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A phase-field model for anisotropic brittle fracturing of piezoelectric ceramics

  • Computational Mechanics
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Abstract

Piezoelectric ceramics are inherently brittle materials that show electromechanical coupling under electrical and mechanical stimuli. In the last decades piezoelectric materials have garnered significant attention due to their established applications as sensors and actuators. In this context the structural reliability of such materials under varied conditions is paramount. In the present work we propose a phase-field approach to model crack propagation in a coupled electromechanical setting. The proposed framework accounts for anisotropic crack propagation by employing appropriate structural tensors that enter the crack-surface-density function as additional arguments. Appropriate choices of degradation functions allow for the accommodation of varied electrical boundary conditions along cracks. Based on experimental results available in the literature, we employ a non-associative dissipative framework in which fracturing processes are driven by the mechanical part of the coupled electromechanical driving force alone. The modeling capabilities of the proposed framework are demonstrated by a set of numerical examples in two and three spatial dimensions.

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Notes

  1. In the present context we do not account for electrostatic Maxwell-stress contributions (Landis 2004; McMeeking et al. 2007; Ricoeur and Kuna 2009) in the crack gap as we consider the crack-gap opening to be small and the crack propagation to be instantaneous. Please refer to Schneider (2007) for a more detailed discussion of this issue.

  2. Damage accumulation occurs when \(\tau = 0\). In that case it follows from (14) that \(f_{em} = c \alpha \). Given (15), the initial condition \(\alpha (t = 0) = 0\) and \(c > 0\), it directly follows that \(f_{em} \ge 0\).

  3. Please note that the vectors \({\varvec{p}}\) and \({\varvec{a}}\) do not coincide in general. While \({\varvec{p}}\) represents the anisotropy of the bulk properties, the vector \({\varvec{a}}\) represents the anisotropy associated with the fracture toughness.

  4. The individual moduli tensors will be specified in Sect. 4.

  5. Note that in the numerical implementation we add a small constant to the degradation functions to prevent ill-posedness of the problem when the material is fully ruptured.

  6. The action of the proper orthogonal tensor \({\varvec{Q}}\in {\mathcal {S}}{\mathcal {O}}(3)\) on vectors \({{\varvec{v}}}\) and tensors \({\varvec{T}}\) of order \(r > 1\) is denoted by the Rayleigh product

    figure a

  7. We recall that alternative approaches to the modeling of anisotropic fracturing are available. For example, Li et al. (2015) have formulated a high-order anisotropic phase-field model of brittle fracturing by including cubic anisotropy in the interfacial free energy. For that, the authors provide an extended Cahn–Hilliard interface model in the form of a tensorial Taylor expansion. The model is numerically implemented into a meshless method, thus allowing for higher-order continuity than standard finite-element discretizations. For further investigations on the modeling of anisotropic fracturing we also again highlight the works cited in Sect. 1.

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Acknowledgements

Funding was provided by German Research Foundation (Grant no. KE 1849/2-2).

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  1. Department of Civil and Environmental Engineering, Institute of Applied Mechanics, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    A. Sridhar & M.-A. Keip

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Sridhar, A., Keip, MA. A phase-field model for anisotropic brittle fracturing of piezoelectric ceramics. Int J Fract 220, 221–242 (2019). https://doi.org/10.1007/s10704-019-00391-9

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