The delamination of a growing elastic sheet with adhesion

We study the onset of delamination blisters in a growing elastic sheet adhered to a flat stiff substrate. When the ends of the sheet are kept fixed, its growth causes residual stresses that lead to delamination. This instability can be viewed as a discontinuous buckling between the complete adhered solution and the buckled solution. We provide an analytic expression for the critical deformation at which the instability occurs. We show that the critical threshold scales with a single dimensionless parameter that comprises information from the geometry of the sheet, the mechanical parameters of material and the adhesive features of the substrate.     

The spreading of nonlinear elastic surface waves

A theory for the lateral spreading of a beam of nonlinear surface acoustic waves across the surface of an arbitrary, homogeneous, elastic half-space is developed. The resulting evolution equation generalizes that obtained for uni-directional waves by replacing an ordinary derivative by a diffusion operator of Schrödinger type. The coefficients arising in the evolution equation are related to partial derivatives of the dispersion relation for linearized surface waves on the half space. Details are given for isotropic materials and for two special cases of beams travelling along axes of high elastic symmetry.     

The effect of inextensibility on elastic surface waves

It is shown that when the complications associated with material anisotropy are absent a simple exact analysis can be given of the effect of unidirectional inextensibility on the propagation of surface waves in a semi-infinite elastic body. Provided that the direction of inextensibility e is not orthogonal to either m or m Λ n (m being the outward unit normal to the traction-free boundary of the body and n the wave normal), a unique surface wave exists with displacement everywhere orthogonal to e. The surface-wave solution is assembled from inhomogeneous plane waves in the usual manner, but a novel feature is the presence of a degenerate wave producing no displacement yet perturbing sinusoidally the tension in the inextensible fibres. When the aforementioned provisos are not met the surface wave either degenerates continuously into a shear wave (when $\text{(}\text{m}\text{Λ}\text{n}\text{)·}\text{e}\text{= 0,}\text{m}\text{·}\text{e}\text{≠ 0)}$, ceases to exist (when $\text{m}\text{·}\text{e}\text{= 0,}\text{n}\text{·}\text{e}\text{≠ 0)}$, or merges smoothly into a Rayleigh wave (when $\text{(}\text{e}\text{=±}\text{m}\text{Λ}\text{n}$, the inextensibility constraint then being inoperative).     

The strain energy function of a metal subjected to finite elastic volume change

The form of the strain energy function for a metal subjected to finite elastic volume change and infinitesimal elastic shearing strain is considered.     

Surface energy and adhesion energy of elastic bodies

We propose a method for computing the surface energy and the adhesion energy of elastic bodies in adhesion state. The method is based on a version of the elastic continuum model originating from the assumption about the multiparticle potential nonlocal interaction of infinitesimal particles of the medium.     

The elastic sphere under arbitrary concentrated surface loads

The elastostatic problem of a sphere subject to a concentrated surface load of arbitrary direction which is equilibrated in a very simple manner by a distribution of surface tractions is solved. Particular attention is placed in the analysis of the singularity at the point of application of the concentrated load.The solution obtained provides a means for reducing problems pertaining to a sphere under arbitrary concentrated and distributed loads to a regular second boundary-value problem for the sphere. For illustration the problem of a sphere acted by two equal, opposite and collinear loads applied at two arbitrary surface points is treated.The paper also contains an exposition and an essential extension of an integration scheme developed by Almansi. Thus, an explicit integral representation of the displacements in an elastic sphere in terms of a vector valued harmonic potential which coincides on the surface with the tractions is obtained.

---

Résumé On présente la résolution du problème élastostatique d'une sphère soumise en surface à une charge ponctuelle de direction arbitraire équilibrée par une distribution trés simple de contraintes superficielles. On s'attache plus spécialement à l'analyse de la singularité au point d'application de la charge ponctuelle.La solution obtenue permet de réduire les problèmes d'une sphère soumise de manière quelconque à des charges ponctuelles et réparties, à un problème avec conditions aux limites régulières. A titre d'exemple, on traite le problème d'une sphère soumise à deux forces ponctuelles, co linéaires, égales et de sens opposé, appliquées en deux points arbitraires de la surface.L'article contient également une exposition et une extension d'un schéma d'intégration développé par Almansi. On obtient ainsi une représentation explicite, sous forme d'intégrale, des déplacements dans une sphère élastique, en termes de potentiel vecteur qui coïncide, à la surface, avec le champ de contraintes.

     

Near-tip fields for a crack growing along an elastic perfectly plastic interface

The stress and deformation fields near the tip of an anti-plane crack growing quasi-statically along an interface of elastic perfectly plastic materials are given in this paper. A family of solutions for the growing crack fields is found covering all admissible crack line shear stress ratios. The project supported by the National Natural Science Foundation of China     

On the surface instability of a highly elastic half-space

The phenomenon of surface instability of an isotropic half-space under biaxial plane stress is studied for compressible elastic materials in finite strain. Euler's method is used to derive the general form of the stability criterion, and analytical details are exhibited by special application to the class of hyperelastic Hadamard materials in two complementary cases: (i) the full solution is derived for the compressible, neo-Hookean members, and (ii) the plane deformation solution is provided for every isotropic, elastic material and specific results are presented for the full Hadamard class. Results appropriate to incompressible Mooney-Rivlin materials are herein obtained as special limit cases. Several theorems are established and some of the conclusions are illustrated graphically.     

Eigenoscillations of an elastic body with a rough surface

Explicit presentations for the initial terms of the asymptotic solution of the spectral problem of the elasticity theory in a plane region with a rapidly oscillating boundary are obtained. Based on asymptotic formulas, two methods for problem modeling are proposed: with the use of Wenzel’s boundary conditions and with the use of the principle of a smooth image of a singularly perturbed boundary. Various approaches to justification of asymptotic presentations are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 103–114, November–December, 2007.     

The energy release rate for hygrothermal coupling elastic materials

In the fracture problems of hydrophilic elastic materials under coupling effects of heat conduction, moisture diffusion and mechanical deformation, the conventional J-integral is no longer path independent. The value of J is unequal to the energy release rate in hygrothermal coupling cases. In the present paper, we derived a general form of the energy release rate for hygrothermal fracture problems of the hydrophilic elastic materials on the basis of energy balance equation in cracked areas. By introducing the constitutive relations and the essential equations of irreversible thermodynamics, a specific expression of the energy release rate was obtained, and the expression can be reformmulated as path independent integrals, which is equivalent to the energy release rate of the fracture body. The path independence of the integrals is then verified numerically. The project supported by the Key Project of Chinese Ministry of Education (03145) and the Science Fund of Southwest Jiaotong University. The English text was polished by Yunming Chen.     

Specific adhesion of a soft elastic body on a wavy surface

This paper aims at developing a stochastic-elastic model of a soft elastic body adhering on a wavy surface via a patch of molecular bonds. The elastic deformation of the system is modeled by using continuum contact mechanics, while the stochastic behavior of adhesive bonds is modeled by using Bell's type of exponential bond association/dissociation rates. It is found that for sufficiently small adhesion patch size or stress concentration index, the adhesion strength is insensitive to the wavelength but decreases with the amplitude of surface undulation, and that for large adhesion patch size or stress concentration index, there exist optimal values of the surface wavelength and amplitude for maximum adhesion strength.     

The determination of elastic parameters of a half-space using a monochromatic vibration field at the surface

The paper presents a method to determine Lamé parameters λ, μ and density ϱ in a layered half-space, using monochromatic vibrations of its surface, excited by a harmonic source which is assumed to be known. The equations governing the vibrations are reduced to the Sturm-Liouville problem in scalar form for Love-type displacements, and in matrix form for the Rayleigh type. The scalar and matrix potentials of the Sturm-Liouville equations can be recovered from the corresponding impedance. Explicit formulas are given to construct the potentials by amplitudes and wavenumbers of normal (progressive) modes and attenuated standing waves. The potentials then are used to determine the elastic parameters and the density. The method can also be used for the acoustic equation.     

Circles on the slowness surface of a cubic elastic material

It is well-known that either the outer or the medial sheet of the slowness surface of an elastic material with cubic symmetry intersects the cube faces in circles. In is shown here that there exist on the next sheet (medial or outer) three pairs of circles centred on the symmetry axes and situated in planes parallel to the cube faces.     

Reflection of a beam of elastic waves by a periodic surface profile

Reflection of elastic waves from a traction-free solid-air boundary of periodic saw-tooth profile is investigated analytically and experimentally. For an incident plane wave the surface displacements on the profile are computed as the solution of a singular integral equation. The reflected field is subsequently obtained by using an integral representation. Incident beams of finite width are represented by Fourier superpositions of plane waves. The dependence of the reflected signal spectra on the incident beam width is examined closely near the fundamental surface resonance frequency. Experimental spectra which were obtained using two different diameter transducers, are compared to the corresponding theoretical spectra. It is found that the depth of the spectral minima depends on the incident beam width. Both analytical and experimental results exhibit the splitting of an incident beam of elastic waves into two reflected beams. The beam splitting is more pronounced for a narrower incident beam and for frequencies close to a resonance frequency of the profile.     

The maximum potential energy of elastic rods under axial gravitational load

An optimal design problem solved previously for an elastic rod hanging under its own weight found the distribution of the cross-sectional area that minimized the total potential energy stored in an equilibrium state, with the admissible designs bounded above and below and also subject to the constraint of prescribed total volume. This work solves the companion problem of the design that stores the maximum potential energy under the same constraint conditions. The method used is based on a comparison theorem for sandwich structures.