Interface crack between different orthotropic media under uniform heat flow


As shown in Fig. 1, the problem under consideration consists of a functionally graded orthotropic strip
(FGOS) of thickness h bonded to two homogeneous semi-infinite orthotropic media with a partially insulated
interface crack of length 2c along the x-axis is considered. The subscript j(j = 1, 2, 3) indicates the FGOS and two semi-infinite orthotropic media respectively.
The remaining thermo-mechanical properties depend on the y-coordinate only and are modeled by an exponential function

(1)

(2)

where are the thermal conductivities for the homogeneous orthotropic substrate II, and ? is an arbitrary nonzero constant.

Fig. 1. Geometry of the layered orthotropic media under steady-state heat flows

The temperature satisfies

(3)

Substituting Eqs. (1) and (2) into the Eq. (3), the heat equation can be given by

(4)

(5)

where .

The heat flux components are written as

(6)

We define the following dimensionless quantities

(7)

(8)

where ?0
and E0
are the typical values of the coefficient of linear thermal expansion and the Young’s
modulus of elasticity for the homogeneous orthotropic substrate, respectively. But
for simplicity, in what follows, the bar appearing with the dimensionless quantities
is omitted.

The Duhamel–Neumann constitutive equations for the plane thermo-elastic problem are
given by Nowinski (1978])

(9)

in which

(10)

The elastic stiffness coefficients and the coefficients of the linear thermal expansion
in dimensionless form are modeled to take the following forms

(11)

where superscripts 1, 2 refer to the FGOS and the homogeneous orthotropic substrate
II, respectively, ? and ? are graded parameters. The properties of material 3 can be found in Eq. (11) when y is taken as h. In Eq. (11), elastic stiffness coefficients in dimensionless form can be represented by the
Young’s moduli and the Poisson’s ratios as

(12)

where ? ij
are the Poisson’s ratios and assumed to be constant. and are Young’s moduli for the homogeneous orthotropic substrate II, respectively.

Substituting Eq. (9) into the equations of equilibrium for the forces reduces these equations to the
forms

(13)

(14)

(15)