Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions

Abstract

a b s t r a c tWe present a family of phase field models for fracture in piezoelectric and ferroelectric materials. These models couple a variational formulation of brittle fracture with, respectively, (1) the linear theory of piezoelectricity, and (2) a Ginzburg Landau model of the ferroelectric microstructure to address the full complexity of the fracture phenomenon in these materials. In these models, both the cracks and the ferroelectric domain walls are represented in a diffuse way by phase fields. The main challenge addressed here is encoding various electromechanical crack models (introduced as crack face boundary conditions in sharp models) into the phase field framework. The proposed models are verified through comparisons with the corresponding sharp crack models. We also perform two dimensional finite element simulations to demonstrate the effect of the different crack face conditions, the electromechanical loading and the media filling the crack gap on the crack propagation and the microstructure evolution. Salient features of the results are compared with experiments.

Phase-Field Modeling of Fracture in Ferroelectric Materials

“…They are also employed in a wide range of applications in applied science and engineering such as fracture [28,40,83], growth of thin films [94] and grain structures [57], image segmentation [11], vesicle bio-membranes [88,97,126] and multi-phase flows [48], to mention a few. This paper also presents another application of phase-field models for microstructure formation and fracture evolution in ferroelectric materials [1,2,3,4,5,6].…”

“…We have introduced briefly two coupled phasefield models in Section 3 considering some of these conditions. In this section, we present a general framework in the context of phase-field models, encompassing all the usual crack-face boundary conditions proposed in the literature for electromechanical materials [5]. The phase-field model of brittle fracture introduced briefly in Section 2.4 is viewed as a regularization of Griffith's sharp-crack model.…”

“…In the context of sharp-crack models in electromechanical materials, the most common crack-face boundary conditions in the literature can be classified as follows [5]: A. Uncoupled electrical/mechanical crack-face conditions Mechanical boundary conditions: These are mainly:…”

“…Recently, phase-field models [69,82,130] and a strong discontinuity approach [96] have been proposed to simulate propagating cracks in linear piezoelectric solids, thus not accounting for the effect of ferroelectric domain microstructures. Towards more realistic approaches, we have proposed a family of coupled phase-field models for both the brittle crack propagation and the microstructure evolution by tackling the full complexity of the fracture phenomenon in ferroelectric materials [1,2,3,4,5,6]. This paper extends the current state of the art in fracture of ferroelectric ceramics by reviewing these models and their developments which are outlined as follows:…”

Finite element simulation of pressure-loaded phase-field fractures

“…Over the last decade phase-field models for fracture [1][2][3][4]-which are closely related to traditional gradient-enhanced damage models [5]-have been successfully applied to a wide range of problems, such as dynamic fracturing [6][7][8][9], large deformation fracturing [10,11], fracturing of electromechanical materials [12,13], cohesive fracturing [14,15], fracturing of thermo-elastic solids [16,17] and many more. The primary advantage of these models is the flexibility with which complex fractures can be simulated, which is a result of the diffuse fracture representation and due to the fact that propagation laws generally follow naturally from energy minimization principles.…”