Phase field fracture in elasto-plastic solids: a length-scale insensitive model for quasi-brittle materials
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ORIGINAL PAPER
Phase field fracture in elasto‑plastic solids: a length‑scale insensitive model for quasi‑brittle materials Jianguang Fang1,2 · Chengqing Wu1 · Timon Rabczuk2 · Chi Wu3 · Guangyong Sun3 · Qing Li3 Received: 15 August 2019 / Accepted: 19 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Phase-field methods for fracture have been integrated with plasticity for better describing constitutive behaviours. In most of the previous phase-field models, however, the length-scale parameter must be interpreted as a material property in order to match the material strength in experiments. This study presents a phase-field model for fracture coupled with plasticity for quasi-brittle materials with emphasis on insensitivity of the length-scale parameter. The proposed model is formulated using variational principles and implemented numerically in the finite element framework. The effective yield stress is calibrated to vary with the length-scale parameter such that the tensile strength remains the same. Moreover, semi-analytical solutions are derived to demonstrate that the length-scale parameter has a negligible effect on the stress–displacement curve. Five representative examples are considered here to validate the phase-field model for fracture in quasi-brittle materials. The simulated force–displacement curves and crack paths agree well with the corresponding experimental results. Importantly, it is found that the global structural response is insensitive to the length scale though it may influence the size of the failure zone. In most cases, a large length-scale parameter can be used for saving the computational cost by allowing the use of a coarse mesh. On the other hand, a sufficiently small length-scale parameter can be selected to prevent overly diffusive damage, making it possible for the proposed phase-field model to simulate the fracture behaviour with 𝛤 -convergence. Keywords Phase-field model · Fracture · Elasto-plastic mechanics · Length-scale parameter · Quasi-brittle material
1 Introduction Computational fracture/damage mechanics has gained extensive interest in the research community for its significant values to predict material failure and guide structural design in engineering practice. Depending upon how they deal with the discontinuity, two groups of numerical approaches have been developed for simulating fracture behaviours: namely, discrete versus continuum/smeared approaches. * Jianguang Fang [email protected]; [email protected] * Chengqing Wu [email protected] 1
School of Civil and Environmental Engineering, University of Technology Sydney, Sydney, NSW 2007, Australia
2
Institute of Structural Mechanics, Bauhaus-University Weimar, 99423 Weimar, Germany
3
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia
Discrete approaches, e.g., Nguyen et al. [1], Zhou and Molinari [2] and Azevedo and Lemos [3], describe a crack in a discrete manner, in which the crack may onl
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