Kink Instability of a Highly Deformable Elastic Cylinder (original) (raw)
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Bending instabilities of soft biological tissues
International Journal of Solids and Structures, 2009
Rubber components and soft biological tissues are often subjected to large bending deformations while ''in service". The circumferential line elements on the inner face of a bent block can contract up to a certain critical stretch ratio k cr (say) before bifurcation occurs and axial creases appear. For several models used to describe rubber, it is found that k cr ¼ 0:56, allowing for a 44% contraction. For models used to describe arteries it is found, somewhat surprisingly, that the strain-stiffening effect promotes instability. For example, the models used for the artery of a seventy-year old human predict that k cr ¼ 0:73, allowing only for a 27% contraction. Tensile experiments conducted on pig skin indicate that bending instabilities should occur even earlier there.
Mechanical Instability of Thin Elastic Rods
Mechanical instability of elastic rods has been subjected to extensive investigations and demonstrated fundamental roles in cytoskeletal mechanics and morphogenesis. Utilizing this instability also has great potential in engineering applications such as stretchable electronics. Here in this review, the fundamental theory underlying twisting and buckling instability of thin elastic rods is described. We then bridge together recent progresses in both theoretical and experimental studies on the topic. The promises and challenges in future studies of large deformation and buckling instability of thin rods are also discussed.
Bending instabilities of soft tissues
Rubber components and soft tissues are often subjected to large bending deformations "in service". The circumferential line elements on the inner face of a bent block can contract up to a certain critical stretch ratio lambda_cr (say) before bifurcation occurs and axial creases appear. For several models used to describe rubber, it is found that lambda_cr = 0.56, allowing for a 44% contraction. For models used to describe arteries it is found, somewhat surprisingly, that the strain-stiffening effect promotes instability. For example, the models used for the artery of a 70 year old human predict that lambda_cr = 0.73, allowing only for a 27% contraction. Tensile experiments conducted on pig skin indicate that bending instabilities should occur even earlier there.
Buckling of an elastic rod embedded on an elastomeric matrix: planar vs. non-planar configurations
Soft Matter, 2014
Please note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal's standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains.
Analytical Solutions of Polymeric Gel Structures under Buckling and Wrinkle
One of the unique property of polymeric gel is that the volume and shape of gel can dramatically change even at mild variation of external stimuli. Though a variety of instability patterns of slender and thin film gel structures due to swelling have been observed in various experimental studies, many are not well understood. This paper presents analytical solutions of swelling induced instability of various slender and thin film gel structures. We adopt the well developed constitutive relation of inhomogeneous field theory of a polymeric network in equilibrium with a solvent and mechanical load or constraint with the incremental modulus concept for slender beam and thin film gel structures. The formulas of buckling and winkle conditions and critical stress values are derived for slender beam and thin film gel structures under swelling induced instability using nonlinear buckling theories of beam and thin film structures. For slender beam structure, we construct the stability diagram with the distinct stable and unstable zones. The critical slenderness ratio and corresponding critical stresses are provided for different dimensionless material parameters. For thin film gel structures, we consider the thin film gel on an elastic foundation with different stiffness. The analytical solutions of critical stress and corresponding wrinkle wavelength, as well as buckling condition (or critical chemical potential) are given. These analytical solutions will provide a guideline for gel structure design used in polymeric gels based MEMS and NEMS structures such as sensors and actuators. More importantly, the work provides a theoretical foundation of gel structure buckling and wrinkle, which instability phenomena are different from normal engineering material buckling.
On Extension and Torsion of Strain-Stiffening Rubber-Like Elastic Circular Cylinders
Journal of Elasticity, 2008
This paper is concerned with investigation of the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic nonlinearly elastic materials. Our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two particular phenomenological constitutive models for such materials that reflect limiting chain extensibility at the molecular level. The axial stretch γ and twist that can be sustained in cylinders composed of such materials are shown to be constrained in a coupled fashion. It is shown that, in the absence of an additional axial force, a transition value γ = γ t of the axial stretch exists such that for γ < γ t , the stretched cylinder tends to elongate on twisting whereas for γ > γ t , the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models such as the Mooney-Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. We also obtain results for materials modeled by the well-known exponential strain-energy widely used in biomechanics applications. This model reflects a strain-stiffening that is less abrupt than that for the limiting chain extensibility models. Surprisingly, it turns out that the results in this case are somewhat more complicated. For a fixed stiffening parameter, provided that the stretch is sufficiently small, the stretched bar always tends to elongate on twisting in the absence of an additional axial force. However, for sufficiently large stretch, the cylinder tends to shorten on undergoing sufficiently small twist but then tends to elongate on further twisting. These results are of interest in view of the widespread use of exponential models in the context of the mechanics of soft biological tissues. The special case of pure torsion is also briefly considered. In this case, the resultant axial force required to maintain pure torsion is compressive for all the models discussed here. In the absence of such a force, the bar would elongate on twisting reflecting the celebrated Poynting effect.
Buckling of rods with spontaneous twist and curvature
We analyze stability of a thin inextensible elastic rod which has non-vanishing spontaneous generalized torsions in its stress-free state. Two classical problems are studied, both involving spontaneously twisted rods: a rectilinear rod compressed by axial forces, and a planar circular ring subjected to uniform radial pressure on its outer perimeter. It is demonstrated that while spontaneous twist stabilizes a rectilinear rod against buckling, its presence has an opposite effect on a closed ring.
Necking, beading, and bulging in soft elastic cylinders
Journal of the mechanics and physics of solids, 2020
Due to surface tension, a beading instability takes place in a long enough fluid column that results in the breakup of the column and the formation of smaller packets with the same overall volume but a smaller surface area. Similarly, a soft elastic cylinder under axial stretching can develop an instability if the surface tension is large enough. This instability occurs when the axial force reaches a maximum with fixed surface tension or the surface tension reaches a maximum with fixed axial force. However, unlike the situation in fluids where the instability develops with a finite wavelength, for a hyperelastic solid cylinder that is subjected to the combined action of surface tension and axial stretching, a linear bifurcation analysis predicts that the critical wavelength is infinite. We show, both theoretically and numerically, that a localized solution can bifurcate subcritically from the uniform solution, but the character of the resulting bifurcation depends on the loading path. For fixed axial stretch and variable surface tension, the localized solution corresponds to a bulge or a depression, beading or necking, depending on whether the axial stretch is greater than a certain threshold value that is dependent on the material model and is equal to 3 √ 2 when the material is neo-Hookean. At this single threshold value, localized solutions cease to exist and the bifurcation becomes exceptionally supercritical. For either fixed surface tension and variable axial force, or fixed axial force and variable surface tension, the localized solution corresponds to a depression or a bulge, respectively. We explain why the bifurcation diagrams in previous numerical and experimental studies look as if the bifurcation were supercritical although it was not meant to. Our analysis shows that beading in fluids and solids are fundamentally different. Fluid beading resulting from the Plateau-Rayleigh instability follows a supercritical linear instability whereas solid beading in general is a subcritical localized instability akin to phase transition.
Modern Physics Letters B, 2007
We derive the shape equations in terms of Euler angles for a uniform elastic rod with isotropic bending rigidity and spontaneous curvature, and study within this model the elasticity and stability of a helical filament under uniaxial force and torque. We find that due to the special requirements on the boundary conditions, a static slightly distorted helix cannot exist in this system except in some special cases. We show analytically that the extension of a helix may undergo a one-step sharp transition. This agrees quantitatively with experimental observations for a stretched helix in a chemically-defined lipid concentrate (CDLC). We predict further that under twisting, the extension of a helix in CDLC may also exhibit similar behavior. We find that a negative twist tends to destabilize a helix.
Surface wrinkling and folding of core-shell soft cylinders
Soft Matter, 2012
Constrained swelling or shrinkage of such soft materials as elastomers, polymeric hydrogels, and biological tissues can create various surface patterns through surface wrinkling and subsequent morphological evolution. Here we study, both theoretically and numerically, the swelling-driven surface wrinkling and pattern evolution of cylindrical elastomers with core-shell structure. The results demonstrate that the system may buckle into different morphologies under different geometric and material parameters. When the swelling of the shell or the shrinkage of the core reaches a threshold, the cylindrical surface will first buckle into a periodically buckling pattern. With further swelling/ shrinkage, wrinkle-to-fold transition may occur, rendering a period-doubling surface topography. The energetic mechanisms underlying this process of wrinkling pattern evolution are analyzed. This study not only benefits the understanding of the morphogenesis in soft materials and tissues (e.g., tumors), but also opens a new avenue for the fabrication of multi-periodic or aperiodic patterns on curved surfaces through self-organization.