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The effect of local inertia around the crack-tip in dynamic fracture of soft materials

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Abstract

Phase-field or gradient-damage approaches offer elegant ways to model cracks. Material stiffness decreases in the cracked region with the evolution of the phase-field or damage variable. This variable and, consequently, the decreased stiffness are spatially diffused, which essentially means the loss of the internal links and the bearing capacity of the material in a finite region. Considering the loss of material stiffness without the loss of inertial mass seems to be an incomplete idea when dynamic fracture is considered. Loss of the inertial mass in the damaged material region may have significant effect on the dynamic failure processes. In the present work, dynamic fracture is analyzed using a theory, which takes into account the local loss of both material stiffness and inertia. Numerical formulation for brittle fracture at large deformations is based on the Cosserat point method, which allows suppressing the hourglass type deformation modes in simulations. Based on the developed algorithms, the effect of the material inertia around a crack tip is studied. Two different problems with single and multiple cracks are considered. Results suggest that in dynamic fracture the localized loss of mass plays an important role at the crack tip. It is found, particularly, that the loss of inertia leads to lower stresses at the crack tip and, because of that, to narrower cracks as compared to the case in which no inertia loss is considered. It is also found that the regularized problem formulation provides global convergence in energy under the mesh refinement. At the same time, the local crack pattern might still depend on the geometry of the unstructured mesh.

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Notes

  1. It should be mentioned here that a higher value of ρcr is used in case II, compared to that in case I. This is because the current value of density is used for the calculation of the stable time increment in case II. Using 𝜖 = 0.001 results in extremely small time steps for elements close to the cracked region, unnecessarily slowing down the simulations. For a few cases, simulations have been performed for similar values of 𝜖 in cases I and II, to ensure that the differences in results between both cases are not only due to different values of ρcr. For S2 results for different values of 𝜖 are shown in Fig. 17 in Appendix 2.

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Acknowledgments

The authors thank Prof. Mahmood Jabareen for introducing them to the CPE formulation and the valuable discussion on it.

Funding

The authors acknowledge the support from the Israel Science Foundation, grant No. ISF-198/15.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Bhilai, Raipur, CG, 492015, India

    Anshul Faye

  2. Faculty of Civil and Environmental Engineering, Technion, Haifa, Israel

    Yoav Lev & K. Y. Volokh

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Appendices

Appendix 1. Results for coarse mesh

Table 4 Details of the meshes S4 and S5
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Fig. 15
figure 15

Coarse meshes for single crack problem. Thick red lines indicate the initial cracks

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Appendix 2. Effect of critical density on crack propagation

Using similar values of 𝜖 in cases I and II, crack propagation for mesh S2 is compared in Fig. 17. For case I, higher value of 𝜖 results in a slightly thicker crack, whereas, for case II, it changes the crack growth pattern.

Fig. 16
figure 16

Mode-I crack problem—case I: contours of ρ/ρ0 for S4 and S5 showing crack growth after the interaction of top and bottom cracks. Initial crack is shown by a thick black line

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Fig. 17
figure 17

Effect of critical density on crack growth pattern in mesh S2 for cases I and II. Initial crack is shown by a thick black line

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Faye, A., Lev, Y. & Volokh, K.Y. The effect of local inertia around the crack-tip in dynamic fracture of soft materials. Mech Soft Mater 1, 4 (2019). https://doi.org/10.1007/s42558-019-0004-2

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