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The propagation of a brittle crack in anisotropic material
Znt. J. Engng Sci. Vol. 3, pp. 77-91. Pergamon Press 1965. Printed in Great Britain.
THE PROPAGATION OF A BRITTLE CRACK ANISOTROPIC MATERIAL
1N
,
C. ATKINSON* University of Leeds, England (Communicated by I. N. SNEDDON) propagation characteristics of a crack in a brittle, lmear elastic material are investigated, The material is supposed to be orthotropic as regards stress-strain relations, and the stresses in the solid, and the normal displacement are found exactly. The result shows that the displacement of the surface is elliptic as in the static case. Also by consideration of the energy flow into the ends of the crack and using a suitable fracture criteria, we can find bounds on the velocity of propagation. In the special case of isotropy the solution reduces to that given by Broberg 121. Abbact-The
1. INTRODUCTION THE problem under discussion is that of a crack, propagating
in one plane, supposed to nucleate from an infinitesimally small micro-crack with maximum velocity from the start. This type of symmetrical crack, moving with constant velocity v. in both the positive and negative directions of the x-axis, has been considered by Broberg [2] and Craggs [3]. Both considered motion in materials which were supposed to be homogeneous and isotropic both as regards stress-strain relations and fracturing characteristics. Broberg used a transform method; and Craggs assuming dynamic similarity reduced the problem to a finite Dirichlet problem and hence gave the solution in a closed form. Both methods were rather complicated algebraically and it seems doubtful whether they could be applied simply and concisely to materials which are orthotropic or of lower orders of symmetry. Here we investigate motion in materials with lower ordered symmetries. However, the crack and fracturing characteristics will not be symmetrical in materials with less than orthotropic symmetry. Nevertheless, the general method of the problem could be used to deal with materials of lower ordered symmetries provided the boundary conditions etc. of the problem were so designed. We investigate the displacements and stresses in a semi-infinite solid, where the plane surface is subject to 1. a constant pressure acting on an infinite strip, the width of which is symmetrically increasing from zero with constant velocity and, 2. to a pressure outside this strip such that the normal displacement of the surface outside this strip is zero. We have now a mixed boundary value problem in an orthotropic body. In an isotropic body Craggs [3] found that outside a circle of radius tit (where ci was the propagation velocity of irrotational waves, and t the time) the effect of the crack was nil because outside this circle the equations of motion became hyperbolic. Consideration of a Cauchy problem using characteristics reduced the boundary conditions at infinity to lie on two circles: one with radius clt, as mentioned above, and the other with radius c,t (where cz is the propagation velocity of equivoluminal waves). Physically, we can reasonably interpret this to mean that, outside these circles, conditions were steady because the crack * At present in the Department of Mathematics, University of Melbourne, Australia. 77
78
C. ATKINSON
had no effect, after all we can only expect the crack to affect appreciably the part of the body which lies in a certain proximity to it. It seems appropriate to mention here that Craggs [3] initially had a uniform constant hydrostatic pressure R at infinity. Nevertheless, this is effectively the same problem as that considered here only now the stress at infinity is zero with constant stress on the surface of the crack. The circular boundaries as mentioned above are closely connected with the envelope of the characteristics of the equations. Fortunately, isotropic materials afford a simple circular boundary; hence, the dynamic similarity method can be readily applied. However, the type of boundary we would obtain for an orthotropic material which is more or less complicated according to the values of the elastic coefficients is not easily amenable to conformate transformation such as that used in [3]. Broberg [2] obtained the mixed boundary value problem, by first stipulating the stress along the whole boundary, the ends of his crack now being x= It vt where v is arbitrary. Then by superposing different pressures q associated with different velocities v, he obtained the original problem. By the method we use here we could, in fact, go straight ahead and solve the mixed boundary value problem; however, we also first solve the problem with the stress stipulated over the whole boundary. 2. STATEMENT
OF PROBLEM
The semi-infinite solid under consideration occupies the space ~20. (Cartesian coordinates x, y, z will be used throughout this paper.) As in [2], we make the following assumptions : The crack is supposed to nucleate from an infinitesimally small micro-crack situated along the z-axis from z= - co to z= + co, and to propagate symmetrically in the positive and negative x directions, with the constant crack tip velocity vO, when an applied tensile stress reaches the nominal value o,,,,=qo. To treat the problem mathematically, we superpose a state of stress cry,,= -q,,, and assume a state of plane strain. Inside the crack, where the y component only is of importance, we may suppose superposition is achieved by the action of a uniform liquid pressure 40. After this state of stress has been superposed, the stresses at infinity are all zero, and we have the following. Problem A: A pressure q. acts on the infinite strip /x( < v,t of the surface y=O of the semi-infinite solid y2 0 (t= time). The vertical displacement uYat the surface y =0 is zero for /xl> v,t. Find the normal stress g,, at the surface y = 0 for [xl> v,,t, and the displacement u,, at the surface y =0 for 1x1~ v,t. After solving this problem we can then solve the original problem by investigation of the stresses in the vicinity of the crack tip using a suitable fracture criterion. Problem B: [Stress given all over boundary.] A pressure q1 is acting on the infinite strip [xl< v1t of the surface y = 0 of the semi-infinite solid y 2 0 whereas no pressure acts on the surface y =0 for 1x1~ vi t. Find the displacement uYat the surface y =O. Because of the symmetry in both problems A and B we simply need consider the region x>o.
79
The propagation of a brittle crack in anisotropic material 3. ANALYSIS
OF MOTION
The equations of motion for an orthotropic
body are:
where p is the density of the solid, (uX, u,,) are the Cartesian components of displacement, and Cll, Ci2, Cz2, C,, =$(C, 1- C, *) are the elastic constants. Now writing X=x- ul, oij=aij(X, y), where cij are the components of stress written in that form, we obtain referring to [l]
ux=&#G+
TlY)+&#o+
T,Y)+4$(X + T*y) +B,$(X+ TIY,
up =&#G+
T,Y)+A,w+
~IY)+qw+
(2) ~*y)+B,~(X+
Tzy>
where $J and 5, are arbitrary functions to be determined from the boundary conditions; bars denote complex conjugates ; and T,,T,, T2, T2 are the solutions of the quartic equation given by Gl-P~2)+~66~Z 1
(Cl2
+
(G*+Gs)~ (C,,-+)+C*2T2
G,P-
I
(3)
=O,
which is the condition for consistency of the equations obtained by substituting (2) in (1). The constants A,, A,; B,, By are given by;
-tc,* +C,,)Tl
A
--xc
A,
(C,, -P~*)+C~&*
B* ’
-(C,2+Gx2
B,=KCll
-pu2)+ c,,T,*]
*
(4)
We can rewrite (2) as uX=~RI(A~#(X+T,Y)+B,~(X+T,Y)) (5)
In writing the solutions as T,, Tl ; T,, T2 we are assuming that equation (3) has complex roots, whereas in the range -co to + co we will have real values of T as solutions of (3). However, equation (6) which follows is still a solution of the equations of motion, We now generalise the solution in the following form. We suggest that uX= 2Rl u,=2Rl
m [-A&)~(x - at + T,(~)Y) + R,($rl/tx - vt + T,(@y)ldr s -CO
(6)
m CA,@)+(X- at + ?;(u)~) + R,(@!& - rt + T&)y)lda s --m
is a solution. Firstly, let us define our integrals, a is to be understood as a complex variable, v = r + iq, say, and our integrals as the limits as q-*0.
C.
80
ATKINSON
When u= 5 is such that T,, Tz are complex, that is D-Ccz* for isotropic materials,? these parts of the integrals are of the same form as that in (5), and as the real part of the sum is equal to the sum of the real parts this part is a solution. Similarly, we can proceed in the same way for values of u which make A, etc. real. Hence, with certain restrictions on the functions, (which are satisfied by our subsequent choice), we could in this way show that (6) above is a solution. However, a more direct approach is simply to substitute (6) in (1) and verify that it is a solution; this we can do quite straightforwardly. Differentiating (6) to obtain the stresses, and referring to [l] for the expressions in the integrands, we find a,,=2Rl
-Co
OD [A,+‘(Xf f
T,y)+B,V(X+
Gy)ld~ (7)
a,,,,=2Rl
m [A&(X + T,y) +BJ/‘(X -I-Tzy)]dv s -m
where +‘(z) = d4(z)/dz and similarly for $. Also we see from [l] that A=C,,A,+C,,A,T,
B= i&B,+
A I = %~(A,TI + AJ
B, = CdBxT,
C,,B,T, (8) + BY) .
We now have two functions c$,$ arbitrary, except for certain conditions to justify differentiation under the integral sign, and two boundary conditions to be satisfied on the surface y=o. 4. PROBLEM
B
For Problem B the boundary conditions are a,,=0
on the surface y=O
(9)
a,,=_ql aYY =o
Forming the first derivatives of the stresses with respect to time, we find, daxy ==2R’ da -$=2Rl
m - v[A#‘(X s -03
+ T, y) + B,$“(X -I-Tzy)]dv
(10)
* s -cc
- o[A#‘(X + TI y) + BJI”(X + T,y)]do .
Let us now take
4
,I_
K,(u) -x+T,y;
*“=-
K2(4 X+T,Y
as our arbitrary functions, where K,, K, are arbitrary and we have to determine them from our boundary conditions; which are now in terms of the stress rates: * We denote by cz the propagation velocity of equivoluminal waves. 7 For orthotropic materials, see Appendix 1.
The propagation of a brittle crack in anisotropic material
81
(11) given on the surface y = 0. We can satisfy the condition on ds,,/dt
straight away, by
A,& +B&=O.
(12)
Then,
X-tT,y X+T,y 1 AK, -+-
BK,
dv,
(13)
In allowing (12), we have to be very careful which values we take for T,, T, for we can choose either TI or T1 ; T, or Tz. However, in Appendix 1, which considers the roots of the quartic (3), we see that when the roots take the form u1 + i/Ii ; CQ# 0. Then the complete set Therefore, if we take ui +ijI1 =T, and will be a,+ipi, a,-&; -c~-i/3~, -a,+@,. -ijI1=T,, then T,= TI and (12) becomes -@1 A&!-B,&=0 which makes ~#y~~t~O on y =0, which is an undesirable condition; so we stipulate that in this case ifa,+i&=T,, then -CQ +i&=T,, i.e. -T,=T,. Also this means that the imaginary part of Tj and T2 can always have a positive sign, i.e. positive sq. root. Interpretation of integrals in (13)
. .
As mentioned previously, in interpreting the integrals, v= Vhlyi (r + iv). We also note that singularities which may be contained in A, K, etc. are all connected with real v, i.e. with e; in the case of K,, K, as yet unknown we make this stipulation, reference to the algebra in Appendix 1 will verify the situation as regards A, A, etc. From these considerations we can write the integral as a Cauchy integral in the upper q plane, the contour being the infinite semi-circle in the upper half plane, and the c axis. The integral around the semi-circle is zero by Jordan’s lemma, which makes the two representations equivalent (Fig. 1).
FIO. 1.
Contour in the <, q plane.
82
C. ATKINSON
The indentations on the 5 axis are made for the branch points and singularities there. For isotropy, these will be closely connected with the wave speeds lcl, +c, (the propagation velocity of irrotational waves is denoted by cr) and there will be similar branch points for orthotropy which will be enumerated in the text and Appendix 1. The singularity associated with X+ T,y=x-- vt+ T,y as y-+0+ and Tl =a1 + iB1 (whatever value cr), fil positive will lie in the upper q plane and hence v=x/t is a pole, provided, of course, v = x/t is such that T, = T,(V) is a complex function. At first sight, there seems a possible contradiction here as we know the integration seems to pass through velocities for which Tl and T2 are real. However, making reference to Craggs [3] and making the necessary adjustments for orthotropy, we can see that the net effect on the body for these values is zero, provided we are moving with a subsonic velocity; so the above representation is shown to be sound. Similarly for Tz. Applying Cauchy’s Theorem to (13) we have d%=2Rz
2~{x/t[~(x/t)K,(x/t)]
1
+x/t[B(x/t)&(x/t)]}
[
on the surface y= 0. Using (12) we have, (14) where D(v)=A,(v)B(v)-A(v)&(v). In the boundary condition (1 i), we can rewrite
on the surface y =O.
Equating this to (14), we find.
-4n:ixD(x/t)Kz(x/t)=q, t2 &(x/t) where 6 is the Dirac delta function. Cauchy integral, we obtain
a2u,
,1,=-2Rl
where
-2u,x 7t [ x2--c@
+: iad(u, -xjt) t2
1
(13
Differentiating (6) to find B2u,#t2, and calculating the
-2ni x ‘D1(x/t) - -K,(x/t) L t 0 t -%(x/t)
D,(x/r) = A 1(xlM$o/t)
1
for y =0
- ~lW)R,(x/t)
(16)
*
The calculations above have been carried out for the region x> 0, ~20; if we desire expressions for the region x 0, so we can write (16) for all x with y =0 simply by replacing x in (16) by 1x1. Th en, substituting (15) into (16), we obtain for all x, y = 0.
The propagation
of a brittle crack in anisotropic
material
83
By substituting for D,fD from the algebra in the appendix, we can show how this a2u,@t2 compares with that given by Broberg in the isotropic case. We thus have solved problem B in the general case of an orthotropic material. However, this is not particularly instructive although it demonstrates the method. From this point we could now apply a method of superposition to solve problem A, but we will show how this is unnecessary. Substituting for D,/D from Appendix 1 into (17), we obtain
The nomenclature used here is the same as that used by Broberg, so that we can compare our results for the special case of isotropy r = c, r. The result (18) is in fact identical to that obtained by Broberg. 5. PROBLEM
A
The boundary conditions on the free surface y=O are
I+-vat CF~~=O all x.
A great deal of the analysis used here, has already been done in problem B; so we will simply refer back to what is required. As in problem B we will satisfy the requirement on cXy= 0 by da,,/dt=O simply by repeating (12) with all the necessary restrictions carried through. Having evaluated the Cauchy iotegral for du,,,/dt on the free surface y = 0, we repeat (14) and similarly (16). Therefore we have
We now have to solve under the boundary conditions, which reduce our problem to the solution of the following two equations, Rl~7$~]=00
for jjcVo
on y=O
(19)
Rl[~(!+~~]=O
for ki>Vo
on y=O.
(20)
We also remember, that these equations arise out of the first derivatives of the original boundary conditions and so will lead to solutions arbitrary up to a constant which can be determined from oyy=q, for 1x1< Yet on y =O. * VI in the above is the same as ~1 used previously.
84
C. ATKINSON
We can satisfy the equations (14) and (20) above by replacing PD
4niK,D -= Al
(21)
iD1(x2/t2-V02)3/2
where P is a constant, to be determined from the condition on aYron the surface y = 0. Also the term ($/t2 - V02)-3/2 gives us the correct order at the end of the crack and at infinity. From Appendix 1, we find that D/D, is purely imaginary for the values of V which we are considering. Hence, we see that (21) satisfies (19) and (20) for the general orthotropic body. 6. THE STRESS
cyy AT THE SURFACE
y=O
From (14), we have
pD
t2 iD,(x2/t2-Vo2)3/2 This is ~0 for /x/t1 < V,,. For /xl> Vet, we know from Appendix Appendix}. So we write
d%_lxl dt--’
*
1, D/D, is purely imaginary
1
(V, c V,; see
p WWh) .
t2 (X2/P- vo2)3’2
Now
dcYY-14 da,* dt -tZdox)lr)’ Then
--d%V_ d(lxl,t)
-1 (x2/t2-
P Irn(~/~~).
vo2)3’2
W
After making the substitutions
we integrate (22) to obtain (23) For the special case of isotropy, equation (23) is the same as that obtained in [2]. The flow of energy into the crack tip can be found by integrating round a small closed curve centre x= Vat and radius r, r being small, then the rate of work done by the stresses on the cylinder, centre x= Vat, radius r, is given by 8 W/at where f3W x =lim
x {6,, cos 8 + o;~ sin B}ry r+OI[ -x
+{0,,sin0+0,,cos8}r$
ci (24) Id Taking the limit of this expression as r-+0 gives us the energy flow into the crack tip. This we can equate to the increase of the surface energy in extending the surface which is the condition that the crack should move steadily.
The propagation of a brittle crack in anisotropic material
US
Then aIV/at=2TV,, gives us a general relation involving VOfrom which we can deduce the range of values of V, for which propagation is likely (see Craggs [3]). 7. THE DISPLACEMENT
uy AT THE SURFACE
y=O
From (16) and (21)
a2u
y~O at2
for Ix/> Vat
for Ix[< Vat a2u,_ __--.-. at2
P
1 x2
(23
t t2 (vg-X2/t2y2
Integrating, we find uY+$-
(26)
$)
with the use of generalized functions, we can write this as
11
P Ziy=- t2-X2/v$H f-7I4 0 v,4 (
.
The shape of the surface displacement is thus a semi-ellipse --x2 +u:IQ vit’ p2t2
$20.
Therefore, the only difference between the displacements in the orthotropic cases is the value of the constant P.
(27) and isotropic
To find the constant P
By comparison with Broberg (for isotropy), we can write P=QIP
Bo= vo/vd
where 2q,B,
1 ’
,/s-B,
Q=-q-- +n s,2Js4WdS s P.Im(D/D,)
Q(s) = Vg(r - B02)3/2 ’
(28)
(2%
8. CONCLUSION
We have found that for propagation in an orthotropic material the shape of the crack is elliptic and differs-from its counterpart in an isotropic material only in its eccentricity. By making calculations, in specific cases, of the general relation concerning energy flow at the crack tip we can deduce the probable bounds of velocity of the crack.
C. ATKNON
86
Acknowledgements-My grateful thanks are due to Prof. J. W. Craggs, who suggested the work and for many helpful criticisms. REFERENCES [l] [2] [3] [4]
C. ATKINSON. In the press. K. B. BROBERG, Arkiv. fiir Fysik, 18, 159 (1960). J. W. CRAGGS, Fracture of Solids, Ed. Drucker and Gilman, lnterscience (1963). M. J. LIGHTHILL, Introduction to Fourier Analysis and generalised function, Cambridge Mechanics and Applied Mathematics. Cambridge University Press (1959). APPENDIX
Monographs
on
1
In this appendix we intend to do the necessary algebra involved in D and D, for orthotropy, and also for isotropy, changing the notation in this latter case in order to make a direct comparison with Broberg. We also intend to show D, ~0 when T1 = -T, as mentioned in the text, and also that D/D, is purely imaginary for the possible crack velocities involved in the problem. In order to show the above, we will work out the algebra in detail so that we can perform the necessary integration to obtain an explicit formula for the stress. Calculation of D, D, From the text we have D=A,B-AB,
64.1)
64.2) (A-3) B= C12Bx+ C,,B,T,
A=C,zA,i-CzzAyTt
64.4) A, = CsdAxT,
BI = CdBxT,
+A,)
+ By)
we obtain: D, = G,(A,B,T,
- A,BJ’,)
and substituting for (A.3)
Ccd,B,(C,z + C,,>CT; - T:IG
1
-N2)
(A.5)
D1=[(C,,-p~2)+C,,T,21[(C,,-p1/2)+C66T221
also A, = CsJ,
-C127’:+(C,, Gl-P~2)+%~12
-9v2)
1
(A.6)
.
From (A.1) and (A.4), we obtain
Investigation of nature of D/D, (for the subsonic speeds) Here we will not go into the algebra in detail but simply consider the real and complex properties of the roots. Later on, however, we will do the detailed algebra from which we could infer the results obtained here. Nevertheless, each provides a check on the other.
The propagation of a brittle crack in anisotropic material
87
Case 1. When the roots of the quartic equation given by (3) are such that T,, T, are purely imaginary. Hence, T:, Ti are both real.
Therefore, from (A.5) above, D, = real quantity x A,&. Forming the quotient DID,,we find that D D, =(real quantity) x (TI - T,) C,, Now, (Tl -T,) is purely imaginary and Ax/A,,- 8,/B,, is purely imaginary, depending essentially on the difference between Tl and Tz. All the other quantities, being the products of purely complex function, are real. Hence, D/D 1 is purely imaginary and can be written as D/Dl=ilf(V,
Cll, . . .)] .
(A-9)
The function f takes real values for subsonic velocities and is determined algebraically from (A.5) and (A.7). Case 2. Where the roots of the quartic have a real and imaginary part. T,=a,+i~,
,
T,=
-al+i~l
For this case
.
Considering the quartic as a quadratic in T2, we can show that T,, T,, - T,, - ii;1 are the roots in this case, since TI # -Tl. We mentioned in the text, that if we accidentally chose T, and -T, as the roots, then the expression (12) reduces to AyKi + B,,K, =O. To see this, we first note that BJB, = -Ax/A, A
B
1
when T, = - Tl. Therefore
_AJi+A, I- B,T,+B,
from (A.3)
(A.lO) Hence the result. Therefore, we choose our Tl and T, as T, =a1 +i& and T,= -al +i&, where al, /-I1#O. We now have T22 - Tt = - 4ia,& which is purely imaginary. We deduce D, = A&, x i x {a real quantity} for subsonic wave speeds. (Tl -T2)=2a, which is purely real, similarly TIT2 is purely real, and Ax/A; B,/B,, is purely real.
which is also purely real. Therefore, the expression (AS) for D/D1 is once again purely imaginary and we can write it as in (A.9).
88
C. ATKINSON
Calculation of D, D,, A, for isotropy. From isotropy, we have
TI = i(1 - P/C:)+ (A.12)
T, = i(1 - V”/Cf)*
CII--CU 2
where pCf = CI1 = 1+2~; pC$ = D
-= D,
= C,, = p. Hence, we deduce
p2Cf(4(1 - V2/Cf*(l - V2/C:)+-(2 - v2/C$)‘} i&l - V2/Cf *V’/C~
A great deal of the relevant algebra used here is contained in [I], so we are simply quoting the result here, however, as a check we will deduce the above result from the algebra for general orthotropy. Simplifying, we can reduce D/D, to the following D
-= Dl
pC;{4(1
-V2/C3*(l-V2/C~)+-(2-V2/C~)2} iV2(1 - V2/Cf)*
(A.13)
Now, changing our nomenclature in order to check algebraically with [2], we make the following substitutions. Write r= C,t, V= Ix[/t= C,lxl/z b= VI/C,, K=C,/C, where C,, C, are the wave velocities in the isotropic medium V= [xl/t, simply gives us D/D, as a function of 1x1/t, and VI is the crack speed. We then obtain
- iz DL_ -si_- 4pC;KZlxl
(A. 14)
F(z2/x2)
where
F(T2/X2)= Furthermore,
&2/X2- 1 {(1/2K2 - ?/x2)2 + ?/x2( 1 - 22/x2)*( l/P
- ?/x2)+) *
(A.15)
substituting r = x2/z2, V2 = x2/t2 = Cfr, we obtain _D =Picf 4
(2K2- r)2 -4K3(K2r)*(lr( 1 - r)‘)+
r)*
(A.16)
which is another form used in [2]. APPENDIX
2
This is virtually a continuation of the algebra contained in Appendix 1, but owing to its magnitude we thought it suitable to separate it. Calculation of D, D, etc. for orthotropy
From (A.5) and (A.7) we have D E=
where
(TI - T&XT, I VI T2) (T~:-T:)(C,,+C~~)(C~~-PV~)
(A.17)
The propagation of a brittle crack in anisotropic material
89
~=I;~,(C,,+C,,)fC,Z,-C,,(C,,-p~Z))+C*,C,,(C,,+C,,)T:Tj: + G,G
2 + Gd(G
1-
(A-18)
P ~2)-~22(~~11-~~2)+C66~:3~(C11-~~2)+C66~iI~
Now, in order to interpret these expressions for general orthotropy, to the quartic (3) and write it as
we are going to refer
MT4+NT2+P=0
(A.19)
where (A.20) We can now write down certain properties of the roots, i.e. the sums and products, etc. From (A.19) T:+T:=-M,
N
(T,-T,)*=(T:+T;-2TrTz)=
Tf T; EL M - ;+2
(A.21) (N-2JPW M
$=-
(A.22)
J N* (T;-TT:)=F2-M=
4P
N2-4PM MZ
.
(A.23)
Note in (A.22) we have taken TIT2 as the negative square root of (A.21). This occurs because of our choice of T,, and T,, whatever the nature of the roots. The expression for L given in (A.18) can be modified to give L=T,T,(C,, +(Cg,
+
G,>{C:,
-
C22(%
-PJ?
+ C22G,C,27’3’; -PV*)}--~~(C~~-PV*)(T:~T~).
-P~2)~Cf3+c,2~66-c22(c,,
(A.24)
We now once again find it desirable to subdivide our algebra into two parts; one, for the situation where N2 >4PM, both roots being purely complex in our configuration, N obviously being positive to make this possible; and two, the situation where N2 <4PM and T,, T2 both having real and imaginary parts. Case (1). N* >4PM= square root. Therefore
> N>2,/Phf;
remembering
T;_TfzdN2-4PM; M
that we have taken the positive
Tl_T2=i(N-$pM)f_
(A.25)
We now have to be careful as to our interpretation of ,/P= (C,, -pV2)*(C,, -pV’)*. For pV2
where
C. ATKINSON
90
+(C11 -Pl/t){Ct,+C,,C,,-C,,(CI,-pV2))
+
CS6(CI 1 -@?N M
(A.26) *
We could henceforth integrate this expression to obtain the stresses in terms of elliptic integrals etc., when C,, < p V2 < C1r. Now JP=i(C,, - p V”)+@V2-Cs6)* = i(P’)+ say. We write (Ti -T+i(N--2E’JP’M)*=
-
-N+,/N2+4PM 2
* > .=y+iS
say.
(A.27)
Hence, for C,, < p Vz < Cl 1 Im(D/D,) =
M{y(P’)*IM+C(C12+Css)(C:,
-C,,(C,,
-pv2))]
+dc
JN2-44PM(C12+C66)(Ci1-~T/‘)
where ~‘=C,~P+(C,,-pv2)(C:,+C~~C~~-C~~(C~~-pV~))+
C&C, I - pV2)N M *
(A.28)
From these algebraic results we could now in theory go on to calculate the integrals, etc. However, it would seem most appropriate to do this numerically, if we wished to calculate an energy or anything of that sort. (Received 25 June 1964)
R6mm&-Les caracteristiques de progation dune f&sure, dans un mat&au fragile et lineairement &stique, sont etudi&es par l’auteur. Le mat&au est suppose We de nature orthotropique, en ce qui conceme les relations effort-deformation, et les contraintes dam la partie solide et le d&%xment, en direction normale, sont d&ermines exactement, Le r&hat fait appara%re que le d&placement de la surface est eiliptique, comme dans ie cas statique. De plus en prenant en considemtion Ie flux d%nergie B I%xtr&mit6 de la fissure et en adoptant un c&&-e de fracture convenabie, l’auteur parvient Btrouver les hmites de la vitesse de propa~tion. Dans Ie cas particulier de l’isotropie, la solution se confond avec celie don&e par Rroberg [2], Zusammen&suug-Die A~b~it~~~h~~ eines Rimes in einem briichigen, linear elastischen Stoff we&n untersucht. Uter der Annahme, dass der Stoff bezt&lich seiner Spannungsdehttungsbeziehungen orthotrop ist werden sowohl die Spanmmgen in Fe&k&per, als such die Normalverschiebung genau bestimmt. Das Rrgebnis zeight, dass, wie im stat&hen Fall, die Oberfi~henverschiebung elhsptisch ist. Ausserdem komxn durch Berucksichtigung des den Rime&en zufkeasenden Energieliusses und Verwendung eines geeigneten Bruchkriteriums die Randwerte fur die Ausbreitungsgeschwindigkeit gefunden werden. Die Bung reduziert sich im Falle der Isotropie zu der von Broberg [2] angegebenen Losung. Sommario-Si studiano le ceratteristiche di propagazione di un’incrinatura in un materiale elastic0 fragile e lineare. Si suppone the il materiale sia ortotropico qua&o ai rapporti sollecitazioni-deformaaione e si scoprono con esattezza le sollecitazioni nel solid0 nonche lo spostamento normale. 11risultato Mica the lo spostamento della superficie &ellittico, come nel case statico. Inoltre, considerando ii flusso d’energia nella estremit& deWincrinatura e adottando opportuni criteri di frattura possiamo trovare limiti sulla velocit& di propagazione. Nel case speciale d’isotropia, la soluzione si riduce a quella avanzata daI Broberg [Z].
The propagation of a brittle crackin anisotropic material
91
THE PROPAGATION OF A BRITTLE CRACK ANISOTROPIC MATERIAL
1N
,
C. ATKINSON* University of Leeds, England (Communicated by I. N. SNEDDON) propagation characteristics of a crack in a brittle, lmear elastic material are investigated, The material is supposed to be orthotropic as regards stress-strain relations, and the stresses in the solid, and the normal displacement are found exactly. The result shows that the displacement of the surface is elliptic as in the static case. Also by consideration of the energy flow into the ends of the crack and using a suitable fracture criteria, we can find bounds on the velocity of propagation. In the special case of isotropy the solution reduces to that given by Broberg 121. Abbact-The
1. INTRODUCTION THE problem under discussion is that of a crack, propagating
in one plane, supposed to nucleate from an infinitesimally small micro-crack with maximum velocity from the start. This type of symmetrical crack, moving with constant velocity v. in both the positive and negative directions of the x-axis, has been considered by Broberg [2] and Craggs [3]. Both considered motion in materials which were supposed to be homogeneous and isotropic both as regards stress-strain relations and fracturing characteristics. Broberg used a transform method; and Craggs assuming dynamic similarity reduced the problem to a finite Dirichlet problem and hence gave the solution in a closed form. Both methods were rather complicated algebraically and it seems doubtful whether they could be applied simply and concisely to materials which are orthotropic or of lower orders of symmetry. Here we investigate motion in materials with lower ordered symmetries. However, the crack and fracturing characteristics will not be symmetrical in materials with less than orthotropic symmetry. Nevertheless, the general method of the problem could be used to deal with materials of lower ordered symmetries provided the boundary conditions etc. of the problem were so designed. We investigate the displacements and stresses in a semi-infinite solid, where the plane surface is subject to 1. a constant pressure acting on an infinite strip, the width of which is symmetrically increasing from zero with constant velocity and, 2. to a pressure outside this strip such that the normal displacement of the surface outside this strip is zero. We have now a mixed boundary value problem in an orthotropic body. In an isotropic body Craggs [3] found that outside a circle of radius tit (where ci was the propagation velocity of irrotational waves, and t the time) the effect of the crack was nil because outside this circle the equations of motion became hyperbolic. Consideration of a Cauchy problem using characteristics reduced the boundary conditions at infinity to lie on two circles: one with radius clt, as mentioned above, and the other with radius c,t (where cz is the propagation velocity of equivoluminal waves). Physically, we can reasonably interpret this to mean that, outside these circles, conditions were steady because the crack * At present in the Department of Mathematics, University of Melbourne, Australia. 77
78
C. ATKINSON
had no effect, after all we can only expect the crack to affect appreciably the part of the body which lies in a certain proximity to it. It seems appropriate to mention here that Craggs [3] initially had a uniform constant hydrostatic pressure R at infinity. Nevertheless, this is effectively the same problem as that considered here only now the stress at infinity is zero with constant stress on the surface of the crack. The circular boundaries as mentioned above are closely connected with the envelope of the characteristics of the equations. Fortunately, isotropic materials afford a simple circular boundary; hence, the dynamic similarity method can be readily applied. However, the type of boundary we would obtain for an orthotropic material which is more or less complicated according to the values of the elastic coefficients is not easily amenable to conformate transformation such as that used in [3]. Broberg [2] obtained the mixed boundary value problem, by first stipulating the stress along the whole boundary, the ends of his crack now being x= It vt where v is arbitrary. Then by superposing different pressures q associated with different velocities v, he obtained the original problem. By the method we use here we could, in fact, go straight ahead and solve the mixed boundary value problem; however, we also first solve the problem with the stress stipulated over the whole boundary. 2. STATEMENT
OF PROBLEM
The semi-infinite solid under consideration occupies the space ~20. (Cartesian coordinates x, y, z will be used throughout this paper.) As in [2], we make the following assumptions : The crack is supposed to nucleate from an infinitesimally small micro-crack situated along the z-axis from z= - co to z= + co, and to propagate symmetrically in the positive and negative x directions, with the constant crack tip velocity vO, when an applied tensile stress reaches the nominal value o,,,,=qo. To treat the problem mathematically, we superpose a state of stress cry,,= -q,,, and assume a state of plane strain. Inside the crack, where the y component only is of importance, we may suppose superposition is achieved by the action of a uniform liquid pressure 40. After this state of stress has been superposed, the stresses at infinity are all zero, and we have the following. Problem A: A pressure q. acts on the infinite strip /x( < v,t of the surface y=O of the semi-infinite solid y2 0 (t= time). The vertical displacement uYat the surface y =0 is zero for /xl> v,t. Find the normal stress g,, at the surface y = 0 for [xl> v,,t, and the displacement u,, at the surface y =0 for 1x1~ v,t. After solving this problem we can then solve the original problem by investigation of the stresses in the vicinity of the crack tip using a suitable fracture criterion. Problem B: [Stress given all over boundary.] A pressure q1 is acting on the infinite strip [xl< v1t of the surface y = 0 of the semi-infinite solid y 2 0 whereas no pressure acts on the surface y =0 for 1x1~ vi t. Find the displacement uYat the surface y =O. Because of the symmetry in both problems A and B we simply need consider the region x>o.
79
The propagation of a brittle crack in anisotropic material 3. ANALYSIS
OF MOTION
The equations of motion for an orthotropic
body are:
where p is the density of the solid, (uX, u,,) are the Cartesian components of displacement, and Cll, Ci2, Cz2, C,, =$(C, 1- C, *) are the elastic constants. Now writing X=x- ul, oij=aij(X, y), where cij are the components of stress written in that form, we obtain referring to [l]
ux=&#G+
TlY)+&#o+
T,Y)+4$(X + T*y) +B,$(X+ TIY,
up =&#G+
T,Y)+A,w+
~IY)+qw+
(2) ~*y)+B,~(X+
Tzy>
where $J and 5, are arbitrary functions to be determined from the boundary conditions; bars denote complex conjugates ; and T,,T,, T2, T2 are the solutions of the quartic equation given by Gl-P~2)+~66~Z 1
(Cl2
+
(G*+Gs)~ (C,,-+)+C*2T2
G,P-
I
(3)
=O,
which is the condition for consistency of the equations obtained by substituting (2) in (1). The constants A,, A,; B,, By are given by;
-tc,* +C,,)Tl
A
--xc
A,
(C,, -P~*)+C~&*
B* ’
-(C,2+Gx2
B,=KCll
-pu2)+ c,,T,*]
*
(4)
We can rewrite (2) as uX=~RI(A~#(X+T,Y)+B,~(X+T,Y)) (5)
In writing the solutions as T,, Tl ; T,, T2 we are assuming that equation (3) has complex roots, whereas in the range -co to + co we will have real values of T as solutions of (3). However, equation (6) which follows is still a solution of the equations of motion, We now generalise the solution in the following form. We suggest that uX= 2Rl u,=2Rl
m [-A&)~(x - at + T,(~)Y) + R,($rl/tx - vt + T,(@y)ldr s -CO
(6)
m CA,@)+(X- at + ?;(u)~) + R,(@!& - rt + T&)y)lda s --m
is a solution. Firstly, let us define our integrals, a is to be understood as a complex variable, v = r + iq, say, and our integrals as the limits as q-*0.
C.
80
ATKINSON
When u= 5 is such that T,, Tz are complex, that is D-Ccz* for isotropic materials,? these parts of the integrals are of the same form as that in (5), and as the real part of the sum is equal to the sum of the real parts this part is a solution. Similarly, we can proceed in the same way for values of u which make A, etc. real. Hence, with certain restrictions on the functions, (which are satisfied by our subsequent choice), we could in this way show that (6) above is a solution. However, a more direct approach is simply to substitute (6) in (1) and verify that it is a solution; this we can do quite straightforwardly. Differentiating (6) to obtain the stresses, and referring to [l] for the expressions in the integrands, we find a,,=2Rl
-Co
OD [A,+‘(Xf f
T,y)+B,V(X+
Gy)ld~ (7)
a,,,,=2Rl
m [A&(X + T,y) +BJ/‘(X -I-Tzy)]dv s -m
where +‘(z) = d4(z)/dz and similarly for $. Also we see from [l] that A=C,,A,+C,,A,T,
B= i&B,+
A I = %~(A,TI + AJ
B, = CdBxT,
C,,B,T, (8) + BY) .
We now have two functions c$,$ arbitrary, except for certain conditions to justify differentiation under the integral sign, and two boundary conditions to be satisfied on the surface y=o. 4. PROBLEM
B
For Problem B the boundary conditions are a,,=0
on the surface y=O
(9)
a,,=_ql aYY =o
Forming the first derivatives of the stresses with respect to time, we find, daxy ==2R’ da -$=2Rl
m - v[A#‘(X s -03
+ T, y) + B,$“(X -I-Tzy)]dv
(10)
* s -cc
- o[A#‘(X + TI y) + BJI”(X + T,y)]do .
Let us now take
4
,I_
K,(u) -x+T,y;
*“=-
K2(4 X+T,Y
as our arbitrary functions, where K,, K, are arbitrary and we have to determine them from our boundary conditions; which are now in terms of the stress rates: * We denote by cz the propagation velocity of equivoluminal waves. 7 For orthotropic materials, see Appendix 1.
The propagation of a brittle crack in anisotropic material
81
(11) given on the surface y = 0. We can satisfy the condition on ds,,/dt
straight away, by
A,& +B&=O.
(12)
Then,
X-tT,y X+T,y 1 AK, -+-
BK,
dv,
(13)
In allowing (12), we have to be very careful which values we take for T,, T, for we can choose either TI or T1 ; T, or Tz. However, in Appendix 1, which considers the roots of the quartic (3), we see that when the roots take the form u1 + i/Ii ; CQ# 0. Then the complete set Therefore, if we take ui +ijI1 =T, and will be a,+ipi, a,-&; -c~-i/3~, -a,+@,. -ijI1=T,, then T,= TI and (12) becomes -@1 A&!-B,&=0 which makes ~#y~~t~O on y =0, which is an undesirable condition; so we stipulate that in this case ifa,+i&=T,, then -CQ +i&=T,, i.e. -T,=T,. Also this means that the imaginary part of Tj and T2 can always have a positive sign, i.e. positive sq. root. Interpretation of integrals in (13)
. .
As mentioned previously, in interpreting the integrals, v= Vhlyi (r + iv). We also note that singularities which may be contained in A, K, etc. are all connected with real v, i.e. with e; in the case of K,, K, as yet unknown we make this stipulation, reference to the algebra in Appendix 1 will verify the situation as regards A, A, etc. From these considerations we can write the integral as a Cauchy integral in the upper q plane, the contour being the infinite semi-circle in the upper half plane, and the c axis. The integral around the semi-circle is zero by Jordan’s lemma, which makes the two representations equivalent (Fig. 1).
FIO. 1.
Contour in the <, q plane.
82
C. ATKINSON
The indentations on the 5 axis are made for the branch points and singularities there. For isotropy, these will be closely connected with the wave speeds lcl, +c, (the propagation velocity of irrotational waves is denoted by cr) and there will be similar branch points for orthotropy which will be enumerated in the text and Appendix 1. The singularity associated with X+ T,y=x-- vt+ T,y as y-+0+ and Tl =a1 + iB1 (whatever value cr), fil positive will lie in the upper q plane and hence v=x/t is a pole, provided, of course, v = x/t is such that T, = T,(V) is a complex function. At first sight, there seems a possible contradiction here as we know the integration seems to pass through velocities for which Tl and T2 are real. However, making reference to Craggs [3] and making the necessary adjustments for orthotropy, we can see that the net effect on the body for these values is zero, provided we are moving with a subsonic velocity; so the above representation is shown to be sound. Similarly for Tz. Applying Cauchy’s Theorem to (13) we have d%=2Rz
2~{x/t[~(x/t)K,(x/t)]
1
+x/t[B(x/t)&(x/t)]}
[
on the surface y= 0. Using (12) we have, (14) where D(v)=A,(v)B(v)-A(v)&(v). In the boundary condition (1 i), we can rewrite
on the surface y =O.
Equating this to (14), we find.
-4n:ixD(x/t)Kz(x/t)=q, t2 &(x/t) where 6 is the Dirac delta function. Cauchy integral, we obtain
a2u,
,1,=-2Rl
where
-2u,x 7t [ x2--c@
+: iad(u, -xjt) t2
1
(13
Differentiating (6) to find B2u,#t2, and calculating the
-2ni x ‘D1(x/t) - -K,(x/t) L t 0 t -%(x/t)
D,(x/r) = A 1(xlM$o/t)
1
for y =0
- ~lW)R,(x/t)
(16)
*
The calculations above have been carried out for the region x> 0, ~20; if we desire expressions for the region x
The propagation
of a brittle crack in anisotropic
material
83
By substituting for D,fD from the algebra in the appendix, we can show how this a2u,@t2 compares with that given by Broberg in the isotropic case. We thus have solved problem B in the general case of an orthotropic material. However, this is not particularly instructive although it demonstrates the method. From this point we could now apply a method of superposition to solve problem A, but we will show how this is unnecessary. Substituting for D,/D from Appendix 1 into (17), we obtain
The nomenclature used here is the same as that used by Broberg, so that we can compare our results for the special case of isotropy r = c, r. The result (18) is in fact identical to that obtained by Broberg. 5. PROBLEM
A
The boundary conditions on the free surface y=O are
I+-vat CF~~=O all x.
A great deal of the analysis used here, has already been done in problem B; so we will simply refer back to what is required. As in problem B we will satisfy the requirement on cXy= 0 by da,,/dt=O simply by repeating (12) with all the necessary restrictions carried through. Having evaluated the Cauchy iotegral for du,,,/dt on the free surface y = 0, we repeat (14) and similarly (16). Therefore we have
We now have to solve under the boundary conditions, which reduce our problem to the solution of the following two equations, Rl~7$~]=00
for jjcVo
on y=O
(19)
Rl[~(!+~~]=O
for ki>Vo
on y=O.
(20)
We also remember, that these equations arise out of the first derivatives of the original boundary conditions and so will lead to solutions arbitrary up to a constant which can be determined from oyy=q, for 1x1< Yet on y =O. * VI in the above is the same as ~1 used previously.
84
C. ATKINSON
We can satisfy the equations (14) and (20) above by replacing PD
4niK,D -= Al
(21)
iD1(x2/t2-V02)3/2
where P is a constant, to be determined from the condition on aYron the surface y = 0. Also the term ($/t2 - V02)-3/2 gives us the correct order at the end of the crack and at infinity. From Appendix 1, we find that D/D, is purely imaginary for the values of V which we are considering. Hence, we see that (21) satisfies (19) and (20) for the general orthotropic body. 6. THE STRESS
cyy AT THE SURFACE
y=O
From (14), we have
pD
t2 iD,(x2/t2-Vo2)3/2 This is ~0 for /x/t1 < V,,. For /xl> Vet, we know from Appendix Appendix}. So we write
d%_lxl dt--’
*
1, D/D, is purely imaginary
1
(V, c V,; see
p WWh) .
t2 (X2/P- vo2)3’2
Now
dcYY-14 da,* dt -tZdox)lr)’ Then
--d%V_ d(lxl,t)
-1 (x2/t2-
P Irn(~/~~).
vo2)3’2
W
After making the substitutions
we integrate (22) to obtain (23) For the special case of isotropy, equation (23) is the same as that obtained in [2]. The flow of energy into the crack tip can be found by integrating round a small closed curve centre x= Vat and radius r, r being small, then the rate of work done by the stresses on the cylinder, centre x= Vat, radius r, is given by 8 W/at where f3W x =lim
x {6,, cos 8 + o;~ sin B}ry r+OI[ -x
+{0,,sin0+0,,cos8}r$
ci (24) Id Taking the limit of this expression as r-+0 gives us the energy flow into the crack tip. This we can equate to the increase of the surface energy in extending the surface which is the condition that the crack should move steadily.
The propagation of a brittle crack in anisotropic material
US
Then aIV/at=2TV,, gives us a general relation involving VOfrom which we can deduce the range of values of V, for which propagation is likely (see Craggs [3]). 7. THE DISPLACEMENT
uy AT THE SURFACE
y=O
From (16) and (21)
a2u
y~O at2
for Ix/> Vat
for Ix[< Vat a2u,_ __--.-. at2
P
1 x2
(23
t t2 (vg-X2/t2y2
Integrating, we find uY+$-
(26)
$)
with the use of generalized functions, we can write this as
11
P Ziy=- t2-X2/v$H f-7I4 0 v,4 (
.
The shape of the surface displacement is thus a semi-ellipse --x2 +u:IQ vit’ p2t2
$20.
Therefore, the only difference between the displacements in the orthotropic cases is the value of the constant P.
(27) and isotropic
To find the constant P
By comparison with Broberg (for isotropy), we can write P=QIP
Bo= vo/vd
where 2q,B,
1 ’
,/s-B,
Q=-q-- +n s,2Js4WdS s P.Im(D/D,)
Q(s) = Vg(r - B02)3/2 ’
(28)
(2%
8. CONCLUSION
We have found that for propagation in an orthotropic material the shape of the crack is elliptic and differs-from its counterpart in an isotropic material only in its eccentricity. By making calculations, in specific cases, of the general relation concerning energy flow at the crack tip we can deduce the probable bounds of velocity of the crack.
C. ATKNON
86
Acknowledgements-My grateful thanks are due to Prof. J. W. Craggs, who suggested the work and for many helpful criticisms. REFERENCES [l] [2] [3] [4]
C. ATKINSON. In the press. K. B. BROBERG, Arkiv. fiir Fysik, 18, 159 (1960). J. W. CRAGGS, Fracture of Solids, Ed. Drucker and Gilman, lnterscience (1963). M. J. LIGHTHILL, Introduction to Fourier Analysis and generalised function, Cambridge Mechanics and Applied Mathematics. Cambridge University Press (1959). APPENDIX
Monographs
on
1
In this appendix we intend to do the necessary algebra involved in D and D, for orthotropy, and also for isotropy, changing the notation in this latter case in order to make a direct comparison with Broberg. We also intend to show D, ~0 when T1 = -T, as mentioned in the text, and also that D/D, is purely imaginary for the possible crack velocities involved in the problem. In order to show the above, we will work out the algebra in detail so that we can perform the necessary integration to obtain an explicit formula for the stress. Calculation of D, D, From the text we have D=A,B-AB,
64.1)
64.2) (A-3) B= C12Bx+ C,,B,T,
A=C,zA,i-CzzAyTt
64.4) A, = CsdAxT,
BI = CdBxT,
+A,)
+ By)
we obtain: D, = G,(A,B,T,
- A,BJ’,)
and substituting for (A.3)
Ccd,B,(C,z + C,,>CT; - T:IG
1
-N2)
(A.5)
D1=[(C,,-p~2)+C,,T,21[(C,,-p1/2)+C66T221
also A, = CsJ,
-C127’:+(C,, Gl-P~2)+%~12
-9v2)
1
(A.6)
.
From (A.1) and (A.4), we obtain
Investigation of nature of D/D, (for the subsonic speeds) Here we will not go into the algebra in detail but simply consider the real and complex properties of the roots. Later on, however, we will do the detailed algebra from which we could infer the results obtained here. Nevertheless, each provides a check on the other.
The propagation of a brittle crack in anisotropic material
87
Case 1. When the roots of the quartic equation given by (3) are such that T,, T, are purely imaginary. Hence, T:, Ti are both real.
Therefore, from (A.5) above, D, = real quantity x A,&. Forming the quotient DID,,we find that D D, =(real quantity) x (TI - T,) C,, Now, (Tl -T,) is purely imaginary and Ax/A,,- 8,/B,, is purely imaginary, depending essentially on the difference between Tl and Tz. All the other quantities, being the products of purely complex function, are real. Hence, D/D 1 is purely imaginary and can be written as D/Dl=ilf(V,
Cll, . . .)] .
(A-9)
The function f takes real values for subsonic velocities and is determined algebraically from (A.5) and (A.7). Case 2. Where the roots of the quartic have a real and imaginary part. T,=a,+i~,
,
T,=
-al+i~l
For this case
.
Considering the quartic as a quadratic in T2, we can show that T,, T,, - T,, - ii;1 are the roots in this case, since TI # -Tl. We mentioned in the text, that if we accidentally chose T, and -T, as the roots, then the expression (12) reduces to AyKi + B,,K, =O. To see this, we first note that BJB, = -Ax/A, A
B
1
when T, = - Tl. Therefore
_AJi+A, I- B,T,+B,
from (A.3)
(A.lO) Hence the result. Therefore, we choose our Tl and T, as T, =a1 +i& and T,= -al +i&, where al, /-I1#O. We now have T22 - Tt = - 4ia,& which is purely imaginary. We deduce D, = A&, x i x {a real quantity} for subsonic wave speeds. (Tl -T2)=2a, which is purely real, similarly TIT2 is purely real, and Ax/A; B,/B,, is purely real.
which is also purely real. Therefore, the expression (AS) for D/D1 is once again purely imaginary and we can write it as in (A.9).
88
C. ATKINSON
Calculation of D, D,, A, for isotropy. From isotropy, we have
TI = i(1 - P/C:)+ (A.12)
T, = i(1 - V”/Cf)*
CII--CU 2
where pCf = CI1 = 1+2~; pC$ = D
-= D,
= C,, = p. Hence, we deduce
p2Cf(4(1 - V2/Cf*(l - V2/C:)+-(2 - v2/C$)‘} i&l - V2/Cf *V’/C~
A great deal of the relevant algebra used here is contained in [I], so we are simply quoting the result here, however, as a check we will deduce the above result from the algebra for general orthotropy. Simplifying, we can reduce D/D, to the following D
-= Dl
pC;{4(1
-V2/C3*(l-V2/C~)+-(2-V2/C~)2} iV2(1 - V2/Cf)*
(A.13)
Now, changing our nomenclature in order to check algebraically with [2], we make the following substitutions. Write r= C,t, V= Ix[/t= C,lxl/z b= VI/C,, K=C,/C, where C,, C, are the wave velocities in the isotropic medium V= [xl/t, simply gives us D/D, as a function of 1x1/t, and VI is the crack speed. We then obtain
- iz DL_ -si_- 4pC;KZlxl
(A. 14)
F(z2/x2)
where
F(T2/X2)= Furthermore,
&2/X2- 1 {(1/2K2 - ?/x2)2 + ?/x2( 1 - 22/x2)*( l/P
- ?/x2)+) *
(A.15)
substituting r = x2/z2, V2 = x2/t2 = Cfr, we obtain _D =Picf 4
(2K2- r)2 -4K3(K2r)*(lr( 1 - r)‘)+
r)*
(A.16)
which is another form used in [2]. APPENDIX
2
This is virtually a continuation of the algebra contained in Appendix 1, but owing to its magnitude we thought it suitable to separate it. Calculation of D, D, etc. for orthotropy
From (A.5) and (A.7) we have D E=
where
(TI - T&XT, I VI T2) (T~:-T:)(C,,+C~~)(C~~-PV~)
(A.17)
The propagation of a brittle crack in anisotropic material
89
~=I;~,(C,,+C,,)fC,Z,-C,,(C,,-p~Z))+C*,C,,(C,,+C,,)T:Tj: + G,G
2 + Gd(G
1-
(A-18)
P ~2)-~22(~~11-~~2)+C66~:3~(C11-~~2)+C66~iI~
Now, in order to interpret these expressions for general orthotropy, to the quartic (3) and write it as
we are going to refer
MT4+NT2+P=0
(A.19)
where (A.20) We can now write down certain properties of the roots, i.e. the sums and products, etc. From (A.19) T:+T:=-M,
N
(T,-T,)*=(T:+T;-2TrTz)=
Tf T; EL M - ;+2
(A.21) (N-2JPW M
$=-
(A.22)
J N* (T;-TT:)=F2-M=
4P
N2-4PM MZ
.
(A.23)
Note in (A.22) we have taken TIT2 as the negative square root of (A.21). This occurs because of our choice of T,, and T,, whatever the nature of the roots. The expression for L given in (A.18) can be modified to give L=T,T,(C,, +(Cg,
+
G,>{C:,
-
C22(%
-PJ?
+ C22G,C,27’3’; -PV*)}--~~(C~~-PV*)(T:~T~).
-P~2)~Cf3+c,2~66-c22(c,,
(A.24)
We now once again find it desirable to subdivide our algebra into two parts; one, for the situation where N2 >4PM, both roots being purely complex in our configuration, N obviously being positive to make this possible; and two, the situation where N2 <4PM and T,, T2 both having real and imaginary parts. Case (1). N* >4PM= square root. Therefore
> N>2,/Phf;
remembering
T;_TfzdN2-4PM; M
that we have taken the positive
Tl_T2=i(N-$pM)f_
(A.25)
We now have to be careful as to our interpretation of ,/P= (C,, -pV2)*(C,, -pV’)*. For pV2
where
C. ATKINSON
90
+(C11 -Pl/t){Ct,+C,,C,,-C,,(CI,-pV2))
+
CS6(CI 1 -@?N M
(A.26) *
We could henceforth integrate this expression to obtain the stresses in terms of elliptic integrals etc., when C,, < p V2 < C1r. Now JP=i(C,, - p V”)+@V2-Cs6)* = i(P’)+ say. We write (Ti -T+i(N--2E’JP’M)*=
-
-N+,/N2+4PM 2
* > .=y+iS
say.
(A.27)
Hence, for C,, < p Vz < Cl 1 Im(D/D,) =
M{y(P’)*IM+C(C12+Css)(C:,
-C,,(C,,
-pv2))]
+dc
JN2-44PM(C12+C66)(Ci1-~T/‘)
where ~‘=C,~P+(C,,-pv2)(C:,+C~~C~~-C~~(C~~-pV~))+
C&C, I - pV2)N M *
(A.28)
From these algebraic results we could now in theory go on to calculate the integrals, etc. However, it would seem most appropriate to do this numerically, if we wished to calculate an energy or anything of that sort. (Received 25 June 1964)
R6mm&-Les caracteristiques de progation dune f&sure, dans un mat&au fragile et lineairement &stique, sont etudi&es par l’auteur. Le mat&au est suppose We de nature orthotropique, en ce qui conceme les relations effort-deformation, et les contraintes dam la partie solide et le d&%xment, en direction normale, sont d&ermines exactement, Le r&hat fait appara%re que le d&placement de la surface est eiliptique, comme dans ie cas statique. De plus en prenant en considemtion Ie flux d%nergie B I%xtr&mit6 de la fissure et en adoptant un c&&-e de fracture convenabie, l’auteur parvient Btrouver les hmites de la vitesse de propa~tion. Dans Ie cas particulier de l’isotropie, la solution se confond avec celie don&e par Rroberg [2], Zusammen&suug-Die A~b~it~~~h~~ eines Rimes in einem briichigen, linear elastischen Stoff we&n untersucht. Uter der Annahme, dass der Stoff bezt&lich seiner Spannungsdehttungsbeziehungen orthotrop ist werden sowohl die Spanmmgen in Fe&k&per, als such die Normalverschiebung genau bestimmt. Das Rrgebnis zeight, dass, wie im stat&hen Fall, die Oberfi~henverschiebung elhsptisch ist. Ausserdem komxn durch Berucksichtigung des den Rime&en zufkeasenden Energieliusses und Verwendung eines geeigneten Bruchkriteriums die Randwerte fur die Ausbreitungsgeschwindigkeit gefunden werden. Die Bung reduziert sich im Falle der Isotropie zu der von Broberg [2] angegebenen Losung. Sommario-Si studiano le ceratteristiche di propagazione di un’incrinatura in un materiale elastic0 fragile e lineare. Si suppone the il materiale sia ortotropico qua&o ai rapporti sollecitazioni-deformaaione e si scoprono con esattezza le sollecitazioni nel solid0 nonche lo spostamento normale. 11risultato Mica the lo spostamento della superficie &ellittico, come nel case statico. Inoltre, considerando ii flusso d’energia nella estremit& deWincrinatura e adottando opportuni criteri di frattura possiamo trovare limiti sulla velocit& di propagazione. Nel case speciale d’isotropia, la soluzione si riduce a quella avanzata daI Broberg [Z].
The propagation of a brittle crackin anisotropic material
91