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In the Phase Field methodology there is a length parameter related to the size of the damage region, which is proportional to the Irwin length. In this work, we apply the Phase Field methodology to the analysis of fracture at the micro-scale in brittle materials, studying the role of this length parameter when the size of the specimen is of the order of the Irwin length. We compare the conclusions obtained with the ones previously presented using Finite Fracture Mechanics. Two different schemes are used, a staggered and a monolithic scheme.

International audience; A large number of advanced finite element shell formulations have been developed, but their adoption is hindered by complexities of transforming mathematical formulations into computer code. Furthermore, it is often not straightforward to adapt existing implementations to emerging frontier problems in thin structural mechanics including nonlinear material behaviour, complex microstructures, multi-physical couplings, or active materials. We show that by using a high-level mathematical modelling strategy and automatic code generation tools, a wide range of advanced plate and shell finite element models can be generated easily and efficiently, including: the linear and non-linear geometrically exact Naghdi shell models, the Marguerre-von Karman shallow shell model, and the Reissner-Mindlin plate model. To solve shear and membrane-locking issues, we use: a novel re-interpretation of the Mixed Interpolation of Tensorial Component (MITC) procedure as a mixed-hybridisable finite element method, and a high polynomial order Partial Selective Reduced Integration (PSRI) method. The effectiveness of these approaches and the ease of writing solvers is illustrated through a large set of verification tests and demo codes, collected in an open-source library, FEniCS-Shells, that extends the FEniCS Project finite element problem solving environment.

International audience; The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill-conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution; and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is 5–6× faster on a majority of the test cases considered.

Shell, plate and beam (thin) structures are widely used in civil, mechanical and aeronautical engineering because they are capable of carrying high loads with a minimal amount of structural mass. Because the out-of-plane dimension is usually much smaller than the two in-plane dimensions, it is possible to asymptotically reduce the full 3D equations of elasticity to a whole variety of equivalent 2D models posed on a manifold embedded in 3D space. This reduction results in massively reduced computational expense and remains a necessity for practical large-scale computation of structures of real engineering interest such as the fuselage of an aircraft. The numerical solution of such mathematical models is a challenging task, especially for very thin shells when shear and membrane locking effects require special attention. As originally noted by [Hale and Baiz, 2013], the high-level form language UFL provides an excellent framework for writing extensible, reusable and pedagogical numerical models of thin structures. To our knowledge fenics-shells represents the first unified open-source implementation of a wide range of thin structural models, including Reissner-Mindlin, Kirchhoff-Love, Von Karman and hierarchical (higher-order) plates, and Madare-Naghdi and Madare-Koiter shell models. Because of the broad scope of fenics-shells, in this talk we will focus on how to cure numerical locking by applying the Mixed Interpolation of Tensorial Components (MITC) approach of [Dvorkin and Bathe, 1986] and [Lee and Bathe, 2010] to a shell with an initially flat reference configuration. The MITC approach consists of an element-by-element interpolation of the degrees of freedom of the rotations onto the degrees of freedom of a reduced rotation space, the latter typically constructed using H(curl) conforming finite elements such as the rotated Raviart-Thomas-Nédélec elements. Then, the bilinear form is constructed on the underlying H(curl) space. Because of the interpolation operator, the original problem is expressed in terms of the degrees of freedom for the rotations only. Within DOLFIN we have implemented this projection operation using two UFL forms within a custom assembler compiled just-in-time using Instant. We show numerical convergence studies that match the apriori bounds available in the literature. E. N. Dvorkin and K.-J. Bathe, “A continuum mechanics based four-node shell element for general non-linear analysis,” Engineering Computations, vol. 1, no. 1, pp. 77–88, 1984. P. S. Lee and K. J. Bathe, “The quadratic MITC plate and MITC shell elements in plate bending,” Advances in Engineering Software, vol. 41, no. 5, pp. 712–728, 2010. J. S. Hale and P. M. Baiz, “Towards effective shell modelling with the FEniCS project” presented at the FEniCS Conference 2013, Jesus College, Cambridge, 19-Mar-2013.

The phase field method has proven to be an important tool in computational mechanics in that it is able to deal naturally with crack nucleation and branching [1]. In this contribution, we demonstrate a large scale phase field model of fracture and cutting of soft tissues undergoing non-linear deformations with a material law defined by a hyperelastic energy density functional. We will also provide some initial thoughts on the how the effect of a porous medium can be incorporated into the phase field model. We implement this work using the FEniCS project and PETSc software packages [2, 3].

Colloque avec actes et comité de lecture. Internationale.; International audience; Les colonnes basaltiques, les septarias, les fissures d'impact en forme de toile d'araignée, les réseaux de fissures dans les boues, les peintures, les céramiques sont autant d'exemples de motifs de rupture faisant apparaitre des fissures en forme d'étoile. Est-il possible de prédire leur forme? La mécanique de la rupture fragile classique basée sur l'utilisation des facteurs d'intensité des contraintes, n'est pas en mesure de le faire. Mais nous verrons à travers une expérience modèle, que l'approche variationnelle de la rupture semble être une alternative rigoureuse pour relever ce challenge. Etant une extension du postulat énergétique de Griffith, cette approche est basée sur le principe que l'évolution quasi-statique d'une fissure est obtenue en minimisant la somme de l'énergie élastique et de l'énergie de rupture. Nous l'appliquerons à des expériences de séchage directionnel d'une suspension colloïdale dans un tube capillaire. Nous constaterons qu'elle est capable de prédire quantitativement les fissures en étoile qui y apparaissent. Enfin, nous l'appliquerons de façon inverse afin d'obtenir certaines informations sur l'origine de motifs naturels, colonnes basaltiques et septarias.

Master Thesis ESM department; In this work passive piezoelectric devices for vibration damping are studied. It is developed the basic idea of synthesizing electrical wave guides to obtain an optimal electro-mechanical energy exchange and therefore to dissipate the mechanical vibrational energy in the electric form. Modular PiezoElectroMechanical (PEM) structures are constituted by continuous elastic beams (or bars) coupled, by means of an array of PZT transducers, to lumped dissipative electric networks. Both refined and homogenized models of those periodic systems are derived by an energetic approach based on the principle of virtual powers. Weak and strong formulation of the dynamical problem are presented having in mind future studies involving the determination of numerical solutions. In this framework the effectiveness of the proposed devices for the suppression of mechanical vibrations is investigated by a wave approach, considering both the extensional and flexural oscillations. The optimal values of the electric parameters for a fixed network topology are derived analytically by a pole placement technique. Their sensitivities on the dimensions of the basic cell of the periodic system and on the design frequency are studied. Moreover the dependence of damping performances on the frequency is analyzed. Comparing the performances of different network topological configurations, the advantages of controlling a mechanical structure with an electric analog are shown. As a consequence of those results, new interconnections of PZT transducers are proposed. An experimental setup for the validation of the analytical and numerical results is proposed and tested. A classical experience on resonant shunted PZT is reproduced. Future experimental work is programmed.

Colloque avec actes et comité de lecture. Internationale.; International audience; Les colonnes basaltiques, les polygones martiens, les réseaux de fractures dans les peintures, les septarias sont des patterns contenant des fissures en forme d'étoile, formés par la contraction frustrée du milieu due à un processus de séchage ou de refroidissement. Je présenterai des expériences de séchage de suspensions colloïdales permettant d'obtenir des fissures en forme d'étoile avec un nombre variable de branches. Je montrerai que ce nombre est régi par le principe de minimisation d'énergie de la mécanique linéaire élastique de la rupture. Pour les thèmes, je pense à S14 ou C6.

National audience; See http://hal.archives-ouvertes.fr/docs/00/59/29/57/ANNEX/r_10LF4Z47.pdf

Colloque avec actes et comité de lecture. Internationale.; International audience; Nous étudions une barre cylindrique élancée, soumise à un chargement en traction et constituée d'un bimatériau: son coeur est élastique incassable tandis que sa périphérie extérieure est faite d'un matériau adoucissant élastique endommageable. On modélise celle-ci à l'aide d'un modèle à gradient d'endommagement. En suivant une approche variationnelle, nous étudions d'un point de vue théorique et numérique l'influence du coeur incassable sur l'évolution et la stabilité des solutions du système.

International audience; Beam models of piezoelectric laminates derived from three-dimensional theories by assuming either the plain-stress or the plain-strain conditions in the beam axis-thickness plane can introduce significant errors in the estimate of the beam constitutive coefficients. In this paper a coupled Euler-Bernoulli like model including 3D effects through a mixed variational formulation is discussed. We report the results of numerical and experimental tests which validate the approach and show the limit of standard modelling techniques.

In this work passive electric damping of structural vibrations by distributed piezoelectric transducers and electric networks is analyzed. Different distributed electric controllers are examined as finite degrees of freedom systems and their performances are compared. Modal reduction is used to optimize the electric parameters.

International audience; 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. Focussing 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-degreesof-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.