Nonlinear plane waves in saturated porous media with incompressible constituents

We consider the propagation of nonlinear plane waves in porous media within the framework of the Biot–Coussy biphasic mixture theory. The tortuosity effect is included in the model, and both constituents are assumed incompressible (Yeoh-type elastic skeleton, and saturating fluid). In this case, the linear dispersive waves governed by Biot’s theory are either of compression or shear-wave type, and nonlinear waves can be classified in a similar way. In the special case of a neo-Hookean skeleton, we derive the explicit expressions for the characteristic wave speeds, leading to the hyperbolicity condition. The sound speeds for a Yeoh skeleton are estimated using a perturbation approach. Then we arrive at the evolution equation for the amplitude of acceleration waves. In general, it is governed by a Bernoulli equation. With the present constitutive assumptions, we find that longitudinal jump amplitudes follow a nonlinear evolution, while transverse jump amplitudes evolve in an almost linearly degenerate fashion.

Total Potential Optimization Using Metaheuristic Algorithms for Solving Nonlinear Plane Strain Systems

Total Potential Optimization using Metaheuristic Algorithms (TPO/MA) is an alternative tool for the analysis of structures. It is shown that this emerging method is advantageous in solving nonlinear problems like trusses, tensegrity structures, cable networks, and plane stress systems. In the present study, TPO/MA, which does not need any specific implementation for nonlinearity, is demonstrated to be successfully applied to the analysis of plane strain structures. A numerical investigation is performed using nine different metaheuristic algorithms and an adaptive harmony search in linear analysis of a structural mechanics problem having 8 free nodes defined as design variables in the minimization problem of total potential energy. For nonlinear stress-strain relation cases, two structural mechanics problems, one being a thick-walled pipe and the other being a cantilever retaining wall, are analyzed by employing adaptive harmony search, which was found to be the best one in linear analyses. The nonlinear stress-strain relations considered in these analyses are hypothetical ones due to the lack of any such relationship in the literature. The results have shown that TPO/MA can solve nonlinear plane strain problems that can be encountered as engineering problems in structural mechanics.

Propagation and interaction of weakly nonlinear plane waves in transversely isotropic elastic materials

The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media. It begins with a definition of stored energy functions of considered hyperelastic models. The equation of elastodynamics as well as the first-order quasilinear hyperbolic system for plane waves are provided. The eigensystem for this system is determined to study three-wave interaction coefficients. The main part of the paper concerns a discussion of these coefficients. Applying the weakly nonlinear asymptotics method, it is shown that in the case of transverse isotropy the inviscid Burgers’ equation describes an evolution of a single quasi-shear wave. The result contradicts the case of isotropy, where the equation with quadratic nonlinearity cannot describe any shear wave propagation. The paper ends with an example of numerical solutions for the obtained evolution equation.

The study of nonlinear deformation of a plane with an elliptical inclusion for harmonic materials

Analytical methods are used to study nonlinear deformation of a plane with an elliptical inclusion. The elastic properties of a material of the plane and inclusion are described with a semi-linear material. The external load is constant nominal (Piola) stresses at infinity. At the inclusion boundary, the conditions of the continuity for stresses and displacements are satisfied. Semi-linear material belongs to the class of harmonic, the methods of the theory of functions of a complex variable are applicable to solving nonlinear plane problems. Stresses and displacements are expressed in terms of two analytical functions of a complex variable, determined by the boundary conditions on the inclusion contour. It is assumed that the stress state of an inclusion is uniform (the tensor of nominal stresses is constant). This hypothesis made it possible to reduce the difficult nonlinear problem of conjugation of two elastic bodies to the solution of two more simpler problems for a plane with an elliptical hole. The validity of this hypothesis is justified by the fact that the constructed solution exactly satisfies all the equations and boundary conditions of the problem. The same hypothesis was used earlier by other authors to solve linear and nonlinear problems of an elliptical inclusion. In the article, a comparative analysis of the stresses and strains is carried out for two models of harmonic materials — semi-linear and John’s. Various variants of values of elasticity parameters of the inclusion and matrix have been considered.