Lorenzo Bardella

On the elastic behavior of syntactic foams

Abstract This work concerns composite materials called “syntactic foams”, i.e., materials made by a polymeric matrix filled with hollow solid inclusions. Explicit formulae for the homogenized values of the elastic moduli of these materials are derived, by means of the physical model and the corresponding elastic solution used by Herve and Pellegrini [Herve, E., Pellegrini, O., 1995. Archives Mechanics. 47 (2), 223–246.]. The morphologically representative patterns theory of Bornert et al. [Bornert, M., Stolz, C., Zaoui, A., 1996. Journal of the Mechanics and Physics of Solids 44, 307–331.] is used to take into account both the influence of the filler gradation and the presence of “unwanted” voids in the matrix, factors that are shown to be important in characterizing the mechanical behavior of syntactic foams. Comparisons with both experimental and numerical results show that the techniques used are capable of predicting, with good accuracy, the elastic moduli of real syntactic foams, i.e., those arising from an actual production process.

Elastic design of syntactic foamed sandwiches obtained by filling of three-dimensional sandwich-fabric panels

A non-conventional sandwich, made by a fabric panel core filled by a syntactic foam, and by resin-impregnated fiberglass skins, is studied in the elastic range, with the aim of giving guidelines to its minimum weight design. Standard homogenization techniques are employed to compute the elastic moduli of the skins, whereas a specifically developed homogenization method has been used to obtain the elastic moduli of the core. A simple but accurate relationship for computing the shear stiffness of the sandwich was used in conjunction with the well-known formulae for the bending stiffness. Comparisons with both experiments and numerical predictions show good accuracy of both the proposed homogenization methods and the overall stiffness evaluation procedure.

A phenomenological constitutive law for the nonlinear viscoelastic behaviour of epoxy resins in the glassy state

Abstract We propose a constitutive model to describe the nonlinear viscoelasticity of epoxy resins in the glassy state. Experimental tests have shown that nonlinear viscoelasticity rules the cyclic behaviour of epoxy resins before the material strength is reached. The proposed model allows the simulation of this cyclic behaviour and, in particular, of the flex which characterises the stress–strain curve upon unloading. The consistent integration algorithm, based on the Central Difference Scheme, of the proposed constitutive law is given. As an example, the model is successfully used to numerically homogenize the cyclic behaviour of hollow sphere-filled epoxy resins (i.e., syntactic foams) by means of unit cell models.

An extension of the Secant Method for the homogenization of the nonlinear behavior of composite materials

Abstract We discuss the consequences of a different application of the principle the Modified Secant Method is [C. R. Acad. Sci. Paris Ser. IIb 320 (1995) 563] based on. In fact, we directly compute the second-order averages of the local fields available from the linear elastic homogenization procedure exploited in order to evaluate the effective elastic moduli. This method, which can be seen either as a simplification of the Modified Secant Method or as an extension of the Secant Method [J. Mech. Phys. Solids 26 (1979) 325], may be useful for any composite whose overall elastic constants need to be estimated by modeling the microstructure through Morphologically Representative Patterns [J. Mech. Phys. Solids 44 (1996) 307], which is for instance the case of syntactic foams [Int. J. Solids and Structures 38/40–41 (2001) 7235]. In order to show the accuracy of the proposed method, we apply it to several examples and compare its results with those obtainable by means of other analytical methods available in the literature, with numerical results of Finite Element simulations, and with experimental results. Closed-form solutions are derived for the effective yield stress of porous metals and incompressible composites reinforced with rigid spheres.

A finite element framework for distortion gradient plasticity with applications to bending of thin foils

Abstract A novel general purpose Finite Element framework is presented to study small-scale metal plasticity. A distinct feature of the adopted distortion gradient plasticity formulation, with respect to strain gradient plasticity theories, is the constitutive inclusion of the plastic spin, as proposed by Gurtin (2004) through the prescription of a free energy dependent on Nye’s dislocation density tensor. The proposed numerical scheme is developed by following and extending the mathematical principles established by Fleck and Willis (2009). The modeling of thin metallic foils under bending reveals a significant influence of the plastic shear strain and spin due to a mechanism associated with the higher-order boundary conditions allowing dislocations to exit the body. This mechanism leads to an unexpected mechanical response in terms of bending moment versus curvature, dependent on the foil length, if either viscoplasticity or isotropic hardening are included in the model. In order to study the effect of dissipative higher-order stresses, the mechanical response under non-proportional loading is also investigated.

Latent hardening size effect in small-scale plasticity

We aim at understanding the multislip behaviour of metals subject to irreversible deformations at small-scales. By focusing on the simple shear of a constrained single-crystal strip, we show that discrete Dislocation Dynamics (DD) simulations predict a strong latent hardening size effect, with smaller being stronger in the range [1.5 µm, 6 µm] for the strip height. We attempt to represent the DD pseudo-experimental results by developing a flow theory of Strain Gradient Crystal Plasticity (SGCP), involving both energetic and dissipative higher-order terms and, as a main novelty, a strain gradient extension of the conventional latent hardening. In order to discuss the capability of the SGCP theory proposed, we implement it into a Finite Element (FE) code and set its material parameters on the basis of the DD results. The SGCP FE code is specifically developed for the boundary value problem under study so that we can implement a fully implicit (Backward Euler) consistent algorithm. Special emphasis is placed on the discussion of the role of the material length scales involved in the SGCP model, from both the mechanical and numerical points of view.

A critical evaluation of mechanical models for sandwich beams

We focus on the description of the stress state of sandwich beams under bending and shear, a non-trivial task if Saint-Venants principle does not hold, as it is the case if the skins are somewhat stiffer than the core. Each of the analytical structural models available in literature turns out to be accurate for a limited range of relative stiffness between core and skins, or sandwich heterogeneity. For a simply supported sandwich beam subject to uniform transversal load, we evaluate the stress by means of (a) the classical theory relying on the linear cross-section kinematics, appropriate if Saint-Venants principle holds, (b) the structural theory based on the zig-zag warping (e.g. Krajcinovic D. Sandwich beam analysis. J Appl Mech, Trans ASME 1972; 39(3): 773–778), and (c) the higher-order theory of Frostig et al. (Frostig Y, Baruch M, Vilnay O, et al. High-order theory for sandwich-beam behavior with transversely flexible core. J Eng Mech, Trans ASCE 1992; 118(5): 1026–1043), the latter usually appropriate when the core is much softer than the skins. The results are compared, for several combinations of material and geometrical parameters, with those of finite element simulations in which the sandwich is modelled as a plane stress continuum. This comparison allows us to provide some graphs which can help in selecting the model appropriate for each sandwich heterogeneity. This is accomplished in terms of non-dimensional material and geometrical parameters the sandwich heterogeneity depends on. We identify and discuss two levels of heterogeneity at which one should switch analytical model: one level is related to the validity of Saint-Venants principle, while the other level is concerned with the definition of antiplane sandwich.

Newmark's time integration method from the discretization of extended functionals

In this note we illustrate how to obtain the full family of Newmark s time integration algorithms within a rigorous variational framework, i.e., by discretizing suitably defined extended functionals, rather than by starting from a weak form (for instance, of the Galerkin type), as done in the past. The availability of functionals as a starting point is useful both as a tool to obtain new families of time integration methods, and as a theoretical basis for error estimates. To illustrate the first issue, here we provide some examples of how to obtain modified algorithms, in some cases significantly more accurate than the basic Newmark one despite having a comparable computational cost.

Time integration errors and some new functionals for the dynamics of a free mass

Abstract We study the numerical integration of the Poisson second-order ordinary differential equation which describes, for instance, the dynamics of a free mass. Classical integration algorithms, when applied to such an equation, furnish solutions affected by a significant “drift” error, apparently not studied so far. In the first part of this work we define measures of such a drift. We then proceed to illustrate how to construct both classical and extended functionals for the equation of motion of a free mass with given initial conditions. These tools allow both the derivation of new variationally-based time integration algorithms for this problem, and, in some cases, the theoretical isolation of the source of the drift. While we prove that this particular error is unavoidable in any algorithmic solution of this problem, we also provide some new time integration algorithms, extensions at little added cost of classical methods, which permit to substantially improve numerical predictions.

A theoretical framework for the study of compression sensing in ionic polymer metal composites

Ionic Polymer Metal Composites (IPMCs) are electro-responsive materials for sensing and actuation, consisting of an ion-exchange polymeric membrane with ionized units, plated within noble metal electrodes. In this work, we investigate the sensing response of IPMCs that are subject to a through-the-thickness compression, by specializing the continuum model introduced by Cha and Porfiri,1 to this one-dimensional problem. This model modifies the classical Poisson-Nernst-Plank system governing the electrochemistry in the absence of mechanical effects, by accounting for finite deformations underlying the actuation and sensing processes. With the aim of accurately describing the IPMC dynamic compressive behavior, we introduce a spatial asymmetry in the properties of the membrane, which must be accounted for to trigger a sensing response. Then, we determine an analytical solution by applying the singular perturbation theory, and in particular the method of matched asymptotic expansions. This solution shows a good agreement with experimental findings reported in literature.