WYPIWYG Hyperelasticity

In this paper we describe a new promising procedure to model hyperelastic materials from given stress-strain data. The main advantage of the proposed method is that the user does not need to have a relevant knowledge of hyperelasticity, large strains or hyperelastic constitutive modelling. The engineer simply has to prescribe some stress strain experimental data (whether isotropic or anisotropic) in also user prescribed stress and strain measures and the model almost exactly replicates the experimental data. The procedure is based on the piece-wise splines model by Sussman and Bathe and may be easily generalized to transversely isotropic and orthotropic materials. The model is also amenable of efficient finite element implementation. In this paper we briefly describe the general procedure, addressing the advantages and limitations. We give predictions for arbitrary ?experimental data? and also give predictions for actual experiments of the behaviour of living soft tissues. The model may be also implemented in a general purpose finite element program. Since the obtained strain energy functions are analytic piece-wise functions, the constitutive tangent may be readily derived in order to be used for implicit static problems, where the equilibrium iterations must be performed and the material tangent is needed in order to preserve the quadratic rate of convergence of Newton procedures.

Métodos computacionales para visco-hiperelasticidad anisótropa en grandes deformaciones

El comportamiento mecánico de muchos materiales biológicos y poliméricos en grandes deformaciones se puede describir adecuadamente mediante formulaciones isocóricas hiperelásticas y viscoelásticas. Las ecuaciones de comportamiento elástico y viscoelástico y las formulaciones computacionales para materiales incompresibles isótropos en deformaciones finitas están ampliamente desarrolladas en la actualidad. Sin embargo, el desarrollo de modelos anisótropos no lineales y de sus correspondientes formulaciones computacionales sigue siendo un tema de investigación de gran interés. Cuando se consideran grandes deformaciones, existen muchas medidas de deformación disponibles con las que poder formular las ecuaciones de comportamiento. Los modelos en deformaciones cuadráticas facilitan la implementación en códigos de elementos finitos, ya que estas medidas surgen de forma natural en la formulación. No obstante, pueden dificultar la interpretación de los modelos y llevar a resultados pocos realistas. El uso de deformaciones logarítmicas permite el desarrollo de modelos más simples e intuitivos, aunque su formulación computacional debe ser adaptada a las exigencias del programa. Como punto de partida, en esta tesis se demuestra que las deformaciones logarítmicas representan la extensión natural de las deformaciones infinitesimales, tanto axiales como angulares, al campo de las grandes deformaciones. Este hecho permite explicar la simplicidad de las ecuaciones resultantes. Los modelos hiperelásticos predominantes en la actualidad están formulados en invariantes de deformaciones cuadráticas. Estos modelos, ya sean continuos o microestructurales, se caracterizan por tener una forma analítica predefinida. Su expresión definitiva se calcula mediante un ajuste de curvas a datos experimentales. Un modelo que no sigue esta metodología fue desarrollado por Sussman y Bathe. El modelo es sólo válido para isotropía y queda definido por una función de energía interpolada con splines, la cual reproduce los datos experimentales de forma exacta. En esta tesis se presenta su extensión a materiales transversalmente isótropos y ortótropos utilizando deformaciones logarítmicas. Asimismo, se define una nueva propiedad que las funciones de energía anisótropas deben satisfacer para que su convergencia al caso isótropo sea correcta. En visco-hiperelasticidad, aparte de las distintas funciones de energía disponibles, hay dos aproximaciones computational típicas basadas en variables internas. El modelo original de Simó está formulado en tensiones y es válido para materiales anisótropos, aunque sólo es adecuado para pequeñas desviaciones con respecto al equilibrio termodinámico. En cambio, el modelo basado en deformaciones de Reese y Govindjee permite grandes deformaciones no equilibradas pero es, en esencia, isótropo. Las formulaciones anisótropas en este último contexto son microestructurales y emplean el modelo isótropo para cada uno de los constituyentes. En esta tesis se presentan dos formulaciones fenomenológicas viscoelásticas definidas mediante funciones hiperelásticas anisótropas y válidas para grandes desviaciones con respecto al equilibrio termodinámico. El primero de los modelos está basado en la descomposición multiplicativa de Sidoroff y requiere un comportamiento viscoso isótropo. La formulación converge al modelo de Reese y Govindjee en el caso especial de isotropía elástica. El segundo modelo se define a partir de una descomposición multiplicativa inversa. Esta formulación está basada en una descripción co-rotacional del problema, es sustancialmente más compleja y puede dar lugar a tensores constitutivos ligeramente no simétricos. Sin embargo, su rango de aplicación es mucho mayor ya que permite un comportamiento anisótropo tanto elástico como viscoso. Varias simulaciones de elementos finitos muestran la gran versatilidad de estos modelos cuando se combinan con funciones hiperelásticas formadas por splines. ABSTRACT The mechanical behavior of many polymeric and biological materials may be properly modelled be means of isochoric hyperelastic and viscoelastic formulations. These materials may sustain large strains. The viscoelastic computational formulations for isotropic incompressible materials at large strains may be considered well established; for example Ogden’s hyperelastic function and the visco-hyperelastic model of Reese and Govindjee are well known models for isotropy. However, anisotropic models and computational procedures both for hyperelasticity and viscohyperelasticity are still under substantial research. Anisotropic hyperelastic models are typically based on structural invariants obtained from quadratic strain measures. These models may be microstructurallybased or phenomenological continuum formulations, and are characterized by a predefined analytical shape of the stored energy. The actual final expression of the stored energy depends on some material parameters which are obtained from an optimization algorithm, typically the Levenberg-Marquardt algorithm. We present in this work anisotropic spline-based hyperelastic stored energies in which the shape of the stored energy is obtained as part of the procedure and which (exactly in practice) replicates the experimental data. These stored energies are based on invariants obtained from logarithmic strain measures. These strain measures preserve the metric and the physical meaning of the trace and deviator operators and, hence, are interesting and meaningful for anisotropic formulations. Furthermore, the proposed stored energies may be formulated in order to have material-symmetries congruency both from a theoretical and from a numerical point of view, which are new properties that we define in this work. On the other hand, visco-hyperelastic formulations for anisotropic materials are typically based on internal stress-like variables following a procedure used by Sim´o. However, it can be shown that this procedure is not adequate for large deviations from thermodynamic equilibrium. In contrast, a formulation given by Reese and Govindjee is valid for arbitrarily large deviations from thermodynamic equilibrium but not for anisotropic stored energy functions. In this work we present two formulations for visco-hyperelasticity valid for anisotropic stored energies and large deviations from thermodynamic equilibrium. One of the formulations is based on the Sidoroff multiplicative decomposition and converges to the Reese and Govindjee formulation for the case of isotropy. However, the formulation is restricted to isotropy for the viscous component. The second formulation is based on a reversed multiplicative decomposition. This last formulation is substantially more complex and based on a corotational description of the problem. It can also result in a slightly nonsymmetric tangent. However, the formulation allows for anisotropy not only in the equilibrated and non-equilibrated stored energies, but also in the viscous behavior. Some examples show finite element implementation, versatility and interesting characteristics of the models.

O 1s excitation and ionization processes in the CO2 molecule studied via detection of low-energy fluorescence emission

Oxygen 1s excitation and ionization processes in the CO2 molecule have been studied with dispersed and non-dispersed fluorescence spectroscopy as well as with the vacuum ultraviolet (VUV) photon?photoion coincidence technique. The intensity of the neutral O emission line at 845 nm shows particular sensitivity to core-to-Rydberg excitations and core?valence double excitations, while shape resonances are suppressed. In contrast, the partial fluorescence yield in the wavelength window 300?650 nm and the excitation functions of selected O+ and C+ emission lines in the wavelength range 400?500 nm display all of the absorption features. The relative intensity of ionic emission in the visible range increases towards higher photon energies, which is attributed to O 1s shake-off photoionization. VUV photon?photoion coincidence spectra reveal major contributions from the C+ and O+ ions and a minor contribution from C2+. No conclusive changes in the intensity ratios among the different ions are observed above the O 1s threshold. The line shape of the VUV?O+ coincidence peak in the mass spectrum carries some information on the initial core excitation

Cost and energy efficient reconfigurable embedded platform using Spartan-6 FPGAs

Modern FPGAs with run-time reconfiguration allow the implementation of complex systems offering both the flexibility of software-based solutions combined with the performance of hardware. This combination of characteristics, together with the development of new specific methodologies, make feasible to reach new points of the system design space, and make embedded systems built on these platforms acquire more and more importance. However, the practical exploitation of this technique in fields that traditionally have relied on resource restricted embedded systems, is mainly limited by strict power consumption requirements, the cost and the high dependence of DPR techniques with the specific features of the device technology underneath. In this work, we tackle the previously reported problems, designing a reconfigurable platform based on the low-cost and low-power consuming Spartan-6 FPGA family. The full process to develop the platform will be detailed in the paper from scratch. In addition, the implementation of the reconfiguration mechanism, including two profiles, is reported. The first profile is a low-area and low-speed reconfiguration engine based mainly on software functions running on the embedded processor, while the other one is a hardware version of the same engine, implemented in the FPGA logic. This reconfiguration hardware block has been originally designed to the Virtex-5 family, and its porting process will be also described in this work, facing the interoperability problem among different families.