Sep 23, 2014
Vojtěch Kulvait, Charles University in Prague, Faculty of Mathematics and Physics, Prague, Czech Republic There is increasing evidence that there exists materials that behave nonlinearly (not according to Hooke’s law) even for small strains. Brittle elastic materials or gum metal alloys are examples of the materials exhibiting such behavior. The poster presented here is focused on the numerical results obtained in FeniCS package when solving models of materials with nonlinear response. The model possesses strain limiting behavior that means that strains are bounded even for high stresses. Antiplane stress problem with V-notch shaped domain in the framework of linearized strain tensor is described. Airy stress function formalism allows us to solve the problem as the second order elliptic PDE. The results show that we are obtaining stress concentration in the vicinity of the tip of the geometry while the strains remain bounded.
MOdelling REvisited + MOdel REduction Modeling, analysis and computing in nonlinear PDEs. September 21-26, 2014, Chateau Liblice, Czech Republic. Recently developed implicit constitutive theory allows one to describe nonlinear response of complex materials in complicated processes and to model phenomena in both fluid and solid mechanics that have hitherto remained unexplained. The theory also provides a thermodynamically consistent framework for technologically important but so far only ad hoc engineering models without sound footing. The overall goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large-scale simulations for the models arising from the new theoretical framework. A natural part of the goal is rigorous mathematical analysis of the models. The nonstandard structure of constitutive relations arising from the new framework requires the reconsideration of many existing approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, basic notions such as the concept of the solution and its well-posedness need to be reconsidered. Further, the complexity of the constitutive relations calls for rigorous investigation on model reduction - the identification of simplified models that capture the chosen (practically relevant) information about the behaviour of the system and disregard irrelevant information. Reliable numerical simulations require the derivation of sharp a posteriori error estimates to control all possible sources of errors, including rarely studied but important algebraic errors. We believe that in solving difficult problems in mathematical modelling the individual aspects discussed above - physics, mathematical analysis and numerical analysis - are so closely interrelated that no breakthrough can be achieved without emphasising the holistic approach as the main principle. Our vision is to follow this principle: the entire process of modelling of complex materials will be revisited in an innovative manner.
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