考虑大型非线性几何变形，使用ANSYS进行所有官员。通常，一端是固定的，而位移边界条件则应用于另一端，从而使我们能够获得von mises应力，应变能量和反作用力。

　　我们考虑了双夹具的压缩屈曲。假定其弯曲挠度为w（x）= a（1 - cos bx），b =2nπ/l，其中l是杆的长度，n表示弯曲屈曲的顺序，x的顺序是沿杆的位置（补充注释1）。最大挠度为WMAX = 2A。手性元素中的杆在n = 1/2模式下弯曲。在适当的约束下，非手续晶格中的细长原始杆通常遵循一阶屈曲模式n = 1，但是如果没有严格的约束，某些全局屈曲模式可能会导致斜杆的n = 1/2（扩展数据图9B）。

　　我们注意到，偏转的假设W（x）= a（1 - cos bx）对于中等变形ε有效< 0.2. When a rod is deeply buckled, it exhibits more complex deformation patterns. In FEA, these transformations in the bending deflection are adaptively accounted for through geometrical nonlinearity. This transformation is not considered in the analytical theory, which explains the increasing deviation from FEA results in Fig. 2 for ε >0.2。

　　在这里，我们简要描述了杆的压缩屈曲的分析模型的结果。完整的建模过程和数学细节在补充注释1中提供。当将轴向压缩位移（δ）应用于半径为R，长度L0和Young的模量ES的双夹杆时，轴向反作力为F1rod。首先，应力σCPR是由纯压缩诱导的。屈曲发生在临界力

　　其中n表示屈曲模式的顺序。屈曲的临界轴向应力是

　　当f1rod< Fc, the rod undergoes uniform strain ε = Δ/L0 and stress σcpr = Esε, storing energy

　　where Vs = πr2L0 denotes rod volume. In this case, von Mises stress is σv = σcpr. Moreover, the critical energy for buckling is

　　This equation shows that the critical energy is proportional to n4. If the buckling order changes from n = 1 to 1/2, the critical energy greatly reduces to 1/16.

　　When buckling occurs, the stress induced by pure compression (σcpr) remains nearly constant. In the general case with n = 1, bending induces maximal tensile or compressive stresses on the surface near the clamped ends

　　where 2a denotes the bending deflection. Consequently, maximal von Mises stress σv = σbend + σcpr is found there. In this case, the strain energy stored in a compressive buckling rod is

　　This equation indicates that σcpr gets an eight times higher increment ratio of energy than σbend when increasing stress. Thus, for a specified moderate stress, such as σv = 0.1Es, a thicker rod obtains higher U1rod/Us until buckling disappears at this stress level. This equation also indicates that combining bending and compression increases the increment ratio of U1rod/Us as strain ε = Δ/h0 increases.

　　We note that when analysing the angled rod and chiral rod, the vertical compression displacement is defined as s. In the above equations, Δ denotes the axial compression displacement. Δ = s for a vertical rod with angle θ = 90°.

　　based on the theory above, we adopt to normalize the compressive force. Moreover, specifying ξ = σcpr/σbend, the energy inside a buckled rod can be simplified as

　　Here, Vs = πr2L0 is the rod volume and λ = σv/Es is the normalized strength. Then, the enthalpy of a single rod is

　　The factor f can be approximated as f ≈ 2.5/1,000 for ξ → 0 and λ < 0.2. For convenience, it is reasonable to use ϕs = Es/1,000 and Us = Vsϕs to normalize the enthalpy and total energy, respectively.

　　The process of deriving the analytical model for chiral buckling is extensive and involves rigorous mathematical calculations. In Supplementary Information section ‘Chiral twist buckling theory’, we systematically elaborate on the geometry, deformation, force, energy, and stress induced by in-plane bending, out-of-plane bending, in-rod twisting, helix and compression modes (Supplementary Notes 2 and 3). Definitions of all variables are also listed there. Here we summery the analytical theory (also in Supplementary Note 3.8), with key parameters labelled in Extended Data Fig. 1.

　　When a vertical compression displacement s is applied to half a chiral metacell, a variable rotation angle θ forms between the two tori. The equivalent strain is ε = s/h0. The reaction force and elastic energy contributed by a single rod are F1rod, U1rod, respectively. Point B is fixed on torus O1. Point A0 is the original point on torus O2, which moves to point A under the compression displacement s. Length is given as

　　The in-plane bending deflection is denoted as 2ain and the out-of-plane bending deflection is denoted as 2aout. Specifying β = ABA0, 2ain = Ls sin β, Lx = Ls cos β. The term 2aout is defined by length |CD| shown in Extended Data Fig. 1b.

　　The moments induced by in-plane bending and in-rod twisting (Supplementary Note 3) are

　　Here b = π/Lx, I = πr4/4 is the second moment of the rod, G is the shear modulus, θr = θ-γ. The bulge-out angle γ arises from the helical deformation of rod, as shown in Extended Data Fig. 1c.

　　The components of the two moments in three-dimensional spaces are

　　where nin is the unit normal vector of the in-plane bending and ntwist is the unit normal vector in the twisting deformation. The concentrated force applied on a single rod denotes (Fx, Fy, Fz). We use the following equations to analyse chiral buckling deformation and calculate energy.

　　We adopt this equivalent stress to evaluate the strength of the rods in the chiral structure.

　　The shear modulus is G = Es/2(1 + υ), where Poisson’s ratio is υ = 0.3. In the chiral metacell, both the in-plane and out-of-plane bending deflection follow the function 1 − cos(πx/L), with their deflection amplitudes denoted as 2ain and 2aout, respectively. In-rod twisting generates shear stress given by τ = Esπr/2.6L0 = σbend(L0 − s)2/2.6L0πa. A proper chiral structure sets L0 ≈ 4R; under large deformation, s ≈ R and the in-plane bending deflection 2a ≈ 2R. Thus, we find that τ ≈ 0.3σbend. Under the same compressive displacement s, σnorm in the chiral rod and the lattice rod (Fig. 1a) are approximately equal.

　　We use the following quantities to evaluate the performance.

　　Here, force F, energy U, volume V, mass m, the projected area in load-bearing direction A and the vertical compression displacement s, with the subscript ‘cell’ are measured on the unit cell. The equivalent elastic modulus E denotes the slope of the ε–σeq curve at a small strain ε < 0.02. The maximal stress on the buckling plateau, σbk = max(σeq), symbolizes the load-bearing strength of the entire metamaterial. These variables are normalized for evaluation as E/Es, σbk/Es, ϕ/ϕs and ρ/ρs, where ρs is the mass density of the rod material, ϕs = Es/1,000 is defined in the section ‘Parameter generalization’.

　　For a chiral metamaterial, there are N rods around the circle with radius R. Here we adopt the integral N = Round(πR/2r) with enough space left to avoid contact between the deformed arms. Thus, Vcell = 4h0(R + r)2, Acell = 4(R + r)2, Fcell = NF1rod, Ucell = NU1rod, mcell = Nmrod, where mrod is the mass of a rod. Here we adopt the maximal cubic volume and square area, instead of the smaller cylinder and circular sizes, to evaluate the performance of chiral metamaterials.

　　For octahedron lattices, we calculate only 1/8 metacell corresponding to one rod. The parameters are defined as follows: Fcell = F1rod, Ucell = U1rod, mcell = mrod, , where θ denotes the oblique angle between the rod and the horizontal. L0r denotes the distance between the neighbour connection nodes.

　　For prism lattices, we calculate only 1/4 metacell corresponding to one rod. Fcell = F1rod, Ucell = U1rod, mcell = mrod, , , where lp denotes the distance between neighbouring rhombuses, that is, the lattice constant along the x-axis in Fig. 4. Here we adopt lp = 4r to form a dense lattice.

　　For octahedron and prism lattices, the useful length of a rod is L0. In the above equations, L0r denotes the distance between two connection nodes in lattices. If L0r = L0, we will obtain ρ/ρs > 1, when the oblique angle θ >70°，这是不切实际的。实际上，连接节点也占据了空间，如图4所示。因此，实用的晶格具有L0R> l0。对于我们的计算，我们对R = 1.5 mm的杆采用合理的值L0R = 1.1L0。

　　图4a中的橡胶手性元素具有r = 1.5 mm的n = 8杆，而钛版的含量为n = 20，r = 0.6 mm。所有样品的H0 = 30 mm或L0 = 30 mm。图4A中的橡胶和钛样品均具有r = 7.5 mm，α0= 5°。棱镜晶格中的斜角为θ= 40°。对于TC4晶格，邻居平行杆之间的距离为3毫米，橡胶晶格的距离为10毫米。八面体和四环体晶格内的杆的长度相等。基于图3，当指定相同的全局菌株ε时，手性，棱镜和八面体格的最大内膜应力近似。尽管理论上可以忍受较高的全局变形，但四核体和张力晶格可以在ε> 0.4时接触到更高的全局变形。我们的绩效评估考虑了这些差异。其他参数和结果在扩展数据表1中列出。

　　在图4C中，手性20°的r = 5.5，2R = 1.8，α0= 20°，H0 = 20 mm。手性50°的r = 6，2R = 1.7，α0= 50°，H0 = 20 mm。图4E中的橡胶样品的Es≈5.5MPa，而其他橡胶样品的ES≈15MPa。

　　进行了环状压缩实验。新制造的金属样品中的杆最初是笔直的。但是，在第一个压缩周期之后，全金属样品表现出一些残留的塑性变形。在这种情况下，α0增加，杆略微弯曲。此外，第一个周期增强了材料，改善了其整体强度。随后的压缩循环是完全可重复的，即不会发生塑性变形。焓被计算为。

　　对于棱镜晶格，扩展数据中显示的单层样本图9a由两个半元素组成。我们测试了两层和四层样品的变形，以确认一致性。对于多层棱镜晶格，如果没有横向约束，则会发生平面外凸出变形（对应于1/2阶屈曲）（参见扩展数据图9B和补充视频5）。与图1A中所需的平面内屈曲模式相比，这种变形将屈曲强度σbk和焓ϕ降低10倍（扩展数据图9H）。因此，如扩展数据图所示。9C和10E，我们将棱镜晶格放在盒子内，以限制横向凸起的变形。