Initiation of decohesion between a flat punch and a thin bonded incompressible layer.

Non-axisymmetric frictionless JKR-type adhesive contact between a rigid body and a thin incompressible elastic layer bonded to a rigid base is considered in the framework of the leading-order asymptotic model, which has the form of an overdetermined boundary value problem. Based on the first-order perturbation of the Neumann operator in the Dirichlet problem for Poisson's equation, the decohesion initiation problem is formulated in the form of a variational inequality. The asymptotic model assumes that the contact zone and its boundary contour during the detachment process are unknown.

AFM-based spherical indentation of a brush-coated soft material: modeling the bottom effect.

It is a common practice in the atomic force microscopy (AFM)-based studies of living cells to differentiate them by values of the elastic (Young's) modulus, which is supposed to be an effective characteristic of the mechanical properties of a cell as a heterogeneous matter. The elastic response of a cell to AFM indentation is known to be affected by a relative distance from an AFM probe to a solid support on to which the cell is cultured. Besides this so-called bottom effect, AFM measurements may carry significant information regarding the effect of molecular brushes covering living cells.

Dynamic fracture regimes for initially prestressed elastic chains.

We study the propagation of a bridge crack in an anisotropic multi-scale system involving two discrete elastic chains that are interconnected by links and possess periodically distributed inertia. The bridge crack is represented by the destruction of every other link between the two elastic chains, and this occurs with a uniform speed. This process is assumed to be sustained by energy provided to the system through its initial configuration, corresponding to the alternating application of compression and tension to neighbouring links.

Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem.

We study the asymptotic behaviour of solutions of a boundary value problem for the Laplace equation in a perforated domain in [Formula: see text], [Formula: see text], with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates in three respects: in the limit case, the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size ϵ of the small hole where we consider the Robin condition collapses to 0. We study how these three singularities interact and affect the asymptotic behaviour as ϵ tends to 0, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters.

The Wiener-Hopf technique, its generalizations and applications: constructive and approximate methods.

This paper reviews the modern state of the Wiener-Hopf factorization method and its generalizations. The main constructive results for matrix Wiener-Hopf problems are presented, approximate methods are outlined and the main areas of applications are mentioned. The aim of the paper is to offer an overview of the development of this method, and demonstrate the importance of bringing together pure and applied analysis to effectively employ the Wiener-Hopf technique.

Exact conditions for preservation of the partial indices of a perturbed triangular 2 × 2 matrix function.

The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class.

Scattering on a square lattice from a crack with a damage zone.

A semi-infinite crack in an infinite square lattice is subjected to a wave coming from infinity, thereby leading to its scattering by the crack surfaces. A partially damaged zone ahead of the crack tip is modelled by an arbitrarily distributed stiffness of the damaged links. While an open crack, with an atomically sharp crack tip, in the lattice has been solved in closed form with the help of the scalar Wiener-Hopf formulation (Sharma 2015 , , 1171-1192 (doi:10.

Waves in elastic bodies with discrete and continuous dynamic microstructure.

This paper presents a unified approach to the modelling of elastic solids with embedded dynamic microstructures. General dependences are derived based on Green's kernel formulations. Specifically, we consider systems consisting of a structure and continuously or discretely distributed oscillators.

Fluid velocity based simulation of hydraulic fracture-a penny shaped model. Part II: new, accurate semi-analytical benchmarks for an impermeable solid.

In the first part of this paper a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off was created for a penny-shaped crack. The power-law rheological model of fluid was assumed and the final scheme was capable of tackling both the viscosity and toughness dominated regimes of crack propagation. The obtained solutions were shown to achieve a high level of accuracy.

Fluid velocity based simulation of hydraulic fracture: a penny shaped model-part I: the numerical algorithm.

In the first part of this paper, a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off, previously demonstrated for the PKN and KGD models, is extended to obtain solutions for a penny-shaped crack. The numerical scheme is capable of dealing with both the viscosity and toughness dominated regimes, with the fracture being driven by a power-law fluid. The computational approach utilizes two dependent variables; the fracture aperture and the reduced fluid velocity.

Articular Contact Mechanics from an Asymptotic Modeling Perspective: A Review.

In the present paper, we review the current state-of-the-art in asymptotic modeling of articular contact. Particular attention has been given to the knee joint contact mechanics with a special emphasis on implications drawn from the asymptotic models, including average characteristics for articular cartilage layer. By listing a number of complicating effects such as transverse anisotropy, non-homogeneity, variable thickness, nonlinear deformations, shear loading, and bone deformation, which may be accounted for by asymptotic modeling, some unsolved problems and directions for future research are also discussed.

Pipette aspiration testing of soft tissues: the elastic half-space model revisited.

The pipette aspiration testing technique is considered, and the elastic half-space model, which was originally introduced in the isotropic incompressible case, is revisited and generalized for the case of transverse isotropy. Asymptotic solutions are obtained in the two limiting cases of a wide and a narrow pipette.

Brittle fracture in a periodic structure with internal potential energy.

We consider a brittle fracture taking account of self-equilibrated distributed stresses existing at microlevel in the absence of external forces. To determine how the latter can affect the crack equilibrium and growth, a model of a structured linearly elastic body is introduced, consisting of two equal symmetrically arranged layers (or half-planes) connected by an interface as a prospective crack path. The interface comprises a discrete set of elastic bonds.