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. Here, we develop a mathematical model for determining the intrinsic effective Young's modulus of a single brush-coated cell from the force-indentation curve with the bottom effect taken into account. The mathematical model is illustrated with the example of AFM data on testing of an eukaryotic cell taken from the literature.

Atomic force microscopy-based indentation of cells: modelling the effect of a pericellular coat

A simple analytical model is built up to account for the interface deformation effect in a spherical atomic force microscopy (AFM)-based quasi-static indentation of a living cell covered with a pericellular brush. The compression behaviour of the pericellular coat is described using the Alexander–de Gennes model that allows for nonlinear deformation. An approximate second-order relation between contact force and indenter displacement is obtained in implicit form, using the Hertzian solution as a first-order approximation. A method of fitting the indentation brush/cell model to experimental data is suggested based on the non-dimensionalized version of the displacement–force relation in the parametric form and illustrated with a specific example of AFM raw data taken from the literature.

Utilization of the ExactMPF package for solving a discrete analogue of the nonlinear Schrödinger equation by the inverse scattering transform method

We are revisiting the problem of solving a discrete nonlinear Schrödinger equation by the inverse scattering transform method, by use of the recently developed ExactMPF package within MAPLE Software. ExactMPF allows for an exact Wiener–Hopf factorization of matrix polynomials regardless of the partial indices of the matrix. The package can be widely used in various problems, where Wiener–Hopf factorization as one of the effective mathematical tools is required, as its code has already been disclosed. The analysis presented in this paper contains not only numerical examples of its use, but is also supported by appropriate and accurate a priori estimations. The procedure itself guarantees that the ExactMPF package produces all computations arithmetically exactly , and a detailed numerical analysis of various aspects of the computational algorithm and approximation strategies is provided in the case of a finite initial impulse.

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. The solution, based on the Wiener-Hopf technique and presented in Ayzenberg-Stepanenko et al. (Ayzenberg-Stepanenko et al. 2014 Proc. R. Soc. A 470, 20140121 (doi:10.1098/rspa.2014.0121)) is used to compute the profile of the medium undergoing failure. We investigate when this solution, representing the steady failure process, is physically acceptable. It is shown that the analytical solution is not always physically applicable and can be used to determine the onset of non-steady failure regimes. These arise from the presence of critical deformations in the wake of the crack front at the sites of the intact links. Additionally, we demonstrate that the structural integrity of the discrete elastic chains can significantly alter the range of speeds for which the bridge crack can propagate steadily. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.

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. This article is part of the theme issue 'Non-smooth variational problems and applications'.

An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package

We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material.

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.

Frictionless Motion of Lattice Defects

Energy dissipation by fast crystalline defects takes place mainly through the resonant interaction of their cores with periodic lattice. We show that the resultant effective friction can be reduced to zero by appropriately tuned acoustic sources located on the boundary of the body. To illustrate the general idea, we consider three prototypical models describing the main types of strongly discrete defects: dislocations, cracks, and domain walls. The obtained control protocols, ensuring dissipation-free mobility of topological defects, can also be used in the design of metamaterial systems aimed at transmitting mechanical information.

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. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.

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 SIAM J. Appl. Math., 75, 1171–1192 (doi:10.1137/140985093); Sharma 2015 SIAM J. Appl. Math.75, 1915–1940. (doi:10.1137/15M1010646)), the problem considered here becomes very intricate depending on the nature of the damaged links. For instance, in the case of a partially bridged finite zone it involves a 2 × 2 matrix kernel of formidable class. But using an original technique, the problem, including the general case of arbitrarily damaged links, is reduced to a scalar one with the exception that it involves solving an auxiliary linear system of N × N equations, where N defines the length of the damage zone. The proposed method does allow, effectively, the construction of an exact solution. Numerical examples and the asymptotic approximation of the scattered field far away from the crack tip are also presented.

Numerical factorization of a matrix-function with exponential factors in an anti-plane problem for a crack with process zone

In this paper, we consider an interface mode III crack with a process zone located in front of the fracture tip. The zone is described by imperfect transmission conditions. After application of the Fourier transform, the original problem is reduced to a vectorial Wiener-Hopf equation whose kernel contains oscillatory factors. We perform the factorization numerically using an iterative algorithm and discuss convergence of the method depending on the problem parameters. In the analysis of the solution, special attention is paid to its behaviour near both ends of the process zone. Qualitative analysis was performed to determine admissible values of the process zone's length for which equilibrium cracks exist. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.

Dynamic fracture of a dissimilar chain

In this paper, we study the dynamic fracture of a dissimilar chain composed of two different mass-spring chains and connected with other springs. The propagation of the fault (crack) is realized under externally applied moving forces. In comparison with a homogeneous double chain, the considered structure displays some new essential features of steady-state crack propagation. Specifically, the externally applied forces are of a different strength, unlike a static case, and should be appropriately chosen to satisfy the equilibrium of the structure. Moreover, there exists a gap in the range of crack speeds where the steady-state fracture cannot occur. We analyse the admissibility of solutions for different model parameters and crack speeds. We complement analytical findings with numerical simulations to validate our results. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.

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. In this paper simple, accurate, semi-analytical approximations of the solution are provided for the zero leak-off case, for a wide range of values of the material toughness and parameters defining the fluid rheology. A comparison with other results available in the literature is undertaken.

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. The latter allows for the application of a local condition of the Stefan type (the speed equation) to trace the fracture front. The obtained numerical solutions are carefully tested using various methods, and are shown to achieve a high level of accuracy.

Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk.XIII, 3-72. (Russian).), it is well known that the set of partial indices of a non-singular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions.

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. Spontaneous crack propagation

Spontaneous brittle fracture is studied based on the model of a body, recently introduced by two of the authors, where only the prospective crack path is specified as a discrete set of alternating initially stretched and compressed bonds. In such a structure, a bridged crack destroying initially stretched bonds may propagate under a certain level of the internal energy without external sources. The general analytical solution with the crack speed–energy relation is presented in terms of the crack-related dynamic Green's function. For anisotropic chains and lattices considered earlier in quasi-statics, the dynamic problems are examined and discussed in detail. The crack speed is found to grow unboundedly as the energy approaches its upper limit. The steady-state sub- and supersonic regimes found analytically are confirmed by numerical simulations. In addition, irregular growth, clustering and crack speed oscillation modes are detected at a lower bound of the internal energy. It is observed, in numerical simulations, that the spontaneous fracture can occur in the form of a pure bridged, partially bridged or fully open crack depending on the internal energy level.

An asymptotic method of factorization of a class of matrix functions

A novel method of asymptotic factorization of n × n matrix functions is proposed. The considered class of matrices is motivated by certain problems originated from the elasticity theory. An example is constructed to illustrate efficiency of the proposed procedure. The quality of approximation and the role of the chosen small parameter are discussed.

Diffusion of curvature on a sheared semi-infinite film

The viscous froth model is used to study the evolution of a long and initially straight soap film which is sheared by moving its endpoint at a constant velocity in a direction perpendicular to the initial film orientation. Film elements are thereby set into motion as a result of the shear, and the film curves. The simple scenario described here enables an analysis of the transport of curvature along the film, which is important in foam rheology, in particular for energy-relaxing ‘topological transformations’. Curvature is shown to be transported diffusively along films, with an effective diffusivity scaling as the ratio of film tension to the viscous froth drag coefficient. Computed (finite-length) film shapes at different times are found to approximate well to the semi-infinite film and are observed to collapse with distances rescaled by the square root of time. The tangent to the film at the endpoint reorients so as to make a very small angle with the line along which the film endpoint is dragged, and this angle decays roughly exponentially in time. The computed results are described in terms of a simple asymptotic solution corresponding to an infinite film that initially contains a right-angled corner.

Dislocations and inclusions in prestressed metals

The effect of prestress on dislocation (and inclusion) fields in nonlinear elastic solids is analysed by extending previous solutions by Eshelby and Willis. Using a plane-strain constitutive model (for incompressible incremental nonlinear elasticity) to describe the behaviour of ductile metals ( J 2 -deformation theory of plasticity), we show that when the level of prestress is high enough that shear band formation is approached, strongly localized strain patterns emerge, when a dislocation dipole is emitted by a source. These may explain cascade activation of dislocation clustering along slip band directions.

The viscous froth model: steady states and the high-velocity limit

The steady-state solutions of the viscous froth model for foam dynamics are analysed and shown to be of finite extent or to asymptote to straight lines. In the high-velocity limit, the solutions consist of straight lines with isolated points of infinite curvature. This analysis is helpful in the interpretation of observations of anomalous features of mobile two-dimensional foams in channels. Further physical effects need to be adduced in order to fully account for these.