PSI - Issue 66

Gabriele Cricrì et al. / Procedia Structural Integrity 66 (2024) 82–86 Author name / Structural Integrity Procedia 00 (2025) 000–000

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In this article, we work out a comparison between the regularized damage formulation contributed by Valoroso and Stolz (2022) and the global energy approach presented by Cricrì (2024). The former is a gradient-based damage model that fits in the generalized standard setting and is augmented with convex constraints; here damage is characterized as an implicit function of distance within interphases that separate the fully damaged regions of a solid body from the sound material. The latter is expressed as the global stationarity of the total free energy of a linear elastic continuum; the equilibrium equations do generalize the Griffith model and the crack propagation conditions emanate from two fracture potentials that are computed via domain integrals. Common to the two approaches is an underlying variational structure, whereby the incremental solution of the problem of an elastic body is sought for as the stationary point of a potential energy, which is separately convex with respect to either displacement and crack extension or displacement and damage fields. Therefore, the governing equations can be conveniently solved by alternating minimization using a staggered Newton-type algorithm that turns out to be quite robust though not preserving the second-order convergence rate. The capabilities of the two approaches are comparatively shown with reference to an L-shaped concrete structure that is often used in the literature as a benchmark. In particular, the assessment is carried out in terms of the crack paths, the overall response curves and the ability of reproducing the whole body of experimental data. 2. Graded damage Regularized damage models allow alleviating spurious mesh sensitivity induced by strain softening; common to all such models is the idea of using extended constitutive equations in which some degree of nonlocality is introduced and a length scale parameter ℓ adds to the usual material data set. The graded damage concept put forward in Valoroso and Stolz (2022) is a generalized standard model endowed with convex constraints. The local state is described by the linearized strain measure ε , a scalar damage variable d and the free energy density function: ��� � , � � � � � ��� �� � � � � � �� � : �� � (1) where is the elasticity tensor and � � is a convex degradation function. The damage variable must comply with two internal constraints. The first one is local and expresses the classical [0,1] bounds whereas the second one is non local and amounts to bound the magnitude of the gradient of damage: � � � � � � 1 �� 0 (2) � � � � ‖∇ ‖ � � ℓ � 0 (3) In particular, the previous equation does implicitly prescribe a linear damage distribution over an interphase of size ℓ , the distance being measured from the sound material. The model is completed by specifying the local dissipation function; for rate-independency it must be a degree-one homogeneous function of the flux variable � : � � � � � � � � � � � � (4) where � � � is an hardening function and � � � � accounts for damage irreversibility. When studying the equilibrium problem of an elastic-damageable solid the field variables are the displacement vector , the damage and a pair of Lagrange multipliers � that serve to enforce the inequality constraints. Owing to non-locality, the thermodynamic potentials are functionals of the state variables. In particular, the total potential energy of the system is a Lagrangian and the dissipation potential is the integral of the local dissipation function over the physical domain Ω : ℰ� , , � � � � � , � Ω � � S �� � � � � � � � � � �� Ω � �� � (5) �� � � � � � � � Ω � (6)