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Strength prediction of tubular composite adhesive joints under torsion
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 67 (2007) 1340–1347 www.elsevier.com/locate/compscitech
Strength prediction of tubular composite adhesive joints under torsion Je Hoon Oh
*
Department of Mechanical Engineering, Hanyang University, 1271 Sa1-dong, Sangrok-gu, Ansan-shi, Gyeonggi-do 426-791, Republic of Korea Received 7 March 2006; received in revised form 26 September 2006; accepted 28 September 2006 Available online 8 December 2006
Abstract This paper addresses prediction of the strength of tubular adhesive joints with composite adherends by combining thermal and mechanical analyses. A finite element analysis was used to calculate the residual thermal stresses generated by cooling down from the adhesive cure temperature, and a nonlinear analysis incorporating the nonlinear adhesive behavior was performed to accurately estimate the mechanical stresses in the adhesive. Joint failure was estimated by three failure criteria: interfacial failure, adhesive bulk failure, and adherend failure. The distributions of residual thermal stresses were investigated for various stacking angles. The effect of residual thermal stresses on joint strength was also taken into consideration. The results indicate that the residual thermal stresses, depending on the stacking angle, have a significant influence on the failure mode and strength of adhesive joints when a subsequent mechanical load is applied. Good agreement is also obtained between the predicted joint strength and the available experimental data. 2006 Elsevier Ltd. All rights reserved. Keywords: A. Adhesive joints; A. Polymer-matrix composites (PMCs); B. Nonlinear behavior; B. Strength; C. Laminates theory
1. Introduction Adhesive joints have been widely used for composite structures because they distribute the load over a larger area than mechanical joints and do not require holes, resulting in a reduction of the stress concentration and an increase in the strength of the joint. Moreover, stress concentrations in joints can be further reduced by adhesives with nonlinear material properties. Several types of adhesive joints are used for joining composite structures: single lap joint, double lap joint, stepped joint, scarf joint, etc. Among these joints, the single lap joint is the most popular because of its manufacturing ease and relatively low cost. Since the performance of joints usually determines the structural efficiency of composite structures, an extensive knowledge of the behavior of adhesive joints and the related effect on joint strength is essential for design purposes. Many researchers have conducted studies on stress distribution in tubular adhesive joints subject to torsion to predict joint strength, given the geometry of the joint and the elastic *
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0266-3538/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.09.021
material properties of the adhesive and adherend [1–7]. Recently, nonlinear analyses including the nonlinear properties of the adhesive have been developed to estimate the torque transmission capability more easily and accurately [8–10]. However, previous analytical studies have focused on adhesive joints with isotropic adherends (e.g., steel). Since composite materials have anisotropic properties that depend on their stacking angle and sequence, the analysis with isotropic adherends is limited in describing the behavior of the adhesive joint with composite adherends. Only a few attempts so far have been made to investigate the stress distribution in tubular adhesive joints under torsion with composite adherends [11–14]. However, the analyses considered linear properties of the adhesive, and the effective properties of composites were calculated from a laminate plate theory even though the composite adherend is not a plate but actually a tube. In order to overcome this limitation and get accurate stress distributions in the joint with composite adherends, a nonlinear solution was developed by the author [15]. The nonlinear analysis incorporated both the nonlinear adhesive behavior and the behavior of the laminated composite tube with different stacking angles and sequences.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
Another important factor that influences the response of adhesive joints is residual thermal stress generated mostly from the different coefficients of thermal expansion (CTEs) of the adhesive and adherend. These thermal stresses may become large enough to initiate adhesive debonding at the interface between the adhesive and adherend or adhesive cracking even under low external loads. Therefore, thermally-induced stresses can be as critical as mechanically-induced stresses in the analysis of tubular adhesive joints. Thorough understanding of the effect of residual thermal stresses on joint strength is essential, especially in the analysis of composite adhesive joints because composites have directionally dependent CTEs along the material axes, unlike isotropic materials. In this study, the strength of adhesive joints under torsion was predicted using the combined thermal and mechanical analyses of joint behavior when the adherends were composite tubes. A finite element analysis was performed to evaluate the residual thermal stresses developed in the joint, and the mechanical stresses in the adhesive layer were calculated using the developed nonlinear analysis [15]. Three different joint failure modes were considered to predict joint failure: interfacial failure, adhesive bulk failure, and adherend failure. The influence of the composite adherend stacking angle on the residual thermal stresses was investigated, and how the residual thermal stresses affect the joint strength was also discussed. Finally, the predicted results were compared with experimental results available in literature [16]. 2. Stress analysis of a tubular adhesive joint Fig. 1(a) shows the geometric configuration of the tubular single lap adhesive joint with laminated composite tube adherends. For numerical calculations, the adhesive layer is assumed to be thin and much more flexible than the adherends. The laminated composite tube of N layers is assumed to be subject to axisymmetric thermomechanical loading as illustrated in Fig. 1(b). Each layer is considered to be a homogenous, orthotropic material in the layer principal coordinates, which means that it becomes a monoclinic material in the global coordinates for off-axis orientations. Perfect bonding between the layers is also assumed in the analysis. Generally, the different CTE and stiffness of the adhesive and the adherend cause internal residual thermal stresses even in the absence of external loads. In order to investigate the effect of such residual thermal stresses on adhesive joint strengths, the following analyses were performed: residual thermal stresses by finite element analysis, and mechanical stresses by nonlinear analysis. 2.1. Analysis of residual thermal stresses The stresses induced by change in temperature DT (due to cool down from the adhesive cure temperature to room
1341
L Adherend 2 Adhesive Adherend 1
T
r1i r1o
r
r2i r2o z
T
F T
z r N
pi
k 1
ri( k )
2 po
1
3
po
ro( k )
Temperature change
Fig. 1. Geometric configuration: (a) tubular single lap adhesive joint and (b) laminated composite tube.
temperature), are discussed in this section. Under axisymmetric conditions of the joint configuration, all displacements, strains, and stresses are independent of the rotational angle h. Thus, the stress–strain–temperature relation has the form r ¼ Dðe aDT Þ
ð1Þ
where r and e are (4 · 4) stress and strain matrices, respectively; D is a (4 · 4) stiffness matrix, and a is the CTE matrix. A finite element method was employed to analyze the residual thermal stresses in the adhesive joint. From the Galerkin formulation, the element stiffness matrix and element force vector due to DT are Z ke ¼ 2p ðBT DBÞr dA ð2aÞ Ze ð2bÞ f e ¼ 2p ðBT DaDT Þr dA e
where B is the element strain–displacement matrix and A is the cross-sectional area of the element. The matrix B is defined as 3 2 N 1;r 0 N 2;r 0 N m;r 0 7 6N N2 Nm 6 1 0 0 0 7 7 6 ð3Þ B¼6 r r r 7 6 0 N 1;z 0 N 2;z 0 N m;z 7 5 4 N 1;z N 1;r N 2;z N 2;r N m;z N m;r
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J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347 Outer adherend Inner adherend r2i r1o r1i
r2o
Bonding Length
Adhesive
Fig. 2. Finite element mesh for the tubular single lap adhesive joint.
where m is the number of nodes for the element and i i N i;r ¼ oN and N i;z ¼ oN are the partial derivatives of shape or oz functions Ni. After assembling the global stiffness matrix and global force vector from the element stiffness and force matrices in Eq. (2), the displacement of each node can be obtained by solving a set of simultaneous algebraic equations. The element stresses are then determined using the relationship in Eq. (1). Fig. 2 illustrates the finite element model for calculation of residual thermal stresses. The axisymmetric model for the analysis was composed of 8-node axisymmetric isoparametric elements. To take into account the stress concentration, the mesh near the edge of the adhesive layer th was made very fine. The radial stress rth r , hoop stress rh , th th axial stress rz , and shear stress srz were obtained from the finite element analysis, and these stress components were used to investigate the effect of thermal stresses on the joint strength. 2.2. Mechanical stress analysis under torsion When the tubular adhesive joint in Fig. 1 is subjected to a torque T, torques T1 and T2 are produced in both adherends 1 and 2 at any given cross-section. Unlike for the case of isotropic adherends, the resultant axial forces F1, F2, and the resultant pressures p1, p2 are also produced in both adherends. Combining the equilibrium equation with the constitutive equation and compatibility condition gives the following system of differential equations including the nonlinear properties of the adhesive [15]: d2 T 2 ¼ fA1 ðcarh Þ þ A2 ðcarh ÞgT 2 þ fB1 ðcarh Þ þ B2 ðcarh ÞgF 2 dz2 þ fC1 ðcarh Þ þ C2 ðcarh Þgp2 þ D1 ðcarh Þ ð4aÞ d2 F 2 ¼ fA3 ðcarz Þ þ A4 ðcarz ÞgT 2 þ fB3 ðcarz Þ þ B4 ðcarz ÞgF 2 dz2 þ fC3 ðcarz Þ þ C4 ðcarz Þgp2 þ D2 ðcarz Þ p2 ¼ E1 ðear Þz þ E2 ðear Þ
ð4bÞ ð4cÞ
BCs :
T 2 ¼ 0;
F2 ¼ 0
and
dF 2 ¼0 dz
at z ¼ 0 ð4dÞ
T2 ¼ T;
F 2 ¼ 0 and
dF 2 ¼0 dz
at
z¼L
ð4eÞ where 2pr21o df ðcarh Þ 1o 2pr21o df ðcarh Þ 2i ðchz ÞT ; A2 ¼ ðchz ÞT ð5aÞ A1 ¼ a wrh g dcrh wrh g dcarh 2pr1o df ðcarz Þ 1o 2pr1o df ðcarz Þ 2i ðe Þ ; A ¼ ðez ÞT ð5bÞ A3 ¼ 4 z T wrz g dcarz wrz g dcarz 2pr21o df ðcarh Þ 1o 2pr21o df ðcarh Þ 2i B1 ¼ ðc Þ ; B ¼ ðchz ÞF ð5cÞ 2 hz F wrh g dcarh wrh g dcarh 2pr1o df ðcarz Þ 1o 2pr1o df ðcarz Þ 2i ðe Þ ; B ¼ ðez ÞF ð5dÞ B3 ¼ 4 z F wrz g dcarz wrz g dcarz 2pr21o df ðcarh Þ 1o 2pr21o df ðcarh Þ 2i C1 ¼ ðc Þ ; C ¼ ðchz Þp ð5eÞ 2 hz p wrh g dcarh wrh g dcarh 2pr1o df ðcarz Þ 1o 2pr1o df ðcarz Þ 2i ðe Þ ; C ¼ ðez Þp ð5fÞ C3 ¼ 4 z p wrz g dcarz wrz g dcarz 2pr21o df ðcarh Þ dwrh a D1 ¼ A1 T þ c ; wrh dcarh dz rh 2pr1o df ðcarz Þ dwrz a c D2 ¼ A3 T þ ð5gÞ dcarz dz rz wrz In Eq. (5), (chz)T, (chz)F, and (chz)p are shear strains in the composite adherend at r = r1o or r = r2i due to unit torque, unit axial force, and unit pressure, respectively. Similarly, (ez)T, (ez)F, and (ez)p are axial strains in the composite adherend at r = r1o or r = r2i due to unit torque, unit axial force, and unit pressure, respectively. The function f(c) represents the relationship between the shear stresses and shear strains in the adhesive layer. The unit load responses (i.e., strains in the composite adherends due to each unit load) in Eq. (5) can be determined from analysis of the laminated composite tube. The radial and shear stress distributions in the adhesive are computed after solving (4) for the torque, axial force, and pressure distributions in adherend 2.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
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3. Failure prediction of the adhesive joint
F 11 r21 þ F 22 r22 þ F 66 r26 þ F 1 r1 þ F 2 r2 þ 2F 12 r1 r2 ¼ 1
There are generally three kinds of joint failure modes: interfacial failure between the adhesive and adherend, bulk failure in the adhesive, and adherend failure. In the interfacial failure mode, the bonding strength at the interface between the adhesive and adherend is assumed to be lower than the adhesive bulk shear strength because of the relatively large residual thermal stresses, resulting in failure at the interface between the adhesive and adherend. The following failure criterion is defined to predict the failure condition of the interfacial failure mode [4,17]. 2 2 2 2 2 2 rr rh rz shz srz srh þ þ þ þ þ ¼1 ST ST ST SS SS SS
where r1 and r2 are the ply stresses in the longitudinal and transverse directions to fibers, r6 is the ply shear stress, and Fi and Fij are strength parameters defined by material strengths:
ð6Þ where ST and SS are the bulk tensile strength and bulk shear strength of the adhesive, respectively. The joint strength in the interfacial failure mode can be estimated by calculating the residual thermal and mechanical stress components that satisfy the failure criterion of Eq. (6). The bulk failure mode is when the bonding strength at the interface is high enough so that joint failure occurs in the adhesive layer, not at the interface. In this failure mode, the residual thermal stresses are not large compared to the adhesive bulk strength. Because the adhesive has nonlinear material properties, adhesive failure does not occur even after the equivalent stress in the adhesive reaches the bulk strength. Therefore, the maximum strain criterion is more suitable to predict adhesive bulk failure of the joint. The effective strain ee of the adhesive is written as [18] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ee ¼ ðe þ e22 þ e23 Þ ð7Þ 3 1 where ei are the three principal strains. The adhesive joint is assumed to fail in the bulk failure mode when the effective strain exceeds the failure strain of the adhesive. If the shear strain in the adhesive is dominant and other strain components are small enough to be neglected, the maximum shear strain criterion, rather than the maximum strain criterion, can be used to predict the adhesive bulk failure. The fracture energy criterion can provide a reasonable prediction of brittle failure load for adhesive joints with isotropic adherends when the bonding length is sufficiently long [19,20]. However, it gives somewhat less accurate predictions for composite adhesive joints and does not consider the variation of failure torques with bonding length [15]. Thus in this work, the maximum strain criterion is mainly used to predict the bulk failure of joints. In the adherend failure mode, it is assumed that the composite adherend fails before the adhesive. Once the torque, axial force, and radial pressure applied at each adherend are computed from Eq. (4), the stresses in the adherend are estimated by analysis of the composite tube [15]. Then, failure of the composite adherend is predicted using the Tsai-Wu failure criterion:
ð8Þ
1 1 1 1 ; F2 ¼ T C ð9aÞ XT XC Y Y 1 1 1 ðF 11 F 22 Þ1=2 F 11 ¼ T C ; F 22 ¼ T C ; F 66 ¼ 2 ; F 12 ¼ 2 X X Y Y S ð9bÞ F1 ¼
where XT and XC represent the tensile and compressive strengths in the fiber direction, YT and YC indicate those in the transverse direction, and S is the shear strength. In order to predict the torque transmission capability T of the adhesive joint which considers the above three failure modes, the transient interpolation function fi is introduced as follows [4]: T ¼ MinðT ad ; T b Þ fi þ T i ð1 fi Þ;
0 6 fi 6 1
ð10Þ
where Tad, Tb and Ti are the torque transmission capability in the adherend failure mode, bulk failure mode, and interfacial failure mode, respectively. Note that fi is dependent on the residual thermal stresses resulting from fabrication. The effect of the residual thermal stresses can be represented by the following initial failure index ki sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi th 2 th 2 th 2 th 2 rr rh rz s ð11Þ ki ¼ þ þ þ rz ST ST ST SS where the superscript th denotes thermal stresses. The relationship between fi and ki is established by comparing analytic results with experiments. 4. Numerical results and discussion Based on the stress and strength analysis in the previous section, a computer program was developed to calculate the stress and strength of the composite adhesive joint. In order to check the validity of the analysis and predict joint strength, the results of the analysis were compared with available experimental results. Choi and Lee [16] tested steel–composite adhesive joints to investigate the effect of the stacking sequence of the composite adherend on the torque transmission capability by varying the stacking angle of the outer composite adherend from [±5]nT to [±45]nT with a ±5 interval. The geometric configuration and material properties of the joint are listed in Tables 1 and 2, respectively. Test results indicated that the torque transmission capabilities of the steel–composite joints increased until [±25]nT, and then decreased with increasing stacking angles as shown in Fig. 3. Moreover, the joint with a [±45]nT stacking angle failed at the lowest torque although the composite tube with that stacking angle has the highest torsional
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J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
Table 1 Configuration of the tubular steel–composite adhesive joint tested in Ref. [16] Inner radius of steel (r1i) Outer radius of steel (r1o) Inner radius of composite (r2i) Outer radius of composite (r2o) Adhesive thickness (g) Bonding length (L)
0 mm 8.4 mm 8.5 mm 10.5 mm 0.1 mm 15 mm
Stacking sequence of composite
[±/]nT (/ = 5,10,. . .,45)
end due to the residual thermal stresses induced during fabrication. The high residual thermal stresses can lead to interfacial debonding, which causes early failure of the joint before the adhesive reaches its failure strain. Prior to analyzing the residual thermal stresses, the variation of CTE of the composite angle-ply tube in the longitudinal, transverse and radial directions was first investigated and presented in Fig. 4. The effective longitudinal CTE is calculated as along ¼
Table 2 Material properties of the adhesive and adherends Adhesive epoxya
E1 (GPa) E2 = E3 (GPa) G12 = G13 (GPa) G23 (GPa) v12 = v13 v23 a1 (106/ C) a2 = a3 (106/ C) XT (MPa) XC (MPa) YT (MPa) YC (MPa) S (MPa) Shear strain limit Ply thickness (mm)
1.30 1.30 0.46 0.46 0.41 0.41 72 72 45 80 – – 30 0.4 –
Adherend Steel
Carbon/epoxyb
200 200 76.9 76.9 0.30 0.30 11.7 11.7 350 – – – – – –
128 10 4.49 3.58 0.28 0.47 0.5 27 1800 1400 45 150 50 – 0.15
ð1Þ
rigidity and strength. These results are the opposite of what would be guessed intuitively, that as the stacking angle increases, the torsional resistance increases, resulting in higher torque transmission capability of the joint. This phenomenon might be explained by the initial strength reduction at the interface between the adhesive and adher-
Torque (N·m)
80
60
40
20
10
15 20 25 30 35 Stacking angle (degree)
40
45
Fig. 3. Torque transmission capabilities of steel–composite joints for various stacking angles / in [±/]nT (The data are taken from Ref. [16]).
ð13Þ
ð1Þ
ri DT ð1Þ
where ri and Dri are the inner radius of the tube and the change in the inner radius after the temperature change DT, respectively. Similar to the effective transverse CTE, the effective radial CTE aradial is 90 out-of-phase with the effective longitudinal CTE along . aradial initially decreases from its large positive transverse CTE of the laminate, passes through zero between the fiber angle / = 40 and / = 45, and attains the maximum negative value near / = 60. The radial CTE then increases to a value somewhat below the longitudinal CTE of the laminate with increasing /. The radial CTE for a tube will become identical to the effective transverse CTE of a flat laminate as the aspect-ratio (ratio of the inner radius to the thickness) of the tube increases. It is interesting to note that the radius exhibits shrinkage for an increase in temperature when the fiber angle / is larger than / 45 in the [±/]nT angle-ply tube. Coefficient of thermal expansion (10-6/ ºC)
100
5
Dri ð1Þ
IPCO 9923, manufactured by Imperial Polychemicals Corporation (USA). b USN 150, manufactured by SK Chemicals (Korea).
0
ð12Þ
where e0z is the constant axial strain that is determined from the previous analysis for the unit temperature change DT. The effective longitudinal and transverse CTEs are quite similar to those for flat laminates calculated from classical lamination theory. However, unlike the flat composite laminates, the change in radius of composite tubes is associated with thermal loading. The effective radial CTE aradial is defined as aradial ¼
a
0
e0z DT
30 Longitudinal CTE Transverse CTE Radial CTE
25 20 15 10 5 0 -5
-10
0
15
30
45
60
75
90
Stacking angle φ (degree) Fig. 4. Variation of the longitudinal, transverse, and radial CTEs of composite tubes.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
Using the effective longitudinal, transverse and radial CTEs of the composite tube, the residual thermal stresses for various stacking angles were calculated through the analysis described in the previous section and shown in Fig. 5. In this calculation, a temperature change of DT = 60 C from the cure temperature of 80 C to room temperature of 20 C was a uniformly applied thermal loading. All the stress components increase progressively with increasing stacking angle. Stress concentrations are developed near the edge of the joint (z = 15 mm) at high stacking angles. The hoop stress (rh) has almost the same distribution as the axial stress (rz), and the shear stress (srz) increases linearly from negative to positive values and becomes maximum near the edge of the joint. It should be noticed that the radial stress (rr) plays a role of the peel stress. A negative peel stress can improve the joint strength by not causing cracks at the interface to open, while a positive peel stress helps the crack opening to induce interfacial failure of the joint. At low stacking angles, negative radial stress occurs at the whole range of the bonding length, but
it becomes positive as the stacking angle increases. There is even a stress concentration near the edge of the joint with a [±45]nT stacking angle. The variation of the radial CTE in Fig. 4 can explain why, under thermal loading, the radial stress in the adhesive increases and changes from compressive to tensile as the stacking angle increases. Since the composite tube has a relatively large positive radial CTE at low stacking angles, radial shrinkage of the composite tube occurs for a negative temperature change, lowering the radial stress in the adhesive. On the contrary, the almost zero or even negative radial CTE over / = 40 causes no change in radius or even radial expansion for thermal loading. This constrains the shrinkage of the adhesive that possesses a large CTE, resulting in the high tensile radial stress in the adhesive layer as shown in Fig. 5. Fig. 6 shows the distribution of the initial failure index ki in Eq. (11) with respect to the stacking angles of the outer composite adherend. The initial failure index increases as the stacking angle increases, and the maximum value of
20
20
= 45° (MPa) th r
Radial stress
th
(MPa)
= 25° = 15° = 5°
10
= 45°
10
= 35°
Hoop stress
1345
0
= 35° = 25° = 15° = 5°
0
-10
-20
-10
0
5
10
-30
15
0
Bonding range (z, mm)
5
10
20
30
= 45° 20
th
Shear stress
z
= 25°
10
rz
10
(MPa)
= 35° = 25° = 15° = 5°
th
(MPa)
= 45°
Axial stress
15
Bonding range (z, mm)
0
= 15° = 5°
0
- 10 - 10
0
5
10
Bonding range (z, mm)
15
- 20
0
5
10
15
Bonding range (z, mm)
Fig. 5. Distribution of residual thermal stresses for various stacking angles / in [±/]nT (DT = 60 C): (a) radial stress, (b) hoop stress, (c) axial stress, and (d) shear stress.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
1
200
0.8
160
0.6
Torque (N·m)
Initial failure index ki
1346
φ = 45° φ = 35°
0.4
φ = 25° φ = 15° φ = 5°
0.2
Experimental data [16]
Adhesive bulk failure Adherend failure
120
Predicted torque by Eq. (10)
80 40 Interfacial failure
0
0
5
0
15
10
1
5
10
15 20 25 30 35 Stacking angle φ (degree)
40
45
0.9
at the adhesive thickness of 0.1 mm [4], their influence depends on the stacking angle of the composite adherend in the steel–composite joint even with the same adhesive thickness. Thus, the influence of residual thermal stresses
0.8 0.7 0.6 0.5
5
10
15
20
25
30
35
40
45
1
Stacking angle φ (degree)
Fig. 6. (a) Distribution of the initial failure index for various stacking angles / in [±/]nT. (b) Variation of the maximum initial failure index with /.
ki occurs near the edge of the joint (z = 15 mm) due to the stress concentration. The increase in the maximum ki with the fiber angle / is very gradual up to / = 15, followed by a much sharper increase as shown in Fig. 6(b). The maximum ki approaches unity at / = 45, which means the initial interfacial failure due to residual thermal stress may occur in the joint with that stacking angle without any external torque. This clearly illustrates that the stacking angle of the composite adherend has a significant influence on the residual thermal stress developed in the joint and eventually the joint strength as well. In order to estimate the torque capacity of the adhesive joint, the test results in Fig. 3 were compared with numerical results considering the three different failure modes. Fig. 7 presents the predicted torque transmission capabilities using the assumptions of interfacial, bulk, and adherend failures as well as the experimental results. The adherend failure mode can explain the experimental data below a fiber angle of / = 10, while the interfacial failure mode can represent a joint failure above / = 30. Notice that there is an abrupt reduction in the interfacial joint strength at high stacking angles due to the effect of residual thermal stresses. Unlike the steel–steel adhesive joint in which the effect of residual thermal stresses is negligible
Failure index
0
0.5
Failure shear strain
0.4 Failure index at r = r2o
0.8
0.3
0.6
0.2 γ raθ
0.4 0.2 0
0.1
γ ≈ε ≈ 0 a rz
a r
0
Failure index at r = r2i
0
10 5 Bonding range (z, mm)
15
Strains in the adhesive
1.2 0.4
b
0
Fig. 7. Predicted torque capacities based on the failure criteria with the experimental data in Fig. 3.
-0.1
35
Stresses in the adhesive (MPa)
Maximum initial failure index ki,max
Bonding range (z, mm)
30 25 τ raθ
20 15 10 5
τ rza
0 -5
σ ra
0
10 5 Bonding range (z, mm)
15
Fig. 8. Failure index, strain, and stress distributions (applied torque T = 91.7 Nm) (a) Failure index of the composite adherend [±15]nT and strains in the adhesive, (b) stresses in the adhesive.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
should be taken into consideration while designing adhesive joints, especially with composite adherends. When adhesive bulk failure is assumed, the torque capacity tends to increase with increasing stacking angle as expected. Under the same assumption, it is also observed that failure of the composite adherend occurs first at the outer radius of the adherend before adhesive failure until the fiber angle / = 20. As an example, Fig. 8 illustrates the failure index of a composite adherend with a stacking angle [±15]nT and stress and strain distributions in the adhesive layer when the applied torque was 91.7 Nm. Adherend failure occurs at the outer radius (r = r2o) before the adhesive shear strain reaches its failure strain of 0.4. The radial and shear stresses ðrar ; sarz Þ and radial and shear strains ðear ; carz Þ are negligibly small compared to the shear stress sarh and shear strain carh , respectively. From Figs. 6(b) and 7, adhesive bulk failure mode can be assumed to be dominant when the maximum initial failure index ki,max is less than 0.52. Therefore, the transient interpolation function fi can be defined as a function of ki,max [4]: fi ¼ 1 fi ¼ e
when
k i; max 6 0:52
30ðk i; max 0:52Þ
when
k i; max > 0:52:
ð14aÞ ð14bÞ
Note that this relationship between fi and ki,max depends on what materials are used for the adhesive and adherends. The torque transmission capabilities predicted by Eqs. (10) and (14) are also plotted in Fig. 7. The predicted results are in good agreement with the experimental results. 5. Conclusions Combined thermal and mechanical analyses were performed to predict the torque transmission capabilities of tubular adhesive joints with composite adherends. The distribution of residual thermal stresses due to fabrication was investigated by varying the stacking angle of the composite adherend. The nonlinear behavior of the adhesive was incorporated into the mechanical analysis for the accurate prediction of joint strength. The validity of the analyses was also checked by comparison with the available experimental results of steel–composite tubular adhesive joints. Since the radial effective CTE of the composite tube is not constant but varies with the stacking angle like the longitudinal and transverse CTEs, the residual thermal stresses in the joint increase with the stacking angle. Especially, the high tensile radial thermal stresses act as peel stresses, thus lowering the joint strength. The adhesive bulk or adherend failure occurs at low stacking angles where the influence of residual thermal stresses is negligible, while the joints with high stacking angles fail at the interface between the adhesive and adherend because of high residual thermal stresses. Unlike a joint with isotro-
1347
pic adherends, the residual thermal stresses have a significant influence on the composite joint strength even though its adhesive thickness is relatively thin. The combined analyses with failure criteria accurately predict the torque capacity of tubular adhesive joints with composite adherends. References [1] Adams RD, Peppiatt NA. Stress analysis of adhesive bonded tubular lap joints. J Adhes 1977;9:1–18. [2] Alwar RS, Nagaraja YR. Viscoelastic analysis of an adhesive tubular joint. J Adhes 1976;8:79–92. [3] Lee DG, Kim KS, Lim YT. An experimental study of fatigue strength for adhesively bonded tubular single lap joints. J Adhes 1991;35: 39–53. [4] Lee SJ, Lee DG. Development of a failure model for the adhesively bonded tubular single lap joint. J Adhes 1992;40:1–14. [5] Lee SJ, Lee DG. Optimal design of the adhesively bonded tubular single lap joint. J Adhes 1995;50:165–80. [6] Choi JH, Lee DG. The torque transmission capabilities of the adhesively bonded tubular single lap joint and the double lap joint. J Adhes 1994;44:197–212. [7] Lee SJ, Lee DG. A closed-form solution for the torque transmission capability of the adhesively bonded tubular double lap joint. J Adhes 1994;44:271–84. [8] Lee SJ, Lee DG. An iterative solution for the torque transmission capability of adhesively bonded tubular single lap joints with nonlinear shear properties. J Adhes 1995;53:217–27. [9] Lee DG, Jeong KS, Choi JH. Analysis of the tubular single lap joint with nonlinear adhesive properties. J Adhes 1995;49: 37–56. [10] Lee DG, Oh JH. Nonlinear Analysis of the torque transmission capability of adhesively bonded tubular lap joints. J Adhes 1999;71:81–106. [11] Thomsen OT, Kildegaard A. Analysis of adhesive bonded generally orthotropic circular shells. in: Proceedings of the fourth European conference of composite materials, Stuttgart, Germany; 1990. p. 723– 29. [12] Chon CT. Analysis of tubular lap joint in torsion. J Compos Mater 1982;16:268–84. [13] Hipol PJ. Analysis and optimization of tubular lap joint subjected to torsion. J Compos Mater 1984;18:298–311. [14] El-hady FA, Kandil N. Optimization of tubular double lap joint configuration. Polym Compos 2002;23:934–41. [15] Oh JH. Nonlinear analysis of adhesively bonded tubular single-lap joints for composites in torsion. Compos Sci Technol 2006; in press. doi:10.1016/j.compscitech.2006.09.020. [16] Choi JK, Lee DG. Torque transmission capabilities of bonded polygonal lap joints for carbon fiber epoxy composites. J Adhes 1995;48:235–50. [17] Kim YG, Oh JH, Lee DG. Strength of adhesively bonded tubular single lap carbon/epoxy composite–steel joints. J Compos Mater 1999;33:1897–917. [18] Hosford WF, Caddell RM. Metal forming mechanics and metallurgy. New York: Prentice-Hall; 1993. pp. 29–48. [19] Pugno N, Carpinteri A. Tubular adhesive joints under axial load. J Appl Mech 2003;70:832–9. [20] Pugno N, Carpinteri A. Strength, stability and size effects in the brittle behaviour of bonded joints under torsion: theory and experimental assessment. Fatigue Fract Engng Mater Struct 2002;25:55–62.
Strength prediction of tubular composite adhesive joints under torsion Je Hoon Oh
*
Department of Mechanical Engineering, Hanyang University, 1271 Sa1-dong, Sangrok-gu, Ansan-shi, Gyeonggi-do 426-791, Republic of Korea Received 7 March 2006; received in revised form 26 September 2006; accepted 28 September 2006 Available online 8 December 2006
Abstract This paper addresses prediction of the strength of tubular adhesive joints with composite adherends by combining thermal and mechanical analyses. A finite element analysis was used to calculate the residual thermal stresses generated by cooling down from the adhesive cure temperature, and a nonlinear analysis incorporating the nonlinear adhesive behavior was performed to accurately estimate the mechanical stresses in the adhesive. Joint failure was estimated by three failure criteria: interfacial failure, adhesive bulk failure, and adherend failure. The distributions of residual thermal stresses were investigated for various stacking angles. The effect of residual thermal stresses on joint strength was also taken into consideration. The results indicate that the residual thermal stresses, depending on the stacking angle, have a significant influence on the failure mode and strength of adhesive joints when a subsequent mechanical load is applied. Good agreement is also obtained between the predicted joint strength and the available experimental data. 2006 Elsevier Ltd. All rights reserved. Keywords: A. Adhesive joints; A. Polymer-matrix composites (PMCs); B. Nonlinear behavior; B. Strength; C. Laminates theory
1. Introduction Adhesive joints have been widely used for composite structures because they distribute the load over a larger area than mechanical joints and do not require holes, resulting in a reduction of the stress concentration and an increase in the strength of the joint. Moreover, stress concentrations in joints can be further reduced by adhesives with nonlinear material properties. Several types of adhesive joints are used for joining composite structures: single lap joint, double lap joint, stepped joint, scarf joint, etc. Among these joints, the single lap joint is the most popular because of its manufacturing ease and relatively low cost. Since the performance of joints usually determines the structural efficiency of composite structures, an extensive knowledge of the behavior of adhesive joints and the related effect on joint strength is essential for design purposes. Many researchers have conducted studies on stress distribution in tubular adhesive joints subject to torsion to predict joint strength, given the geometry of the joint and the elastic *
Tel.: +82 31 400 5252; fax: +82 31 406 5550. E-mail address: [email protected].
0266-3538/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.09.021
material properties of the adhesive and adherend [1–7]. Recently, nonlinear analyses including the nonlinear properties of the adhesive have been developed to estimate the torque transmission capability more easily and accurately [8–10]. However, previous analytical studies have focused on adhesive joints with isotropic adherends (e.g., steel). Since composite materials have anisotropic properties that depend on their stacking angle and sequence, the analysis with isotropic adherends is limited in describing the behavior of the adhesive joint with composite adherends. Only a few attempts so far have been made to investigate the stress distribution in tubular adhesive joints under torsion with composite adherends [11–14]. However, the analyses considered linear properties of the adhesive, and the effective properties of composites were calculated from a laminate plate theory even though the composite adherend is not a plate but actually a tube. In order to overcome this limitation and get accurate stress distributions in the joint with composite adherends, a nonlinear solution was developed by the author [15]. The nonlinear analysis incorporated both the nonlinear adhesive behavior and the behavior of the laminated composite tube with different stacking angles and sequences.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
Another important factor that influences the response of adhesive joints is residual thermal stress generated mostly from the different coefficients of thermal expansion (CTEs) of the adhesive and adherend. These thermal stresses may become large enough to initiate adhesive debonding at the interface between the adhesive and adherend or adhesive cracking even under low external loads. Therefore, thermally-induced stresses can be as critical as mechanically-induced stresses in the analysis of tubular adhesive joints. Thorough understanding of the effect of residual thermal stresses on joint strength is essential, especially in the analysis of composite adhesive joints because composites have directionally dependent CTEs along the material axes, unlike isotropic materials. In this study, the strength of adhesive joints under torsion was predicted using the combined thermal and mechanical analyses of joint behavior when the adherends were composite tubes. A finite element analysis was performed to evaluate the residual thermal stresses developed in the joint, and the mechanical stresses in the adhesive layer were calculated using the developed nonlinear analysis [15]. Three different joint failure modes were considered to predict joint failure: interfacial failure, adhesive bulk failure, and adherend failure. The influence of the composite adherend stacking angle on the residual thermal stresses was investigated, and how the residual thermal stresses affect the joint strength was also discussed. Finally, the predicted results were compared with experimental results available in literature [16]. 2. Stress analysis of a tubular adhesive joint Fig. 1(a) shows the geometric configuration of the tubular single lap adhesive joint with laminated composite tube adherends. For numerical calculations, the adhesive layer is assumed to be thin and much more flexible than the adherends. The laminated composite tube of N layers is assumed to be subject to axisymmetric thermomechanical loading as illustrated in Fig. 1(b). Each layer is considered to be a homogenous, orthotropic material in the layer principal coordinates, which means that it becomes a monoclinic material in the global coordinates for off-axis orientations. Perfect bonding between the layers is also assumed in the analysis. Generally, the different CTE and stiffness of the adhesive and the adherend cause internal residual thermal stresses even in the absence of external loads. In order to investigate the effect of such residual thermal stresses on adhesive joint strengths, the following analyses were performed: residual thermal stresses by finite element analysis, and mechanical stresses by nonlinear analysis. 2.1. Analysis of residual thermal stresses The stresses induced by change in temperature DT (due to cool down from the adhesive cure temperature to room
1341
L Adherend 2 Adhesive Adherend 1
T
r1i r1o
r
r2i r2o z
T
F T
z r N
pi
k 1
ri( k )
2 po
1
3
po
ro( k )
Temperature change
Fig. 1. Geometric configuration: (a) tubular single lap adhesive joint and (b) laminated composite tube.
temperature), are discussed in this section. Under axisymmetric conditions of the joint configuration, all displacements, strains, and stresses are independent of the rotational angle h. Thus, the stress–strain–temperature relation has the form r ¼ Dðe aDT Þ
ð1Þ
where r and e are (4 · 4) stress and strain matrices, respectively; D is a (4 · 4) stiffness matrix, and a is the CTE matrix. A finite element method was employed to analyze the residual thermal stresses in the adhesive joint. From the Galerkin formulation, the element stiffness matrix and element force vector due to DT are Z ke ¼ 2p ðBT DBÞr dA ð2aÞ Ze ð2bÞ f e ¼ 2p ðBT DaDT Þr dA e
where B is the element strain–displacement matrix and A is the cross-sectional area of the element. The matrix B is defined as 3 2 N 1;r 0 N 2;r 0 N m;r 0 7 6N N2 Nm 6 1 0 0 0 7 7 6 ð3Þ B¼6 r r r 7 6 0 N 1;z 0 N 2;z 0 N m;z 7 5 4 N 1;z N 1;r N 2;z N 2;r N m;z N m;r
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J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347 Outer adherend Inner adherend r2i r1o r1i
r2o
Bonding Length
Adhesive
Fig. 2. Finite element mesh for the tubular single lap adhesive joint.
where m is the number of nodes for the element and i i N i;r ¼ oN and N i;z ¼ oN are the partial derivatives of shape or oz functions Ni. After assembling the global stiffness matrix and global force vector from the element stiffness and force matrices in Eq. (2), the displacement of each node can be obtained by solving a set of simultaneous algebraic equations. The element stresses are then determined using the relationship in Eq. (1). Fig. 2 illustrates the finite element model for calculation of residual thermal stresses. The axisymmetric model for the analysis was composed of 8-node axisymmetric isoparametric elements. To take into account the stress concentration, the mesh near the edge of the adhesive layer th was made very fine. The radial stress rth r , hoop stress rh , th th axial stress rz , and shear stress srz were obtained from the finite element analysis, and these stress components were used to investigate the effect of thermal stresses on the joint strength. 2.2. Mechanical stress analysis under torsion When the tubular adhesive joint in Fig. 1 is subjected to a torque T, torques T1 and T2 are produced in both adherends 1 and 2 at any given cross-section. Unlike for the case of isotropic adherends, the resultant axial forces F1, F2, and the resultant pressures p1, p2 are also produced in both adherends. Combining the equilibrium equation with the constitutive equation and compatibility condition gives the following system of differential equations including the nonlinear properties of the adhesive [15]: d2 T 2 ¼ fA1 ðcarh Þ þ A2 ðcarh ÞgT 2 þ fB1 ðcarh Þ þ B2 ðcarh ÞgF 2 dz2 þ fC1 ðcarh Þ þ C2 ðcarh Þgp2 þ D1 ðcarh Þ ð4aÞ d2 F 2 ¼ fA3 ðcarz Þ þ A4 ðcarz ÞgT 2 þ fB3 ðcarz Þ þ B4 ðcarz ÞgF 2 dz2 þ fC3 ðcarz Þ þ C4 ðcarz Þgp2 þ D2 ðcarz Þ p2 ¼ E1 ðear Þz þ E2 ðear Þ
ð4bÞ ð4cÞ
BCs :
T 2 ¼ 0;
F2 ¼ 0
and
dF 2 ¼0 dz
at z ¼ 0 ð4dÞ
T2 ¼ T;
F 2 ¼ 0 and
dF 2 ¼0 dz
at
z¼L
ð4eÞ where 2pr21o df ðcarh Þ 1o 2pr21o df ðcarh Þ 2i ðchz ÞT ; A2 ¼ ðchz ÞT ð5aÞ A1 ¼ a wrh g dcrh wrh g dcarh 2pr1o df ðcarz Þ 1o 2pr1o df ðcarz Þ 2i ðe Þ ; A ¼ ðez ÞT ð5bÞ A3 ¼ 4 z T wrz g dcarz wrz g dcarz 2pr21o df ðcarh Þ 1o 2pr21o df ðcarh Þ 2i B1 ¼ ðc Þ ; B ¼ ðchz ÞF ð5cÞ 2 hz F wrh g dcarh wrh g dcarh 2pr1o df ðcarz Þ 1o 2pr1o df ðcarz Þ 2i ðe Þ ; B ¼ ðez ÞF ð5dÞ B3 ¼ 4 z F wrz g dcarz wrz g dcarz 2pr21o df ðcarh Þ 1o 2pr21o df ðcarh Þ 2i C1 ¼ ðc Þ ; C ¼ ðchz Þp ð5eÞ 2 hz p wrh g dcarh wrh g dcarh 2pr1o df ðcarz Þ 1o 2pr1o df ðcarz Þ 2i ðe Þ ; C ¼ ðez Þp ð5fÞ C3 ¼ 4 z p wrz g dcarz wrz g dcarz 2pr21o df ðcarh Þ dwrh a D1 ¼ A1 T þ c ; wrh dcarh dz rh 2pr1o df ðcarz Þ dwrz a c D2 ¼ A3 T þ ð5gÞ dcarz dz rz wrz In Eq. (5), (chz)T, (chz)F, and (chz)p are shear strains in the composite adherend at r = r1o or r = r2i due to unit torque, unit axial force, and unit pressure, respectively. Similarly, (ez)T, (ez)F, and (ez)p are axial strains in the composite adherend at r = r1o or r = r2i due to unit torque, unit axial force, and unit pressure, respectively. The function f(c) represents the relationship between the shear stresses and shear strains in the adhesive layer. The unit load responses (i.e., strains in the composite adherends due to each unit load) in Eq. (5) can be determined from analysis of the laminated composite tube. The radial and shear stress distributions in the adhesive are computed after solving (4) for the torque, axial force, and pressure distributions in adherend 2.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
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3. Failure prediction of the adhesive joint
F 11 r21 þ F 22 r22 þ F 66 r26 þ F 1 r1 þ F 2 r2 þ 2F 12 r1 r2 ¼ 1
There are generally three kinds of joint failure modes: interfacial failure between the adhesive and adherend, bulk failure in the adhesive, and adherend failure. In the interfacial failure mode, the bonding strength at the interface between the adhesive and adherend is assumed to be lower than the adhesive bulk shear strength because of the relatively large residual thermal stresses, resulting in failure at the interface between the adhesive and adherend. The following failure criterion is defined to predict the failure condition of the interfacial failure mode [4,17]. 2 2 2 2 2 2 rr rh rz shz srz srh þ þ þ þ þ ¼1 ST ST ST SS SS SS
where r1 and r2 are the ply stresses in the longitudinal and transverse directions to fibers, r6 is the ply shear stress, and Fi and Fij are strength parameters defined by material strengths:
ð6Þ where ST and SS are the bulk tensile strength and bulk shear strength of the adhesive, respectively. The joint strength in the interfacial failure mode can be estimated by calculating the residual thermal and mechanical stress components that satisfy the failure criterion of Eq. (6). The bulk failure mode is when the bonding strength at the interface is high enough so that joint failure occurs in the adhesive layer, not at the interface. In this failure mode, the residual thermal stresses are not large compared to the adhesive bulk strength. Because the adhesive has nonlinear material properties, adhesive failure does not occur even after the equivalent stress in the adhesive reaches the bulk strength. Therefore, the maximum strain criterion is more suitable to predict adhesive bulk failure of the joint. The effective strain ee of the adhesive is written as [18] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ee ¼ ðe þ e22 þ e23 Þ ð7Þ 3 1 where ei are the three principal strains. The adhesive joint is assumed to fail in the bulk failure mode when the effective strain exceeds the failure strain of the adhesive. If the shear strain in the adhesive is dominant and other strain components are small enough to be neglected, the maximum shear strain criterion, rather than the maximum strain criterion, can be used to predict the adhesive bulk failure. The fracture energy criterion can provide a reasonable prediction of brittle failure load for adhesive joints with isotropic adherends when the bonding length is sufficiently long [19,20]. However, it gives somewhat less accurate predictions for composite adhesive joints and does not consider the variation of failure torques with bonding length [15]. Thus in this work, the maximum strain criterion is mainly used to predict the bulk failure of joints. In the adherend failure mode, it is assumed that the composite adherend fails before the adhesive. Once the torque, axial force, and radial pressure applied at each adherend are computed from Eq. (4), the stresses in the adherend are estimated by analysis of the composite tube [15]. Then, failure of the composite adherend is predicted using the Tsai-Wu failure criterion:
ð8Þ
1 1 1 1 ; F2 ¼ T C ð9aÞ XT XC Y Y 1 1 1 ðF 11 F 22 Þ1=2 F 11 ¼ T C ; F 22 ¼ T C ; F 66 ¼ 2 ; F 12 ¼ 2 X X Y Y S ð9bÞ F1 ¼
where XT and XC represent the tensile and compressive strengths in the fiber direction, YT and YC indicate those in the transverse direction, and S is the shear strength. In order to predict the torque transmission capability T of the adhesive joint which considers the above three failure modes, the transient interpolation function fi is introduced as follows [4]: T ¼ MinðT ad ; T b Þ fi þ T i ð1 fi Þ;
0 6 fi 6 1
ð10Þ
where Tad, Tb and Ti are the torque transmission capability in the adherend failure mode, bulk failure mode, and interfacial failure mode, respectively. Note that fi is dependent on the residual thermal stresses resulting from fabrication. The effect of the residual thermal stresses can be represented by the following initial failure index ki sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi th 2 th 2 th 2 th 2 rr rh rz s ð11Þ ki ¼ þ þ þ rz ST ST ST SS where the superscript th denotes thermal stresses. The relationship between fi and ki is established by comparing analytic results with experiments. 4. Numerical results and discussion Based on the stress and strength analysis in the previous section, a computer program was developed to calculate the stress and strength of the composite adhesive joint. In order to check the validity of the analysis and predict joint strength, the results of the analysis were compared with available experimental results. Choi and Lee [16] tested steel–composite adhesive joints to investigate the effect of the stacking sequence of the composite adherend on the torque transmission capability by varying the stacking angle of the outer composite adherend from [±5]nT to [±45]nT with a ±5 interval. The geometric configuration and material properties of the joint are listed in Tables 1 and 2, respectively. Test results indicated that the torque transmission capabilities of the steel–composite joints increased until [±25]nT, and then decreased with increasing stacking angles as shown in Fig. 3. Moreover, the joint with a [±45]nT stacking angle failed at the lowest torque although the composite tube with that stacking angle has the highest torsional
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J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
Table 1 Configuration of the tubular steel–composite adhesive joint tested in Ref. [16] Inner radius of steel (r1i) Outer radius of steel (r1o) Inner radius of composite (r2i) Outer radius of composite (r2o) Adhesive thickness (g) Bonding length (L)
0 mm 8.4 mm 8.5 mm 10.5 mm 0.1 mm 15 mm
Stacking sequence of composite
[±/]nT (/ = 5,10,. . .,45)
end due to the residual thermal stresses induced during fabrication. The high residual thermal stresses can lead to interfacial debonding, which causes early failure of the joint before the adhesive reaches its failure strain. Prior to analyzing the residual thermal stresses, the variation of CTE of the composite angle-ply tube in the longitudinal, transverse and radial directions was first investigated and presented in Fig. 4. The effective longitudinal CTE is calculated as along ¼
Table 2 Material properties of the adhesive and adherends Adhesive epoxya
E1 (GPa) E2 = E3 (GPa) G12 = G13 (GPa) G23 (GPa) v12 = v13 v23 a1 (106/ C) a2 = a3 (106/ C) XT (MPa) XC (MPa) YT (MPa) YC (MPa) S (MPa) Shear strain limit Ply thickness (mm)
1.30 1.30 0.46 0.46 0.41 0.41 72 72 45 80 – – 30 0.4 –
Adherend Steel
Carbon/epoxyb
200 200 76.9 76.9 0.30 0.30 11.7 11.7 350 – – – – – –
128 10 4.49 3.58 0.28 0.47 0.5 27 1800 1400 45 150 50 – 0.15
ð1Þ
rigidity and strength. These results are the opposite of what would be guessed intuitively, that as the stacking angle increases, the torsional resistance increases, resulting in higher torque transmission capability of the joint. This phenomenon might be explained by the initial strength reduction at the interface between the adhesive and adher-
Torque (N·m)
80
60
40
20
10
15 20 25 30 35 Stacking angle (degree)
40
45
Fig. 3. Torque transmission capabilities of steel–composite joints for various stacking angles / in [±/]nT (The data are taken from Ref. [16]).
ð13Þ
ð1Þ
ri DT ð1Þ
where ri and Dri are the inner radius of the tube and the change in the inner radius after the temperature change DT, respectively. Similar to the effective transverse CTE, the effective radial CTE aradial is 90 out-of-phase with the effective longitudinal CTE along . aradial initially decreases from its large positive transverse CTE of the laminate, passes through zero between the fiber angle / = 40 and / = 45, and attains the maximum negative value near / = 60. The radial CTE then increases to a value somewhat below the longitudinal CTE of the laminate with increasing /. The radial CTE for a tube will become identical to the effective transverse CTE of a flat laminate as the aspect-ratio (ratio of the inner radius to the thickness) of the tube increases. It is interesting to note that the radius exhibits shrinkage for an increase in temperature when the fiber angle / is larger than / 45 in the [±/]nT angle-ply tube. Coefficient of thermal expansion (10-6/ ºC)
100
5
Dri ð1Þ
IPCO 9923, manufactured by Imperial Polychemicals Corporation (USA). b USN 150, manufactured by SK Chemicals (Korea).
0
ð12Þ
where e0z is the constant axial strain that is determined from the previous analysis for the unit temperature change DT. The effective longitudinal and transverse CTEs are quite similar to those for flat laminates calculated from classical lamination theory. However, unlike the flat composite laminates, the change in radius of composite tubes is associated with thermal loading. The effective radial CTE aradial is defined as aradial ¼
a
0
e0z DT
30 Longitudinal CTE Transverse CTE Radial CTE
25 20 15 10 5 0 -5
-10
0
15
30
45
60
75
90
Stacking angle φ (degree) Fig. 4. Variation of the longitudinal, transverse, and radial CTEs of composite tubes.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
Using the effective longitudinal, transverse and radial CTEs of the composite tube, the residual thermal stresses for various stacking angles were calculated through the analysis described in the previous section and shown in Fig. 5. In this calculation, a temperature change of DT = 60 C from the cure temperature of 80 C to room temperature of 20 C was a uniformly applied thermal loading. All the stress components increase progressively with increasing stacking angle. Stress concentrations are developed near the edge of the joint (z = 15 mm) at high stacking angles. The hoop stress (rh) has almost the same distribution as the axial stress (rz), and the shear stress (srz) increases linearly from negative to positive values and becomes maximum near the edge of the joint. It should be noticed that the radial stress (rr) plays a role of the peel stress. A negative peel stress can improve the joint strength by not causing cracks at the interface to open, while a positive peel stress helps the crack opening to induce interfacial failure of the joint. At low stacking angles, negative radial stress occurs at the whole range of the bonding length, but
it becomes positive as the stacking angle increases. There is even a stress concentration near the edge of the joint with a [±45]nT stacking angle. The variation of the radial CTE in Fig. 4 can explain why, under thermal loading, the radial stress in the adhesive increases and changes from compressive to tensile as the stacking angle increases. Since the composite tube has a relatively large positive radial CTE at low stacking angles, radial shrinkage of the composite tube occurs for a negative temperature change, lowering the radial stress in the adhesive. On the contrary, the almost zero or even negative radial CTE over / = 40 causes no change in radius or even radial expansion for thermal loading. This constrains the shrinkage of the adhesive that possesses a large CTE, resulting in the high tensile radial stress in the adhesive layer as shown in Fig. 5. Fig. 6 shows the distribution of the initial failure index ki in Eq. (11) with respect to the stacking angles of the outer composite adherend. The initial failure index increases as the stacking angle increases, and the maximum value of
20
20
= 45° (MPa) th r
Radial stress
th
(MPa)
= 25° = 15° = 5°
10
= 45°
10
= 35°
Hoop stress
1345
0
= 35° = 25° = 15° = 5°
0
-10
-20
-10
0
5
10
-30
15
0
Bonding range (z, mm)
5
10
20
30
= 45° 20
th
Shear stress
z
= 25°
10
rz
10
(MPa)
= 35° = 25° = 15° = 5°
th
(MPa)
= 45°
Axial stress
15
Bonding range (z, mm)
0
= 15° = 5°
0
- 10 - 10
0
5
10
Bonding range (z, mm)
15
- 20
0
5
10
15
Bonding range (z, mm)
Fig. 5. Distribution of residual thermal stresses for various stacking angles / in [±/]nT (DT = 60 C): (a) radial stress, (b) hoop stress, (c) axial stress, and (d) shear stress.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
1
200
0.8
160
0.6
Torque (N·m)
Initial failure index ki
1346
φ = 45° φ = 35°
0.4
φ = 25° φ = 15° φ = 5°
0.2
Experimental data [16]
Adhesive bulk failure Adherend failure
120
Predicted torque by Eq. (10)
80 40 Interfacial failure
0
0
5
0
15
10
1
5
10
15 20 25 30 35 Stacking angle φ (degree)
40
45
0.9
at the adhesive thickness of 0.1 mm [4], their influence depends on the stacking angle of the composite adherend in the steel–composite joint even with the same adhesive thickness. Thus, the influence of residual thermal stresses
0.8 0.7 0.6 0.5
5
10
15
20
25
30
35
40
45
1
Stacking angle φ (degree)
Fig. 6. (a) Distribution of the initial failure index for various stacking angles / in [±/]nT. (b) Variation of the maximum initial failure index with /.
ki occurs near the edge of the joint (z = 15 mm) due to the stress concentration. The increase in the maximum ki with the fiber angle / is very gradual up to / = 15, followed by a much sharper increase as shown in Fig. 6(b). The maximum ki approaches unity at / = 45, which means the initial interfacial failure due to residual thermal stress may occur in the joint with that stacking angle without any external torque. This clearly illustrates that the stacking angle of the composite adherend has a significant influence on the residual thermal stress developed in the joint and eventually the joint strength as well. In order to estimate the torque capacity of the adhesive joint, the test results in Fig. 3 were compared with numerical results considering the three different failure modes. Fig. 7 presents the predicted torque transmission capabilities using the assumptions of interfacial, bulk, and adherend failures as well as the experimental results. The adherend failure mode can explain the experimental data below a fiber angle of / = 10, while the interfacial failure mode can represent a joint failure above / = 30. Notice that there is an abrupt reduction in the interfacial joint strength at high stacking angles due to the effect of residual thermal stresses. Unlike the steel–steel adhesive joint in which the effect of residual thermal stresses is negligible
Failure index
0
0.5
Failure shear strain
0.4 Failure index at r = r2o
0.8
0.3
0.6
0.2 γ raθ
0.4 0.2 0
0.1
γ ≈ε ≈ 0 a rz
a r
0
Failure index at r = r2i
0
10 5 Bonding range (z, mm)
15
Strains in the adhesive
1.2 0.4
b
0
Fig. 7. Predicted torque capacities based on the failure criteria with the experimental data in Fig. 3.
-0.1
35
Stresses in the adhesive (MPa)
Maximum initial failure index ki,max
Bonding range (z, mm)
30 25 τ raθ
20 15 10 5
τ rza
0 -5
σ ra
0
10 5 Bonding range (z, mm)
15
Fig. 8. Failure index, strain, and stress distributions (applied torque T = 91.7 Nm) (a) Failure index of the composite adherend [±15]nT and strains in the adhesive, (b) stresses in the adhesive.
J.H. Oh / Composites Science and Technology 67 (2007) 1340–1347
should be taken into consideration while designing adhesive joints, especially with composite adherends. When adhesive bulk failure is assumed, the torque capacity tends to increase with increasing stacking angle as expected. Under the same assumption, it is also observed that failure of the composite adherend occurs first at the outer radius of the adherend before adhesive failure until the fiber angle / = 20. As an example, Fig. 8 illustrates the failure index of a composite adherend with a stacking angle [±15]nT and stress and strain distributions in the adhesive layer when the applied torque was 91.7 Nm. Adherend failure occurs at the outer radius (r = r2o) before the adhesive shear strain reaches its failure strain of 0.4. The radial and shear stresses ðrar ; sarz Þ and radial and shear strains ðear ; carz Þ are negligibly small compared to the shear stress sarh and shear strain carh , respectively. From Figs. 6(b) and 7, adhesive bulk failure mode can be assumed to be dominant when the maximum initial failure index ki,max is less than 0.52. Therefore, the transient interpolation function fi can be defined as a function of ki,max [4]: fi ¼ 1 fi ¼ e
when
k i; max 6 0:52
30ðk i; max 0:52Þ
when
k i; max > 0:52:
ð14aÞ ð14bÞ
Note that this relationship between fi and ki,max depends on what materials are used for the adhesive and adherends. The torque transmission capabilities predicted by Eqs. (10) and (14) are also plotted in Fig. 7. The predicted results are in good agreement with the experimental results. 5. Conclusions Combined thermal and mechanical analyses were performed to predict the torque transmission capabilities of tubular adhesive joints with composite adherends. The distribution of residual thermal stresses due to fabrication was investigated by varying the stacking angle of the composite adherend. The nonlinear behavior of the adhesive was incorporated into the mechanical analysis for the accurate prediction of joint strength. The validity of the analyses was also checked by comparison with the available experimental results of steel–composite tubular adhesive joints. Since the radial effective CTE of the composite tube is not constant but varies with the stacking angle like the longitudinal and transverse CTEs, the residual thermal stresses in the joint increase with the stacking angle. Especially, the high tensile radial thermal stresses act as peel stresses, thus lowering the joint strength. The adhesive bulk or adherend failure occurs at low stacking angles where the influence of residual thermal stresses is negligible, while the joints with high stacking angles fail at the interface between the adhesive and adherend because of high residual thermal stresses. Unlike a joint with isotro-
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pic adherends, the residual thermal stresses have a significant influence on the composite joint strength even though its adhesive thickness is relatively thin. The combined analyses with failure criteria accurately predict the torque capacity of tubular adhesive joints with composite adherends. References [1] Adams RD, Peppiatt NA. Stress analysis of adhesive bonded tubular lap joints. J Adhes 1977;9:1–18. [2] Alwar RS, Nagaraja YR. Viscoelastic analysis of an adhesive tubular joint. J Adhes 1976;8:79–92. [3] Lee DG, Kim KS, Lim YT. An experimental study of fatigue strength for adhesively bonded tubular single lap joints. J Adhes 1991;35: 39–53. [4] Lee SJ, Lee DG. Development of a failure model for the adhesively bonded tubular single lap joint. J Adhes 1992;40:1–14. [5] Lee SJ, Lee DG. Optimal design of the adhesively bonded tubular single lap joint. J Adhes 1995;50:165–80. [6] Choi JH, Lee DG. The torque transmission capabilities of the adhesively bonded tubular single lap joint and the double lap joint. J Adhes 1994;44:197–212. [7] Lee SJ, Lee DG. A closed-form solution for the torque transmission capability of the adhesively bonded tubular double lap joint. J Adhes 1994;44:271–84. [8] Lee SJ, Lee DG. An iterative solution for the torque transmission capability of adhesively bonded tubular single lap joints with nonlinear shear properties. J Adhes 1995;53:217–27. [9] Lee DG, Jeong KS, Choi JH. Analysis of the tubular single lap joint with nonlinear adhesive properties. J Adhes 1995;49: 37–56. [10] Lee DG, Oh JH. Nonlinear Analysis of the torque transmission capability of adhesively bonded tubular lap joints. J Adhes 1999;71:81–106. [11] Thomsen OT, Kildegaard A. Analysis of adhesive bonded generally orthotropic circular shells. in: Proceedings of the fourth European conference of composite materials, Stuttgart, Germany; 1990. p. 723– 29. [12] Chon CT. Analysis of tubular lap joint in torsion. J Compos Mater 1982;16:268–84. [13] Hipol PJ. Analysis and optimization of tubular lap joint subjected to torsion. J Compos Mater 1984;18:298–311. [14] El-hady FA, Kandil N. Optimization of tubular double lap joint configuration. Polym Compos 2002;23:934–41. [15] Oh JH. Nonlinear analysis of adhesively bonded tubular single-lap joints for composites in torsion. Compos Sci Technol 2006; in press. doi:10.1016/j.compscitech.2006.09.020. [16] Choi JK, Lee DG. Torque transmission capabilities of bonded polygonal lap joints for carbon fiber epoxy composites. J Adhes 1995;48:235–50. [17] Kim YG, Oh JH, Lee DG. Strength of adhesively bonded tubular single lap carbon/epoxy composite–steel joints. J Compos Mater 1999;33:1897–917. [18] Hosford WF, Caddell RM. Metal forming mechanics and metallurgy. New York: Prentice-Hall; 1993. pp. 29–48. [19] Pugno N, Carpinteri A. Tubular adhesive joints under axial load. J Appl Mech 2003;70:832–9. [20] Pugno N, Carpinteri A. Strength, stability and size effects in the brittle behaviour of bonded joints under torsion: theory and experimental assessment. Fatigue Fract Engng Mater Struct 2002;25:55–62.