By Nasser Alkmim
Introducion
In this post, I will talk about an experience that I had in Prague with a course on an advanced mechanics topic in 2021 with Prof. Milan Jirásek. By the way, I am Nasser Alkmim and I am doing my Ph.D. at the University of Innsbruck in the unit of Strength of Materials and Structural Analysis.
A little about my background
Before talking about Prof. Jirásek’s course, I will briefly provide some context with my background. I did my bachelor’s in Civil Engineering, which has many different focuses. For example, structural and geotechnical engineering, water supply and treatment, transportation systems and construction management. It is a very broad and general course. I did a little of all, however, my main interest end up narrowing to structural engineering. I think I was fascinated by the mathematical treatment of it. So I end up focusing my studies on modeling structures using mathematics and computers. My master’s topic was modeling mechanical nonlinear behavior considering the microstructures of the material with the finite element method. My current work on the Ph.D. is also related to the modeling of complex material behavior but with more focus on the numerical and computational aspects of it, more specifically the linear solver optimization with appropriate preconditioners.
Course on modeling localized inelastic deformation
This course takes place yearly in Prague. After a one-year break from the pandemic, the course happened in presence in September 2021. It was my first time in Prague and the city, of course, impressed me.
Sunset picture that I took near the Vyšehradské sady. We, me and my course colleagues, stroll around this area after the course. The walk along this park is nice, but it is a bit far from the center where all the city attractions are.
The idea of the course is to provide a sum- mary of an advanced topic which is usually not taught in a regular curriculum because of its specificity. The main topic discussed is regu- larization techniques for modeling the failure of quasi-brittle materials. Usually, the pre-requisite for comprehending this topic is knowledge of con- tinuum mechanics, plasticity theory, damage me- chanics and finite element method.
Quasi-brittle materials are a class of materi- als that have a specific mode of failure. Brittle materials fail suddenly, while ductile mate- rials experience large deformations before fail- ure. Quasi-brittle experiences two types of failure, brittle and ductile, depending on the load. Under high compressive loads, for instance, the failure is characterized by the formation of mi- crocracks which ultimately result in the "softening" (Bažant, Zdeněk P., 2004). Softening in this context is the reduction of load-carrying capac- ity after the maximum load was reached. Understanding the softening behavior can be valuable in assessing the lifetime of structures and informing the development of new materials. One example of such material is concrete, which is abundantly used in small-scale construction like houses and large-scale structures like a dam1, used for electricity generation.
The failure of such material is hard to model. With the standard continuum formulation, when the material reaches its peak load, the subsequent material response is not unique. This means multiple answers can be considered acceptable, the model is then "nonobjective" or "ill-posed" (Jirásek, Milan,). In practical terms, this means that if you refine your finite element mesh, your result will not converge and will result in a different response. This is not good and indeed a problem that many researchers face from the eighties until today. At the core of the problem is the relation between the localization of strains and the finite element mesh2.
The solution is to regularize the continuum model. Many techniques were studied in the last three decades, and the course gives an overview of their advantages and disadvantages of them. One particular technique, which is the one I use in my work, is the gradient nonlocal formulation.
A little more about Prague
After the course, we (me and the others from the course) went out to explore the city. Our strategy was simple: a moderate walk along a chosen neighborhood, then stop at a restaurant for a meal. On the list of places we visited was The Prague Castle, which is impressive because of its size and it was walking distance from the university.
Another place we visited was the Prague astronomical clock, which is an interesting seeing. The location is where most of the tourism is concentrated, which I usually tend to avoid. Other than visiting historical buildings and sites, we went to restaurants where I tried some typical Czech cuisine and also some trendy restaurant places (for burgers).
Overall I had a nice experience with the course and the city. I recommend the course to anyone that is starting a Ph.D. and whose work is related to the mechanical simulation of quasi- brittle materials.
Prague astronomical clock
(600 years old) in the Old Town
square, where most tourists go
for a walk.
1World largest dam in Chine Three Gorges dam and third largest in Brazil Itaipu.
2Maybe I can make a blog post about this later with a more in-depth explanation.