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The penny-shaped crack problem in micropolar elasticity
hf. 1. Engmg Sri. Vol. IS, pp.651-6fd @ Pergamon Press Ltd., 1980. Printed in Great Britain
THE PENNY-SHAPED CRACK PROBLEM IN MICROP~LAR ELASTICITY H. S. PAUL and K. SRIDHARAN Department of Mathematics, Indian Institute of Technology,Madras-~34, India A-t--In the linear theory of micropoiar elasticity, the problem of a penny-shaped crack in a transverse field of constant uniaxial tension is studied. By means of Hankel transforms and dual integral equations the problem is reduced to a regular Fredholm integral equation of the second kind and is then solved numerically. The singular fields arising at the crack-tip are studied in detail and the results are compared with those of the couplestress theory. Classical results are derived as limiting case. The stress environment at the periphery of the crack is found to depend on, apart from Poisson’s ratio and a material length-parameter, another parameter which characterises the coupling of the microstructure with the displacement field. This parameter does not occur in the analogous problem in couple stress theory.
1. INTRODUCTION
THE EFFECTof couple stresses on singular stress concentrations has been studied in a series of
papers by Sternberg and others. A summary of these papers can be found in{l]. The static stress dist~bution at the tip of a Griffith crack in transverse uniaxial tension field is treated in [2] and the same for a penny-shaped crack is studied in[3]. In these studies the linearized couple stress theory of Mindlin and Tiersten[4] has been employed and the stress concentration or intensity factors have been found to depend on Poisson’s ratio and a material lengthparameter. In this paper the effect of couple stresses on the singular stress concentration in the neighbourhood of a penny-shaped crack in transverse tension field is studied by employing the linear theory of micropolar elasticityt51. This theory, which is a special case of the micromorphic theory developed by Eringen and his co-workers, assigns each point of the continuum six degrees of freedom-three components of the displacement field u and three components of the microrotation field p-and consequently the tensor fields of force stress and couple stress become fuIIy dete~inate. The couple stress theory can be deduced from micropolar theory by neglecting the inertia of the microconstituents and introducing the kinematic constraint Q = l/2 curl u. This deduction brings to light certain inherent limitations of the couple stress theory [6]. It is shown, in this paper, that the stress environment at the periphery of the crack depends on three parameters, v’, N and M; v’ is micropolar Poisson’s ratio, N is a nondimensional number characterising the interaction between the microstructure and the displacement field and M is the ratio of N to 7 where T is material length-parameter/crack radius. The parameter N does not occur in the analysis of the crack problem based on the couple stress theory[3]. The micropolar results coincide with the classical results in the limit N +O. 2. GOVERNING
EQUATIONS
For the linear isotropic micropolar medium the constitutive equations for the force stress tensor tii and the couple stress tensor mii are[5]. tii = Aegg
+ cfi + qeS
mii = ay;gij
+
p,$j
+ CLeii + ,$i
(2.1) (2.2)
where eii and y’j are the strain measures expressible in terms of the displacement vector U’ and the microrotation vector cpi as eij = I.1.E.S. t 8/5-A
Uj;i
-
Tij =
(Fij&
cPi;j
651
m
(2.3 (2.4)
652
H.S.PAULandK.SRIDHARAN
and g” is the metric tensor and eiim is the alternating tensor; A, CLare the classical Lame constants while k, (r, p, y are additional elastic modulii which arise due to the microstructure of the medium. These elastic modulii are subjected to the following restrictions due to the requirement that the internal energy be nonnegative. Os3A+2p+k, OG3a+p+y,
0<2p+k, -ySPSy,
Ock OCy’
(2.5)
In the absence of body forces and couples, the momentum balance equations are
tijzj= poii mii,j+ Eir*t,s =
(2.6) Jg
(2.7)
where ~0 and J are, respectively, mass and moment of inertia per unit volume; a dot denotes partial differentiation with respect to time. Equations (2.0-o-(.4), (2.6) and (2.7) form a complete system. Physically, a certain class of materials with fibrous and elongated grains are thought to be described by this theory. 3.THECRACKPROBLEM
Equations (2.6) and (2.7) can be written in terms of the vectors u and 4p.For the static case we have[S] (A+~)graddivu+(~+k)V*u+kcurl4p=O
(3.1)
(cu+p)graddiv~++V’~+kcurlu-2kq=O
(3.2)
In cylindrical polar coordinate system (r, 8, z), when there is axisymmetry, the above equations decompose into two independent sets[7]. The set corresponding to the deformation field
u = W, z),O,wk 2))
(3.3)
cp= (0,dr, z), 0) is [(u + k)D* + (A + 2p’)P&]u + (A + u)Dw,r- kDq = 0
(3.4)
(A + u)D[r-‘(ru),,] + [(A+ 2u’)D*+ (p + k)Wo]w + kr-‘(rep),, = 0
(3.5)
kDu - kw,, + [y@, + D*) - 2k](p = 0
(3.6)
where D = alaz, 5%= a*lar’+ r-‘alar - nr-*, n = 0,l and
The parameter p’ may be called micropolar shear modulus since CL’-+p in the limit k + 0. The stress fields accompanying the deformation field (3.3) are
(t
g
ij
and [:,
x
:zj
(3*7)
where tIn. . *. tI??,mti, . . . , mze denote the physical components. Expressions for these components
The penny-shaped crack problem in micropolar elasticity
653
can be obtained from eqns (2.1)-(2.4) as[8] t, = Z@‘U.,-I”he tee= 2p’r-‘u + Ae t,, = 2~‘w,~+ he
(t,, t*,) = p’(w,, + u,z) 7 3U.z - w,,) -e kp (n&e,me,) = (r, B)cpJ- (S,yv9 hz,
m,B) = 0% ~19,~
where e = div u = u,, + r-‘u + w,,. From (3.3) we see that 4p is solenoidal and so the first term in (3.2) vanishes. As k+O the field eqns (3.1) and (3.2) decouple and the former reduces to classical Lame’s equation. As y +O (k# 0) eqns (3.1) and (3.2) reduce to (h f EL’)grad div u + p’V2u = 0 which is Lame’s equation with enhanced shear modulus II’. In this limit, however, the order of the d~erenti~ eqn (3.2) is lowered. This suggests the possible emergence of funny layer effects. It is surprising to note that (3.1) and (3.2) reduce to Lame’s‘ equation (with body force field) also in the limit y -+ w Consider homogeneous isotropic medium uninterrupted except for the penny-shaped crack: z = 0 + , 0 s r 6 c with stress free faces. Let there be constant tension tzz = POat i~nity. The general solution to this problem is the superposition of the solutions to the following problems: (i) the problem of the crack-free region subjected to uniaxial tension tzz = po and (ii) the problem of the crack opened out by constant normal pressure with no loading at infinity, Problem (i) is trivial. Its solution is: u=
’
Q
=o.
It is easily verified that the only nonvanishing stress generated by this field is tzz = PO.It may be seen that the classical solution[9] is recovered as k +O. Now we consider problem (ii). By symmetry, the dist~bution of stress in the nei~~~h~ of a crack in an infinite medium is identical with that produced in a semi-infinite medium fz B 0) when its boundary is subjected to the following conditions.
f*z(J,0) = -po mdr, 0) = 0
I
w(r,O)=O cp(r,Q=O
I
t,,tr,0) = 0
OSFGC
(3.8)
c
(3.9)
Odr
(3.10)
The components of stress, displacement and microrotation must vanish as (r2 + z2)“+@. Further, these conditions are to be supplemented by an “edge condition”. In conformity with the classical result (p. 139[IO]) we take this condition as w(r,Oo)=OW(c-r))
as r-+c-.
4,lNTEGRALEQUATlONS
If we take u(r, 2) = om5W, Wditd d5 I
(3.11)
654
H. S. PAUL and K. SRIDHARAN w(r, z) =
I
cp(c z) =
I
,,= 5W,
z)Jo@)
dt
ox &&5 zV1@9
dt
and substitute in eqns (3.4)-(3.6), we find that [ 1l] [(CL+ k)IY- (A + 2/.&‘)52]rz -(A + /.L)@rV-k&5 = 0 (A + /.L)@X+ [(A + 2$)D2 -(/J + k)t2]ti + k& = 0 kDii + k&V + [r(D’-
l*) - 2k]@ = 0.
The solution of this system appropriate for the half space .z 20 is AC
a([, z)=
(4.1)
G(t,z)=
c
A+~zB]~-~~-+?
(4.2)
+(,$, z) = Bedgz+ CeeRZ
(4.3)
where n* = 5*+ 2p’W[y(cL+ k)l. Components of stresses on planes z = const. are given by OE
I [(-&+CL)B-k&zB
MC z) = 21L’
A +2~’
0
+ -?- Cheeq’
I
2P’
A +2j.~’
dt
&(&)
(4.4)
co
Mr, 2) = 2pt
I [( 0
-~A+B-~zB)~-~‘+~C~~~-“‘]~~~(~~)~C
mze(r, z) =
Iow +
(BteMgz Cve-'?tJd5r)
dy
(4.5)
dE
(4.6)
In eqns (4.1)-(4.6) A, B, C are arbitrary functions of 5. The boundary condition (3.10) gives.
5A=B+$J2c.
(4.7)
For convenience, we introduce nondimensional variables p, x, y defined by r = cp,
5 = x/c,
lj = ylc.
Eliminating A with the help of (4.7), eqns (4.4), (4.6), (4.2) and (4.3) give t,,(p, 0) = -/.dc-2 MP,
- [( 1 - v’)-‘B(x) + $(c2p’) I0 0) = -w-3
I
- bB(x)
0
+
x(x - y)C(x)]xJo(xp)
YC(~WI(~P)
d-x
dx
(4.8) (4.9)
m w(p,,O)=
c-1
W)JO(XP)
d.x
(4.10)
655
The penny-shapedcrack problemin,micropolarelasticity
Q(p, 0) = C-2
[B(x) + cw1x.w)
(4.11)
dx
where y = (x2 + M2)“2,
M2 =
2cc’k Y
~2
p+k
Poisson’s ratio given by the relation 2(1- v’) = (A + 2p’)/(A + 11’).The micropolar modulii y and CL’have the dimensions of force and stress respectivley. We may define an internal characteristic length I of the medium by
and Y’ is the micropolar
I=
h/m')1"2
Then (4.12) Now boundary conditions (3.8) and (3.9) provide the following simultaneous duel integral equations - [(1 - v’)-‘B(x) + -& x(x - y)C(x)]xJO(xp) dx = c2;, I0 - [xB(x) + yC(x)]xJ,(xp) dx = 0, 0 S p S 1 I0 OD B(x)J-,(xp) dx = 0, p > 1 I0 -[B(x) + C(x)]xJ,(xp) dx = 0, p > 1. I0
0s p G1
(4.13) (4.14) (4.15) (4.16)
The above system can be reduced to a single pair of dual integral equations involving one unknown. Multiplying (4.13) by p and integrating with respect to b from 0 to p we obtain (p.
wm ~[(1-v’)-‘s(x)+-$x(x-y)C(x)]h(x~)dx=$X2~,oS~rl. 0
(4.17)
Similarly, eqn (4.10) can be expressed as m P x-‘B(x)pTi(xp) dx = c p’w(p’, 0)dp’. I0 I0 If p > 1, the integral on the right can be split into sum of two integrals, JO+Jp in which the latter integral is zero in view of the boundary condition (3.9). Therefore cc I0
x-'B(x)J,(xp)dx = Cop+ p > 1
(4.18)
where COis a constant given by Co= c
I
o1 pw(p, 0)dp.
(4.19)
The linear combinations $(c2p’) times (4.14) added to (4.17) and $(c2p’) times (4.16) added to
656
H.S.PAULandK.SRIDHARAN
(1 - Y’)-’ times (4.18) lead to the following well known dual integral equations[l3]
I
m
xV(x)J,(xp) dx = ; PC*;,
0s p <
0
V(x)J,(xp) dx = -j-v'p' -
I
(4.20)
(4.21)
P'l
in which V(x) = [(l - Y’)_‘x-’+ x~/(c*y’)lB(x) + -$
XC(X).
(4.22)
The solution of (4.20), (4.21) is vtx)
Ah
3j.L’
COsinx (2 >“z~5,2(x)+ 1-y’
7rx
x
(4.23)
or
VW=;
2czpO sinx-xcosx x3 C p’
-_sinx 3x
1
(4.24)
c-~B(x)
(4.25)
--Co sinx +1-V’ x *
For convenience, we introduce nondimensional function -1
where N is a nondimensional material parameter defined by (4.26) The number N, which is essentially a measure of the relative strengths of k and CL,characterises the coupling of the microstructure with the displacement field (see eqns (3.1), (3.2)). For this reason it may be called micropolar coupling number. In order to obtain an integral equation representation for B,(x) let G(x) = ox-‘y + bx(y -x) - 1
(4.27)
where a=[l+(l-v’)N2]-‘,
b =l$;;,;;2$.
(4.28)
We note that, for finite M G(x) =0(x-‘)
x(y-x)=0(1)
as x+m
(4.29)
and G(X) = 0(x-‘)
as x + 0.
It can be shown from eqns (4.13), (4.15), (4.22), (4.25) and (4.27), after some manipulation, that B,(x) is govemed by the following dual integral equations.
651
The penny-shaped crack problem in micropolar elasticity m
I
0
x[l + G(x)]B,(x)J,,(xP> dx = F(P),
(4.30)
0d p 6 1
m
I
0
B,(x)Jo(xp)
dx = 0,
P
(4.31)
’ 1
where (4.32) We find that as N+O the pair (4.30), (4.31) reduces to the pair of dual integral equations obtained by Sneddon (p. 136 of [lo]) for the corresponding porblem in classical elasticity. The fact that (4.30), (4.31) reduce to their classical counterpart in the limit k +O is in agreement with the smooth transition of micropolar equations to classical Lame’s equations pointed out in Section 3. To obtain an integral equation from (4.30), (4.31), we introduce auxiliary function Jl(t) by B,(x) = I,’ Jl(t) sin (xt) dt
(4.33)
and require that Jl(t) has continuous derivative in [0,11.Let $(O) be zero. This will be verified later. Integrating (4.33) by parts and applying Riemann-Lebesgue Lemma[l4] it can be shown that xB,(x) = O(1) as x +m
(4.34)
Equation (4.33) satisfies (4.31) identically. Using eqns (4.33) and (4.25) in (4.10) we obtain (4.35) which, on integrating by parts, shows that w@, 0) satisfies the edge condition (3.11). The representation for B,(x) in (4.33) also enables us to express the constant Co in terms of #(t). From (4.10), (4.19) and (4.25) we have c
0
Applying Dirichlet’s formula this takes the compact form (4.36) It can further be shown that the rotation cp(p,O)vanishes at the crack-tip, p = 1. From (4.11), (4.22), (4.23), (4.25) and (4.33) we have (p. 99,47[12]).
tJl(t) dt + $
$(t)q(t* - p*) dt
From eqns (4.28), (4.36) and (4.37) we see that cp@,O)+O asp+l-.
,
P
(4.37)
658
H.S.PAULand K.SRIDHARAN
Now multiplying both sides of equation (4.30) by p/q(t’ from 0 to t, we obtain
-
p’) and integrating with respect to p
x[l + G(x)]B,(x)J,,(xp) dx =
Substituting p = t sin 8, changing the order of integration and using the first entry in Section 3.8.2 of [ 121we get
(”(l+G(x))B~(x)sin(xt)dx=
Jo
t
PF(P)
-dp,
octal.
(4.38)
Substituting for B,(x ) from (4.33) we finally arrive at the regular Fredholm integral equation of the second kind 4(t) +
1’K(u, th,b(u) do = f,(t) + Wz(f),
Osttl,
M
(4.39)
where K(u, t) = K(u, t; a, b) = a oXG(x) sin (xv) sin (xt) dx I
(4.40)
f,(t) = t + h(t)
(4.41)
h(~)=~~(~(x2+M2)-x)[sinx~~Cosx-~]sin(xf)dx
(4.42)
(4.43) and (4.44) The above integrals converge absolutely and uniformly and therefore they represent continuous functions. Now the limit t +O of (4.39) confirms the earlier assumption 4(O) = 0. Whatever may be the constant C& the function $(t) given by (4.45)
40) = h(t) + GW) is a solution of (4.39) provided K(u, t)$i(U) du = fi(t),
0 C t s I,
i =
1,2.
(4.46)
These two integral equations have the same kernel which is symmetric and continuous. This facilitates the numerical computation of $i(t). Having determined &(t)y CA can be computed from the following formula which is easily derivable from (4.36), (4.44) and (4.45). G=
[al,’ W)dt]/[l-nlt+,(t)dt].
(4.47)
$,(t), ((12(t)are first obtained by solving (4.46) numerically, then COis obtained from (4.47). 4(t) can be calculated from (4.45). Once t)(t) is known, the problem can be regarded as solved.
The penny-shaped crack problem in micropolar elasticity 5. STRESS CONCENTRATION
659
AT THE CRACK-TIP
Equation (4.8) gives, on using (4.22), (4.25)
L(p, 0) =
-2m
U + W)lx&(xVo(xp) dx Im
PO
0
$ I om (x - y)xV(x)J,,(xp) dx.
-
Substitution of (4.33) in this yields
t,,(p,O)=~pelL(l)~-?rpo
If@-1)
I
mini (p. I)
2 I
o
.*j
oz
I xG(x)B,(x)Jo(xp)
-$I,
dx - $
0
I
om(x - y)xV(x)Jo(xP) dx.
(5.1)
From (4.23), (4.29) and (4.34) it is clear that the second and the third terms above are regular at p = 1. In the neighbourhood of p 4 1, therefore t,,(p, 0) = (2/~)p~~(l)(j? - 1))“’ + terms regular at p = 1, p > 1.
(5.2)
From eqns (4.9), (4.22), (4.25) and (4.33), we have l$(r)dfrG(x)sin(xr)
x J,(xp) dx - $
YV(x)J,(xP) dx,
P ’ 1.
(5.3)
The first two terms are bounded at p = 1. The last term, with the help of (4.23) can be put in the form
[1 + O(X-~)]~(X)J~,~(X)J,(~P)dx - CL’ It follows that the singular behavior of m&,0) continuous integrals ~(x)J&)Jr(xP)
at the crack-tip is governed by the dis-
pjco
=
dr and -- c -1 _ v, o JI(XP) sin x dx I
whose values (p. 99[12]) are, respectively,
2 CPO - 3 fi
r(W)
I(7/2)I(- l/2)
and
_.&I c
1
1-v’ptl@2Y
p’l.
The Hypergeometric function appearing above can be expressed as: 2FI(~,~;~;p~2)=(l-P~z)-‘~2~~~~~~~~~~~+termswhichvanishasp~l+.
660
H.S.PAULandK.SRIDHARAN
In the neighbourhood of p = 1, therefore,
m&,O)=~p c l O&7@?)
co ---p1 c l-l&~
+termsregularatp=I+,
1 p>l.
(5.4)
It may be observed that both the force stress and couple stress have the same order of singularity viz. (p - l)- “* . We may define the force and couple stress intensity factors as
Equations (5.2) and (5.4) provide the following formulas. KI = 0’2/7r)po~‘(l)Vc
(5.6)
K* = (V2/7r)poc3’* ; - c;, . (
(5.7)
>
Referring to eqns (4.45)-(4.47), (4.39)-(4&l) we see that K1 and K2 depend on v’, N and M(= N/(//c)). In couple stress theory these intensity factors depend on Poisson’s ratio and length-ratio (I/c) only[3]. In classical elasticity KI is free from material parameters while K2 does not exist. It may be noted that the formulas (5.6) and (5.7) are also true for the case of crack with stress-free faces situated in the transverse tension field p. because the solution for Problem (i) presented in Section 3 has no contribution to the singular terms in eqns (5.2) and (5.4). A quantity of physical importance is the work done in opening the crack
Using eqns (4.19), (4.36) and (4.44) W can be expressed as w=4$j(l+)CQ.
(5.8)
As k +O, N, M and K(u, t) +O and y +x; v’ and CL’tend to their classical values v and t.~. Therefore, in the limit k ‘0, G,(t) = t, $2(f) = 0 and Ch = l/3. It readily follows that KI, Kz and W coincide with their classical values (pp. 138, 139[10]) as k +O. 6.NUMERICALRESULTSANDDISCUSSION
The symmetric and continuous kernel K(u, t) defined in (4.40) can be expressed as an integral with finite limits. For this purpose we write K(u, t) as K(u, t) =
K*(v, t), K*(t, u),
if t 2 u if t G u
(6.1)
where m K*(r, s) = 27T o G(X) sin (rX) sin (sX) dX, I
r G s.
Replacing the sine functions by Bessel functions K*(r, s) = k A/(rs)[l, + I21
(6.2)
661
The penny-shaped crack problem in micropolar elasticity
where G,(X)J,,~(rX)Hi’),(sX)
dX,
r >O,
s > 0,
i = 1,2
in which G,(X) = XG(X). The integrals 1, and Zzcan be evaluated by contour integration in the 2 = X + iY plane. The integrands have branch points at ?iM due to the radical y(Z) = d/(2 + M?. When the branch cuts and the branches of y(Z) are chosen as shown in Fig. 1, +i(a - bY*)(Y*-M*)“*+.ibY’TiY, = + (a - bY*)(M* - Y*)“* 2 ibY T iY,
G,(+iY)
if Y >M if Y
and, therefore,
I
z,+z*=QM (a TO
bY*)(M*-
Y2)“*11,2(rY)K,,2(sY) dY
where 1,,2(.) and K,,*( .) are modified Bessel functions. Hence, substituting this in (6.2) we have K*(r,s)=aloM
(a-bY*)(M*-
Y2)1’2e-“YY-‘sinh(rY)dY,
TCS.
(6.3)
It may be seen that the restriction s 3 r secures the limiting condition that K(r, S) +O, as s +O. By a similar procedure the inlinite integral defining h(t), (4.42), can be expressed as a finite integral. But such a representation can be seen to be valid only for t > 1, whereas, for the numerical evaluation of $i(t) from (4.46) h(t) is to be computed for values of t in[O, 11. With the aid of the identity
and a Weber-Schafheitlin discontinuous integral (p. 100[12]) eqn (4.42) can. be expressed as h(t)=$y*t(l-t*)H(l-t)-&M4cP(x)sin(xt)dx
(6.4)
where P(x) =
3(sinx-xcosx)-x*sinx x4(x + v(x’ + M*))*
*
Fig. 1. Contours Cl, C, for integrals I,, &-branch out.
H. S. PAUL and K. SRIDHARAN
662
The second term on the right hand side of eqn (6.4) can be expressed as
M4A,(S,r)+&M4A2(S.t)
P(x) sin(xt)dx +&
+O[K9M4(1 +M')], 6 > M
(6.5)
where &(S, t) = [C4(S, 1 - t) - C4(S, 1 + t)] - (;
M2 + 3)
[GAS, 1- t) - Ce(6, 1 + t)]
+;M2(;if2+3)[C,@,l-t)-Cs(B,l+t)]
&(a, t) = -[&(S,
-;
1 - t) - Ss(6, 1 + t)] +;
M*[S,(S, 1 - t) - ST(S,I+ t)]
M4[S9(S, 1- t) - Sg(S, 1 + t)]
in which Ci(S, 0) =
I
=x-~ cos (0x) dx, i = 4,6,8 6 x-j sin (Ox)dx, j = 5,7,9.
The last mentioned integrals have simple expressions which are suitable for numerical evahration e3 COS(~~) i sin(6e) ---+4 21 cos (se)
c4@y e,=T [
(seJ3 -2 (sej2
1 I6
7~ -3.283 0 sin (ex) dx
w
x
and C,+,(6, e) = F
[
cos (se) 1 sin (se) (se)n-----T-in - 1 (se) - I -&
s,+,(s,
e)=~~+~c.(S,e),
c,-dk
e),
n = 57
n =4,6,8.
The integral equations for $i(r) in (4.46) have been solved numerically by the method of Fox and Goodwin[ 151. Their common kernel has been evaluated by applying 24-point Gaussian quadrature formula for the integral in (6.3). The expressions (6.4), (6.5) have been used for h(t). V’is taken as 0.25 for all the cases considered. Figure 2 shows the variation of the normalized force stress intensity factor (FIF), namely, ~Kr/(I/(2c)po) against the length ratio l/c for M = 0.2, 0.5, 1.0 and 1.5. The FIF for classical case is free from material parameters (p. 138[lo]) and the dashed line in the figure corresponds to this. Figure 3 shows the variation of the nondimensional couple stress intensity factor (CIF), namely, ?~Kz/(~(~c)c~o) against l/c for the same four values of M. The work done by the constant pressure POin opening the crack, i.e. p'W/(c3p$ is plotted against l/c in Fig. 4 for the same values of M. Figure 5 depicts the transverse displacement of the crack face for I/c = 0.2, 0.5 and 1.0. M is taken as 1. The curve for the classical case, the dashed line, is an ellipse (p. 139[10]). The curves for increasing values of l/c are seen to be ellipses with diminishing minor axes. The same is the case in couple stress theory (p. 960[3]). It can be seen that as I/c increases the normal displacement w(p, O), for fixed p, decreases; and W also decreases as one can expect.
663
The penny-shaped crack problem in micropolar elasticity
I
Jx
PO
= 1.0
1.058
\
1.0
-y
zo.5 =
-----_------
__-----____----
-‘;;--
1.017 1.003 I
I 0x2
0.0
CLASSICAL
I 0.6
04
I 0.8
J 1.0
-
0,
Fig. 2. Force stress intensity factor vs micropolar length-parameter. Poisson’s ratio = 0.25.
1
‘/c -
0.0
,
I
0.2
04
I
1 1.0
1
0.6
0.8
Fig. 3. Couple stress intensity factor vs micropolar length-parameter. Poisson’s ratio = 0.25.
@6p’w k3p
2)
0*4-
a
0.2-
1 l/c
0.0
1 0.2
00
1 0.6
I 04
I 1.0
I 0.8
Fig. 4. Work done in opening the crack vs micropolar length-parameter. Poisson’s ratio = 0.25.
O.J-
0.2-
0.1 r/c o.o0.0
0.2
-
0.6
0.8
Fig. 5. Shape of the crack. M = 1, Poisson’s ratio = 0.25.
1.0
H. S. PAUL and K. SRIDHARAN
664
Now we consider the two Emits: (1) M-+0 while I/c is fixed and (2) iic 40 while M is fixed. in limit (1) k+O; and so this limit should yield classical results. Accordingly the curves for decreasing vatues of M, in Figs. 24, can be seen to approach the respective classical curves. In limit (2) CIF, W and w attain their classical values whereas FIF does not attain its classical value 1. This happens also in the couple stress theory. In couple stress theory the FIF+2.5 as UC+0[3]. This number 2.5 may be the upperbound of the limiting values of FIF in the present theory, namely, 1.003, 1.017,. . . ‘(Fig. 2). It may be pointed out that, for indefinitely large M, the integral equation (4.39) ceases to be one of Fredholm’s second type. Actuaily, in the limit M + a, eqn (4.39), on using (4.44) and (4.36), becomes I
where
This appears to be due to the boundary layer effect mentioned in sec. 3 since M+-m may be regarded as y -0. The brake down of the integral equation has also been evident in the numerical results obtained. The FIF for fixed l/c has been found to diverge quite rapidly with increasing M. For example, the FIF(for l/c = 1) for M = 2, 2.5, 3, 6, 9 were found to be, 0.1631 x IO’, 0.1940 x IO’, 0.1003 x IO*,0.3084 x 104,0.3267 x l@, respectively. For the reason mentioned above we may take M = o(l). The ~ont~uum hypothesis requires that I/c 4 I (p. 2,205 [SJ). These two conditions imply that k % p and c-*y G CL.Then, form (2.5), we have c-*p G CL.Thus the micropolar modulii k, p and y are all smaII (a does not appear in the axisymmetric problem under consideration). The present analysis, therefore, pertains to what may be called weaMy micropolar material. Ac~no~fe~e~en~s-me authors wish to express their sincere thanks to Prof. A. C. Eringen for his valuable su~estions. One of the authors (KS.) is grateful to the University Grauts Commission, New Delhi, for offering Research Fellowship under the Faculty Improvement Programme. His thanks are also due to the Indian Institute of Technology, Madras, for providing facilities to carry out this work and the Pachaiyappa’s Trust Board, Madras, for granting study leave. REFERENCES 111E. STENBERG, Mechanics of Generafized Confinua, IUTAM-Sump. 1967,p. 9.5,Freudenstadt~f~f~ga~. SpringerVerIag, Berlin (1968). [Z] E. STERNBERG and R. MUKI, fnt. J. Solids Stmctum. 3,69 (1967). [3] U. B. C. 0. EJIKE, Int. /. Solids Strachres. 7,947 (1969). [4] R. D. MINDLIN and H. F. TIERSTEN, Arch. Ration. Mech. Analysis 11,415 (1962). [5] A. C. ERINGEN (editor), Continuum Physics, Vol. IV. Academic Press, New York (1976). (61 A. C. ERINGEN, Fracfure: An Aduanced Treatise, Vol. II Chap. 7 (Edited by H. Liebowitz). Academic Press, New York (1968). [7] W. NOWACKI and W. K. NOWACKI, Proc. Vib. Probl~s 2, IO,97 (1969). [8] FLUGGE (editor), En&y.of Physics Vol. III/I, p. 802. Spoor-Verl~, Be&n (1960). [9] I. N. SOKOLNIKOFF, ~uf~uf~cu~ 77zmy of Elasticity. McGraw-Hill, New York (1956). [lo] I. N. SNEDDON and M. LOWENGRUB, Crack Problems in Classical Theory of Elasticity. Wiley, New York (1969). [I II R. S. DHALIWAL and S. M. KHAN, fnf. 1. Eugng. Sci. 14,769 (1976). [12] W. MAGNUS, F. OBERHE’ITINGER and R. P. SONI, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Edn. Springer-Verlag, Berlin (1966). [13] I. N. SNEDDON, Mi*cd Bomdury V&e Problems in Poteafiaf Tkory, p. 87.WiIey, New York (1%6). f14] T. M. APOSTOL, Maf~a~ic~ Analysis, p. 469. Addiso~WesIey, New York (1971). [IS] L. FOX and E. G~D~N, Phi. Trans. R. Sot. A245,.501(1953). (Rcwived 2 May 1979)
THE PENNY-SHAPED CRACK PROBLEM IN MICROP~LAR ELASTICITY H. S. PAUL and K. SRIDHARAN Department of Mathematics, Indian Institute of Technology,Madras-~34, India A-t--In the linear theory of micropoiar elasticity, the problem of a penny-shaped crack in a transverse field of constant uniaxial tension is studied. By means of Hankel transforms and dual integral equations the problem is reduced to a regular Fredholm integral equation of the second kind and is then solved numerically. The singular fields arising at the crack-tip are studied in detail and the results are compared with those of the couplestress theory. Classical results are derived as limiting case. The stress environment at the periphery of the crack is found to depend on, apart from Poisson’s ratio and a material length-parameter, another parameter which characterises the coupling of the microstructure with the displacement field. This parameter does not occur in the analogous problem in couple stress theory.
1. INTRODUCTION
THE EFFECTof couple stresses on singular stress concentrations has been studied in a series of
papers by Sternberg and others. A summary of these papers can be found in{l]. The static stress dist~bution at the tip of a Griffith crack in transverse uniaxial tension field is treated in [2] and the same for a penny-shaped crack is studied in[3]. In these studies the linearized couple stress theory of Mindlin and Tiersten[4] has been employed and the stress concentration or intensity factors have been found to depend on Poisson’s ratio and a material lengthparameter. In this paper the effect of couple stresses on the singular stress concentration in the neighbourhood of a penny-shaped crack in transverse tension field is studied by employing the linear theory of micropolar elasticityt51. This theory, which is a special case of the micromorphic theory developed by Eringen and his co-workers, assigns each point of the continuum six degrees of freedom-three components of the displacement field u and three components of the microrotation field p-and consequently the tensor fields of force stress and couple stress become fuIIy dete~inate. The couple stress theory can be deduced from micropolar theory by neglecting the inertia of the microconstituents and introducing the kinematic constraint Q = l/2 curl u. This deduction brings to light certain inherent limitations of the couple stress theory [6]. It is shown, in this paper, that the stress environment at the periphery of the crack depends on three parameters, v’, N and M; v’ is micropolar Poisson’s ratio, N is a nondimensional number characterising the interaction between the microstructure and the displacement field and M is the ratio of N to 7 where T is material length-parameter/crack radius. The parameter N does not occur in the analysis of the crack problem based on the couple stress theory[3]. The micropolar results coincide with the classical results in the limit N +O. 2. GOVERNING
EQUATIONS
For the linear isotropic micropolar medium the constitutive equations for the force stress tensor tii and the couple stress tensor mii are[5]. tii = Aegg
+ cfi + qeS
mii = ay;gij
+
p,$j
+ CLeii + ,$i
(2.1) (2.2)
where eii and y’j are the strain measures expressible in terms of the displacement vector U’ and the microrotation vector cpi as eij = I.1.E.S. t 8/5-A
Uj;i
-
Tij =
(Fij&
cPi;j
651
m
(2.3 (2.4)
652
H.S.PAULandK.SRIDHARAN
and g” is the metric tensor and eiim is the alternating tensor; A, CLare the classical Lame constants while k, (r, p, y are additional elastic modulii which arise due to the microstructure of the medium. These elastic modulii are subjected to the following restrictions due to the requirement that the internal energy be nonnegative. Os3A+2p+k, OG3a+p+y,
0<2p+k, -ySPSy,
Ock OCy’
(2.5)
In the absence of body forces and couples, the momentum balance equations are
tijzj= poii mii,j+ Eir*t,s =
(2.6) Jg
(2.7)
where ~0 and J are, respectively, mass and moment of inertia per unit volume; a dot denotes partial differentiation with respect to time. Equations (2.0-o-(.4), (2.6) and (2.7) form a complete system. Physically, a certain class of materials with fibrous and elongated grains are thought to be described by this theory. 3.THECRACKPROBLEM
Equations (2.6) and (2.7) can be written in terms of the vectors u and 4p.For the static case we have[S] (A+~)graddivu+(~+k)V*u+kcurl4p=O
(3.1)
(cu+p)graddiv~++V’~+kcurlu-2kq=O
(3.2)
In cylindrical polar coordinate system (r, 8, z), when there is axisymmetry, the above equations decompose into two independent sets[7]. The set corresponding to the deformation field
u = W, z),O,wk 2))
(3.3)
cp= (0,dr, z), 0) is [(u + k)D* + (A + 2p’)P&]u + (A + u)Dw,r- kDq = 0
(3.4)
(A + u)D[r-‘(ru),,] + [(A+ 2u’)D*+ (p + k)Wo]w + kr-‘(rep),, = 0
(3.5)
kDu - kw,, + [y@, + D*) - 2k](p = 0
(3.6)
where D = alaz, 5%= a*lar’+ r-‘alar - nr-*, n = 0,l and
The parameter p’ may be called micropolar shear modulus since CL’-+p in the limit k + 0. The stress fields accompanying the deformation field (3.3) are
(t
g
ij
and [:,
x
:zj
(3*7)
where tIn. . *. tI??,mti, . . . , mze denote the physical components. Expressions for these components
The penny-shaped crack problem in micropolar elasticity
653
can be obtained from eqns (2.1)-(2.4) as[8] t, = Z@‘U.,-I”he tee= 2p’r-‘u + Ae t,, = 2~‘w,~+ he
(t,, t*,) = p’(w,, + u,z) 7 3U.z - w,,) -e kp (n&e,me,) = (r, B)cpJ- (S,yv9 hz,
m,B) = 0% ~19,~
where e = div u = u,, + r-‘u + w,,. From (3.3) we see that 4p is solenoidal and so the first term in (3.2) vanishes. As k+O the field eqns (3.1) and (3.2) decouple and the former reduces to classical Lame’s equation. As y +O (k# 0) eqns (3.1) and (3.2) reduce to (h f EL’)grad div u + p’V2u = 0 which is Lame’s equation with enhanced shear modulus II’. In this limit, however, the order of the d~erenti~ eqn (3.2) is lowered. This suggests the possible emergence of funny layer effects. It is surprising to note that (3.1) and (3.2) reduce to Lame’s‘ equation (with body force field) also in the limit y -+ w Consider homogeneous isotropic medium uninterrupted except for the penny-shaped crack: z = 0 + , 0 s r 6 c with stress free faces. Let there be constant tension tzz = POat i~nity. The general solution to this problem is the superposition of the solutions to the following problems: (i) the problem of the crack-free region subjected to uniaxial tension tzz = po and (ii) the problem of the crack opened out by constant normal pressure with no loading at infinity, Problem (i) is trivial. Its solution is: u=
’
Q
=o.
It is easily verified that the only nonvanishing stress generated by this field is tzz = PO.It may be seen that the classical solution[9] is recovered as k +O. Now we consider problem (ii). By symmetry, the dist~bution of stress in the nei~~~h~ of a crack in an infinite medium is identical with that produced in a semi-infinite medium fz B 0) when its boundary is subjected to the following conditions.
f*z(J,0) = -po mdr, 0) = 0
I
w(r,O)=O cp(r,Q=O
I
t,,tr,0) = 0
OSFGC
(3.8)
c
(3.9)
Odr
(3.10)
The components of stress, displacement and microrotation must vanish as (r2 + z2)“+@. Further, these conditions are to be supplemented by an “edge condition”. In conformity with the classical result (p. 139[IO]) we take this condition as w(r,Oo)=OW(c-r))
as r-+c-.
4,lNTEGRALEQUATlONS
If we take u(r, 2) = om5W, Wditd d5 I
(3.11)
654
H. S. PAUL and K. SRIDHARAN w(r, z) =
I
cp(c z) =
I
,,= 5W,
z)Jo@)
dt
ox &&5 zV1@9
dt
and substitute in eqns (3.4)-(3.6), we find that [ 1l] [(CL+ k)IY- (A + 2/.&‘)52]rz -(A + /.L)@rV-k&5 = 0 (A + /.L)@X+ [(A + 2$)D2 -(/J + k)t2]ti + k& = 0 kDii + k&V + [r(D’-
l*) - 2k]@ = 0.
The solution of this system appropriate for the half space .z 20 is AC
a([, z)=
(4.1)
G(t,z)=
c
A+~zB]~-~~-+?
(4.2)
+(,$, z) = Bedgz+ CeeRZ
(4.3)
where n* = 5*+ 2p’W[y(cL+ k)l. Components of stresses on planes z = const. are given by OE
I [(-&+CL)B-k&zB
MC z) = 21L’
A +2~’
0
+ -?- Cheeq’
I
2P’
A +2j.~’
dt
&(&)
(4.4)
co
Mr, 2) = 2pt
I [( 0
-~A+B-~zB)~-~‘+~C~~~-“‘]~~~(~~)~C
mze(r, z) =
Iow +
(BteMgz Cve-'?tJd5r)
dy
(4.5)
dE
(4.6)
In eqns (4.1)-(4.6) A, B, C are arbitrary functions of 5. The boundary condition (3.10) gives.
5A=B+$J2c.
(4.7)
For convenience, we introduce nondimensional variables p, x, y defined by r = cp,
5 = x/c,
lj = ylc.
Eliminating A with the help of (4.7), eqns (4.4), (4.6), (4.2) and (4.3) give t,,(p, 0) = -/.dc-2 MP,
- [( 1 - v’)-‘B(x) + $(c2p’) I0 0) = -w-3
I
- bB(x)
0
+
x(x - y)C(x)]xJo(xp)
YC(~WI(~P)
d-x
dx
(4.8) (4.9)
m w(p,,O)=
c-1
W)JO(XP)
d.x
(4.10)
655
The penny-shapedcrack problemin,micropolarelasticity
Q(p, 0) = C-2
[B(x) + cw1x.w)
(4.11)
dx
where y = (x2 + M2)“2,
M2 =
2cc’k Y
~2
p+k
Poisson’s ratio given by the relation 2(1- v’) = (A + 2p’)/(A + 11’).The micropolar modulii y and CL’have the dimensions of force and stress respectivley. We may define an internal characteristic length I of the medium by
and Y’ is the micropolar
I=
h/m')1"2
Then (4.12) Now boundary conditions (3.8) and (3.9) provide the following simultaneous duel integral equations - [(1 - v’)-‘B(x) + -& x(x - y)C(x)]xJO(xp) dx = c2;, I0 - [xB(x) + yC(x)]xJ,(xp) dx = 0, 0 S p S 1 I0 OD B(x)J-,(xp) dx = 0, p > 1 I0 -[B(x) + C(x)]xJ,(xp) dx = 0, p > 1. I0
0s p G1
(4.13) (4.14) (4.15) (4.16)
The above system can be reduced to a single pair of dual integral equations involving one unknown. Multiplying (4.13) by p and integrating with respect to b from 0 to p we obtain (p.
wm ~[(1-v’)-‘s(x)+-$x(x-y)C(x)]h(x~)dx=$X2~,oS~rl. 0
(4.17)
Similarly, eqn (4.10) can be expressed as m P x-‘B(x)pTi(xp) dx = c p’w(p’, 0)dp’. I0 I0 If p > 1, the integral on the right can be split into sum of two integrals, JO+Jp in which the latter integral is zero in view of the boundary condition (3.9). Therefore cc I0
x-'B(x)J,(xp)dx = Cop+ p > 1
(4.18)
where COis a constant given by Co= c
I
o1 pw(p, 0)dp.
(4.19)
The linear combinations $(c2p’) times (4.14) added to (4.17) and $(c2p’) times (4.16) added to
656
H.S.PAULandK.SRIDHARAN
(1 - Y’)-’ times (4.18) lead to the following well known dual integral equations[l3]
I
m
xV(x)J,(xp) dx = ; PC*;,
0s p <
0
V(x)J,(xp) dx = -j-v'p' -
I
(4.20)
(4.21)
P'l
in which V(x) = [(l - Y’)_‘x-’+ x~/(c*y’)lB(x) + -$
XC(X).
(4.22)
The solution of (4.20), (4.21) is vtx)
Ah
3j.L’
COsinx (2 >“z~5,2(x)+ 1-y’
7rx
x
(4.23)
or
VW=;
2czpO sinx-xcosx x3 C p’
-_sinx 3x
1
(4.24)
c-~B(x)
(4.25)
--Co sinx +1-V’ x *
For convenience, we introduce nondimensional function -1
where N is a nondimensional material parameter defined by (4.26) The number N, which is essentially a measure of the relative strengths of k and CL,characterises the coupling of the microstructure with the displacement field (see eqns (3.1), (3.2)). For this reason it may be called micropolar coupling number. In order to obtain an integral equation representation for B,(x) let G(x) = ox-‘y + bx(y -x) - 1
(4.27)
where a=[l+(l-v’)N2]-‘,
b =l$;;,;;2$.
(4.28)
We note that, for finite M G(x) =0(x-‘)
x(y-x)=0(1)
as x+m
(4.29)
and G(X) = 0(x-‘)
as x + 0.
It can be shown from eqns (4.13), (4.15), (4.22), (4.25) and (4.27), after some manipulation, that B,(x) is govemed by the following dual integral equations.
651
The penny-shaped crack problem in micropolar elasticity m
I
0
x[l + G(x)]B,(x)J,,(xP> dx = F(P),
(4.30)
0d p 6 1
m
I
0
B,(x)Jo(xp)
dx = 0,
P
(4.31)
’ 1
where (4.32) We find that as N+O the pair (4.30), (4.31) reduces to the pair of dual integral equations obtained by Sneddon (p. 136 of [lo]) for the corresponding porblem in classical elasticity. The fact that (4.30), (4.31) reduce to their classical counterpart in the limit k +O is in agreement with the smooth transition of micropolar equations to classical Lame’s equations pointed out in Section 3. To obtain an integral equation from (4.30), (4.31), we introduce auxiliary function Jl(t) by B,(x) = I,’ Jl(t) sin (xt) dt
(4.33)
and require that Jl(t) has continuous derivative in [0,11.Let $(O) be zero. This will be verified later. Integrating (4.33) by parts and applying Riemann-Lebesgue Lemma[l4] it can be shown that xB,(x) = O(1) as x +m
(4.34)
Equation (4.33) satisfies (4.31) identically. Using eqns (4.33) and (4.25) in (4.10) we obtain (4.35) which, on integrating by parts, shows that w@, 0) satisfies the edge condition (3.11). The representation for B,(x) in (4.33) also enables us to express the constant Co in terms of #(t). From (4.10), (4.19) and (4.25) we have c
0
Applying Dirichlet’s formula this takes the compact form (4.36) It can further be shown that the rotation cp(p,O)vanishes at the crack-tip, p = 1. From (4.11), (4.22), (4.23), (4.25) and (4.33) we have (p. 99,47[12]).
tJl(t) dt + $
$(t)q(t* - p*) dt
From eqns (4.28), (4.36) and (4.37) we see that cp@,O)+O asp+l-.
,
P
(4.37)
658
H.S.PAULand K.SRIDHARAN
Now multiplying both sides of equation (4.30) by p/q(t’ from 0 to t, we obtain
-
p’) and integrating with respect to p
x[l + G(x)]B,(x)J,,(xp) dx =
Substituting p = t sin 8, changing the order of integration and using the first entry in Section 3.8.2 of [ 121we get
(”(l+G(x))B~(x)sin(xt)dx=
Jo
t
PF(P)
-dp,
octal.
(4.38)
Substituting for B,(x ) from (4.33) we finally arrive at the regular Fredholm integral equation of the second kind 4(t) +
1’K(u, th,b(u) do = f,(t) + Wz(f),
Osttl,
M
(4.39)
where K(u, t) = K(u, t; a, b) = a oXG(x) sin (xv) sin (xt) dx I
(4.40)
f,(t) = t + h(t)
(4.41)
h(~)=~~(~(x2+M2)-x)[sinx~~Cosx-~]sin(xf)dx
(4.42)
(4.43) and (4.44) The above integrals converge absolutely and uniformly and therefore they represent continuous functions. Now the limit t +O of (4.39) confirms the earlier assumption 4(O) = 0. Whatever may be the constant C& the function $(t) given by (4.45)
40) = h(t) + GW) is a solution of (4.39) provided K(u, t)$i(U) du = fi(t),
0 C t s I,
i =
1,2.
(4.46)
These two integral equations have the same kernel which is symmetric and continuous. This facilitates the numerical computation of $i(t). Having determined &(t)y CA can be computed from the following formula which is easily derivable from (4.36), (4.44) and (4.45). G=
[al,’ W)dt]/[l-nlt+,(t)dt].
(4.47)
$,(t), ((12(t)are first obtained by solving (4.46) numerically, then COis obtained from (4.47). 4(t) can be calculated from (4.45). Once t)(t) is known, the problem can be regarded as solved.
The penny-shaped crack problem in micropolar elasticity 5. STRESS CONCENTRATION
659
AT THE CRACK-TIP
Equation (4.8) gives, on using (4.22), (4.25)
L(p, 0) =
-2m
U + W)lx&(xVo(xp) dx Im
PO
0
$ I om (x - y)xV(x)J,,(xp) dx.
-
Substitution of (4.33) in this yields
t,,(p,O)=~pelL(l)~-?rpo
If@-1)
I
mini (p. I)
2 I
o
.*j
oz
I xG(x)B,(x)Jo(xp)
-$I,
dx - $
0
I
om(x - y)xV(x)Jo(xP) dx.
(5.1)
From (4.23), (4.29) and (4.34) it is clear that the second and the third terms above are regular at p = 1. In the neighbourhood of p 4 1, therefore t,,(p, 0) = (2/~)p~~(l)(j? - 1))“’ + terms regular at p = 1, p > 1.
(5.2)
From eqns (4.9), (4.22), (4.25) and (4.33), we have l$(r)dfrG(x)sin(xr)
x J,(xp) dx - $
YV(x)J,(xP) dx,
P ’ 1.
(5.3)
The first two terms are bounded at p = 1. The last term, with the help of (4.23) can be put in the form
[1 + O(X-~)]~(X)J~,~(X)J,(~P)dx - CL’ It follows that the singular behavior of m&,0) continuous integrals ~(x)J&)Jr(xP)
at the crack-tip is governed by the dis-
pjco
=
dr and -- c -1 _ v, o JI(XP) sin x dx I
whose values (p. 99[12]) are, respectively,
2 CPO - 3 fi
r(W)
I(7/2)I(- l/2)
and
_.&I c
1
1-v’ptl@2Y
p’l.
The Hypergeometric function appearing above can be expressed as: 2FI(~,~;~;p~2)=(l-P~z)-‘~2~~~~~~~~~~~+termswhichvanishasp~l+.
660
H.S.PAULandK.SRIDHARAN
In the neighbourhood of p = 1, therefore,
m&,O)=~p c l O&7@?)
co ---p1 c l-l&~
+termsregularatp=I+,
1 p>l.
(5.4)
It may be observed that both the force stress and couple stress have the same order of singularity viz. (p - l)- “* . We may define the force and couple stress intensity factors as
Equations (5.2) and (5.4) provide the following formulas. KI = 0’2/7r)po~‘(l)Vc
(5.6)
K* = (V2/7r)poc3’* ; - c;, . (
(5.7)
>
Referring to eqns (4.45)-(4.47), (4.39)-(4&l) we see that K1 and K2 depend on v’, N and M(= N/(//c)). In couple stress theory these intensity factors depend on Poisson’s ratio and length-ratio (I/c) only[3]. In classical elasticity KI is free from material parameters while K2 does not exist. It may be noted that the formulas (5.6) and (5.7) are also true for the case of crack with stress-free faces situated in the transverse tension field p. because the solution for Problem (i) presented in Section 3 has no contribution to the singular terms in eqns (5.2) and (5.4). A quantity of physical importance is the work done in opening the crack
Using eqns (4.19), (4.36) and (4.44) W can be expressed as w=4$j(l+)CQ.
(5.8)
As k +O, N, M and K(u, t) +O and y +x; v’ and CL’tend to their classical values v and t.~. Therefore, in the limit k ‘0, G,(t) = t, $2(f) = 0 and Ch = l/3. It readily follows that KI, Kz and W coincide with their classical values (pp. 138, 139[10]) as k +O. 6.NUMERICALRESULTSANDDISCUSSION
The symmetric and continuous kernel K(u, t) defined in (4.40) can be expressed as an integral with finite limits. For this purpose we write K(u, t) as K(u, t) =
K*(v, t), K*(t, u),
if t 2 u if t G u
(6.1)
where m K*(r, s) = 27T o G(X) sin (rX) sin (sX) dX, I
r G s.
Replacing the sine functions by Bessel functions K*(r, s) = k A/(rs)[l, + I21
(6.2)
661
The penny-shaped crack problem in micropolar elasticity
where G,(X)J,,~(rX)Hi’),(sX)
dX,
r >O,
s > 0,
i = 1,2
in which G,(X) = XG(X). The integrals 1, and Zzcan be evaluated by contour integration in the 2 = X + iY plane. The integrands have branch points at ?iM due to the radical y(Z) = d/(2 + M?. When the branch cuts and the branches of y(Z) are chosen as shown in Fig. 1, +i(a - bY*)(Y*-M*)“*+.ibY’TiY, = + (a - bY*)(M* - Y*)“* 2 ibY T iY,
G,(+iY)
if Y >M if Y
and, therefore,
I
z,+z*=QM (a TO
bY*)(M*-
Y2)“*11,2(rY)K,,2(sY) dY
where 1,,2(.) and K,,*( .) are modified Bessel functions. Hence, substituting this in (6.2) we have K*(r,s)=aloM
(a-bY*)(M*-
Y2)1’2e-“YY-‘sinh(rY)dY,
TCS.
(6.3)
It may be seen that the restriction s 3 r secures the limiting condition that K(r, S) +O, as s +O. By a similar procedure the inlinite integral defining h(t), (4.42), can be expressed as a finite integral. But such a representation can be seen to be valid only for t > 1, whereas, for the numerical evaluation of $i(t) from (4.46) h(t) is to be computed for values of t in[O, 11. With the aid of the identity
and a Weber-Schafheitlin discontinuous integral (p. 100[12]) eqn (4.42) can. be expressed as h(t)=$y*t(l-t*)H(l-t)-&M4cP(x)sin(xt)dx
(6.4)
where P(x) =
3(sinx-xcosx)-x*sinx x4(x + v(x’ + M*))*
*
Fig. 1. Contours Cl, C, for integrals I,, &-branch out.
H. S. PAUL and K. SRIDHARAN
662
The second term on the right hand side of eqn (6.4) can be expressed as
M4A,(S,r)+&M4A2(S.t)
P(x) sin(xt)dx +&
+O[K9M4(1 +M')], 6 > M
(6.5)
where &(S, t) = [C4(S, 1 - t) - C4(S, 1 + t)] - (;
M2 + 3)
[GAS, 1- t) - Ce(6, 1 + t)]
+;M2(;if2+3)[C,@,l-t)-Cs(B,l+t)]
&(a, t) = -[&(S,
-;
1 - t) - Ss(6, 1 + t)] +;
M*[S,(S, 1 - t) - ST(S,I+ t)]
M4[S9(S, 1- t) - Sg(S, 1 + t)]
in which Ci(S, 0) =
I
=x-~ cos (0x) dx, i = 4,6,8 6 x-j sin (Ox)dx, j = 5,7,9.
The last mentioned integrals have simple expressions which are suitable for numerical evahration e3 COS(~~) i sin(6e) ---+4 21 cos (se)
c4@y e,=T [
(seJ3 -2 (sej2
1 I6
7~ -3.283 0 sin (ex) dx
w
x
and C,+,(6, e) = F
[
cos (se) 1 sin (se) (se)n-----T-in - 1 (se) - I -&
s,+,(s,
e)=~~+~c.(S,e),
c,-dk
e),
n = 57
n =4,6,8.
The integral equations for $i(r) in (4.46) have been solved numerically by the method of Fox and Goodwin[ 151. Their common kernel has been evaluated by applying 24-point Gaussian quadrature formula for the integral in (6.3). The expressions (6.4), (6.5) have been used for h(t). V’is taken as 0.25 for all the cases considered. Figure 2 shows the variation of the normalized force stress intensity factor (FIF), namely, ~Kr/(I/(2c)po) against the length ratio l/c for M = 0.2, 0.5, 1.0 and 1.5. The FIF for classical case is free from material parameters (p. 138[lo]) and the dashed line in the figure corresponds to this. Figure 3 shows the variation of the nondimensional couple stress intensity factor (CIF), namely, ?~Kz/(~(~c)c~o) against l/c for the same four values of M. The work done by the constant pressure POin opening the crack, i.e. p'W/(c3p$ is plotted against l/c in Fig. 4 for the same values of M. Figure 5 depicts the transverse displacement of the crack face for I/c = 0.2, 0.5 and 1.0. M is taken as 1. The curve for the classical case, the dashed line, is an ellipse (p. 139[10]). The curves for increasing values of l/c are seen to be ellipses with diminishing minor axes. The same is the case in couple stress theory (p. 960[3]). It can be seen that as I/c increases the normal displacement w(p, O), for fixed p, decreases; and W also decreases as one can expect.
663
The penny-shaped crack problem in micropolar elasticity
I
Jx
PO
= 1.0
1.058
\
1.0
-y
zo.5 =
-----_------
__-----____----
-‘;;--
1.017 1.003 I
I 0x2
0.0
CLASSICAL
I 0.6
04
I 0.8
J 1.0
-
0,
Fig. 2. Force stress intensity factor vs micropolar length-parameter. Poisson’s ratio = 0.25.
1
‘/c -
0.0
,
I
0.2
04
I
1 1.0
1
0.6
0.8
Fig. 3. Couple stress intensity factor vs micropolar length-parameter. Poisson’s ratio = 0.25.
@6p’w k3p
2)
0*4-
a
0.2-
1 l/c
0.0
1 0.2
00
1 0.6
I 04
I 1.0
I 0.8
Fig. 4. Work done in opening the crack vs micropolar length-parameter. Poisson’s ratio = 0.25.
O.J-
0.2-
0.1 r/c o.o0.0
0.2
-
0.6
0.8
Fig. 5. Shape of the crack. M = 1, Poisson’s ratio = 0.25.
1.0
H. S. PAUL and K. SRIDHARAN
664
Now we consider the two Emits: (1) M-+0 while I/c is fixed and (2) iic 40 while M is fixed. in limit (1) k+O; and so this limit should yield classical results. Accordingly the curves for decreasing vatues of M, in Figs. 24, can be seen to approach the respective classical curves. In limit (2) CIF, W and w attain their classical values whereas FIF does not attain its classical value 1. This happens also in the couple stress theory. In couple stress theory the FIF+2.5 as UC+0[3]. This number 2.5 may be the upperbound of the limiting values of FIF in the present theory, namely, 1.003, 1.017,. . . ‘(Fig. 2). It may be pointed out that, for indefinitely large M, the integral equation (4.39) ceases to be one of Fredholm’s second type. Actuaily, in the limit M + a, eqn (4.39), on using (4.44) and (4.36), becomes I
where
This appears to be due to the boundary layer effect mentioned in sec. 3 since M+-m may be regarded as y -0. The brake down of the integral equation has also been evident in the numerical results obtained. The FIF for fixed l/c has been found to diverge quite rapidly with increasing M. For example, the FIF(for l/c = 1) for M = 2, 2.5, 3, 6, 9 were found to be, 0.1631 x IO’, 0.1940 x IO’, 0.1003 x IO*,0.3084 x 104,0.3267 x l@, respectively. For the reason mentioned above we may take M = o(l). The ~ont~uum hypothesis requires that I/c 4 I (p. 2,205 [SJ). These two conditions imply that k % p and c-*y G CL.Then, form (2.5), we have c-*p G CL.Thus the micropolar modulii k, p and y are all smaII (a does not appear in the axisymmetric problem under consideration). The present analysis, therefore, pertains to what may be called weaMy micropolar material. Ac~no~fe~e~en~s-me authors wish to express their sincere thanks to Prof. A. C. Eringen for his valuable su~estions. One of the authors (KS.) is grateful to the University Grauts Commission, New Delhi, for offering Research Fellowship under the Faculty Improvement Programme. His thanks are also due to the Indian Institute of Technology, Madras, for providing facilities to carry out this work and the Pachaiyappa’s Trust Board, Madras, for granting study leave. REFERENCES 111E. STENBERG, Mechanics of Generafized Confinua, IUTAM-Sump. 1967,p. 9.5,Freudenstadt~f~f~ga~. SpringerVerIag, Berlin (1968). [Z] E. STERNBERG and R. MUKI, fnt. J. Solids Stmctum. 3,69 (1967). [3] U. B. C. 0. EJIKE, Int. /. Solids Strachres. 7,947 (1969). [4] R. D. MINDLIN and H. F. TIERSTEN, Arch. Ration. Mech. Analysis 11,415 (1962). [5] A. C. ERINGEN (editor), Continuum Physics, Vol. IV. Academic Press, New York (1976). (61 A. C. ERINGEN, Fracfure: An Aduanced Treatise, Vol. II Chap. 7 (Edited by H. Liebowitz). Academic Press, New York (1968). [7] W. NOWACKI and W. K. NOWACKI, Proc. Vib. Probl~s 2, IO,97 (1969). [8] FLUGGE (editor), En&y.of Physics Vol. III/I, p. 802. Spoor-Verl~, Be&n (1960). [9] I. N. SOKOLNIKOFF, ~uf~uf~cu~ 77zmy of Elasticity. McGraw-Hill, New York (1956). [lo] I. N. SNEDDON and M. LOWENGRUB, Crack Problems in Classical Theory of Elasticity. Wiley, New York (1969). [I II R. S. DHALIWAL and S. M. KHAN, fnf. 1. Eugng. Sci. 14,769 (1976). [12] W. MAGNUS, F. OBERHE’ITINGER and R. P. SONI, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Edn. Springer-Verlag, Berlin (1966). [13] I. N. SNEDDON, Mi*cd Bomdury V&e Problems in Poteafiaf Tkory, p. 87.WiIey, New York (1%6). f14] T. M. APOSTOL, Maf~a~ic~ Analysis, p. 469. Addiso~WesIey, New York (1971). [IS] L. FOX and E. G~D~N, Phi. Trans. R. Sot. A245,.501(1953). (Rcwived 2 May 1979)