Eccentric annular crack under general nonuniform internal pressure

1 Introduction

An annular crack consists of two concentric circular edges that are coplanar; such a crack is the simplest idealization of a toroidal crack, which is frequently observed at material interfaces or in homogeneous media after fabrication. For this reason, many researchers considered an annular crack inside isotropic materials under uniform far-field loading by employing the asymptotic and approximate methods used by Smetanin [1], Moss and Kobayashi [2], and Shibuya et al. [3]. Perhaps, the most significant attempt that has been made in this area belongs to Fabrikant [4], who reduced the problem of a mode I annular crack with non-axisymmteric loading over the faces of the crack to a pair of non-singular Fredholm integral equations of the second kind. Saxena and Dhaliwal [5] examined the problem of an annular crack surrounding an elastic fiber in a transversely isotropic matrix. They reduced the mixed-boundary value problem to a singular integral equation. Shindo et al. [6] considered the problem of a permeable annular crack in a piezoelectric fiber enclosed by an infinite isotropic solid subjected to mode I far-field loading. Their formulation lead to a pair of coupled singular integral equations. Recently, Shodja et al. [7] considered a concentric set of energetically consistent annular and penny-shaped cracks in an infinite, transversely isotropic piezoelectric medium under a uniform far-field electromechanical loading parallel to the poled direction and perpendicular to the crack faces. They reduced the problem to a Fredholm integral equation of the second kind.

A more realistic idealization of a toroidal crack encountered in materials is inferred by considering the angular dependence of crack size, that is the inner and outer crack edges, generally, consist of two eccentric circles. It is proposed to give an accurate analytical solution for such an eccentric annular crack within an isotropic infinite body under nonuniform (non-axisymmetric) internal pressure. The solution of the problem is expressed in the form of a set of two decoupled Fredholm integral equations of the second kind. Subsequently, by employing a numerical scheme suggested by Atkinson [8], the algorithm for the solution of these equations is provided. The solution of the eccentric annular crack under uniform mode I loading is readily obtained as a limiting case and presented in this work. Furthermore, the stress intensity factors are delineated in dimensionless plots for different values corresponding to (1) ratio of the inner to the outer radius and (2) ratio of the eccentricity to inner or outer radius.

2 Governing equations and statement of the problem

The equilibrium equations in the absence of body forces can be expressed as

where σ denotes the stress tensor. The corresponding constitutive relations for isotropic materials may be written as follows:

where u is the displacement vector; I, the identity matrix; μ, the shear modulus; and ν, the Poisson’s ratio.

Consider an eccentric annular crack with eccentricity e embodied by an isotropic matrix. Let r₁ and r₂ denote the inner and the outer radii of the crack edges, respectively. The origin of the Cartesian coordinate system O(x₁, x₂, x₃) is set in such a way that the crack lies in the x₁x₂-plane and the center of the inner circle is located at (c₁, 0, 0), where

and a is a constant depending on r₁, r₂ and e which will be introduced shortly. The faces of the crack are subjected to an arbitrary pressure, p(x₁, x₂), as elucidated in Figure 1.

Figure 1:

The geometry of the eccentric annular crack.

Suppose that the crack plane lies in ℝ², coinciding with the x₁x₂-plane. Let 𝔻₁ be the domain bounded by the inner crack edge, and 𝔻₂, the complement of the domain enclosed by the outer crack edge. It can easily be understood that the crack occupies the region ℝ²-(𝔻₁∪𝔻₂) in the x₁x₂-plane, where

Due to symmetry of the problem with respect to x₃=0, the mixed-boundary conditions may be written as

where u_i and σ_ij (i, j=1, 2, 3) denote the components of displacement and stress fields, respectively.

3 The Papkovich-Neuber solution and three-dimensional mapping

In order to solve the mixed boundary value problem stated in Section 2, the well-known Papkovich-Neuber potential function is employed:

where ψ and φ are the harmonic functions with respect to (x₁, x₂, x₃). One can conventionally take

In view of Eqs. (11), (2) and (6)–(8), it can be shown that

on x₃=0. Thus, the original problem has been reduced to a three-part mixed boundary value problem for ψ₃. The eccentricity of the annular crack has lead to the above conditions, which are difficult to work with. Therefore, a three-dimensional mapping similar to that of the two-dimensional case introduced by Muskhelishvili [9] is defined, which transforms the set of eccentric circles to a set of concentric circles, as follows:

These relations are expressible in a compact form as

where δ_ij (i, j=1, 2, 3) denotes the Kronecker delta. The details of the mapping are shown in Figure 2.

Figure 2:

Top view of the three-dimensional mapping given by Eq. (18).

As is shown in the Appendix, in the transformed space, ψ₃ is a harmonic function:

A cylindrical coordinate system (ρ, ϕ, ξ₃) is defined in the transformed space. Therefore, boundary conditions (13)–(15) are transformed to

on ξ₃=0 and 0<ϕ<2π, where ρ₁ and ρ₂ are the respective radii of the inner and outer crack-tips in the transformed space,

At this point, by the well-known method proposed by Fabrikant [4], the three-part mixed boundary value problem defined by (20)–(22) reduces to

with

where γ̅ is the complex conjugate of γ:

F₊(ρ, ϕ) and F_-(ρ, ϕ) can be interpreted through the following equations:

or

on ξ₃=0, and L is an integral operator defined as

If (r₁, θ₁) and (r₂, θ₂) denote the polar coordinates with origins at the center of the inner and outer crack edges in the untransformed space, then the stress intensity factors can be written as

where superscripts 1 and 2 pertain to the inner and outer borders of the crack, respectively. Alternatively,

where

For the sake of completeness, a brief discussion on the numerical methodology for the two-dimensional Fredholm integral equations (24) is given in the remainder of this section.

By virtue of the numerical method introduced by Atkinson [8], the governing Fredholm integral equations can be converted to a set of linear algebraic equations. The intervals [0, ρ1/ρ2] and [0, 2π] are divided into N₁ and N₂ equal segments, respectively, so that

In view of these definitions, the unknown functions F_±(λ_i, ω_j) are obtained from

where

4 Results and discussion

At first, for the sake of verification, two problems of a concentric annular crack subjected to uniform (Section 4.1) and non-axisymmetric (Section 4.2) internal pressures that are available in the literature are readily re-examined as the special cases of the current work. Subsequently, the problem of an eccentric annular crack under uniform far-field tension is addressed in Section 4.3.

4.1 Concentric annular crack under uniform tension

The numerical values of the stress intensity factors pertinent to the concentric annular crack under uniform tension are available in the Handbook by Tada et al. [10] as well as in the work of Noda et al. [11]. As it is evident from Table 1, the numerical values of the normalized stress intensity factors KI(1)/(σ0πc) and KI(2)/(σ0πc) reproduced for different ratios of c/r_c via the present theory with zero eccentricity (e=0) are in good agreement with their results; note that c=(r₂-r₁)/2 and r_c=(r₂+r₁)/2. Furthermore, the stress intensity factors for the special cases of center crack and penny-shaped crack can readily be calculated in the limits as r₁/r₂→1(c/r_c→0) and r₁/r₂→0(c/r_c→1), respectively. For the center crack, the current theory yields KI(1)/(σ0πc)=1.004 and KI(2)/(σ0πc)=1.002, which are in reasonable agreement with the exact values of 1. For the case of the penny-shaped crack, the current theory results in KI(2)/(σ0πc)=0.901 as compared to the exact value of 22/π∼0.900.

Table 1:

Comparison of the values of the inner and outer stress intensity factors for various ratios of c/r_c for the case of a concentric annular crack with those of Noda et al. [11] and Tada et al. [10].

c/rc	KI(1)/(σ0πc)			KI(2)/(σ0πc)
	Current study	Ref. [11]	Ref. [10]	Current study	Ref. [11]	Ref. [10]
0.1	1.030	1.030	1.031	0.979	0.979	0.979
0.2	1.068	1.068	1.069	0.963	0.963	0.963
0.3	1.118	1.118	1.117	0.950	0.950	0.950
0.4	1.182	1.182	1.180	0.939	0.939	0.939
0.5	1.269	1.269	1.265	0.930	0.930	0.930
0.6	1.390	1.390	1.385	0.922	0.922	0.922
0.7	1.572	1.573	1.567	0.916	0.916	0.915
0.8	1.887	1.885	1.881	0.910	0.910	0.910
0.9	2.616	2.577	2.610	0.905	0.903	0.904

Figure 3A and B compare, respectively, the variations of the normalized stress intensity factors KI(i)/(σ0πc), i=1, 2 in terms of r₁/r₂ obtained using the current theory with those from the Handbook by Tada et al. [10]. The results are in good agreement, and moreover, for the above-mentioned limiting cases of a center crack (r₁/r₂→1) and a penny-shaped crack (r₁/r₂→0), the results converge to the exact values of 1 [both KI(1)/(σ0πc) and KI(2)/(σ0πc)] and 22/π(KI(2)/(σ0πc)), respectively. It is evident that the normalized stress intensity factor KI(i)/(σ0πc) along the inner crack-tip (i=1) decreases with increasing r₁/r₂, while the trend for that of the outer crack-tip (i=2) is reversed.

Figure 3:

Comparison of the variations of the (A) inner and (B) outer stress intensity factors in terms of r₁/r₂ for the case of concentric annular crack with those of Tada et al. [10].

4.3 Eccentric annular crack under uniform far-field tension

In Sections 2 and 3, the problem of an eccentric annular crack under arbitrary pressure over the faces of the crack was formulated in terms of two decoupled Fredholm integral equations. One of the most practical cases is the case of an eccentric annular crack under uniform far-field loading which is equivalent to uniform pressure over the crack faces; therefore, the stress intensity factors corresponding to this problem are examined for various crack configurations.

On setting p(x₁, x₂)=σ₀, the right-hand side of the Fredholm integral equations (24) degenerate, so that

(45)g1(ρ, ϕ)=-iσ0a2ρ(z2z¯2)2[z1 tanh-11-ρ22/ρ2z1-z1 tanh-11-ρ12/ρ2z1], (45)

(46)g2(ρ, ϕ)=-2σ0a2ρ(z2z¯2)2Re[z2 tanh-11-ρ22/ρ2z2-z2 tanh-11-ρ12/ρ2z2]+ig1(ρ, ϕ), (46)

where

(47)z1=1-1a2ρ2, z2=1-eiϕaρ. (47)

Figure 4A–C which correspond, respectively, to r₁/r₂=0.1, 0.2, 0.4 show the variation of the normalized stress intensity factor associated with the inner crack edge, KI(1)/(σ0πr2),, with an angle θ₁; each figure displays four curves which are pertinent to e/r₂=0, 0.1, 0.2, 0.4. Similar sets of plots are provided for the outer crack edge in Figure 5A–C. It is observed that for all cases the stress intensity factors decrease with increasing r₁/r₂. For a given value of r₁/r₂ and e/r₂, the stress intensity factor for the inner crack edge is larger than that of the outer crack edge. From comparison of the figures pertinent to the outer crack edge, it is seen that a curve for a given value e/r₂≠0 crosses the line e/r₂=0 at an angle θ̅₂, which remains unchanged as r₁/r₂ varies. In other words for an annular crack with eccentricity, e/r₂≠0, there exists an angle, 0°<θ̅₂=θ̅₂(e/r₂)<180° independent of the ratio r₁/r₂ along the outer crack edge at which KI(2)/(σ0πr2) becomes equal to the stress intensity factor pertinent to the outer crack edge of the concentric annular crack for which r₁ and r₂ are kept the same as those of the eccentric annular crack but e/r₂=0.

Figure 4:

Effect of the eccentricity on the inner stress intensity factor for the cases: (A) r₁/r₂=0.1, (B) r₁/r₂=0.2, and (C) r₁/r₂=0.4.

Figure 5:

Effect of the eccentricity on the outer stress intensity factor for the cases: (A) r₁/r₂=0.1, (B) r₁/r₂=0.2, and (C) r₁/r₂=0.4.

From Figure 4A–C, it is seen that for e>0 the maximum and minimum values of the stress intensity factor associated with the inner crack edge occur at θ₁=0 and π, respectively. Figure 5A–C reveal that similar observation holds for the outer crack edge.

Figure 6 shows the variation of the normalized value of the maximum stress intensity factor, KI(1)/(σ0πr2), pertinent to the inner crack edge as a function of the normalized eccentricity, e/r₁, for different values of r₁/r₂=0.1, 0.2, 0.4. Similarly, KI(2)/(σ0πr2) versus e/r₁ pertinent to the outer crack edge for r₁/r₂=0.1, 0.2, 0.4 has been plotted in Figure 7. From Figure 6, it is observed that KI(1)/(σ0πr2) follows an increasing trend with the normalized eccentricity and reaches its peak value at about e/r₁≈1. KI(1)/(σ0πr2), then decreases with further increase in e/r₁. However, Figure 7 shows that KI(2)/(σ0πr2) increases monotonically with e/r₁; for larger values of r₁/r₂, there is a greater sensitivity to the change in the value of the normalized eccentricity, e/r₁.

Figure 6:

Variation of the maximum inner stress intensity factor in terms of eccentricity for the three cases of r₁/r₂=0.1, 0.2, and 0.4.

Figure 7:

Variation of the maximum outer stress intensity factor in terms of eccentricity for the three cases of r₁/r₂=0.1, 0.2, and 0.4.