J.
Eng.
Technol.
Sci.
Vol.
No.
5, 2020, 665-676
Stress Intensity Factors for Crack Problems in Bonded Dissimilar Materials Khairum Hamzah1,4.
Nik Mohd Asri Nik Long1,2*.
Norazak Senu1,2 & Zainidin Eshkuvatov3 Laboratory of Computational Sciences and Mathematical Physics.
Institute for Mathematical Research.
Universiti Putra Malaysia, 43400 Serdang.
Selangor.
Malaysia Mathematics Department.
Faculty of Science.
Universiti Putra Malaysia, 43400 Serdang.
Selangor.
Malaysia Faculty of Ocean Engineering Technology and Informatics.
Universiti Malaysia Terengganu, 21030 Kuala Terengganu.
Malaysia Fakulti Teknologi Kejuruteraan Mekanikal dan Pembuatan.
Universiti Teknikal Malaysia Melaka.
Hang Tuah Jaya, 76100 Durian Tunggal.
Melaka.
Malaysia *E-mail: nmasri@upm.
Highlights:
The problem of inclined cracks subjected to normal and shear stress in bonded dissimilar materials was formulated.
The modified complex potentials function method was used to formulate the hypersingular integral equations.
The obtained system of hypersingular integral equations was solved numerically using the appropriate quadrature formula.
The stress intensity factors at the crack tips depend on the elastic constantAos ratio and crack geometries.
Abstract.
The inclined crack problem in bonded dissimilar materials was considered in this study.
The system of hypersingular integral equations (HSIE.
was formulated using the modified complex potentials (MCP) function method, where the continuity conditions of the resultant force and the displacement are In the equations, the crack opening displacement (COD) serves as the unknown function and the traction along the cracks as the right-hand terms.
applying the curved length coordinate method and the appropriate quadrature formulas, the HSIEs are reduced to the system of linear equations.
It was found that the nondimensional stress intensity factors (SIF) at the crack tips depend on the ratio of elastic constants, the crack geometries and the distance between the crack and the boundary.
Keywords: complex variable function.
bonded dissimilar materials.
integral equation.
stress intensity factor.
Introduction The stress intensity factors (SIF) at the crack tip are among the physical quantities that can be used to analyze crack problems in engineering structures.
Systems of Received July 22nd, 2020.
Revised August 4th, 2020.
Accepted for publication August 13th, 2020.
Copyright A2020 Published by ITB Institute for Research and Community Services.
ISSN: 2337-5779.
DOI: 10.
5614/j.
Khairum Hamzah, et al.
HSIEs or singular integral equations have been proposed to find the SIF for crack problems in an infinite plane by Nik Long and Eshkuvatov in .
and Denda and Dong in .
, and for half plane elasticity by Chen, et al.
and Elfakhakhre, et al.
Crack problems in bonded dissimilar materials are discussed in .
The SIF for two inclined cracks in bonded dissimilar materials were calculated using Fredholm integral equations with the density distribution as undetermined function in .
The body forces method and traction free conditions of the cracks were used in finding the solution of inclined, kinked and branched cracks in bonded dissimilar materials in .
The nondimensional SIF for two- and threedimensional crack problems in bonded dissimilar materials were computed using the finite element procedure based on the ratio of COD in .
The HSIEs were used to calculate the nondimensional SIF for multiple cracks in the upper part of bonded dissimilar materials in .
The mixed-mode dynamic of SIF for an interface crack in two bonded half planes was investigated by summing the extended finite element method and a domain independent interaction integrated method in .
The nondimensional SIF for the collinear interface cracks in two bonded half planes were calculated by combining the solution for an inner and an outer collinear crack in .
The objective of this paper was to determine the behavior of nondimensional SIF at the crack tips for crack problems in upper and lower parts of bonded dissimilar materials subjected to remote shear stress A x A A x A p or normal stress A y A A y A p by using the MCP function method.
Problem Formulation The stress components AA x ,A y ,A xy A , the resultant force function A X .
Y A , and the displacements A u , v A are expressed in terms of the two complex potentials AI AA A A A ' AA A and Ao AA A A A ' AA A as follows:
A y A A x A 2iA xy A 2 EEAAI ' j AA A A Ao j AA A EE f A AY j A iX j A A j AA A A A AI j AA A A A j AA A .
2G j A u j A iv j A A A jA j AA A A A AI j AA A AA j AA A .
Stress Intensity factor for Cracks Problems where A A x A iy is a complex variable.
G j is the shear modulus of elasticity.
A j A A 3 A v j A A1 A v j A for plane stress.
A j A 3 A 4v j for plane strain, v j is PoissonAos ratio and j A 1,2 .
The derivative of the resultant force Eq.
with respect to A , yields:
AAYj A iX j A A AI j AA A A AI j AA A A ddAA EEAAI ' j AA A A Ao j AA AEE A Au N A iT Ay j where the normal (N) and tangential (T) components of traction along the segment A.
A A dA depend on the position of a point A and the direction of the segment d A d A .
The complex potentials for the crack L in an infinite plane can be expressed as .
A AA A A
1 g A t A dt 2A EL t A A .
A AA A A 1 g A t Adt 1 g A t A dt 1 g A t A tdt 2A EL t A A 2A EL t A A 2A EL A t A A A2 where g A t A is COD function defined by:
i AA A 1A g A t A A 2G A u A t A A iv A t A A , t Ea L and A u A t A A iv A t A A A A u A t A A iv A t A A A A u A t A A iv A t A A denote the displacements at A A point t and superscript and Ae are the upper and lower crack faces, respectively.
Consider two cracks L1 and L2 in the upper and lower parts of a bonded dissimilar material, respectively, and the conditions for remote shear stress and normal stress are:
Ax A Ax .
Ay A Ay where E1 A 2G1 A1 A v1 A and E2 A 2G2 A1 A v2 A are YoungAos modulus of elasticity for upper and lower parts of bonded dissimilar materials, respectively, and assuming other stress is zero.
The MCP function for crack L1 can be described Khairum Hamzah, et al.
by summation of the principal AA AA A ,A AA A A and complementary AA AA A ,A AA A A parts of the complex potentials as follows:
A1 AA A A A1 p AA A A A1c AA A .
A 1 AA A AA 1 p AA A AA 1c AA A .
where the principal parts of complex potentials are referred to an infinite plane For crack L2 the complex potentials are represented by A2 AA A and A 2 AA A .
Applying continuity conditions to the resultant force Eq.
and displacement functions Eq.
, then substitute Eqs.
and after some manipulations the following complex potentials are obtainable:
A1c AA A A 1 EEA AI1 p AA A A A 1 p AA A EE .
A Ea S1 A Lb A 1c AA A A 2 A1 p AA A A 1 EEA AI1 p AA A .
A A 2 AI '1 p AA A A AA '1 p AA A EE .
A Ea S1 A Lb
A2 AA A A A1 A 2 AA1 p AA A ,A Ea S2 A Lb
A 2 AA A A A 1 A 2 aI1 p AA A A A1 A 1 AA1 p AA A ,A Ea S2 A Lb
A A
where A1 p AA A A A1 p A .
Lb is the boundary.
S1 and S 2 are the upper and lower parts of bonded dissimilar materials, respectively, and constants defined as:
G2 A G1
A G A A 2G1
, 2A 1 2
G1 A A1G2
G2 A A 2G1
, 2 are bi-elastic .
The HSIEs for the cracks in both the upper and lower parts of bonded dissimilar materials involve four traction components A N A t0 A A iT A t0 AA jk A j A 1,2, k A 1,2 A , which can be divided into two groups.
The first two tractions A N A t10 A A iT A t10 AA11 and A N A t20 A A iT A t20 AA21 are obtained when the observation point is placed at points t10 Ea L1 and t20 Ea L2 , respectively, caused by g1 A t1 A at t1 Ea L1 .
The traction A N A t A A iT A t AA can be obtained by summing the principal and complementary parts.
Substituting Eqs.
into Eq.
yields the Stress Intensity factor for Cracks Problems principal part, and substituting Eqs.
into Eq.
gives the complementary part of the traction.
Then, letting point A approaches t10 on the crack and changing d A d A into d t 10 dt10 , yields:
A N A t A A iT A t AA A AN A t A A iT A t AA A AN At A A iT At AA
g1 A t1 A dt1
A t1 A t10 A A A1 A t1 , t10 Ag1 A t1 A dt1 A A2 A t1 , t10 A g1 A t1 Adt1
2A L
2A LE
d t 10
A1 A t1 , t10 A A B1 A t1 , t10 A A 1 E B3 A t1 , t10 A A A B5 A t1 , t10 A A B4 At1 , t10 A E
d t1
d t1 d t10
d t10
B4 A t1 , t10 A A B4 A t1 , t10 A A B6 A t1 , t10 A E A 2
B4 A t1 , t10 A dt1 dt10
d t10
d t1
A2 A t1 , t10 A A B2 A t1 , t10 A A 1 E B4 A t1 , t10 A A B4 A t1 , t10 A A B6 A t1 , t10 A A B3 A t1 , t10 A E
A t1 A t10 A dt1 dt10 E
B1 A t1 , t10 A A A1
A t1 A t10 A EE t1 A t10 dt1 dt10 EE
B2 A t1 , t10 A A A A A t1 A t10 E t1 A t10 E dt 1 dt 10 E
dt 1 dt 10 E
EE A 2
dt1 dt10 EE t 1 A t10 EE t1 A t10 E dt1 dt10 E B3 A t1 , t10 A A EEt 1 A 2t1 A t10 EE At A t A B4 A t1 , t10 A A At A t A 2 A 3t A 2t A t A 6 A t A t AA t A t A B At , t A A A At A t A At A t A B6 A t1 , t10 A A EEt1 A t 10 A 2t10 EE t1 A t10 Khairum Hamzah, et al.
Eq.
represents a single crack in the upper part of a bonded dissimilar Substituting Eqs.
into Eq.
and applying Eqs.
yields the traction for A N A t20 A A iT A t20 AA21 as follows:
A N A t A A iT A t AA A A1 A A A E
g1 A t1 A dt1
A t1 A t20 A A A3 A t1 , t20 Ag1 A t1 A dt1
2A LE
A4 A t1 , t20 A g1 A t1 Adt1
2A LE
A3 A t1 , t20 A A A1 A 1 A At A t A A4 A t1 , t20 A A At A t A dt1 dt 20
A A1 A 2 A
dt1 dt20
A t1 A t20 A E
dt 20 E
A A1 A 1 A
EA1 A 2 A
A EA1 A 2 A 2t20 A A1 A 1 A 2t1 A A 1 A 2 A t 1 A t 20 E At A t A The second two tractions A N A t A A iT A t AA dt1 dt 20 dt1 dt20 A N A t A A iT A t AA obtained when the observation points are placed at t10 Ea L1 and t20 Ea L2 , respectively, caused by g 2 A t2 A at t2 Ea L2 .
In this process we need to introduce two bi-elastic constants defined as follows:
G1 A G2
A G AA G
, 4A 2 1 1 2
G2 A A 2G1
G1 A A1G2
which is evaluated by changing the subscript 1 to 2 and 2 to 1 in Eq.
The system of HSIEs for two cracks L1 and L2 in both the upper and the lower parts of a bonded dissimilar material is obtained as follows:
AN At A A iT At AA A AN At A A iT At AA A AN At A A iT At AA AN At A A iT At AA A AN At A A iT At AA A AN At A A iT At AA .
In solving the system of HSIEs for a single crack in the upper part Eq.
and two cracks in both the upper and the lower parts Eqs.
of a bonded dissimilar material, it is well known that the curved length coordinate method can Stress Intensity factor for Cracks Problems be used to transform the integral along the cracks into real axis s j with an interval of 2a j .
The COD function g A t A is defined as follows:
g j A t j A t At A s A A a 2j A s 2j H j A s j A .
where H j A s j A A H j1 A s j A A iH j 2 A s j A .
A j A 1, 2 A .
Results and Discussions The SIF at the crack tips A j and B j of the crack L j A j A 1, 2 A are defined as A K1 A iK 2 A A A 2A tlim t A t A g '1 A t1 A A aA FA A K1 A iK 2 A B A 2A tlim t A t B g '2 A t2 A A bA FB where FA A F1 A A iF2 A and FB A F1B A iF2 B are the nondimensional SIF at crack tips A j and B j , respectively.
Consider an inclined crack with length 2R in the upper part of a bonded dissimilar material subjected to remote stress A y A A y A p as defined in Figure 1.
Figure 1 An inclined crack in a bonded dissimilar material.
Khairum Hamzah, et al.
Table 1 shows the nondimensional SIF when A A 90o and h 2 R varies for different elastic constant ratios G2 G1 .
Our numerical results are completely in agreement with those of Isida and Noguchi .
It is found that the Mode I nondimensional SIF.
F1 , at crack tip A1 is equal to F1 at tip A2 , whereas the Mode II nondimensional SIF.
F2 , at crack tip A1 is equal to the negative of F2 at tip A2 .
Table 1
G2 G1
Nondimensional SIF for a crack parallel to the interface (Figure .
SIF
F1A2
F1A2.
F2A2
F2A2.
F1A2
F1A2.
F2A2
F2A2.
F1A2
F1A2.
F2A2
F2A2.
h 2R Figure 2 shows the nondimensional SIF when R h A 0.
9 and A varies.
It is observed that at crack tip A1 (Figure 2.
), as A increases F1 increases and F2 increases for A A 50 , whereas F1 decreases and F2 increases as G2 G1 .
SIF at crack tip A1 .
SIF at crack tip A2 Figure 2 SIF when R h A 0.
9 and A varies (Figure .
Stress Intensity factor for Cracks Problems At crack tip A2 (Figure 2.
), as A increases F1 increases and F2 increases for A A 500 .
As G2 G1 increases F1 decreases at crack tip A2 and.
F2 increases for A A 600 and decreases for A A 600 .
Consider the two inclined cracks in the upper and lower parts of a bonded dissimilar material subjected to remote stress A x A A x A p displayed in Figure Figure 3 Two inclined cracks in a bonded dissimilar material.
Figure 4 shows the nondimensional SIF at all cracks tips for two inclined cracks for different values of G2 G1 when A1 A A 2 A 20o and R h varies (Figure .
is observed that as R h and G2 G1 increase F1 increases at crack tips B1 and B2, whereas at crack tips A1 and A2.
F1 increases as R h increases and F1 decreases as G2 G1 increases.
As G2 G1 increases F2 decreases at crack tip A1 and increases at crack tip B1.
As R h and G2 G1 increase F2 does not show any significant difference at crack tips A2 and B2.
The nondimensional SIF when A1 varies for R h A 0.
9 and A 2 A 45o at all cracks tips are presented in Figure 5.
It is found that as A1 and G2 G1 increase F1 Khairum Hamzah, et al.
decreases at crack tips A1 and A2.
But F2 increases for A1 A 45o at crack tips A1 and A2, and as G2 G1 increases F2 decreases at crack tip A1, and does not show any significant difference at crack tip A2.
At crack tip B1.
F1 increases for A1 A 20o and decreases for A1 A 20o , and F1 increases as G2 G1 increases.
At crack tip B2.
F1 decreases for A1 A 40o and increases as G2 G1 increases for A1 A 40o .
However F2 does not show any significant differences at crack tips B1 and B2 as A1 and G2 G1 increase.
SIF at crack tip A1 .
SIF at crack tip A2 .
SIF at crack tip B1 .
SIF at crack tip B2 Figure 4 SIF when A1 A A 2 A 20o and R h varies (Figure .
Stress Intensity factor for Cracks Problems .
SIF at crack tip A1 .
SIF at crack tip A2 .
SIF at crack tip B1 .
SIF at crack tip B2 Figure 5 SIF when A 2 A 45o .
R h A 0.
9 and A1 varies (Figure .
Conclusion An inclined crack in the upper part and two inclined cracks in the upper and lower parts of a bonded dissimilar material subjected to remote stress with different elastic constants G1 and G 2 were studied.
The systems of HSIEs for these problems were formulated by using the MCP function method.
The behavior of the nondimensional SIF at all crack tips depends on the ratio of elastic constants, the crack geometries and the distance between the crack and the boundary.
Acknowledgement The first author would like to acknowledge the generous support of the Ministry of Education Malaysia.
Universiti Putra Malaysia and Universiti Teknikal Malaysia Melaka for financing the research of this project.
The second author Khairum Hamzah, et al.
would like to thank the Universiti Putra Malaysia for Putra Grant project number
References