Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: Non-linear model and stability analysis

The stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end is analyzed. The beam is herein modeled as a non-linear Cosserat rod model. Non-linear, partial integro-differential equations of motion are derived expanded up to cubic terms in the transversal displacement and torsional angle of the beam. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Interaction between conservative and non-conservative loads with respect to the stability problem is discussed. The critical spectral properties are derived and the corresponding critical eigenspace is evaluated.     

Surface instability of a semi-infinite anisotropic elastic body under surface van der Waals forces

We investigate the surface instability of an anisotropic elastic half-plane subjected to surface van der Waals forces due to the influence of another rigid contactor by means of the Stroh formalism. It is observed that the surface of a generally anisotropic elastic half-plane subjected to van der Waals forces from another rigid flat is always unstable. The wave number of the surface wrinkling is only reliant on the positive M₂₂ component of the 3 × 3 surface admittance tensor M, the van der Waals interaction coefficient β and the surface energy γ of the elastic half-plane. The decay rate of surface perturbation along the direction normal to the surface of the anisotropic half-plane is different from the wave number, a phenomenon different from that observed for an isotropic half-plane.     

Contact with friction of a rigid cylinder with an elastic half-space

An exact solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space is presented. The corresponding boundary-value problem is formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation is constructed using the Wiener-Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full stick between the cylinder and half-space.     

Beading instability in soft cylindrical gels with capillary energy: Weakly non-linear analysis and numerical simulations

Soft cylindrical gels can develop a long-wavelength peristaltic pattern driven by a competition between surface tension and bulk elastic energy. In contrast to the Rayleigh–Plateau instability for viscous fluids, the macroscopic shape in soft solids evolves toward a stable beading, which strongly differs from the buckling arising in compressed elastic cylinders.This work proposes a novel theoretical and numerical approach for studying the onset and the non-linear development of the elasto-capillary beading in soft cylinders, made of neo-Hookean hyperelastic material with capillary energy at the free surface, subjected to axial stretch. Both a theoretical study, deriving the linear and the weakly non-linear stability analyses for the problem, and numerical simulations, investigating the fully non-linear evolution of the beaded morphology, are performed. The theoretical results prove that an axial elongation can not only favour the onset of beading, but also determine the nature of the elastic bifurcation. The fully non-linear phase diagrams of the beading are also derived from finite element numerical simulations, showing two peculiar morphological transitions when varying either the axial stretch or the material properties of the gel. Since the bifurcation is found to be subcritical for very slender cylinders, an imperfection sensitivity analysis is finally performed. In this case, it is shown that a surface sinusoidal imperfection can resonate with the corresponding marginally stable solution, thus selecting the emerging beading wavelength.In conclusion, the results of this study provide novel guidelines for controlling the beaded morphology in different experimental conditions, with important applications in micro-fabrication techniques, such as electrospun fibres.     

Chaos of liquid surface waves in a vessel under vertical excitation with slowly modulated amplitude

Free surface waves in a cylinder of liquid under vertical excitation with slowly modulated amplitude are investigated in the current paper. It is shown by both theoretical analysis and numerical simulation that chaos may occur even for a single mode with modulation which can be used to explain Gollub and Meyer's experiment. The implied resonant mechanism accounting for this phenomenon is further elucidated.     

Electro-mechanical coupling wave propagating in a locally resonant piezoelectric/elastic phononic crystal nanobeam with surface effects

The model of a spring-mass resonator periodically attached to a piezoelectric/elastic phononic crystal(PC) nanobeam with surface effects is proposed, and the corresponding calculation method of the band structures is formulized and displayed by introducing the Euler beam theory and the surface piezoelectricity theory to the plane wave expansion(PWE) method. In order to reveal the unique wave propagation characteristics of such a model, the band structures of locally resonant(LR) elastic PC Eul...     

O(2)‐symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

The steady, axisymmetric laminar flow of a Newtonian fluid past a centrally‐located sphere in a pipe first loses stability with increasing flow rate at a steady O(2)‐symmetry breaking bifurcation point. Using group theoretic results, a number of authors have suggested techniques for locating singularities in branches of solutions that are invariant with respect to the symmetries of an arbitrary group. These arguments are presented for the O(2)‐symmetry encountered here and their implementation for O(2)‐symmetric problems is discussed. In particular, how a bifurcation point may first be detected and then accurately located using an ‘extended system’ is described. Also shown is how to decide numerically if the bifurcating branch is subcritical or supercritical. The numerical solutions were obtained using the finite element code ENTWIFE. This has enabled the computation of the symmetry breaking bifurcation point for a range of sphere‐to‐pipe diameter ratios. A wire along the centerline of the pipe downstream of the sphere is also introduced, and its effect on the critical Reynolds number is shown to be small. Copyright © 2000 John Wiley & Sons, Ltd.     

Fracture of a body with a periodic set of coaxial cracks under forces directed along them: an axisymmetric problem

Linearized solid mechanics is used to solve an axisymmetric problem for an infinite body with a periodic set of coaxial cracks. Two nonclassical fracture mechanisms are considered: fracture of a body with initial stresses acting in parallel to crack planes and fracture of materials compressed along cracks. Numerical results are obtained for highly elastic materials described by the Bartenev–Khazanovich, Treloar, and harmonic elastic potentials. The dependence of the fracture parameters on the loading conditions, the physical and mechanical characteristics of the material, and the geometrical parameters is analyzed Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 3–18, February 2009.     

Adhesive contact of an elastic semi-infinite solid with a rigid rough surface: Strength of adhesion and contact instabilities

The effect of adhesion on the contact behavior of elastic rough surfaces is examined within the framework of the multi-asperity contact model of Greenwood and Williamson (1966), known as the GW model. Adhesive surface interaction is modeled by nonlinear springs with a force–displacement relation governed by the Lennard–Jones (LJ) potential. Constitutive models are presented for contact systems characterized by low and high Tabor parameters, exhibiting continuous (stable) and discontinuous (unstable) surface approach, respectively. Constitutive contact relations are obtained by integrating the force–distance relation derived from the LJ potential with a finite element analysis of single-asperity adhesive contact. These constitutive relations are then incorporated into the GW model, and the interfacial force and contact area of rough surfaces are numerically determined. The development of attractive and repulsive forces at the contact interface and the occurrence of instantaneous surface contact (jump-in instability) yield a three-stage evolution of the contact area. It is shown that the adhesion parameter introduced by Fuller and Tabor (1975) governs the strength of adhesion of contact systems with a high Tabor parameter, whereas the strength of adhesion of contact systems with a low Tabor parameter is characterized by a new adhesion parameter, defined as the ratio of the surface roughness to the equilibrium interatomic distance. Applicable ranges of aforementioned adhesion parameters are interpreted in terms of the effective surface separation, obtained as the sum of the effective distance range of the adhesion force and the elastic deformation induced by adhesion. Adhesive strength of rough surfaces in the entire range of the Tabor parameter is discussed in terms of a generalized adhesion parameter, defined as the ratio of the surface roughness to the effective surface separation.     

Numerical analysis of stress distribution in the upper arm tissues under an inflatable cuff: Implications for noninvasive blood pressure measurement

An inflatable cuff wrapped around the upper arm is widely used in noninvasive blood pressure measurement. However, the mechanical interaction between cuff and arm tissues, a factor that potentially affects the accuracy of noninvasive blood pressure measurement, remains rarely addressed. In the present study, finite element (FE) mod-els were constructed to quantify intra-arm stresses generated by cuff compression, aiming to provide some theoretical evi-dence for identifying factors of importance for blood pressure measurement or explaining clinical observations. Obtained results showed that the simulated tissue stresses were highly sensitive to the distribution of cuff pressure on the arm sur-face and the contact condition between muscle and bone. In contrast, the magnitude of cuff pressure and small variations in elastic properties of arm soft tissues had little influence on the efficiency of pressure transmission in arm tissues. In par-ticular, it was found that a thickened subcutaneous fat layer in obese subjects significantly reduced the effective pres-sure transmitted to the brachial artery, which may explain why blood pressure overestimation occurs more frequently in obese subjects in noninvasive blood pressure measurement.