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Edge crack in an elastic layer resting on Winkler foundation
Engineering Fracture Mechanics 70 (2003) 2353–2361 www.elsevier.com/locate/engfracmech
Edge crack in an elastic layer resting on Winkler foundation S.J. Matysiak a
a,*
, V.J. Pauk
b
Faculty of Geology, Institute of Hydrogeology and Engineering Geology, University of Warsaw, Al. Z_ wirki i Wigury 93, Warsaw 02-089, Poland b Faculty of Civil and Environment Engineering, Kielce University of Technology, Al. 1000-lecia Pa nstwa Polskiego 7, Kielce 25-314, Poland Received 8 April 2002; received in revised form 18 December 2002; accepted 8 January 2003
Abstract The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Elastic layer; Crack; Stress intensity factor; Winkler foundation
1. Introduction The problems of stress distributions in an elastic layer (stratum) weakened by cracks are of significant interest in geology and geophysics as well as in engineering constructions. The elastic strata are frequently situated on substrates which can be treated as Winkler foundation. The knowledge of stresses permits to forecast fracture processes of the bodies. This paper is a continuation of our studies [1] concerning the problem of a ponderable elastic layer fixed to a rigid foundation and weakened by an edge crack normal to the boundary plane. The plane problem of stress distribution in an elastic layer resting on Winkler foundation is considered. The layer contains a Griffith edge crack normal to the lower boundary plane and is loaded by normal forces acting on the upper boundary surface of the body. The considerations are restricted to the case of symmetric distribution of boundary loads with respect to the extension of crack line. The problem is solved by using the Fourier transform method, dual integral equations [2], and Fredholm integral equations of the second kind. The stress intensity factor (SIF) is expressed in terms of the solution of the derived Fredholm integral equation.
*
Corresponding author. Tel.: +48-22-554-00-11; fax: +48-22-554-00-01. E-mail address: [email protected] (S.J. Matysiak).
0013-7944/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-7944(03)00007-9
2354
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
The numerical results for the SIF are presented for the concentrated loading acting with a constant intensity on the upper boundary. The considered crack problem is of interest in geophysical and rock mechanics as well as plate–subsoil interactions. The analysis of stresses in layers with crack have received wide attention (see, for example, monographs [3–6]).
2. Formulation of problem Consider the plane static problem of elastic layer resting on a Winkler foundation. The layer is weakened by an edge crack normal to the boundary (see Fig. 1). Let the problem be related to a Cartesian coordinate system ðx; yÞ such that the lower boundary of layer is situated along the axis 0x and the crack is placed along the axis 0y (Fig. 1). Let h denote the thickness of layer and a be the length of crack. Let k, l be the Lame constants of elastic layer and k be the Winkler stiffness of foundation. The upper plane of layer is loaded symmetrically with respect to the axis 0y. Let ðuðx; yÞ; vðx; yÞÞ denote the displacement vector and rxx ðx; yÞ, ryy ðx; yÞ, rxy ðx; yÞ denote the components of stress tensor at the point ðx; yÞ. The considered problem is described by equations of the plane theory of elasticity and the following boundary conditions: rxy ðx; 0Þ ¼ 0;
x 2 R;
ð2:1Þ
rxy ðx; hÞ ¼ 0;
x 2 R;
ð2:2Þ
ryy ðx; 0Þ ¼ kvðx; 0Þ
x 2 R;
ð2:3Þ
ryy ðx; hÞ ¼ pðxÞ;
x 2 R;
ð2:4Þ
rxx ð0; yÞ ¼ 0; uð0; yÞ ¼ 0;
y 2 ð0; aÞ;
ð2:5Þ
y 2 ða; hÞ;
ð2:6Þ
Fig. 1. The scheme of elastic layer with crack.
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
rxy ð0; yÞ ¼ 0;
y 2 ð0; hÞ;
2355
ð2:7Þ
where pðÞ is a given function defining the intensity of loadings. According to the results of paper [1] the solution of equations for the static state of plane strain within the framework of linear elasticity can be written in the form rffiffiffi Z 1 2 1 uðx; yÞ ¼ f½A þ Bay þ 2ð1 mÞD sinhðayÞ þ ½C þ Day þ 2ð1 mÞB 2l p 0 rffiffiffi Z 1 2 1 1 uðbÞ½2ð1 mÞ þ bxebx cosðbyÞdb; coshðayÞg sinðaxÞda 2l p 0 b ð2:8Þ rffiffiffi Z 1 2 1 f½A þ Bay ð1 2mÞD coshðayÞ þ ½C þ Day ð1 2mÞB vðx; yÞ ¼ 2l p 0 rffiffiffi Z 1 2 1 1 uðbÞ½ð1 2mÞ þ bxebx sinðbyÞdb; sinhðayÞg cosðaxÞda 2l p 0 b and rffiffiffi Z 1 2 rxx ðx; yÞ ¼ af½A þ Bay þ 2D sinhðayÞ þ ½C þ Day þ 2B coshðayÞg cosðaxÞda p 0 rffiffiffi Z 1 2 þ uðbÞ½1 þ bxebx cosðbyÞdb; p 0 rffiffiffi Z 1 2 af½A þ Bay sinhðayÞ þ ½C þ Day coshðayÞg cosðaxÞda ryy ðx; yÞ ¼ p rffiffiffi Z0 1 2 uðbÞ½1 bxebx cosðbyÞdb; þ p 0 rffiffiffi Z 1 2 rxy ðx; yÞ ¼ af½A þ Bay þ D coshðayÞ þ ½C þ Day þ B sinhðayÞg sinðaxÞda p 0 rffiffiffi Z 1 2 uðbÞbxebx sinðbyÞdb; þ p 0
ð2:9Þ
where m denotes PoissonÕs ratio m¼
k 2ðk þ lÞ
and A, B, C, D, uðbÞ are unknowns which should be determined by using boundary conditions (2.1)–(2.6) and Eqs. (2.8) and (2.9). The boundary condition (2.7) is satisfied by the expression for rxy given in Eq. (2.9).
3. Reduction of the problem to integral Fredholm equation By satisfying boundary conditions (2.1)–(2.4) by Eqs. (2.8) and (2.9) after some calculations (using the integrals given in Appendix A) we arrive at the following system of algebraic equations:
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S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
A þ D ¼ 0; A
2ð1 mÞ 2ð1 mÞ ahC ð1 2mÞD ¼ ahF0 ðaÞ; h h
1 ½A þ Bah þ D coshðahÞ þ ½C þ Dah þ B sinhðahÞ ¼ F2 ðaÞ; a
ð3:1Þ
1 ½A þ Bah sinhðahÞ þ ½C þ Dah coshðahÞ ¼ P ðaÞ þ F1 ðaÞ; a where h denotes the dimensionless stiffness coefficient of foundation h¼
ð1 mÞkh l
ð3:2Þ
and F0 ðaÞ ¼
4 p
4 F1 ðaÞ ¼ p F2 ðaÞ ¼
4 p
Z
1
uðbÞ 0
Z
1
uðbÞ 0
Z
1
uðbÞ 0
ab ða2
þ b2 Þ
2
db;
ab cosðbhÞ ða2 þ b2 Þ
2
b2 sinðbhÞ ða2 þ b2 Þ
2
db;
ð3:3Þ
db;
as well as rffiffiffi Z 1 2 P ðaÞ ¼ pðxÞ cosðaxÞdx: p 0
ð3:4Þ
Denoting by ah ½sinh2 ðahÞ a2 h2 ; h ah U1 ðaÞ ¼ G1 ðaÞ coshðahÞ þ ½sinhðahÞ ah coshðahÞ h a2 h2 G2 ðaÞ 1 sinhðahÞ; h W ðaÞ ¼ sinhðahÞ coshðahÞ þ ah þ
U2 ðaÞ ¼ G1 ðaÞah sinhðahÞ þ G2 ðaÞ½sinhðahÞ þ ah coshðahÞ; 1 a2 h2 p1 ðaÞ ¼ 1 sinhðahÞ; a h p2 ðaÞ ¼
1 ½sinhðahÞ þ ah coshðahÞ; a
ð3:5Þ
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
2357
the solution of linear algebraic equations (3.1) can be written in the form ah U2 ðaÞ þ p2 ðaÞP ðaÞ A¼ F0 ðaÞ ; h W ðaÞ B¼
U1 ðaÞ þ p1 ðaÞP ðaÞ ; W ðaÞ
ð3:6Þ
U2 ðaÞ þ p2 ðaÞP ðaÞ C¼ ; W ðaÞ D ¼ A: The last unknown in Eqs. (2.8) and (2.9) is the function uðÞ, which should be determined by using Eqs. (2.8), (2.9) and (3.6) and boundary conditions (2.5) and (2.6). From Eqs. (2.9)1 and (3.6) it follows that rffiffiffi Z 1 rffiffiffi Z 1 rffiffiffi Z 1 2 2 2 uðbÞ cosðbyÞdb þ uðbÞRðb; yÞdb þ P ðbÞSðb; yÞdb; ð3:7Þ rxx ð0; yÞ ¼ p 0 p 0 p 0 where Z 4b 1 da Rðb; yÞ ¼ ½aS0 ða; yÞ þ aS1 ða; yÞ cosðbhÞ þ S2 ða; yÞb sinðbhÞ ; 2 2 p 0 ða þ b2 Þ 1 a2 h2 Sða; yÞ ¼ 1 ½ay sinhðayÞ þ 2 coshðayÞ sinhðahÞ W ðaÞ h ah þ ½sinhðahÞ þ ah coshðahÞ coshðayÞ ðsinhðayÞ þ ay coshðayÞÞ ; h ah a S0 ða; yÞ ¼ ½sinhðayÞ þ ay coshðayÞW ðaÞ ½ay sinhðayÞ þ 2 coshðayÞ sinh2 ðahÞ h W ðaÞ ah 2 2 2 ð3:8Þ þ coshðahÞ ðsinhðayÞ þ ay coshðayÞÞ ½sinh ðahÞ a h ; h a ah coshðayÞ ðsinhðayÞ þ ay coshðayÞÞ ½sinhðahÞ þ ah coshðahÞ S1 ða; yÞ ¼ W ðaÞ h a2 h2 ½ay sinhðayÞ þ 2 coshðayÞ 1 sinhðahÞ ; h a ah coshðayÞ ðsinhðayÞ þ ay coshðayÞÞ ah sinhðahÞ S2 ða; hÞ ¼ W ðaÞ h ah ½ah sinhðayÞ þ 2 coshðayÞ coshðahÞ þ ðsinhðahÞ ah coshðahÞÞ : h By using Eqs. (3.7) and (2.8)1 the boundary conditions (2.5) and (2.6) lead to the following dual integral equations for unknown function uðÞ: Z 1 Z 1 Z 1 uðbÞ cosðbyÞdb þ uðbÞRðb; yÞdb ¼ P ðbÞSðb; yÞdb; for 0 6 y < a; 0
Z
0
0 1
1
b uðbÞ cosðbyÞdb ¼ 0;
0
for a < y 6 h:
ð3:9Þ
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S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
Taking into account function uðbÞ in the form [2]: uðbÞ ¼ b
Z
a
gðtÞJ0 ðbtÞdt;
ð3:10Þ
0
where gðÞ is an unknown function, J0 ðÞ is the Bessel function, we arrive at the following Abel integral equation: Z y Z a Z a d dt gðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ gðtÞR1 ðt; yÞdt ¼ P ðbÞSðb; yÞdb; for y 2 ð0; aÞ; ð3:11Þ dy 0 y 2 t2 0 0 where
Z 4 1 ½S0 ða; yÞk0 ða; tÞ þ S1 ða; yÞk1 ða; tÞ þ S2 ða; yÞk2 ða; tÞda; R1 ðt; yÞ ¼ p 0 Z 1 2 b J0 ðbtÞ k0 ða; tÞ ¼ a db; 2 ða2 þ b2 Þ 0 Z 1 2 b J0 ðbtÞ cosðbhÞ k1 ða; tÞ ¼ a db; 2 ða2 þ b2 Þ 0 Z 1 3 b J0 ðbtÞ sinðbhÞ k2 ða; tÞ ¼ db: 2 ða2 þ b2 Þ 0
ð3:12Þ
By using the inverse Abel transformation given in [2] AbelÕs integral equation (3.11) is reduced to the following Fredholm integral equation of the second kind (after some calculations and integrals given in [7]): 2t gðtÞ þ p
Z
a 0
2t gðtÞR^1 ðt0 ; tÞdt0 ¼ p
Z
t
Z
1
P ðbÞS^ðb; tÞdb;
t 2 ð0; aÞ;
ð3:13Þ
0
where R^1 ðt0 ; tÞ ¼
0
S^ðb; tÞ ¼
Z
t
0
dy R1 ðt0 ; yÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 t y2
dy Sðb; yÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 t y2
ð3:14Þ
The kernel R^1 and the function S^ given by (3.14) can be rewritten by using Eqs. (3.12) and (3.8) in the form 4 R^1 ðt0 ; tÞ ¼ p
Z
1 0
2 X
S^i ða; tÞki ða; t0 Þda;
i¼0
1 a2 h2 S^ða; tÞ ¼ 1 ½f3 ðatÞ þ 2f2 ðatÞ sinhðahÞ W ðaÞ h ah þ f2 ðatÞ ðf1 ðatÞ þ f4 ðatÞÞ ½sinhðahÞ þ ah coshðahÞ ; h
ð3:15Þ
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
2359
where
ah ah ^ S0 ða; tÞ ¼ ½f1 ðatÞ: þ f4 ðatÞW ðaÞ ½f3 ðatÞ þ 2f2 ðatÞ sinh2 ðahÞ h W ðaÞ ah þ f2 ðatÞ ðf1 ðatÞ þ f4 ðatÞÞ ½sinh2 ðahÞ a2 h2 ; h a ah f2 ðatÞ ðf1 ðatÞ þ f4 ðatÞÞ ½sinhðahÞ þ ah coshðahÞ S^1 ða; tÞ ¼ W ðaÞ h a2 h2 ½f3 ðatÞ þ 2f2 ðatÞ 1 2 sinhðahÞ ; h a ah f2 ðatÞ ðf1 ðatÞ þ f4 ðatÞÞ ah sinhðahÞ S^2 ða; tÞ ¼ W ðaÞ h ah ½f3 ðatÞ þ 2f2 ðatÞ coshðahÞ þ ðsinhðahÞ ah coshðahÞÞ : h
ð3:16Þ
Functions f1 ðÞ; . . . ; f4 ðÞ are given in Appendix A. 4. Numerical results The obtained integral Fredholm equation (3.13) will be solved numerically. From the standpoint of the linear fracture mechanics the behavior of stresses near the crack tip is of special interest. By using (2.8), (2.9), (3.7) and (3.10) it follows that rffiffiffi 2 ygðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 0ð1Þ: rxx ð0; yÞ ¼ ð4:1Þ p a y 2 a2 The SIF KI KI ¼ limþ y!a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðy aÞrxx ð0; yÞ
can be expressed in term of gðaÞ: pffiffiffi 2gðaÞ KI ¼ pffiffiffi : a
ð4:2Þ
ð4:3Þ
To determine the SIF KI and finding the solution of the Fredholm integral equation we assume that the function pðÞ is given for example in the form ð4:4Þ
pðxÞ ¼ p0 dðxÞ;
where dðÞ is the Dirac function and p0 is given constant point load. For the use of numerical calculations the following nondimensional variables and parameters are introduced: t s¼ ; a
t0 s0 ¼ ; a
KI KI ¼ pffiffiffi ; ap0
a k1 ¼ : h
ð4:5Þ
The Fredholm integral equation (3.13) with the kernel and the right-hand side described by (3.14) should rewritten in the nondimensional form (the details will be omitted). The obtained numerical results are
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S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
presented in Figs. 2 and 3. The graphs of the dimensionless SIF KI as a function of nondimensional parameter k1 ¼ a=h for the following values of the dimensionless stiffness coefficient of foundation h ¼ 0:1, 0.5, 5.0, 10.0 are presented in Fig. 2. The graphs of the nondimensionless SIF KI as a function of parameter h for the following values of k1 ¼ 0:1, 0.2, 0.3, 0.4, 0.5 are shown in Fig. 3.
Fig. 2. Effects of the crack length on SIF for some values of the Winkler foundation stiffness.
Fig. 3. Effects of the Winkler foundations on SIF for some values of the crack length.
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
2361
5. Conclusion The obtained numerical results presented in Figs. 2 and 3 show that: (1) The nondimensional SIF KI is considerably increasing with an increase of nondimensional crack length, defined by parameter k1 and given by Eq. (4.5). (2) The SIF KI is decreasing with an increase of the dimensionless stiffness coefficient h of Winkler foundation. (3) The SIF KI defined by (4.2) can be calculated from the results presented in Figs. 2 and 3 for some geological structures (stratum) weakened by edge cracks normal to boundary planes. Appendix A The following integrals have been applied [7]: Z 1 b ebx cosðaxÞdx ¼ ; for b > 0; 2 þ b2 a 0 Z
1
x ebx cosðaxÞdx ¼
0
Z 0
1
x2 cosðaxÞ ðb2 þ
f1 ðatÞ ¼
2 x2 Þ
Z
f2 ðatÞ ¼
Z Z 0
for b > 0;
;
p ð1 abÞeab ; 4b
1
coshðatzÞ p pffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ I0 ðatÞ; 2 2 1z
1
atz sinhðatzÞ p pffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ atI1 ðatÞ; 2 2 1z
1
1 X atz coshðatzÞ 2k þ 1 2kþ1 pffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ ðatÞ ; 2 ½ð2k 1Þ!! 1 z2 k¼0
0
f4 ðatÞ ¼
2
1 sinhðatzÞ X ðatÞ pffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; 2 2 1z k¼0 ½ð2k þ 1Þ!!
0
f3 ðatÞ ¼
ða2 þ b2 Þ
1
0
Z
dx ¼
b2 a2
2kþ1
where In ðÞ, n ¼ 0; 1 are the modified Bessel function of the first kind of order n.
References [1] [2] [3] [4] [5]
Matysiak SJ, Pauk VJ. On crack problem in an elastic ponderable layer. Int J Fract 1999;96:371–80. Sneddon IN. Mixed boundary value problems in potential theory. Amsterdam: North Holland Publ. Co.; 1966. Sneddon IN, Lowengrub M. Crack problems in the classical theory of elasticity. New York: John Wiley and Sons Inc.; 1969. Cherepanov GP. Brittle fracture mechanics. Moscow: Science; 1974 [in Russian]. Savruk MP. Stress intensity factors in bodies with cracks. In: Panasiuk VV, editor. Fracture mechanics, vol. II. Kiev: Naukova Dumka; 1988 [in Russian]. [6] Sih GC. Handbook of stress intensity factors. Bethlehem: Leigh University Press; 1993. [7] Gradstein IS, Ryzik IM. Tables of integrals, sum, series and products. Moscow: Science; 1971 [in Russian].
Edge crack in an elastic layer resting on Winkler foundation S.J. Matysiak a
a,*
, V.J. Pauk
b
Faculty of Geology, Institute of Hydrogeology and Engineering Geology, University of Warsaw, Al. Z_ wirki i Wigury 93, Warsaw 02-089, Poland b Faculty of Civil and Environment Engineering, Kielce University of Technology, Al. 1000-lecia Pa nstwa Polskiego 7, Kielce 25-314, Poland Received 8 April 2002; received in revised form 18 December 2002; accepted 8 January 2003
Abstract The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Elastic layer; Crack; Stress intensity factor; Winkler foundation
1. Introduction The problems of stress distributions in an elastic layer (stratum) weakened by cracks are of significant interest in geology and geophysics as well as in engineering constructions. The elastic strata are frequently situated on substrates which can be treated as Winkler foundation. The knowledge of stresses permits to forecast fracture processes of the bodies. This paper is a continuation of our studies [1] concerning the problem of a ponderable elastic layer fixed to a rigid foundation and weakened by an edge crack normal to the boundary plane. The plane problem of stress distribution in an elastic layer resting on Winkler foundation is considered. The layer contains a Griffith edge crack normal to the lower boundary plane and is loaded by normal forces acting on the upper boundary surface of the body. The considerations are restricted to the case of symmetric distribution of boundary loads with respect to the extension of crack line. The problem is solved by using the Fourier transform method, dual integral equations [2], and Fredholm integral equations of the second kind. The stress intensity factor (SIF) is expressed in terms of the solution of the derived Fredholm integral equation.
*
Corresponding author. Tel.: +48-22-554-00-11; fax: +48-22-554-00-01. E-mail address: [email protected] (S.J. Matysiak).
0013-7944/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-7944(03)00007-9
2354
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
The numerical results for the SIF are presented for the concentrated loading acting with a constant intensity on the upper boundary. The considered crack problem is of interest in geophysical and rock mechanics as well as plate–subsoil interactions. The analysis of stresses in layers with crack have received wide attention (see, for example, monographs [3–6]).
2. Formulation of problem Consider the plane static problem of elastic layer resting on a Winkler foundation. The layer is weakened by an edge crack normal to the boundary (see Fig. 1). Let the problem be related to a Cartesian coordinate system ðx; yÞ such that the lower boundary of layer is situated along the axis 0x and the crack is placed along the axis 0y (Fig. 1). Let h denote the thickness of layer and a be the length of crack. Let k, l be the Lame constants of elastic layer and k be the Winkler stiffness of foundation. The upper plane of layer is loaded symmetrically with respect to the axis 0y. Let ðuðx; yÞ; vðx; yÞÞ denote the displacement vector and rxx ðx; yÞ, ryy ðx; yÞ, rxy ðx; yÞ denote the components of stress tensor at the point ðx; yÞ. The considered problem is described by equations of the plane theory of elasticity and the following boundary conditions: rxy ðx; 0Þ ¼ 0;
x 2 R;
ð2:1Þ
rxy ðx; hÞ ¼ 0;
x 2 R;
ð2:2Þ
ryy ðx; 0Þ ¼ kvðx; 0Þ
x 2 R;
ð2:3Þ
ryy ðx; hÞ ¼ pðxÞ;
x 2 R;
ð2:4Þ
rxx ð0; yÞ ¼ 0; uð0; yÞ ¼ 0;
y 2 ð0; aÞ;
ð2:5Þ
y 2 ða; hÞ;
ð2:6Þ
Fig. 1. The scheme of elastic layer with crack.
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
rxy ð0; yÞ ¼ 0;
y 2 ð0; hÞ;
2355
ð2:7Þ
where pðÞ is a given function defining the intensity of loadings. According to the results of paper [1] the solution of equations for the static state of plane strain within the framework of linear elasticity can be written in the form rffiffiffi Z 1 2 1 uðx; yÞ ¼ f½A þ Bay þ 2ð1 mÞD sinhðayÞ þ ½C þ Day þ 2ð1 mÞB 2l p 0 rffiffiffi Z 1 2 1 1 uðbÞ½2ð1 mÞ þ bxebx cosðbyÞdb; coshðayÞg sinðaxÞda 2l p 0 b ð2:8Þ rffiffiffi Z 1 2 1 f½A þ Bay ð1 2mÞD coshðayÞ þ ½C þ Day ð1 2mÞB vðx; yÞ ¼ 2l p 0 rffiffiffi Z 1 2 1 1 uðbÞ½ð1 2mÞ þ bxebx sinðbyÞdb; sinhðayÞg cosðaxÞda 2l p 0 b and rffiffiffi Z 1 2 rxx ðx; yÞ ¼ af½A þ Bay þ 2D sinhðayÞ þ ½C þ Day þ 2B coshðayÞg cosðaxÞda p 0 rffiffiffi Z 1 2 þ uðbÞ½1 þ bxebx cosðbyÞdb; p 0 rffiffiffi Z 1 2 af½A þ Bay sinhðayÞ þ ½C þ Day coshðayÞg cosðaxÞda ryy ðx; yÞ ¼ p rffiffiffi Z0 1 2 uðbÞ½1 bxebx cosðbyÞdb; þ p 0 rffiffiffi Z 1 2 rxy ðx; yÞ ¼ af½A þ Bay þ D coshðayÞ þ ½C þ Day þ B sinhðayÞg sinðaxÞda p 0 rffiffiffi Z 1 2 uðbÞbxebx sinðbyÞdb; þ p 0
ð2:9Þ
where m denotes PoissonÕs ratio m¼
k 2ðk þ lÞ
and A, B, C, D, uðbÞ are unknowns which should be determined by using boundary conditions (2.1)–(2.6) and Eqs. (2.8) and (2.9). The boundary condition (2.7) is satisfied by the expression for rxy given in Eq. (2.9).
3. Reduction of the problem to integral Fredholm equation By satisfying boundary conditions (2.1)–(2.4) by Eqs. (2.8) and (2.9) after some calculations (using the integrals given in Appendix A) we arrive at the following system of algebraic equations:
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A þ D ¼ 0; A
2ð1 mÞ 2ð1 mÞ ahC ð1 2mÞD ¼ ahF0 ðaÞ; h h
1 ½A þ Bah þ D coshðahÞ þ ½C þ Dah þ B sinhðahÞ ¼ F2 ðaÞ; a
ð3:1Þ
1 ½A þ Bah sinhðahÞ þ ½C þ Dah coshðahÞ ¼ P ðaÞ þ F1 ðaÞ; a where h denotes the dimensionless stiffness coefficient of foundation h¼
ð1 mÞkh l
ð3:2Þ
and F0 ðaÞ ¼
4 p
4 F1 ðaÞ ¼ p F2 ðaÞ ¼
4 p
Z
1
uðbÞ 0
Z
1
uðbÞ 0
Z
1
uðbÞ 0
ab ða2
þ b2 Þ
2
db;
ab cosðbhÞ ða2 þ b2 Þ
2
b2 sinðbhÞ ða2 þ b2 Þ
2
db;
ð3:3Þ
db;
as well as rffiffiffi Z 1 2 P ðaÞ ¼ pðxÞ cosðaxÞdx: p 0
ð3:4Þ
Denoting by ah ½sinh2 ðahÞ a2 h2 ; h ah U1 ðaÞ ¼ G1 ðaÞ coshðahÞ þ ½sinhðahÞ ah coshðahÞ h a2 h2 G2 ðaÞ 1 sinhðahÞ; h W ðaÞ ¼ sinhðahÞ coshðahÞ þ ah þ
U2 ðaÞ ¼ G1 ðaÞah sinhðahÞ þ G2 ðaÞ½sinhðahÞ þ ah coshðahÞ; 1 a2 h2 p1 ðaÞ ¼ 1 sinhðahÞ; a h p2 ðaÞ ¼
1 ½sinhðahÞ þ ah coshðahÞ; a
ð3:5Þ
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
2357
the solution of linear algebraic equations (3.1) can be written in the form ah U2 ðaÞ þ p2 ðaÞP ðaÞ A¼ F0 ðaÞ ; h W ðaÞ B¼
U1 ðaÞ þ p1 ðaÞP ðaÞ ; W ðaÞ
ð3:6Þ
U2 ðaÞ þ p2 ðaÞP ðaÞ C¼ ; W ðaÞ D ¼ A: The last unknown in Eqs. (2.8) and (2.9) is the function uðÞ, which should be determined by using Eqs. (2.8), (2.9) and (3.6) and boundary conditions (2.5) and (2.6). From Eqs. (2.9)1 and (3.6) it follows that rffiffiffi Z 1 rffiffiffi Z 1 rffiffiffi Z 1 2 2 2 uðbÞ cosðbyÞdb þ uðbÞRðb; yÞdb þ P ðbÞSðb; yÞdb; ð3:7Þ rxx ð0; yÞ ¼ p 0 p 0 p 0 where Z 4b 1 da Rðb; yÞ ¼ ½aS0 ða; yÞ þ aS1 ða; yÞ cosðbhÞ þ S2 ða; yÞb sinðbhÞ ; 2 2 p 0 ða þ b2 Þ 1 a2 h2 Sða; yÞ ¼ 1 ½ay sinhðayÞ þ 2 coshðayÞ sinhðahÞ W ðaÞ h ah þ ½sinhðahÞ þ ah coshðahÞ coshðayÞ ðsinhðayÞ þ ay coshðayÞÞ ; h ah a S0 ða; yÞ ¼ ½sinhðayÞ þ ay coshðayÞW ðaÞ ½ay sinhðayÞ þ 2 coshðayÞ sinh2 ðahÞ h W ðaÞ ah 2 2 2 ð3:8Þ þ coshðahÞ ðsinhðayÞ þ ay coshðayÞÞ ½sinh ðahÞ a h ; h a ah coshðayÞ ðsinhðayÞ þ ay coshðayÞÞ ½sinhðahÞ þ ah coshðahÞ S1 ða; yÞ ¼ W ðaÞ h a2 h2 ½ay sinhðayÞ þ 2 coshðayÞ 1 sinhðahÞ ; h a ah coshðayÞ ðsinhðayÞ þ ay coshðayÞÞ ah sinhðahÞ S2 ða; hÞ ¼ W ðaÞ h ah ½ah sinhðayÞ þ 2 coshðayÞ coshðahÞ þ ðsinhðahÞ ah coshðahÞÞ : h By using Eqs. (3.7) and (2.8)1 the boundary conditions (2.5) and (2.6) lead to the following dual integral equations for unknown function uðÞ: Z 1 Z 1 Z 1 uðbÞ cosðbyÞdb þ uðbÞRðb; yÞdb ¼ P ðbÞSðb; yÞdb; for 0 6 y < a; 0
Z
0
0 1
1
b uðbÞ cosðbyÞdb ¼ 0;
0
for a < y 6 h:
ð3:9Þ
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Taking into account function uðbÞ in the form [2]: uðbÞ ¼ b
Z
a
gðtÞJ0 ðbtÞdt;
ð3:10Þ
0
where gðÞ is an unknown function, J0 ðÞ is the Bessel function, we arrive at the following Abel integral equation: Z y Z a Z a d dt gðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ gðtÞR1 ðt; yÞdt ¼ P ðbÞSðb; yÞdb; for y 2 ð0; aÞ; ð3:11Þ dy 0 y 2 t2 0 0 where
Z 4 1 ½S0 ða; yÞk0 ða; tÞ þ S1 ða; yÞk1 ða; tÞ þ S2 ða; yÞk2 ða; tÞda; R1 ðt; yÞ ¼ p 0 Z 1 2 b J0 ðbtÞ k0 ða; tÞ ¼ a db; 2 ða2 þ b2 Þ 0 Z 1 2 b J0 ðbtÞ cosðbhÞ k1 ða; tÞ ¼ a db; 2 ða2 þ b2 Þ 0 Z 1 3 b J0 ðbtÞ sinðbhÞ k2 ða; tÞ ¼ db: 2 ða2 þ b2 Þ 0
ð3:12Þ
By using the inverse Abel transformation given in [2] AbelÕs integral equation (3.11) is reduced to the following Fredholm integral equation of the second kind (after some calculations and integrals given in [7]): 2t gðtÞ þ p
Z
a 0
2t gðtÞR^1 ðt0 ; tÞdt0 ¼ p
Z
t
Z
1
P ðbÞS^ðb; tÞdb;
t 2 ð0; aÞ;
ð3:13Þ
0
where R^1 ðt0 ; tÞ ¼
0
S^ðb; tÞ ¼
Z
t
0
dy R1 ðt0 ; yÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 t y2
dy Sðb; yÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 t y2
ð3:14Þ
The kernel R^1 and the function S^ given by (3.14) can be rewritten by using Eqs. (3.12) and (3.8) in the form 4 R^1 ðt0 ; tÞ ¼ p
Z
1 0
2 X
S^i ða; tÞki ða; t0 Þda;
i¼0
1 a2 h2 S^ða; tÞ ¼ 1 ½f3 ðatÞ þ 2f2 ðatÞ sinhðahÞ W ðaÞ h ah þ f2 ðatÞ ðf1 ðatÞ þ f4 ðatÞÞ ½sinhðahÞ þ ah coshðahÞ ; h
ð3:15Þ
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
2359
where
ah ah ^ S0 ða; tÞ ¼ ½f1 ðatÞ: þ f4 ðatÞW ðaÞ ½f3 ðatÞ þ 2f2 ðatÞ sinh2 ðahÞ h W ðaÞ ah þ f2 ðatÞ ðf1 ðatÞ þ f4 ðatÞÞ ½sinh2 ðahÞ a2 h2 ; h a ah f2 ðatÞ ðf1 ðatÞ þ f4 ðatÞÞ ½sinhðahÞ þ ah coshðahÞ S^1 ða; tÞ ¼ W ðaÞ h a2 h2 ½f3 ðatÞ þ 2f2 ðatÞ 1 2 sinhðahÞ ; h a ah f2 ðatÞ ðf1 ðatÞ þ f4 ðatÞÞ ah sinhðahÞ S^2 ða; tÞ ¼ W ðaÞ h ah ½f3 ðatÞ þ 2f2 ðatÞ coshðahÞ þ ðsinhðahÞ ah coshðahÞÞ : h
ð3:16Þ
Functions f1 ðÞ; . . . ; f4 ðÞ are given in Appendix A. 4. Numerical results The obtained integral Fredholm equation (3.13) will be solved numerically. From the standpoint of the linear fracture mechanics the behavior of stresses near the crack tip is of special interest. By using (2.8), (2.9), (3.7) and (3.10) it follows that rffiffiffi 2 ygðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 0ð1Þ: rxx ð0; yÞ ¼ ð4:1Þ p a y 2 a2 The SIF KI KI ¼ limþ y!a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðy aÞrxx ð0; yÞ
can be expressed in term of gðaÞ: pffiffiffi 2gðaÞ KI ¼ pffiffiffi : a
ð4:2Þ
ð4:3Þ
To determine the SIF KI and finding the solution of the Fredholm integral equation we assume that the function pðÞ is given for example in the form ð4:4Þ
pðxÞ ¼ p0 dðxÞ;
where dðÞ is the Dirac function and p0 is given constant point load. For the use of numerical calculations the following nondimensional variables and parameters are introduced: t s¼ ; a
t0 s0 ¼ ; a
KI KI ¼ pffiffiffi ; ap0
a k1 ¼ : h
ð4:5Þ
The Fredholm integral equation (3.13) with the kernel and the right-hand side described by (3.14) should rewritten in the nondimensional form (the details will be omitted). The obtained numerical results are
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presented in Figs. 2 and 3. The graphs of the dimensionless SIF KI as a function of nondimensional parameter k1 ¼ a=h for the following values of the dimensionless stiffness coefficient of foundation h ¼ 0:1, 0.5, 5.0, 10.0 are presented in Fig. 2. The graphs of the nondimensionless SIF KI as a function of parameter h for the following values of k1 ¼ 0:1, 0.2, 0.3, 0.4, 0.5 are shown in Fig. 3.
Fig. 2. Effects of the crack length on SIF for some values of the Winkler foundation stiffness.
Fig. 3. Effects of the Winkler foundations on SIF for some values of the crack length.
S.J. Matysiak, V.J. Pauk / Engineering Fracture Mechanics 70 (2003) 2353–2361
2361
5. Conclusion The obtained numerical results presented in Figs. 2 and 3 show that: (1) The nondimensional SIF KI is considerably increasing with an increase of nondimensional crack length, defined by parameter k1 and given by Eq. (4.5). (2) The SIF KI is decreasing with an increase of the dimensionless stiffness coefficient h of Winkler foundation. (3) The SIF KI defined by (4.2) can be calculated from the results presented in Figs. 2 and 3 for some geological structures (stratum) weakened by edge cracks normal to boundary planes. Appendix A The following integrals have been applied [7]: Z 1 b ebx cosðaxÞdx ¼ ; for b > 0; 2 þ b2 a 0 Z
1
x ebx cosðaxÞdx ¼
0
Z 0
1
x2 cosðaxÞ ðb2 þ
f1 ðatÞ ¼
2 x2 Þ
Z
f2 ðatÞ ¼
Z Z 0
for b > 0;
;
p ð1 abÞeab ; 4b
1
coshðatzÞ p pffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ I0 ðatÞ; 2 2 1z
1
atz sinhðatzÞ p pffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ atI1 ðatÞ; 2 2 1z
1
1 X atz coshðatzÞ 2k þ 1 2kþ1 pffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ ðatÞ ; 2 ½ð2k 1Þ!! 1 z2 k¼0
0
f4 ðatÞ ¼
2
1 sinhðatzÞ X ðatÞ pffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; 2 2 1z k¼0 ½ð2k þ 1Þ!!
0
f3 ðatÞ ¼
ða2 þ b2 Þ
1
0
Z
dx ¼
b2 a2
2kþ1
where In ðÞ, n ¼ 0; 1 are the modified Bessel function of the first kind of order n.
References [1] [2] [3] [4] [5]
Matysiak SJ, Pauk VJ. On crack problem in an elastic ponderable layer. Int J Fract 1999;96:371–80. Sneddon IN. Mixed boundary value problems in potential theory. Amsterdam: North Holland Publ. Co.; 1966. Sneddon IN, Lowengrub M. Crack problems in the classical theory of elasticity. New York: John Wiley and Sons Inc.; 1969. Cherepanov GP. Brittle fracture mechanics. Moscow: Science; 1974 [in Russian]. Savruk MP. Stress intensity factors in bodies with cracks. In: Panasiuk VV, editor. Fracture mechanics, vol. II. Kiev: Naukova Dumka; 1988 [in Russian]. [6] Sih GC. Handbook of stress intensity factors. Bethlehem: Leigh University Press; 1993. [7] Gradstein IS, Ryzik IM. Tables of integrals, sum, series and products. Moscow: Science; 1971 [in Russian].