**The department of computational mechanics of deformable systems**

The department was founded in 2010 on the basis of department of mathematical methods of fracture mechanics and contact phenomena. Head of Department - Doctor of physico-mathematical sciences, Professor Viktor Myhas’kiv. Till 2010 the department was headed by Associated member of the National Academy of Sciences of Ukraine, Professor Hryhoryj Kit

*Scientific direction of the department**:*

- development and application based on the boundary integral and T-matrix formulation of mathematical models of effective numerical methods and algorithms, software for local and global analysis of the mechanical behavior complexly strukturized deformable environments and systems for static and dynamic disturbances of different nature.

*Department research priorities:*

- mathematical modeling and numerical boundary-integral analysis of static elastic and thermoelastic fields in 3D and 2D bodies with cracks and inclusions (H.S.Kit);
- mathematical modeling and numerical boundary-integral analysis of dynamic elastic fields in 3D solids with cracks and inclusions for the time and frequency domains (V.V. Myhas’kiv);
- mathematical modeling and numerical T-matrix analysis of wave field strength and electrical origin in 2D bodies with inclusions, Inverse problems wave scattering by elastic objects (Ya.I. Kunets);
- multilevel analysis of elastic waves propagation in 3D and 2D matrix composites, including nanosized structures with stochastic and orderly distribution of fillers (V.V. Myhas’kiv, Ya.I. Kunets).

**The main results:**

The principles of heat conduction and thermoelasticity theory of homogeneous and piecewise-homogeneous isotropic bodies with cracks were developed; they are based on the potential method and reducing the corresponding boundary value problems to singular integral (SIE) or integral-differential (SIDE) equations. The unknown stationary temperature field is presented in the form of harmonic potentials of simple and double layer densities of which are the sources and dipoles of heat at crack location. On the cracks the temperature, heat flows or conditions of non-ideal thermal contact are given; they ensure accounting head conduction and thermal resistance of crack filler After satisfying the boundary conditions system of integral equations for determining the density potentials is obtained.

To solve the problem of thermoelasticity is constructed integral image of the component displacements and stresses in the case of plane problems through complex potentials Kolosov-Mushelishvili, in the case of spatial tasks - in Newton potentials, the density of which is jumping movements opposing surface cracks. After satisfying the boundary conditions on the cracks obtained SIE or SIDE to determine these densities, which are due to the stress intensity factors. A line of exact and approximate methods of solution integral equations is proposed.

Based on the developed methods conducted important research influence the size and configuration of cracks, their thermal characteristics of heat transfer plates with the external environment, the contact surface cracks, interact with alien inclusions of the boundary of the body at its maximum and the thermoelastic equilibrium under the action of force and temperature factors.

The analytical - numerical method of boundary integral equation (boundary element) is generalized on 3D dynamic problems of the cracks and thin inclusions theory. For infinite, semi-infinite and bimaterial body with cracks and disk rigid inclusions under conditions of elastic wave propagation an integral representation of displacement and stress components is constructed in terms of Helmholtz potentials in the case of steady-state process or in the term of wave potentials - in the case of the transition process. The potential density are functions that characterize the crack opening or power influence on the inclusion during dynamic deformation. To determine them a system of boundary integral equations with a kernel of Helmholtz potential or wave potential is obtained.

To prove the3D dynamic problems of the theory of cracks and thin inclusions among developed effective methods for solving boundary integral equations obtained in the time and frequency regions are developed. They base on their regularization by integral operators with static kernels with subsequent boundary-element constructing the discrete analogues in the form of linear algebraic equations in the case of time marsh analysis – in the form of recurrent systems of linear algebraic equations.

Complex analysis of stress concentration near spatial cracks and disk rigid inclusions is carried out with regard for inertial and resonance effects due to power factors. In particular, the change of stress intensity factors with time in the body with a singular crack or inclusion is studied under harmonic loading, load in the form of functions Heviside or Dirac delta function. We describe the wave field on the interaction of cracks and inclusions between them, from the boundary of the body and the interface material.

The method of boundary integral equations together with the method of merging asymptotic expansions extended to 2D and 3D problems of wave diffraction on thin elastic inclusions. This approach allowed a detailed description of the stress state of the structure near the edges of the inclusion, where the boundary layer phenomena occur, depending on these edges shape. Besides, the peculiarities of the scattering amplitude and directivity characteristics in the Fraunhofer zone are studied for different geometrical and physical - mechanical parameters of thin walled inhomogeneities what made it possible to formulate and solve the corresponding inverse problems.

Methods for determining the geometrical, physical and mechanical characteristics of local inhomogeneities (plane cracks, thin tunnel inclusion, volumetric body) in elastic-deformed medium are developed. They base on solution the inverse boundary value problems of elastic waves scattering.

Combining the method of boundary integral equations and effective medium , the phase velocities and elastic wave attenuation coefficients in composites with blocs of disk rigid inclusions and fibers of non-canonical cross section investigated with regard for the possibility of their partial delamination from matrix material.

Doctors of Sciences Miroslav Khay and Ostap Poberezhny contributed much into the researches of departments.

The results of the research in the department published 6 books and reserved 6 doctoral dissertations (H.S.Kit, M.V.Khay, O.V.Poberezhnyy, V.F.Yemets, V.V.Myhaskiv, Ya.I.Kunets ).

For a series of papers "Theory and methods of stress state and strength of solid deformable bodies with stress concentrators" H.S.Kit and V.V.Myhaskiv awarded the State Prize of Ukraine in Science and Technology in 2011.

Active research conducted by young scientists department. For a course of developments Ph.D. Ivanna Butrak and Ph.D. Igor Zhbadynsky were given the Prize of the President of Ukraine for young scientists (2008), Ph.D. Joanna Butrak received a grant of the President of Ukraine for young scientists (2010).

The department has accumulated rich experience in international cooperation project by ISF (1994-1995), project DFG (2008-2009), projects DAAD (2004, 2010), projects INTAS (1999-2001, 2006-2009, intas.iapmm.lviv.ua). In 2012, the group of Prof. V.V.Myhaskiv won a grant STCU and the Academy of Sciences of Ukraine #5726 "Methodology-software complex for seismoacoustic prospecting of shale gases based on adequate models of structural mechanics".