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In this paper we identify a set of two-dimensional variational inequalities governing the displacement of a linearly elastic flexural shell subjected to a confinement condition, expressing that all the points of the admissible deformed configurations remain in a given half-space. The action of friction is neglected. [ABSTRACT FROM AUTHOR]

Invaginations are partial enclosures formed by surfaces. Typically formed by biological membranes; they abound in nature. In this paper, we consider fundamentally different structures: elastically stabilized invaginations. Focusing on spherical invaginations formed by elastic membranes, we carried out experiments and mathematical modeling to understand the stress and strain fields underlying stable structures. Friction plays a key role in stabilization, and consequently the required force balance is an inequality. Using a novel scheme, we were able to find stable solutions of the balance equations for different models of elasticity, with reasonable agreement with experiments. [ABSTRACT FROM AUTHOR]

In this paper we analyze a virtual element method for the two dimensional elasticity problem allowing small edges. With this approach, the classic assumptions on the geometrical features of the polygonal meshes can be relaxed. In particular, we consider only star-shaped polygons for the meshes. Suitable error estimates are presented, where a rigorous analysis on the influence of the Lamé constants in each estimate is presented. We report numerical tests to assess the performance of the method. [ABSTRACT FROM AUTHOR]

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model, which cannot describe certain nonequilibrium phenomena by finite collisions of particles, decreasing the fidelity of the solution. Based on an alternative formulation of the targeted essentially non-oscillatory (TENO) scheme, a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems. The numerical flux is constructed by the TENO interpolation of the solution and its derivatives, instead of being reconstructed from the physical flux. The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations. The numerical methods are verified by several classical plasma problems. The results show that compared with the original TENO scheme, the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation. [ABSTRACT FROM AUTHOR]

Wetting of a liquid droplet on another liquid substrate is governed by the well-known Neumann equations. The present work aims to develop an explicit modified version of the Neumann equations for axisymmetric wetting of a liquid droplet on a highly stretched elastic membrane of non-zero bending rigidity. An explicit modified form of the Neumann equations is derived to determine the two contact angles, which is reduced to Young's equation for a liquid droplet on a rigid membrane or to the Neumann equations for a liquid droplet on another liquid substrate. Further implications of the modified Neumann equations are examined by comparison with some previous results reported in the recent literature, particularly considering the ranges of material and geometrical parameters of the liquid droplet-membrane system which have not been recently addressed in the literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with longtime dynamics of semilinear Lamé systems ∂ t 2 u - μ Δ u - (λ + μ) ∇ div u + α ∂ t u + f (u) = b , defined in bounded domains of R 3 with Dirichlet boundary condition. Firstly, we establish the existence of finite dimensional global attractors subjected to a critical forcing f(u). Writing λ + μ as a positive parameter ε , we discuss some physical aspects of the limit case ε → 0 . Then, we show the upper-semicontinuity of attractors with respect to the parameter when ε → 0 . To our best knowledge, the analysis of attractors for dynamics of Lamé systems has not been studied before. [ABSTRACT FROM AUTHOR]

We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing thin infinite elastic rods with material coefficients that rapidly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the rod thickness and the period of material oscillations, which are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under the assumption on material symmetries as well as in the general case when these two types of displacements are intertwined. [ABSTRACT FROM AUTHOR]

The surface fracture toughness is an important mechanical parameter for studying the failure behavior of air plasma sprayed (APS) thermal barrier coatings (TBCs). As APS TBCs are typical multilayer porous ceramic materials, the direct applications of the traditional single edge notched beam (SENB) method that ignores those typical structural characters may cause errors. To measure the surface fracture toughness more accurately, the effects of multilayer and porous characters on the fracture toughness of APS TBCs should be considered. In this paper, a modified single edge V-notched beam (MSEVNB) method with typical structural characters is developed. According to the finite element analysis (FEA), the geometry factor of the multilayer structure is recalculated. Owing to the narrower V-notches, a more accurate critical fracture stress is obtained. Based on the Griffith energy balance, the reduction of the crack surface caused by micro-defects is corrected. The MSEVNB method can measure the surface fracture toughness more accurately than the SENB method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the interfacial behavior of a thin one-dimensional (1D) hexagonal quasicrystal (QC) film bonded on an elastic substrate subjected to a mismatch strain due to thermal variation. The contact interface is assumed to be non-slipping, with both perfectly bonded and debonded boundary conditions. The Fourier transform technique is adopted to establish the integral equations in terms of interfacial shear stress, which are solved as a linear algebraic system by approximating the unknown phonon interfacial shear stress via the series expansion of the Chebyshev polynomials. The expressions are explicitly obtained for the phonon interfacial shear stress, internal normal stress, and stress intensity factors (SIFs). Finally, based on numerical calculations, we briefly discuss the effects of the material mismatch, the geometry of the QC film, and the debonded length and location on stresses and SIFs. [ABSTRACT FROM AUTHOR]

A fuzzy arithmetic framework for the efficient possibilistic propagation of shape uncertainties based on a novel fuzzy edge detection method is introduced. The shape uncertainties stem from a blurred image that encodes the distribution of two phases in a composite material. The proposed framework employs computational homogenisation to upscale the shape uncertainty to a effective material with fuzzy material properties. For this, many samples of a linear elasticity problem have to be computed, which is significantly sped up by a highly accurate low-rank tensor surrogate. To ensure the continuity of the underlying mapping from shape parametrisation to the upscaled material behaviour, a diffeomorphism is constructed by generating an appropriate family of meshes via transformation of a reference mesh. The shape uncertainty is then propagated to measure the distance of the upscaled material to the isotropic and orthotropic material class. Finally, the fuzzy effective material is used to compute bounds for the average displacement of a non-homogenized material with uncertain star-shaped inclusion shapes. [ABSTRACT FROM AUTHOR]

We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes. [ABSTRACT FROM AUTHOR]

The Steigmann–Ogden (S–O) model has been widely applied in the literature to study the surface/interface bending effects arising from surface/interface energy on the elastic behavior of nanostructures and nanocomposites. In the existing analytic solutions for an elastic inclusion-matrix system with linearized versions of the S–O model, however, the interface bending moment was consistently defined based on an incomplete curvature formula which neglects the contribution of interface stretch to the change in the curvature of the interface during deformation. In this paper, we are concerned with the plane deformation of an elastic interface-bulk system incorporating a modified linearized version of the S–O model in which the interface bending moment is defined by an accurate first-order formula describing the change in the curvature of the interface under arbitrary small deformation. We formulate the corresponding boundary condition for an arbitrary curved interface in terms of the complex potential functions of the surrounding bulk. We then apply this complex-variable formulation of the boundary condition to the plane deformation of a circular inclusion within an elastic infinite matrix undergoing a uniform far-field loading and derive a closed-form solution for the stress field around the inclusion and explicit expressions for the effective properties of the composite structure homogenized by the dilute and Mori–Tanaka methods. We identify a revised result that the stress distribution inside the circular inclusion keeps uniform for all types of uniform remote loading when the normalized interface stretching rigidity is six times the normalized interface bending rigidity. Numerical examples are given to compare the current results of the stress field and effective properties with those corresponding to the use of the incomplete curvature formula. [ABSTRACT FROM AUTHOR]

This paper studies surface waves called Bleustein–Gulyaev (BG) waves in piezoelectricity. They propagate along the surface of a homogeneous piezoelectric half-space whose constituent material has C 6 hexagonal symmetry, where the surface is subject to the mechanically-free and electrically-closed condition. We revisit the Barnett–Lothe integral formalism for general piezoelectricity and give straightforward proofs, which only use the positive definiteness of the elasticity tensor and of the dielectric tensor, to derive fundamental properties of the Barnett–Lothe tensors. This leads us to obtain a criterion for the existence of subsonic surface waves. Moreover, when the waves propagate in the direction of the 1-axis along the surface of the piezoelectric half-space x 2 ≤ 0 of C 6 hexagonal symmetry whose 6-fold axis of rotational symmetry coincides with the 3-axis, we compute explicitly the phase velocity of the BG waves and investigate its perturbation, i.e., the shift in the velocity due to a perturbation of the material constants which need not have any symmetry. [ABSTRACT FROM AUTHOR]

We consider a special approach to investigate a mixed boundary value problem (BVP) for the Lamé system of elasticity in the case of three-dimensional bounded domain Ω ⊂ R 3 , when the boundary surface S = ∂ Ω is divided into two disjoint parts, S D and S N , where the Dirichlet and Neumann type boundary conditions are prescribed respectively for the displacement vector and stress vector. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations which do not contain neither extensions of the Dirichlet or Neumann data, nor the Steklov–Poincaré type operator. Moreover, the right hand sides of the resulting pseudodifferential system are vectors coinciding with the Dirichlet and Neumann data of the problem under consideration. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space [ W 2 1 (Ω) ] 3 and representability of solutions in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that the operator is invertible in the L p -based Besov spaces with 4 3 < p < 4 , which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ‾ with α = 1 2 − ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

Approximations for the elastic properties of dilute solid suspensions with imperfect interfacial bonding are derived and assessed. A variational procedure is employed in such a way that the resulting approximations reproduce exact results for weakly anisotropic but otherwise arbitrarily large interfacial compliances. Two approximations are generated which display the exact same format but differ in the way the interfacial compliance is averaged over the interfaces: the first approximation depends on an 'arithmetic' mean while the second approximation depends on a 'harmonic' mean. Both approximations allow for arbitrary elastic anisotropy of the constitutive phases but are restricted to suspended inclusions of spherical shape. The approximations are applied to a class of isotropic suspensions and confronted to full-field numerical simulations for assessment. Simulations are performed by means of a Fast Fourier Transform algorithm suitably implemented to handle dilute suspensions with imperfect interfaces. Also included in the comparisons are available results for suspensions with extremely anisotropic bondings. Overall, the 'harmonic' approximation is found to be much more precise than the 'arithmetic' approximation. The finding is of practical relevance given the widespread use of 'arithmetic' approximations in existing descriptions based on modified Eshelby tensors. [ABSTRACT FROM AUTHOR]

The three-dimensional traction problem for steady elastic oscillations equations is studied. Representability of its solution by means of a double layer potential is considered instead of the more usual simple layer potential. [ABSTRACT FROM AUTHOR]

This paper develops a family of new weak Galerkin (WG) finite element methods (FEMs) for solving linear elasticity in the primal formulation. For a convex quadrilateral mesh, degree k ≥ 0 vector-valued polynomials are used independently in element interiors and on edges for approximating the displacement. No penalty or stabilizer is needed for these new methods. The methods are free of Poisson-locking and have optimal order (k + 1) convergence rates in displacement, stress, and dilation (divergence of displacement). Numerical experiments on popular test cases are presented to illustrate the theoretical estimates and demonstrate efficiency of these new solvers. Extension to cuboidal hexahedral meshes is briefly discussed. [ABSTRACT FROM AUTHOR]

In mathematical physics and beyond, one encounters many beautiful inequalities that relate geometric or physical quantities describing the shape or size of a set. Such isoperimetric inequalities often have a long history and many important applications. For instance, the eponymous and most classical of all isoperimetric inequalities was known already in antiquity. It asserts that among all closed planar curves of a given length, the circles with perimeter equal to that length, and only they, enclose the largest area. Though not nearly as well-known, an isoperimetric inequality conjectured by Saint-Venant in the 1850s and first proved by Pólya almost a century later, is also very beautiful and important. By presenting a short proof as well as two simple physical interpretations, this article illustrates why the result deserves to be cherished by every student of applied analysis. [ABSTRACT FROM AUTHOR]

We study the homogenization of a linear elastic solid reinforced by a small volume fraction of very stiff fibers. The effective material is characterized by the emergence of concentrations of strain energies around and within the fibers. The former is expressed in terms of the discrepancy appearing between the averaged effective displacements in the fibers and in the matrix, by means of a tensor-valued capacity of the cross-sections of the fibers. The latter corresponds to a combination of stretching, bending and torsional energies, where the torsional contribution is specific to anisotropic constitutive components. [ABSTRACT FROM AUTHOR]

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations. [ABSTRACT FROM AUTHOR]

The use of one theorem of spectral analysis proved by Bordoni on a model of linear anisotropic beam proposed by the author allows the determination of the variation range of vibration frequencies of a beam in two typical restraint conditions. The proposed method is very general and allows its use on a very wide set of problems of engineering practice and mathematical physics. [ABSTRACT FROM AUTHOR]

We consider the conductivity problem with a simply connected or multi-coated inclusion in two dimensions. The potential perturbation due to an inclusion admits a classical multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). The GPTs have been fundamental building blocks in conductivity inclusion problems. In this paper, we present a new concept of geometric multipole expansion and its expansion coefficients, named the Faber polynomial polarization tensors (FPTs), using the conformal mapping and the Faber polynomials associated with the inclusion. The proposed expansion leads us to a series solution method for a simply connected or multi-coated inclusion of general shape, while the classical expansion leads us to a series solution only for a single- or multilayer circular inclusion. We also provide matrix expressions for the FPTs using the Grunsky matrix of the inclusion. In particular, for the simply connected inclusion with extreme conductivity, the FPTs admit simple formulas in terms of the conformal mapping associated with the inclusion. As an application of the concept of the FPTs, we construct semi-neutral inclusions of general shape that show relatively negligible field perturbations for low-order polynomial loadings. These inclusions are of the multilayer structure whose material parameters are determined such that some coefficients of geometric multipole expansion vanish. [ABSTRACT FROM AUTHOR]

It is well-known that residual thermal stress poses a significant threat to the stability and reliability of brazed structures. However, theoretical studies focusing on the effects of residual thermal stress on even simple brazed structures remain largely absent from the literature. In this paper, we use complex variable methods to derive theoretical solutions characterizing residual stress fields for annular and bilayer structures. In addition, using a series of numerical examples, we investigate the main factors determining residual thermal stress in brazing. Our results show that both radial and hoop stress are present in the annular structure while the stress component parallel to the interface is the unique principal stress in the bilayer structure. In an annular structure in which the total thickness of materials coincides with the length of the inner radius, the residual thermal stress in the outer base material is four times higher than that of the inner base material. This is attributed to the fact that the two materials have different curvatures. In the case of a thin-wall structure in which the inner radius is ten times that of the total thickness of materials, the corresponding residual thermal stress has no more than 15% error when compared to that from the bilayer structure. Since the distortion of brazed structures is due mainly to the effects of residual thermal stress within the corresponding base materials, it would seem practical to minimize the thickness of the solder used in the brazing process. [ABSTRACT FROM AUTHOR]

In this paper, we obtain Green's functions of two-dimensional (2D) piezoelectric quasicrystal (PQC) in half-space and bimaterials. Based on the elastic theory of QCs, the Stroh formalism is used to derive the general solutions of displacements and stresses. Then, we obtain the analytical solutions of half-space and bimaterial Green's functions. Besides, the interfacial Green's function for bimaterials is also obtained in the analytical form. Before numerical studies, a comparative study is carried out to validate the present solutions. Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force, the line dislocation, the line charge, and the phason line force. As a result, the coupling effect among the phonon field, the phason field, and the electric field is prominent, and the butterfly-shaped contours are characteristic in 2D PQCs. In addition, the changes of material parameters cause variations in physical quantities to a certain degree. [ABSTRACT FROM AUTHOR]

A transient multi-physics model incorporated with an electromagneto-thermomechanical coupling is developed to capture the multi-field behavior of a single-pancake (SP) insert no-insulation (NI) coil in a hybrid magnet during the charging and discharging processes. The coupled problem is resolved by means of the finite element method (FEM) for the magneto-thermo-elastic behaviors and the Runge-Kutta method for the transient responses of the electrical circuits of the hybrid superconducting magnet system. The results reveal that the transient multi-physics responses of the insert NI coil primarily depend on the charging/discharging procedure of the hybrid magnet. Moreover, a reverse azimuthal current and a compressive hoop stress are induced in the insert NI coil during the charging process, while a forward azimuthal current and a tensile hoop stress are observed during the discharging process. The induced voltages in the insert NI coil can drive the currents flowing across the radial turns where the contact resistance exists. Therefore, it brings forth significant Joule heat, causing a temperature rise and a uniform distribution of this heat in the coil turns. Accordingly, a thermally/mechanically unstable or quenching event may be encountered when a high operating current is flowing in the insert NI coil. It is numerically predicted that a quick charging will induce a compressive hoop stress which may bring a risk of buckling instability in the coil, while a discharging will not. The simulations provide an insight of hybrid superconducting magnets under transient start-up or shutdown phases which are inevitably encountered in practical applications. [ABSTRACT FROM AUTHOR]

This paper presents the three-dimensional analysis of the stress distribution arising in an isotropic infinite slab with a triaxial ellipsoidal cavity, the surface of which is subjected to the three principal stresses σ 1 , σ 2 , and σ 3 . To satisfy both boundary conditions on the surface of slab and the cavity, harmonic functions in rectangular coordinates are used and double Fourier transform is applied. The problem is reduced to the solution of three integro-differential equations. The existence and uniqueness of the solution is investigated. [ABSTRACT FROM AUTHOR]

We establish a non-local integral difference quotient representation for symmetric gradient semi-norms in B D (Ω) and L D (Ω) , which does not require the manipulation of distributional derivatives. Our representation extends the formulas for the symmetric gradient established by Mengesha for vector-fields in W 1 , p (Ω ; R d) , which are inspired by the gradient semi-norm formulas introduced by Bourgain, Brezis and Mironescu in W 1 , p (Ω) and by Dávila in B V (Ω) . [ABSTRACT FROM AUTHOR]

A new mathematical formulation for the constitutive laws governing elastic perfectly plastic materials is proposed here. In particular, it is shown that the elastic strain rate and the plastic strain rate form an orthogonal decomposition with respect to the tangent cone and the normal cone of the yield domain. It is also shown that the stress rate can be seen as the projection on the tangent cone of the elastic stress tensor. This approach leads to a coherent mathematical formulation of the elasto-plastic laws and simplifies the resulting system for the associated flow evolution equations. The cases of one or two yield functions are treated in detail. The practical examples of the von Mises and Tresca yield criteria are worked out in detail to demonstrate the usefulness of the new formalism in applications. [ABSTRACT FROM AUTHOR]

We investigate indentation of a linear viscoelastic half-space with periodically distributed heterogeneities by a smooth, rigid, axisymmetric indenter of an arbitrary profile. For this, we first show how such materials may be homogenized directly in the time-domain to yield a homogeneous,anisotropic, linear viscoelastic medium. For the layered half-space that we then focus upon homogenization leads to a transversely isotropic, homogeneous, viscoelastic medium. We then replace the layered half-space in the original indentation problem by its homogenized counterpart. For axisymmetric indenters, for loadings in which the contact area is non-decreasing, we solve this indentation problem analytically by extending the correspondence principle of Graham (in Q Appl Math 26:167–174, 1968. https://doi.org/10.1090/qam/99860). Finally, we compare the results of homogenization with direct finite element computations of both displacement and force controlled indentation tests. We find that the contact pressure, the depth of penetration and the displacement of the top surface of the layered half-space are well approximated by those obtained through homogenization, and the approximation improves as the layer thickness is reduced, which is precisely when FE computations become harder. The latter aspect makes the development of the present approach useful. [ABSTRACT FROM AUTHOR]

In this paper, the three-dimensional (3D) interfacial fracture is analyzed in a one-dimensional (1D) hexagonal quasicrystal (QC) coating structure under mechanical loading. A planar interface crack with arbitrary shape is studied by a displacement discontinuity method. Fundamental solutions of interfacial concentrated displacement discontinuities are obtained by the Hankel transform technique, and the corresponding boundary integral-differential equations are constructed with the superposition principle. Green's functions of constant interfacial displacement discontinuities within a rectangular element are derived, and a boundary element method is proposed for numerical simulation. The singularity of stresses near the crack front is investigated, and the stress intensity factors (SIFs) as well as energy release rates (ERRs) are determined. Finally, relevant influencing factors on the fracture behavior are discussed. [ABSTRACT FROM AUTHOR]

The general theory of slender structure buckling by Grabovsky and Truskinovsky (Cont. Mech. Thermodyn. 19(3–4):211-243, 2007), (later extended in J. Nonlinear Sci. 26(1):83–119, 2016 by Grabovsky and the author), predicts that the critical buckling load of a thin shell under dead loading is closely related to the Korn's constant (in Korn's first inequality) of the shell under the Dirichlet boundary conditions resulting from the loading program. It is known that under zero Dirichlet boundary conditions on the thin part of the boundary of positive, negative, and zero (one principal curvature vanishing, and one apart from zero) Gaussian curvature shells, the optimal Korn constant in Korn's first inequality scales like the thickness to the power of −1, − 4 / 3 , and − 3 / 2 respectively. In this work we analyse the scaling of the optimal constant in Korn's first inequality for elliptic shells that contain a finite number of points where both principal curvatures vanish. We prove that the presence of at least one such point on the shell leads to the scaling drop from the thickness to the power of −1 to the thickness to the power of − 3 / 2 . To our best knowledge, this is the first result in the direction for constant-sign curvature shells, that do not contain a developable region. In addition, under the assumption that a suitable trivial branch exists, we prove that in fact the buckling deformation of such shells under dead loading, should be localized at the vanishing curvature points, as the shell thickness h goes to zero. [ABSTRACT FROM AUTHOR]

Most often there is a great disparity between experimental results and analytic results. Before now under great disparity, researchers kept suspecting the experimental procedures without investigating whether their analytic solution actually satisfy the governing equation. When a body in a plane is under loading, the loading splits the plane into regions, and the governing equation must be satisfied at these regions for the derived solution to be true. In this paper, I considered a homogeneous infinite orthotropic material containing a semi-infinite crack. A longitudinal shear load of magnitude Q is applied on the crack front. The displacement field in closed form is obtained. A verification of this solution at the split regions is carried out and shown to satisfy the governing differential equation. [ABSTRACT FROM AUTHOR]

In this paper we analyze a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem. A posteriori estimators for the nearly and perfectly compressible elasticity spectral problems are proposed. With a post-process argument, we are able to prove reliability and efficiency for the proposed estimators. The numerical method is based in Raviart-Thomas elements to approximate the pseudostress and piecewise polynomials for the displacement. We illustrate our results with numerical tests in two and three dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we show that the so-called Korn inequality holds for vector fields with a zero normal or tangential trace on a subset (of positive measure) of the boundary of Lipschitz domains. We further show that the validity of this inequality depends on the geometry of this subset of the boundary. We then consider three eigenvalue problems for the Lamé operator: we constrain the traction in the tangential direction and the normal component of the displacement, the related problem of constraining the normal component of the traction and the tangential component of the displacement, and a third eigenproblem that considers mixed boundary conditions. We show that eigenpairs for these eigenproblems exist on a broad variety of domains. Analytic solutions for some of these eigenproblems are given on simple domains. [ABSTRACT FROM AUTHOR]

Rare earth barium copper oxide (REBCO) is the most researched and commercialized second-generation high-temperature superconducting material. Due to the anisotropic structure, strong deformation sensitivity, and central field errors caused by screening current effects, it is still a challenge for commercialization applications. In this study, the transversely isotropic constitutive relationship is selected as the mechanical model based on the structural characteristics of REBCO tapes, and suitable microelements are selected to equate the elastic constants using their average stress-strain relationships. Then, a two-dimensional axisymmetric model for coils wound by single-layer tapes is constructed to analyze the dependence of the electric-magnetic-force distribution in the tape on the strain. Finally, the anisotropic approximation of the homogenized bulk method is used to equate large-turn high-field coils, and the electric-magnetic-force distribution characteristics of the coils with/without screening effects and mechanical strain conditions are investigated, respectively. The results reveal that the mechanical strain has a weakening effect on the electromagnetic field distribution of superconducting tapes, but causes a significant enhancement in the force field distribution. In the presence of 0.5% mechanical strain, the maximum weakening of the peak value of the current density and the critical current density inside the high-field coil can reach about 8% and 13%, respectively, with a nearly 5 times increase in the peak stress. The screening current makes the current field distribution inside the coil improve by about 10 times. The screening current induced magnetic field can reach up to 0.8 T, making the relative error of the high-field coil center up to 7.8%. [ABSTRACT FROM AUTHOR]

We present an elementary and self-contained proof that there are exactly four symmetry classes of the elasticity tensor in two dimensions: oblique, rectangular, square, and isotropic. In two dimensions, orthogonal transformations are either reflections or rotations. The proof is based on identification of constraints imposed by reflections and rotations on the elasticity tensor, and it simply employs elementary tools from trigonometry, making the proof accessible to a broad audience. For completeness, we identify the sets of transformations (rotations and reflections) for each symmetry class and report the corresponding equations of motions in classical linear elasticity. [ABSTRACT FROM AUTHOR]

This investigation is focused on the nonreciprocal transmission of the non-collinear mixing wave in a nonlinear elastic wave metamaterial. The nonlinear elastic wave metamaterial with asymmetric structure is used to break the traditional reciprocal transmission, which is composed of a nonlinear material and a linear phononic crystal. When two non-collinear shear waves following the resonant condition interact with each other, the sum frequency longitudinal wave can be generated and propagate together with fundamental and second harmonics. Based on the transfer and stiffness matrices, band gaps and transmission coefficients are derived. The changing of band structures results in manipulating elastic waves with different frequencies. In the nonreciprocal frequency region, elastic waves can propagate along the positive direction while the reverse case is prohibited. The mixing of two incident SV waves in the nonlinear elastic wave metamaterial is verified by experiments and the results can support the theoretical analysis and numerical calculations. Because of material nonlinearity and asymmetric structure, the nonlinear elastic wave metamaterial can behave as a mechanical diode. The present work shows the possibility to design nonreciprocal elastic wave device by nonlinear wave mixing methods. [ABSTRACT FROM AUTHOR]

Solutions of the Lamé–Navier system in R 3 have been represented in quaternion analysis by several generalized Kolosov–Muskhelishvili formulae. These representations depend on the shape of the domains such as star-like domain, normal domain with respect to one direction and complement of a star-like domain. In this paper, we consider the Lamé–Navier system in a ball shell domain Ω = { x ∈ R 3 | r 1 < | x | < r 2 } . Using the Laurent series expansion of monogenic functions, we prove a generalized Kolosov–Muskhelishvili formula in Ω . [ABSTRACT FROM AUTHOR]

In linear elasticity, universal displacements for a given symmetry class are those displacements that can be maintained by only applying boundary tractions (no body forces) and for arbitrary elastic constants in the symmetry class. In a previous work, we showed that the larger the symmetry group, the larger the space of universal displacements. Here, we generalize these ideas to the case of anelasticity. In linear anelasticity, the total strain is additively decomposed into elastic strain and anelastic strain, often referred to as an eigenstrain. We show that the universality constraints (equilibrium equations and arbitrariness of the elastic constants) completely specify the universal elastic strains for each of the eight anisotropy symmetry classes. The corresponding universal eigenstrains are the set of solutions to a system of second-order linear PDEs that ensure compatibility of the total strains. We show that for three symmetry classes, namely triclinic, monoclinic, and trigonal, only compatible (impotent) eigenstrains are universal. For the remaining five classes universal eigenstrains (up to the impotent ones) are the set of solutions to a system of linear second-order PDEs with certain arbitrary forcing terms that depend on the symmetry class. [ABSTRACT FROM AUTHOR]

A kind of Riemann problem of (λ , k) bi-analytic functions are studied by using the theory of boundary value problems for analytic functions. The solutions of Riemann problem of (λ , k) bi-analytic function are obtained. The conclusion may be applied to the quasi-static system of thermoelasticity. [ABSTRACT FROM AUTHOR]

In our previous papers (Griso et al. in J. Elast. 141:181–225, 2020; J. Elast., 2021, https://doi.org/10.1007/s10659-021-09816-w), we considered thick periodic structures (first paper) and thin stable periodic structures (second paper) made of small cylinders (length of order ε and cross-sections of radius r ). In the first paper r = κ ε with κ a fixed constant, ε → 0 , while in the second ε → 0 and r / ε → 0 . In this paper, our aim is to give the asymptotic behavior of thin periodic unstable structures, when ε → 0 , r / ε → 0 and ε 2 / r → 0 . Our analysis is again based on decompositions of displacements. As for stable periodic structures, Korn type inequalities are proved. Several classes of unstable and auxetic structures are introduced. The unfolding and limit homogenized problems are really different of those obtained for the thin stable periodic structures. The limit homogenized operators are anisotropic, the spaces containing the macroscopic limit displacements depend on the periodicity cells. It was not the case in the two previous studies. Some examples are given. [ABSTRACT FROM AUTHOR]

In this paper we provide some more details on the numerical analysis and we present some enlightening numerical results related to the spectrum of a finite element least-squares approximation of the linear elasticity formulation introduced recently. We show that, although the formulation is robust in the incompressible limit for the source problem, its spectrum is strongly dependent on the Lamé parameters and on the underlying mesh. [ABSTRACT FROM AUTHOR]

This paper made a revisit on the contact behavior and application of a beam (rail) on a half-plane/half-space. It is found that when the beam is subjected to a concentrated force, the shear deformation of the beam must always be considered, because the deformation only occurs within a limited length of the beam. The second finding of this paper is that the reactions and the surface settlements of the half-plane/half-space are significantly affected by the load position on the cross-section of the beam (on the top of rail in the concerned areas of application) and must be considered in practice. The Flamant/Boussinesq solutions are used to establish integral equations for the surface settlement of the half-plane/half-space below the beam. The Fourier Transform is applied to the first derivative of these integral equations and to the differential equations of the Timoshenko beam. Solving these two equations and employing the inverse Fourier Transform, the deflection and the reaction can be obtained in a compact format. After dispersing the wheel pressure on top of the rail to the centroid, the deflection and reaction are in agreement with the finite element analysis. Finally, simplified equations for equivalent bearing lengths are proposed. Nonuniform distribution of the reactions along the width of the beam is treated by using an equivalent uniform width. If the shear deformation is ignored, the problem becomes the classic one solved by Biot (J Appl Mech 4:A1–A7, 1937). Compared with Biot solution, the current way of solution is compact, their minor differences are the characteristic length. The current solution is a fourth root while the Biot solution is a cube root. [ABSTRACT FROM AUTHOR]

The adhesion failure has become one dominant factor in determining the reliability and service life of miniaturized devices subject to loadings with arbitrary orientations. This article establishes an adhesive full stick contact model between an elastic half-space and a rigid cylinder loaded in any direction. Using the Papkovich-Neuber functions, the Fourier integral transform, and the asymmetric bipolar coordinates, the exact solution is obtained. Unlike the Johnson-Kendall-Roberts (JKR) model, the present adhesive contact model takes into account the effects of the load direction as well as the coupling of the normal and tangential contact stresses. Besides, it considers the full stick contact which has large values of the friction coefficient between contacting surfaces, contrary to the frictionless contact supposed in the JKR model. The result shows that suitable angles can be found, which makes the contact surfaces difficult to be peeled off or easy to be pressed into. [ABSTRACT FROM AUTHOR]