Interaction between Interfacial Collinear Griffith Cracks in Composite Media under Thermal Loading

1 Introduction

It looks that microscopic flows do not lead to safe structure to fail. Sometimes, it becomes very expensive to replace the component of engineering structures. In fact, on one hand, due to increasing demands for energy and material conservations, the safety margins assigned to structures have to be smaller. On the other hand, the detection of a flaw in a structure does not automatically mean that it is not safe to use anymore. This is particularly relevant in the case of expensive materials or components of structures whose usage it would be inconvenient to interrupt. In this setting fracture, mechanics plays a key role during the analysis of materials which exhibits cracks and also to predict whether and in which manner failure may occur.

A property of a structure relating to its ability to sustain defects until repair is called damage tolerance. During design of engineering structures, the damage tolerance is always taken in account as it is assumed that flaws can exist in any structure and such flaws propagate with usage. In aerospace engineering structures, this approach is necessary to avoid the extension of cracks. In fracture mechanics, crack growth is exponential in nature, that is, the crack growth rate is a function of an exponent of crack size according to the Paris law. The exponential crack growths led to the development of non-destructive testing methods through which the structural engineers may inspect invisible cracks occur in structures which grow slowly. So amounts of maintenance checks are reduced by nondestructive inspections. Crack propagation and arrest have become important topics in a structure containing isolated region of an unstable crack growth. So emergence of an unstable crack from bad region can still be arrested using the surrounding of good materials, provided good materials have sufficiently high fracture toughness i.e., materials have large resistance to protect the structure from crack propagation. This clearly exhibits the importance of studying propagation of cracks occur in structures and the arrest of crack propagation for the safety of the structure. The physical quantities such as stress intensity factor, crack energy, stress magnification factor (SMF) play important roles during the study of crack arrest [1–8].

Problems consist of heat and deformation that has attracted much interest to the scientists and engineers for last couple of decades. The thermal stress concentration near the crack tips has become an interesting topic of research nowadays. In the formation of structural members of airplanes, motor vehicles and high speed trains, composite materials are used widely due to their light weight and strong nature. When a cracked structural member is subject to different temperature fields, then the evaluation of stress intensity factors becomes essential due to disturbance in heat flux. The study of thermal stress around the cracked surface becomes important for the prediction of stability and service life of cracked engineering materials and structures. In linear elastic fracture mechanics, the study of geometry of collinear cracks has practical importance as pre-existing cracks lead to fracture due to interaction of cracks which forms a major crack in a medium. In 2002, Noda and Wang [9] have studied the interaction between collinear cracks situated in an inhomogeneous medium under transient loading. During thermo elastic analysis of a cracked solid, a considerable effort has been given by the researchers based on the theory of the thermo-elasticity. In 1962, Sih [10] observed the singular character of thermal stress near a crack in an infinite plate when heat flows perpendicular to the crack. A solution of a thermo-elastic crack problem had been given by Atkinson and Clement [11] in an anisotropic medium with single crack. Applying the method given by Muskhelishvili [12], Clement [13] solved the thermo-elastic crack problem bonded between dissimilar anisotropic materials. He assumed that the heat flows through the two surfaces of a crack equally but opposite in direction. In 1977, Sekine [14] calculated the thermal stresses near the crack tips of an isolated line crack in a semi-infinite medium subjected to uniform heat flow. In 1979, same author studied thermo-elastic interaction between two cracks [15]. In 1991, Itou [16] has calculated the thermal stresses around an isolated crack in an infinite elastic strip in which the surfaces of the strip are maintained at different temperature. In 1993, Itou and Rengen [17] studied the thermal stresses around two parallel cracks situated at the interface positions of two bonded dissimilar elastic half planes. In 1995, Itou and Rengen [18] have solved a problem of two collinear cracks in an adhesive layer sandwiched between two dissimilar elastic half planes. In 2001, thermal stresses in an infinite orthotropic plate around two parallel cracks under uniform heat flow were evaluated by Itou [19]. In 2007, Zhou et al. [20] have investigated transient two-dimensional thermal crack problem in a functionally graded orthotropic strip using Laplace and Fourier transform techniques. In 2007, Baksi et al. [21] have determined the thermal stresses and displacement fields in an orthotropic plane containing a pair of equal collinear Griffith cracks using integral transform technique based on displacement potential functions under steady-state temperature field. In 2013, Zhong et al. [22] have investigated the thermal stress around two collinear Griffith cracks in an orthotropic solid subjected to thermo-mechanical loading using Fourier transform technique. Recently, Itou [23] has calculated the thermal stress in an infinite orthotropic plane around two upper collinear cracks placed parallel to a lower crack. Problem related to thermal stress can also be found in the research articles [24–29]. But to the best of authors’ knowledge, the problem related to interaction between interfacial cracks under thermo-mechanical loading are not yet been done by any researcher.

The main goal of this article is to analyze the interaction among three collinear Griffith cracks situated at the interface of two orthotropic thermo-elastic half planes under uniform heat flux. To study the effect of temperature on displacements and stresses, an integral technique has been applied. The problem is reduced to a dual form of the integral equations, which is solved numerically using Jacobi polynomials. The expressions of SIFs at the tips of the cracks are found analytically. The graphical presentations of the effect of outer cracks on the propagation of central crack and also the propagation tendency of outer crack due to the presence of central one for different particular cases are the key feature of this article.

2 Problem Formulation

Consider two bonded homogeneous orthotropic elastic half planes y≥0 and y≤0 containing three collinear Griffith cracks at the interface of y=0 when Cartesian co-ordinate axes coincide with the axes of symmetry of the elastic material. When thermal conditions are applied to the surface of an arbitrary two-dimensional orthotropic half planes, then temperature field only depends on in-plane co-ordinates under steady-state condition. The temperature distribution functions T⁽ⁱ⁾ (x,y) are assumed to satisfy the following heat conduction equation in the orthotropic media as

where (K(i))2 = Ky(i)/Kx(i) and Ky(i),Kx(i) (i=1, 2) are the thermal conductive coefficients of half planes along y- and x-directions, respectively.

The general solution of T^(j) (x,y) is (c.f., Akoz and Tauchert [30])

where i = −1,j=1, 2 and A^(j)(p) and A̅^(j)(p) are the arbitrary functions of p.

Here, we have assumed that

and hence the Fourier integral form of temperature distribution may be written as

From (2) and (4), we get

From (2) and (5), the temperature distribution T⁽ⁱ⁾(x,y) can be expressed as

Consider

where h(x) is the prescribed temperature distribution, which becomes line source along y-axis, and δ(x) is Dirac delta function. Therefore, resultant temperature distribution is reduced to

The relations between plane stress induced by the distributions of temperature and displacement components u⁽ⁱ⁾ and v⁽ⁱ⁾ are given by

The elastic constants are given by C11(i) = Exx(i)1 − υxy(i)υyx(i), C22(i) = Eyy(i)1 − υxy(i)υyx(i), C12(i) = Eyy(i)υxy(i)1 − υxy(i)υyx(i) = Exx(i)υyx(i)1 − υxy(i)υyx(i), C66(i) = Gxy(i) and stress–temperature coefficients are defined by

βx(i) = C12(i)αyy(i) + C11(i)αxx(i), βy(i) = C12(i)αxx(i) + C22(i)αyy(i), where Exx(i), Eyy(i) are the Young’s moduli, Gxy(i) is shear modulus, υxy(i) and υyx(i) are Poisson’s ratios, and αxx(i) and αyy(i) are linear expansion coefficients. The quantities with superscripts i=1, 2 refer to those for the half plane-1 and half plane-2, respectively. It is to be noted that the unit of Cjk(i)’s is taken in GPa and units of βx(i) and βy(i) are considered as GPa/deg.

The displacement equations of equilibrium are given by

in which u⁽ⁱ⁾=u⁽ⁱ⁾(x, y), v⁽ⁱ⁾=v⁽ⁱ⁾(x, y) are the displacement components along x- and y-directions. It is assumed that at the interface y=0, the central crack defined by |x|<a and the outer defined by b<|x|<1 are opened by internal normal and shearing tractions p₁(x) and p₂(x)^, respectively (Figure 1). The boundary conditions on y=0 are given by

Figure 1:

Geometry of the problem.

In Figure 1, the inclined arrows represent the regions for the semi-infinite half planes, the vertical arrows denote the direction of the applied normal stress and the horizontal arrows denote the direction of shearing stress for our considered mixed mode type problem.

3 Solution of the Problem

During solution of the problem, we first introduce displacement potentials ψ⁽ⁱ⁾(x, y) and ϕj(i)(x, y) as (Sharma [31])

Potential functions for the half planes are given by

The displacement components u⁽ⁱ⁾ and v⁽ⁱ⁾ may be written as

The corresponding thermal stresses are

The displacement equations (12) and (13) are satisfied by (20) for nontrivial ϕj(i) if

Here, potential functions ϕj(i) satisfy the following differential equation:

(∂2∂x2 + ∂2∂y2)ϕj(i)(x, y) = 0, i = 1, 2; j = 1, 2,

where μ1(i) and μ2(i) are the real roots of the equation

A1(i)(s) and A2(i)(s) (i=1, 2) are the undetermined functions. Applying the boundary conditions (18) and (19), we get

A1(2)(s)  = k(2)(1 + η(2))μ1(2)2(μ2(2) − μ1(2))(1 + k1(2))(K(2) − μ2(2)K(2)) + C66(1)C66(2)k(1)(1 + η(1))2K(1)μ1(2)(μ22(2) − K(1))(1 + k1(2))(μ2(2) − μ1(2))+ A1(1)(s)μ1(2)(1 + k1(1))(μ2(2) − μ1(1))μ1(1)(1 + k1(2))(μ2(2) − μ1(2)) + A2(1)(s)μ1(2)(1 + k2(1))(μ2(2) − μ2(1))μ2(1)(1 + k1(2))(μ2(2) − μ1(2))A2(2)(s) = k(2)(1 + η(2))μ2(2)2(μ1(2) − μ2(2))(1 + k2(2))(K(2) − μ1(2)K(2)) + C66(1)C66(2)k(1)(1 + η(1))2K(1)μ2(2)(μ21(2) − K(1))(1 + k2(2))(μ1(2) − μ2(2))+ A1(1)(s)μ2(2)(1 + k1(1))(μ1(2) − μ1(1))μ1(1)(1 + k2(2))(μ1(2) − μ2(2)) + A2(1)(s)μ2(2)(1 + k2(1))(μ1(2) − μ2(1))μ2(1)(1 + k2(2))(μ1(2) − μ2(2)).

Boundary conditions (16) and (17) with the help of above-mentioned equations give rise to

∫0∞{α1 + α2A1(1)(s) + α3A2(1)(s)}sin(sx)s=0, a < x < b, 1 < x < ∞,

∫0∞{β1 + β2A1(1)(s) + β3A2(1)(s)}cos(sx)s = 0, a < x < b, 1 < x < ∞.

Now setting

α1 + α2A1(1)(s) + α3A2(1)(s) = ∫0af1(t)cos(st)dt +  ∫b1f1(t)cos(st)dt,

β1 + β2A1(1)(s) + β3A2(1)(s) = ∫0af2(t)sin(st)dt + ∫b1f2(t)sin(st)dt,

where

α1 = (k(1)−k(2)2) − (k(2)(1 + η(2))μ1(2)2(μ2(2) − μ1(2))(1 + k1(2))(K(2) − μ2(2)K(2))+ C66(1)C66(2)k(1)(1 + η(1))2K(1)μ1(2)(μ22(2) − K(1))(1 + k1(2))(μ2(2) − μ1(2)))− (k(2)(1 + η(2))μ2(2)2(μ1(2) − μ2(2))(1 + k2(2))(K(2) − μ1(2)K(2))+ C66(1)C66(2)k(1)(1 + η(1))2K(1)μ2(2)(μ21(2) − K(1))(1 + k2(2))(μ1(2) − μ2(2)))

α2 = (1 − μ1(2)(1 + k1(1))(μ2(2) − μ1(1))μ1(1)(1 + k1(2))(μ2(2) − μ1(2))− μ2(2)(1 + k1(1))(μ1(2) − μ1(1))μ1(1)(1 + k2(2))(μ1(2) − μ2(2))),

α3 = (1 − μ1(2)(1 + k2(1))(μ2(2) − μ2(1))μ2(1)(1 + k1(2))(μ2(2) − μ1(2))− μ2(2)(1 + k2(1))(μ1(2) − μ2(1))μ2(1)(1 + k2(2))(μ1(2) − μ2(2))),

β1 = (k(1)η(1)2K(1) − k(2)η(2)2K(2)),

β2 = (k1(1)μ1(1) + k1(2)μ1(2)μ1(2)(1 + k1(1))(μ2(2) − μ1(1))μ1(1)(1 + k1(2))(μ2(2) − μ1(2))+ k2(2)μ2(2)μ2(2)(1 + k1(1))(μ1(2) − μ1(1))μ1(1)(1 + k2(2))(μ1(2) − μ2(2)))

β3 = (k2(1)μ2(1) + k1(2)μ1(2)μ1(2)(1 + k2(1))(μ2(2) − μ2(1))μ2(1)(1 + k1(2))(μ2(2) − μ1(2))+ k2(2)μ2(2)μ2(2)(1 + k2(1))(μ1(2) − μ2(1))μ2(1)(1 + k2(2))(μ1(2) − μ2(2))),

and after lengthy process of mathematical manipulations, boundary conditions (14) and (15) finally lead to the following singular integral equations

where

a1 = 2πC66(1)[(1 + k1(1))(β3β3α2 − α3β2) + (1 + k2(1))(β2β2α3 − α2β3)]

1b1 = 2πC66(1)[(1 + k1(1))(α3β3α2 − α3β2) + (1 + k2(1))(α2β2α3 − α2β3)]

c1 = 2πC66(1)[(1 + k1(1))μ1(1)(α3β3α2 − α3β2) + (1 + k2(1))μ2(1)(α2β2α3 − α2β3)]

1d1 = 2πC66(1)[(1 + k1(1))μ1(1)(β3β3α2 − α3β2) + (1 + k2(1))μ2(1)(β2β2α3 − α2β3)]

c = 2πC66(1)[(1 + k1(1))μ1(1)(α3β1 − α1β3β3α2 − α3β2) + (1 + k2(1))μ2(1)(α2β1 − α1β2β2α3 − α2β3)+ (1 + η(1))2k(1)K(1)].

Equations (31) and (32) are reduced to the following singular integral equations for the determination of unknown functions f_i(x) satisfying the conditions

where

ϕk(x) = a1b1f1(x) + irkc1d1f2(x), k = 1, 2

ε = a1b1c1d1, rk = (−1)k + 1, k = 1, 2

gk(x) = 2π[b1/a1′p1(x) + rkd1/c1′p2(x)], k = 1, 2.

The solution of above integral equations (34) may be assumed as

where ωk(x) = (1 − x)αk(1 + x)βk,

αk = − 12 + iωk,   βk = − 12 − iωk,   ωk = rkω

and ω = 12πln|1 + ε1 − ε|,

with c_kn are unknown constants. Now using (33), we get ∫−11φi(t)dt = 0, and ∫b/a1/aφi(t)dt = 0,   i = 1, 2.