The stress field around a tensile crack interacting with a circular inclusion in a thermally stressed material

Pergamon

Mechanics Research Communications,Vol. 22, No. 4, pp. 387-392, 1995 Copyright © 1995 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/95 $9.50 + .00 0093-6413(95)00040-2

THE STRESS FIELD AROUND A TENSILE C R A C K INTERACTING WITH A CIRCULAR INCLUSION IN A THERMALLY STRESSED MATERIAL

K.P. Herrmann and R. Wang Laboratorium for Technische Mechanik, UniversiOt Paderborn, 33098 Paderborn, Germany

(Received 17 November 1994, accepted for print 17 April 1995)

Introduction Crack-inclusion interactions are of importance in engineering constructions, for example consisting of fiber reinforced composites or ceramics. A close-formed solution of the stress field of a crack interacting with a circular misfitting inclusion was given by Bhargava [1]. However, the reference [ 1] contains only the stress field in the inclusion. Further, it is not clear whether the given solution satisfies the stress-free boundary conditions at the crack faces. The present paper investigates the interaction problem between a plane tensile crack and a circular inclusion with a thermal misfit at an arbitrary position of an infinite plane. The thermal misfit may be caused by a thermal expansion of the inclusion or by a steady cooling process. The close-formed solution of the stresses in the inclusion and the matrix, respectively, are obtained by combining the dislocation model of a crack with Eshelby's point-force method for inclusions and by using the complex function technique. The stress field satisfies the stressfree boundary conditions at the crack faces as well as the continuity conditions for the stresses at the interface of the inclusion and the matrix. The results of the calculations are given in the graphs at the end of the paper.

Dislocation Density Function of a Crack The geometry and coordinate system of the problem are shown in Fig. 1. An unbounded isotropic elastic solid in a state of plane strain contains a crack located at - c <_x 1(or x) < c,

x2(or y) = 0, - ~o < x 3 < oo, where x k (k = 1, 2, 3) are the rectangular coordinates. Further, there exists a cylindrical inclusion with the radius R having its cylindrical axis parallel to the x 3 - a x i s and its centre of the circular cross section at the point ~" = ~ + i I/. The inclusion tends to undergo a thermal misfit which may be caused by a thermal expansion of the inclusion or by a steady cooling process. Owing to the elastic constraints of the surrounding medium or the difference of the linear thermal expansion coefficients between the matrix and the inclusion (the mismatch of the elastic properties between the matrix and the inclusion has not been considered in this case), thermal stresses develop in the inclusion and the matrix, respectively, and an interaction occurs between the crack and the inclusion. 387

388

K.P. H E R R M A N N and R. W A N G

x z (ory) ~

g(~,n) R

z(x,y) j ,/

/

/ ~ /J~f

zf ~

!

i

X1(or x)

FIG. 1 A crack with a circular inclusion at an arbitrary position ~".

For a thermally mismatched inclusion under plane strain, the thermal misfit strain is

E l l = 5 2 2 = E',

(l)

'¢12 = 0.

with

e=(l+v)c~.~T

or

e=(l+v)A~T

(2)

where v is the Poisson's ratio, a is the linear thermal expansion coefficient and AT is the temperature raise from an initially stress-free state, or Ac~ is the difference of the linear thermal expansion coefficients between the matrix and the inclusion and AT is the temperature change from an initially stress-free state. The stress component

t722(x,O)

for a thermally

mismatched inclusion, which will act at the crack surface on the x - axis reads as follows [2] 4#eR 2 (x-~)2-rl 2 tY22(x'O)= (l+r) [(x_~)2 +rl2]2

where ~7= 3 - 4 v

(3)

and # is the shear modulus. The solution of the problem consists in the

determination of the dislocation density function for a crack which is disturbed by a thermal misfit inclusion. This function is the solution of the equilibrium equation of the forces acting on the crack surface: Aj_CcD(x') x'-x dx" --rr22(x',O),

where

D(x) is

A= 2tr(lI~- v)

the dislocation density function and

bD(x) is

(4)

the Burgers vector of the

dislocations in the unit space at point x. Further, the integral in (4) denotes a Cauchy principal value, and the solution of the integral equation (4) is well known as [3]

INTERACTION OF A TENSILE CRACK WITH AN INCLUSION

2

1

D(x)=-

#2

fc . q c ' - x " . o 2 2 ( x

lr2A cU~_x2 ,-c

#

,O)dx

•

-x---~

389

B

(5)

~ 4c 2 _ x 2

where the constant B has to be zero because of the requirement JCcD(X)dx = O,

(6)

that means, the relative displacements of the two crack faces must vanish for x = +c. By substituting equation (3) into equation (5) with B = 0 and by performing the integration gives the dislocation density function of the crack as follows [4] 2/.it~R 2 . i D(x)

= z~(I+lc)A--~[

[

~x - c 2

~c2 _¢2 (¢_ x)2

~x-c2

1

(7)

+ ~c 2 _ ~2 ( ~ _ x)2

Determination of the Stress Fields The stress field due to the crack disturbed by the misfitting inclusion can be calculated from the dislocation density function D(x) by superposing stresses of the single edge dislocation, which can be written as the following integrals:

o'iCi + io'iC2 =

JCc(crd+icrd2JD(x)dx (8)

o.f2 _ o.Cl + 2iac7 = S_. ~- (o.22d _alla+2iadl2)D(x)dx d

where o"0 is the stress field of a single edge dislocation. By substituting the equation (7) into equation (8) and by performing the integration leads to

('+'
+
1

-

'-(~.Z - C2 )(~ - -Z) + (Z - "Z)(( ~ - Z)

1

I

+ ' Z ( ~ - Z ) ( Z - "Z)(~'Z - C 2 ) / ( C

2-

~2

390

K.P. HERRMANN and R. WANG 1

- ( ¢ z - c ' ) ( ¢ - z ) + ( z - z K ( ¢ - z) 1

+'i(-~ - "i)(z - "i)(~-'i - c 2)/(c

2 - -i 2)

(~-¢)-2(z-¢) (~-g)-2(z-~)[ (~_¢), , ~ j

0"C2_0"C+2i0"C=2/ze-R2 l

2(Z-'Z)

I (9)

2(Z-~)

(l+~c)[ ( z - O 3 ( z - ~ ) 3 z___)+

-

c ' ) a c ' - z2)+ 2( z-

]]

00)

with z = x + i y . T h e actual stresses can be obtained by superposing the stresses from equations (9) and (10) and the stresses of the mismatched inclusion. Finally, the stresses in the inclusion are given by in . in C + i0-c12 0-11 +/0-12 =0"11

4/.uz (l+r)

(ll)

0-~2 --0-11 in + ^2"tO'12 , in =0-22 C _ 0-C] + 2i0-C2

and the stresses in the matrix read as follows

out . out = 0"1Cl+i0"1C2

0"11 +ta12

4/-teR2

(l+r)(~_~) 2

(12) out out +2/O'12 A. out =0"22-0"11 C C +2i0-C12 0-22-0-11

8uee 2 (l + r ) ( z - 0 2

Numerical Examples Further, the inclusion has been assumed in several different positions and the stresses are drawn in a nondimensional style. The black solid lines denote the crack and the dotted lines mean the contour of the inclusion. Figure 2 gives the stress distribution in polar coordinates with the original point at the center of the inclusion. It shows that the stress components O'rr , O'ro are continuous at the inclusionmatrix interface. The stresses in the inclusion have undergone a great change as the inclusion is above the crack (also so under the crack). The stress field in the neighborhood of a crack

I N T E R A C T I O N OF A TENSILE C R A C K WITH AN INCLUSION

Stress intensity: <-2.8

Stress intensity: <-0.94

>0.67

>0. 94

FIG. 2 Distributions of stresses O'rr and O'r0 of the crack-inclusion interaction (parameters: c=l, R=0.5, 0 < x < 2.5, -1 < y < 1.5,/.te/(l + lc)=0.3).

tip can be a pressure or a tension state dependent on the relative location of the inclusion. It may be obviously seen from the figures that the interaction phenomenon is controlled by two conditions, namely the continuity of the stresses at the inclusion-matrix interface and the stress concentration at the crack tip.

Stress intensity: <-1.17 ~

>1.37

FIG. 3 Distribution of the stresses 0"22 of the crack-inclusion interaction (parameters: c= 1, R=0.5, 0 < x < 2.5, - l < y < 1.5, ,t/e ] (1 + to)=0.3).

391

392

K.P. HERRMANN and R. WANG Figure 3 gives the distribution of the stress 0"22 in dependence on the rectangular coordinates out

x, y. It shows that the stress components 0"22 are continuous at the inclusion-matrix interface. in

The stresses in the inclusion 0-22 have undergone a great change as the inclusion is above the crack (also so under the crack). The stress field in the neighborhood of a crack tip can be a pressure or a tension state dependent on the relative location of the inclusion. This result coincides with having a positive or a negative sign of the stress intensity factor [2, 5]. It may be obviously seen from the figures that the interaction phenomenon is controlled by several factors, namely the stress-free crack faces, the continuity of the stress at the top and the bottom of the inclusion-matrix interface, and the stress concentration at the crack tip.

Summary A closed-form solution of the stress field for the interaction problem of a tensile crack with a circular misfitting inclusion has been obtained. The stress field satisfies the stress-free boundary conditions on the crack faces as well as the continuity conditions at the interface of the inclusion and the matrix. The obtained results show clearly the interaction between the crack and the inclusion. The solutions allow to analyze essential properties of the crackinclusion interaction by varying important parameters of the problem, and can be applied for a basic understanding of the strengthening and toughening mechanisms of particle reinforced materials.

Acknowledgement The financial support of the Alexander von Humboldt-Foundation for one of the authors (R. Wang) is gratefully acknowledged.

References 1. R.R. Bhargava, J. of Elasticity 7, 201(1977) 2. K.P. Herrmann and R. Wang, ZAMM, accepted for publication 3. R.W. Lardner, Mathematical Theory of Dislocations and Fracture, University of Toronto Press, Toronto (1974) 4. R. Wang, Acta Physica Sinica 39, 1908(1990) 5. K.P. Herrmann and R. Wang, Engng Fracture Mech., submitted for publication