The linear constraints in Poincar\'{e} and Korn type inequalities

We investigate the character of the linear constraints which are needed for Poincar\'e and Korn type inequalities to hold. We especially analyze constraints which depend on restriction on subsets of positive measure and on the trace on a portion of the boundary.Comment: Revised versio

On doubling inequalities for elliptic systems

We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lame' system of isotropic linearized elasticity. These inequalities depend on global properties of the solutions.Comment: 13 pages, submitted for publicatio

Stable determination of an inclusion in an elastic body by boundary measurements (unabridged)

We consider the inverse problem of identifying an unknown inclusion contained in an elastic body by the Dirichlet-to-Neumann map. The body is made by linearly elastic, homogeneous and isotropic material. The Lam\'e moduli of the inclusion are constant and different from those of the surrounding material. Under mild a-priori regularity assumptions on the unknown defect, we establish a logarithmic stability estimate. For the proof, we extend the approach used for electrical and thermal conductors in a novel way. Main tools are propagation of smallness arguments based on three-spheres inequality for solutions to the Lam\'e system and refined local approximation of the fundamental solution of the Lam\'e system in presence of an inclusion.Comment: 58 pages, 4 figures. This is the extended, and revised, version of a paper submitted for publication in abridged for

Two new Horaiclavus (Horaiclavidae, Conoidea) species from the Indo-Pacific region

The genus Horaiclavus includes eight Holocene Indo-Pacific species (Appeltans et al. 2012). Herein, we describe two new species that resemble members of this genus in some aspects of shell morphology, but otherwise show features that suggest that they differ from “typical” Horaiclavus species

Stable determination of an inclusion in an inhomogeneous elastic body by boundary measurements

In this paper we consider the stability issue for the in-verse problem of determining an unknown inclusion contained in an elastic body by all the pairs of measurements of displacement and trac-tion taken at the boundary of the body. Both the body and the inclusion are made by inhomogeneous linearly elastic isotropic material. Under mild a priori assumptions about the smoothness of the inclusion and the regularity of the coefficients, we show that the logarithmic stability estimate proved in [3] in the case of piecewise constant coefficients continues to hold in the inhomogeneous case. We introduce new arguments which allow to simplify some technical aspects of the proof given in [3]

Uniqueness in the determination of loads in multi-span beams and plates

Most of the results available on the inverse problem of determining loads acting on elastic beams or plates under transverse vibration refer to single beam or single plate. In this paper, we consider the determination of sources in multi-span systems obtained by connecting either two Euler-Bernoulli elastic beams or two rectangular Kirchhoff-Love elastic plates. The material of the structure is assumed to be homogeneous and isotropic. The transverse load is of the form g(t)f(x), where g(t) is a known function of time and f(x) is the unknown term depending on the position variable x. Under slight a priori assumptions, we prove a uniqueness result for f(x) in terms of observations of the dynamic response taken at interior points of the structure in an arbitrary small interval of time. A numerical implementation of the method is included to show the possible application of the results in the practical identification of the source term

On Isospectral Composite Beams

We consider a composite system consisting of two identical straight elastic beams under longitudinal vibration connected by an elastic interface capable of counteracting the relative vibration of the two beams with its shearing stiffness. We construct examples of isospectral composite beams, i.e., countable one-parameter families of beams having different shearing stiffness but exactly the same eigenvalues under a given set of boundary conditions. The construction is explicit and is based on the reduction to a one-dimensional Sturm–Liouville eigenvalue problem and the application of a Darboux’s lemma

Reconstructing Loads in Nanoplates from Dynamic Data

It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components (Formula presented.) of the transverse load (Formula presented.), where (Formula presented.) and (Formula presented.) is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff–Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads (Formula presented.). The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction