Journal of Theoretical and Applied Mechanics, Sofia, vol. 56 Issue 2 (2026)

Academician Petar Radoev Popivanov was born on 06.04.1946 in Sofia, into a family of prominent contributors to Bulgarian independence and builders of the Third Bulgarian state... Since his young years, Petar Popivanov has shown a serious interest in mathematics, participated in and won mathematical competitions and Olympiads. His first publication was in the journal “Mathematics” in 1963, before graduating from secondary education with excellent grades and a gold medal. Following his interests, Petar Popivanov became a student at the Faculty of Mathematics and Mechanics at Sofia University. There, Petar Popivanov orientated himself towards the field of partial differential equations and defended a thesis on non-elliptic boundary value problems...

doi: https://doi.org/10.55787/jtams.2026.2.AI00001

This work is a state-of-the-art review on the mechanical modeling of macro- and nano-heterogeneous quasi-crystals (QCs)\textbf{, }and specifically on elastic, piezoelectric and magneto-electro-elastic (MEE) materials that containing cracks, holes and inclusions. QCs exhibit unique properties, including high stiffness and brittle behavior under both static and dynamic loading, making them promising for advanced technological applications. The paper first summarizes available fundamental solutions and half-space Green's functions in elastostatics and elastodynamics. These provide the foundation for modeling the mechanical response of QCs using advanced computational techniques such as the boundary element method (BEM), meshless methods, Green's function methods and the method of fundamental solutions, all of which serve as a basis for numerical solutions involving various mechanical models. Next, models based on Bak's theory are discussed, highlighting the macro- and nano-scale approaches and including the Gurtin and Murdoch surface elasticity model for nanoscale heterogeneities. The review systematically discusses elastostatic and elastodynamic formulations across different material classes and defect configurations, providing an overview of current modeling capabilities. Finally, open questions and challenges are identified, offering guidance for future research aimed at bridging the gap between theoretical models and practical applications of QCs in high-tech industries.

doi: https://doi.org/10.55787/jtams.2026.2.AI00253

In this study, a novel formulation of rod structures for dynamic analysis by the finite element force method using the element forces is proposed. The general equation of natural oscillation in which the unknowns are nodal forces is used. The modified flexibility matrix is developed via the combination of consistent and lumped flexibility matrices with a specific ratio. The rate of convergence of force mode shapes and frequencies using the modified flexibility matrix is considered. Numerical examples for the structural rods and frames are given to verify the effectiveness and practical applicability of the present study. It has been found that for a rod system, the use of a modified flexibility formulation provides a good rate of convergence.

doi: https://doi.org/10.55787/jtams.2026.2.AI00062

This paper investigates Lie symmetries and analytical solutions of the governing equations of classical elasticity and couple-stress elasticity. The equilibrium equations for both models are formulated in terms of displacement fields and analyzed by the Lie symmetry method. The admitted symmetry generators are determined and used to derive invariant solutions through symmetry reductions. A comparison of the symmetry structures of the two theories shows that classical elasticity admits a scaling symmetry in the spatial variables, whereas this symmetry is absent in couple-stress elasticity due to the presence of higher-order terms and an intrinsic material length scale. This difference reflects the role of microstructural effects in the couple-stress model. In addition, explicit general solutions of both systems are computed using the computer algebra system Macaulay2 for linear partial differential equations with constant coefficients. For classical elasticity, the solutions are expressed in terms of harmonic functions, while in the couple-stress case additional functions satisfying Helmholtz-type equations appear. The results demonstrate how Lie symmetry analysis and symbolic computation complement each other in the study of generalized elasticity models.

doi: https://doi.org/10.55787/jtams.2026.2.AI00252

This paper is focused on analyzing the dissipated energy in mechanisms performing spatial motion. The components of the mechanisms are functionally graded along the length. Furthermore, the components have non-linear viscoelastic behavior. The viscoelastic behavior of the mechanism components is treated by a model having a spring and two dashpots (one of the dashpots has non-linear behavior). The viscoelastic model parameters and the specific mass change continuously along the length of the mechanism components. The dissipated energy is determined and checked by a method previously published. The influence of parameters of spatial motion and viscoelastic model parameters distribution on the dissipated energy is elucidated. An application of dissipated energy analysis using a failure criterion is presented.

doi: https://doi.org/10.55787/jtams.2026.2.AI00254

In this paper, the linear theory of Moore-Gibson-Thompson thermoviscoelasticity {based on Darcy's law} for Kelvin-Voigt porous materials is introduced and the steady vibration problems of this theory are studied. The governing {systems of} equations of motion and steady vibrations are proposed. {These systems are formulated in terms of the displacement vector field and the changes of the following two mechanical values: the pore pressures and the temperature of a porous material.} The fundamental solution to the system of steady vibration equations is constructed explicitly using four elementary functions, and its basic properties are established. Then, Green's identity is obtained and the uniqueness theorems for classical solutions of the basic boundary value problems are proved. Finally, existence theorems for classical solutions of the boundary value problems of steady vibrations are established using the potential method.

doi: https://doi.org/10.55787/jtams.2026.2.AI00259

We present an analytical solution for the steady-state stress field of a rotating elastic disk under gravity, supported by a frictionless localized point contact with a rigid ground. The problem is formulated within two-dimensional linear elastodynamics under plane-stress conditions, with gravity and centrifugal effects treated as body forces and the contact as a singular boundary traction. The steady-state solution is obtained as the long-time limit of the dynamic equations and exhibits a natural modal decomposition. Monopole and dipole modes originate from torque balance and uniform body force, while higher-order modes describe localized stress transmission. The formulation recovers the classical solution for a freely rotating disk in the absence of gravity and contact.

doi: https://doi.org/10.55787/jtams.2026.2.AI00271

This study evaluates the efficacy of a novel skin-displacement measuring device used as a real-time feedback tool for developing palpatory skills in therapeutic massage education. Utilizing two Chronojump® linear encoders, skin displacement was quantified in millimeters during practical training sessions with second-year students. This objective data served as an initial feedback mechanism for students during subsequent unassisted palpation attempts on various subjects. The assessment demonstrated a progressive increase in manual accuracy, rising from 61% on the first unassisted attempt to 84% and 89% on the second and third attempts, respectively. These findings indicate that integrating objective mechanical feedback significantly enhances the acquisition and precision of palpatory skills in clinical training.

doi: https://doi.org/10.55787/jtams.2026.2.AI00277