Phase-field fracture models for elastomers

The primary objective of this work is to examine, present, and suggest phase-field damage models that are suitable for simulating the fracture behavior of soft materials, with a specific focus on rubbers. To achieve this, it is essential to comprehend the various ways in which fracture occurs in rubbers through different case studies, such as simple tension tests or tests with pre-etched specimens subjected to quasi-static loading conditions. Along with the specimen shape and testing mode, it is crucial to understand how the material’s viscosity influences the rate and mode of fracture and the propagation of fracture. Next, this study will concentrate on three specific phenomena, aiming to replicate them numerically using the proposed models. To illustrate brittle fracture phenomena, the experiments detailed in Hocine et al. [2002], investigating the behavior of pre-incised specimens, will be replicated. The investigation into cohesive fracture phenomena will draw from the findings in Millereau et al. [2018], which explored double network elastomers and their pseudo ductility. Finally, the examination of dynamic fracture behavior will reference experiments from Corre et al. [2020], emphasizing the pivotal role of fracture propagation. After analyzing the experimental data, it will be necessary to introduce finite elastic and viscoelastic models for soft materials, like rubber, hydrogels, biological tissues and others that are able to undergo significant deformations (even up to 700%) under low stresses. Additionally, phase-field models of damage will be incorporated. The existing literature on finite elasticity models will be explored, including well-known models summarized in Dal et al. [2021], as well as less known models such as the one proposed by Lopez-Pamies [2010], which will also be utilized in the proposed models. The utilization of these models will contribute to formulating a Maxwell rheological scheme, as outlined in Reese and Govindjee [1998], Holzapfel [1996], and Kumar and Lopez-Pamies [2016], in order to depict viscoelastic characteristics. Subsequently, a damage state variable will be introduced to create a viscoelastic phase field model. In these phase-field models, the material’s characteristics, affecting its elastic parameters, are represented by fields of state variables. In the case of damage models, the state variable considered is damage, which progressively degrades the material parameters until failure. A literature review of damage models, including the works of Pham et al. [2011], Lancioni and Corinaldesi [2018], Wu [2017], and others, will also be conducted. Following the introduction and literature review, an elastic rate independent model capable of reproducing both brittle (Hocine et al. [2002]) and pseudo-ductile (Millereau et al. [2018]) behaviors, by incorporating a cohesive damage model into the Ogden’s (Ogden [2003]) finite elastic model will be proposed. By this way, an interpretation of failure phenomena at large damage displacements will be proposed. This model will be thoroughly explained from theoretical and numerical perspectives, elucidating the role of different constitutive parameters. In this scenario, three parameters within the degra- dation function (representing the impact of damage on the material’s elastic moduli) will hold significant importance. These parameters will respectively define the fracture’s onset, the progression during the initial post-elastic phase, and the material’s behavior at the point of complete failure. Numerical simulations will be performed to replicate the actual behavior observed in experimental tests. The model’s potential and limitations, particularly the absence of viscosity, will be analyzed and discussed. To address the influence of viscosity on fracture propagation and failure modes, a second rate-dependent formulation will be proposed, utilizing a generalized Maxwellrheological model similar to the works of Reese and Govindjee [1998] and Kumar and Lopez-Pamies [2016]. In this theory, a damage model representing brittle fracture will be considered. In this formulation, a parameter will be introduced to assess the energy dissipation resulting from the rate of damage and, thus, associated with crack propagation, as Loew et al. [2019], Levitas et al. [2010], Hakim and Karma [2009]. This parameter will serve as a characteristic time of the damage variable and it plays a crucial role in reproducing experiments of Corre et al. [2020]. This damage characteristic time will interact with the material characteristic time, which is the ratio between the viscous parameter and the stiffness. The complete model, along with the interpretation of constituent parameters, will be presented. Furthermore, considering the extreme scenarios of infinite and infinitesimal material characteristic times, pertaining to the damage characteristic time, as well as finite viscosity, will be investigated to understand their effects. The rate dependence will enable the study of loading rate influence on rupture and crack propagation in soft materials. The model will be properly calibrate to reproduce the behavior of Corre’s material, to replicate crack propagation findings. Finite element modeling will be employed to examine the material’s state around the fracture tip in terms of energies involved, dissipations, and stresses, observing the local response to crack formation. Subsequently, these two models will be analyzed to highlight their strengths and weaknesses regarding their ability to reproduce experimental results and their numerical complexities. The conclusions will summarize the potential of the presented models, the applicability of phase-field damage models to finite viscoelasticity, and the limitations of the proposed approaches. Future developments will be suggested, considering the limitations discussed and taking into account existing models in the literature.