Paolo Bisegna

Variational bounds for the overall properties of piezoelectric composites

In this paper, variational bounds for the overall properties of periodic heterogeneous media with nonlinear and nonlocal piezoelectric constitutive relationships are obtained. First, elementary bounds are developed by extending to piezoelectric materials the well-known Voigt and Reuss bounds. Then, by generalizing the two Hashin-Shtrikman principles, eight new variational principles are derived and applied to obtain bounds. In fact, the variational principles developed are based on auxiliary electroelastic equilibrium problems, which can be solved by a transformation from the space domain to the Fourier domain. As an application, fibre-reinforced linear piezoelectric composites are considered, and expressions of upper and lower bounds for the overall properties of these composites are developed in closed form.

Diffusion of the Second Messengers in the Cytoplasm Acts as a Variability Suppressor of the Single Photon Response in Vertebrate Phototransduction

The single photon response in vertebrate phototransduction is highly reproducible despite a number of random components of the activation cascade, including the random activation site, the random walk of an activated receptor, and its quenching in a random number of steps. Here we use a previously generated and tested spatiotemporal mathematical and computational model to identify possible mechanisms of variability reduction. The model permits one to separate the process into modules, and to analyze their impact separately. We show that the activation cascade is responsible for generation of variability, whereas diffusion of the second messengers is responsible for its suppression. Randomness of the activation site contributes at early times to the coefficient of variation of the photoresponse, whereas the Brownian path of a photoisomerized rhodopsin (Rh*) has a negligible effect. The major driver of variability is the turnoff mechanism of Rh*, which occurs essentially within the first 2-4 phosphorylated states of Rh*. Theoretically increasing the number of steps to quenching does not significantly decrease the corresponding coefficient of variation of the effector, in agreement with the biochemical limitations on the phosphorylated states of the receptor. Diffusion of the second messengers in the cytosol acts as a suppressor of the variability generated by the activation cascade. Calcium feedback has a negligible regulatory effect on the photocurrent variability. A comparative variability analysis has been conducted for the phototransduction in mouse and salamander, including a study of the effects of their anatomical differences such as incisures and photoreceptors geometry on variability generation and suppression.

Evaluation of higher-order theories of piezoelectric plates in bending and in stretching

Many models for the flexural and membranal behaviour of piezoelectric plates are available in the literature. They are based on different assumptions concerning the strain, stress, electric and electric-displacement fields inside the plate. A critical comparison among such models is presented here in a completely analytic way, in order to assess the accuracy of the results they provide and determine their range of applicability. The comparison is made by using a class of case-study problems, whose analytical solutions in the framework of the linear theory of piezoelectricity are available, as benchmarks for the solutions supplied by the plate models. The evaluated models are also here rationally derived from the three-dimensional theory of piezoelectricity, and a consistent treatment of the stress and electric-displacement relaxation conditions is proposed.

Bounds on the overall properties of composites with debonded frictionless interfaces

In the present work the homogenization problem of periodic composites with nonlinear hyperelastic constituents and debonded frictionless interfaces is formulated as a minimum problem of the free- or complementary-energy of the unit cell of the composite over the convex set of the admissible strain or stress fields, respectively. Because of the unilateral contact condition at the debonded interfaces, the overall constitutive behavior of the composite turns out to be nonlinear even when the constituents are linearly hyperelastic. In this particular case, the homogenized free- and complementary-energy density functionals of the composite are positively homogeneous of degree two, i.e. from a mechanical point of view, the overall constitutive behavior of the composite is nonlinear conewise multimodular. Approximations of the proposed variational principles, suitable for numerical computations, are built up by adopting the finite-element method. In this way, rigorous upper bounds on the homogenized free- and complementary-energy density functionals are obtained. Then, lower bounds are derived by using the conjugacy relation between the homogenized energy density functionals.

On methods for bounding the overall properties of periodic piezoelectric fibrous composites

Abstract The homogenization problem of linearly piezoelectric fibrous composites with a periodic microstructure is formulated by adopting the free- and complementary-energy minimum principles, as well as the Hashin Shtrikman principles. The auxiliary electroelastic problems involved in the Hashin-Shtrikman principles are solved by a transformation from the space domain to the Fourier domain. Approximations of those variational principles, suitable for numerical computations, are built up by adopting the Fourier-series method and the finite-element method. In this way, rigorous upper and lower bounds on the overall material moduli are obtained. An example is presented, in order to show the effectiveness of the proposed approximation techniques and to compare them.

A consistent theory of thin piezoelectric plates

In this paper a theory of thin piezoelectric plates is obtained throigh a rational derivation from the three-dimensional linear theory of piezoelectricity. The coupling between the ekictric and mechanical fields is taken into account, leading to a consistent definition of the bending and stretching stiffnesses. In particular, it is shown that a piezoelectric plate has a different stretching stiffness when it is used as an actuator or as a sensor. The procedure used to derive the field equations governing the piezoelectric plate problem is based on the initial functions method, in conjunction with a rescaling of the applied loads. The field equations are then rewritten in a variational form, according to a generalized statement of the virtual work principle, in order to deduce the compatible boundary conditions. The theory established here is used to find closed-form expressions of the solutions of some technical problems, involving piezoelectric plates used as sensors or actuators.

Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character. As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u0, which keeps the prescribed boundary data, and solves the equation . This is an elliptic equation containing a term depending on the history of the gradient of u0; the matrices B0, A1 in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term ℱ keeping trace of the initial data.

An impedance-based flow microcytometer for single cell morphology discrimination.

Cell shape is a fundamental biological feature, providing specific information about physiological or pathological cellular conditions. Most of the state-of-the-art microfluidic cytometers, however, only allow simple cell analysis, including viability studies, cell counting and sorting. In this work, we present a non-invasive, label-free device capable of single cell morphology discrimination in continuous flow. The device is based on the principle of liquid electrodes, fabricated in a cross configuration around a sensing zone. This arrangement allows measurement of cell impedance along orthogonal orientations and extraction of an index describing cell shape anisotropy. By adding prior to the sensing volume a series of lateral liquid electrodes, the particle stream was focused toward the channel midline and each cell was oriented in a specific direction before shape sensing. We demonstrate the proof of concept by performing spherical and elongated particle discrimination. As an application, we show that the shape changes experienced during cell division can be monitored and characterized. In particular, budding yeasts at different stages of the mitotic cycle were identified by extracting their anisotropy index.

Identification of key factors that reduce the variability of the single photon response

Rod photoreceptors mediate vision in dim light. Their biological function is to discriminate between distinct, very low levels of illumination, i.e., they serve as reliable photon counters. This role requires high reproducibility of the response to a particular number of photons. Indeed, single photon responses demonstrate unexpected low variability, despite the stochastic nature of the individual steps in the transduction cascade. We analyzed individual system mechanisms to identify their contribution to variability suppression. These include: (i) cooperativity of the regulation of the second messengers; (ii) diffusion of cGMP and Ca2+ in the cytoplasm; and (iii) the effect of highly localized cGMP hydrolysis by activated phosphodiesterase resulting in local saturation. We find that (i) the nonlinear relationships between second messengers and current at the plasma membrane, and the cGMP hydrolysis saturation effects, play a major role in stabilizing the system; (ii) the presence of a physical space where the second messengers move by Brownian motion contributes to stabilization of the photoresponse; and (iii) keeping Ca2+ at its dark level has only a minor effect on the variability of the system. The effects of diffusion, nonlinearity, and saturation synergize in reducing variability, supporting the notion that the observed high fidelity of the photoresponse is the result of global system function of phototransduction.

Mindlin-Type Finite Elements for Piezoelectric Sandwich Plates

New finite-element formulations are developed for the analysis of a plate having thin piezoelectric actuators bonded on its upper and/or lower surfaces. The proposed finite elements are two-dimensional, quadrangular, four-node, Mindlin-type, locking-free, and have five degrees of freedom per node. These are the deflection of the middle plane of the plate, the rotations of the fibers normal to the middle plane and the actuation electric potentials of the piezoelectric actuators. The effectiveness of the proposed finite-element formulations is shown by studying some case problems, whose analytical solutions are available.