Transient Behavior of Multiple Interface Cracks in Two Non-Homogeneous Half-Layers


In this paper, the analytical solution of two dissimilar orthotropic functionally graded half-layers with interface Volterra-type screw dislocation under anti-plane transient loading is studied using linear elasticity theory. The energy loss is displayed by viscous damping, and the rate of gradual alteration of elastic stiffness, damping constant and mass density of two dissimilar non-homogeneous half-layers are modeled with exponential law. In the first stage, a single dislocation, Laplace and Fourier transform are applied to obtain the stress components with simple Cauchy kernel in the interface of the layers. Then, by means of the image method, the stress fields are derived in the interface of two half-layers. The dynamic stress intensity factors (DSIFs) are calculated in time domain by using numerical Laplace inversion and the distributed dislocation technique. Eventually, various examples are reported to demonstrate the effects of the crack length, cracks location, geometrical parameters, material specifications, viscous damping and the interaction of cracks on the DSIFs.

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\(A_{1} (\zeta ,s),A_{2} (\zeta ,s),A_{3} (\zeta ,s),A_{4} (\zeta ,s)\) :

Unknown coefficients

b mz :

Burger vector

B mzj :

Dislocation density

c :

Wave slowness

g :

Ratio of shear moduli

gmzj (q,t):

Regular terms of dislocation density

h 1 :

Thickness of lower half-layer

h 2 :

Thickness of upper half-layer


Heaviside step function

K(t)Li :

Mode-III stress intensity factor of left side of crack

K(t)Ri :

Mode-III stress intensity factor of right side of crack

K 0 :

Mode-III stress intensity factor of a crack in infinite plane

\(\bar{K}_{ij} (q,p,s)\) :

Kernels of integral equations in Laplace domain

l :

Half-lengths of crack

N :

Total number of cracks

s :

Laplace variable

w :

Out-of-plane displacement component


Functions describing the geometry of cracks

\(\lambda_{1} ,\lambda_{2}\) :

FGM exponent of the half-layers

\(\mu_{x} (y),\mu_{y} (y)\) :

Shear modules in the x- and y-directions

\(\gamma (y)\) :

Viscous damping coefficient per unit volume

\(\rho (y)\) :

Mass density

\(\sigma_{zx} ,\sigma_{zy}\) :

Out-of-plane stress components

\(\zeta\) :

Fourier variable


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Bagheri, R. Transient Behavior of Multiple Interface Cracks in Two Non-Homogeneous Half-Layers. Iran J Sci Technol Trans Mech Eng 44, 619–629 (2020).

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  • Dissimilar half-layers
  • Dislocation technique
  • Image method
  • Multiple cracks
  • Transient loading
  • Dynamic stress intensity factors