Issues Concerning Isometric Deformations of Planar Regions to Curved Surfaces
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Abstract
In this work, we make a distinction between the differential geometric notion of an isometry relationship among two dimensional surfaces embedded in threedimensional point space and the continuum mechanical notion of an isometric deformation of a two dimensional material surface. We illustrate the importance of separating the abstract theory of surfaces in differential geometry and their related differential geometric features from the physical notion of a material surface which is subject to a deformation from a given reference configuration. In differential geometry, while two surfaces may be isometric, the mapping between them that characterizes the isometry is simply a mapping between the points of the surfaces and not necessarily between corresponding material particles of a single deformed material surface.
We review two equivalent characterizations of a smooth isometric deformation of a flat material surface into a curved surface, and emphasize the requirement that the referential directrix and rulings, and their deformed counterparts, must provide a basis for establishing a complete curvilinear coordinate covering of the material surface in both the reference and deformed states. Because this covering requirement has been overlooked in recent publications concerning the isometric bending of ribbons, we illustrate its importance in properly defining the deformation of a ribbon in the two examples of a flat rectangular material strip that is isometrically deformed into either (i) a portion of a circular cylindrical surface, or (ii) a portion of a circular conical surface. We then show how the accurate calculation of the bending energy in these two examples is influenced by this oversight. In example (i), the curvature along the generators of the deformed surface, generally helical in form, is constant. In this special circumstance overlooking the covering requirement, as has been done in the literature by integrating the specific bending energy, dependent only on the curvature, over a domain on the supporting circular cylindrical surface equal in area, though not equal in geometric form, to that of the deformed ribbon, gives the correct bending energy result. In example (ii), the curvature along a generator of the cone is not constant and the calculation of the bending energy is, indeed, compromised by this oversight.
The historically important dimensional reductions that Sadowsky and Wunderlich introduced to study the bending energy and the equilibrium configurations of isometrically deformed rectangular strips have gained classical notoriety within the subject of elastic ribbons and Möbius bands. We show that the Sadowsky and Wunderlich functionals also overlook the covering requirement and that the exact bending energy is underestimated by these functionals, the Sadowsky functional being the lowest. We then show that the error in using these functionals can be great for a rectangular strip of given length \(\ell\) and width \(w\), depending on the form of the isometric deformation and the size of the halflengthtowidth ratio \(w/2\ell\). The Sadowsky functional is meant to apply to strips for which \(w/2\ell\) is sufficiently small, in which case the covering requirement is of little consequence, and for such strips it yields an acceptable approximation of the actual bending energy. In such cases the Wunderlich functional shows only an incremental improvement over the Sadowsky calculation. While the Wunderlich functional is meant to apply accurately for all strips, without restricting the size of \(w/2\ell\), we show that in overlooking the covering requirement it greatly underestimates the actual bending energy for many isometrically deformed ribbons. In particular, we show relative errors between the exact, the Wunderlich, and the Sadowsky calculations of the bending energy as a function of \(w/2\ell\) for the case of a rectangular strip which is isometrically deformed into a portion of a right circular conical surface, and we observe that the error in approximating the exact bending energy by the Wunderlich functional for reasonable ratios \(w/2\ell\) is large and unacceptable. We then give an example of the isometric deformation of a rectangular strip whose Wunderlich functional predicts zero bending energy but for which the exact bending energy can be as large as one pleases.
Finally, contrary to suggestions in the literature, we argue that Kirchhoff rod theory does not generally apply to the study of the isometric deformation of a thin rectangular strip because for this class of problems the through thickness dimension of the strip is assumed to be infinitesimal as compared to its width \(w\). For Kirchhoff rod theory to apply, these dimensions must be comparable.
Keywords
Isometry Unstretchable Inextensional Twodimensional Riemannian manifold Embedding DevelopableMathematics Subject Classification
53A05 74K15 74K35 57R40 53A451 Introduction
In classical continuum mechanics, a body ℬ may be viewed, geometrically, as a compact threedimensional Riemannian manifold endowed with a Riemannian metric. A configuration of a body makes an identification between each body element \(x\) belonging to ℬ and a point \(\boldsymbol{x}\) in a subset \({\mathcal{R}}_{*}\) of threedimensional Euclidean point space \(\mathbb{E}^{3}\). In this setting, \({\mathcal{R}}_{*}\) is called the reference configuration of the body. A (smooth) deformation of the ‘body’ from \({\mathcal{R}}_{*}\) to some other subset ℛ of \(\mathbb{E}^{3}\) is then a mapping \(\boldsymbol{x}\mapsto\boldsymbol{y}\), where \(\boldsymbol{y}\) denotes a generic element of ℛ. In this setting, ℛ is called the spatial (or deformed) configuration of the body. A deformation is isometric if lengths between all points in \({\mathcal{R}}_{*}\) and corresponding points in ℛ are preserved under the mapping. Because a body cannot generally be embedded in \(\mathbb{E}^{3}\), a threedimensional Euclidean observer cannot discern its geometrical structure without recourse to special instruments.
An analogous level of clarity is absent from a significant portion of the literature concerned with ribbons and sheets conceived of as twodimensional bodies, which we refer to as material surfaces. The difference between the abstract mathematical notion of a surface in twodimensional Euclidean point space \(\mathbb{E}^{2}\) and its embedding into \(\mathbb{E}^{3}\) and the physical notion of the configuration of a material surface as viewed in \(\mathbb{E}^{3}\) appears to have gone unappreciated in much of the literature dealing with configurations of ribbons and sheets in \(\mathbb{E}^{3}\).
As an abstract mathematical entity, a material surface \({\mathcal{P}}\) may be considered, geometrically, to be a compact twodimensional manifold endowed with a Riemannian metric. On this basis, an isometry between two distinct material surfaces can be defined and studied. A configuration of a material surface \({\mathcal{P}}\) rests on an identification between each element \(x\) of \({\mathcal{P}}\) and a point \(\boldsymbol{x}\) in a subset \({\mathcal{D}}\) of \(\mathbb{E}^{2}\); \({\mathcal{D}}\) is called the reference configuration of the material surface. A (smooth) deformation of the ‘material surface’ from \({\mathcal{D}}\) to \({\mathcal{S}}\) is then a mapping \(\boldsymbol {x}\mapsto\boldsymbol{y}\), where \(\boldsymbol{y}\) is an element of \(\mathbb{E}^{3}\). In this setting, \({\mathcal{S}}\) is called the spatial (or deformed) configuration of the material surface. A deformation from \({\mathcal{D}}\) to \({\mathcal{S}}\) is isometric if the length of any curve of points in \({\mathcal{D}}\) is preserved under the mapping. This stands in contrast to the prevailing view in classical differential geometry, where two surfaces are said to be isometric if they have the same Gaussian curvature at corresponding points and, in particular, a curved surface that is isometric to a planar region must be developable.
What perhaps makes things confusing when considering material surfaces and threedimensional bodies is that, because of the embedding property, the Riemannian manifold and configurations of a material surface can be discerned by a threedimensional Euclidean observer. Unless special care is taken in defining an isometry between two surfaces and an isometric deformation of a material surface, the distinction may therefore be easily missed. In particular, the pitfall of misconstruing the notion (and condition) of conservation of Gaussian curvature as the constraint appropriate to characterizing an unstretchable (or inextensional) twodimensional body must be avoided.
We explain below, by analogy and at a reasonably fundamental level, our understanding of why much of the literature on unstretchable material surfaces is marred by confusion surrounding the distinction between the isometric deformation of a material surface and the differential geometric idea of an isometry between two surfaces. Section 2 is devoted to background. Drawing on the work of Chen, Fosdick and Fried [1], we define precisely what we mean by a smooth isometric deformation of a flat material surface into a curved surface, provide two equivalent conditions that are necessary and sufficient that a deformation of a flat material surface to a curved surface is isometric, and highlight the roles of the geometric objects central to our description. Most crucial among these are the referential and spatial directrices and the referential rulings and the spatial generators, which together provide a basis for establishing meaningful correspondences between parametrizations of the material surface in its reference and deformed configurations. In Sect. 3, we study a class of mappings introduced by Dias and Audoly [2] and show that all such mappings are isometric in the sense defined in Sect. 2. In Sect. 4, we consider issues that arise in connection with parametrizations of the reference and deformed configurations of a material surface. These issues hinge on the importance of ensuring a surjective correspondence between material points and parameter pairs. Dias and Audoly [2] overlook this issue and we find that their approach yields a complete covering of the referential and deformed surfaces only in the simple degenerate case where the reference configuration of the material surface is rectangular and is deformed into a (not necessarily circular) right cylindrical ring. Moreover, we discover that the “edge functions” of Dias and Audoly [2] fail to cure this difficulty. In Sect. 5 we explore how calculations of bending energy may be affected by failing to ensure a surjective correspondence between material points and parameter pairs. To illustrate our point, we compute the bending energy for a rectangular material strip bent to conform to a portion of a right circular conical surface. We then compare that energy to the analogous energy obtained by evaluating Wunderlich’s [3, 4] dimensionallyreduced energy functional, proving that that functional is strictly bounded above by the bending energy. We also find that Wunderlich’s [3, 4] functional provides an accurate estimate of the bending energy only in the limit in which the halfwidthtolength ratio of the strip is infinitesimal and that Sadowsky’s [5, 6] functional performs just as well as Wunderlich’s [3, 4] in that regime. This stems from the absence of a surjective correspondence between material points and parameters that is inherent to the parametric approach employed by Wunderlich [3, 4] and those who have emulated his work or utilized his functional. In Sect. 6, we consider the implications of a key assumption in the theory of Kirchhoff rods, namely the assumption that the cross sections of such a rod are rigid and, thus, in particular, cannot sustain inplane deformations. Observing that the theory of Dias and Audoly [2] violates this assumption, we argue that their theory applies only to striplike bodies that have widths comparable to their thicknesses and, thus, to Kirchhoff rods with infinitesimal cross sectional area or framed curves. Finally, in Sect. 7, we discuss and summarize our findings.
2 Preliminaries
Consider a material surface that is identified with an open, connected subset \({\mathcal{D}}\) of twodimensional Euclidean point space \(\mathbb {E}^{2}\). Suppose that \({\mathcal{D}}\) is deformed isometrically into a surface \({\mathcal{S}}\) in threedimensional point space \(\mathbb {E}^{3}\). Let \(\{\boldsymbol{\imath}_{1},\boldsymbol{\imath}_{2}\}\) be a fixed, positively oriented, orthonormal basis for the translation space \(\mathbb{V}^{2}\) of \(\mathbb{E}^{2}\) and define \(\boldsymbol {\imath}_{3}:=\boldsymbol{\imath}_{1}\times\boldsymbol{\imath}_{2}\) so that \(\{\boldsymbol{\imath}_{1},\boldsymbol{\imath}_{2},\boldsymbol {\imath}_{3}\}\) is a fixed, positively oriented, orthonormal basis for the translation space \(\mathbb{V}^{3}\) of \(\mathbb{E}^{3}\).
2.1 Deformations

There exist mappings \(\tilde{\alpha}\) and \(\tilde {\beta}\) such that each point \(\boldsymbol{x}\) in \({\mathcal{D}}\) and its image \(\boldsymbol{y}\) on \({\mathcal{S}}\) obey (2.7) and (2.8), where \(\hat{\boldsymbol{x}}\) and \(\hat {\boldsymbol{y}}\) are as defined in (2.3) and (2.6).
 The partial derivatives of \(\hat{\boldsymbol{x}}\) and \(\hat{\boldsymbol{y}}\) defined in (2.3) and (2.6) satisfyfor each parameter pair \((\alpha,\beta)\) corresponding to a point \(\boldsymbol{x}\) in \({\mathcal{D}}\) or, equivalently, by (2.4)–(2.6), \(\boldsymbol{R}\) and \(\boldsymbol{b}\) satisfy$$ \biggl\dfrac { \partial \hat{\boldsymbol{y}}}{ \partial {\alpha } } \biggr^{2}= \biggl\dfrac { \partial \hat{\boldsymbol{x}}}{ \partial {\alpha } } \biggr^{2}, \qquad \dfrac { \partial \hat{\boldsymbol{y}}}{ \partial {\alpha } } \cdot \dfrac { \partial \hat{\boldsymbol{y}}}{ \partial {\beta } } =\dfrac { \partial \hat{\boldsymbol{x}}}{ \partial {\alpha } } \cdot \dfrac { \partial \hat{\boldsymbol{x}}}{ \partial {\beta } } , \quad\text{and}\quad \biggl\dfrac { \partial \hat{\boldsymbol{y}}}{ \partial {\beta } } \biggr^{2}= \biggl\dfrac { \partial \hat{\boldsymbol{x}}}{ \partial {\beta } } \biggr^{2} $$(2.12)for each admissible choice of the only argument \(\alpha\) on which they depend.$$ \boldsymbol{R}^{\prime}\boldsymbol{b}=\boldsymbol{0}, $$(2.13)
2.2 Alternative Conditions for Isometry
2.3 Framing of the Spatial Directrix
2.4 Darboux Vector of the Spatial Directrix
3 Interpretation of the Class of Mappings Considered by Dias and Audoly [2]
Dias and Audoly [2] refer to \(S\) and \(V\) as “longitudinal” and “transverse” coordinates. From (3.4), \(S\) provides a onetoone correspondence between points on the referential directrix, which Dias and Audoly [2] identify as a locus of material points,^{7} and their images on the spatial directrix. An analogous statement does not, however, apply to \(V\). This raises a need to properly characterize the inescapably material points that lie on the edges of the material surface. To address this need, Dias and Audoly [2] introduce “edge functions” \(V_{\pm}(\eta,S)\) and in this regard state that “\(S\) varies in the interval \(0\le S\le L\), where \(L\) is the curvilinear length of the centerline,” and that “\(V\) varies in a domain \(V_{}(\eta,S)\le V\le V_{+}(\eta,S)\)”, where “\(V_{\pm}(\eta,S)\) encode the relative positions of the edges of the material surface with respect to the centerline”.^{8} However, they do not explain whether—or, if so, how—the ordered pairs \((S,V)\) correspond to material points.
4 Importance of the Correspondence Between Curvilinear Coordinate Pairs and Material Points
We next focus on the requirement that each coordinate pair \((\alpha ,\beta)\) corresponds to a unique material point \(\boldsymbol{x}\) in \({\mathcal{D}}\) and, moreover, that the underlying correspondence between coordinate pairs and points provides a complete covering of \({\mathcal{D}}\).^{12} Together, these requirements ensure that the relationship between coordinate pairs and material points is surjective.
4.1 General Considerations
Although Dias and Audoly [2] do not mention this requirement, an appreciation of its importance is evident from their introduction of “edge functions” \(V_{\pm}(\eta,S)\). For the particular case in which the reference configuration of the material surface is a rectangular strip, Dias and Audoly [2] choose \(V_{\pm}(\eta,S)\) so that the collection of parameter pairs \((S,V)\) covers the region occupied by the material surface in its reference configuration, but that choice restricts the referential rulings to be perpendicular to the referential directrix, which they take to be midway between the long edges of the strip, and consequently limits the parametrizations (3.1) and (3.2) to describe only the bending of a rectangular material strip into a right circular cylindrical form. It does not, for instance, encompass the bending of a rectangular material strip into a helical ribbon coincident with a portion of a right cylindrical surface or into a conical ribbon coincident with a portion of a right circular conical surface, wherein the referential rulings must be inclined relative to the referential directrix (as discussed later in this section).
4.2 Example: Isometric Deformation of a Rectangular Material Strip to a Helical Sector of a Cylindrical Surface
We consider the isometric deformation of a rectangular material strip \({\mathcal{D}}\) onto a helical sector \({\mathcal{S}}\) of a right circular cylindrical surface \(\mathcal{Y}\) of radius \(r_{0}\). To simplify the discussion, we identify the spatial manifestation of \({\mathcal{D}}\) with \({\mathcal{S}}\) and refer to \({\mathcal{S}}\) as a ‘helical ribbon’.
In the bending of a rectangular material strip into a helical ribbon, we therefore conclude that the parametric description of Dias and Audoly [2] fails to completely cover both the planar subset \({\mathcal{D}}\) of \(\mathbb{E}^{2}\) occupied by the strip in its reference configuration and the curved surface \({\mathcal{S}}\) in \(\mathbb{E}^{3}\) identified with the helical ribbon. Moreover, that description includes extraneous coordinate pairs that do not correspond to material points of \({\mathcal{D}}\). Although we have exposed these drawbacks in the context of a special example, we emphasize that, as discussed in the previous section, their generic nature is evident from Fig. 1 of Dias and Audoly [2].
4.3 Example: Isometric Deformation of a Rectangular Strip to a Sector of a Conical Surface
4.3.1 Characterization of the Conical Surface
4.3.2 Characterization of the Deformation
4.3.3 Coverage of the Strip by Curvilinear Coordinates
 A trapezoid within which all rulings extend from the lower edge \({\mathcal{L}}_{}\) of \({\mathcal{D}}\) and the upper edge \({\mathcal{L}}_{+}\) of \({\mathcal{D}}\). This subregion of \({\mathcal {D}}\) consists of all coordinate pairs \((\alpha,\beta)\) belonging to$$ {\mathcal{T}}_{c}=\bigl\{ (\alpha,\beta):\alpha_{0}\le \alpha\le\alpha _{\ell},\beta_{}(\alpha)\le\beta\le \beta_{+}(\alpha)\bigr\} . $$(4.38)
 A triangle within which all rulings extend from the left edge \({\mathcal{L}}_{0}\) of \({\mathcal{D}}\) and the upper edge \({\mathcal{L}}_{+}\) of \({\mathcal{D}}\). This subregion of \({\mathcal {D}}\) consists of all coordinate pairs \((\alpha,\beta)\) belonging to$$ {\mathcal{T}}_{0}=\bigl\{ (\alpha,\beta):\alpha_{}\le \alpha\le\alpha _{0},\beta_{0}(\alpha)\le\beta\le \beta_{+}(\alpha)\bigr\} . $$(4.39)
 A triangle within which all rulings extend from the right edge \({\mathcal{L}}_{\ell}\) of \({\mathcal{D}}\) and the upper edge \({\mathcal{L}}_{+}\) of \({\mathcal{D}}\). This subregion of \({\mathcal{D}}\) consists of all coordinate pairs \((\alpha,\beta)\) belonging to$$ {\mathcal{T}}_{\ell}=\bigl\{ (\alpha,\beta):\alpha_{\ell}\le \alpha\le \alpha_{+},\beta_{\ell}(\alpha)\le\beta\le \beta_{+}(\alpha)\bigr\} . $$(4.40)
4.3.4 Incomplete Coverage of the Strip by Curvilinear Coordinates
5 Bending Energies
To illustrate the importance of the foregoing statement, we next consider three particular examples involving isometric deformations of rectangular material strips.
5.1 Example 1: Bending Energy of a Rectangular Strip Isometrically Deformed to a Ribbon Coincident with a Sector of a Conical Surface
In this first example, we consider the calculation of the stored energy for a rectangular strip that is isometrically deformed into a portion of a right circular conical surface and compare this calculation to those predicted by the related Sadowsky and Wunderlicht functionals.
5.1.1 Rectilinear Parametrization of the Bending Energy
5.1.2 Curvilinear Parametrization of the Bending Energy
5.1.3 Wunderlich’s Functional
Wunderlich’s [3, 4] functional is a dimensionally reduced version of the bending energy \(E\) defined in (5.1), with its domain of integration being the midline of the spatial manifestation \({\mathcal{S}}\) of the material surface. We next specialize that functional to the problem of isometrically deforming the rectangular material strip \({\mathcal{D}}\) to the conical ribbon \({\mathcal{S}}\) and compare the resulting expression with the bending energy \(E\) of \({\mathcal{S}}\) in either the form (5.7) determined by the rectilinear parametrization or its equivalent alternative form (5.11) determined by the curvilinear parametrization.
5.1.4 Sadowsky’s Functional
Like Wunderlich’s [3, 4] functional (5.15), Sadowsky’s [5, 6] functional (5.17) involves integrating only over the material portion of the spatial directrix \({\mathcal{C}}_{0}\), namely the midline of \({\mathcal{S}}\). However, granted that \(w\ll\ell\), the portions of the directrices that extend beyond the strip and its image are of negliglible length. In this sense, the range of integration in (5.17) is consistent with the hypothesis \(w\ll\ell\). An analogous statement does not apply to Wunderlich’s [3, 4] functional, which is meant to provide an accurate approximation for the bending energy in situations where \(w\) need not be infinitesimal small in comparison to \(\ell\). We anticipate that, for \(w\ll\ell\), the error incurred by using Sadowsky’s [5, 6] functional should be comparable to that incurred by using Wunderlich’s [3, 4] functional. Granted that these observations are accurate, Wunderlich’s [3, 4] functional would appear to be of negligible utility. For \(w\ll\ell\), Sadowsky’s [5, 6] functional should suffice and otherwise—the error associated with failing to completely cover \({\mathcal{D}}\)—neither of the functionals in question is generally fit to provide a good approximation to the bending energy.^{18}
5.1.5 Bounds on the Wunderlich and Sadowsky Functionals
Our analysis demonstrates that Wunderlich’s [3, 4] functional \(E_{\text{W}}\) underestimates the bending energy \(E\) of a conical ribbon \({\mathcal{S}}\) obtained by applying the isometric deformation \(\tilde{\boldsymbol{y}}\) defined in (4.19) to the rectangular material strip \({\mathcal{D}}\) for all admissible choices of the length and width, \(\ell\) and \(w\), of \({\mathcal{D}}\), the angle \(\theta_{0}\) determining the inclination of the referential ruling that emanates from the lefthand endpoint of the midline of the strip, and the length \(\varLambda\) of the generator of the portion of the right circular conical surface \({\mathcal{K}}\) that lies above the \((x_{1},x_{2})\)plane. Additionally, Sadowsky’s [5, 6] functional underestimates Wunderlich’s [3, 4] functional in an analogous fashion.
5.1.6 Illustrative Comparisons
Values of the scaled relative errors \(I\) and \(J\) defined in (5.55) at \(w/2\ell=1.0\times10^{2}\) and \(w/2\ell=3.5\times 10^{2}\) and for \(\theta_{0}=\pi/33\), \(\theta_{0}=\pi/27\), \(\pi /21\), \(\pi/15\), \(\pi/9\), and \(\pi/3\)
\(\theta_{0}\)  w/2ℓ = 1.0 × 10^{−2}  w/2ℓ = 3.5 × 10^{−2}  

I (×10^{6})  J (×10^{7})  I (×10^{3})  J (×10^{5})  
π/33  1.01003  6.71129  6.09847  3.11603 
π/27  1.00669  6.69650  4.89521  3.02511 
π/21  1.00403  6.68473  3.78157  2.95680 
π/15  1.00205  6.67594  2.73486  2.90794 
π/9  1.00074  6.67009  1.74625  2.87643 
π/3  1.00012  6.66720  0.78801  2.86114 
A more significant quantitative distinction between \(I\) and \(J\) is apparent from the insets of Figs. 10 and Fig. 11, from which we see that, for each value of \(\theta_{0}\) considered, \(I\) begins to deviate from the cubic scaling (5.47) at a lower value of \(w/2\ell\) than that at which \(J\) begins to deviate from (5.44). In particular, whereas the values of \(I\) for the given choices of \(\theta_{0}\) at the lefthand endpoint of the inset interval span nearly an order of magnitude, the corresponding values of \(J\) are graphically indistinguishable. The values of \(I\) and \(J\) at \(w/2\ell=3.5\times10^{2}\) appear in Table 1. For each choice of \(\theta_{0}\), the threshold of \(w/2\ell\) above which \(E_{\text{S}}\) fails to afford an accurate approximation to \(E_{\text{W}}\) thus exceeds the threshold of \(w/2\ell\) which fails to afford an accurate approximation to \(E\).
Although the foregoing observations are specific to the particular problem of isometrically bending a rectangular material strip \({\mathcal {D}}\) of length \(\ell\) and width \(w\) to coincide with a conical ribbon, they demonstrate that within the context of that problem Wunderlich’s [3, 4] functional \(E_{\text{W}}\) does not provide an accurate approximation to the bending energy \(E\) unless the halfwidthtolength ratio \(w/2\ell\) is surprisingly small. Moreover, if \(w/2\ell\) is sufficiently small to ensure that \(E_{\text{W}}\) provides an accurate approximation to \(E\), our findings show that Sadowsky’s [5, 6] functional \(E_{\text{S}}\) provides an accurate approximation to \(E_{\text{W}}\) and, thus, a reasonably accurate approximation to \(E\). With reference to the discussion in Sects. 4.3.3 and 4.3.4, this difficulty stems from a failure to ensure a complete covering of the reference region. Only for rectangular material strips of sufficiently small halfwidthtolength aspects ratios is it possible to neglect the presence of unruled corners of the strip without compromising accuracy.
5.2 Example 2: Bending Energy of a Rectangular Strip Isometrically Deformed to a Helical Ribbon Coincident with a Portion of a Cylindrical Surface
5.3 Example 3: Bending Energy Resulting from Isometrically Deforming a Triangular Corner of a Rectangular Material Strip
The deformation required to roll up the unruled corners of \({\mathcal{D}}\) as described above is not onetoone and, thus, does not satisfy all conditions set forth in Sect. 2.1. Alternatively, it is possible to define a deformation in which the unruled corners of \({\mathcal{D}}\) are rolled into spirals that can be tightened to drive up the bending energy in a manner analogous to that achieved by decreasing the radius \(r\) of the rods involved in the deformation described above.
6 Applicability of Kirchhoff Rod Theory to the Modeling of StripLike Material Surfaces
It is well known that the shape of a Kirchhoff rod endowed with a material frame \(\{\boldsymbol{d}_{1},\boldsymbol{d}_{2},\boldsymbol {d}_{3}\}\), as introduced in Sect. 3, is completely determined by its midline and the orientations of its crosssections, which are necessarily perpendicular to the directrix with tangent \(\boldsymbol{d}_{3}\), and that this geometrical information is completely encoded in the Darboux vector \(\boldsymbol{\omega}\) defined such that \(\boldsymbol{d}_{i}^{\prime}=\boldsymbol{\omega}\times \boldsymbol{d}_{i}\), \(i=1,2,3\). Although the directrix of a Kirchhoff rod may bend and the cross sections of the rod may rotate about \(\boldsymbol{d}_{3}\), those cross sections are by hypothesis rigid and thus cannot change shape.^{19} In the present context, the cross sections of a sheetlike object with midsurface \({\mathcal{S}}\) modeled as a rod with spatial material frame \(\{\boldsymbol{d}_{1},\boldsymbol{d}_{2},\boldsymbol{d}_{3}\}\) should be rectangular with the sides parallel to \(\boldsymbol{d}_{1}\) being long (i.e., wide) in comparison to those parallel to \(\boldsymbol {d}_{2}\) (i.e., throughthickness). Granted the interpretations of the components of the Darboux vector of such a rod stated and supplied immediately after (2.23), it follows from (2.30)_{2} that the cross sections of a rod remain rigid only in the presence of the highly restrictive requirement \(\kappa_{g}=0\), which, by (2.32) is met only if the referential directrix \({\mathcal{C}}_{0}\) of \({\mathcal{D}}\) is straight. Otherwise, the rectangular cross sections bend about \(\boldsymbol{d}_{1}\) and consequently do not remain rigid. This effect is evident in the sequence of deformations shown in Fig. 13. There, the short ends of the material surface, which are at the ends of the material portion of the referential directrix, are material lines and would represent the terminal cross sections of a rodbased description. To describe the referential material surface depicted in the top left of Fig. 13 as a Kirchhoff rod, it would be necessary to ensure that those terminal cross sections be rigid and, thus, deform only to the extent that they would only be allowed to rotate about the spatial directrix. This would be overly restrictive for any striplike material surface and in particular would not allow for a deformation of the kind depicted in, say, the sequence shown in Fig. 13. This would drastically undermine any basis for modeling a sheetlike object as a Kirchhoff rod unless, of course, the conventional slenderness hypothesis underlying all rod theories known to the present authors were also in place. In the current setting, that hypothesis amounts to restricting attention to situations with rulings of characteristic length comparable to the throughthickness dimension of the sheetlike object. Although such a hypothesis is met by certain polymer and biopolymer filaments, it is only relevant to a small and not particularly significant class of problems involving thin unstretchable media like paper.
Dias and Audoly [2] place considerable emphasis on a purported connection between their approach to the description of striplike material surfaces and the Kirchhoff theory of rods. In view of the foregoing discussion, such a connection is possible only if the width and the throughthickness of the ribbon are comparable. Since the through thickness of a ribbon or sheet is assumed to be negligible, the approach of Dias and Audoly [2] is more aptly connected to the theory of framed curves, which, as Giusteri and Fried [14] explain, can be viewed as the specialization of Kirchhoff rod theory that arises on restricting attention to rods of infinitesimally vanishing crosssectional thickness.
7 Discussion and Summary
We have emphasized that in characterizing the representation of all isometric deformations of a flat twodimensional material region into a curved surface in three dimensional Euclidean space it is important that the referential directrix and rulings, and the corresponding deformed directrix and generatrices, provide for the establishment of material curvilinear coordinates which completely cover the reference and distorted surfaces. Otherwise, the deformation of parts of the material surface is undefined. Also, we have discussed, by example, the importance of distinguishing between the differential geometric notion of an isometry between two or more twodimensional surfaces embedded in three dimensional Euclidean space and the continuum mechanical notion of the isometric deformation of a single material surface.
The covering requirement noted above has not been fully appreciated in publications which deal with the isometric deformation of ribbons and, as a result, the oversight has led to misunderstandings and questionable claims. We have tracked the details of this issue in two examples of a rectangular flat material strip that is deformed into (i) a portion of a circular cylindrical surface, and (ii) a portion of a circular conical surface. We have observed that this oversight also is prevalent in the important onedimensional reductions of Sadowsky [5, 6] and Wunderlich [3, 4] for representing the bending energy of a flat undistorted material surface that is isometrically deformed into a bent shape.
We proceeded to determine, in complete detail, the bending energy \(E\) of a rectangular material strip \({\mathcal{D}}\) of length \(\ell\) and width \(w\) that is isometrically deformed to coincide with a portion \({\mathcal {S}}\) of a right circular conical surface \({\mathcal{K}}\). Further, we compared the results of that calculation to energies obtained from the corresponding onedimensional energy functionals \(E_{ \text{W}}\) and \(E_{\text{S}}\) of Wunderlich [3, 4] and Sadowsky [5, 6], finding that \(E_{ \text{S}}< E_{\text{W}}< E\) for all relevant choices of \(\ell\), \(w\), cone apex angle \(2\varphi\), and certain parameters associated with \({\mathcal{K}}\) and the placement of the bent strip on \({\mathcal{K}}\). The discrepancy between \(E_{\text{S}}\) and \(E_{\text{W}}\) arises simply because Sadowsky’s [5, 6] functional is supposed to apply only to situations in which the halfwidthtolength ratio \(w/2\ell\) of \({\mathcal{D}}\) is vanishingly small whereas Wunderlich’s [3, 4] functional is designed to apply to situations in which that ratio is finite. The discrepancy between \(E_{\text{W}}\) and \(E\) arises instead from failing to ensure a surjective correspondence between material points and curvilinear coordinate parameter pairs and is, thus, far more relevant to our concerns regarding the covering requirement.
In some cases the Wunderlich functional underestimates the exact bending energy by a sizable and unacceptable amount. As an extreme example, we considered a rectangular material strip whose midline is identified as the referential directrix, and we marked the strip with parallel rulings at an angle \(\theta_{0}\) satisfying \(0 < \theta_{0} < \pi/2\) with the midline. In this case there are two triangular regions, one in the upper right corner and one in the lower left corner of the rectangle, corners cut off by the far right and far left rulings which terminate at the ends of the midline that are not covered by the coordinate system generated by the midline and the set of rulings. We isometrically deformed the strip by rolling up the corners on cylinders of equal and sufficiently small radii whose generators are parallel to the rulings, as shown in Fig. 13. The remaining central portion of the strip was untouched and remained flat in its undistorted state. For this deformation, the Wunderlich functional \(E_{\text{W}}\) records zero, while the bending energy \(E\) of the deformed strip is as large as one pleases depending on the radii of the cylinders over which the corners are wrapped. We concluded that the Wunderlich functional does not account for the covering requirement and, therefore, contains a fundamental and potentially significant deficiency; for many situations it does not wellrepresent the bending energy. As it turns out, for cases in which the halfwidthtolength ratio \(w/2\ell\) of \({\mathcal{D}}\) is sufficiently small, it is only incrementally better than the Sadowsky functional.
In the literature, Kirchhoff rod theory has been suggested as a possible theory for modeling the isometric deformation of material ribbons. In the last short section of this work, we gave reasons why this approach is viable only if the width and the throughthickness dimensions of the ribbon are comparable. An association of Kirchhoff rod theory with the study of the deformation of striplike surfaces is more aptly connected to the theory of framed curves wherein a Kirchhoff rod is assumed to have infinitesimally vanishing crosssectional thickness.
Footnotes
 1.
In the present work, we replace the coordinates \((\eta^{1},\eta^{2})\) used by Chen, Fosdick and Fried [1] with \((\alpha,\beta)\).
 2.
Moreover, it is possible and sometimes even necessary to choose \({\mathcal{C}}_{0}\) to be disjoint from \({\mathcal{D}}\).
 3.
Here, we use \(p\) in place of the symbol \(\lambda\) used by Chen, Fosdick and Fried [1].
 4.
See, for example, Hartman [9].
 5.
Dias and Audoly [2] refer to the directrices as “centerlines”.
 6.
 7.
See the third sentence in the first paragraph of Sect. 2.1 of Dias and Audoly [2].
 8.
See the discussion beginning in the final paragraph on page 52 of Dias and Audoly [2].
 9.
The conditions (3.6)_{1} and (3.6)_{2} are conveyed in constraints (10*) and (11*) of Dias and Audoly [2], where their (7*) is used to express the components of their Darboux vector \(\boldsymbol{\omega}\) in terms of the triad \(\{ \boldsymbol{d}_{1},\boldsymbol{d}_{2},\boldsymbol{d}_{3}\}\). Notice, however, that (3.6)_{1} involves the additional condition \(\boldsymbol{D}_{3}^{\prime}\cdot\boldsymbol{D}_{1}=\kappa_{g}\) which appears in the penultimate sentence of the second paragraph in Sect. 2.3 of the paper of Dias and Audoly [2].
 10.
 11.
The presentation of Dias and Audoly [2] is not convincing as it stands because it emphasizes the differential geometric notion of isometry and does not clearly define the deformation of material points from a reference form to a spatial form and explicitly distinguish and characterize the deformation as isometric.
 12.
See the first bullet item in the final paragraph of Sect. 2.1.
 13.
 14.
See the discussion in the second paragraph of Sect. 4. For a helical ribbon with spatial directrix being a helix of lead angle \(\pi/2\theta_{0}\), the referential rulings must be inclined at an angle \(\theta_{0}\) when measured counterclockwise from the referential directrix. In the present setting, the edge functions \(\pm w/2\) that Dias and Audoly [2] introduce in their discussion of rectangular material strips must therefore be replaced by \(\pm w\csc\theta_{0}/2\).
 15.
For a complete description of that deformation and derivations of all kinematical objects used in the subsequent calculations, we refer the reader to Chen, Fosdick and Fried [1, Sect. 9].
 16.
These subregions are complementary to the extent that their interiors are disjoint.
 17.It is also possible to directly obtain (5.16) from the version of Wunderlich’s [3, 4] functional that appears in (30*) of Dias and Audoly [2]. To do that, we require the appropriate specializations of their quantities \(\eta\) and \(\omega_{1}\). With this in mind, we recall from (3.11)_{1} that the quantity \(\nu\) introduced in (2.2) is equal to \(\eta\) of Dias and Audoly [2]. Since \(\nu=\cot\theta\) for the deformation \(\tilde {\boldsymbol{y}}\) of \({\mathcal{D}}\) to \({\mathcal{S}}\) defined in (4.19), we thus see from (4.22) that \(\eta\) of Dias and Audoly [2] is given byFrom (3.11)_{2}, we find moreover that our Darboux vector \(\boldsymbol{\delta}\) is equal to the Darboux vector \(\boldsymbol {\omega}\) of Dias and Audoly [2]. Thus, by (2.30)_{1}, (4.23), and (9.22) of Chen, Fosdick and Fried [1], we determine that \(\omega_{1}\) of Dias and Audoly [2] is given by$$ \eta=\cot\theta. $$(†)Using (†) and (‡) in (30*) of Dias and Audoly [2], we obtain (5.15) and, therefore, upon integration with respect to \(\alpha\), arrive at (5.16).$$ \omega_{1}=\frac{p}{\sqrt{1+\eta^{2}}} =\frac{\cot\varphi}{\varLambda\sin\theta_{0}}\frac{1}{\sqrt{1+\eta ^{2}}} \frac{1}{\rho^{2}}. $$(‡)
 18.
 19.
A geometrically exact rod model that carefully incorporates inplane cross sectional deformation was recently developed by Kumar and Mukherjee [11]. Previously, Hodges [12] proposed a constitutively based strategy to account for cross sectional deformation and Gould and Burton [13] developed a rod theory in which each cross section is itself viewed as a rod.
Notes
Acknowledgements
The work of Eliot Fried was supported by the Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan. Yichao Chen thanks the Okinawa Institute of Science and Technology for hospitality and generous support during a sabbatical and subsequent visits. The authors express their gratitude to Michael Grunwald for his help in preparing preliminary versions of the figures.
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