Roman Pasternak’s research while affiliated with Lutsk National Technical University and other places

The paper presents studies on the Green’s function for thermomagnetoelectroelastic medium and its reduction to the contour integral. Based on the previous studies the thermomagnetoelectroelastic Green’s function is presented as a surface integral over a half-sphere. The latter is then reduced to the double integral, which inner integral is evaluated explicitly using the complex variable calculus and the Stroh formalism. Thus, the Green’s function is reduced to the contour integral. Since the latter is evaluated over the period of the integrand, the paper proposes to use trapezoid rule for its numerical evaluation with exponential convergence. Several numerical examples are presented, which shows efficiency of the proposed approach for evaluation of Green’s function in thermomagnetoelectroelastic anisotropic solids.

The paper presents a general boundary element approach for analysis of 3D cracks in anisotropic thermomagnetoelectroelastic solids. Dual boundary integral equations are derived, which kernels are explicitly written. These equations do not contain volume integrals in the absence of distributed body heat and extended body forces, which is advantageous comparing to the existing approaches. The issues on the boundary element solution of these equations are discussed in details. The efficient numerical evaluation of kernels based on the trapezoid rule is proposed. Modified Kutt's quadrature with Chebyshev nodes is derived for integration of singular and hypersingular integrals. Nonlinear polynomial mappings are adopted for smoothing the integrand at the crack front, which is advantageous for accurate evaluation of field intensity factors. Special shape functions are introduced, which account for a square-root singularity of extended stress and heat flux at the crack front. The issues on numerical determination of field intensity factors are discussed. Several numerical examples are presented, which show the efficiency (low computational time and high precision) of the proposed boundary element formulation.

The paper derives Somigliana type boundary integral equations for 3D thermomagnetoelectroelasticity of anisotropic solids. In the absence of distributed volume heat and body forces these equations contain only boundary integrals. Besides all of the obtained terms of integral equations are to be calculated in the real domain, which is advantageous to the known equations that can contain volume integrals or whose terms should be calculated in the mapped temperature domain. All kernels of the derived integral equations and the 3D thermomagnetoelectroelastic Green׳s function for a point heat are obtained explicitly based on the Radon transform technique. Verification of the obtained equations and fundamental solutions is provided.

This paper utilizes the Stroh formalism and the complex variable approach to derive the integral formulae and boundary integral equations of anisotropic thermoelectroelasticity for a bimaterial solid with Kapitza-type interface. Obtained integral formulae and boundary integral equations do not contain domain integrals, thus, the boundary element approach based on them does not require any additional procedures accounting for the stationary temperature field acting in the solid. All kernels of the boundary integral equations are written explicitly in a closed form. Verification for limiting values of thermal resistance of the interface is provided. Obtained boundary integral equations are incorporated into the boundary element analysis procedure. Several problems are considered, which shows the influence of thermal resistance of the bimaterial interface on fields’ intensity at the tips of electrically permeable and impermeable cracks.

This paper presents a comprehensive study on the 2D boundary integral equations, Green׳s functions and boundary element method for thermoelectroelastic bimaterials containing cracks and thin inclusions. Based on the extended Stroh formalism, complex variable approach and the Cauchy integral formula, the paper derives integral formulae for the Stroh complex functions, and Somigliana type integral identities for 2D thermoelectroelastic bimaterial. The kernels arising in the integral formulae are obtained explicitly and in a closed-form. It is proved that these kernels are fundamental solutions for a line extended force and a line heat. The far-field mechanical, electric and thermal load and internal volume load are accounted for in the obtained integral formulae. The latter allow to derive boundary integral equations for a bimaterial containing holes, cracks and thin inclusions, and to develop the corresponding boundary element approach. Special tip boundary elements used in the analysis allow accurate determination of the stress and electric displacement intensity factors for cracks and thin deformable inclusions. Several numerical examples are considered that show the validity and efficiency of the developed boundary element approach in the analysis of defective thermoelectroelastic anisotropic bimaterials.

The paper derives the equations, which should be satisfied by the temperature field that does not induce stress and electric displacement in an anisotropic thermoelectroelastic solid. It is shown that these equations are satisfied identically only if the pyroelectric solid is heated or cooled by a constant temperature. Due to the tertiary pyroelectric effect a free thermoelectroelastic solid, which temperature is a linear function of spatial coordinates, can undergo nonzero internal stress and electric displacement. Sufficient conditions are obtained, which satisfaction vanishes stress and electric displacement in a free pyroelectric solid under the action of a steady-state uniform heat flow.

The paper presents a complex variable approach for obtaining of the integral formulae and integral equations for plane thermoelectroelasticity of an anisotropic bimaterial with thermally insulated interface. Obtained relations do not contain domain integrals and incorporate only physical boundary functions such as temperature, heat flux, extended displacement and traction, which are the main advances of these relations.

The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelectroelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and Stroh orthogonality relations to obtain the integral formulae for the Stroh complex functions, which are piecewise-analytic in the complex half-plane with holes and opened mathematical cuts. Further application of the Stroh formalism allows derivation of the Somigliana type integral formulae and boundary integral equations for a thermoelectroelastic half-space. The kernels of these equations correspond to the fundamental solutions of heat transfer, electroelasticity and thermoelectroelasticity for a half-space. It is shown that the difference between the obtained fundamental solution of thermoelectroelasticity and those presented in literature is due to the fact, that present solution additionally accounts for extended displacement and stress continuity conditions, thus, it is physically correct. Obtained integral equations are introduced into the boundary element approach. Numerical examples validate derived boundary integral equations, show their efficiency and accuracy.

This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.