# Matteo NegriUniversity of Pavia | UNIPV · Department of Mathematics "F. Casorati"

Matteo Negri

Associate Professor

## About

51

Publications

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Introduction

**Skills and Expertise**

Additional affiliations

November 2014 - present

October 2002 - November 2014

October 2001 - September 2002

## Publications

Publications (51)

The evolution of brittle fracture in a material can be conveniently investigated by means of the phase-field technique introducing a smooth crack density functional. Following Borden et al. (2014), two distinct types of phase-field functional are considered: (i) a second-order model and (ii) a fourth-order one. The latter approach involves the bi-L...

In the present work, a computationally efficient and explicit algorithm for the rigorous enforcement of the irreversibility constraint in the phase-field modeling of brittle fracture is presented. The proposed approach is staggered and relies on the alternate minimization of the total energy functional. The phase-field evolution turns out to be gov...

We consider a periodic, linear elastic laminate with a brittle crack, evolving along a prescribed path according to Griffith's criterion. We study the homogenized limit of this evolution, as the size of the layers vanishes. The limit evolution is governed again by Griffith's criterion, in terms of the energy release (of the homogenized elastic ener...

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and then we prove existence (employing time-discrete schemes with different implementations of the constraint), uniqueness, power and energy identity, comp...

We consider the quasi-static evolution of a brittle layer on a stiff substrate; adhesion between layers is assumed to be elastic. Employing a phase-field approach we obtain the quasi-static evolution as the limit of time-discrete evolutions computed by an alternate minimization scheme. We study the limit evolution, providing a qualitative discussio...

We study the dynamics of visco-elastic materials coupled by a common cohesive interface (or, equivalently, {two single domains separated by} a prescribed cohesive crack) in the anti-plane setting. We consider a general class of traction-separation laws featuring an activation threshold on the normal stress, softening and elastic unloading. In stron...

We consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. After...

We consider high order phase field functionals introduced in Borden et al. (2014) and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy. We take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in Sobolev spaces and in spaces of tensor product B...

We consider the quasi-static evolution of a brittle layer on a stiff substrate; adhesion between layers is assumed to be elastic. Employing a phase-field approach we obtain the quasi-static evolution as the limit of time-discrete evolutions computed by an alternate minimization scheme. We study the limit evolution, providing a qualitative discussio...

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraint in time and natural regularity assumptions. We provide first a notion of weak solution, inspired by the theory of curves of maximal slope, and then existence (employing time-discrete schemes with different "implementations" of the constraint), uniquenes...

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraint in time and natural regularity assumptions. We provide first a notion of weak solution, inspired by the theory of curves of maximal slope, and then existence (employing time-discrete schemes with different "implementations" of the constraint), uniquenes...

We consider second order phase field functionals, in the continuum setting, and their discretization with isogeometric tensor product B-splines. We prove that these functionals, continuum and discrete, $\Gamma$-converge to a brittle fracture energy, defined in the space $GSBD^2$. In particular, in the isogeometric setting, since the projection oper...

We consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. After...

We consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic L 2 {L^{2...

We study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the minimization of the phase-field variable; the advantage of this choice, from a computational stand point, is in the efficiency of the numerical implementa...

In this paper we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can de...

We consider the quasi-static evolution of a prescribed cohesive interface: dissipative under loading and elastic under unloading. We provide existence in terms of parametrized BV -evolutions, employing a discrete scheme based on local minimization, with respect to the H¹-norm, of a regularized energy. Technically, the evolution is fully characteriz...

We deal with a model for an elastic material with a cohesive crack along a prescribed fracture set. We consider two n-dimensional elastic bodies and a cohesive law, on their common interface, with incompenetrability constraint and general loading–unloading regimes. We first provide a time-discrete evolution by means of local minimizers of the energ...

We consider time-discrete evolutions for a phase-field model (for fracture and damage) obtained by alternate minimization schemes. First, we characterize their time-continuous limit in terms of parametrized BV-evolutions, introducing a suitable family of "intrinsic energy norms". Further, we show that the limit evolution satisfies Griffith's criter...

We consider configurational variations of a homogeneous (anisotropic) linear elastic material \(\Omega \subset \mathbb {R}^n\) with a crack K. First, we provide a simple way to compute configurational variations of energy by means of a volume integral. Then, under increasing information on the regularity of the displacement field we show how to obt...

In this paper we will describe how gradient flows, in a suitable norm, are natural and helpful to generate quasi-static evolutions in brittle fracture. First, we will consider the case of a brittle crack running along a straight line according to Griffith’s law. Then, we will see how the same approach leads to quasi-static evolutions in the phase f...

We consider crack propagation in brittle nonlinear elastic materials in the context of quasi-static evolutions of energetic type. Given a sequence of self-similar domains nΩ on which the imposed boundary conditions scale according to Bažant's law, we show, in agreement with several experimental data, that the corresponding sequence of evolutions co...

We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals F of class C-1 in reflexive separable Banach spaces. We provide a couple of constructiv...

In the setting of finite elasticity we study the asymptotic behaviour of a crack that propagates quasi-statically in a brittle material. With a natural scaling of size and boundary conditions we prove that for large domains the evolution with finite elasticity converges to the evolution with linearized elasticity. In the proof the crucial step is t...

We consider the quasi-static evolution of a straight crack within the recently developed phase-field approach and the classical sharp crack approach, and we show a strong correlation between the outcomes from the two approaches: the corresponding energies, minimizers, energy release rates and quasi-static evolutions converge as the internal length...

The paper concerns the control of rigid inclusion shapes in elastic bodies with cracks. Cracks are located on the boundary of rigid inclusions and in the bulk. Inequality type boundary conditions are imposed at the crack faces to guarantee mutual non-penetration. Inclusion shapes are considered as control functions. First we provide the problem for...

For planar mixed mode crack propagation in brittle materials many similar criteria have been proposed. In this work the Principle of Local Symmetry together with Griffith Criterion will be the governing equations for the evolution. The Stress Intensity Factors, a crucial ingredient in the theory, will be employed in a ‘non-local’ (regularized) fash...

We propose a model for a 2D elastic body with a thin elastic inclusion in which delamination of the inclusion may take place, thus forming a crack. Non-linear boundary conditions at the crack faces are imposed to prevent mutual penetration. We prove existence and uniqueness of the equilibrium configuration, considering both the variational and the...

The classical approach to the simulation of cracks in deformable solids is characterized by linear boundary conditions on the crack edges, which frequently results in physical contradictions. For describing the exfoliation, boundary conditions in the form of inequalities excluding the mutual penetration of crack edges are used. Within the framework...

We prove that a suitable rescaling of biased Perona–Malik energies, defined in the discrete setting, Γ-converges to an anisotropic version of the Mumford–Shah functional. Numerical results are discussed.

We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier by A. J. Lew and G. C. Buscaglia [Int. J. Numer. Methods Eng. 76, No. 4, 427–454 (2008; Zbl 1195.76258)]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that...

Free discontinuity problems arising in the variational theory for fracture mechanics are considered. A Γ -convergence proof for an r-adaptive 3D finite element discretization is given in the case of a brittle material. The optimal displacement field, crack pattern and mesh geometry are obtained through a variational procedure that encompasses both...

Considering anti-plane elasticity we provide an existence result for the energy release rate along a piecewise C1, 1 path that admits a kink. We provide two representations: an asymptotic one in terms of the stress intensity factor and an integral one in terms of the Eshelby tensor. Both the formulas make use of an implicit coefficient, depending o...

We consider a brittle material under displacement control with a mode III crack running in quasi-static regime along a straight line. Several definitions of propagation, based on variational criteria, have been proposed in the last decade. Our aim is a detailed study of their properties, in particular as far as energy, energy release rate and jump...

On the base of many experimental results, e.g., Ravi-Chandar and Knauss (Int. J. Fract. 26:65–80, 1984), Sharon et al. (Phys. Rev. Lett. 76(12):2117–2120, 1996), Hauch and Marder (Int. J. Fract. 90:133–151, 1998), the object of our analysis is a rate-dependent model for the propagation of a crack in brittle materials. Restricting ourselves
to the q...

Prompted by recent works on line tension effects upon the stability of liquid droplets laid on a rigid substrate, we prove
existence and stability of equilibrium configurations when the substrate is either planar or spherical. For positive line
tension our argument involves only set symmetrization, while for negative line tension a stability criter...

Abstract We consider the propagation of a crack in a brittle material along a prescribed crack path and define a quasi-static evolution by means of stationary points of the free energy. We show that this evolution satisfies Griffith’s criterion in a suitable form which takes into account both stable and unstable propagation, as well as an energy ba...

Our analysis focuses on the mechanical energies involved in the propagation of fractures: the elastic energy, stored in the bulk, and the fracture energy, concentrated in the crack. We consider a finite element model based on a smeared crack approach: the fracture is approximated geometrically by a stripe of elements and mechanically by a softening...

Usually, smeared crack techniques are based on the following features: the fracture is represented by means of a band of finite elements and by a softening constitutive law of damage type. Often, these methods are implemented with nonlocal operators that control the localization effects and reduce the mesh bias. We consider a nonlocal smeared crack...

Besides efficient techniques allowing for the finite-element modeling of propagating displacement discontinuities, the numerical simulation of fracture processes in quasibrittle materials requires the definition of criteria for crack initiation and propagation. Among several alternatives proposed in the literature, the possibility to characterize e...

We consider a class of functionals which are defined in the spaces SBV and SBD and which do not depend on the traces u
± on the set of discontinuity points. In this work we prove that it is possible to approximate these energies, in the sense of Γ-convergence, by means of a family of non-local functionals defined in Sobolev spaces. Moreover we illu...

Cohesive-crack models, pioneered by Barenblatt [1], are commonly used for the simulation of fracture in quasi-brittle materials. In this area, among various other directions of recent research, we mention here: (a) development of efficient finite-element formulations such as extended finite elements [2], (b) coupling between continuum damage modell...

In this work, the problem of inception and growth of a cohesive crack in an elastic bar is considered. The position where the crack actually forms is obtained from the minimality conditions of an energy functional which includes the surface energy. The progressive damage of the cohesive interface is taken into account by means of a step by step pro...

The Griffith model for the mechanics of fractures in brittle materials is consider in the weak formulation of SBD spaces. We suggest an approximation, in the sense of –convergence, by a sequence of discrete functionals defined on finite elements spaces over structured and adaptive triangulations. The quasi-static evolution for boundary value proble...

Linearized elastic energies are derived from rescaled nonlinear energies by means of Γ-convergence. For Dirichlet and mixed boundary value problems in a Lipschitz domain Ω, the convergence of minimizers takes place in the weak topology of H
1(Ω,R
n
) and in the strong topology of W
1,q
(Ω,R
n
) for 1≤q<2.

Linearized elastic energies are derived from rescaled nonlinear energies by means of Gamma-convergence. For Dirichlet and mixed boundary value problems in a Lipschitz domain Omega, the convergence of minimizers takes place in the weak topology of H-1(Omega,R-n) and in the strong topology of W-1,q(Omega,R-n) for 1less than or equal toq<2.

We consider a discrete approximation of the Mumford–Shah functional defined on finite element spaces. For a relevant model
case we study global and local minima; from their behavior, we are led to propose two minimization methods, based on the quasi-Newton
algorithm, which can find the absolute minimum of the functional.

We compute explicitly the anisotropy effect in the H 1 term, generated in the approximation of the Mumford-Shah functional by finite element spaces defined on structured triangulations. 1 Introduction As a mathematical model for image segmentation, Mumford and Shah proposed the variational problem min u;K n a Z Omega nK jruj 2 dx + bH 1 (K) + Z Ome...