Emergent perversions in the buckling of heterogeneous elastic strips

Abstract

A perversion in an otherwise uniform helical structure, such as a climbing plant tendril, refers to a kink that connects two helices with opposite chiralities. Such singularity structures are widely seen in natural and artificial mechanical systems, and they provide the fundamental mechanism of helical symmetry breaking. However, it is still not clear how perversions arise in various helical structures and which universal principles govern them. As such, a heterogeneous elastic bistrip system provides an excellent model to address these questions. Here, we investigate intrinsic perversion properties which are independent of strip shapes. This study reveals the rich physics of perversions in the 3D elastic system, including the condensation of strain energy over perversions during their formation, the repulsive nature of the perversion-perversion interaction, and the coalescence of perversions that finally leads to a linear defect structure. This study may have implications for understanding relevant biological motifs and for use of perversions as energy storers in the design of micromuscles and soft robotics.S pontaneous symmetry breaking provides a unifying conceptual understanding of emergent ordered structures arising in various condensed matters (1). In an elastic medium, which is one of the simplest organizations of matter, symmetry-breaking instabilities via buckling can lead to extraordinarily rich patterns and generate a wealth of shapes at multiple length scales that can be exploited in many scientific disciplines (2). A prototype of elastic buckling is the Euler instability of a homogeneous elastic rod under uniaxial compression at the ends that finally breaks the rotational symmetry (3). Introduction of extra structures in an elastic medium like mechanical heterogeneities (4), nonlinearity of materials (2), geometric asymmetry (5), or intrinsic curvature (6) provides new dimensions that can produce even richer buckling modes, including helices and perversions (6, 7), wavy structures (8), regular networks of ridges (9), and even selfsimilar fractal patterns (2, 10). Of these emergent symmetry broken structures, the helical shapes are of particular interest due to their ubiquitousness in nature and the strong connection with biological motifs, as noticed by Darwin in his 1875 book describing the curl of plant tendrils (11). Remarkably, biological helical structures permeate over several length scales from the developed helical valve on opening seed pods (12), to the regular chiral structures in the flagella of bacteria (13), the spiral ramps of rough endoplasmic reticulum (14), and the chromosome of Escherichia coli (15, 16).The proliferation of perversions in an otherwise uniform helical structure can further break the helical symmetry (Fig. 1A shows a typical perversion in the helix) (4, 6, 17). Here, a perversion refers to a kink that connects two helices with opposite chiralities. Therefore, perversions belong to a large class of fundamental defects in systems with discrete symmetry which have the n...

Modeling the formation of double rolls from heterogeneously patterned gels

“…This violation arises because of the axisymmetric assumption, Eq. (10), that prohibits shear strains, and thereby, the dependence of the displacements on the θ-coordinates. Second, because the minimizing configuration acts to restore the prescribed rest length on each strip, and thereby to relax the in-plane stresses, we would expect σ θθ to obtain a maximum value at z = L and then decay to zero away from that maximum.…”

“…Morphogenesis is the process by which biological organisms develop their shape. This process typically encompasses the growth of new tissue [1][2][3][4], which enables such events as: the reshaping of epithelial sheets [5,6], the development of plants and leafs [7][8][9][10], and the emergence of surface morphologies on living organs [11][12][13]. The growth of biological tissue involves the addition of mass to an evolving structure; if the structure is formed from an elastic material, the growth commonly drives a two-dimensional (2D) shape to transition into a three-dimensional (3D) configuration [1,14].…”

Fabricating 3D Structures by Combining 2D Printing and Relaxation of Strain

“…Bilayers may also form 3D structures when materials with “mismatching” strain or internal stress are bonded to one another. For example, bonding a stretched elastomeric material to an unstretched elastomeric material will cause bending in the resulting bilayer due to residual stresses . This concept has also been applied at the nanoscale: when two thin layers of materials with different lattice constants are deposited on a substrate, the bilayer “rolls” into a nanotube upon release from the substrate …”

“…In the final state, the shorter strip is under tensional stress and the longer strip under compressive stress. Although hyperelastic bistrips have been studied extensively, all previously reported hyperelastic bistrip systems have been fabricated by gluing together two preformed elastomers . We fabricated the bistrips by stretching the shorter strip, and then we extruded uncured elastomer onto the stretched strip (Figure b), which, after curing, forms the longer strip—a process that is compatible with 2D printing techniques.…”

Flapping, swirling and flipping: Non-linear dynamics of pre-stressed active filaments

“…In many important contexts however, rods and filaments that are subject to deforming forces and torques start off from shapes that are neither planar nor stress-free. Such prestressed and twisted three-dimensional shapes abound in nature at all scales; examples include buckling growing tendrils [58,59], the curling of ropes hitting a surface [60], torsionally constrained DNA looping mediated by protein binding [61,62], self-contact driven DNA buckling [63,64], and relaxation of DNA supercoils by topoisomerases [65]. As a typical example, we note that a pre-stressed filament clamped at both ends -being both pre-stressed and strongly constrained at the boundaries -is expected to have different dynamics than stress-free cantilevers.…”