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Virtual clustering analysis for phase field model of quasi-static brittle fracture

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Abstract

In this paper, we develop a Virtual Clustering Analysis method for a phase field model of brittle fracture. In addition to the strain/stress field, we treat the phase field variable via clusters as well, based on Green’s function of the governing Helmholtz equation. Around the crack path, we assign one cluster per cell. We detect the crack tip and recluster accordingly as the crack propagates. Three examples are presented to demonstrate the numerical efficiency of the proposed method, including either straight or curved crack, under tension or shear.

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Acknowledgements

This work is supported by National Key R &D Program of China under Grant Number 2023YFA1008901, and National Natural Science Foundation of China under Grant Numbers 11832001 and 11988102.

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Authors and Affiliations

  1. HEDPS and LTCS, College of Engineering, Peking University, Beijing, 100871, China

    Shaoqiang Tang & Jingcheng Miao

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  1. Shaoqiang Tang
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  2. Jingcheng Miao
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Correspondence to Shaoqiang Tang.

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Formulation of damaged modulus

Formulation of damaged modulus

$$\begin{aligned}{} & {} \varvec{C} {:=}\frac{\partial \varvec{\sigma }}{\partial \varvec{\varepsilon }}=(1-d)^2 \left[ 3\lambda \tilde{H}(\textrm{tr}(\varvec{\varepsilon }))\mathbb {I}^{vol} + 2\mu \mathbb {P}^+\right] \nonumber \\{} & {} \qquad +\left[ 3\lambda \tilde{H}(-\textrm{tr}(\varvec{\varepsilon }))\mathbb {I}^{vol} +2\mu \mathbb {P}^-\right] \end{aligned}$$
(A.1)
$$\begin{aligned}{} & {} \mathbb {P}^{\pm } {:=} \frac{\partial \varvec{\varepsilon }^{\pm }}{\partial \varvec{\varepsilon }} = \sum _a^3 \tilde{H}(\pm \varepsilon _a) (\varvec{M}_a)_{ij} (\varvec{M}_{a})_{kl} + \frac{1}{2} \sum _a^3 \sum _{b\ne a}^3 \theta _{ab} \nonumber \\{} & {} \qquad \left[ (\varvec{M}_a)_{ik} (\varvec{M}_b)_{jl} + (\varvec{M}_b)_{il} (\varvec{M}_b)_{jk} \right] \end{aligned}$$
(A.2)

Here \(\tilde{H}(\cdot )\) is the Heaviside function and \(\theta _{ab}\) is defined as

$$\begin{aligned} \theta _{ab}=\left\{ \begin{aligned}&\frac{\langle \varepsilon _a \rangle _{\pm } -\langle \varepsilon _b \rangle _{\pm } }{\varepsilon _a -\varepsilon _b}&, \textrm{if} \ \varepsilon _a \ne \varepsilon _b, \\&\tilde{H}(\pm \varepsilon _a)&, \textrm{if} \ \varepsilon _a =\varepsilon _b . \end{aligned} \right. \end{aligned}$$
(A.3)

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Tang, S., Miao, J. Virtual clustering analysis for phase field model of quasi-static brittle fracture. Comput Mech (2024). https://doi.org/10.1007/s00466-024-02459-3

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