A characterization of ℓ1double bubbles with general interface interaction - Friedrich, Manuel; Górny, Wojciech; Stefanelli, Ulisse (Advances in Calculus of Variations, 2025)
We investigate the optimal arrangements of two planar sets of given volume which are minimizing the ℓ 1 _1 double-bubble interaction functional. The latter features a competition between the minimization of the ℓ 1 _1 perimeters of the two sets and the maximization of their ℓ 1 _1 interface. We investigate the problem in its full generality for sets of finite perimeter, by considering the whole range of possible interaction intensities and all relative volumes of the two sets. The main result is the complete classification of minimizers.
CRIS-Team Scopus Importer:2025-04-11
Keywords: Double bubble; Wulff shape

A compactness result in GSBVpand applications to Γ -convergence for free discontinuity problems - Friedrich, Manuel (Calculus of Variations and Partial Differential Equations, 2019)
We present a compactness result in the space GSBV p which extends the classical statement due to Ambrosio (Arch Ration Mech 111:291–322, 1990) to problems without a priori bounds on the functions. As an application, we revisit the Γ-convergence results for free discontinuity functionals established recently by Cagnetti et al. [Ann Inst H Poincaré Anal Non Linéaire (to appear)]. We investigate sequences of boundary value problems and show convergence of minimum values and minimizers.
CRIS-Team Scopus Importer:2021-10-21

A Derivation of Linearized Griffith Energies from Nonlinear Models - Friedrich, Manuel (Archive for Rational Mechanics and Analysis, 2017)
We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame indifferent energies in a brittle fracture via Γ -convergence. The convergence is given in terms of rescaled displacement fields measuring the distance of deformations from piecewise rigid motions. The configurations of the limiting model consist of partitions of the material, corresponding piecewise rigid deformations and displacement fields which are defined separately on each component of the cracked body. Apart from the linearized Griffith energy the limiting functional also comprises the segmentation energy, which is necessary to disconnect the parts of the specimen.
CRIS-Team Scopus Importer:2021-10-21

A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations - Almi, Stefano; Caponi, Maicol; Friedrich, Manuel; Solombrino, Francesco (Mathematische Annalen, 2024)
We derive a strain-gradient theory for plasticity as the Γ-limit of discrete dislocation fractional energies, without the introduction of a core-radius. By using the finite horizon fractional gradient introduced by Bellido et al. (Adv Calc Var 17:1039–1055, 2024), we consider a nonlocal model of semi-discrete dislocations, in which the stored elastic energy is computed via the fractional gradient of order 1-α. As α goes to 0, we show that suitably rescaled energies Γ-converge to the macroscopic strain-gradient model of Garroni et la. (J Eur Math Soc (JEMS) 12:1231–1266, 2010).
CRIS-Team Scopus Importer:2024-11-01
Keywords: 26A33; 35R11; 49J45; 74C05

A Korn-type inequality in SBD for functions with small jump sets - Friedrich, Manuel (Mathematical Models & Methods in Applied Sciences, 2017)
We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in SBD2 with a sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly, the result shows that each function in SBD2 has bounded variation away from an arbitrarily small part of the domain.
CRIS-Team Scopus Importer:2021-10-21
Keywords: brittle materials; Functions of bounded deformation; Korn's inequality; Korn-Poincaré inequality; variational fracture

An Analysis of Crystal Cleavage in the Passage from Atomistic Models to Continuum Theory - Friedrich, Manuel; Schmidt, Bernd (Archive for Rational Mechanics and Analysis, 2015)
We study the behavior of brittle atomistic models in general dimensions under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to-continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. In particular, our results apply to mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.
CRIS-Team Scopus Importer:2021-10-21

An atomistic-to-continuum analysis of crystal cleavage in a two-dimensional model problem - Friedrich, Manuel; Schmidt, Bernd (Journal of Nonlinear Science, 2014)
A two-dimensional atomic mass spring system is investigated for critical fracture loads and its crack path geometry. We rigorously prove that, in the discreteto- continuum limit, the minimal energy of a crystal under uniaxial tension leads to a universal cleavage law and energy minimizers are either homogeneous elastic deformations or configurations that are completely cracked and do not store elastic energy. Beyond critical loading, the specimen generically cleaves along a unique optimal crystallographic hyperplane. For specific symmetric crystal orientations, however, cleavage might fail. In this case a complete characterization of possible limiting crack geometries is obtained. © Springer Science+Business Media New York 2013.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Atomistic models; Brittle materials; Discrete-to-continuum limits; Free discontinuity problems; Variational fracture

A piecewise korn inequality in SBD and applications to embedding and density results - Friedrich, Manuel (SIAM Journal on Mathematical Analysis, 2018)
We present a piecewise Korn inequality for generalized special functions of bounded deformation (GSBD2) in a planar setting generalizing the classical result in elasticity theory to the setting of functions with jump discontinuities. We show that for every configuration there is a partition of the domain such that on each component of the cracked body the distance of the function from an infinitesimal rigid motion can be controlled solely in terms of the linear elastic strain. In particular, the result implies that GSBD2 functions have bounded variation after subtraction of a piecewise infinitesimal rigid motion. As an application we prove a density result in GSBD2 in dimension two. Moreover, for all d = 2 we show GSBD2(ω) C (GBV (ω; R))d and the embedding SBD2(ω) L1(ω; Rd) → SBV (ω; Rd) into the space of special functions of bounded variation (SBV). Finally, we present a KornPoincaŕe inequality for functions with small jump sets in arbitrary space dimension.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Brittle materials; Coarea formula; Functions of bounded deformation; Korn's inequality; Korn-Poincaŕe inequality; Variational fracture

A Proof of Finite Crystallization via Stratification - Friedrich, Manuel; Kreutz, Leonard (Journal of Statistical Physics, 2023)
We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body short-ranged particle interactions and three-body angular potentials favoring bond-angles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality in ℓ1 . As a byproduct, we also prove a fluctuation estimate for minimizers of the configurational energy, known as the n3 / 4 -law.
CRIS-Team Scopus Importer:2023-12-22
Keywords: Atomic interaction potentials; Crystallization; Edge isoperimetric inequality; Square lattice; Stratification

Atomistic potentials and the cauchy-born rule for carbon nanotubes: A review - Mainini, Edoardo; Piovano, Paolo; Friedrich, Manuel (Rendiconti del Seminario Matematico, 2019)
Carbon nanotubes are modeled as point particle configurations in the framework of Molecular Mechanics, where interactions are described by means of short range attractiverepulsive potentials. The identification of local energy minimizers yields a variational description for the stability of rolled-up hexagonal-lattice structures. Optimality of periodic configurations is preserved under moderate tension, hence justifying the elastic behavior of carbon nanotubes in the axial traction regime.
CRIS-Team Scopus Importer:2021-10-21

Atomistic-to-continuum convergence for quasi-static crack growth in brittle materials - Friedrich, Manuel; Seutter, Joscha (Mathematical Models & Methods in Applied Sciences, 2025)
We study the atomistic-to-continuum limit for a model of quasi-static crack evolution driven by time-dependent boundary conditions. We consider a two-dimensional atomic mass spring system whose interactions are modeled by classical interaction potentials, supplemented by a suitable irreversibility condition accounting for the breaking of atomic bonding. In a simultaneous limit of vanishing interatomic distance and discretized time step, we identify a continuum model of quasistatic crack growth in brittle fracture [G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998) 1319–1342] featuring an irreversibility condition, a global stability, and an energy balance. The proof of global stability relies on a careful adaptation of the jump-transfer argument in [G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Commun. Pure Appl. Math. 56 (2003) 1465–1500] to the atomistic setting.
CRIS-Team Scopus Importer:2025-04-11
Keywords: atomistic systems; Brittle materials; discrete-to-continuum limits; free-discontinuity problems; irreversibility condition; quasistatic crack propagation; variational fracture; Γ-convergence

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule - Friedrich, Manuel; Mainini, Edoardo; Piovano, Paolo; Stefanelli, Ulisse (Archive for Rational Mechanics and Analysis, 2019)
Carbon nanotubes are modeled as point configurations and investigated by minimizing configurational energies including two- and three-body interactions. Optimal configurations are identified with local minima and their fine geometry is fully characterized in terms of lower-dimensional problems. Under moderate tension, we prove the existence of periodic local minimizers, which indeed validates the so-called Cauchy–Born rule in this setting.
CRIS-Team Scopus Importer:2021-10-21

Crystallization in a One-Dimensional Periodic Landscape - Friedrich, Manuel; Stefanelli, Ulisse (Journal of Statistical Physics, 2020)
We consider the crystallization problem for a finite one-dimensional collection of identical hard spheres in a periodic energy landscape. This issue arises in connection with the investigation of crystalline states of ionic dimers, as well as in epitaxial growth on a crystalline substrate in presence of lattice mismatch. Depending on the commensurability of the radius of the sphere and the period of the landscape, we discuss the possible emergence of crystallized states. In particular, we prove that crystallization in arbitrarily long chains is generically not to be expected.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Crystallization; Epitaxial growth; Hard spheres; Ionic dimers; Periodic landscape

Crystallization in the hexagonal lattice for ionic dimers - Friedrich, Manuel; Kreutz, Leonard (Mathematical Models & Methods in Applied Sciences, 2019)
We consider finite discrete systems consisting of two different atomic types and investigate ground-state configurations for configurational energies featuring two-body short-ranged particle interactions. The atomic potentials favor some reference distance between different atomic types and include repulsive terms for atoms of the same type, which are typical assumptions in models for ionic dimers. Our goal is to show a two-dimensional crystallization result. More precisely, we give conditions in order to prove that energy minimizers are connected subsets of the hexagonal lattice where the two atomic types are alternately arranged in the crystal lattice. We also provide explicit formulas for the ground-state energy. Finally, we characterize the net charge, i.e. the difference of the number of the two atomic types. Analyzing the deviation of configurations from the hexagonal Wulff shape, we prove that for ground states consisting of n particles the net charge is at most of order O(n1/4) where the scaling is sharp.
CRIS-Team Scopus Importer:2021-10-21
Keywords: configurational energy minimization; crystallization; ground-state; hexagonal lattice; Ionic dimers; net charge

Derivation of a von Kármán plate theory for thermoviscoelastic solids - Badal, Rufat; Friedrich, Manuel; Machill, Lennart (Mathematical Models & Methods in Applied Sciences, 2024)
In this paper, we derive a von Kármán plate theory from a three-dimensional quasistatic nonlinear model for nonsimple thermoviscoelastic materials in the Kelvin-Voigt rheology, in which the elastic and the viscous stress tensor comply with a frame indifference principle [A. Mielke and T. Roubíček, Thermoviscoelasticity in Kelvin-Voigt rheology at large strains, Arch. Ration. Mech. Anal. 238 (2020) 1-45]. In a dimension-reduction limit, we show that weak solutions to the nonlinear system of equations converge to weak solutions of an effective two-dimensional system featuring mechanical equations for viscoelastic von Kármán plates, previously derived in Friedrich and Kružík [Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity, Arch. Ration. Mech. Anal. 238 (2020) 489-540], coupled with a linear heat-transfer equation. The main challenge lies in deriving a priori estimates for rescaled displacement fields and temperatures, which requires the adaptation of generalized Korn's inequalities and bounds for heat equations with L1-data to thin domains.
CRIS-Team Scopus Importer:2025-02-04
Keywords: dimension reduction; frame indifferent viscous stresses; generalized Korn's inequality; Thermoviscoelasticity

Derivation of effective theories for thin 3D nonlinearly elastic rods with voids - Friedrich, Manuel; Kreutz, Leonard; Zemas, Konstantinos (Mathematical Models & Methods in Applied Sciences, 2024)
We derive a dimension-reduction limit for a three-dimensional rod with material voids by means of Γ-convergence. Hereby, we generalize the results of the purely elastic setting [M. G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-convergence, Calc. Var. Partial Differential Equations 18 (2003) 287-305] to a framework of free discontinuity problems. The effective one-dimensional model features a classical elastic bending-torsion energy, but also accounts for the possibility that the limiting rod can be broken apart into several pieces or folded. The latter phenomenon can occur because of the persistence of voids in the limit, or due to their collapsing into a discontinuity of the limiting deformation or its derivative. The main ingredient in the proof is a novel rigidity estimate in varying domains under vanishing curvature regularization, obtained in [M. Friedrich, L. Kreutz and K. Zemas, Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces, preprint (2021), arXiv:2107.10808].
CRIS-Team Scopus Importer:2024-04-05
Keywords: curvature regularization; dimension reduction; fracture; Geometric rigidity; rod theories; variable domains

Derivation of von Kármán Plate Theory in the Framework of Three-Dimensional Viscoelasticity - Friedrich, Manuel; Kruzik, Martin (Archive for Rational Mechanics and Analysis, 2020)
We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in Kelvin’s-Voigt’s rheology to derive a viscoelastic plate model of von Kármán type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity considered in Friedrich and Kružík (SIAM J Math Anal 50:4426–4456, 2018) where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. Combining the derivation of nonlinear plate theory by Friesecke, James and Müller (Commun Pure Appl Math 55:1461–1506, 2002; Arch Ration Mech Anal 180:183–236, 2006), and the abstract theory of gradient flows in metric spaces by Sandier and Serfaty (Commun Pure Appl Math 57:1627–1672, 2004), we perform a dimension-reduction from three dimensions to two dimensions and identify weak solutions of viscoelastic form of von Kármán plates.
CRIS-Team Scopus Importer:2021-10-21

DISCRETE-TO-CONTINUUM LINEARIZATION IN ATOMISTIC DYNAMICS - Friedrich, Manuel; Seitz, Manuel; Stefanelli, Ulisse (Discrete and Continuous Dynamical Systems, 2025)
In the stationary case, atomistic interaction energies can be proved to Γ-converge to classical elasticity models in the simultaneous atomistic-to-continuum and linearization limit [19, 41]. The aim of this note is that of extending the convergence analysis to the dynamic setting. Moving within the framework of [41], we prove that solutions of the equation of motion driven by atomistic deformation energies converge to the solutions of the momentum equation for the corresponding continuum energy of linearized elasticity. By recasting the evolution problems in their equivalent energy-dissipation-inertia-principle form, we directly argue at the variational level of evolutionary Γ-convergence [33, 37]. This in particular ensures the pointwise in time convergence of the energies.
CRIS-Team Scopus Importer:2024-11-22
Keywords: Discrete-to-continuum and linearization limit; equation of motion; evolutive Γ-convergence; variational evolution

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy - Friedrich, Manuel; Kreutz, Leonard; Schmidt, Bernd (Archive for Rational Mechanics and Analysis, 2021)
We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the ‘sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of Γ -convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, that is, on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum–solid and solid–solid phase transitions. The latter relies fundamentally on a structure result for grain boundaries showing that, due to the extremely brittle setup, interpolating boundary layers near cracks are energetically not favorable.
CRIS-Team Scopus Importer:2021-10-21

Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity - Crismale, Vito; Friedrich, Manuel (Archive for Rational Mechanics and Analysis, 2020)
We extend the results about the existence of minimizers, relaxation, and approximation proven by Bonnetier and Chambolle (SIAM J Appl Math 62:1093–1121, 2002), Chambolle and Solci (SIAM J Math Anal 39:77–102, 2007) for an energy related to epitaxially strained crystalline films, and by Braides et al. (ESAIM Control Optim Calc Var 13:717–734, 2007) for a class of energies defined on pairs of function-set. We study these models in the framework of three-dimensional linear elasticity, where a major obstacle to overcome is the lack of any a priori assumption on the integrability properties of displacements. As a key tool for the proofs, we introduce a new notion of convergence for (d- 1) -rectifiable sets that are jumps of GSBDp functions, called σsymp-convergence.
CRIS-Team Scopus Importer:2021-10-21

EQUILIBRIUM of IMMERSED HYPERELASTIC SOLIDS - Friedrich, Manuel; Kruzik, Martin; Stefanelli, Ulisse (Discrete and Continuous Dynamical Systems. Series S, 2021)
We discuss different equilibrium problems for hyperelastic solids immersed in a fluid at rest. In particular, solids are subjected to gravity and hydrostatic pressure on their immersed boundaries. By means of a variational approach, we discuss free-floating bodies, anchored solids, and floating vessels. Conditions for the existence of local and global energy minimizers are presented.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Equilibrium; Hydrostatic pressure; Hyperelastic solid

Existence of quasi-static crack evolution for atomistic systems - Badal, Rufat; Friedrich, Manuel; Seutter, Joscha (Forces in Mechanics, 2022)
We consider atomistic systems consisting of interacting particles arranged in atomic lattices whose quasi-static evolution is driven by time-dependent boundary conditions. The interaction of the particles is modeled by classical interaction potentials where we implement a suitable irreversibility condition modeling the breaking of atomic bonding. This leads to a delay differential equation depending on the complete history of the deformation at previous times. We prove existence of solutions and provide numerical tests for the prediction of quasi-static crack growth in particle systems.
CRIS-Team Scopus Importer:2022-11-25
Keywords: Atomistic systems; Delay differential equations; Irreversibility condition; Minimizing movements; Quasi-static crack growth

Finite Crystallization and Wulff shape emergence for ionic compounds in the square lattice - Friedrich, Manuel; Kreutz, Leonard (Nonlinearity, 2020)
We present two-dimensional crystallization results in the square lattice for finite particle systems consisting of two different atomic types. We identify energy minimizers of configurational energies featuring two-body short-ranged particle interactions which favor some reference distance between different atomic types and contain repulsive contributions for atoms of the same type. We first prove that ground states are connected subsets of the square lattice with alternating arrangement of the two atomic types in the crystal lattice, and address the emergence of a square macroscopic Wulff shape for an increasing number of particles. We then analyze the signed difference of the number of the two atomic types, the so-called net charge, for which we prove the sharp scaling O(n1/4) in terms of the particle number n. Afterwards, we investigate the model under prescribed net charge. We provide a characterization for the minimal energy and identify a critical net charge beyond which crystallization in the square lattice fails. Finally, for this specific net charge we prove a crystallization result and identify a diamond-like Wulff-shape of energy minimizers which illustrates the sensitivity of the macroscopic geometry on the net charge.
CRIS-Team Scopus Importer:2021-10-21
Keywords: configurational energy minimization; crystallization; ground state; ionic dimers; net charge; square lattice; Wulff shape

From atomistic systems to linearized continuum models for elastic materials with voids - Friedrich, Manuel; Kreutz, Leonard; Zemas, Konstantinos (Nonlinearity, 2023)
We study an atomistic model that describes the microscopic formation of material voids inside elastically stressed solids under an additional curvature regularization at the discrete level. Using a discrete-to-continuum analysis, by means of a recent geometric rigidity result in variable domains (Friedrich et al 2021 arXiv:2107.10808) and Γ-convergence tools, we rigorously derive effective linearized continuum models for elastically stressed solids with material voids in three-dimensional elasticity.
CRIS-Team Scopus Importer:2023-01-06
Keywords: atomistic systems; discrete-to-continuum limits; free discontinuity problems; functions of bounded deformation; material voids; Γ-convergence

Functionals Defined on Piecewise Rigid Functions: Integral Representation and Γ -Convergence - Friedrich, Manuel; Solombrino, Francesco (Archive for Rational Mechanics and Analysis, 2020)
We analyse integral representation and Γ-convergence properties of functionals defined on piecewise rigid functions, that is, functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our results are based on localization techniques for Γ-convergence and a careful adaption of the global method for relaxation (Bouchitté et al. in Arch Ration Mech Anal 165:187–242, 2002; Bouchitté et al. in Arch Ration Mech Anal 145:51–98, 1998), to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation.
CRIS-Team Scopus Importer:2021-10-21

Geometric rigidity on Sobolev spaces with variable exponent and applications - Almi, Stefano; Caponi, Maicol; Friedrich, Manuel; Solombrino, Francesco (Nodea-Nonlinear Differential Equations and Applications, 2025)
We present extensions of rigidity estimates and of Korn’s inequality to the setting of (mixed) variable exponents growth. The proof techniques, based on a classical covering argument, rely on the log-Hölder continuity of the exponent to get uniform regularity estimates on each cell of the cover, and on an extension result à laNitsche in Sobolev spaces with variable exponents. As an application, by means of Γ-convergence we perform a passage from nonlinear to linearized elasticity under variable subquadratic energy growth far from the energy well.
CRIS-Team Scopus Importer:2025-02-17
Keywords: Korn inequality; Mixed growth; Nonlinear and linear elasticity; Rigidity estimates; Variable exponent; Γ-convergence

Graphene ground states - Friedrich, Manuel; Stefanelli, Ulisse (Zeitschrift für Angewandte Mathematik und Physik, 2018)
Graphene is locally two-dimensional but not flat. Nanoscale ripples appear in suspended samples and rolling up often occurs when boundaries are not fixed. We address this variety of graphene geometries by classifying all ground-state deformations of the hexagonal lattice with respect to configurational energies including two- and three-body terms. As a consequence, we prove that all ground-state deformations are either periodic in one direction, as in the case of ripples, or rolled up, as in the case of nanotubes.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Graphene; Ground states; Nonflatness; Periodicity; Three-dimensional structures

Griffith energies as small strain limit of nonlinear models for nonsimple brittle materialsy - Friedrich, Manuel (Mathematics in Engineering, 2020)
We consider a nonlinear, frame indifferent Griffith model for nonsimple brittle materials where the elastic energy also depends on the second gradient of the deformations. In the framework of free discontinuity and gradient discontinuity problems, we prove existence of minimizers for boundary value problems. We then pass to a small strain limit in terms of suitably rescaled displacement fields and show that the nonlinear energies can be identified with a linear Griffith model in the sense of Γ- convergence. This complements the study in [39] by providing a linearization result in arbitrary space dimensions.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Brittle materials; Free discontinuity problems; Functions of bounded variation and deformation; Griffith energies; Nonsimple materials; Variational fracture; Γ-convergence

In situ control of graphene ripples and strain in the electron microscope - Ludacka, U.; Monazam, M. R. A.; Rentenberger, Christian; Stefanelli, Ulisse; Meyer, J. C.; Kotakoski, Jani; Friedrich, Manuel (npj 2D Materials and Applications, 2018)
We demonstrate control over the three-dimensional (3D) structure of suspended 2D materials in a transmission electron microscope. The shape of our graphene samples is measured from the diffraction patterns recorded at different sample tilts while applying tensile strain on the sample carrier. The changes in the shape of the pattern and in individual diffraction spots allow us to analyze both corrugations and strain in the lattice. Due to the significant effect of ripples and strain on the properties of 2D materials, our results may lead to new ways for their engineering for applications.
CRIS-Team Scopus Importer:2021-10-21

Integral representation for energies in linear elasticity with surface discontinuities - Crismale, Vito; Friedrich, Manuel; Solombrino, Francesco (Advances in Calculus of Variations, 2020)
In this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation (GSBD p GSBDp) in arbitrary space dimensions. Functionals of this type naturally arise in the modeling of linear elastic solids with surface discontinuities including phenomena as fracture, damage, surface tension between different elastic phases, or material voids. Our approach is based on the global method for relaxation devised in [G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145 1998, 1, 51-98] and a recent Korn-type inequality in GSBD p GSBDp, cf. [F. Cagnetti, A. Chambolle and L. Scardia, Korn and Poincaré-Korn inequalities for functions with a small jump set, preprint 2020]. Our general strategy also allows to generalize integral representation results in SBD pSBDp, obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on SBD p SBDp in dimension two, Arch. Ration. Mech. Anal. 223 2017, 3, 1337-1374], to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation (GSBV pGSBVp).
CRIS-Team Scopus Importer:2021-10-21
Keywords: free discontinuity problems; generalized special functions of bounded deformation; global method for relaxation; Integral representation; Korn-type inequalities

Lattice ground states for embedded-atom models in 2D and 3D - Betermin, Laurent; Friedrich, Manuel; Stefanelli, Ulisse (Letters in Mathematical Physics, 2021)
The Embedded-Atom Model (EAM) provides a phenomenological description of atomic arrangements in metallic systems. It consists of a configurational energy depending on atomic positions and featuring the interplay of two-body atomic interactions and nonlocal effects due to the corresponding electronic clouds. The purpose of this paper is to mathematically investigate the minimization of the EAM energy among lattices in two and three dimensions. We present a suite of analytical and numerical results under different reference choices for the underlying interaction potentials. In particular, Gaussian, inverse-power, and Lennard-Jones-type interactions are addressed.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Embedded-atom model; Epstein zeta function; Lattice energy minimization

Lower semicontinuity for functionals defined on piecewise rigid functions and on GSBD - Friedrich, Manuel; Perugini, Matteo; Solombrino, Francesco (Journal of Functional Analysis, 2021)
In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called BD-ellipticity, which is in the spirit of BV-ellipticity defined by Ambrosio and Braides [5]. By specific examples we show that this novel concept is in fact stronger compared to its BV analog. We further provide a sufficient condition implying BD-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of GSBDp functions.
CRIS-Team Scopus Importer:2021-10-21
Keywords: BD-ellipticity; GSBD; Lower semicontinuity; Piecewise rigid functions

Non-interpenetration conditions in the passage from nonlinear to linearized Griffith fracture - Almi, Stefano; Davoli, Elisa; Friedrich, Manuel (Journal De Mathematiques Pures Et Appliquees, 2023)
We characterize the passage from nonlinear to linearized Griffith-fracture theories under non-interpenetration constraints. In particular, sequences of deformations satisfying a Ciarlet-Nečas condition in SBV2 and for which a convergence of the energies is ensured, are shown to admit asymptotic representations in GSBD2 satisfying a suitable contact condition. With an explicit counterexample, we prove that this result fails if convergence of the energies does not hold. We further prove that each limiting displacement satisfying the contact condition can be approximated by an energy-convergent sequence of deformations fulfilling a Ciarlet-Nečas condition. The proof relies on a piecewise Korn-Poincaré inequality in GSBD2, on a careful blow-up analysis around jump points, as well as on a refined GSBD2-density result guaranteeing enhanced contact conditions for the approximants.
CRIS-Team Scopus Importer:2023-05-26
Keywords: Ciarlet-Nečas; Contact condition; Griffith fracture; Linearization; Non-interpenetration

Nonlinear and Linearized Models in Thermoviscoelasticity - Badal, Rufat; Friedrich, Manuel; Kružík, Martin (Archive for Rational Mechanics and Analysis, 2023)
We consider a quasistatic nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin–Voigt rheology, where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. The force balance is formulated in the reference configuration by resorting to the concept of nonsimple materials, whereas the heat transfer equation is governed by the Fourier law in the deformed configurations. Weak solutions are obtained by means of a staggered in-time discretization where the deformation and the temperature are updated alternatingly. Our result refines a recent work by Mielke and Roubíček (Arch Ration Mech Anal 238:1–45, 2020) since our approximation does not require any regularization of the viscosity term. Afterwards, we focus on the case of deformations near the identity and small temperatures, and we show by a rigorous linearization procedure that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. The same property holds for time-discrete approximations and we provide a corresponding commutativity result.
CRIS-Team Scopus Importer:2023-01-13

Nonlocal-to-local limit in linearized viscoelasticity - Friedrich, Manuel; Seitz, Manuel; Stefanelli, Ulisse (Communications in Applied and Industrial Mathematics, 2024)
We study the quasistatic evolution of a linear peridynamic Kelvin-Voigt viscoelastic material. More specifically, we consider the gradient flow of a nonlocal elastic energy with respect to a nonlocal viscous dissipation. Following an evolutionary Γ-convergence approach, we prove that the solutions of the nonlocal problem converge to the solution of the local problem, when the peridynamic horizon tends to 0, that is, in the nonlocal-to-local limit.
CRIS-Team Scopus Importer:2024-06-07
Keywords: evolutionary Γ-convergence; Kelvin-Voigt rheology; nonlocal-to-local limit; Peridynamics; viscoelasticity

Numerical approximation of von K-ARM-AN viscoelastic plates - Friedrich, Manuel; Kruzik, Martin; Valdman, Jan (Discrete and Continuous Dynamical Systems. Series S, 2021)
We consider metric gradient ows and their discretizations in time and space. We prove an abstract convergence result for time-space discretizations and identify their limits as curves of maximal slope. As an application, we consider a_nite element approximation of a quasistatic evolution for viscoelastic von K_arm_an plates [44]. Computational experiments exploiting C1_nite elements are provided, too.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Dissipative distance; Metric gradient ows; Minimizing movements; Numerical approximation; Viscoelasticity; ô-convergence

On a decomposition of regular domains into John domains with uniform constants - Friedrich, Manuel (Esaim-Control Optimisation and Calculus of Variations, 2018)
We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain Ω ⊂ ℝ2 with C1-boundary there is a corresponding partition Ω = Ω1 ∪ ⋯ ∪ ΩN with ΣNj=1 H1(∂Ωj \∂Ω) ≤ θ such that each component is a John domain with a John constant only depending on θ. The result implies that many inequalities in Sobolev spaces such as Poincaré's or Korn's inequality hold on the partition of Ω for uniform constants, which are independent of Ω.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Free discontinuity problems; John domains; Korn's inequality; Shape optimization problems

On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime - Friedrich, Manuel; Schmidt, Bernd (Networks and Heterogeneous Media, 2015)
We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of G-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Atomistic models; Brittle materials; Discrete-tocontinuum limits; Free discontinuity problems; Variational fracture

One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams - Friedrich, Manuel; Machill, Lennart (Calculus of Variations and Partial Differential Equations, 2023)
We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Γ -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Γ -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).
CRIS-Team Scopus Importer:2023-07-21

On the passage from nonlinear to linearized viscoelasticity - Friedrich, Manuel; Kruzik, Martin (SIAM Journal on Mathematical Analysis, 2018)
We formulate a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the KelvinVoigt rheology where the viscosity stress tensor complies with the principle of time-continuous frame indifference. We identify weak solutions in the nonlinear framework as limits of time-incremental problems for vanishing time increment. Moreover, we show that linearization around the identity leads to the standard system for linearized viscoelasticity and that solutions of the nonlinear system converge in a suitable sense to solutions of the linear one. The same property holds for time-discrete approximations, and we provide a corresponding commutativity result. Our main tools are the theory of gradient flows in metric spaces and σ-convergence.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Curves of maximal slope; Dissipative distance; Metric gradient flows; Minimizing movements; Viscoelasticity; σ-convergence

Quasistatic crack growth in 2d-linearized elasticity - Solombrino, Francesco; Friedrich, Manuel (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2018)
In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation (GSBD). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the GSBD setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Brittle materials; Crack propagation; Free discontinuity problems; Quasistatic evolution; Variational fracture

QUASISTATIC GROWTH OF CAVITIES AND CRACKS IN THE PLANE - Bresciani, Marco; Friedrich, Manuel (SIAM Journal on Mathematical Analysis, 2025)
We propose a model for the quasistatic growth of cavities and cracks in two-dimensional nonlinear elasticity. Cavities and cracks are modeled as discrete and compact subsets of a planar domain, respectively, and deformations are defined only outside of cracks. The model accounts for the irreversibility of both processes of cavitation and fracture, and it allows for the coalescence of cavities into cracks. Our main result shows the existence of quasistatic evolutions in the case of a finite number of cavities under an a priori bound on the number of connected components of the cracks.
CRIS-Team Scopus Importer:2025-05-30
Keywords: cavitation; fracture; nonlinear elasticity; quasistatic evolution

Ripples in Graphene: A Variational Approach - Friedrich, Manuel; Stefanelli, Ulisse (Communications in Mathematical Physics, 2020)
Suspended graphene samples are observed to be gently rippled rather than being flat. In Friedrich et al. (Z Angew Math Phys 69:70, 2018), we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of next-to-nearest neighbor interaction energies and classified their fine nonflat geometries. In this second paper on graphene nonflatness, we refine the analysis further and prove the emergence of wave patterning. Moving within the frame of Friedrich et al. (2018), rippling formation in graphene is reduced to a two-dimensional problem for one-dimensional chains. Specifically, we show that almost minimizers of the configurational energy develop waves with specific wavelength, independently of the size of the sample. This corresponds remarkably to experiments and simulations.
CRIS-Team Scopus Importer:2021-10-21

Stability of Z(2) configurations in 3D - Betermin, Laurent; Friedrich, Manuel; Stefanelli, Ulisse (Nonlinearity, 2021)
Inspired by the issue of stability of molecular structures, we investigate the strict minimality of point sets with respect to configurational energies featuring two- and three-body contributions. Our main focus is on characterizing those configurations which cannot be deformed without changing distances between first neighbours or angles formed by pairs of first neighbours. Such configurations are called angle-rigid. We tackle this question in the class of finite configurations in Z(2), seen as planar three-dimensional point sets. A sufficient condition preventing angle-rigidity is presented. This condition is also proved to be necessary when restricted to specific subclasses of configurations.
CRIS-Team WoS Importer:2021-11-26

Stochastic homogenisation for functionals defined on asymptotically piecewise rigid functions - Donnarumma, Antonio Flavio; Friedrich, Manuel (Calculus of Variations and Partial Differential Equations, 2025)
We study stochastic homogenisation of free-discontinuity surface functionals defined on piecewise rigid functions which arise in the study of fracture in brittle materials. In particular, under standard assumptions on the density, we show that there exists a Γ-limit almost surely and that it can be represented by a surface integral. In addition, the effective density can be characterised via a suitable asymptotic cell formula and is deterministic under an ergodicity assumption. We also show via Γ-convergence that the homogenised functional defined on piecewise rigid functions can be recovered from a Griffith-type model by passing to the limit of vanishing elastic deformations.
CRIS-Team Scopus Importer:2025-02-28

STRONG EXISTENCE FOR FREE DISCONTINUITY PROBLEMS IN LINEAR ELASTICITY - Friedrich, Manuel; Labourie, Camille; Stinson, Kerrek (SIAM Journal on Mathematical Analysis, 2025)
In this note we show Ahlfors-regülärity for ä lärge cläss of qüäsiminimizers of the Griffith fünctionäl. This ällows üs to prove thät, for ä ränge of free discontinüity problems in lineär elästicity with änisotropic, cohesive, or heterogeneoüs behävior, minimizers häve än essentiälly closed jümp set änd äre thüs strong minimizers. Oür notion of qüäsiminimälity is inspired by änd generälizes previoüs notions in the literätüre for the Mümford-Shäh fünctionäl änd comprises fünctions which locälly close to the cräck häve ät most ä fixed percentäge of excess cräck relätive to minimizers. As for minimizers of the Griffith fünctionäl treäted by Chämbolle, Conti, änd Iürläno, oür proof of Ahlfors-regülärity relies on conträdiction-compäctness änd än äpproximätion resült for GSBD fünctions, showing the robüstness of this äpproäch with respect to generälizätion of bülk änd sürfäce densities.
CRIS-Team Scopus Importer:2025-04-18
Keywords: Ahlfors-regülärity; free discontinüity problem; Griffith fünctionäl; lineär elästicity

The double-bubble problem on the square lattice - Friedrich, Manuel; Górny, Wojciech; Stefanelli, Ulisse (Interfaces and Free Boundaries, 2023)
We investigate the minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of minimisers, and, in some parameter regime, we compute the optimal perimeter as a function of the size of the point sets. Moreover, we provide a sharp bound on the difference between two minimisers, which are generally not unique, and use it to rigorously identify their Wulff shape as the size of the point sets scales up.
CRIS-Team Scopus Importer:2024-12-16
Keywords: Double bubble; optimal point configuration; square lattice; Wulff shape

The geometry of C60: A rigorous approach via molecular mechanics - Friedrich, Manuel; Piovano, Paolo; Stefanelli, Ulisse (SIAM Journal on Applied Mathematics, 2016)
Molecular Mechanics describes molecules as particle configurations interacting via classical potentials. These configurational energies usually consist of the sum of different phenomenological terms which are tailored to the description of specific bonding geometries. This approach is followed here to model the fullerene C60, an allotrope of carbon corresponding to a specific hollow spherical structure of sixty atoms. We rigorously address different modeling options and advance a set of minimal requirements on the configurational energy able to deliver an accurate prediction of the fine three-dimensional geometry of C60 as well as of its remarkable stability. In particular, the experimentally observed truncated-icosahedron structure with two different bond lengths is shown to be a strict local minimizer.
CRIS-Team Scopus Importer:2021-10-21
Keywords: C; Configurational energy minimization; Fullerene; Local stability

Tilings with Nonflat Squares: A Characterization - Friedrich, Manuel; Seitz, Manuel; Stefanelli, Ulisse (Milan Journal of Mathematics, 2022)
Inspired by the modelization of 2D materials systems, we characterize arrangements of identical nonflat squares in 3D. We prove that the fine geometry of such arrangements is completely characterized in terms of patterns of mutual orientations of the squares and that these patterns are periodic and one-dimensional. In contrast to the flat case, the nonflatness of the tiles gives rise to nontrivial geometries, with configurations bending, wrinkling, or even rolling up in one direction.
CRIS-Team WoS Importer:2022-04-29

Two-well linearization for solid-solid phase transitions - Davoli, Elisa; Friedrich, Manuel (Journal of the European Mathematical Society, 2025)
In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus undetected by the limiting deformations. By Γ-convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [Calc. Var. Partial Differential Equations 59, art. 44 (2020)] for a model with anisotropic second-order perturbation.
CRIS-Team Scopus Importer:2025-02-28
Keywords: linearization; sharp-interface; Solid-solid phase transitions; two-well rigidity

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions - Davoli, Elisa; Friedrich, Manuel (Calculus of Variations and Partial Differential Equations, 2020)
We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid–solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of Γ -convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases.
CRIS-Team Scopus Importer:2021-10-21

Variational Models with Eulerian–Lagrangian Formulation Allowing for Material Failure - Bresciani, Marco; Friedrich, Manuel; Mora-Corral, Carlos (Archive for Rational Mechanics and Analysis, 2025)
We investigate the existence of minimizers of variational models featuring Eulerian–Lagrangian formulations. We consider energy functionals depending on the deformation of a body, defined on its reference configuration, and an Eulerian map defined on the unknown deformed configuration in the actual space. Our existence theory moves beyond the purely elastic setting and accounts for material failure by addressing free-discontinuity problems where both deformations and Eulerian fields are allowed to jump. To do this, we build upon the work of Henao and Mora-Corral regarding the variational modeling of cavitation and fracture in nonlinear elasticity. Two main settings are considered by modeling deformations as Sobolev and SBV-maps, respectively. The regularity of Eulerian maps is specified in each of these two settings according to the geometric and topological properties of the deformed configuration. We present some applications to specific models of liquid crystals, phase transitions, and ferromagnetic elastomers. Effectiveness and limitations of the theory are illustrated by means of explicit examples.
CRIS-Team Scopus Importer:2025-02-17

Γ-convergence for Free-Discontinuity Problems in Linear Elasticity: Homogenization and Relaxation - Friedrich, Manuel; Perugini, Matteo; Solombrino, Francesco (Indiana University Mathematics Journal, 2023)
We analyze the Γ -convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to Γ -convergence and represent the Γ -limit in an integral form defined on the space of generalized special functions of bounded deformation (GSBDp). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow us to characterize relaxations of functionals on GSBDp, and cover the classical case of periodic homogenization.
CRIS-Team Scopus Importer:2024-12-16
Keywords: free-discontinuity problems; functions of bounded deformation; homogenization; relaxation; Variational fracture; Γ -convergence