PIERLUIGI CESANA | Last modified date：2021.06.21 |

Associate Professor /
Australia Branch /
Institute of Mathematics for Industry

**Papers**

1. | Pierluigi Cesana, Andrés Alessandro León Baldelli, Gamma-convergence results for nematic elastomer bilayers: relaxation and actuation, ArXiv, ArXiv preprint (currently under consideration for publication on international journal), 2020.12, [URL], We compute effective energies of thin bilayer structures composed by soft nematic elastic-liquid crystals in various geometrical regimes and functional configurations. Our focus is on order-strain interaction in elastic foundations composed of an isotropic layer attached to a nematic substrate. We compute Gamma-limits as the layers thickness vanishes in two main scaling regimes exhibiting spontaneous stress relaxation and shape-morphing, allowing in both cases out-of-plane displacements. This extends the plane strain modelling of [*], showing the asymptotic emergence of fully coupled macroscopic active-nematic foundations. Subsequently, we focus on actuation and compute asymptotic configurations of an active plate on nematic foundation interacting with an applied electric field. From the analytical standpoint, the presence of an electric field and its associated electrostatic work turns the total energy into a non-convex and non-coercive functional. We show that equilibrium solutions are min-max points of the system, that min-maximising sequences pass to the limit and, that the limit system can exert mechanical work under applied electric fields.[*]: P. Cesana and A. A. León Baldelli. "Variational modelling of nematic elastomer foundations". In: Mathematical Models and Methods in Applied Sciences 14 (2018). |

2. | Pierluigi Cesana, Patrick van Meurs, Discrete-to-continuum limits of planar disclinations, ESAIM:COCV, 10.1051/cocv/2021025, 27, 23, 2021.03, [URL], In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a nearest-neighbor-type energy for the atomic bonds with an additional term to penalize change in volume. We enforce the appearance of disclinations by means of a special boundary condition.Our main result is the discrete-to-continuum limit of this energy as the lattice size tends to zero. Our proof method is relaxation of the energy. The main mathematical novelty of our proof is a density theorem for the special boundary condition. In addition to our limit theorem, we construct examples of planar disclinations as solutions to numerical minimization of the model and show that classical results for wedge disclinations are recovered by our analysis.. |

3. | P. Cesana, F.D. Porta, A. Rueland, C. Zillinger, B. Zwicknagl, Exact constructions in the (non-linear) planar theory of elasticity: From elastic crystals to nematic elastomers, Archive for Rational Mechanics and Analysis, https://doi.org/10.1007/s00205-020-01511-9, 237, 383-445, 2020.04, [URL], In this article we deduce necessary and sufficient conditions for the presence of “Conti-type”, highly symmetric, exactly stress-free constructions in the geometrically non-linear, planar n-well problem, generalising results of CONTI et al. (Proc R Soc A 473(2203):20170235, 2017). Passing to the limit $n\to\infty$, this allows us to treat solid crystals and nematic elastomer differential inclusions simultaneously. In particular, we recover and generalise (non-linear) planar tripole star type deformations which were experimentally observed in KITANO and KIFUNE (Ultramicroscopy 39(1–4):279–286, 1991), MANOLIKAS and AMELINCKX (Physica Status Solidi (A) 60(2):607–617, 1980; Physica Status Solidi (A) 61(1):179–188, 1980). Furthermore, we discuss the corresponding geometrically linearised problem.. |

4. | Dilruk Gallage, Dimetre Triadis, Philip Broadbridge, Pierluigi Cesana, Solution for 4th-order nonlinear axisymmetric surface diffusion by inverse method, Physica D, 10.1016/j.physd.2019.132288, 405, 132288, 2020.02, [URL], We present a method for constructing similarity solutions of a fourth-order nonlinear partial differential equation for axisymmetric surface diffusion by extending an inverse method previously used for the second-order one-dimensional nonlinear diffusion equation. After imposing a solution profile, both a feasible surface tension, and a compatible mobility function are deduced simultaneously. Although the profile is not one-to-one, an optimization algorithm is implemented to construct a mobility function that is a function of surface orientation, with no practical difference in mobility between different arms of the many-to-one profile. It is shown that the solution of the linear model well approximates the solution of the nonlinear model, in which the surface tension and mobility are close to constant for a wide range of surface angles, even when nonlinear geometric terms are included.. |

5. | Pierluigi Cesana, A. Baldelli, Variational modelling of nematic elastomer foundations, Mathematical Models and Methods in Applied Sciences, 10.1142/S021820251850063X, 28, 14, 2863-2904, 2018.12, [URL], We compute the Gamma-limit of energy functionals describing mechanical systems composed of a thin nematic liquid crystal elastomer sustaining a homogeneous and isotropic elastic membrane. We work in the regime of infinitesimal displacements and model the orientation of the liquid crystal according to the order tensor theories of both Frank and De Gennes. We describe the asymptotic regime by analysing a family of functionals parametrised by the vanishing thickness of the membranes and the ratio of the elastic constants, establishing that, in the limit, the system is represented by a two-dimensional integral functional interpreted as a linear membrane on top of a nematic active foundation involving an effective De Gennes optic tensor which allows for low order states. The latter can suppress shear energy by formation of microstructure as well as act as a pre-strain transmitted by the foundation to the overlying film.. |

6. | P. Cesana, B.M. Hambly, A probabilistic model for martensitic interfaces, preprint (under review), 2018.10, [URL], We analyse features of the patterns formed from a simple model for a martensitic phase transition. This is a fragmentation model that can be encoded by a general branching random walk. An important quantity is the distribution of the lengths of the interfaces in the pattern and we establish limit theorems for some of the asymptotics of the interface profile. We are also able to use a general branching process to show almost sure power law decay of the number of interfaces of at least a certain size. We discuss the numerical aspects of determining the behaviour of the density profile and power laws from simulations of the model.. |

7. | P. Broadbridge, D. Triadis, D. Gallage, Pierluigi Cesana, Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion, Symmetry, 10.3390/sym10030072, 10, 3, 1-18, 2018.03, [URL], Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing phase of cylindrical or spherical inclusion, allowing for a “mushy” zone with a mixed state that is controlled by imposing a pure state at the boundary. The diffusion coefficients for transport of one phase through the mixture depend on the phase field value, since the physical structure of the mixture depends on the relative proportions of the two phases. A source term promotes stability of both of the pure phases but this tendency may be controlled or even reversed through the boundary conditions. |

8. | Pierluigi Cesana, Relaxation of an energy model for the triangle-to-centred rectangle transformation, The Role and Importance of Mathematics in Innovation
Proceedings of the Forum “Math-for-Industry” 2015, 10.1007/978-981-10-0962-4_11, 117-126, 2016.04, We model and analyze the two-dimensional triangle-to-centred rectangle transformation of elastic crystals. By considering a Ginzburg–Landau type model, we compute the relaxation of the total energy both in the case of compressible and incompressible materials and construct some possible explicit microstructures as the approximate solutions of a non-convex minimization problem.. |

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