- Email: [email protected]
Theory of linear, nonlinear and dynamic fracture for quasicrystals
Engineering Fracture Mechanics 82 (2012) 185–194
Contents lists available at SciVerse ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Theory of linear, nonlinear and dynamic fracture for quasicrystals Tian-You Fan a,⇑, Zhi-Yi Tang a, Wei-Qiu Chen b a b
School of Physics, Beijing Institute of Technology, Beijing 100081, PR China Institute of Applied Mechanics, Zhejiang University, Hangzhou 310027, PR China
a r t i c l e
i n f o
Article history: Received 3 May 2010 Received in revised form 21 December 2011 Accepted 22 December 2011
Keywords: Quasicrystals Phonon Phason Phonon–phason coupling Linear fracture Nonlinear fracture Dynamic fracture
a b s t r a c t In this paper, fracture theory of quasicrystalline material is presented, concerning with linear, nonlinear and dynamic fracture problems for different quasicrystal systems observed to date. Main attention is however paid to three-dimensional icosahedral quasicrystals and two-dimensional decagonal quasicrystals, which play the central role in the new kind of materials. Relevant experimental results are also discussed. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction D. Shechtman was awarded the Nobel Prize of Chemistry 2011 due to the discovery of quasicrystal in 1982, which has aroused the great interest on the structure and material again. After the discovery people are interested in the study of solid mechanics, including fracture theory, of the quasicrystalline material. Fracture theory of solids was initiated by Griffith [1] almost a century ago. Since then its scope extends from glass, ceramic and other absolute brittle solids in the early period to metallic material, further to composite material etc. These materials are structural or engineering materials. The main feature of their deformation can be described by one displacement field u(ux, uy, uz) (or u(u1, u2, u3)) and one strain field eij = (oui/oxj + ouj/oxi)/2, which is the basis of their fracture theory. It is evident that the discussion is given in the conventional three-dimensional space. Such a fracture theory is conventional and classical. The discovery of quasicrystals makes it be changed. Quasicrystals bear a new structure of solid as well as a new kind of novel materials, for which high dimensional space, i.e. the six-dimensional space E6 must be introduced. It conk sists of two subspaces, one is the physical space (or parallel space) E3 , and the other is the complementary space to the phys? ical space (or vertical space) E3 , an inner space, this means E6 ¼ Ek3 E? 3 , in which denotes the direct sum. To describe deformation of quasicrystals it is necessary to introduce two displacement fields u(u1, u2, u3) and w(w1, w2, w3), this means ~ ¼ uðu1 ; u2 ; u3 Þ wðw1 ; w2 ; w3 Þ. The displacement field u(u1, u2, u3) is similar to that in the classical the total displacement u k elasticity, and is called as the phonon field according to the physical terminology, which is in the parallel space E3 , while w(w1, w2, w3) is an unusual displacement field, called as the phason field and in the vertical space E? . It follows that there 3 are two strain fields eij = (oui/oxj + ouj/oxi)/2 and wij = owi/oxj, the latter being asymmetric. The associate stress tensor to eij is
⇑ Corresponding author. E-mail address: [email protected] (T.-Y. Fan). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2011.12.009
186
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
denoted by rij, and that to wij is denoted by Hij. In general, the stresses rij and Hij are coupled each other, which can be revealed by the generalized Hooke’s law such as
rij ¼ C ijkl ekl þ Rijkl wkl Hij ¼ Rklij ekl þ K ijkl wkl
ð1Þ
in which Cijkl is the phonon elastic tensor, Kijkl the phason one, and Rklij the phonon–phason coupling one. The determination of these constants is related to the symmetry of the quasicrystals, which can be done by group theory and group representation theory. This results in that the elastic constants of different quasicrystal systems are quite different. The phason field and the phonon–phason coupling field present important effects in elasticity, plasticity and dynamics of quasicrystalline material. The unusual nature of quasicrystals challenges the theories of classical elasticity and fracture. The huge number of field variables and the complexity of field equations lead to greater difficulty in the fracture analyses of this novel material than that of the classical fracture theory. In recent years, however, many crack problems have been investigated, and exact analytic, approximate and numerical solutions have been obtained. These studies provide a basis for further establishing the whole system of fracture theory. The exact solutions for crack problems of quasicrystals are of particular importance, revealing the essential nature which describes the fracture behaviour of this kind of materials. The first exact analytic solution for a Griffith crack in decagonal quasicrystals was derived by Li and Fan [2] in 1999. Afterward Fan [3], and Fan and Mai [4] developed the idea of linear elastic fracture mechanics of quasicrystals based on the common feature of crack solutions. Rudhart et al. [5] discussed the relevant topics from another point of view. Recently Fan and Guo [6], Zhu and Fan [7], and Li and Fan [8] derived analytic solutions of cracks in three-dimensional icosahedral quasicrystals. Furthermore Fan et al. [9], Fan and Fan [10], Fan and Fan [11] and Li and Fan [20] presented the analytic solutions of plastic cracks, though there is lack of plastic constitutive equation for quasicrystals so far. The elastodynamics and fracture dynamics are of particular interests. The phason modes and phonon modes present quite different behaviour in the dynamic process, in which phason modes behave like diffusion rather than wave propagation, refer to Fan et al. [12]. Zhu and Fan [13,14], and Wang and Fan [15,16] carried out dynamic crack analyses of quasicrystals. Meanwhile, the measurement of fracture toughness of quasicrystals was also reported [17]. With the fruitful information mentioned above, it is time to give a summary of the fracture study of quasicrystals. This is down in the present paper, which also put forward some new views on the fracture mechanics of quasicrystals. 2. Linear fracture theory of quasicrystals The crack solutions obtained in the works cited above reveal some common features of stress field and displacement field around the crack tip in quasicrystals. In terms of exact solutions for cracks in one-, two- and three-dimensional quasicrystals in the regime of linear elasticity, the results have a general sense. In addition, the solutions for cracks in dynamic state and with nonlinear behaviour of quasicrystals also exhibit some common features. An evident character is that the fracture behaviour is related to the symmetry of quasicrystal system. Another is that the phason and phonon–phason coupling play an important role. First we consider the linear elastic solutions. As indicted in the above works, within the framework of linear elasticity and for the infinitely sharp crack model, the stresses near the crack-tip, referring to Fig. 1, i.e.
Fig. 1. A Griffith crack in a decagonal quasicrystal subjected to a tensile stress (pulling stress) and the crack tip coordinate systems.
187
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
r1 =a 1
ð2Þ
have been obtained. It is shown that both phonon stresses and phason stresses exhibit singularity of the order r1=2 (r1 ? 0), 1 and other terms can be ignored when compared to this singular term. Stress singularity is practically implausible, and it is the result of an ideal mathematical model. A few researchers indicated its severe weakness in theory and the paradox of its methodology [2,3]. However, before the establishment of more reasonable fracture theory, we still follow this theory, which actually has demonstrated its power in engineering applications (e.g. K-based fracture resistance design). Now we focus on the field variables near the crack tip. For two-dimensional decagonal quasicrystals, by keeping the term in order of (r1/a)1/2 only in the crack solution for the Mode I loading condition, we have
8 k > I > rxx ¼ pKffiffiffiffiffiffiffi cos 12 h1 1 sin 12 h1 sin 32 h1 > > 2 p r 1 > > > k > > KI > p ffiffiffiffiffiffiffi > r ¼ cos 12 h1 1 þ sin 12 h1 sin 32 h1 > > yy 2pr 1 > > > k > > I > > rxy ¼ ryx ¼ pKffiffiffiffiffiffiffi cos 12 h1 cos 32 h1 > 2 pr1 > > > < k d KI Hxx ¼ p21ffiffiffiffiffiffiffi sin h1 2 sin 32 h1 þ 32 sin h1 cos 52 h1 2 p r > 1 > > > k > d21 K I 3 >H ¼ p > ffiffiffiffiffiffiffi 2 sin2 h1 cos 52 h1 > yy > 2pr 1 > > > > k > d21 K I 3 > > Hxy ¼ p ffiffiffiffiffiffiffi 2 sin2 h1 sin 52 h1 > > 2 p r 1 > > > > k > d KI > : Hyx ¼ p21ffiffiffiffiffiffiffi sin h1 2 cos 32 h1 32 sin h1 sin 52 h1
ð3Þ
2pr 1
where d21 = R(K1 K2)/4(MK1 R2), M = (C11 C12)/2 is elastic constants of phonon, K1, K2 the elastic constants of phason, R the phonon–phason coupling elastic constants, and k
K I ¼ limþ x!a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2pðx aÞryy ðx; 0Þ ¼ pap
ð4Þ
( x p pffiffiffiffiffiffiffiffiffi 1 jxj > a x2 a2
ð5Þ
in which
ryy ðx; 0Þ ¼
jxj < a
p
Here Eq. (4) represents a physical parameter describing the fracture behaviour of quasicrystals under the Mode I loading condition. The physical meaning of the generalized surface tractions hi = Hijnj is clear, but have not been measured so far and is not considered at the physical boundary (simply assumed to be zero). Therefore, we only obtain the K kI , whereas the stress intensity factor for the phason field is still unavailable. Can we use the stress intensity factor K kI in the parallel space (physical space) as the physical parameter to characterize the crack stability (instability) in quasicrystals? This shall depend upon tests. It can be found that K kI given by Eq. (4) is not directly related to the material constants of quasicrystals when the state of applied stress is self-equilibrium. This is similar to the conventional fracture mechanics for elastic materials. Nevertheless, it does not mean that it cannot be used as the physical parameter to characterize the crack stability/instability in quasicrystals. Further study on this topic is needed. The displacement field at the crack tip is strongly related to material constants, which is also similar to the conventional fracture mechanics for elastic materials. Of course, we must distinguish the results among different quasicrystal systems. For point groups 5 m and 10 mm quasicrystals it can be found from the solution that
( uy ðx; 0Þ ¼
0
p 2
K1 MK 1 R2
þ
1 LþM
jxj > a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a x jxj < a
ð6Þ
wy ðx; 0Þ ¼ 0 jxj < 1
ð7Þ
where L = C12, M = (C11 C12)/2 represent the elastic constants of phonon field, K1 the phason one, and R the phonon–phason coupling one. Because of the existence of crack, the variation of strain energy of the system is
WI ¼ 2
Z 0
a
ðryy ðx; 0Þ Hyy ðx; 0Þðuy ðx; 0Þ wy ðx; 0ÞÞdx ¼ 2
Z
a
ryy ðx; 0Þuy ðx; 0Þdx ¼
0
which is called crack strain energy with the suffix ‘‘I’’ indicating the Mode I crack.
pa2 p2 4
1 K1 þ L þ M MK 1 R2
ð8Þ
188
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
It is found that although under the assumption that the generalized surface traction hi = Hijnj is zero, the crack strain energy is still related to the elastic constant K1 of the phason field and the phonon–phason coupling elastic constant R, in addition to the phonon elastic constants L = C12 and M = (C11 C12)/2. Similar to conventional structural materials for point groups 5 m and 10 mm quasicrystals we define the strain energy release rate (crack growth force)
GI ¼
1 @W 1 pap2 1 K1 1 1 K1 ¼ þ þ ¼ ðK kI Þ2 2 @a L þ M MK 1 R2 4 L þ M MK 1 R 4
ð9Þ
for point groups 5; 5 and 10; 10 quasicrystals
GI ¼
LðK 1 þ K 2 Þ þ 2ðR21 þ R22 Þ k 2 ðK I Þ 8ðL þ MÞc
ð10Þ
where c ¼ MðK 1 þ K 2 Þ 2ðR21 þ R22 Þ, and for three-dimensional icosahedral quasicrystals
GI ¼
1 2
1 c7 ðK k Þ2 þ l þ k c3 I
ð11Þ
where
c1 ¼
Rð2K 2 K 1 ÞðlK 1 þ lK 2 3R2 Þ 2ðlK 1 2R2 Þ
c3 ¼ lðK 1 K 2 Þ R2
c7 ¼
ðlK 2 R2 Þ2
lK 1 2R2
c3 K 1 þ 2c1 R
lK 1 2R2
It can be seen that GI also depends on the phason elastic constants and phonon–phason coupling constants. In the above relations, for point groups 5 m and 10 mm, due to L + M > 0, MK1 R2 > 0, M + L > 0, the crack energy WI and icosahedral and 10; 10, the crack energy release rate GI are all positive and hence physically meaningful. For point groups 5; 5 and other quasicrystals, this fact still holds. Considering the vivid physical meaning of GI, we recommend
GI ¼ GIC
ð12Þ
as the crack initiation criterion, where GIC is the critical value, a material constant determined experimentally. With the explicit expression GI, the measurement of GIC is convenient, and will be discussed in the next section. The above results have been documented in relevant references mentioned earlier. With these common features of crack solutions for quasicrystalline materials, the fundamentals of fracture theory for this new kind of materials can be set up. 3. Crack growth force expressions of standard quasicrystal specimens and related testing strategy for determining the critical value GIC Mong et al. [17] measured the fracture toughness for Al65Cu20Co15 decagonal quasicrystal by using nonstandard specimens, because there are lack of expressions for stress intensity factors and energy release rate. In addition, nonstandard approach (e.g. indentation approach) was used for the measurement of fracture toughness. During the characterization of mechanical properties of quasicrystals, similar to the conventional structural materials, standard specimens (e.g. cracked specimens) are expected to use. Here we recommend using the three-point bending specimen to determine GIC. The corresponding GI expressions can be obtained by extending the Eq. (9) and others. 3.1. Characterization of GI and GIC of three-point bending quasicrystal specimen k
Since K I is independent of material constants, according to the conventional fracture mechanics, the stress intensity factor for the three-point bending specimen as shown in Fig. 2 is k
KI ¼
PS BW 3=2
a 1=2 a 3=2 a 5=2 a 7=2 a 9=2 29 4:6 þ 21:8 37:6 þ 38:7 W W W W W
ð13Þ
189
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
Fig. 2. Three-point bending specimen.
Fig. 3. Compact tension specimen for measuring fracture toughness of quasicrystalline material.
Therefore we can obtain by the extension of (1.7) as
GI ¼
1=2 a 3=2 a 5=2 a 7=2 a 9=2 1 1 K1 PS a þ 29 4:6 þ 21:8 37:6 þ 38:7 4 L þ M MK 1 R2 BW 3=2 W W W W W
ð14Þ
For point groups 5 m and 10 mm, where S is the specimen span, B the specimen thickness, W the specimen width, a the crack length including the size of the machined notch, and P the external force (per unit length). Finally, the GIC value can be determined by measuring the critical external force PC. For other quasicrystal systems the results are similar. 3.2. Characterization of GI and GIC of compact tension quasicrystal specimen It can be found in fracture mechanics that the stress intensity factor of compact tension specimen as shown in Fig. 3 is
K kI ¼
PS BW 3=2
29:6
a 1=2 W
185:5
a 3=2 W
þ 655:7
a 5=2 W
a 7=2 a 9=2 1017:0 þ 638:9 W W
ð15Þ
Thus for point groups 5 m and 10 mm quasicrystals,
GI ¼
a 1=2 a 3=2 a 5=2 a 7=2 a 9=2 1 1 K1 P þ 29:6 185:5 þ 655:7 1017:0 þ 638:9 2 3=2 4 L þ M MK 1 R BW W W W W W ð16Þ
where B, W, a, and P have the same meanings above. The GIC value can be determined by measuring the critical external force PC. For other quasicrystal systems the results are similar. 4. Nonlinear fracture mechanics At medium and low temperature, quasicrystals exhibit brittleness, but they dramatically present plasticity–ductility at high temperature. In addition, near the high stress concentration zone, e.g. around dislocation core or crack tip, plastic flow
190
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
appears too. The study on plasticity of quasicrystals has aroused a great deal of attention [20–29]. In the regime of nonlinear deformation of quasicrystals, the stress intensity factor and energy release rate cannot be used as fracture parameters. Therefore we must carry out elasto-plastic analysis. This difficult topic has been touched in some references quoted above. Here only some key points are mentioned. Because there is lack of enough data of macroscopic experiments as well as micro-mechanism of quasicrystal plasticity, the constitutive equation of plastic deformation of the material is still unknown. To overcome the difficulty, people developed some effective models, e.g. The generalized cohesive force model (or generalized DB model), generalized Eshelby principle and dislocation pile-up model (or generalized BCS model). The interesting point is the nonlinear solutions constructed based on these three different models are in exact agreement. Instead of stress intensity factor and energy release rate, the crack tip opening displacement or generalized Eshelby integral may be a quantity characterizing the mechanic behaviour of crack tip under nonlinear deformation of quasicrystals. These quantities are strongly related to material constants, and hence discussions must be given for different quasicrystal systems. For one-dimensional hexagonal quasicrystals, we have obtained the crack tip tearing displacement for mode III crack
dIII ¼
4K 2 sc a
pðC 44 K 2 R23 Þ
ln sec
ps1 2sc
ð17Þ
and for two-dimensional quasicrystals with point groups 5 m and 10 mm, the crack tip opening displacement for mode I crack is
dI ¼
2rc a
p
1 K1 þ L þ M MK 1 R2
ln sec
pp 2rc
ð18Þ
For Mode II crack, the crack tip sliding displacement dII was also exactly determined by Fan and Fan [11]. For icosahedral and other quasicrystal systems, the results are similar, which can also be found in the reference by Fan and Fan [11]. h i 1 The size of plastic zone around the crack tip is d ¼ a sec ps 1 for one-dimensional hexagonal quasicrystals, and 2sc h i d ¼ a sec 2prpc 1 for two-dimensional quasicrystals with point groups 5 m and 10 mm. It is similar for three-dimensional quasicrystals. We have proposed the following fracture criterion for mode I crack
dI ¼ dIc
ð19Þ
and for mode II and mode III cracks there are similar criterion, which have been discussed in the above-mentioned references. As pointed out in Ref. [10], the generalized Eshelby integral can also be a fracture parameter, and one can set up the corresponding fracture criterion based on the integral. A full discussion can be found in Refs. [9,10]. The experimental measurement of nonlinear fracture toughness of quasicrystals also has been introduced in Ref. [11], and is omitted here for brevity. 5. Dynamic fracture As the work of Fan et al. [12] shows, the study on elastodynamics for quasicrystals is very difficult, one of the reasons lies in the different points of view in the field [30–38]. This leads to the difficulty on dynamic fracture study. In spite of the difficulty, the available studies have provided us some useful information. In the dynamic process the phasons and phonons play quite different roles compared to the static process. For the socalled elasto-/hydro-dynamic equation system for quasicrystals, the dynamic crack initiation problem can be solved by the finite difference method. In the linear case, the crack dynamic initiation can be described by dynamic stress intensity factor, and some results of specimens, which are sensitive functions of loading type, loading rate and specimen geometry including crack geometry, have been obtained in our previous works. One of the results is shown in Fig. 4 for central crack specimen of icosahedral Al–Pd–Mn quasicrystal under the action of impact loading. The results indicate the effects of phason and phonon-phason coupling are more evident than that in the static case. With the results we can propose the fracture criterion for dynamic crack initiation as follows
K I ðtÞ ¼ K Id ðr_ Þ
ð20Þ
in which KI(t) is the dynamic stress intensity factor evaluated by different approaches, and K Id ðr_ Þ, a material constant which is the function of loading rate r_ and measured by test, represents the dynamic fracture toughness for initiation of crack growth of the material. While for fast crack propagation/crack arrest problems we have other results. Shown in Fig. 5 is the one for a central crack specimen for icosahedral Al–Pd–Mn quasicrystals. Accordingly, there is a fracture criterion such as
K I ðtÞ 6 K ID ðVÞ
ð21Þ
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
191
Fig. 4. Normalized dynamic stress intensity factor of central stationary crack specimen under Heaviside impact loading.
Fig. 5. Normalized dynamic stress intensity factor of propagating crack with constant crack speed of central crack specimen (for icosahedral Al–Pd–Mn quasicrystals).
where KI(t) is also the dynamic stress intensity factor, a computational quantity, while KID(V),a material constant which is the function of crack speed V = da/dt and must be measured by tests, denotes the fracture toughness for fast propagating crack. In Eq. (21), the equality sign represents crack propagation condition, and the inequality sign marks the crack arrest condition. 6. Measurement of fracture toughness and relevant mechanical parameters of quasicrystalline material Ref. [17] reported the measurement of fracture toughness of two-dimensional decagonal quasicrystal Al65Cu20Co15 as well as three-dimensional icosahedral Al–Li–Cu quasicrystal. Here the authors used the indentation approach. 6.1. Fracture toughness When the material is partially pressed, the crack around the indenter will appear as the compressible stress reaches a certain value, which describes the ability of fracture of the material along the direction of the compressible stress. When the crack length 2a is greater than 2.5 times 2c, the length of diagonal of the indenter, the fracture toughness can be evaluated by
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi K IC ¼ 0:203HV cð 3 ðc=aÞÞ in which HV denotes hardness of the material. Their results of measurement for decagonal Al–Ni–Co quasicrystal is
ð22Þ
192
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
pffiffiffiffiffi K IC ¼ 1:0 1:2MPa m
ð23Þ
with HV = 11.0 11.5 GPa, and for icosahedral Al–Li–Cu quasicrystal is
pffiffiffiffiffi K IC ¼ 0:94MPa m
ð24Þ
in which HV = 4.10 GPa. The values of fracture toughness for general alloys among black metals measured by Ma [18] are much greater than the pffiffiffiffiffi above data, and the same for aluminium alloys and other coloured metals, e.g. For aluminium it is 33MPa m referred to Fan [19]. Hence, one can find that quasicrystals are very brittle. The authors consider the indirect measurement by indentation approach of fracture toughness of quasicrystals probably is not so exact. Because Eq. (22) is empirical, the stress intensity factor formula should be used for the exact measurement. Due to the high brittleness of the material, maybe it is easy to use the indentation approach. 6.2. Tensile strength The tensile strength rc is measured through the formula
rc ¼ 0:187P=a2
ð25Þ
Meng et al. [17] obtained
rc ¼ 450 MPa
ð26Þ
before annealing, and
rc ¼ 550 MPa
ð27Þ
after annealing for decagonal Al–Ni–Co quasicrystal. There are much work on the testing for the material constants, referring to Edagawa et al. [37,38]. Fig. 6 shows the SEM morphology of grain interior containing large hole. Fig. 7 shows diagram of indentation crack under applied load 100 g before annealing and after annealing. Fig. 8 shows the SEM fractograph and fracture feature for decagonal Al65Cu20Co15 quasicrystals.
Fig. 6. SEM morphology of grain interior hole (a) prismatic grain with perfect basic plane, (b) grain with prefect prismatic plane.
Fig. 7. Morphology of cracks near impression under 100 g load (a) before annealing, (b) after 850 °C, 36 h annealing.
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
193
Fig. 8. (a) SEM fractograph, fracture feature of decagonal Al65Cu20Co15 quasicrystals grain without plastic deformation, (b) magnification.
7. Conclusion and discussion The fracture of quasicrystals is one of the most interesting topics in both theory and practice. At present, quite rich results have been obtained for the linear elastic fracture study, whereas the nonlinear and dynamic fracture studies are still in their infant stages, which must be paid more attention so that more relevant works can be done. The difficulty in studying plastic fracture of quasicrystals lies in the lack of constitutive equations, so the present work is based on some simple physical models, the discussion presents some limitations. The dynamic fracture study introduced in Section 5 is based on the elasto-/hydro-dynamic model, which may be one of the possible models. There are different points of view on the phason dynamics so far, and the experimental data are not sufficient to confirm which point is more precise. The study carried out in our group may provide probe for the further study. Acknowledgements The work is supported by the National Natural Science Foundation of China through Grants 10372016, 10672022 and partly supported by the Alexander von Humboldt Foundation of Germany. References [1] Griffith AA. The phenomena of rupture and flow in solids. Phil Trans Roy Soc A 1920;221(1):163–97. [2] (a) Li XF. Defects problems and analytic solutions of quasicrystals, Dissertation, Beijing Institute of Technology; 1999 [in Chinese].; (b) Li XF, Fan TY, Sun YF. A decagonal quasicrystal with a Griffith crack. Phil Mag A 1999;79(8):1943–52. [3] Fan TY. Mathematical theory of elasticity and defects of quasicrystals. Adv Mech 2000;30(2):161–74 [in Chinese]. [4] Fan TY, Mai YW. Theory of elasticity, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Appl Mech Rev 2004;57(5):235–44. [5] Rudhart C, Gumsch, Trebin HR. Crack propagation in quasicrystals. In: Trebin HR, editor. Quasicrystals. Berlin: Wiley; 2003. [6] (a) Fan TY, Guo LH. The final governing equation of plane elasticity of icosahedral quasicrystals. Phys Lett A 2005;341(5):235–9; (b) Li LH, Fan TY. Final governing equation of plane elasticity of icosahedral quasicrystals—stress potential method, Chin Phys Lett 24(3): 2519–21. [these references developed the potential theory for three-dimensional quasicrystals, and the potential theory for one-dimensional quasicrystals, can refer to Chen WQ, Ma YL, Ding HJ. On the three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mech Re Commun 2004; 31(5): 633–641].; (c) Peng YZ, Fan TY. Elastic theory of 1-D quasiperiodic stacking 2-D crystals. J Phys: Condens Matter 2000;12(45):9381–7; (d) Liu GT, Fan TY, Guo RP. Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals. Int J Solid and Structures 2004;41(14):3949–59. [7] Zhu AY, Fan TY. Elastic analysis of Mode II Griffith crack in an icosahedral quasicrystal. Chin Phys 2007;16(4):1111–8. [8] Li LH, Fan TY. Complex variable method for plane elasticity of icosahedral quasicrystals and elliptic notch problem. Sci China, G 2008;51(6):1–8. [9] Fan TY, Tang ZY, Li LH, Li Wu. The strict theory of complex variable function method of sextuple harmonic equation and applications. J Math Phys 2010;51(5):053519. [10] Fan TY, Fan L. Plastic fracture of quasicrystals. Phil Mag 2008;88(4):323–35. [11] Fan TY, Fan L. The relation between generalized Eshelby energy-momentum tensor and generalized BCS and DB models. Chin Phys B 2011;20(3):036102. [12] (a) Fan TY, Wang XF, Li W, et al. Elasto-hydrodynamics of quasicrystals. Phil Mag 2009;89(6):501–12; (b) Fan TY, Tang ZY. Elasto-hydrodynamics of quasicrystalline materials and its applications. In: Hydrodynamics. berlin: springer; 2011. [13] Zhu AY, Fan TY. Dynamic crack propagation of decagonal Al–Ni–Co decagonal quasicrystals. J Phys: Condens Matter 2008;20(29):295217. [14] Zhu AY, Fan TY. Fast crack propagation in three-dimensional icosahedral Al–Pd–Mn quasicrystals, unpublished work, or see Zhu AY. 2009, Fourier analysis on higher-order partial differential equations of elliptic type in elasticity and elasto-hydro dynamics of quasicrystals, Dissertation, Beijing Institute of Technology, Beijing, China; 2009 [in Chinese]. [15] Wang XF, Fan TY. Dynamic behaviour of the icosahedral Al–Pd–Mn quasicrystal with a Griffith crack. Chin Phys B 2009;18(2):709–14. [16] Wang XF, Fan TY. Study on the dynamics of the double cantilever-beam specimen of decagonal Al–Ni–Co quasicrystals. Appl Math Applic 2009;211(2):336–46. [17] Meng XM, Tong BY, Wu YK. Mechanical behaviour of Al65Cu20Co15 quasicrystal. Acta Metallurgy Sinica 1994;30(1):61–4 [in Chinese]. [18] Ma DL, editor. Handbook for parameters of fracture mechanics behaviour of general black metals. Beijing: Industrial Press; 1994 [in Chinese]. [19] Fan TY. Foundation of fracture theory. Beijing: Science Press; 2003 [in Chinese]. [20] Li W, Fan TY. Plastic analysis of crack in icosahedral quasicrystals. Phil Mag 2009;89(31):2823–31.
194
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
[21] Messerschmidt U, Bartisch M, Geyer B, et al. Plastic deformation of icosahedral Al–Pd–Mn single quasicrystals II, interpretation of experimental results. Phil Mag A 2000;80(7):1165–81. [22] Urban K, Wollgarten M. Dislocation and plasticity of quasicrystals. Mater Sci Forum 1994;150–151(2):315–22. [23] Wollgarten M et al. In-situ observation of dislocation motion in icosahedral Al–Pd–Mn single quasicrystals. Phil Mag Lett 1995;71(1):99–105. [24] Feuerbacher M, Bartsch B, Grushk B, et al. Plastic deformation of decagonal Al–Ni–Co quasicrystals. Phil Mag Lett 1997;76(4):375–96. [25] Messerschmidt U, Bartsch M, Feuerbacher M, et al. Friction mechanism of dislocation motion in icosahedral Al–Pd–Mn single quasicrystals. Phil Mag A 1999;79(9):2123–35. [26] Fan TY, Trebin H-R, Messerschmidt U, Mai YW. Plastic flow coupled with a Griffith crack in some one- and two-dimensional quasicrystals. J Phys: Condens Matter 2004;16(47):5229–40. [27] Feuerbacher M, Urban K. Plastic behaviour of quasicrystalline materials. In: Trebin HR, editor. Quasicrystals. Berlin: Wiley Press; 2003. [28] Ding DH, Yang WG, Hu CZ, et al. Generalized elasticity theory of quasicrystals. Phys Rev B 1993;48(10):7003–10. [29] Hu CZ, Wang RH, Ding DH. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep Prog Phys 2000;63(1):1–39. [30] Fan TY, Li XF, Sun YF. A moving screw dislocation in one-dimensional hexagonal quasicrystal. Acta Physica Sinica (Overseas Edition) 1999;8(3):288–95. [31] Fan TY. A study on special heat of one-dimensional hexagonal quasicrystals. J Phys: Condense Matter 1999;11(45):L513–7. [32] Fan TY, Mai YW. Partition function and state equation of point group 12 mm two-dimensional quasicrystals. Euro Phys J B 2003;31(1):25–7. [33] Li CL, Liu YY. Phason-strain influences on low-temperature specific heat of the decagonal Al–Ni–Co quasicrystal. Chin Phys Lett 2001;18(4):570–2. [34] Li CL, Liu YY. Low-temperature lattice excitation of icosahedral Al–Mn–Pd quasicrystals. Phys Rev B 2001;63(6):064203. [35] Rochal SB, Lorman VL. Anisotropy of acoustic-phonon properties of an icosahedral quasicrystals at high temperature due to phonon-phason coupling. Phys Rev B 2000;62(2):874–9. [36] Rochal SB, Lorman VL. Minimal model of the phonon-phason dynamics on icosahedral quasicrystals and its application for the problem of internal friction in the i-AlPdMn alloys. Phys Rev B 2002;66(14):144204. [37] Edagawa K, Takeuchi S. Elasticity, dislocations and their motion in quasicrystals, Dislocations in Solids. In: Nabarro ERN, Hirth JP. editors. 2007. p. 365– 417 [Chapter 76]. [38] Edagawa K, Giso Y. Experimental evaluation of phonon-phason coupling in icosahedral quasicrystals. Phil Mag 2007;87(1):77–95.
Contents lists available at SciVerse ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Theory of linear, nonlinear and dynamic fracture for quasicrystals Tian-You Fan a,⇑, Zhi-Yi Tang a, Wei-Qiu Chen b a b
School of Physics, Beijing Institute of Technology, Beijing 100081, PR China Institute of Applied Mechanics, Zhejiang University, Hangzhou 310027, PR China
a r t i c l e
i n f o
Article history: Received 3 May 2010 Received in revised form 21 December 2011 Accepted 22 December 2011
Keywords: Quasicrystals Phonon Phason Phonon–phason coupling Linear fracture Nonlinear fracture Dynamic fracture
a b s t r a c t In this paper, fracture theory of quasicrystalline material is presented, concerning with linear, nonlinear and dynamic fracture problems for different quasicrystal systems observed to date. Main attention is however paid to three-dimensional icosahedral quasicrystals and two-dimensional decagonal quasicrystals, which play the central role in the new kind of materials. Relevant experimental results are also discussed. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction D. Shechtman was awarded the Nobel Prize of Chemistry 2011 due to the discovery of quasicrystal in 1982, which has aroused the great interest on the structure and material again. After the discovery people are interested in the study of solid mechanics, including fracture theory, of the quasicrystalline material. Fracture theory of solids was initiated by Griffith [1] almost a century ago. Since then its scope extends from glass, ceramic and other absolute brittle solids in the early period to metallic material, further to composite material etc. These materials are structural or engineering materials. The main feature of their deformation can be described by one displacement field u(ux, uy, uz) (or u(u1, u2, u3)) and one strain field eij = (oui/oxj + ouj/oxi)/2, which is the basis of their fracture theory. It is evident that the discussion is given in the conventional three-dimensional space. Such a fracture theory is conventional and classical. The discovery of quasicrystals makes it be changed. Quasicrystals bear a new structure of solid as well as a new kind of novel materials, for which high dimensional space, i.e. the six-dimensional space E6 must be introduced. It conk sists of two subspaces, one is the physical space (or parallel space) E3 , and the other is the complementary space to the phys? ical space (or vertical space) E3 , an inner space, this means E6 ¼ Ek3 E? 3 , in which denotes the direct sum. To describe deformation of quasicrystals it is necessary to introduce two displacement fields u(u1, u2, u3) and w(w1, w2, w3), this means ~ ¼ uðu1 ; u2 ; u3 Þ wðw1 ; w2 ; w3 Þ. The displacement field u(u1, u2, u3) is similar to that in the classical the total displacement u k elasticity, and is called as the phonon field according to the physical terminology, which is in the parallel space E3 , while w(w1, w2, w3) is an unusual displacement field, called as the phason field and in the vertical space E? . It follows that there 3 are two strain fields eij = (oui/oxj + ouj/oxi)/2 and wij = owi/oxj, the latter being asymmetric. The associate stress tensor to eij is
⇑ Corresponding author. E-mail address: [email protected] (T.-Y. Fan). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2011.12.009
186
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
denoted by rij, and that to wij is denoted by Hij. In general, the stresses rij and Hij are coupled each other, which can be revealed by the generalized Hooke’s law such as
rij ¼ C ijkl ekl þ Rijkl wkl Hij ¼ Rklij ekl þ K ijkl wkl
ð1Þ
in which Cijkl is the phonon elastic tensor, Kijkl the phason one, and Rklij the phonon–phason coupling one. The determination of these constants is related to the symmetry of the quasicrystals, which can be done by group theory and group representation theory. This results in that the elastic constants of different quasicrystal systems are quite different. The phason field and the phonon–phason coupling field present important effects in elasticity, plasticity and dynamics of quasicrystalline material. The unusual nature of quasicrystals challenges the theories of classical elasticity and fracture. The huge number of field variables and the complexity of field equations lead to greater difficulty in the fracture analyses of this novel material than that of the classical fracture theory. In recent years, however, many crack problems have been investigated, and exact analytic, approximate and numerical solutions have been obtained. These studies provide a basis for further establishing the whole system of fracture theory. The exact solutions for crack problems of quasicrystals are of particular importance, revealing the essential nature which describes the fracture behaviour of this kind of materials. The first exact analytic solution for a Griffith crack in decagonal quasicrystals was derived by Li and Fan [2] in 1999. Afterward Fan [3], and Fan and Mai [4] developed the idea of linear elastic fracture mechanics of quasicrystals based on the common feature of crack solutions. Rudhart et al. [5] discussed the relevant topics from another point of view. Recently Fan and Guo [6], Zhu and Fan [7], and Li and Fan [8] derived analytic solutions of cracks in three-dimensional icosahedral quasicrystals. Furthermore Fan et al. [9], Fan and Fan [10], Fan and Fan [11] and Li and Fan [20] presented the analytic solutions of plastic cracks, though there is lack of plastic constitutive equation for quasicrystals so far. The elastodynamics and fracture dynamics are of particular interests. The phason modes and phonon modes present quite different behaviour in the dynamic process, in which phason modes behave like diffusion rather than wave propagation, refer to Fan et al. [12]. Zhu and Fan [13,14], and Wang and Fan [15,16] carried out dynamic crack analyses of quasicrystals. Meanwhile, the measurement of fracture toughness of quasicrystals was also reported [17]. With the fruitful information mentioned above, it is time to give a summary of the fracture study of quasicrystals. This is down in the present paper, which also put forward some new views on the fracture mechanics of quasicrystals. 2. Linear fracture theory of quasicrystals The crack solutions obtained in the works cited above reveal some common features of stress field and displacement field around the crack tip in quasicrystals. In terms of exact solutions for cracks in one-, two- and three-dimensional quasicrystals in the regime of linear elasticity, the results have a general sense. In addition, the solutions for cracks in dynamic state and with nonlinear behaviour of quasicrystals also exhibit some common features. An evident character is that the fracture behaviour is related to the symmetry of quasicrystal system. Another is that the phason and phonon–phason coupling play an important role. First we consider the linear elastic solutions. As indicted in the above works, within the framework of linear elasticity and for the infinitely sharp crack model, the stresses near the crack-tip, referring to Fig. 1, i.e.
Fig. 1. A Griffith crack in a decagonal quasicrystal subjected to a tensile stress (pulling stress) and the crack tip coordinate systems.
187
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
r1 =a 1
ð2Þ
have been obtained. It is shown that both phonon stresses and phason stresses exhibit singularity of the order r1=2 (r1 ? 0), 1 and other terms can be ignored when compared to this singular term. Stress singularity is practically implausible, and it is the result of an ideal mathematical model. A few researchers indicated its severe weakness in theory and the paradox of its methodology [2,3]. However, before the establishment of more reasonable fracture theory, we still follow this theory, which actually has demonstrated its power in engineering applications (e.g. K-based fracture resistance design). Now we focus on the field variables near the crack tip. For two-dimensional decagonal quasicrystals, by keeping the term in order of (r1/a)1/2 only in the crack solution for the Mode I loading condition, we have
8 k > I > rxx ¼ pKffiffiffiffiffiffiffi cos 12 h1 1 sin 12 h1 sin 32 h1 > > 2 p r 1 > > > k > > KI > p ffiffiffiffiffiffiffi > r ¼ cos 12 h1 1 þ sin 12 h1 sin 32 h1 > > yy 2pr 1 > > > k > > I > > rxy ¼ ryx ¼ pKffiffiffiffiffiffiffi cos 12 h1 cos 32 h1 > 2 pr1 > > > < k d KI Hxx ¼ p21ffiffiffiffiffiffiffi sin h1 2 sin 32 h1 þ 32 sin h1 cos 52 h1 2 p r > 1 > > > k > d21 K I 3 >H ¼ p > ffiffiffiffiffiffiffi 2 sin2 h1 cos 52 h1 > yy > 2pr 1 > > > > k > d21 K I 3 > > Hxy ¼ p ffiffiffiffiffiffiffi 2 sin2 h1 sin 52 h1 > > 2 p r 1 > > > > k > d KI > : Hyx ¼ p21ffiffiffiffiffiffiffi sin h1 2 cos 32 h1 32 sin h1 sin 52 h1
ð3Þ
2pr 1
where d21 = R(K1 K2)/4(MK1 R2), M = (C11 C12)/2 is elastic constants of phonon, K1, K2 the elastic constants of phason, R the phonon–phason coupling elastic constants, and k
K I ¼ limþ x!a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2pðx aÞryy ðx; 0Þ ¼ pap
ð4Þ
( x p pffiffiffiffiffiffiffiffiffi 1 jxj > a x2 a2
ð5Þ
in which
ryy ðx; 0Þ ¼
jxj < a
p
Here Eq. (4) represents a physical parameter describing the fracture behaviour of quasicrystals under the Mode I loading condition. The physical meaning of the generalized surface tractions hi = Hijnj is clear, but have not been measured so far and is not considered at the physical boundary (simply assumed to be zero). Therefore, we only obtain the K kI , whereas the stress intensity factor for the phason field is still unavailable. Can we use the stress intensity factor K kI in the parallel space (physical space) as the physical parameter to characterize the crack stability (instability) in quasicrystals? This shall depend upon tests. It can be found that K kI given by Eq. (4) is not directly related to the material constants of quasicrystals when the state of applied stress is self-equilibrium. This is similar to the conventional fracture mechanics for elastic materials. Nevertheless, it does not mean that it cannot be used as the physical parameter to characterize the crack stability/instability in quasicrystals. Further study on this topic is needed. The displacement field at the crack tip is strongly related to material constants, which is also similar to the conventional fracture mechanics for elastic materials. Of course, we must distinguish the results among different quasicrystal systems. For point groups 5 m and 10 mm quasicrystals it can be found from the solution that
( uy ðx; 0Þ ¼
0
p 2
K1 MK 1 R2
þ
1 LþM
jxj > a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a x jxj < a
ð6Þ
wy ðx; 0Þ ¼ 0 jxj < 1
ð7Þ
where L = C12, M = (C11 C12)/2 represent the elastic constants of phonon field, K1 the phason one, and R the phonon–phason coupling one. Because of the existence of crack, the variation of strain energy of the system is
WI ¼ 2
Z 0
a
ðryy ðx; 0Þ Hyy ðx; 0Þðuy ðx; 0Þ wy ðx; 0ÞÞdx ¼ 2
Z
a
ryy ðx; 0Þuy ðx; 0Þdx ¼
0
which is called crack strain energy with the suffix ‘‘I’’ indicating the Mode I crack.
pa2 p2 4
1 K1 þ L þ M MK 1 R2
ð8Þ
188
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
It is found that although under the assumption that the generalized surface traction hi = Hijnj is zero, the crack strain energy is still related to the elastic constant K1 of the phason field and the phonon–phason coupling elastic constant R, in addition to the phonon elastic constants L = C12 and M = (C11 C12)/2. Similar to conventional structural materials for point groups 5 m and 10 mm quasicrystals we define the strain energy release rate (crack growth force)
GI ¼
1 @W 1 pap2 1 K1 1 1 K1 ¼ þ þ ¼ ðK kI Þ2 2 @a L þ M MK 1 R2 4 L þ M MK 1 R 4
ð9Þ
for point groups 5; 5 and 10; 10 quasicrystals
GI ¼
LðK 1 þ K 2 Þ þ 2ðR21 þ R22 Þ k 2 ðK I Þ 8ðL þ MÞc
ð10Þ
where c ¼ MðK 1 þ K 2 Þ 2ðR21 þ R22 Þ, and for three-dimensional icosahedral quasicrystals
GI ¼
1 2
1 c7 ðK k Þ2 þ l þ k c3 I
ð11Þ
where
c1 ¼
Rð2K 2 K 1 ÞðlK 1 þ lK 2 3R2 Þ 2ðlK 1 2R2 Þ
c3 ¼ lðK 1 K 2 Þ R2
c7 ¼
ðlK 2 R2 Þ2
lK 1 2R2
c3 K 1 þ 2c1 R
lK 1 2R2
It can be seen that GI also depends on the phason elastic constants and phonon–phason coupling constants. In the above relations, for point groups 5 m and 10 mm, due to L + M > 0, MK1 R2 > 0, M + L > 0, the crack energy WI and icosahedral and 10; 10, the crack energy release rate GI are all positive and hence physically meaningful. For point groups 5; 5 and other quasicrystals, this fact still holds. Considering the vivid physical meaning of GI, we recommend
GI ¼ GIC
ð12Þ
as the crack initiation criterion, where GIC is the critical value, a material constant determined experimentally. With the explicit expression GI, the measurement of GIC is convenient, and will be discussed in the next section. The above results have been documented in relevant references mentioned earlier. With these common features of crack solutions for quasicrystalline materials, the fundamentals of fracture theory for this new kind of materials can be set up. 3. Crack growth force expressions of standard quasicrystal specimens and related testing strategy for determining the critical value GIC Mong et al. [17] measured the fracture toughness for Al65Cu20Co15 decagonal quasicrystal by using nonstandard specimens, because there are lack of expressions for stress intensity factors and energy release rate. In addition, nonstandard approach (e.g. indentation approach) was used for the measurement of fracture toughness. During the characterization of mechanical properties of quasicrystals, similar to the conventional structural materials, standard specimens (e.g. cracked specimens) are expected to use. Here we recommend using the three-point bending specimen to determine GIC. The corresponding GI expressions can be obtained by extending the Eq. (9) and others. 3.1. Characterization of GI and GIC of three-point bending quasicrystal specimen k
Since K I is independent of material constants, according to the conventional fracture mechanics, the stress intensity factor for the three-point bending specimen as shown in Fig. 2 is k
KI ¼
PS BW 3=2
a 1=2 a 3=2 a 5=2 a 7=2 a 9=2 29 4:6 þ 21:8 37:6 þ 38:7 W W W W W
ð13Þ
189
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
Fig. 2. Three-point bending specimen.
Fig. 3. Compact tension specimen for measuring fracture toughness of quasicrystalline material.
Therefore we can obtain by the extension of (1.7) as
GI ¼
1=2 a 3=2 a 5=2 a 7=2 a 9=2 1 1 K1 PS a þ 29 4:6 þ 21:8 37:6 þ 38:7 4 L þ M MK 1 R2 BW 3=2 W W W W W
ð14Þ
For point groups 5 m and 10 mm, where S is the specimen span, B the specimen thickness, W the specimen width, a the crack length including the size of the machined notch, and P the external force (per unit length). Finally, the GIC value can be determined by measuring the critical external force PC. For other quasicrystal systems the results are similar. 3.2. Characterization of GI and GIC of compact tension quasicrystal specimen It can be found in fracture mechanics that the stress intensity factor of compact tension specimen as shown in Fig. 3 is
K kI ¼
PS BW 3=2
29:6
a 1=2 W
185:5
a 3=2 W
þ 655:7
a 5=2 W
a 7=2 a 9=2 1017:0 þ 638:9 W W
ð15Þ
Thus for point groups 5 m and 10 mm quasicrystals,
GI ¼
a 1=2 a 3=2 a 5=2 a 7=2 a 9=2 1 1 K1 P þ 29:6 185:5 þ 655:7 1017:0 þ 638:9 2 3=2 4 L þ M MK 1 R BW W W W W W ð16Þ
where B, W, a, and P have the same meanings above. The GIC value can be determined by measuring the critical external force PC. For other quasicrystal systems the results are similar. 4. Nonlinear fracture mechanics At medium and low temperature, quasicrystals exhibit brittleness, but they dramatically present plasticity–ductility at high temperature. In addition, near the high stress concentration zone, e.g. around dislocation core or crack tip, plastic flow
190
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
appears too. The study on plasticity of quasicrystals has aroused a great deal of attention [20–29]. In the regime of nonlinear deformation of quasicrystals, the stress intensity factor and energy release rate cannot be used as fracture parameters. Therefore we must carry out elasto-plastic analysis. This difficult topic has been touched in some references quoted above. Here only some key points are mentioned. Because there is lack of enough data of macroscopic experiments as well as micro-mechanism of quasicrystal plasticity, the constitutive equation of plastic deformation of the material is still unknown. To overcome the difficulty, people developed some effective models, e.g. The generalized cohesive force model (or generalized DB model), generalized Eshelby principle and dislocation pile-up model (or generalized BCS model). The interesting point is the nonlinear solutions constructed based on these three different models are in exact agreement. Instead of stress intensity factor and energy release rate, the crack tip opening displacement or generalized Eshelby integral may be a quantity characterizing the mechanic behaviour of crack tip under nonlinear deformation of quasicrystals. These quantities are strongly related to material constants, and hence discussions must be given for different quasicrystal systems. For one-dimensional hexagonal quasicrystals, we have obtained the crack tip tearing displacement for mode III crack
dIII ¼
4K 2 sc a
pðC 44 K 2 R23 Þ
ln sec
ps1 2sc
ð17Þ
and for two-dimensional quasicrystals with point groups 5 m and 10 mm, the crack tip opening displacement for mode I crack is
dI ¼
2rc a
p
1 K1 þ L þ M MK 1 R2
ln sec
pp 2rc
ð18Þ
For Mode II crack, the crack tip sliding displacement dII was also exactly determined by Fan and Fan [11]. For icosahedral and other quasicrystal systems, the results are similar, which can also be found in the reference by Fan and Fan [11]. h i 1 The size of plastic zone around the crack tip is d ¼ a sec ps 1 for one-dimensional hexagonal quasicrystals, and 2sc h i d ¼ a sec 2prpc 1 for two-dimensional quasicrystals with point groups 5 m and 10 mm. It is similar for three-dimensional quasicrystals. We have proposed the following fracture criterion for mode I crack
dI ¼ dIc
ð19Þ
and for mode II and mode III cracks there are similar criterion, which have been discussed in the above-mentioned references. As pointed out in Ref. [10], the generalized Eshelby integral can also be a fracture parameter, and one can set up the corresponding fracture criterion based on the integral. A full discussion can be found in Refs. [9,10]. The experimental measurement of nonlinear fracture toughness of quasicrystals also has been introduced in Ref. [11], and is omitted here for brevity. 5. Dynamic fracture As the work of Fan et al. [12] shows, the study on elastodynamics for quasicrystals is very difficult, one of the reasons lies in the different points of view in the field [30–38]. This leads to the difficulty on dynamic fracture study. In spite of the difficulty, the available studies have provided us some useful information. In the dynamic process the phasons and phonons play quite different roles compared to the static process. For the socalled elasto-/hydro-dynamic equation system for quasicrystals, the dynamic crack initiation problem can be solved by the finite difference method. In the linear case, the crack dynamic initiation can be described by dynamic stress intensity factor, and some results of specimens, which are sensitive functions of loading type, loading rate and specimen geometry including crack geometry, have been obtained in our previous works. One of the results is shown in Fig. 4 for central crack specimen of icosahedral Al–Pd–Mn quasicrystal under the action of impact loading. The results indicate the effects of phason and phonon-phason coupling are more evident than that in the static case. With the results we can propose the fracture criterion for dynamic crack initiation as follows
K I ðtÞ ¼ K Id ðr_ Þ
ð20Þ
in which KI(t) is the dynamic stress intensity factor evaluated by different approaches, and K Id ðr_ Þ, a material constant which is the function of loading rate r_ and measured by test, represents the dynamic fracture toughness for initiation of crack growth of the material. While for fast crack propagation/crack arrest problems we have other results. Shown in Fig. 5 is the one for a central crack specimen for icosahedral Al–Pd–Mn quasicrystals. Accordingly, there is a fracture criterion such as
K I ðtÞ 6 K ID ðVÞ
ð21Þ
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
191
Fig. 4. Normalized dynamic stress intensity factor of central stationary crack specimen under Heaviside impact loading.
Fig. 5. Normalized dynamic stress intensity factor of propagating crack with constant crack speed of central crack specimen (for icosahedral Al–Pd–Mn quasicrystals).
where KI(t) is also the dynamic stress intensity factor, a computational quantity, while KID(V),a material constant which is the function of crack speed V = da/dt and must be measured by tests, denotes the fracture toughness for fast propagating crack. In Eq. (21), the equality sign represents crack propagation condition, and the inequality sign marks the crack arrest condition. 6. Measurement of fracture toughness and relevant mechanical parameters of quasicrystalline material Ref. [17] reported the measurement of fracture toughness of two-dimensional decagonal quasicrystal Al65Cu20Co15 as well as three-dimensional icosahedral Al–Li–Cu quasicrystal. Here the authors used the indentation approach. 6.1. Fracture toughness When the material is partially pressed, the crack around the indenter will appear as the compressible stress reaches a certain value, which describes the ability of fracture of the material along the direction of the compressible stress. When the crack length 2a is greater than 2.5 times 2c, the length of diagonal of the indenter, the fracture toughness can be evaluated by
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi K IC ¼ 0:203HV cð 3 ðc=aÞÞ in which HV denotes hardness of the material. Their results of measurement for decagonal Al–Ni–Co quasicrystal is
ð22Þ
192
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
pffiffiffiffiffi K IC ¼ 1:0 1:2MPa m
ð23Þ
with HV = 11.0 11.5 GPa, and for icosahedral Al–Li–Cu quasicrystal is
pffiffiffiffiffi K IC ¼ 0:94MPa m
ð24Þ
in which HV = 4.10 GPa. The values of fracture toughness for general alloys among black metals measured by Ma [18] are much greater than the pffiffiffiffiffi above data, and the same for aluminium alloys and other coloured metals, e.g. For aluminium it is 33MPa m referred to Fan [19]. Hence, one can find that quasicrystals are very brittle. The authors consider the indirect measurement by indentation approach of fracture toughness of quasicrystals probably is not so exact. Because Eq. (22) is empirical, the stress intensity factor formula should be used for the exact measurement. Due to the high brittleness of the material, maybe it is easy to use the indentation approach. 6.2. Tensile strength The tensile strength rc is measured through the formula
rc ¼ 0:187P=a2
ð25Þ
Meng et al. [17] obtained
rc ¼ 450 MPa
ð26Þ
before annealing, and
rc ¼ 550 MPa
ð27Þ
after annealing for decagonal Al–Ni–Co quasicrystal. There are much work on the testing for the material constants, referring to Edagawa et al. [37,38]. Fig. 6 shows the SEM morphology of grain interior containing large hole. Fig. 7 shows diagram of indentation crack under applied load 100 g before annealing and after annealing. Fig. 8 shows the SEM fractograph and fracture feature for decagonal Al65Cu20Co15 quasicrystals.
Fig. 6. SEM morphology of grain interior hole (a) prismatic grain with perfect basic plane, (b) grain with prefect prismatic plane.
Fig. 7. Morphology of cracks near impression under 100 g load (a) before annealing, (b) after 850 °C, 36 h annealing.
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
193
Fig. 8. (a) SEM fractograph, fracture feature of decagonal Al65Cu20Co15 quasicrystals grain without plastic deformation, (b) magnification.
7. Conclusion and discussion The fracture of quasicrystals is one of the most interesting topics in both theory and practice. At present, quite rich results have been obtained for the linear elastic fracture study, whereas the nonlinear and dynamic fracture studies are still in their infant stages, which must be paid more attention so that more relevant works can be done. The difficulty in studying plastic fracture of quasicrystals lies in the lack of constitutive equations, so the present work is based on some simple physical models, the discussion presents some limitations. The dynamic fracture study introduced in Section 5 is based on the elasto-/hydro-dynamic model, which may be one of the possible models. There are different points of view on the phason dynamics so far, and the experimental data are not sufficient to confirm which point is more precise. The study carried out in our group may provide probe for the further study. Acknowledgements The work is supported by the National Natural Science Foundation of China through Grants 10372016, 10672022 and partly supported by the Alexander von Humboldt Foundation of Germany. References [1] Griffith AA. The phenomena of rupture and flow in solids. Phil Trans Roy Soc A 1920;221(1):163–97. [2] (a) Li XF. Defects problems and analytic solutions of quasicrystals, Dissertation, Beijing Institute of Technology; 1999 [in Chinese].; (b) Li XF, Fan TY, Sun YF. A decagonal quasicrystal with a Griffith crack. Phil Mag A 1999;79(8):1943–52. [3] Fan TY. Mathematical theory of elasticity and defects of quasicrystals. Adv Mech 2000;30(2):161–74 [in Chinese]. [4] Fan TY, Mai YW. Theory of elasticity, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Appl Mech Rev 2004;57(5):235–44. [5] Rudhart C, Gumsch, Trebin HR. Crack propagation in quasicrystals. In: Trebin HR, editor. Quasicrystals. Berlin: Wiley; 2003. [6] (a) Fan TY, Guo LH. The final governing equation of plane elasticity of icosahedral quasicrystals. Phys Lett A 2005;341(5):235–9; (b) Li LH, Fan TY. Final governing equation of plane elasticity of icosahedral quasicrystals—stress potential method, Chin Phys Lett 24(3): 2519–21. [these references developed the potential theory for three-dimensional quasicrystals, and the potential theory for one-dimensional quasicrystals, can refer to Chen WQ, Ma YL, Ding HJ. On the three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mech Re Commun 2004; 31(5): 633–641].; (c) Peng YZ, Fan TY. Elastic theory of 1-D quasiperiodic stacking 2-D crystals. J Phys: Condens Matter 2000;12(45):9381–7; (d) Liu GT, Fan TY, Guo RP. Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals. Int J Solid and Structures 2004;41(14):3949–59. [7] Zhu AY, Fan TY. Elastic analysis of Mode II Griffith crack in an icosahedral quasicrystal. Chin Phys 2007;16(4):1111–8. [8] Li LH, Fan TY. Complex variable method for plane elasticity of icosahedral quasicrystals and elliptic notch problem. Sci China, G 2008;51(6):1–8. [9] Fan TY, Tang ZY, Li LH, Li Wu. The strict theory of complex variable function method of sextuple harmonic equation and applications. J Math Phys 2010;51(5):053519. [10] Fan TY, Fan L. Plastic fracture of quasicrystals. Phil Mag 2008;88(4):323–35. [11] Fan TY, Fan L. The relation between generalized Eshelby energy-momentum tensor and generalized BCS and DB models. Chin Phys B 2011;20(3):036102. [12] (a) Fan TY, Wang XF, Li W, et al. Elasto-hydrodynamics of quasicrystals. Phil Mag 2009;89(6):501–12; (b) Fan TY, Tang ZY. Elasto-hydrodynamics of quasicrystalline materials and its applications. In: Hydrodynamics. berlin: springer; 2011. [13] Zhu AY, Fan TY. Dynamic crack propagation of decagonal Al–Ni–Co decagonal quasicrystals. J Phys: Condens Matter 2008;20(29):295217. [14] Zhu AY, Fan TY. Fast crack propagation in three-dimensional icosahedral Al–Pd–Mn quasicrystals, unpublished work, or see Zhu AY. 2009, Fourier analysis on higher-order partial differential equations of elliptic type in elasticity and elasto-hydro dynamics of quasicrystals, Dissertation, Beijing Institute of Technology, Beijing, China; 2009 [in Chinese]. [15] Wang XF, Fan TY. Dynamic behaviour of the icosahedral Al–Pd–Mn quasicrystal with a Griffith crack. Chin Phys B 2009;18(2):709–14. [16] Wang XF, Fan TY. Study on the dynamics of the double cantilever-beam specimen of decagonal Al–Ni–Co quasicrystals. Appl Math Applic 2009;211(2):336–46. [17] Meng XM, Tong BY, Wu YK. Mechanical behaviour of Al65Cu20Co15 quasicrystal. Acta Metallurgy Sinica 1994;30(1):61–4 [in Chinese]. [18] Ma DL, editor. Handbook for parameters of fracture mechanics behaviour of general black metals. Beijing: Industrial Press; 1994 [in Chinese]. [19] Fan TY. Foundation of fracture theory. Beijing: Science Press; 2003 [in Chinese]. [20] Li W, Fan TY. Plastic analysis of crack in icosahedral quasicrystals. Phil Mag 2009;89(31):2823–31.
194
T.-Y. Fan et al. / Engineering Fracture Mechanics 82 (2012) 185–194
[21] Messerschmidt U, Bartisch M, Geyer B, et al. Plastic deformation of icosahedral Al–Pd–Mn single quasicrystals II, interpretation of experimental results. Phil Mag A 2000;80(7):1165–81. [22] Urban K, Wollgarten M. Dislocation and plasticity of quasicrystals. Mater Sci Forum 1994;150–151(2):315–22. [23] Wollgarten M et al. In-situ observation of dislocation motion in icosahedral Al–Pd–Mn single quasicrystals. Phil Mag Lett 1995;71(1):99–105. [24] Feuerbacher M, Bartsch B, Grushk B, et al. Plastic deformation of decagonal Al–Ni–Co quasicrystals. Phil Mag Lett 1997;76(4):375–96. [25] Messerschmidt U, Bartsch M, Feuerbacher M, et al. Friction mechanism of dislocation motion in icosahedral Al–Pd–Mn single quasicrystals. Phil Mag A 1999;79(9):2123–35. [26] Fan TY, Trebin H-R, Messerschmidt U, Mai YW. Plastic flow coupled with a Griffith crack in some one- and two-dimensional quasicrystals. J Phys: Condens Matter 2004;16(47):5229–40. [27] Feuerbacher M, Urban K. Plastic behaviour of quasicrystalline materials. In: Trebin HR, editor. Quasicrystals. Berlin: Wiley Press; 2003. [28] Ding DH, Yang WG, Hu CZ, et al. Generalized elasticity theory of quasicrystals. Phys Rev B 1993;48(10):7003–10. [29] Hu CZ, Wang RH, Ding DH. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep Prog Phys 2000;63(1):1–39. [30] Fan TY, Li XF, Sun YF. A moving screw dislocation in one-dimensional hexagonal quasicrystal. Acta Physica Sinica (Overseas Edition) 1999;8(3):288–95. [31] Fan TY. A study on special heat of one-dimensional hexagonal quasicrystals. J Phys: Condense Matter 1999;11(45):L513–7. [32] Fan TY, Mai YW. Partition function and state equation of point group 12 mm two-dimensional quasicrystals. Euro Phys J B 2003;31(1):25–7. [33] Li CL, Liu YY. Phason-strain influences on low-temperature specific heat of the decagonal Al–Ni–Co quasicrystal. Chin Phys Lett 2001;18(4):570–2. [34] Li CL, Liu YY. Low-temperature lattice excitation of icosahedral Al–Mn–Pd quasicrystals. Phys Rev B 2001;63(6):064203. [35] Rochal SB, Lorman VL. Anisotropy of acoustic-phonon properties of an icosahedral quasicrystals at high temperature due to phonon-phason coupling. Phys Rev B 2000;62(2):874–9. [36] Rochal SB, Lorman VL. Minimal model of the phonon-phason dynamics on icosahedral quasicrystals and its application for the problem of internal friction in the i-AlPdMn alloys. Phys Rev B 2002;66(14):144204. [37] Edagawa K, Takeuchi S. Elasticity, dislocations and their motion in quasicrystals, Dislocations in Solids. In: Nabarro ERN, Hirth JP. editors. 2007. p. 365– 417 [Chapter 76]. [38] Edagawa K, Giso Y. Experimental evaluation of phonon-phason coupling in icosahedral quasicrystals. Phil Mag 2007;87(1):77–95.