In recent years, there has been a burgeoning interest in elasto-capillary instability in soft solids. Given that most soft matter is compressible, elucidating the effect of bulk compressibility on the post-bifurcation behaviors of Plateau–Rayleigh (PR) instability in solid-like materials is essential, yet this issue remains unresolved. In this paper, we investigate elasto-capillary necking/bulging instabilities in a compressible soft hyperelastic cylinder subjected to axial force and strain-dependent surface stresses. A one-dimensional (1D) reduced model is employed to derive the fully nonlinear post-bifurcation solutions. The validity of the 1D asymptotic results is confirmed by three-dimensional (3D) finite element (FE) simulations. Based on the bifurcation and post-bifurcation results across different loading scenarios, we systematically analyze and elucidate the role of bulk compressibility in the complex bulk–surface interactions. The thorough systematic analyses reveal that bulk compressibility not only plays a pivotal role in mediating bulk–surface deformation but also dictates the sensitivity of instability behaviors to surface parameters, axial force, and geometrical size. Specific guidelines to modulate elasto-capillary necking/bulging in solid-like materials through the rational utilization of bulk compressibility are provided. These findings are beneficial for the design and application of soft solid materials and structures.
StyleRWJagotaAHuiCY, et al. Elastocapillarity: surface tension and the mechanics of soft solids. Annu Rev Condens Matter Phys2017; 8(1): 99–118.
2.
MatsuoESTanakaT. Patterns in shrinking gels. Nature1992; 358(6386): 482–485.
3.
KurokiYSekimotoK. Arrested-volume phase transition of a gel around a hole. Europhysics Letters1994; 26(3): 227.
4.
MeiHLandisCMHuangR. Concomitant wrinkling and buckle-delamination of elastic thin films on compliant substrates. Mech Mater2011; 43(11): 627–642.
5.
SeifiSParkHS. Computational modeling of electro-elasto-capillary phenomena in dielectric elastomers. Int J Solids Struct2016; 87: 236–244.
6.
ChangCCWangZShengYJ, et al. Nanostructure collapse by elasto-capillary instability. Soft Matter2014; 10(42): 8542–8547.
7.
MoraSPhouTFromentalJM, et al. Capillarity driven instability of a soft solid. Phys Rev Lett2010; 105(21): 214301.
8.
XuanCBigginsJ. Plateau–Rayleigh instability in solids is a simple phase separation. Phys Rev E2017; 95(5): 053106.
9.
TaffetaniMCiarlettaP. Beading instability in soft cylindrical gels with capillary energy: weakly non-linear analysis and numerical simulations. J Mech Phys Solids2015; 81: 91–120.
10.
FuYJinLGorielyA. Necking, beading, and bulging in soft elastic cylinders. J Mech Phys Solids2021; 147: 104250.
11.
EmeryDFuY. Post-bifurcation behaviour of elasto-capillary necking and bulging in soft tubes. Proc R Soc A2021; 477(2254): 20210311.
12.
WangMFuY. Necking of a hyperelastic solid cylinder under axial stretching: evaluation of the infinite-length approximation. Int J Eng Sci2021; 159: 103432.
13.
YeYLiuYAlthobaitiA, et al. Localized bulging in an inflated bilayer tube of arbitrary thickness: effects of the stiffness ratio and constitutive model. Int J Solids Struct2019; 176: 173–184.
14.
LiuYYangLXieYX. Inflation-induced bulge initiation and evolution in graded cylindrical tubes of arbitrary thickness. Mech Mater2023; 178: 104561.
15.
LiuYDorfmannL. Localized necking and bulging of finitely deformed residually stressed solid cylinder. Math Mech Solids2024; 29(6): 1153–1175.
16.
LiuYYuXDorfmannL. Reduced model and nonlinear analysis of localized instabilities of residually stressed cylinders under axial stretch. Math Mech Solids2024; 29(9): 1879–1899.
17.
YuXFuY. On the incremental equations in surface elasticity. Math Mech Solids2025; 30(2): 470–489.
18.
HenannDLBertoldiK. Modeling of elasto-capillary phenomena. Soft Matter2014; 10(5): 709–717.
19.
YanYLiMZhaoZL, et al. An energy method for the bifurcation analysis of necking. Extreme Mech Lett2022; 55: 101793.
20.
ZhangXYangYXuF. A combined finite-discrete element model for elasto-capillary phenomena. Int J Mech Sci2023; 251: 108305.
21.
LestringantCAudolyB. A one-dimensional model for elasto-capillary necking. Proc R Soc A2020; 476(2240): 20200337.
22.
XuQJensenKEBoltyanskiyR, et al. Direct measurement of strain-dependent solid surface stress. Nat Commun2017; 8(1): 555.
23.
BainNJagotaASmith-MannschottK, et al. Surface tension and the strain-dependent topography of soft solids. Phys Rev Lett2021; 127(20): 208001.
HeydenSVlahovskaPMDufresneER. A robust method for quantification of surface elasticity in soft solids. J Mech Phys Solids2022; 161: 104786.
26.
BakilerDAJaviliADortdivanliogluB. Surface elasticity and area incompressibility regulate fiber beading instability. J Mech Phys Solids2023; 176: 105298.
27.
ZhuPLiDYuX, et al. Surface elasticity effect on Plateau–Rayleigh instability in soft solids. Int J Mech Sci2025; 305: 110750.
28.
AudolyBHutchinsonJW. Analysis of necking based on a one-dimensional model. J Mech Phys Solids2016; 97: 68–91.
29.
LestringantCAudolyB. A one-dimensional model for elasto-capillary necking. Proc R Soc A2020; 476(2240): 20200337.
30.
LestringantCAudolyB. Asymptotically exact strain-gradient models for nonlinear slender elastic structures: a systematic derivation method. J Mech Phys Solids2020; 136: 103730.
31.
YuXFuY. A one-dimensional model for axisymmetric deformations of an inflated hyperelastic tube of finite wall thickness. J Mech Phys Solids2023; 175: 105276.
32.
YuXChenX. An asymptotically consistent morphoelastic shell model for compressible biological structures with finite-strain deformations. J Mech Phys Solids2024; 191: 105768.
33.
YuXFuY. Analysis of axisymmetric necking of a circular dielectric membrane based on a one-dimensional model. J Mech Phys Solids2025; 198: 106071.
34.
CarewTEVaishnavRNPatelDJ. Compressibility of the arterial wall. Circ Res1968; 23(1): 61–68.
35.
JurvelinJBuschmannMDHunzikerE. Optical and mechanical determination of Poisson’s ratio of adult bovine humeral articular cartilage. J Biomech1997; 30(3): 235–241.
36.
FattI. Dynamics of water transport in the corneal stroma. Exp Eye Res1968; 7(3): 402–412.
37.
MowVCHolmesMHLaiWM. Fluid transport and mechanical properties of articular cartilage: a review. J Biomech1984; 17(5): 377–394.
38.
KyriacouSKMohamedAMillerK, et al. Brain mechanics for neurosurgery: modeling issues. Biomech Model Mechanobiol2002; 1(2): 151–164.
39.
MarshJLBuckwalterJGelbermanR, et al. Articular fractures: does an anatomic reduction really change the result?JBJS2002; 84(7): 1259–1271.
40.
YamadaH. Strength of biological materials. Williams & Wilkins, 1970.
41.
WahlstenAPensalfiniMStracuzziA, et al. On the compressibility and poroelasticity of human and murine skin. Biomech Model Mechanobiol2019; 18(4): 1079–1093.
42.
MoranRSmithJHGarcíaJJ. Fitted hyperelastic parameters for human brain tissue from reported tension, compression, and shear tests. J Biomech2014; 47(15): 3762–3766.
43.
LiYHuZLiC. New method for measuring Poisson’s ratio in polymer gels. J Appl Polym Sci1993; 50(6): 1107–1111.
44.
ChippadaUYurkeBLangranaNA. Simultaneous determination of Young’s modulus, shear modulus, and Poisson’s ratio of soft hydrogels. J Mater Res2010; 25(3): 545–555.
45.
CappelloJd’HerbemontVLindnerA, et al. Microfluidic in-situ measurement of Poisson’s ratio of hydrogels. Micromachines2020; 11(3): 318.
46.
BaiYBiSWangW, et al. Biocompatible, stretchable, and compressible cellulose/mxene hydrogel for strain sensor and electromagnetic interference shielding. Soft Mater2022; 20(4): 444–454.
47.
BriotNChagnonGConnessonN, et al. Experimental device to measure the compressibility coefficient of soft materials. PLoS ONE2025; 20(5): e0322716.
48.
DortdivanliogluBJaviliA. Plateau–Rayleigh instability of soft elastic solids. Effect of compressibility on pre and post bifurcation behavior. Extreme Mech Lett2022; 55: 101797.
49.
EmeryD. Elasto-capillary necking, bulging and Maxwell states in soft compressible cylinders. Int J Nonlin Mech2023; 148: 104276.
50.
GurtinMEMurdochIA. A continuum theory of elastic material surfaces. Arch Ration Mech Anal1975; 57(4): 291–323.
51.
SteigmannDJOgdenRW. Elastic surface-substrate interactions. Proc R Soc A Math Phys Eng Sci1999; 455(1982): 437–474.
52.
DuanHWangJKarihalooBL. Theory of elasticity at the nanoscale. Adv Appl Mech2009; 42: 1–68.
53.
LiuZJagotaAHuiCY. Modeling of surface mechanical behaviors of soft elastic solids: theory and examples. Soft Matter2020; 16(29): 6875–6889.
54.
GadAIGaoXL. An extended Hill’s lemma for non-Cauchy continua based on the modified couple stress and surface elasticity theories. Math Mech Solids2023; 28(7): 1652–1670.
