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Concept# Constitutive equation

Summary

In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations.
Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, it is often ne

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The research work reported in this dissertation is aimed to develop efficient and stable numerical schemes in order to obtain accurate numerical solution for viscoelastic fluid flows within the spectral element context. The present research consists in the transformation of a large class of differential constitutive models into an equation where the main variable is the logarithm of the conformation tensor or a quantity related to it in a simple way. Particular cases cover the Oldroyd-B fluid and the FENE-P model. Applying matrix logarithm formulation in the framework of the spectral element method is a new type of approach that according to our knowledge no one has implemented before. The reformulation of the classical constitutive equation using a new variable namely the logarithmic formulation, enforces the eigenvalues of the conformation tensor to remain positive for all steps of the simulation. However, satisfying the symmetric positive definiteness of the conformation tensor during the simulation is the necessary condition for stability; but definitely, it is not the sufficient condition to reach meaningful results. The main effort of this research is devoted to introduce a new algorithm in order to overcome the drawback of direct reformulating the classical constitutive equation to the logarithmic one. To evaluate the capability of the extended matrix logarithm formulation, comprehensive studies have been done based on the linear stability analysis to show the influence of this method on the resulting eigenvalue spectra and explain its success to tackle high Weissenberg numbers. With this new method one can treat high Weissenberg number flows at values of practical interest. One of the worst obstacles for numerical simulation of viscoelastic fluids is the presence of spurious modes during the simulation. At high Weissenberg number, many schemes suffer from instabilities and numerical convergence may not be attainable. This is often attributed to the presence of solution singularities due to the geometry, the dominant non-linear terms in the constitutive equations, or the change of type of the underlying mixed-form differential system. Refining the mesh proved to be not very helpful. In this study, to understand more deeply the mechanism of instability generation a comprehensive study about the growth of spurious modes with time evolution, mesh refinement, boundary conditions and Weissenberg number or any other affected parameters has been performed. Then to get rid of these spurious modes the filter based stabilization of spectral element methods proposed by Boyd was applied with success.

Thin Film Bulk Acoustic Wave Resonators (TFBARs) had been developed a decade ago and since then were implemented extensively in mobile communications devices. The "heart" of a TFBAR consists of a piezoelectric film that operates as an acousto-electric transducer, stabilizing the transmission at a given predetermined frequency. For reasons such as space economy in hand-held devices, it is of interest to make these TFBARs tunable, so that a single TFBAR is multi-band responsive. This thesis demonstrates for the first time electrically tunable, single-component TFBARs. A theory describing the tuning behavior of dc bias induced acoustic resonances was developed. Then we made the hypothesis that dc bias induced piezoelectric BaxSr1-xTiO3 (BST) thin films – namely, paraelectric, non-piezoelectric films operating under dc bias – can be used to make electrically tunable TFBARs and that the devices can be switched on or off depending on the dc bias state. The devices were then fabricated: We integrated BST lms onto silicon substrates, micromachined the substrate to create the TFBARs and a new type of suspended planar capacitor which were then characterized, analyzed, and modeled, demonstrating successfully the new concept: We developed a theory describing the electrical tuning behavior of the dc bias induced acoustic resonances in paraelectric thin lms in terms of material parameters. The field dependent constitutive piezoelectric equations were derived from the Landau free energy P-expansion by taking the linear and nonlinear electrostrictive terms as well as the background permittivity into account. We considered two modes of excitation for the tuning of the acoustic resonances, namely the thickness excitation (TE) mode and the lateral field excitation (LFE) mode. The tuning behavior of the two types of resonators based on BST thin films was modeled and discussed. For the modeling we calculated the relevant tensor components controlling the tuning of the BST resonators from the available literature data. The fabrication of the membrane-type TFBARs was realized by integrating BST thin films onto silicon substrates and using micromaching technologies. We showed that the developed TFBARs can be switched on or off with a dc bias. At a dc electric field of 615 kV/cm we observed a tuning of -2.4% (-66 MHz) and -0.6% (-16 MHz) for the resonance and antiresonance frequencies of the device, while the resonance frequency at a dc electric field extrapolated to 0 kV/cm was 2.85 GHz. The effective electromechanical coupling factor k2eff of the device increased up to 4.4%. The tuning was non-hysteretic. The Quality-factor (Q-factor) of the device was about 200. The developed micromaching processes for the TFBARs were used to fabricate coplanar BST capacitors on silicon. Micromaching was used to remove the Si substrate under the active area of the device. Comparing this new micromachined coplanar capacitor with conventional non-micromachined capacitors, we demonstrated that the removal of the substrate from the active device area resulted in a reduction of parasitic effects. The micromachined coplanar capacitor showed an increased tunability and a reduced loss tangent in comparison to the non-micromachined capacitor. The micromachined capacitor showed a relative tunability of 37% at a dc electric field of 1100 kV/cm. The loss tangent was 0.08 and 0.06 at zero and maximum dc bias, respectively, at a measurement frequency of 20 GHz. The integration of epitaxial BST thin films on silicon was studied as well because of its potential improvement of the device performance in comparison to devices based on polycrystalline films. Two different electrode/buffer layer systems, YBa2Cu3O7-x (YBCO) / CeO2 / YxZr1-xO2-x/2 (YSZ) and TiN were used for the integration. Epitaxial BST thin films were grown with both structures. For the BST / YBCO / CeO2 / YSZ / Si structure, the BST unit cell was rotated by 45° in the in-plane dimension with respect to the substrate. The BST layer exhibited good structural quality as indicated by a Full Width at Half Maximum (FWHM) of 0.5° of the rocking curve of the BST(002) diffraction peak. For the BST / TiN / Si structure, the BST layer was grown with a cube-on-cube epitaxial relationship on the silicon substrate, thus demonstrating epitaxially grown BST on silicon using conventional bottom electrode that is easily acceptable by the microelectronic industry. However, in this case, the structural quality of the BST layer was reduced in comparison to the BST / YBCO / CeO2 / YSZ / Si structure. The FWHM of the rocking curve of the BST(002) diffraction peak was 2.2°. We established the temperature (T=550 to 600 °C) and pressure (p ≈ 10-7 to 5 × 10-4 Torr) conditions for the growth of epitaxial BST thin films on TiN-buffered Si. At too high temperatures and/or oxygen pressures epitaxial BST thin film growth was impeded due to the oxidation of the TiN layer. By introducing experimental results from the electrical characterization of the micromachined devices with the polycrystalline BST into our developed theory, we could successfully model the tuning behavior of our fabricated TFBARs. The modeling allowed us to de-embed the intrinsic electromechanical properties of a freestanding BST layer. The effect of increasing mechanical load on the tuning performance of the device was modeled and studied experimentally. Under strong mechanical load, the tuning of both resonance and antiresonance frequency was reduced. The effect was attributed to a reduction in the tuning of k2eff of the device and of the sound velocity of the BST layer.

Over the last two decades, we have witnessed an increasing application of mathematical models and numerical simulations for the study of the cardiovascular system. Indeed, both tools provide an important contribution to the analysis of the functioning of the different components of the cardiovascular system (i.e. heart, vessels and blood) and of their interactions either in physiological and pathological conditions. For this reason, reliable constitutive models for the cardiac, arterial and venous tissues as well as for the blood are an essential prerequisite for a number of different objectives that range from the improved diagnostic to the study of the onset and development of cardiovascular diseases (e.g atherosclerosis or aneurysms). This work focuses on the mathematical and numerical modeling of healthy and unhealthy cerebral arterial tissue. In particular, it presents a detailed analysis of different constitutive models for the arterial tissue by means of finite element numerical simulations of arterial wall mechanics and fluid-structure interaction problems occurring in hemodynamics. Hyperelastic isotropic and anisotropic constitutive laws are considered for the description of the passive mechanical behavior of the vessels. An anisotropic multi-mechanism model, specifically proposed for the cerebral arterial tissue, for which the activation of the collagen fibers occurs at finite strains is employed. Firstly, the constitutive laws are numerically validated by considering numerical simulations of static inflation tests on a cylindrical geometry representing a specimen of anterior cerebral artery. With this regard, the material parameters for the constitutive law are obtained from the data fitting of experimental measurements obtained on the same vessel. The constitutive models are critically discussed according to their capability of describing the physiogical highly nonlinear behavior of arteries and on other numerical aspects related to the computational simulation of arterial wall mechanics. Afterwards, simulations of the blood flow and vessel wall interactions are carried out on idealized blood vessels in order to analyze the influence of the modeling choice for the arterial wall on hemodynamic and mechanical quantities that are commonly considered as indicators of physiological or pathological conditions of arteries. We also consider the numerical simulations of unhealthy cerebral arterial tissues by taking into account the mechanical weakening of the vessel wall that occurs during early development stages of cerebral aneurysms by means of static inflation and FSI simulations. We employ both isotropic and anisotropic models study the effects of the mechanical degradation on hemodynamic and mechanical quantities of interest. The FSI simulations are carried out both on idealized geometries of blood vessels and on domains representing idealized and anatomically realistic cerebral aneurysms.