A Theory of Fracture with a Polygonal Shape Crack

T. Mura    Department of Civil Engineering, Northwestern University, Evanston, IL 60208, USA

THE EQUIVALENT INCLUSION METHOD

An applied stress σij0 is distributed by σij due to the existence of the inhomogeneity Ω. This stress disturbance σij can be equivalent to the eigenstress σij in Ω. when proper eigenstrain ∈ ij* is chosen, which is determined from the following expression (see Fig. 2)

ij0+σij=Cijkl*∈kl0+∈kl=Cijkl∈kl0+∈kl−∈kl*

where

kl=Sklmn∈mn*

In Fig. 2, when Cijkl* = 0, the inhomogeneity Ω is a void or a crack.

ELASTIC FIELD OF A POLYGONALSTAR SHAPED INCLUSION

The displacement ui(x) in a inclusions Ω subject to uniform eigenstrain ∈. * ij(x ') is given in Mura [1987]

ix=Cjlmn∈mn*∫ΩGij,lx′−xdx′=Cilmn∈mn*∫ΩGij,lx−x′dx′

where Gij (x′ – x) is Green’s function for isotropic materials,

ilmn∂∂xiGijx¯=18π1−ν1−2νΔmjx¯n+Δjnx¯m−Δmnx¯jr3+3x¯nx¯mx¯jr5

Where ¯j=xj′−x¯j,r:=|x′−x|, and ¯i=rℓi where ℓ is the unit vector on ∑ as shown in Fig. 3 The locations of points x and x′ are also illustrated in Fig. 3.

When x is located inside the inclusion the integral (3) can be explicitly performed. The volume element dx′ in (3) can be written as (see Fig. 3):

x′=drdS=r2drdw

where dw is a surface element of a unit sphere ∑ centered at point x (Eshelby [1957]). Since multiplying r2 to the Eq. (4) is independent of r, integrating of (3) with respect to r, we have

ix=∈jk*8π1−ν∫∑rℓgijkℓdw

where

ijkℓ=1−2νΔijℓk+Δikℓj−Δjkℓi+3ℓiℓjℓk

and the vector ℓ is defined as

:=x′−xx′−x

An arbitrary point x′ on an m-pointed polygonal inclusion can be expressed by the following polar coordinate expression

1′=r′ϕcosϕsinθ

2′=r′ϕsinϕsinθ

3′=ccosθ

′ϕ=r0′+r1′2+4r1′−r0′π2∑n=1∞sinmπ2n2sinmnϕ

where 0 ≤ ϕ ≤ 2π,0 ≤ θ < π, n are odd integers, and r′(ϕ) = ((x′1)2+(x′2)2) 1/2 at θ = π/2. The periphery of the two-dimensional inclusion projected into x′1–x′2 plane (See Figure 3-5). r′ can be expressed by a periodic function of ϕ which is measured from the coordinate axis O′X′1. The coordinate system is so taken, such that X′1 axis passes through the point of r′0. As a periodic function of ϕ, r′ is plotted in Fig. 6. The period of an m-pointed polygonal star is 2π/m as shown in Fig. 5, in which a special case of m = 5 is plotted. Also, r′ is an even function of ϕ as shown in Fig. 6, where the shortest radius r′ is denoted by r′ 0 and the longest radius r′ by r′1. r′(ϕ) in (9)-(12) can be expressed as the Fourier series as shown follows,

′ϕ=a02+∑n=1∞ancosnπϕL+bnsinnπϕL

where

0=1L∫−LLr′ϕdϕ

n=1L∫−LLr′ϕcosnπϕLdϕ

n=1L∫−LLr′ϕsinnπϕLdϕ

where 2 L is the period of the periodic function r′(ϕ) in Fig. 6, and

:=πm

radius variation r′(ϕ) is a piecewise linear function in the interval of one period, which is

′ϕ=r0′−r1′−r0′Lϕ∀−L≤ϕ<0r0′+r1′−r0′Lϕ∀0≤ϕ

Then, one can show that

0=mπ∫−L0r0′−r1′−r0′Lϕdϕ+mπ∫0Lr0′+r1′−r0′Lϕdϕ=r0′+r1′

n=mπ∫−L0r0′−r1′−r0′Lϕcosnmϕdϕ+mπ∫0Lr0′+r1′−r0′Lϕcosnmϕdϕ=−4r1′−r0′n2π2

and

n=mπ∫−L0r0′−r1′−r0′Lϕsinnmϕdϕ+mπ∫0Lr0′+r1′−r0′Lϕsinnmϕdϕ=0

consequently

′ϕ=r0′+r1′2−4r1′−r0′π2∑n=1∞1n2cosnmϕ

or

′ϕ=r0′+r1′2−4r1′−r0′π2∑n=1∞1n2cosnmϕ−π2n+π2n=r0′+r1′2−4r1′−r0′π2.∑n=1∞1n2cosnmϕ−π2ncosmπ2−sinnmϕ−π2nsinmπ2

If we denote −π2n again as ϕ, by considering the fact that cos(mπ/2) = 0

because m is odd, Eq. (23) can be written...