Research

My current research interests include stability of structures, fracture and damage mechanics, computational mechanics, plates and shells, large deformations and instabilities in soft solids.

Fracture and damage mechanics

I study fracture and damage mechanics in the framework of the variational approach to fracture. My interest focus on the links between gradient damage models and brittle fracture, and the use of damage models as phase-field models of fracture. In a series of works, we have shown how crack nucleation can be studied as a problem of structural instability and how the morphogenesis of complex crack patterns can be predicted by a bifurcation analysis. I work also on numerical developments to improve the efficiency of the available HPC techniques for phase-field models of fracture. I distribute several open-source codes to reproduce the results of my papers and, hopefully, serve as examples for others.

Some publications on the topic

1. León Baldelli, A. A., & Maurini, C. (2021). Numerical bifurcation and stability analysis of variational gradient-damage models for phase-field fracture. Journal of the Mechanics and Physics of Solids, 152, 104424. https://doi.org/https://doi.org/10.1016/j.jmps.2021.104424 [Publisher] [Pdf]

   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.

2. Li, B., & Maurini, C. (2019). Crack kinking in a variational phase-field model of brittle fracture with strongly anisotropic surface energy. Journal of the Mechanics and Physics of Solids, 125, 502–522. https://doi.org/https://doi.org/10.1016/j.jmps.2019.01.010 [Publisher] [Pdf]

   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-field model. The model includes in the energy functional a quadratic term on the second gradient of the phase-field. This term has a regularizing effect, energetically penalizing the crack curvature. We provide analytical formulas for the dependence of the surface energy on the crack direction and develop an open-source finite-element solver for the higher-order phase-field problem. Quantitative numerical experiments for the crack kinking problem show that the crack kinking directions observed in our phase-field simulations are in close agreement with the generalized maximum energy release rate criterion. Finally, we revisit a thermal quenching experiment in the case of slabs with strongly anisotropic surface energies. We show that the anisotropy can strongly affect the observed crack patterns, either by stabilizing straight cracks or by inducing zig-zag crack patterns. In the case of zig-zag cracks, we observe that crack kinking is always associated with an unstable propagation of a finite length add-crack in a single time-step.

3. Le, D. T., Marigo, J.-J., Maurini, C., & Vidoli, S. (2018). Strain-gradient vs damage-gradient regularizations of softening damage models. Computer Methods in Applied Mechanics and Engineering, 340, 424–450. https://doi.org/10.1016/j.cma.2018.06.013 [Publisher] [Pdf]
4. Tanné, E., Li, T., Bourdin, B., Marigo, J.-J., & Maurini, C. (2018). Crack nucleation in variational phase-field models of brittle fracture. Journal of the Mechanics and Physics of Solids, 110(Supplement C), 80–99. https://doi.org/10.1016/j.jmps.2017.09.006 [Publisher] [Pdf]
5. Alessi, R., Marigo, J.-J., Maurini, C., & Vidoli, S. (2018). Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: one-dimensional examples. International Journal of Mechanical Sciences. https://doi.org/10.1016/j.ijmecsci.2017.05.047 [Publisher] [Pdf]
6. Farrell, P., & Maurini, C. (2016). Linear and nonlinear solvers for variational phase-field models of brittle fracture. International Journal for Numerical Methods in Engineering, 5(109), 648–667. https://doi.org/10.1002/nme.5300 [Publisher] [Pdf]
7. Marigo, J.-J., Maurini, C., & Pham, K. (2016). An overview of the modelling of fracture by gradient damage models. Meccanica, 1–22. https://doi.org/10.1007/s11012-016-0538-4 [Publisher] [Pdf]

   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 problem. The total energy of the body is defined in terms of the state variables which are the displacement field and the damage field in the case of quasi-brittle materials. That energy contains in particular gradient damage terms in order to avoid too strong damage localizations. The formulation of the damage evolution problem is then based on the concepts of irreversibility, stability and energy balance. That allows us to construct homogeneous as well as localized damage solutions in a closed form and to illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Moreover, the variational formulation leads to a natural numerical method based on an alternate minimization algorithm. Several numerical examples illustrate the ability of this approach to account for all the process of fracture including a 3D thermal shock problem where the crack evolution is very complex.

8. Leon Baldelli, A. A., Babadjian, J.-F., Bourdin, B., Henao, D., & Maurini, C. (2014). A variational model for fracture and debonding of thin films under in-plane loadings. Journal of the Mechanics and Physics of Solids, 70(0), 320–348. https://doi.org/10.1016/j.jmps.2014.05.020 [Publisher] [Pdf]

   Abstract 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 mechanical response of such systems under increasing loads, we formulate a dimension-reduced, rate-independent, irreversible evolution law accounting for both transverse fracture and debonding. We propose a numerical implementation based on a regularized formulation of the fracture problem via a gradient damage functional, and provide an illustration of its capabilities exploring complex crack patterns, showing a qualitative comparison with geometrically involved real life examples. Moreover, we justify the underlying dimension-reduced model in the setting of scalar-valued displacement fields by a rigorous asymptotic analysis using Γ -convergence, starting from the three-dimensional variational fracture (free-discontinuity) problem under precise scaling hypotheses on material and geometric parameters.

9. Bourdin, B., Marigo, J.-J., Maurini, C., & Sicsic, P. (2014). Morphogenesis and Propagation of Complex Cracks Induced by Thermal Shocks. Physical Review Letters, 112, 014301. https://doi.org/10.1103/PhysRevLett.112.014301 [Publisher] [Pdf]
10. Sicsic, P., Marigo, J.-J., & Maurini, C. (2014). Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling. Journal of the Mechanics and Physics of Solids, 63(0), 256–284. https://doi.org/10.1016/j.jmps.2013.09.003 [Publisher] [Pdf]
11. Seffen, K. A., & Maurini, C. (2013). Growth and shape control of disks by bending and extension. Journal of the Mechanics and Physics of Solids, 61(1), 190–204. https://doi.org/10.1016/j.jmps.2012.08.003 [Publisher] [Pdf]

    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 structures. Such behaviour inspires us to distinguish and to compare two possible modes of differential growth not normally compared to each other, in order to reveal the full range of out-of-plane shapes of an initially flat disk. The first growth mode, frequently employed by engineers, is characterised by direct bending strains through the thickness, and the second mode, mainly apparent in biological systems, is driven by extensional strains of the middle surface. When each mode is considered separately, it is shown that buckling is common to both modes, leading to bistable shapes: growth from bending strains results in a double-curvature limit at buckling, followed by almost developable deformation in which the Gaussian curvature at buckling is conserved; during extensional growth, out-of-plane distortions occur only when the buckling condition is reached, and the Gaussian curvature continues to increase. When both growth modes are present, it is shown that, generally, larger displacements are obtained under in-plane growth when the disk is relatively thick and growth strains are small, and vice versa. It is also shown that shapes can be mono-, bi-, tri- or neutrally stable, depending on the growth strain levels and the material properties: furthermore, it is shown that certain combinations of growth modes result in a free, or natural, response in which the doubly curved shape of disk exactly matches the imposed strains. Such diverse behaviour, in general, may help to realise more effective actuation schemes for engineering structures. \copyright 2012 Elsevier Ltd. All rights reserved.

Morphing structures

Slender structures may experience large global changes of their shape with small local deformations of the material. An emerging community of researchers proposes to benefit from geometrical nonlinearities to conceive structures able to hold multiple configurations of largely different shapes, each one associated to a specific functional regime. Similar systems are currently denoted as morphing, or shape-changing, structures. Their careful design can exploit geometric nonlinear effects to obtain great changes in shape through active materials with limited actuation power. Potential applications encompass aeronautics (shape-changing aerodynamic panels for flow control), energy (flexible and deployable solar cells), electronics (flexible and folding electronic devices), civil engineering (adaptive architecture including morphing functional elements), optics (shape-changing mirrors for active focusing), and microelectromechanical systems (micro-switches, mechanical memory cells, valves, micro-pumps).

In a series of works, we studied the multistable of shells and its dependence on on the material properties, the initial shape and the prestresses. We show how active materials (piezoelectric actuators) can be effectively used to control their shape.

For a recent paper, see here

Some publications on the topic

1. Corsi, G., De Simone, A., Maurini, C., & Vidoli, S. (2019). A neutrally stable shell in a Stokes flow: a rotational Taylor’s sheet. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2227), 20190178. https://doi.org/10.1098/rspa.2019.0178 [Publisher] [Pdf]
2. Hale, J. S., Brunetti, M., Bordas, S. P. A., & Maurini, C. (2018). Simple and extensible plate and shell finite element models through automatic code generation tools. Computers & Structures, 209, 163–181. https://doi.org/10.1016/j.compstruc.2018.08.001 [Publisher] [Pdf]
3. Hamouche, W., Maurini, C., Vidoli, S., & Vincenti, A. (2017). Multi-parameter actuation of a neutrally stable shell: a flexible gear-less motor. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 473(2204). https://doi.org/10.1098/rspa.2017.0364 [Publisher] [Pdf]

   We have designed and tested experimentally a morphing structure consisting of a neutrally stable thin cylindrical shell driven by a multi-parameter piezoelectric actuation. The shell is obtained by plastically deforming an initially flat copper disc, so as to induce large isotropic and almost uniform inelastic curvatures. Following the plastic deformation, in a perfectly isotropic system, the shell is theoretically neutrally stable, having a continuous set of stable cylindrical shapes corresponding to the rotation of the axis of maximal curvature. Small imperfections render the actual structure bistable, giving preferred orientations. A three-parameter piezoelectric actuation, exerted through micro-fibre-composite actuators, allows us to add a small perturbation to the plastic inelastic curvature and to control the direction of maximal curvature. This actuation law is designed through a geometrical analogy based on a fully nonlinear inextensible uniform-curvature shell model. We report on the fabrication, identification and experimental testing of a prototype and demonstrate the effectiveness of the piezoelectric actuators in controlling its shape. The resulting motion is an apparent rotation of the shell, controlled by the voltages as in a ‘gear-less motor’, which is, in reality, a precession of the axis of principal curvature.

4. Hamouche, W., Maurini, C., Vincenti, A., & Vidoli, S. (2016). Basic criteria to design and produce multistable shells. Meccanica, 51, 2305–2320. https://doi.org/10.1007/s11012-016-0375-5 [Publisher] [Pdf]

   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 allow to easily provide guidelines to build shells with different stability properties.

Nonlinear elasticity

Buckling of an elastic ridge

See (Lestringant et al., 2017)

Deformation of a soft solid by capillary effects

See (Mora et al., 2013)

Some publications on the topic

1. Lestringant, C., Maurini, C., Lazarus, A., & Audoly, B. (2017). Buckling of an Elastic Ridge: Competition between Wrinkles and Creases. Physical Review Letters, 118(16), 165501. https://doi.org/10.1103/PhysRevLett.118.165501 [Publisher] [Pdf]
2. Mora, S., Maurini, C., Phou, T., Fromental, J.-M., Audoly, B., & Pomeau, Y. (2013). Solid Drops: Large Capillary Deformations of Immersed Elastic Rods. Physical Review Letters, 111(11), 114301. https://doi.org/10.1103/PhysRevLett.111.114301 [Publisher] [Pdf]
3. Annaidh, A. N., Bruyère, K., Destrade, M., Gilchrist, M. D., Maurini, C., Otténio, M., & Saccomandi, G. (2012). Automated estimation of collagen fibre dispersion in the dermis and its contribution to the anisotropic behaviour of skin. Annals of Biomedical Engineering, 40(8), 1666–1678. https://doi.org/10.1007/s10439-012-0542-3 [Publisher] [Pdf]

   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 identify the orientation of collagen fibres using inexpensive and relatively simple techniques. The method uses established histological techniques and an algorithm implemented in the MATLAB image processing toolbox. It takes an average of 15 s to evaluate one image, compared to several hours if assessed visually. The technique was applied to histological sections of human skin with different Langer line orientations and a definite correlation between the orientation of Langer lines and the preferred orientation of collagen fibres in the dermis (p<0:001;R 2=0:95) was observed. The structural parameters of the Gasser-Ogden- Holzapfel (GOH) model were all successfully evaluated. The mean dispersion factor for the dermis was k = 0:1404\pm 0:0028: The constitutive parameters μ, k 1 and k 2 were evaluated through physically-based, least squares curvefitting of experimental test data. The values found for μ, k 1 and k 2 were 0.2014 MPa, 243.6 and 0.1327, respectively. Finally, the above model was implemented in ABAQUS/ Standard and a finite element (FE) computation was performed of uniaxial extension tests on human skin. It is expected that the results of this study will assist those wishing to model skin, and that the algorithm described will be of benefit to those who wish to evaluate the collagen dispersion of other soft tissues. \copyright 2012 Biomedical Engineering Society.

Piezoelectric structure and vibration control

During my Ph.D. I study the passive vibration control of mechanical structures trhough distributed piezoelectric transducers and resonant electric network. It the comporary terminology, we designed, fabricated and tested electromechanical meta-materials for dissipating mechanical vibrations in electric circuits. This kind of ideas recently received a renovated interest.

1. Porfiri, M., Maurini, C., & Pouget, J. (2007). Identification of electromechanical modal parameters of linear piezoelectric structures. Smart Materials and Structures, 16(2), 323–331. https://doi.org/10.1088/0964-1726/16/2/010 [Publisher] [Pdf]

   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 easily implementable and rely on elementary vibration tests. We show the application of these methods to a sample structure hosting multiple transducers. \copyright IOP Publishing Ltd.

2. Maurini, C., Porfiri, M., & Pouget, J. (2006). Numerical methods for modal analysis of stepped piezoelectric beams. Journal of Sound and Vibration, 298(4-5), 918–933. https://doi.org/10.1016/j.jsv.2006.05.041 [Publisher] [Pdf]

   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 eigenvalue problem for the infinite dimensional system. An accurate and manageable novel method, that enriches the assumed modes basis functions with special jump functions, is presented. Numerical results are compared with experimental data and the accuracy of the adopted beam model is validated. \copyright 2006 Elsevier Ltd. All rights reserved.

3. Maurini, C., Pouget, J., & dell’Isola, F. (2006). Extension of the Euler-Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach. Computers and Structures, 84(22-23), 1438–1458. https://doi.org/10.1016/j.compstruc.2006.01.016 [Publisher] [Pdf]

   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 typical of standard beam models is weakened by imposing vanishing stress resultants. This integral condition is enforced by adopting a mixed variational principle and Lagrange multiplier method. Explicit expressions for the beam constitutive coefficients are given and the sandwich and bimorph piezoelectric benders are studied in details. The model is assessed through comparisons with standard models and 3D finite element results, showing an important enhancement of standard beam theories. \copyright 2006 Civil-Comp Ltd. and Elsevier Ltd.

4. Dell’Isola, F., Maurini, C., & Porfiri, M. (2004). Passive damping of beam vibrations through distributed electric networks and piezoelectric transducers: prototype design and experimental validation. Smart Materials and Structures, 13, 299. https://doi.org/10.1088/0964-1726/13/2/008 [Publisher] [Pdf]
5. Maurini, C., Dell’Isola, F., & Del Vescovo, D. (2004). Comparison of piezoelectronic networks acting as distributed vibration absorbers. Mechanical Systems and Signal Processing, 18(5), 1243–1271. https://doi.org/10.1016/S0888-3270(03)00082-7 [Publisher] [Pdf]

   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 analysed and compared. Their optimal parameters are found by adopting a criterion for critical damping of k̄-waves: the parameters are suitably chosen to have the quickest temporal vibration decay for a single wave number k̄. The analysis is based on homogenized models of the modular piezo-electromechanical systems, i.e. they are regarded as continuous systems by assuming that the number of modules per unit length is high enough with respect to the considered wave numbers. Calling k̄-absorbers the corresponding optimal absorbers, we show that: (i) k̄-waves are damped in k̄-absorbers with an optimal decay time which is independent of the absorber interconnecting topology, while it depends only on the piezoelectric coupling coefficient; (ii) the efficiency of k̄-absorbers depends significantly on the absorber interconnecting topology for k different from k̄; (iii) one of the proposed absorbers (which is made of a fourth-order electric transmission line with a second-order electric dissipation) equally performs for all the wave numbers and accomplishes an effective multi-modal damping for the mechanically forced response; (iv) the optimal values of the electric parameters differently depend on the number n of used circuit modules for different interconnecting topologies and, in particular, the optimal inductance per module needed in a fourth-order electric transmission line is proportional 1/n3. \copyright 2003 Elsevier Ltd. All rights reserved.

6. Maurini, C., Pouget, J., & dell’Isola, F. (2004). On a model of layered piezoelectric beams including transverse stress effect. International Journal of Solids and Structures, 41(16-17), 4473–4502. https://doi.org/10.1016/j.ijsolstr.2004.03.002 [Publisher] [Pdf]

   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 deductive approach based on a mixed variational formulation is adopted and distributions of deformation, stress, electric field and electric displacement are simultaneously prescribed. The attention is focused on the choice of the most fitting assumptions to recover complex 3D cross-sectional field distributions. In particular, transverse interactions between different layers are taken into account by enforcing specific conditions on transverse stress through the Lagrange multipliers method. The estimate of electromechanical beam constitutive coefficients is discussed and comparison with standard modelling approaches, which assume either vanishing transverse stresses or vanishing transverse strains, is emphasized. For a sandwich piezoelectric beam and for a two-layer beam, expressions of the beam constitutive coefficients are provided and the main features of the proposed model are highlighted by presenting the through-the-thickness distribution of the 3D state fields associated to beam-axis deformations and applied voltage. As a main peculiarity, the proposed beam model is able to coherently estimate the equivalent piezoelectric capacitance also when the thickness of elastic and piezoelectric layers is comparable. \copyright 2004 Elsevier Ltd. All rights reserved.

Collaborations

Some co-authors and friends:

• B. Audoly, Laboratoire de Mécanique des Solides, Ecole Polytechnique/CNRS
• J.F. Babadjian, Laboratoire Jacques-Louis Lions, UPMC/CNRS
• B. Bourdin, Dept. of Mathematics, Lousiana State University, US
• M. Destrade, Nationa University of Galway, Irland
• P. Farrell, Univ. of Oxford
• G. Gauthier, FAST, Paris 11/UPMC/CNRS
• J. Hale, Univ. of Luxembourg
• D. Henao Manrique, Fac. de Matemáticas, Pont. Univ. Cat. de Chile
• A. Lazarus, Institut Jean Le Rond d’Alembert, UMPC/CNRS
• V. Lazarus, FAST, Paris 11/UPMC/CNRS
• J.-J. Marigo, Laboratoire de Mécanique des Solides, Ecole Polytechnique
• S. Neukirch, Institut Jean Le Rond d’Alembert, UMPC/CNRS
• S. Mora, Institut Coulomb, Université de Montpellier
• K. Seffen, Engineering Department, University of Cambridge, UK
• S. Vidoli, Dip. Ing. Strutturale e Geotecnica, La Sapienza, Rome, Italy
• A. Vincenti, Institut Jean Le Rond d’Alembert, UMPC/CNRS