Wednesday, February 22th, 2023. Time:  12 noon
Hybrid! O.C. Zienkiewicz Conference Room, C1 Building, UPC Campus Nord, Barcelona 

See video on Youtube

ABSTRACT

The mathematical problem corresponding to classical continuum damage mechanics can be defined as a non-cooperative equilibrium point. This concept is also named after its inventor, the mathematician John Forbes Nash, as Nash's point.

Nash's formulations are incorporated into a wide range of disciplines, especially in economics and game theory. Be 𝑓𝑖(𝑥𝑖,𝑥𝑗) the profit function of a player 𝑖; be 𝑥𝑖 the variables controlled by this player and 𝑥𝑗 (with 𝑗≠𝑖) the remaining factors that affect his/her earnings. The optimization process of the profit function can be written as: 𝑓𝑖(𝑥𝑖,𝑥𝑗)≥𝑓𝑖(𝑥𝑖∗,𝑥𝑗) for any admissible 𝑥𝑖∗. Consider now a game with 𝑛 players. Each of them will try to optimize a different gain function𝑓𝑖(𝑥𝑖,𝑥𝑗) considering its own strategy 𝑥𝑖 and the strategies 𝑥𝑗 of the remaining players. A Nash equilibrium is reached when the 𝑛 inequalities are verified simultaneously, thereby optimizing all the profit functions and leaving the players with no incentive to change their strategies.

In the case of elasticity coupled to continuous damage, the Nash point depends on only two "players". The independent variables of the problem are the displacement and continuous damage fields. The first of the functional inequalities corresponds to the principle of the minimum potential energy for a non-homogeneous structure whose elastic properties depend on local damage values. The second inequality is a variational formulation of the damage evolution law. Therefore, the first inequality depends on the free energy accumulated in the structure and the second on the energy dissipated by damage. The equilibrium state is the result of a "game” in which one of the "players” tries to transform mechanical work into reversible energy, and the other tries to dissipate it.

As it is well known, continuum damage mechanics gives objective numerical results only before localization as defined by the Rice and Rudniki criterion. This criterion describes the conditions for the appearance of discontinuities in the displacement and damage fields. Lumped damage mechanics (LDM) is based on the following description of the process of energy dissipation. In a first phase, the dissipation is continuous and can be described using some continuum damage model. This phase ends when the location criterion is verified. In the second phase, the dissipation can be considered as concentrated in areas of zero volume (lumped damage) and is a consequence of the evolution of the discontinuities. Nash's formulation offers an interpretation of the non-objectivity of classical damage models and a procedure for regularize them: the variational inequality of energy dissipation lacks a term of lumped dissipation; as the classical models do not specify how much and how this dissipation evolves, the problem may exhibit an infinite number of solutions; in LDM, an “extra player” is added whose variational inequality describes the lumped damage evolution law, therefore completing the description of the phenomenon. In this seminar, the numerical implementation of LDM models in conventional finite element programs is also described, objectivity is shown in all the phases of the process and the formulation results are compared with experimental analyses. Additionally, it is shown that classical damage and cohesive fracture mechanics can be considered as particular cases of the LDM formulation.

LDM, as the theory of plasticity, can also be used for the analysis of plates, shells, beams and frames. In those cases, plastic hinges or hinge lines are generalized including lumped damage variables. In the last part of the seminar, a simplified model of these characteristics is presented and its consequences in the field of reliability and stochastic analysis of complex structures are analyzed. It is demonstrated that the use of LDM allows to significantly increase the limits of validity of any numerical statistical method, for example, the Monte Carlo one.

SPEAKER CV

Julio Flórez López is a Full professor at the Civil Engineering School of the Chongqing University (China) and a Emeritus Professor of the Structural Engineering Department at the School of Engineering of the University of Los Andes (Venezuela). He is a specialist in mechanics applied to civil engineering structures. He worked for the University of Los Andes in Venezuela for more than thirty years. 

His main accomplishment is the development of a new branch of Structural Mechanics called Lumped Damage Mechanics. This theory is a generalization and extension of Fracture and Damage Mechanics that includes those theories as particular cases. Lumped Damage Mechanics can be applied to solids, plates, RC and steel frame structures. Lumped Damage Mechanics is based on an innovative variational formulation that defines the problem as a Nash non-cooperative equilibrium point.

He also has contributions on Localization Theory and Boundary Element Methods, he have worked on the modelling and experimental analysis of Masonry and Origami Structures amongst others.
He was the leader of an interdisciplinary group that developed a web-based nonlinear finite element programs for the analysis of frames subjected to earthquake loadings, probably the first of its class. The term “Cloud Computing” was not yet used at that time.

Seminar Programme 2023