Raffaella Rizzoni

Pressurized Shape Memory Thin Films

We study the behavior of a martensitic thin film with a hydrostatic pressure applied underneath the film. The problem is formulated in 3-D for a single crystal film of thickness h, and a Cosserat membrane theory is derived by Γ-convergence techniques in the limit h→0. The membrane theory is further simplified using a second Γ-convergence argument based on hard moduli. The resulting theory supports energy minimizing “tunnels”: structures having the shape of part of a cylinder cut by a plane parallel to its axis, obtained by releasing the film from the substrate along a strip with a certain orientation. As the temperature is raised (at fixed pressure) the energy minimizing shape collapses gradually to the substrate, accompanied by a martensite-to-austenite phase transformation. During this process the tunnel supports a microstructure consisting of fine bands of austenite parallel to the axis of the tunnel, alternating with bands of a single variant of martensite. Formulas for the associated volume–temperature–pressure relation are given: in these the latent heat of transformation plays an important role.

Asymptotic Study on a Soft Thin Layer: The Non-Convex Case

It is proposed to model the adhesive bonding of elastic bodies when the adhesive is a phase-transforming material. For this purpose, the (isothermal) Frémond model is adopted, including only two variants of martensite. In the first part of this paper, asymptotic expansions are used to study the asymptotic behavior of the adhesive as its thickness and elastic coefficients tend toward zero. In the second part, the energy minimization approach is used and the equilibrium of a one-dimensional bar is studied in detail. The simplified one-dimensional context adopted here makes it possible to compute contact laws taking nucleation and the kinetics of the phase transformation explicitly into account.

Comparative Assessment of Two Constitutive Models for Superelastic Shape-Memory Wires Against Experimental Measurements

Two constitutive models representative of two well-known modeling techniques for superelastic shape-memory wires are reviewed. The first model has been proposed by Kim and Aberayatne in the framework of finite thermo-elasticity with non-convex energy [1]. In the present article this model has been modified in order to take into account the difference between elastic moduli of austenite and martensite and to introduce the isothermal approximation proposed in [1]. The second model has been developed by Auricchio et al. within the theory of irreversible thermodynamics with internal variables [2]. Both models are temperature and strain rate dependent and they take into account thermal effects. The focus in this article is on investigating how the two models compare with experimental data obtained from testing superelastic NiTi wires used in the design of a prototypal anti-seismic device [3, 4]. After model calibration and numerical implementation, numerical simulations based on the two models are compared with data obtained from uniaxial tensile tests performed at two different temperatures and various strain rates.

On twinning and domain switching in ferroelectric Pb(Zr1−xTix)O3—part I: twins and domain walls

Abstract We study the electromechanical behavior of lead zirconate titanate ferroelectric ceramics (PZT), by means of a three-dimensional continuum model for deformable ferroelectric bodies in their polar phase characterized by spontaneous polarization and strain. Spontaneous polarization and strain organize into a domain structure which minimizes electrostatic and elastic energies and which can be modified by the application of electromechanical loads. Such process, which is called “domain switching”, is associated with electrical and mechanical hysteresis and can be studied as a minimization problem for a functional which reminds the micromagnetic energy of deformable ferromagnetics. In this paper, which is the first of two, we deal with the electromechanical model and related constitutive assumptions, as well as with the analysis of domain structure in PZT. In particular, following the discover of a new monoclinic phase in PZT carried by Noheda and co-workers, we analyze twinning between spontaneous strain at the various phase boundaries and show that both non-generic, non-conventional twins and finely-twinned laminates are possible, and also that the presence of a monoclinic phase may explain PZT exceptional properties.

Multiscale Modeling of Imperfect Interfaces and Applications

Modeling interfaces between solids is of great importance in the fields of mechanical and civil engineering. Solid/solid interface behavior at the microscale has a strong influence on the strength of structures at the macroscale, such as gluing, optical systems, aircraft tires, pavement layers and masonry, for instance. In this lecture, a deductive approach is used to derive interface models, i.e. the thickness of the interface is considered as a small parameter and asymptotic techniques are introduced. A family of imperfect interface models is presented taking into account cracks at microscale. The proposed models combine homogenization techniques for microcracked media both in three-dimensional and two-dimensional cases, which leads to a cracked orthotropic material, and matched asymptotic method. In particular, it is shown that the Kachanov type theory leads to soft interface models and, alternatively, that Goidescu et al. theory leads to stiff interface models. A fully nonlinear variant of the model is also proposed, derived from the St. Venant-Kirchhoff constitutive equations. Some applications to elementary masonry structures are presented.

Assessment of models and methods for pressurized spherical composites

The elastic properties of a spherical heterogeneous structure with isotropic periodic components is analyzed and a methodology is developed using the two-scale asymptotic homogenization method (AHM) and spherical assemblage model (SAM). The effective coefficients are obtained via AHM for two different composites: (a) composite with perfect contact between two layers distributed periodically along the radial axis; and (b) considering a thin elastic interphase between the layers (intermediate layer) distributed periodically along the radial axis under perfect contact. As a result, the derived overall properties via AHM for homogeneous spherical structure have transversely isotropic behavior. Consequently, the homogenized problem is solved. Using SAM, the analytical exact solutions for appropriate boundary value problems are provided for different number of layers for the cases (a) and (b) in the spherical composite. The numerical results for the displacements, radial and circumferential stresses for both methods are compared considering a spherical composite material loaded by an inside pressure with the two cases of contact conditions between the layers (a) and (b).

On modelling brick/mortar interface via a St. Venant-Kirchhoff orthotropic soft interface. Part II: in silico analysis

A new strategy for the numerical modelling of brick/mortar interfaces at the macro-scale taking into account finite strains and evolving microcracking phenomena is introduced. A numerical validation of the non-linear-imperfect interface model formulated in Part I of the present paper, is proposed. Well-established experimental data concerning diagonal compression of masonry walls are simulated within the finite element method. The localisation zones of highest displacement jumps and stresses obtained through the numerical simulations are in good agreement with the experimental findings. The proposed non-linear interface model is also compared with a linear interface model for masonry structures.

On modelling brick/mortar interface via a St. Venant-Kirchhoff orthotropic soft interface. Part I: theory

In this paper, a nonlinear-imperfect interface model is proposed in order to model brick/mortar-interface behavior in small masonry assemblies. The proposed model, formulated according a micromechanical strategy, derives from a consolidated approach coupling arguments of asymptotic analysis and homogenisation method. The adopted asymptotic technique is extended to the finite strain theory. The homogenisation strategy, under the non-interacting approximation, is extended to microcracked-orthotropic-hyperelastic materials. Simple numerical simulations, developed within the framework of finite element method, highlight the model soundness and its applicability in finite-strain problems.

Two-Sided Estimates for Local Minimizers in Compressible Elasticity

In this communication we anticipate some results of a research in progress [DR07], whose purpose is to find necessary conditions and sufficient conditions for local energy minima in finite elasticity. Though our analysis includes both compressible and incompressible continua, the present account is restricted to the compressible case. Consider a three-dimensional continuous body, occupying a region Ω in the reference configuration. Assume that the body is hyperelastic and homogeneous, that is, that there is a strain energy density w which is the same at all points of Ω. The function w is assumed to be frame-indifferent

Numerical Analysis of Two Non-linear Soft Thin Layers

In a first part, we consider a bar with extremities subject to a given displacement and made by two elastic bodies with linear stress-strain relation separated by an adhesive layer of thickness h. The material of the adhesive is characterized by a non convex (piecewise quadratic) strain energy density with elastic modulus k. After considering the equilibrium problem of the bar and determining the stable and metastable solutions, we let (h,k) tending to zero and we obtain the corresponding asymptotic contact laws, linking the stress to the jump of the displacement at the adhesive interface. The second part of the paper is devoted to the bi-dimensional problem of two elastic bodies separated by a thin soft adhesive. The behaviour of the adhesive is non associated elastic-plastic. As in the first part, we study the asymptotic contact laws.