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In this paper, we study anisotropic Hooke’s tensor: we propose a factorization of its fourth-order harmonic part into second-order tensors. We obtain moreover explicit equivariant reconstruction formulas, using second-order covariants, for transverse isotropic and orthotropic fourth-order harmonic tensors, and for trigonal and tetragonal fourth-order harmonic tensors up to a cubic fourth order covariant remainder. [ABSTRACT FROM AUTHOR]

This study aims to characterize the elastic parameters in a functionally graded material. To implement this work, a parameter identification algorithm is proposed by combining a meshless method with the Nelder–Mead simplex (NMS) approach. The meshless method is based on the method of fundamental solutions and a radial basis function approximation, while the NMS approach is adopted to minimize the objective function and at the same time to obtain the unknown parameters. The objective function in this study characterizes the difference between the observed and the numerically predicted displacements under the estimated parameters. The robustness and effectiveness of the proposed scheme are verified by three numerical examples. [ABSTRACT FROM PUBLISHER]

The two-dimensional elastic Green’s function is calculated for a general anisotropic elastic bimaterial containing a line dislocation and a concentrated force while accounting for the interfacial structure by means of a generalized interfacial elasticity paradigm. The introduction of the interface elasticity model gives rise to boundary conditions that are effectively equivalent to those of a weakly bounded interface. The equations of elastic equilibrium are solved by complex variable techniques and the method of analytical continuation. The solution is decomposed into the sum of the Green’s function corresponding to the perfectly bonded interface and a perturbation term corresponding to the complex coupling nature between the interface structure and a line dislocation/concentrated force. Such construct can be implemented into the boundary integral equations and the boundary element method for analysis of nano-layered structures and epitaxial systems where the interface structure plays an important role. [ABSTRACT FROM AUTHOR]

A variational model for epitaxially strained films accounting for the presence of dislocations is considered. Existence, regularity and some qualitative properties of solutions are addressed. [ABSTRACT FROM AUTHOR]

A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks (see [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201-232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201-232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I, 350, 2012, 861-865]. [ABSTRACT FROM AUTHOR]

Quasicrystals (QCs) are sensitive to the piezoelectric (PE) effect. This paper studies static deformation of a multilayered one-dimensional (1D) hexagonal QC plate with the PE effect. The exact closed-form solutions of the extended displacement and traction for a homogeneous piezoelectric quasicrystal (PQC) plate are derived from an eigensystem. The general solutions for multilayered PQC plates are then obtained using the propagator matrix method when mechanical and electrical loads are applied on the top surface of the plate. Numerical examples for several sandwich plates made up of PQC, PE, and QC materials are provided to show the effect of stacking sequence on phonon, phason, and electric fields under mechanical and electrical loads, which is useful in designing new composites for engineering structures. [ABSTRACT FROM AUTHOR]

The aim of this article is to present and rigorously analyze topological sensitivity based algorithms for detection of diametrically small inclusions in an isotropic homogeneous elastic formation using single and multiple measurements of the far-field scattering amplitudes. A L 2 -cost functional is considered and a location indicator is constructed from its topological derivative. The performance of the indicator is analyzed in terms of the topological sensitivity for location detection and stability with respect to measurement and medium noises. It is established that the location indicator does not guarantee inclusion detection and achieves only a low resolution when there is mode-conversion in an elastic formation. Accordingly, a weighted location indicator is designed to tackle the mode-conversion phenomenon. It is substantiated that the weighted function renders the location of an inclusion stably with resolution as per Rayleigh criterion. [ABSTRACT FROM AUTHOR]

We verify the objectivity (invariance to rigid body rotations) ordinary state-based peridynamic models published in the literature that differ in their formulas. We find and explain the sources for the differences between these published formulas. We demonstrate that a primary cause leading to these differences is the way in which the peridynamic volume dilatation is defined in the different formulations. We show that the equations of motion derived from one approach apply to deformations with small or large rotations and is objective. The other approach is valid only for deformations with zero/infinitesimal rotations, thus it is not objective. We also show that both state-based models recover the correct bond-based formulations with the appropriate Poisson ratio values when the term containing the volume dilatation vanishes. [ABSTRACT FROM AUTHOR]

We employ conformal mapping techniques to study the existence of internal uniform hydrostatic stresses inside two non-elliptical inclusions when the surrounding matrix is simultaneously subjected to a concentrated couple and remote uniform in-plane stresses. The unknown complex coefficients appearing in the corresponding mapping function can be determined analytically for a given pair of loading, one material and three geometric parameters. This allows us to subsequently identify the shapes of the two inclusions. Our analysis further reveals that the shapes of the inclusions depend on the concentrated couple, whereas the corresponding internal uniform hydrostatic stresses do not. [ABSTRACT FROM AUTHOR]

We consider a family of linearly elastic shells with thickness $2\varepsilon$ (where $\varepsilon$ is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface $S$ , and may enter in contact with a rigid foundation along the bottom face. We are interested in studying the limit behavior of both the three-dimensional problems, given in curvilinear coordinates, and their solutions (displacements $\boldsymbol{u}^{\varepsilon}$ of covariant components $u_{i}^{\varepsilon}$ ) when $\varepsilon$ tends to zero. To do that, we use asymptotic analysis methods. On one hand, we find that if the applied body force density is $O(1)$ with respect to $\varepsilon$ and surface tractions density is $O(\varepsilon)$ , a suitable approximation of the variational formulation of the contact problem is a two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem for an elastic membrane. On the other hand, if the applied body force density is $O(\varepsilon^{2})$ and surface tractions density is $O(\varepsilon^{3})$ , the corresponding approximation is a different two-dimensional inequality which can be identified as the variational formulation of the obstacle problem for an elastic flexural shell. We finally discuss the existence and uniqueness of solution for the limit two-dimensional variational problems found. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded and connected open subset of ℝ with a Lipschitz-continuous boundary, the set Ω being locally on the same side of ∂Ω. A vector version of a fundamental lemma of J. L. Lions, due to C. Amrouche, the first author, L. Gratie and S. Kesavan, asserts that any vector field v = ( u) ∈ (D′(Ω)), such that all the components $$\frac{1}{2}({\partial _j}{v_i} + {\partial _i}{v_j})$$ , 1 ≤ i, j ≤ N, of its symmetrized gradient matrix field are in the space H(Ω), is in effect in the space (L(Ω)). The objective of this paper is to show that this vector version of J. L. Lions lemma is equivalent to a certain number of other properties of interest by themselves. These include in particular a vector version of a well-known inequality due to J. Nečas, weak versions of the classical Donati and Saint-Venant compatibility conditions for a matrix field to be the symmetrized gradient matrix field of a vector field, or a natural vector version of a fundamental surjectivity property of the divergence operator. [ABSTRACT FROM AUTHOR]

We show that the reasoning in favor of a symmetric couple stress tensor in Yang et al.'s introduction of the modified couple stress theory contains a gap, but we present a reasonable physical hypothesis, implying that the couple stress tensor is traceless and may be symmetric anyway. To this aim, the origin of couple stress is discussed on the basis of certain properties of the total stress itself. In contrast to classical continuum mechanics, the balance of linear momentum and the balance of angular momentum are formulated at an infinitesimal cube considering the total stress as linear and quadratic approximation of a spatial Taylor series expansion. [ABSTRACT FROM AUTHOR]

In this paper a new numerical method for the shape reconstruction of obstacles in elastic scattering is proposed. Initially, the direct scattering problem for a rigid body and the mathematical setting for the corresponding inverse one are presented. Inverse uniqueness issues for the general case of mixed boundary conditions on the boundary of our obstacle, which are valid for a rigid body as well are established. The inversion algorithm based on the factorization method is presented into a suitable form and a new numerical scheme for the reconstruction of the shape of the scatterer, using far-field measurements, is given. In particular, an efficient Tikhonov parameter choice technique, calledImproved Maximum Product Criterion(IMPC) and its linchpin within the framework of the factorization method is exploited. Our regularization parameter is computed via a fast iterative algorithm which requires noa prioriknowledge of the noise level in the far-field data. Finally, the effectiveness of IMPC is illustrated with various numerical examples involving a kite, an acorn, and a peanut-shaped object. [ABSTRACT FROM PUBLISHER]

In this paper, we derive W and piecewise C estimates for solutions, and their t-derivatives, of divergence form parabolic systems with coefficients piecewise Hölder continuous in space variables x and smooth in t. This is an extension to parabolic systems of results of Li and Nirenberg [Comm Pure Appl Math, 2003, 56: 892-925] on elliptic systems. These estimates depend on the shape and the size of the surfaces of discontinuity of the coefficients, but are independent of the distance between these surfaces. [ABSTRACT FROM AUTHOR]

Based on von Karman's plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material (FGM) circular plates with inplane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail. [ABSTRACT FROM AUTHOR]

The stability of a strained cuboidal island deposited on a substrate has been numerically investigated by means of finite element simulations in the case where the structure is submitted to misfit strain resulting from the lattice mismatch between the island and the substrate. In the hypothesis where the surface energy is isotropic, it is found that, depending on the island volume, the formation of a truncated or inverted truncated pyramid can be favored by the misfit strain with respect to the cuboidal shape. A shape diagram is finally provided as a function of the misfit strain and island volume. [ABSTRACT FROM AUTHOR]

A hybrid method involving boundary analysis and boundary collocation is used to obtain an approximate solution for a plane problem of uncoupled thermoelasticity with mixed thermal and mechanical boundary conditions in a square domain with one curved side. The unknown functions in the cross-section are obtained in the form of series expansions in Cartesian harmonics. A boundary analysis reveals the singular behavior of the solution at the transition points. In order to simulate the weak discontinuities of the temperature function and the discontinuities of stress, these expansions are enriched with proper harmonic functions with a singular behavior at the transition points. The results are discussed, and the functions of practical interest are represented on the boundary and also inside the domain. The locations where possible debonding of the fixed part of the boundary may take place are noted. [ABSTRACT FROM AUTHOR]

We establish interior Lipschitz estimates at the macroscopic scale for solutions to systems of linear elasticity with rapidly oscillating periodic coefficients and mixed boundary conditions in domains periodically perforated at a microscopic scale ε by establishing H 1 -convergence rates for such solutions. The interior estimates are derived directly without the use of compactness via an argument presented in [3] that was adapted for elliptic equations in [2] and [11] . As a consequence, we derive a Liouville type estimate for solutions to the systems of linear elasticity in unbounded periodically perforated domains. [ABSTRACT FROM AUTHOR]

Unlike in complex analysis, in all hypercomplex function theories a hypercomplex variable is not monogenic (regular) and there exists a problem to define analogues of positive and negative powers of the complex variable. R. Fueter firstly introduces a system of symmetric regular quaternion polynomials as analogues of positive powers of a complex variable and proves the Taylor theorem in his theory. In Clifford analysis an analogical idea is used. The Fueter symmetric polynomials are both left- and right-regular, the symmetric polynomials in Clifford analysis are also both left- and right-monogenic. In this paper we construct only left-regular quaternion polynomials and show that the theory of regular quaternion functions of a vector valued quaternion variable can be developed using these polynomials. [ABSTRACT FROM AUTHOR]

A new type of displacement pile, the X-section cast-in-place concrete (XCC) pile, has recently been developed in China. Extensive field tests and laboratory experiments are undertaken to evaluate its performance and quantify the non-uniform deformation effect (NUDE) of the X-shaped cross section during installation. This paper develops a simplified theoretical model that attempts to capture the NUDE. Based on the theory of complex variable plane elasticity, closed-form solutions of the stress and displacement for the X-shaped cavity boundary value problem are given. Subsequently, the analytical solution is used to evaluate the NUDE, the concrete filling index (CFI), and the perimeter reduction coefficient of the XCC pile cross section. The computed results are compared with field test results, showing reasonable agreement. The present simplified theoretical model reveals the deformation mechanism of the X-shaped cavity and facilitates application of the newly developed XCC pile technique in geotechnical engineering. [ABSTRACT FROM AUTHOR]

The composite voxel technique was developed in the framework of linear elasticity and hyperelasticity for regular voxel grid discretizations which cannot resolve material interfaces exactly in general. In this work, we study the inelastic behavior of two-phase laminates. In particular, we derive an analytical nonlinear formula for the unknown rank one jump of the strain across the interface. This enables us to extend the composite voxel technique to account for inelastic material behavior at small strains. We demonstrate by numerical experiments on continuously and discontinuously reinforced thermoplastics with elastoplastic matrix behavior that FFT-based computational homogenization on downsampled microstructures equipped with composite voxels produces stress–strain curves mimicking those obtained for the full resolution. For industrial sized microstructures it turns out that the computations can be accelerated by a factor of up to 40 compared to a direct parallelization of the fully resolved problem. [ABSTRACT FROM AUTHOR]

The concept of scattering coefficients has played a pivotal role in a broad range of inverse scattering and imaging problems in acoustic, and electromagnetic media. In view of their promising applications in inverse problems related to mathematical imaging and elastic cloaking, the notion of elastic scattering coefficients of an inclusion is presented from the perspective of boundary layer potentials and a few properties are discussed. A reconstruction algorithm is developed and analyzed for extracting the elastic scattering coefficients from multi-static response measurements of the scattered field in order to cater to inverse scattering problems. The decay rate, stability and error analyses, and the estimate of maximal resolving order in terms of the signal-to-noise ratio are discussed. Moreover, scattering-coefficients-vanishing structures are designed and their utility for enhancement of nearly elastic cloaking is elucidated. [ABSTRACT FROM AUTHOR]

The orientation dependence of the plane strain Young's modulus, $\tilde{E}$ , of cubic materials has been analysed as a function of the direction along which a uniaxial stress is applied to a single crystal and the perpendicular direction in the single crystal along which the strain is constrained to be zero. The locus of $\tilde{E}$ in the plane perpendicular to the axis of uniaxial stress is shown to be a circle when this stress is applied along $\langle111\rangle$ . For materials with anisotropy ratios $A > 1$ , global minima in $\tilde{E}$ occur when the stress is applied along $\langle 001\rangle$ and when the strain along one of the two perpendicular $\langle100\rangle $ directions is set to zero. Identical global maxima in $\tilde{E}$ are found when the stress is applied along two different families of $\langle\mathit {uuw} \rangle$ directions and the direction of zero strain is along either a perpendicular $\langle1\bar{1}0\rangle$ or $\langle \mathit{ww} \overline{2u}\rangle$ direction. For materials with $A < 1$ , the global maxima in $\tilde{E}$ occur when the stress is applied along $\langle001\rangle$ and when the strain along one of the two perpendicular $\langle100\rangle$ directions is set to zero, and identical global minima are found when the stress is applied along two different families of $\langle\mathit {uuw} \rangle$ directions and the direction of zero strain is along either a perpendicular $\langle1\bar{1}0\rangle$ or $\langle \mathit{ww} \overline{2u}\rangle$ direction. [ABSTRACT FROM AUTHOR]

We consider a thin heterogeneous layer consisting of thin beams (of radius $r$ ) and study the limit behavior of this problem as the period $\varepsilon $ , the thickness $\delta$ and the radius $r$ of the beams tend to zero. The decomposition of the displacement field into beams developed by Griso (J. Math. Pures Appl. 89:199-223, 2008) is used, which allows to obtain a priori estimates. Two types of unfolding operators are introduced to deal with different parts of the decomposition. In conclusion, we obtain the limit problem together with transmission conditions across the interface. [ABSTRACT FROM AUTHOR]

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite wedge-shaped inclusion, which meets the interface at a right angle and is loaded with normal forces is considered. The normal contact stresses along the contact line are determined and the behavior of the contact stresses in the neighborhood of singular points is established. By using methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite interval. Using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. [ABSTRACT FROM AUTHOR]

The shape transition of a one-layer thick strained island deposited on a semi-infinite substrate has been theoretically investigated from an energy variation calculation. It is found that depending on the misfit strain, the initially circular island may become unstable beyond a critical size and can evolve toward an elliptical shape. As the surface of the island increases, another transition is expected to occur which consists in the ellipse splitting into two identical circular islands of smaller size. A shape diagram is finally displayed for the island as a function of the misfit strain and island surface and a scenario of nanostructure evolution is discussed. [ABSTRACT FROM AUTHOR]

We provide a framework for simultaneous homogenization and dimension reduction in the setting of linearized elasticity as well as non-linear elasticity for the derivation of homogenized von Kármán plate and bending rod models. The framework encompasses even perforated domains and domains with oscillatory boundary, provided that the corresponding extension operator can be constructed. Locality property of $$\varGamma $$ -closure is established, i.e. every energy density obtained by the homogenization process can be in almost every point obtained as the limit of periodic energy densities. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the 2D vector displacement field in a heterogeneous elastic solid deforming under plane stress conditions. The problem is motivated by applications in quasistatic elastography. From precise and accurate measurements of one component of the 2D vector displacement field and very limited information of the second component, the method reconstructs the second component quite accurately. No a priori knowledge of the heterogeneous distribution of material properties is required. This method relies on using a special form of the momentum equations to filter ultrasound displacement measurements to produce more precise estimates. We verify the method with applications to simulated displacement data. We validate the method with applications to displacement data measured from a tissue mimicking phantom, and in-vivo data; significant improvements are noticed in the filtered displacements recovered from all the tests. In verification studies, error in lateral displacement estimates decreased from about 50% to about 2%, and strain error decreased from more than 250% to below 2%. [ABSTRACT FROM AUTHOR]

In this paper we prove asymptotically sharp weighted 'first-and-a-half' $2D$ Korn and Korn-like inequalities with a singular weight occurring from Cartesian to cylindrical change of variables. We prove some Hardy and the so-called 'harmonic function gradient separation' inequalities with the same singular weight. Then we apply the obtained $2D$ inequalities to prove similar inequalities for washers with thickness $h$ subject to vanishing Dirichlet boundary conditions on the inner and outer thin faces of the washer. A washer can be regarded in two ways: As the limit case of a conical shell when the slope goes to zero, or as a very short hollow cylinder. While the optimal Korn constant in the first Korn inequality for a conical shell with thickness $h$ and with a positive slope scales like $h^{1.5}$ , e.g., (Grabovsky and Harutyunyan in , 2016), the optimal Korn constant in the first Korn inequality for a washer scales like $h^{2}$ and depends only on the outer radius of the washer as we show in the present work. The Korn constant in the first and a half inequality scales like $h$ and depends only on $h$ . The optimal Korn constant is realized by a Kirchhoff Ansatz. This results can be applied to calculate the critical buckling load of a washer under in plane loads, e.g., (Antman and Stepanov in J. Elast. 124(2):243-278, 2016). [ABSTRACT FROM AUTHOR]

Stress field solutions and Stress Intensity Factors (SIFs) are found for $n$ -cusped hypocycloidal shaped voids and rigid inclusions in an infinite linear elastic plane subject to nonuniform remote antiplane loading, using complex potential and conformal mapping. It is shown that a void with hypocycloidal shape can lead to a higher SIF than that induced by a corresponding star-shaped crack; this is counter intuitive as the latter usually produces a more severe stress field in the material. Moreover, it is observed that when the order $m$ of the polynomial governing the remote loading grows, the stress fields generated by the hypocycloidal-shaped void and the star-shaped crack tend to coincide, so that they become equivalent from the point of view of a failure analysis. Finally, special geometries and loading conditions are discovered for which there is no stress singularity at the inclusion cusps and where the stress is even reduced with respect to the case of the absence of the inclusion. The concept of Stress Reduction Factor (SRF) in the presence of a sharp wedge is therefore introduced, contrasting with the well-known definition of Stress Concentration Factor (SCF) in the presence of inclusions with smooth boundary. The results presented in this paper provide criteria that will help in the design of ultra strong composite materials, where stress singularities always promote failure. Furthermore, they will facilitate finding the special conditions where resistance can be optimized in the presence of inclusions with non-smooth boundary. [ABSTRACT FROM AUTHOR]

The elastic displacement in an isotropic elastic half-space with free surface is calculated for a point tensor force which may arise from the seismic moment of seismic sources concentrated at an inner point of the half-space. The starting point of the calculation is the decomposition of the displacement by means of the Helmholtz potentials and a simplified version of the Grodskii-Neuber-Papkovitch procedure. The calculations are carried out by using generalized Poisson equations and in-plane Fourier transforms, which are convenient for treating boundary conditions. As a general result we compute the displacement in the isotropic elastic half-space with free surface caused by point forces with arbitrary structure and orientation, localized either beneath the surface (generalized Mindlin problem) or on the surface (generalized Boussinesq-Cerruti problems). The inverse Fourier transforms are carried out by means of Sommerfeld-type integrals. For forces buried in the half-space explicit results are given for the surface displacement, which may exhibit finite values at the origin, or at distances on the surface of the order of the depth of the source. The problem presented here may be viewed as an addition to the well-known static problems of elastic equilibrium of a half-space under the action of concentrated loads. The application of the method to similar problems and another approach to the starting point of the general solution are discussed. [ABSTRACT FROM AUTHOR]

In many sectors of today’s industry it is of utmost importance to detect defects in elastic structures contained in technical devices to guarantee their failure-free operation. As currently used signal processing techniques have natural limits with respect to accuracy and significance, modern mathematical methods are crucial to improve current algorithms. We consider in this paper a parameter identification approach for isotropic and linear elastic structures described by their Lamé parameters and a material density. This approach can be employed for non-destructive defect detection, location and characterization from time-dependent measurements of one elastic wave. To this end, we show that the operator linking the static parameters with the wave measurements is Fréchet differentiable, which allows to set up Newton-like methods for the non-linear parameter identification problem. We indicate the performance of a specific inexact Newton-like regularization method by numerical examples for a testing problem of a thin plate from measurements of the normal component of the displacement field on the boundary. As an extension, we further augment this method with a total variation regularization and thereby improve reconstructed parameters that feature edges. [ABSTRACT FROM PUBLISHER]

In this article, we derive the inside–outside duality for two time-harmonic, elastic scattering problems. First, we consider a rigid scattering object inside an isotropic, homogeneous background medium and second, we consider a penetrable, inhomogeneous scattering object inside this background medium. For the first scattering problem, we make use of the particular behavior or certain eigenvalues of the corresponding far field operator to characterize interior Dirichlet eigenvalues of the negative Navier operator. Then we adapt this technique to determine interior transmission eigenvalues that correspond to the second scattering problem. [ABSTRACT FROM AUTHOR]

In this paper, we design two classes of stabilized mixed finite element methods for linear elasticity on simplicial grids. In the first class of elements, we use - and - to approximate the stress and displacement spaces, respectively, for , and employ a stabilization technique in terms of the jump of the discrete displacement over the edges/faces of the triangulation under consideration; in the second class of elements, we use - to approximate the displacement space for , and adopt the stabilization technique suggested by Brezzi, Fortin, and Marini []. We establish the discrete inf-sup conditions, and consequently present the a priori error analysis for them. The main ingredient for the analysis are two special interpolation operators, which can be constructed using a crucial bubble function space of polynomials on each element. The feature of these methods is the low number of global degrees of freedom in the lowest order case. We present some numerical results to demonstrate the theoretical estimates. [ABSTRACT FROM AUTHOR]

In this paper, a new linear shell model of Naghdi type is formulated for shells with little regularity, namely the shells whose middle surface is parameterized by a W 1, ∞ function. Thus corners in the undeformed geometry are inherent in the formulation. Unknowns in the model are the displacement u ˜ of the middle surface of the shell and the infinitesimal rotation ω ˜ of the shell cross-section. In difference to the classical shell models, the existence and uniqueness of the solution is obtained for ( u ˜ , ω ˜ ) ∈ H 1 × H 1 with a very simple proof without usage of almost any differential geometry of surfaces. We relate the new model with known shell models in two ways. In the first we show that asymptotically, with respect to small thickness of the shell, the model behaves as the membrane model or the flexural shell model in the corresponding regime. In the second, for smooth enough middle surface, we relate the terms in the weak formulation of the model with terms in the classical Naghdi shell model. Further, we prove continuous dependence of the solution of the model on the change of the undeformed middle surface. At the end, we also present a numerical approximation of the model for the middle surface with a corner at the joint of two pieces that dominantly behave differently (one as a membrane and the other one as a flexural shell). [ABSTRACT FROM AUTHOR]

The asymptotic approximation of the displacement of a linear elastic beam with prescribed non-homogeneous Dirichlet conditions at an end is constructed. The non-homogeneous Dirichlet conditions considered in this work have the form of rigid displacements. The six values needed to state an auxiliary one-dimensional system are obtained and the error of the approximation is estimated. A numerical example illustrates the high precision of the approximation. [ABSTRACT FROM PUBLISHER]

One of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the far-field. Of specific importance is the uniformity of Eshelby's tensor $\mathbf{S}$ . Following Eshelby's seminal work, a vast literature has been generated using and developing Eshelby's result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor $\mathbf{P}=\mathbf{S}\mathbf{D}^{0}$ arises, where $\mathbf{D}^{0}$ is the host medium compliance tensor. The tensor $\mathbf{P}$ is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill tensor, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases. Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of, for example, layered media and fibre reinforced composites and also to the cases when voids and cracks are present. In the main, results for the Hill and Eshelby tensors are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently non-trivial due to the requirement of the associated Green's tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green's function. The Concentration tensor $\boldsymbol{\mathcal{A}}$ linking interior fields to those imposed in the far-field is derived for a wide variety of problems. These results can therefore be used in the various micromechanical schemes. Directly related to the tensors of Eshelby and Hill is the so-called Moment tensor $\mathbf{M}$ . As well as arising in the literature on micromechanics, this tensor is important in the vast area of research associated with inverse problems and specifically with the problem of identifying an object inside some domain given the application of a specific set of boundary conditions. Due to its fundamental importance and direct link to the Eshelby and Hill tensors, here we state the connection between $\mathbf{M}, \mathbf{P}$ and $\mathbf{S}$ in an effort to ensure that the work is of use to as wide a community as possible. Both tensor and matrix formulations are considered and contrasted throughout. Appendices give various details that illustrate the implementation of both approaches. [ABSTRACT FROM AUTHOR]

The peridynamic theory of solid mechanics is a nonlocal reformulation of the classical continuum mechanics theory. At the continuum level, it has been demonstrated that classical (local) elasticity is a special case of peridynamics. Such a connection between these theories has not been extensively explored at the discrete level. This paper investigates the consistency between nearest-neighbor discretizations of linear elastic peridynamic models and finite difference discretizations of the Navier–Cauchy equation of classical elasticity. Although nearest-neighbor discretizations in peridynamics have been numerically observed to present grid-dependent crack paths or spurious microcracks, this paper focuses on a different, analytical aspect of such discretizations. We demonstrate that, even in the absence of cracks, such discretizations may be problematic unless a proper selection of weights is used. Specifically, we demonstrate that using the standard meshfree approach in peridynamics, nearest-neighbor discretizations do not reduce, in general, to discretizations of corresponding classical models. We study nodal-based quadratures for the discretization of peridynamic models, and we derive quadrature weights that result in consistency between nearest-neighbor discretizations of peridynamic models and discretized classical models. The quadrature weights that lead to such consistency are, however, model-/discretization-dependent. We motivate the choice of those quadrature weights through a quadratic approximation of displacement fields. The stability of nearest-neighbor peridynamic schemes is demonstrated through a Fourier mode analysis. Finally, an approach based on a normalization of peridynamic constitutive constants at the discrete level is explored. This approach results in the desired consistency for one-dimensional models, but does not work in higher dimensions. The results of the work presented in this paper suggest that even though nearest-neighbor discretizations should be avoided in peridynamic simulations involving cracks, such discretizations are viable, for example for verification or validation purposes, in problems characterized by smooth deformations. Moreover, we demonstrate that better quadrature rules in peridynamics can be obtained based on the functional form of solutions. [ABSTRACT FROM AUTHOR]

We study the contribution of surface elasticity to the two-dimensional contact problem in a generally anisotropic material using the Stroh sextic formalism. Surface elasticity is incorporated into the model of deformation using an anisotropic version of the continuum-based surface/interface model of Gurtin and Murdoch. Full-field analytic solutions are obtained in terms of exponential integrals for an anisotropic half-space when the contact surface is subjected to two particular types of loading: first, we consider the case of a uniform load (shearing and pressure) applied to an infinitely long strip of the contact surface and second, by reducing the strip to zero width, we deduce the corresponding result for a concentrated line force acting on the contact surface. The analysis indicates that the surface deformation gradient is finite in the first case of uniform loading of the strip and exhibits a weak logarithmic singularity at the location of the applied concentrated line force in the second case. [ABSTRACT FROM AUTHOR]