Effective high-order approximations of layered coatings and linings of anisotropic elastic, viscoelastic and nematic materials

Beginning at low frequencies, asymptotically exact models of anisotropic coatings and linings with a small ratio of the half-thickness to the longitudinal deformation scale are constructed. The requirement on the conditions of contact with the substrate, where at least one of the boundary conditions must contain a displacement component of the strain in explicit form is “non-classical” here. The action of the coating/lining on a thicker body is approximated by impedance boundary conditions at the interface. The error of the model is reduced to the third order for layered packets and to the sixth order for a single layer. The physical limit of the applicability is the frequency of the first quasi-resonance in the corresponding deformed system. A comparison with the propagator matrix and numerical testing for partial waves shows satisfactory accuracy, comparable with the accuracy of the theory of classical plates of similar order. The results can be used in contact problems and for rapid algorithms for calculating the spectrum of the eigenwaves in half-spaces and thick layered plates with any number of coatings and linings. An extension to the case of viscoelastic materials and nematic elastomers is given.