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A phase-field model for fracture in biological tissues

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Abstract

This work presents a recently developed phase-field model of fracture equipped with anisotropic crack driving force to model failure phenomena in soft biological tissues at finite deformations. The phase-field models present a promising and innovative approach to thermodynamically consistent modeling of fracture, applicable to both rate-dependent or rate-independent brittle and ductile failure modes. One key advantage of diffusive crack modeling lies in predicting the complex crack topologies where methods with sharp crack discontinuities are known to suffer. The starting point is the derivation of a regularized crack surface functional that \({\varGamma }\)-converges to a sharp crack topology for vanishing length-scale parameter. A constitutive balance equation of this functional governs the crack phase-field evolution in a modular format in terms of a crack driving state function. This allows flexibility to introduce alternative stress-based failure criteria, e.g., isotropic or anisotropic, whose maximum value from the deformation history drives the irreversible crack phase field. The resulting multi-field problem is solved by a robust operator split scheme that successively updates a history field, the crack phase field and finally the displacement field in a typical time step. For the representative numerical simulations, a hyperelastic anisotropic free energy, typical to incompressible soft biological tissues, is used which degrades with evolving phase field as a result of coupled constitutive setup. A quantitative comparison with experimental data is provided for verification of the proposed methodology.

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Notes

  1. Generalized Ginzburg–Landau Equation. Equation (6) was defined in Miehe et al. (2010b) as a generalized Ginzburg–Landau equation for the phase-field evolution, when it is represented

    figure a

    in terms of the variational derivative of a total energy density function \(\widehat{\psi }_{\text {tot}}\) by the phase field d. In contrast to classical phase-field equations as reviewed in Gurtin (1996), the above equation accounts for the irreversibility due to the Macauley bracket. The total energy must contain contributions due to the degrading bulk response and the regularized fracture surface energy. A simple example for finite elasticity at fracture is

    figure b

    where the parameter \(g_c\) is a critical energy release rate. For this ansatz, (7) results in the evolution Eq. (6) with the crack driving state function

    figure c

    that contains the nominal energy \(\widetilde{\psi }({\varvec{ F }})\) of the undamaged material. In this variationally consistent setting, the crack driving state function is derived from a total energy, see Miehe et al. (2014) for more advanced definitions. In contrast, when starting from (2) and using (4), the definition of the crack driving is state function \(\widetilde{D}\) is not constrained to be related to a variational derivative of an energy. This allows the incorporation of the convenient stress-based function outlined in (19) below.

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Acknowledgments

Support for this research was provided by the German Research Foundation (DFG) for the Cluster of Excellence Exc 310 Simulation Technology at the University of Stuttgart.

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  1. Institute of Applied Mechanics (CE), Chair I, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    Arun Raina & Christian Miehe

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  1. Arun Raina
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  2. Christian Miehe
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Correspondence to Christian Miehe.

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Raina, A., Miehe, C. A phase-field model for fracture in biological tissues. Biomech Model Mechanobiol 15, 479–496 (2016). https://doi.org/10.1007/s10237-015-0702-0

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