Transactions of A. Razmadze Mathematical Institute Vol. 173 (2019), issue 3, 103–130 INVESTIGATION OF NONCLASSICAL TRANSMISSION PROBLEMS OF THE THERMO-ELECTRO-MAGNETO ELASTICITY THEORY FOR COMPOSED BODIES BY THE INTEGRAL EQUATION METHOD MAIA MREVLISHVILI 1 AND DAVID NATROSHVILI 1,2 Abstract. We investigate multi-field problems for complex elastic anisotropic structures when in different adjacent components of the composed body different refined models of elasticity theory are considered. In particular, we analyse the case when we have the generalized thermo-electro-magneto elasticity model (GTEME model) in one region of the composed body and the generalized thermo- elasticity model (GTE model) in the other adjacent region. This type of mechanical problem is described mathematically by systems of partial differential equations with appropriate transmission and boundary conditions. In the GTEME model part we have six-dimensional unknown physical field (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function), while in the GTE model part we have four- di- mensional unknown physical field (three components of the displacement vector and temperature distribution function). The diversity in dimensions of the interacting physical fields are taken into consideration in mathematical formulation and analysis of the corresponding boundary-transmission problems. We apply the potential method and the theory of pseudodifferential equations and prove the uniqueness and existence theorems of solutions to different type boundary-transmission problems in appropriate Sobolev spaces. 1. Introduction Modern industrial and technological processes apply widely, on the one hand, composite materials with complex microstructure and, on the other hand, complex composed structures consisting of materials having essentially different physical properties (for example, piezoelectric, piezomagnetic, hemitropic materials, two- and multi-component mixtures, nano-materials, bio-materials, and solid structures constructed by composition of these materials, such as, e.g., Smart Materials and other meta-materials). Therefore the investigation and analysis of mathematical models describing the mechanical, thermal, electric, magnetic and other physical properties of such materials are of crucial importance for both fundamental research and practical applications. In the study of active material systems, there is significant interest in the coupling effects be- tween elastic, electric, magnetic and thermal fields. For example, piezoelectric materials (electro- elastic coupling) have been used as ultrasonic transducers and micro-actuators; pyroelectric materials (thermal-electric coupling) have been applied in thermal imaging devices; piezomagnetic materials (elastic-magnetic coupling) are pursued for health monitoring of civil structures (see [9, 12, 13, 15, 24– 32, 39, 45–47, 50, 51, 53, 55], and the references therein). Although natural materials rarely show full coupling between elastic, electric, magnetic, and ther- mal fields, some artificial materials do. In [54], it was reported that the fabrication of BaTiO 3 -CoFe 2 O 4 composite had the electro-magnetic effect not existing in either constituent. Other examples of similar complex coupling can be found in [3–6, 16–18, 34–36, 40, 41, 48, 56]. For more detailed historical and bibliographic data see [1, 7, 49]. In the present paper, we investigate multi-field problems for complex elastic anisotropic structures when in different adjacent components of the composed body different refined models of elasticity 2010 Mathematics Subject Classification. 31B10, 35B65, 35C15, 35D30, 35J47, 35J57, 35S05, 47G10, 47G30, 47G40, 74E10, 74F05, 74F15, 74G30, 74G40, 74G55. Key words and phrases. Thermo-electro-magneto-elasticity; Thermo-elasticity; Green-Lindsay’s model; Boundary value problem; Transmission problem; Potential method; Pseudodifferential equations.