \mathcal {E}\,\) drawn by the body motion in spacetime, is governed by a control system which prescribes increments of the force system \(\,\varvec{\varDelta }\textbf{f}_{\varvec{\varOmega }}\in (\mathcal {V}_{\varvec{\varOmega }})'\,\) and of driven displacement field \(\,\varvec{\varDelta }\textbf{w}_{\varvec{\varOmega }}\in \mathcal {V}_{\varvec{\varOmega }}\,\) and also the update of the conforming kinematic subspace from the initial \(\,L_{\varvec{\varOmega }}\subset \mathcal {V}_{\varvec{\varOmega }}\,\) to the final one \(\,L_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}\subset \mathcal {V}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}\,\) after the time lapse \(\,\alpha \).
Let us assume for simplicity a smooth quasi-static evolution with the material not leaving the elastic range.
A initial guess on the realisation of the body configuration and of the control upgrade at the end of the incremental step can be performed by evaluating increment \(\,\varvec{\varDelta }\textbf{u}\in \varvec{\varDelta }\textbf{w}_{\varvec{\varOmega }}+L_{\varvec{\varOmega }}\,\) of displacement from the configuration \(\,{\varvec{\varOmega }}\,\) associated with the data increment \(\,\{\varvec{\varDelta }\textbf{f}_{\varvec{\varOmega }},\varvec{\varDelta }\textbf{w}_{\varvec{\varOmega }}\}\,\) at the beginning of the time step.
This evaluation of the incremental displacement is based on the rate formulation of equilibrium [50, 55] in which test fields are assumed to be parallel transported by the motion along the trajectory, in accord with Euler law of Dynamics.
Replacing time rates with finite increments, the Rate Virtual Power Principle (RVPP) takes the form of an Incremental Virtual Power Principle (IVPP):This variational principle holds for any \(\,\delta \textbf{v}\in L_{\varvec{\varOmega }}\,\), being:The incremental stretch mixed tensor \(\,{\dot{\textbf{D}}}(\textbf{v}_{\varvec{\varphi }},\delta \textbf{v})\,\) is associated with the covariant tensor [55]:so that:Elasticity is expressed by the incremental constitutive relation:The elastic compliance \(\,\textbf{H}(\varvec{\varSigma })\,\) is the Hesse operator of a smooth strictly convex scalar stress potential \(\,\varvec{\varXi }\,\):20Then \(\,\textbf{H}(\varvec{\varSigma })\,\) is positive definite, hence invertible, and such is the elastic stiffness \(\,\textbf{K}(\varvec{\varSigma })=\textbf{H}(\varvec{\varSigma })^{-1}\,\) so that:In a purely elastic range:and the incremental elastic equilibrium problem writes:with \(\,\varvec{\varDelta }\textbf{u}\in \varvec{\varDelta }\textbf{w}_{\varvec{\varOmega }}+L_{\varvec{\varOmega }}\subset \mathcal {V}_{\varvec{\varOmega }}\,\), for all test fields \(\,\delta \textbf{v}\in L_{\varvec{\varOmega }}\).
$$\begin{aligned} \begin{aligned} {\langle }\varvec{\varDelta }\textbf{f},\delta \textbf{v}{\rangle }_{\varvec{\varOmega }}&\,=\int _{{\varvec{\varOmega }}}^{}{\langle }\varvec{\varDelta }\varvec{\varSigma },\textbf{D}(\delta \textbf{v}){\rangle }\,\textbf{m}+\int _{{\varvec{\varOmega }}}^{}{\langle }\varvec{\varSigma },\varvec{\varDelta }\textbf{D}(\varvec{\varDelta }\textbf{u},\delta \textbf{v}){\rangle }\,\textbf{m}. \end{aligned} \end{aligned}$$
(90)
$$\begin{aligned} \left\{ \begin{aligned}&\,\varvec{\varDelta }\textbf{m}:=\mathcal {L}_{(\varvec{\varDelta }\textbf{u})}\textbf{m}=\textbf{0}, \quad \text {by conservation of mass}, \\&\,\varvec{\varDelta }(\delta \textbf{v}):=\nabla _{(\varvec{\varDelta }\textbf{u})}\delta \textbf{v}=\textbf{0},\quad \text {by construction}. \end{aligned} \right. \end{aligned}$$
(91)
$$\begin{aligned} \tfrac{1}{2}\,\mathcal {L}_{\varvec{\varDelta }\textbf{u}}\mathcal {L}_{\delta \textbf{v}}(\textbf{g})=\textbf{g}\cdot \varvec{\varDelta }\textbf{D}(\varvec{\varDelta }\textbf{u},\delta \textbf{v}). \end{aligned}$$
(92)
$$\begin{aligned} \varvec{\varDelta }\textbf{D}(\varvec{\varDelta }\textbf{u},\delta \textbf{v})=\text {sym}_\textbf{g}\Big ((\nabla \varvec{\varDelta }\textbf{u})^A\cdot \nabla \delta \textbf{v}\Big ). \end{aligned}$$
(93)
$$\begin{aligned} \varvec{\varDelta }\textbf{E}=\textbf{H}(\varvec{\varSigma })\cdot \varvec{\varDelta }\varvec{\varSigma }. \end{aligned}$$
(94)
$$\begin{aligned} \textbf{H}=d_F^2\varvec{\varXi }. \end{aligned}$$
(95)
$$\begin{aligned} \varvec{\varDelta }\varvec{\varSigma }=\textbf{K}(\varvec{\varSigma })\cdot \varvec{\varDelta }\textbf{E}. \end{aligned}$$
(96)
$$\begin{aligned} \varvec{\varDelta }\textbf{E}=\textbf{D}(\varvec{\varDelta }\textbf{u}), \end{aligned}$$
(97)
$$\begin{aligned} \begin{aligned} {\langle }\varvec{\varDelta }\textbf{f},\delta \textbf{v}{\rangle }_{\varvec{\varOmega }}&=\int _{{\varvec{\varOmega }}}^{} {\langle }\textbf{K}(\varvec{\varSigma })\cdot \textbf{D}(\varvec{\varDelta }\textbf{u}),\textbf{D}(\delta \textbf{v}){\rangle }\,\textbf{m}\\&\quad +\int _{{\varvec{\varOmega }}}^{}{\langle }\varvec{\varSigma },\varvec{\varDelta }\textbf{D}(\varvec{\varDelta }\textbf{u},\delta \textbf{v}){\rangle }\,\textbf{m}, \end{aligned}\nonumber \\ \end{aligned}$$
(98)
Provided the variational problem in Eq. (98) admits a displacement solution \(\,\varvec{\varDelta }\textbf{u}\,\), a first trial \(\,\varvec{\varphi }_\alpha ({\varvec{\varOmega }})\,\) for the deformed configuration is available.21
Then, with \(\,\varvec{\varDelta }\varvec{\varSigma }\,\) given by Eqs. (96)–(97), the incremental elastic response \(\,\varvec{\varDelta }\textbf{r}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}\in \mathcal {F}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}\,\) is given by:Note that conservation of mass along the motion is expressed by:The response in Eq. (99) is compared with the incremental forcein which \(\,\delta \textbf{v}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}\,\) is parallel transported by the movement \(\,\varvec{\varphi }_\alpha :{\mathcal {T}}\mapsto {\mathcal {T}}\,\):where \(\,{\Uparrow }\,\) denotes the forward distant parallel transport in the Euclid space \(\,S\,\) from the spatial configuration \(\,{\varvec{\varOmega }}\,\) to the displaced one \(\,\varvec{\varphi }_\alpha ({\varvec{\varOmega }})\).
$$\begin{aligned} \begin{aligned}&\,{\langle }\varvec{\varDelta }\textbf{r},\delta \textbf{v}{\rangle }_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})} \\&\,=\int _{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}^{} \varvec{\varphi }_\alpha {\uparrow }{\langle }\textbf{K}(\varvec{\varSigma }+\varvec{\varDelta }\varvec{\varSigma })\cdot \textbf{D}(\varvec{\varDelta }\textbf{u}),\textbf{D}(\delta \textbf{v}){\rangle }\cdot \textbf{m}\\&\quad \,+\int _{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}^{} \varvec{\varphi }_\alpha {\uparrow }{\langle }\varvec{\varSigma }+\varvec{\varDelta }\varvec{\varSigma },\varvec{\varDelta }\textbf{D}(\varvec{\varDelta }\textbf{u},\delta \textbf{v}){\rangle }\cdot \textbf{m}. \end{aligned}\nonumber \\ \end{aligned}$$
(99)
$$\begin{aligned} \varvec{\varphi }_\alpha {\uparrow }\textbf{m}=\textbf{m}. \end{aligned}$$
(100)
$$\begin{aligned} {\langle }\varvec{\varDelta }\textbf{f},\delta \textbf{v}{\rangle }_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}, \end{aligned}$$
(101)
$$\begin{aligned} \delta \textbf{v}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}:=\varvec{\varphi }_\alpha {\Uparrow }\,\delta \textbf{v}_{{\varvec{\varOmega }}}, \end{aligned}$$
(102)
At this stage the force increment \(\,\varvec{\varDelta }\textbf{f}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}\in \mathcal {F}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}\,\) is updated by the control system.
The incremental equilibrium gap \(\,\varvec{\varDelta }\textbf{r}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}-\varvec{\varDelta }\textbf{f}_{\varvec{\varphi }_\alpha ({\varvec{\varOmega }})}\,\) is applied to the trial configuration \(\,\varvec{\varphi }_\alpha ({\varvec{\varOmega }})\,\) corresponding to the running iteration.
The elastic equilibrium Eq. (98) with \(\,\varvec{\varphi }_\alpha ({\varvec{\varOmega }})\,\) taking the place of \(\,{\varvec{\varOmega }}\,\), updates the current guess and another iteration for the elastic incremental displacement solution is performed. The iterative loop comes to a stop provided that a suitably chosen norm of the equilibrium gap becomes smaller than a prescribed tolerance.
6 Referential equilibria: critical notes
In application-oriented treatments, a referential configuration \(\,{\varvec{\varOmega }}_{\textsc {ref}}\,\) of a 3D body is a domain of simple shape in \(\,\Re ^3\,\) (simplexes or parallelepides) which are put in diffeomorphic correspondence with similar elements of an approximate finite partition of the actual body configuration \(\,{\varvec{\varOmega }}\,\) under investigation.
The one-to-one differentiable placement map \(\,\textbf{p}:{\varvec{\varOmega }}_{\textsc {ref}}\mapsto {\varvec{\varOmega }}\,\) does not admits an explicit inverse map, as a rule.
This is the case, for instance, in the highly successful Finite Element Method (FEM) developed since the sixties of the past century.
In theoretical treatments exposed in the literature, the situation is different and the referential configuration is considered to be occupied by the whole body, or by an element of it, at some instant of time in the relevant ambient Euclid space \(\,S\).
There is clear evidence that equilibrium cannot be investigated in referential configurations other than isometric or uniformly scaled avatars considered in structural calculations, an opinion shared by any well-trained structural engineer.
In spite of this, the confusion-illusion induced by scholars who considered referential equilibrium as a valid convenient alternative, quickly became viral and is nowadays widespread in the relevant literature, as witnessed by the list in Sect. 1.
An analysis of the original treatment by Truesdell and Noll in [3, Eq. (44.12–15)], revisited by Morton Edward Gurtin in [6, Ch.IX \(\mathsection \)27], and by Paolo Podio-Guidugli in [11, Ch.II \(\mathsection \)10], clarifies some basic aspects of the matter.
Therein external forces acting on the actual configuration \(\,{\varvec{\varOmega }}\,\) of the body are assumed to consist of boundary forces per unit area \(\,\textbf{s}\in T_{\partial {\varvec{\varOmega }}}S\,\) and of bulk forces per unit volume \(\,\textbf{b}\in T_{{\varvec{\varOmega }}}S\).
According to Eq. (10.5)\({}_1\) of [11], these force densities are (tacitly) parallel transported from the actual configuration \(\,{\varvec{\varOmega }}\,\) to a reference configuration along the map \(\,\textbf{p}^{-1}:{\varvec{\varOmega }}\mapsto {\varvec{\varOmega }}_{\textsc {ref}}\,\) and (explicitly) scaled in such a way that the force resultant is left invariant.
In [11, Eqs. (10.5)] the configuration \(\,{\varvec{\varOmega }}\,\) is labelled as “deformed”, with \(\,{\varvec{\varOmega }}_{\textsc {ref}}=\textbf{p}^{-1}({\varvec{\varOmega }})\,\) named “reference” configuration.
The portrayed surficial and volumetric forces in Eq. (10.5) of [11]:are, in our notation, given by:With a common abuse in Euclid spacetime, the foreword parallel transport \(\,{\Uparrow }\,\) and its backward inverse \(\,{\Downarrow }\,\) are usually assumed to be the identity and therefore ignored. In [6, Ch.IX \(\mathsection \)27] Gurtin says:
$$\begin{aligned} \left\{ \begin{aligned}&\,\textbf{s}_\textsc {ref}:\partial {\varvec{\varOmega }}_{\textsc {ref}}\mapsto T_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}\Re ^3, \\&\,\textbf{b}_\textsc {ref}:{\varvec{\varOmega }}_{\textsc {ref}}\mapsto T_{{\varvec{\varOmega }}_{\textsc {ref}}}\Re ^3, \end{aligned} \right. \end{aligned}$$
(103)
$$\begin{aligned} \left\{ \begin{aligned}&\,\textbf{s}_\textsc {ref}\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}=\textbf{p}\,{\Downarrow }\,(\textbf{s}\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}}), \\&\,\textbf{b}_{\textsc {ref}}\cdot \mathcal {V}_{{\varvec{\varOmega }}_{\textsc {ref}}}\;\;=\textbf{p}\,{\Downarrow }\,(\textbf{b}\cdot \mathcal {V}_{{\varvec{\varOmega }}}). \end{aligned} \right. \end{aligned}$$
(104)
“It is not convenient to work with
\(\,\textbf{T}\,\)
(
Cauchy
stress) since the deformed configuration is not known in advance.”
22
In [11], after Eqs. (10.5), Podio-Guidugli says:
“With these definitions we can portray the surface-force and volume-force fields
\(\,\mathrm {{\textbf {s}},{\textbf {b}}}\,\)
in the reference shape.”
The portrait adopted in [11] and here reported in Eq. (104) is motivated by the change of integration domain from \(\,{\varvec{\varOmega }}_{\textsc {ref}}\,\) to \(\,{\varvec{\varOmega }}\,\) and from \(\,\partial {\varvec{\varOmega }}_{\textsc {ref}}\,\) to \(\,\partial {\varvec{\varOmega }}\,\) and by the corresponding changes of volumetric and surficial Jacobi determinants.
Resultant force and resultant moment are evaluated in the actual configuration \(\,{\varvec{\varOmega }}\,\) by the integrals:with \(\,\textbf{r}:S_{\varvec{\varOmega }}\mapsto {{\varvec{\varOmega }}}\,\) position vector field with respect to a given pole \(\,\mathcal {O}\in S_{\varvec{\varOmega }}\).
$$\begin{aligned} \left\{ \begin{aligned}&\,\oint _{\partial {\varvec{\varOmega }}}^{}\textbf{s}\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}} +\int _{{\varvec{\varOmega }}}^{}\textbf{b}\cdot \mathcal {V}_{{\varvec{\varOmega }}}, \\&\,\oint _{\partial {\varvec{\varOmega }}}^{}(\textbf{r}\times \textbf{s})\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}} +\int _{{\varvec{\varOmega }}}^{}(\textbf{r}\times \textbf{b})\cdot \mathcal {V}_{{\varvec{\varOmega }}}, \end{aligned} \right. \end{aligned}$$
(105)
In the reference manifold \(\,{\varvec{\varOmega }}_{\textsc {ref}}\,\) a formulation in a convected frame gives:with \(\,(\textbf{p}^{-1}\circ \textbf{r}):S_{\varvec{\varOmega }}\mapsto {{\varvec{\varOmega }}}\mapsto {{\varvec{\varOmega }}_{\textsc {ref}}}\,\) position vector field with respect to the pole \(\,\mathcal {O}\in S_{\varvec{\varOmega }}\). According to the portrait in Eq. (104) the resultant force integrals in Eqs. (105)\(_1\) and (106)\(_1\) are equal.
$$\begin{aligned} \left\{ \begin{aligned}&\,\oint _{\partial {\varvec{\varOmega }}_{\textsc {ref}}}^{}\textbf{s}_{\textsc {ref}}\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}} +\int _{{\varvec{\varOmega }}_{\textsc {ref}}}^{}\textbf{b}_{\textsc {ref}}\cdot \mathcal {V}_{{\varvec{\varOmega }}_{\textsc {ref}}}, \\&\,\oint _{\partial {\varvec{\varOmega }}_{\textsc {ref}}}^{}\Big ((\textbf{p}^{-1}\circ \textbf{r})\times \textbf{s}_{\textsc {ref}}\Big )\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}} +\int _{{\varvec{\varOmega }}_{\textsc {ref}}}^{}\Big ((\textbf{p}^{-1}\circ \textbf{r})\times \textbf{b}_{\textsc {ref}}\Big )\cdot \mathcal {V}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}, \end{aligned} \right. \end{aligned}$$
(106)
On the contrary, in general the relation between the resultant moments in Eq. (105)\(_2\) and (106)\(_2\) depends on the change of integration domain. Therefore vanishing of Eq. (106)\(_2\) and vanishing of the resultant moment in Eq. (105)\(_2\) are independent assumptions.
Indeed, as remarked in [3, Eq. (44.12–15)], according to Antonio Signorini [59, 60] the vanishing of Eq. (106)\(_2\) is to be considered as a compatibility condition to be verified a posteriori, that is once the placement map will eventually be available for the evaluation of the deformed configuration. No strategy to get this goal was, however, envisaged.
In the treatment originally exposed in [3, Eq. (44.12–15)] the integration domains \(\,{\varvec{\varOmega }}_{\textsc {ref}}\,\) and \(\,{\varvec{\varOmega }}=\textbf{p}({\varvec{\varOmega }}_{\textsc {ref}})\,\) were improperly denoted by a same symbol \(\,\mathcal {B}\,\) and in the treatments [6, 11] the position vector field \(\,\textbf{p}^{-1}\circ \textbf{r}:S_{\varvec{\varOmega }}\mapsto {{\varvec{\varOmega }}_{\textsc {ref}}}\,\) is deceptively denoted by the same symbol as the position vector \(\,\textbf{r}:S_{\varvec{\varOmega }}\mapsto {{\varvec{\varOmega }}}\,\) in the actual configuration \(\,{\varvec{\varOmega }}\).
As described in Sect. 5 dealing with evolutive equilibrium, in a nonlinear structural analysis at each instant of time the control system assigns loads and kinematic constraints acting on the actual configuration of the body at that time instant. Familiar examples are a bicycle being driven along a country path or a car moving along city streets.
A shape-independent system of loads is therefore conceivable only in the trivial case of spatially standing motions.23
Remark 2
In [3, Eq. (44.20) p. 128] an attempt is suggested to recover a referential equilibrium condition in the standard form:where \(\,\textbf{r}_{\textsc {ref}}:S_{{\varvec{\varOmega }}_{\textsc {ref}}}\mapsto {\varvec{\varOmega }}_{\textsc {ref}}\,\) is a field of position vectors with respect to a given pole \(\,\mathcal {O}_\textsc {ref}\in S_{{\varvec{\varOmega }}_{\textsc {ref}}}\).
$$\begin{aligned} \left\{ \begin{aligned}&\,\oint _{\partial {\varvec{\varOmega }}_{\textsc {ref}}}^{}\textbf{s}_{\textsc {ref}}\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}} +\int _{{\varvec{\varOmega }}_{\textsc {ref}}}^{}\textbf{b}_{\textsc {ref}}\cdot \mathcal {V}_{{\varvec{\varOmega }}_{\textsc {ref}}}=\textbf{0}, \\&\,\oint _{\partial {\varvec{\varOmega }}_{\textsc {ref}}}^{}\Big (\textbf{r}_{\textsc {ref}}\times \textbf{s}_{\textsc {ref}}\Big )\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}} +\int _{{\varvec{\varOmega }}_{\textsc {ref}}}^{}\Big (\textbf{r}_{\textsc {ref}}\times \textbf{b}_{\textsc {ref}}\Big )\cdot \mathcal {V}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}} =\textbf{0}, \end{aligned} \right. \end{aligned}$$
(107)
The passage from the compatibility condition Eq. (106)\(_2\) to the vanishing of the resultant moment in Eq. (107)\(_2\), was attempted by appealing to a theorem due to Daniel Augusto da Silva [61], exposed in [3, Eq. (44.18) p. 128] and based on the notion of astatic load tensor [3, Eq. (44.17) p. 127] and [62, Prop. (14.1) p. 296].
The conclusion in [3, Eq. (44.20) p. 128] was that, for any given loading and any given pole \(\,\mathcal {O}_\textsc {ref}\in S_{{\varvec{\varOmega }}_{\textsc {ref}}}\,\), it is possible to rotate rigidly the reference configuration in such a way that Eq. (107)\(_2\) be fulfilled. This fair geometric result is however not effective because the rotation depends on the choice of the pole \(\,\mathcal {O}_\textsc {ref}\in S_{{\varvec{\varOmega }}_{\textsc {ref}}}\).
The whole procedure is of no interest in Structural Mechanics where kinematic constraints are usually considered acting on the body and further conditions of admissibility, in addition to equilibrium, are imposed on the stress field, as usually made for instance in constitutive relations of perfect plasticity.
For completeness sake, let us reproduce hereafter the treatment usually described in the literature. In the sequel the tangent placement is denoted by:and Eq. (38) is rewritten as:Putting \(\,\varSigma =\partial {\varvec{\varOmega }}_{\textsc {ref}}\,\) in Nanson formula Eq. (52)\(_1\) we get:Equation (79) provides for the traction the formula \(\,\textbf{s}:=\textbf{T}\cdot \textbf{n}_{\partial {\varvec{\varOmega }}}\,\) in terms of the Cauchy stress \(\,\textbf{T}\,\) and of the normal versor \(\,\textbf{n}_{\partial {\varvec{\varOmega }}}\,\) to \(\,\partial {\varvec{\varOmega }}\).
$$\begin{aligned} \textbf{F}:=T\textbf{p}:T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto T{{\varvec{\varOmega }}}. \end{aligned}$$
(108)
$$\begin{aligned} \textrm{cof}(\textbf{F}):=J_{\textbf{p}}\cdot \textbf{F}^{-A}:T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto {T{\varvec{\varOmega }}}. \end{aligned}$$
(109)
$$\begin{aligned} \mathcal {A}_{\partial {\varvec{\varOmega }}}\cdot \textbf{n}_{\partial {\varvec{\varOmega }}} =\textrm{cof}(\textbf{F})\cdot (\mathcal {A}_{{\varvec{\varOmega }}_{\textsc {ref}}}\cdot \textbf{n}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}} ). \end{aligned}$$
(110)
Then Nanson formula Eq. (110) leads to the evaluation:a computation carried out in coordinates in [2, Eq. (210.5), p. 553].
$$\begin{aligned} \begin{aligned} \textbf{s}\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}}&\,=\textbf{T}\cdot (\mathcal {A}_{\partial {\varvec{\varOmega }}}\cdot \textbf{n}_{\partial {\varvec{\varOmega }}}) \\&\,=\textbf{T}\cdot \textrm{cof}(\textbf{F})\cdot ( \mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}\cdot \textbf{n}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}} ) \\&\,=\textbf{P}\cdot (\mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}\cdot \textbf{n}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}) \\&\,=(\textbf{p}\,{\Uparrow }\mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}})\cdot (\textbf{P}\cdot \textbf{n}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}), \end{aligned}\nonumber \\ \end{aligned}$$
(111)
In Eq. (111) Gabrio Piola two-point tensor \(\,\textbf{P}\,\),24 also qualified as pseudo-stress in [2, \(\mathsection \)210, p. 553], is defined by:Moreover, the portrait proposed in Eq. (10.5) of [11] and reproduced here in Eq. (104)\(_1\) with an explicit parallel transport:when substituted in Eq. (111), gives:Under the special assumption that the reference placement belongs to the dynamical trajectory,25 the spatial metric tensor \(\,\textbf{g}:{V\mathcal {E}}\mapsto {({V\mathcal {E}})^*}\,\) is well defined therein and the following analysis may be applied.
$$\begin{aligned} \textbf{P}:=\textbf{T}\cdot \textrm{cof}(\textbf{F}):T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto {T{\varvec{\varOmega }}}. \end{aligned}$$
(112)
$$\begin{aligned} \textbf{s}\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}}=\textbf{p}\,{\Uparrow }(\textbf{s}_\textsc {ref}\cdot \mathcal {A}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}), \end{aligned}$$
(113)
$$\begin{aligned} \textbf{p}\,{\Uparrow }\textbf{s}_\textsc {ref}=\textbf{P}\cdot \textbf{n}_{\partial {\varvec{\varOmega }}_{\textsc {ref}}}. \end{aligned}$$
(114)
By definition of cofactor map Eq. (109) and of Piola two-point tensor Eq. (112), the \(\textbf{g}\)-symmetry of \(\,\textbf{T}\,\) gives:The \(\textbf{g}\)-symmetry of second Piola tensor \(\textbf{S}:T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto T{\varvec{\varOmega }}_{\textsc {ref}}\) follows:26This equation, given in [3, Eq. (43A.9)], appears also in [17, Eq. (25.10)].27
$$\begin{aligned} \textbf{P}\cdot \textbf{F}^{A}=J_{\textbf{p}}\cdot \textbf{T}=\textbf{F}\cdot \textbf{P}^{A}. \end{aligned}$$
(115)
$$\begin{aligned} \begin{aligned} \textbf{S}:=&\,\textbf{F}^{-1}\textbf{P}=\textbf{P}^{A}\textbf{F}^{-A} =J_{\textbf{p}}\,\textbf{F}^{-1}\textbf{T}\textbf{F}^{-A} :T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto T{\varvec{\varOmega }}_{\textsc {ref}}. \end{aligned} \end{aligned}$$
(116)
The following flawed expression is, however, found in [64, p. 139]:Piola’s early treatment of referential equilibria and subsequent developments due to G.R.G. Kirchhoff [65, 66] and to C.G. Neumann [67], were reproduced in [2, \(\mathsection \)210, p. 553], repeated in [6, Ch.IX \(\mathsection \)27, Eq. (6)], [11, Ch.II, Eq. (10.12)\(_1\)] and in [17, Ch.24].
$$\begin{aligned} \textbf{S}:=J_{\textbf{p}}\Big (\textbf{p}{\downarrow }\textbf{T}\Big ) =J_{\textbf{p}}\,\textbf{F}^{-1}\textbf{T}\textbf{F}:T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto T{\varvec{\varOmega }}_{\textsc {ref}}. \end{aligned}$$
(117)
Piola’s method consists of a reshaping of Cauchy differential equation:This reshaping, carried out on the basis of (106)\(_1\) and according to Eqs. (104) and (114), gives in terms of the divergence \(\,\textrm{Div}\,\) of Piola pseudo-stress \(\,\textbf{P}\,\) in the manifold \(\,{\varvec{\varOmega }}_{\textsc {ref}}\,\), the condition:Equivalence between Eqs. (118) and (119) is inferred from equivalence of Eq. (105)\(_1\) and (106)\(_1\) expressing translational equilibria, respectively, in the deformed and initial configuration, according to the load reshaping assumed in Eq. (104).
$$\begin{aligned} \textrm{div}(\textbf{T})+\textbf{b}=\textbf{0}, \quad \textrm{in}\,{\varvec{\varOmega }}. \end{aligned}$$
(118)
$$\begin{aligned} \begin{aligned}&\,\textrm{Div}(\textbf{P})+\textbf{b}_{\textsc {ref}}=\textbf{0},\quad \textrm{in}\;{\varvec{\varOmega }}_{\textsc {ref}}. \end{aligned} \end{aligned}$$
(119)
This reshaping was aimed to preserve translational equilibria in rewriting the equilibrium conditions pertaining to the deformed configuration \(\,{\varvec{\varOmega }}\,\) in terms of fields defined in the initial configuration \(\,{\varvec{\varOmega }}_{\textsc {ref}}\). As observed above, the argument is not applicable when dealing with rotatory equilibria where the placement map \(\,\textbf{p}:{\varvec{\varOmega }}_{\textsc {ref}}\mapsto {\varvec{\varOmega }}\,\) plays an essential role.
As a final observation, Piola two-point tensor \(\,\textbf{P}:T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto {T{\varvec{\varOmega }}}\,\) Eq. (112) and the \(\textbf{g}\)-symmetry of second Piola tensor \(\,\textbf{S}:T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto T{\varvec{\varOmega }}_{\textsc {ref}}\,\) Eq. (116) cannot enter in constitutive relations because their definition involves an arbitrary purely geometric tangent map \(\,\textbf{F}:=T\textbf{p}:T{\varvec{\varOmega }}_{\textsc {ref}}\mapsto T{{\varvec{\varOmega }}}\). Independence of the arbitrary placement map can only be achieved by rate formulations in terms of a natural stress rate and of a dual stretching [68].
7 Concluding remarks
In computational contexts, local reference manifolds are alien geometric elements conceived to perform constitutive evaluations by means of forth and back local transformations of material fields between actual and reference shapes, with the choice of a local reference element dictated by convenience.
Physical evidence shows that a material body is a time-parametrised family of configurations, each made of simultaneous events dragged by the motion along the trajectory and characterised by mass invariance.
Adoption of this physically based definition eliminates the need for the introduction of an alien (that is, not included in the trajectory) body configuration \(\,\mathcal {B}\,\), whose points are candidates for being called particles, and the need for endowing it with specific topological properties. Rather, a particle is what experiments reveal:Only invertible smooth correspondences between referential elements and elements of a partition of the current configuration are usually available.
-
a one-parameter family (a time-parametrised curve) of motion-related material events along the trajectory.
Push-pull transformation of material tensor fields to referential ones, as specified in Sect. 2.3, can be performed at corresponding points and may be convenient for numerical integration in computational procedures, such as the finite element method (FEM) and similar ones.
Parallel transport of spatial fields is not applicable when the reference configuration is an alien manifold, a convenient finite subdomain with simple shape in \(\,\Re ,\Re ^2\,\) or \(\,\Re ^3\). This is the case occurring in engineering methods of analysis and their applications. Simple direct questions arise:
“What structural engineer would ever carry out the calculation of an arch stone bridge by writing the equilibrium equations on a straight beam of reference?”.28
“How to compare equilibrium features of a compressed column under gravitational loads with the ones of a referential cantilever (with a 90 degrees rotated axis) under parallel transported loads?”.
Equilibrium conditions ought to be imposed in variational form on the current configuration.
The geometry of this configuration plays a decisive role in detecting uniqueness, bifurcation and dynamical instability (e.g. snap-through) phenomena.
Equilibrium in Continuum Mechanics, with a discussion about symmetry of Cauchy stress, has been revisited in Sect. 4.
Incremental equilibrium is based on formulation of rate equilibrium illustrated in geometric terms in [50, 55].
The investigation is motivated by and aimed at improving theoretical formulations in Nonlinear Structural Mechanics.
The target is a critical analysis of previous treatments in order to promote an elimination of still persisting gaps and contradictions between formal presentations and applicative methodologies devised to render directly available to structural engineers effective computational tools of nonlinear analysis.
Declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
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