Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

Computer and Robot Vision Vol. 1, by R.M. Haralick and Linda G. Shapiro, Addison-Wesley, 1992, ISBN 0-201-10887-1.

Computer and Robot Vision

A phase-field description of dynamic brittle fracture

Journal of Physics Condensed Matter: Preface

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Linear And Nonlinear Programming

Gradient damage models used in phase-field approaches to brittle fracture are characterised by material softening and instabilities. We present novel numerical techniques for the bifurcation and stability analysis along quasi-static evolution paths as well as practical tools to select stable evolutions. Our approach stems from the variational approach to fracture and the theory of rate-independent irreversible processes whereby a quasi-static evolution is formulated in terms of incremental energy minimisation under unilateral constraints. Focusing on the discrete setting obtained with finite elements techniques, we discuss the links between bifurcation criteria for an evolution and stability of equilibrium states. Key concepts are presented through the analytical solution of a two-degrees-of-freedom model featuring a continuum family of bifurcation branches. We introduce numerical methods to (i) assess (second-order) stability conditions for time-discrete evolutions subject to damage irreversibility, and (ii) to select possible stable evolutions based on an energetic criterion. Our approach is based on the solution of a coupled eigenvalue problem which accounts for the time-discrete irreversibility constraint on damage. Several numerical examples illustrate that this approach allows us to filter out unstable solutions provided by standard (first-order) minimisation algorithms as well as to effectively compute stable evolution paths. We demonstrate our purpose on a multifissuration problem featuring complex fracture patterns, to show how the eigenvalue analysis enables to compute and retrieve morphological properties of emerging cracks.

https://hal.archives-ouvertes.fr/hal-03106368/document

Numerical bifurcation and stability analysis of variational gradient-damage models for phase-field fracture

Bistable structures, such as buckled beams, are characterized by a two-well potential Their nonlinear properties are currently exploited in actuators to produce relatively high displacements and forces with low actuation energies We investigate the use of distributed multiparameter actuation to control the buckling and postbuckling behaviour of a three-layer piezoelectric beam pinned at either end A two-parameter bending actuation controls the transversal motion, whilst an axial actuation modulates the tangent bending stiffness The postbuckling behaviour is studied by reducing to a 2 dof system a nonlinear extensible elastica model When the bending actuation is spatially symmetric, the postbuckling phenomena are characterized by a snapthrough instability The use of a two-parameter actuation opens new transition scenarios, where it is possible to get quasi-static transitions between the two equilibria of the buckled beam, without any instability phenomenon

Bistable Buckled Beam: Modelling and Piezoelectric Actuation

In this paper, we examine and compare four different techniques for modal analysis of stepped piezoelectric
beams. The first technique is based on the solution of the exact transcendental eigenvalue problem, formulated
in terms of the dynamic stiffness matrix. The other three techniques are based on the Galerkin method for
obtaining a finite-dimensional version of the system. Besides the classical assumed modes method and finite-element
method, we propose a novel enhanced version of the assumed modes method, which introduces special
jump functions to enrich the standard basis functions.

Modal analysis of stepped piezoelectric beams

Modal analysis of stepped piezoelectric beams with applications to electric vibration damping

Formation of basalt columns during cooling of lava may be modeled by the drying of colloidal silica suspension confined in capillary cells (Allain and Limat 1995, Gauthier et al. 2007). During the drying process, particles aggregate at the open edge forming a growing drained gelled porous medium. High negative capillary pressure in the draining fluid (Dufresne et al., 2003) and adhesion to the walls of the cells generates high tensile stresses in the gel leading to crack formation. Depending on the experimental conditions and the shape of the cell (rectangular or circular), several crack morphologies appear. Here the aim is to compare the experimental morphologies with the ones predicted by fracture mechanics. For this purpose, the drained gelled porous medium is modeled by a linear elastic medium subjected to tensile prestresses and the cracks by the variational approach to fracture of Bourdin, Francfort and Marigo (1998, 2000, 2008).

Corrado Maurini

Papers

Numerical bifurcation and stability analysis of variational gradient-damage models for phase-field fracture

Bistable Buckled Beam: Modelling and Piezoelectric Actuation

Modal analysis of stepped piezoelectric beams

Modal analysis of stepped piezoelectric beams with applications to electric vibration damping

Basalt Columns and Crack Formation during Directional Drying of Colloidal Suspensions in Capillary Tubes