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A linear elastic circular disc is analyzed under a self-equilibrated system of loads applied along its boundary. A distinctive feature of the investigation, conducted using complex variable analysis, is the assumption that the material is incompressible (in its linearized approximation), rendering the governing equations formally identical to those of Stokes flow in viscous fluids. After deriving a general solution to the problem, an isoperimetric constraint is introduced at the boundary to enforce inextensibility. This effect can be physically realized, for example, by attaching an inextensible elastic rod with negligible bending stiffness to the perimeter. Although the combined imposition of material incompressibility and boundary inextensibility theoretically prevents any deformation of the disc, it is shown that the problem still admits non-trivial solutions. This apparent paradox is resolved by recognizing the approximations inherent in the linearized theory, as confirmed by a geometrically nonlinear numerical analysis. Nonetheless, the linear solution retains significance: it may represent a valid stress distribution within a rigid system and can identify critical conditions of interest for design applications. 

The finite element algorithm is developed to solve antiplane problems involving elastic domains whose boundaries or their parts are coated with thin and relatively stiff layers. These layers are modeled by the vanishing thickness Gurtin–Murdoch material surfaces that could be open or closed, and smooth or non-smooth. The governing equations for the problems are derived using variational arguments. The domains are discretized using triangular finite elements. In general, standard linear elements are used to approximate displacements in the domain. However, to capture the singular behavior of the elastic fields near the tips of the open Gurtin–Murdoch surfaces, a novel blended singular element is devised. Numerical examples are presented to demonstrate the accuracy and robustness of the algorithm developed. 

The plane strain problem of an isotropic elastic matrix subjected to uniform far-field load and containing multiple stiff prestressed arcs located on the same circle is considered. The boundary conditions for the arcs are described by those of either Gurtin–Murdoch or Steigmann–Ogden theories in which the arcs are endowed with their own elastic energies. The material parameters for each arc can in general be different. The problem is reduced to the system of real variables hypersingular boundary integral equations in terms of two scalar unknowns expressed via the components of the stress tensors of the arcs. The unknowns are approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The coefficients in series are found from the system of linear algebraic equations that are solved using the collocation method. The expressions for the stress intensity factors are derived and numerical examples are presented to illustrate the influence of governing dimensionless parameters. 

An analytical solution is derived for the bifurcations of an elastic disc that is constrained on the boundary with an isoperimetric Cosserat coating. The latter is treated as an elastic circular rod, either perfectly or partially bonded (with a slip interface in the latter case) and is subjected to three different types of uniformly distributed radial loads (including hydrostatic pressure). The proposed solution technique employs complex potentials to treat the disc’s interior and incremental Lagrangian equations to describe the prestressed elastic rod modelling the coating. The bifurcations of the disc occur with modes characterized by different circumferential wavenumbers, ranging between ovalization and high-order waviness, as a function of the ratio between the elastic stiffness of the disc and the bending stiffness of its coating. The presented results find applications in various fields, such as coated fibres, mechanical rollers, and the growth and morphogenesis of plants and fruits. 

A difficulty in the theory of a thin elastic interface is that series expansions in its thickness become disordered in the high-contrast limit, i.e. when the interface is much softer or much stiffer than the media on either side. We provide a mathematical analysis of such series for an annular coating around a cylindrical fibre embedded in an elastic matrix subject to biaxial forcing. We determine the order of magnitude of successive terms in the series, and hence the terms that need to be retained in order to ensure that every neglected term is smaller in order of magnitude than at least one retained term. In this way, we obtain uniform approximations for quantities such as the jump in the displacement and stress across the coating, and explain some peculiarities that have been observed in numerical work. A key finding is that it is essential to distinguish three types of boundary-value problem, corresponding to ‘distant forcing’, ‘localized forcing’ and ‘the homogeneous problem’, since they give different patterns of disorder in the corresponding series expansions. This provides a meaningful correspondence between physical principles and our mathematical results.