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Axial compression of circular cylindrical bar with coaxial subsurface cylindrical crack of finite length
Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
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Axial compression of circular cylindrical bar with coaxial subsurface cylindrical crack of finite length A.N. Guz, Yu.I. Khoma * Department of Dynamics and Stability of Continuum Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Nesterov Str. 3, Kyiv 03057, Ukraine
a r t i c l e
i n f o
Article history: Available online 20 May 2009 Keywords: Local stability loss Axial compression Coaxial subsurface cylindrical crack Loading along a defect Non-classical fracture mechanics
a b s t r a c t A fracture stability of a circular cylindrical bar with a coaxial surface cylindrical crack subjected to an axial compression is considered. A state of subcritical initial strain is assumed. A non-classical fracture criterion is based on a local stability loss near the defect. The theory of integral Fourier transforms and series expansions are used to reduce these problems to a system of paired integral equations and then to a system of linear algebraic equations with respect to the contraction parameter. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The problems of fracture mechanics of materials under compression along the cracks represent a special class of problems because they cannot be adequately described by the relations of the linear fracture mechanics. In the case of the forces are applied along the crack the stress intensity factors calculated on the basis of the linear fracture mechanics are equal to zero [1]. Consequently, the classical fracture criterion of the Griffith–Irwin, the crack propagation criterion and also their generalizations used in the linear fracture mechanics [1–3] are not applicable to this scheme of loading. It was shown as early as in fundamental studies carried out by Euler on the critical force for compressed bars that the loss of stability, i.e. when the load reaches the critical force can be regarded as the beginning of failure. Therefore, in loading along the cracks it is rational to use, the stability criterion within the framework of the linearized stability theory proposed in [4]. According to this criterion the fracture process is caused by a local stability loss near the crack and quantified in terms of a critical eigenvalue. The local instability of the materials is examined on the basis of relations of the three-dimensional linearized mechanics of deformed bodies. While an influence of the loads acting along the cracks on the fracture parameters have been confirmed experimentally. Symmetric and nonaxisymmetric compressive failure of the half-space with cracks were analyzed in [5,6]. Fracture problems for an infinite solid with a circular cylindrical crack under compression have been discussed previously [7] and for a cylindrical * Corresponding author. E-mail address: [email protected] (Yu.I. Khoma). 0167-8442/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2009.05.001
cavity in infinite solid and a finite length coaxial cylindrical crack compressed axially [8]. Some fracture problems have been considered in [9–11]. In this paper we describe a method for studying the initial phase of a failure of a circular cylindrical bar weakened with a coaxial subsurface cylindrical crack in uniform axial compression along the defect. The analytical solution has been carried out in common form for different models of materials (elastic, composite), the theory of large or small initial strains. The model of solid has to be specified on the final stage of the examination, i.e. the numerical solution.
2. Basic equations To solve the mechanical fracture problem a common approach is used for compressible and incompressible elastic bodies with an arbitrary elastic potential, both for large subcritical deformations and for two variants of theory of small subcritical deformations. The case of unequal roots of the characteristic equation [12,13] is considered. In formulating the linearized problem, Lagrangian coordinates xj that coincide with Cartesian coordinates in the undeformed state, as well as perturbations in the displacement vector uj and stress tensor t relative to unit area of the body in the undeformed state are used. It assumed that a circular cylindrical bar weakened with a coaxial subsurface cylindrical crack is subjected to a uniform uniaxial compression along the x3 axis. The initial forces applied at infinity create a homogeneous stressed-deformed subcritical state in a neighborhood of the defect, which is determined by the following relationships [14]
203
A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
u0j ¼ djm ðkj 1Þxm ; S011
¼
S022
¼ 0;
S033
kj ¼ const;
k1 ¼ k2 – k3 ;
ðk3 < 1Þ;
¼ const – 0:
ð1Þ
S0ij
Here are the components of the symmetric stress tensor relative to a unit area of the body in the underformed state, u0j are the displacements corresponding to the initial stresses S0ij , kj ðj ¼ 1; 2; 3Þ are the contractions along the axes, and dij is the Kronecker symbol. A superscript ‘‘0” in Eq. (1) and in the following indicates quantities pertaining to the initial state. The corresponding perturbed quantities are indicated without any additional indices. The linearized equilibrium equations in the underformed state coordinates for compressible and incompressible bodies are [14]
Lma ua ¼ 0;
@2 ; @xi @xb @ua qma ¼ 0; @xm
Lma ¼ ximab
Lma
ð3Þ
2e0ij ¼ dij ðk2j 1Þ;
ð4Þ
while in the theory of small initial deformations are related to the contractions kj by
e0ij ¼ dij ðkj 1Þ:
ð5Þ
In the case of an uniform subcritical state (1) the linearized equilibrium Eq. (2) for compressible solids have the form [14]
@ 2 ua ¼ 0: @xi @xb
ð6Þ
Here the components of the tensor x for the theory of large and small initial deformations are given by [14], respectively
xijab ¼ ka kj ½dij dab Aib þ ð1 dij Þðdia djb þ dib dja Þlij þ dib dja S0bb ; xijab ¼ dij dab Aib þ ð1 dij Þðdia djb þ dib dja Þlij þ djb dja S0bb ; 0 Sbb 0 rbb :
ð7Þ
@ 2 ua @p þ qij ¼ 0; @xi @xi @xb
qij
@uj ¼ 0; @xi
ð8Þ
ð9Þ
where the components of the tensor for the theory of large and small initial deformations are given by [14], respectively
ijab ¼ kj ka ½dij dab Aib þ ð1 dij Þðdia djb þ dib dja Þlij þ dib dja S0bb ; ijab ¼ dij dab Aib þ ð1 dij Þðdia djb þ dib dja Þlij þ
djb dja S0bb :
r0bb ¼
ð13Þ
e0i ¼ ki 1:
j
For a hyperelastic body, where the elastic potential is given as a function 0 0 of0 the principal values of the Green deformation tensor U0 e1 ; e2 ; e3 in the theory of large initial deformations S0ij , Aib , and lij for compressible solids have the form [14]
@ 2 U0 ; @ e0i @ e0j
S0ii ¼
!
1 @ @ lij ¼ U0 : ð14Þ @ e0i @ e0j 2 e0i e0j
@ U0 ; @ e0i
The system of linearized equilibrium Eqs. (6) and (9) for compressible and incompressible solids respectively consists of partial differential equations with constant coefficients and in the case of uniform subcritical deformations, its general solutions can be constructed in terms of potential functions. Let the coordinate system xm is now chosen to be a cylindrical polar system ðr; h; x3 Þ with the axis x3 as one of the cylindrical bar. The general solutions of the static linearized Eqs. (6) and (9) for axially symmetric problems in the case of unequal roots n01 – n02 of the governing characteristic equation are given in terms of two potential functions ui ðr; x3 Þði ¼ 1; 2Þ which obey the equations
! @2 1 @ @2 þ þ ni 2 ui ðr; x3 Þ ¼ 0 ði ¼ 1; 2Þ; @r2 r @r @x3
ð15Þ
and have the following form [12]
ur ¼
@ u1 @ u2 þ ; @r @r
u3 ¼ m1
@ u1 @ u2 þ m2 : @r @r
ð16Þ
The representations for the components of the Kirchhoff stress tensor t on the cylindrical surface are given by [7]
"
In the case of an uniform subcritical state (1) the linearized equilibrium Eq. (2) for incompressible solids are [14]
ijab
i
Aij ¼
The components of the Green deformation tensor e0 in the theory of large initial deformations are related to the contractions kj by [14]
xijab
lij
@ 2 U0 ; @ e0i @ e0j
@ U0 þ p0 ; @ e0i ! 1 @ @ ¼ U0 ; @ e0i @ e0j 2 e0 e0
Aij ¼
ð2Þ
@u4 ¼ 0; @xa @2 ¼ imab ; u4 p: @xi @xb
Lma ua þ qam
The expressions for the theory of small initial deformations are as follows
ð10Þ ð11Þ
# @ 2 u1 @ 2 u2 ; t r3 ¼ C 44 d1 þ d2 @r@x3 @r@x3 " ! ! # @ @2 @ @2 t rr ¼ C 44 p1 l1 u1 p2 l2 u2 : þ þ r@r @x23 r@r @x23
ð17Þ
The quantities in Eqs. (16) and (17) depend on the choice of material (form of the elastic potential) and the initial stress-deformed state. For compressible solids, they can be written in terms of the components of the tensor x as
mi ¼ ðx1111 ni x3113 Þðx1133 þ x1313 Þ1 ;
C 44 ¼ x1313 ;
di ¼ 1 þ mi x1331 x1 1313 ;
pi ¼ ðx1111 ni þ x1133 mi Þx1 1313 ;
ð18Þ
The values lij , Aib and, consequently, components of the tensors x and are determined accordingly to the type of the elastic potential. For a hyperelastic body, where the elastic potential is given as a function of the principal values of the Green deformation tensor U0 e01 ; e02 ; e03 in the theory of large initial deformations the values 0 Sij , Aib , and lij for incompressible solids are defined as [14]
For incompressible solids, they are given in terms of the components of the tensor as follows
@ 2 U0 Aij ¼ 0 0 2p0 dij k4 j ; @ ei @ ej
1 lj ¼ ð1111 1122 Þ½q11 q1 33 ð1133 1313 Þnj 1133 mj þ 3113
lij
1
@ U0 S0ii ¼ 0 þ k2 i p0 ; @ ei !
C 44 ¼ 1313 ; pj ¼
mj ¼ q11 q1 33 nj ;
½q11 q1 33 ð1133
ði ¼ 1; 2Þ:
dj ¼ 1 þ 1331 1 1313 mj ;
1313 Þnj 1133 mj þ 3113 1 1313 ;
ðj ¼ 1; 2Þ;
@ @ 2 1 ¼ 0 U0 p0 k2 i kj ; k1 k2 k3 ¼ 1; qi ¼ ki : 0 0 0 @ @ e ej 2 e e i i
lj ¼ ðx1111 x1122 Þðx1111 ni þ x1133 mi Þ
1
j
ð12Þ
ð19Þ where the roots n1 and n2 of the governing characteristic equation are for compressible solids
204
n1;2
A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ c c2 x3333 x3113 x1 1111 x1331 ;
2x1111 x1331 c ¼ x1111 x3333 þ x1331 x3113 ðx1133 þ x1313 Þ2 ; ð20Þ and for incompressible solids
n1;2 ¼ c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c2 q233 q2 11 3113 1331 ;
where vi ðkÞ, wi ðkÞ and yi ðkÞ ði ¼ 1; 2Þ are unknown weighting functions; I0 and K 0 are the modified Bessel functions. When there is no the lateral cylindrical surface ðr ¼ b þ hÞ, i.e. h ! þ1 (infinite solid), it is necessary to set wi ðkÞ 0 in the integral representations [7]. Using expansion (26), we obtain the components of displacements (16) and stresses (17) for the region 1
ð21Þ
2q211 1331 c ¼ q211 3333 þ q233 1111 2q11 q33 ð1133 þ 1313 Þ:
ð1Þ
C 44 bur ¼
2 Z X
ð1Þ
C 44 bu3 ¼
Consider a circular cylindrical bar of radius b þ h containing a subsurface circular cylindrical crack of radius b and length 2a, located on the lateral part of the cylindrical surface fr ¼ b; 0 6 h < 2p; a 6 x3 6 ag, whose axis coincides with the bar axis x3 and which is compressed along the x3 axis by uniform forces applied at infinity. Here ðr; h; x3 Þ be the cylindrical coordinates and h is the distance between the lateral part of the surface of the cylindrical bar and the surface of the circular cylindrical crack. As a result of the compression parallel to the crack axis, a homogeneous initial stress and strain state (1) occurs near the crack-like defect. Separate an area of the bar into two regions: r < b and b < r < b þ h. Denote by the superscript 1 the quantities referred to the region r < b and by the superscript 2 the ones referred to the region b < r < b þ h. Assume that the edges of the crack, as well as the surface of the bar, are free of stresses. Here it assumed that the surfaces of the defect do not come into a contact interaction. In addition, at the boundary of regions 1ðr < bÞ and 2ðb < r < b þ hÞ the continuity conditions for the displacements and stresses must be satisfied outside the crack. Given these remarks, consider the boundary conditions for the linearized problem in the following form
2 ð1Þ
b t r3 ¼ 2
ð2Þ
t r3 ¼ 0;
when q ¼ q ;
ð1Þ ð2Þ ¼ ¼ 0; t r3 ¼ t r3 ¼ 0; ð1Þ ð2Þ ð1Þ ð2Þ ur ur ¼ 0; u3 u3 ¼ 0; ð1Þ ð2Þ ð1Þ ¼ tð2Þ t r3 ¼ tr3 ; when t rr rr ;
ð1Þ t rr
tð2Þ rr
0 6 j1j < þ1; j1j < b;
when q ¼ 1; when q ¼ 1;
j1j > b;
q ¼ 1; 0 6 j1j < þ1:
ð22Þ
i¼1
0
2 Z X
þ1
þ1
pffiffiffiffi vi ðkÞmi kI0 ð ni kqÞ sin 1kdk;
pffiffiffiffi pffiffiffiffi vi ðkÞdi ni k2 I1 ð ni kqÞ sin 1kdk;
0
2 Z X
þ1
pffiffiffiffi pffiffiffiffi li ni pffiffiffiffi vi ðkÞpi k kI0 ð ni kqÞ I1 ð ni kqÞ cos 1kdk:
q
0
i¼1
ð28Þ Similarly, for the region 2 the components of displacements (16) and stresses (17) are ð2Þ
C 44 bur ¼
2 Z X
ð1Þ
C 44 bu3
þ1 0
i¼1
2 ð2Þ
b t r3
0
sin 1kdk; 2 Z þ1 X
pffiffiffiffi pffiffiffiffi pffiffiffiffi ¼ di ni k2 wi ðkÞI1 ð ni kqÞ yi ðkÞK 1 ð ni kqÞ 0
i¼1
b t ð2Þ rr
pffiffiffiffi pffiffiffiffi pffiffiffiffi ni k wi ðkÞI1 ð ni kqÞ yi ðkÞK 1 ð ni kqÞ
cos 1kdk; 2 Z þ1 X
pffiffiffiffi pffiffiffiffi ¼ mi k wi ðkÞI0 ð ni kqÞ þ yi ðkÞK 0 ð ni kqÞ i¼1
2
ð2Þ t rr ¼ 0;
Z
2 X
i¼1
b t ð1Þ rr ¼
pffiffiffiffi pffiffiffiffi vi ðkÞ ni kI1 ð ni kqÞ cos 1kdk;
0
i¼1
3. Problem formulation
þ1
sin 1kdk pffiffiffiffi 2 Z þ1 X pffiffiffiffi li ni pffiffiffiffi ¼ wi ðkÞpi k kI0 ð ni kqÞ I1 ð ni kqÞ
q
0
i¼1
pffiffiffiffi pffiffiffiffi pffiffiffiffi li n i þ yi ðkÞpi k kK 0 ð ni kqÞ þ K 1 ð ni kqÞ cos 1kdk :
ð23Þ
q
ð24Þ
ð29Þ
ð25Þ
where q ¼ r=b; q ¼ ðb þ hÞ=b; 1 ¼ x3 =b and b ¼ a=b are dimensionless quantities and the superscripts 1 and 2 refer to regions 1 and 2, respectively. Thus, the boundary conditions include conditions at the surface of the bar (22), conditions at the edges of the cracks (23), and continuity conditions for the displacements (24) and stresses (25) at the interface between the regions 1ðr < bÞ and 2ðb < r < b þ hÞ.
Introducing the following notations
pffiffiffiffi k i ðkqÞ ¼ vi ðkÞ pffiffiffiffi I1 ð ni kqÞ; v ni pffiffiffiffi k i ðkqÞ ¼ wi ðkÞ pffiffiffiffi I1 ð ni kqÞ; w ni
pffiffiffiffi pffiffiffiffi I0 ð ni kqÞ li ni ni k pffiffiffiffi ; I1 ð ni kqÞ q pffiffiffiffi pffiffiffiffi K 0 ð ni kqÞ li ni pffiffiffiffi þ ei ðkqÞ ¼ ni k ; q K 1 ð ni kqÞ
g i ðkqÞ ¼
pffiffiffiffi k i ðkqÞ ¼ yi ðkÞ pffiffiffiffi K 1 ð ni kqÞ; y ni ð30Þ
4. Derivation of dual integral equations To obtain a system of dual integral equations, the theory of the Fourier integral transforms is used. Represent the potential functions u1 ðr; x3 Þ and u2 ðr; x3 Þ separately in each of the regions 1ðr < bÞ and 2ðb < r < b þ hÞ applying the subscripts 1 and 2 to them in the corresponding regions, in the form of Fourier integral transforms
C 44 u C 44 u
ð1Þ i ð2Þ i
¼
Z
1
pffiffiffiffi vi ðkÞ I0 ð ni kqÞ cos 1kdk;
ði ¼ 1; 2Þ;
ð26Þ
0
¼
Z
0
þ
1
Eqs. (28) and (29) can be rewritten in the simple form ð1Þ
C 44 bur ¼ ð1Þ
C 44 bu3 ¼
0
pffiffiffiffi yi ðkÞ K 0 ð ni kqÞ cos 1kdk;
2 Z X
2
b t ð1Þ rr ¼
2 Z X i¼1
b t r3 ¼ ði ¼ 1; 2Þ;
ð27Þ
i ðkqÞni cos 1kdk; v
i ðkqÞmi k1 g i ðkqÞ þ li ni q1 sin 1kdk; v
þ1
i ðkqÞpi g i ðkqÞ cos 1kdk; v
0
2 Z X i¼1
þ1
0
i¼1
2 ð1Þ
1
þ1 0
i¼1
pffiffiffiffi wi ðkÞ I0 ð ni kqÞ cos 1kdk
Z
2 Z X
þ1
i ðkqÞkni di sin 1kdk; v
0
ð31Þ
205
A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
and ð2Þ
C 44 bur ¼
2 X
Z
i¼1 ð2Þ C 44 bu3
2
b tð2Þ rr
¼
i ðkqÞni y i ðkqÞni cos 1kdk; ½w
0
þ1
i ðkqÞmi k1 fg i ðkqÞ þ li ni q1 g w
i¼1
i ðkqÞmi k1 ei ðkqÞ li ni q1 sin 1kdk þy Z 2 þ1 X i ðkqÞpi g i ðkqÞ þ y i ðkqÞpi ei ðkqÞ cos 1kdk ¼ ½w
¼
þ1
0
Z ð32Þ
fw11 ðkÞDr ðkÞ þ w12 ðkÞ D3 ðkÞg cos 1kdk ¼ 0;
ð0 6 1 < bÞ;
fw21 ðkÞDr ðkÞ þ w22 ðkÞ D3 ðkÞg sin 1kdk ¼ 0;
ð0 6 1 < bÞ;
þ1
0
ð38Þ
0
and
2 Z X i¼1
Z
0
2 Z X
i¼1 2 ð2Þ b tr3
Without going into details the rest boundary conditions (23) and (24) yields the following system of the dual integral equations with respect to the new unknown functions Dr ðkÞ and D3 ðkÞ
þ1
þ1
Z
i ðkqÞkni di y i ðkqÞkni di sin 1kdk: ½w 0
Dr ðkÞ cos 1kdk ¼ 0;
ðb < 1 < 1Þ;
D3 ðkÞ sin 1kdk ¼ 0;
ðb < 1 < 1Þ:
0
Taking into account Eqs. (31) and (32), from the boundary coni ðkÞ in the ditions (22) under q ¼ q , we determine the functions y i ðkÞði ¼ 1; 2Þ terms of w
1 ðkqÞ þ e22 ðkqÞw 2 ðkqÞ; 1 ðkqÞ ¼ e12 ðkqÞw y 1 ðkqÞ e21 ðkqÞw 2 ðkqÞ; 2 ðkqÞ ¼ e11 ðkqÞw y
þ1
ðq ¼ q Þ;
ð33Þ
where
n1 d1 p1 g 1 ðkqÞ þ n1 d1 p1 e1 ðkqÞ ; n1 d1 p2 e2 ðkqÞ n2 d2 p1 e1 ðkqÞ n d p g ðkqÞ þ n2 d2 p1 e1 ðkqÞ e21 ðkqÞ ¼ 1 1 2 2 ; n1 d1 p2 e2 ðkqÞ n2 d2 p1 e1 ðkqÞ n d p g ðkqÞ þ n1 d1 p2 e2 ðkqÞ e12 ðkqÞ ¼ 2 2 1 1 ; n1 d1 p2 e2 ðkqÞ n2 d2 p1 e1 ðkqÞ n d p g ðkqÞ þ n2 d2 p2 e2 ðkqÞ e22 ðkqÞ ¼ 2 2 2 2 : n1 d1 p2 e2 ðkqÞ n2 d2 p1 e1 ðkqÞ
e11 ðkqÞ ¼
ð34Þ
ð35Þ
g11 ðkÞ ¼
ð36Þ
where
pffiffiffiffiffi pffiffiffiffiffi I1 ð n1 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi f1 ðkÞ ¼ p1 g 1 ðkÞ þ p1 e1 ðkÞe12 ðkq Þ I 1 ð n 1 kÞ K 1 ð n 1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I 1 ð n1 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi ; p2 e2 ðkÞe11 ðkq Þ I1 ð n1 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi f2 ðkÞ ¼ p2 g 2 ðkÞ þ p1 e1 ðkÞe22 ðkq Þ I 1 ð n 2 kÞ K 1 ð n 1 kq Þ pffiffiffiffiffi pffiffiffiffiffi n k I ð q Þ K 1 ð n2 kÞ 1 2 pffiffiffiffiffi pffiffiffiffiffi ; p2 e2 ðkÞe21 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n1 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi h1 ðkÞ ¼ n1 d1 n1 d1 e12 ðkq Þ I1 ð n1 kÞ K 1 ð n1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I 1 ð n1 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi ; þ n2 d2 e11 ðkq Þ I1 ð n1 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi h2 ðkÞ ¼ n2 d2 n1 d1 e22 ðkq Þ I1 ð n2 kÞ K 1 ð n1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I ð n 1 2 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi : þ n2 d2 e21 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ
0
Here f1 ðkÞfr2 ðkÞ z2 ðkÞg f2 ðkÞfr1 ðkÞ z1 ðkÞg ; fx1 ðkÞ q1 ðkÞgfr2 ðkÞ z2 ðkÞg fx2 ðkÞ q2 ðkÞgfr1 ðkÞ z1 ðkÞg f1 ðkÞfx2 ðkÞ q2 ðkÞg f2 ðkÞfx1 ðkÞ q1 ðkÞg k; w12 ðkÞ ¼ fx1 ðkÞ q1 ðkÞgfr2 ðkÞ z2 ðkÞg fx2 ðkÞ q2 ðkÞgfr1 ðkÞ z1 ðkÞg h1 ðkÞfr2 ðkÞ z2 ðkÞg h2 ðkÞfr1 ðkÞ z1 ðkÞg k; w21 ðkÞ ¼ fx1 ðkÞ q1 ðkÞgfr2 ðkÞ z2 ðkÞg fx2 ðkÞ q2 ðkÞgfr1 ðkÞ z1 ðkÞg h1 ðkÞfx2 ðkÞ q2 ðkÞg h2 ðkÞfx1 ðkÞ q1 ðkÞg w22 ðkÞ ¼ k2 ; fx1 ðkÞ q1 ðkÞgfr2 ðkÞ z2 ðkÞg fx2 ðkÞ q2 ðkÞgfr1 ðkÞ z1 ðkÞg w11 ðkÞ ¼
ð40Þ
x1 ðkÞ ¼ g11 ðkÞn2 g21 ðkÞn1 ;
Here
n1 d1 f1 ðkÞ p1 g 1 ðkÞh1 ðkÞ ; n1 d1 p1 g 2 ðkÞ n2 d2 p1 g 1 ðkÞ n d f ðkÞ p1 g 1 ðkÞh2 ðkÞ g12 ðkÞ ¼ 1 1 2 ; n1 d1 p2 g 2 ðkÞ n2 d2 p1 g 1 ðkÞ n d f ðkÞ p2 g 2 ðkÞh1 ðkÞ g21 ðkÞ ¼ 2 2 1 ; n1 d1 p2 g 2 ðkÞ n2 d2 p1 g 1 ðkÞ n d f ðkÞ p2 g 2 ðkÞh2 ðkÞ g22 ðkÞ ¼ 2 2 2 ; n1 d1 p2 g 2 ðkÞ n2 d2 p1 g 1 ðkÞ
ð39Þ
þ1
where
Taking into account Eqs. (31) and (32), from the boundary coni ðkÞ in the terms of ditions (25) we determine the functions v i ðkÞði ¼ 1; 2Þ as follows w
1 ðkÞ g22 ðkÞw 2 ðkÞ; 1 ðkÞ ¼ g21 ðkÞw v v2 ðkÞ ¼ g11 ðkÞw1 ðkÞ þ g12 ðkÞw2 ðkÞ:
Z
x2 ðkÞ ¼ g12 ðkÞn2 g22 ðkÞn1 ;
z1 ðkÞ ¼ g11 ðkÞm2 ½g 2 ðkÞ þ l2 n2 g21 ðkÞm1 ½g 1 ðkÞ þ l1 n1 ; z2 ðkÞ ¼ g12 ðkÞm2 ½g 2 ðkÞ þ l2 n2 g22 ðkÞm1 ½g 1 ðkÞ þ l1 n1 ; r 1 ðkÞ ¼ m1 fg 1 ðkÞ þ l1 n1 g þ m1 fe1 ðkÞ l1 n1 ge12 ðkq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kÞ K 1 ð n1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi ; m2 fe2 ðkÞ l2 n2 ge11 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ r 2 ðkÞ ¼ m2 fg 2 ðkÞ þ l2 n2 g þ m1 fe1 ðkÞ l1 n1 ge22 ðkq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kÞ K 1 ð n1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi ; m2 fe2 ðkÞ l2 n2 ge21 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi n k I ð q Þ K 1 ð n1 kÞ 1 1 pffiffiffiffiffi pffiffiffiffiffi q1 ðkÞ ¼ n1 þ n1 e12 ðkq Þ I 1 ð n 1 kÞ K 1 ð n 1 kq Þ pffiffiffiffiffi pffiffiffiffiffi n1 kq Þ K 1 ð n2 kÞ I ð 1 pffiffiffiffiffi pffiffiffiffiffi ; þ n2 e11 ðkq Þ I1 ð n1 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi q2 ðkÞ ¼ n2 n1 e22 ðkq Þ I 1 ð n 2 kÞ K 1 ð n 1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I ð n2 kq Þ K 1 ð n2 kÞ 1 pffiffiffiffiffi pffiffiffiffiffi : þ n2 e21 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ
ð41Þ
The obtained paired integral equations are reduced to a system of linear algebraic equations by use of the series expansion method.
ð37Þ
5. Solution of the system of paired integral equations To solve the system of the paired integral equations, we choose a solution in a form such that the integral Eq. (39) specified over the interval b < 1 < 1 are satisfied identically. The remaining equations are reduced to a system of algebraic equations. To do this we expand the unknown functions Dr ðkÞ and D3 ðkÞ into infinite series of Bessel functions of the first kind J i and with the unknown coefficients aj and bj [15]
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A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
Dr ðkÞ ¼
1 X
aj k1 J 2jþ1 ðbkÞ;
D3 ðkÞ ¼
j¼0
1 X
bj k1 J2jþ2 ðbkÞ:
ð42Þ
þ1
(
j¼0
0
Z
þ1 X
aj w11 ðkÞJ 2jþ1 ðbkÞ þ
j¼0
þ1
0
þ1 X
) bj w12 ðkÞJ2jþ2 ðbkÞ k1 cos 1kdk ¼ 0;
j¼0
ð0 6 1 < bÞ ( ) þ1 þ1 X X aj w21 ðkÞJ 2jþ1 ðbkÞ þ bj w22 ðkÞJ2jþ2 ðbkÞ k1 sin 1kdk ¼ 0: j¼0
Using the representations of the functions cos 1k and sin 1k in the form of series as follows [15]
sin 1k ¼
i¼0 1 X
ð44Þ 2J 2iþ1 ðbkÞ sinð2i þ 1Þu;
u ¼ arcsinð1=bÞ;
j¼0
aj Q 11 ij ðbkÞ þ
bj Q 12 ij ðbkÞ ¼ 0;
i ¼ 0; 1; 2; . . . ;
j¼0
Xþ1
a Q 21 ðbkÞ þ j¼0 j ij
þ1 X
ð45Þ bj Q 22 ij ðbkÞ ¼ 0;
i ¼ 0; 1; 2; . . . ;
Q 21 ij ðbkÞ ¼ Q 22 ij ðbkÞ ¼ Q 12 ij ðbkÞ ¼
Z
þ1
w11 ðkÞk1 J 2i ðbkÞJ 2jþ1 ðbkÞdk;
Z
0
Z
0
Z
0
k1 k2 k3 ¼ 1;
ð50Þ
q 1 ¼ q2 – q3 :
In the case of an uniform uniaxial compression along the x3 axis, it can be shown that [14]
k1 ¼ k2 ¼ k3
ð51Þ
:
Combining Eqs. (50), (51), and (10), there results
1111 ¼ 4C 10 ; n1 ¼ 1;
3333
1313 ¼ 2C 10 k3 ; ¼ 2C 10 1 þ k3 3
m1 ¼ k3=2 3 ; 3 1 l2 ¼ 2 1 þ k3 : n2 ¼
3=2
1122 ¼ 1133 ¼ 0; k3 3 ;
;
m2 ¼ k3=2 ; 3
For a compressible body, where the elastic potential is given as [14]
1 2
U ¼ kA21 þ lA2 ;
ð53Þ
where A1 , A2 are algebraic invariants of the Green deformation tensor and k, l are the Lame elastic constants, the components of the tensor x for the theory of small initial deformations in the case of an uniform uniaxial compression along the x3 axis are
x1111 ¼ x2222 ¼ k þ 2l; x1221 ¼ x1331 ¼ x2112 ¼ x2332 ¼ l; x1122 ¼ x1133 ¼ x2211 ¼ x2233 ¼ x3311 ¼ x3322 ¼ k; x1212 ¼ x1313 ¼ x2121 ¼ x2323 ¼ x3131 ¼ x3232 ¼ l; x3113 ¼ l þ r033 ; x3223 ¼ l þ r033 ; x3333 ¼ l þ r033 : ð54Þ
þ1
w21 ðkÞk1 J 2iþ1 ðbkÞJ 2jþ1 ðbkÞdk;
For the case of axially symmetric loading, i.e, k1 ¼ k2 – k3 , there results
þ1
w22 ðkÞk1 J 2iþ1 ðbkÞJ 2jþ2 ðbkÞdk; þ1
0
ð52Þ
j¼0
12 21 22 where the values Q 11 ij ðbkÞ, Q ij ðbkÞ, Q ij ðbkÞ, and Q ij ðbkÞ denote the following expression
Q 11 ij ðbkÞ ¼
S0bb ¼ 2C 10 þ k2 b p0 ;
l1 ¼ 1;
and equating the coefficients of the harmonics in Eq. (43) to zero, we obtain the following homogeneous system of linear algebraic equations with respect to the unknown aj and bj þ1 X
ð49Þ
2 lij ¼ p0 k2 i kj ;
3113 ¼ 1331 ¼ 2C 10 ;
ei J2i ðbkÞ cos 2iu; e0 ¼ 1; ei ¼ 2; ði P 1Þ;
i¼0
Xþ1
Aib ¼ 2p0 dib k4 b ;
1=2
ð43Þ
cos 1k ¼
e01 þ e02 þ e03 ;
where C 10 is a constant of the body; e0i ¼ 1Þ=2 are the principal values of the Green’s deformation tensor. The values lij , Aib in Eq. (12) for the solid with the Treloar potential (49) are
j¼0
1 X
ðk2i
Direct substitution of expansions (42) into (39) causes them to be satisfied identically [15], while Eq. (38) take the form
Z
W 0 ¼ 2C 10 A1 ¼ 2C 10
w12 ðkÞk1 J 2i ðbkÞJ 2jþ2 ðbkÞdk;
r033 ¼
ði; j ¼ 0; 1; 2; . . .Þ: ð46Þ
The homogeneous system of algebraic Eq. (45) has a non-trivial solution if the determinant of the system equals to zero, i.e for incompressible solids
det kdkl ðb; q ; k3 ; Þk ¼ 0 ðk; l ¼ 1; 2; 3; . . .Þ;
ð47Þ
and compressible solids
det kdkl ðb; q ; k3 ; xÞk ¼ 0 ðk; l ¼ 1; 2; 3; . . .Þ:
ð48Þ
Here the components of the tensors x and for the theory of large and small initial deformations are defined are determined by Eqs. (7), (8), (10), and (11), respectively. Therefore, we have reduced the linearized problem formulated above to an eigen-value problem for Eq. (45) in terms of the parameter k3 < 1, which should be solved numerically. 6. Some models of solid The components of the tensors x and in Eqs. (47) and (48) are determined accordingly to the type of the elastic potential. The elastic potential (Treloar potential) for incompressible Neo– Hookean type solids is given as [14]
lð3k þ 2lÞ ðk3 1Þ: kþl
ð55Þ
Substituting Eqs. (54) and (55) into Eqs. (18) and (20), the results are
k 3k þ 2l ðk3 1Þ; m2 ¼ 1; kþl kþl k 3k þ 2l ðk3 1Þ; d2 ¼ 2; d1 ¼ 2 þ kþl kþl 2k þ l 3k þ 2l 3k þ 2l ðk3 1Þ; p2 ¼ 2 þ ðk3 1Þ; p1 ¼ 2 þ kþl kþl kþl 1 1 l1 ¼ ; l2 ¼ ; 1 2kþl 3kþ2l 1 3kþ2l 1 þ 2 kþl kþl ðk3 1Þ 1 þ 2 kþl ðk3 1Þ m1 ¼ 1 þ
n1 ¼ 1 þ
3k þ 2l ðk3 1Þ; k þ 2l
n2 ¼ 1 þ
l 3k þ 2l ðk3 1Þ: k þ l k þ 2l ð56Þ
Described is a method for studying the initial phase of failure of a circular cylindrical bar weakened with a coaxial subsurface cylindrical crack in uniform axial compression along the defect. The analytical solution has been carried out in common form for different models of materials (elastic, composite), the theory of large or small initial strains. The model of solid has to be specified on the final stage of the examination, i.e. the numerical solution.
A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
The linearized problem formulated above is reduced to an eigen-value problem for Eq. (45) in terms of the parameter k3 < 1 in common form for different models of materials, which should be solved numerically.
References [1] M.K. Kassir, G.C. Sih, Mechanics of Fracture. Three-Dimensional Crack Problems, vol. 2, Noordhoff, Leyden, 1975. [2] G.P. Cherepanov, Mechanics of Brittle Fracture, Nauka, Moscow, 1974. [3] A.A. Wells, Application of fracture mechanics at and beyond general yielding, Brit. Weld. J. 10 (1963) 563–570. [4] A.N. Guz, On one criterion of fracture of solids in compression along the cracks. Spatial problem, Dokl. Academ. Nauk USSR 261 (1981) 42–45. [5] A.N. Guz, V.M. Nazarenko, Symmetric failure of the half-space with pennyshaped cracks in compression, Theor. Appl. Fract. Mech. 3 (1985) 233–245. [6] V.L. Bogdanov, A.N. Guz, V.M. Nazarenko, Nonaxisymmetric compressive failure of a circular crack parallel to a surface of half space, Theor. Appl. Fract. Mech. 22 (1995) 239–247.
207
[7] A.N. Guz, Yu.I. Khoma, Stability of an infinite solid with a circular cylindrical crack under compression using the Treloar potential, Theor. Appl. Fract. Mech. 39 (2003) 275–280. [8] A.N. Guz, Yu.I. Khoma, Integral formulation for a circular cylindrical cavity in infinite solid and a finite length coaxial cylindrical crack under compressed axially, Theor. Appl. Fract. Mech. 45 (2006) 204–211. [9] A.N. Guz, T.V. Rudnitskii, Contact interaction of an elastic punch and an elastic half-space with initial (residual) stresses, Int. Appl. Mech. 43 (2007) 1325– 1335. [10] S.Yu. Babich, Yu.P. Glukhov, A.N. Guz, A dynamic problem for a prestressed compressible layered half-space, Int. Appl. Mech. 44 (2008) 388–405. [11] S.Yu. Babich, Yu.P. Glukhov, A.N. Guz, Using complex potentials to determine the reaction of a prestressed two-layer elastic half-space to a moving load, Int. Appl. Mech. 44 (2008) 81–492. [12] A.N. Guz, Mechanics of Brittle Fracture of Materials with Initial Stresses, Naukova Dumka, Kyiv, 1983. [13] A.N. Guz, Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies, Springer-Verlag, 1999. [14] A.N. Guz, M.Sh. Dyshel, V.M. Nazarenko, Fracture and Stability of Materials with Cracks, vol. 4, Kiev, Book 1, 1992. [15] I.S. Gradshtein, I.M. Ryzhik, Tables of Integrals, Sums, Series and Products, 4th ed., Moscow, 1963.
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Axial compression of circular cylindrical bar with coaxial subsurface cylindrical crack of finite length A.N. Guz, Yu.I. Khoma * Department of Dynamics and Stability of Continuum Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Nesterov Str. 3, Kyiv 03057, Ukraine
a r t i c l e
i n f o
Article history: Available online 20 May 2009 Keywords: Local stability loss Axial compression Coaxial subsurface cylindrical crack Loading along a defect Non-classical fracture mechanics
a b s t r a c t A fracture stability of a circular cylindrical bar with a coaxial surface cylindrical crack subjected to an axial compression is considered. A state of subcritical initial strain is assumed. A non-classical fracture criterion is based on a local stability loss near the defect. The theory of integral Fourier transforms and series expansions are used to reduce these problems to a system of paired integral equations and then to a system of linear algebraic equations with respect to the contraction parameter. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The problems of fracture mechanics of materials under compression along the cracks represent a special class of problems because they cannot be adequately described by the relations of the linear fracture mechanics. In the case of the forces are applied along the crack the stress intensity factors calculated on the basis of the linear fracture mechanics are equal to zero [1]. Consequently, the classical fracture criterion of the Griffith–Irwin, the crack propagation criterion and also their generalizations used in the linear fracture mechanics [1–3] are not applicable to this scheme of loading. It was shown as early as in fundamental studies carried out by Euler on the critical force for compressed bars that the loss of stability, i.e. when the load reaches the critical force can be regarded as the beginning of failure. Therefore, in loading along the cracks it is rational to use, the stability criterion within the framework of the linearized stability theory proposed in [4]. According to this criterion the fracture process is caused by a local stability loss near the crack and quantified in terms of a critical eigenvalue. The local instability of the materials is examined on the basis of relations of the three-dimensional linearized mechanics of deformed bodies. While an influence of the loads acting along the cracks on the fracture parameters have been confirmed experimentally. Symmetric and nonaxisymmetric compressive failure of the half-space with cracks were analyzed in [5,6]. Fracture problems for an infinite solid with a circular cylindrical crack under compression have been discussed previously [7] and for a cylindrical * Corresponding author. E-mail address: [email protected] (Yu.I. Khoma). 0167-8442/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2009.05.001
cavity in infinite solid and a finite length coaxial cylindrical crack compressed axially [8]. Some fracture problems have been considered in [9–11]. In this paper we describe a method for studying the initial phase of a failure of a circular cylindrical bar weakened with a coaxial subsurface cylindrical crack in uniform axial compression along the defect. The analytical solution has been carried out in common form for different models of materials (elastic, composite), the theory of large or small initial strains. The model of solid has to be specified on the final stage of the examination, i.e. the numerical solution.
2. Basic equations To solve the mechanical fracture problem a common approach is used for compressible and incompressible elastic bodies with an arbitrary elastic potential, both for large subcritical deformations and for two variants of theory of small subcritical deformations. The case of unequal roots of the characteristic equation [12,13] is considered. In formulating the linearized problem, Lagrangian coordinates xj that coincide with Cartesian coordinates in the undeformed state, as well as perturbations in the displacement vector uj and stress tensor t relative to unit area of the body in the undeformed state are used. It assumed that a circular cylindrical bar weakened with a coaxial subsurface cylindrical crack is subjected to a uniform uniaxial compression along the x3 axis. The initial forces applied at infinity create a homogeneous stressed-deformed subcritical state in a neighborhood of the defect, which is determined by the following relationships [14]
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A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
u0j ¼ djm ðkj 1Þxm ; S011
¼
S022
¼ 0;
S033
kj ¼ const;
k1 ¼ k2 – k3 ;
ðk3 < 1Þ;
¼ const – 0:
ð1Þ
S0ij
Here are the components of the symmetric stress tensor relative to a unit area of the body in the underformed state, u0j are the displacements corresponding to the initial stresses S0ij , kj ðj ¼ 1; 2; 3Þ are the contractions along the axes, and dij is the Kronecker symbol. A superscript ‘‘0” in Eq. (1) and in the following indicates quantities pertaining to the initial state. The corresponding perturbed quantities are indicated without any additional indices. The linearized equilibrium equations in the underformed state coordinates for compressible and incompressible bodies are [14]
Lma ua ¼ 0;
@2 ; @xi @xb @ua qma ¼ 0; @xm
Lma ¼ ximab
Lma
ð3Þ
2e0ij ¼ dij ðk2j 1Þ;
ð4Þ
while in the theory of small initial deformations are related to the contractions kj by
e0ij ¼ dij ðkj 1Þ:
ð5Þ
In the case of an uniform subcritical state (1) the linearized equilibrium Eq. (2) for compressible solids have the form [14]
@ 2 ua ¼ 0: @xi @xb
ð6Þ
Here the components of the tensor x for the theory of large and small initial deformations are given by [14], respectively
xijab ¼ ka kj ½dij dab Aib þ ð1 dij Þðdia djb þ dib dja Þlij þ dib dja S0bb ; xijab ¼ dij dab Aib þ ð1 dij Þðdia djb þ dib dja Þlij þ djb dja S0bb ; 0 Sbb 0 rbb :
ð7Þ
@ 2 ua @p þ qij ¼ 0; @xi @xi @xb
qij
@uj ¼ 0; @xi
ð8Þ
ð9Þ
where the components of the tensor for the theory of large and small initial deformations are given by [14], respectively
ijab ¼ kj ka ½dij dab Aib þ ð1 dij Þðdia djb þ dib dja Þlij þ dib dja S0bb ; ijab ¼ dij dab Aib þ ð1 dij Þðdia djb þ dib dja Þlij þ
djb dja S0bb :
r0bb ¼
ð13Þ
e0i ¼ ki 1:
j
For a hyperelastic body, where the elastic potential is given as a function 0 0 of0 the principal values of the Green deformation tensor U0 e1 ; e2 ; e3 in the theory of large initial deformations S0ij , Aib , and lij for compressible solids have the form [14]
@ 2 U0 ; @ e0i @ e0j
S0ii ¼
!
1 @ @ lij ¼ U0 : ð14Þ @ e0i @ e0j 2 e0i e0j
@ U0 ; @ e0i
The system of linearized equilibrium Eqs. (6) and (9) for compressible and incompressible solids respectively consists of partial differential equations with constant coefficients and in the case of uniform subcritical deformations, its general solutions can be constructed in terms of potential functions. Let the coordinate system xm is now chosen to be a cylindrical polar system ðr; h; x3 Þ with the axis x3 as one of the cylindrical bar. The general solutions of the static linearized Eqs. (6) and (9) for axially symmetric problems in the case of unequal roots n01 – n02 of the governing characteristic equation are given in terms of two potential functions ui ðr; x3 Þði ¼ 1; 2Þ which obey the equations
! @2 1 @ @2 þ þ ni 2 ui ðr; x3 Þ ¼ 0 ði ¼ 1; 2Þ; @r2 r @r @x3
ð15Þ
and have the following form [12]
ur ¼
@ u1 @ u2 þ ; @r @r
u3 ¼ m1
@ u1 @ u2 þ m2 : @r @r
ð16Þ
The representations for the components of the Kirchhoff stress tensor t on the cylindrical surface are given by [7]
"
In the case of an uniform subcritical state (1) the linearized equilibrium Eq. (2) for incompressible solids are [14]
ijab
i
Aij ¼
The components of the Green deformation tensor e0 in the theory of large initial deformations are related to the contractions kj by [14]
xijab
lij
@ 2 U0 ; @ e0i @ e0j
@ U0 þ p0 ; @ e0i ! 1 @ @ ¼ U0 ; @ e0i @ e0j 2 e0 e0
Aij ¼
ð2Þ
@u4 ¼ 0; @xa @2 ¼ imab ; u4 p: @xi @xb
Lma ua þ qam
The expressions for the theory of small initial deformations are as follows
ð10Þ ð11Þ
# @ 2 u1 @ 2 u2 ; t r3 ¼ C 44 d1 þ d2 @r@x3 @r@x3 " ! ! # @ @2 @ @2 t rr ¼ C 44 p1 l1 u1 p2 l2 u2 : þ þ r@r @x23 r@r @x23
ð17Þ
The quantities in Eqs. (16) and (17) depend on the choice of material (form of the elastic potential) and the initial stress-deformed state. For compressible solids, they can be written in terms of the components of the tensor x as
mi ¼ ðx1111 ni x3113 Þðx1133 þ x1313 Þ1 ;
C 44 ¼ x1313 ;
di ¼ 1 þ mi x1331 x1 1313 ;
pi ¼ ðx1111 ni þ x1133 mi Þx1 1313 ;
ð18Þ
The values lij , Aib and, consequently, components of the tensors x and are determined accordingly to the type of the elastic potential. For a hyperelastic body, where the elastic potential is given as a function of the principal values of the Green deformation tensor U0 e01 ; e02 ; e03 in the theory of large initial deformations the values 0 Sij , Aib , and lij for incompressible solids are defined as [14]
For incompressible solids, they are given in terms of the components of the tensor as follows
@ 2 U0 Aij ¼ 0 0 2p0 dij k4 j ; @ ei @ ej
1 lj ¼ ð1111 1122 Þ½q11 q1 33 ð1133 1313 Þnj 1133 mj þ 3113
lij
1
@ U0 S0ii ¼ 0 þ k2 i p0 ; @ ei !
C 44 ¼ 1313 ; pj ¼
mj ¼ q11 q1 33 nj ;
½q11 q1 33 ð1133
ði ¼ 1; 2Þ:
dj ¼ 1 þ 1331 1 1313 mj ;
1313 Þnj 1133 mj þ 3113 1 1313 ;
ðj ¼ 1; 2Þ;
@ @ 2 1 ¼ 0 U0 p0 k2 i kj ; k1 k2 k3 ¼ 1; qi ¼ ki : 0 0 0 @ @ e ej 2 e e i i
lj ¼ ðx1111 x1122 Þðx1111 ni þ x1133 mi Þ
1
j
ð12Þ
ð19Þ where the roots n1 and n2 of the governing characteristic equation are for compressible solids
204
n1;2
A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ c c2 x3333 x3113 x1 1111 x1331 ;
2x1111 x1331 c ¼ x1111 x3333 þ x1331 x3113 ðx1133 þ x1313 Þ2 ; ð20Þ and for incompressible solids
n1;2 ¼ c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c2 q233 q2 11 3113 1331 ;
where vi ðkÞ, wi ðkÞ and yi ðkÞ ði ¼ 1; 2Þ are unknown weighting functions; I0 and K 0 are the modified Bessel functions. When there is no the lateral cylindrical surface ðr ¼ b þ hÞ, i.e. h ! þ1 (infinite solid), it is necessary to set wi ðkÞ 0 in the integral representations [7]. Using expansion (26), we obtain the components of displacements (16) and stresses (17) for the region 1
ð21Þ
2q211 1331 c ¼ q211 3333 þ q233 1111 2q11 q33 ð1133 þ 1313 Þ:
ð1Þ
C 44 bur ¼
2 Z X
ð1Þ
C 44 bu3 ¼
Consider a circular cylindrical bar of radius b þ h containing a subsurface circular cylindrical crack of radius b and length 2a, located on the lateral part of the cylindrical surface fr ¼ b; 0 6 h < 2p; a 6 x3 6 ag, whose axis coincides with the bar axis x3 and which is compressed along the x3 axis by uniform forces applied at infinity. Here ðr; h; x3 Þ be the cylindrical coordinates and h is the distance between the lateral part of the surface of the cylindrical bar and the surface of the circular cylindrical crack. As a result of the compression parallel to the crack axis, a homogeneous initial stress and strain state (1) occurs near the crack-like defect. Separate an area of the bar into two regions: r < b and b < r < b þ h. Denote by the superscript 1 the quantities referred to the region r < b and by the superscript 2 the ones referred to the region b < r < b þ h. Assume that the edges of the crack, as well as the surface of the bar, are free of stresses. Here it assumed that the surfaces of the defect do not come into a contact interaction. In addition, at the boundary of regions 1ðr < bÞ and 2ðb < r < b þ hÞ the continuity conditions for the displacements and stresses must be satisfied outside the crack. Given these remarks, consider the boundary conditions for the linearized problem in the following form
2 ð1Þ
b t r3 ¼ 2
ð2Þ
t r3 ¼ 0;
when q ¼ q ;
ð1Þ ð2Þ ¼ ¼ 0; t r3 ¼ t r3 ¼ 0; ð1Þ ð2Þ ð1Þ ð2Þ ur ur ¼ 0; u3 u3 ¼ 0; ð1Þ ð2Þ ð1Þ ¼ tð2Þ t r3 ¼ tr3 ; when t rr rr ;
ð1Þ t rr
tð2Þ rr
0 6 j1j < þ1; j1j < b;
when q ¼ 1; when q ¼ 1;
j1j > b;
q ¼ 1; 0 6 j1j < þ1:
ð22Þ
i¼1
0
2 Z X
þ1
þ1
pffiffiffiffi vi ðkÞmi kI0 ð ni kqÞ sin 1kdk;
pffiffiffiffi pffiffiffiffi vi ðkÞdi ni k2 I1 ð ni kqÞ sin 1kdk;
0
2 Z X
þ1
pffiffiffiffi pffiffiffiffi li ni pffiffiffiffi vi ðkÞpi k kI0 ð ni kqÞ I1 ð ni kqÞ cos 1kdk:
q
0
i¼1
ð28Þ Similarly, for the region 2 the components of displacements (16) and stresses (17) are ð2Þ
C 44 bur ¼
2 Z X
ð1Þ
C 44 bu3
þ1 0
i¼1
2 ð2Þ
b t r3
0
sin 1kdk; 2 Z þ1 X
pffiffiffiffi pffiffiffiffi pffiffiffiffi ¼ di ni k2 wi ðkÞI1 ð ni kqÞ yi ðkÞK 1 ð ni kqÞ 0
i¼1
b t ð2Þ rr
pffiffiffiffi pffiffiffiffi pffiffiffiffi ni k wi ðkÞI1 ð ni kqÞ yi ðkÞK 1 ð ni kqÞ
cos 1kdk; 2 Z þ1 X
pffiffiffiffi pffiffiffiffi ¼ mi k wi ðkÞI0 ð ni kqÞ þ yi ðkÞK 0 ð ni kqÞ i¼1
2
ð2Þ t rr ¼ 0;
Z
2 X
i¼1
b t ð1Þ rr ¼
pffiffiffiffi pffiffiffiffi vi ðkÞ ni kI1 ð ni kqÞ cos 1kdk;
0
i¼1
3. Problem formulation
þ1
sin 1kdk pffiffiffiffi 2 Z þ1 X pffiffiffiffi li ni pffiffiffiffi ¼ wi ðkÞpi k kI0 ð ni kqÞ I1 ð ni kqÞ
q
0
i¼1
pffiffiffiffi pffiffiffiffi pffiffiffiffi li n i þ yi ðkÞpi k kK 0 ð ni kqÞ þ K 1 ð ni kqÞ cos 1kdk :
ð23Þ
q
ð24Þ
ð29Þ
ð25Þ
where q ¼ r=b; q ¼ ðb þ hÞ=b; 1 ¼ x3 =b and b ¼ a=b are dimensionless quantities and the superscripts 1 and 2 refer to regions 1 and 2, respectively. Thus, the boundary conditions include conditions at the surface of the bar (22), conditions at the edges of the cracks (23), and continuity conditions for the displacements (24) and stresses (25) at the interface between the regions 1ðr < bÞ and 2ðb < r < b þ hÞ.
Introducing the following notations
pffiffiffiffi k i ðkqÞ ¼ vi ðkÞ pffiffiffiffi I1 ð ni kqÞ; v ni pffiffiffiffi k i ðkqÞ ¼ wi ðkÞ pffiffiffiffi I1 ð ni kqÞ; w ni
pffiffiffiffi pffiffiffiffi I0 ð ni kqÞ li ni ni k pffiffiffiffi ; I1 ð ni kqÞ q pffiffiffiffi pffiffiffiffi K 0 ð ni kqÞ li ni pffiffiffiffi þ ei ðkqÞ ¼ ni k ; q K 1 ð ni kqÞ
g i ðkqÞ ¼
pffiffiffiffi k i ðkqÞ ¼ yi ðkÞ pffiffiffiffi K 1 ð ni kqÞ; y ni ð30Þ
4. Derivation of dual integral equations To obtain a system of dual integral equations, the theory of the Fourier integral transforms is used. Represent the potential functions u1 ðr; x3 Þ and u2 ðr; x3 Þ separately in each of the regions 1ðr < bÞ and 2ðb < r < b þ hÞ applying the subscripts 1 and 2 to them in the corresponding regions, in the form of Fourier integral transforms
C 44 u C 44 u
ð1Þ i ð2Þ i
¼
Z
1
pffiffiffiffi vi ðkÞ I0 ð ni kqÞ cos 1kdk;
ði ¼ 1; 2Þ;
ð26Þ
0
¼
Z
0
þ
1
Eqs. (28) and (29) can be rewritten in the simple form ð1Þ
C 44 bur ¼ ð1Þ
C 44 bu3 ¼
0
pffiffiffiffi yi ðkÞ K 0 ð ni kqÞ cos 1kdk;
2 Z X
2
b t ð1Þ rr ¼
2 Z X i¼1
b t r3 ¼ ði ¼ 1; 2Þ;
ð27Þ
i ðkqÞni cos 1kdk; v
i ðkqÞmi k1 g i ðkqÞ þ li ni q1 sin 1kdk; v
þ1
i ðkqÞpi g i ðkqÞ cos 1kdk; v
0
2 Z X i¼1
þ1
0
i¼1
2 ð1Þ
1
þ1 0
i¼1
pffiffiffiffi wi ðkÞ I0 ð ni kqÞ cos 1kdk
Z
2 Z X
þ1
i ðkqÞkni di sin 1kdk; v
0
ð31Þ
205
A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
and ð2Þ
C 44 bur ¼
2 X
Z
i¼1 ð2Þ C 44 bu3
2
b tð2Þ rr
¼
i ðkqÞni y i ðkqÞni cos 1kdk; ½w
0
þ1
i ðkqÞmi k1 fg i ðkqÞ þ li ni q1 g w
i¼1
i ðkqÞmi k1 ei ðkqÞ li ni q1 sin 1kdk þy Z 2 þ1 X i ðkqÞpi g i ðkqÞ þ y i ðkqÞpi ei ðkqÞ cos 1kdk ¼ ½w
¼
þ1
0
Z ð32Þ
fw11 ðkÞDr ðkÞ þ w12 ðkÞ D3 ðkÞg cos 1kdk ¼ 0;
ð0 6 1 < bÞ;
fw21 ðkÞDr ðkÞ þ w22 ðkÞ D3 ðkÞg sin 1kdk ¼ 0;
ð0 6 1 < bÞ;
þ1
0
ð38Þ
0
and
2 Z X i¼1
Z
0
2 Z X
i¼1 2 ð2Þ b tr3
Without going into details the rest boundary conditions (23) and (24) yields the following system of the dual integral equations with respect to the new unknown functions Dr ðkÞ and D3 ðkÞ
þ1
þ1
Z
i ðkqÞkni di y i ðkqÞkni di sin 1kdk: ½w 0
Dr ðkÞ cos 1kdk ¼ 0;
ðb < 1 < 1Þ;
D3 ðkÞ sin 1kdk ¼ 0;
ðb < 1 < 1Þ:
0
Taking into account Eqs. (31) and (32), from the boundary coni ðkÞ in the ditions (22) under q ¼ q , we determine the functions y i ðkÞði ¼ 1; 2Þ terms of w
1 ðkqÞ þ e22 ðkqÞw 2 ðkqÞ; 1 ðkqÞ ¼ e12 ðkqÞw y 1 ðkqÞ e21 ðkqÞw 2 ðkqÞ; 2 ðkqÞ ¼ e11 ðkqÞw y
þ1
ðq ¼ q Þ;
ð33Þ
where
n1 d1 p1 g 1 ðkqÞ þ n1 d1 p1 e1 ðkqÞ ; n1 d1 p2 e2 ðkqÞ n2 d2 p1 e1 ðkqÞ n d p g ðkqÞ þ n2 d2 p1 e1 ðkqÞ e21 ðkqÞ ¼ 1 1 2 2 ; n1 d1 p2 e2 ðkqÞ n2 d2 p1 e1 ðkqÞ n d p g ðkqÞ þ n1 d1 p2 e2 ðkqÞ e12 ðkqÞ ¼ 2 2 1 1 ; n1 d1 p2 e2 ðkqÞ n2 d2 p1 e1 ðkqÞ n d p g ðkqÞ þ n2 d2 p2 e2 ðkqÞ e22 ðkqÞ ¼ 2 2 2 2 : n1 d1 p2 e2 ðkqÞ n2 d2 p1 e1 ðkqÞ
e11 ðkqÞ ¼
ð34Þ
ð35Þ
g11 ðkÞ ¼
ð36Þ
where
pffiffiffiffiffi pffiffiffiffiffi I1 ð n1 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi f1 ðkÞ ¼ p1 g 1 ðkÞ þ p1 e1 ðkÞe12 ðkq Þ I 1 ð n 1 kÞ K 1 ð n 1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I 1 ð n1 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi ; p2 e2 ðkÞe11 ðkq Þ I1 ð n1 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi f2 ðkÞ ¼ p2 g 2 ðkÞ þ p1 e1 ðkÞe22 ðkq Þ I 1 ð n 2 kÞ K 1 ð n 1 kq Þ pffiffiffiffiffi pffiffiffiffiffi n k I ð q Þ K 1 ð n2 kÞ 1 2 pffiffiffiffiffi pffiffiffiffiffi ; p2 e2 ðkÞe21 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n1 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi h1 ðkÞ ¼ n1 d1 n1 d1 e12 ðkq Þ I1 ð n1 kÞ K 1 ð n1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I 1 ð n1 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi ; þ n2 d2 e11 ðkq Þ I1 ð n1 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi h2 ðkÞ ¼ n2 d2 n1 d1 e22 ðkq Þ I1 ð n2 kÞ K 1 ð n1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I ð n 1 2 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi : þ n2 d2 e21 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ
0
Here f1 ðkÞfr2 ðkÞ z2 ðkÞg f2 ðkÞfr1 ðkÞ z1 ðkÞg ; fx1 ðkÞ q1 ðkÞgfr2 ðkÞ z2 ðkÞg fx2 ðkÞ q2 ðkÞgfr1 ðkÞ z1 ðkÞg f1 ðkÞfx2 ðkÞ q2 ðkÞg f2 ðkÞfx1 ðkÞ q1 ðkÞg k; w12 ðkÞ ¼ fx1 ðkÞ q1 ðkÞgfr2 ðkÞ z2 ðkÞg fx2 ðkÞ q2 ðkÞgfr1 ðkÞ z1 ðkÞg h1 ðkÞfr2 ðkÞ z2 ðkÞg h2 ðkÞfr1 ðkÞ z1 ðkÞg k; w21 ðkÞ ¼ fx1 ðkÞ q1 ðkÞgfr2 ðkÞ z2 ðkÞg fx2 ðkÞ q2 ðkÞgfr1 ðkÞ z1 ðkÞg h1 ðkÞfx2 ðkÞ q2 ðkÞg h2 ðkÞfx1 ðkÞ q1 ðkÞg w22 ðkÞ ¼ k2 ; fx1 ðkÞ q1 ðkÞgfr2 ðkÞ z2 ðkÞg fx2 ðkÞ q2 ðkÞgfr1 ðkÞ z1 ðkÞg w11 ðkÞ ¼
ð40Þ
x1 ðkÞ ¼ g11 ðkÞn2 g21 ðkÞn1 ;
Here
n1 d1 f1 ðkÞ p1 g 1 ðkÞh1 ðkÞ ; n1 d1 p1 g 2 ðkÞ n2 d2 p1 g 1 ðkÞ n d f ðkÞ p1 g 1 ðkÞh2 ðkÞ g12 ðkÞ ¼ 1 1 2 ; n1 d1 p2 g 2 ðkÞ n2 d2 p1 g 1 ðkÞ n d f ðkÞ p2 g 2 ðkÞh1 ðkÞ g21 ðkÞ ¼ 2 2 1 ; n1 d1 p2 g 2 ðkÞ n2 d2 p1 g 1 ðkÞ n d f ðkÞ p2 g 2 ðkÞh2 ðkÞ g22 ðkÞ ¼ 2 2 2 ; n1 d1 p2 g 2 ðkÞ n2 d2 p1 g 1 ðkÞ
ð39Þ
þ1
where
Taking into account Eqs. (31) and (32), from the boundary coni ðkÞ in the terms of ditions (25) we determine the functions v i ðkÞði ¼ 1; 2Þ as follows w
1 ðkÞ g22 ðkÞw 2 ðkÞ; 1 ðkÞ ¼ g21 ðkÞw v v2 ðkÞ ¼ g11 ðkÞw1 ðkÞ þ g12 ðkÞw2 ðkÞ:
Z
x2 ðkÞ ¼ g12 ðkÞn2 g22 ðkÞn1 ;
z1 ðkÞ ¼ g11 ðkÞm2 ½g 2 ðkÞ þ l2 n2 g21 ðkÞm1 ½g 1 ðkÞ þ l1 n1 ; z2 ðkÞ ¼ g12 ðkÞm2 ½g 2 ðkÞ þ l2 n2 g22 ðkÞm1 ½g 1 ðkÞ þ l1 n1 ; r 1 ðkÞ ¼ m1 fg 1 ðkÞ þ l1 n1 g þ m1 fe1 ðkÞ l1 n1 ge12 ðkq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kÞ K 1 ð n1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi ; m2 fe2 ðkÞ l2 n2 ge11 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ r 2 ðkÞ ¼ m2 fg 2 ðkÞ þ l2 n2 g þ m1 fe1 ðkÞ l1 n1 ge22 ðkq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kÞ K 1 ð n1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n2 kÞ pffiffiffiffiffi pffiffiffiffiffi ; m2 fe2 ðkÞ l2 n2 ge21 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi n k I ð q Þ K 1 ð n1 kÞ 1 1 pffiffiffiffiffi pffiffiffiffiffi q1 ðkÞ ¼ n1 þ n1 e12 ðkq Þ I 1 ð n 1 kÞ K 1 ð n 1 kq Þ pffiffiffiffiffi pffiffiffiffiffi n1 kq Þ K 1 ð n2 kÞ I ð 1 pffiffiffiffiffi pffiffiffiffiffi ; þ n2 e11 ðkq Þ I1 ð n1 kÞ K 1 ð n2 kq Þ pffiffiffiffiffi pffiffiffiffiffi I1 ð n2 kq Þ K 1 ð n1 kÞ pffiffiffiffiffi pffiffiffiffiffi q2 ðkÞ ¼ n2 n1 e22 ðkq Þ I 1 ð n 2 kÞ K 1 ð n 1 kq Þ pffiffiffiffiffi pffiffiffiffiffi I ð n2 kq Þ K 1 ð n2 kÞ 1 pffiffiffiffiffi pffiffiffiffiffi : þ n2 e21 ðkq Þ I1 ð n2 kÞ K 1 ð n2 kq Þ
ð41Þ
The obtained paired integral equations are reduced to a system of linear algebraic equations by use of the series expansion method.
ð37Þ
5. Solution of the system of paired integral equations To solve the system of the paired integral equations, we choose a solution in a form such that the integral Eq. (39) specified over the interval b < 1 < 1 are satisfied identically. The remaining equations are reduced to a system of algebraic equations. To do this we expand the unknown functions Dr ðkÞ and D3 ðkÞ into infinite series of Bessel functions of the first kind J i and with the unknown coefficients aj and bj [15]
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A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
Dr ðkÞ ¼
1 X
aj k1 J 2jþ1 ðbkÞ;
D3 ðkÞ ¼
j¼0
1 X
bj k1 J2jþ2 ðbkÞ:
ð42Þ
þ1
(
j¼0
0
Z
þ1 X
aj w11 ðkÞJ 2jþ1 ðbkÞ þ
j¼0
þ1
0
þ1 X
) bj w12 ðkÞJ2jþ2 ðbkÞ k1 cos 1kdk ¼ 0;
j¼0
ð0 6 1 < bÞ ( ) þ1 þ1 X X aj w21 ðkÞJ 2jþ1 ðbkÞ þ bj w22 ðkÞJ2jþ2 ðbkÞ k1 sin 1kdk ¼ 0: j¼0
Using the representations of the functions cos 1k and sin 1k in the form of series as follows [15]
sin 1k ¼
i¼0 1 X
ð44Þ 2J 2iþ1 ðbkÞ sinð2i þ 1Þu;
u ¼ arcsinð1=bÞ;
j¼0
aj Q 11 ij ðbkÞ þ
bj Q 12 ij ðbkÞ ¼ 0;
i ¼ 0; 1; 2; . . . ;
j¼0
Xþ1
a Q 21 ðbkÞ þ j¼0 j ij
þ1 X
ð45Þ bj Q 22 ij ðbkÞ ¼ 0;
i ¼ 0; 1; 2; . . . ;
Q 21 ij ðbkÞ ¼ Q 22 ij ðbkÞ ¼ Q 12 ij ðbkÞ ¼
Z
þ1
w11 ðkÞk1 J 2i ðbkÞJ 2jþ1 ðbkÞdk;
Z
0
Z
0
Z
0
k1 k2 k3 ¼ 1;
ð50Þ
q 1 ¼ q2 – q3 :
In the case of an uniform uniaxial compression along the x3 axis, it can be shown that [14]
k1 ¼ k2 ¼ k3
ð51Þ
:
Combining Eqs. (50), (51), and (10), there results
1111 ¼ 4C 10 ; n1 ¼ 1;
3333
1313 ¼ 2C 10 k3 ; ¼ 2C 10 1 þ k3 3
m1 ¼ k3=2 3 ; 3 1 l2 ¼ 2 1 þ k3 : n2 ¼
3=2
1122 ¼ 1133 ¼ 0; k3 3 ;
;
m2 ¼ k3=2 ; 3
For a compressible body, where the elastic potential is given as [14]
1 2
U ¼ kA21 þ lA2 ;
ð53Þ
where A1 , A2 are algebraic invariants of the Green deformation tensor and k, l are the Lame elastic constants, the components of the tensor x for the theory of small initial deformations in the case of an uniform uniaxial compression along the x3 axis are
x1111 ¼ x2222 ¼ k þ 2l; x1221 ¼ x1331 ¼ x2112 ¼ x2332 ¼ l; x1122 ¼ x1133 ¼ x2211 ¼ x2233 ¼ x3311 ¼ x3322 ¼ k; x1212 ¼ x1313 ¼ x2121 ¼ x2323 ¼ x3131 ¼ x3232 ¼ l; x3113 ¼ l þ r033 ; x3223 ¼ l þ r033 ; x3333 ¼ l þ r033 : ð54Þ
þ1
w21 ðkÞk1 J 2iþ1 ðbkÞJ 2jþ1 ðbkÞdk;
For the case of axially symmetric loading, i.e, k1 ¼ k2 – k3 , there results
þ1
w22 ðkÞk1 J 2iþ1 ðbkÞJ 2jþ2 ðbkÞdk; þ1
0
ð52Þ
j¼0
12 21 22 where the values Q 11 ij ðbkÞ, Q ij ðbkÞ, Q ij ðbkÞ, and Q ij ðbkÞ denote the following expression
Q 11 ij ðbkÞ ¼
S0bb ¼ 2C 10 þ k2 b p0 ;
l1 ¼ 1;
and equating the coefficients of the harmonics in Eq. (43) to zero, we obtain the following homogeneous system of linear algebraic equations with respect to the unknown aj and bj þ1 X
ð49Þ
2 lij ¼ p0 k2 i kj ;
3113 ¼ 1331 ¼ 2C 10 ;
ei J2i ðbkÞ cos 2iu; e0 ¼ 1; ei ¼ 2; ði P 1Þ;
i¼0
Xþ1
Aib ¼ 2p0 dib k4 b ;
1=2
ð43Þ
cos 1k ¼
e01 þ e02 þ e03 ;
where C 10 is a constant of the body; e0i ¼ 1Þ=2 are the principal values of the Green’s deformation tensor. The values lij , Aib in Eq. (12) for the solid with the Treloar potential (49) are
j¼0
1 X
ðk2i
Direct substitution of expansions (42) into (39) causes them to be satisfied identically [15], while Eq. (38) take the form
Z
W 0 ¼ 2C 10 A1 ¼ 2C 10
w12 ðkÞk1 J 2i ðbkÞJ 2jþ2 ðbkÞdk;
r033 ¼
ði; j ¼ 0; 1; 2; . . .Þ: ð46Þ
The homogeneous system of algebraic Eq. (45) has a non-trivial solution if the determinant of the system equals to zero, i.e for incompressible solids
det kdkl ðb; q ; k3 ; Þk ¼ 0 ðk; l ¼ 1; 2; 3; . . .Þ;
ð47Þ
and compressible solids
det kdkl ðb; q ; k3 ; xÞk ¼ 0 ðk; l ¼ 1; 2; 3; . . .Þ:
ð48Þ
Here the components of the tensors x and for the theory of large and small initial deformations are defined are determined by Eqs. (7), (8), (10), and (11), respectively. Therefore, we have reduced the linearized problem formulated above to an eigen-value problem for Eq. (45) in terms of the parameter k3 < 1, which should be solved numerically. 6. Some models of solid The components of the tensors x and in Eqs. (47) and (48) are determined accordingly to the type of the elastic potential. The elastic potential (Treloar potential) for incompressible Neo– Hookean type solids is given as [14]
lð3k þ 2lÞ ðk3 1Þ: kþl
ð55Þ
Substituting Eqs. (54) and (55) into Eqs. (18) and (20), the results are
k 3k þ 2l ðk3 1Þ; m2 ¼ 1; kþl kþl k 3k þ 2l ðk3 1Þ; d2 ¼ 2; d1 ¼ 2 þ kþl kþl 2k þ l 3k þ 2l 3k þ 2l ðk3 1Þ; p2 ¼ 2 þ ðk3 1Þ; p1 ¼ 2 þ kþl kþl kþl 1 1 l1 ¼ ; l2 ¼ ; 1 2kþl 3kþ2l 1 3kþ2l 1 þ 2 kþl kþl ðk3 1Þ 1 þ 2 kþl ðk3 1Þ m1 ¼ 1 þ
n1 ¼ 1 þ
3k þ 2l ðk3 1Þ; k þ 2l
n2 ¼ 1 þ
l 3k þ 2l ðk3 1Þ: k þ l k þ 2l ð56Þ
Described is a method for studying the initial phase of failure of a circular cylindrical bar weakened with a coaxial subsurface cylindrical crack in uniform axial compression along the defect. The analytical solution has been carried out in common form for different models of materials (elastic, composite), the theory of large or small initial strains. The model of solid has to be specified on the final stage of the examination, i.e. the numerical solution.
A.N. Guz, Yu.I. Khoma / Theoretical and Applied Fracture Mechanics 51 (2009) 202–207
The linearized problem formulated above is reduced to an eigen-value problem for Eq. (45) in terms of the parameter k3 < 1 in common form for different models of materials, which should be solved numerically.
References [1] M.K. Kassir, G.C. Sih, Mechanics of Fracture. Three-Dimensional Crack Problems, vol. 2, Noordhoff, Leyden, 1975. [2] G.P. Cherepanov, Mechanics of Brittle Fracture, Nauka, Moscow, 1974. [3] A.A. Wells, Application of fracture mechanics at and beyond general yielding, Brit. Weld. J. 10 (1963) 563–570. [4] A.N. Guz, On one criterion of fracture of solids in compression along the cracks. Spatial problem, Dokl. Academ. Nauk USSR 261 (1981) 42–45. [5] A.N. Guz, V.M. Nazarenko, Symmetric failure of the half-space with pennyshaped cracks in compression, Theor. Appl. Fract. Mech. 3 (1985) 233–245. [6] V.L. Bogdanov, A.N. Guz, V.M. Nazarenko, Nonaxisymmetric compressive failure of a circular crack parallel to a surface of half space, Theor. Appl. Fract. Mech. 22 (1995) 239–247.
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[7] A.N. Guz, Yu.I. Khoma, Stability of an infinite solid with a circular cylindrical crack under compression using the Treloar potential, Theor. Appl. Fract. Mech. 39 (2003) 275–280. [8] A.N. Guz, Yu.I. Khoma, Integral formulation for a circular cylindrical cavity in infinite solid and a finite length coaxial cylindrical crack under compressed axially, Theor. Appl. Fract. Mech. 45 (2006) 204–211. [9] A.N. Guz, T.V. Rudnitskii, Contact interaction of an elastic punch and an elastic half-space with initial (residual) stresses, Int. Appl. Mech. 43 (2007) 1325– 1335. [10] S.Yu. Babich, Yu.P. Glukhov, A.N. Guz, A dynamic problem for a prestressed compressible layered half-space, Int. Appl. Mech. 44 (2008) 388–405. [11] S.Yu. Babich, Yu.P. Glukhov, A.N. Guz, Using complex potentials to determine the reaction of a prestressed two-layer elastic half-space to a moving load, Int. Appl. Mech. 44 (2008) 81–492. [12] A.N. Guz, Mechanics of Brittle Fracture of Materials with Initial Stresses, Naukova Dumka, Kyiv, 1983. [13] A.N. Guz, Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies, Springer-Verlag, 1999. [14] A.N. Guz, M.Sh. Dyshel, V.M. Nazarenko, Fracture and Stability of Materials with Cracks, vol. 4, Kiev, Book 1, 1992. [15] I.S. Gradshtein, I.M. Ryzhik, Tables of Integrals, Sums, Series and Products, 4th ed., Moscow, 1963.