Two-dimensional analytical stress distribution model for unbalanced FRP composite single-lap joints

Accepted Manuscript Two-dimensional analytical stress distribution model for unbalanced FRP composite single-lap joints Zhengwen Jiang, Shui Wan, Thomas Keller, Anastasios P. Vassilopoulos PII:

S0997-7538(17)30470-9

DOI:

10.1016/j.euromechsol.2017.07.011

Reference:

EJMSOL 3464

To appear in:

European Journal of Mechanics / A Solids

Received Date: 13 June 2017 Revised Date:

19 July 2017

Accepted Date: 26 July 2017

Please cite this article as: Jiang, Z., Wan, S., Keller, T., Vassilopoulos, A.P., Two-dimensional analytical stress distribution model for unbalanced FRP composite single-lap joints, European Journal of Mechanics / A Solids (2017), doi: 10.1016/j.euromechsol.2017.07.011. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Two-dimensional analytical stress distribution model for unbalanced FRP composite single-lap joints

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Zhengwen Jiang a, b, Shui Wan a*, Thomas Keller b, Anastasios P. Vassilopoulos b a School of Transportation, Southeast University, Nanjing 210096, China b Composite Construction Laboratory (CCLab), Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 16, Bâtiment BP, CH-1015 Lausanne, Switzerland

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Abstract: A two-dimensional analytical stress distribution model taking into account the effects of large deformation, bending-tension coupling and interfacial compliance on the quasi-static behavior of unbalanced FRP composite single-lap joints (SLJs) under tension is introduced in this paper. The model can accurately estimate the stress distributions in the adhesive layer of unbalanced composite SLJ. The analytical results are compared to numerical simulations from a nonlinear finite element model and results from existing models validating the accuracy of the introduced two-dimensional model. A parametric study investigating the influence of the tensile load, and the material properties on the stress distributions in the adhesive bond-line is performed with the newly introduced model. The results demonstrate that the geometrical nonlinearity of the SLJ configuration leads to the reduction of the peak values of normalized adhesive stress distributions, and the difference between two peak values of normalized adhesive stress distributions in the adhesive middle plane can be reduced by appropriate selection of the joint adherends.

1. Introduction

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Keywords: unbalanced composite single-lap joint; adhesively bonded interface; interfacial stress distributions; large deformation; bending-tension coupling.

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Adhesively-bonded FRP composite single-lap joints (SLJs) have been widely used in structural applications (including aeronautics, automobile and construction industries, etc.) as they offer the ability to join efficiently and simply dissimilar materials without affecting the geometry of the structure [1, 2]. In order to further understand the mechanical behavior of unbalanced FRP composite adhesively bonded SLJ, it is necessary to obtain the adhesive shear and peel stresses. The main challenge of the theoretical analyses of adhesively-bonded SLJs under tension is the eccentric loading path, causing large deformations, and therefore the relationship between the edge moment and the applied tensile load (i.e., the edge moment factor) is nonlinear [3]. As the applied tensile load increases, geometrical nonlinearity can pronouncedly affect the adhesive stress distributions [3-5]. When an unbalanced SLJ is composed of composite laminates with unsymmetrical stacking sequence, coupling of the bending and tension behavior (i.e., the coefficient of bending-tension coupling B11≠0) and the relatively lower transverse stiffness of composite laminates have a considerable effect on the adhesive stress distributions [5, 6]. Hence, the effects of lager deformation, bending-tension coupling and relatively lower transverse stiffness should be incorporated into theoretical models for the unbalanced composite SLJ. 1

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During the past decades, many theoretical efforts have been allocated for the derivation of the adhesive stress distributions in balanced isotropic SLJs [4, 7-10]. Nevertheless, there is only a very limited number of theoretical studies for balanced composite SLJs [5, 11, 12]. Yang and Pang provided a one-dimensional beam model for composite SLJs based on the first-order laminated anisotropic plate theory [11]. Zou et al [12] presented a one-dimensional beam model for balanced composite SLJs, while Luo and Tong [5] extended their own one-dimensional beam model of the balanced isotropic SLJ [4] to derive a model for the simulation of the behavior of balanced composite SLJs taking into account the effect of large deformation in the overlap region. Compared to the balanced SLJs, the theoretical analysis of unbalanced SLJs is more complicated due to dissimilar materials and/or geometrical properties between the upper and the lower adherends. Only a few models have been introduced for the analysis of unbalanced SLJs [13-16]. Mortensen and Thomsen [13] provided an approach for determining the adhesive stress distributions of unbalanced composite SLJs neglecting the variation of the adhesive stress distribution along the thickness direction and the effect of large deformation in the overlap region. Jiang et al [14] extended the model of Luo and Tong [4] to model unbalanced isotropic SLJs incorporating the effect of large deformation in the overlap region providing more accurate estimations of the edge moment factors and the interfacial stress distributions compared to those from other one-dimensional models, e.g [15, 16]. All of the above-mentioned solutions are one-dimensional beam models that neglect the stress-free condition in the free ends of the adhesive layer and the variation of the adhesive stress distribution along the thickness direction [17, 18]. In order to alleviate the disadvantages of the one-dimensional models, quasi-two dimensional and two-dimensional models, assuming adhesive stresses variation through the thickness direction were introduced for balanced [19-21] and unbalanced SLJ [22-24]. Quasi-two dimensional models considered only one of the adhesive stress components variating through the thickness of unbalanced SLJs, e.g. the models of Allman [22] and Zhao et al [23] assumed peel stress as a variant across the thickness direction. Based on the edge moment factors determined by the Cheng model [18], Jiang et al [24] introduced a two-dimensional model for unbalanced SLJ with incorporating the effect of interfacial compliance, however, still ignoring the effects of overlap large rotation and bending-tension coupling. Based on the above-mentioned literature review, it is concluded that no two-dimensional model taking into account the effects of bending-tension coupling and the large deformation in the overlap region, which can significantly affect the interfacial stress distribution as shown in [5, 25], was provided for SLJ in published literature, especially for unbalanced long FRP composite SLJ. A new, two-dimensional model for the analysis of the behavior of unbalanced composite SLJs taking into account the effects of bending-tension coupling, large deformation and interfacial compliance is introduced in this paper. The applicability of the new model is demonstrated by simulating the quasi-static tensile behavior of unbalanced composite SLJs with different relative adhesive/adherend stiffness ratios (including inflexible, intermediate flexibility and flexible adhesives [26]). The obtained results are compared with those from existing theoretical models and from nonlinear finite element analysis. The introduced model is also

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ACCEPTED MANUSCRIPT used for a parametric study, investigating the effects of tensile load and material properties of the adherends on the stress distributions along the adhesive bond-line. 2. Theoretical model

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The new two-dimensional model is based on the previously introduced one-dimensional and two-dimensional models from the authors [14, 24]. The boundary conditions at the ends of the overlap region are necessary requirements for obtaining the adhesive stress distributions with the two-dimensional model. In the published literature, no solutions were provided for such boundary conditions of unbalanced composite SLJ with simultaneously considering the effects of large deformation and bending-tension coupling. Therefore, the one-dimensional beam model [14] is modified herein in order to take into account the effect of the bending-tension coupling and be used for the estimation of the bending moment induced at the overlap edge and the transverse deformation distributions for the overlap region of the unbalanced SLJs with non-symmetric adherends. The governing differential equations for the upper and lower interfaces, considering the geometrical nonlinearity of the SLJ, are constructed based on the displacement compatibility condition of flexible interface. Through employing the coefficients of interfacial compliance, the effect of relatively low transverse stiffness is incorporated. Using the boundary conditions provided by the improved one-dimensional beam model and equations of equilibrium, the stress distributions for the adhesive layer can be obtained. The unbalanced SLJ composed of composite laminates with unsymmetrical stacking sequence was studied in the present paper. In order to analyze the unbalanced composite SLJ with width b, shown in Fig. 1, the joint is divided into four sections: the upper (section 1) and lower adherends (section 2) in the overlap region, and the upper (section 3) and lower adherends (section 4) in the outer region. The adhesive with thickness ta is used to bond the upper adherend with thickness t1 and the lower adherend with thickness t2 following the SLJ configuration. The length of the overlap region is L. The lengths of section 3 and section 4 are l1 and l2, respectively. The two ends of the adherends in the outer regions are simply supported and a tensile load P is applied.

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ACCEPTED MANUSCRIPT Fig.1 Schematic configuration for an unbalanced composite SLJ under tensile load: (a) geometrical parameters; (b) force equilibrium free-body diagram

2.1 Determination of boundary conditions

(i = 1, 2 )

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 dui d 2 wi N = A − B i i  i dx dx 2  2  M = B dui − D d wi i i  i dx dx 2

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The one-dimensional beam model [14] is used to determine the boundary conditions for the introduced two-dimensional model. In order to characterize the coupling bending and tension behavior induced by the non-symmetric stacking sequence of the composite adherends, the effect of bending-tension coupling, which was neglected in the previously presented one-dimensional model [14], is incorporated into the one-dimensional beam model by introducing the bending-tension coupling stiffness Bi (i=1, 2) in the definition of axial force Ni (i=1, 2) and bending moment Mi (i=1, 2):

(i = 1, 2 )

(1)

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where Ai and Di (i=1, 2) denote the axial extension- and bending stiffness of sections 1 and 2, respectively; while ui, wi (i=1, 2) are the axial displacements, transverse deformations for sections 1 and 2, respectively.

By implementing the solutions for the axial forces and bending moments of Eq. (1) into the analysis described in [14], the edge moments and the transverse deformation distributions

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along the overlap length can be estimated.

2.2 Adhesive stress distributions

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2.2.1 Equilibrium equations In [24], a two-dimensional model without considering the effects of large deformation in the overlap region and bending-tension coupling was provided for unbalanced SLJ. In order to provide more accurate predictions for the adhesive stress distributions, the effects of bending-tension coupling and large deformation are incorporated into the present improved two-dimensional model. The bending-tension coupling effect induced by non-symmetric adherends is considered by implementing Eq. (1). When the large rotation in the overlap region is taken into account, the associated stress equilibrium diagram is that shown in Fig. 2. Compared to the global equilibrium equations of [24], the nonlinear terms τi(x)dwi/dx, Nidwi/dx (i=1, 2, a) are added into the present global equilibrium equations:

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Fig. 2 Stress equilibrium of infinitesimal isolated body for the geometrically nonlinear analysis dN 1 dQ1 d w1 = b σ 1 ( x ) − bτ 1 ( x ) = bτ 1 ( x ) ; dx dx dx

dN 2 dQ2 dw2 = − bτ 2 ( x ) ; = − bσ 2 ( x ) + bτ 2 ( x ) dx dx dx dN a = b τ 2 ( x ) − τ 1 ( x )  dx

;

dM 1 bt dw 1 = Q1 − 1 τ 1 ( x ) + N 1 dx 2 dx

;

;

dM 2 bt dw2 = Q2 − 2 τ 2 (x ) + N 2 dx 2 dx

dQa dwa = b  σ 2 ( x ) − σ 1 ( x )  + b τ 1 ( x ) − τ 2 ( x )  dx dx

(2-b)

; (2-c)

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dM a bt dw = Qa − a τ 1 ( x ) + τ 2 ( x )  + N a a dx 2 dx

(2-a)

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where Qi (i= 1, 2, a) are the transverse shear forces for sections 1, 2, and adhesive layer, respectively; Na, Ma are the axial forces and bending moments for adhesive layer, respectively; τi(x), σi(x) (i=1, 2) are the adhesive shear stress, peel stress along the upper and lower interfaces, respectively; wa is the transverse deformation of the adhesive layer. By considering the effect of the relatively lower transverse stiffness of the composite laminates compared to their longitudinal stiffness, the displacement continuity conditions along the upper and the lower interfaces shown in Fig. 3 are obtained from [27, 28]: dw dw w1 − Cn1σ1 ( x ) + Cs1τ1 ( x ) 1 = wa + Cnaσ1 ( x ) − Csaτ1 ( x ) a (3-a) dx dx dw dw w2 + Cn 2σ 2 ( x ) − Cs 2τ 2 ( x ) 2 = wa − Cnaσ 2 ( x ) + Csaτ 2 ( x ) a (3-b) dx dx t dw t dw u1 + 1 1 − Cs1τ1 ( x ) = ua − a a + Csaτ1 ( x ) (3-c) 2 dx 2 dx t dw t dw u2 − 2 2 + Cs 2τ 2 ( x ) = ua + a a − Csaτ 2 ( x ) (3-d) 2 dx 2 dx where Cni, Csi (i=1, 2, a) are coefficients of interfacial compliance for upper, lower adherends

(

)

(i) and adhesive layer under interfacial peel and shear stress, respectively, and Cni = ti 10E33 ,

Csi = ti

(15G ) (i = 1, 2 ) (i )

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(

( a)

)

(

( a)

[27, 28]; Cna = ta 2E33 , Csa = ta 3G13

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)

[29].

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Fig. 3 Displacement along interface induced by highly concentrated interfacial stresses

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2.2.2 Stress distributions along adhesive interfaces Based on the sets of Eqs. (2)-(3), the fully coupled differential equations can be obtained as follows (the detailed derivation process is shown in appendix A): d 4 M1 d 4 N1 d 2 M1 d 2 N1 (4-a) + a11u + a12u + a13u + a14u M 1 + a15u N1 = a16u w1 + f u ( x ) 4 4 2 2 dx dx dx dx 4 2 2 d 4 M1 u d N1 u d M1 u d N1 + b + b + b + b14u M 1 + b15u N1 = b16u w1 + g u ( x ) (4-b) 11 12 13 dx 4 dx 4 dx 2 dx 2 where au1i (i= 1, 2, …,6), bu1i (i= 1, 2, …,6), f u(x) and gu(x) are shown in appendix A. N1 and M1 can be obtained as follows (the detailed derivation process is shown in appendix A):

N1 = c1ueR1 x + c2ue−R1 x +c3ueR2 x + c4ue− R2 x + eR3 x c5u cos ( R4u x ) + c6u sin ( R4u x ) u

u

u

u

u

+e−R3 x c7u cos ( R4u x ) + c8u sin ( R4u x ) + N1uc

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u

M1 = T1uc1ueR1 x + T2uc2ue−R1 x + T3uc3ueR2 x + T4uc4ue−R2 x + eR3 x T5uc5u cos ( R4u x ) + T6uc6u sin ( R4u x)  u

u

u

u

u

+e−R3 x T7uc7u cos ( R4u x ) + T8uc8u sin ( R4u x)  + M1uc

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(5-a)

u

(5-b)

Then, substituting Eqs. (5-a) and (5-b) into Eq. (2-a) yields:

Q1 = V1uc1ueR1 x + V2uc2ue−R1 x +V3uc3ueR2 x +V4uc4ue−R2 x + eR3 x V5uc5u cos ( R4u x ) +V6uc6u sin ( R4u x ) u

u

u

u

u

+e−R3 x V7uc7u cos ( R4u x ) +V8uc8u sin ( R4u x ) + Q1uc u

(5-c)

where cui (i= 1, 2,…., 9) are unknown coefficients; Nu1c and Rui (i= 1, 2,…., 9) are shown in appendix A; Mu1c and Qu1c are the functions of Nu1c; Tui (i= 1, 2,…, 8) and Vui (i= 1, 2,…, 8) are the functions of Rui (i= 1, 2,…., 9). Substituting Eq. (5-a) into Eq. (2-a) and substituting Eqs. (5-a) and (5-b) into Eqs. (2-a) and (2-b) yield, respectively: 6

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(

τ 1 = R1u c1u e R x − c2u e − R u 1

u 1x

) +R ( c e u 2

u R2u x 3

− c4u e − R2 x u

)

+ e R3 x  ( R3u c5u + R4u c6u ) cos ( R4u x ) + ( R3u c6u − c5u R4u ) sin ( R4u x )  u

+ e − R3 x  ( R4u c8u − R3u c7u ) cos ( R4u x ) − ( R3u c8u + R4u c7u ) sin ( R4u x ) + u

(6-a) d ( N1uc ) dx

σ 1 = W1u c1u e R x + W2u c2u e − R x + W3u c3u e R x + W4u c4u e − R u 1

u 1

u 2

u 2x

+ e R3 x W5u c5u cos ( R4u x ) + W6u c6u sin ( R4u x ) u

(6-b)

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+e

− R3u x

W7u c7u cos ( R4u x ) + W8u c8u sin ( R4u x )  + σ 1uc  

where σu1c is a function of Nu1c; Wui (i= 1, 2,…, 8) is a function of Rui (i= 1, 2,…., 9). cui (i= 1, 2,…., 9) can be determined by the following boundary conditions:  1  P ( t1 + t 2 + 2 t a ) − M 1* − M 2*  ， M 1 ( − L / 2 ) = − M 1*  2 L 

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N1 ( −L / 2) = P ， Q1 ( − L / 2 ) = − Ma ( −L / 2) = 0 , τ1 ( L / 2) = 0

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N1 ( L / 2) = 0 , Q1 ( L / 2 ) = 0 ， M1 ( L / 2) = 0

(7-a)

(7-b) (7-c)

Similar to the solution proposed for the upper interface, the adhesive stress distributions for the lower interface can be obtained, as shown in appendix A.

σ xxa =

Na 12Ma ρa + bta bta2

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2.2.3 Stress distributions through adhesive thickness According to the classical beam-plate theory, the longitudinal normal stress σaxx of the adhesive layer is assumed to vary linearly across its thickness direction. (8)

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where ρa=za/ta (-1/2≤ρa≤1/2). Based on the two-dimensional elasticity theory, the stress continuity in the upper and lower interfaces, and the adhesive stress distributions in the upper and lower interfaces (Eq. (6) and Eq. (A-15)), the adhesive stress distributions can be obtained as follows: (12 ρ a2 − 1) (τ 1 + τ 2 ) + ( 3 − 12 ρ a2 ) Qa + PAa ( 3 − 12 ρ a2 ) dwa τ xza = − ρ a (τ 2 − τ 1 ) + (9-a) 4 2 bta 2bta ( A1 + A2 + Aa ) dx σ zza =

ta ( 4 ρ a2 − 1)  dτ 2 dτ 1  ta ( 4 ρ a3 − ρ a )  dτ 1 dτ 2  ( 3ρ a − 4 ρ a3 ) (σ 2 − σ 1 ) (σ 1 + σ 2 ) − + +  −  − 8 dx  4 dx  2 2  dx  dx

(9-b)

The developed model has been implemented in a Matlab code allowing the derivation of the adhesive stress distributions for the unbalanced composite SLJ when their material and geometrical parameters are known. The Matlab code was used to estimate the adhesive stress distributions of the unbalanced composite SLJ in the next section for the model validation.

3. Model validation and discussion In this section, the new model is validated through comparisons with numerical results from non-linear finite element modeling. The comparison between the new model and the existing models was further used to demonstrate the advancements of the new model. Compared to

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one-dimensional models and quasi-two dimensional models for unbalanced SLJ, the Jiang et al model incorporating the effect of interfacial compliance was proved more accurate [24], and therefore was selected here for the comparisons. Different SLJs configurations have been selected for the comparison; their material and geometrical parameters are given in Table 1. The cases shown in Table 1 correspond to different joint configurations described in the following list: • Case 1: Unbalanced composite SLJ with relatively intermediate flexibility adhesive (i.e., (10ta/t1> Ea/ E1 and 10Ea/E1 >ta/t1) or (10ta/t2> Ea/ E2 and 10Ea/E2 >ta/t2) [26]); • Case 2: Unbalanced composite SLJ with relatively inflexible adhesive (i.e., 10ta/t1≤ Ea/ E1 or 10ta/t2≤ Ea/ E2 [26]); Case 3: Unbalanced composite SLJ with relatively flexible adhesive (i.e., 10Ea/E1 • ≤ta/t1 or 10Ea/E2 ≤ta/t2 [26]). According to [25] and [30], the effect of large rotation in the overlap region on the adhesive stress distributions cannot be neglected when (8P/D)0.5L/2≥1 (with D the flexural stiffness of the overlap region). The material and geometrical characteristics of each joint configuration were selected so that the parameter (8P/D)0.5L/2 for all cases was larger than 1. The comparison of the used models is performed on the basis of the comparisons of the adhesive stress distributions. The adhesive stresses shown in all figures were normalized by τ0=P/(bL). Table 1 Material and geometrical parameters for composite SLJ composite ply: EL=32.5GPa, ET=7.29GPa, µLT=0.258, GLT=6.7GPa material properties

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[±15°/±30°],

[±45°/±60°],

Ea=3.5GPa,

B1=3.65N·m

B2=2.55N·m

ν=0.4

[±15°/±30°],

[±45°/±60°],

Ea=7.0GPa,

B1=3.65N·m

B2=2.55N·m

ν=0.4

[±45°/±60°],

Ea=0.05GPa,

B2=2.55N·m

ν=0.4

[±15°/±30°], B1=3.65N·m

adhesive layer

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lower adherend

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upper adherend

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case

geometrical parameters

P (N)

(8P/D)0.5L/2

l1=l2=100mm, L=80mm, t1=t2=2mm, ta=0.2mm,

1000

10.338

1000

7.932

300

5.471

b=1mm l1=l2=30mm, L=60mm, t1=t2=2mm, ta=0.135mm, b=1mm l1=l2=20mm, L=80mm, t1=t2=2mm, ta=0.3mm, b=1mm

The geometrically nonlinear finite element model of the unbalanced composite SLJ was constructed in ANSYS (academic version 15.0). The 2D plane strain eight-node element, plane 82, with large deflection capability was utilized to mesh the composite adherends and the adhesive layer. There are five and four elements across the thickness direction of each ply and the adhesive layer, respectively. Due to the fact that the adhesive stress distributions near the two free ends are sensitive to the element length, convergence studies were performed with changing the element length (le) in the regions near the two free edges and the element length le was set to 0.02mm. In the geometrically nonlinear analysis, 50 equal load increments were chosen (based on a preliminary study, in order to guarantee the convergence of the non-linear analysis) and the Newton’s method was employed for the solution of the NFEM. 8

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3.1 Case 1 0

0

0

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-10

-5 -5 -10

-25

NFEM Jiang model [24] Present model

-30 -40 -36 -32 -28 28 x (mm)

32

-30

NFEM Jiang model [24] Present model

-15

36

40

-20 -40 -36 -32 -28 28 x (mm)

(a)

32

NFEM Jiang model [24] Present model

-40

36

(b)

40

-50 -40 -36 -32 -28 28 x (mm)

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-20

τ1 / τ0

τ1 / τ0

τm / τ0

-20 -10

-15

32

36

40

(c)

40 30

30 NFEM Jiang model [24] Present model

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Fig. 4 Shear stress distributions for Case 1: (a) upper interface (b) middle plane (c) lower interface 40

NFEM Jiang model [24] Present model

30

NFEM Jiang model [24] Present model

10

10

σ2 / τ0

σm / τ0

σ1 / τ0

20

20

10 0

0 -10 -40 -36 -32 -28 28 x (mm)

(a)

32

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40

-10

-10 -40 -36 -32 -28 28 x (mm)

(b)

32

36

40

-20 -40 -36 -32 -28 28 x (mm)

32

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40

(c)

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Fig. 5 Peel stress distributions for Case 1: (a) upper interface (b) middle plane (c) lower interface

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Figs. 4-5 show the adhesive stress distributions in the unbalanced composite SLJ of Case 1, as derived by using Jiang et al model [24], the present model, and NFEM. The x-axis in each graph is truncated in the middle, between -28mm＜x＜28mm, since stress values in this range are almost constant. The shear stresses in the upper interface, as determined by NFEM, show a high peak value at x=-40mm and then decreases significantly to approximate zero in less than 5 mm from the left overlap end (i.e., about 0.05L). The shear stress in the upper interface is zero at the right end, while it reaches its peak value near to this overlap end. It decreases rapidly to nearly zero near the end, in the length of about 0.05L. The shear stress in the middle plane shown in Fig. 4(b) satisfies the stress-free end condition at the two free ends of the overlap, and reaches the peak value near the two free ends of the overlap. Similar to the shear stress distribution in the upper interface, the shear stress distribution in the lower interface, shown in Fig. 4(c), presents peaks at the right overlap end and is almost zero in the middle. 9

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The shear stress distributions for Case 1 as estimated by the present new model agree remarkably well with those derived by the NFEM. The relative errors between the peak values of the normalized shear stresses in the middle plane of adhesive layer determined by the present new model and NFEM are less than 3%, while the estimated values from Jiang et al model [24], deviate significantly from both other results, especially near the free ends of the overlap. For the peak values of the normalized shear stresses in the middle plane, the maximum relative error between the results of Jiang et al model [24] and NFEM is 9.7%. The comparison of the normal (peel) stress distributions, is presented in Figs. 5a-c. The peel stress along the upper interface determined by NFEM obtains the maximum (positive) value at the left overlap end, while it decreases rapidly to a compressive stress and then increases to an approximate zero stress in the length of about 0.05L. At the other overlap end, the peel stress is negative, however it obtains a positive peak value very near the end, before reducing as approaching the middle of the overlap length. The peel stress in the lower interface shows the opposite behavior from that in the upper interface (see Fig. 5c). It obtains a negative peak value (compressive stress) and a peak value (tensile stress) at the left end, and the right end of the overlap, respectively. The peel stress in the middle plane shown in Fig. 5(b) reaches the peak value (tensile stress) at the two ends of the overlap and then decreases rapidly to a compressive stress (i.e., at about 0.03L). All models’ estimations are in agreement, except the peak values, where the new model provides closer results with NFEM. The maximum relative errors between the peak values of the normalized peel stresses in the middle plane determined by NFEM and the present model, Jiang et al model [24] are 9.2% and 54.80% respectively.

-10

NFEM Jiang model [24] Present model

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-20 -30 -27 -24 -21 21 x (mm)

24

40 30

τm / τ0

τm / τ0

-5

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0

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3.2 Case 2

NFEM Jiang model [24] Present model

20 10 0

27

-10 -30 -27 -24 -21 21 x (mm)

30

(a)

24

27

30

(b)

Fig. 6 Stress distributions for the adhesive mid-plane for Case 2: (a) shear stress (b) peel stress

Similar comments apply to the results for Case 2. The adhesive stress distributions of the unbalanced composite SLJ with inflexible adhesive have the same variation trend as the adhesive stress distributions shown in Figs. 4-5. Only the stress distributions in the middle plane of adhesive layer are shown in Fig. 6 for the purpose of concision. The shear stress distributions in the range of 21mm≤x≤29mm and the peak values of peel stress determined 10

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0

0

-1

-1

-1

τm / τ0

τ1 / τ0

NFEM Jiang model [24] Present model

-4 -40 -36 -32 -28 28 x (mm)

32

-3

36

40

-2

NFEM Jiang model [24] Present model

-4 -40 -36 -32 -28 28 x (mm)

(a)

32

SC

-2

-3

36

40

NFEM Jiang model [24] Present model

-4 -40 -36 -32 -28 28 x (mm)

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-2

-3

τ2 / τ0

0

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3.3 Case 3

(b)

32

36

40

(c)

Fig. 7 Shear stress distributions for Case 3: (a) upper interface (b) middle plane (c) lower interface

6

5

4

12

TE D

10

8

NFEM Jiang model [24] Present model

-5

σ2 / τ0

σm / τ0

σ1 / τ0

2

0

0

4

-2

NFEM Jiang model [24] Present model

-4

EP

-10

-15 -40 -36 -32 -28 28 x (mm)

32

40

-6 -40 -36 -32 -28 28 x (mm)

(b)

AC C

(a)

36

NFEM Jiang model [24] Present model 32

0

36

40

-4 -40 -36 -32 -28 28 x (mm)

32

36

40

(c)

Fig. 8 Peel stress distributions for Case 3: (a) upper interface (b) middle plane (c) lower interface

The shear and peel stress distributions in the adhesive for Case 3 are presented in Figs. 7-8. Compared to the two previous cases, both the shear and peel stress distributions, are more uniform, with less stress peaks near the ends of the overlap length, due to the relatively lower adhesive stiffness. As can be seen in Fig. 7, shear stress peaks less than 4τ0 have been estimated for all stress paths. In addition, the shear stress at the middle of the overlap length does not tend to zero but keeps a value, around 25-30% of the peak stress. Peel stress distributions for all stress paths presented in Fig. 8 show a trend similar to the previous two examined cases, although much smoother, with considerably less stress peaks near to the ends of the overlap length. 11

ACCEPTED MANUSCRIPT

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In this case, where a flexible adhesive is considered, the difference between the adhesive stress distributions determined by the present solution and NFEM results are larger than for the two previously examined cases. The reason for these deviations may be the non-linear terms ignored in Eq. (4), which have greater effects when the ratio between the adherend and the adhesive stiffness is larger than in Case 3. Nevertheless, the newly introduced model is more consistent with NFEM than the previous two-dimensional model that ignores the effects of the overlap large rotation and bending-tension coupling.

4. Parametric study

M AN U

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The comparisons presented in the previous section confirmed that the newly introduced two-dimensional model is capable of estimating accurately the stress distributions along the adhesive bond-line in unbalanced composite SLJs with relatively intermediate flexibility or inflexible adhesives. The validated model can be used for the investigation of the effects of the applied tensile load P and the material properties (i.e., EL, ET and GLT) on the adhesive stress distributions of unbalanced composite SLJs. It is convenient to obtain adhesive stress distributions of the unbalanced composite SLJ for every case by changing load and material parameters in the Matlab code. 4.1 Effect of load

AC C

EP

TE D

The stress distributions along the middle adhesive plane of the unbalanced composite SLJ under different tensile loads are shown in Fig. 9. The unbalanced composite SLJ of Case 1 is chosen for the demonstration. The peak values for the normalized shear stresses near the two overlap ends decrease sharply and then decrease slightly as the tensile load gradually increases. The similar variation trend can be observed in the peak values of the normalized peel stress distribution. The variation trend of peak values for the normalized adhesive stress distribution indicate the growth ratio of adhesive stress is less than that from increasing tensile load. This phenomenon is due to the eccentricity of the load path caused by the geometrical nonlinearity of the SLJ configuration, shown in Fig. 1. Specifically, when load increases, the rotation in the overlap region increases and the edge moment factor reduces, as is also reported in [14]. Due to these effects, the peak adhesive stress values would decrease.

50

0

40

-5

30

τm / τ0

σm / τ0

-10 -15 -20

P=0.093N P=149.7N P=458.5N P=1000N

20

P=0.093N P=149.7N P=458.5N P=1000N

-25 -40 -36 -32 -28 28 x (mm)

32

10 0 36

-10 -40 -36 -32 -28 28 x (mm)

40

12

32

36

40

ACCEPTED MANUSCRIPT (a)

(b)

Fig. 9 Stress distributions for different load levels along the adhesive mid-plane (a) shear stress (b) peel stress

4.2 Effect of material property

TE D

M AN U

SC

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The effects of elastic modulus ratio and shear modulus ratio of composite plies between the upper and lower laminates on the normalized stress distributions along the adhesive mid-plane of the unbalanced composite SLJ are presented in Fig. 10. The SLJ of Case 1 corresponds to the case of E2L/E1L =E2T/E1T=G2LT/G1LT=1. The geometrical characteristics of this SLJ are kept constant for all examples, while the elastic properties vary. For all the cases in the present parametric study, E2L/E1L =E2T/E1T=G2LT/G1LT. When E2L/E1L increases from 1 to 4, the peak values of the normalized shear stresses near the left end gradually increases and the peak values of normalized shear stresses near the right end gradually decreases, as shown in Fig. 10(a). When E2L/E1L =2, the stiffness of the upper adherend is similar to that of the lower adherend and the difference between the two peak values of the normalized adhesive shear stresses in the middle plane is the smallest. When E2L/E1L varies from 2 to 4 or from 2 to 1, the difference between the two peak values of the normalized adhesive stress distributions in the middle plane increases as the stiffness difference between upper adherend and lower adherend increases. A similar variation trend can be obtained from the peak values of the normalized peel stresses. This conclusion agrees with the corresponding conclusions obtained by Zhao [23], showing that the stress intensities of unbalanced composite SLJ can be reduced by selecting optimal combinations of the materials elastic properties, i.e., E2L/E1L, E2T/E1T and G2LT/G1LT.

0

-10

AC C

0

E2L / E1L =4

-32

-28

28

E2L / E1L =4

5

E2L / E1L =3

-36

E2L / E1L =3

10

E2L / E1L =2

-20 -40

E2L / E1L =2

15

E2L / E1L =1

-15

E2L / E1L =1

20

σm / τ0

τm / τ0

EP

-5

25

32

36

-5 -40

40

x (mm)

-36

-32

-28

28

32

36

40

x (mm)

(a)

(b)

Fig. 10 Material effect on the stress distributions along the adhesive mid-plane (a) shear stress (b) peel stress

5. Conclusions In this paper, a new two-dimensional model considering the effects of large deformations in the overlap, bending-tension coupling and interfacial compliance is introduced for

13

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determining the adhesive stress distributions in unbalanced SLJs. A matlab code has been developed for the model implementation. The new model can be used for the estimation of the adhesive stress distributions for the unbalanced composite SLJ with relatively flexible, intermediate flexibility and inflexible adhesive layers. The accuracy of the new model is confirmed by comparisons to NFEM results. Comparisons between the new model and the existing model were further performed. The applicability of the validated model has been demonstrated by performing a parametric study, investigating the effects of the applied tensile load as well as the material elastic properties on the adhesive stress distributions of SLJs. The following conclusions are drawn: • The improved two-dimensional model can accurately predict the adhesive stress distributions for the unbalanced composite SLJ with relatively inflexible (i.e., 10ta/t1≤ Ea/ E1 or 10ta/t2≤ Ea/E2) or intermediate flexibility adhesive (i.e., (10ta/t1> Ea/E1 and 10Ea/E1 >ta/t1) or (10ta/t2> Ea/E2 and 10Ea/E2 >ta/t2)). • For the unbalanced composite SLJ with relatively flexible adhesive (i.e., 10Ea/E1 ≤ta/t1 or 10Ea/E2 ≤ta/t2), the new two-dimensional model has some deviations from the NFEM results, although always performs better compared to existing models. The deviations have been attributed to non-linear terms missing from the displacement continuity conditions for upper and lower interfaces. • The model, in the form of a Matlab code, can be used for the investigation of the effects of several parameters on the adhesive stress distributions. It has been shown that the geometrical nonlinearity of the SLJ configuration leads to the reduction of the peak values of the normalized adhesive stress distributions in the middle plane, and that an appropriate selection of the joint adherends can reduce the difference between two peak values of normalized adhesive stress distributions in the middle plane. • The new two-dimensional model provides the basis for better understanding of the mechanical behavior of the adhesively-bonded FRP composite SLJs, and can be utilized as a tool for design stages of unbalanced FRP composite SLJs with relatively inflexible or intermediate flexibility adhesives.

AC C

ACKNOWLEDGEMENTS

This research is sponsored by the Major State Basic Research Development Program of China (973 Program) (No.2012CB026200), the Fundamental Research Funds for the Central Universities (No.CXZZ13_0110), the Fundamental Research Funds for outstanding Doctoral Dissertation of Southeast university (No.YBJJ-1422) and the Scientific Research Foundation of Graduated School of Southeast University (No.YBPY-1609).

References: [1] Magalhaes A. G., de Moura M. F. S. F., Goncalves J. P. M. Evaluation of stress concentration effects in single-lap bonded joints of laminates composite materials. Int J Adhes Adhes. 2005;25(4):313-319.

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[2] Budhe S., Banea M.D., de Barros S., da Silva L.F.M. An updated review of adhesively bonded joints in composite materials. Int J Adhes Adhes. 2017; 72:30-42. [3] Tong L., Steven G. P. Analysis and design of structural bonded joints. Boston: Kluwer Academic Publishers, 1999. [4] Luo Q. T., Tong L. Y. Fully-coupled nonlinear analysis of single lap adhesive joints. Int J Solids Struct. 2006;44(7-8):2349-2370. [5] Luo Q. T., Tong L. Y. Analytical solutions for nonlinear analysis of composite single-lap adhesive joints. Int J Adhes Adhes. 2009;29(2):144-154. [6] Tsai M. Y., Morton J., Matthews F. L. Experimental and numerical studies of a laminated composite single-lap joint. J Compos Mater. 1995;29(9):1254-1275. [7] Goland M., Reissner E. The stresses in cemented joints. Journal of applied mechanics. 1944;11:A17-A27. [8] Hart-Smith L. J. Adhesive-bonded single-lap joints.: NASA langley Research Center; 1973. [9] Chen D., Cheng S. An analysis of adhesive-bonded single-lap joints. J Appl Mech. 1983;50(1):109-115. [10] Oplinger D. W. Effects of adherend deflections in single lap joints. Int J Solids Struct. 1994;31(18):2565-2587. [11] Yang C., Pang S. S. Stress-strain analysis of single-lap composite joints under tension. J Eng Mater-T ASME. 1996;118(2):247-255. [12] Zou G. P., Shahin K., Taheri F. An analytical solution for the analysis of symmetric composite adhesively bonded joints. Compos Struct. 2004;65(3-4):499-510. [13] Mortensen F., Thomsen O. T. Analysis of adhesive bonded joints: a unified approach. Compos Sci and Technol. 2002;62(7-8):1011-1031. [14] Jiang Z. W., Wan S., Zhong Z. P.. Geometrically nonlinear analysis for unbalanced adhesively bonded single-lap joint based on flexible interface theory. Arch Appl Mech. 2016;86(7):1273-1294. [15] Cheng S., Chen D., Shi Y. Analysis of adhesive-bonded joints with nonidentical adherends. J Eng Mech-ASCE. 1991;117(3):605-623. [16] Bigwood D. A., Crocombe A. D. Elastic analysis and engineering design formula for bonded joints. Int J Adhes Adhes. 1989;9(4):229-242. [17] Da Silva L. F. M., Das Neves P. J. C., Adams R. D., et al. Analytical models of adhesively bonded joints—Part I: Literature survey. Int J Adhes Adhes. 2009;29(3):319-330. [18] Da Silva L. F. M., Das Neves P. J. C., Adams R. D., et al. Analytical models of adhesively bonded joints—Part II: Comparative study. Int J Adhes Adhes. 2009;29(3):331-341. [19] Sawa T., Nakano K., Toratani H. A two-dimensional stress analysis of single-lap adhesive joints subjected to tensile loads. J Adhes Sci Technol. 1997;11(8):1039-1062. [20] Wang J.L., Zhang C. Three-parameter, elastic foundation model for analysis of adhesively bonded joints. Int J Adhes Adhes. 2009;29(5):495-502. [21] Zhao B., Lu Z., Lu Y. Two-dimensional analytical solution of elastic stresses for balanced single-lap joints—Variational method. Int J Adhes Adhes. 2014;49:115-126. [22] Allman D. J. A theory for elastic stresses in adhesive bonded lap joints. Q J Mech Appl Math. 1977;30(4):415-436. 15

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Appendix A

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[23] Zhao B., Lu Z. H., Yi N. Closed-form solutions for elastic stress–strain analysis in unbalanced adhesive single-lap joints considering adherend deformations and bond thickness. Int J Adhes Adhes. 2011;31(6):434-445. [24] Jiang Z. W., Wan S., Zhong Z. P. Two dimensional analysis for an adhesively bonded composite single-lap joint based on flexible interface theory. J Adhes Sci Technol. 2015;29(22):2408-2432. [25] Osnes H., Andersen A. Computational analysis of geometric nonlinear effects in adhesively bonded single lap composite joints. Composite Part B-Eng. 2003;34(5):417-427. [26] Tsai M. Y., Morton J. An evaluation of analytical and numerical solutions to the single-lap joint. Int J Solids Struct. 1994;31(18):2537-2563. [27] Wang J. L., Qiao P. Z. Novel beam analysis of end notched flexure specimen for mode-II fracture. Eng Fract Mech. 2004;71(2):219-231. [28] Qiao P. Z., Chen F. L. An improved adhesively bonded bi-material beam model for plated beams. Eng Struct. 2008;30(7):1949-1957. [29] Suhir E. Stress in bi-material thermostats. J Appl Mech. 1986;53:657-660. [30] Jiang Z. W., Wan S., Song A. M. An alternative solution for the edge moment factors of the unbalanced adhesive single-lap joint in tension. Int J Adhes Adhes. 2017;75:1-16.

The overall equilibrium equations for an infinitesimal isolated body shown in Fig. 2 can be obtained as:

N T = N1 + N 2 + N a = P Q T = Q1 + Q 2 + Q a = − Q1* = − Q 2*

(A-1a) (A-1b)

t +t   MT = M1 + M2 + Ma + N1r1 + Na  r1 − 1 a  − N2 r2 = P −w − α ( l1 + L 2 + x ) + r1  2  

TE D

(A-1c)

where NT, QT and MT are the total axial force, transverse shear force and bending moment of the cross section in overlap region, respectively; Q1* and Q2* are the transverse shear forces in two ends of the overlap region, respectively; r1, r2 are the distances between the neutral planes of section 1 and 2, respectively, and the overlap

EP

region, and r1=(t1+2ta+t2)/[2(1+A1/A2)] and r2=(t1+2ta+t2)/[2(1+A2/A1)]; w is the transverse deformation of the overlap region; α is the rotation angle of the composite SLJ, and α=(t1+2ta+t2)/[2(l1+L+l2)]. A.1 Differential governing equations for upper interface

AC C

According to Euler-Bernoulli beam theory, the constitutive equations for the adhesive layer are obtained as: N a = Aa

du a d 2 wa , M a = − Da dx dx 2

(A-2)

where Da is bending stiffness of the adhesive layer; ua is axial displacements of the adhesive layer. Based on Eqs. (2-a), (2-b) and principle of stiffness distribution, peel stresses σ1(x) and σ2(x) can be expressed as:  A1 P d 2 w1 1 d 2 M 1 t1 d 2 N1 + − σ 1 ( x ) = 2 2 b dx 2b dx ( A1 + A2 + Aa ) b dx 2   2 2 A2 P d 2 w2 σ ( x ) = − 1 d M 2 + t2 d N 2 + 2 2  2 b dx 2b dx ( A1 + A2 + Aa ) b dx 2 

(A-3)

Neglecting the nonlinear terms in Eq. (3-a), differentiating Eq. (3-a) with respect to x twice and combining with Eq. (A-3) yields: d 2 wa d 2 w1 ( Cn1 + Cna ) d 4 M 1 t1 ( Cn1 + Cna ) d 4 N1 A1 P ( Cn1 + Cna ) d 4 w1 = − − + dx 2 dx 2 b dx 4 2b dx 4 b ( A1 + A2 + Aa ) dx 4

16

(A-4)

ACCEPTED MANUSCRIPT Differentiating Eq. (3-c) with respect to x once and combining with Eqs. (2), (A-1) and Eq. (A-3) yields:

t ( C + Cna ) d 4 M1 t1ta ( Cn1 + Cna ) d 4 N1 ( Cs1 + Csa ) d 2 N1 dua = − a n1 − − dx 2b dx 4 4b dx 4 b dx 2 t A P ( Cn1 + Cna ) d 4 w1 ( t1 + ta ) d 2 w1 du1 + a 1 + + 2b ( A1 + A2 + Aa ) dx 4 2 dx 2 dx

(A-5)

Neglecting the nonlinear terms in Eq. (3-b), differentiating Eq. (3-b) with respect to x twice, substituting Eq. (A-4) and Eq. (A-5) into Eq. (3-b) and combining with Eq. (1), Eq. (2) and Eq. (A-1) (here, w=w1 is adopted in Eq.

RI PT

(A-1c)), yields Eq. (4-a). In Eq. (4-a),  ( C n 2 + C na ) ( C n1 + Cna )  A2 ( Da + D2 ) − B22    −  2 b b B − A D   ( ) 2 2 2  γu =  + 2 − 2 + + + A t C C B A r r A t t    a a ( n1 na )  2 2( 1 2) 2( 1 a )  −  2 4b ( B2 − A2 D2 )  

 A2 P B2 − A2 ( r1 + r2 )  ( Cn2 + Cna ) PB ( C + C ) A1B1P ( Cn1 + Cna )  A2 ( Da + D2 ) − B22         − 1 2 n2 na − 2 b ( B1 − AD b( B12 − A1D1 )( B22 − A2 D2 ) ( A1 + A2 + Aa )  1 1) 1  b ( B2 − A2 D2 ) ( A1 + A2 + Aa ) u a13 = u   γ   ta AB  (C + C )  Aa 2B2 − 2A2 ( r1 + r2 ) + A2 (t1 + ta ) 1 1 P ( Cn1 + Cna ) + s1 sa   −   2 2 b 2( B2 − A2 D2 )    2b ( B1 − AD 1 1 ) ( A1 + A2 + Aa ) 

SC

;

 A1  A2 ( Da + D2 ) − B22   A1 A22 P 2 ( Cn 2 + Cna )  −  2  2 2 2 1  ( B1 − A1 D1 )( B2 − A2 D2 ) b ( B1 − A1 D1 )( B2 − A2 D2 ) ( A1 + A2 + Aa )  u a14 = u   γ  Aa ( ta A1 + t1 A1 + 2 B1 ) 2 B2 − 2 A2 ( r1 + r2 ) + A2 ( t1 + ta )  A2 + − 2  2 2 B − A D 4 B − A D B − A D ( 1 1 1 )( 2 2 2 ) 2 2)  ( 2 

a16u =

1

PA2

γ u ( B22 − A2 D2 )

;

f

u

(x) =

;

;

M AN U

u a12

 PA1 ( C n 2 + C na )  A22 P ( C n 2 + C na ) −   2 b ( B22 − A2 D 2 ) ( A1 + A2 + Aa )  b ( B1 − A1 D1 )    2 2   A P C + C A D + D − B ( n1 na )  2 ( a 1  1 2) 2   = u +  2 2 γ  b ( B1 − A1 D1 )( B2 − A2 D2 ) ( A1 + A2 + Aa )    2  Aa t a A1 P ( C n 1 + C na )  2 B2 − 2 A2 ( r1 + r2 ) + A2 ( t1 + ta )   +  2 2 4 b B − A D B − A D A + A + A ( ) ( )( ) a 1 1 1 2 2 2 1 2  

;

  2 ( r1 + r2 ) − t2  ( Cn 2 + C na ) t1 ( Cn1 + Cna )  A2 ( Da + D2 ) − B22    −  2b 2b ( B22 − A2 D2 )  1  a11u = u   γ  Aa t1ta ( Cn1 + C na )  2 B2 − 2 A2 ( r1 + r2 ) + A2 ( t1 + ta )   −  8b ( B22 − A2 D2 )  

;

 B2 − A2 ( r1 + r2 ) Aa ( ta B1 + t1B1 + 2D1 ) 2B2 − 2 A2 ( r1 + r2 ) + A2 ( t1 + ta )  −   2 4( B12 − A1D1 )( B22 − A2 D2 )  1  ( B2 − A2 D2 ) a15u = u   2 2 2 γ    B A D + D − B ( ) B1 A2 P (Cn2 + Cna )  1 2 a 2 2 − 2 +  2 2 2  b( B1 − A1D1 )( B2 − A2 D2 ) ( A1 + A2 + Aa ) ( B1 − A1D1 )( B2 − A2 D2 ) 

P  B − A ( r + r )  P α A2 1  ( l1 + L 2 + x ) +  2 2 2 1 2   u  2 ( B 2 − A2 D 2 )   ( B 2 − A2 D 2 )

γ

;

.

Adding Eq. (3-c) and Eq. (3-d), differentiating the result equation with respect to x once and combining with Eq.

TE D

(1), Eq. (2) and Eq. (A-1) (here, w=w1 is adopted in Eq. (A-1c)), Eq. (A-4) and Eq. (A-5) yields Eq. (4-b). In Eq. (4-b),

 t (C + C )  A ( t B − 2D2 ) Aa ( 2 B2 − t2 A2 )  2 ( r1 + r2 ) − ( t1 + ta )   Da ( Cn1 + Cna )( 2B2 − t2 A2 )  a n1 na  2 + a 22 2 + −  2b 2 ( B2 − A2 D2 ) 4 ( B22 − D2 A2 ) 2b ( B22 − D2 A2 )    

ηu = 

A12 P b = ( A1 D1 − B12 ) ( A1 + A2 + Aa )

;

AC C 1 P ( 2 B2 − t2 A2 )

η 2 ( B22 − D2 A2 )

;

g u (x ) =

t1 ; 2

( Cs2 − Cs1 ) A1B1PDa ( 2B2 − t2 A2 )( Cn1 + Cna )  −   b 2b( B12 − A1D1 )( B22 − D2 A2 ) ( A1 + A2 + Aa ) 1   b =   η  ta A1B1P( Cn1 + Cna ) A ( t B − 2D2 ) Aa ( 2B2 − t2 A2 ) 2( r1 + r2 ) − ( t1 + ta )  ( Cs1 + Csa )  2 + a 22 2  + +  2b B2 − A D A + A + A + 2 b 2 B − A D 4 B − D A  ( 2 2 2) ( 2 2 2) 1 1 )( 1 2 a)     ( 1

 ( 2B1 + t1 A1 ) A1 Da ( 2B2 − t2 A2 )  ( 2B2 − t2 A2 ) − +   2 B2 − D1 A1 ) 2 ( B22 − D2 A2 ) 2 ( B12 − A1 D1 )( B22 − D2 A2 )  1  ( 1 b14u =   η  ( ta A1 + t1 A1 + 2B1 )  Aa ( t2 B2 − 2D2 ) Aa ( 2B2 − t2 A2 ) 2 ( r1 + r2 ) − ( t1 + ta )   2 +  + − 2 ( B12 − A1 D1 )  2 ( B22 − A2 D2 ) 4 ( B22 − D2 A2 )    

b16u =

b11u =

u 13

EP

u 12

;

;

;

 ( t B + t B + 2D )  A ( t B − 2D ) Aa ( 2B2 − t2 A2 ) 2 ( r1 + r2 ) − ( t1 + ta )   1 2    a 1 1 1  2 + a 22 2 + 2  2 ( B12 − A1 D1 )   − −  2 B A D 4 B D A ( ) ( ) 2 2 2 2 2 2 1   b15u =   η  ( 2D + t B ) ( t2 B2 − 2D2 ) + ( r1 + r2 )( 2B2 − t2 A2 )  B1 Da ( 2B2 − t2 A2 ) 1 1 1  − − −  2 2 2 2 ( B22 − A2 D2 )  2 ( B1 − D1 A1 ) 2 ( B1 − A1 D1 )( B2 − D2 A2 ) 

P  ( r + r )( 2 B 2 − t 2 A 2 ) + ( t 2 B 2 − 2 D 2 )   1  P α ( 2 B 2 − t 2 A 2 ) ( l1 + L 2 + x ) −  1 2   η u  2 ( B 22 − D 2 A 2 ) 2 ( B 22 − D 2 A2 ) 

;

.

Using the elimination method, M1 is removed from the fully coupled nonlinear equation and the differential

governing equation concerning N1 for the unbalanced composite SLJ can be determined as: 6 4 2 d 8N1 u d N1 u d N1 u d N1 u u u + g 10 + g 11 + g 13 + g 15 N 1 = g 16 w 1 + g 17 f dx 8 dx 6 dx 4 dx 2

M 1 = − f 01u

u

(x) +

u g 18 g u (x )

d 8N1 d 6N1 d 4 N1 d 2N1 − f 10u − f 11u − f 13u − f 15u N 1 − f 16u w 1 + f 17u f 8 6 4 dx dx dx dx 2

where

17

u

(x)

(A-6a)

(A-6b)

ACCEPTED MANUSCRIPT b1ui − a 1ui ( i = 1, 3, 4 , 5 ) b1u2 − a 1u2

c16u B1 B12 − A1 D1 a12u − c14u

a15u + d

u 15

=

f10u =

g 1ui =

e11u − d10u A1 e16u − d14u B12 − A1 D1

; ;

;

d16u =

f 1 ui =

f 1 ui − e 1ui ( i = 1, 3, 5, 6 , 7 ) f 0u1

;

u c16 =

u b16u − a 16 u a 12 − b12u

a16u c − a12u u 14

;

;

c17u =

d17u =

1 a − c14u

d 1ui − e 1u( i + 2 ) d 1u4 −

A 1 e 1u6 B 12 − A 1 D 1

u g18 =−

u e18 f 01u

1 a − b12u

u 12

(i

u e10 =

;

= 1, 3 )

;

c18u =

;

u 12

1 b − a12u u 12

d 10u u d − c14

f 1 u5 =

d 1u5 d 1u4

e 1ui =

;

u 14

(B (B

;d

2 1

− A1 D 1

2 1

− A1 D 1

u 10

=

c 1u1 c - a 1u2 u 14

;

d1ui =

a1ui − c1u(i +2) a −c

c 1ui − d 1ui ( i = 1, 3, 5, 6 , 7 ) c 1u4 − d 1u4

)+ )−

B 1 e 1u6 A 1 e 1u6

;

f1ui = d 14u −

u 12

u 14

;e

u 18

=

( i = 1,3)

;d

u c18 c − d 14u u 14

;

e10u A1 e16u − d14 B12 − A1 D1

;

= f 01u =

;

d 1ui ( i = 6,7 ) A1 e16u 2 B − A D ( 1 1 1)

c16u A1 B12 − A1 D1 a12u − c14u

a14u − u 14

;

g 1u0 =

f 1 u0 − e 1u0 f 0u1

;

.

RI PT

c 1ui =

The characteristic equation of Eq. (A-6a) is:

R8 + g10u R6 + g11u R4 + g13u R2 + g15u = 0

(A-7)

Through numerical calculation, the roots of the Eq. (A-7) can be obtained for one case as: ±Ru1, ±Ru2 , ±Ru3± iRu4.

SC

The particular solution of the Eq. (A-6a) is:

N1uc = g16u w1 + g17u f u ( x) + g18u gu ( x) g15u

(A-8)

A.2 Stress distributions for lower interface

M AN U

Substituting Eq. (5) into Eq. (A-4) and then substituting the results into Eq. (A-2), Ma can be obtained.

Neglecting the nonlinear terms in Eq. (3-b), differentiating Eq. (3-b) with respect to x twice and combining with Eq. (A-3) yields:

d 2 wa d 2 w2 ( Cn 2 + Cna ) d 4 M 2 t2 ( Cn 2 + Cna ) d 4 N 2 A2 P ( Cn 2 + Cna ) d 4 w2 = − + + dx 2 dx 2 b dx 4 2b dx 4 b ( A1 + A2 + Aa ) dx 4

(A-9)

Differentiating Eq. (3-d) with respect to x once and combining with Eq. (2), Eq. (A-1) and Eq. (A-3) yields:

TE D

dua ta ( Cn 2 + Cna ) d 4 M 2 ta t2 ( Cn 2 + Cna ) d 4 N 2 ( Cs 2 + Csa ) d 2 N 2 = − − dx 2b dx 4 4b dx 4 b dx 2 4 2 t A P ( Cn 2 + Cna ) d w2 ( t2 + ta ) d w2 du2 − a 2 − + dx 2 dx 2b ( A1 + A2 + Aa ) dx 4 2

(A-10)

Neglecting the nonlinear terms in Eq. (3-a), differentiating Eq. (3-a) with respect to x twice, substituting Eq. (A-10)

(A-1c)) yields:

EP

and Eq. (A-10) into Eq. (3-a) and combining with Eq. (1), Eq. (2) and Eq. (A-1) (here, w=w2 is adopted in Eq.

d 4M 2 d 4N2 d 2M 2 d 2N2 + a 1l 1 + a 1l 2 + a 1l 3 + a 1l 4 M 2 + a 1l 5 N 2 = a 1l 6 w 2 + f 4 4 2 dx dx dx dx 2

l

(x)

(A-11a)

AC C

where

 ( C + C ) ( Cn 2 + Cna )  A1 ( Da + D1 ) − B12   na    n1 − b b ( B12 − A1 D1 )   γl =   ta ( Cn 2 + Cna )  A1 Aa ( t1 + ta ) + 2 B1 Aa   +  2 4b ( B1 − A1 D1 )  

;

t1 + t2 − 2 ( r1 + r2 ) ( Cn1 + Cna ) t2 a11l =  − 2bγ l 2

  A1 B1 P ( Cn1 + Cna ) A12 P ( r1 + r2 )( Cn1 + Cna )   +  b ( B12 − A1 D1 ) ( A1 + A2 + Aa ) b ( B12 − A1 D1 ) ( A1 + A2 + Aa )    2   B2 P ( Cn1 + Cna ) 1  A2 B2 P ( Cn 2 + Cna )  A1 ( Da + D1 ) − B1   l − a13 = l −  2 2 2 γ  b ( B1 − A1 D1 )( B2 − A2 D2 ) ( A1 + A2 + Aa ) b ( B2 − A2 D2 )      A1 Aa ( t1 + ta ) + 2 B1 Aa   ta A2 B2 P ( Cn 2 + Cna ) Cs 2 + Csa )   (   − + 2 2 b 2 B − A D 2 b B − A D A + A + A   ( ) ( ) ( ) 1 1 1 2 2 2 1 2 a    

  B1 + A1 ( r1 + r2 )   B2 ( ta + t2 ) − 2 D2   A1 Aa ( t1 + ta ) + 2 B1 Aa   +   2 2 2 B − A D 4 B − A D B − A D   ( ) ( )( ) 1 1 1 1 1 1 2 2 2 1  a15l = l   2 2 2 γ  B2  A1 ( Da + D1 ) − B1   B2 A1 P ( Cn1 + Cna ) − 2 +  2 2 2  b ( B1 − A1 D1 )( B2 − A2 D2 ) ( A1 + A2 + Aa ) ( B1 − A1 D1 )( B2 − A2 D2 ) 

；

l 16

；a

=

 A2 1 P ( Cn1 + Cna )  A2 ( A1 + 2 A2 + Aa )   − 2 1 2 − − B A D B A D  ( ) ( ) 2 2 2 1 1 1  2 A2 P − 2 ( B2 − A2 D2 ) ( A1 + A2 + Aa ) a12l =

;

γ l b ( A1 + A2 + Aa ) 

;

 A2  A1 ( Da + D1 ) − B12  2 B2 − A2 ( t2 + ta )   A1 Aa ( t1 + ta ) + 2 B1 Aa    +    2 2 4 ( B12 − A1 D1 )( B22 − D2 A2 ) 1  ( B1 − A1 D1 )( B2 − A2 D2 )  a = l  γ  A2 A12 P 2 ( Cn1 + Cna ) A1  − −   2 2 2 B − A D b B − A D B − A D A + A + A ( ) ( 1 1 1 )( 2 2 2 ) 1 2 a 1 1)  ( 1  l 14

1

PA1

γ l ( B12 − A1 D1 )

；

f l (x) =

；

 A1 Pα PB 1   (l + L 2 + x ) + 2 1  γ l  ( B12 − A1 D1 ) 1 ( B1 − A1 D1 ) 

Adding Eq. (3-c) and Eq. (3-d), differentiating the result equation with respect to x once and combining with Eq.

18

ACCEPTED MANUSCRIPT (1), Eq. (2) and Eq. (A-1) (here, w=w2 is adopted in Eq. (A-1c)), Eq. (A-9) and Eq. (A-10) yields: d 4M 2 d 4N2 d 2M 2 d 2N2 + b1l1 + b1l 2 + b1l3 + b1l 4 M 2 + b15l N 2 = b1l6 w 2 + g l ( x ) 4 4 2 dx dx dx dx 2

(A-11b)

where  ta ( Cn 2 + C na )  Aa ( t1 + ta )( 2 B1 + t1 A1 ) + 2 Aa ( 2 D1 + t1 B1 ) − 8 ( B12 − D1 A1 )        8b ( B12 − D1 A1 ) l η =   Da ( Cn 2 + C na )( 2 B1 + t1 A1 )  −  2 2b ( B1 − D1 A1 )  

; b11l = −

 ( Cs1 − Cs 2 ) A2 B2 PDa ( 2 B1 + t1 A1 )( C n 2 + C na )  −   b 2b ( B12 − D1 A1 )( B22 − A2 D2 ) ( A1 + A2 + Aa )     1   ta A2 B2 P (C n 2 + C na )   b13l = l    η   2b ( B22 − A2 D2 ) ( A1 + A2 + Aa )   Aa ( t1 + ta ) ( 2 B1 + t1 A1 ) + 2 Aa ( 2 D1 + t1 B1 ) − 8 ( B12 − D1 A1 )  +   2  4 B − D A ( ) C + C 1 1 1 ( )    s2 sa   −  b    

 A2 Da ( 2B1 + t1 A1 )  ( 2B2 − t2 A2 ) ( 2B1 + t1 A1 ) + −   2 ( B12 − D1 A1 )( B22 − A2 D2 ) 2 ( B22 − D2 A2 ) 2 ( B12 − D1 A1 )   1   b14l = l   η  2B2 − A2 ( t2 + ta )  Aa ( t1 + ta )( 2B1 + t1 A1 ) + 2 Aa ( 2D1 + t1 B1 ) − 8 ( B12 − D1 A1 )  +  2 2 8 ( B1 − D1 A1 )( B2 − D2 A2 )  

b 1l 6 =

P

1

η

l

2

(2

(B

B 1 + t1 A 1 2 1

− D

1

A1

)

;

)

(B

gl ( x) =

2 2

A22 P − A2 D 2 ) ( A1 + A2 + Aa )

;

RI PT

;

b12l = −

P ( 2 D1 + t1 B1 )  1  Pα ( 2 B1 + t1 A1 )   (l + L 2 + x ) + η l  2 ( B12 − D1 A1 ) 1 2 ( B12 − D1 A1 ) 

;

.

SC

 ( t2 B2 − 2 D2 ) ( 2 D1 + t1 B1 ) + ( r1 + r2 )( 2B1 + t1 A1 )  B2 Da ( 2 B1 + t1 A1 ) + −   2 2 ( B12 − D1 A1 ) 2 ( B12 − D1 A1 )( B22 − A2 D2 )  1  2 ( B2 − A2 D2 )  b = l  2 η   B2 ( ta + t2 ) − 2D2   Aa ( t1 + ta )( 2 B1 + t1 A1 ) + 2 Aa ( 2D1 + t1 B1 ) − 8 ( B1 − D1 A1 )    +  2 2 8 B − D A B − A D ( 1 1 1 )( 2 2 2 )   l 15

;

t2 ; 2

Using the elimination method, M2 is removed from the fully coupled nonlinear equations (i.e., Eq. (A-11)) and the differential governing equation concerning N2 for the unbalanced composite SLJ can be determined as:

M 2 = − f 0l1

(x) +

g 1l 8 g l ( x )

M AN U

2 d 8N2 d 6N2 d 4N2 l d N2 l l + g 1l 0 + g 1l 1 + g 13 + g 15 N 2 = g 16 w 2 + g 1l 7 f dx 8 dx 6 dx 4 dx 2

l

d 8N2 d 6N2 d 4N2 d 2N2 − f 10l − f 1l1 − f 13l − f 15l N 2 − f 16l w 2 + f 17l f 8 6 4 dx dx dx dx 2

l

(x)

(A-12a) (A-12b)

where the relationship between gl10, gl1i (i= 1, 3, 5, 6, 7, 8), f l01, f l10, f l1i (i= 1, 3, 5, 6, 7) and al1i (i= 1, 2, 3, 4, 5, 6), bl1i (i= 1, 2, 3, 4, 5, 6) follow the relationship between gu10, gu1i (i= 1, 3, 5, 6, 7, 8), f u01, f u10, f u1i (i= 1, 3, 5, 6, 7) and au1i (i= 1, 2, 3, 4, 5, 6), bu1i (i = 1, 2, 3, 4, 5, 6).

TE D

The characteristic equation of Eq. (A-12a) is:

R8 + g10l R6 + g11l R4 + g13l R2 + g15l = 0

(A-13)

Through numerical calculation, the roots of the Eq. (A-12) can be obtained for one case as: ±Rl1, ±Rl2 , ±Rl3± iRl4. N2, M2 can be obtained as follows:

N2 = c1l eR1 x + c2l e− R1 x +c3l eR2 x + c4l e− R2 x + eR3x c5l cos ( R4l x ) + c6l sin ( R4l x )  l

l

l

l

EP

l

+e− R3x c7l cos ( R4l x) + c8l sin ( R4l x )  + N2l c l

M2 = T1l c1l eR1 x + T2l c2l e−R1 x + T3l c3l eR2 x + T4l c4l e−R2 x + eR3x T5l c5l cos( R4l x) + T6l c6l sin ( R4l x) l

l

l

l

l

+e−R3x T7l c7l cos ( R4l x) + T8l c8l sin ( R4l x) + M2l c

AC C

(A-14a)

l

(A-14b)

Then, substituting Eq. (A-14) into Eq. (2-b) yields:

Q2 = V1l c1l eR1 x +V2l c2l e− R1 x +V3l c3l eR2 x +V4l c4l e−R2 x + eR3x V5l c5l cos ( R4l x ) +V6l c6l sin ( R4l x)  l

l

l

l

l

+e−R3x V7l c7l cos( R4l x) +V8l c8l sin ( R4l x)  + Q2l c l

(A-14c)

where cli (i= 1, 2,…., 9) are unknown coefficients; Nl2c is the particular solution for the Eq. (A-12a), and

N2lc =g16l w1 +g17l f l ( x) +g18l gl ( x) g15l ; Ml2c and Ql2c are the functions of Nl2c; Tli (i= 1, 2,…, 8) and Vli (i= 1, 2,…, 8) are the functions of Rli (i= 1, 2,…., 9). Substituting Eq. (A-14a) into Eq. (2-b) and substituting Eqs. (A-14a) and (A-14b) into Eq. (A-3) yield, respectively:

19

ACCEPTED MANUSCRIPT

(

) (

τ 2 ( x ) = R1l c1l e R x − c2l e− R x +R2l c3l e R x − c4l e− R x +e

R3l x

l 1

l 1

l 2

l 2

)

( R3l c5l + R4l c6l ) cos ( R4l x ) + ( R3l c6l − c5l R4l ) sin ( R4l x )  

(A-15a)

+e− R3 x ( R4l c8l − R3l c7l ) cos ( R4l x ) − ( R3l c8l + R4l c7l ) sin ( R4l x ) + l

d ( N 2l c ) dx

σ 2 = W1l c1l eR x + W2l c2l e− R x + W3l c3l eR x + W4l c4l e− R x l 1

l 1

l 2

l 2

+eR3x W5l c5l cos ( R4l x ) + W6l c6l sin ( R4l x ) l

+e

(A-15b)

W c cos ( R x ) + W c sin ( R x ) + σ   l l 7 7

l 4

l l 8 8

l 4

l 1c

RI PT

− R3l x

where cli (i= 1, 2,…., 9) are unknown coefficients; σl1c is a function of Nl1c; Wli (i= 1, 2,…, 8) is a function of Rli (i= 1, 2,…., 9). Substituting Eq. (A-14) into Eq. (A-9) and then substituting the results into Eq. (A-2), Ma can be obtained. cli (i= 1, 2,…., 9) can be determined by the following boundary conditions:

τ 2 ( − L 2) = 0

Ma ( L 2) = 0

(A-16a)

(A-16b)

 1  P ( t + t + 2ta ) Q2 ( L 2) = −  1 2 − M1* − M2*  L 2 

AC C

EP

TE D

N2 ( L 2) = P

M2 ( − L 2) = 0

SC

Q2 ( − L 2) = 0

M AN U

N2 ( − L 2) = 0

20

M2 ( L 2) = M2*

(A-16c)

ACCEPTED MANUSCRIPT Highlights

AC C

EP

TE D

M AN U

SC

RI PT

Two-dimensional model considering simultaneously the effects of large deformation, bending-tension coupling and interfacial compliance was proposed for FRP unbalanced SLJ. Solutions for interface stress distributions of the unbalanced FRP composite SLJ with different relative adhesive/adherend stiffness ratios were presented. The effects of tensile load P, material properties of composite adherend on the adhesive stress distributions in middle plane were studied