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Static and dynamic mechanical behaviors of cracked Mindlin plates in ordinary state-based peridynamic framework

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Abstract

An ordinary state-based peridynamic (PD) model based on the Mindlin plate theory is presented to deal with fracture mechanics problems. Static and dynamic stress resultant intensity factors regarded as the primary fracture parameters for plate structures are evaluated by the displacement extrapolation method. Owing to the PD surface effect, however, the accuracy of the displacement field near crack surfaces is significantly affected. Therefore, the arbitrary horizon domain method is adopted to correct the surface effect. It derives the variable PD parameters to properly describe mechanical behaviors for each material point. Several numerical examples are investigated to examine the performance of the presented method. It indicates that the PD Mindlin plate model incorporated with the arbitrary horizon domain method validly minimizes the influence of the PD surface effect and provides an effective approach to evaluate static and dynamic moment intensity factors.

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Acknowledgements

The first author gratefully acknowledges financial support from Japan–Taiwan Exchange Association.

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Authors and Affiliations

  1. Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima, Japan

    M. J. Dai & S. Tanaka

  2. Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, UK

    S. Oterkus & E. Oterkus

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  1. M. J. Dai
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  2. S. Tanaka
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  3. S. Oterkus
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Appendix A: Examination of boundary conditions in the nonlocal theory

Appendix A: Examination of boundary conditions in the nonlocal theory

The peridynamic method regarded as a nonlocal continuum theory requires some special treatments to impose boundary conditions, such as the weak form of peridynamics [42], the modified fictitious node method [43], and the variable horizon method [44]. In the present paper, volumetric regions (nonlocal boundary) are employed to implement external load and constraint boundary conditions (see Chap. 2 in Ref. [31]). For external load boundary conditions, the external load computed as body force density is applied on a real material layer along the boundary possessing nonzero volume. For constraint boundary conditions, the prescribed constraint is adopted on a fictitious layer with the particular length scale, \(\delta \), outside the physical boundary. Several numerical examples with constraint boundary conditions are successfully examined in Sect. 4. In “Appendix”, a numerical example with external load and constraint boundary conditions is investigated.

Fig. 18
figure 18

Cantilever square plate with a central crack under dynamic transverse shear load

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Fig. 19
figure 19

Fracture behaviors of the cantilever square plate, a displacement \(u_z\) and rotation \(\theta _y\) nearby the crack tip, b normalized moment intensity factor for different PD approaches

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A cantilever square plate with a central crack under dynamic transverse shear load is considered. The length L, width W, and thickness h of the square plate are 2.0, 2.0, and 0.1 m, respectively. The half crack length a is set to 0.1 m. The square plate is subjected to a transverse shear force of \(QH(t)=1.0\) MPa on the right edge with clamped BCs imposed on the left edge (see Fig. 18). Young’s modulus E, Poisson’s ratio \(\nu \), and density \(\rho \) are set to 200 GPa, 0.3, and 7850 kg/\(\mathrm {m}^3\), respectively.

As illustrated in Fig. 19a, b, the displacement nearby the crack tip and dynamic \(F_1\) results are in good agreement between the PD and FEM solutions. It demonstrates that the proposed nonlocal method can well simulate fracture behaviors under different types of boundary conditions.

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Dai, M.J., Tanaka, S., Oterkus, S. et al. Static and dynamic mechanical behaviors of cracked Mindlin plates in ordinary state-based peridynamic framework. Acta Mech 233, 299–316 (2022). https://doi.org/10.1007/s00707-021-03127-w

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