Generalized Korn's inequality and conformal Killing vectors

Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity.Comment: 8 page

IoT sensors for modern structural health monitoring. A new frontier

The problem of determining the structural safety level of buildings and civil engineering infrastructures (CEIs) is raising growing concern worldwide. Most of the reinforced concrete constructions have a design life not greater than 100 years, and today it is necessary to face the problem of assessing their level of safety and structural integrity. Such problem is even more pressing when a construction is subjected to extreme environmental conditions. The long-term goal of this study is the realization of wireless low- cost devices, and a data management software, for the structural health monitoring of buildings and CEIs, with remotely controlled sensors embedded in, or installed on, the structural elements, to measure stresses together with accelerations. Once equipped with such system, each construction can become part of the Internet of Things, permitting users and authorities to be alerted in case structural safety is diminished or compromised. A crucial aspect is the unaltered preservation of measurement data over time, which cannot just rely on third parties, and for which it is necessary the exploitation of suitable data-protection technologies. This study have been carried out by experimental testing and validation, both in lab and on site, of the monitoring devices designed and realized. Results show that it is possible to realize low-cost monitoring systems, and related installation techniques, for integration in every new or existing buildings and CEIs

Permanent monitoring of thin structures with low-cost devices

Recently, structural monitoring technology invested in methodologies that give direct information on structures' stress state. Optic fibers, strain gauges, pressure cells give real-time data on the stress condition of a structural element, often determining the area where peak stresses have been reached, with a clear advantage over other less direct monitoring methodologies, such as, e.g., the use of accelerometers and inverse analysis to estimate internal forces. In addition, stresses can be recorded in a data log for analysis after a loading event, as well as for taking into account the lifelong stress state of the structure. Beams and columns of a reinforced concrete frame can be effectively monitored for flexural loads. Differently, thin shells are most of their lifespan under membrane regime, and, when properly designed, they rarely move to the bending regime. Our proposal is to monitor the stress in thin structures by small-sized low- cost devices able to record the stress history at key locations, sending alerts when necessary, with the aim of ensuring safety against the risk of collapse, or simply to perform maintenance/repairing activities. Such devices are realized with cheap off-the-shelf electronics and traditional strain gauges. The application examples are given as laboratory tests performed on a reinforced concrete plate, a masonry panel, and a steel beam. Results shows that the permanent monitoring control of stresses can be conveniently carried out on new structures using low-cost devices of the type we designed and realized in-house

Exact solutions for linearly electroelastic plate-like bodies

A three-dimensional, explicit and exact solution is derived for a transversely Isotropic, linearly electroelastic body in the form of a right cylinder of arbitrary cross section, being simply supported and connected to ground over its lateral boundary, and subject to an arbitrary distribution of force and charge over its end faces. When electric phenomena are ignored, this solution reduces to the solution given in for linearly elastic plate-like bodies

On morphoelastic rods

Morphoelastic rods are thin bodies which can grow and can change their intrinsic curvature and torsion. We deduce a system of equations that rule accretion and remodeling in a morphoelastic rod by combining balance laws involving non-standard forces with constitutive prescriptions filtered by a dissipation principle that takes into account both standard and non-standard working. We find that, as in the theory of three-dimensional bulk growth proposed [DiCarlo, A and Quiligotti, S. Mech Res Commun 2002; 29: 449-456], it is possible to identify a universal coupling mechanism between stress and growth, conveyed by an Eshelbian driving force

On Variational Approaches to Plate Models

The three basic functionals of potential energy, complementary energy and Hellinger-Prange-Reissner are used to obtain a rational derivation of Reissner-Mindlin plate models, starting from the three-dimensional theory. We show that the models so obtained are instances of the same plate theory; nevertheless, due to the different constitutive relations governing their response, they mimic the three-dimensional behaviour in three different manners

Marcus integration method for shearable plates

A method proposed by Marcus [5] to integrate the classical biharmonic equation of simply supported, unshearable plates with polygonal contour is extended to apply to shearable plates as well, provided the supporting device is of the 'hard' type

Lie Groups and the Compatibility Conditions for Continua with Rigid Structure

A derivation of the compatibility conditions for a continuum with rigid structure undergoing finite deformations is proposed. The geometry of the Lie group of frame changes is used. The compatibility conditions are deduced by requiring the involutiveness of a distribution associated to the strain: their relationship with the Maurer-Cartan equations of the group of frame changes is demonstrated. Applications to micropolar and Cauchy continua are given