Angelo Marcello Tarantino

Thin Hyperelastic sheets of compressible material: Field equations, airy stress function and an application in fracture mechanics

The equations of the equilibrium theory of thin hyperelastic sheets under plane stress condition and the associated Airy stress function are deduced for a compressible Mooney-Rivlin material. Such an analysis is then employed to formulate a nonlinear fracture mechanics problem. By means of an asymptotic procedure, the deformation and stress singular fields in proximity of the crack-tip are computed.

On the Finite Motions Generated by a Mode I Propagating Crack

The motion field surrounding a rapidly propagating crack, loaded symmetrically about the plane of the crack, is investigated. The problem is formulated within the framework of finite elastodynamics for thin slabs composed of compressible hyperelastic material. Writing the motion equations, the initial and the internal boundary conditions, with respect to a coordinate system that translates with the moving crack tip, we perform an asymptotic local analysis for a traction-free straight crack that suddenly grows at constant velocity. Moreover, the asymptotic Piola–Kirchhoff and Cauchy stress fields are computed, and we discuss the order of singularity of the dynamic stresses.

Crack Propagation in Finite Elastodynamics

Within the framework of finite elastodynamics, a crack propagation analysis, for sheets of compressible hyperelastic material, is formulated. By exploiting a dynamic generalization of the Stephensons result, general far-field loading conditions are considered. Through an asymptotic singular analysis, the motion and the stress fields around a dynamically moving crack tip are then computed. Emphasis is placed on the order of singularity in the asymptotic Piola-Kirchhoff and Cauchy stresses, on the determination of crack profile and of the vector energy flux at the moving crack tip. Moreover, the most important differences with respect to the classical predictions of linear elastodynamic theory are evidenced.

Homoclinic and heteroclinic bifurcations in the non-linear dynamics of a beam resting on an elastic substrate

Abstract The non-linear dynamics of a slender “elastica”, fixed at its base and free at the top, resting on an elastic substrate, axially loaded and subjected to periodic excitation, has been analyzed. Taking into account the non-linear inertial terms, the single-mode dynamics of the systems is governed by a Duffing equation with fifth-order non-linearities. In the considered range of parameters, two qualitatively different phase portraits exist. When the axial load p is less than the Eulerian critical value, there are three centers and two saddles (with the related stable and unstable manifolds). After the pitchfork bifurcation, the two saddles and the middle center coalesce in an unique new saddle which has a pair of symmetric homoclinic solutions. Melnikov criteria on the chaotic dynamics of the system are derived on the basis of analytical expressions for the homoclinic and the heteroclinic orbits. They involve transverse intersections of the stable and unstable manifolds that represent the starting point for a subsequent route to a chaotic dynamics. Numerical simulations which aim to show some effects of the global bifurcations on the actual dynamics are presented.

Chaotic dynamics of an elastic beam resting on a winkler-type soil

The statical and dynamical qualitative behaviour of a slender ‘elastica’, fixed at its base and free at the top, resting on an elastic foundation, axially loaded and subjected to a vibration at the base, is analysed. It has been shown that, depending on the foundation parameters, the static bifurcation may be supercritical or subcritical. Initially, the unperturbed dynamics is analysed, showing that there are four different behaviours for the four combinations, p ≶ pcr, supercritical/ subcritical bifurcation (p is the loading parameter, pcr its buckling value). By considering the effects of nonlinear inertia, chaotic dynamics is then examined by means of Melnikovs method. In the case of supercritical bifurcation it has been shown that chaos occurs for p > pcr due to homoclinic transverse intersections, while in the other case chaos due to double transverse heteroclinic intersection occurs for p < pcr. These types of chaos furnish qualitatively different chaotic zone diagrams in the parameter space (Φ, αf), where Φ is the frequency of the excitation, f its amplitude and α the damping parameter, which are illustrated and discussed. Emphasis is placed on the physical interpretation of the mathematical tools employed.

Magnetostriction of a Hard Ferromagnetic and Elastic Thin-Film Structure

In this paper, the magnetostriction of a thin film ferromagnetic and elastic structure is investigated. The structure is composed of an elastic layer confined between two perfectly bonded thin ferromagnetic layers in a plane strain setting. The ferromagnetic layers are saturated insulators and their magnetization is assumed at every stage parallel to the relevant layers axis (latent microstructure). The structure is studied within the theory of magneto-elastic interaction and focus is set on reducing the problem to a manageable one-dimensional form. This is accomplished employing the Euler—Bernoulli kinematics in conjunction with Maxwell equations. Once the model is stated, a system of coupled nonlinear integro-differential equations is obtained in the deformation components. Integral terms spring from the non-local character of the magnetic interaction. Some features of the solution are pointed out through an asymptotic analysis. The system is solved through the spectral method and a collocation technique, employing Gaussian quadrature at a second grid to assess the magnetic field and an iterative solver for nonlinear systems. Plots of the deformed configuration, of the internal action and of the layerss interaction are given. A parametric analysis brings out the role of some dimensionless quantities. Finally, solutions are compared with experimental results on Nickel magnetostriction.

Asymmetric Equilibrium Configurations of Symmetrically Loaded Isotropic Square Membranes

The homogeneous deformations provided by the equilibrium problem of nonlinear isotropic hyperelastic symmetrically loaded membranes are analyzed. Besides the universal symmetric solutions, the problem considered, depending on the form of the stored energy function, may admit asymmetric solutions. For general incompressible materials, the mathematical conditions governing the global development of these asymmetric solutions are investigated. Explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a Valanis–Landel and an Ogden material. For this last case, which is frequently used to model rubberlike materials, a broad numerical analysis was performed. The more qualitatively interesting branches of asymmetric equilibria are shown and the influence of the material parameters is discussed. Finally, using the energy criterion a number of considerations are made on stability.

Assessment of Stone Columns as a Mitigation Technique of Liquefaction-Induced Effects during Italian Earthquakes (May 2012)

Soil liquefaction has been observed worldwide during recent major earthquakes with induced effects responsible for much of the damage, disruption of function, and considerable replacement expenses for structures. The phenomenon has not been documented in recent time with such damage in Italian context before the recent Emilia-Romagna Earthquake (May 2012). The main lateral spreading and vertical deformations affected the stability of many buildings and impacted social life inducing valuable lessons on liquefaction risk assessment and remediation. This paper aims first of all to reproduce soil response to liquefaction-induced lateral effects and thus to evaluate stone column mitigation technique effectiveness by gradually increasing the extension of remediation, in order to achieve a satisfactory lower level of permanent deformations. The study is based on the use of a FE computational interface able to analyse the earthquake-induced three-dimensional pore pressure generation adopting one of the most credited nonlinear theories in order to assess realistically the displacements connected to lateral spreading.

Asymmetric equilibrium configurations of hyperelastic cylindrical bodies under symmetric dead loads

Homogeneous deformations provided by the nonlinear equilibrium problem of symmetrically loaded isotropic hyperelastic cylindrical bodies are investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions, besides the expected symmetric solutions. For general compressible materials, the mathematical condition allowing the assessment of these asymmetric solutions, which describe the global path of equilibrium branches, is given. Explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a Mooney-Rivlin and a neo-Hookean material. A broad numerical analysis is performed and the qualitatively more interesting asymmetric equilibrium branches are shown. The influence of the constitutive parameters is discussed, and, using the energy criterion, a number of considerations are carried out concerning the stability of the equilibrium solutions.

Degrading bouc-wen model parameters identification under cyclic load

The purpose of this article is to describe the Bouc–Wen model of hysteresis for structural engineering which is used to describe a wide range of nonlinear hysteretic systems, as a consequence of its capability to produce a variety of hysteretic patterns. This article focuses on the application of the Bouc–Wen model to predict the hysteretic behaviour of reinforced concrete bridge piers. The purpose is to identify the optimal values of the parameters so that the output of the model matches as well as possible the experimental data. Two repaired, retrofitted and reinforced concrete bridge pier specimens (in a 1:6 scale of a real bridge pier) are tested in a laboratory and used for experiments in this article. An identification of Bouc–Wen models parameters is performed using the force–displacement experimental data obtained after cyclic loading tests on these two specimens. The original model involves many parameters and complex pinching and degrading functions. This makes the identification solution unmanageable and with numerical problems. Furthermore, from a computational point of view, the identification takes too much time. The novelty of this work is the proposal of a simplification of the model allowed by simpler pinching and degrading functions and the reduction of the number of parameters. The latter innovation is effective in reducing computational efforts and is performed after a deep study of the mechanical effects of each parameter on the pier response. This simplified model is implemented in a MATLAB code and the numerical results are well fit to the experimental results and are reliable in terms of manageability, stability, and computational time.