Non-potential curl forces, such as those explored by Berry and Shukla, challenge the standard Hamiltonian mechanics necessary for quantization. This paper addresses this limitation by deploying a tiered geometric scheme: applying Birkhoffian mechanics for forces that admit a variational representation, and generalizing to Santilli's Lie-admissible mechanics for those that do not. By categorizing these forces based on their variationality, the proposed framework preserves a generalized geometric structure that yields path-dependent constants of motion. This method formalizes Hamilton's concept of accommodating non-potential forces through a direct deformation of the underlying symplectic structure. Furthermore, to bridge these generalized algebraic structures with established physics, the manuscript demonstrates that the standard Peierls bracket of covariant field theory naturally generalizes into a Lieadmissible algebra when applied to dissipative, non-conservative systems. The paper is written pedagogically to be accessible to graduate physics students.
This study investigates the two-dimensional sliding frictional contact problem of a functionally graded (FG) elastic layer pressed by a rigid cylindrical punch within the framework of couple-stress elasticity (CSE) to account for microstructural size effects. Coulomb friction conditions are assumed to prevail across the contact interface. The Poisson's ratio of the layer is kept constant, while its modulus of elasticity varies exponentially through the thickness. Utilizing integral transform techniques along with the boundary conditions, the contact problem is formulated as a singular integral equation of the second kind, where the contact stress function and the contact width are the primary unknowns. This singular integral equation is solved numerically via Jacobi polynomials and the Gauss{Jacobi integration formula, ensuring compliance with the equilibrium and consistency conditions. Finally, a comprehensive parametric analysis is conducted to evaluate the in uences of the characteristic material length, material inhomogeneity parameter, friction coefficient, punch radius, indentation load, and Poisson's ratio on both the contact stress distribution and the underlying in-plane stress fields.
Understanding the mechanical behaviour of isotropic points (IPs)-regions of hydrostatic stress with zero shear-is essential for interpreting photoelastic fringe patterns and analysing load transfer in solid bodies. This study investigates IP formation and evolution in circular disks subjected to four-point loading using a combination of photoelastic experiments, digital image processing, and analytical modelling. A generalised stress framework based on superposition is developed to predict isochromatic fringe fields and IP locations for arbitrary combinations of normal and tangential contact forces. Beyond case-specific solutions, the study establishes a unified physical interpretation of fringe patterns based on the concepts of zero-shear loci, symmetry constraints, and stress-topology transitions. It is shown that symmetry governs the localisation of IPs along principal axes, while symmetry breaking leads to their migration into the disk interior. Under normal loading, IPs form along boundary-induced zero-shear regions and undergo coalescence and splitting, indicating transitions in stress topology. In contrast, tangential loading produces a persistent central isotropic point corresponding to a null stress state, resulting in qualitatively different fringe evolution. A novel inverse methodology is proposed to determine contact force components from experimentally measured IP coordinates. The results demonstrate that IP locations are invariant to the load magnitude but highly sensitive to the load direction, making them reliable indicators of the load geometry. The findings highlight isotropic points as fundamental descriptors of stress redistribution and fringe topology. While the present study focuses on a circular disk under four-point loading, the framework may be extended to more complex systems such as granular assemblies and contact-driven structures, subject to further investigation.
This paper investigates the propagation of shear horizontal (SH) waves in piezoelectric nanoplates by coupling the Gurtin-Murdoch surface model with the nonlocal strain gradient theory. The Legendre polynomial expansion method with analytical integration is developed, which reformulates the complex acoustic wave partial differential equation (PDE) solution problem into a standard generalized eigenvalue problem, thereby obtaining its dispersion relation. In contrast to conventional polynomial techniques, this approach obviates the necessity of redundant integration, thereby facilitating the derivation of comprehensive solutions over the full frequency spectrum. The validity of the proposed method is verified by calculating the phase velocities of SH waves in a piezoelectric nanoplate without surface and strain gradient effects and comparing the results with literature data. The convergence and computational efficiency of the method are also demonstrated. Results indicate that the stiffness softening dominated by the surface effect and the hardening induced by the strain gradient effect form a competitive mechanical response. The surface effect increases wave velocity, while the nonlocal strain gradient effect decreases it. This interplay is frequency-dependent, with the surface effect dominating at low frequencies and the nonlocal strain gradient effect becoming dominant at higher frequencies. These findings offer valuable insights for the design of smart nano-devices.
This study examines the linear instability in a channel flow of the Navier-Stokes-Voigt viscoelastic fluid, influenced by couple stresses. It confirms the applicability of Squire's theorem and develops a generalized eigenvalue problem for two-dimensional modes using two Chebyshev collocation methods. This problem is then solved with the QZ algorithm. Despite the base flow maintaining characteristics of the Newtonian fluid, the instability of the fluid flow is significantly affected by the presence of the Kelvin-Voigt parameter and the couple stresses parameter. Numerical results showed the effect of increasing the couple stresses and the Kelvin-Voigt parameters on the stability of the system. As the value of these parameters increases, the critical values of the Reynolds number begin to increase, which initially indicates a stabilising effect.
