# The Effect of Geometric Configurations on the Elastic Behavior of an Edge-Cracked Bonded Strip

Authors: Xin Lan; Shaobo Ji; Nao-Aki Noda; Yong Cheng
Journal: Advances in Materials Science and Engineering
 (2017)

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## Abstract

Interface cracks often initiate around the bonding free-edge corner due to the high-stress concentration. In this research, the elastic behavior of an edge-cracked dissimilar bonded strip subjected to remote tensile load is investigated using the proportional crack tip opening displacement method based on FE analysis for arbitrary material combinations. The stress intensity factor, energy release rate, and mode mixity are computed and compared systematically with varying geometrical configurations and material combinations. Then, the combined effects of the relative height of the bonded component and material combination are discussed for the typical engineering materials.


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## Body

## 1. Introduction

There is an increasing demand on the use of bonded joints in modern structural applications. Interfacial cracks are normally observed around the bonding edge region in joints and areas of discontinuities due to the high-stress concentration, residual stresses, and interfacial flaws. The presence of a crack affects the performance of a structure, and the propagation may eventually cause a through thickness crack which results in the failure of a structure.The problem of interface crack was firstly investigated by Williams [1] who discovered the asymptotic solution of the elastic field around an open crack; then, his work was followed and extended by Rice and Sih [2], Erdogan [3, 4], England [5], Willis [6], and many others. In linear elastic fracture mechanics, the stress intensity factor (SIF) and energy release rate (ERR) are used to predict the stress state and the stable crack growth caused by the remote load. Therefore, many researchers have tried to solve the singular fields for various geometries and material combinations. Recently, many studies were found to accurately determine the order of the stress singularity [7–12]. Then, quite a lot of researches have been devoted to the analysis of the intensity of the singular stress fields for interfacial crack problems [13, 14]. Akisanya and Fleck [15] evaluated the singular stress fields at the free edge of a long bimaterial strip subjected to uniform tension. Lan et al. [16] investigated the effect of material combinations and the relative crack size on the SIFs at the crack tip of a bimaterial bonded strip. Noda and Lan [17] analyzed the SIFs of the shallow edge interface cracks for arbitrary material combinations and proposed an empirical formula for the equal pair zone in the α-β space.However, all the aforementioned studies concentrate their discussions on the case of bonded long strips and ignore the effect of geometrical configurations. In the practical industrial applications, the geometrical configuration of the bonded strip varies greatly with the actual demands. The different heights of the two bonded components may exhibit longitudinal symmetry or asymmetry characteristics, thus producing the variation of the mixed mode for a specific material combination. As known, the fracture toughness (i.e., critical energy release rate or critical stress intensity factor) depends strongly on the mode mixity. In order to predict the onset of the delamination at the bimaterial interface, one must investigate the elastic behaviors for various geometric configurations. In this research, the SIFs, ERR, and mix-mode angle are systematically computed for various geometric configurations and material combinations based on FE analysis using the proportional crack tip opening displacement (CTOD) method [18]. The relative height of the two components of the bonding strip varies from l/W=2 to l/W=0.333, and the geometric configurations are composed of longitudinal symmetrical and asymmetrical cases. Then, the effects of the heights of the bonded components together with material combinations are thoroughly investigated for the most fundamental edge interface crack problems. The conclusion may help to understand the dependence of the fracture mechanism on the geometric configurations and may also contribute to the guidance for material selection and structural design for the modern layered materials.


## 2. Analysis Method

### 2.1. Formulation for the Interface Crack Problems

Consider two isotropic elastic materials joined along thex-axis as indicated in Figure 1 with material 1 above the interface and material 2 below. The stress distributions along the interface are defined as shown in [19](1)σy+iτxy=KI+iKII2πrr2aiε,r⟶0.Here, σy and τxy denote the normal and shear stress components near the crack tip, respectively, r is the radial distance behind the crack tip, a is the half crack length, and ε is the bielastic constant given by(2)ε=12πln⁡κ1/μ1+1/μ2κ2/μ2+1/μ1,κm=3-4νmplanestrain3-νm1+νmplanestress,where μmm=1,2 and νmm=1,2 are the shear moduli and Poisson’s ratios of either respective material. The associated relative crack opening displacement values δd=udr,θ=π-udr,θ=-π and (d=x,y) for nodes i, i′ at a distance r behind the crack tip, shown in Figure 1, are given by [20](3)δy+iδx=KI+iKII21+2iεcosh⁡επκ1+1μ1+κ2+1μ2r2π1/2rliε,where l is an arbitrary reference length which scales with specimen size or crack length; for the definition of (1), we have l=2a without loss of generality.Figure 1
Stress distribution and relative crack displacement of an interface crack.Consideringr/liε=cos⁡(εln⁡r/l)+isin⁡(εln⁡r/l) and rearranging (3), then the stress intensity factor components KI,KII can be separated as(4)KI=Sδy-2εδxcos⁡εln⁡rl+δx+2εδysin⁡εln⁡rl,(5)KII=Sδx+2εδycos⁡εln⁡rl-δy-2εδxsin⁡εln⁡rl,(6)S=2cosh⁡επr/2π-1/2κ1+1/μ1+κ2+1/μ2.We can rewrite (4) and (5) as(7)KIδy=Scos⁡Q+2εsin⁡Q+sin⁡Q-2εcos⁡Qδxδy,(8)KIIδx=Scos⁡Q+2εsin⁡Q-sin⁡Q-2εcos⁡Qδyδx,(9)Q=εln⁡rl.From (7) and (8), when Q, ε, and δy/δx are kept the same for two different interface cracks, then we get a relationship as (10)KIδy=const,KIIδx=const.Considering two interface crack ProblemsA and B (say problems in Figure 2), by satisfying the preconditions as shown in (11) and (12), then the stress intensity factors KI, KII exhibit a proportional relationship with δy, δx as depicted in (13). The relative crack opening displacement values δy and δx can be computed by FE analysis, assuming one of the two problems is analytically well solved in advance; KI and KII of Problem A are given in advance, and then the SIFs of Problem B can be easily obtained from (13). Hence,(11)QA=QBεA=εB⟶εln⁡rlA=εln⁡rlBεA=εB,(12)δyδxA=δyδxB,(13)KIδyA=KIδyB,KIIδxA=KIIδxB,and the strain energy release rate and mode angle for the crack advance in the interface are (14)G=116cosh2⁡επκ1+1μ1+κ2+1μ2KI2+KII2,ψ=arctan⁡KIIKI.Figure 2
Geometric configuration for (a) the reference ProblemA and (b) the given unknown Problem B.
(a)
(b)


### 2.2. Application of the Proportional COD Method

The problem where SIFs have been solved in advance can be treated as the reference. Therefore, a central cracked dissimilar bonded half plane subjected to remotely uniform tensile and shear stresses as shown in Figure2(a) is selected as the reference problem for generality. Its analytical solution was firstly derived by Rice and Sih [2] and takes the form(15)KI∗+iKII∗=σy∞+iτxy∞πa1+2iε,where an asterisk (∗) is employed to denote the SIFs for the reference problem. σy∞ and τxy∞ are the remote uniform tension and shear applied to the bonded half planes. a is the half crack length of the center crack. Furthermore, the transversal tension values σx1∞ and  σx2∞ in Figure 2(a) behave as(16)σx2∞=11+κ2μ2μ11+κ1σx1∞+3-κ2-μ2μ13-κ1σy∞.As previously mentioned, the preconditions in (11) and (12) should be firstly met to ensure the current method is available. Equation (11) can be easily satisfied by making the bielastic constant ε and the relative distance behind the crack tip r/l the same for the two problems. Here, some extra techniques should be employed to make (12) satisfied. We consider the reference problem shown in Figure 2(a); the relative CODs δy and δx can be solved in an indirect manner using the principle of linear superposition. Let δy,A∗ and δx,A∗ denote the CODs of Problem A subjected to combined T and S; δy,A1T=1∗ and δx,A1T=1∗ denote those of Problem A subjected to pure unit tension T=1; and δy,A2S=1∗ and δx,A2S=1∗ denote those of Problem A subjected to pure unit shear S=1, respectively. Using the theory of linear superposition, then the relative CODs δy,A∗ and δx,A∗ of the reference problem (Problem A) take the following form:(17)δy,A∗=δy,A1T=1∗×T+δy,A2S=1∗×S,(18)δx,A∗=δx,A1T=1∗×T+δx,A2S=1∗×S.Recall (12) and substitute δy and δx with δy,A∗ and δx,A∗ for Problem A; then, we have(19)δy,A∗δx,A∗A=δy,A1T=1∗×T+δy,A2S=1∗×Sδx,A1T=1∗×T+δx,A2S=1∗×SA=δy,Bδx,BB.Rearranging (18) gives the solution of S/T:(20)ST=δx,B·δy,A1T=1∗-δy,B·δx,A1T=1∗δy,B·δx,A2S=1∗-δx,B·δy,A2S=1∗.Using T and S in (19) as the boundary condition for Problem A, then (12) is satisfied and eventually (13) is set up. Finally, the SIFs for the target unknown problem (Problem B) can be obtained using the proportional relationship as given in (20):(21)KI,B=δy,Bδy,A×KI,A,KII,B=δx,Bδx,A×KII,A.


## 2.1. Formulation for the Interface Crack Problems

Consider two isotropic elastic materials joined along thex-axis as indicated in Figure 1 with material 1 above the interface and material 2 below. The stress distributions along the interface are defined as shown in [19](1)σy+iτxy=KI+iKII2πrr2aiε,r⟶0.Here, σy and τxy denote the normal and shear stress components near the crack tip, respectively, r is the radial distance behind the crack tip, a is the half crack length, and ε is the bielastic constant given by(2)ε=12πln⁡κ1/μ1+1/μ2κ2/μ2+1/μ1,κm=3-4νmplanestrain3-νm1+νmplanestress,where μmm=1,2 and νmm=1,2 are the shear moduli and Poisson’s ratios of either respective material. The associated relative crack opening displacement values δd=udr,θ=π-udr,θ=-π and (d=x,y) for nodes i, i′ at a distance r behind the crack tip, shown in Figure 1, are given by [20](3)δy+iδx=KI+iKII21+2iεcosh⁡επκ1+1μ1+κ2+1μ2r2π1/2rliε,where l is an arbitrary reference length which scales with specimen size or crack length; for the definition of (1), we have l=2a without loss of generality.Figure 1
Stress distribution and relative crack displacement of an interface crack.Consideringr/liε=cos⁡(εln⁡r/l)+isin⁡(εln⁡r/l) and rearranging (3), then the stress intensity factor components KI,KII can be separated as(4)KI=Sδy-2εδxcos⁡εln⁡rl+δx+2εδysin⁡εln⁡rl,(5)KII=Sδx+2εδycos⁡εln⁡rl-δy-2εδxsin⁡εln⁡rl,(6)S=2cosh⁡επr/2π-1/2κ1+1/μ1+κ2+1/μ2.We can rewrite (4) and (5) as(7)KIδy=Scos⁡Q+2εsin⁡Q+sin⁡Q-2εcos⁡Qδxδy,(8)KIIδx=Scos⁡Q+2εsin⁡Q-sin⁡Q-2εcos⁡Qδyδx,(9)Q=εln⁡rl.From (7) and (8), when Q, ε, and δy/δx are kept the same for two different interface cracks, then we get a relationship as (10)KIδy=const,KIIδx=const.Considering two interface crack ProblemsA and B (say problems in Figure 2), by satisfying the preconditions as shown in (11) and (12), then the stress intensity factors KI, KII exhibit a proportional relationship with δy, δx as depicted in (13). The relative crack opening displacement values δy and δx can be computed by FE analysis, assuming one of the two problems is analytically well solved in advance; KI and KII of Problem A are given in advance, and then the SIFs of Problem B can be easily obtained from (13). Hence,(11)QA=QBεA=εB⟶εln⁡rlA=εln⁡rlBεA=εB,(12)δyδxA=δyδxB,(13)KIδyA=KIδyB,KIIδxA=KIIδxB,and the strain energy release rate and mode angle for the crack advance in the interface are (14)G=116cosh2⁡επκ1+1μ1+κ2+1μ2KI2+KII2,ψ=arctan⁡KIIKI.Figure 2
Geometric configuration for (a) the reference ProblemA and (b) the given unknown Problem B.
(a)
(b)


## 2.2. Application of the Proportional COD Method

The problem where SIFs have been solved in advance can be treated as the reference. Therefore, a central cracked dissimilar bonded half plane subjected to remotely uniform tensile and shear stresses as shown in Figure2(a) is selected as the reference problem for generality. Its analytical solution was firstly derived by Rice and Sih [2] and takes the form(15)KI∗+iKII∗=σy∞+iτxy∞πa1+2iε,where an asterisk (∗) is employed to denote the SIFs for the reference problem. σy∞ and τxy∞ are the remote uniform tension and shear applied to the bonded half planes. a is the half crack length of the center crack. Furthermore, the transversal tension values σx1∞ and  σx2∞ in Figure 2(a) behave as(16)σx2∞=11+κ2μ2μ11+κ1σx1∞+3-κ2-μ2μ13-κ1σy∞.As previously mentioned, the preconditions in (11) and (12) should be firstly met to ensure the current method is available. Equation (11) can be easily satisfied by making the bielastic constant ε and the relative distance behind the crack tip r/l the same for the two problems. Here, some extra techniques should be employed to make (12) satisfied. We consider the reference problem shown in Figure 2(a); the relative CODs δy and δx can be solved in an indirect manner using the principle of linear superposition. Let δy,A∗ and δx,A∗ denote the CODs of Problem A subjected to combined T and S; δy,A1T=1∗ and δx,A1T=1∗ denote those of Problem A subjected to pure unit tension T=1; and δy,A2S=1∗ and δx,A2S=1∗ denote those of Problem A subjected to pure unit shear S=1, respectively. Using the theory of linear superposition, then the relative CODs δy,A∗ and δx,A∗ of the reference problem (Problem A) take the following form:(17)δy,A∗=δy,A1T=1∗×T+δy,A2S=1∗×S,(18)δx,A∗=δx,A1T=1∗×T+δx,A2S=1∗×S.Recall (12) and substitute δy and δx with δy,A∗ and δx,A∗ for Problem A; then, we have(19)δy,A∗δx,A∗A=δy,A1T=1∗×T+δy,A2S=1∗×Sδx,A1T=1∗×T+δx,A2S=1∗×SA=δy,Bδx,BB.Rearranging (18) gives the solution of S/T:(20)ST=δx,B·δy,A1T=1∗-δy,B·δx,A1T=1∗δy,B·δx,A2S=1∗-δx,B·δy,A2S=1∗.Using T and S in (19) as the boundary condition for Problem A, then (12) is satisfied and eventually (13) is set up. Finally, the SIFs for the target unknown problem (Problem B) can be obtained using the proportional relationship as given in (20):(21)KI,B=δy,Bδy,A×KI,A,KII,B=δx,Bδx,A×KII,A.


## 3. Method Robustness Study

The computational accuracy of the current procedure is demonstrated by pursuing a method robustness study. The test example is an edge interface crack in plane stress condition. The crack length is set to2a=2mm. A plate width of W=1620×2a=3240mm and a length of L=2W=6480mm are used to model the reference problem. The crack length for the given unknown Problem B is fixed to a=1mm, and the width of the bonded strip varies in the range a/W= 0.1~0.4. The length is set to 2 times the width of the bonded strip. Furthermore, the minimum finite element sizes emin=36 are kept the same for the reference and the given unknown problems. The same elastic parameters E1/E2=2,4,10,100 and ν1=ν2=0.3 which were adopted by other researchers [21] are also assumed in the computation. Figure 3 shows finite element idealization with linear quadrilateral elements. The singular regions around the crack tip are well refined in a self-similar manner by increasing the number of layers, and the element size for each inferior layer is one-third the superior one. All the CODs are computed using the MSC.MARC [22] finite element analysis package in this research. The normalized SIFs KI/σπa and KII/σπa are tabulated in Table 1 together with those predicted by Matsumto et al. [21]. As shown in this table, the results of the current procedure coincide with those predicted by Matsumto et al. [21].Table 1
Normalized SIFsKI/σπa and KII/σπa for the edge interface crack (ν1=ν2=0.3, plane stress).
E
1
/
E
2
a
/
W
Edge interface crack
NormalizedKI
NormalizedKII
Present
Matsumto et al. [21]
Present
Matsumto et al. [21]
2
0.1
1.195
1.190
−0.129
−0.127
0.2
1.367
1.367
−0.137
−0.137
0.3
1.658
1.657
−0.158
−0.156
0.4
2.108
2.109
−0.198
−0.195
4
0.1
1.209
1.199
−0.239
−0.237
0.2
1.368
1.368
−0.251
−0.251
0.3
1.653
1.655
−0.288
−0.288
0.4
2.100
2.102
−0.359
−0.358
10
0.1
1.229
1.222
−0.340
−0.336
0.2
1.369
1.366
−0.349
−0.348
0.3
1.648
1.648
−0.399
−0.394
0.4
2.089
2.090
−0.495
−0.491
100
0.1
1.252
1.251
−0.425
−0.424
0.2
1.370
1.376
−0.429
−0.429
0.3
1.642
1.647
−0.485
−0.470
0.4
2.078
2.083
−0.598
−0.569Figure 3
Finite element idealization around the crack tip.


## 4. Results and Discussions

### 4.1. Geometric Configurations and Material Elastic Parameters

The geometric configurations of the edge-cracked bimaterial strips are shown in Figure4. Two elastic, isotropic, and homogenous strips are perfectly bonded along the interface with material 1 above and material 2 below. Let a denote the interface crack length, W denote the width of the bonded strip, and L1 and L2 denote the height of each material, respectively. There are two geometric configurations considered in this research. Namely, they are the symmetrically and asymmetrically bonded strip shown in Figures 4(a) and 4(b), respectively. The heights of the two bonded components 1 and 2 in Figure 4(a) are kept the same and are variable L1=L2=0.333W~2W. However, in Figure 4(b), only the height of the upper component 1 varies in the range L1=0.333W~2W, leaving the lower component 2 fixed to L2=2W. Furthermore, in this research, the relative crack length is fixed to a constant of a/W=0.1 to make the effect of geometric configurations obvious. Then, the SIFs, ERRs, and mode angles are computed for the two different geometric configurations with varying height of the bonded materials.Figure 4
Geometric configurations for (a) longitudinal symmetrically bonded strip and (b) longitudinal asymmetrically bonded strip.
(a)
(b)The singular stress fields around the crack tip for the aforementioned problems with a given geometric configuration are only determined on the two elastic mismatch parametersα,β [24]. The physically admissible values are allowed to lie within a parallelogram dominated by α=±1 and β=α/4±0.25. The representative material combinations for the typical engineering materials were firstly compiled by Suga et al. [23]. They pointed out that the α,β values for typical material combinations are concentrated along β=0 and β=α/4 lines in α-β space. Here, we plot the material combinations using the data compiled by Suga et al. [23] together with those provided by others in Figure 5. It can be noted that most of the combinations fall within a narrow band in β=α/2±0.1, and the α,β values of typical material combinations are distributed in the vicinity of β=0.325α and β=0.25α for plane stress and plane strain conditions, respectively. Therefore, in this research, the SIFs, ERRs, and mode angles are systematically computed for the whole physical admissible α,β as well as the typical engineering material combinations β=0.325α and β=0.25α.Figure 5
Dundurs parameters for typical engineering materials [23].


### 4.2. Solutions of the Bimaterial Bonded Long Strip

As a reference, the bimaterial bonded long strip subjected to tension shown in Figure4(a) is firstly investigated. The heights of the two components are long enough (L/W≥2) to make sure that the crack tip fields are not affected by the length of the strip. The SIFs and ERRs are systematically investigated for the whole range of material combinations α∈-1,1, β∈-0.5,0.5. The 3D SIF distributions as well as the contour plots are shown in Figure 6. It should be noted that all the SIFs are normalized using σπa in this graph. As can be seen from Figure 6(a), the maximum value of KI is located in the lower left and upper right corner of the α-β space, and the minimum KI is located at the other two corners α=±1,β=0 of the parallelogram. In the good pair region, KI decreases from 1.189 of the homogeneous crack α=0,β=0 to the diagonal lower left and upper right endpoints α=±1,β=±0.5 in the α-β space; conversely, it presents an increasing tendency from the origin to the other pair of diagonal corners α=±1,β=0 in the bad pair zone. In Figure 6(b), the distributions of KII/σπa behave almost like a plane, and the values increase from the left to the right in the contour plot monotonously. Furthermore, switching materials 1 and 2 upside down which contributes to reversed α,β merely changes the signs of KII/σπa, and the minimum absolute value 0 of KII/σπa locates on the line β=2.8627α in the whole material space.Figure 6
3D normalized SIFs and contour plots of (a)KI/σπa and (b) KII/σπa.
(a)
(b)The energy methods are widely employed in the prediction of the onset and propagation of the interface cracks or delaminations. Therefore, the strain energy release rates and their corresponding mode angles are also investigated and plotted in Figures7(a) and 7(b), respectively. As can be seen from this graph, the normalized ERR GE∗/σ2πa and mode angle ψ exhibit a similar varying tendency to those of the normalized SIFs KI/σπa and KII/σπa. This is due to the fact that KII is relatively smaller than KI in the whole α-β space for the remote tensile loading case. It is known that the interface toughness depends strongly on the mode mixity; thus, in order to thoroughly and clearly investigate the effect of geometric configuration on the mode mixity, only the normalized ERR GE∗/σ2πa and mode angle ψ will be demonstrated in the following sections due to the similar varying tendency between KI/σπa, KII/σπa, and GE∗/σ2πa, ψ. Without special notifications, one can get similar conclusions for the case of KI/σπa, KII/σπa.Figure 7
3D distributions and contour plots of (a) normalized ERRGE∗/σ2πa and (b) mode angle ψ.
(a)
(b)


### 4.3. Solutions of the Symmetrically Bonded Strip

The SIFs of the symmetrically bonded strip shown in Figure4(a) for typical engineering materials are computed by varying the height of the two materials L1=L2=0.333~2W. It is seen that GE∗/σ2πa and ψ exhibit similarly varying tendency for β=0.25α and β=0.325α in Figures 8(a) and 8(b), respectively. Furthermore, the effect of the height of the bonded component is divided into three different zones: L1/W≤0.1, 0.1<L1/W<0.2, and L1/W≥0.2. The dependence of ERR and mode angle on Dundurs parameter α exhibits different varying tendency for L1/W≤0.1 and L1/W>0.2, and the zone of 0.1<L1/W≤0.2 can be denoted as the transition region which connects the two different variations smoothly. In order to observe the effect of the height of the bonded component, GE∗/σ2πa and ψ are plotted against α for various L1/W in Figures 9(a) and 9(b), respectively. The values of GE∗/σ2πa for L1/W≤0.1 and β=0.325α are plotted in red in the right ordinates in Figure 9(a) to make the variations obvious. Figure 9(a) shows that the maximum and minimum values of GE∗/σ2πa for L1/W≤0.1 are located at α=0 and α→±1, respectively. However, opposite trends are observed when L1/W>0.2. Furthermore, Figure 9(b) shows that the maximum and minimum absψ are always located at α→±1 and α=0, respectively, for a given height of L1/W>0.2. This means that the severe mismatch in the plane tensile modulus across the interface would lead to a bigger mode angle in this case, and ψ does not change monotonically with α for the other two zones of L1/W≤0.1 and 0.1<L1/W<0.2.Figure 8
Normalized ERRGE∗/σ2πa and mode angle ψ of a symmetrically bonded strip for (a) β=0.25α and (b) β=0.325α.
(a)
(b)Figure 9
Variations of (a) normalized ERRGE∗/σ2πa and (b) mode angle ψ of a symmetrically bonded strip for β=0.325α.
(a)
(b)


### 4.4. Solutions of the Asymmetrically Bonded Strip

The normalized ERRGE∗/σ2πa and its corresponding mode angle ψ of the edge-cracked asymmetrically bonded strip shown in Figure 4(b) under the material combinations of β=0.25α and β=0.325α are computed by varying the height of the two materials L1=0.333~2W. As demonstrated in Figure 10, a similar varying tendency for β=0.25α and β=0.325α also exists in the asymmetrically bonded strip. Similarly, GE∗/σ2πa and ψ for some typical L1/W are plotted against α in Figures 11(a) and 11(b), respectively. It should be noted that GE∗/σ2πa of L1/W≤0.1 and β=0.325α are plotted in red in the right ordinates of Figure 11(a) to make the variations obvious. Three different zones, that is, L1/W≤0.1, 0.1<L1/W<0.2, and L1/W≥0.2, are also employed in the discussion. Firstly, considering the case where L1/W≤0.1, when material 1 in Figure 4(b) is extremely stiff compared to material 2, α approaches 1, GE∗/σ2πa in Figure 11(a) reaches its minimum value, and abs(ψ) in Figure 11(b) comes closer to its maximum one. However, when material 1 is extremely compliant compared to material 2, α approaches -1 and GE∗/σ2πa and ψ reach their maximum and minimum values, respectively. Furthermore, the maximum absψ is always located around α→-1. Secondly, considering the case where L1/W≥0.2, Figure 11(a) shows that the minimum GE∗/σ2πa is located at α∈0,1, and increasing L1/W moves the minimum GE∗/σ2πa towards α=0. Furthermore, when material 1 (lower height) is extremely compliant compared to material 2 (α→-1), GE∗/σ2πa reaches its maximum value for a specific L1/W. Finally, for the case of 0.1<L1/W<0.2, the normalized energy release rate GE∗/σ2πa is determined by the combined effect of the aforementioned discussions. The exact size of the transition zone is dominated by the constraint of the remote load on the singular stress field around the crack tip.Figure 10
Normalized ERRGE∗/σ2πa and mode angle ψ of an asymmetrically bonded strip for (a) β=0.25α and (b) β=0.325α.
(a)
(b)Figure 11
Variations of (a) normalized ERRGE∗/σ2πa and (b) mode angle ψ of an asymmetrically bonded strip under β=0.325α.
(a)
(b)


### 4.5. Effect of the Height of the Bonded Components on ERR and Mode Mixity

The variations of the normalized ERRGE∗/σ2πa and mode angle ψ are plotted against logarithmic L1/W with varying α= −0.9, 0.45, 0, 0.45, and 0.9 in Figures 12(a) and 12(b), respectively. The lines of the symmetrical models are plotted in black and those of the asymmetrical models are in red. The effects of the height of the bonded components on ERR and mode mixity for each model are investigated. Then, a schematic of the simple elliptic rule fracture criterion shown in Figure 13 is employed to predict the possibility of crack generation for a demonstration. Furthermore, the elastic behaviors of the symmetrical and asymmetrical models are compared and discussed.Figure 12
Variations of (a) normalized ERRGE∗/σ2πa and (b) mode angle ψ against logarithmic L1/W.
(a)
(b)Figure 13
Variation ofGCψ with ψ for the typically elliptic rule fracture criterion.Firstly, let us consider the symmetrically bonded strip shown in Figure4(a). Figure 12(a) shows that GE∗/σ2πa decreases monotonously and sharply with increasing height L1 for fixed α when L1/W≤0.1, and it then gradually increases and reaches a constant when L1≥W. Furthermore, as demonstrated in Figure 12(b), the mode angles ψ are opposite in sign and equal in magnitude for the symmetric model, and the absolute values absψ decrease with decreasing L1 from a constant value at L1≥W.Secondly, considering the other case of the asymmetrical model shown in Figure4(b), GE∗/σ2πa increases with decreasing L1 for fixed α, and ψ exhibits a simple decreasing tendency with decreasing L1; Figure 12(b) also demonstrates that the absolute value absψ does not exhibit a uniform varying relationship for all the material combinations. Specifically, when α≥0, the mode angle ψ≤0 and its magnitude increases with decreasing L1; however, when α<0, in the process of L1 getting smaller, the sign of ψ changes from positive to negative and the magnitude of ψ increases after an initial decrease.Finally, the elastic behaviors of the symmetrical and asymmetrical models are compared.GE∗/σ2πa of the symmetrical model is always bigger than that of the asymmetrical model for fixing L1, α. However, the interface toughness (say critical ERR or critical SIFs) depends strongly on the mode mixity ψ. The variation of the critical ERR GCψ with ψ for the typically elliptic rule fracture criterion is demonstrated in Figure 13. It should be noted that the asymmetry in the interface toughness as well as the minimum toughness deviated from the origin (ψ≠0) may exist for specific material combinations. Here, we assume that small magnitude in ψ means smaller interface toughness for this specific rough criterion. Therefore, the elastic behaviors for different geometric configurations should be compared together with the mode mixity. When material 1 becomes stiffer α≥0, GE∗/σ2πa of the symmetrical model gets higher, and its mode angle ψ becomes smaller which means smaller interface toughness. Therefore, the symmetrical model faces a higher risk of crack initiation compared with the asymmetrical model. On the other hand, when material 1 is more compliant α<0, the magnitude of ψ of the symmetrical model increases after an initial decrease with decreasing L1. Therefore, in the first stage of decreasing ψ, the symmetrical model is also more dangerous than the asymmetrical one. However, in the next stage of increasing ψ, the elastic behavior is dominated by the combined effect of material combination and geometrical configuration. The relationship between the two models should be verified through a detailed comparison of the specific values between the ERR and the critical interface toughness.


## 4.1. Geometric Configurations and Material Elastic Parameters

The geometric configurations of the edge-cracked bimaterial strips are shown in Figure4. Two elastic, isotropic, and homogenous strips are perfectly bonded along the interface with material 1 above and material 2 below. Let a denote the interface crack length, W denote the width of the bonded strip, and L1 and L2 denote the height of each material, respectively. There are two geometric configurations considered in this research. Namely, they are the symmetrically and asymmetrically bonded strip shown in Figures 4(a) and 4(b), respectively. The heights of the two bonded components 1 and 2 in Figure 4(a) are kept the same and are variable L1=L2=0.333W~2W. However, in Figure 4(b), only the height of the upper component 1 varies in the range L1=0.333W~2W, leaving the lower component 2 fixed to L2=2W. Furthermore, in this research, the relative crack length is fixed to a constant of a/W=0.1 to make the effect of geometric configurations obvious. Then, the SIFs, ERRs, and mode angles are computed for the two different geometric configurations with varying height of the bonded materials.Figure 4
Geometric configurations for (a) longitudinal symmetrically bonded strip and (b) longitudinal asymmetrically bonded strip.
(a)
(b)The singular stress fields around the crack tip for the aforementioned problems with a given geometric configuration are only determined on the two elastic mismatch parametersα,β [24]. The physically admissible values are allowed to lie within a parallelogram dominated by α=±1 and β=α/4±0.25. The representative material combinations for the typical engineering materials were firstly compiled by Suga et al. [23]. They pointed out that the α,β values for typical material combinations are concentrated along β=0 and β=α/4 lines in α-β space. Here, we plot the material combinations using the data compiled by Suga et al. [23] together with those provided by others in Figure 5. It can be noted that most of the combinations fall within a narrow band in β=α/2±0.1, and the α,β values of typical material combinations are distributed in the vicinity of β=0.325α and β=0.25α for plane stress and plane strain conditions, respectively. Therefore, in this research, the SIFs, ERRs, and mode angles are systematically computed for the whole physical admissible α,β as well as the typical engineering material combinations β=0.325α and β=0.25α.Figure 5
Dundurs parameters for typical engineering materials [23].


## 4.2. Solutions of the Bimaterial Bonded Long Strip

As a reference, the bimaterial bonded long strip subjected to tension shown in Figure4(a) is firstly investigated. The heights of the two components are long enough (L/W≥2) to make sure that the crack tip fields are not affected by the length of the strip. The SIFs and ERRs are systematically investigated for the whole range of material combinations α∈-1,1, β∈-0.5,0.5. The 3D SIF distributions as well as the contour plots are shown in Figure 6. It should be noted that all the SIFs are normalized using σπa in this graph. As can be seen from Figure 6(a), the maximum value of KI is located in the lower left and upper right corner of the α-β space, and the minimum KI is located at the other two corners α=±1,β=0 of the parallelogram. In the good pair region, KI decreases from 1.189 of the homogeneous crack α=0,β=0 to the diagonal lower left and upper right endpoints α=±1,β=±0.5 in the α-β space; conversely, it presents an increasing tendency from the origin to the other pair of diagonal corners α=±1,β=0 in the bad pair zone. In Figure 6(b), the distributions of KII/σπa behave almost like a plane, and the values increase from the left to the right in the contour plot monotonously. Furthermore, switching materials 1 and 2 upside down which contributes to reversed α,β merely changes the signs of KII/σπa, and the minimum absolute value 0 of KII/σπa locates on the line β=2.8627α in the whole material space.Figure 6
3D normalized SIFs and contour plots of (a)KI/σπa and (b) KII/σπa.
(a)
(b)The energy methods are widely employed in the prediction of the onset and propagation of the interface cracks or delaminations. Therefore, the strain energy release rates and their corresponding mode angles are also investigated and plotted in Figures7(a) and 7(b), respectively. As can be seen from this graph, the normalized ERR GE∗/σ2πa and mode angle ψ exhibit a similar varying tendency to those of the normalized SIFs KI/σπa and KII/σπa. This is due to the fact that KII is relatively smaller than KI in the whole α-β space for the remote tensile loading case. It is known that the interface toughness depends strongly on the mode mixity; thus, in order to thoroughly and clearly investigate the effect of geometric configuration on the mode mixity, only the normalized ERR GE∗/σ2πa and mode angle ψ will be demonstrated in the following sections due to the similar varying tendency between KI/σπa, KII/σπa, and GE∗/σ2πa, ψ. Without special notifications, one can get similar conclusions for the case of KI/σπa, KII/σπa.Figure 7
3D distributions and contour plots of (a) normalized ERRGE∗/σ2πa and (b) mode angle ψ.
(a)
(b)


## 4.3. Solutions of the Symmetrically Bonded Strip

The SIFs of the symmetrically bonded strip shown in Figure4(a) for typical engineering materials are computed by varying the height of the two materials L1=L2=0.333~2W. It is seen that GE∗/σ2πa and ψ exhibit similarly varying tendency for β=0.25α and β=0.325α in Figures 8(a) and 8(b), respectively. Furthermore, the effect of the height of the bonded component is divided into three different zones: L1/W≤0.1, 0.1<L1/W<0.2, and L1/W≥0.2. The dependence of ERR and mode angle on Dundurs parameter α exhibits different varying tendency for L1/W≤0.1 and L1/W>0.2, and the zone of 0.1<L1/W≤0.2 can be denoted as the transition region which connects the two different variations smoothly. In order to observe the effect of the height of the bonded component, GE∗/σ2πa and ψ are plotted against α for various L1/W in Figures 9(a) and 9(b), respectively. The values of GE∗/σ2πa for L1/W≤0.1 and β=0.325α are plotted in red in the right ordinates in Figure 9(a) to make the variations obvious. Figure 9(a) shows that the maximum and minimum values of GE∗/σ2πa for L1/W≤0.1 are located at α=0 and α→±1, respectively. However, opposite trends are observed when L1/W>0.2. Furthermore, Figure 9(b) shows that the maximum and minimum absψ are always located at α→±1 and α=0, respectively, for a given height of L1/W>0.2. This means that the severe mismatch in the plane tensile modulus across the interface would lead to a bigger mode angle in this case, and ψ does not change monotonically with α for the other two zones of L1/W≤0.1 and 0.1<L1/W<0.2.Figure 8
Normalized ERRGE∗/σ2πa and mode angle ψ of a symmetrically bonded strip for (a) β=0.25α and (b) β=0.325α.
(a)
(b)Figure 9
Variations of (a) normalized ERRGE∗/σ2πa and (b) mode angle ψ of a symmetrically bonded strip for β=0.325α.
(a)
(b)


## 4.4. Solutions of the Asymmetrically Bonded Strip

The normalized ERRGE∗/σ2πa and its corresponding mode angle ψ of the edge-cracked asymmetrically bonded strip shown in Figure 4(b) under the material combinations of β=0.25α and β=0.325α are computed by varying the height of the two materials L1=0.333~2W. As demonstrated in Figure 10, a similar varying tendency for β=0.25α and β=0.325α also exists in the asymmetrically bonded strip. Similarly, GE∗/σ2πa and ψ for some typical L1/W are plotted against α in Figures 11(a) and 11(b), respectively. It should be noted that GE∗/σ2πa of L1/W≤0.1 and β=0.325α are plotted in red in the right ordinates of Figure 11(a) to make the variations obvious. Three different zones, that is, L1/W≤0.1, 0.1<L1/W<0.2, and L1/W≥0.2, are also employed in the discussion. Firstly, considering the case where L1/W≤0.1, when material 1 in Figure 4(b) is extremely stiff compared to material 2, α approaches 1, GE∗/σ2πa in Figure 11(a) reaches its minimum value, and abs(ψ) in Figure 11(b) comes closer to its maximum one. However, when material 1 is extremely compliant compared to material 2, α approaches -1 and GE∗/σ2πa and ψ reach their maximum and minimum values, respectively. Furthermore, the maximum absψ is always located around α→-1. Secondly, considering the case where L1/W≥0.2, Figure 11(a) shows that the minimum GE∗/σ2πa is located at α∈0,1, and increasing L1/W moves the minimum GE∗/σ2πa towards α=0. Furthermore, when material 1 (lower height) is extremely compliant compared to material 2 (α→-1), GE∗/σ2πa reaches its maximum value for a specific L1/W. Finally, for the case of 0.1<L1/W<0.2, the normalized energy release rate GE∗/σ2πa is determined by the combined effect of the aforementioned discussions. The exact size of the transition zone is dominated by the constraint of the remote load on the singular stress field around the crack tip.Figure 10
Normalized ERRGE∗/σ2πa and mode angle ψ of an asymmetrically bonded strip for (a) β=0.25α and (b) β=0.325α.
(a)
(b)Figure 11
Variations of (a) normalized ERRGE∗/σ2πa and (b) mode angle ψ of an asymmetrically bonded strip under β=0.325α.
(a)
(b)


## 4.5. Effect of the Height of the Bonded Components on ERR and Mode Mixity

The variations of the normalized ERRGE∗/σ2πa and mode angle ψ are plotted against logarithmic L1/W with varying α= −0.9, 0.45, 0, 0.45, and 0.9 in Figures 12(a) and 12(b), respectively. The lines of the symmetrical models are plotted in black and those of the asymmetrical models are in red. The effects of the height of the bonded components on ERR and mode mixity for each model are investigated. Then, a schematic of the simple elliptic rule fracture criterion shown in Figure 13 is employed to predict the possibility of crack generation for a demonstration. Furthermore, the elastic behaviors of the symmetrical and asymmetrical models are compared and discussed.Figure 12
Variations of (a) normalized ERRGE∗/σ2πa and (b) mode angle ψ against logarithmic L1/W.
(a)
(b)Figure 13
Variation ofGCψ with ψ for the typically elliptic rule fracture criterion.Firstly, let us consider the symmetrically bonded strip shown in Figure4(a). Figure 12(a) shows that GE∗/σ2πa decreases monotonously and sharply with increasing height L1 for fixed α when L1/W≤0.1, and it then gradually increases and reaches a constant when L1≥W. Furthermore, as demonstrated in Figure 12(b), the mode angles ψ are opposite in sign and equal in magnitude for the symmetric model, and the absolute values absψ decrease with decreasing L1 from a constant value at L1≥W.Secondly, considering the other case of the asymmetrical model shown in Figure4(b), GE∗/σ2πa increases with decreasing L1 for fixed α, and ψ exhibits a simple decreasing tendency with decreasing L1; Figure 12(b) also demonstrates that the absolute value absψ does not exhibit a uniform varying relationship for all the material combinations. Specifically, when α≥0, the mode angle ψ≤0 and its magnitude increases with decreasing L1; however, when α<0, in the process of L1 getting smaller, the sign of ψ changes from positive to negative and the magnitude of ψ increases after an initial decrease.Finally, the elastic behaviors of the symmetrical and asymmetrical models are compared.GE∗/σ2πa of the symmetrical model is always bigger than that of the asymmetrical model for fixing L1, α. However, the interface toughness (say critical ERR or critical SIFs) depends strongly on the mode mixity ψ. The variation of the critical ERR GCψ with ψ for the typically elliptic rule fracture criterion is demonstrated in Figure 13. It should be noted that the asymmetry in the interface toughness as well as the minimum toughness deviated from the origin (ψ≠0) may exist for specific material combinations. Here, we assume that small magnitude in ψ means smaller interface toughness for this specific rough criterion. Therefore, the elastic behaviors for different geometric configurations should be compared together with the mode mixity. When material 1 becomes stiffer α≥0, GE∗/σ2πa of the symmetrical model gets higher, and its mode angle ψ becomes smaller which means smaller interface toughness. Therefore, the symmetrical model faces a higher risk of crack initiation compared with the asymmetrical model. On the other hand, when material 1 is more compliant α<0, the magnitude of ψ of the symmetrical model increases after an initial decrease with decreasing L1. Therefore, in the first stage of decreasing ψ, the symmetrical model is also more dangerous than the asymmetrical one. However, in the next stage of increasing ψ, the elastic behavior is dominated by the combined effect of material combination and geometrical configuration. The relationship between the two models should be verified through a detailed comparison of the specific values between the ERR and the critical interface toughness.


## 5. Conclusions

In this research, the elastic behavior of an edge-cracked dissimilar bonded strip subjected to remote tensile load is investigated with varying different geometric configurations for arbitrary material combinations. The stress intensity factors, energy release rates, and mode angles are computed and compared systematically. It was found that the dependence of SIF, ERR, and mode angle on material combination should be discussed in three different zones according to the height of the bonded component. Furthermore, the combined effects of material combination and the geometric configurations on the elastic behavior of the bimaterial bonded strip are also investigated. Then, the combined effects of the relative height of each bonded component and material combination are discussed for the typical engineering materials. This may help the guidance for material selection and structural design for the modern layered materials.



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*Source: 2380740-2017-03-26.xml*