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Structural Mechanics and Design of Active Origami Structures

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Book cover Active Origami

Abstract

A model for the mechanics of active origami structures with smooth folds is presented in this chapter. The model entails the integration of the surface kinematics model for origami with smooth folds developed in Chap. 5 and existing plate theories to obtain a structural representation for folds of non-zero thickness. The implementation of the model in a computational environment is also addressed. We provide examples including origami structures comprised of both elastic materials and active materials. Afterwards, the unfolding polyhedra and tuck-folding methods studied in Chaps. 6 and 7 are extended to develop frameworks for the design of active origami structures. The extensions account for the folding deformation achievable by smooth folds of specified thickness and constituent materials, which is not considered in the purely kinematic formulations of Chaps. 6 and 7.

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Notes

  1. 1.

    This additive decomposition of strain is valid only in the case of linearized strains. The work presented in this chapter considers such a case.

  2. 2.

    (⋅ : ⋅) denotes the inner product of two second-order tensors. If such tensors are expressed in an orthonormal coordinate system, their inner product is given as \(\mathbf {Y}:\mathbf {Z} = \textstyle \sum _{i=1}^3\textstyle \sum _{j=1}^3 Y_{ij} Z_{ij}\).

  3. 3.

    Integration over the volume of the smooth fold domain is given as \(\textstyle \int _{\mathfrak {F}^i}(\cdot )\,\mathrm {d}v =\textstyle \int _{\mathcal {F}^i} \textstyle \int _{-\textstyle \frac {\mathfrak {h}_i}{2}}^{\textstyle \frac {\mathfrak {h}_i}{2}}(\cdot )\,\mathrm {d}s_3\,\mathrm {d}a =\textstyle \int _{-\textstyle \frac {\hat {w}^0_i}{2}}^{\textstyle \frac {\hat {w}^0_i}{2}} \textstyle \int _{0}^{\hat {L}_i}\textstyle \int _{-\textstyle \frac {\mathfrak {h}_i}{2}}^{\textstyle \frac {\mathfrak {h}_i}{2}}(\cdot )\,\mathrm {d}s_3\,\mathrm {d}s_2\,\mathrm {d}s_1\).

  4. 4.

    For each face of the goal mesh \(\mathcal {M}\), the position vectors of its nodes are denoted \(\tilde {\mathbf {y}}^{j1},\tilde {\mathbf {y}}^{j2},\tilde {\mathbf {y}}^{j3} \in \mathbb {R}^3\) and its unit normal vector is calculated via (3.5).

  5. 5.

    The reader can readily verify the normalization coefficient in (8.70) since \(\left ( (-\eta _1R\cos {}(2\pi \eta _2))^2 + (-\eta _1R\sin {}(2\pi \eta _2))^2 + (2p)^2\right )^{1/2} = \left (\eta _1^2R^2(\cos ^2(2\pi \eta _2) + \sin ^2(2\pi \eta _2)) +\right .\)\(\left . 4p^2\right )^{1/2} = \left (\eta _1^2R^2 + 4p^2\right )^{1/2}\).

  6. 6.

    The unit normal vector \({\mathbf {n}}_{\mathcal {G}}\) of the parabolic surface is determined as \({\mathbf {n}}_{\mathcal {G}} = \left (\textstyle \frac {\partial \mathbf {q}}{\partial \eta _1}\times \textstyle \frac {\partial \mathbf {q}}{\partial \eta _2}\right )\|\textstyle \frac {\partial \mathbf {q}}{\partial \eta _1}\times \textstyle \frac {\partial \mathbf {q}}{\partial \eta _2}\|{ }^{-1}\).

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  1. Department of Aerospace Engineering, Texas A&M University, College Station, TX, USA

    Edwin A. Peraza Hernandez, Darren J. Hartl & Dimitris C. Lagoudas

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Peraza Hernandez, E.A., Hartl, D.J., Lagoudas, D.C. (2019). Structural Mechanics and Design of Active Origami Structures. In: Active Origami. Springer, Cham. https://doi.org/10.1007/978-3-319-91866-2_8

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