Evaluation of debonding strength of single lap joint by the intensity of singular stress field

In this paper, the similarity of the singular stress field of the single lap joint (SLJ) is discussed to evaluate the debonding fracture by the intensity of the singular stress field (ISSF). The practical method is proposed for analyzing the ISSF for the SLJ. The analysis method focuses on the FEM stress at the interface end by applying the same mesh pattern to the unknown and reference models. It is found that the independent technique useful for the bonded plate and butt joint cannot be applied to the SLJ because the singular stress field of the SLJ consists of two singular stress terms. The FEM stress is divided to two FEM stresses by applying the unknown and reference models to different minimum element sizes. Then, the practicality of the present method is examined by applying to the previous tensile test results of the SLJ composed of the aluminum alloy and the epoxy resin. The ISSFs for the SLJ were calculated by changing the adhesive thickness t2 and the overlap length l2. In the case of the SLJ with 225 mm in total length and 7 mm in adherend thickness, it was found that the similar singular stress fields are formed in the range of 0.15 mm ≤ t2 ≤ 0.9mm and 15 mm ≤ l2 ≤ 50 mm. It is shown that the critical ISSFs at the fracture are constant in the range.


Introduction
The intensity of singular stress field (ISSF) is useful for evaluating the debonding strength [1][2][3][4]. Generally, the ISSF cannot be calculated directly by the finite element method (FEM) [5][6][7][8]. The authors proposed the method for calculating the ISSF easily and accurately by the FEM [3,4]. The method does not require the complex calculation and can be applied to various bonded structures [9][10][11][12]. In the previous studies, the butt joint was analyzed under all material combination by using the bonded plate as the reference solution [3,4]. The singular stress field of the butt joint is expressed with a singular stress term. On the other hand, for many material combinations, the singular stress field of the single lap joint (SLJ) consists of two singular stress terms and is not discussed sufficiently. The similarity of the singular stress field needs be discussed to evaluate the debonding strength by the ISSF [10,13]. The method for analyzing two ISSFs easily and conveniently is required.
In this paper, the practical method for calculating two ISSFs for SLJ from the stress at the interface end by FEM is proposed. When the FE analyses are performed on the reference and unknown models under the same mesh patter and the same material combination, the ratio of the FEM stresses at the interface end of the unknown model to that of the reference model  Figure 1. Bonded plate used as the reference model. corresponds to the ratio of the ISSF of the unknown model to that of the reference model. Since the singular stress field of the SLJ consists on two singular terms, the sum of two FEM stresses is output as the nodal solution. Therefore, the FEM stress is divided to two FEM stresses by applying the unknown and reference models to different minimum element sizes. Then, two ISSFs are calculated by the divided FEM stresses. Then, the present method is applied to the previous experimental results of the SLJ. The similarity of the singular stress field and the debonding fracture criterion are discussed.

Mesh-independent technique useful for evaluating the ISSF for butt joint
The authors proposed the method for calculating the ISSF for the butt joint ( Fig. 1) accurately by using the ISSF for the bonded plate (Fig. 2) as the reference solution [3,4]. The real singular stresses of the bonded plate and the butt joint, σ PLT ij and σ BJ ij , are given by the following equations, respectively.
Here, r is the distance on the interface from the corner edge, λ is the singular index, K PLT σ ij and K BJ σ ij are ISSFs for the bonded plate and the butt joint, respectively. When the FE analyses are performed on the bonded plate and the butt joint under the same mesh pattern and the same material combination, the ratio of the FEM stresses, σ BJ ij0,FEM /σ PLT ij0,FEM , corresponds to the ratio of the ISSFs, K BJ σ ij /K PLT σ ij , as follows [3,4].
The real singular stress of the SLJ is given by the following equation under many material combinations [10,13].
The K SLJ σ ij ,λ 2 /K SLJ σ ij ,λ 1 is necessary to discuss the similarity of the singular stress field. However, the K SLJ σ ij ,λ 2 /K SLJ * σ ij ,λ 2 cannot be calculated from the FEM stress ratio.
3. Mesh-independent technique useful for evaluating the ISSF for SLJ Figure 3 shows the schematic illustrations of the single lap joint models. The model (a) is subdivided by the minimum element size e min = e 0 . The FEM stress at the interface end and the ISSF are denoted with σ SLJ-a ij0,FEM = σ SLJ ij0,FEM and K SLJ-a σ ij ,λ k = K SLJ σ ij ,λ k , respectively. The model (b) is as large as the model (a) and subdivided by e min = ne 0 . The FEM stress at the interface end and the ISSF are denoted with σ SLJ-b ij0,FEM = σ SLJ ij0,FEM | e min =n e 0 and K SLJ-b σ ij ,λ k , respectively. The FEM stress of the model (a), σ SLJ ij0,FEM , is expressed as follows.

Division of the FEM stress
The σ SLJ ij0,FEM has to be divided into σ SLJ ij0,FEM,λ 1 and σ SLJ ij0,FEM,λ 2 in order to calculate the K SLJ σ ij ,λ k . Since the minimum element size of the model (b) is n times as large as that of the model (a), the FEM stress of the model (b), σ SLJ ij0,FEM e min =n e 0 , is also expressed as follows [14,15].   (6) and (7) are solved on the σ SLJ ij0,FEM,λ 1 and σ SLJ ij0,FEM,λ 2 , the following equations are obtained.

Mesh-independent technique
The ratio of the ISSFs can be obtained from the ratios of the FEM stresses divided by Eqs. (8) and (9) as follows.
As shown in Eq. (10), the ISSFs for the unknown model can be determined by those for the only one reference model. That is the utmost advantage obtained by dividing the FEM stresses.  Figure 5 shows the fracture load P af under (a) t 2 constant condition and (b) l 2 constant condition. The P af increases with increasing the l 2 as shown in Fig. 5(a). Then, the P af is almost independent of the t 2 under l 2 constant condition. Figure 6 shows the average shear stress at the fracture, τ c = P af /(l 2 W ), obtained from Fig. 5(a). When l 2 < 15 mm, the τ c becomes constant at about 28.7 MPa. When the overlap length is short, the cohesive fracture occurs and the τ c becomes constant. In this study, it is supposed that debonding fracture occurs when l 2 > 15 mm. Figure 4 shows the schematic illustration of the analysis model. Dundurs' parameters are α = −0.8699 and β = −0.06642 [10,13]. The SLJ has two different real singular indexes λ 1 = 0.6062 and λ 2 = 0.9989 at point O. In this analysis, all models were subdivided by the same mesh pattern (Fig. 7). The minimum element size e min is changed to confirme the mesh    Table 1 shows the FEM stresses of the models with (l 2 , t 2 ) = (25, 0.15), (50, 0.15) and (25, 0.90). The FEM stresses are quite different depending on the mesh size e min . Table 2 shows the K SLJ σ ij0 ,λ 1 /K SLJ * σ ij0 ,λ 1 and the K SLJ σ ij0 ,λ 2 /K SLJ * σ ij0 ,λ 2 obtained from the FEM stress in Table 1, where the specimen A25 model with (l 2 , t 2 ) = (25, 0.15) is used as the reference solution and * is added in the superscript. The K SLJ σ ij0 ,λ 1 /K SLJ * σ ij0 ,λ 1 by the present method is independent of the mesh size e min and has the same value as the K SLJ σ ij0 ,λ 1 /K SLJ * σ ij0 ,λ 1 by th RWCIM [10]. The K SLJ σ ij0 ,λ 2 /K SLJ * σ ij0 ,λ 2 are little different depending on the e min . That is because the |σ SLJ ij0,F EM,λ 2 | is much smaller than the |σ SLJ ij0,F EM,λ 1 |. Since the K SLJ σ x0 ,λ 2 /K SLJ * σ x0 ,λ 2 by the present method has the same value as the K SLJ σ ij0 ,λ 2 /K SLJ * σ ij0 ,λ 2 by th RWCIM, it is found that the FEM stress in the x direction on the material 1 is the most suitable for the present method in this material combination. Figure 8 shows the K SLJ σ ij0 ,λ 1 /K SLJ * σ ij0 ,λ 1 and the K SLJ σ ij0 ,λ 2 /K SLJ * σ ij0 ,λ 2 obtained by changing the l 2 and the t 2 variously. When 0.15 mm ≤ t 2 ≤ 0.9 mm and 15 mm ≤ l 2 ≤ 50 mm, the K SLJ σ ij0 ,λ 1 /K SLJ * σ ij0 ,λ 1 and the K SLJ σ ij0 ,λ 2 /K SLJ * σ ij0 ,λ 2 decrease linearly with increasing the l 2 . Figure 9 shows the C SLJ σ ij0 /C SLJ * σ ij0 obtained from the K SLJ σ ij0 ,λ 1 /K SLJ * σ ij0 ,λ 1 and the K SLJ σ ij0 ,λ 2 /K SLJ * σ ij0 ,λ 2 in Fig. 8 .0549] n = 3 is used in all analyses. The smallest element size of the coarse model, ne 0 , is three times as large as that of the fine model. Table 2. Mesh-independent ISSF ratio K SLJ σ ij ,λ 1 K SLJ * σ ij ,λ 1 and K SLJ σ ij ,λ 2 K SLJ * σ ij ,λ 2 obtained from the FEM stress in Table 1. When 0.15 mm ≤ t 2 ≤ 0.9 mm and 15 mm ≤ l 2 ≤ 50 mm, the C SLJ σ ij0 /C SLJ * σ ij0 is almost constant and varies from 0.9 to 1.1. It can be confirmed that the similar singular stress fields are formed in the range. Figure 10 shows the critical ISSFs at the fracture, K SLJ σc /K SLJ * σc , in the range of 0.15 mm ≤ t 2 ≤ 0.9 mm and 10 mm ≤ l 2 ≤ 50 mm. The solid line is the average K SLJ σc /K SLJ * σc . The K SLJ σc /K SLJ * σc values are constant within about 10 % error.

Conclusion
In this paper, the ISSFs for the SLJ were calculated by changing the adhesive thickness t 2 and the overlap length l 2 and the similarity of the singular stress field of the SLJ was discussed. Then, it was shown that the debonding strength can be expressed as the constant value of the ISSF. The following conclusion can be drawn.  . Relation between K SLJ σ ij ,λ 1 /K SLJ * σ ij ,λ 1 , K SLJ σ ij ,λ 2 /K SLJ * σ ij ,λ 2 and l 2 .  (i) The analysis method for calculating the ISSF is applied to the previous tensile test results of the SLJ composed of the aluminum alloy and the epoxy resin. It was found that the similar singular stress fields are formed in the range of 0.15 mm ≤ t 2 ≤ 0.9 mm and 15 mm ≤ l 2 ≤ 50 mm in the case of the SLJ with 225 mm in total length and 7 mm in adherend thickness. (ii) When the specimens are satisfied with 0.15 mm ≤ t 2 ≤ 0.9 mm and 15 mm ≤ l 2 ≤ 50 mm, the critical ISSFs at the fracture were constant within 10 % error. (iii) It was found that the FEM stress can be divided to two FEM stresses by applying the unknown and reference models to different minimum element sizes. Two ISSFs for the SLJ can be obtained by using the divided FEM stresses.