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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For given damage and Lagrange multipliers, the displacement at equilibrium is the minimizer of the potential energy (5) whereas the evolution of the damage field at given displacement is governed by a normality rule: �� � � 0; � � 0; � �� � � � � 0 (7) For the problem at hand, the energy release rate is a variational derivative because of nonlocality originating from (3). For damage evolution � � � 0 � one obtains a differential problem plus a boundary condition: � � � � � �� � div � � ∇ ��� � , ∀ ; � ‖ ∇ ∇ � �‖ � � 0, ∀ (8) being the outward unit normal. Since the strains and damage gradients are allowed to be discontinuous, the displacement and the damage fields are discretized using standard C 0 finite element interpolations in the respective weak forms. As for Lagrange multipliers, they are evaluated at quadrature points of the elements, whereby the total number of degrees of freedom of the structural problem are not affected by the multipliers themselves. 3. Global energy approach The total potential energy obtained as the sum of the bulk elastic energy, the virtual work of applied forces and the surface energy dissipated in the fracture process of a linear elastic continuum is the only required ingredient of the Griffith-like energy approach presented by Cricrì (2024). Here the structural problem is considered as an ordered sequence of static equilibrium configurations where the displacement field , the load parameter and the crack configuration are the state variables. Owing to the path-dependent character of the resulting boundary value problem, the latter is recast in incremental form with the additional condition that the magnitude of crack increments should be small enough to preserve the solution accuracy. The necessary condition for the equilibrium at a given load level � � � is the stationarity of the total potential energy that is achieved by alternating minimizations with respect to incremental displacements Δ and crack increments , which are in turn represented in polar coordinates as � � , � . The crack propagation step amounts to update the crack configuration � based on two fracture potentials. In particular, the fracture extension potential � and the kink potential � are defined as follows. The potential � is the (opposite of) mechanical energy increment due to a unit increment of the crack magnitude in the tangent direction, whereas the kink potential � is the (opposite of) mechanical energy increment due to a unit rotation of the crack with respect to the tangent direction. Under the homogeneity and isotropy assumptions of the fracture energy � one obtains the following equilibrium conditions for the unknowns and ∆ : � � 1 � � � � � � � � � � , � � � � � � � � , � � � � 0 (9) The fracture potentials are the domain integrals � ��� � : ∇ �� ; � � ��� � : ∇ �� � (10) where � and � are test vector functions and is the Eshelby energy-momentum tensor: � � � �� � ∇ (11) which in turn depends only on the Cauchy stress and the elastic energy � that are solution of the elastic problem at fixed crack configuration � . A very efficient way to solve the nonlinear system (9) amounts to prescribe the crack increment magnitude ∆ and keep as unknowns the kink angle and the incremental load parameter ∆ . In this way,