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Numerical simulation of crack deflection and penetration at an interface in a bi-material under dynamic loading by time-domain boundary element method

Abstract

The hybrid time-domain boundary element method (BEM), together with the multi-region technique, is applied to simulate the dynamic process of crack deflection/ penetration at an interface in a bi-material. The whole bi-material is divided into two regions along the interface. The traditional displacement boundary integral equations (BIEs) are employed with respect to the exterior boundaries; meanwhile, the non-hypersingular traction BIEs are used with respect to the part of the crack in the matrix. Crack propagation along the interface is numerically modelled by releasing the nodes in the front of the moving crack tip and crack propagation in the matrix is modeled by adding new elements of constant length to the moving crack tip. The dynamic behaviours of the crack deflection/penetration at an interface, propagation in the matrix or along the interface and kinking out off the interface, are controlled by criteria developed from the quasi-static ones. The numerical results of the crack growth trajectory for different inclined interface and bonded strength are computed and compared with the corresponding experimental results. Agreement between numerical and experimental results implies that the present time-domain BEM can provide a simulation for the dynamic propagation and deflection of a crack in a bi-material.

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Abbreviations

Γ1(t ), Γ2(t ):

Part of the crack faces in matrix

Γin (t ):

Part of the crack faces at the interface

\({\partial B_{\rm in}^1,\ \partial B_{\rm in}^2}\) :

Interface

\({\partial B_{ex}^1,\ \partial B_{ex}^2}\) :

External boundaries

\({\partial B_{\rm t}^1,\ \partial B_{\rm t}^2}\) :

Part of the external boundaries with prescribed traction

\({\partial B_{\rm u}^1,\ \partial B_{\rm u}^2}\) :

Part of the external boundaries with prescribed displacements

\({\hat{{t}}_\alpha({\bf x},t)}\) :

Prescribed boundary traction

\({\hat{{u}}_\alpha({\bf x},t)}\) :

Prescribed boundary displacements

μ j :

Shear modulus

λ j :

Lamé constants

ν j :

Poisson’s ratio

ρ j :

Mass density

\({c_{\rm T}^j}\) :

Shear wave velocity

\({c_{\rm L}^j}\) :

Longitudinal wave velocity

\({C_{\alpha \beta \gamma \delta}^j}\) :

Component of elasticity tensor

\({\sigma_{\alpha \beta}^j}\) :

Component of stress tensor

\({u_\alpha^j}\) :

Component of displacement vector

e ɛ δ :

2-Dimensional permutation tensor

δ α β :

Kronecker delta

φ :

Kinking angle of an interface crack or penetrating angle of a matrix crack

ψ :

Phase angle or crack tip mode mixity

η :

Traction resolution factor for the interface

\({\Gamma_{\rm in}^{\rm d}}\) :

Dynamic critical energy release rate of the interface

\({\Gamma_1^{\rm d},\ \Gamma_2^{\rm d}}\) :

Dynamic critical energy release rates of the materials

\({K_{\rm ID}(\dot{a})}\) :

Dynamic fracture toughness of the materials

Ga, φ):

Dynamic energy release rate of a crack penetrating the interface by Δa at an angle of φ or of an interface crack kinking from the interface by Δa at an angle of φ

G d  = Ga, 0°):

Dynamic energy release rate of a crack deflecting to the interface by Δa

G m  = max (G(Δ, φ)):

maximum value of dynamic energy release rate of a crack penetrating the interface by Δa or of an interface crack Kinking from the interface by Δa with φ varying from 0° to 180°

\({G_{\rm in} (\dot {a})}\) :

Dynamic energy release rate for a propagating interfacial crack

\({G\left({t;\dot {a}} \right)}\) :

Dynamic energy release rate for a propagating matrix crack

\({{\bf K}(t;\dot {a})}\) :

Complex dynamic stress intensity factor (DSIF)

\({{K}_I(t;\dot {a}), {K_{II}}(t;\dot {a})}\) :

the first and second dynamic stress intensity factors (DSIFs)

δ 1, δ 2 :

Tangential and normal crack opening displacements (CODs)

Δy :

Element length of the external boundaries

Δy in :

Element length of the interface

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Correspondence to Yue-Sheng Wang.

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Lei, J., Wang, YS. & Gross, D. Numerical simulation of crack deflection and penetration at an interface in a bi-material under dynamic loading by time-domain boundary element method. Int J Fract 149, 11 (2008). https://doi.org/10.1007/s10704-008-9215-5

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Keywords

  • Dynamic fracture
  • Bi-material
  • Interface
  • Crack
  • Crack deflection and penetration
  • Fracture criterion
  • Time-domain boundary element method