## Abstract

The two-dimensional model for a thin-walled structure in the plane strain time-harmonic case is discussed. A model of a surface crack defect subjected to oscillatory thermal loading in the middle part of the thin-walled bridges is presented. As a first part of the asymptotic model, the effective transmission condition in the vicinity of a deep surface-breaking crack in a thin-walled bridge is discussed. Then, the two-term boundary layer model determining the asymptotic behaviour of the field near the tip of the crack is constructed.

## 1. Introduction

In this paper, we study the asymptotic model of a structure containing thin ligaments weakened by deep surface-breaking cracks. A simple lattice model can be used to approximate displacements away from cracks. Introduction of boundary layers describing physical fields near the regions of the structure weakened by cracks leads to the analysis of the singular behaviour of stress near ‘vertices’ of cracks.

The boundary layers around junctions and defects within thin-walled structures in the case of anti-plane shear deformations were constructed in Zalipaev *et al*. (2007, 2008). General singular perturbation techniques and models of multi-structures are presented in Kozlov *et al*. (1999) and Maz'ya *et al*. (2001). The mathematical models of physical fields around cracks can be found in Koiter (1956), Liebowitz (1972) and Movchan & Movchan (1995).

In the present paper, we extend the results of the earlier work of Zalipaev *et al*. (2007, 2008) to the case of plain strain. Special attention is given to the evaluation of the stress intensity factors for deep surface-breaking cracks within thin ligaments of the structure.

The triangular structure is shown in figure 1*a*. The axis *x*_{3} is orthogonal to the plane of the figure. We study the plane strain deformation in a time-harmonic case of thermal load. The lattice is subjected to a thermal load *T*=*T*_{0}e^{iωt} with frequency *ω* applied to the central triangular cell. The other cells of the lattice are maintained at zero temperature. We focus our attention on the analysis of the physical fields within the thin ligament containing a deep surface-breaking crack. In particular, the cracked region can be modelled as an additional junction that allows for the discontinuity in the displacement.

The emphasis of the analysis is on the boundary layers. This provides both the effective transmission spring-like conditions near a defect and the stress intensity factor for a deep surface-breaking crack. The boundary layers are described in scaled coordinates and, to the leading order, are static. This is consistent with the quasi-static formulation of the thermoelastic problem. We note that for very high frequencies, the asymptotic approach used here would not be valid and additional studies involving dynamic boundary layers would be required. Our asymptotic results may be regarded as low frequency approximations.

The displacement amplitude inside the walls of the structure ** u**=(

*u*

_{1},

*u*

_{2})

^{T}satisfies the Lame equation(1.1)and the boundary conditions on the interior part of the boundary of the elementary cell where the displacement and temperature satisfy(1.2)together with the Dirichlet boundary condition on the exterior boundary. Here,

*γ*=

*α*(3

*λ*+3

*μ*), where

*α*is the linear thermal expansion coefficient, and

*λ*and

*μ*are the Lame constants. Away from the lattice junctions, the temperature satisfies(1.3)and the boundary condition(1.4)along the thin bridge length, where

*k*is the thermal diffusivity coefficient.

A horizontal thin bridge within the structure contains a deep surface-breaking crack *Γ*_{ϵ}, as shown in figure 1*b*; the detailed description of the geometry is given in §2*a*.

The structure of the paper can be described as follows. In §2, a lower dimensional beam approximation for a thin-walled bridge with surface-breaking crack is introduced for the plane strain problem, and the effective transmission spring-like conditions near the defect are derived. The uniform asymptotic approximation of the displacement field near the crack tip is described in §3, together with the two-term boundary layer model, which is required for the evaluation of the stress intensity factor. Our main aim is to derive a connection between the boundary layer problem involving the stress intensity factor and the lower dimensional model of a beam. This is represented at the end of the paper by a theorem that relates the stress intensity factor to a displacement jump in the lower dimensional model. On the practical side, we note that this jump is frequency dependent.

## 2. Reduction to a lower dimensional model within a thin bridge

We use a small parameter *ϵ*, which is the ratio of width to length of a thin-walled bridge. Hence, we obtain the following problem for a thin rectangle *Ω*_{ϵ}={(*x*_{1}, *x*_{2}):−*L*<*x*_{1}<*L*, 0<*x*_{2}<*ϵL*} with the crack representing a damaged thin-walled bridge (vertical surface-breaking crack; figure 2). The notations will be used for the thin ligaments on the left and right of the crackAway from the junctions of the triangular structure, the displacement satisfies the Lame equation(2.1)where *e*_{2} is the unit vector pointing direction of the axis *x*_{2}. Here, *T* is an *x*_{1}-independent solution of (1.3) and (1.4) given byThe boundary conditions are given by(2.2)(2.3)(2.4)and on the crack faces are(2.5)whereandwhere *h*_{ϵ} is the length of the intact part of the thin-walled bridge (figure 2). The longitudinal displacements ±*q* in (2.4) correspond to the displacements of the junction points within the triangular lattice.

### (a) Effective transmission condition

Away from the left and right ends *x*_{1}=±*L* of the rectangle, and from the crack *x*_{1}=0, the leading-order approximation of the displacement is given by(2.6)where −*L*<*x*_{1}<0 and 0<*x*_{1}<*L* within and , respectively, as shown in figure 2. Next, we deal with the boundary layer near the junction containing the crack.

#### (i) The first boundary layer and weight functions in a semi-infinite strip

The first boundary layer is described in the scaled coordinatesas *ϵ*→0. This transforms the domain *Ω*_{ϵ} into the union of two half-strips *Π*_{±} (*h*_{ϵ}=0) (figure 3), where

The asymptotic solution inside both half-strips *Π*_{±} is sought in the formThe boundary layer terms *w*^{±}(*ξ*_{1}, *ξ*_{2}) satisfy the Lame equation in *ξ*_{1}, *ξ*_{2} coordinates(2.7)and the following boundary conditions in the scaled coordinates:(2.8)(2.9)(2.10)whereand *C*_{ϵ}=*h*_{ϵ}/(ϵ*L*).

Consider the weight functions ^{±}(*ξ*_{1}, *ξ*_{2}) in *Π*_{±} determined bywith the following behaviour at infinity (*ξ*_{1}→±∞):(2.11)where the vector function ^{*}(*ξ*_{1}, *ξ*_{2}) vanishes exponentially as *ξ*_{1}→±∞.

Now, we employ the Betti formula for *w*^{±} and the weight function ^{±} to obtain(2.12)

Applying the Betti formula for (1, 0)^{T} and the weight function ^{±}, the second integral in (2.12) can be evaluated as follows:with the constant *γ*^{*}=4*μ*(*μ*+*λ*)/(2*μ*+*λ*). To the leading order, it yields(2.13)The integral term in (2.13) will be analysed in §2*a*(ii).

#### (ii) Derivation of the effective transmission condition

Now, we estimate the integralConsider the weight function ^{+} represented in the form (a similar construction applies to ^{−})where *Γ*_{1,2} are the constants depending on log *C*_{ϵ}. Here, *Χ*(*r*) is the infinitely differentiable cut-off function such thatand for 1<*r*<2, *Χ*(*r*) determines a smooth transition from 0 to 1. The vector function ^{(0)}(*ξ*_{1}, *ξ*_{2}) is determined by the boundary value problem(2.14)

As , the vector function ^{(0)}(*ξ*_{1}, *ξ*_{2}) satisfies the asymptotic representation(2.15)corresponding to a point force applied at the vertex of the wedge. Here, ** Φ** is the regular part, vanishing at the corner and depending only on

**/|**

*ξ***|, and**

*ξ**κ*=3−4

*ν*with

*ν*being Poisson's ratio.

The condition at infinity, as *ξ*_{1}→∞, is given by (2.11).

The function ^{(1)}(*η*_{1}, *η*_{2}) in new coordinates(2.16)is defined as a solution of the following boundary value problem in the first quadrant *Q*_{I}={(*η*_{1}, *η*_{2}): *η*_{1}>0, *η*_{2}>0}:(2.17)

(2.18)

(2.19)

(2.20)

Implicitly, log *C*_{ϵ} is present on the right-hand side of (2.17) due to the logarithmic term log *r*=log *C*_{ϵ}*ρ* in the expression for ^{(0)}(*ξ*_{1}, *ξ*_{2}) (see formula (2.15)). When *C*_{ϵ}→0, the terms containing log *C*_{ϵ} on the right-hand side of (2.17) can be removed if the constants *Γ*_{1,2} are chosen as follows:Thus, we obtain the asymptotic formulawhere *R*^{(1)}(*C*_{ϵ}) is the remainder that is bounded as *C*_{ϵ}→0.

Substitution of this approximation into (2.13) leads to the effective transmission condition.

*The leading term of the uniform asymptotic approximation for the solution* *u*_{ϵ}(*x*) *to the problem* (*2.1*)–(*2.5*) *satisfies the following inhomogeneous effective transmission condition*:(2.21)

The junction condition (2.21) represents an effective ‘spring’ whose stiffness is controlled by *ϵL*, *C*_{ϵ} and the model solution in a wedge. It also includes a ‘force term’ associated with the applied temperature.

## 3. Asymptotics near the crack tip

The construction of the uniform asymptotic solution valid near the asymptotically small region of the crack tip requires the second boundary layer problem. This model problem is to be described in the scaled coordinates (2.16) in the upper half-plane (figure 4*a*). The crack *Γ* is given by *η*_{1}=0, *η*_{2}>1. We seek the asymptotic solution valid near the crack tip in the form(3.1)where *Χ*(|** η**|) is a smooth cut-off function introduced in §2

*a*(ii).

### (a) First boundary layer: definition of *w*^{±}(*ξ*_{1}, *ξ*_{2})

*w*

In the asymptotic solution (3.1), the functions *w*^{±}(*ξ*_{1}, *ξ*_{2}) are defined in the scaled coordinates (*ξ*_{1}, *ξ*_{2}) as solutions to the problem defined in (2.7)–(2.10).

For , the vector function *w*^{±}(*ξ*_{1}, *ξ*_{2}) replicates the displacement corresponding to a concentrated force applied to a wedge corner. It is determined by the following asymptotic representation at |*ξ*|→0 (see Theotokoglou & Stampouloglou 2004):(3.2)(3.3)where *Φ*^{(0)} and *Φ*^{(1)} in the polar coordinates are given by(3.4)(3.5)The corresponding stress components *σ*_{rr}, *σ*_{rθ} are(3.6)(3.7)where and should be used when ** w**=

*w*^{±}.

### (b) Second boundary layer: definition of *v*(*η*_{1}, *η*_{2})

*v*

The function ** v**(

*η*

_{1},

*η*

_{2}) is the boundary layer term satisfying the following problem:(3.8)where

(3.9)

In the small neighbourhood of the crack tip, the displacement ** v** and its stress components in the polar coordinates

*ρ*and

*θ*with respect to the crack tip (figure 4

*a*)are given by the asymptotic formulae (see, for example, Parton & Perlin 1981)(3.10)(3.11)as

*ρ*→0. The coefficient

*K*(

*C*

_{ϵ}) is the stress intensity factor.

### (c) Definition of the weight function *Z*(*η*_{1}, *η*_{2})

*Z*

Consider the weight function ** Z**(

*η*

_{1},

*η*

_{2}), which is singular (of order

*O*(

*ρ*

^{−1/2})) at the vertex of the crack and satisfies the following problem:(3.12)

(3.13)

In the polar coordinates, the vector function ** Z**(

*η*

_{1},

*η*

_{2}) and its stress components in the small neighbourhood of the crack tip are given by (see, for example, Parton & Perlin 1981)(3.14)(3.15)as

*ρ*→0.

### (d) Evaluation of the stress intensity factor

In this section, we assume that *η*_{1}=*r* cos *θ* and *η*_{2}=*r* sin *θ*. Let us apply the Betti formula for ** v**(

*η*

_{1},

*η*

_{2}) and

**(**

*Z**η*

_{1},

*η*

_{2}) in the domain interior with respect to the contour

*γ*, whereand the contours ,

*γ*

_{δ},

*γ*

_{c}are shown in figure 4

*a*. Then passing to the limits

*R*→∞,

*δ*→0 and taking into account the asymptotic expansion of

**(**

*v**η*

_{1},

*η*

_{2}) and

**(**

*Z**η*

_{1},

*η*

_{2}) near the crack tip ((3.10), (3.11), (3.14) and (3.15)), we obtain

Thus, *K*(*C*_{ϵ}) can be obtained as(3.16)It is important to note that, owing to the presence of the cut-off function, the support of the integrand lies inside the quarter rings between the arcs and , where *R*_{1}<2 and *R*_{2}>3 (figure 4*b*). Thus, the above integral on the right-hand side must be treated inside these finite domains. Using the Betti formula for the right-hand side in (3.16), and taking into account the homogeneous traction boundary conditions on the crack faces and on the boundary of the half-plane, we can write the integral (3.16), to the leading order, in the formUsing the estimatesand passing to the limit *R*_{2}→∞, to the leading order, we obtain that(3.17)where on the right-hand side of (3.17) for *Φ*^{(0)}(*θ*, *a*, *b*) and *Φ*^{(1)}(*θ*, *a*, *b*), the following arguments must be taken:and in (3.17)are assumed. With reference to formulae (3.4) and (3.5), we note that at infinity the weight functions *Z*^{±} tend to constant vectors with the following polar components:where *ζ* is a constant. To evaluate *ζ*, we use the result of Panasuk *et al*. (1976) and Savruk (1988), where the weight function is considered as the stress intensity factor produced by a pair of point forces of the magnitude 1/2 applied at a distance *x* from the boundary of the half-plane in the normal direction to the crack faces, and this stress intensity is approximated in the formand *a* is the length of the ligament separating the vertex of the crack from the half-plane boundary. Since *ζ* is obtained via the asymptotics at infinity, we evaluate *K*_{I} in the limit as *x*→∞In particular, this illustrates the dependence on the ligament length *a*. However, for our purpose, we use the scaled coordinates *η*_{1} and *η*_{2}, and hence the ligament length equals 1, which gives the value of *ζ*

Thus, we obtain the following expression for the stress intensity factor in the auxiliary problem (in coordinates *η*_{1}, *η*_{2}) for the half-plane with the semi-infinite crack,(3.18)

Now, we formulate the results for the uniform asymptotic approximation of the solution *u*_{ϵ}(*x*).

*The leading term* *u*_{ϵ}(*x*) *of the uniform asymptotic approximation for the solution to the problems* (*2.1*)–(*2.5*) *is given by*(3.19)*where the non-smooth boundary layer term* *v**has the asymptotic representation* (*3.10*) *near the crack tip,* *with*(3.20)

We note that formula (3.20) incorporates the solution of the boundary layer type together with the effective transmission condition (2.21).

Owing to the scaling of coordinates in the boundary layer problem, the physical stress intensity factor corresponding to the displacement (3.19) is given bywhere *K*(*C*_{ϵ}) is given in (3.20) and *h*_{ϵ} is the length of the small ligament.

## 4. Conclusion

We have constructed a uniform asymptotic approximation of the displacement field in a thin elastic body with a thin ligament owing to the presence of a deep surface-breaking crack. The method of compound asymptotic expansions was used to develop the asymptotic approximation, whose particular feature is in the two-term boundary layer occurring near the vertex of the deep crack. This results in the approximate junction condition that replicates an elastic string modelling a weak junction, and furthermore the second boundary layer contains the information about the stress singularity and hence the stress intensity factor at the crack tip. The explicit analytical representations have been derived both for the stiffness of the ‘effective’ spring in the junction condition and for the stress intensity factor.

## Acknowledgments

The financial support through the grant EP/D035082/1 from the UK Engineering and Physical Science Research Council is gratefully acknowledged.

## Footnotes

- Received November 3, 2008.
- Accepted December 12, 2008.

- © 2009 The Royal Society