Stress concentration coefficient in a composite double lap adhesively bonded joint

International Journal of Adhesion & Adhesives 63 (2015) 102–107

Contents lists available at ScienceDirect

International Journal of Adhesion & Adhesives journal homepage: www.elsevier.com/locate/ijadhadh

Stress concentration coefficient in a composite double lap adhesively bonded joint Pascale Saleh a, Georges Challita a,n, Khaled Khalil a,b a b

Equipe MMC, Université Libanaise, Faculté de génie, Campus Hadath, Beyrouth, Lebanon EMM-GeM, Institut de Recherche en Génie Civil et Mécanique UMR CNRS 6183, 58 rue Michel Ange, BP 420, 44600 Saint-Nazaire, France

art ic l e i nf o

a b s t r a c t

Article history: Accepted 11 August 2015 Available online 20 August 2015

The main target of this paper is to investigate the effect of peak stress at the extremities of the adhesive layer of a bonded assembly subjected to dynamic shear impact. It is known, that under both static and dynamic loadings such joints endure at their extremities high level of stresses, an aspect known as edge effects. Double lap joint assembly was considered with unidirectional carbon–epoxy substrates and Araldite 2031 adhesive. To quantify this edge effect, a specific coefficient, named coefficient of stress concentration was defined: it is the ratio of the maximum shear stress to the average shear stress. This coefficient helps to calculate maximum strength of the joint since experimentally, only average shear stress could be measured. A numerical analysis at the midplane of the joint was carried out to investigate the effect of geometrical and material parameters on this stress concentration factor. It was found that this factor is constant with the time once the equilibrium is established. Moreover, this stress concentration coefficient decreases with higher Young's modulus of the adherents, lower Young's modulus of the adhesive, thicker and shorter adhesive layer. A unified parameter involving geometrical and mechanical parameters of the specimen was established to quantify this stress concentration factor. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Stress analysis Impact shear Adhesive Stress concentration Finite element

1. Introduction The peak of shear stress at the free extremities of an adhesive layer in a bonded assembly, known as edge effect, was one of the main problems encountered in bonding field. This effect appeared for both static and dynamic loadings. Since then, huge efforts were carried out by many researchers to investigate deeply this effect. Others tried to minimize as possible the level of this peak; in other words to homogenize as possible the shear stress field along the overlap length. One of the first ever works to solve the edge effect was presented by Mylonas and De Bruyne [1] who proposed a tapered adherent from outside near the joint edge in order to decrease the stiffness in the neighborhood of the extremity and thus decrease the peak of stress. Henning [2] used mixed-adhesive: a brittle one in the middle and ductile one at the edges since ductile is softer. This solution was tested experimentally by Da Silva et al. [3]. Adams and Peppiatt [4] showed that the existence of spew fillet at the edges decreases the stress concentration. The approach of Sage [5] was a variable thickness joint with thin layer at the middle and thick one at the edges which limits remarkably the edge effect while Adams et al. [6] found n

Corresponding author. Tel.: þ 961 3 188734. E-mail addresses: [email protected], [email protected] (G. Challita). http://dx.doi.org/10.1016/j.ijadhadh.2015.08.005 0143-7496/& 2015 Elsevier Ltd. All rights reserved.

an approximate solution of a differential equation function of the variable thickness that reduces or eliminates shear stress concentration at the ends adhesive layer. They used shear lag model, which states that the adhesive shear strain is proportional to the difference of displacements of the corresponding adherents. Theoretical and experimental results were compared to show good agreement. Adams and Wake [7] carried out FEM analysis to prove that tapered substrates in the neighborhood of the adhesive ends improve the stress homogeneity while a spew fillet in the layer increases the joint's strength. Adams and Harris [8] proposed an impact block technique to test dynamically bonded assemblies; the plots of normal and shear stresses in the joint showed no uniformity at all. Similarly, Wada et al. [9] found stress singularity at the edges of dissimilar cylindrical bonded assembly under both static and dynamic tensile loadings. They also simulated the dynamic tests through FEM using ANSYS software and established an analytical model to calculate the impact tensile strength. Cognard et al. [10] and [11] proposed special substrate geometry to overcome the edge effect and validated it through FEM analysis for non-linear adhesive: each substrate had a T-shape with particular geometry at the beak. Many static loadings were applied for both metallic and composite adherents. In addition, a modified experimental set-up (Arcan fixture) was designed. Vable and Maddi [12] proposed a numerical method called boundary element method (BEM) and compared it with classical FEM for single and double lap joint. Many values of spew angle were taken to

P. Saleh et al. / International Journal of Adhesion & Adhesives 63 (2015) 102–107

examine the edge effect. Zou and Taheri [13] investigated bonded composite pipes under static torsion. They applied numerical and analytical studies to obtain shear stress distribution and to choose the best configuration that minimizes the edge effect. Challita and Othman [14] simulated on Abaqus software the SHPB technique applied on double lap joints with steel adherents and epoxy adhesive and proposed many correction parameters; one among them was the stress concentration factor described earlier in the abstract. They used it to evaluate the shear strength in the joint after measuring experimentally the average shear in [15]. Hua et al. [16] carried out simulations on Abaqus, with a crack initiation analysis and dissimilar single lap joint adherents (titanium and composites). They applied tensile load while studying the influence of varying recess lengths and material and geometrical nonlinearities of the adhesive. The results are that adding a spew fillet will reduce the peak stress concentrations in the adhesive layer, and improve the strength. Hazimeh et al. [17] extended to work done in [14] but for unidirectional PEEK– carbon substrates. They investigated uniquely the maximum average shear stress and the coefficient of homogeneity of the shear stress distribution without quantifying the stress concentration effect. Huge efforts have been done to study in details edge effects; some have tried to change the geometry at the joints' ends to minimize this effect, others have tried to change the configuration of the whole specimen. The majority of the studies have just considered static loading while the dynamic loading cases were not so frequent. In this paper, a parametric numerical study will be carried out on the explicit standard module of Abaqus to determine the stress concentration factor at the joint edges in a double lap joint with unidirectional carbon–epoxy substrates and Araldite 2031 adhesive in terms of geometrical and mechanical parameters of the specimen. The set-up used to apply the impact shear is the Split Hopkinson Pressure Bar technique (SHPB). Since this technique allows only measuring the average shear strength in the adhesive joint [14], the numerically established stress concentration factor will allow evaluating the peak shear stress in the adhesive joint.

The material used for the adherent used is unidirectional carbon fiber reinforced epoxy composite laminates while the adhesive is an epoxy type Araldite 2031 [18]. The mechanical properties of elastic orthotropic carbon/epoxy composite adherents found using the mixing law for Vf ¼39.6% are summarized in Table 1. The adhesive is elastic isotropic polymer having a Young's modulus of 1 GPa and a Poisson's ratio of 0.4. This work will involve first a reference study by simulating the dynamic test using values summarized in both Tables 1 and 2. Second, a parametric study will be carried out; in other words, the influence of different geometrical and material parameters will be investigated: adhesive's thickness, adhesive's Young modulus, and overlap length, adherents' material properties (fiber volume fraction), and adherents' thicknesses. 2.2. Finite element model The explicit standard module of Abaqus will be adopted to run the numerical simulations. The Split Hopkinson Pressure Bar (SHPB) technique will be used. Numerically, the loading is represented by a velocity impact pulse applied on the central adherent as shown in Fig. 2a. The velocity input to the commercial FEA software is depicted in Fig. 2b. All details of the concept of the SHPB set-up are found in [14]. Besides the specimen, the output bar of the SHPB will be modeled, in order to avoid the formation of a reflected wave and to make sure the measured wave represents only the transmitted one. Same model of Hazimeh et al. [17] will be considered in the present study. A quarter of the model will be modeled, due to the presence of two planes of symmetry for the double lap joint specimen and applied loading. This output bar is a long elastic bar made of steel, of cylindrical shape, having a 200 GPa Young's modulus, and 0.3 Poisson ratio, and a density of 7800 kg/m3. A displacement of zero in the direction of the load, x direction in our case, was imposed at the free end of the output bar. Table 1 Properties of the unidirectional composite adherent. Quantity Unit

2. Numerical model 2.1. Specimen description The double lap joint shown in Fig. 1 is proposed in this study. The horizontal shift between the central adherent and the extreme one is 2 mm. The overlap length is 14 mm. Each of the extreme adherents is 2 mm thick while the central adherent has a thickness twice as much as the extreme adherents. The adhesive layer's thickness is 100 μm. The width of the whole assembly is 12 mm.

103

Density Exx Eyy Ezz Gxy Gxz Gyz νxy νxz νyz

kg/m3 GPa GPa GPa GPa GPa GPa Dimensionless Dimensionless Dimensionless

Fig. 1. Double lap joint specimen.

Fiber, Carbon T300-J, 396Tex, Torayca, SPFICAR [19]

Resin-RTM 6 Hexcel [19]

UD

1780 230 15 15 50 50 5.78 0.278 0.278 0.300

1100 2.89 2.89 2.89 1.07 1.07 1.07 0.350 0.350 0.350

1360 92.8 4.25 4.25 1.75 1.75 1.57 0.321 0.321 0.350

104

P. Saleh et al. / International Journal of Adhesion & Adhesives 63 (2015) 102–107

Table 2 Values of the reference model's parameters. Adhesive Parameters Values

L0 (mm) 14

Adherent e0 (mm) 0.1

w (mm) 12

E0 (GPa) 1

νa 0.4

Ls (mm) 16

eextreme (mm) 2

ecentral (mm) 4

w (mm) 12

Exx, Eyy, Ezz, Gxy, Gxz, Gyz νxy, νxz, νyz Values in Table 1

vertical plane defined by (y¼ w/2) which serves also as one of the two planes of symmetry of the sample, the vertical one. Thus av τxz (t ) =

1 L 0 × Ta

L0

∫0 ∫0

Ta

⎛ ⎞ w τxz ⎜ x, y = , z, t ⎟ dxdz ⎝ ⎠ 2

(2)

Assuming further the stress to be constant in the direction of the thickness We conclude av τxz (t ) =

1 L0

∫0

L0

⎛ ⎞ w T τxz ⎜ x, y = , z = a , t ⎟ dx ⎝ 2 2 ⎠

(3)

A coefficient quantifying the shear stress concentration in the specimen is computed by the ratio of the peak shear stress (at the edges) to the average shear stress in the adhesive joint

Θ=

The tied node-to-surface constraint is used between adherents and adhesive, while a frictionless interaction is imposed between the output bar and the specimen. The element C3D8R element, explicit linear, from the family of 3D stress, with reduced integration and hourglass control is used for the different parts. The mesh size is 1 mm for the bar's elements, 0.2 mm for the adherent's elements and 25  500  500 μm3 for the adhesive's element. A refinement of 25  10  500 μm3 same to the one done in [17] is applied at the edges of the adhesive because of the peak of stress concentration at these zones. Fig. 3 illustrates the FEM model in Abaqus for the system specimen-bar. Adherents were assigned a material having anisotropic properties with a specified material orientation, since the module standard of the software Abaqus does not have the composite layup functionality.

3. Stress concentration coefficient The objective of the paper being the shear stress state in the adhesive layer, stress distributions were plotted along the overlap length. Average shear stress in the adhesive layer is computed using the following formula:

1 L 0 × Ta × W

L0

W

∫0 ∫0 ∫0

Ta

τxz (x, y , z, t ) dxdydz

(4)

Since it is already known that the stress is not homogeneous along the overlap length, it is expected that the value of this coefficient exceeds one in the zones close to the edges of the joint and becomes inferior to unity in the zones around the middle of the joint's length. In the ideal case, when the stress field is perfectly homogeneous, this coefficient will always be equal to one. The target is to establish numerically this coefficient in terms of the geometry and the materials of the specimen and since the SHPB technique measures easily the average shear stress in the joint, Eq. (4) allows calculating the peak shear stress in the adhesive joint.

Fig. 2. (a) Loaded specimen and (b) impact pulse.

av τxz (t ) =

max τxz (x, y , z, t ) av τxz (t )

(1)

The shear stress being constant in the direction of the width, from previous studies [14] and [17], values can be extracted at the

4. Reference study The target is to study the influence of many geometrical and material parameters related to the specimen on this stress concentration coefficient. E0 and νa are the adhesive's Young's modulus and Poisson's ratio respectively. To arrange this study, a set of default parameters is defined; then for each simulation, one parameter is changed while the others remain as set by default. This allows examining the influence exclusively of the changed parameter on the stress concentration coefficient. This set of default parameters is known as reference model. The corresponding values are found in Table 2. Fig. 4 below shows the evolution of the stress concentration coefficient during 40 μs, duration which is twice the input velocity duration. This duration is sufficient to ensure that the pulse has traveled through the specimen and the equilibrium is well established. This is the same duration considered by Hazimeh et al. in [17]. The data are recorded each 0.25 μs.The shear stress concentration coefficient increases and decreases sharply at the first few microseconds then it decreases at a slower rate, reaching an almost constant value. In a previous study carried out by the authors [20], it has been shown that the evolution of the average shear stress in the joint with respect to the time reaches its peak at about 19.5 μs. Thus the denominator of Eq. (4) will be measured at this instant. After 6 μs, the dynamic equilibrium is established; at that point, the dynamic heterogeneity due to the parameters of the test setup and

P. Saleh et al. / International Journal of Adhesion & Adhesives 63 (2015) 102–107

105

Fig. 3. Numerical model using ABAQUS Explicit Standard module.

Stress concentration coefficient

25

Table 3 Values of adhesive's parameters.

20 15 10 5 0

0

10

20 30 Time (micro-second)

40

50

Fig. 4. Stress concentration coefficient versus time.

wave propagation has disappeared, this is explained by the stabilization of θ to a constant value in the graph of Fig. 4. However, the structural heterogeneity, intrinsic to the body of the specimen, remains till the end this is shown in the same graph that in the permanent regime, the shear stress concentration coefficient is about 1.45. If the structural heterogeneity was not intrinsic to the specimen, one should have obtained θ equal one and thus a homogeneous shear stress field. The coming step is to change different parameters related to the specimen and measure the value of θ for each configuration at the permanent regime (19.5 μs especially) in order to establish a relationship between this coefficient and the specimen's parameters.

5. Parametric study 5.1. Adhesive parameters As listed in Table 3, three adhesive related parameters will be studied besides the reference values: joint's thickness, overlap length and adhesive's Young's modulus. Fig. 5 shows that the stress concentration coefficient decreases with a thicker adhesive. Indeed, the softness of the adhesive increases with the increase of the layer's thickness and thus the

Parameters

Values

e0 (mm) L0 (mm) E0 (GPa)

0.02, 0.05, 0.2, 0.3, 0.5 10, 12, 16, 18, 20 0.2, 0.5, 2, 5

shear stress field tends to become more homogenous which yields to a decrease in the values of the stress concentration coefficient. Graphs in Figs. 6 and 7 show that the stress concentration factor decreases for shorter overlap lengths and softer adhesive and thus the shear stress field homogeneity improves. When the overlap length increases, the stiffness of the joint increases and hence the peak values of shear stress at the joint's edge increase which means that the values of the stress concentration coefficients increase similarly. 5.2. Adherents parameters In addition, two adherent related parameters will be studied besides the reference values: adherents' thicknesses and fiber volume fraction. The values are summarized in Tables 4a and 4b. Top and bottom adherents refer exactly to extreme plates in Fig. 1 while the central adherent is not other than the middle plate. The configuration of the specimen is always such as the thickness of the middle adherent is twice as much as the extreme adherents. Investigating the adherents' thicknesses, we can conclude that the stress concentration coefficient decreases with thicker adherents. The increase thickness of the adherents induces an increase in the longitudinal stiffness and thus the shear stress field becomes more homogenous which appears by an increase in the stress concentration coefficient as shown in Fig. 8. On the other hand, increasing the fiber volume fraction induces an increase in the longitudinal stiffness of the adherents and hence a decrease in the stress concentration coefficient (Fig. 9).

106

P. Saleh et al. / International Journal of Adhesion & Adhesives 63 (2015) 102–107

1.54

Stress concentration coefficient

Stress concentration coefficient

2.5 2 1.5 1 0.5 0

0

0.1

0.2

0.3

1.52 1.5 1.48 1.46 1.44 1.42 1.4 1.38 1.36

0

2

0.4

4

6

8

10

12

Central adherent's thickness (mm)

Adhesive's thickness (mm)

Fig. 8. Effect of adherent's thicknesses on the stress concentration coefficient.

Fig. 5. Effect of adhesive's thickness on the stress concentration. 1.6 Sstress concentration coefficient

Stress concentration coefficient

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4

1.5 1.45 1.4 1.35 1.3 1.25

0.2 0

1.55

0

5

10

15

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fiber volume fraction

25

Overlap length (mm)

Fig. 9. Effect of the carbon fiber volume on the stress concentration coefficient.

Fig. 6. Effect of overlap length on stress concentration. 2.5

Stress concentration coefficient

2.5 2

2 1.5

1.5 θ

1

Theta Linear (Theta)

1

0.5 0.5

0

0

1000

2000

3000

4000

5000

6000 0

Adhesive's Young’s modulus (MPa)

0

2

Fig. 7. Effect of the adhesive's young modulus on the stress concentration.

4 lnλ

6

8

Fig. 10. Variation of the stress concentration coefficient in terms of the unified parameter. Table 4a Values of adherents' thicknesses for four configurations.

6. Unified parameter

Similar adherents thickness (mm) Bottom eextreme

Central ecentral

Top eextreme

1 3 4 5

2 6 8 10

1 3 4 5

Table 4b Values of adherents' fiber volume fraction. Vf (%)

0.2

0.3

0.4

0.5

0.6

It has been shown in [14] that the SHPB technique gives good estimation of the average shear stress in the joint. However, practically, the edges of the joint suffer high peaks of stresses, higher than the values measured by the SHPB technique hence the need of the stress concentration coefficient θ to correct the values measured by the SHPB technique. In [14], a unified parameter was proposed to evaluate analytically θ for steel adherents. Later on, those values were used in [15] to correct the experimental results of SHPB set-up. Upon the results found in the parametric study carried out above, one can define a unified parameter λ whose mathematical expression involves the parameters changed in the numerical study. The defined expression for this unified parameter is

P. Saleh et al. / International Journal of Adhesion & Adhesives 63 (2015) 102–107

λ=

E0·L 0 Vf ·e0 ·e

(5)

Since E0 and L0 have the same tendency to increase the stress concentration coefficient, they were multiplied together in the numerator, while Vf, e0 and e have the tendency to decrease the stress concentration factor and hence they were multiplied together in the denominator. The final dimension of λ is GPa/mm which is a linear stiffness. The plot of Fig. 10 shows the evolution of θ as function of the natural logarithm of the unified parameter lambda. A linear regression helps to deduce a mathematical equation to calculate θ knowing that λ is already calculated from the mechanical and geometrical configuration of the bonded double lap joint. The approximate linear equation is

θ ≈ 0.324 ln λ − 0.195

(6)

This latter equation is valid for λ ranging between 35 GPa/mm and 875 GPa/mm. When the configuration of the specimen leads to values of λ close to 35 GPa/mm, it could be deduced that the shear stress field in the joint is quasi-homogenous and the value of θ is tending to 1. In other words, when the type of adhesive and the unidirectional laminates adherents are chosen, the geometry of the specimen could be designed according to the following equation:

L0 V ≈ 35· f e ·e0 E0

(7)

On the other hand, for heterogeneous shear stress field, the shear strength could be calculated by multiplying the value of the average shear stress measured from the SHPB technique by the value of θ calculated from Eq. (6).

7. Conclusion The edge effect aspect in a double lap unidirectional composite bonded joint was quantified through a 3D FEM analysis of dynamic shear loading applying the SHPB technique. A stress concentration coefficient was defined as the ratio of the maximum shear stress in the joint, which is at the edge, over the average shear stress in the joint. For stiffer substrates and softer adhesive, the stress field homogeneity improves and thus the stress concentration coefficient becomes closer to the unity. This stiffness is geometrical, through the thickness and overlap length, and mechanical through the Young's modulus and the fiber volume fraction. A unified parameter, involving all the geometrical and mechanical parameters of the specimen, is defined. An approximate linear mathematical equation is established to calculate θ

107

from the unified parameter. This has the advantage to define the geometry of the specimen that produces the best homogeneity of the stress field. Also, the joint shear strength could be estimated from the experimental measurement of the average shear stress.

References [1] Mylonas CA, De Bruyne NA. Adhesion and adhesives. The Netherlands: Elsevier; 1951. p. 96. [2] G. Henning Plastie Kautschuk 12 1965 459. [3] Da Silva Lucas FM, Lopes Maria João CQ. Joint strength optimization by the mixed-adhesive technique. Int J Adhes Adhes 2009;29:509–14. [4] Adams RD, Peppiatt NA. Stress analysis of adhesive bonded lap joints. J Strain Anal 1974;9(3):185–96. [5] Sage GN. Aspects of bonded joint design in carbon fiber reinforced plastic. Adhesion 1979;3:123–47. [6] Adams RD, Chambers SH, Strother PJA, Peppiatt NA. Rubber model for adhesive lap joints. J Strain Anal 1973;8(1):52–7. [7] Adams RD, Wake WC. Structural adhesive joints in engineering. London: Applied Science Publishers; 1984. [8] Adams RD, Harris JA. A critical assessment of the block impact test for measuring impact strength of adhesive bonds. Int J Adhes Adhes 1996;16:61–71. [9] Wada H, Suzuki K, Murase K, Kennedy TC. Evaluation of impact tensile strength for PMMA/Al butt adhesive joint impact engineering application. In: Proceedings of the 4th International symposium on impact engineering. Kumamoto, Japan; 16–18 July 2001. p. 463–8. [10] Cognard JY, Davies P, Gineste B, Sohier L. Development of an improved adhesive test method for composite assembly design. Compos Sci Technol 2005;65:359–68. [11] Cognard JY, Davies P, Sohier L, Créachcadec R. A study of the non-linear behavior of adhesively-bonded composite assemblies. Compos Struct 2006;76:34–46. [12] Vable M, Maddi JR. Boundary element analysis of adhesively bonded joints. Int J Adhes Adhes 2006;26:133–44. [13] Zou GP, Taheri F. Stress analysis of adhesively bonded sandwich pipe joints subjected to torsional loading. Int J Solids Struct 2006;43:5953–68. [14] Challita G, Othman R. Finite-element analysis of SHPB tests on double-lap adhesive joints. Int J Adhes Adhes 2010;30:236–44. [15] Challita G, Othman R, Casari P, Khalil K. Experimental investigation of the shear dynamic behaviour of double-lap adhesively bonded joints on a wide range of strain rates. Int J Adhes Adhes 2011;31:146–53. [16] Hua Y, Gu L, Trogdon M. Three-dimensional modeling of carbon/epoxy to titanium single-lap joints with variable adhesive recess length. Int J Adhes Adhes 2012;38:25–30. [17] Hazimeh R, Challita G, Khalil K, Othman R. Finite element analysis of adhesively bonded composite joints subjected to impact loadings. Int J Adhes Adhes 2015;56:24–31. [18] HUNSTMAN Adhesives, Araldite 2031; February 2009. [19] Hallal A, Younes R, Fardoun F, Nehme S. Improved analytical model to predict the effective elastic properties of 2.5D interlock woven fabrics composite. Compos Struct 2012;94:3009–28. [20] Saleh P, Challita G, Hazimeh R, Khalil K. Influence of composite adherents properties on the dynamic behavior of double lap bonded joint. Int J Chem Nucl Mater Met Eng 2015;9(4):410–6.