Giovanni Formica

A mixed solution strategy for the nonlinear analysis of brick masonry walls

The paper presents a discrete mechanical model for masonry walls based on a Lagrangean description where each brick is described as a rigid body and each mortar joint as an interface element. Constitutive assumptions, characterized by elasticity, damage and friction, are associated to the joints only. A numerical solution strategy, based on a mixed path-following approach in terms of stresses, strains, displacements, damage and load parameters, is proposed for avoiding convergence problems related to the joint softening behaviour. Some numerical results are also presented showing the robustness and effectiveness of this proposal.

Numerical size estimates of inclusions in elastic bodies

We consider the problem of detecting elastic inclusions in elastic bodies by means of mechanical boundary data only, that is measurements of boundary displacement and traction. In previous work of some of the present authors, upper and lower bounds on the size (area or volume) of the inclusions were proven analytically. Following the guidelines drawn up in such previous theoretical study, an extended numerical investigation has been performed in order to prove the effectiveness of this approach. The sensitivity with respect to various relevant parameters is also analysed.

Nonlinear finite element-based path following of periodic solutions

A computational framework is proposed to path follow the periodic solutions of nonlinear spatially continuous systems and more general coupled multiphysics problems represented by systems of partial differential equations with time-dependent excitations. The set of PDEs is cast in first order differential form (in time) u = f (u ,s,t;c ) where u (s,t) is the vector collecting all state variables including the velocities/time rates, s is a space coordinate (here, one-dimensional systems are considered without lack of generality for the space dependence) and t denotes time. The vector field f depends, in general, not only on the classical state variables (such as positions and velocities) but also on the space gradients of the leading unknowns. The space gradients are introduced as part of the state variables. This is justified by the mathematical and computational requirements on the continuity in space up to the proper differential order of the space gradients associated with the unknown position vector field. The path following procedure employs, for the computation of the periodic solutions, only the evaluation of the vector field f . This part of the path following procedure within the proposed combined scheme was formerly implemented by Dankowicz and coworkers in a MATLAB software package called COCO. The here proposed procedure seeks to discretize the space dependence of the variables using finite elements based on Lagrangian polynomials which leads to a discrete form of the vector field f . A concurrent bifurcation analysis is carried out by calculating the eigenvalues of the monodromy matrix. A hinged-hinged nonlinear beam subject to a primary-resonance harmonic transverse load or to a parametric-resonance horizontal end displacement is considered as a case study. Some primary-resonance frequency-response curves are calculated along with their stability to assess the convergence of the discretization scheme. The frequency-response curves are shown to be in close agreement with those calculated by direct integration of the PDEs through the FE software called COMSOL Multiphysics. Besides primary-resonance direct forcing conditions, also parametric forcing causing the principal parametric resonance of the lowest two bending modes is considered through construction of the associated transition curves. The proposed approach integrates algorithms from the finite element and bifurcation domains thus enabling an accurate and effective unfolding of the bifurcation and post-bifurcation scenarios of nonautonomous PDEs with the underlying structures.Copyright

Nonlinear Dynamic Response of Carbon Nanotube Nanocomposite Microbeams

The nonlinear dynamic response of nanocomposite microcantilevers is investigated. The microbeams are made of a polymeric hosting matrix (e.g., epoxy, polyether ether ketone (PEEK), and polycarbonate) reinforced by longitudinally aligned carbon nanotubes (CNTs). The 3D transversely isotropic elastic constitutive equations for the nanocomposite material are based on the equivalent inclusion theory of Eshelby and the Mori–Tanaka homogenization approach. The beam-generalized stress resultants, obtained in accordance with the Saint-Venant principle, are expressed in terms of the generalized strains making use of the equivalent constitutive laws. These equations depend on both the hosting matrix and CNTs elastic properties as well as on the CNTs volume fraction, geometry, and orientation. The description of the geometry of deformation and the balance equations for the microbeams are based on the geometrically exact Euler–Bernoulli beam theory specialized to incorporate the additional inextensibility constraint due to the relevant boundary conditions of microcantilevers. The obtained equations of motion are discretized via the Galerkin method retaining an arbitrary number of eigenfunctions. A path following algorithm is then employed to obtain the nonlinear frequency response for different excitation levels and for increasing volume fractions of carbon nanotubes. The fold lines delimiting the multistability regions of the frequency responses are also discussed. The volume fraction is shown to play a key role in shifting the linear frequencies of the beam flexural modes to higher values. The CNT volume fraction further shifts the fold lines to higher excitation amplitudes, while it does not affect the backbones of the modes (i.e., oscillation frequency–amplitude curves). [DOI: 10.1115/1.4034736]

In-plane Strain and Stress Fields in Theories of Shearable Laminated Plates Subject to Transverse Loads

The predictions of a new theory of orthotropic laminated plates are compared with those of two other theories equally based on a Reissner-Mindlin Ansatz for the displacement field, either layer by layer [5] or for the whole plate [4]. A well-known merit of such an Ansatz is to allow and account for transverse shearings. What we are after here is to determine how well in-plane strain and stress fields are described. For definiteness, we consider circular plates that are axi-symmetrically loaded, whose layers are made of transversely isotropic materials and are symmetrically located with respect to the midplane of the plate. The new theory allows for an explicit analytic solution of this problem, as the simpler of the two theories considered for comparison does, but shows an accuracy closer to the other more complex theory, whose governing equations we solve numerically; as a benchmark, we use a numerical solution of the corresponding three-dimensional equilibrium problem; the results of our comparison are summarized graphically in the final section. While the new theory allows for an explicit analytic solution of this simple problem, the governing equations of the other two theories are solved numerically; as a benchmark, we use a numerical solution of the corresponding three-dimensional equilibrium problem; the results of our comparison are summarized graphically in the final section.

Geometric Constructive Traces in Drawings by Francesco Borromini

Here some results are presented from the ongoing research of the analysis of the drawings by Francesco Borromini, preserved at the Albertina in Vienna, relating to designs of cupolas, in order to highlight the geometric constructions and the proportional relationships in function of the realization of the projects. In fact, besides the symbolic geometries traceable especially in the most famous buildings, such as the San Carlo alle Quattro Fontane or Sant’Ivo alla Sapienza, evidenced even by Borromini himself, other and more decisive geometries are revealed by carefully observing the signed drawings. The research methodology adopted is based on comparing what emerged from the analysis of the drawings, the study of the treatises concerning the construction methods of the cupolas and the results obtained from the survey by means of a 3D laser scanner. The survey, in fact, becomes the tool to ascertain the correspondence between the theoretical form underlying the drawings, and the realized form, including the provisional works that determine the geometric conformation, verifying the link between the geometric-graphic reasoning related to the load distribution and the consequent dimension of the components.

Computational efficiency and accuracy of sequential nonlinear cyclic analysis of carbon nanotube nanocomposites

Abstract The accuracy and efficiency of a numerical strategy for sequential nonlinear cyclic analyses of carbon nanotube nanocomposites are investigated. The computational approach resorts to a nonlinear 3D finite element implementation that seeks to solve the cyclic hysteretic response of the nanocomposite. A variant of the Newton-Raphson method within a time integration scheme is proposed whereby the elastic tangent matrix is chosen as iteration matrix without paying the price of its iterative update. This is especially rewarding in the context of the employed mechanical model which exhibits hysteresis manifested through a discontinuous change in the stiffness at the reversal points where the loading direction is reversed. Key implementation aspects – such as the integration of the nonlinear 3D equations of motion, the numerical accuracy/efficiency as a function of the time step or the mesh size – are discussed. In particular, efficiency is regarded as performing fast computations especially when the number of cyclic analyses becomes large. By making use of laptop CPU cores, a good speed of computations is achieved not only through parallelization but also employing a caching procedure for the iteration matrix.