Being one of the most promising candidates for the modeling of localized failure in solids, so far the phase-field method has been applied only to brittle fracture with very few exceptions. In this work, a unified phase-field theory for the mechanics of damage and quasi-brittle failure is proposed within the framework of thermodynamics. Specifically, the crack phase-field and its gradient are introduced to regularize the sharp crack topology in a purely geometric context. The energy dissipation functional due to crack evolution and the stored energy functional of the bulk are characterized by a crack geometric function of polynomial type and an energetic degradation function of rational type, respectively. Standard arguments of thermodynamics then yield the macroscopic balance equation coupled with an extra evolution law of gradient type for the crack phase-field, governed by the aforesaid constitutive functions. The classical phase-field models for brittle fracture are recovered as particular examples. More importantly, the constitutive functions optimal for quasi-brittle failure are determined such that the proposed phase-field theory converges to a cohesive zone model for a vanishing length scale. Those general softening laws frequently adopted for quasi-brittle failure, e.g., linear, exponential, hyperbolic and Cornelissen et al. (1986) ones, etc., can be reproduced or fit with high precision. Except for the internal length scale, all the other model parameters can be determined from standard material properties (i.e., Young’s modulus, failure strength, fracture energy and the target softening law). Some representative numerical examples are presented for the validation. It is found that both the internal length scale and the mesh size have little influences on the overall global responses, so long as the former can be well resolved by sufficiently fine mesh. In particular, for the benchmark tests of concrete the numerical results of load versus displacement curve and crack paths both agree well with the experimental data, showing validity of the proposed phase-field theory for the modeling of damage and quasi-brittle failure in solids.

A unified phase-field theory for the mechanics of damage and quasi-brittle failure

Weakly Differentiable Functions

Progress in nonlinear differential equations and their applications, Vol. 80

Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Spatial tessellations. Concepts and Applications of Voronoi diagrams

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書評 L.Ambrosio, N.Fusco and D.Pallara: Functions of Bounded Variation and Free Discontinuity Problems

We study the asymptotic behavior, as the mesh size $\varepsilon$ tends to zero, of a general class of discrete energies defined on functions $u:\alpha\in\varepsilon\mathbb Z^N\cap\ \Omega\mapsto u(\alpha)\in{{\mathbb R}^d}$ of the form \begin{eqnarray*} F_{\varepsilon}(u)=\sum\limits_{ \begin{array}{ll} {\scriptstyle \alpha, \beta \in \varepsilon\mathbb Z^N} \\ \scriptstyle \symbol{91}\alpha ,\beta \symbol{93} \subset\Omega \end{array} }\hskip-0.3cm g_{\varepsilon}(\alpha,\beta,u(\alpha)-u(\beta)) \end{eqnarray*} and satisfying superlinear growth conditions. We show that all the possible varia\-tional limits are defined on $W^{1,p}(\Omega;{{\mathbb R}^d})$ of the local type $$ \int_\Omega f(x,\nabla u)\, dx. $$ We show that, in general, f may be a quasi-convex nonconvex function even if very simple interactions are considered. We also treat the case of homogenization, giving a general asymptotic formula that can be simplified in many situations (e.g., in the case of nearest neighbor interactions or under ...

A General Integral Representation Result for Continuum Limits of Discrete Energies with Superlinear Growth

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg–Landau framework.

/pdf/metastability-and-dynamics-of-discrete-topological-4oa8ae6dl1.pdf

Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

We provide a variational description of nearest-neighbours and
next-to-nearest neighbours binary lattice systems By studying the
$\Gamma$-limit of proper scaling of the energies of the
systems, we highlight phase and anti-phase boundary phenomena and
show how they depend on the geometry of the lattice

/pdf/phase-and-anti-phase-boundaries-in-binary-discrete-systems-a-26iwof7aun.pdf

Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint

This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under polynomial growth assumptions, we prove that the energy functionals ${F_\varepsilon}$ stored in the deformation of an ${{\varepsilon}}$-scaling of a stochastic lattice Γ-converge to a continuous energy functional when ${{\varepsilon}}$ goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic. We also generalize, to systems and nonlinear settings, well-known results on stochastic homogenization of discrete elliptic equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance, isotropy and natural states.

/pdf/integral-representation-results-for-energies-defined-on-3yng114u6y.pdf

Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity

https://cvgmt.sns.it/media/doc/paper/1681/RenEneDisSys10.pdf

Roberto Alicandro

Papers

A General Integral Representation Result for Continuum Limits of Discrete Energies with Superlinear Growth

Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint

Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity

Ginzburg-Landau functionals and renormalized energy: A revised Γ-convergence approach