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Seismic wave propagation is a complex process that is influenced by various factors, including the elastic properties of the medium and the initial stress condition. In this study, we have investigated the reflection and refraction phenomena of plane waves at orthotropic layer between two triclinic half-spaces with initial stresses. It has been assumed that the lower and upper half-spaces are composed of highly anisotropic triclinic crystalline material. The calculations have been performed to determine the phase velocity of the quasi waves. The theoretical expression of reflection/refraction coefficients and energy ratios of each reflected and refracted quasi waves has been derived in closed form. It has been shown that the reflection/refraction coefficients, phase velocity and energy ratios depend on the incident angle, initial stress, propagation vector, displacement vector and elastic properties of media. Numerical results demonstrate the significant impact of initial stresses on reflection and refraction coefficients and energy ratios. The slowness surface diagrams are established for various waves and energy conservation law is verified to ensure correctness of the study. This research contributes to understanding seismic wave propagation in complex anisotropic media, with applications in forward and inverse geophysical reflection/refraction problems. The findings can be used to improve our understanding of the Earth's internal structure and seismic exploration and monitoring. [ABSTRACT FROM AUTHOR]

The paper considers an axisymmetric stress state of a homogeneous elastic space with a circular disc-shaped crack, one of the edges of which is pressed into a cylindrical circular stamp with static friction. It is assumed that the contact zone is considered under the generalized law of dry friction, i.e. tangential contact stresses are proportional to normal contact pressure, while the proportionality coefficient depends on the radial coordinates of the points of the contacting surfaces and is directly proportional to them. Considering the fact that in this case the Abel images of contact stresses are also related in a similar way, the solution of the problem, with the help of rotation operators and theory of analytical functions, is reduced to an inhomogeneous Riemann problem for two functions and the closed solution in quadratures is constructed. A numerical analysis was carried out and regularities of changes in both normal and shear real contact stresses, as well as rigid displacement of the stamp depending on the physical and geometric parameters were revealed. [ABSTRACT FROM AUTHOR]

For a one dimensional 2-lattice we write down the momentum equation and the equation ruling the shift vector for the dynamic case as a first order system and characterize it in terms of its hyperbolicity. We use similar arguments for problems of increasing difficulty and tackle: one dimensional 3-lattices, three dimensional 2-lattices, three dimensional 3-lattices and finally the most general case of three dimensional (n + 1) -lattices. Our approach is confined to the geometrically and materially linear elastodynamic theory and is valid for generic anisotropic materials that have the multilattice structure. The main finding is that, with the assumptions adopted, the presence of each shift vector is related with zero eigenvalues when the system is written as a first order system. [ABSTRACT FROM AUTHOR]

The elastic properties of dilute solid suspensions with imperfectly bonded inclusions of ellipsoidal shape are estimated. The imperfect bonding is regarded as a sharp interface across which the displacement jumps in proportion to the surface traction. Elastic compliances of the matrix and inclusion phases can exhibit arbitrary anisotropy while that of the bonding exhibits an anisotropy that depends on the interface normal only. A variational framework is employed to generate pairs of elementary bounds, asymptotically exact results, and approximations for the effective elasticity tensor. Each member of the pair differs in the way the bonding compliance is averaged over the interfacial surface: an 'arithmetic' mean in one case and a 'harmonic' mean in the other case. The results are used to infer the most convenient approximation for a given range of material parameters. [ABSTRACT FROM AUTHOR]

We study the effective properties of a linear elastic or perfectly viscoplastic composite consisting of a soft matrix reinforced by possibly very stiff fibers. The effective material is characterized by the emergence of a discrepancy between the displacement in the fibers and the overall averaged displacement, giving rise to a concentration strain in the matrix. The effective energy stored in the fibers is a combination of stretching, bending and torsional energies. This work completes and corrects results previously obtained by the author and coworkers, who failed to notice the torsional contribution. [ABSTRACT FROM AUTHOR]

There are many situations where groundwater flow has high flow rates, causing large seepage forces. Examples are flows around highly conductive fractures and tunnels. We present a new analytic element for a tunnel in an elastic medium. We combined the analytic element method for groundwater flow with that for linear elasticity, and include the seepage force as a body force in the linearly elastic model. We represent tunnels and fractures as analytic elements. The solution for the case considered is limited to steady state flow and fluid-to-solid coupling. We present examples of the computed seepage forces around a tunnel and a fracture as well as a comparison with another numerical model. [ABSTRACT FROM AUTHOR]

The paper addresses the problem of finding the necessary and sufficient conditions to be satisfied by the engineering moduli of an anisotropic material for the elastic energy to be positive for each state of strain or stress. The problem is solved first in the most general case of a triclinic material and then each possible case of elastic syngony is treated as a special case. The method of analysis is based upon a rather forgotten theorem of linear algebra and, in the most general case, the calculations, too much involved, are carried out using a formal computation code. New, specific bounds, concerning some of the technical constants, are also found. [ABSTRACT FROM AUTHOR]

Nonlinear effects can enrich the propagation of elastic waves in mechanical metamaterials, which makes it possible to extend classical phenomena and functions in linear systems to nonlinear ones. In this work, rather than monochromatic waves in similar linear structures, the negative refraction is realized by mixing waves which are generated in nonlinear elastic wave metamaterials. Based on the stiffness matrix and plane wave expansion methods, dispersion curves of in–plane modes resulting from the collinear and non–linear mixings of two longitudinal waves are calculated. In the frequency spectrum, two propagating modes coalesce at exceptional points due to the coupling of in–plane modes, and those points at which the refraction type changes are also exceptional ones. Two kinds of negative refraction can be found in the mixing modes near exceptional points, but each of them needs to be induced in a specific configuration. Moreover, experiments are performed to support the pure negative refraction and beam splitting of the nonlinear elastic waves. Particularly, the parallel configuration is able to separate and extract the nonlinear mode when the single–mode negative refraction occurs, which shows the possibility to design elastic wave device by the negative refraction of nonlinear mixing waves. [ABSTRACT FROM AUTHOR]

Motivated from standard procedures in linear wave equations, we write the equations of classical elastodynamics as a linear symmetric hyperbolic system in terms of the displacement gradient ( u x ) and the velocity ( u t ); this is in contrast with common practice, where the stress tensor and the velocity are used as the basic variables. We accomplish our goal by a judicious use of the compatibility equations. The approach using the stress tensor and the velocity requires use of the time differentiated constitutive law as a field equation; the present approach is devoid of this need. The symmetric form presented here is based on a Cartesian decomposition of the variables and the differential operators that does not alter the Hamiltonian structure of classical elastodynamics. We comment on the differences of our approach with that using the stress tensor in terms of the differentiability of the coefficients and the differentiability of the solution. Our analysis is confined to classical elastodynamics, namely geometrically and materially linear anisotropic elasticity which we treat as a linear theory per se and not as the linearization of the nonlinear theory. We, nevertheless, comment on the symmetrization processes of the nonlinear theories and the potential relation of them with the present approach. [ABSTRACT FROM AUTHOR]

An elastic map T associates stress with strain in some material. A symmetry of T is a rotation of the material that leaves T unchanged, and the symmetry group of T consists of all such rotations. The symmetry class of T describes the symmetry group but without the orientation information. With an eye toward geophysical applications, Browaeys & Chevrot developed a theory which, for any elastic map T and for each of six symmetry classes Σ , computes the " Σ -percentage" of T . The theory also finds a "hexagonal approximation"—an approximation to T whose symmetry class is at least transverse isotropic. We reexamine their theory and recommend that the Σ -percentages be abandoned. We also recommend that the hexagonal approximations to T be replaced with the closest transverse isotropic maps to T . [ABSTRACT FROM AUTHOR]

We present a novel numerical method for calculating optimal design in topology optimization problems for 3D linear elastic structures. The algorithm is based on necessary conditions of optimality for problem which was obtained by relaxing the original one via the homogenization method in the sense of operators (G- or H-convergence), and can be implemented for self-adjoint problems. The method relies on recently obtained explicit expressions for the lower Hashin–Shtrikman bound on complementary energy and information on the microstructure that saturates the bound. We tested the algorithm on two benchmark examples, namely the cantilever and the bridge problem. The algorithm provides the solution in a first few iterations and the true composites appear in the optimal design. We also implement a penalization procedure to obtain classical design with slight increase of the cost functional. [ABSTRACT FROM AUTHOR]

We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Bayesian method for nonlinear elastic inverse problems. As a working model, we are interested in the inverse problem of restoring elastic properties from measured tissue displacement. In order to reduce the computational cost, we will use the following multi-fidelity model approach. First, we construct a surrogate low-fidelity DNNs-based model in the prior distribution, then use a certain number of simulations of high fidelity model associated with an adaptive strategy online to update the low-fidelity model locally. Numerical examples show that the proposed method can solve nonlinear elastic inverse problems efficiently and accurately. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new vectorized MATLAB-based algorithm for efficient serial computation of global matrix/force arising from finite element method (FEM) for meshes of any type and approximation order in linear elasticity. Because for-loops in MATLAB are very slow, we propose a modified process that takes advantage of vectorization and sparse assembly to achieve good performance while using the same memory as the standard algorithm. For this purpose, by using good programming practices, the implementation of this scheme is succinctly described and can be integrated into any MATLAB package dealing with FEM. Specifically, attention is paid to the calculation of the triplet (row index, column index, matrix components) as well as the assembly of the global stiffness matrix, mass matrix and force vector. Additionally, an extension of the proposed approach for Mindlin plate theory and functionally graded materials is outlined. Finally, the accuracy of this strategy is verified on selected numerical tests after comparing the obtained results with those of ABAQUS. In terms of performance, the study conducted on a set of meshes considering the standard algorithm and two other well-known MATLAB vectorized algorithms revealed that: (i) for a 2D beam problem meshed with P 1 -triangle elements, a speedup of about 8 and 15 is achieved with sparse and fsparse, respectively. (ii) for a 3D plate problem meshed with P 1 -tetrahedral elements, a speedup of about 4 and 8 is achieved with sparse and fsparse, respectively. When compared to ABAQUS performance, the proposed scheme results in a computational time that is about five times smaller. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive investigation of stress propagation in a two-dimensional elastic circular disk. To accurately describe the displacements and stress fields within the disk, we employ a scalar and vector potential approach, representing them as sums of Bessel functions. The determination of the coefficients for these expansions is accomplished in the Laplace space, where we compare the boundary conditions. By converting the inverse Laplace transforms into complex integrals using residue calculus, we successfully derive explicit expressions for the displacements and stress fields. Notably, these expressions encompass primary, secondary, and surface waves, providing a thorough characterization of the stress propagation phenomena within the disk. Our findings contribute to the understanding of mechanical behavior in disk-shaped components and can be valuable in the design and optimization of such structures across various engineering disciplines. [ABSTRACT FROM AUTHOR]

We show that the linear brittle Griffith energy on a thin rectangle Γ -converges after rescaling to the linear one-dimensional brittle Euler–Bernoulli beam energy. In contrast to the existing literature, we prove a corresponding sharp compactness result, namely a suitable weak convergence after subtraction of piecewise rigid motions with the number of jumps bounded by the energy. [ABSTRACT FROM AUTHOR]

Fiber orientation tensors (FOT) are widely used to approximate statistical orientation distributions of fibers within fiber-reinforced polymers. The design process of components made of such fiber-reinforced composites is usually accompanied by a virtual process chain. In this virtual process chain, process-induced FOT are computed in a flow simulation and transferred to the structural simulation. Within the structural simulation, effective macroscopic properties are identified based on the averaged information contained in the FOT. Solving the field equations in flow simulations as well as homogenization of effective stiffnesses necessitates the application of a closure scheme, computing higher-order statistical moments based on assumptions. Additionally, non-congruent spatial discretizations require an intermediate mapping operation. This mapping operation is required, if the discretization, i.e., mesh, of the flow simulation differs from the discretization of the structural simulation. The main objective of this work is to give an answer to the question: Does the sequence of closure and mapping influence the achieved results? It will turn out, that the order influences the result, raising the consecutive question: Which order is beneficial? Both questions are addressed by deriving a quantification of the closure-related uncertainty. The two possible sequences, mapping followed by closure and closure followed by mapping, yield strongly different results, with the magnitude of the deviation even exceeding the magnitude of a reference result. Graphical consideration reveals that for both transversely isotropic and planar FOT-input, invalid results occur if the mapping takes place prior to closure. This issue is retrieved by orientation averaging stiffness tensors. As a by-product, we explicitly define for the first time the admissible parameter space of orthotropic fourth-order fiber orientation tensors and define a distance measure in this parameter space. [ABSTRACT FROM AUTHOR]

Generically, a fully measured elasticity tensor has no material symmetry. For single crystals with a cubic lattice, or for the aeronautics turbine blades superalloys such as Nickel-based CMSX-4, cubic symmetry is nevertheless expected. It is in practice necessary to compute the nearest cubic elasticity tensor to a given raw one. Mathematically formulated, the problem consists in finding the distance between a given tensor and the cubic symmetry stratum. It has recently been proved that closed symmetry strata are affine algebraic sets (for any tensorial representation of the rotation group): they are defined by polynomial equations without requirement to polynomial inequalities. Such equations have furthermore been derived explicitly for the closed cubic elasticity stratum. We propose to make use of this mathematical property to formulate the distance to cubic symmetry problem as a polynomial (in fact quadratic) optimization problem, and to derive its quasi-analytical solution using the technique of Gröbner bases. The proposed methodology also applies to cubic Hill elasto-plasticity (where two fourth-order constitutive tensors are involved). [ABSTRACT FROM AUTHOR]

Within the framework of linear elasticity, we show that any displacement of a straight rod is the sum of a Bernoulli–Navier displacement and two terms, one for shearing and the other for warping. Then, we load a straight rod so that bending and shear contribute the same to the rotations of the cross-section. [ABSTRACT FROM AUTHOR]

The Cauchy relations distinguish between rari- and multi-constant linear elasticity theories. These relations are treated in this paper in a form that is invariant under two groups of transformations: indices permutation and general linear transformations of the basis. The irreducible decomposition induced by the permutation group is outlined. The Cauchy relations are then formulated as a requirement of nullification of an invariant subspace. A successive decomposition under rotation group allows to define the partial Cauchy relations and two types of elastic materials. We explore several applications of the full and partial Cauchy relations in physics of materials. The structure's deviation from the basic physical assumptions of Cauchy's model is defined in an invariant form. The Cauchy and non-Cauchy contributions to Hooke's law and elasticity energy are explained. We identify wave velocities and polarization vectors that are independent of the non-Cauchy part for acoustic wave propagation. Several bounds are derived for the elasticity invariant parameters. [ABSTRACT FROM AUTHOR]

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate. [ABSTRACT FROM AUTHOR]

We show we can derive the so called polar representation of 2D symmetric second and fourth order real tensors essentially just relying on the spectral theorem for unitary tensors in the complex field. The use of a coordinate-free approach allows us to clearly and methodically detect rotation-invariant quantities and to readily (and directly) deduce the representation theorems. [ABSTRACT FROM AUTHOR]

We review some works on non-classical boundary conditions, and loading devices inspired by discussions with late Professor J.L. Ericksen. For a circular contact area between a body and a linearly elastic, isotropic and homogeneous half space as a loading device, we estimate the constant in the classical Winkler's expression of the normal surface traction at a point of the contact region in terms of the normal to the surface displacement of the same point. We also provide expressions for the tangential tractions at points of the common interface between the body and the loading device. The surface tractions are in general functionals of the contact surface displacements and their tangential derivatives. It is shown that many boundary conditions fall outside the usual definition of natural boundary conditions and may require special treatments in numerically solving boundary-value problems. [ABSTRACT FROM AUTHOR]

We explicitly characterize boundary conditions that are compatible with low order variational principles. The freedom afforded by adding in a null Lagrangian without altering the Euler–Lagrange equation significantly expands the range of variationally admissible boundary conditions, although not all possibilities are permitted. Applications to several fundamental problems arising in elastostatics, including bars, beams, and plates, are presented. [ABSTRACT FROM AUTHOR]

The Steklov eigenvalues of the Lamé operator arise from the analysis of surface waves and vibration modes in a linearly elastic structure. In this paper, based on the Lagrange conforming finite element method, we propose and analyze an a posteriori error indicator of the residual type for the Steklov eigenvalue problem of the Lamé operator in linear elasticity. The a posteriori error indicator is computed locally from the approximate eigenpair. We prove the reliability and the efficiency of the error indicator for eigenfunctions as well as the reliability of the error indicator for eigenvalues. Finally, we present numerical experiments to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

Porous systems with different damping mechanisms have been extensively studied in the literature. In recent research, exponential decay of the energy has been achieved by assuming equal wave velocities, which may not hold in practice. In this study, we investigate the one-dimensional porous elastic system free of the second spectrum with Kelvin–Voigt damping. First, we establish the global well-posedness of the system using the Faedo–Galerkin technique, then we show that the exponential stability of the system does not require the assumption of equal wave speed for exponential decay to occur. A numerical scheme is proposed and analyzed. Furthermore, under additional regularity assumptions on the solution, we establish a priori error estimates and present numerical results demonstrating the exponential behavior of the energy. [ABSTRACT FROM AUTHOR]

Second-generation high-temperature superconducting (HTS) conductors, specifically rare earth-barium-copper-oxide (REBCO) coated conductor (CC) tapes, are promising candidates for high-energy and high-field superconducting applications. With respect to epoxy-impregnated REBCO composite magnets that comprise multilayer components, the thermomechanical characteristics of each component differ considerably under extremely low temperatures and strong electromagnetic fields. Traditional numerical models include homogenized orthotropic models, which simplify overall field calculation but miss detailed multi-physics aspects, and full refinement (FR) ones that are thorough but computationally demanding. Herein, we propose an extended multi-scale approach for analyzing the multi-field characteristics of an epoxy-impregnated composite magnet assembled by HTS pancake coils. This approach combines a global homogenization (GH) scheme based on the homogenized electromagnetic T-A model, a method for solving Maxwell's equations for superconducting materials based on the current vector potential T and the magnetic field vector potential A, and a homogenized orthotropic thermoelastic model to assess the electromagnetic and thermoelastic properties at the macroscopic scale. We then identify "dangerous regions" at the macroscopic scale and obtain finer details using a local refinement (LR) scheme to capture the responses of each component material in the HTS composite tapes at the mesoscopic scale. The results of the present GH-LR multi-scale approach agree well with those of the FR scheme and the experimental data in the literature, indicating that the present approach is accurate and efficient. The proposed GH-LR multi-scale approach can serve as a valuable tool for evaluating the risk of failure in large-scale HTS composite magnets. [ABSTRACT FROM AUTHOR]

The analysis of the mathematical and mechanical properties of thermoelastic coupling tensors in anisotropic laminates is the topic of this paper. Some theoretical results concerning the compliance tensors are shown and their mechanical consequences analyzed. Moreover, the case of thermally stable laminates, important for practical applications, is also considered. The study is carried out in the framework of the polar method, a mathematical formalism particularly well-suited for the analysis of planar anisotropic problems, introduced by Prof. G. Verchery in 1979. [ABSTRACT FROM AUTHOR]

Recently in [V. Markaki, D. Kourounis and A. Charalambopoulos, A dual self-monitored reconstruction scheme on the TV -regularized inverse conductivity problem, IMA J. Appl. Math. 86 2021, 3, 604–630], a novel reconstruction scheme has been developed for the solution of the inclusion problem in the inverse conductivity problem on the basis of a weighted self-guided regularization process generalizing the total variation approach. The present work extends this concept by implementing and investigating its applicability in the two-dimensional elasticity setting. To this end, we employ the framework of the reconstruction of linear and isotropic elastic structures described by their Lamé parameters. Numerical examples of increasingly challenging geometric complexities illustrate the enhanced accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

Under antiplane deformation, the problem of the contact interaction of a thin infinite stringer of finite width with an elastic 3D wedge-shaped body of an arbitrary opening angle is considered. It is assumed that one face of the wedge is rigidly clamped while a stringer is attached to the other face, which does not extend to the top of the wedge. The deformation of a stringer is described by the well-known Melan model for antiplane deformation, and the stress–strain state of an elastic wedge is described by the equations of elasticity theory. Two formulations of the contact problem are discussed according to the two models of stringer deformation. In the first formulation, as usual, external tangential loads and concentrated forces acting on the stringer are specified, and it is required to determine the tangential contact stresses arising under the stringer and axial stresses in it. In the second formulation of the problem, the elastic displacements of the stringer points are predetermined; it is required to determine the force factors acting on the stringer that provide the specified mode of elastic displacements. Using the Mellin integral transform, solving the contact problem under consideration is reduced to solving a singular integral equation (SIE) with a kernel in the form of a sum of the Cauchy kernel and a regular kernel in the first formulation and to solving a Fredholm integral equation (IE) of the first kind with a symmetric logarithmic kernel in the second formulation. The solution of both integral equations is constructed by the method of mechanical quadratures, based on the use of well-known Gauss-type quadrature formulas for calculating singular integrals at Chebyshev nodes, as well as Gauss-type quadrature formulas for calculating integrals containing logarithmic and signum regular kernels. Ultimately, solving the equation is reduced to solving a finite system of linear algebraic equations (SLAE). A numerical analysis of the problem was carried out, and regularities in the change in the main mechanical quantities of the problem were revealed in a quite wide range of changes in the characteristic physical and geometric parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse coefficients problem of recovering a shear modulus μ and density ρ of a medium from the Neumann-to-Dirichlet map. This inverse problem is motivated by the reconstruction of mechanical properties of tissues in non-destructive testing. We prove Lipschitz stability results for any dimension $ d \geq 2 $ d ≥ 2 , provided that the parameters μ and ρ have upper and lower bounds and belong to a known finite dimensional subspace. The proofs rely on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, combined with the techniques of localized potentials. [ABSTRACT FROM AUTHOR]

Nano-coatings hold the key to unlocking the vital potential of three-phase nanocomposites by urgently addressing thermal stress concentration. Consequently, the characteristics of double-interfacial heat flux and thermal stress have garnered considerable attention. In this paper, we introduce a set of interfacial conditions aimed at modeling double-interfacial phonon scattering and double-interfacial elasticity in three-phase nanocomposites. We derive closed-form solutions for the thermo-elastic field surrounding a circular nano-inclusion containing the nano-coatings. Through control over the nano-coating thickness and their properties, including thermal conductivity and interfacial phonon scattering coefficient, we can achieve a "neutral inclusions and coatings" temperature field, promoting uniform heat transfer. However, it is important to note that this approach leads to a noticeable increase in thermal stresses at the interfaces and the emergence of stress discontinuities due to the double-interfacial phonon scattering effect. Furthermore, the impact of coating thickness, coating thermal conductivity, and coating thermal expansion coefficient on the maximum interfacial thermal stress exhibits significant variability. Crucially, our findings reveal that adjusting the coating parameters can result in a substantial reduction in the overall thermal stress field's maximum value. This discovery carries significant implications for the design of nanocomposites and holds great promise for various applications. [ABSTRACT FROM AUTHOR]

We consider an optimization problem for the eigenvalues of a multi-material elastic structure that was previously introduced by Garcke et al. (Adv. Nonlinear Anal. 11:159–197, 2022). There, the elastic structure is represented by a vector-valued phase-field variable, and a corresponding optimality system consisting of a state equation and a gradient inequality was derived. In the present paper, we pass to the sharp-interface limit in this optimality system by the technique of formally matched asymptotics. Therefore, we derive suitable Lagrange multipliers to formulate the gradient inequality as a pointwise equality. Afterwards, we introduce inner and outer expansions, relate them by suitable matching conditions and formally pass to the sharp-interface limit by comparing the leading order terms in the state equation and in the gradient equality. Furthermore, the relation between these formally derived first-order conditions and results of Allaire and Jouve (Comput. Methods Appl. Mech. Eng. 194:3269–3290, 2005) obtained in the framework of classical shape calculus is discussed. Eventually, we provide numerical simulations for a variety of examples. In particular, we illustrate the sharp-interface limit and also consider a joint optimization problem of simultaneous compliance and eigenvalue optimization. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a low-order robust numerical method for the linear elasticity problem. The method is based on a Bernardi–Raugel-like H (div) -conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. 42 (2022) 3711–3734]. Therein, the lowest-order H (div) -conforming Raviart–Thomas space ( RT 0 ) was added to the classical conforming P 1 × P 0 pair to meet the inf-sup condition, while preserving the divergence constraint and some important features of conforming methods. Due to the inf-sup stability of the P 1 ⊕ RT 0 × P 0 pair, a locking-free elasticity discretization with respect to the Lamé constant λ can be naturally obtained. Moreover, our scheme is gradient-robust for the pure and homogeneous displacement boundary problem, that is, the discrete H 1 -norm of the displacement is O (λ - 1) when the external body force is a gradient field. We also consider the mixed displacement and stress boundary problem, whose P 1 ⊕ RT 0 discretization should be carefully designed due to a consistency error arising from the RT 0 part. We propose both symmetric and nonsymmetric schemes to approximate the mixed boundary case. The optimal error estimates are derived for the energy norm and/or L 2 -norm. Numerical experiments demonstrate the accuracy and robustness of our schemes. [ABSTRACT FROM AUTHOR]

In modern large-scale storage systems, locally repairable codes(LRCs) improve the repair efficiency when node failure occurs. This work discusses several new methods for constructing LRCs with small localities using the generator and parity-check matrices approach. Many new LRCs have been found, all distance-optimal (d-optimal) or best-known (BK). Based on the newly obtained LRCs and known results, a database presenting all d-optimal (or BK) binary LRCs with a relatively small locality where n ≤ 40 is given. The majority of them are r-optimal or near r-optimal. [ABSTRACT FROM AUTHOR]

Korn's inequality has been at the heart of much exciting research since its first appearance in the beginning of the 20th century. Many are the applications of this inequality to the analysis and construction of discretizations of a large variety of problems in continuum mechanics. In this paper, we prove that the classical Korn inequality holds true in anisotropic Sobolev spaces. We also prove that an extension of Korn's inequality, involving non-linear continuous maps, is valid in such spaces. Finally, we point out that another classical inequality, namely Poincaré's inequality, also holds in anisotropic Sobolev spaces. [ABSTRACT FROM AUTHOR]

Surface cracks are commonly observed in coatings and films. When structures with coatings are subject to stretching, opening mode cracks are likely to form on the surface, which may further lead to other forms of damage, such as interfacial delamination and substrate damage. Possible crack forms include cracks extending towards the interface and channeling across the film. In this paper, a two-dimensional numerical model is proposed to obtain the structural strain energy at arbitrary crack lengths for bilayer structures under uniaxial tension. The energy release rate and structural stress intensity factors can be obtained accordingly, and the effects of geometry and material features on fracture characteristics are investigated, with most crack patterns being confirmed as unstable. The proposed model can also facilitate the analysis of the stress distribution in periodic crack patterns of films. The results from the numerical model are compared with those obtained by the finite element method (FEM), and the accuracy of the theoretical results is demonstrated. [ABSTRACT FROM AUTHOR]

We study the numerical approximation of the inverse scattering problem in the two-dimensional homogeneous isotropic linear elasticity with an unknown linear load given by a square matrix. For both backscattering data and fixed-angle scattering data, we show how to obtain numerical approximations of the so-called Born approximations and propose new iterative algorithms that provide sequences of approximations to the unknown load. Numerical evidences of the convergence for not too large loads are also given. [ABSTRACT FROM AUTHOR]

We achieve neutrality of a double coated circular inhomogeneity embedded in an infinite matrix subjected to uniform in-plane stresses at infinity. The introduction of the double coated circular inhomogeneity does not disturb the original uniform in-plane stress distribution in the matrix. Consequently, we obtain exact representations of the effective transverse shear modulus and effective transverse Poisson's ratio of double coated disk assemblages of various sizes completely replacing the matrix. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze a virtual element method for the two dimensional elasticity spectral problem, where the polygonal meshes allow for the presence of small edges. Under this approach and with the aid of the theory of compact operators, we prove convergence for the proposed VEM and error estimates. We report a series of numerical tests in order to assess the performance of the method where we analyze the effects of the Poisson ratio on the computation of the order of convergence, together with the effects of the stabilization term on the arising of spurious eigenvalues. [ABSTRACT FROM AUTHOR]

To improve the thermoelectric converting performance in applications such as power generation, reutilization of heat energy, refrigeration, and ultrasensitive sensors in scramjet engines, a thermoelectric film/substrate system is widely designed and applied, whose interfacial behavior dominates the strength and service life of thermoelectric devices. Herein, a theoretical model of a thermoelectric film bonded to a graded substrate is proposed. The interfacial shear stress, the normal stress in the thermoelectric film, and the stress intensity factors affected by various material and geometric parameters are comprehensively studied. It is found that adjusting the inhomogeneity parameter of the graded substrate, thermal conductivity, and current density of the thermoelectric film can reduce the risk of interfacial failure of the thermoelectric film/graded substrate system. Selecting a stiffer and thicker thermoelectric film is advantageous to the reliability of the thermoelectric film/graded substrate system. The results should be of great guiding significance for the present and upcoming applications of thermoelectric materials in various fields. [ABSTRACT FROM AUTHOR]

Korn's first inequality states that there exists a constant such that the L 2 -norm of the infinitesimal displacement gradient is bounded above by this constant times the L 2 -norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body ℬ. This inequality is known to hold when the L 2 -norm is replaced by the L p -norm for any p ∈ (1 , ∞) . However, if p = 1 or p = ∞ the resulting inequality is false. It was previously shown that if one replaces the L ∞ -norm by the BMO -seminorm (Bounded Mean Oscillation) then one maintains Korn's inequality. (Recall that L ∞ (B) ⊂ BMO (B) ⊂ L p (B) ⊂ L 1 (B) , 1 < p < ∞ .) In this manuscript it is shown that Korn's inequality is also maintained if one replaces the L 1 -norm by the norm in the Hardy space H 1 , the predual of BMO . One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of ℬ. [ABSTRACT FROM AUTHOR]

The classical problem of a two-dimensional infinite linear elastic wedge, with opening angle of 2 α , subjected to the action of a concentrated couple at its tip, was considered by Carothers. The solution, which presents a quadratic singularity in the stress field, suffers a spurious behavior at the critical angle α = α ∗ ≃ 0.715 π , when the stress grows unboundedly within the body. This inconsistency is usually referred to as the "wedge paradox". Here, relying on rather intuitive arguments, we present a representation for the stress field in the wedge which captures other states characterized by a quadratic singularity. This is first obtained by considering an auxiliary problem, in which the wedge is ideally split into three wedges under tip couples, whose values are determined by compatibility conditions in terms of stress and displacement on the common lines. Remarkably, we find that the state of stress varies continuously on α ; at α = α ∗ the stress does not vanish, although the action at the wedge tip has zero resultant and zero moment resultant. An alternative derivation of the elastic solution in the wedge, which relies upon the notion of nuclei of strain, suggests that, at the critical angle, the applied action is that of dipoles without moment at the tip point. We conclude that it is the form of Carothers' solution that cannot account for all states of stress characterized by a quadratic singularity at the vertex. The paradox naturally disappears in the proposed representation. [ABSTRACT FROM AUTHOR]

We establish the uniformity of stresses and strains inside an imperfectly bonded elastic ellipsoidal inhomogeneity embedded within an infinite elastic matrix subjected to uniform remote stresses and strains. Both the inhomogeneity and the matrix can be either isotropic elastic or generally anisotropic elastic. The imperfect interface is described by a spring-type imperfect interface characterized by a single imperfect interface function. The same degree of imperfection of the ellipsoidal interface is realized in both the normal and tangential directions. We identify the imperfect interface function leading to an internal uniform field. The internal uniform strains and stresses within the ellipsoidal inhomogeneity are obtained with the aid of Eshelby's equivalent inclusion method. [ABSTRACT FROM AUTHOR]

Critical velocities of a two-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are analytically derived by using a first-order shear deformation shell theory incorporating the transverse shear, rotary inertia and material anisotropy. The composite tube consists of two perfectly bonded axisymmetric circular cylindrical layers of dissimilar materials, which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form expressions for four critical velocities are first derived for the general case by including the effects of transverse shear, rotary inertia, material orthotropy and radial stress. The formulas for composite tubes without the transverse shear, rotary inertia or radial stress effect and with simpler anisotropy are then obtained as special cases. In addition, it is shown that the model for a single-layer, homogeneous tube is included in the current model as a special case. To illustrate the newly derived closed-form formulas, a composite tube with an isotropic inner layer and an orthotropic outer layer is analyzed as an example. The numerical values of the lowest critical velocity of the two-layer composite tube predicted by the new formulas compare well with existing data. [ABSTRACT FROM AUTHOR]