We construct a new operator splitting scheme to describe a fluid-driven brittle fracture propagation in a Biot medium based on a time-scale separation assumption. In this context, we propose an alternate hybrid time-stepping scheme, where in the injection step, prior to fracture advance, we explore the framework of the fixed stress split scheme with a hydrodynamic subsystem solved ahead of the geomechanics for a frozen total mean stress. Conversely, after convergence of the fixed stress split iterations, when the pore pressure exceeds a critical value, the coupling between poromechanics and fracture propagation is accomplished by considering a fast time scale, with frozen pore-pressure and Darcy velocity fields. Such a latter step is performed in a separate iteration loop with the elasticity subsystem incorporating the Francfort Marigo variational model for a thin damage region. In this setting, the evolution of the damaged zone is governed by the sensitivity of the associated shape functional with respect to the nucleation of a small damaged zone, which is computed within the framework of the topological derivative method. The resulting approach is algorithmically described in detail. A numerical assessment of the model is constructed by performing a series of benchmark examples, showing different features of the proposed approach, such as characterization of fracture-activation pressure, crack path forecast, and the ability to capture kinking and bifurcations and quantifying the effects of the in situ stress field on the crack path.

Purpose The purpose of this paper is to experimentally validate the crack growth control based on the topological derivative of the famous Rice's integral. Design/methodology/approach Single edge notch tensile specimens with two configurations were tested. Displacement fields near notch were experimentally obtained using the digital image correlation method. These displacements were used to verify the minimization of the associated shape functional, which is defined in terms of the Rice's integral, when a set of controls (holes) positioned according to the topological derivative information, is inserted. Based on the Griffth's energy criterion, this minimization represents an improvement in the fracture toughness of cracked bodies. Findings The experimental tests confirmed that a decrease around 27% in the value of the associated shape functional can be obtained following this approach. Therefore, the results allow us to conclude that the predictive methodology for crack growth control based on the topological derivative is feasible. Originality/value This is the first work concerning experimental validation of crack growth control based on the topological derivative method.

PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.Design/methodology/approachThe Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.FindingsNumerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.Originality/valueThis is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.

The process of fluid-driven crack propagation in permeable rocks is investigated by a simple hydro-mechanical model and the concept of topological derivative. Analytical and numerical computations are made to propose a promising model tested and validated on a series of bidimensional benchmark examples. The main guidelines for this model are the use of simple finite elements with a minimal number of user-defined algorithmic parameters.

In fracture mechanics, an important question concerns the useful life of mechanical components. Such components are, usually, submitted to the actions of external forces and/or degrading agents which can trigger the crack nucleation and propagation process. In particular, when a mechanical component is already partially cracked, the question is how to extend its remaining useful life. In this work, a simple and efficient methodology aiming to extend the remaining useful life of cracked elastic bodies is proposed. More precisely, we want to find a way to retard or even avoid the triggering of the crack propagation process by nucleating hard and/or soft inclusions far from the crack tip. The main idea consists in minimize a shape functional based on the Rice’s integral with respect to the nucleation of inclusions by using the concept of topological derivative. The obtained sensitivity, which corroborates with the famous Eshelby theorem, is used to indicate the regions where the controls have to be inserted. According to the Griffith’s energy criterion, this simple procedure allows for increasing the remaining useful life of the cracked body. Finally, some numerical experiments are presented showing the applicability of the proposed methodology.

This paper deals with a simplified hydraulic fracture model based on the concept of topological derivatives. It means that we consider a two dimensional idealization in which the rock is assumed to be impermeable, while the fracturing process is activated by a given pressure acting within the existing geological faults. The basic idea consists in adapting the Francfort–Marigo damage model to the context of hydraulic fracture. The Francfort–Marigo damage model is a variational approach to describe the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damaged regions under an irreversibility constraint. In our model problem, the loading comes out from a pressurized damaged region embedded into the rock, which is used to trigger the hydraulic fracturing process. In particular, a shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipation energy term is minimized with respect to a set of ball-shaped pressurized inclusions by using the topological derivative concept. Thus, the topological asymptotic expansion of the shape functional with respect to the nucleation of a circular inclusion endowed with non-homogeneous transmission condition on its boundary is obtained. The associated topological derivative, which corroborates with the famous Eshelby theorem, is used to devise a simple topology optimization algorithm specifically designed to simulate the whole nucleation and propagation process of hydraulic fracturing. To assess our model, some numerical examples are presented, showing typical features of hydraulic fracture phenomenon, including the characterization of the fault-activation pressure and specific crack path growth, allowing for kinking and bifurcations.

This paper deals with a novel hydraulic fracture model based on the concept of topological derivative. The basic idea consists in adapting the Francfort-Marigo damage model to the context of hydraulic fracture. The Francfort-Marigo damage model is a variational approach to describe the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damage regions under an irreversibility constraint. In our problem, the loading comes out from a prescribed pressure acting within the damage region, which is used to trigger the hydraulic fracturing process. A shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipation energy term is minimized with respect to a set of ball-shaped pressurized inclusions by using the topological derivative concept. In particular, the topological asymptotic expansion of the shape functional with respect to the nucleation of a circular inclusion endowed with non-homogeneous transmission condition on its boundary is rigorously developed. The associated topological derivative, which corroborates with the famous Eshelby theorem, is used to devise a simple topology optimization algorithm specifically designed to simulate the whole nucleation and propagation process of hydraulic fracturing. To assess our model, some numerical examples are presented, showing typical features of hydraulic fracture phenomenon, including the characterization of the fault-activation pressure and specific crack path growth, allowing for kinking and bifurcations.

A análise de sensibilidade topológica fornece uma função escalar, chamada derivada topológica, que mede a sensibilidade de um dado funcional de forma em relação a uma perturbação singular infinitesimal no domı́nio, tal como a inserção de furos, inclusões, termos fonte ou até mesmo trincas. Este conceito tem se mostrado extremamente relevante no tratamento de uma ampla gama de problemas da fı́sica e da engenharia. Neste trabalho, é apresentada a derivada topológica no contexto de fraturamento hidráulico tridimensional. Inicialmente, será introduzido um modelo de evolução de dano submetido a pressão hidrostática. Tal modelo é obtido, basicamente, incorporando o conceito de dano pressurizado ao modelo de Francfort-Marigo. Na sequência, é apresentada, de fato, a derivada topológica associada ao novo modelo. Este resultado é o termo principal da expansão assintótica topológica da energia potencial total associada a um problema de elasticidade linear tridimensional considerando como perturbação topológica a nucleação de uma inclusão esférica com condição de transmissão não homogênea. Objetiva-se, futuramente, a partir dos resultados aqui obtidos, construir um algoritmo de nucleação e propagação de dano submetido à pressão hidrostática em três dimensões.

In this paper the topological derivative concept is applied in the context of compliance topology optimization of structures subject to design-dependent hydrostatic pressure loading under volume constraint. The topological derivative represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations, such as holes, inclusions, source-terms and cracks. In particular, the topological asymptotic expansion of the total potential energy associated with plane stress or plane strain linear elasticity, taking into account the nucleation of a circular inclusion with non-homogeneous transmission condition on its boundary, is rigorously developed. Physically, there is a hydrostatic pressure acting on the interface of the topological perturbation, allowing to naturally deal with loading-dependent structural topology optimization. The obtained result is used in a topology optimization algorithm based on the associated topological derivative together with a level-set domain representation method. Finally, some numerical examples are presented, showing the influence of the hydrostatic pressure on the topology of the structure.

The Griffith-Francfort-Marigo damage model describes the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damage regions. It is based on the minimization of a shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipated energy, with respect to the distribution of the healthy and damaged phases, under an irreversibility constraint. A natural approach to deal with such a minimization problem consists in considering the topological derivative concept to nucleate small damaged regions and the shape gradient to propagate them. In contrast to such an approach, in this paper the Griffith-Francfort-Marigo damage model is revisited by using the sole tool of topological derivative. In particular, we propose a striking simple numerical scheme based on the computation of the topological derivative field to determine damage nucleation as well as crack/damage propagation. In other words, the topological derivative is used as descent direction to minimize the Francfort-Marigo functional indicating, in each iteration, the regions that have to be damaged. Therefore, the proposed topology optimization algorithm is able to capture the whole nucleation and propagation damaging process, including important features like kinking and bifurcations. These properties are confirmed through several numerical experiments and by comparison with available laboratory experiments. (c) 2017 Elsevier Ltd. All rights reserved.