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

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.


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][8][9][10][11][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 - 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 2 Advances in Materials Science and Engineering 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 / = 2 to / = 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.

Analysis Method
2.1.Formulation for the Interface Crack Problems.Consider two isotropic elastic materials joined along the -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] Here,   and   denote the normal and shear stress components near the crack tip, respectively,  is the radial distance behind the crack tip,  is the half crack length, and  is the bielastic constant given by where   ( = 1, 2) and ]  ( = 1, 2) are the shear moduli and Poisson's ratios of either respective material.The associated relative crack opening displacement values   =   (,  = ) −   (,  = −) and ( = , ) for nodes ,   at a distance  behind the crack tip, shown in Figure 1, are given by [20] where  is an arbitrary reference length which scales with specimen size or crack length; for the definition of (1), we have  = 2 without loss of generality.Considering (/)  = cos( ln(/)) +  sin( ln(/)) and rearranging (3), then the stress intensity factor components  I ,  II can be separated as We can rewrite (4) and (5) as From ( 7) and (8), when , , and   /  are kept the same for two different interface cracks, then we get a relationship as Considering two interface crack Problems  and  (say problems in Figure 2), by satisfying the preconditions as shown in (11) and the strain energy release rate and mode angle for the crack advance in the interface are 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 Figure 2(a) is selected as the reference problem for generality.Its analytical solution was firstly derived by Rice and Sih [2] and takes the form where an asterisk ( * ) is employed to denote the SIFs for the reference problem. ∞  and  ∞  are the remote uniform tension and shear applied to the bonded half planes. is the half crack length of the center crack.Furthermore, the transversal tension values  ∞ 1 and  ∞ 2 in Figure 2(a) behave as 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  and the relative distance behind the crack tip / 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 Recall ( 12) and substitute   and   with  * , and  * , for Problem ; then, we have Rearranging (18) gives the solution of /: Using  and  in (19) as the boundary condition for Problem , then ( 12) is satisfied and eventually ( 13) is set up.Finally, the SIFs for the target unknown problem (Problem ) can be obtained using the proportional relationship as given in (20):

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 to 2 = 2 mm.A plate width of  = 1620 × 2 = 3240 mm and a length of  = 2 = 6480 mm are used to model the reference problem.The crack length for the given unknown Problem  is fixed to  = 1mm, and the width of the bonded strip varies in the range / = 0.1∼0.4.The length is set to 2 times the width of the bonded strip.Furthermore, the minimum finite element sizes  min = 3 6 are kept the same for the reference and the given unknown problems.The same elastic parameters  1 / 2 = 2, 4, 10, 100 and ] 1 = ] 2 = 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]   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 parameters (, ) [24].The physically admissible values are allowed to lie within a parallelogram dominated by  = ±1 and  = /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 (, ) values for typical material combinations are concentrated along  = 0 and  = /4 lines in - 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  = /2 ± 0.1, and the (, ) values of typical material combinations are distributed in the vicinity of  = 0.325 and  = 0.25 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 (, ) as well as the typical engineering material combinations  = 0.325 and  = 0.25.

Solutions of the Bimaterial Bonded Long Strip.
As a reference, the bimaterial bonded long strip subjected to tension shown in Figure 4(a) is firstly investigated.The heights of the two components are long enough (/ ≥ 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  ∈ (−1, 1),  ∈ (−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 √ in this graph.As can be seen from Figure 6(a), the maximum value of  I is located in the lower left and upper right corner of the - space, and the minimum  I is located at the other two corners ( = ±1,  = 0) of the parallelogram.In the good pair region,  I decreases from 1.189 of the homogeneous crack  ( = 0,  = 0) to the diagonal lower left and upper right endpoints ( = ±1,  = ±0.5) in the - space; conversely, it presents an increasing tendency from the origin to the other pair of diagonal corners ( = ±1,  = 0) in the bad pair zone.In Figure 6(b), the distributions of  II /√ 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 (, ) merely changes the signs of  II /√, and the minimum absolute value 0 of  II /√ locates on the line  = 2.8627 in the whole material space.
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 Figures 7(a) and 7(b), respectively.As can be seen from this graph, the normalized ERR  * /( 2 ) and mode angle  exhibit a similar varying tendency to those of the normalized SIFs  I /√ and  II /√.This is due to the fact that  II is relatively smaller than  I in the whole - 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  * /( 2 ) and mode angle  will be demonstrated in the following sections due to the similar varying tendency between  I /√,  II /√, and  * /( 2 ), .Without special notifications, one can get similar conclusions for the case of  I /√,  II /√.height of the two materials  1 =  2 = 0.333 ∼ 2.It is seen that  * /( 2 ) and  exhibit similarly varying tendency for  = 0.25 and  = 0.325 in Figures 8(a) and 8(b), respectively.Furthermore, the effect of the height of the bonded component is divided into three different zones:  1 / ≤ 0.1, 0.1 <  1 / < 0.2, and  1 / ≥ 0.2.The dependence of ERR and mode angle on Dundurs parameter  exhibits different varying tendency for  1 / ≤ 0.1 and  1 / > 0.2, and the zone of 0.1 <  1 / ≤ 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,  * /( 2 ) and  are plotted against  for various  1 / in Figures 9(a) and 9(b), respectively.The values of  * /( 2 ) for  1 / ≤ 0.1 and  = 0.325 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  * /( 2 ) for  1 / ≤ 0.1 are located at  = 0 and  → ±1, respectively.However, opposite trends are observed when  1 / > 0.2.Furthermore, Figure 9(b) shows that the maximum and minimum abs() are always located at  → ±1 and  = 0, respectively, for a given height of  1 / > 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  does not change monotonically with  for the other two zones of  1 / ≤ 0.1 and 0.1 <  1 / < 0.2.

Solutions of the Asymmetrically Bonded Strip.
The normalized ERR  * /( 2 ) and its corresponding mode angle  of the edge-cracked asymmetrically bonded strip shown in Figure 4(b) under the material combinations of  = 0.25 and  = 0.325 are computed by varying the height of the two materials  1 = 0.333 ∼ 2.As demonstrated in Figure 10, a similar varying tendency for  = 0.25 and  = 0.325 also exists in the asymmetrically bonded strip.Similarly,  * /( 2 ) and  for some typical  1 / are plotted against  in Figures 11(a) and 11(b), respectively.It should be noted that  * /( 2 ) of  1 / ≤ 0.1 and  = 0.325 are plotted in red in the right ordinates of Figure 11(a) to make the variations obvious.Three different zones, that is,  1 / ≤ 0.1, 0.1 <  1 / < 0.2, and  1 / ≥ 0.2, are also employed in the discussion.Firstly, considering the case where  1 / ≤ 0.1, when material 1 in Figure 4(b) is extremely stiff compared to material 2,  approaches 1,  * /( 2 ) in Figure 11(a) reaches its minimum value, and abs() in Figure 11(b) comes closer to its maximum one.However, when material 1 is extremely compliant compared to material 2,  approaches −1 and  * /( 2 ) and  reach their maximum and minimum values, respectively.Furthermore, the maximum abs() is always located around  → −1.Secondly, considering    the case where  1 / ≥ 0.2, Figure 11(a) shows that the minimum  * /( 2 ) is located at  ∈ (0, 1), and increasing  1 / moves the minimum  * /( 2 ) towards  = 0. Furthermore, when material 1 (lower height) is extremely compliant compared to material 2 ( → −1),  * /( 2 ) reaches its maximum value for a specific  1 /.Finally, for the case of 0.1 <  1 / < 0.2, the normalized energy release rate  * /( 2 ) 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.

Effect of the Height of the Bonded Components on
ERR and Mode Mixity.The variations of the normalized ERR  * /( 2 ) and mode angle  are plotted against logarithmic  1 / with varying  = −0.9,0.45, 0, 0.45, and 0.9 in Figures 12(a  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. Firstly, let us consider the symmetrically bonded strip shown in Figure 4(a).Figure 12(a) shows that  * /( 2 ) decreases monotonously and sharply with increasing height  1 for fixed  when  1 / ≤ 0.1, and it then gradually increases and reaches a constant when  1 ≥ .Furthermore, as demonstrated in Figure 12 tendency with decreasing  1 ; Figure 12(b) also demonstrates that the absolute value abs() does not exhibit a uniform varying relationship for all the material combinations.Specifically, when  ≥ 0, the mode angle  ≤ 0 and its magnitude increases with decreasing  1 ; however, when  < 0, in the process of  1 getting smaller, the sign of  changes from positive to negative and the magnitude of  increases after an initial decrease.Finally, the elastic behaviors of the symmetrical and asymmetrical models are compared. * /( 2 ) of the symmetrical model is always bigger than that of the asymmetrical model for fixing  1 , .However, the interface toughness (say critical ERR or critical SIFs) depends strongly on the mode mixity .The variation of the critical ERR   () with  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 ( ̸ = 0) may exist for specific material combinations.Here, we assume that small magnitude in  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  ≥ 0,  * /( 2 ) of the symmetrical model gets higher, and its mode angle  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  < 0, the magnitude of  of the symmetrical model increases after an initial decrease with decreasing  1 .Therefore, in the first stage of decreasing , the symmetrical model is also more dangerous than the asymmetrical one.However, in the next stage of increasing , 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.

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.

Figure 1 :
Figure 1: Stress distribution and relative crack displacement of an interface crack.

Figure 2 :
Figure 2: Geometric configuration for (a) the reference Problem  and (b) the given unknown Problem .
(a); the relative CODs   and   can be solved in an indirect manner using the principle of linear superposition.Let  * , and  * , denote the CODs of Problem  subjected to combined  and ;  =1 * ,1 and  =1 * ,1 denote those of Problem  subjected to pure unit tension  = 1; and  =1 * ,2 and  =1 * ,2 denote those of Problem  subjected to pure unit shear  = 1, respectively.Using the theory of linear superposition, then the relative Advances in Materials Science and Engineering CODs  * , and  * , of the reference problem (Problem ) take the following form :

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  denote the interface crack length,  denote the width of the bonded strip, and  1 and  2 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 Figures4(a) and 4(b), respectively.The heights of the two bonded components 1 and 2 in Figure4(a) are kept the same and are variable  1 =  2 = 0.333 ∼ 2.However, in Figure4(b), only the height of the upper component 1 varies in the range  1 = 0.333 ∼ 2, leaving the lower component 2 fixed to  2 = 2.Furthermore, in this research, the relative crack length is fixed to a constant of / = 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 3 :
Figure 3: Finite element idealization around the crack tip.
Symmetrically Bonded Strip.The SIFs of the symmetrically bonded strip shown in Figure 4(a) for typical engineering materials are computed by varying the a
) and 12(b), respectively.The lines of the symmetrical models are plotted in black and those
(13)12), then the stress intensity factors  I ,  II exhibit a proportional relationship with   ,   as depicted in(13).The relative crack opening displacement values   and   can be computed by FE analysis, assuming one of the two problems is analytically well solved in advance;  I and  II of Problem  are given in advance, and then the SIFs of Problem