Corrado Maurini

• Ph.D.
• Professor (Full) at Sorbonne University

We argue that nucleation of brittle cracks in initially flawless soft elastic solids is preceded by a nonlinear elastic instability, which cannot be captured without accounting for geometrical precise description of finite elastic deformation. As a prototypical problem we consider a homogeneous elastic body subjected to tension and assume that it i...

This webpage provides basic examples on computational methods to solve fracture mechanics problems using DOLFINx. This work includes the contribution of the Early Stage Researchers (ESR) of the ITN project Newfrac funded by the European Commission under a Marie Skłodowska-Curie Actions Grant Agreement n. 861061.

Phase-field models of brittle fracture are typically endowed with a decomposition of the elastic strain energy density in order to realistically describe fracture under multi-axial stress states. In this contribution, we identify the essential requirements for this decomposition to correctly describe both nucleation and propagation of cracks. Discu...

We argue that nucleation of brittle cracks in initially flawless soft elastic solids is preceded by a continuum instability which cannot be captured without accounting for geometrically and physical nonlinearities of the constitutive response. To corroborate this somewhat counterintuitive claim, we present a theoretical and numerical study of the s...

Damage Mechanics and Local Approach to Fracture

Phase-field models of brittle fracture are typically endowed with a decomposition of the elastic strain energy density in order to realistically describe fracture under multi-axial stress states. In this contribution, we identify the essential requirements for this decomposition to correctly describe both nucleation and propagation of cracks. Discu...

We study multiple transverse cracking of symmetric laminates in the framework of the variational approach to fracture. Considering the Griffith model, we assume that several cracks can appear instantaneously through the whole thickness of the core layer, separating the bar in n elastic segments. We show that the energy minimization implies the bifu...

We study the nonlinear forced dynamics of a bistable buckled beam. Depending on the forcing frequency and amplitude, we observe three different regimes: (i) small intra-well oscillations in the neighborhood of one of the equilibria, (ii) transient snap-through ending into intra-well oscillations, (iii) persistent dynamic snap-through. We build expe...

Thin elastic shells are almost inextensible but easy to bend. In the presence of prestresses, geometric frustrations can produce complex elastic energetic landscapes, which have been tailored for the design of morphing structures with multiple stable equilibria or neutrally stable manifolds. We show that the co-existence of stiff and floppy modes l...

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 ord...

Phase-field models of brittle fracture can be regarded as gradient damage models including an intrinsic internal length. This length determines the stability threshold of solutions with homogeneous damage and thus the strength of the material, and is often tuned to retrieve the experimental strength in uniaxial tensile tests. In this paper, we focu...

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 appro...

In a seminal paper published in 1951, Taylor studied the interactions between a viscous fluid and an immersed flat sheet which is subjected to a travelling wave of transversal displacement. The net reaction of the fluid over the sheet turned out to be a force in the direction of the wave phase-speed. This effect is a key mechanism for the swimming...

In strongly anisotropic materials the orientation-dependent fracture surface energy is a non-convex function of the crack angle. In this context, the classical Griffith model becomes ill-posed and requires a regularization. We revisit the crack kinking problem in materials with strongly anisotropic surface energies by using a variational phase-fiel...

Local damage models with softening needs localization limiters to preserve the mathematical and physical consistency. In this paper we compare the properties of strain-gradient and damage-gradient regularizations. Gradient-damage models introduce a quadratic dependency of the dissipated energy on the gradient of the damage field and are nowadays ex...

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 m...

Phase-field models, sometimes referred to as gradient damage or smeared crack models, are widely used methods for the numerical simulation of crack propagation in brittle materials. Theoretical results and numerical evidences show that they can predict the propagation of a pre-existing crack according to Griffith' criterion. For a one-dimensional p...

We have designed and tested experimentally a morphing structure consisting of a neutrally stable thin cylindrical shell driven by a multiparameter piezoelectric actuation. The shell is obtained by plastically deforming an initially flat copper disk, so as to induce large isotropic and almost uniform inelastic curvatures. Following the plastic defor...

We have designed and tested experimentally a morphing structure consisting of a neutrally stable thin cylindrical shell driven by a multiparameter piezoelectric actuation. The shell is obtained by plastically deforming an initially flat copper disk, so as to induce large isotropic and almost uniform inelastic curvatures. Following the plastic defor...

Plasticity and damage are two fundamental phenomena in nonlinear solid mechanics associated to the development of inelastic deformations and the reduction of the material stiffness. Alessi et al. [5] have recently shown, through a variational framework, that coupling a gradient-damage model with plasticity can lead to macroscopic behaviours assimil...

We investigate the elastic buckling of a triangular prism made of a soft elastomer. A face of the prism is bonded to a stiff slab that imposes an average axial compression. We observe two possible buckling modes which are localized along the free ridge. For ridge angles $\phi$ below a critical value $\phi^\star\approx 90^\circ$ experiments reveal a...

We investigate the elastic buckling of a triangular prism made of a soft elastomer. A face of the prism is bonded to a stiff slab that imposes an average axial compression. We observe two possible buckling modes which are localized along the free ridge. For ridge angles $\phi$ below a critical value $\phi^\star\approx 90^\circ$ experiments reveal a...

The paper is devoted to gradient damage models which allow us to describe all the process of degradation of a body including the nucleation of cracks and their propagation. The construction of such model follows the variational approach to fracture and proceeds into two stages: (1) definition of the energy; (2) formulation of the damage evolution p...

A shell can have multiple stable equilibria either if its initial curvature is sufficiently high or if a suitably strong pre-stress is applied. Under the hypotheses of a thin and shallow shell, we derive closed form results for the critical values of curvatures and pre-stresses leading to bistability and tristability. These analytical expressions a...

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,...

This paper presents a gradient approach for the quasi-static macroscopic modeling of superelasticity in softening shape memory alloys bars. The model is assumed to be rate-independent and to depend on a single internal variable. Regularization of the model is achieved through the free energy by assuming a quadratic dependance with respect to the gr...

These notes give a short introduction to the methods for the study of stability of elastic structures. We consider only the finite-dimensional case, where the state of the system is represented by a discrete set of variables. The core of the exposition focuses on the illustration of energetic methods where equilibrium and stability are found by stu...

We study fracture and debonding of a thin stiff film bonded to a rigid substrate through a thin compliant layer, introducing a two-dimensional variational fracture model in brittle elasticity. Fractures are naturally distinguished between transverse cracks in the film (curves in 2D) and debonded surfaces (2D planar regions). In order to study the m...

We study the genesis and the selective propagation of complex crack networks induced by thermal shock or drying of brittle materials. We use a quasistatic gradient damage model to perform large-scale numerical simulations showing that the propagation of fully developed cracks follows Griffith criterion and depends only on the fracture toughness, wh...

Basalt columns, septarias, and mud cracks possess beautiful and intriguing crack patterns that are hard to predict because of the presence of cracks intersections and branches. The variational approach to brittle fracture provides a mathematically sound model based on minimization of the sum of bulk and fracture energies. It does not require any a...

We study fracture and delamination of a thin stiff film bonded on a rigid substrate through a thin compliant bonding layer. Starting from the three-dimensional system, upon a scaling hypothesis, we provide an asymptotic analysis of the three-dimensional variational fracture problem as the thickness goes to zero, using Gamma-convergence. We deduce a...

Under the effect of surface tension, a blob of liquid adopts a spherical shape when immersed in another fluid. We demonstrate experimentally that soft, centimeter-size elastic solids can exhibit a similar behavior: when immersed into a liquid, a gel having a low elastic modulus undergoes large, reversible deformations. We analyze three fundamental...

The extrusion of the Anatolian plate toward the Aegean domain is accommodated by the North Anatolian fault. For almost 10 Myr, the North Anatolian fault has been propagating over about 1000 km toward the West, therefore the different segments of the fault are not the same age. However, the displacement along the different segments that are older th...

Differential growth of thin elastic bodies furnishes a surprisingly simple explanation of the complex and intriguing shapes of many biological systems, such as plant leaves and organs. Similarly, inelastic strains induced by thermal effects or active materials in layered plates are extensively used to control the curvature of thin engineering struc...

Collagen fibres play an important role in the mechanical behaviour of many soft tissues. Modelling of such tissues now often incorporates a collagen fibre distribution. However, the availability of accurate structural data has so far lagged behind the progress of anisotropic constitutive modelling. Here, an automated process is developed to identif...

Collagen fibres play an important role in the mechanical behaviour of many soft tissues. Modelling of such tissues now often incorporates a collagen fibre distribution. However, the availability of accurate structural data has so far lagged behind the progress of anisotropic constitutive modelling. Here, an automated process is developed to identif...

The small-amplitude in-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible, shearable, planar Kirchhoff elastic rod under large displacements and rotations, and the vibration frequencies are computed both analytically and numerically as a function of the loading. Of particular interest is...

We study multifissuration and debonding phenomena of a thin film bonded to a stiff substrate us-ing the variational approach to fracture mechanics. We consider a reduced one-dimensional membrane model where the loading is introduced through uniform inelastic (e.g. thermal) strains in the film or imposed dis-placements of the substrate. Fracture phe...

We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction t...

Résumé — Une coque mince orthotrope peut exhiber plusieurs configurations d'équilibre stable en fonction des paramètres tels que sa géométrie, ses propriétés matériau et quand elle est soumise à des champs de déformations anélastiques, telles qu'induites par effet thermique ou piézoélectrique. Dans ce travail, on détermine les régions de bi-et tri-...

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 variationnel...

The present work attempts to present a consistent and efficient approach to piezoelectric laminated beams. The influence of hypotheses on three-dimensional sectional deformations and stress distributions on the estimate of the beam electromechanical properties is analyzed. By exploiting a mixed variational formulation and Lagrange multipliers metho...

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 Comment: ICTAM04

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.

The aim of this work is two-fold: to design devices for passive electric damping of structural vibrations by distributed piezoelectric transducers and electric networks, and to experimentally validate the effectiveness of such a damping concept. Two different electric networks are employed, namely a purely resistive network and an inductive-resisti...

In its numerical implementation, the variational approach to brittle fracture approximates the crack evolution in an elastic solid through the use of gradient damage models. In the present paper, we first formulate the quasi-static evolution problem for a general class of such dam-age models. Then, we introduce a stability criterion in terms of the...

This paper studies the stable equilibrium shapes of free multilayered orthotropic plates loaded by inelastic deformations induced by thermal and piezoelectric effects. Starting with a Von Kármán plate kinematics and an energetic formulation, a discrete intrinsic nonlinear model in terms of curvatures only is deduced. The model has 3 degrees of free...

We present a mechanistically faithful and mathematically sound approach to the numerical simulation of secondary thermal cracks propagation in EGS based on the variational approach to fracture [1-4]. While remaining compatible with classical theories, this approach provides a unified framework to crack nucleation, full path identification, includin...

This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319–1342]. We focus on th...

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 flu...

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...

Composite shells show a rich multistable behaviour of interest for the design of shape-changing (morphing) structures. Previous studies have investigated how the initial shape determines the shell stability properties. For uniform initial curvatures and orthotropic material behaviour, not more than two stable equilibria have been reported. In this...

We propose two identification techniques for estimating the piezoelectric couplings and the piezoelectric capacitances of reduced order modal models of linear piezoelectric structures. The two methods are easily implementable and demand few input data, which can be obtained both with experimental testing and numerical models (e.g. finite elements)....

Bistable structures, such as buckled beams or plates, are characterized by a two-well po-tential. Their nonlinear properties are currently exploited in actuators design (e.g. MEMS micropumps, switches, memory cells) to produce relatively high displacements and forces with low actuation energies. We investigate the use of distributed multiparameter...

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-dimens...

Reduced-order modal models of linear piezoelectric structures are useful in vibration control and health monitoring. We study experimental identification of the fundamental parameters of these modal models. We propose two identification techniques for estimating piezoelectric modal couplings and piezoelectric modal capacitances. Both methods are ea...

This paper analyzes different numerical methods for modal analysis of stepped piezoelectric beams modeled by the Euler–Bernoulli beam theory. Results from standard numerical approaches, that rely on the discretization of the stepped beam (assumed modes and finite-element methods), are compared with the solution of the exact transcendental eigenvalu...

In this paper a coupled Euler–Bernoulli model of laminated piezoelectric beams is proposed. It is characterized by accounting for the influence of 3D distribution of mechanical stresses and strains through corrected electromechanical constitutive equations. In particular, the hypothesis of vanishing transverse (width direction) normal stress typica...

COTUTELLE FRANCO-ITALIENNE Décembre 2005

Different numerical and experimental methods for modal analysis of beams hosting a set of piezoelectric transducers are presented. The critical analysis of the achieved estimations of natural frequencies and mode shapes allows for distinguishing among the errors arising from coarse modeling, inaccurate numerical solutions and improper experimental...

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 m...

Electric vibration absorbers made of distributed piezoelectric devices for the control of beam vibrations are studied. The absorbers are obtained by interconnecting an array of piezoelectric transducers uniformly distributed on a beam with different modular electric networks. Five different topologies are considered and their damping performance is...

In this paper a Euler–Bernoulli-like model of layered piezoelectric beams is presented. It describes more accurately than the others already presented in the literature both transverse (Poisson and piezoelectrically induced) cross-sectional deformations and through-the-thickness variations of the electric field and electric displacement. A deductiv...

In this paper models of layered piezoelectric beams are discussed. The attention is focused on the analysis of the assumptions on transversal stress and strain distribution and their influence on the deduction of the beam constitutive equations from a three dimensional description. A model accounting for non trivial transversal interactions between...

The aim of this work is two-fold: to design devices for passive electric damping of structural vibrations by distributed piezoelectric transducers and electric networks, and to experimentally validate the effectiveness of such a damping concept. Two different electric networks are employed, namely a purely resistive network and an inductive–resisti...

Several electric vibration absorbers based on distributed piezoelectric control of beam vibrations are studied. The damping devices are conceived by interconnecting with different modular electric networks an array of piezoelectric transducers uniformly distributed on a beam. Five different vibration absorbers made of five different network interco...

Several electric vibration absorbers based on distributed piezoelectric control of beam vibrations are studied. The damping devices are conceived by interconnecting with different modular electric networks an array of piezoelectric transducers uniformly distributed on a beam. Five different vibration absorbers made of five different network interco...

Master Thesis ESM department

Résumé — On se propose, à partir d'un modèle de plaques multicouches orthotropes sous l'hypo-thèse de courbure uniforme, d'étudier la conception d'un actionnement multiparamétrique efficace pour ces structures non linéaires. Ce modèle sera validé par des simulations numériques éléments finis. Mots clés — Contrôle de forme, plaque composite, stabili...

Résumé — Nous étudions un cylindre mince que l'on soumet à une torsion croissante. Celui-ci est composé d'un bimatériaux adoucissant-incassable. Le matériau adoucissant est modélisé à l'aide d'un modèle d'endommagement à gradient. En suivant une approche variationelle, on s'in-téresse alors à l'évolution et à la stabilité de la structure au cours d...

We present recent results on the modeling and nu-merical simulation of reservoir stimulation in Hot Dry Rocks. Our approach is based on a mechanisti-cally faithful yet mathematically sound model, gener-alizing Griffith's idea of competition between bulk and surface energies. At each time increment, the fracture and displacement configuration of a r...

Cracks sometimes form beautiful, complex, intriguing, but hard to predict patterns. Stable propagation of preexisting simple cracks is nowadays quite well described by the traditional linear elastic fracture mechanics theory. However, crack nucleation, unstable propagation, or complicated fracture pattern identification are still poorly understood....