Professor Anindya Chatterjee
Indian Institute of Technology (IIT) at Kanpur, India
Plenary lecture: A two-state hysteresis model obtained from a high-dimensional frictional system.
Anindya Chatterjee got his bachelor's degree in mechanical engineering from the Indian Institute of Technology (IIT) at Kharagpur, and his PhD from Cornell University. He has taught at the Indian Institute of Science at Bangalore and also at IIT Kharagpur and IIT Kanpur. Chatterjee's research interests are in the areas of dynamics, applied mechanics and applied mathematics. His prior research has included rigid body impact models, walking machines, rotors, vehicle dynamics, fatigue damage modeling, delay differential equations, parametrically forced systems including ion dynamics in Paul traps, fractional order derivatives, various linear and nonlinear vibration problems, material damping, and hysteresis models.
Rate-independent hysteresis in energy dissipation within many materials and structures has been empirically observed by many researchers over more than a century. In such materials and structures the use of viscous damping in vibrations is therefore incorrect in principle, although it is analytically convenient.
In hysteresis, the load-displacement or stress-strain curve is irreversible and changes slope whenever the direction of loading is changed, leading to loops with sharp corners. Small reversals of loading within larger loading paths lead to minor loops in the load-displacement curve. Under complex loading, the load-displacement curve can have many loops within loops.
One can attempt to describe the load-displacement curve in terms of evolution equations. In structural mechanics, the famous Bouc-Wen model has been used for decades to describe hysteresis. This model is a differential equation with one internal state, captures single loops well, but cannot capture minor loops. One can show that, in order to capture minor loops within such larger loops, the hysteresis model should have at least two internal states.
In this talk, I will describe our mechanistic approach toward purposefully constructing a simple two-state model that captures such hysteresis loops. Our approach differs from other conceivable empirical approaches where ad hoc relations are guessed and shown to produce hysteresis loops. Here, we start with a physically based micromechanical model with many springs and sliding elements, i.e., a high dimensional frictional system resembling smaller systems studied by Iwan half a century ago.
We first solve the resulting equations incrementally via the linear complementarity problem. Then, using new approximation methods suitable for such a system, we are able to reduce the order of the system to the theoretical minimum requirement of two internal states. This reduction involves identification of shape functions, use of a work inequality, a simple fortuitous approximation of a complicated dissipation function, solution of a slightly offbeat minimization problem, reduction of the resulting equations to easily tractable form, elimination of superfluous fitted parameters through coordinate changes, and finally, separation of the roles of the remaining fitted parameters through a combination of further analytical simplification and approximation.
In summary, I will start with initial motivation, present the high dimensional model, and trace the route of reduction, approximation and simplification leading to a final novel and simple two-state hysteresis model.