Elastic fibers lose stability under compression and undergo a buckling phenomenon to transition into a new minimum energy state. The buckling of fibers was previously considered as a form of failure, which was undesired due to the loss of the original configurations. However, this was later on demonstrated as a strategy to tailor the performance of soft and smart materials. Recently, shrinkage buckling has been employed in the preparation of conductors to improve tensile and ductile strengths [1–4]. Elastic fibers can be prepared to have various geometries, including U-shaped, spiral, and others [5–7], and to be extremely ductile. In recent work, it was observed that elastic fibers embedded in an external elastic matrix suffered from a wavy buckling configuration, resulting from shrinkage after the release of pre-stretch, as shown in Figure 1A. Through high-magnification SEM images, it can be observed that the elastic fiber is buckled into a wavy configuration superimposed with smaller waves, establishing a hierarchical buckling phenomenon [7], as illustrated in the conceptual diagram of Figure 1B. The hierarchy of buckling provides the option to customize the buckling phenomenon to achieve the functional requirements specific to certain scenarios. From a practical perspective, such buckling structures have been widely applied in electronic devices, including wearable electronics[8,9], flexible displays [10], energy devices, sensors [11], resonators, and electromagnetic wave absorbers [12–15]. By adopting a suitable buckling structure, the functionality of the devices can be further enhanced.

Until present, several analytical and numerical models have been developed to investigate the buckling and instability of fibrous materials. Dayan et al. [16] investigated the behavior of an elastic fiber constrained by flexible and rigid tubes under a compressive force through theoretical, numerical, and experimental methods. They emphasized the deviation in behavior between a compressed fiber constrained by a deformable tube and a fiber confined within a rigid cylinder. Su et al. [17] utilized the classical Euler beam theory to investigate the buckling problem of a slender elastic rod embedded in an elastic matrix. They determined the critical conditions for the transition of the rod from a planar wave to a non-planar coiled structure. Xiao et al. [18] categorized the buckling of an elastic rod under cylindrical transverse confinement into four specific deformation modes: initial two-dimensional (2D) shape, small three-dimensional (3D) shape, 3D helix shape and 3D alpha shape. They also investigated the critical axial displacement at the transitional point between those four deformation modes. Interestingly, a film [19,20] or ribbon [21–23] attached to an elastic substrate can also produce a buckling structure. The selection of substrates in these models is diverse, ranging from single-layered substrates and multilayered solid elastic substrates [20] to fluid substrates [23,24]. They exhibited similarities to the buckling of fibers under transverse confinement [6] and illustrated the varied structural responses among systems with different substrates. In addition, similar phenomena can be observed in studies with the same confinement conditions, such as cell microscopy [25,26], oil push rod [27] and plant root growth [28]. Most of the aforementioned studies focus on single-level buckling, with only a few analyzing hierarchical buckling. Based on this situation, in order to better explain the problem of hierarchical buckling of elastic fibers, the dimensionless theory is utilized in this paper for the analytical derivation. The dimensionless theory is a widely employed analytical method in the fields of mathematics and physics. It is utilized to enhance the comprehension and description of various problems. There has been considerable recent work on reducing the complexity of problems by removing the effects of scale in physical phenomena through the use of dimensionless theories, and this approach has yielded excellent results [29–35]. Therefore, we developed a dimensionless analytical solution to investigate the buckling hierarchy of elastic fibers with using elliptical cross-sections under transverse confinement, through the superposition of buckling of various buckling modes. For the establishment of a theoretical model, the fiber shape and elasticity are the critical parameters for the simulation of their counterparts. Moreover, in recent studies, the cross-section selection for the elastic fibers was basically the ideal circular or rectangular shape, which may affect the accuracy of the calculated results. Consequently, an irregular elliptical shape in this paper instead of the ideal circular shape to better reflect the actual situation.

In this paper, a theoretical model is established for the buckling phenomenon in sheath-core structural systems. The remainder of the paper is organized as follows: In Section 2, an analytical derivation of the hierarchical buckling phenomenon for elastic fibers with elliptical/circular cross-sections is conducted for the first time. The effect of the external solid/fluid substrates was investigated through their transformation to equivalent stiffness. A finite element (FE) model is established in Section 3 to verify the theoretical derivation. Finally, we present numerical examples to investigate several factors that may impact the hierarchical buckling effect observed in elastic fibers, and the mode transition law of buckling. We found that the morphology of buckling and its associated characteristic scales can be tailored by varying the geometric and material parameters of the system.

FE simulations were performed to validate the theoretical derivation through the commercial software package ABAQUS. In the simulations, small deformations were considered and the elastic fiber was modeled using element type C3D8R (an 8-node linear brick). Reduced integration and enhanced hourglass control are adopted to enable the elements to have an enhanced tolerance of distortion. The elastic Young’s modulus and Poisson’s ratio of the fiber were E _f = 59 GPa and v _m = 0.22 [5], respectively. The mean diameter and length of the fiber were R = 0.643 mm and L = 100 mm. Articulated boundary conditions were established at both ends of the elastic fibers (only the axial displacement being released) and a compressive tendency being imposed. To simulate the effect of transverse constraints on the elastic fibers, an array of springs was applied to the outside of the fibers using the elastic foundation module in the interaction. By utilizing the transformation relationship between the matrix stiffness and the spring stiffness discussed in the previous section, we were able to impose different spring stiffness values. To ensure accuracy and convergence reliability, we continuously debugged the number and size of cells. Ultimately, we used 22,000 meshes and 33,033 nodes with a mesh size of 0.1 mm. We checked the quality of the meshes using the mesh quality checking system in ABAQUS, which showed that the number of wrong and warning meshes was zero, indicating that our mesh setup was reasonable and correct. Finally, the buckling analysis was performed using a linear perturbation algorithm (via the BUCKLE module in ABAQUS).

Figure 6 illustrates the hierarchical buckling modes of the elastic fibers through the FEA results. In Figure 6, we show the buckling modes of elastic fibers with four representative configurations, which combinations of modal superpositions, say c ² = 64π ⁴, 5625π ⁴, 4624π ⁴ and 3600π ⁴ (respectively, a, b, c, and d) in Figure 6. According to Eq. 9, it can be inferred that Figure 6 present joint buckling phenomenon by superposition of modes 1 and 8 (Figure 6A), modes 5 and 15 (Figure 6B), modes 4 and 17 (Figure 6C) as well as modes 2 and 30 (Figure 6D), respectively. It is observed that the hierarchical buckling phenomenon becomes more significant with the introduction of higher-order buckling modes. The FE simulations are employed to validate the present result by generating the dimensionless critical buckling loads as functions of dimensionless spring stiffness. Figure 7 compares the theoretical prediction and numerical ones, and it is found that the FE results do not closely fit with the theoretical calculations. We speculate that the reasons for the differences between the reduced model (Wrinkler foundation model) and the FE results is due to shear deformation in the matrix, which the Wrinkler model does not take into account. However, these simulation results are qualitatively consistent with the experimental results and supporting the theoretical analysis in Section 2.

Based on the experimental observation, the hierarchical buckling of elastic fibers with elliptical and circular cross-sections, under transverse confinement, was investigated through analytical derivations. It was demonstrated that the elastic fibers’ hierarchical buckling phenomenon could be generated through triggering of superposition of buckling at two modes at the same time when the transverse confinement was satisfied c ² = π ⁴ m ² v ² . In addition, the hierarchical buckling phenomenon becomes more significant with the introduction of higher-order buckling modes. Moreover, hierarchical buckling can be effectively controlled by changing the Young’s modulus ratio (e = E _m L ⁴/E _f I _f ) between the matrix and the elastic fibers.

The analyses are in good qualitative agreement with the experimental results. However, a accurate quantitative comparison is challenging due to the important role played by imperfections and measurement uncertainties. We expect to take the following measures to improve the accuracy of the calculations in our subsequent work: Firstly, refining the computational grid to increase its density enables better capture of local details and structural deformations. Secondly, optimizing the accuracy of boundary conditions is crucial for precise calculation results. Additionally, selecting a more accurate material model that effectively describes the structural response during loading is essential. By combining these methods, the accuracy of the calculations can be significantly improved.

In addition, we have also found that the cross-sectional geometric characteristics of the fibers influence in the buckling configuration. In Figure 8, we plotted the relationship between dimensionless critical buckling loads for different b/a ratios, based on the theoretical results presented in this paper. For elliptical cross-sections elastic fibers, the ratio b/a of the short and long semi-axes of the elliptical cross-sections also affects the critical buckling load and buckling configuration of the elastic fiber. The critical buckling load increased as the cross-sections approached a circular shape. We speculate that when the fiber cross-sections are sufficiently close to a circular shape, the fibers undergo a transition from hierarchical buckling to 3D helical buckling [16,17]. However, the ratio of the short to the long axis of the elliptical cross-section mainly affects the buckling of the elastic fibers in the form of hierarchical buckling or 3D helical buckling. It is only a secondary factor affecting the modal superposition of the fibers. With the focus of this study being only the explanation of the hierarchical buckling phenomenon, the specific transition conditions were not thoroughly examined.

In summary, the analysis we proposed can be employed to predict the hierarchical buckling phenomenon observed in the experiment. The main reason in influencing the buckling phenomenon is the interaction of the matrix with the elastic fiber. By controlling the stiffness of the external matrix, the degree of hierarchical buckling of the elastic fibers could be tailored. It is shown that for a particular spring stiffness C, two different buckling modes can be simultaneously triggered at the onset of buckling, and the formation of hierarchical buckling is a result of the joint action of the two modes. This unique buckling phenomenon can motivate the development of new path for novel flexible electronic devices. Moreover, the present work provides theoretical guidance for the production, design and application of elastic/hyper-elastic fibers for the future use of buckling mechanisms to produce devices with controllable and sophisticated hierarchical structures.