Abstract
We consider the problem of characterizing the smooth, isometric deformations of a planar material region identified with an open, connected subset {mathcal{D}} of two-dimensional Euclidean point space mathbb{E}^{2} into a surface {mathcal{S}} in three-dimensional Euclidean point space mathbb{E}^{3}. To be isometric, such a deformation must preserve the length of every possible arc of material points on {mathcal{D}}. Characterizing the curves of zero principal curvature of {mathcal{S}} is of major importance. After establishing this characterization, we introduce a special curvilinear coordinate system in mathbb{E}^{2}, based upon an à priori chosen pre-image form of the curves of zero principal curvature in {mathcal{D}}, and use that coordinate system to construct the most general isometric deformation of {mathcal{D}} to a smooth surface {mathcal{S}}. A necessary and sufficient condition for the deformation to be isometric is noted and alternative representations are given. Expressions for the curvature tensor and potentially nonvanishing principal curvature of {mathcal{S}} are derived. A general cylindrical deformation is developed and two examples of circular cylindrical and spiral cylindrical form are constructed. A strategy for determining any smooth isometric deformation is outlined and that strategy is employed to determine the general isometric deformation of a rectangular material strip to a ribbon on a conical surface. Finally, it is shown that the representation established here is equivalent to an alternative previously established by Chen, Fosdick and Fried (J. Elast. 119:335–350, 2015).
Highlights
In the present paper, we describe an alternative strategy designed to mitigate the aforementioned difficulties
The major problem related to the characterization of all isometric deformations yfrom D ⊂ E2 to S ⊂ E3 stems from the necessary and sufficient condition (4.14), which requires the determination of Q : R → Orth+ as the solution of the following tensor initialvalue problem: Qη1 = W η1 Q η1, with Q(0) = Q0 ∈ Orth+, (8.1)
We shall summarize in six steps a strategy for the characterization of every isometric deformation from a region in E2 to a surfaces in E3 and illustrate how the solution to problem (8.1) is the key element: 1. Recall from (4.14) that the fundamental proper orthogonal linear transformation Q, which is at the basis for constructing any isometric deformation, must satisfy ax Q Q = λa2, where λ is scalar-valued and where the unit vector-valued field a2 defines the direction of the straight lines of zero principal curvature on the deformed surface S. 2
Summary
We describe an alternative strategy designed to mitigate the aforementioned difficulties This strategy produces a different, but equivalent, necessary and sufficient representation for the class of isometric deformations of planar material regions and it corrects a fundamental misunderstanding concerning an interpretation of the coordinate representation that has circulated in the mainstream literature on the subject. Our primary objective is to determine a representation for the most general smooth isometric deformation ythat takes each point x in an open, connected subset D of E2 to a point y on a surface S in E3:. Our approach hinges on a characterization, provided, of the generators of any surface S determined by such a deformation. This characterization leads naturally to the introduction, in Sect.
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