A CRACK APPROACHING AN INTERFACE BETWEEN
Whether the crack is going to deflect single or double sided into the interface depends on the load conditions at the crack tip. The more probable and frequent form is the double sided deflection, which occurs when the load phase angle ψ is <
45°. Single sided crack deflection into the interface occurs in cases when the Mode II load is dominant at the crack tip. In that case, the load phase angle ψ is > 45°.
The "competition" between the crack penetrating or deflecting into the interface is considered by comparing the "driving forces" needed for those cases - energy release rates for crack penetration of the interface Gp to energy release rate for crack deflection into the interface Gd. Those results are then further used for determination of the interface toughness Γi and the base material toughness Γ, which ensures that the crack would deflect into the interface.
The crack will deflect into the interface if the following holds:
Γ
> Γ
d i
p
G
G . (1)
The crack will cross the interface if inequality (1) is reversed. The objective of this paper is to determine the materials characteristics Γi and Γ and the crack diving forces Gd and Gp. The results presented in this paper were obtained by the symbolic programming routine Mathematica.
2. Problem formulation
For analyzing the problem of a crack on the interface between the two anisotropic materials, the 3×3 matrix H is used, which depends on elastic constants of both materials and which has the modulus of elasticity dimensions. Definition of the H-matrix is given in Appendix of [9], following [7]
and [8].
For the case of the two different orthotropic materials, with mutually perpendicular principal axes and an interface, which lies along the x-axis, components of the H matrix, are:
λ λ
= 4 + 4
11 2 11 22 1 2 11 22 2,
H n s s n s s
λ λ
= +
22 4 11 22 4 11 22
1 2
1 1
2 2 ,
H n s s n s s
= = + − +
12 21 11 22 12 1 11 22 12 2
H H i s s s s s s
(2)
where [ ]1 denotes variables of material 1, while [ ]2
denotes variables of material 2, and where:
11=
1
s 1
E , 22 =
2
s 1
E , 66 =
12
s 1
G , (3)
ν ν
= − 12 = − 21
12 1 2
s E E ,λ = 11 = 2
22 1
s E
s E , ρ
= (1+ )
n 2 ,
ρ + ν ν
= 12 66 = 1 2 − 12 21
11 22 12
2 2 2
s s E E
s s G .
Here sij are the compliances, which correspond to the Young's and shear moduli and Poisson's ratio. Parameters λ and ρ measure anisotropy in the sense that for λ = 1 material has the cubic symmetry and for λ = ρ = 1 material is isotropic.
The Dundurs' [12] parameters are:
α Σ Σ
= − +
1
1, β = 12
11 22
iH
H H , (4)
where: Σ = s s11 22 2 s s11 22 1 .
Stress field in the vicinity of the interfacial crack tip has the oscillatory character, which is measured by the complex stress intensity factor K, defined by Rice [13]:
σ σ ε
+ 11 = π
22 2
i
yy i H xy Kr
H r , (5)
where:
ε β
π β
= −
+ 1 ln1
2 1 . (6)
Energy release rate for the crack on the interface is:
= 222πε 2
4 ( )
G H K
ch , (7)
For the case of the crack deflection into the inter-face, based on dimensional analysis, one obtains:
a a
γ ε ε
+ −
+ = +
1
1 2 I 2 ( i i )
K iK k a d e . (8)
where: d and e are dimensionless complex func-tions of α, β, λ1, λ2, ρ1 and ρ2. Factor kI is proportional to load, while γ is a real variable that depends on α, β, λ1, λ2, ρ1 and ρ2. For analysis in this paper explicit knowledge of those parameters is not necessary.
By substituting equation (8) into (7), one obtains the energy release rate for the crack that deflects into the interface as:
Γ
> Γ
d i
p
G
G . (9)
A Crack Approaching an Interface Between the Two Orthotropic Materials 116
For the case of the crack crossing the interface, the stress field ahead of the crack tip corresponds to pure Mode I of crack propagation. Based on the dimensional analysis, one obtains the stress intensity factor for this case as:
a +γ
= 21
I I
K ck , (10)
where c is the dimensionless function of α, β, λ1, λ2, ρ1 and ρ2. Energy release rate for the crack crossing the interface is, [8]:
a γ
λ λ
= 11 1 2 = 11 1 2 2 1 2+
4 4
1 1
( ) ( )
p s n I s n I
G K c k . (11)
Ratio of energy release rates Gd/Gp depends neither on a nor on kI, i.e.:
πε λ
= ⋅
+ +
⋅
222 4
11 1 1
2 2
2
1 4 ( ) ( )
[ 2Re( )]
d p
G H
G ch s n
d e de
c
. (12)
Relative tendency of the crack to deflect into the interface or to continue to propagate across it can be determined based on equation (12).
Knowing the dimensionless complex functions c, d and e one can determine the ratio Gd/Gp. Solu-tion proposed by [5] for the isotropic materials is here applied to orthotropic materials.
3. Results and discussion
In Figure 2 the variation of the two energy rele-ase rates ratio is shown in terms of parameter α and for the three different values of parameter β. The ratio Gd/Gp is shown for two cases: for the crack that is deflecting into the interface in the single direction (solid lines) and for the crack that is deflecting into the interface in two (double) direc-tions (broken lines), for the same crack lengths.
From Figure 2 one can see that the variation of parameter β does not significantly influence the variation of the Gd/Gp ratio, so in the further analysis it was adopted that β = 0. It can also be seen that values of the Gd/Gp ratio for the double-sided deflected crack are for 3 to 4 % larger than those for the crack that deflects into the interface in the single direction. This is why, in further considerations, only the double-sided deflection was analyzed.
If the values presented in Figure 2 were compared to results presented in [5] and [6], which were obtained for the isotropic materials, the relatively good agreement could be seen, what confirms the correctness of the conducted analysis.
Figure 2. Influence of parameter β on ratio Gd/Gp
variation in terms of parameter α.
Due to large number of factors that influence the Gd/Gp ratio variation with α, it is not possible to consider this variation in combination of all the factors simultaneously. This is why is in Figure 3 shown variation of the Gd/Gp ratio as a function of α, for the three different values of the anisotropic parameter ρ1.
From Figure 3 can be seen that with increase of parameter ρ1 values, the value of ratio Gd/Gp
decreases, what points to the fact that anisotropy that corresponds to factor ρ1, does not influence favorably the strengthening of the interfacial toughness.
Figure 3. Influence of parameter ρ1 on ratio Gd/Gp
variation in terms of parameter α.
J. Djoković, R. Nikolić, A. Sedmak 117
In Figure 4 is shown variation of the Gd/Gp ratio as a function of α, for the three different values of the anisotropic parameter λ1.
From Figure 4 can be seen that the variation of the Gd/Gp ratio is very sensitive to variation of λ1. Ratio Gd/Gp increases with increase of λ1. It can be noticed that the interface toughness is for 3 to 4 times higher than the substrate toughness, for the case of the crack deflecting into the interface.
Figure 4. Influence of parameter λ1 on ratio Gd/Gp
dependence on α .
In Figure 5 is shown variation of the Gd/Gp ratio as a function of α, for the three different values of the anisotropic parameter ρ2.
Figure 5. Influence of parameter ρ2 on ratio Gd/Gp
dependence on α .
In Figure 6 is presented variation of the Gd/Gp
ratio as a function of α, for the three different values of the anisotropic parameter λ2,
From Figure 5 can be noticed that ratio Gd/Gp
increases with increase of ρ2 and the same is valid for variation of the ratio Gd/Gp with increase of λ2, what can be seen from Figure 6.
Figure 6. Influence of parameter λ2 on ratio Gd/Gp
dependence on α .
In Figure 7 is presented the variation of the load phase angle ψ in terms of parameter α, for four different values of parameter λ1.
Figure 7. Variation of the load phase angle ψ in terms of α, for different values of λ1
A Crack Approaching an Interface Between the Two Orthotropic Materials 118
Considering that the load phase angle measures relative value of the Mode I with respect to Mode II, one can see from Figure 7 that with variation of parameter λ1 value of the Mode II component changes within interval 40 to 80 % of the Mode I component.
4. Conclusion
Based on results presented in Figures 3 to 6, one can conclude that the ratio of energy release rate needed for the crack deflection into the interface Gd
and the energy release rate needed for the crack to penetrate the interface Gp depends on variation of the anisotropic parameters λ1, λ2, ρ1 and ρ2. The Gd/Gp ratio changes within interval 0.2 to 5. When the value of this ratio exceeds 1 that means that crack approaching interface at the right angle will deflect into it even if the interface toughness is higher than the toughness of the base material.
From Figure 7 can be seen that, for the majority of cases, the Mode II load component, for the crack that has deflected into the interface (double-sided) varies within range 40 to 80 % of the Mode I load component.
Results presented in this paper enable the com-parison of the interface toughness to the base ma-terial (substrate) toughness, thus concluding whet-her the incoming crack would deflect into the inter-face or would it penetrate the interinter-face and con-tinue to propagate in the second material across it.
If the ratio of the interface fracture toughness to fracture toughness of the material into which the crack continues to propagate is less than the ratio of the energy release rate for the crack that deflects into the interface and the energy release rate for the crack that penetrates the interface, the crack will deflect into the interface. If this relation was reversed, the crack will penetrate the interface.
5. Acknowledgement
This research was partially financially supported by European regional development fund and Slovak state budget by the project "Research Center of the University of Žilina" - ITMS 26220220183 and by the Ministry of Education, Science and Techno-logical Development of Republic of Serbia through Grants ON174001, ON174004 and TR32036.
Authors gratefully acknowledge their support.
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J. Djoković, R. Nikolić, A. Sedmak 119