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We propose a boundary-element-based method for crack problems in multilayered elastic medium that consists of a set of individually homogeneous strata. The method divides the medium along the slit-like crack surface so that the effects of the elements placed along one crack surface become distinguishable from those placed along the other surface. As a result, the direct method that cannot be directly applied for crack problems turns out to be applicable. After that, we derive a recursive formula that obtains a “stiffness matrix” for each layer by exploiting the chain-like structure of the system, enabling a sequential computation to solve the displacements on the crack surface in each layer “consecutively” in a descending order from the very top layer to the very bottom one. In our method, the final system of equations only contains the unknown displacements on the crack surface, ensuring the efficiency of the method. The numerical examples demonstrate better accuracy and broader applicability of our method compared to the displacement discontinuity method and more-acceptable efficiency of our method compared to the conventional direct method.

Crack problems in multilayered elastic media have attracted extensive attention for years due to their wide applications in both a theoretical analysis and practical industry. The boundary element method (BEM) is widely chosen among various numerical methods to solve the crack problems. Compared to other numerical methods, such as the phase field method (PFM) or the finite element method (FEM), the BEM ensures satisfying accuracy, broad applicability, and satisfactory efficiency. Therefore, this paper reviews the state-of-the-art progress in a boundary-element analysis of the crack problems in multilayered elastic media by concentrating on implementations of the two branches of the BEM: the displacement discontinuity method (DDM) and the direct method (DM). The review shows limitation of the DDM in applicability at first and subsequently reveals the inapplicability of the conventional DM for the crack problems. After that, the review outlines a pre-treatment that makes the DM applicable for the crack problems and presents a DM-based method that solves the crack problems more efficiently than the conventional DM but still more slowly than the DDM. Then, the review highlights a method that combines the DDM and the DM so that it shares both the efficiency of the DDM and broad applicability of the DM after the pre-treatment, making it a promising candidate for an analysis of the crack problems. In addition, the paper presents numerical examples to demonstrate an even faster approximation with the combined method for a thin layer, which is one of the challenges for hydraulic-fracturing simulation. Finally, the review concludes with a comprehensive summary and an outlook for future study.

Topologically protected waves in classical media provide unique opportunities for one-way wave transport and immunity to defects. Contrary to acoustics and electromagnetics, their observation in elastic solids has so far been elusive because of the presence of multiple modes and their tendency to hybridize at interfaces. Here, we report on the experimental investigation of topologically protected helical edge modes in elastic plates patterned with an array of triangular holes, along with circular holes that produce an accidental degeneracy of two Dirac cones. Such a degeneracy is subsequently lifted by careful breaking of the symmetry along the thickness direction, which emulates the spin orbital coupling in the quantum spin Hall effect. The joining of two plates that are mirror-symmetric copies of each other about the plate midthickness introduces a nontrivial interface that supports helical edge waves. The experimental observation of these topologically protected wave modes in elastic continuous plates opens avenues for the practical realization of structural components with topologically nontrivial waveguiding properties and their application to elastic waveguiding and confinement.

In marine seismic exploration, ocean-bottom cable techniques accurately record the multicomponent seismic wavefield; however, the seismic wave propagation in fluid–solid media cannot be simulated by a single wave equation. In addition, when the seabed interface is irregular, traditional finite-difference schemes cannot simulate the seismic wave propagation across the irregular seabed interface. Therefore, an acoustic–elastic forward modeling and vector-based P- and S-wave separation method is proposed. In this method, we divide the fluid–solid elastic media with irregular interface into orthogonal grids and map the irregular interface in the Cartesian coordinates system into a horizontal interface in the curvilinear coordinates system of the computational domain using coordinates transformation. The acoustic and elastic wave equations in the curvilinear coordinates system are applied to the fluid and solid medium, respectively. At the irregular interface, the two equations are combined into an acoustic–elastic equation in the curvilinear coordinates system. We next introduce a full staggered-grid scheme to improve the stability of the numerical simulation. Thus, separate P- and S-wave equations in the curvilinear coordinates system are derived to realize the P- and S-wave separation method.

Using a three-variable higher-order shear deformation theory (HSDT), this research proposes an analytical method for studying the free vibration and stability of perfect and imperfect functionally graded (FG) beams resting on variable elastic foundations (VEFs). Unlike the other HSDTs, in this study, the number of unknown functions involved is only three, while the other HSDTs include four unknown functions. Besides, this theory meets the boundary requirements of zero tension on the beam surfaces and allows for hyperbolic distributions of transverse shear stresses without the need for shear correction factors. The elastic medium is supposed to have two parameters (i.e., Winkler–Pasternak foundations), with the Winkler parameter in the longitudinal direction being variable variations (linear, parabolic, sinusoidal, cosine, exponential, and uniform) and the Pasternak parameter being fixed, at first.1 The effective material characteristics of the FG beam are assumed to follow a simple power-law distribution in the thickness direction. Furthermore, the influence of porosity is investigated by considering four distinct types of porosity distribution patterns. First, the equations of motion are derived using Hamilton’s principle, and then Navier’s method is used to solve the system of equations for the FG beam with simply supported ends analytically. The correctness of the current formulation is demonstrated by comparing them with the results of open literature. Finally, parametric studies are done to explore the impacts of various parameters on the free vibration and buckling behaviors of FG beams. The new theory is shown to be not only correct but also simple in predicting the free vibration and buckling responses of FG beams resting on VEFs.

A finite element model using four-unknown shear deformation theory integrated with the nonlocal theory is proposed for the bending and free vibration analysis of functionally graded (FG) nanoplates resting on elastic foundations. The present study developed the four-node quadrilateral element using Lagrangian and Hermitian interpolation functions for analysis of the membrane and bending displacement fields of FG nanoplates. Such a finite element formulation is suitable to investigate for the FG nanoplates resting on the elastic medium foundation with the stiffness matrices, the mass matrices and the load vectors using the second derivatives. The material properties of FG nanoplates are assumed to vary through the thickness direction by a power rule distribution of volume-fractions of the constituents. The equation of motion for FG nanoplates resting on the elastic foundation is obtained through Hamilton’s principle. Several numerical results are presented to demonstrate the accuracy and reliability of the present approach in comparison with other existing methods. In addition, the effects of geometrical parameters, material parameters, nonlocal parameters on the static bending and the free vibration responses of the nanoplates is also investigated in detail.

In this study, free vibration analyses of embedded carbon and silica carbide nanotubes lying on an elastic matrix are performed based on Eringen’s nonlocal elasticity theory. These nanotubes are modeled as nanobeam and nanorod. Elastic matrix is considered as Winkler–Pasternak elastic foundation and axial elastic media for beam and rod models, respectively. The vibration formulations of the beam and rod are derived by utilizing Hamilton’s principle. The obtained equations of motions are solved by the method of separation of variables and finite element-based Hermite polynomials for various boundary conditions. The effects of boundary conditions, system modeling, structural sizes such as length, cross-sectional sizes, elastic matrix, mode number, and nonlocal parameters on the natural frequencies of these nanostructures are discussed in detail. Moreover, the availability of size-dependent finite element formulation is investigated in the vibration problem of nanobeams/rods resting on an elastic matrix.

The two-dimensional scattering of elastic waves by a cluster of aligned circular cylindrical elastic inclusions embedded in a circular cylindrical domain of an infinitely extended elastic medium is analysed numerically. The analysis is based on a semi-analytical method where the scattered waves are expressed as the eigenfunction expansions of the time-harmonic displacement potentials and the expansion coefficients are determined by a numerical collocation technique. The wave fields within and around the cluster are demonstrated for different spatial arrangements of inclusions in the cluster. It is also examined if the scattering characteristics of the cluster can be simulated by a single equivalent scatterer which has the effective homogenized properties of the matrix/inclusion composite. As a result, it is shown that for sufficiently small inclusion radius-wavelength ratios, the scattering characteristics of the cluster are relatively insensitive to the spatial inclusion arrangements therein, and can be reasonably simulated by those of the equivalent scatterer.

Torsional vibration response of a circular nanoshaft, which is restrained by the means of elastic springs at both ends, is a matter of great concern in the field of nano-/micromechanics. Hence, the complexities arising from the deformable boundary conditions present a formidable obstacle to the attainment of closed-form solutions. In this study, a general method is presented to calculate the torsional vibration frequencies of functionally graded porous tube nanoshafts under both deformable and rigid boundary conditions. Classical continuum theory, upgraded with nonlocal strain gradient elasticity theory, is employed to reformulate the partial differential equation of the nanoshaft. First, torsional vibration equation based on the nonlocal strain gradient theory is derived for functionally graded porous nanoshaft embedded in an elastic media via Hamilton’s principle. The ordinary differential equation is found by discretizing the partial differential equation with the separation of variables method. Then, Fourier sine series is used as the rotation function. The necessary Stokes' transformation is applied to establish the general eigenvalue problem including the different parameters. For the first time in the literature, a solution that can analyze the torsional vibration frequencies of functionally graded porous tube shafts embedded in an elastic media under general (elastic and rigid) boundary conditions on the basis of nonlocal strain gradient theory is presented in this study. The results obtained show that while the increase in the material length scale parameter, elastic media and spring stiffnesses increase the frequencies of nanoshafts, the increase in the nonlocal parameter and functionally grading index values decreases the frequencies of nanoshafts. The detailed effects of these parameters are discussed in the article.

Nondestructive testing (NDT) is a major method in practical applications such as material exploration and safety assessment. The interaction of ultrasonic waves with roughness material is crucial in NDT, thus the related ultrasonic diffraction theory at the rough interface is widely investigated and analyzed, especially for the periodic elastic medium interface. However, the actual surface is usually porous due to the long-term moisture infiltration. In this paper, the reflection and transmission characteristics of ultrasonic waves on the fluid-porous medium periodic rough surfaces are studied in details.