The Hertzian contact theory, as well as most of the other classical theories of normal and tangential contact, provides displacements and the distribution of normal and tangential stress components directly in the contact surface. However, other components of the full stress tensor in the material may essentially influence the material behaviour in contact. Of particular interest are principal stresses and the equivalent von Mises stress, as well as the gradient of the hydrostatic pressure. For many engineering and biomechanical problems, it would be important to find these stress characteristics at least in the contact plane. In the present paper, we show that the complete stress state in the contact plane can be easily found for axially symmetric contacts under very general assumptions. We provide simple explicit equations for all stress components and the normal component of the gradient of hydrostatic pressure in the form of one-dimensional integrals.

Different types of resonators are used to create acoustic metameterials and metasurfaces. Recent studies focused on the use of multiple resonators of dipole, quadrupole, octupole and even hexadecapole types. This paper considers the theory of an acoustic metasurface, which is a flat surface with a periodic arrangement of multiple resonators. The sound field reflected by the metasurface is determined. If the distance between the resonators is less than a half the wavelength of the incident plane wave, the far field can be described by a reflection coefficient that depends on the angle of incidence. This allows to characterize the acoustic properties of the metasurface by a homogenized boundary condition, which is a high order tangential impedance boundary condition. The tangential impedance depending on the multiple order of the resonators is introduced. In addition, we analyze sound absorption properties of these metasurfaces, which are a critical factor in determining their performance. The paper presents a theoretical model that accounts for the multiple orders of resonators and their impact on sound absorption. The maximum absorption coefficient for a diffuse sound field, as well as the optimal value for the homogenized impedance, are calculated for arbitrary multipole orders.