The nonlinear bending-torsion theory for curved rods as Gamma-limit of three-dimensional elasticity

Asymptotic models for curved rods derived from nonlinear elasticity by Gamma-convergence

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dimension reduction starting from linearized elasticity. Finally we also address the case of thin elastic rings.

Continuum Elasticity Theory Approach for Spontaneous Bending and Twisting of Ribbons Induced by Mechanical Anisotropy

Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initially flat, straight ribbon, with prescribed magnitudes and orientations of the principal curvatures when subjected to arbitrary surface stress and/or internal residual strain distribution. The helix angle, radius, axis and chirality of the deformed helical ribbons are predicted with…

不可伸长杆的多材料形状优化的相场模型

Phase field model for multi-material shape optimization of inextensible rods

本文提出一种优化非均匀弹性杆弯曲与扭转刚度的相场模型，通过 $Γ$-收敛证明其收敛至清晰界面模型，并利用最速下降法数值求解，将最优形状与植物茎形态发育相联系。

Nonlinear dynamic intertwining of rods with self-contact

Twisted marine cables on the sea floor can form highly contorted three-dimensional loops that resemble tangles. Such tangles or hockles are topologically equivalent to the plectomenes that form in supercoiled DNA molecules. The dynamic evolution of these intertwined loops is studied herein using a computational rod model that explicitly accounts for dynamic self-contact. Numerical solutions are presented for an illustrative example of a long rod subjected to increasing twist at one end. The solutions reveal the dynamic evolution of the rod from an initially straight state, through a buckled st…

一维变分问题的充要条件及其在弹性力学中的应用

Necessary and sufficient conditions for one-dimensional variational problems with applications to elasticity

本文推导一维变分泛函弱、强极小元更简洁的充要条件，并应用于弹性稳定性，给出各类端点约束下自然笔直 Kirchhoff 杆稳定的最优条件，解决了既有文献中的公开问题。

Pseudomomentum: origins and consequences

The balance of pseudomomentum is discussed and applied to simple elasticity, ideal fluids, and the mechanics of inextensible rods and sheets. A general framework is presented in which the simultaneous variation of an action with respect to position, time, and material labels yields bulk balance laws and jump conditions for momentum, energy, and pseudomomentum. The example of simple elasticity of space-filling solids is treated at length. The pseudomomentum balance in ideal fluids is shown to imply conservation of vorticity, circulation, and helicity, and a mathematical similarity is noted betw…