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Boundary element analysis of cracks normal to bimaterial interfaces
~~~gineering FracrureMechanicsVol. 40, No. 3, pp. 487491, 199I F’rinted in Great Britain.
oo13-7944/91 53.00+ 0.00 Pergamon Prcas pk.
BOUNDARY ELEMENT ANALYSIS OF CRACKS NORMAL TO BIMATERIAL INTERFACES Y. W. KWON Dept. of Mechanical Engineering, Naval Postgraduate School, Monterey, CA 93943, U.S.A. and R. DUTfON Dept. of Ceramic Engineering, University of Missouri-Rolla, Rolla, MO 65401, U.S.A. Ah&act-The singularity of stress at the tip of a crack, which is normal to the interface of two different materials, varies depending on the material properties of the two materials. A subdomain boundary element technique is used to solve the interface crack problems. A shape function, containing the same order of singularity as that in the interface cracks, is used for the interpolation of tractions, and a correct order of shape function is also used for displacements. Some numerical examples show that the present analyses yield good solutions for these interface crack problems.
INTRODUCTION CRACKS IN A structure reduce the load-carrying capacity of the structure because of the stress-concentration at the crack tips. Linear elastic fracture mechanics show that the stress at the crack tip is infinite. In order to compute the stress distribution near the crack tip using the finite element method or the boundary element method, the mesh near the crack tip should be very refined or a special shape function should be used to model the crack tip. The special shape function includes the singularity of stress at the crack tip. The singularity of stress at the crack tip is 0.5 in a linear elastic homogeneous body. Thus, many shape functions have been developed to model the stress singularity of 0.5 in the displacement-based finite element method. One of the popular shape functions is the quarter-point isoparametric shape function[ 1,2]. In the displacement-based finite element analysis, stresses are computed by differentiating shape functions for displacements. Thus, the quarter-point isoparametric shape function yields the correct order of interpolations for both displacements and stresses. However, in the boundary element formulation the displacements and tractions are interpolated independently. The quarter-point isoparametric shape function does not represent the singularity for tractions without differentiation. Thus, Blandford et a1.[33 modified the quarter-point shape function so that it can represent the singularity of 0.5 for tractions. The order of stress singularity at the crack tip, normal to the interface of two different materials, is different from that at the crack tip in a homogeneous material. It varies depending on the material properties of the two materials. References[4,5] show the variation of order of stress singularity for different ratios of GJG,, where G denotes the shear modulus and subscripts 1 and 2 indicate the two different materials. Therefore, the quarter-point shape function cannot represent the order of stress singularity in cracks perpendicular to bimaterial interfaces correctly. The present study uses the correct order of interpolation functions for tractions and displacements to solve such interface crack problems using a subdomain boundary element technique.
BOUNDARY
ELEMENT ANALYSIS
The boundary element analysis is performed by discretizing the boundary of the problem domain in the boundary integral equation. The equation for the two-dimensional elasticity problem is written as:
488
Y. W. KWON and R. DUlTON
where u and t denote displacement and traction, and r is the boundary of the given domain. Tti and r& are the fundamental singular solutions. These solutions are provided in refs [6,7]. Both displacement and traction are interpolated by a shape function and nodal values on each boundary as given below:
where di is either displacement or traction, 4’ is a shape function, d’; is a nodal value, and N is the number of nodes per element. If the element is located at the crack tip, the shape function has to include the singularity, r-4, for traction, where r is the distance from the crack tip. For a crack in a homogeneous material, q is 0.5. The quarter-point isoparametric shape function can represent the traction and displacement as follows: d, = a; + affi
+ a;r.
(3)
This is the correct representation for displacement. However, for traction eq. (3) needs to be divided by fi to model the stress singularity of r-O.‘. This technique was used in ref. [3]. Cracks perpendicular to the interface of two different materials have stress singularity other than 0.5. If a crack is located in the softer material, i.e. GJG, > 1 in Fig. 1, the singularity is less than 0.5, while a crack in the stiffer material has a singularity larger than 0.5. In these cases the quarter-point shape function cannot represent the correct order of singularity. Instead a shape function given below can be used: 4’ = {1 + (2P - l)< - (< + 1)“}/(2 - 2’)
w
~*={l+r-(~+ly}/(2-2p)
W-9
43={-2P-2P5
(JW
+2(l5 +1)p}/(2-29
where the corresponding nodes are shown in Fig. 2. This shape function was developed in ref. [8]. In order to model the crack tip singularity rmq in an interface crack, p in eqs (4a-c) is set to -q for traction and to (1 - q) for displacement. Then both traction and displacement have the correct order of interpolation at the crack tip. Because eq. (1) is derived for a homogeneous material, a subdomain technique is used to solve the interface crack problems. As shown in Fig. 3, the whole domain is divided into two subdomains, which are respectively homogeneous. The boundary element technique is applied to each subdomain resulting in the following matrix equations:
for subdomain I, and
(5b)
NODE 1
Fig. 1. Crack normal to the interface of a bimaterial.
NODE 3
NODE 2
Fig. 2. Location of nodes in natural coordinate system.
Boundary element analysis of cracks normal to bimaterial interfaces
6L INTERFACE
INTERFACE
L SUBDOMAIN I
SUBDOMAIN II
=
v
,
/ *
I I L
c-a
L
I I .
CRACK LENGTH
t
Fig. 3. Subdomain technique for a nonhomogeneous body.
Fig. 4. A quarter of a plate with double edge cracks.
for subdomain II. Here subscript 2 indicates the interface boundary of each subdomain. The equilibrium and continuity conditions at the common interface give u: = u:’
(6a)
II t:- t2.
RW
and Incorporating eqs (6a) and (6b) into eqs @a) and (Sb) results in the following matrix equations:
The shape function, eqs (4a-c), is used at the crack tip of both subdomains because the crack tip is located at the interface. NUMERICAL
RIZXJLTS
The first example was a homogeneous plate with double edge cracks. The plane strain condition was assumed for this and all other example problems. The geometry and boundary discretixation are shown in Fig. 4. Because of symmetry, a quarter of the plate was analyzed using nine three-noded elements. The stress intensity factor was computed using a single point displacement next to the crack tip. Table 1 shows the comparison. Both the quarter-point shape function and present shape function were used. In addition, a single domain and two subdomains were considered, respectively. The quarter-point shape function and present shape function gave comparable solutions. The subdomain analysis gave a better solution than the single domain analysis. Table 1. Stress intensity factor for a double cracked plate Analytical solution Quarter-point shape function with single domain Present shape function with single domain Present shape function with two ~ubdomaina
2.731 2.665 2.650 2.7u(
490
Y. W. KWON and R. DUlTON
BODY
I
BODY
61
2
B2
5L
Table 2. Normalized shearing stress at the interface of a bimaterial
0.05
CRACK
NORMALIZED SHEAR NEAR CRACK TIP ANALY. C
0.10 0.50 2.00 i
singularity order
Analytical solution
BEM solution
0.8173 0.7536 0.5745 0.4338
0.8156 0.7285 0.4561 0.2774
0.7899 0.7107 0.4580 0.2513
CRACK TIP
STRESS
AT INTERFACE
SOLN = 0.7285
BE SOLN. (COARSE MESH) = 0.8365 BE SOLN. (REFINED MESH) = 0.7107
Fig. 5. A quarter of a bimaterial plate with a crack normal to the interface.
The following example was a crack in a bimaterial. The crack was perpendicular to the interface of two different materials. The case of study was a bimaterial with G2/G, = 0.1, where G, is the shear modulus of the material containing the crack. That is, the crack is located in the stiffer material. Both materials have the same Poisson’s ratio of 0.3, The singularity of stress for that case is t - o.7536. It has stronger singularity compared to that in a homogeneous material. Both coarse and refined meshes were used for computation. The coarse mesh has five elements along the regular boundary and three elements along the interface boundary of each subdomain. The refined mesh was obtained by halving the elements in the coarse mesh. The normalized major stress component near the crack tip along the interface, which is the shear stress component normalized by the external uniform traction, was computed and compared with the analytical solution[4] in Fig. 5. The coarse and refined meshes gave 15% and 2.4% errors, respectively, in comparing that stress. The previous refined mesh was used for the following computation. Various cases of bimaterials were studied. The ratios of the two shear moduli GJG, were 0.05, 0.5, and 2.0, respectively, along with the same Poisson’s ratio of 0.3. The stress singularity and normalized major stress component at the interface of each case are compared in Table 2. The solutions are comparable to the analytical solutions even with a rather coarse mesh. CONCLUSION Interface crack problems were solved using the boundary element method. The crack tip, located perpendicular to the interface of two different materials, has the stress singularity of r-4, where q varies depending on the ratio of shear moduli of the two materials. Thus, a shape function, which has correct interpolations for both traction and displacement, was used to model the crack tip singularity. The shape function contains the r-* variation for traction and the r(‘-q) variation for displacement at the crack tip. A subdomain technique was also used to handle the nonhomogeneity. Analyses of interface crack problems using the present approach yielded comparable solutions compared to the analytical solutions.
Boundary element analysis of cracks normal to bimaterial interfaces
491
REFERENCES [l] R. D. Henshell and K. G. Shaw, Crack tip elements are unnecea9ary. far. J. numer. Meth. Ettgng 9, 495-507 (1975). [2] R. S. Barsoum, On the use of isoparametric 6nite elements in linear fracture mechanics. Znt. J. namer. Me& Eqrrg 10, 25-37 (1976). [3] G. E. Blandford, A. R. Ingralfea and J. A. Liggctt, Twodimensional stress intensity factor computations using the boundary element method. Inr. J. anmet. Mesh. Engug 17, 387-404 (1981). [4] D. 0. Swenson and C. A. Rau, Jr., The stress distribution around a crack perpendicular to an interface between materials. Znr. J. Fracture Mech. 6, 357-365 (1970). [!Jl K. Y. Lin and J. W. Mar, Finite element analysis of stress intensity factors for cracks at a bi-material interface. Inr. J. Frucfure 12, 521431 (1976). [6] C. A. Brebbia and J. Dominguez, Boundary Elements, An IntroductoryCourse. Computational Mechanica Publications, Southampton (1989). [I P. K. Banetjee and R. Butterfield, BounaiwyElement Methods in Engineering Science. McGraw-Hill, London (1981). [S] Y. W. Kwon, Development of finite element shape functions with derivative singularity. Compur. Srruct. 381159-l 163 (1988). (Received 16 October 1990)
oo13-7944/91 53.00+ 0.00 Pergamon Prcas pk.
BOUNDARY ELEMENT ANALYSIS OF CRACKS NORMAL TO BIMATERIAL INTERFACES Y. W. KWON Dept. of Mechanical Engineering, Naval Postgraduate School, Monterey, CA 93943, U.S.A. and R. DUTfON Dept. of Ceramic Engineering, University of Missouri-Rolla, Rolla, MO 65401, U.S.A. Ah&act-The singularity of stress at the tip of a crack, which is normal to the interface of two different materials, varies depending on the material properties of the two materials. A subdomain boundary element technique is used to solve the interface crack problems. A shape function, containing the same order of singularity as that in the interface cracks, is used for the interpolation of tractions, and a correct order of shape function is also used for displacements. Some numerical examples show that the present analyses yield good solutions for these interface crack problems.
INTRODUCTION CRACKS IN A structure reduce the load-carrying capacity of the structure because of the stress-concentration at the crack tips. Linear elastic fracture mechanics show that the stress at the crack tip is infinite. In order to compute the stress distribution near the crack tip using the finite element method or the boundary element method, the mesh near the crack tip should be very refined or a special shape function should be used to model the crack tip. The special shape function includes the singularity of stress at the crack tip. The singularity of stress at the crack tip is 0.5 in a linear elastic homogeneous body. Thus, many shape functions have been developed to model the stress singularity of 0.5 in the displacement-based finite element method. One of the popular shape functions is the quarter-point isoparametric shape function[ 1,2]. In the displacement-based finite element analysis, stresses are computed by differentiating shape functions for displacements. Thus, the quarter-point isoparametric shape function yields the correct order of interpolations for both displacements and stresses. However, in the boundary element formulation the displacements and tractions are interpolated independently. The quarter-point isoparametric shape function does not represent the singularity for tractions without differentiation. Thus, Blandford et a1.[33 modified the quarter-point shape function so that it can represent the singularity of 0.5 for tractions. The order of stress singularity at the crack tip, normal to the interface of two different materials, is different from that at the crack tip in a homogeneous material. It varies depending on the material properties of the two materials. References[4,5] show the variation of order of stress singularity for different ratios of GJG,, where G denotes the shear modulus and subscripts 1 and 2 indicate the two different materials. Therefore, the quarter-point shape function cannot represent the order of stress singularity in cracks perpendicular to bimaterial interfaces correctly. The present study uses the correct order of interpolation functions for tractions and displacements to solve such interface crack problems using a subdomain boundary element technique.
BOUNDARY
ELEMENT ANALYSIS
The boundary element analysis is performed by discretizing the boundary of the problem domain in the boundary integral equation. The equation for the two-dimensional elasticity problem is written as:
488
Y. W. KWON and R. DUlTON
where u and t denote displacement and traction, and r is the boundary of the given domain. Tti and r& are the fundamental singular solutions. These solutions are provided in refs [6,7]. Both displacement and traction are interpolated by a shape function and nodal values on each boundary as given below:
where di is either displacement or traction, 4’ is a shape function, d’; is a nodal value, and N is the number of nodes per element. If the element is located at the crack tip, the shape function has to include the singularity, r-4, for traction, where r is the distance from the crack tip. For a crack in a homogeneous material, q is 0.5. The quarter-point isoparametric shape function can represent the traction and displacement as follows: d, = a; + affi
+ a;r.
(3)
This is the correct representation for displacement. However, for traction eq. (3) needs to be divided by fi to model the stress singularity of r-O.‘. This technique was used in ref. [3]. Cracks perpendicular to the interface of two different materials have stress singularity other than 0.5. If a crack is located in the softer material, i.e. GJG, > 1 in Fig. 1, the singularity is less than 0.5, while a crack in the stiffer material has a singularity larger than 0.5. In these cases the quarter-point shape function cannot represent the correct order of singularity. Instead a shape function given below can be used: 4’ = {1 + (2P - l)< - (< + 1)“}/(2 - 2’)
w
~*={l+r-(~+ly}/(2-2p)
W-9
43={-2P-2P5
(JW
+2(l5 +1)p}/(2-29
where the corresponding nodes are shown in Fig. 2. This shape function was developed in ref. [8]. In order to model the crack tip singularity rmq in an interface crack, p in eqs (4a-c) is set to -q for traction and to (1 - q) for displacement. Then both traction and displacement have the correct order of interpolation at the crack tip. Because eq. (1) is derived for a homogeneous material, a subdomain technique is used to solve the interface crack problems. As shown in Fig. 3, the whole domain is divided into two subdomains, which are respectively homogeneous. The boundary element technique is applied to each subdomain resulting in the following matrix equations:
for subdomain I, and
(5b)
NODE 1
Fig. 1. Crack normal to the interface of a bimaterial.
NODE 3
NODE 2
Fig. 2. Location of nodes in natural coordinate system.
Boundary element analysis of cracks normal to bimaterial interfaces
6L INTERFACE
INTERFACE
L SUBDOMAIN I
SUBDOMAIN II
=
v
,
/ *
I I L
c-a
L
I I .
CRACK LENGTH
t
Fig. 3. Subdomain technique for a nonhomogeneous body.
Fig. 4. A quarter of a plate with double edge cracks.
for subdomain II. Here subscript 2 indicates the interface boundary of each subdomain. The equilibrium and continuity conditions at the common interface give u: = u:’
(6a)
II t:- t2.
RW
and Incorporating eqs (6a) and (6b) into eqs @a) and (Sb) results in the following matrix equations:
The shape function, eqs (4a-c), is used at the crack tip of both subdomains because the crack tip is located at the interface. NUMERICAL
RIZXJLTS
The first example was a homogeneous plate with double edge cracks. The plane strain condition was assumed for this and all other example problems. The geometry and boundary discretixation are shown in Fig. 4. Because of symmetry, a quarter of the plate was analyzed using nine three-noded elements. The stress intensity factor was computed using a single point displacement next to the crack tip. Table 1 shows the comparison. Both the quarter-point shape function and present shape function were used. In addition, a single domain and two subdomains were considered, respectively. The quarter-point shape function and present shape function gave comparable solutions. The subdomain analysis gave a better solution than the single domain analysis. Table 1. Stress intensity factor for a double cracked plate Analytical solution Quarter-point shape function with single domain Present shape function with single domain Present shape function with two ~ubdomaina
2.731 2.665 2.650 2.7u(
490
Y. W. KWON and R. DUlTON
BODY
I
BODY
61
2
B2
5L
Table 2. Normalized shearing stress at the interface of a bimaterial
0.05
CRACK
NORMALIZED SHEAR NEAR CRACK TIP ANALY. C
0.10 0.50 2.00 i
singularity order
Analytical solution
BEM solution
0.8173 0.7536 0.5745 0.4338
0.8156 0.7285 0.4561 0.2774
0.7899 0.7107 0.4580 0.2513
CRACK TIP
STRESS
AT INTERFACE
SOLN = 0.7285
BE SOLN. (COARSE MESH) = 0.8365 BE SOLN. (REFINED MESH) = 0.7107
Fig. 5. A quarter of a bimaterial plate with a crack normal to the interface.
The following example was a crack in a bimaterial. The crack was perpendicular to the interface of two different materials. The case of study was a bimaterial with G2/G, = 0.1, where G, is the shear modulus of the material containing the crack. That is, the crack is located in the stiffer material. Both materials have the same Poisson’s ratio of 0.3, The singularity of stress for that case is t - o.7536. It has stronger singularity compared to that in a homogeneous material. Both coarse and refined meshes were used for computation. The coarse mesh has five elements along the regular boundary and three elements along the interface boundary of each subdomain. The refined mesh was obtained by halving the elements in the coarse mesh. The normalized major stress component near the crack tip along the interface, which is the shear stress component normalized by the external uniform traction, was computed and compared with the analytical solution[4] in Fig. 5. The coarse and refined meshes gave 15% and 2.4% errors, respectively, in comparing that stress. The previous refined mesh was used for the following computation. Various cases of bimaterials were studied. The ratios of the two shear moduli GJG, were 0.05, 0.5, and 2.0, respectively, along with the same Poisson’s ratio of 0.3. The stress singularity and normalized major stress component at the interface of each case are compared in Table 2. The solutions are comparable to the analytical solutions even with a rather coarse mesh. CONCLUSION Interface crack problems were solved using the boundary element method. The crack tip, located perpendicular to the interface of two different materials, has the stress singularity of r-4, where q varies depending on the ratio of shear moduli of the two materials. Thus, a shape function, which has correct interpolations for both traction and displacement, was used to model the crack tip singularity. The shape function contains the r-* variation for traction and the r(‘-q) variation for displacement at the crack tip. A subdomain technique was also used to handle the nonhomogeneity. Analyses of interface crack problems using the present approach yielded comparable solutions compared to the analytical solutions.
Boundary element analysis of cracks normal to bimaterial interfaces
491
REFERENCES [l] R. D. Henshell and K. G. Shaw, Crack tip elements are unnecea9ary. far. J. numer. Meth. Ettgng 9, 495-507 (1975). [2] R. S. Barsoum, On the use of isoparametric 6nite elements in linear fracture mechanics. Znt. J. namer. Me& Eqrrg 10, 25-37 (1976). [3] G. E. Blandford, A. R. Ingralfea and J. A. Liggctt, Twodimensional stress intensity factor computations using the boundary element method. Inr. J. anmet. Mesh. Engug 17, 387-404 (1981). [4] D. 0. Swenson and C. A. Rau, Jr., The stress distribution around a crack perpendicular to an interface between materials. Znr. J. Fracture Mech. 6, 357-365 (1970). [!Jl K. Y. Lin and J. W. Mar, Finite element analysis of stress intensity factors for cracks at a bi-material interface. Inr. J. Frucfure 12, 521431 (1976). [6] C. A. Brebbia and J. Dominguez, Boundary Elements, An IntroductoryCourse. Computational Mechanica Publications, Southampton (1989). [I P. K. Banetjee and R. Butterfield, BounaiwyElement Methods in Engineering Science. McGraw-Hill, London (1981). [S] Y. W. Kwon, Development of finite element shape functions with derivative singularity. Compur. Srruct. 381159-l 163 (1988). (Received 16 October 1990)