Pavel Krejčí

Grants
- Phenomenological modeling of polymeric smart foams with behavior controlled by the magnetic field (CNR-16-08)
  from 05/04/2016 to 31/12/2018 main investigator
- The aim of the collaboration is to develop and test experimentally a reliable mathematical model for magnetic foams taking into account hysteresis effects in the material, with the goal to provide, as output, parameters needed to obtain optimal performance in both "passive" (structural reinforcement and gradient properties) and "active" modes (actuation and control through magnetic field) of lightweight porous polymeric structures reinforced with aligned magnetic particles. The average distribution, size and shape of particles, their magnetic characteristics, and mechanical properties of the foam will be considered as processing parameters. Homogenization techniques will be adopted in order to derive simple constitutive relationships for low computational requirements. The developed constitutive model will be phenomenological, while the governing equations will be based on the magnetic and mechanical balance laws. The development and validation of the model will be performed by comparing predictions with experiments.
- Performance and thermodynamic aspects of incrementally non-linear constitutive equations of the rate type (7AMB16AT035)
  from 01/01/2016 to 31/12/2017 main investigator
- Objectives: Although first models of hypoplasticity were proposed some decades ago, their properties have not yet been sufficiently understood, neither from the viewpoint of general applicability to large classes of technical problems, nor from the viewpoint of their mathematical well-posedness and numerical complexity. In special cases, however, there is evidence of their strong modeling potential for describing irreversible deformations, in particular in the case of granular or porous solids. The goal of this project is to exploit the complementary knowledge and experience of its participants, and classify the individual models within a larger thermomechanical context. The models will be investigated both analytically and numerically, and the results will be compared with known experimental data published in the literature, which will enable us to decide about their applicability for solving practical problems.
- Analysis of mathematical models of multifunctional materials with hysteresis (GA15-12227S)
  from 01/01/2015 to 31/12/2017 main investigator
- Objectives: The project topic is mathematical modeling, analysis, and numerical simulations of processes taking place in multifunctional materials with hysteresis. The results will include (1) a rigorous derivation of systems of ordinary and partial differential equations based on physical principles and experimentally verified constitutive relations, (2) proofs of existence, and possibly also uniqueness and stability of solutions to the equations, and (3) their numerical approximation including error bounds. The main applications will involve piezoelectric and magnetostrictive materials used as sensors, actuators, and energy harvestors, as well as thermoelastoplastic materials subject to material fatigue. The presence of hysteresis makes all these steps challenging, also because hysteresis nonlinearities are non-differentiable, which creates difficulties both in the analysis and in the numerics. New algorithms will have to be developed to treat the problems in maximal complexity.
- Analyse mathématique et numérique de problèmes de contact pour des matériaux à mémoire (CNRS - F-13-14-01)
  from 01/01/2013 to 31/12/2014 main investigator
- Účastníci projektu hodlají soustředit kapacity výzkumných pracovníků Institutu Camille Jordana v Lyonu a Matematického ústavu AV ČR v Praze při vývoji matematických a numerických metod kvalitativního a kvantitativního popisu nevratných procesů v materiálech s pamětí. Přesněji řečeno, jde o porovnání různých přístupů k popisu disipativních systémů, které nejsou v rovnovážném stavu, jako jsou například energetická metoda, metoda (kvazi-)variačních nerovnic a metoda hysterezních operátorů. Cílem projektu je nalézt a následně využít syntézy těchto přístupů pro formulaci a řešení nových úloh v teorii nevratných dějů. Kromě toho budou vyvíjeny numerické metody řešení těchto úloh, které mohou napomoci k pochopení příčiny vzniku nejednoznačnosti řešení.
- Mathematical modeling of processes in hysteresis materials (P201/10/2315)
  from 01/01/2010 to 31/12/2014 main investigator
- Objectives: Hysteresis, i.e., nonlinear relations exhibiting a complicated input-output behavior in form of nested loops that cannot be described by functions or graphs, occurs in many fields of science, e.g., in ferromagnetism, micromagnetics, solid-solid phase transitions, and elastoplasticity. Hysteretic systems carry a memory of their former states, which renders their input-output mapping both nondifferentiable and nonlocal in time, so that conventional weak convergence techniques for solving evolution systems fail. Therefore, dynamical elastoplastic processes with hysteresis are found in the mathematical literature much less frequently than quasistatic ones, and a substantial progress in this direction is necessary. In a recent breakthrough, it was shown that the three-dimensional single-yield von Mises constitutive law leads, after a dimensional reduction to beams or plates, to a multi-yield Prandtl-Ishlinskii hysteresis operator. It is in fact quite natural that the lower dimensional observer does not see any sharp transition from the purely elastic to the purely plastic regime as in the von Mises model: if a plate is bent then small plasticized zones start forming first near the boundary and then propagate to the interior, which still preserves a partial elasticity. This gradual plasticizing is reflected by the Prandtl-Ishlinskii superposition of single-yield elements that are successively activated. This new groundbreaking theory will be expanded to more complex structures like Mindlin-Reissner plates, and curved rods and shells. Temperature and material fatigue effects will be included. A thermodynamically consistent theory of temperature and fatigue dependent Prandtl-Ishlinskii operators will be developed, along with efficient and reliable numerical methods. Questions of theoretical and numerical stability, and the long time behavior of the system of energy and momentum balance laws are central objectives.