The Gol'denveizer problem of a torus was studied analytically by Audoly and Pomeau (2002), and the accuracy of the Audoly and Pomeau linear law was confirmed numerically by Sun (2021). However, the law does not include the major radius R of the torus. To find the influence of the major radius, we used finite element numerical simulation to simulate different cases, and we propose a modified Audoly and Pomeau linear law for vertical deformation, which includes R. A linear law of horizontal deformation is presented as well. Our studies show that the Audoly and Pomeau linear law has high accuracy. With modified vertical and horizontal deformation, a displacement-compatible relation between them is formulated.

Gol'denveizer's problem of a torus has been analyzed by Audoly and Pomeau (2002) and Sun (2021). However, all of the investigations of Gol'denveizer's problem of an elastic torus have been linear. In this paper, the finite element method is used to more accurately address this problem. Furthermore, Sun (2021) cannot be solved by nonlinear analysis. We research the nonlinear mechanical properties of Gol'denveizer's problem of circular and elliptic tori, and relevant nephograms are given. We study the buckling of Gol'denveizer's problem of an elastic torus, and propose failure patterns and force-displacement curves of tori in the nonlinear range. Investigations reveal that circular tori have more rich buckling phenomena as the parameter a increases. Gol'denveizer's problem of the buckling of an elliptic torus is analyzed, and we find a new buckling phenomenon called a "skirt." As a/b increases, the collapse load of an elliptic torus of the Gol'denveizer problem is enhanced gradually.

Weaving is an ancient and effective structural forming technique characterized by the ability to convert two-dimensional ribbons to three-dimensional structures. However, most 3D structures woven from straight ribbons have topological defects. Baek et al. proposed a method to weave smoother continuous 3D surface structures using naturally curved (in-plane) ribbons, obtained a new surface structure with relatively continuous variation of Gaussian curvature, and analyzed its geometric properties. We believe that this new 3D surface structure with smooth geometric properties must correspond to new mechanical properties. To this end, we investigated a 3D surface structure using naturally curved (in-plane) ribbon weaving, and the results of calculations and experiments show that such structures have better buckling stability than those woven with straight ribbons. It is observed that the number of ribbons influences the buckling behavior of different types of woven structures.

Weaving technology can convert two-dimensional structures such as ribbons into three-dimensional structures by specific connections. However, most of the 3D structures fabricated by conventional weaving methods using straight ribbons have some topological defects. In order to obtain smoother continuous 3D surface structures, Baek et al. proposed a novel weaving method using naturally curved (in-plane) ribbons to fabricated three-dimensional curved structures and using this method to weave new spherical weave structures that are closer to perfect spheres. We believe that this new spherical weave structure with smooth geometric properties must correspond to new mechanical properties. To this end, we investigated the buckling characteristics of different types of spherical weave structures by the combination of test and finite element method. The results of calculations and experiments show that the failure mode of the spherical weave structure under vertical loading can be divided into two stages: a flat contact region forms between the spherical weave structure and the rigid plate and inward dimple of ribbons. The spherical weave structures using naturally curved (in-plane) ribbon weaving have better buckling stability than those woven with straight ribbons. The vertical buckling load of spherical weave structures using naturally curved ribbon increases with the width and thickness of the ribbon. In addition, this paper combines test, theoretical and finite element analysis to propose the buckling load equation and buckling correction factor equation for the new spherical weave structure under vertical compression load.