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This work presents numerical solution for wave motion in a functionally graded piezoelectric half-plane that includes contributions of incident time-harmonic SH waves, waves reflected by the traction-free surface and scattered by multiple curvilinear cracks. A special type of material gradient is studied, where material properties vary exponentially with respect to the depth coordinate. A non-hypersingular traction Boundary Integral Equation Method based on analytically derived Green’s function of a graded half-plane is developed and verified. A series of numerical results show the influence of the material gradient characteristics, the properties of the applied dynamic load, the cracks geometry, the cracks interaction phenomenon and the coupled character of the electromechanical continuum on the wave motions and on the local mechanical and electrical stress concentration fields developing in the graded half-plane.

A semi-analytical method for computation of stress and displacement field due to surface loading on a linear elastic, isotropic and homogeneous half-plane containing a hole is presented. The method relies on the principle of superposition and comprises of three parts, viz. half-plane with no hole subjected to the intended surface loading, infinite plane containing a hole subjected to unspecified traction on the hole surface and lastly half-plane with no hole subjected to corrective loading on the surface. The loadings in the latter two cases are sought to ensure that all the boundary conditions are satisfied. Michell solution and combination of Airy stress function approach and Fourier transforms are the tools utilized for the second and third part, respectively. The complete method is thoroughly validated through three distinct examples from literature.

The study is devoted to some applied aspects of the Dirac delta function. On the basis of this function, an integral representation was found for the deviation of the functions of the Holder class H α (0 < α < 1) from their Poisson integrals in the upper half-plane. In the current research, exact equalities of the upper bounds for the deviations of the functions of the Holder class H α from the Poisson operators in the upper half-plane were found by applying the known properties of the Dirac delta function.

In this paper, the Poisson operator is developed on the basis of solutions to polyharmonic equations. It is based on the triharmonic equation in Cartesian coordinates in the presence of classical boundary conditions. As a result, it is shown that the constructed triharmonic function is a sign-changing operator. The integral kernel of the triharmonic Poisson integral in the upper half-plane changes its sign twice within the definition domain. The triharmonic Poisson–Taylor operator discovered in the paper is determined by the Taylor formulas for two different integrals that form the triharmonic function.

The paper is devoted to the study of the approximation properties of the biharmonic Poisson integral in the upper half-plane. The problem of approximating functions by biharmonic Poisson operators in the upper half-plane in the metric space L p (−∞, +∞) is considered. The main result of the paper is based on the representation of the integral kernel of the biharmonic Poisson integral obtained by applying the parameterization approach. It was found that the considered integral kernel belongs to the class of delta-shaped kernels. The upper bound is obtained for the approximation of functions by biharmonic Poisson operators in terms of the first-order modulus of continuity.

By using modular functions on the upper complex half-plane, we study a class of strain energies for crystalline materials whose global invariance originates from the full symmetry group of the underlying lattice. This follows Ericksen’s suggestion which aimed at extending the Landau-type theories to encompass the behavior of crystals undergoing structural phase transformation, with twinning, microstructure formation, and possibly associated plasticity effects. Here we investigate such Ericksen-Landau strain energies for the modelling of reconstructive transformations, focusing on the prototypical case of the square-hexagonal phase change in 2D crystals. We study the bifurcation and valley-floor network of these potentials, and use one in the simulation of a quasi-static shearing test. We observe typical effects associated with the micro-mechanics of phase transformation in crystals, in particular, the bursty progress of the structural phase change, characterized by intermittent stress-relaxation through microstructure formation, mediated, in this reconstructive case, by defect nucleation and movement in the lattice.

The plane problem on the effect of a moving load on an incompressible half-plane with inhomogeneity in the form of a thin surface layer is considered. The effect of the moving load, initial stresses, and mechanical parameters of elements of a layered foundation on the main characteristics of its stress-strain state is studied.

Within the context of Gurtin–Murdoch surface elasticity theory, closed-form analytical solutions are derived for an isotropic elastic half-plane subjected to a concentrated/uniform surface load. Both the effects of residual surface stress and surface elasticity are included. Airy stress function method and Fourier integral transform technique are used. The solutions are provided in a compact manner that can easily reduce to special situations that take into account either one surface effect or none at all. Numerical results indicate that surface effects generally lower the stress levels and smooth the deformation profiles in the half-plane. Surface elasticity plays a dominant role in the in-plane elastic fields for a tangentially loaded half-plane, while the effect of residual surface stress is fundamentally crucial for the out-of-plane stress and displacement when the half-plane is normally loaded. In the remaining situations, combined effects of surface elasticity and residual surface stress should be considered. The results for a concentrated surface force serve essentially as fundamental solutions of the Flamant and the half-plane Cerruti problems with surface effects. The solutions presented in this work may be helpful for understanding the contact behaviors between solids at the nanoscale.

The effect of visco-elastic slip interface on the dynamic response of a shallow tunnel is investigated. The model with elastic, viscosity and slip coefficients is adopted to analyze the effect of the imperfect interface between the tunnel lining and the surrounding rock mass. In order to satisfy the traction free boundary condition, the half-plane above the shallow tunnel is simplified to an arc with large radius. The tunnel is mapped into an annular region to derive the dynamic response of the non-circular lined shallow tunnel with imperfect interface. The expanded coefficients of the complex function are determined based on the imperfect boundary condition and the traction free boundary condition of the half-plane. The effects of stiffness parameter, viscosity and slip coefficients on the responses of tunnel under different depths and different wave frequencies are studied.

In this study, a seismic analysis of semi-sine shaped alluvial hills above a circular underground cavity subjected to propagating oblique SH-waves using the half-plane time domain boundary element method (BEM) was carried out. By dividing the problem into a pitted half-plane and an upper closed domain as an alluvial hill and applying continuity/boundary conditions at the interface, coupled equations were constructed and ultimately, the problem was solved step-by-step in the time domain to obtain the boundary values. After solving some verification examples, a semi-sine shaped alluvial hill located on an underground circular cavity was successfully analyzed to determine the amplification ratio of the hill surface. For sensitivity analysis, the effects of the impedance factor and shape ratio of the hill were also considered. The ground surface responses are illustrated as three-dimensional graphs in the time and frequency domains. The results show that the material properties of the hill and their heterogeneity with the underlying half-space had a significant effect on the surface response.