arXiv subjects

Fabio Vallejo

Publications and source records attributed to Fabio Vallejo.

4 recordsLinked to original sources

arXiv2026

Dynamical stability of planar phase boundaries for hyperelastic materials of Hadamard type

The dynamical stability of laminates or planar phase boundaries for hyperelastic materials of Hadamard type in two space dimensions is studied. For that purpose, the stability function, known as the Lopatinskii determinant, is computed for states of deformation at both sides of the planar interface that account for the generalized Legendre-Hadamard conditions derived by Grabovsky and Truskinovsky (J. Elast. 123 (2016), 225--243). The sufficient conditions for the dynamical stability of such configurations are described in terms of the physical parameters of the model, such as the shear modulus, and computed under kinetic conditions across the interface of both Maxwell (conservation of energy) or Abeyaratne and Knowles (dissipation of energy) types.

math.AP↗

arXiv2025

Revisiting the problem of existence of surface Rayleigh waves with impedance boundary conditions

This paper considers the problem of surface waves in an isotropic elastic half-space endowed with impedance boundary conditions as first proposed by Godoy et al. [Wave Motion 49 (2012), 585-594]. These conditions are controlled by two impedance parameters, where the standard stress-free boundary condition is retrieved for zero impedance. While the existence of a unique surface wave (called Rayleigh wave) is well-established for the standard stress-free boundary condition, the introduction of more general boundary conditions may lead to the absence of surface waves or even cause the PDE boundary value problem to become ill-posed. For the case of Godoy's impedance boundary conditions, the problem of existence and uniqueness of a surface wave of Rayleigh type was investigated by means of the complex function method based on Cauchy-type integrals. However, this method is quite cumbersome and hard to apply. In this work, we present an alternative method based on elementary tools from calculus to deal with the problem. We consider a particular case where both impedance parameters are non-zero and demonstrate the existence and uniqueness of the surface wave for all material and boundary parameter values. Numerical examples are presented to illustrate the effect of the impedance parameter on the speed of the surface wave.

physics.class-ph↗

arXiv2024

The secular equation for elastic surface waves under non standard boundary conditions of impedance type: A perspective from linear algebra

The study of elastic surface waves under impedance boundary conditions has become an intensive field of research due to their potential to model a wide range of problems. However, even when the secular equation, which provides the speed of the surface wave, can be explicitly derived, the analysis is limited to specific cases due to its cumbersome final expression. In this work, we present an alternative method based on linear algebra tools, to deal with the secular equation for surface waves in an isotropic elastic half-space subjected to non-standard boundary conditions of impedance type. They are defined by proportional relationships between both the stress and velocity components at the surface, with complex proportional ratios. Our analysis shows that the associated secular equation does not vanish in the upper complex half-plane including the real axis. Interestingly, the full impedance boundary conditions proposed by Godoy et al. [Wave Motion 49 (2012), 585-594] arise as a particular limit case. An approximation technique is introduced, in order to extend the analysis from the original problem to Godoy's impedance boundary conditions. As a result, it is shows that the secular equation with full Godoy's impedance boundary condition does not vanish outside the real axis. This is a crucial property for the well-posedness of the boundary value problem of partial differential equations, and thus crucial for the model to explain surface wave propagation. Due to the cumbersome secular equation, this property has been verified only for particular cases of the impedance boundary condition, namely the stress-free boundary condition (zero impedance) and when either one of the impedance parameter is set to zero (normal and tangential impedance cases).

math-ph↗

arXiv2022

Stability of classical shock fronts for compressible hyperelastic materials of Hadamard type

This paper studies the uniform and weak Lopatinski\uı conditions associated to classical (Lax) shock fronts of arbitrary amplitude for compressible hyperelastic materials of Hadamard type in several space dimensions. Thanks to the seminal works of Majda (Mem. Amer. Math. Soc. 43 (1983), no. 281; Mem. Amer. Math. Soc. 41 (1983), no. 275) and Métivier (Trans. Amer. Math. Soc. 296 (1986), pp. 431-479; Comm. Partial Diff. Eqs. 15 (1990), no. 7, pp. 983-1028), the uniform Lopatinski\uı condition ensures the local-in-time, multidimensional, nonlinear stability of such fronts. The stability function (also called Lopatinski\uı determinant) for shocks of arbitrary amplitude in this large class of hyperelastic materials is computed explicitly. This information is used to establish the conditions for uniform and weak shock stability in terms of the parameters of the shock and of the elastic moduli of the material.

math.AP↗