Influence of Reinforcement Length on Singularity of Single-Lap Joints

In order to enhance the strength of single-lap joints, the single-lap joints with reinforcements were proposed. *e influence of reinforcement length on the singular behavior near to the interface point of single-lap joints was investigated theoretically and numerically. *e theoretical strength of singularity point was calculated by Bogy determinant. Stresses along the interface close to the singularity points were calculated with finite element analyses (FEAs). Results showed that the singular stress intensity factor of single-lap joints can be decreased by the reinforcement. However, the singular stress intensity factor of single-lap joints with reinforcements was decreased slightly with increasing reinforcement length.


Introduction
Recently, adhesive bonding is widely used in light-weight design such as aerospace industry and automobile industry [1].Adhesive bonding is one of these new methods that are becoming common, especially in situations where the joints are between materials that are dissimilar, difficult to weld, or have been precoated [2][3][4].However, the debonding easily happens at the interface point of the adhesive bonding because of the singular behavior, and it is useful for assessing the strength of adhesive bonding through analyzing singular behavior.
e failure loads of adhesive joints with the variation of bondline thickness were predicted by the critical stress intensity in Van Tooren et al.'s stress singularity model [5].
ey found good correlation between the failure loads versus bondline thickness and the experimental results.
In Breto et al.'s study [6], two independent methodologies were proposed for selecting the intermediate material between adhesive bands in mixed adhesive joints attending to the singularity impact.
e first one is based on an analytical approach for calculating the root locus varying Young's modulus, and the second one is based on the influence of distance (instead of the order) on the singular term.e same result obtained from two different methodologies was summarized, and the feasibility of different methodologies was also discussed.e influence of bondline thickness on the static strength and fatigue life of single-lap joints was reported in Tang et al.'s paper [7] using generalized stress singularity approach.e test results were analyzed and compared to the simulation results.Results showed that the static strength decreased with increasing bondline thickness.
e failure initiated at the interface of thick composite laminate singlelap joints with the thick bondline.
e generalized stress singularity approach provided a viable method for predicting the failure initiation of thick composite laminate single-lap joints.
e effect of different material and geometric parameters (the ratio of the adherend lengths, the ratio of the adherend thickness, and Young's modulus ratio between different adherends) on stress state near the interface point of the single-lap joints using the finite element analysis (FEA) and experimental measurements were studied in Sawa et al.'s paper [8].It was observed that the singular behavior occurs near the interface point, and the stress increased with decreasing Young's modulus of adherend and increased with decreasing thickness.
e effect of various geometric parameters (adhesive thickness, bond length, and angle and shape of the adhesive edge) on the singular behavior of single-lap joints for a total of 30 cases was reported in Goglio et al.'s paper [9].Results showed that the less influencing parameter is the edge shape (straight edge and fillet edge) and the most influencing parameter is edge angle.
e singular stress field increases or reduces uniformly with increasing bond length and amplifies evidently with the bondline increasing in case of square edge.
Zhang et al. [10] investigated the variation of stress intensity factors with different material combinations.Dundurs' parameters (α and β) were used for calculating the stress intensity factors with different cracks (central internal interface cracks, periodic interface cracks, and edge interface cracks) by the FEA.When Dundurs' parameters α and β were defined value 0.2 and 0.3, respectively, the stress intensity factors reached maximum value.When Dundurs' parameters α and β were defined value 1 and 0, respectively, the stress intensity factors reached minimum value.
Noda et al. [11] explored the debonding strength of adhesive joints with and without fictitious crack using the singular stress intensity at the interface point.e critical stress intensity at the interface point can be used to evaluate the strength of adhesive joints.
is approach provided a convenient method for predicting the debonding strength of adhesive joints with and without fictitious crack.
e influence of bondline thickness on singular behavior of single-lap joints was discussed in Gleich et al.'s study [12].In order to enhance the singularity accurately, considerable mesh refinement was necessary in the region around the singularity point.e size of the smallest element was only 0.00005 mm around the singularity point.ey found that the linear region identifies the singularity field appeared in 0.00001 mm < r < 0.001 mm. e stress intensity factor of the singularity point reached maximum value when the bondline thickness was defined 0.02 mm.
Zhou et al. [13] explored the influence of singular behavior on the stress distribution at the crack tip of rectangular bond specimen by three-dimensional finite element models.e influence of singular behavior increased with the decreasing distance from the interface point.
Kilic et al. [14] discussed the influence of the adhesive overflow on the singular stress state of single-lap joints by a global-local finite element technique.
is method has obvious advantages in description of the stress state in the critical regions of single-lap joints without a fine mesh.e results showed that the adhesive overflow can improve the adhesive strength.e energy release rate and the strain energy density criterion may be beneficial in identifying adhesive strength and failure sites.
In many cases, the singularity parameters (stress intensity factor and singularity order) cannot be calculated strictly.e apparent singularity parameters were used for calculating the endurance limits of adhesive joints including and excluding the crack in Imanaka et al.'s study [15].e fatigue strength of adhesive joints decreased with decreasing lap length.However, the fatigue strength of adhesive joints excluding the crack decreased sharply.
ere are many articles focusing the influence of various geometric parameters on singular behavior at the interface point of single-lap joints.In the authors' previous study [16], the effects of stiffness at the end of the overlap on the strength of single-lap joints were studied and reinforcement of the single-lap joint was proposed.
e effects of reinforcements with different lengths on the singularity at the interface point of single-lap joints were investigated through Bogy determinant and FEA in this paper.

Determination of the Singularity Parameters around the Singularity Point
Bogy and Wang [17,18] derived characteristic equations to determine the singularity order λ. e strength of the singularity points of single-lap joint can be obtained by the characteristic equations.
For the wedge of Figure 1(b), the transcendental equation that determines the singularity strength taken from Bogy is as follows: where p is the characteristic value that satisfies the condition D � 0, c 1 and c 2 are the angles of the free edges of the two materials, and α and β are the Dundurs constants.
in which the auxiliary function ϕ(p, x) is defined by where x is in radians.
For plane strain, 2

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For plane stress, e singularity order is de ned by λ 1 − p, and λ depends only on the corner geometry (de ned by c 1 and c 2 ) and elastic constants of the materials around the singularity.e range of values lies within 0 < λ < 1. λ is not generally equal to 0.5 as it is for a sharp crack in isotropic materials.E ectively, λ describes the shape of the stress eld around the singularity and H describes the magnitude of the stress eld.
e solution for the generic stress component σ ij (with ij r, θ) is written in the following form [11]:

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where H k is the SIF corresponding to the kth root, f ijk (λ, θ) is the angular function, and σ ij (θ) are thermal stresses.More typically, this equation is simpli ed to one term with thermal stresses ignored: For the peel stresses and the shear stresses, (8) can be written as σ θθ r, θ 0 Hr −λ f θθ θ 0 , where H σ Hf θθ , ( τ rθ r, θ 0 Hr −λ f rθ θ 0 , where H τ Hf rθ .(10) By using the peel and shear stresses date around the singularity, such as that obtained by nite element analysis, two expressions can be derived to determine both λ and H. Taking lg of ( 9) and (10) yields the following equations: Equation ( 11) implies that a plot of lg σ FE (y-axis) versus lg(r) (x-axis) will yield a linear graph with slope λ. e stress intensity, H σ , can be calculated from the intercept on the y-axis.e same can be done for H τ using (12).

Finite Element Analysis and Results
A linear 2D FEA was performed on a single-lap joint by the software ANSYS (ANSYS Inc, Pittsburgh, United States) with material parameters de ned in Table 1.e substrates and reinforcements are de ned as aluminium alloy.e adhesive is de ned as structural acrylate adhesive [16].Boundary conditions and con guration are illustrated in Figure 2. e numerical models were adopted with 4-node plane strain elements with 20 MPa stress.Although two singularities exist at each end of the overlap points A, B, C, and D of lap joints, only the strongest singularity, which is of most interest, was investigated [5].Although the reinforcement adds new corner points, the points A and D are still the strongest singularity points.e failure initiated at the strongest singularity points [7].Points A and D are both the strongest singularity points and the symmetry points.So point A was selected for singularity analysis.To enhance the singularity accurately, considerable mesh re nement was necessary in the region around the point A. e size of the smallest element was only 1.0 × 10 −6 mm. e following nomenclature was used in this study: RL 0 joints (single-lap joints without reinforcements), RL 1 joints (R 30 mm), RL 2 joints (R 40 mm), RL 3 joints (R 50 mm), RL 4 joints (R 60 mm), RL 5 joints (R 70 mm), and RL 6 joints (R 80 mm).
Stresses along the interface (line AB in Figure 2) close to the singularity point A were investigated for the range of reinforcement length given in Table 1.In this way, a stress eld equation can be derived based on the FEA results for each reinforcement length.Figures 3 and 4 plot the peel and shear stresses date, s-stress, and r-distance, respectively, close to the point A along the interface AB.It can be seen that there is a linear region in the range 5 < lg(r) < 3 or 0.00001 mm < r < 0.001 mm and nonlinear region where  Advances in Materials Science and Engineering lg(r) > −3. e linear region identi es the singularity eld and the nonlinear region identi es the region outside the singularity eld where (7) is no longer valid [5].From these plots, the parameters λ, H σ , and H τ can be calculated by taking the linear region of the graph where the singularity eld exits as shown in Figure 5. e values of the strength of the singularity, λ, compare well with the analytical prediction of 0.3222 given in Figure 6 using (1).e values of singularity parameters are shown in Table 2.
Figures 7 and 8 show that the stress intensity factor of RL 0 joints (H σ 3.3984, H τ 1.2145) is greater than singlelap joints with reinforcements and that the stress intensity factor of RL 6 joints (H σ 2.5213, H τ 0.8905) is smaller than other single-lap joints with reinforcements.From the singularity analysis, the stress intensity factor of lap joints with reinforcements decreased with increasing reinforcement length.But the in uence of reinforcement length on stress intensity factor and joints strength is limited.e stress intensity factor is found to decrease slightly with increasing reinforcement length.

Conclusions
e reinforcements are often used to improve the strength of adhesive bonding and repair the surface of the damaged structures. is paper has investigated the in uence of the reinforcement length on the singular behavior of single-lap joints.e following results were obtained: (1) e stress intensity factor of single-lap joints was in uenced by the reinforcements.e stress intensity factor of single-lap joints can be decreased by the reinforcement.Advances in Materials Science and Engineering (2) e stress intensity factor of single-lap joints with reinforcements was in uenced slightly by the reinforcement length.e stress intensity factor of single-lap joints with reinforcements decreased slightly with increasing reinforcement length.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.

Table 1 :
Material parameters and single-lap geometry.

Table 2 :
Stress intensity factor in single-lap joints.