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A directional-dependent localized mass-field damage model for anisotropic brittle fracture

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A Correction to this article was published on 01 April 2022

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Abstract

The nature of fracture in most engineering materials exhibits the inherent anisotropy. Assuming isotropic evolution of fracture in such anisotropic materials is no longer consistent. It is because that the principles and fundamental issues that behind the directional-dependent fracture mechanism are still interrogative. In this paper, the recently developed mass-field damage model for brittle fracture in isotropic materials is extended to account for anisotropic crack propagation in transversely isotropic materials. To model a unidirectional fiber-reinforced composite, the fiber orientation characterizing deformation between the reference and current configurations is thus defined in terms of a scalar fiber-stretch factor through the anisotropic strain energy density (SED). The significance of the study lies in the development of a new variant for the constitutive law for mass flux to be able to deal with directional-dependent fracture in solids. To this end, the pure forms of mass source and mass flux are thus revisited so as to possibly capture the anisotropy, i.e., a novel constitutive law for the spatial mass flux is defined by additionally introducing a second-order structural tensor, which is a function of the anisotropic coefficient and fiber directions controlling the anisotropy of directional fracture. Consequently, in contrast to the traditional local mass-balance equation for isotropic materials, the new form of the locally diffusive mass-balance equation controlling the evolution of mass density is now incorporated an anisotropic spatial mass flux function instead. In addition, we also explore the characteristics of the anisotropic length scale parameter in the context of directional-dependent fracture analysis. Another novelty is devoted to the energy decomposition for the mass-balance equation of the SED for anisotropic materials, i.e., the anisotropic part of the SED is described in terms of the first invariant of the right Cauchy–Green deformation tensor and another material parameter related to the reinforced fibers. The finite element discretization associated with a staggered solution algorithm for anisotropic brittle fracture is described. The merits of the developed anisotropic fracture model are illustrated via numerical examples under quasi-static loading condition.

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Notes

  1. Here, as the same in our previous study [26], the model \(M_I\) introduced by Tang et al. [27] is again used in this paper.

  2. Since the deleted element method is adopted in this work, the term \(\mathcal {H}\text {exp}\left[ -\left( \frac{{W^e}^+}{\Phi } \right) ^m \right] \) in the constitutive laws for mass source and mass flux represent the variations of density at element level. Another thing should be noticed that the evolution of the switch parameter in Eq. (10) can be interpreted as \(\dot{\zeta } = -H\left( \epsilon - \frac{\psi _e}{\psi _\mathrm{f}}\right) , \zeta \left( t = 0\right) = 0\), [36] with \(\psi _\mathrm{f}\) being the constant bulk failure energy and \(\psi _{e}(\varvec{F})\) describing the elastic energy at element level, see [36] for their definitions.

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Acknowledgements

Hung Thanh Tran is gratefully acknowledged the Japanese Government MEXT scholarship for his Integrated Doctoral Education Program.

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  1. Department of Civil and Environmental Engineering, Tokyo Institute of Technology, 2-12-1-W8-22, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    Hung Thanh Tran & Tinh Quoc Bui

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  1. Hung Thanh Tran
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Correspondence to Tinh Quoc Bui.

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The original online version of this article was revised: In the original article few corrections were missed and this has been corrected now.

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Tran, H.T., Bui, T.Q. A directional-dependent localized mass-field damage model for anisotropic brittle fracture. Acta Mech 233, 1317–1336 (2022). https://doi.org/10.1007/s00707-022-03147-0

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