Discrete approximations of the Föppl–Von Kármán shell model: From coarse to more refined models

AbstractThe problem of deducing, from the Föppl–Von Kármán energy functional, a sequence of reduced discrete models having few degrees of freedom is analyzed. Similar discrete models have been recently intensively studied to analyze the multistable behavior of shallow shells, the bifurcations of composite laminates under temperature loads or the wrinkling in soft tissues.In particular three relevant examples are discussed and compared among them, where the curvature is assumed uniform, linearly and quadratically varying through the shell. While the uniform-curvature assumption dates back to Mansfield (1962), linear variations of the shell curvatures can describe smooth transitions between everted configurations, while quadratic variations can account for the, usually disregarded, bending boundary conditions.For their deduction we revisit the Maxwell–Mohr method: accordingly, a sequence of auxiliary elliptic problems of plane elasticity is solved to determine the statically unknown membranal stresses. This is a key ingredient for the presented models to compare extremely well with Finite Element approximations or with literature models with far more degrees of freedom

Multiparameter actuation of a neutrally-stable shell: a flexible gear-less motor

We have designed and tested experimentally a morphing structure consisting of a neutrally stable thin cylindrical shell driven by a multiparameter piezoelectric actuation. The shell is obtained by plastically deforming an initially flat copper disk, so as to induce large isotropic and almost uniform inelastic curvatures. Following the plastic deformation, in a perfectly isotropic system, the shell is theoretically neutrally stable, owning a continuous manifold of stable cylindrical shapes corresponding to the rotation of the axis of maximal curvature. Small imperfections render the actual structure bistable, giving preferred orientations. A three-parameter piezoelectric actuation, exerted through micro-fiber-composite actuators, allows us to add a small perturbation to the plastic inelastic curvature and to control the direction of maximal curvature. This actuation law is designed through a geometrical analogy based on a fully non-linear inextensible uniform-curvature shell model. We report on the fabrication, identification, and experimental testing of a prototype and demonstrate the effectiveness of the piezoelectric actuators in controlling its shape. The resulting motion is an apparent rotation of the shell, controlled by the voltages as in a "gear-less motor", which is, in reality, a precession of the axis of principal curvature.Comment: 20 pages, 9 figure

Towards a phase-field model for thin structures: a coarse-grained constitutive law for brittle fracture of beams

Damage gradient models approximate fracture mechanics using a modulation of the material stiffness. To this aim a single scalar field, the damage, is used to degrade as a whole the elastic energy. If applied to the structural models of beams and shells, where the elastic energy is the sum of the stretching and bending contributions, a similar approach is not able to capture some important features. For instance, the coupling between axial and bending strains induced by cracks non-symmetric with respect to the center line is completely missed. In this contribution, we deduce a constitutive law for a beam having a crack non-symmetric with respect to the center line. To achieve this, we perform an asymptotic coarse-grained procedure from a 2D problem, using a sharp interface model. We deduce a homogenized 1D elastic energy, coupling the axial and bending strains, the constitutive parameters of which depend in a different way on the crack depth and we state how precisely they do. This should pave the way to a rational development of a phase-field gradient model for thin structures

Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks

In this paper a novel device aimed at controlling the mechanical vibrations of plates by means of a set of electrically-interconnected piezoelectric actuators is described. The actuators are embedded uniformly in the plate wherein they connect every node of an electric network to ground, thus playing the two-fold role of capacitive element in the electric network and of couple suppliers. A mathematical model is introduced to describe the propagation of electro-mechanical waves in the device; its validity is restricted to the case of wave-forms with wave-length greater than the dimension of the piezoelectric actuators used. A self-resonance criterion is established which assures the possibility of electro-mechanical energy exchange. Finally the problem of vibration control in simply supported and clamped plates is addressed; the optimal net-impedance is determined. The results indicate that the proposed device can improve the performances of piezoelectric actuationComment: 22 page

Mohr's Circle

Once assigned a stress tensor (a 2x2 symmetric matrix) and a direction (a unit vector n)Componente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemátic