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Enhanced resistance of mode II fracture by nonlocal interactions in 2D locally resonant elastic wave metamaterials

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Abstract

As a significant extension of classical structures, elastic wave metamaterials are widely applied to the vibration isolation and turning wave propagation. However, little attention has been paid on their fracture and arrest properties. In this work, the dispersion curves and arrest property of mode II crack in 2D elastic wave metamaterials are studied, in which both nonlocal interactions and local resonators are considered. Because every unit cell is connected to the second nearest neighboring one with the massless spring along the x-direction, the nonlocal interaction is achieved. The dynamic effective mass is derived and influences of nonlocal interactions on the dispersion relation are analyzed. Moreover, the energy release ratio which characterizes the arrest ability is derived and the effects of structural parameters are discussed. The theoretical predictions are also compared to the finite element simulation. Numerical results show that the crack propagation resistance of mode II crack can be improved significantly by introducing proper nonlocal springs within a finite steady-state region. But in the oscillation region, the energy release ratio G0/G is unstable. Furthermore, an additional energy barrier can be generated before the region in which the crack propagates stably.

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Abbreviations

m :

node number along the x-direction

n :

node number along the y-direction

k i :

stiffness of spring i

M 1 :

main resonator mass

M 2 :

local resonator mass

a :

lattice constant

u j,m,n :

displacement of the particle j at (m, n)

A j :

wave amplitude.

v :

crack propagation speed

c :

shear wave speed

ω 0 :

local resonance frequency

ω :

wave frequency

q :

wave number

M st :

static mass

M eff :

dynamic effective mass

w m,n :

displacement of the dynamic effective mass

t :

time variable

η :

continuous variable

Q(η):

compressed component between n = 0 and n = − 1

H(η):

Heaviside unit step function

L(k):

kernel function

Q(0):

limit elongation

p(+0):

limit force

G 0 :

local energy release rate

G :

global energy release rate

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Acknowledgements

The authors wish to express gratitude for the support provided by the National Natural Science Foundation of China (Grant Nos. 12021002 and 11991031).

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Authors and Affiliations

  1. Department of Mechanics, Tianjin University, Tianjin, 300350, China

    Xuan Zhang & Yi-Ze Wang

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  1. Xuan Zhang
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  2. Yi-Ze Wang
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XZ wrote the main manuscript text and Y-ZW designed and supervised the research. Both authors reviewed the manuscript.

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Correspondence to Yi-Ze Wang.

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Zhang, X., Wang, YZ. Enhanced resistance of mode II fracture by nonlocal interactions in 2D locally resonant elastic wave metamaterials. Int J Fract 242, 1–22 (2023). https://doi.org/10.1007/s10704-023-00699-7

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