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Abstract A modified incremental harmonic balance (IHB) method is used to determine periodic solutions of wave propagation in discrete, strongly nonlinear, periodic structures, and solutions are found to be in a two-dimensional hyperplane. A novel method based on the Hill’s method is developed to analyze stability and bifurcations of periodic solutions. A simplified model of wave propagation in a strongly nonlinear monatomic chain is examined in detail. The study reveals the amplitude-dependent property of nonlinear wave propagation in the structure and relationships among the frequency, the amplitude, the propagation constant, and the nonlinear stiffness. Numerous bifurcations are identified for the strongly nonlinear chain. Attenuation zones for wave propagation that are determined using an analysis of results from the modified IHB method and directly using the modified IHB method are in excellent agreement. Two frequency formulae for weakly and strongly nonlinear monatomic chains are obtained by a fitting method for results from the modified IHB method, and the one for a weakly nonlinear monatomic chain is consistent with the result from a perturbation method in the literature.
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Quadratic-stretch elasticity
A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed.
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- Award ID(s):
- 2001262
- PAR ID:
- 10322620
- Date Published:
- Journal Name:
- Mathematics and Mechanics of Solids
- Volume:
- 27
- Issue:
- 3
- ISSN:
- 1081-2865
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1177/10812865211022417
