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Design of functionally graded joints using a polyurethane-based adhesive with varying amounts of acrylate
International Journal of Adhesion & Adhesives 76 (2017) 38–46
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International Journal of Adhesion and Adhesives journal homepage: www.elsevier.com/locate/ijadhadh
Design of functionally graded joints using a polyurethane-based adhesive with varying amounts of acrylate
MARK
⁎
Scott E. Stapletona, , Julia Weimerb, Jan Spenglerb a b
Department of Mechanical Engineering, University of Massachusetts Lowell, Lowell, MA 01854 USA Plastics Division, Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt 64289, Germany
A R T I C L E I N F O
A BS T RAC T
Keywords: Finite element stress analysis Stress distribution Mechanical properties of adhesives Joint design Functionally graded bondline
Adhesives with graded properties along the bondline are being developed to increase the strength of adhesively bonded joints. Efforts to do this in the past have resulted in mixed results. Two adhesive parameters need to be considered: the geometry of the gradation and the material properties of the adhesive at different gradation levels. In order to consider both of these aspects, a computational model was created to aid in not only the design of adhesive gradations but also judge whether a specific adhesive gradation method will be able to result in strength increases. In this study, the model was introduced and compared with published results. A new adhesive gradation system was created by using a polyurethane-based adhesive with varying amounts of acrylate, and a numerical analysis was performed to determine the potential advantages of the adhesive gradation.
1. Introduction Adhesively bonded joints have been receiving increased attention with the rise of fiber reinforced composite materials. Adhesively bonded joints generally allow a more gradual transfer of shear load from one adherend to another than bolted or riveted joints and do not require holes, which may interrupt fiber paths. However, peel stress concentrations in the corners of joints often causes a bulk of the adhesive to remain underutilized and can even result in premature failure. Many methods have been proposed to distribute stress more evenly in joints. Most involve altering the geometry of the joint [1], including tapering the adherend [2], increasing the thickness of the adhesive at the end [3], fillets [4], rounded adherend corners [5], novel joint geometries [6], and joint insertions [7]. More recently, grading the adhesive properties along the length of the joint has become a popular focus of researchers towards relieving stress concentrations and increasing joint strength. Bi-adhesive joints were the first to be widely studied, with most theoretical findings showing positive results and experimental studies showing mixed results, with many important design guidelines identified [8–15]. More recently, continuously graded adhesively bonded joints have been studied theoretically [16– 22], with very few experimental studies [23–25]. One of the broad lessons to be learned from these studies is that grading the adhesive does not universally result in performance
⁎
Corresponding author. E-mail address: [email protected] (S.E. Stapleton).
http://dx.doi.org/10.1016/j.ijadhadh.2017.02.006
Available online 06 February 2017 0143-7496/ © 2017 Elsevier Ltd. All rights reserved.
increases. However, it is probably safe to say that grading the adhesive universally results in increased complication, cost, and/or time. Therefore, it seems necessary that the development of functionally graded adhesives be tightly coupled with design models if it is to ever find industrial application. Although linear elastic models can provide valuable insights in the pre-yielding stress distribution and even stress allowables for adhesive joints, models which include in some way the nonlinear nature of most adhesives is necessary. Therefore, design models which consider material nonlinearities coupled with experimental development of adhesion gradation systems are needed to realize beneficial functionally graded adhesive joints. The current study uses a design model previously developed [26], which is a combination of a structural model of cylindrical plates on an elastic foundation and a finite element approach. The model requires one element through the thickness of the joint, and a co-rotational formulation includes geometric nonlinearities [27] while adaptive shape functions and an internal adaptive mesh include the effects of material nonlinearities and crack growth [28]. A formulation is presented here which includes a modified Von-Mises plasticity formulation [29] in the framework of a thin adhesive layer constrained by two stiff adherends, along with the interpolation strategy between data curves for the continuously graded adhesive. A few numerical examples are shown to provide insights in considering nonlinearities for graded adhesive systems, and the model is compared with experimental data in the literature [24].
International Journal of Adhesion & Adhesives 76 (2017) 38–46
S.E. Stapleton et al.
compared to the peel and shear stress components, or εx = εy=0 . Assuming this, the stiffness matrix can be written as
Finally, a novel adhesive system, which can be graded by changing acrylate content or cure temperature is presented. The difference between the conventional adhesive systems and the adhesive system used in this study is the application of only one formulation that is able to generate graded properties along the bondline for stress peak reduction due to the curing temperature. The adhesive system is based on a polyurethane adhesive in combination with a hydroxyl-terminated acrylate, which is responsible for the adjustable mechanical properties. Due to the hydroxyl-termination, the acrylate is permanently integrated into the polyurethane network after the first global curing process at moderate temperatures. The next locally acting curing step at higher temperatures, the acrylate polymerization proceeds, resulting in an increase in the network density and consequently an increasing stiffness. The material properties of the acrylate content grading are used to model a single lap joint configuration and, with a linear gradation, the design process is demonstrated.
C1 =
2. Method
where Jσ2 is the second invariant of the deviatoric stress tensor, Iσ1 is the first invariant of the stress tensor, and
2.1. Computational model
Cs =
⎡C E 0 ⎤ D=⎢ 1 a ⎥ ⎣ 0 Ga ⎦
where Ea and Ga are the elastic Young's modulus and shear modulus and
seff = Cs Jσ 2 + CvIσ1
(7)
(8)
Cs Cv Jε2 + Iε1 ν+1 1 − 2ν
(9)
where Jε2 is the second invariant of the deviatoric strain tensor and Iε1 is the first invariant of the strain tensor. Considering the assumptions about the x and y strain components, the effective stress becomes
Cs 2 1/2 (σz (C1−1)2 + τxz2 ) + Cvσz(2C1+1) 3
seff =
(10)
and the effective strain is 1/2 Cs ⎛ 1 2 1 ⎞ Cv εz . ⎜ εz + τxz2 ⎟ + ν+1 ⎝ 3 4 ⎠ 1 − 2ν
eeff =
(11)
A tensile test was used to characterize the adhesive, so the tensile stress and strain had to be tied to the effective stress and strain. The effective stress yield stress from a tensile test, Yeff , can be written as
⎛ C ⎞ Yeff = ⎜ s + Cv⎟Y (εt , ε p ) ⎝ 3 ⎠
(12)
ε p)
is a function of the tensile strain at initial yield, εt , and where Y (εt , the accumulated plastic strain, ε p . A bar over a value indicates a value from the tensile stress-strain curve or a value which has been converted into that space. Finally, the effective accumulated plastic strain, eeffp , can be found by the equation
⎛ C ⎞ eeffp = ⎜ s + Cv⎟ε p . ⎝ 3 ⎠
(13)
The yield function, f , is defined as
(1)
f = seff − Yeff .
(14)
(2)
If f ≤ 0 , then the stress calculated was correct. When the initial stress was not correct, an iterative predictor/corrector method was utilized to find the plastic strain which satisfies the yield function. For iteration n + 1, the flow rule was defined as
(3)
εnp+1 = εnp + ndλ .
The stress can be calculated based on the elastic strain as
where
σ = [ σz τxz ]T .
3 (S +1) S −1 σ , Cv = , and S = c 2λ 2λ σt
eeff =
where ε p is the plastic strain, εel is the elastic strain, and all strains are in the vector form as
σ = Dε el
(6)
where σc and σt are the compressive and tension yield stresses. Similarly, an effective strain, eeff , is described by
2.1.2. Adhesive plasticity The highly nonlinear nature of most adhesives requires the use of some sort of nonlinear material model for the adhesive. Previous versions of the bonded joint element model used a nonlinear elastic model with the shear and normal modes decoupled, which was intended to be used with characterization tests such as double cantilever beam (DCB) and end notch flexure (ENF) tests. However, the most common method for characterizing adhesives has remained a simple tensile test on a pure adhesive specimen. Therefore, a plasticity model was introduced along with a method to use tensile test data to characterize a thin adhesive layer. At a material point, we assume that the total strain, ε,can be broken up into a plastic and elastic portion
ε = [ εz γxz ]T .
1−νa (1 − 2νa )(1+νa )
and νa is the Poisson's ratio of the adhesive. A modified Von Mises plasticity theory has been introduced by Gali et al. [29] where the yield behavior is dependent on both deviatoric and hydrostatic stress which causes a difference between uniaxial tension and compression. An effective stress, seff , is defined as
2.1.1. Joint Element model The bonded joint element model was used as the basis for the analysis. This model uses the linear elastic solution of a structural model to determine the exact shape functions for two overlapping adhesively bonded adherends [26]. Using this method, the overlap section can be represented by a single element for a linear elastic analysis. Furthermore, geometric nonlinearity has been considered by using a co-rotational formulation to capture large rotations [27]. Material nonlinearities were also included, with an increase in the number of elements needed for a converged solution. Finally, to enable a coarse mesh even when using nonlinear materials and considering progressive failure, an adaptive mesh along with adaptive shape functions were derived and applied [28]. The maximum number of joint elements used during this study was six, with a mesh convergence study conducted for each example. All simulations were run on an inhouse finite element software.
ε = ε p + εel
(5)
(15)
where (4)
n=
Assuming that the adhesive is much softer than the adherends, the adhesive strain parallel to the adherends can be considered negligible
σ′ , σ′
(16)
λ is a plastic multiplier, and σ′ is a vector of the stress, with the shear 39
International Journal of Adhesion & Adhesives 76 (2017) 38–46
S.E. Stapleton et al.
component divided by two to so that the flow direction is transferred to engineering strain space. The accumulated plastic strain accumulation was found by taking a small step from the previous iteration by the equation
(17)
The plastic strain vector for iteration n + 1 was found by linearizing about the previous step, giving Furthermore, the value of the yield function at iteration n + 1 was also found by linearizing about the previous step:
⎛ ∂f ⎞T ⎛ ∂f ⎞T fn +1 = fn + ⎜ ⎟ dσ + ⎜ p ⎟ dε p ⎝ ∂ε ⎠ ⎝ ∂σ ⎠
Interpolated Curves
120
Stress (MPa)
⎛ ∂f ⎞T εnp+1 = εnp + ⎜ ⎟ ndλ . ⎝ ∂σ ⎠
Input Curves
80
40
0
(18)
0
0.2
which can be rewritten as
fn +1
⎛ C ⎞ ∂Y ⎛ ∂f ⎞ ⎛ ∂f ⎞T = fn − ⎜ ⎟ Dndλ − ⎜ s + Cv⎟ p nT ⎜ ⎟dλ ⎝ ∂σ ⎠ ⎝ ∂σ ⎠ ⎝ 3 ⎠ ∂ε
fn . ⎛ ∂f T ⎞ ⎛ ∂f ⎞ ⎛ C ⎞ ∂Y ⎛ ∂f ⎞ ⎜ ⎟D⎜ ∂σ ⎟ + ⎜ 3s + Cv⎟ ∂ε p nT ⎜ ∂σ ⎟ ⎠ ⎝ ⎠ ⎝ ∂σ ⎠ ⎝ ⎠ ⎝
0.6
0.8
1
Strain (mm/mm) Fig. 1. Stress vs strain curves showing the interpolation between experimental curves for the graded adhesive analysis for a thin adhesive layer with constrained modulus effect.
(19)
and assuming that the next step results in a satisfied yield function ( fn+1 =0 ), the accumulated plastic strain increment was found by
dλ =
0.4
2.3. Materials 2.3.1. Adhesive system The adhesive formulation consists of Desmophen® 1380BT from Bayer, PolyTHF2000 from BASF and Desmodur® N3400 from Bayer. Desmophen® 1380BT as the crosslinking component of the basic adhesive formulation is a tri-functional polypropylene ether polyole. PolyTHF2000 is a di-functional polybutyleneoxide, which has the function to expand the network of the polyurethane systeme. Desmodur®. N3400 is an aliphatic poly isocyanate. SR399 is a di-pentaerythritol pentaacrylate with five acrylate functionalities. The acrylate SR399 is provided by Sartomer. Due to the humidity sensitiveness of the isocyanate isolithpowder as a moisture absorbance was added to the formulation. Furthermore, zinc-bis-(2ethylhexanoate) as a reaction accelerator for the polyurethanes is also part of the adhesive system.
(20)
Using previous equations, the current variables can be found for the n + 1 increment, and the process can be repeated until the yield function is satisfied. Finally, the adhesive was considered failed and was deleted from the analysis when the accumulated plastic strain exceeded the accumulated plastic strain of the tensile specimen. 2.1.3. Property interpolation The input for the plasticity model for a functionally graded adhesive is a collection of curves taken from tensile tests of single-adhesive specimens for different properties within the gradation. For each curve, a tensile yield stress (σy ), tensile to compression yield stress ratio (S ), Poisson's ratio (ν ), and maximum tensile strain (εfail ), was collected. Each curve was identified by its initial modulus, and a modulus gradation function was also required as an input. A piecewise-linear function was created to interpolate between the input for the all of the input variables vs. initial modulus. For a material point, the initial modulus was found using the gradation function, and the corresponding variables (σy,λ , ν,εfail ) were found using the interpolation function. The state variables and current strain state were given, and the value of the yield stress (Y (εt , ε p )) was found by interpolating between values of the neighboring curves. An illustration of the interpolation between a collection of input curves for a thin adhesive layer under tensile loading is shown in Fig. 1. Finally, the situation arises when an interpolation must be made past the failure strain of one of the curves. For this situation, the curves are assumed to extend linearly based on their final slopes. Therefore, if an input curve has a sudden slope change near failure, the curve must be slightly smoothed so that the final slope is representative of the curve as a whole.
2.4. Chemical formulation The experiment was carried out by Desmophen® 1380BT and PolyTHF2000 dried at 60 ° C under vacuum and subsequently the polyol components were mixed with Desmodur® N3400 and with the acrylate SR399 in different amounts. 1 Wt.-% isolithpowder and 1 Wt.% zinc-bis-(2ethylhexanoate) was added to the ingredients in a 30 mL polyethylene disposable cup. The formulation was mixed under vacuum in a speed mixer DAC 400.a VAC P Hausschild Germany for 90 seconds. Afterwards, tensile specimen were produced according to DIN EN 527-2. The formulation was poured into a silicone mold for the tensile specimens and cured at 80 °C for the basic system and at 200 °C for the acrylate polymerization. The concrete adhesive composition is shown in Table 1. It was necessary to investigate the curing behavior of the adhesive system. Therefore, a Differential Scanning Calorimetry measurement (DSC measurement) of the system, which was precured at 80 °C, was Table 1 Composition of the adhesive formulation with a different amount of acrylate SR399.
2.2. Experimental method To validate the theoretical model of graded properties along the bondline, an adhesive system with an adjustable module for reduction of stress peaks was developed. In order to vary the stiffness within the adhesive layer, the network density of the adhesive needs to be changed locally. The gradient can be achieved by a combination of different curing mechanisms which increase the network density by local triggering. In the areas in which only one of the curing reactions accesses, the network density is low and the material is less rigid. 40
Desmophen ® 1380 BT (Wt.-%)
PolyTHF 2000 (Wt.-%)
Acrylate SR 399 (Wt.-%)
Desmodur® N3400 (Wt.-%)
30.96 28.05 21.93 18.85 16.53
20.12 18.23 14.25 12.25 10.74
5.57 10.24 24.12 31.10 36.36
43.34 43.47 39.69 37.79 36.36
International Journal of Adhesion & Adhesives 76 (2017) 38–46
S.E. Stapleton et al.
2
heat flow [mW]
numerical examples using were analyzed. Two different joint configurations were investigated: double cantilever beam (DCB) and single lap joint, with the geometric parameters shown in Fig. 4. The adherends were aluminum, with a Young's modulus of 70 GPa. The adhesive was a fictitious adhesive system shown in Fig. 5a. Three curves were used as inputs, with the initial stiffness being 3, 3.5, and 4 GPa, maximum stress of 35, 40, and 45 MPa and failure strain of 0.025, 0.05, and 0.075. The tensile curves were described using a tanh function as outlined in [30]. The input curves along with a few interpolated curves are shown in Fig. 5b for an adhesive layer (with constrained modulus). First, DCB specimens were analyzed and linearly graded adhesives between the maximum and minimum moduli were compared in Fig. 6. “Max” denotes a single adhesive system made from the stiffest modulus material, “Min” is single-adhesive with the most compliant adhesive, “Lin+” is a linear curve between the most compliant adhesive on the left of the joint to the stiffest adhesive on the right, and “Lin-“ is the opposite as shown in Fig. 6a. The load-displacement curves in Fig. 6b show the single adhesive joint with the most compliant adhesive is a maximum bound, since it has the highest failure strain and the highest strain energy under the curve. The “Max” adhesive is a lower bound, since it's stress-strain curve has the lowest area under the curve. As can be expected, the two linear curves, “Lin+” and “Lin-“ seem two connect the two single-adhesive systems. This is not necessarily useful information, but the fact that the curves look just as one would expect can serve as a form of validation that the model. The comparison in Fig. 7 provides a little more insight. The same curves for the single adhesive systems are present, but the transition from one adhesive to the other for the graded adhesive joints is done using a Tanh function rather than a linear function. The result is that the “Tanh+“ specimen holds slightly more load than the “Min” specimen before initial cracking. This is due to the fact that the transition is less gradual, and the stiffer adhesive takes some of the load off of the corner, making the maximum load higher. For a DCB specimen with progressive cracking, this may seem to be of little importance since the load quickly drops after initial cracking through the graded region and the joint behaves as the lower bound adhesive “Max”, but for other joint types without progressive failure, the pre-cracking behavior is all that matters in joint performance. Similar to the analysis of the DCB joint, Fig. 8 shows the modulus along the adhesive for two single-adhesive joints making a lower and upper bound, and two linear symmetric gradations. The results show that the linearly graded adhesives generally held loads similar to the outer-most adhesive type, but the “Lin+” joint held slightly more load before catastrophic failure. The stress was spread slightly more evenly within the adhesive, resulting in a slightly higher load. The tanh graded adhesives shown in Fig. 9 showed even more potential for improvement in maximum load over the outer-most adhesive based system. The more abrupt gradient allowed the stress to be distributed even more evenly throughout the adhesive, resulting in even higher failure loads. The “Tanh-“ specimen even displayed some crack growth over the graded region before hitting the peak load. These numerical examples seem to indicate that grading adhesive joints which typically display catastrophic failure such as single lap joints can have a positive impact on maximum load and the amount of load increase is dependent on the shape of the gradient in the adhesive.
164,06 °C
exo up 3
1
0
-1
-2
-3 50
100
150
200
250
T [°C] Fig. 2. DSC-curve of the acrylate's reaction peak of the adhesive system with 36 Wt.-% acrylate cured at 80 °C.
made. The physical-chemical analysis was performed with a Mettler DSC822e/700 differential scanning calorimeter. The heat flow during the curing reaction was measured while the heating rate was set to 10 K/min from a starting temperature of 0 °C up to 270 °C (Fig. 2). The result shows that the reaction start of the acrylate is at 120 °C and the peak maximum is located at 164 °C. The shoulder following the maximum peak of the acrylate comes from the uretdione group of the isocyanate, but has just a minor influence on the mechanical characteristics. Nevertheless, the DSC curves show that the curing temperature of 200 °C for the acrylate system is sufficient. 2.5. Tensile test setup The mechanical tests to measure the tensile properties of the adhesive system were performed with a Zwick Z 2,5 (Zwick Roell Deutschland GmbH & Co.KG) testing machine (load cell of 2,5kN) based on DIN EN 527-4. The test was performed with a testing speed of 2.5 mm/min. At least two specimens were tested for each adhesive formulation and curing temperature. The dimensions of the specimens are illustrated in Fig. 3. 3. Results In order to demonstrate the design model for functionally graded adhesives, analysis on two numerical examples was performed. Second, the model was compared with experimental results obtained from literature for a functionally graded adhesive joint. Finally, the tensile test results of the developed adhesive are displayed for varying acrylate content and curing temperature. 3.1. Numerical examples In order to become familiar with the model and explore the characteristics of joints with functionally graded adhesives, a few
3.2. Comparison with experiments
50 mm 30 mm
The current model was compared with experiments conducted by Carbas et al. [24], in which Araldite 2011 adhesive was cured at different temperatures to obtain different material properties. The geometric parameters of the single lap joint specimens tested are shown in Fig. 10, the stress-strain curves obtained for single adhesive tensile specimens cured at different temperatures is shown in Fig. 11a, and the temperature distributed along the adhesive layer reported by
8 mm 4 mm 20 mm Fig. 3. Geometric parameters of tensile specimens.
41
International Journal of Adhesion & Adhesives 76 (2017) 38–46
S.E. Stapleton et al.
10 mm
1 mm
10 mm
F, D
1 mm F, D
1 mm 0.2 mm 1 mm
0.2 mm
a)
10 mm
b)
10 mm
10 mm
Fig. 4. Geometric parameters for the a) DCB and b) single lap joint examples. 60
50
Stress (MPa)
Stress (MPa)
40
30
20
40
20
10
Input Curves Interpolated Curves 0
0 0
a)
0.02
0.04
0.06
0.08
0
b)
Strain (mm/mm)
0.02
0.04
0.06
0.08
Strain (mm/mm)
Fig. 5. Adhesive constitutive relations for numerical examples: a) tensile stress/strain curves and b) interpolations for a fictional adhesive system with constrained modulus effect.
x
4500
6
F, D
5
l0
4000
Lin+ Lin Min Max
F (kN)
Ea (MPa)
4
3500
3
2 3000
Lin +
Lin -
Min
Max
1
0
2500 0
0.2
0.4
0.6
0.8
x/l0 (mm/mm)
a)
0
1
0.5
1
1.5
2
D (mm)
b)
Fig. 6. DCB examples: a) initial modulus along the length of the adhesive and b) resultant force-displacement curves for linearly graded adhesives.
Carbas et al. [24] strength steel with the adherends was The results for
with the temperature profile shown in Fig. 11b are shown in Fig. 12. The experimental results and results for the current model are compared. For the model, a spring with a compliance of 80 MN/m was inserted on the boundary to match the initial stiffness of the
is shown in Fig. 11b. The adherends were high a reported stiffness of 220 GPa, and no yielding of reported. specimens uniformly cured at 40 °C, 100 °C, and
4500
x
6
F, D l0
4000
Tanh + Tanh Min Max
5
F (kN)
Ea (MPa)
4
3500
3
2
3000
Tanh +
Tanh -
Min
Max
1
0
2500 0
0.2
0.4
0.6
0.8
0
1
x/l0 (mm/mm)
b)
0.5
1
1.5
2
D (mm)
Fig. 7. DCB examples: a) initial modulus along the length of the adhesive and b) resultant force-displacement curves for tanh graded adhesives.
42
International Journal of Adhesion & Adhesives 76 (2017) 38–46
S.E. Stapleton et al. 4500
x
Lin+ Lin Min Max
300
l0
4000
250
F (kN)
Ea (MPa)
350
F, D
3500
200 150 100
3000
Lin +
Lin -
Min
Max
50
0
2500 0
0.2
0.4
0.6
0.8
0
1
x/l0 (mm/mm)
a)
0.05
0.1
0.15
0.2
D (mm)
b)
Fig. 8. Single lap joint examples: a) initial modulus along the length of the adhesive and b) resultant force-displacement curves for linearly graded adhesives.
x
4500
350
F, D
l0 4000
250
F (kN)
Ea (MPa)
Tanh + Tanh Min Max
300
3500
200 150 100
3000
Tanh +
Tanh +
Min
Max
50
0
2500 0
0.2
0.4
0.6
0.8
0
1
x/l0 (mm/mm)
a)
0.05
0.1
0.15
0.2
D (mm)
b)
Fig. 9. Single lap joint examples: a) initial modulus along the length of the adhesive and b) resultant force-displacement curves for tanh graded adhesives.
25 mm
uniformly cured specimens did not match between the model and experiments, but this was expected. In the experiments, these specimens were reported to have failed by failure at the interface between the adhesive and adherend. The strength of this interface was not characterized nor captured by the model, therefore it is expected that the model would have a higher peak load. The graded adhesive specimen showed a much lower peak load (24% lower) in the model than in the experiments. This could be attributed to several factors. There could have been a fair amount of variability in the experimental local material properties of the adhesive, which could have played a large role in the prediction of failure loads. Additionally, the model is very sensitive to the failure strain of the adhesive, and this value is often known to vary significantly for tensile adhesive specimens.
F, D
2 mm 1 mm
70 mm
50 mm
70 mm
Fig. 10. Geometric parameters for single lap joints tested by Carbas et al. [24].
experiments to account for possible compliance in the load train. The results show that for the 40 °C uniformly cured specimens, the initial softening, general shape, and failure displacement matched up well. The initial peak was predicted with an 8% error. However, the model increased in load after initial softening while the experiments remained at a rather constant load, which is probably just due to the difference between the bulk adhesive tests and the joint tests. The 100 °C
140 120 100
30
20
T (°C)
Stress (MPa)
40
Cure Temp. 120°C 100°C 80°C 60°C 40°C
10
60 40
20 0
0 0
a)
80
0.1
0.2
0.3
0.4
Strain (mm/mm)
0.5
0.6
0.7
b)
0
10
20
30
x (mm)
40
50
Fig. 11. Plots showing a) tensile stress-strain curves for adhesive cured at different temperatures and b) temperature distribution within joint for specimens tested by Carbas et al. [24].
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International Journal of Adhesion & Adhesives 76 (2017) 38–46
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Fig. 12. Comparison of a) force vs. displacement curves for from experiments conducted by Carbas et al. [24] (Exp) and the present model (JE) along with b) failure sequence of the functionally graded joint.
70
2.5
6% 10% 24% 31% 36%
1.5
6%
60
10% 24%
50
Stress (MPa)
2
Stress (MPa)
Wt. -% Acrylate
Wt. -% Acrylate
1
31% 40
36%
30 20
0.5
10
T=80 ºC
0 0
0.2
a)
0.4
0.6
0.8
T=200 ºC
0 0
1
b)
Strain (mm/mm)
0.2
0.4
0.6
0.8
1
1.2
Strain (mm/mm)
Fig. 13. Stress strain curves for the polyurethane specimens with different amounts of acrylate cured at a) 80 °C and b) 200 °C.
The results of the specimens cured at 200 °C show that the adhesive systems with 6 wt. % and 10 wt. % behave like elastomers. In this case, the effect of the acrylate is insufficient. Taking the adhesive formulations with higher acrylate contents into account, the effect of the acrylate becomes clear. The most significant difference of the modulus (Δ 916.32 MPa) and of the tensile strength (Δ56.85 MPa) between specimens cured at different temperatures can be found in the adhesive system with the highest acrylate content of 36%. It can be concluded that a higher acrylate content cured at 80 °C leads to higher strains and on the other side to higher tensile strengths and lower strains when cured at 200 °C. The results of the tensile test confirm that it is possible to formulate an adhesive system, which is able to show different mechanical properties depending on the curing cycle. For the realization of graded
Therefore, a monte-carlo type simulation might be able to more accurately predict the graded joint behavior. 3.3. Adhesive development Fig. 13 and Table 2 show the results of the tensile test of the different amounts of the acrylate in the basic adhesive system. By means of the stress-strain curves, certain trends can be noted. There is a clear difference in the maximum tensile strength and modulus between the adhesive system cured by 80 °C and cured by 200 °C independently from the acrylate content. The tensile specimens cured at 80 °C show stressstrain curves characteristic for elastomers. Furthermore, all specimens cured at 80 °C have a similar curve, and the specimen with the lowest acrylate content possesses the lowest modulus.
Table 2 Summary of the tensile tests of the adhesive, contrasting acrylate content and cure temperature.
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International Journal of Adhesion & Adhesives 76 (2017) 38–46
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25 mm
for joints with progressive failure and those with catastrophic failures. A comparison with experimental data from literature was made, although the model underpredicted the graded joint strength, which may be the result of variability in material properties. Finally, an adhesive system was formulated based on polyurethane and different weight percentages of acrylate to display varying amounts of elastomeric characteristics. The curing temperature was defined by DSC measurement and the different mechanical properties due to the acrylate content and curing temperature were tested by single lap shear tests. A design study was conducted for a given single lap joint configuration which showed that the strength could be increased with the graded adhesive, if only by a small amount. However, more extensive studies are required to judge fully how effective the proposed adhesive system would be as a graded adhesive. This study should be seen as an initial work, meant to set the stage for further advances in functionally graded adhesives. The results of the mechanical testing have clearly shown that the adhesive has a high potential to realize graded properties along the bondline. But it is necessary to confirm this statement with experimental results. The challenge will be to implement the gradient into the joining area. There are two different opportunities. Firstly, it might be feasible to vary the amount of acrylate along the bondline in the fusion zone while applying the basic formulation with varying acrylate content. The challenge for this application is the precise application of the formulation, although this could be feasible by adapting 3-D printing technology. Secondly, it is conceivable to create the gradient with the help of a local heat-transfer on the lap shear specimen. For this method, some work has already been done. The source for the local heat-transfer was performed by a laser-radiation after the first global curing step for the polyurethane system. The results of the DSC measurements have shown that the heat of the laser was able to start the reaction of the acrylate. However, the mechanical testing of the system cured at 80 °C, of the system cured at 200 °C and of the system with the laser-radiation lead to no useable results due to the poor adhesion. To gain a better adhesion, the use of some adhesion promotor seems reasonable. Finally, there is much to be done using joint models to gain further insight into the design of bonds with functionally graded adhesives. With the further development of material science and manufacturing capabilities, models such as the one presented here can to be used to develop requirements for characteristic material properties to achieve the most beneficial response, and materials can then be designed to meet these requirements. In order to achieve this, extensive parametric studies need to be carried out, varying joint geometry, adherend stiffness, and adhesive material properties. Key variables need to be identified and simplified design rules need to be established to “certify” an adhesive gradation method to avoid costly development with only
F, D
5 mm 1 mm
70 mm
50 mm
70 mm
Fig. 14. Geometric parameters and boundary conditions for single lap joint specimens.
properties along the bondline for stress peak reduction, this difference in the mechanical properties must be reached. 3.4. Theoretical study of acrylic+polyurethane joints Finally, a theoretical study was performed using the adhesive system developed in the current research. A single lap joint was analyzed, with boundary conditions and geometric parameters shown in Fig. 14. The adherends were given a modulus of 220 GPa, similar to C-12 steel. A linear gradation was chosen for its simplicity, with a constant modulus zone and a graded zone. The length of the graded zone, l , was varied as shown in Fig. 15a (and defined in Figure 16a). The resulting force-displacement curves are shown in Fig. 15b. Similarly, the gradation and results of the best performing gradation (l /lo=0.1) are compared with constant adhesive joints with 6% and 36% acrylate in Fig. 16. If the graded zone (l as defined in Fig. 16a) was too short, the response approached that of the ungraded specimens with 36% acrylate. If the zone was too large, the specimens tended towards the response of the 6% acrylate, which was extremely soft with a low strength. The best performing gradation had a larger peak stress than using any of the adhesives singly (5.7% larger than 36% acrylate, 377% larger than 6% acrylate). However, the gradation was far from optimum, and a more exhaustive optimization study should be run to find the best gradation for the joint. This study does demonstrate the type of analysis that could be conducted for a potential adhesive gradation system before investing time and money in pursuing a gradation manufacturing method. For functionally graded adhesive joints, the design and application should be tightly coupled with predictive models because the gains can be either negligible or nonexistent without the right design. 4. Summary and future work In this study, a design model is demonstrated to aid in the design of functionally graded adhesives for bonded joints. A plasticity formulation is formulated for the case of a thin adhesive layer constrained by two stiff adherend layers, and is introduced into an already existing model. An interpolation scheme is outlined for interpolating between adhesive tensile curves, and numerical examples are analyzed for DCB and single lap joints, showing that the effects of grading are different
1400
40
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0.4
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Fig. 15. Single lap joint results for different shapes of grading, showing a) the graded modulus for different specimens and b) the resulting force-displacement plots.
45
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Fig. 16. Single lap joint results comparing single adhesive systems with 6 and 36% acrylate, and the best performing graded adhesive specimen (l/lo=0.1): a) modulus along adhesive and b) force vs. displacement plot. and Technology, vol. 24, no. 7, pp. 1251–1281. [14] das Neves PJC, da Silva LFM, Adams RD. Analysis of mixed adhesive bonded joints Part I: theoretical formulation. J Adhes Sci Technol 2009;23(1):1–34. [15] das Neves PJC, da Silva LFM, Adams RD. Analysis of mixed adhesive bonded joints Part II: parametric study. J Adhes Sci Technol 2009;23(1):35–61. [16] Kumar S. Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads. Int J Adhes Adhes 2009;29(8):785–95. [17] Kumar S, Scanlan JP. Stress analysis of shaft-tube bonded joints using a Variational method. J Adhes 2010;86(4):369–94. [18] Spaggiari A, Dragoni E. Regularization of torsional stresses in tubular lap bonded joints by means of functionally graded adhesives. Int J Adhes Adhes 2014;53:23–8. [19] Stein N, Mardani H, Becker W. An efficient analysis model for functionally graded adhesive single lap joints. Int J Adhes Adhes 2016;70:117–25. [20] Nimje, SV, Panigrahi, SK. Numerical Simulation for Stress and Failure of Functionally Graded Adhesively Bonded Tee Joint of Laminated FRP Composite Plates, Inter J Adhes Adhes. [21] Nimje SV, Panigrahi SK. Interfacial failure analysis of functionally graded adhesively bonded double supported tee joint of laminated FRP composite plates. Int J Adhes Adhes 2015;58:70–9. [22] Carbas RJC, da Silva LFM, Madureira ML, Critchlow GW.Modelling of Functionally Graded Adhesive Joints, J Adhes, vol. 0, no. ja, p. null. [23] Stapleton SE, Waas AM, Arnold SM. Functionally graded adhesives for composite joints. Int J Adhes Adhes 2012;35:36–49. [24] Carbas RJC, da Silva LFM, Critchlow GW. Adhesively bonded functionally graded joints by induction heating. Int J Adhes Adhes 2014;48:110–8. [25] Sancaktar E, Kumar S. Selective use of rubber toughening to optimize lap-joint strength. J Adhes Sci Technol 2000;14(10):1265–96. [26] Stapleton SE, Waas AM. The Analysis of Adhesively Bonded Advanced Composite Joints Using Joint Finite Elements [NASA/CR-2012-217606, Apr.]. Cleveland, OH: NASA Glenn Research Center; 2012. [27] Stapleton SE, Waas AM, Arnold SM, Bednarcyk BA. Corotational formulation for bonded joint finite elements. AIAA J 2014;52(6):1280–93. [28] Stapleton SE, Pineda EJ, Gries T,Waas AM Adaptive shape functions and internal mesh adaptation for modeling progressive failure in adhesively bonded joints, International Journal of Solids and Structures. [29] Gali S, Dolev G, Ishai O. An effective stress/strain concept in the mechanical characterization of structural adhesive bonding. Int J Adhes Adhes 1981;1(3):135–40. [30] Stapleton SE, Waas AM, Bednarcyk BA. Modeling progressive failure of bonded joints using a single joint finite element. AIAA J 2011;49:1740–9.
minimal gains. Finally, robust and reliable automated design methods need to be developed in order to enable the industrial design of functionally graded joints. Acknowledgements The authors gratefully acknowledge Prof. Dr.-Ing. Jürgen Wieser and his team from the Fraunhofer Institute for Structural Durability and System Reliability LBF for their support and use of manufacturing and testing facilities. References [1] da Silva LFM, Adams RD. Techniques to reduce the peel stresses in adhesive joints with composites,. Int J Adhes Adhes 2007;27(3):227–35. [2] Hart-Smith LJ, Company DA, Center LR. Analysis and design of advanced composite bonded joints. National Aeronautics and Space Administration; 1974. [3] Hart-Smith LJ. Adhesive-bonded double-lap joints. NASA CR-112235 1973. [4] Lang TP, Mallick PK. Effect of spew geometry on stresses in single lap adhesive joints. Int J Adhes Adhes 1998;18(3):167–77. [5] Zhao X, Adams RD, da Silva LFM. Single Lap Joints with Rounded Adherend Corners: stress and Strain Analysis. J Adhes Sci Technol 2011;25(8):819–36. [6] Zeng QG, Sun CT. Novel design of a bonded lap joint. AIAA J 2001;39(10):1991–6. [7] Turaga UVRS, Sun CT. Improved design for metallic and composite single-lap joints. J Aircr 2008;45(2):440–7. [8] Pires I, Quintino L, Durodola JF, Beevers A. Performance of bi-adhesive bonded aluminium lap joints. Int J Adhes Adhes 2003;23(3):215–23. [9] Fitton MD, Broughton JG. Variable modulus adhesives: an approach to optimised joint performance. Int J Adhes Adhes 2005;25(4):329–36. [10] Broughton JG, Fitton MD. Science of Mixed-Adhesive Joints. In: da Silva LFM, Pirondi A, Öchsner A, editors. Hybrid Adhesive Joints. Springer Berlin Heidelberg; 2011. p. 257–81. [11] Temiz Ş. Application of bi-adhesive in double-strap joints subjected to bending moment. J Adhes Sci Technol 2006;20(14):1547–60. [12] Özer H, Öz Ö. Three dimensional finite element analysis of bi-adhesively bonded double lap joint. Int J Adhes Adhes 2012;37:50–5. [13] Kumar S, Pandey PC. Behaviour of Bi-adhesive Joints, Journal of Adhesion Science
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Contents lists available at ScienceDirect
International Journal of Adhesion and Adhesives journal homepage: www.elsevier.com/locate/ijadhadh
Design of functionally graded joints using a polyurethane-based adhesive with varying amounts of acrylate
MARK
⁎
Scott E. Stapletona, , Julia Weimerb, Jan Spenglerb a b
Department of Mechanical Engineering, University of Massachusetts Lowell, Lowell, MA 01854 USA Plastics Division, Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt 64289, Germany
A R T I C L E I N F O
A BS T RAC T
Keywords: Finite element stress analysis Stress distribution Mechanical properties of adhesives Joint design Functionally graded bondline
Adhesives with graded properties along the bondline are being developed to increase the strength of adhesively bonded joints. Efforts to do this in the past have resulted in mixed results. Two adhesive parameters need to be considered: the geometry of the gradation and the material properties of the adhesive at different gradation levels. In order to consider both of these aspects, a computational model was created to aid in not only the design of adhesive gradations but also judge whether a specific adhesive gradation method will be able to result in strength increases. In this study, the model was introduced and compared with published results. A new adhesive gradation system was created by using a polyurethane-based adhesive with varying amounts of acrylate, and a numerical analysis was performed to determine the potential advantages of the adhesive gradation.
1. Introduction Adhesively bonded joints have been receiving increased attention with the rise of fiber reinforced composite materials. Adhesively bonded joints generally allow a more gradual transfer of shear load from one adherend to another than bolted or riveted joints and do not require holes, which may interrupt fiber paths. However, peel stress concentrations in the corners of joints often causes a bulk of the adhesive to remain underutilized and can even result in premature failure. Many methods have been proposed to distribute stress more evenly in joints. Most involve altering the geometry of the joint [1], including tapering the adherend [2], increasing the thickness of the adhesive at the end [3], fillets [4], rounded adherend corners [5], novel joint geometries [6], and joint insertions [7]. More recently, grading the adhesive properties along the length of the joint has become a popular focus of researchers towards relieving stress concentrations and increasing joint strength. Bi-adhesive joints were the first to be widely studied, with most theoretical findings showing positive results and experimental studies showing mixed results, with many important design guidelines identified [8–15]. More recently, continuously graded adhesively bonded joints have been studied theoretically [16– 22], with very few experimental studies [23–25]. One of the broad lessons to be learned from these studies is that grading the adhesive does not universally result in performance
⁎
Corresponding author. E-mail address: [email protected] (S.E. Stapleton).
http://dx.doi.org/10.1016/j.ijadhadh.2017.02.006
Available online 06 February 2017 0143-7496/ © 2017 Elsevier Ltd. All rights reserved.
increases. However, it is probably safe to say that grading the adhesive universally results in increased complication, cost, and/or time. Therefore, it seems necessary that the development of functionally graded adhesives be tightly coupled with design models if it is to ever find industrial application. Although linear elastic models can provide valuable insights in the pre-yielding stress distribution and even stress allowables for adhesive joints, models which include in some way the nonlinear nature of most adhesives is necessary. Therefore, design models which consider material nonlinearities coupled with experimental development of adhesion gradation systems are needed to realize beneficial functionally graded adhesive joints. The current study uses a design model previously developed [26], which is a combination of a structural model of cylindrical plates on an elastic foundation and a finite element approach. The model requires one element through the thickness of the joint, and a co-rotational formulation includes geometric nonlinearities [27] while adaptive shape functions and an internal adaptive mesh include the effects of material nonlinearities and crack growth [28]. A formulation is presented here which includes a modified Von-Mises plasticity formulation [29] in the framework of a thin adhesive layer constrained by two stiff adherends, along with the interpolation strategy between data curves for the continuously graded adhesive. A few numerical examples are shown to provide insights in considering nonlinearities for graded adhesive systems, and the model is compared with experimental data in the literature [24].
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compared to the peel and shear stress components, or εx = εy=0 . Assuming this, the stiffness matrix can be written as
Finally, a novel adhesive system, which can be graded by changing acrylate content or cure temperature is presented. The difference between the conventional adhesive systems and the adhesive system used in this study is the application of only one formulation that is able to generate graded properties along the bondline for stress peak reduction due to the curing temperature. The adhesive system is based on a polyurethane adhesive in combination with a hydroxyl-terminated acrylate, which is responsible for the adjustable mechanical properties. Due to the hydroxyl-termination, the acrylate is permanently integrated into the polyurethane network after the first global curing process at moderate temperatures. The next locally acting curing step at higher temperatures, the acrylate polymerization proceeds, resulting in an increase in the network density and consequently an increasing stiffness. The material properties of the acrylate content grading are used to model a single lap joint configuration and, with a linear gradation, the design process is demonstrated.
C1 =
2. Method
where Jσ2 is the second invariant of the deviatoric stress tensor, Iσ1 is the first invariant of the stress tensor, and
2.1. Computational model
Cs =
⎡C E 0 ⎤ D=⎢ 1 a ⎥ ⎣ 0 Ga ⎦
where Ea and Ga are the elastic Young's modulus and shear modulus and
seff = Cs Jσ 2 + CvIσ1
(7)
(8)
Cs Cv Jε2 + Iε1 ν+1 1 − 2ν
(9)
where Jε2 is the second invariant of the deviatoric strain tensor and Iε1 is the first invariant of the strain tensor. Considering the assumptions about the x and y strain components, the effective stress becomes
Cs 2 1/2 (σz (C1−1)2 + τxz2 ) + Cvσz(2C1+1) 3
seff =
(10)
and the effective strain is 1/2 Cs ⎛ 1 2 1 ⎞ Cv εz . ⎜ εz + τxz2 ⎟ + ν+1 ⎝ 3 4 ⎠ 1 − 2ν
eeff =
(11)
A tensile test was used to characterize the adhesive, so the tensile stress and strain had to be tied to the effective stress and strain. The effective stress yield stress from a tensile test, Yeff , can be written as
⎛ C ⎞ Yeff = ⎜ s + Cv⎟Y (εt , ε p ) ⎝ 3 ⎠
(12)
ε p)
is a function of the tensile strain at initial yield, εt , and where Y (εt , the accumulated plastic strain, ε p . A bar over a value indicates a value from the tensile stress-strain curve or a value which has been converted into that space. Finally, the effective accumulated plastic strain, eeffp , can be found by the equation
⎛ C ⎞ eeffp = ⎜ s + Cv⎟ε p . ⎝ 3 ⎠
(13)
The yield function, f , is defined as
(1)
f = seff − Yeff .
(14)
(2)
If f ≤ 0 , then the stress calculated was correct. When the initial stress was not correct, an iterative predictor/corrector method was utilized to find the plastic strain which satisfies the yield function. For iteration n + 1, the flow rule was defined as
(3)
εnp+1 = εnp + ndλ .
The stress can be calculated based on the elastic strain as
where
σ = [ σz τxz ]T .
3 (S +1) S −1 σ , Cv = , and S = c 2λ 2λ σt
eeff =
where ε p is the plastic strain, εel is the elastic strain, and all strains are in the vector form as
σ = Dε el
(6)
where σc and σt are the compressive and tension yield stresses. Similarly, an effective strain, eeff , is described by
2.1.2. Adhesive plasticity The highly nonlinear nature of most adhesives requires the use of some sort of nonlinear material model for the adhesive. Previous versions of the bonded joint element model used a nonlinear elastic model with the shear and normal modes decoupled, which was intended to be used with characterization tests such as double cantilever beam (DCB) and end notch flexure (ENF) tests. However, the most common method for characterizing adhesives has remained a simple tensile test on a pure adhesive specimen. Therefore, a plasticity model was introduced along with a method to use tensile test data to characterize a thin adhesive layer. At a material point, we assume that the total strain, ε,can be broken up into a plastic and elastic portion
ε = [ εz γxz ]T .
1−νa (1 − 2νa )(1+νa )
and νa is the Poisson's ratio of the adhesive. A modified Von Mises plasticity theory has been introduced by Gali et al. [29] where the yield behavior is dependent on both deviatoric and hydrostatic stress which causes a difference between uniaxial tension and compression. An effective stress, seff , is defined as
2.1.1. Joint Element model The bonded joint element model was used as the basis for the analysis. This model uses the linear elastic solution of a structural model to determine the exact shape functions for two overlapping adhesively bonded adherends [26]. Using this method, the overlap section can be represented by a single element for a linear elastic analysis. Furthermore, geometric nonlinearity has been considered by using a co-rotational formulation to capture large rotations [27]. Material nonlinearities were also included, with an increase in the number of elements needed for a converged solution. Finally, to enable a coarse mesh even when using nonlinear materials and considering progressive failure, an adaptive mesh along with adaptive shape functions were derived and applied [28]. The maximum number of joint elements used during this study was six, with a mesh convergence study conducted for each example. All simulations were run on an inhouse finite element software.
ε = ε p + εel
(5)
(15)
where (4)
n=
Assuming that the adhesive is much softer than the adherends, the adhesive strain parallel to the adherends can be considered negligible
σ′ , σ′
(16)
λ is a plastic multiplier, and σ′ is a vector of the stress, with the shear 39
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component divided by two to so that the flow direction is transferred to engineering strain space. The accumulated plastic strain accumulation was found by taking a small step from the previous iteration by the equation
(17)
The plastic strain vector for iteration n + 1 was found by linearizing about the previous step, giving Furthermore, the value of the yield function at iteration n + 1 was also found by linearizing about the previous step:
⎛ ∂f ⎞T ⎛ ∂f ⎞T fn +1 = fn + ⎜ ⎟ dσ + ⎜ p ⎟ dε p ⎝ ∂ε ⎠ ⎝ ∂σ ⎠
Interpolated Curves
120
Stress (MPa)
⎛ ∂f ⎞T εnp+1 = εnp + ⎜ ⎟ ndλ . ⎝ ∂σ ⎠
Input Curves
80
40
0
(18)
0
0.2
which can be rewritten as
fn +1
⎛ C ⎞ ∂Y ⎛ ∂f ⎞ ⎛ ∂f ⎞T = fn − ⎜ ⎟ Dndλ − ⎜ s + Cv⎟ p nT ⎜ ⎟dλ ⎝ ∂σ ⎠ ⎝ ∂σ ⎠ ⎝ 3 ⎠ ∂ε
fn . ⎛ ∂f T ⎞ ⎛ ∂f ⎞ ⎛ C ⎞ ∂Y ⎛ ∂f ⎞ ⎜ ⎟D⎜ ∂σ ⎟ + ⎜ 3s + Cv⎟ ∂ε p nT ⎜ ∂σ ⎟ ⎠ ⎝ ⎠ ⎝ ∂σ ⎠ ⎝ ⎠ ⎝
0.6
0.8
1
Strain (mm/mm) Fig. 1. Stress vs strain curves showing the interpolation between experimental curves for the graded adhesive analysis for a thin adhesive layer with constrained modulus effect.
(19)
and assuming that the next step results in a satisfied yield function ( fn+1 =0 ), the accumulated plastic strain increment was found by
dλ =
0.4
2.3. Materials 2.3.1. Adhesive system The adhesive formulation consists of Desmophen® 1380BT from Bayer, PolyTHF2000 from BASF and Desmodur® N3400 from Bayer. Desmophen® 1380BT as the crosslinking component of the basic adhesive formulation is a tri-functional polypropylene ether polyole. PolyTHF2000 is a di-functional polybutyleneoxide, which has the function to expand the network of the polyurethane systeme. Desmodur®. N3400 is an aliphatic poly isocyanate. SR399 is a di-pentaerythritol pentaacrylate with five acrylate functionalities. The acrylate SR399 is provided by Sartomer. Due to the humidity sensitiveness of the isocyanate isolithpowder as a moisture absorbance was added to the formulation. Furthermore, zinc-bis-(2ethylhexanoate) as a reaction accelerator for the polyurethanes is also part of the adhesive system.
(20)
Using previous equations, the current variables can be found for the n + 1 increment, and the process can be repeated until the yield function is satisfied. Finally, the adhesive was considered failed and was deleted from the analysis when the accumulated plastic strain exceeded the accumulated plastic strain of the tensile specimen. 2.1.3. Property interpolation The input for the plasticity model for a functionally graded adhesive is a collection of curves taken from tensile tests of single-adhesive specimens for different properties within the gradation. For each curve, a tensile yield stress (σy ), tensile to compression yield stress ratio (S ), Poisson's ratio (ν ), and maximum tensile strain (εfail ), was collected. Each curve was identified by its initial modulus, and a modulus gradation function was also required as an input. A piecewise-linear function was created to interpolate between the input for the all of the input variables vs. initial modulus. For a material point, the initial modulus was found using the gradation function, and the corresponding variables (σy,λ , ν,εfail ) were found using the interpolation function. The state variables and current strain state were given, and the value of the yield stress (Y (εt , ε p )) was found by interpolating between values of the neighboring curves. An illustration of the interpolation between a collection of input curves for a thin adhesive layer under tensile loading is shown in Fig. 1. Finally, the situation arises when an interpolation must be made past the failure strain of one of the curves. For this situation, the curves are assumed to extend linearly based on their final slopes. Therefore, if an input curve has a sudden slope change near failure, the curve must be slightly smoothed so that the final slope is representative of the curve as a whole.
2.4. Chemical formulation The experiment was carried out by Desmophen® 1380BT and PolyTHF2000 dried at 60 ° C under vacuum and subsequently the polyol components were mixed with Desmodur® N3400 and with the acrylate SR399 in different amounts. 1 Wt.-% isolithpowder and 1 Wt.% zinc-bis-(2ethylhexanoate) was added to the ingredients in a 30 mL polyethylene disposable cup. The formulation was mixed under vacuum in a speed mixer DAC 400.a VAC P Hausschild Germany for 90 seconds. Afterwards, tensile specimen were produced according to DIN EN 527-2. The formulation was poured into a silicone mold for the tensile specimens and cured at 80 °C for the basic system and at 200 °C for the acrylate polymerization. The concrete adhesive composition is shown in Table 1. It was necessary to investigate the curing behavior of the adhesive system. Therefore, a Differential Scanning Calorimetry measurement (DSC measurement) of the system, which was precured at 80 °C, was Table 1 Composition of the adhesive formulation with a different amount of acrylate SR399.
2.2. Experimental method To validate the theoretical model of graded properties along the bondline, an adhesive system with an adjustable module for reduction of stress peaks was developed. In order to vary the stiffness within the adhesive layer, the network density of the adhesive needs to be changed locally. The gradient can be achieved by a combination of different curing mechanisms which increase the network density by local triggering. In the areas in which only one of the curing reactions accesses, the network density is low and the material is less rigid. 40
Desmophen ® 1380 BT (Wt.-%)
PolyTHF 2000 (Wt.-%)
Acrylate SR 399 (Wt.-%)
Desmodur® N3400 (Wt.-%)
30.96 28.05 21.93 18.85 16.53
20.12 18.23 14.25 12.25 10.74
5.57 10.24 24.12 31.10 36.36
43.34 43.47 39.69 37.79 36.36
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2
heat flow [mW]
numerical examples using were analyzed. Two different joint configurations were investigated: double cantilever beam (DCB) and single lap joint, with the geometric parameters shown in Fig. 4. The adherends were aluminum, with a Young's modulus of 70 GPa. The adhesive was a fictitious adhesive system shown in Fig. 5a. Three curves were used as inputs, with the initial stiffness being 3, 3.5, and 4 GPa, maximum stress of 35, 40, and 45 MPa and failure strain of 0.025, 0.05, and 0.075. The tensile curves were described using a tanh function as outlined in [30]. The input curves along with a few interpolated curves are shown in Fig. 5b for an adhesive layer (with constrained modulus). First, DCB specimens were analyzed and linearly graded adhesives between the maximum and minimum moduli were compared in Fig. 6. “Max” denotes a single adhesive system made from the stiffest modulus material, “Min” is single-adhesive with the most compliant adhesive, “Lin+” is a linear curve between the most compliant adhesive on the left of the joint to the stiffest adhesive on the right, and “Lin-“ is the opposite as shown in Fig. 6a. The load-displacement curves in Fig. 6b show the single adhesive joint with the most compliant adhesive is a maximum bound, since it has the highest failure strain and the highest strain energy under the curve. The “Max” adhesive is a lower bound, since it's stress-strain curve has the lowest area under the curve. As can be expected, the two linear curves, “Lin+” and “Lin-“ seem two connect the two single-adhesive systems. This is not necessarily useful information, but the fact that the curves look just as one would expect can serve as a form of validation that the model. The comparison in Fig. 7 provides a little more insight. The same curves for the single adhesive systems are present, but the transition from one adhesive to the other for the graded adhesive joints is done using a Tanh function rather than a linear function. The result is that the “Tanh+“ specimen holds slightly more load than the “Min” specimen before initial cracking. This is due to the fact that the transition is less gradual, and the stiffer adhesive takes some of the load off of the corner, making the maximum load higher. For a DCB specimen with progressive cracking, this may seem to be of little importance since the load quickly drops after initial cracking through the graded region and the joint behaves as the lower bound adhesive “Max”, but for other joint types without progressive failure, the pre-cracking behavior is all that matters in joint performance. Similar to the analysis of the DCB joint, Fig. 8 shows the modulus along the adhesive for two single-adhesive joints making a lower and upper bound, and two linear symmetric gradations. The results show that the linearly graded adhesives generally held loads similar to the outer-most adhesive type, but the “Lin+” joint held slightly more load before catastrophic failure. The stress was spread slightly more evenly within the adhesive, resulting in a slightly higher load. The tanh graded adhesives shown in Fig. 9 showed even more potential for improvement in maximum load over the outer-most adhesive based system. The more abrupt gradient allowed the stress to be distributed even more evenly throughout the adhesive, resulting in even higher failure loads. The “Tanh-“ specimen even displayed some crack growth over the graded region before hitting the peak load. These numerical examples seem to indicate that grading adhesive joints which typically display catastrophic failure such as single lap joints can have a positive impact on maximum load and the amount of load increase is dependent on the shape of the gradient in the adhesive.
164,06 °C
exo up 3
1
0
-1
-2
-3 50
100
150
200
250
T [°C] Fig. 2. DSC-curve of the acrylate's reaction peak of the adhesive system with 36 Wt.-% acrylate cured at 80 °C.
made. The physical-chemical analysis was performed with a Mettler DSC822e/700 differential scanning calorimeter. The heat flow during the curing reaction was measured while the heating rate was set to 10 K/min from a starting temperature of 0 °C up to 270 °C (Fig. 2). The result shows that the reaction start of the acrylate is at 120 °C and the peak maximum is located at 164 °C. The shoulder following the maximum peak of the acrylate comes from the uretdione group of the isocyanate, but has just a minor influence on the mechanical characteristics. Nevertheless, the DSC curves show that the curing temperature of 200 °C for the acrylate system is sufficient. 2.5. Tensile test setup The mechanical tests to measure the tensile properties of the adhesive system were performed with a Zwick Z 2,5 (Zwick Roell Deutschland GmbH & Co.KG) testing machine (load cell of 2,5kN) based on DIN EN 527-4. The test was performed with a testing speed of 2.5 mm/min. At least two specimens were tested for each adhesive formulation and curing temperature. The dimensions of the specimens are illustrated in Fig. 3. 3. Results In order to demonstrate the design model for functionally graded adhesives, analysis on two numerical examples was performed. Second, the model was compared with experimental results obtained from literature for a functionally graded adhesive joint. Finally, the tensile test results of the developed adhesive are displayed for varying acrylate content and curing temperature. 3.1. Numerical examples In order to become familiar with the model and explore the characteristics of joints with functionally graded adhesives, a few
3.2. Comparison with experiments
50 mm 30 mm
The current model was compared with experiments conducted by Carbas et al. [24], in which Araldite 2011 adhesive was cured at different temperatures to obtain different material properties. The geometric parameters of the single lap joint specimens tested are shown in Fig. 10, the stress-strain curves obtained for single adhesive tensile specimens cured at different temperatures is shown in Fig. 11a, and the temperature distributed along the adhesive layer reported by
8 mm 4 mm 20 mm Fig. 3. Geometric parameters of tensile specimens.
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10 mm
1 mm
10 mm
F, D
1 mm F, D
1 mm 0.2 mm 1 mm
0.2 mm
a)
10 mm
b)
10 mm
10 mm
Fig. 4. Geometric parameters for the a) DCB and b) single lap joint examples. 60
50
Stress (MPa)
Stress (MPa)
40
30
20
40
20
10
Input Curves Interpolated Curves 0
0 0
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0.04
0.06
0.08
0
b)
Strain (mm/mm)
0.02
0.04
0.06
0.08
Strain (mm/mm)
Fig. 5. Adhesive constitutive relations for numerical examples: a) tensile stress/strain curves and b) interpolations for a fictional adhesive system with constrained modulus effect.
x
4500
6
F, D
5
l0
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Lin+ Lin Min Max
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Max
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a)
0
1
0.5
1
1.5
2
D (mm)
b)
Fig. 6. DCB examples: a) initial modulus along the length of the adhesive and b) resultant force-displacement curves for linearly graded adhesives.
Carbas et al. [24] strength steel with the adherends was The results for
with the temperature profile shown in Fig. 11b are shown in Fig. 12. The experimental results and results for the current model are compared. For the model, a spring with a compliance of 80 MN/m was inserted on the boundary to match the initial stiffness of the
is shown in Fig. 11b. The adherends were high a reported stiffness of 220 GPa, and no yielding of reported. specimens uniformly cured at 40 °C, 100 °C, and
4500
x
6
F, D l0
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Tanh + Tanh Min Max
5
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Max
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1
x/l0 (mm/mm)
b)
0.5
1
1.5
2
D (mm)
Fig. 7. DCB examples: a) initial modulus along the length of the adhesive and b) resultant force-displacement curves for tanh graded adhesives.
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x
Lin+ Lin Min Max
300
l0
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Max
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0
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0
1
x/l0 (mm/mm)
a)
0.05
0.1
0.15
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D (mm)
b)
Fig. 8. Single lap joint examples: a) initial modulus along the length of the adhesive and b) resultant force-displacement curves for linearly graded adhesives.
x
4500
350
F, D
l0 4000
250
F (kN)
Ea (MPa)
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200 150 100
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x/l0 (mm/mm)
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0.05
0.1
0.15
0.2
D (mm)
b)
Fig. 9. Single lap joint examples: a) initial modulus along the length of the adhesive and b) resultant force-displacement curves for tanh graded adhesives.
25 mm
uniformly cured specimens did not match between the model and experiments, but this was expected. In the experiments, these specimens were reported to have failed by failure at the interface between the adhesive and adherend. The strength of this interface was not characterized nor captured by the model, therefore it is expected that the model would have a higher peak load. The graded adhesive specimen showed a much lower peak load (24% lower) in the model than in the experiments. This could be attributed to several factors. There could have been a fair amount of variability in the experimental local material properties of the adhesive, which could have played a large role in the prediction of failure loads. Additionally, the model is very sensitive to the failure strain of the adhesive, and this value is often known to vary significantly for tensile adhesive specimens.
F, D
2 mm 1 mm
70 mm
50 mm
70 mm
Fig. 10. Geometric parameters for single lap joints tested by Carbas et al. [24].
experiments to account for possible compliance in the load train. The results show that for the 40 °C uniformly cured specimens, the initial softening, general shape, and failure displacement matched up well. The initial peak was predicted with an 8% error. However, the model increased in load after initial softening while the experiments remained at a rather constant load, which is probably just due to the difference between the bulk adhesive tests and the joint tests. The 100 °C
140 120 100
30
20
T (°C)
Stress (MPa)
40
Cure Temp. 120°C 100°C 80°C 60°C 40°C
10
60 40
20 0
0 0
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80
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0.5
0.6
0.7
b)
0
10
20
30
x (mm)
40
50
Fig. 11. Plots showing a) tensile stress-strain curves for adhesive cured at different temperatures and b) temperature distribution within joint for specimens tested by Carbas et al. [24].
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Fig. 12. Comparison of a) force vs. displacement curves for from experiments conducted by Carbas et al. [24] (Exp) and the present model (JE) along with b) failure sequence of the functionally graded joint.
70
2.5
6% 10% 24% 31% 36%
1.5
6%
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10% 24%
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Stress (MPa)
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Stress (MPa)
Wt. -% Acrylate
Wt. -% Acrylate
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31% 40
36%
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10
T=80 ºC
0 0
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0 0
1
b)
Strain (mm/mm)
0.2
0.4
0.6
0.8
1
1.2
Strain (mm/mm)
Fig. 13. Stress strain curves for the polyurethane specimens with different amounts of acrylate cured at a) 80 °C and b) 200 °C.
The results of the specimens cured at 200 °C show that the adhesive systems with 6 wt. % and 10 wt. % behave like elastomers. In this case, the effect of the acrylate is insufficient. Taking the adhesive formulations with higher acrylate contents into account, the effect of the acrylate becomes clear. The most significant difference of the modulus (Δ 916.32 MPa) and of the tensile strength (Δ56.85 MPa) between specimens cured at different temperatures can be found in the adhesive system with the highest acrylate content of 36%. It can be concluded that a higher acrylate content cured at 80 °C leads to higher strains and on the other side to higher tensile strengths and lower strains when cured at 200 °C. The results of the tensile test confirm that it is possible to formulate an adhesive system, which is able to show different mechanical properties depending on the curing cycle. For the realization of graded
Therefore, a monte-carlo type simulation might be able to more accurately predict the graded joint behavior. 3.3. Adhesive development Fig. 13 and Table 2 show the results of the tensile test of the different amounts of the acrylate in the basic adhesive system. By means of the stress-strain curves, certain trends can be noted. There is a clear difference in the maximum tensile strength and modulus between the adhesive system cured by 80 °C and cured by 200 °C independently from the acrylate content. The tensile specimens cured at 80 °C show stressstrain curves characteristic for elastomers. Furthermore, all specimens cured at 80 °C have a similar curve, and the specimen with the lowest acrylate content possesses the lowest modulus.
Table 2 Summary of the tensile tests of the adhesive, contrasting acrylate content and cure temperature.
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25 mm
for joints with progressive failure and those with catastrophic failures. A comparison with experimental data from literature was made, although the model underpredicted the graded joint strength, which may be the result of variability in material properties. Finally, an adhesive system was formulated based on polyurethane and different weight percentages of acrylate to display varying amounts of elastomeric characteristics. The curing temperature was defined by DSC measurement and the different mechanical properties due to the acrylate content and curing temperature were tested by single lap shear tests. A design study was conducted for a given single lap joint configuration which showed that the strength could be increased with the graded adhesive, if only by a small amount. However, more extensive studies are required to judge fully how effective the proposed adhesive system would be as a graded adhesive. This study should be seen as an initial work, meant to set the stage for further advances in functionally graded adhesives. The results of the mechanical testing have clearly shown that the adhesive has a high potential to realize graded properties along the bondline. But it is necessary to confirm this statement with experimental results. The challenge will be to implement the gradient into the joining area. There are two different opportunities. Firstly, it might be feasible to vary the amount of acrylate along the bondline in the fusion zone while applying the basic formulation with varying acrylate content. The challenge for this application is the precise application of the formulation, although this could be feasible by adapting 3-D printing technology. Secondly, it is conceivable to create the gradient with the help of a local heat-transfer on the lap shear specimen. For this method, some work has already been done. The source for the local heat-transfer was performed by a laser-radiation after the first global curing step for the polyurethane system. The results of the DSC measurements have shown that the heat of the laser was able to start the reaction of the acrylate. However, the mechanical testing of the system cured at 80 °C, of the system cured at 200 °C and of the system with the laser-radiation lead to no useable results due to the poor adhesion. To gain a better adhesion, the use of some adhesion promotor seems reasonable. Finally, there is much to be done using joint models to gain further insight into the design of bonds with functionally graded adhesives. With the further development of material science and manufacturing capabilities, models such as the one presented here can to be used to develop requirements for characteristic material properties to achieve the most beneficial response, and materials can then be designed to meet these requirements. In order to achieve this, extensive parametric studies need to be carried out, varying joint geometry, adherend stiffness, and adhesive material properties. Key variables need to be identified and simplified design rules need to be established to “certify” an adhesive gradation method to avoid costly development with only
F, D
5 mm 1 mm
70 mm
50 mm
70 mm
Fig. 14. Geometric parameters and boundary conditions for single lap joint specimens.
properties along the bondline for stress peak reduction, this difference in the mechanical properties must be reached. 3.4. Theoretical study of acrylic+polyurethane joints Finally, a theoretical study was performed using the adhesive system developed in the current research. A single lap joint was analyzed, with boundary conditions and geometric parameters shown in Fig. 14. The adherends were given a modulus of 220 GPa, similar to C-12 steel. A linear gradation was chosen for its simplicity, with a constant modulus zone and a graded zone. The length of the graded zone, l , was varied as shown in Fig. 15a (and defined in Figure 16a). The resulting force-displacement curves are shown in Fig. 15b. Similarly, the gradation and results of the best performing gradation (l /lo=0.1) are compared with constant adhesive joints with 6% and 36% acrylate in Fig. 16. If the graded zone (l as defined in Fig. 16a) was too short, the response approached that of the ungraded specimens with 36% acrylate. If the zone was too large, the specimens tended towards the response of the 6% acrylate, which was extremely soft with a low strength. The best performing gradation had a larger peak stress than using any of the adhesives singly (5.7% larger than 36% acrylate, 377% larger than 6% acrylate). However, the gradation was far from optimum, and a more exhaustive optimization study should be run to find the best gradation for the joint. This study does demonstrate the type of analysis that could be conducted for a potential adhesive gradation system before investing time and money in pursuing a gradation manufacturing method. For functionally graded adhesive joints, the design and application should be tightly coupled with predictive models because the gains can be either negligible or nonexistent without the right design. 4. Summary and future work In this study, a design model is demonstrated to aid in the design of functionally graded adhesives for bonded joints. A plasticity formulation is formulated for the case of a thin adhesive layer constrained by two stiff adherend layers, and is introduced into an already existing model. An interpolation scheme is outlined for interpolating between adhesive tensile curves, and numerical examples are analyzed for DCB and single lap joints, showing that the effects of grading are different
1400
40
x
F, D
1200
l0
l/l0 0.01 0.1 0.2 0.3 0.4 0.48
30
F (kN)
Ea (MPa)
1000 800
l/l0 600
0.01 0.1 0.2 0.3 0.4 0.48
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0 0
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0.4
20
0
0.5
b)
x/l0 (mm/mm)
0.2
0.4
0.6
0.8
1
D (mm)
Fig. 15. Single lap joint results for different shapes of grading, showing a) the graded modulus for different specimens and b) the resulting force-displacement plots.
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Graded
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6% Acrylate
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Fig. 16. Single lap joint results comparing single adhesive systems with 6 and 36% acrylate, and the best performing graded adhesive specimen (l/lo=0.1): a) modulus along adhesive and b) force vs. displacement plot. and Technology, vol. 24, no. 7, pp. 1251–1281. [14] das Neves PJC, da Silva LFM, Adams RD. Analysis of mixed adhesive bonded joints Part I: theoretical formulation. J Adhes Sci Technol 2009;23(1):1–34. [15] das Neves PJC, da Silva LFM, Adams RD. Analysis of mixed adhesive bonded joints Part II: parametric study. J Adhes Sci Technol 2009;23(1):35–61. [16] Kumar S. Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads. Int J Adhes Adhes 2009;29(8):785–95. [17] Kumar S, Scanlan JP. Stress analysis of shaft-tube bonded joints using a Variational method. J Adhes 2010;86(4):369–94. [18] Spaggiari A, Dragoni E. Regularization of torsional stresses in tubular lap bonded joints by means of functionally graded adhesives. Int J Adhes Adhes 2014;53:23–8. [19] Stein N, Mardani H, Becker W. An efficient analysis model for functionally graded adhesive single lap joints. Int J Adhes Adhes 2016;70:117–25. [20] Nimje, SV, Panigrahi, SK. Numerical Simulation for Stress and Failure of Functionally Graded Adhesively Bonded Tee Joint of Laminated FRP Composite Plates, Inter J Adhes Adhes. [21] Nimje SV, Panigrahi SK. Interfacial failure analysis of functionally graded adhesively bonded double supported tee joint of laminated FRP composite plates. Int J Adhes Adhes 2015;58:70–9. [22] Carbas RJC, da Silva LFM, Madureira ML, Critchlow GW.Modelling of Functionally Graded Adhesive Joints, J Adhes, vol. 0, no. ja, p. null. [23] Stapleton SE, Waas AM, Arnold SM. Functionally graded adhesives for composite joints. Int J Adhes Adhes 2012;35:36–49. [24] Carbas RJC, da Silva LFM, Critchlow GW. Adhesively bonded functionally graded joints by induction heating. Int J Adhes Adhes 2014;48:110–8. [25] Sancaktar E, Kumar S. Selective use of rubber toughening to optimize lap-joint strength. J Adhes Sci Technol 2000;14(10):1265–96. [26] Stapleton SE, Waas AM. The Analysis of Adhesively Bonded Advanced Composite Joints Using Joint Finite Elements [NASA/CR-2012-217606, Apr.]. Cleveland, OH: NASA Glenn Research Center; 2012. [27] Stapleton SE, Waas AM, Arnold SM, Bednarcyk BA. Corotational formulation for bonded joint finite elements. AIAA J 2014;52(6):1280–93. [28] Stapleton SE, Pineda EJ, Gries T,Waas AM Adaptive shape functions and internal mesh adaptation for modeling progressive failure in adhesively bonded joints, International Journal of Solids and Structures. [29] Gali S, Dolev G, Ishai O. An effective stress/strain concept in the mechanical characterization of structural adhesive bonding. Int J Adhes Adhes 1981;1(3):135–40. [30] Stapleton SE, Waas AM, Bednarcyk BA. Modeling progressive failure of bonded joints using a single joint finite element. AIAA J 2011;49:1740–9.
minimal gains. Finally, robust and reliable automated design methods need to be developed in order to enable the industrial design of functionally graded joints. Acknowledgements The authors gratefully acknowledge Prof. Dr.-Ing. Jürgen Wieser and his team from the Fraunhofer Institute for Structural Durability and System Reliability LBF for their support and use of manufacturing and testing facilities. References [1] da Silva LFM, Adams RD. Techniques to reduce the peel stresses in adhesive joints with composites,. Int J Adhes Adhes 2007;27(3):227–35. [2] Hart-Smith LJ, Company DA, Center LR. Analysis and design of advanced composite bonded joints. National Aeronautics and Space Administration; 1974. [3] Hart-Smith LJ. Adhesive-bonded double-lap joints. NASA CR-112235 1973. [4] Lang TP, Mallick PK. Effect of spew geometry on stresses in single lap adhesive joints. Int J Adhes Adhes 1998;18(3):167–77. [5] Zhao X, Adams RD, da Silva LFM. Single Lap Joints with Rounded Adherend Corners: stress and Strain Analysis. J Adhes Sci Technol 2011;25(8):819–36. [6] Zeng QG, Sun CT. Novel design of a bonded lap joint. AIAA J 2001;39(10):1991–6. [7] Turaga UVRS, Sun CT. Improved design for metallic and composite single-lap joints. J Aircr 2008;45(2):440–7. [8] Pires I, Quintino L, Durodola JF, Beevers A. Performance of bi-adhesive bonded aluminium lap joints. Int J Adhes Adhes 2003;23(3):215–23. [9] Fitton MD, Broughton JG. Variable modulus adhesives: an approach to optimised joint performance. Int J Adhes Adhes 2005;25(4):329–36. [10] Broughton JG, Fitton MD. Science of Mixed-Adhesive Joints. In: da Silva LFM, Pirondi A, Öchsner A, editors. Hybrid Adhesive Joints. Springer Berlin Heidelberg; 2011. p. 257–81. [11] Temiz Ş. Application of bi-adhesive in double-strap joints subjected to bending moment. J Adhes Sci Technol 2006;20(14):1547–60. [12] Özer H, Öz Ö. Three dimensional finite element analysis of bi-adhesively bonded double lap joint. Int J Adhes Adhes 2012;37:50–5. [13] Kumar S, Pandey PC. Behaviour of Bi-adhesive Joints, Journal of Adhesion Science
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