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Thermo-mechanical behavior of a viscoelastic FGMs coating containing an interface crack

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Abstract

In this paper the plane thermo-mechanical behavior of a crack in a viscoelastic functionally graded materials (FGMs) coating with arbitrary material properties bonded to a homogeneous substrate is studied. In order to avoid the complex forms that describe the viscoelastic properties of FGMs, a multi-layered model for the FGMs coating is developed. The compliance and thermal conductivity in the multi-layered model linearly vary in each layer. In this mixed boundary value problem, the system is reduced to singular integral equations and solved numerically with the Lobatto-Chebyshev collocation technique. Using the correspondence principle and Laplace transform, the problem of an interface crack between a homogeneous substrate and a viscoelastic FGMs is solved. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the multi-layered model. The numerical results confirm that the fracture toughness of materials can be greatly improved by the graded variation of material parameters. It is also confirmed that the specific variation of material parameters greatly influences the fracture behavior of viscoelastic FGMs coating.

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Abbreviations

a(y, t), b(y, t):

Compliances of FGM

c :

Crack length

e ij :

Deviator components of strains

E(x, t):

Relaxation function in extension

E h :

Young’s modulus of the FGMs coating at y = h

E 0 :

Young’s modulus of the homogeneous substrate

Ei(t):

Exponential integral function

F :

Airy stress function

h :

Thickness of the FGM coating

i :

The imaginary unit

j :

Layer No.

G1(X, t), G2(X, t):

Relaxation functions

L :

Number of layers

k(y, t):

Thermal conductivity of FGM

\({K_{\rm I}^\pm, K_\Pi^\pm}\) :

Mode I and II stress intensity factor

p :

Laplace transform variable

Q 0 :

Heat flux at y = h

s ij :

Deviator components of stresses

s :

Fourier variable

T(x, t):

Temperature

t :

Time

u x , u y :

Displacement components

x, y:

Cartesian coordinates

δ ij :

Kronecker delta

δ(s):

Delta function

ν :

Possion’s ratio

α(y, t):

Thermal expansion coefficient of the FGM coating

α h :

Thermal expansion coefficient of the FGMs coating at y = h

α 0 :

Thermal expansion coefficient of the homogeneous substrate

σ xx , σ yy , σ xy :

Stress components

\({\varepsilon_{ij}}\) :

Strain tensors

β j , γ j :

The linear graded parameters in the jth layer

\({\varphi}\) :

Relaxation function related to the thermal expansion coefficient

χ :

Real constant

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Authors and Affiliations

  1. Engineering Mechanics and Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ON, M5S 3G8, Canada

    Z. Q. Cheng & S. A. Meguid

  2. School of Civil Engineering, Zhengzhou University, 450001, Zhengzhou, China

    Z. Q. Cheng

  3. School of Aerospace Engineering and Applied Mechanics, Tongji University, 200092, Shanghai, China

    Z. Zhong

Authors
  1. Z. Q. Cheng
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  2. S. A. Meguid
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  3. Z. Zhong
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Correspondence to S. A. Meguid.

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Cheng, Z.Q., Meguid, S.A. & Zhong, Z. Thermo-mechanical behavior of a viscoelastic FGMs coating containing an interface crack. Int J Fract 164, 15–29 (2010). https://doi.org/10.1007/s10704-010-9452-2

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  • DOI: https://doi.org/10.1007/s10704-010-9452-2

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