On the elastic far-field response of a two-dimensional coated circular inhomogeneity: Analysis and applications

Displacements representations for the problems with spherical and circular material surfaces

“…in which the elastic parameters µ and κ = 3 − 4ν are the parameters of the specific phase: (µ, κ) for the matrix or (µI , κI ) for the inhomogeneity. In Mogilevskaya et al (2008) and Zemlyanova and Mogilevskaya (2018), it was shown that the potentials inside and outside of circular inhomogeneity are inside the inhomogeneity (z = r exp (iϑ) , r < R)…”

“…In Mogilevskaya et al (2008) and Zemlyanova and Mogilevskaya (2018), it was shown that the potentials inside and outside of circular inhomogeneity are inside the inhomogeneity (z = r exp (iϑ) , r < R)…”

Elastic disk with isoperimetric Cosserat coating

“…The external load will be modelled as a constant normal traction p (q = 0) applied along two diametrically opposed small arcs of equal amplitude, s = α R, where α is an angle centred at the vertical diameter of the disk. Using the complex Fourier series expansion (28), multiplying both sides of it by e −miθ , m = ±1, ±2, ... ± n and integrating over the whole circle leads to (68) The right hand side of equation ( 68) is different from zero and equal to 2π if and only if n = m. Hence, at every fixed m, one non-null coefficient is determined. Collecting terms with the same power of e ±niθ and inverting equation ( 68) yield the values of all coefficients D ±n .…”

“…The advent of nanotechnologies has strongly fueled the development of interface models. These have been used to estimate thermo-mechanical properties of nano-composites, of mono-layered-graphene based materials [24,25], to analyze the interaction between nano-inhomogeneities [26] and the mechanics of reinforcements, such as coated fibers [27,28] or nanoplatelets embedded in a core matrix [29,30]. The presence of coatings or thin layers strongly influences failure mechanisms, such as crack nucleation and propagation inside the coating/substrate [31,32,33], or delamination induced by mechanical/thermal mismatch and residual stresses existing between coating and substrate [34,35,36,37,38,39,40,41,42], or by curvature changes or buckling [43,44,45].…”

On the Bövik–Benveniste methodology and related approaches for modelling thin layers

“…The solutions for the two models are compared with the exact solution of the original problem of figure 2 a that is available in, e.g. [63,64]. The normalized jumps in radial and tangential traction components are plotted in figure 3 as functions of polar angle θ=false[0,π/2false].…”