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An antiplane shear crack moving in one-dimensional hexagonal quasicrystals
International Journal of Solids and Structures 71 (2015) 255–261
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An antiplane shear crack moving in one-dimensional hexagonal quasicrystals G.E. Tupholme ⇑ Faculty of Engineering and Informatics, University of Bradford, Bradford BD7 1DP, UK
a r t i c l e
i n f o
Article history: Received 4 February 2015 Received in revised form 26 May 2015 Available online 2 July 2015 Keywords: Moving antiplane shear cracks Quasicrystals Dislocation layers
a b s t r a c t An investigation is presented of an antiplane shear strip crack subjected to representative general non-constant crack-face loading conditions which is moving uniformly in one-dimensional hexagonal quasicrystals, using an extended technique of continuous dislocation layers. Readily-calculable explicit closed-form representations are determined and discussed for the components of the phonon and phason stress fields created throughout the media. Illustrative numerical results are presented graphically. As a special case, the corresponding expressions are deduced explicitly for a non-uniformly loaded stationary crack. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The discovery of new complex structures with unusual properties in solid materials was first revealed when Shechtman et al. (1984) reported upon their seminal experimental observations of aluminium–manganese alloys having aperiodic symmetry. Subsequently, these became known as quasicrystals, which are aperiodic materials that display quasiperiodic translational symmetry and non-classical crystallographic rotational symmetry. Many experimental observations have shown quasicrystals to be quite brittle and thus prone to failure due to defects. From a practical point of view, it is therefore most desirable to study the propagation of cracks within them. Ding et al. (1993) and Fan (2011, 2013), for example and the references therein, give comprehensive accounts of the now well-documented physical and mathematical theories of quasicrystals which have been developed. The concept of not only a phonon but also a phason displacement is introduced and theoretical descriptions of their deformations necessitates a detailed study of the interaction of the phonon and phason stress fields created. Investigations of various stationary cracks subject to uniform constant loads within one-dimensional hexagonal quasicrystals have been undertake. Liu et al. (2003) used the complex function method to analyze a semi-infinite crack and to discuss its interaction with a dislocation. Complex variable theory and conformal transformations enabled Li and Fan (2008) to derive representations for the phonon and phason stress intensity factors at the tips ⇑ Tel.: +44 1274 234273. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijsolstr.2015.06.027 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.
of two semi-infinite cracks. Shi (2009) introduced harmonic functions to reduce the antiplane sliding mode models of collinear periodic cracks and/or rigid line inclusions to solvable Riemann– Hilbert problems. A development of a Stroh-type formulation was utilized by Guo and Lu (2011) to examine the fracture mechanics of four mode III cracks originating from an elliptical hole, using a new mapping function to reduce the boundary value problem to Cauchy integral equations. Then more recently, Guo et al. (2013a) used Fourier transforms, dual integral equations and complete elliptic integrals to analyze a crack in a quasicrystal strip containing an antiplane crack and Guo et al. (2013b) proposed a semi-inverse method by introducing Westergaard’s stress functions for studying the behavior of a Griffith crack in an infinite material. Li (2014) considered the three mode I problems of a penny-shaped crack, an external circular crack and a semi-infinite plane crack embedded in an infinite space of quasicrystals. Extensive additional references are also conveniently provided to the available literature in these references. However, the now well-established technique of using continuous distributions of dislocations to model slit-like cracks was first devised for isotropic elastic media; as reviewed by, for example, Bilby and Eshelby (1968) and Lardner (1974). Subsequently, Barnett and Asaro (1972) used a six-dimensional eigenvalue extended Stroh formalism for thereby studying cracks in general anisotropic elastic media and it is noteworthy that they suggested that the technique of dislocation layers appears ‘‘to be more straightforward and to facilitate computational convenience’’ more easily than the approaches using complex variable methods and integral transforms.
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The provision of mechanical descriptions of quasicrystals is an evolving area of investigation and the implications of more sophisticated models are currently under consideration for adoption. Recently, for example, Colli and Mariano (2011) suggested that in particular situations the basic quasicrystal linear elasticity formulation may lead to non-physical anomalies. Further, attention had been expressed by, for example, Fan et al. (2012) in considering nonlinear effects in quasicrystals to account for their plasticity– ductility at high temperatures. These extremely interesting fields of study however are beyond the scope here. The purpose of the present paper is to demonstrate to the quasicrystal community that the so-called dislocation layer technique is indeed especially suited for extension in a particularly convenient and quite straightforward manner to derive analytically explicit versatile expressions for the components of the fields created by a mode III crack moving through one-dimensional hexagonal quasicrystals under arbitrary general crack-face loading conditions. The foundations of this current analysis are provided by the results of Fan et al. (1999) for a dislocation moving uniformly through such materials. In Section 2, the basic formulation of the situation under consideration is presented, together with the required components of the phonon and phason displacement and stress fields around a moving quasicrystal screw dislocation. An analysis and derivation of the resulting fields created by a moving arbitrarily-loaded antiplane shear crack are then given and discussed in Section 3. Some representative numerical results for the variation of the stress component with speed around the crack tip are provided in Section 4. Then, the corresponding results, which have not been reported previously, for a stationary crack subjected to general non-uniform loading in a quasicrystal are derived in Section 5 explicitly as a special case. Finally, in Section 6, the conclusions from this investigation are drawn together. 2. Physical and theoretical formulation A homogeneous one-dimensional hexagonal quasicrystal with point group 6 mm is supposed to be everywhere at rest and stress-free in a natural reference state initially, with a uniform density, q. It is periodic in the x–y plane and quasiperiodic in the positive direction of the z-axis, relative to a fixed system of rectangular Cartesian coordinates (x, y, z). An embedded loaded plane Griffith-type strip crack of width 2c moves through the material parallel to its axis in its own plane. At a time t, it is assumed that the crack occupies the region y = 0, vt c < x < vt + c, 1 < z < 1 of the x–z plane, as depicted in Fig. 1, where v is its speed of propagation. y
Here, cij are the elastic constants in the phonon field, using the conventional contracted Voigt’s notation with i and j taking integer values, K 1 and K 2 are two elastic constants in the phason field and R1 , R2 and R3 are three phonon–phason coupling elastic constants. A moving coordinate n is defined by
n ¼ x vt
ð2Þ
and a mode III antiplane deformation is created by applying prescribed general non-constant phonon and phason stresses symmetrically to the two faces of the crack. Thus the boundary conditions are taken to be
ryz ðn; 0Þ ¼ T ðnÞ; Hzy ðn; 0Þ ¼ HðnÞ; for jnj < c;
ð3Þ
with the medium remaining undisturbed at infinity. Analogously, an antiplane deformation created by instead specifying the phonon and phason strain components eyz and wzy on the crack faces could be studied. The detailed analysis corresponding to that presented here for the conditions (3) can be developed by an interested reader. All the field variables are independent of z in this case. Hence the relations between the relevant non-zero phonon and phason strain components and the phonon displacement component uz and the phason displacement wz are given by
exz ¼
1 @uz ; 2 @x
eyz ¼
@wz ; @x
wzy ¼
wzx ¼
1 @uz ; 2 @y
ð4Þ
@wz @y
ð5Þ
and the generalized Hooke’s laws required reduce, from Eq. (1), to
ryz ¼ 2c44 ezy þ Rwzy ; rxz ¼ 2c44 ezx þ Rwzx ;
ð6Þ
Hzy ¼ 2Rezy þ Kwzy ;
ð7Þ
Hzx ¼ 2Rezx þ Kwzx ;
where, for brevity of presentation here and henceforth, the constants R3 and K 2 are abbreviated to simply R and K, respectively. As an antecedent to studying this moving mode III quasicrystal crack, it is appropriate to consider first a ‘‘quasicrystal screw dislocation’’ moving in the material with extended Burgers vector ð0; 0; b; dÞ in four-dimensional space which has a discontinuity b in the phonon displacement u (the traditional Burgers vector, b, of an elastic screw dislocation) within a ‘‘phonon dislocation’’ and also an analogous discontinuity d in the phason displacement within a ‘‘phason dislocation’’ across the slip plane. These jumps are given, respectively, for n > 0, by
2c
O
The constitutive equations relating the phonon stress components rij , phason stress components Hzi , phonon strain components eij and phason strain components wzi , where i, j = x, y or z, within the quasicrystal can be written in matrix form as 2 3 2 32 3 rxx exx c11 c12 c13 0 0 0 R1 0 0 6r 7 6c 6 7 0 0 0 R1 0 0 7 6 yy 7 6 12 c11 c13 7 6 eyy 7 6 7 6 76 7 6 rzz 7 6 c13 c13 c33 6 7 7 e 0 0 0 R 0 0 2 6 7 6 7 6 zz 7 6r 7 6 76e 7 0 0 0 0 R3 7 6 yz 7 0 0 2c44 6 yz 7 6 0 6 7 6 76 7 6 rxz 7 ¼ 6 0 6 7 0 0 0 2c44 0 0 R3 0 7 6 7 6 7 6 exz 7: 6 7 6 76 7 0 0 0 0 c11 c12 0 0 0 7 6 exy 7 6 rxy 7 6 0 6 7 6 76 7 6 Hzz 7 6 R1 R1 R2 6 7 0 0 0 K1 0 0 7 6 wzz 7 6 7 6 7 6 7 6 76 7 0 0 K 2 0 5 4 wzx 5 0 0 0 2R3 4 Hzx 5 4 0 Hzy wzy 0 0 0 2R3 0 0 0 0 K2 ð1Þ
x
uIII ðn; 0þÞ uIII ðn; 0Þ ¼ ð0; 0; bÞ;
III wIII z ðn; 0þÞ wz ðn; 0Þ ¼ d:
ð8Þ vt Fig. 1. A strip crack moving with constant speed
v in the x-direction.
Throughout, the superscript III refers to quantities associated with a mode III deformation.
G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
A discussion of the dynamics of such a straight screw dislocation in a one-dimensional hexagonal quasicrystal with the Laue classes 6/mh and 6/mh mm has been given by Fan et al. (1999), during which expressions for the induced phonon and phason displacement components were presented. It is these which stimulated the impetus for the current analysis. From them, it follows that for a moving screw dislocation situated at the origin in a one-dimensional hexagonal quasicrystal
1 2 2 1 b1 y 1 b2 y þ R b uIII ðn; yÞ ¼ a tan tan z n n 2pða2 þ R2 Þ b y b y tan1 2 d þRa tan1 1 n n wIII z ðn; yÞ ¼
b y b y tan1 2 b Ra tan1 1 n n 2pða2 þ R Þ b y b y þ a2 tan1 2 d þ R2 tan1 1 n n 2
a ¼ c44 K þ
ð10Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc44 KÞ2 þ 4R2 2
ð11Þ
for i ¼ 1 and 2;
ð12Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2 s2i
in which the two wave speeds, s1 and s2 , under antiplane shear conditions are given by
si ¼
pffiffiffiffiffiffiffiffiffiffi ei =q;
ð13Þ
with
e1 e2
2
h n o b b1 aðc44 a þ R2 Þ þ b2 R2 ðc44 aÞ
2pða2 þ R Þn n oi þdR b1 ðc44 a þ R2 Þ b2 aðc44 aÞ
HIII zy ðn; 0Þ ¼
h
1 2
ð19Þ
n o bR b1 aða þ KÞ b2 ðaK R2 Þ ð20Þ
The analogous phonon and phason strain components can be derived from Eqs. (4), (5), (9) and (10) if desired. 3. Moving antiplane shear crack
and
1
1
rIIIyz ðn; 0Þ ¼
1
where
bi ¼
Crucially it can be observed that on the boundary y = 0, where the boundary conditions (3) are to be imposed,
2pða2 þ R Þn n oi þd b1 R2 ða þ KÞ þ b2 aðaK R2 Þ
ð9Þ
257
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c44 þ K þ ðc44 KÞ2 þ 4R2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c44 þ K ðc44 KÞ2 þ 4R2 2:
ryz ðn; 0Þ ¼ ð14Þ
The corresponding non-zero components of the phonon and phason stresses then follow from the constitutive equations (6) and (7) in the forms
" ( ) b1 aðc44 a þ R2 Þ b2 R2 ðc44 aÞ b þ n2 þ b21 y2 n2 þ b22 y2 2pða2 þ R2 Þ ( )# b ðc44 a þ R2 Þ b2 aðc44 aÞ ð15Þ 2 þdR 1 2 n þ b21 y2 n þ b22 y2
rIIIxz ðn; yÞ ¼
y
" ( ) n b1 aðc44 a þ R2 Þ b2 R2 ðc44 aÞ III ryz ðn; yÞ ¼ b þ n2 þ b21 y2 n2 þ b22 y2 2pða2 þ R2 Þ ( )# b ðc44 a þ R2 Þ b2 aðc44 aÞ 2 þdR 1 2 n þ b21 y2 n þ b22 y2 ( ) y b1 aða þ KÞ b2 ðaK R2 Þ III Hzx ðn; yÞ ¼ bR 2 n2 þ b21 y2 n þ b22 y2 2pða2 þ R2 Þ ( )# b R2 ða þ KÞÞ b2 aðaK R2 Þ þ þd 1 2 n þ b21 y2 n2 þ b22 y2
ð16Þ
c
ð17Þ Z
c
c
ð18Þ
2
h n o b b1 aðc44 a þ R2 Þ þ b2 R2 ðc44 aÞ
It is apparent in evaluating ryz ðn; 0Þ and Hzy ðn; 0Þ from the Plemelj formulae that the integrals throughout Eqs. (21) and (22) should be interpreted as Cauchy principal value integrals. The pair of simultaneous equations which arises by equating the expressions (21) and (22) to the imposed boundary conditions (3) can be solved to yield
c
"
1
2pða2 þ R Þ Z c n o f ðn0 Þ 0 2 0 dn þ dR b1 ðc 44 a þ R Þ b2 aðc 44 aÞ n n c Z c gðn0 Þ 0 ð21Þ 0 dn ; c n n h n o 1 Hzy ðn; 0Þ ¼ bR b1 aða þ KÞ b2 ðaK R2 Þ 2 2 2pða þ R Þ Z c n o f ðn0 Þ 0 2 2 0 dn þ d b1 R ða þ KÞ þ b2 aðaK R Þ c n n Z c gðn0 Þ 0 : ð22Þ dn 0 c n n
Z
"
( ) n b1 aða þ KÞ b2 ðaK R2 Þ HIII ðn; yÞ ¼ bR zy n2 þ b21 y2 n2 þ b22 y2 2pða2 þ R2 Þ ( )# b R2 ða þ KÞÞ b2 aðaK R2 Þ þ þd 1 2 n þ b21 y2 n2 þ b22 y2
The well-established fundamental dislocation layer method, which was devised for a purely elastic material, whereby a loaded crack is analyzed by replacing it by an equivalent continuous planar distribution of dislocations is now developed correspondingly for the crack under consideration here within a quasicrystal. For isotropic materials, the details of its basic implementation have been conveniently given by, for example, Bilby and Eshelby (1968) and Lardner (1974). The basic technique is adapted by spreading an array of moving quasicrystal screw dislocations of the type prescribed in Section 2 over the region jnj < c, y ¼ 0; 1 < z < 1. To model the mode III crack subject to conditions (3), the densities of the proposed distributions of phonon and phason dislocations are taken to be f ðnÞ and gðnÞ, respectively. Then it follows, from Eqs. (19) and (20), that the corresponding phonon and phason stress components at a point on the n-axis are given by
f ðn0 Þ 0 2p dn ¼ n n0 bb1 b2 ðc44 K R2 Þða2 þ R2 Þ hn o b1 R2 ða þ KÞ þ b2 aðaK R2 Þ T ðnÞ n o i R b1 ðc44 a þ R2 Þ b2 aðc44 aÞ HðnÞ ;
ð23Þ
gðn0 Þ 0 2p dn ¼ n n0 db1 b2 ðc44 K R2 Þða2 þ R2 Þ h n o R b1 aða þ KÞ b2 ðaK R2 Þ T ðnÞ n o i þ b1 aðc44 a þ R2 Þ þ b2 R2 ðc44 aÞ HðnÞ :
ð24Þ
258
G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
The appropriate solutions of these integral equations are deducible from the results of Muskhelishvili (1953) and Gakhov (1966) in the forms
2
f ðnÞ ¼
1
n
pbb1 b2 ðc44 K R2 Þða2 þ R2 Þ ðc2 n2 Þ12 Z
c
2
ð33Þ ð25Þ
n o H Hzy ðn; yÞ ¼ K4 F Tb1 ðhb1 Þ F Tb2 ðhb2 Þ þ K2 F H b1 ðhb1 Þ þ K1 F b2 ðhb2 Þ;
1
ð34Þ
1 2
c o ðc2 n02 Þ h n R b1 aða þ KÞ b2 ðaK R2 Þ T ðn0 Þ 0 n n n c o i þ b1 aðc44 a þ R2 Þ þ b2 R2 ðc44 aÞ Hðn0 Þ dn0 ;
ryz ðn; yÞ Z
b 2
2pða þ R Þ 2
(
c
b1 aðc44 a þ R2 Þ
ðn n00 Þ
00 2
b21 y2
þ
b2 R2 ðc44 aÞ 00 2
ð26Þ
)
b22 y2
f ðn00 Þdn00
ðn n Þ þ ðn n Þ þ ( ) b1 ðc44 a þ R2 Þ b2 aðc44 aÞ þ ðn n Þ gðn00 Þdn00 : 2 2 2pða2 þ R2 Þ c ðn n00 Þ þ b21 y2 ðn n00 Þ þ b22 y2 c
Z
dR
c
00
ð27Þ
Upon substitution of the expressions (25) and (26) this can be shown, after simplification, to take the convenient form
ryz ðn; yÞ ¼
Z
1
p2
Z
c
1
ðc2 n002 Þ2 ½fK1 T ðn0 Þ þ K3 Hðn0 Þg
c
ðn n00 Þdn00 0 n o þ fK2 T ðn Þ c ðc2 n002 Þ ðn0 n00 Þ ðn n00 Þ2 þ b2 y2 1 3 Z c 00 00 ðn n Þdn 7 0 K3 Hðn0 Þg n o5dn ; 1 c ðc 2 n002 Þ2 ðn0 n00 Þ ðn n00 Þ2 þ b2 y2 2
K1 T p K2 T p F ðhb Þ F ðhb Þ b1 b1 1 2 b 2 b2 2 2 1 H p 1 p F b1 ðhb1 Þ F H ðh Þ K3 b b1 b2 b2 2 2 2
rxz ðn; yÞ ¼
with the relative phonon and phason displacements of the two crack faces assumed to vanish at n = ± c. Now that the necessary densities f ðnÞ and gðnÞ have been determined, all the components of the phonon and phason fields can be directly calculated explicitly as desired from Eqs. (15)–(18), (25) and (26). The expression for ryz ðn; yÞ is derived here as a typical example, to illustrate the type of detailed analyses which are necessary. It is seen from Eq. (16) that
1 T p 1 p F ðhb Þ F Tb2 ðhb2 Þ b1 b1 1 2 b2 2 K2 H p K1 H p F ðhb Þ F ðhb Þ b1 b1 1 2 b2 b2 2 2
1 2
where the dimensionless constants K1 ; K2 and K3 can be expressed as
ðc44 a þ R2 ÞðaK R2 Þ 2
2
ða2 þ R Þðc44 K R Þ
;
K2 ¼
R2 ðc44 aÞða þ KÞ ða2 þ R2 Þðc44 K R2 Þ
;
ð29Þ
Rðc44 a þ R Þðc44 aÞ ða2 þ R2 Þðc44 K R2 Þ
:
ð30Þ
The functions F Fk ðhk Þ, Rk ðn; yÞ and hk ðn; yÞ, for k ¼ b1 ; b2 and 1, and F ¼ T and H, are now defined for the sake of notational simplification by
F Fk ðhk Þ ¼
1
p
Z
c
c
1 ky cos hk þ ðn n0 Þ sin hk 2 n o ðc n002 Þ2 Fðn0 Þdn0 ; 2 2 0 2 Rk ðn n Þ þ k y
n o1 2 2 Rk eihk ¼ c2 ðn þ ikyÞ ;
ð31Þ
ð32Þ
ð36Þ
where
K4 ¼
Rða þ KÞðaK R2 Þ ða2 þ R2 Þðc44 K R2 Þ
:
ð37Þ
At this stage, it is worthy of note that, with the general boundary conditions (3) imposed, Eqs. (33)–(36) exhibit explicitly that all the stress components depend upon both T and H together with the quasicrystal material constants and the speed of the crack. The important features of the distributions of these components near a crack tip become apparent by setting
n ¼ c þ r cos /;
y ¼ r sin /
ð38Þ
and considering situations where r c, into Eqs. (33)–(36). Then, from Eq. (32), the corresponding approximations to Rk and hk as r ? 0 can be shown to be
Rk
12 1 2 2 2crðcos2 / þ k sin /Þ2 ;
hk ðp Uk Þ=2;
ð39Þ ð40Þ
with
Uk ¼ tan1 ðk tan /Þ;
ð41Þ
1
where tan ð:::Þ is understood to indicate the principal value of the inverse tangent for 0 / p=2 and p plus the principal value for p=2 / p. Substituting these into Eqs. (33)–(36) and (31), and putting 2
K3 ¼
2
1
Dk ¼ ðcos2 / þ k sin /Þ4
2
ð35Þ
Hzx ðn; yÞ ¼ K4
c
ð28Þ
K1 ¼
Similarly, Eqs. (15), (17), (18), (A.1) and (A.2) yield the representations
pdb1 b2 ðc44 K R2 Þða2 þ R2 Þ ðc2 n2 Þ12 Z
¼
o
ryz ðn; yÞ ¼ K1 F Tb1 ðhb1 Þ þ K2 F Tb2 ðhb2 Þ þ K3 F Hb1 ðhb1 Þ F Hb2 ðhb2 Þ :
1 o ðc2 n02 Þ2 hn b1 R2 ða þ KÞ þ b2 aðaK R2 Þ T ðn0 Þ 0 n n nc o i R b1 ðc44 a þ R2 Þ b2 aðc44 aÞ Hðn0 Þ dn0 ;
gðnÞ ¼
with the branches of the square root function determined by choosing hk to be zero for jnj < c, y = 0+ and defined by analytic continuation elsewhere. Then, using the result (A.2) in the Appendix, it follows from the expression (28) that
ð42Þ
for k ¼ b1 ; b2 and 1, yields
Ub1 Ub2 K1 K2 þ cos cos 2 Db 2 2 r Db 1 Ub1 Ub2 KH 1 1 ; cos cos þ pffiffiffi K3 Db 1 2 Db 2 2 r
K
ryz ðr; /Þ pTffiffiffi
Ub1 Ub2 KT 1 1 Hzy ðr; /Þ pffiffiffi K4 cos cos Db 2 Db 2 2 r 1 Ub1 Ub2 K H K2 K1 þ ; cos cos þ pffiffiffi 2 Db2 2 r Db 1
ð43Þ
ð44Þ
259
G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
K
Ub1 Ub2 K1 K2 þ sin sin 2 b2 Db2 2 r b1 Db1 Ub1 Ub2 KH 1 1 ; pffiffiffi K3 sin sin b1 Db1 b2 Db2 2 2 r
rxz ðr; /Þ pTffiffiffi
Ub1 Ub2 KT 1 1 Hzx ðr; /Þ pffiffiffi K4 sin sin b1 Db1 b2 Db2 2 2 r Ub1 Ub2 KH K2 K1 þ ; sin sin pffiffiffi 2 b2 Db2 2 r b 1 Db 1
ð45Þ
ð46Þ
as r ? 0, where K T and K H , which are defined for F ¼ T and H by
1 K F ¼ pffiffiffiffiffiffi p 2c
Z
c
c
c þ n0 c n0
12
Fðn0 Þdn0
ð47Þ
are the phonon and phason stress intensity factors corresponding to that at the end of an isotropic elastic crack. Finally, the Eqs. (43) and (45), and (44) and (46), respectively, yield expressions for the phonon and phason stress components, r/z and Hz/ , near the crack tip of the form
r/z ðr; /Þ
Hz/ ðr; /Þ
Ub1 Ub1 K1 K T þ K3 K H 1 pffiffiffi sin / þ cos cos / sin b1 2 2 r Db 1 Ub2 Ub2 K2 K T K3 K H 1 pffiffiffi sin / þ cos cos / ; þ sin b2 2 2 r Db2 ð48Þ Ub1 Ub1 K4 K T þ K2 K H 1 pffiffiffi sin / þ cos cos / sin b1 2 2 r Db 1 Ub2 Ub2 K4 K T K1 K H 1 pffiffiffi sin / þ cos cos / : sin b2 2 2 r Db 2 ð49Þ
Clearly, as in the analogous classical isotropic elastic situation, all the field components have been shown to exhibit pffiffiffi a 1= r crack-tip behavior and to depend upon the imposed non-constant crack-face excitations, T (n) and H(n), only through the intensity factors defined by Eq. (47). It is instructive to observe that in practice therefore the interesting magnitudes of the stress and displacement concentrations around the crack tip can be increased or decreased as desired by varying appropriately, depending upon the material constants of the quasicrystal, the applied phonon and phason loads. Further, it should be noted from Eqs. (25) and (26) that the present analysis is not applicable when b1 ¼ 0 or b2 ¼ 0 and when c44 K R2 ¼ 0. From the definition (12), these values of b1 and b2 are attained when the speed of the crack, v, equals that of the two wave speeds, s1 and s2 , under antiplane shear conditions given by Eq. (13). The measurements of the values of the relevant material constants of quasicrystals are at present not completely reliable. For one-dimensional hexagonal quasicrystals none have been reported yet. However, Guo et al. (2013a) suggest that a reasonable approximation could be the data for the icosahedral Al–Pd–Mn
alloy 8
for
which
c44 ¼ 7:49 1010 Nm2
and
2
K ¼ 1:25 10 Nm . Significant experimental indeterminancy is apparent in the value for R. But, as stated by Fan et al. (1999) ‘‘the phonon–phason coupling constant R is, in general, quite small’’, so R2 will be significantly smaller than the product of c44 and K.
Fig. 2. Variation around the crack tip of the scaled phonon stress component pffiffiffi r r/z =K T for the scaled speed v =s2 ¼ 0:5.
pffiffiffi r r/z =K T in the illustrative case of H(n) = 0, for example, can be calculated for a subsonically moving crack from the representation (48) for a range of values of the scaled speed v =s2 , with the wave speed s2 given by Eq. (13). These are dependent upon having reliable values for the material constants of the medium of interest. But to indicate the general trends here it is convenient to utilize the aforementioned typical representative values for the icosahedral Al–Pd–Mn alloy which include (from, for example, Fan (2011, 2013), Guo et al. (2013a) and c44 ¼ 7:49 1010 Nm2 , K ¼ 1:25 108 Nm2 ,
R ¼ 7:24 108 Nm2
and
q ¼ 5:1
3
10 kg m3 . With this data and the speed of the crack within the range 0 v < s2 , the graphs are almost coincident. But it is seen from Fig. 2 from the representative curve for the mid-range speed v =s2 ¼ 0:5 that as would be expected the magnitude of this stress component decreases as the angle from the forward direction of motion increases. It is found that this decrease becomes very slightly smaller when the speed of the crack increases within this range. This is shown in Fig. 3, where the variation around the crack tip of the pffiffiffi scaled phonon stress component rs/z =K T is displayed for a range of values of the scaled speed v =s2 . Here s/z ¼ 105 ðr/z r/z jv ¼0 Þ, which thus represents a measure of the difference in value between r/z for a given speed v and its value for a stationary crack when v ¼ 0. 5. Stationary antiplane shear crack Putting v ¼ 0 throughout the analysis considered in Section 3 will obviously yield the corresponding results for the fields created around a stationary quasicrystal crack subject to the general non-uniform loadings prescribed in Eq. (3). These have not been derived elsewhere and it is thus interesting to explicitly record them here briefly. Clearly, Eq. (12) shows that b1 ¼ b2 ¼ 1 when v ¼ 0. Therefore, by noting from Eqs. (29) that K1 þ K2 ¼ 1 and recalling the definition of F F1 ðh1 Þ in Eq. (31), it follows from Eqs. (33)–(36) that for a stationary crack
4. Numerical results
ryz ðx; yÞ ¼ F T1 ðh1 Þ; Hzy ðx; yÞ ¼ F H1 ðh1 Þ;
Demonstrative numerical results for the variation around the crack tip at n ¼ c of the scaled phonon stress component
p rxz ðx; yÞ ¼ F T1 h1 ; 2
Hzx ðx; yÞ ¼ F H 1
ð50Þ
p h1 : 2
ð51Þ
260
G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
Fig. 3. Variation around the crack tip of the scaled phonon stress component
Similarly, with v ¼ 0, Eqs. (43)–(46) demonstrate that near to a crack tip the practically interesting distribution of these fields are given, as r ? 0, by
K
ryz ðr; /Þ pTffiffiffi cos r
K
/ ; 2
rxz ðr; /Þ pTffiffiffi sin r
/ ; 2
KH / Hzy ðr; /Þ pffiffiffi cos ; 2 r KH / Hzx ðr; /Þ pffiffiffi sin 2 r
ð52Þ
ð53Þ
and then, from Eqs. (48) and (49),
K / ; r/z ðr; /Þ pTffiffiffi cos 2 r
KH / Hz/ ðr; /Þ pffiffiffi cos : 2 r
rffiffiffi c T; 2
KT ¼
rffiffiffi c H: 2
ð54Þ
rffiffiffiffiffi c / ; T cos 2r 2
Contour integrations can be used to verify that
Z
c
c
dn00 n o 2 ðc2 n002 Þ ðn0 n00 Þ ðn n00 Þ þ j2 y2 1 2
¼ Z
c
c
¼ rffiffiffiffiffi c / ; H cos 2r 2
Hzy ðr; /Þ
ð56Þ
which, as would be expected, agree with the solutions derived by Guo et al. (2013b) using a semi-inverse method (when their remote 1 loadings r1 yz and Hzy are replaced by –T and –H, respectively). 6. Conclusions Explicit closed-form expressions for the components of the stress fields around a mode III Yoffe-type crack subjected to arbitrary non-constant phonon and phason stress loads moving within quasicrystals are derived. The analysis is based upon the fundamental linear elastic formulation of one-dimensional hexagonal quasicrystals, using an extended continuous distribution of dislocations technique.
pfjy sin H ðn n0 Þ cos Hg n o ; 2 yjR ðn n0 Þ þ j2 y2
ðA:1Þ
ðn n00 Þdn00 n o 2 ðc2 n002 Þ ðn0 n00 Þ ðn n00 Þ þ j2 y2
ð55Þ
The expressions (52) then reduce to
ryz ðr; /Þ
There is currently still a dearth of accurate experimental data for the values of the relevant material constants. But reasonable representative values are introduced to enable the illustrative variations in the phonon stress component with the angle around the crack-tip and the speed to be readily calculated and presented graphically. Expressions for the field around a similarly-loaded, corresponding stationary crack, which have not been available previously, are deduced as a special case. Appendix A
For the moving crack of Section 3, it was found that r/z ðr; /Þ and Hz/ ðr; /Þ both depended upon both T ðnÞ and HðnÞ, whereas, in contrast, it is interesting to observe that the above r/z ðr; /Þ depends only upon T ðxÞ and likewise Hz/ ðr; /Þ depends only upon HðxÞ. Finally, in the special case of a stationary crack when the imposed loadings in Eq. (3) are simply constants, so that T ðxÞ ¼ T = constant and HðxÞ ¼ H = constant, the intensity factors in Eq. (47) can be evaluated as
KT ¼
pffiffiffi r s/z =K T for a range of values of the scaled speed v =s2 .
1 2
pfjy cos H þ ðn n0 Þ sin Hg n o 2 R ðn n0 Þ þ j2 y2
for constant
n
ðA:2Þ
j, where the branches of
ReiH ¼ c2 ðn þ ijyÞ
2
o12
are chosen analogously to those in Eq. (39). References Barnett, D.M., Asaro, R.J., 1972. The fracture mechanics of slit-like cracks in anisotropic elastic media. J. Mech. Phys. Solids 20, 353–366. Bilby, B.A., Eshelby, J.D., 1968. Dislocations and the theory of fracture. In: Liebowitz, H. (Ed.), Fracture, vol. 1. Academic Press, New York, pp. 99–182. Colli, S., Mariano, P.M., 2011. The standard descriptions of quasicrystal linear elasticity may produce non-physical results. Phys. Lett. A 375, 3335–3339. Ding, D.H., Yang, W.G., Hu, C.Z., Wang, R.H., 1993. Generalized elasticity theory of quasicrystals. Phys. Rev. B 48, 7003–7009. Fan, T.Y., 2011. The Mathematical Theory of Elasticity of Quasicrystals and its Applications. Science Press, Springer-Verlag, Beijing, Heidelberg.
G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261 Fan, T.Y., 2013. Mathematical theory and methods of mechanics of quasicrystalline materials. Engineering 5, 407–448. Fan, T.-Y., Li, X.-F., Sun, Y.-F., 1999. A moving screw dislocation in a one-dimensional hexagonal quasicrystal. Acta Phys. Sin. (Overseas Edn.) 8, 288–295. Fan, T.-Y., Tang, Z.-Y., Chen, W.-Q., 2012. Theory of linear, nonlinear and dynamic fracture for quasicrystals. Eng. Fract. Mech. 82, 185–194. Gakhov, F.D., 1966. Boundary Value Problems. Pergamon, Oxford. Guo, J.-H., Lu, Z.-X., 2011. Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals. Appl. Math. Comput. 217, 9397–9403. Guo, J.H., Yu, J., Xing, Y.M., 2013a. Anti-plane analysis on a finite crack in a onedimensional hexagonal quasicrystal strip. Mech. Res. Commun. 52, 40–45. Guo, J.-H., Yu, J., Si, R., 2013b. A semi-inverse method of a Griffith crack in onedimensional hexagonal quasicrystals. Appl. Math. Comput. 219, 7445–7449.
261
Lardner, R.W., 1974. Mathematical Theory of Dislocations and Fracture. University of Toronto Press, Toronto. Li, X.-Y., 2014. Elastic field in an infinite medium of one-dimensional hexagonal quasicrystal with a planar crack. Int. J. Solids Struct. 51, 1442–1455. Li, L.-H., Fan, T.-Y., 2008. Exact solutions of two semi-infinite collinear cracks in a strip of one-dimensional hexagonal quasicrystal. Appl. Math. Comput. 196, 1–5. Liu, G.-T., Guo, R.-P., Fan, T.-Y., 2003. On the interaction between dislocations and cracks in one-dimensional hexagonal quasi-crystals. Chin. Phys. 12, 1149–1155. Muskhelishvili, N.I., 1953. Singular Integral Equations. Noordhoff Int. Pub, Leyden. Shechtman, D., Blech, I., Gratias, D., Cahn, J.W., 1984. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953. Shi, W., 2009. Collinear periodic cracks and/or rigid line inclusions of antiplane sliding mode in one-dimensional hexagonal quasicrystal. Appl. Math. Comput. 215, 1062–1067.
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
An antiplane shear crack moving in one-dimensional hexagonal quasicrystals G.E. Tupholme ⇑ Faculty of Engineering and Informatics, University of Bradford, Bradford BD7 1DP, UK
a r t i c l e
i n f o
Article history: Received 4 February 2015 Received in revised form 26 May 2015 Available online 2 July 2015 Keywords: Moving antiplane shear cracks Quasicrystals Dislocation layers
a b s t r a c t An investigation is presented of an antiplane shear strip crack subjected to representative general non-constant crack-face loading conditions which is moving uniformly in one-dimensional hexagonal quasicrystals, using an extended technique of continuous dislocation layers. Readily-calculable explicit closed-form representations are determined and discussed for the components of the phonon and phason stress fields created throughout the media. Illustrative numerical results are presented graphically. As a special case, the corresponding expressions are deduced explicitly for a non-uniformly loaded stationary crack. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The discovery of new complex structures with unusual properties in solid materials was first revealed when Shechtman et al. (1984) reported upon their seminal experimental observations of aluminium–manganese alloys having aperiodic symmetry. Subsequently, these became known as quasicrystals, which are aperiodic materials that display quasiperiodic translational symmetry and non-classical crystallographic rotational symmetry. Many experimental observations have shown quasicrystals to be quite brittle and thus prone to failure due to defects. From a practical point of view, it is therefore most desirable to study the propagation of cracks within them. Ding et al. (1993) and Fan (2011, 2013), for example and the references therein, give comprehensive accounts of the now well-documented physical and mathematical theories of quasicrystals which have been developed. The concept of not only a phonon but also a phason displacement is introduced and theoretical descriptions of their deformations necessitates a detailed study of the interaction of the phonon and phason stress fields created. Investigations of various stationary cracks subject to uniform constant loads within one-dimensional hexagonal quasicrystals have been undertake. Liu et al. (2003) used the complex function method to analyze a semi-infinite crack and to discuss its interaction with a dislocation. Complex variable theory and conformal transformations enabled Li and Fan (2008) to derive representations for the phonon and phason stress intensity factors at the tips ⇑ Tel.: +44 1274 234273. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijsolstr.2015.06.027 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.
of two semi-infinite cracks. Shi (2009) introduced harmonic functions to reduce the antiplane sliding mode models of collinear periodic cracks and/or rigid line inclusions to solvable Riemann– Hilbert problems. A development of a Stroh-type formulation was utilized by Guo and Lu (2011) to examine the fracture mechanics of four mode III cracks originating from an elliptical hole, using a new mapping function to reduce the boundary value problem to Cauchy integral equations. Then more recently, Guo et al. (2013a) used Fourier transforms, dual integral equations and complete elliptic integrals to analyze a crack in a quasicrystal strip containing an antiplane crack and Guo et al. (2013b) proposed a semi-inverse method by introducing Westergaard’s stress functions for studying the behavior of a Griffith crack in an infinite material. Li (2014) considered the three mode I problems of a penny-shaped crack, an external circular crack and a semi-infinite plane crack embedded in an infinite space of quasicrystals. Extensive additional references are also conveniently provided to the available literature in these references. However, the now well-established technique of using continuous distributions of dislocations to model slit-like cracks was first devised for isotropic elastic media; as reviewed by, for example, Bilby and Eshelby (1968) and Lardner (1974). Subsequently, Barnett and Asaro (1972) used a six-dimensional eigenvalue extended Stroh formalism for thereby studying cracks in general anisotropic elastic media and it is noteworthy that they suggested that the technique of dislocation layers appears ‘‘to be more straightforward and to facilitate computational convenience’’ more easily than the approaches using complex variable methods and integral transforms.
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G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
The provision of mechanical descriptions of quasicrystals is an evolving area of investigation and the implications of more sophisticated models are currently under consideration for adoption. Recently, for example, Colli and Mariano (2011) suggested that in particular situations the basic quasicrystal linear elasticity formulation may lead to non-physical anomalies. Further, attention had been expressed by, for example, Fan et al. (2012) in considering nonlinear effects in quasicrystals to account for their plasticity– ductility at high temperatures. These extremely interesting fields of study however are beyond the scope here. The purpose of the present paper is to demonstrate to the quasicrystal community that the so-called dislocation layer technique is indeed especially suited for extension in a particularly convenient and quite straightforward manner to derive analytically explicit versatile expressions for the components of the fields created by a mode III crack moving through one-dimensional hexagonal quasicrystals under arbitrary general crack-face loading conditions. The foundations of this current analysis are provided by the results of Fan et al. (1999) for a dislocation moving uniformly through such materials. In Section 2, the basic formulation of the situation under consideration is presented, together with the required components of the phonon and phason displacement and stress fields around a moving quasicrystal screw dislocation. An analysis and derivation of the resulting fields created by a moving arbitrarily-loaded antiplane shear crack are then given and discussed in Section 3. Some representative numerical results for the variation of the stress component with speed around the crack tip are provided in Section 4. Then, the corresponding results, which have not been reported previously, for a stationary crack subjected to general non-uniform loading in a quasicrystal are derived in Section 5 explicitly as a special case. Finally, in Section 6, the conclusions from this investigation are drawn together. 2. Physical and theoretical formulation A homogeneous one-dimensional hexagonal quasicrystal with point group 6 mm is supposed to be everywhere at rest and stress-free in a natural reference state initially, with a uniform density, q. It is periodic in the x–y plane and quasiperiodic in the positive direction of the z-axis, relative to a fixed system of rectangular Cartesian coordinates (x, y, z). An embedded loaded plane Griffith-type strip crack of width 2c moves through the material parallel to its axis in its own plane. At a time t, it is assumed that the crack occupies the region y = 0, vt c < x < vt + c, 1 < z < 1 of the x–z plane, as depicted in Fig. 1, where v is its speed of propagation. y
Here, cij are the elastic constants in the phonon field, using the conventional contracted Voigt’s notation with i and j taking integer values, K 1 and K 2 are two elastic constants in the phason field and R1 , R2 and R3 are three phonon–phason coupling elastic constants. A moving coordinate n is defined by
n ¼ x vt
ð2Þ
and a mode III antiplane deformation is created by applying prescribed general non-constant phonon and phason stresses symmetrically to the two faces of the crack. Thus the boundary conditions are taken to be
ryz ðn; 0Þ ¼ T ðnÞ; Hzy ðn; 0Þ ¼ HðnÞ; for jnj < c;
ð3Þ
with the medium remaining undisturbed at infinity. Analogously, an antiplane deformation created by instead specifying the phonon and phason strain components eyz and wzy on the crack faces could be studied. The detailed analysis corresponding to that presented here for the conditions (3) can be developed by an interested reader. All the field variables are independent of z in this case. Hence the relations between the relevant non-zero phonon and phason strain components and the phonon displacement component uz and the phason displacement wz are given by
exz ¼
1 @uz ; 2 @x
eyz ¼
@wz ; @x
wzy ¼
wzx ¼
1 @uz ; 2 @y
ð4Þ
@wz @y
ð5Þ
and the generalized Hooke’s laws required reduce, from Eq. (1), to
ryz ¼ 2c44 ezy þ Rwzy ; rxz ¼ 2c44 ezx þ Rwzx ;
ð6Þ
Hzy ¼ 2Rezy þ Kwzy ;
ð7Þ
Hzx ¼ 2Rezx þ Kwzx ;
where, for brevity of presentation here and henceforth, the constants R3 and K 2 are abbreviated to simply R and K, respectively. As an antecedent to studying this moving mode III quasicrystal crack, it is appropriate to consider first a ‘‘quasicrystal screw dislocation’’ moving in the material with extended Burgers vector ð0; 0; b; dÞ in four-dimensional space which has a discontinuity b in the phonon displacement u (the traditional Burgers vector, b, of an elastic screw dislocation) within a ‘‘phonon dislocation’’ and also an analogous discontinuity d in the phason displacement within a ‘‘phason dislocation’’ across the slip plane. These jumps are given, respectively, for n > 0, by
2c
O
The constitutive equations relating the phonon stress components rij , phason stress components Hzi , phonon strain components eij and phason strain components wzi , where i, j = x, y or z, within the quasicrystal can be written in matrix form as 2 3 2 32 3 rxx exx c11 c12 c13 0 0 0 R1 0 0 6r 7 6c 6 7 0 0 0 R1 0 0 7 6 yy 7 6 12 c11 c13 7 6 eyy 7 6 7 6 76 7 6 rzz 7 6 c13 c13 c33 6 7 7 e 0 0 0 R 0 0 2 6 7 6 7 6 zz 7 6r 7 6 76e 7 0 0 0 0 R3 7 6 yz 7 0 0 2c44 6 yz 7 6 0 6 7 6 76 7 6 rxz 7 ¼ 6 0 6 7 0 0 0 2c44 0 0 R3 0 7 6 7 6 7 6 exz 7: 6 7 6 76 7 0 0 0 0 c11 c12 0 0 0 7 6 exy 7 6 rxy 7 6 0 6 7 6 76 7 6 Hzz 7 6 R1 R1 R2 6 7 0 0 0 K1 0 0 7 6 wzz 7 6 7 6 7 6 7 6 76 7 0 0 K 2 0 5 4 wzx 5 0 0 0 2R3 4 Hzx 5 4 0 Hzy wzy 0 0 0 2R3 0 0 0 0 K2 ð1Þ
x
uIII ðn; 0þÞ uIII ðn; 0Þ ¼ ð0; 0; bÞ;
III wIII z ðn; 0þÞ wz ðn; 0Þ ¼ d:
ð8Þ vt Fig. 1. A strip crack moving with constant speed
v in the x-direction.
Throughout, the superscript III refers to quantities associated with a mode III deformation.
G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
A discussion of the dynamics of such a straight screw dislocation in a one-dimensional hexagonal quasicrystal with the Laue classes 6/mh and 6/mh mm has been given by Fan et al. (1999), during which expressions for the induced phonon and phason displacement components were presented. It is these which stimulated the impetus for the current analysis. From them, it follows that for a moving screw dislocation situated at the origin in a one-dimensional hexagonal quasicrystal
1 2 2 1 b1 y 1 b2 y þ R b uIII ðn; yÞ ¼ a tan tan z n n 2pða2 þ R2 Þ b y b y tan1 2 d þRa tan1 1 n n wIII z ðn; yÞ ¼
b y b y tan1 2 b Ra tan1 1 n n 2pða2 þ R Þ b y b y þ a2 tan1 2 d þ R2 tan1 1 n n 2
a ¼ c44 K þ
ð10Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc44 KÞ2 þ 4R2 2
ð11Þ
for i ¼ 1 and 2;
ð12Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2 s2i
in which the two wave speeds, s1 and s2 , under antiplane shear conditions are given by
si ¼
pffiffiffiffiffiffiffiffiffiffi ei =q;
ð13Þ
with
e1 e2
2
h n o b b1 aðc44 a þ R2 Þ þ b2 R2 ðc44 aÞ
2pða2 þ R Þn n oi þdR b1 ðc44 a þ R2 Þ b2 aðc44 aÞ
HIII zy ðn; 0Þ ¼
h
1 2
ð19Þ
n o bR b1 aða þ KÞ b2 ðaK R2 Þ ð20Þ
The analogous phonon and phason strain components can be derived from Eqs. (4), (5), (9) and (10) if desired. 3. Moving antiplane shear crack
and
1
1
rIIIyz ðn; 0Þ ¼
1
where
bi ¼
Crucially it can be observed that on the boundary y = 0, where the boundary conditions (3) are to be imposed,
2pða2 þ R Þn n oi þd b1 R2 ða þ KÞ þ b2 aðaK R2 Þ
ð9Þ
257
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c44 þ K þ ðc44 KÞ2 þ 4R2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c44 þ K ðc44 KÞ2 þ 4R2 2:
ryz ðn; 0Þ ¼ ð14Þ
The corresponding non-zero components of the phonon and phason stresses then follow from the constitutive equations (6) and (7) in the forms
" ( ) b1 aðc44 a þ R2 Þ b2 R2 ðc44 aÞ b þ n2 þ b21 y2 n2 þ b22 y2 2pða2 þ R2 Þ ( )# b ðc44 a þ R2 Þ b2 aðc44 aÞ ð15Þ 2 þdR 1 2 n þ b21 y2 n þ b22 y2
rIIIxz ðn; yÞ ¼
y
" ( ) n b1 aðc44 a þ R2 Þ b2 R2 ðc44 aÞ III ryz ðn; yÞ ¼ b þ n2 þ b21 y2 n2 þ b22 y2 2pða2 þ R2 Þ ( )# b ðc44 a þ R2 Þ b2 aðc44 aÞ 2 þdR 1 2 n þ b21 y2 n þ b22 y2 ( ) y b1 aða þ KÞ b2 ðaK R2 Þ III Hzx ðn; yÞ ¼ bR 2 n2 þ b21 y2 n þ b22 y2 2pða2 þ R2 Þ ( )# b R2 ða þ KÞÞ b2 aðaK R2 Þ þ þd 1 2 n þ b21 y2 n2 þ b22 y2
ð16Þ
c
ð17Þ Z
c
c
ð18Þ
2
h n o b b1 aðc44 a þ R2 Þ þ b2 R2 ðc44 aÞ
It is apparent in evaluating ryz ðn; 0Þ and Hzy ðn; 0Þ from the Plemelj formulae that the integrals throughout Eqs. (21) and (22) should be interpreted as Cauchy principal value integrals. The pair of simultaneous equations which arises by equating the expressions (21) and (22) to the imposed boundary conditions (3) can be solved to yield
c
"
1
2pða2 þ R Þ Z c n o f ðn0 Þ 0 2 0 dn þ dR b1 ðc 44 a þ R Þ b2 aðc 44 aÞ n n c Z c gðn0 Þ 0 ð21Þ 0 dn ; c n n h n o 1 Hzy ðn; 0Þ ¼ bR b1 aða þ KÞ b2 ðaK R2 Þ 2 2 2pða þ R Þ Z c n o f ðn0 Þ 0 2 2 0 dn þ d b1 R ða þ KÞ þ b2 aðaK R Þ c n n Z c gðn0 Þ 0 : ð22Þ dn 0 c n n
Z
"
( ) n b1 aða þ KÞ b2 ðaK R2 Þ HIII ðn; yÞ ¼ bR zy n2 þ b21 y2 n2 þ b22 y2 2pða2 þ R2 Þ ( )# b R2 ða þ KÞÞ b2 aðaK R2 Þ þ þd 1 2 n þ b21 y2 n2 þ b22 y2
The well-established fundamental dislocation layer method, which was devised for a purely elastic material, whereby a loaded crack is analyzed by replacing it by an equivalent continuous planar distribution of dislocations is now developed correspondingly for the crack under consideration here within a quasicrystal. For isotropic materials, the details of its basic implementation have been conveniently given by, for example, Bilby and Eshelby (1968) and Lardner (1974). The basic technique is adapted by spreading an array of moving quasicrystal screw dislocations of the type prescribed in Section 2 over the region jnj < c, y ¼ 0; 1 < z < 1. To model the mode III crack subject to conditions (3), the densities of the proposed distributions of phonon and phason dislocations are taken to be f ðnÞ and gðnÞ, respectively. Then it follows, from Eqs. (19) and (20), that the corresponding phonon and phason stress components at a point on the n-axis are given by
f ðn0 Þ 0 2p dn ¼ n n0 bb1 b2 ðc44 K R2 Þða2 þ R2 Þ hn o b1 R2 ða þ KÞ þ b2 aðaK R2 Þ T ðnÞ n o i R b1 ðc44 a þ R2 Þ b2 aðc44 aÞ HðnÞ ;
ð23Þ
gðn0 Þ 0 2p dn ¼ n n0 db1 b2 ðc44 K R2 Þða2 þ R2 Þ h n o R b1 aða þ KÞ b2 ðaK R2 Þ T ðnÞ n o i þ b1 aðc44 a þ R2 Þ þ b2 R2 ðc44 aÞ HðnÞ :
ð24Þ
258
G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
The appropriate solutions of these integral equations are deducible from the results of Muskhelishvili (1953) and Gakhov (1966) in the forms
2
f ðnÞ ¼
1
n
pbb1 b2 ðc44 K R2 Þða2 þ R2 Þ ðc2 n2 Þ12 Z
c
2
ð33Þ ð25Þ
n o H Hzy ðn; yÞ ¼ K4 F Tb1 ðhb1 Þ F Tb2 ðhb2 Þ þ K2 F H b1 ðhb1 Þ þ K1 F b2 ðhb2 Þ;
1
ð34Þ
1 2
c o ðc2 n02 Þ h n R b1 aða þ KÞ b2 ðaK R2 Þ T ðn0 Þ 0 n n n c o i þ b1 aðc44 a þ R2 Þ þ b2 R2 ðc44 aÞ Hðn0 Þ dn0 ;
ryz ðn; yÞ Z
b 2
2pða þ R Þ 2
(
c
b1 aðc44 a þ R2 Þ
ðn n00 Þ
00 2
b21 y2
þ
b2 R2 ðc44 aÞ 00 2
ð26Þ
)
b22 y2
f ðn00 Þdn00
ðn n Þ þ ðn n Þ þ ( ) b1 ðc44 a þ R2 Þ b2 aðc44 aÞ þ ðn n Þ gðn00 Þdn00 : 2 2 2pða2 þ R2 Þ c ðn n00 Þ þ b21 y2 ðn n00 Þ þ b22 y2 c
Z
dR
c
00
ð27Þ
Upon substitution of the expressions (25) and (26) this can be shown, after simplification, to take the convenient form
ryz ðn; yÞ ¼
Z
1
p2
Z
c
1
ðc2 n002 Þ2 ½fK1 T ðn0 Þ þ K3 Hðn0 Þg
c
ðn n00 Þdn00 0 n o þ fK2 T ðn Þ c ðc2 n002 Þ ðn0 n00 Þ ðn n00 Þ2 þ b2 y2 1 3 Z c 00 00 ðn n Þdn 7 0 K3 Hðn0 Þg n o5dn ; 1 c ðc 2 n002 Þ2 ðn0 n00 Þ ðn n00 Þ2 þ b2 y2 2
K1 T p K2 T p F ðhb Þ F ðhb Þ b1 b1 1 2 b 2 b2 2 2 1 H p 1 p F b1 ðhb1 Þ F H ðh Þ K3 b b1 b2 b2 2 2 2
rxz ðn; yÞ ¼
with the relative phonon and phason displacements of the two crack faces assumed to vanish at n = ± c. Now that the necessary densities f ðnÞ and gðnÞ have been determined, all the components of the phonon and phason fields can be directly calculated explicitly as desired from Eqs. (15)–(18), (25) and (26). The expression for ryz ðn; yÞ is derived here as a typical example, to illustrate the type of detailed analyses which are necessary. It is seen from Eq. (16) that
1 T p 1 p F ðhb Þ F Tb2 ðhb2 Þ b1 b1 1 2 b2 2 K2 H p K1 H p F ðhb Þ F ðhb Þ b1 b1 1 2 b2 b2 2 2
1 2
where the dimensionless constants K1 ; K2 and K3 can be expressed as
ðc44 a þ R2 ÞðaK R2 Þ 2
2
ða2 þ R Þðc44 K R Þ
;
K2 ¼
R2 ðc44 aÞða þ KÞ ða2 þ R2 Þðc44 K R2 Þ
;
ð29Þ
Rðc44 a þ R Þðc44 aÞ ða2 þ R2 Þðc44 K R2 Þ
:
ð30Þ
The functions F Fk ðhk Þ, Rk ðn; yÞ and hk ðn; yÞ, for k ¼ b1 ; b2 and 1, and F ¼ T and H, are now defined for the sake of notational simplification by
F Fk ðhk Þ ¼
1
p
Z
c
c
1 ky cos hk þ ðn n0 Þ sin hk 2 n o ðc n002 Þ2 Fðn0 Þdn0 ; 2 2 0 2 Rk ðn n Þ þ k y
n o1 2 2 Rk eihk ¼ c2 ðn þ ikyÞ ;
ð31Þ
ð32Þ
ð36Þ
where
K4 ¼
Rða þ KÞðaK R2 Þ ða2 þ R2 Þðc44 K R2 Þ
:
ð37Þ
At this stage, it is worthy of note that, with the general boundary conditions (3) imposed, Eqs. (33)–(36) exhibit explicitly that all the stress components depend upon both T and H together with the quasicrystal material constants and the speed of the crack. The important features of the distributions of these components near a crack tip become apparent by setting
n ¼ c þ r cos /;
y ¼ r sin /
ð38Þ
and considering situations where r c, into Eqs. (33)–(36). Then, from Eq. (32), the corresponding approximations to Rk and hk as r ? 0 can be shown to be
Rk
12 1 2 2 2crðcos2 / þ k sin /Þ2 ;
hk ðp Uk Þ=2;
ð39Þ ð40Þ
with
Uk ¼ tan1 ðk tan /Þ;
ð41Þ
1
where tan ð:::Þ is understood to indicate the principal value of the inverse tangent for 0 / p=2 and p plus the principal value for p=2 / p. Substituting these into Eqs. (33)–(36) and (31), and putting 2
K3 ¼
2
1
Dk ¼ ðcos2 / þ k sin /Þ4
2
ð35Þ
Hzx ðn; yÞ ¼ K4
c
ð28Þ
K1 ¼
Similarly, Eqs. (15), (17), (18), (A.1) and (A.2) yield the representations
pdb1 b2 ðc44 K R2 Þða2 þ R2 Þ ðc2 n2 Þ12 Z
¼
o
ryz ðn; yÞ ¼ K1 F Tb1 ðhb1 Þ þ K2 F Tb2 ðhb2 Þ þ K3 F Hb1 ðhb1 Þ F Hb2 ðhb2 Þ :
1 o ðc2 n02 Þ2 hn b1 R2 ða þ KÞ þ b2 aðaK R2 Þ T ðn0 Þ 0 n n nc o i R b1 ðc44 a þ R2 Þ b2 aðc44 aÞ Hðn0 Þ dn0 ;
gðnÞ ¼
with the branches of the square root function determined by choosing hk to be zero for jnj < c, y = 0+ and defined by analytic continuation elsewhere. Then, using the result (A.2) in the Appendix, it follows from the expression (28) that
ð42Þ
for k ¼ b1 ; b2 and 1, yields
Ub1 Ub2 K1 K2 þ cos cos 2 Db 2 2 r Db 1 Ub1 Ub2 KH 1 1 ; cos cos þ pffiffiffi K3 Db 1 2 Db 2 2 r
K
ryz ðr; /Þ pTffiffiffi
Ub1 Ub2 KT 1 1 Hzy ðr; /Þ pffiffiffi K4 cos cos Db 2 Db 2 2 r 1 Ub1 Ub2 K H K2 K1 þ ; cos cos þ pffiffiffi 2 Db2 2 r Db 1
ð43Þ
ð44Þ
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G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
K
Ub1 Ub2 K1 K2 þ sin sin 2 b2 Db2 2 r b1 Db1 Ub1 Ub2 KH 1 1 ; pffiffiffi K3 sin sin b1 Db1 b2 Db2 2 2 r
rxz ðr; /Þ pTffiffiffi
Ub1 Ub2 KT 1 1 Hzx ðr; /Þ pffiffiffi K4 sin sin b1 Db1 b2 Db2 2 2 r Ub1 Ub2 KH K2 K1 þ ; sin sin pffiffiffi 2 b2 Db2 2 r b 1 Db 1
ð45Þ
ð46Þ
as r ? 0, where K T and K H , which are defined for F ¼ T and H by
1 K F ¼ pffiffiffiffiffiffi p 2c
Z
c
c
c þ n0 c n0
12
Fðn0 Þdn0
ð47Þ
are the phonon and phason stress intensity factors corresponding to that at the end of an isotropic elastic crack. Finally, the Eqs. (43) and (45), and (44) and (46), respectively, yield expressions for the phonon and phason stress components, r/z and Hz/ , near the crack tip of the form
r/z ðr; /Þ
Hz/ ðr; /Þ
Ub1 Ub1 K1 K T þ K3 K H 1 pffiffiffi sin / þ cos cos / sin b1 2 2 r Db 1 Ub2 Ub2 K2 K T K3 K H 1 pffiffiffi sin / þ cos cos / ; þ sin b2 2 2 r Db2 ð48Þ Ub1 Ub1 K4 K T þ K2 K H 1 pffiffiffi sin / þ cos cos / sin b1 2 2 r Db 1 Ub2 Ub2 K4 K T K1 K H 1 pffiffiffi sin / þ cos cos / : sin b2 2 2 r Db 2 ð49Þ
Clearly, as in the analogous classical isotropic elastic situation, all the field components have been shown to exhibit pffiffiffi a 1= r crack-tip behavior and to depend upon the imposed non-constant crack-face excitations, T (n) and H(n), only through the intensity factors defined by Eq. (47). It is instructive to observe that in practice therefore the interesting magnitudes of the stress and displacement concentrations around the crack tip can be increased or decreased as desired by varying appropriately, depending upon the material constants of the quasicrystal, the applied phonon and phason loads. Further, it should be noted from Eqs. (25) and (26) that the present analysis is not applicable when b1 ¼ 0 or b2 ¼ 0 and when c44 K R2 ¼ 0. From the definition (12), these values of b1 and b2 are attained when the speed of the crack, v, equals that of the two wave speeds, s1 and s2 , under antiplane shear conditions given by Eq. (13). The measurements of the values of the relevant material constants of quasicrystals are at present not completely reliable. For one-dimensional hexagonal quasicrystals none have been reported yet. However, Guo et al. (2013a) suggest that a reasonable approximation could be the data for the icosahedral Al–Pd–Mn
alloy 8
for
which
c44 ¼ 7:49 1010 Nm2
and
2
K ¼ 1:25 10 Nm . Significant experimental indeterminancy is apparent in the value for R. But, as stated by Fan et al. (1999) ‘‘the phonon–phason coupling constant R is, in general, quite small’’, so R2 will be significantly smaller than the product of c44 and K.
Fig. 2. Variation around the crack tip of the scaled phonon stress component pffiffiffi r r/z =K T for the scaled speed v =s2 ¼ 0:5.
pffiffiffi r r/z =K T in the illustrative case of H(n) = 0, for example, can be calculated for a subsonically moving crack from the representation (48) for a range of values of the scaled speed v =s2 , with the wave speed s2 given by Eq. (13). These are dependent upon having reliable values for the material constants of the medium of interest. But to indicate the general trends here it is convenient to utilize the aforementioned typical representative values for the icosahedral Al–Pd–Mn alloy which include (from, for example, Fan (2011, 2013), Guo et al. (2013a) and c44 ¼ 7:49 1010 Nm2 , K ¼ 1:25 108 Nm2 ,
R ¼ 7:24 108 Nm2
and
q ¼ 5:1
3
10 kg m3 . With this data and the speed of the crack within the range 0 v < s2 , the graphs are almost coincident. But it is seen from Fig. 2 from the representative curve for the mid-range speed v =s2 ¼ 0:5 that as would be expected the magnitude of this stress component decreases as the angle from the forward direction of motion increases. It is found that this decrease becomes very slightly smaller when the speed of the crack increases within this range. This is shown in Fig. 3, where the variation around the crack tip of the pffiffiffi scaled phonon stress component rs/z =K T is displayed for a range of values of the scaled speed v =s2 . Here s/z ¼ 105 ðr/z r/z jv ¼0 Þ, which thus represents a measure of the difference in value between r/z for a given speed v and its value for a stationary crack when v ¼ 0. 5. Stationary antiplane shear crack Putting v ¼ 0 throughout the analysis considered in Section 3 will obviously yield the corresponding results for the fields created around a stationary quasicrystal crack subject to the general non-uniform loadings prescribed in Eq. (3). These have not been derived elsewhere and it is thus interesting to explicitly record them here briefly. Clearly, Eq. (12) shows that b1 ¼ b2 ¼ 1 when v ¼ 0. Therefore, by noting from Eqs. (29) that K1 þ K2 ¼ 1 and recalling the definition of F F1 ðh1 Þ in Eq. (31), it follows from Eqs. (33)–(36) that for a stationary crack
4. Numerical results
ryz ðx; yÞ ¼ F T1 ðh1 Þ; Hzy ðx; yÞ ¼ F H1 ðh1 Þ;
Demonstrative numerical results for the variation around the crack tip at n ¼ c of the scaled phonon stress component
p rxz ðx; yÞ ¼ F T1 h1 ; 2
Hzx ðx; yÞ ¼ F H 1
ð50Þ
p h1 : 2
ð51Þ
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G.E. Tupholme / International Journal of Solids and Structures 71 (2015) 255–261
Fig. 3. Variation around the crack tip of the scaled phonon stress component
Similarly, with v ¼ 0, Eqs. (43)–(46) demonstrate that near to a crack tip the practically interesting distribution of these fields are given, as r ? 0, by
K
ryz ðr; /Þ pTffiffiffi cos r
K
/ ; 2
rxz ðr; /Þ pTffiffiffi sin r
/ ; 2
KH / Hzy ðr; /Þ pffiffiffi cos ; 2 r KH / Hzx ðr; /Þ pffiffiffi sin 2 r
ð52Þ
ð53Þ
and then, from Eqs. (48) and (49),
K / ; r/z ðr; /Þ pTffiffiffi cos 2 r
KH / Hz/ ðr; /Þ pffiffiffi cos : 2 r
rffiffiffi c T; 2
KT ¼
rffiffiffi c H: 2
ð54Þ
rffiffiffiffiffi c / ; T cos 2r 2
Contour integrations can be used to verify that
Z
c
c
dn00 n o 2 ðc2 n002 Þ ðn0 n00 Þ ðn n00 Þ þ j2 y2 1 2
¼ Z
c
c
¼ rffiffiffiffiffi c / ; H cos 2r 2
Hzy ðr; /Þ
ð56Þ
which, as would be expected, agree with the solutions derived by Guo et al. (2013b) using a semi-inverse method (when their remote 1 loadings r1 yz and Hzy are replaced by –T and –H, respectively). 6. Conclusions Explicit closed-form expressions for the components of the stress fields around a mode III Yoffe-type crack subjected to arbitrary non-constant phonon and phason stress loads moving within quasicrystals are derived. The analysis is based upon the fundamental linear elastic formulation of one-dimensional hexagonal quasicrystals, using an extended continuous distribution of dislocations technique.
pfjy sin H ðn n0 Þ cos Hg n o ; 2 yjR ðn n0 Þ þ j2 y2
ðA:1Þ
ðn n00 Þdn00 n o 2 ðc2 n002 Þ ðn0 n00 Þ ðn n00 Þ þ j2 y2
ð55Þ
The expressions (52) then reduce to
ryz ðr; /Þ
There is currently still a dearth of accurate experimental data for the values of the relevant material constants. But reasonable representative values are introduced to enable the illustrative variations in the phonon stress component with the angle around the crack-tip and the speed to be readily calculated and presented graphically. Expressions for the field around a similarly-loaded, corresponding stationary crack, which have not been available previously, are deduced as a special case. Appendix A
For the moving crack of Section 3, it was found that r/z ðr; /Þ and Hz/ ðr; /Þ both depended upon both T ðnÞ and HðnÞ, whereas, in contrast, it is interesting to observe that the above r/z ðr; /Þ depends only upon T ðxÞ and likewise Hz/ ðr; /Þ depends only upon HðxÞ. Finally, in the special case of a stationary crack when the imposed loadings in Eq. (3) are simply constants, so that T ðxÞ ¼ T = constant and HðxÞ ¼ H = constant, the intensity factors in Eq. (47) can be evaluated as
KT ¼
pffiffiffi r s/z =K T for a range of values of the scaled speed v =s2 .
1 2
pfjy cos H þ ðn n0 Þ sin Hg n o 2 R ðn n0 Þ þ j2 y2
for constant
n
ðA:2Þ
j, where the branches of
ReiH ¼ c2 ðn þ ijyÞ
2
o12
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