- Email: [email protected]
Stress-based fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions
Journal Pre-proof Stress-based fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions S. Çavdar, D. Teutenberg, G. Meschut, A. Wulf, O. Hesebeck, M. Brede, B. Mayer PII:
S0143-7496(19)30232-5
DOI:
https://doi.org/10.1016/j.ijadhadh.2019.102483
Reference:
JAAD 102483
To appear in:
International Journal of Adhesion and Adhesives
Please cite this article as: Çavdar S, Teutenberg D, Meschut G, Wulf A, Hesebeck O, Brede M, Mayer B, Stress-based fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions, International Journal of Adhesion and Adhesives, https://doi.org/10.1016/ j.ijadhadh.2019.102483. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Stress-based fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions S. Çavdara*, D. Teutenberga, G. Meschuta, A. Wulfb**, O. Hesebeckb, M. Bredeb, B. Mayerb a
Laboratory for Materials and Joining Technology (LWF), Paderborn University,
Paderborn, Germany; bFraunhofer Institute for Manufacturing Technology and Advanced Materials (IFAM), Bremen, Germany
Provide full correspondence details here including e-mail for the *corresponding author *) Serkan Çavdar, Pohlweg 47-49, 33098 Paderborn, Fon: +49 5251 60-5276, E-Mail: [email protected] **) Andreas Wulf, Wiener Strasse 12, 28359 Bremen, Fon: +49 421 2246 7368, EMail: [email protected]
Stress-based fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions The joining of dissimilar materials like aluminium and carbon fiber reinforced polymer using hyperelastic adhesives to reduce high stress gradients at the overlapping ends in bonded joints is becoming progressively important. The design of these adhesively bonded joints with complex mechanical behaviour requires a valid and efficient method for virtual lifetime prediction. In following, a concept for stress-based fatigue life prediction with associated experimental parameter identification of adhesively bonded joints using hyperplastic adhesives under multiaxial stress conditions is presented. Furthermore, a possibility of optical crack monitoring for adhesively bonded test specimens is demonstrated and a method for crack detection based on amplitude compliance is proposed. Based on these results, the presented lifetime prognosis concept will be validated experimentally and numerically for inhomogeneous stress fields in the adhesive layer using technological test specimens. In addition to the number of cycles to failure, the expected cycles to crack initiation is considered as the prediction variable.
Keywords: lifetime prediction; fatigue (D); finite element stress analysis (C); polyurethane (A); mechanical properties of adhesives (D); adhesive bonding
Introduction Adhesive joining technologies are particularly used in bodyshell constructions to apply innovative lightweight designs to reduce fuel consumption and CO2 emissions. In context of the increasingly implemented multi-material construction concepts, the joining of dissimilar materials like aluminium (Al) and carbon fiber reinforced polymer (CFRP) using adhesive technology is becoming progressively important. Elastic
adhesive systems appear to be suitable, which according to [1] are used with adhesive layer thicknesses between 1 and 5 mm for bonding of attachements like panes, planking and structural components in vehicle and building constructions. Due to the hyperelastic material behaviour, these adhesive systems reduce high stress gradients at the overlapping ends in bonded joints, which appears to be particularly advantageous with regard to mechanical fatigue in cyclically loaded components. Accordingly, the design of these bonded joints, which are subjected to a cyclic local stress field due to cyclic external loads, requires a valid and efficient method for virtual lifetime prediction within the framework of the product development process. For the practical application of the prediction methods, low model complexity and resulting short computation times are important. The relevance of knowledge about adhesive fatigue is reflected in the large number of research projects. These range, for example, from the investigation of environmental influences [2] to interface topology [3] in addition to the consideration of statically superimposed mean loads [4] and variable amplitudes [5]. The complex mechanical behaviour of hyperelastic adhesives and CFRP adherends calls the validity of existing prediction concepts like [6–9],which are mostly developed and validated for structural adhesives and subdivided in stress-life and fatigue initiation/ propagation approaches [10], into question. Out of this motivation, a concept for stress-based lifetime prediction of adhesively bonded hyperelastic joints with CFRP and Al adherends under fatigue loading with constant amplitudes is presented. A polyurethane adhesive is characterized by quasi-static, cyclical load-increasing and load-controlled fatigue tests partly with optical crack monitoring. Furthermore, a stiffness-based method for crack detection is shown. Based on these results, an S-N curve approach for numerical fatigue life
prediction under consideration of the stress multiaxiality is presented and validated by experiments on technological specimens.
Experimental investigations Experimental Details The following experimental investigations are performed with a two-component semistructural polyurethane adhesive (BETAFORCE™ 2850L by DuPont de Nemours, Wilmington, USA; tensile modulus 27 MPa, tensile strength 10 MPa), which is suitable for CFRP materials. In order to ensure complete curing, all adhesive substances and adhesively bonded specimens are initially stored for 7 days at 23 °C and 50 % RH. The subsequent 2-day storage at 40 °C and 50 % RH causes an increase in stiffness and strength due to the moisture-induced post-crosslinking. The final reconditioning takes also place for 7 days at 23 °C and RH. To achieve a cohesive failure in the adhesive layer, all metallic adherends are cathodic dip coated (CDC) and cleaned with isopropanol.
Adhesive characterization The adhesive characterization includes strain controlled tensile, compression and shear tests under quasi-static as well as cyclic quasi-static loading on a uniaxial testing machine with an extensometer (Zwick Z50 with makroXtens by ZwickRoell, Ulm, DE). Sample geometries and dimensions according to DIN 6701-3 are shown in Figure 1. Tensile and compression tests are carried out at a nominal strain rate of ε 1 = 0.05 s-1 whereas shear tests take place at a shear rate of 1 = 0.1 s-1 according to DIN 6701-3.
Figure 1. Geometry and dimensions in mm of samples for adhesive characterization. The adhesive characterization of the adhesive under fatigue loading is carried out by use of shear-specimens of type b on a hydro pulse system (MTS Landmark 50 kN by MTS, Eden Prairie, USA). Tests are carried out with sinusoidal load and load control. Load ratios are 0.1 and 1. The frequency is 5 Hz for tests with high load amplitudes and 7 Hz for all other load amplitudes. To study long time mechanical behaviour creep tests at room temperature and at three different static load levels are performed over a time period of approximately three months using sample type b. The loads are applied by means of steel weights. The masses are 36.9, 55.4 and 73.9 kg which induce nominal shear stresses of 0.5, 0.75 and 1 MPa.
Adhesive bonding characterization For the study of the adhesive bonding properties, thick adherend shear (TASS) and butt joint specimens (BJS) are produced from EN-AW 6060 (Al-Al) in according to DIN EN 14869-2 and DIN EN 15870. In order to investigate the influence of the stress multiaxiality on the adhesive layer strength, angle joint specimens with an inclination of 45° (AJS-45°) are additionally derived from TASS and BJS. The influence of CFRP joining partner on the failure behaviour of bonded hybrid joints is investigated by substitute an adherend with a CFRP laminated (HexPly® M79/34%/UD300/CHS by
Hexcel Corporation, Stamford, USA; 8-layer quasi-isotropic structure with peel ply, laminate thickness of 2.3 mm) adherend (Al-CFRP) in according to [11]. The CFRP laminates are bonded on one side to the aluminium parts with structural adhesive system (SikaPower®-1277 by Sika, Baar, CH; not considered furthermore). The width and overlap length of the TASS and AJS-45° are 20 mm. The BJS has an adhesive layer diameter of 15 mm. The adhesive layer thickness is due to its application relevance 2 mm in all cases. The geometry of the Al-CFRP specimens are shown in Figure 2.
Figure 2. Adhesive layer dimensions of thick adherend shear (TASS), angle joint (AJS45°), butt joint (BJS) and LWF-KS2 specimens, CFRP laminated adherends (Al-CFRP). For the model validation, the LWF-KS2 specimen in according to [12] made of EN AW-6016-T6 (sheet thickness of 1.5 mm) is used for 0°, 45° and 90° (tension dominated) load introduction angles. In order to prevent fatigue-induced failure of specimens in the notch radii, the adherends for the 90° fatigue tests are made of HX420 steel. One-sided CFRP laminates are applied over the bond area (cf. Figure 2) to represent an Al-CFRP joint. The quasi-static tests are carried out with at least 5 repetitions per test series on a tensile testing machine (Zwick Z100 by ZwickRoell, Ulm, DE) in according to DIN 6701-3 with nominal principal strain rate of ε 1 = 0.05 s-1 . This corresponds for the TASS with initial shear strain rate of = 0.1 s-1 (test speed 12 mm/min) and for the BJS
with initial tensile strain rate of = 0.05 s-1 (test speed 6 mm/min). Assuming a state of plane strain in the adhesive layer with inclination of = 45°, the test speed for the AJS45° is 7 mm/min. In order to prevent transverse forces during the tests, the load is introduced with cardanic bearings. The adhesive displacements in the basic tests (TASS, BJS, AJS-45°) are measured with an extensometer (makroXtens by ZwickRoell, Ulm, DE) and in the validation tests (LWF-KS2) with digital image correlation (ARAMIS Adjustable 2D/3D 4M by GOM, Braunschweig, DE) close to the adhesive layer. The evaluation of the parameter is carried out in form of nominal tensile and shear stress as well as tensile and shear strain, which can be calculated from test load , adhesive layer width , overlap length and adhesive layer displacement ∆:
TASS:
∆!
tan
#
⋅
∆! "
⋅ sin
⋅ sin
(3)
&
∆!
and and
(2)
"
$⋅%
BJS:
AJS:
(1)
⋅
(4)
"
⋅ cos
(5)
∆!
⋅ cos
(6)
⋅
tan
"
The load controlled fatigue tests are realized on a hydro pulse system (Hydropuls PSA by Schenk, Darmstadt, DE) according to DIN 50100. The pulsating tests with stress ratio of R = 0.1 are done with constant amplitudes and load frequency of f = 8 Hz. The results of the fatigue tests are evaluated under the assumption of a logarithmic normal distribution based on the standard deviations and resulting scattering margins TN. The relationship between the load amplitude + (force , shear or tensile stress amplitude) and the endured number of cycles ,- is described with equation (7) and
slope parameter .. /
/ 0
1
,-
2 3
(7)
Experimental Results The experimental results are divided into adhesive properties, which are decisive to model the hyperelastic material, and properties of the adhesive bond, which serve to identify and describe the quasi-static and fatigue failure behaviour.
Adhesive characterization Figure 3 shows the results of all quasi-static tests. To ensure the comparability of all cyclic quasi-static tests nominal compression strain levels are set to 10, 20, 40 and 60 %. Nominal tensile strain levels include 10, 20, 40, 60, 80 and 100 %. Nominal shear levels are 19.75, 37.5, 68.5, 97.5, 124.4 and 150 %. These nominal shear levels are chosen to yield maximum principle strains of 10, 20, 40, 60, 80 and 100 % within the shear sample. The results are shown in Figure 4.
Figure 3. Experimental results of quasi-static tests (a. tension, b. compression, c. shear).
Figure 4. Experimental results of cyclic quasi-static tests (a. tension, b. compression, c. shear). Scatter occurs due to small bubbles in the crack surfaces in case of quasistatic tension tests and due to partial adhesive failure in case of cyclic shear tests. The experimental creep test results are fit to a creep compliance model (8) that depends on time 4 and applied nominal shear stress J6, 48 9: ; < 9; < 9= < >? @ 4 A B? 68 < >? @ 4 A
(8)
In this equation the initial static compliance B? is depending on the shear stress. This yields a better overall fit for nonlinear elastic materials than a constant initial compliance. The individual parameters 9C are taken directly from the quasi-static shear tests and are identified as -0.0296, 0.0654 and 0.0468 MPa. The parameter >? , D and E are identified using a non-linear multi regression least square procedure in Python (version 2.7) as 0.0303, 0.1181 and 0.0829. Figure 5 shows the experimental results in comparison to the compliance model (dotted lines).
Figure 5. Experimental results vs. creep model. Figure 6 shows the S-N-curves from the fatigue tests and the experimental results.
Figure 6. Experimental results of shear test under fatigue loading (a. R=0.1, b. R=-1). All individual test results are used to identify the curve parameters in (7) for 0.1 (. 9.261, I? 3.71 LMN) and 1 (. 9.354, I? 5.54 LMN).
Figure 7. Relevant experimental values of fatigue tests. Figure 7 shows the relevant experimental quantities which are measured during a fatigue test or can be evaluated from these results respectively. Maximum O and minimum P loads as well as maximum QO and minimum QP displacements are measured directly and can be used to calculate mean load R and displacement QR , load and displacement amplitudes Q , slope E and creep displacement QST which is the intersection of each individual cycle with the displacement axis
R 0.5 6O < P 8
(9)
QR 0.5 6QO < QP 8
(10)
0.5 6O P 8
(11)
Q 0.5 6QO QP 8
(12)
U QST
(13)
V
W X
< Q=
(14)
Using equations (9) - (14) together with the bond area > and bond thickness 4 the initial stiffness U? , total stiffness UYY , creep stiffness UST and amplitude stiffness U and their inverse B? , BYY , BST and B can be calculated by 6?8
V 6?8
U? 608 VW6?8 → B? 608 W 6?8 W
W
6\8
V 6\8
UYY 6,8 V[6\8 → BYY 6,8 [ 6\8 [
[ 6\8 ]^ 6\8_V[ 6?8
UST 6,8 V
U 6,8
6\8 V 6\8
[
→ BST 6,8 → B 6,8
V]^ 6\8_V[ 6?8 [ 6\8
V 6\8
6\8
(15) (16) (17) (18)
Figure 8 shows the evaluation of different compliances for a dynamic shear load amplitude of 0.99 MPa, mean shear load of 1.21 MPa, 0.1 and a frequency of 7 Hz.
Figure 8. Evaluation of compliances. The creep compliance B`YYaS 6,8 from the static model equation (8) is calculated from the mentioned mean shear load of 1.21 MPa. The static creep compliance does not fit the creep compliance from the fatigue test. The reason for this behaviour is due to the nonlinear static behaviour of the adhesive. Since the maximum load of the first load cycle exceeds the mean load during the static creep tests the initial dynamic compliance must exceed the static initial compliance. Changing the initial compliance B? from the static creep test in equation (8) to the initial compliance B? 608 from the fatigue test ̅ 6,8. yields a much better fit as can be seen for B`YYaS Another observation is that the dynamic creep compliance exceeds the static creep compliance significantly after about 35000 load cycles. From that point on the creep compliance rate seems to be constant until final rupture. The reason for this behaviour is not obviously clear from the test results. To deal with this problem a camera setup is installed to observe the overlap ends of the shear specimens during fatigue tests at a load ratio of 0.1. The image
acquisition is triggered always at the point of maximum deflection for selected load cycle numbers.
Figure 9. Evidently evaluation of crack initiation from image evaluation. The images for each load cycle are then merged together with a plot of dynamic stiffness vs. load cycle number into a movie. Figure 9 shows this evaluation at a cycle number of 31955. At this point one can see from the movie that first cracks occur at both overlap ends and then start to propagate. This observation motivates the assumption that the change in creep compliance rate results from the initiation and propagation of crack(s) which decrease the bonded area and increase the creep load and as result accelerate the creep compliance rate. Crack propagation is stable until approx. 230000 load cycles. Figure 10 shows a amplitude compliance plot for a test at a load ratio of 1. In this case creep compliance should not be relevant since no mean load is applied. Instead the amplitude compliance shows a behaviour which is very similar to creep although no creep load is applied.
Figure 10. Amplitude compliance of fatigue test at a load ratio of 1. It seems to be that the increase of amplitude compliance and decrease of its rate is related to strain weakening that is well known from elastomeric materials and can be seen in Figure 4. Normally the increase of compliance should stop after a particular number of load cycles and saturate. In this case after approximately 40000 load cycles one can observe an inflexion point where the amplitude compliance starts to increase. This can be related by visual evaluation to crack initiation and beginning of crack propagation. Due to the similarity to creep behaviour the amplitude compliance is fit to the creep model (8). Figure 11 shows on the right the same amplitude compliance curve as before and a selection rectangle (blue) which represents the set of experimental data that have to be fit to the model. On the right side one can see the model fit (magenta) for the best parameter set and a transition point at 32000 load cycles.
Figure 11. Selection rectangle (right), model fit (left). The transition point is evaluated by means of a recursive fit procedure in Python (version 2.7). The blue selection rectangle is divided iteratively in sub rectangles of equal width. The number of experimental data inside of any individual sub rectangle is fixed and evenly spaced for the cycle axis. Therefore only the increment in load cycles increases. Corresponding compliance values are calculated by means of linear interpolation. The data within each sub rectangle are then fit to the model and model parameters as well as the sum of least squares as a fit quality indicator are stored. The transition point now represents the width of that sub rectangle in which the model best fits.
Figure 12. Evaluation of transition point for a fatigue test at a load ratio of 0.1. Figure 12 shows results of this evaluation procedure for a fatigue test at a load ratio of 0.1. Consequently all fatigue tests at 0.1 and 1 are evaluated with this method and Figure 13 shows the results.
Figure 13. S-N-curve for cycles to crack initiation for fatigue test at different load ratios (a. R=0.1, b. R=-1). In comparison to Figure 6, the slope of the S-N-curve for crack initiation is nearly the same for both load-ratios (.?.= 9.343 and .1= 9.683). The stress amplitude I? decreases from 3.71 to 2.86 MPa for 0.1 and from 5.54 to 4.84 MPa for 1.
Adhesive bonding characterization The investigated specimens show cohesive failure in the experiments, which are decisive to characterize an adhesive bond. Representative curves for the Al-Al and AlCFRP joints in the stress-strain diagram in according to equations (1) - (6) with associated fracture patterns are shown in Figure 14, left. Due to the 45° inclination of the adhesive layer to the load introduction, the shear and normal nominal stress-strain components of the AJS-45° are equal in magnitude as result of Mohr's coordinate transformation. Furthermore, the stiffness of the butt joint test is higher than that of the shear test due to the lateral strain obstruction.
Figure 14. Stress-strain curves (nominal strain rate 0.05 s-1) and quasi-static strength diagram with elliptical/ circular failure curve for TASS, AJS-45° and BJS, ta = 2 mm. A comparison of the quasi-static test results (Al-Al vs. Al-CFRP) shows identical stiffness and strength with a decrease in ultimate strain on the CFRP side and cohesive failure zone close to the interface (cf. Figure 14, left). The multiaxial strength behaviour in nominal stresses can be described by an elliptical failure curve (cf. Figure 14,
right),
which
here
in
the
first
quadrant
τn-max = σn-max = 9.0 MPa into a circular equation:
merges
with
E2
and
d
ef
efg[ h
A
i
jf
jfg[ h
A
i 1
(19)
Figure 15 shows the nominal shear and tensile fatigue strengths in form of S-N curves. A comparison of the S-N curves between the test series with Al-Al and Al-CFRP confirms the effect that the substitution of the adherend materials has no significant influence on the endurable load level and thus to the endurable number of cycles. The S-N curve parameter ., I? and the scatter ranges kl for Al-Al and Al-CFRP are mostly of a similar order of magnitude. Differences in test series (especially AJS-45°) can be attributed to the specimen manufacturing tolerances as well as to the general scatter of
10.0
10.0
Nominal shear stress amplitude [MPa]
Nominal tensile stress amplitude [MPa]
fatigue tests.
1.0
1.0
Probability of failure 10% 50% 90%
0.1 1,000 103
TASS (Al-Al) AJS-45° (Al-Al) TASS (Al-CFRP) AJS-45° (Al-CFRP)
10,000 100,000 104 105 Cycles to failure Nf
1,000,000 106
Probability of failure 10% 50% 90%
0.1 1,000 103
BJS (Al-Al) AJS-45° (Al-Al) BJS (Al-CFRP) AJS-45° (Al-CFRP)
10,000 100,000 104 105 Cycles to failure Nf
1,000,000 106
Figure 15. Comparison of nominal shear and tensile stress S-N curves for TASS, BJS and AJS-45°, pulsating (R = 0.1) load conditions. The results of the LWF-KS2 validation tests are evaluated due to inhomogeneous stress fields in the adhesive layer in form of the applied forces and displacements measured locally near the adhesive layer (cf. Figure 16). Analogous to basic tests with stiff test specimens, an increasing stiffness with increasing tensile component can be recorded. Nevertheless, in contrast to basic tests, the larger deformation of adherends and the resulting inhomogeneity of the stress in the adhesive layer at 90° load compared to 0°
load leads to a decrease in the quasi-static ultimate load (cf. Figure 16, left). Due to the preventive steel adherends with higher stiffness (avoidance of joining part failure in notch radius) of the 90°-loaded LWF-KS2 specimen in the fatigue tests, the endurable load amplitudes of the 90° test series are higher than those of the 0° test series (cf. Figure 16, right). Furthermore, the slope of the F-N curves for all load conditions are of a similar order of magnitude.
Figure 16. Force-displacement curves (nominal strain rate 0.05 s-1) and F-N curves for 0°, 45° and 90° loaded LWF-KS2 specimens, pulsating (R = 0.1) load conditions. Stress based approaches for multiaxial failure hypothesis In following a criterion for upper stress-based lifetime prognosis of multiaxial stressed adhesive joints with hyperelastic material behaviour is shown initially under several simplifying assumptions in idealised nominal stress space. These findings as well as numerical simulations of the basic tests lead to a local stress analysis.
Nominal stress analysis The failure-based S-N curves in Figure 15 in combination with the procedure to determine the corresponding numbers of cycles to crack initiation shown in Figure 11 are used to describe the multiaxial fatigue strength in form of a failure- and crack-based
master S-N curve. In order to idealize the stress states, the following simplifying assumptions are made analogous to structural adhesive joints: (1) In first approximation, a homogeneous stress field in the adhesive layer is assumed due to stiff adherends. (2) Failure hypotheses are considered, which describe the stress multiaxiality by means of coordinate independent invariants of the stress tensor. It is assumed that a quasi-static strength criteria is also suitable to describe the fatigue behaviour. (3) The stress state in the adhesive layer is idealized assuming the validity of linear elasticity theory. With BJS and AJS-45°, in addition to the externally applied load stresses , additional transverse stresses (m and n ) occur as a result of the lateral strain obstruction of adhesive layers. These are taken in to account with a poisson ratio of o 0.488 and linear elasticity equations: p
m n =1p
(20)
The considered failure hypothesis is a combination of the second invariant of the deviatoric part of the Cauchy stress J2 (cf. equation (23)) with the first invariant of the Cauchy stress I1 (cf. equation (22)). The second invariant of the stress deviator represents the distortion energy and can be equated with the von Mises criterion. An extension by the hydrostatic part I1 extends this to a pressure-dependent model, which can be found for example as generalized Mohr-Coulomb criterion in linear form in the Drucker-Prager-model. Under linear elastic aspects, the hydrostatic weighting factor r can also be understood as lateral strain parameter, so that equation (21) can be transferred to the linear elastic strain energy density and thus to the failure criteria according to Beltrami. An extension of this approach in form of a yield function as well
as equivalent stress st for fatigue loading with plastically compressible adhesive joints can be found in [13] and [9].
=
st uB; < rv=;
(21)
v= w < m < n
(22)
;
;
; ; ; B; yzw m { < zm n { < 6n w 8; | < wm < mn < wn x
(23)
The creation of the nominal upper stress based master S-N curve is carried out in a regression analysis. For this purpose, the data points (upper nominal stress, number of cycles to failure) of different test series (TASS, BJS, AJS-45°) converted into an upper equivalent stress by means of equation (21) with start parameter c0 and interpreted as the result of a common S-N curve with scattering span kl , the associated standard deviation and RMSE in an Excel spreadsheet. The optimization result (failure and initial crack-based
master
rf = 0.377 and ri = 0.406,
S-N slope
curve)
with
parameter
hydrostatic
weighting
.f = 9.0 and .i = 9.3
with
factor minimum
scattering span of kNf = 1:3.1 and kNi = 1:4.7 is shown in Figure 17, left. Equation (21) can also be interpreted as isochronous elliptical equation that can be used to represent fatigue strength by ellipses with constant number of cycles (cf. Figure 17, right). Based on previously made assumptions, the strengths of the TASS are on the ordinate axis with vanishing hydrostatic component. As the tensile component increases, the hydrostatic component increases as result of lateral strain obstruction.
10
Equivalent upper nominal stress [MPa]
10 ~
TASS (Al-CFRP) AJS-45° (Al-CFRP) BJS (Al-CFRP) Crack initiation
1
Probability of failure
;
v=O stO
;
1
6
TASS (Al-CFRP) AJS-45° (Al-CFRP) BJS (Al-CFRP) Nf=10^3 Nf=10^4 Nf=10^5 Nf=10^6 Nf=10^7
4
2
10% 50% 90%
k = 9.0 TN = 1 : 3.1
10,000 104
B;O
stO
8
(J2u)1/2 [MPa]
stO B;O < r v=O
~
;
0 100,000 105
1,000,000 106
Cycles to failure Nf
10,000,000 107
0
2
4 6 I1u [MPa]
8
10
Figure 17. Upper nominal stress-based master S-N curve for pulsating (R = 0.1) load conditions and stress invariant representation of isochronous failure curve for ta = 2 mm. Local stress concept Adhesive layers of real components with flexible adherends are characterized by inhomogeneous stress fields due to elastic and plastic deformations of the parts to be joined. For this reason, a local master S-N curve will be introduced below based on the local stress concept (peak stress or hot-spot concept). For the evaluation of the local adhesive layer stresses in the basic tests, the geometries in Figure 2 are modelled using finite elements in Abaqus/CAE (version R2017x). The discretization of the adherends, the CFRP coupon and the auxiliary adhesive layer is carried out with continuum elements of form C3D8. A hybrid C3D8H formulation with an element edge length of 0.25 mm (8 elements above adhesive layer thickness) is used for the adhesive layer. The parts are joined by tied contacts. According to experimental investigations, the force introduction is realized by means of kinematic coupling via reference points. A detailed view of the assembled models is shown in Figure 18. The adherends and the auxiliary adhesive layer are isotropic and the CFRP coupon orthotropically linear elastic. The hyperelasticity of the adhesive is
modeled with the Marlow model [14] implemented in Abaqus/Standard (version R2017x), neglecting the quasi-static failure. Only the nominal stress-strain curve of the tensile tests (cf. Figure 3) with a poisson ratio of 0.488 (residual compressibility) serve as input variables. The cyclic stress softening with permanent residual deformation can be considered by coupling the Marlow model with a fictive von Mises plasticity and a damage approach according to Ogden-Roxburgh [15] based on [16] and [17].
Figure 18. Simulation models of basic test specimens. Based on the assumption that the cohesive failure occurs in middle of the adhesive layer, the stress distribution is evaluated in middle of the adhesive layer. The curves of the stress invariants v= and B; are compared in Figure 19 as examples of load levels, which lead with 50 % probability of failure to 105 load cycles to failure in experiments.
25
25
TASS BJS AJS-45°
FBJS = 771 N
20
TASS BJS AJS-45°
20
FTASS = 1480 N
FAJS−45° = 1548 N
15
I1 [MPa]
J2 [MPa²]
15
10
10
FAJS−45° = 1548 N
5
5
FTASS = 1480 N
FBJS = 771 N 0
0 0.0
0.2
0.4 0.6 x / lo [-]
0.8
1.0
0.0
0.2
0.4 0.6 x / lo [-]
0.8
1.0
Figure 19. Distribution of second deviatoric invariant J2 (left) and first invariant of CAUCHY stress (right) in symmetric plane and middle of the adhesive layer over the normalized overlap length lo for TASS, BJS and AJS-45° with load equivalence of 105 . In order to consider these stress inhomogeneities, the previously introduced method for creating a master S-N curve is transferred to local stress conditions. For this purpose, the experimental tested upper forces (TASS, BJS and AJS-45°) are defined as loads in the corresponding FE models. The failure critical location in the adhesive layer is determined by the local maximum of B; (shear dominating) for TASS and by v= (hydrostatic dominating) for BJS. The value pairs of B; and v= in highest stressed elements are assigned to the respective number of cycles to failure and crack-initiation. The identification of the master S-N curve parameter is carried out in two steps in Excel and initially only using the test series of the TASS and BJS. The following application of equation (21) with first parameter set to the FE results in Abaqus/CAE (version R2017x) confirms the assumption of the failure critical location for the TASS and BJS. Furthermore, by evaluating the scalar failure criterion at the AJS-45°, unique locations of maximum stress are identified. The value pairs of B; and v= in highest stressed elements allow a second parameter optimization with inclusion of experimental results of AJS-45°. This procedure results in the local master S-N curve shown in Figure 20
with
hydrostatic
weighting
factor
rf = 0.038 and ri = 0.039,
slope
parameter
.f = 8.1 and .i = 8.3 and minimum scattering span of kNf = 1:3.3 and kNi = 1:4.9, analogous to Figure 17. 10
10 ~
TASS (Al-CFRP) AJS-45° (Al-CFRP) BJS (Al-CFRP) Crack initiation
1
Probability of failure
;
v < r =O stO
;
1
6
TASS (Al-CFRP) AJS-45° (Al-CFRP) BJS (Al-CFRP) Nf=10^3 Nf=10^4 Nf=10^5 Nf=10^6 Nf=10^7
4
2
10% 50% 90%
k = 8.1 TN = 1 : 3.3
10,000 104
B;O
stO
8
(J2u)1/2 [MPa]
Equivalent upper stress [MPa]
stO B;O < r v=O
~
;
0 100,000 105
1,000,000 106
Cycles to failure Nf
10,000,000 107
0
10
20 I1u [MPa]
30
40
Figure 20. Upper local stress-based master S-N curve for pulsating (R = 0.1) load conditions and stress invariant representation of isochronous failure curve for ta = 2 mm. In order to verify the failure criterion and the resulting methodology of predicting the fatigue lifetime, the previously generated FE models are simulated quasistatically at different forces. The projection of the results against the master S-N curve provides the 50 % S-N curve of the respective specimen. Furthermore, the sensitivity of the mesh size on the fatigue life to be predicted is investigated. The successive coarsening of the adhesive layer discretization from 8 elements up to 2 elements above the adhesive layer thickness leads to comparable results with the identical parameter r. For example of the AJS-45°, a doubling of the element edge length from 8 to 4 elements over the adhesive layer thickness only leads to an increase of the predicted lifetime by about 4 % and a coarsening from 8 to 2 elements only to an increase of 9 %.
Validation The validation of the stress-based lifetime prediction concept is carried out in the following on basis of the CFRP laminated LWF-KS2 specimens (cf. Figure 16). In contrast to basic test specimens, the stress inhomogeneity in the adhesive layer is caused by load-induced adherend deformation (elastic/ plastic) in addition to the adhesive necking at overlap ends with increasing lateral strain obstruction in middle of the specimen. The further procedure provides a quasi-static simulation (with expected upper forces) of the geometry to be prognosticated with subsequent evaluation of the stress states and projection against an S-N curve, which is usual in the application-oriented lifetime prognosis. Therefore the LWF-KS2 specimen shown in Figure 2 is modelled analogous to the basic test specimens. The discretization of metallic parts to be joined is performed with shell elements of the form S4 with an assigned thickness of 1.5 mm. In order to consider possible plastifications of sheet metals, an elasto-plastic material behaviour with isotropic hardening von Mises plasticity is considered according to a corresponding yield curve. The element form and material definition of the CFRP coupon, the auxiliary (EP) and investigation (PU) adhesive layer and further boundary conditions correspond to the models of the basic test specimens. A detailed view of the assembled models is shown in Figure 21.
Figure 21. Simulation model of LWF-KS2 specimen.
The resulting stress components in the adhesive layer elements for different load levels (analogous to tested load horizons) are converted with equation (21) into failurerelevant equivalent stress with hydrostatic weighting factor rf = 0.038 / ri = 0.039 in post-processing. The distribution of the scalar failure criterion in middle of the adhesive layer are compared in Figure 22 as examples for a load level which leads with 50 % probability of occurrence to a prediction of 105 load cycles to failure. The projection of the exemplary marked maxima of the curves with the master S-N curve in Figure 20 leads to the endurable number of failure cycles.
Figure 22. Distribution of equivalent upper stresses in middle of adhesive layer vs. normalized overlap length lo and width wo for LWF-KS2 specimen with 105 load cycles. The predicted lifetimes (cycles to crack initiation and adhesive failure) of the simulated load stages are compared to the experimental S-N curves in Figure 23. The resulting prediction accuracies for the highest and lowest experimentally tested load horizons (a-max and a-min) are illustrated in Table 1 in form of the ,pre ⁄,exp ratio. ,i-exp represents the number of cycles to crack initiation and ,f-exp the number of cycles to adhesive failure with 50 % probability of occurrence. Based on the basic experiments the amplitude compliance is fit to the creep model (8) to identify the fatigue initiation life. The transition point is assessed using the recursive adjustment method shown in
Figure 11 with Matlab (R2018b). ,i-pre and ,f-pre represent the predicted quantities. ,pre ⁄,exp 1 is thus an underestimation (conservative design) and ,pre ⁄,exp 1 an overestimation of the expected lifetime. The predicted number of cycles to crack initiation and failure deviates from the F-N curves by up to factor 2.5 (conservative), which is due to scatter in fatigue tests and the indirect determination of crack initiation by means of stiffness evaluation, especially for relatively small deformations.
Figure 23. Comparison of experimental and numerical predicted F-N curves (crack initiation and failure) for 0°, 45° and 90° loaded LWF-KS2 specimen, R = 0.1. Table 1. Comparison of experimental cycles to crack initiation/ failure with numerical predicted lifetimes for 0°, 45° and 90° loaded LWF-KS2 specimen. KS2-0°
KS2-45°
KS2-90°
Ni-pre ⁄Ni-exp Nf-pre ⁄Nf-exp Ni-pre ⁄Ni-exp Nf-pre ⁄Nf-exp Ni-pre ⁄Ni-exp Nf-pre ⁄Nf-exp Fa-max ax 0.77
0.43
0.59
0.80
1.42
0.58
Fa-min 0.71
0.42
2.29
1.85
2.16
0.40
The results confirm the applicability of the shown method for lifetime prognosis to inhomogeneous stressed adhesive layers and thus to industrially manufactured and hyperelastically bonded sheet metal and CFRP components.
Conclusions Use of hyperelastic adhesives for structural applications e.g. in transportation industry requires robust and reliable design methods. Especially lifetime prognosis is of strong demand. The deformation of semi-structural polyurethane adhesives can be characterized with uniaxial tensile tests and modelled in according to Marlow with sufficient accuracy. For fatigue investigations of hyperelastic adhesives on CFRP adherends, geometrically adapted (adhesive layer thickness, overlapping length, adhesive layer diameter) and with CFRP laminated test specimens in form of the Thick Adherend Shear, Butt Joint as wells as Angle Joint Specimen can be used. In contrast to structural adhesives with comparatively small adhesive layer thicknesses, flexible adhesive systems do not have a homogeneous stress state due to the adhesive layer necking in a thicker layer. Accordingly, a numerical evaluation of the local stress distribution and strength evaluation appears inevitably useful. The combination of the investigated adhesive and material reacts under quasi-static load with constant strength and decreasing rupture strain. There is no influence on the fatigue strength. The influence of stress multiaxiality on fatigue strength are described with an isotropic and pressure dependent equivalent stress hypothesis. The obligatory parameter
identification and optimization in combination leads to the local master S-N curve and thus enables the lifetime prognosis of multi-axially stressed adhesive joints. This concept is based on the assumption of absence of crack initiation and propagation during tests. However, it is shown by means of image acquisition and evaluation that cracks already occur in early stages of the tests and change the nature of the test into a fracture mechanics problem. Use of cycles to failure is then technically speaking only valid for the individual tests and sample types. Since image acquisition during tests is rather extensive a method is proposed to estimate the first occurrence of cracks from the results of load/ displacement curves. Consequently all test results are reevaluated and compared to former evaluation. As result S-N curves are shifted to less cycle numbers without changing the slope. Using numbers of cycles to crack initiation result in more generality of the method. The local stress analysis on the LWF-KS2 specimen under consideration of the experimental boundary conditions from the fatigue tests reflects a stress inhomogeneity due to the joint part deformation. This effect supports the suitability of the specimen for the validation of predicted results. The shown validation process confirms the prognostic capability of the published computationally efficient method for stress-based lifetime prediction of adhesively bonded joints with hyperelastic material behaviour under fatigue loading.
Acknowledgements All shown research results are taken from IGF research project 19187 BG by Research Association for Automotive Technology (FAT) and Research Association on Welding and Allied Processes (DVS), which have been founded by the AiF under the program for promotion of industrial research (IGF) by the Federal Ministry of Economics and Energy based on a decision of the German Bundestag.
References [1] Lutz A, Droste A, Brändli C. Strukturkleben im Fahrzeugleichtbau: Eigenschaften und Haftungsspektrum moderner Strukturklebstoffe, Simulation und Anwendung im Karosserierohbau und in der Montage. Landsberg: Verl. Moderne Industrie; 2013. [2] Costa M, Viana G, da Silva LFM, Campilho RDSG. Environmental effect on the fatigue degradation of adhesive joints: A review. The Journal of Adhesion 2016;93(1-2):127–46. https://doi.org/10.1080/00218464.2016.1179117. [3] Razavi SMJ, Ayatollahi, Samari M, da Silva LFM. Effect of interface non-flatness on the fatigue behavior of adhesively bonded single lap joints. Proceedings of the IMechE 2019;233(7):1277–86. https://doi.org/10.1177/1464420717739551. [4] Pereira AM, Reis PNB, Ferreira JAM. Effect of the mean stress on the fatigue behaviour of single lap joints. The Journal of Adhesion 2017;93(6):504–13. https://doi.org/10.1080/00218464.2015.1110699. [5] Shenoy V, Ashcroft IA, Critchlow GW, Crocombe AD, Abdel Wahab MM. An evaluation of strength wearout models for the lifetime prediction of adhesive joints subjected to variable amplitude fatigue. International Journal of Adhesion and Adhesives 2009;29(6):639–49. https://doi.org/10.1016/j.ijadhadh.2009.02.008. [6] Hennemann O-D, Hahn O, Schlimmer M et al. Methodenentwicklung zur Berechnung und Auslegung geklebter Stahlbauteile im Fahrzeugbau bei schwingender Beanspruchung: Schriftenreihe Forschung für die Praxis P 653. Düsseldorf: Verlag und Vertriebsgesellschaft mbH; 2012. [7] Matzenmiller A, Hanselka H, Mayer B et al. Schwingfestigkeitsauslegung von geklebten Stahlbauteilen des Fahrzeugbaus unter Belastung mit variablen
Amplituden: Schriftenreihe Forschung für die Praxis P 796. Düsseldorf: Verlag und Vertriebsgesellschaft mbH; 2012. [8] Schmidt H. Schwingfestigkeitsanalyse struktureller Klebverbindungen unter Belastung mit variablen Amplituden. Zugl.: Darmstadt, Techn. Univ., Diss., 2014. Stuttgart: Fraunhofer-Verl; 2014. [9] Mayer B, Meschut G, Matzenmiller A, Hanselka H et al. Analyse der Schwingfestigkeit geklebter Stahlverbindungen unter mehrkanaliger Belastung: Schriftenreihe Forschung für die Praxis P 1028. Düsseldorf: Verlag und Vertriebsgesellschaft mbH; Forthcoming 2020. [10] Abdel Wahab MM. Fatigue in Adhesively Bonded Joints: A Review. ISRN Materials Science 2012;2012(3):1–25. http://dx.doi.org/10.5402/2012/746308. [11] Meschut G, Teutenberg D, Wünsche M. Prüfkonzept für geklebte Stahl/CFKStrukturen. Adhaes Kleb Dicht 2015;59(3):16–21. https://doi.org/10.1007/s35145015-0513-6. [12] DVS - Deutscher Verband für Schweissen und verwandte Verfahren e.V., EFB Europäische Forschungsgesellschaft für Blechverarbeitung e.V. Testing of Properties of Joints: Testing of Properties of mechanical and hybrid (mechanical/bonded) joints (DVS/EFB 3480-1). Düsseldorf: Verlag für Schweißen und verwandte Verfahren DVS-Verlag GmbH; 2007. [13] Schlimmer M. Anstrengungshypothese für Metallklebverbindungen. Mat.-wiss. u. Werkstofftech. 1982;13(6):215–21. https://doi.org/10.1002/mawe.19820130606. [14] Marlow RS. A general first-invariant hyperelastic constitutive model. In: Busfield J, editor. Constitutive models for rubber III: Proceedings of the third European Conference on Constitutive Models for Rubber 2003, London, UK. Lisse: Balkema; 2003, p. 157–160.
[15] Ogden RW, Roxburgh DG. A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Sciences
1999;455(1988):2861–77.
https://doi.org/10.1098/rspa.1999.0431. [16] Govindarajan SM, Hurtado JA, Mars WV (eds.). Simulation of Mullins effect and permanent set in filled elastomers using multiplicative decomposition: Constitutive models for rubber V. Proceedings of the 5th European Conference on Constitutive Models for Rubber, ECCMR 2007, Paris, France. London: Taylor & Francis; 2008. [17] Kassner M. Entwicklung einer Methodik zur Simulation der Lebensdauer von Klebverbindungen mit Polyurethan-Klebstoffen. Zugl.: Paderborn, Univ., Diss., 2015. Aachen: Shaker Verlag; 2016.
IJAA - Joint design AB19_67: H.L Wan, J.Y Min, J. Zhang, J.P Lin, C.C Sun, Effect of adherend deflection on lap-shear tensile strength of laser-treated adhesive-bonded joints AB19_163: J Weiland, MZ Sadeghi, A Schiebahn, U Reisgen, KU Schroeder, Analysis of back-face strain measurement for adhesively bonded single lap joints using Strain Gauge, Digital Image Correlation and Finite Element Method AB19_158: MZ Sadeghi, J Weiland, A Preisler, J Zimmermann, A Schiebahn, U Reisgen, KU Schroeder, Damage detection in adhesively bonded single lap joints by using backface strain: proposing a new position for backface strain gauges AB19_38: A Barroso, JC Marín, V Mantič, F París, Premature failures in standard test specimens with composite materials induced by stress singularities in adhesive joints AB19_45: Ioannis Katsivalis, Ole Thybo Thomsen, Stefanie Feih, Mithila Achinth, Development of Cohesive Zone Models for the Prediction of Damage and Failure of Glass/Steel Adhesive Joints AB19_26: K Sekmen, E Paroissien, F Lachaud, Simplified stress analysis of multilayered adhesively bonded structures AB19_27: B Ordonneau, E Paroissien, M Salaün, J Malrieu, A Guigue, S Schwartz, A methodology for the computation of the macro-element stiffness matrix for the stress analysis of a lap joint with functionally graded adhesive properties AB19_52: Mohamed Nasr Saleh, Milad Saeedifar, Dimitrios Zarouchas, Sofia Teixeira De Freitas, Stress analysis of double-lap bi-material joints bonded with thick adhesive AB19_159: MZ Sadeghi, J Zimmermann, J Weiland, A Gabener, U Reisgen, KU Schroeder, Failure load prediction of adhesively bonded single lap joints by using various FEM techniques AB19_13: M. A. Morgado, R.J.C Carbas, D.G. dos Santos, L.F.M da Silva, Strength of CFRP joints reinforced with adhesive layers AB19_146: H Grefe, MW Kandula, K Dilger, Influence of the fibre orientation on the lap shear strength and fracture behaviour of adhesively bonded composite metal joints at high strain rates AB19_160: JPA Valente, RDSG Campilho, EAS Marques, JJM Machado, LFM da Silva, Geometrical optimization of adhesive joints under tensile impact loads using cohesive zone modelling AB19_166: T Tannert, A Gerber, T Vallée, Hybrid adhesively bonded timber-concrete-composite floors AB19_189: A. Rudawska, J.W. Sikora, M. Müller, P. Valasek: Effect of Environmental Ageing at Lower and Sub-zero Temperatures on Adhesive Joint Strength AB19_36: RCM Sales, AF de Sousa, CBG Brito, JLS Sena, NNA Silveira, GM Cândido, MV Donadon, Analysis of hygrothermal effects on mixed mode I/II interlaminar fracture toughness of carbon composite joints AB19_197: Y Tan, X Li, G Chen, Q Gao, G-Q Lu, X Chen, Effects of thermal aging on long-term reliability and failure modes of sintered-silver lap-shear joint AB19_33: K.V. Machalická, M. Vokáč, P. Pokorný, M. Pavlíková, Effect of various artificial ageing procedures on adhesive joints for civil engineering applications AB19_135: J Zimmermann, MZ Sadeghi, KU Schroeder, Exposure of structural epoxy adhesive to combination of tensile stress and γ-radiation
AB19_103: C Thongchom, A Lenwari, RS Aboutaha, Bond properties between CFRP plate and firedamaged concrete AB19_75: JR Ni, JY Min, JP Lin, S Wang, HL Wan, Effect of adhesive type on mechanical properties of galvanized steel/CFRP adhesive-bonded joints AB19_176: Barbora Nečasová, Pavel Liška, Jiří Šlanhof, Barbora Kovářová, Effect of Adhesive Joint Stiffness on Maximal Size of Large-format Cladding Comparison of Artificial and Real Environment AB19_83: S Çavdar, D Teutenberg, G Meschut, A Wulf, O Hesebeck, M Brede, B Mayer, Stressbased fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions AB19_94: F Moroni, F Musiari, C Favi, Effect of the surface morphology over the fatigue performance of metallic single lap-shear joints AB19_7: D. V. Srinivasan, A. Aggarwal, S. Idapalapati, Temperature-dependent Peel Performance of Adhesively Rebonded Hybrid Joints AB19_154: S Psarras, G Sotiriadis, T Loutas, V Kostopoulos, Evaluating experimentally and numerically different scarf-repair methodologies
S0143-7496(19)30232-5
DOI:
https://doi.org/10.1016/j.ijadhadh.2019.102483
Reference:
JAAD 102483
To appear in:
International Journal of Adhesion and Adhesives
Please cite this article as: Çavdar S, Teutenberg D, Meschut G, Wulf A, Hesebeck O, Brede M, Mayer B, Stress-based fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions, International Journal of Adhesion and Adhesives, https://doi.org/10.1016/ j.ijadhadh.2019.102483. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Stress-based fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions S. Çavdara*, D. Teutenberga, G. Meschuta, A. Wulfb**, O. Hesebeckb, M. Bredeb, B. Mayerb a
Laboratory for Materials and Joining Technology (LWF), Paderborn University,
Paderborn, Germany; bFraunhofer Institute for Manufacturing Technology and Advanced Materials (IFAM), Bremen, Germany
Provide full correspondence details here including e-mail for the *corresponding author *) Serkan Çavdar, Pohlweg 47-49, 33098 Paderborn, Fon: +49 5251 60-5276, E-Mail: [email protected] **) Andreas Wulf, Wiener Strasse 12, 28359 Bremen, Fon: +49 421 2246 7368, EMail: [email protected]
Stress-based fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions The joining of dissimilar materials like aluminium and carbon fiber reinforced polymer using hyperelastic adhesives to reduce high stress gradients at the overlapping ends in bonded joints is becoming progressively important. The design of these adhesively bonded joints with complex mechanical behaviour requires a valid and efficient method for virtual lifetime prediction. In following, a concept for stress-based fatigue life prediction with associated experimental parameter identification of adhesively bonded joints using hyperplastic adhesives under multiaxial stress conditions is presented. Furthermore, a possibility of optical crack monitoring for adhesively bonded test specimens is demonstrated and a method for crack detection based on amplitude compliance is proposed. Based on these results, the presented lifetime prognosis concept will be validated experimentally and numerically for inhomogeneous stress fields in the adhesive layer using technological test specimens. In addition to the number of cycles to failure, the expected cycles to crack initiation is considered as the prediction variable.
Keywords: lifetime prediction; fatigue (D); finite element stress analysis (C); polyurethane (A); mechanical properties of adhesives (D); adhesive bonding
Introduction Adhesive joining technologies are particularly used in bodyshell constructions to apply innovative lightweight designs to reduce fuel consumption and CO2 emissions. In context of the increasingly implemented multi-material construction concepts, the joining of dissimilar materials like aluminium (Al) and carbon fiber reinforced polymer (CFRP) using adhesive technology is becoming progressively important. Elastic
adhesive systems appear to be suitable, which according to [1] are used with adhesive layer thicknesses between 1 and 5 mm for bonding of attachements like panes, planking and structural components in vehicle and building constructions. Due to the hyperelastic material behaviour, these adhesive systems reduce high stress gradients at the overlapping ends in bonded joints, which appears to be particularly advantageous with regard to mechanical fatigue in cyclically loaded components. Accordingly, the design of these bonded joints, which are subjected to a cyclic local stress field due to cyclic external loads, requires a valid and efficient method for virtual lifetime prediction within the framework of the product development process. For the practical application of the prediction methods, low model complexity and resulting short computation times are important. The relevance of knowledge about adhesive fatigue is reflected in the large number of research projects. These range, for example, from the investigation of environmental influences [2] to interface topology [3] in addition to the consideration of statically superimposed mean loads [4] and variable amplitudes [5]. The complex mechanical behaviour of hyperelastic adhesives and CFRP adherends calls the validity of existing prediction concepts like [6–9],which are mostly developed and validated for structural adhesives and subdivided in stress-life and fatigue initiation/ propagation approaches [10], into question. Out of this motivation, a concept for stress-based lifetime prediction of adhesively bonded hyperelastic joints with CFRP and Al adherends under fatigue loading with constant amplitudes is presented. A polyurethane adhesive is characterized by quasi-static, cyclical load-increasing and load-controlled fatigue tests partly with optical crack monitoring. Furthermore, a stiffness-based method for crack detection is shown. Based on these results, an S-N curve approach for numerical fatigue life
prediction under consideration of the stress multiaxiality is presented and validated by experiments on technological specimens.
Experimental investigations Experimental Details The following experimental investigations are performed with a two-component semistructural polyurethane adhesive (BETAFORCE™ 2850L by DuPont de Nemours, Wilmington, USA; tensile modulus 27 MPa, tensile strength 10 MPa), which is suitable for CFRP materials. In order to ensure complete curing, all adhesive substances and adhesively bonded specimens are initially stored for 7 days at 23 °C and 50 % RH. The subsequent 2-day storage at 40 °C and 50 % RH causes an increase in stiffness and strength due to the moisture-induced post-crosslinking. The final reconditioning takes also place for 7 days at 23 °C and RH. To achieve a cohesive failure in the adhesive layer, all metallic adherends are cathodic dip coated (CDC) and cleaned with isopropanol.
Adhesive characterization The adhesive characterization includes strain controlled tensile, compression and shear tests under quasi-static as well as cyclic quasi-static loading on a uniaxial testing machine with an extensometer (Zwick Z50 with makroXtens by ZwickRoell, Ulm, DE). Sample geometries and dimensions according to DIN 6701-3 are shown in Figure 1. Tensile and compression tests are carried out at a nominal strain rate of ε 1 = 0.05 s-1 whereas shear tests take place at a shear rate of 1 = 0.1 s-1 according to DIN 6701-3.
Figure 1. Geometry and dimensions in mm of samples for adhesive characterization. The adhesive characterization of the adhesive under fatigue loading is carried out by use of shear-specimens of type b on a hydro pulse system (MTS Landmark 50 kN by MTS, Eden Prairie, USA). Tests are carried out with sinusoidal load and load control. Load ratios are 0.1 and 1. The frequency is 5 Hz for tests with high load amplitudes and 7 Hz for all other load amplitudes. To study long time mechanical behaviour creep tests at room temperature and at three different static load levels are performed over a time period of approximately three months using sample type b. The loads are applied by means of steel weights. The masses are 36.9, 55.4 and 73.9 kg which induce nominal shear stresses of 0.5, 0.75 and 1 MPa.
Adhesive bonding characterization For the study of the adhesive bonding properties, thick adherend shear (TASS) and butt joint specimens (BJS) are produced from EN-AW 6060 (Al-Al) in according to DIN EN 14869-2 and DIN EN 15870. In order to investigate the influence of the stress multiaxiality on the adhesive layer strength, angle joint specimens with an inclination of 45° (AJS-45°) are additionally derived from TASS and BJS. The influence of CFRP joining partner on the failure behaviour of bonded hybrid joints is investigated by substitute an adherend with a CFRP laminated (HexPly® M79/34%/UD300/CHS by
Hexcel Corporation, Stamford, USA; 8-layer quasi-isotropic structure with peel ply, laminate thickness of 2.3 mm) adherend (Al-CFRP) in according to [11]. The CFRP laminates are bonded on one side to the aluminium parts with structural adhesive system (SikaPower®-1277 by Sika, Baar, CH; not considered furthermore). The width and overlap length of the TASS and AJS-45° are 20 mm. The BJS has an adhesive layer diameter of 15 mm. The adhesive layer thickness is due to its application relevance 2 mm in all cases. The geometry of the Al-CFRP specimens are shown in Figure 2.
Figure 2. Adhesive layer dimensions of thick adherend shear (TASS), angle joint (AJS45°), butt joint (BJS) and LWF-KS2 specimens, CFRP laminated adherends (Al-CFRP). For the model validation, the LWF-KS2 specimen in according to [12] made of EN AW-6016-T6 (sheet thickness of 1.5 mm) is used for 0°, 45° and 90° (tension dominated) load introduction angles. In order to prevent fatigue-induced failure of specimens in the notch radii, the adherends for the 90° fatigue tests are made of HX420 steel. One-sided CFRP laminates are applied over the bond area (cf. Figure 2) to represent an Al-CFRP joint. The quasi-static tests are carried out with at least 5 repetitions per test series on a tensile testing machine (Zwick Z100 by ZwickRoell, Ulm, DE) in according to DIN 6701-3 with nominal principal strain rate of ε 1 = 0.05 s-1 . This corresponds for the TASS with initial shear strain rate of = 0.1 s-1 (test speed 12 mm/min) and for the BJS
with initial tensile strain rate of = 0.05 s-1 (test speed 6 mm/min). Assuming a state of plane strain in the adhesive layer with inclination of = 45°, the test speed for the AJS45° is 7 mm/min. In order to prevent transverse forces during the tests, the load is introduced with cardanic bearings. The adhesive displacements in the basic tests (TASS, BJS, AJS-45°) are measured with an extensometer (makroXtens by ZwickRoell, Ulm, DE) and in the validation tests (LWF-KS2) with digital image correlation (ARAMIS Adjustable 2D/3D 4M by GOM, Braunschweig, DE) close to the adhesive layer. The evaluation of the parameter is carried out in form of nominal tensile and shear stress as well as tensile and shear strain, which can be calculated from test load , adhesive layer width , overlap length and adhesive layer displacement ∆:
TASS:
∆!
tan
#
⋅
∆! "
⋅ sin
⋅ sin
(3)
&
∆!
and and
(2)
"
$⋅%
BJS:
AJS:
(1)
⋅
(4)
"
⋅ cos
(5)
∆!
⋅ cos
(6)
⋅
tan
"
The load controlled fatigue tests are realized on a hydro pulse system (Hydropuls PSA by Schenk, Darmstadt, DE) according to DIN 50100. The pulsating tests with stress ratio of R = 0.1 are done with constant amplitudes and load frequency of f = 8 Hz. The results of the fatigue tests are evaluated under the assumption of a logarithmic normal distribution based on the standard deviations and resulting scattering margins TN. The relationship between the load amplitude + (force , shear or tensile stress amplitude) and the endured number of cycles ,- is described with equation (7) and
slope parameter .. /
/ 0
1
,-
2 3
(7)
Experimental Results The experimental results are divided into adhesive properties, which are decisive to model the hyperelastic material, and properties of the adhesive bond, which serve to identify and describe the quasi-static and fatigue failure behaviour.
Adhesive characterization Figure 3 shows the results of all quasi-static tests. To ensure the comparability of all cyclic quasi-static tests nominal compression strain levels are set to 10, 20, 40 and 60 %. Nominal tensile strain levels include 10, 20, 40, 60, 80 and 100 %. Nominal shear levels are 19.75, 37.5, 68.5, 97.5, 124.4 and 150 %. These nominal shear levels are chosen to yield maximum principle strains of 10, 20, 40, 60, 80 and 100 % within the shear sample. The results are shown in Figure 4.
Figure 3. Experimental results of quasi-static tests (a. tension, b. compression, c. shear).
Figure 4. Experimental results of cyclic quasi-static tests (a. tension, b. compression, c. shear). Scatter occurs due to small bubbles in the crack surfaces in case of quasistatic tension tests and due to partial adhesive failure in case of cyclic shear tests. The experimental creep test results are fit to a creep compliance model (8) that depends on time 4 and applied nominal shear stress J6, 48 9: ; < 9; < 9= < >? @ 4 A B? 68 < >? @ 4 A
(8)
In this equation the initial static compliance B? is depending on the shear stress. This yields a better overall fit for nonlinear elastic materials than a constant initial compliance. The individual parameters 9C are taken directly from the quasi-static shear tests and are identified as -0.0296, 0.0654 and 0.0468 MPa. The parameter >? , D and E are identified using a non-linear multi regression least square procedure in Python (version 2.7) as 0.0303, 0.1181 and 0.0829. Figure 5 shows the experimental results in comparison to the compliance model (dotted lines).
Figure 5. Experimental results vs. creep model. Figure 6 shows the S-N-curves from the fatigue tests and the experimental results.
Figure 6. Experimental results of shear test under fatigue loading (a. R=0.1, b. R=-1). All individual test results are used to identify the curve parameters in (7) for 0.1 (. 9.261, I? 3.71 LMN) and 1 (. 9.354, I? 5.54 LMN).
Figure 7. Relevant experimental values of fatigue tests. Figure 7 shows the relevant experimental quantities which are measured during a fatigue test or can be evaluated from these results respectively. Maximum O and minimum P loads as well as maximum QO and minimum QP displacements are measured directly and can be used to calculate mean load R and displacement QR , load and displacement amplitudes Q , slope E and creep displacement QST which is the intersection of each individual cycle with the displacement axis
R 0.5 6O < P 8
(9)
QR 0.5 6QO < QP 8
(10)
0.5 6O P 8
(11)
Q 0.5 6QO QP 8
(12)
U QST
(13)
V
W X
< Q=
(14)
Using equations (9) - (14) together with the bond area > and bond thickness 4 the initial stiffness U? , total stiffness UYY , creep stiffness UST and amplitude stiffness U and their inverse B? , BYY , BST and B can be calculated by 6?8
V 6?8
U? 608 VW6?8 → B? 608 W 6?8 W
W
6\8
V 6\8
UYY 6,8 V[6\8 → BYY 6,8 [ 6\8 [
[ 6\8 ]^ 6\8_V[ 6?8
UST 6,8 V
U 6,8
6\8 V 6\8
[
→ BST 6,8 → B 6,8
V]^ 6\8_V[ 6?8 [ 6\8
V 6\8
6\8
(15) (16) (17) (18)
Figure 8 shows the evaluation of different compliances for a dynamic shear load amplitude of 0.99 MPa, mean shear load of 1.21 MPa, 0.1 and a frequency of 7 Hz.
Figure 8. Evaluation of compliances. The creep compliance B`YYaS 6,8 from the static model equation (8) is calculated from the mentioned mean shear load of 1.21 MPa. The static creep compliance does not fit the creep compliance from the fatigue test. The reason for this behaviour is due to the nonlinear static behaviour of the adhesive. Since the maximum load of the first load cycle exceeds the mean load during the static creep tests the initial dynamic compliance must exceed the static initial compliance. Changing the initial compliance B? from the static creep test in equation (8) to the initial compliance B? 608 from the fatigue test ̅ 6,8. yields a much better fit as can be seen for B`YYaS Another observation is that the dynamic creep compliance exceeds the static creep compliance significantly after about 35000 load cycles. From that point on the creep compliance rate seems to be constant until final rupture. The reason for this behaviour is not obviously clear from the test results. To deal with this problem a camera setup is installed to observe the overlap ends of the shear specimens during fatigue tests at a load ratio of 0.1. The image
acquisition is triggered always at the point of maximum deflection for selected load cycle numbers.
Figure 9. Evidently evaluation of crack initiation from image evaluation. The images for each load cycle are then merged together with a plot of dynamic stiffness vs. load cycle number into a movie. Figure 9 shows this evaluation at a cycle number of 31955. At this point one can see from the movie that first cracks occur at both overlap ends and then start to propagate. This observation motivates the assumption that the change in creep compliance rate results from the initiation and propagation of crack(s) which decrease the bonded area and increase the creep load and as result accelerate the creep compliance rate. Crack propagation is stable until approx. 230000 load cycles. Figure 10 shows a amplitude compliance plot for a test at a load ratio of 1. In this case creep compliance should not be relevant since no mean load is applied. Instead the amplitude compliance shows a behaviour which is very similar to creep although no creep load is applied.
Figure 10. Amplitude compliance of fatigue test at a load ratio of 1. It seems to be that the increase of amplitude compliance and decrease of its rate is related to strain weakening that is well known from elastomeric materials and can be seen in Figure 4. Normally the increase of compliance should stop after a particular number of load cycles and saturate. In this case after approximately 40000 load cycles one can observe an inflexion point where the amplitude compliance starts to increase. This can be related by visual evaluation to crack initiation and beginning of crack propagation. Due to the similarity to creep behaviour the amplitude compliance is fit to the creep model (8). Figure 11 shows on the right the same amplitude compliance curve as before and a selection rectangle (blue) which represents the set of experimental data that have to be fit to the model. On the right side one can see the model fit (magenta) for the best parameter set and a transition point at 32000 load cycles.
Figure 11. Selection rectangle (right), model fit (left). The transition point is evaluated by means of a recursive fit procedure in Python (version 2.7). The blue selection rectangle is divided iteratively in sub rectangles of equal width. The number of experimental data inside of any individual sub rectangle is fixed and evenly spaced for the cycle axis. Therefore only the increment in load cycles increases. Corresponding compliance values are calculated by means of linear interpolation. The data within each sub rectangle are then fit to the model and model parameters as well as the sum of least squares as a fit quality indicator are stored. The transition point now represents the width of that sub rectangle in which the model best fits.
Figure 12. Evaluation of transition point for a fatigue test at a load ratio of 0.1. Figure 12 shows results of this evaluation procedure for a fatigue test at a load ratio of 0.1. Consequently all fatigue tests at 0.1 and 1 are evaluated with this method and Figure 13 shows the results.
Figure 13. S-N-curve for cycles to crack initiation for fatigue test at different load ratios (a. R=0.1, b. R=-1). In comparison to Figure 6, the slope of the S-N-curve for crack initiation is nearly the same for both load-ratios (.?.= 9.343 and .1= 9.683). The stress amplitude I? decreases from 3.71 to 2.86 MPa for 0.1 and from 5.54 to 4.84 MPa for 1.
Adhesive bonding characterization The investigated specimens show cohesive failure in the experiments, which are decisive to characterize an adhesive bond. Representative curves for the Al-Al and AlCFRP joints in the stress-strain diagram in according to equations (1) - (6) with associated fracture patterns are shown in Figure 14, left. Due to the 45° inclination of the adhesive layer to the load introduction, the shear and normal nominal stress-strain components of the AJS-45° are equal in magnitude as result of Mohr's coordinate transformation. Furthermore, the stiffness of the butt joint test is higher than that of the shear test due to the lateral strain obstruction.
Figure 14. Stress-strain curves (nominal strain rate 0.05 s-1) and quasi-static strength diagram with elliptical/ circular failure curve for TASS, AJS-45° and BJS, ta = 2 mm. A comparison of the quasi-static test results (Al-Al vs. Al-CFRP) shows identical stiffness and strength with a decrease in ultimate strain on the CFRP side and cohesive failure zone close to the interface (cf. Figure 14, left). The multiaxial strength behaviour in nominal stresses can be described by an elliptical failure curve (cf. Figure 14,
right),
which
here
in
the
first
quadrant
τn-max = σn-max = 9.0 MPa into a circular equation:
merges
with
E2
and
d
ef
efg[ h
A
i
jf
jfg[ h
A
i 1
(19)
Figure 15 shows the nominal shear and tensile fatigue strengths in form of S-N curves. A comparison of the S-N curves between the test series with Al-Al and Al-CFRP confirms the effect that the substitution of the adherend materials has no significant influence on the endurable load level and thus to the endurable number of cycles. The S-N curve parameter ., I? and the scatter ranges kl for Al-Al and Al-CFRP are mostly of a similar order of magnitude. Differences in test series (especially AJS-45°) can be attributed to the specimen manufacturing tolerances as well as to the general scatter of
10.0
10.0
Nominal shear stress amplitude [MPa]
Nominal tensile stress amplitude [MPa]
fatigue tests.
1.0
1.0
Probability of failure 10% 50% 90%
0.1 1,000 103
TASS (Al-Al) AJS-45° (Al-Al) TASS (Al-CFRP) AJS-45° (Al-CFRP)
10,000 100,000 104 105 Cycles to failure Nf
1,000,000 106
Probability of failure 10% 50% 90%
0.1 1,000 103
BJS (Al-Al) AJS-45° (Al-Al) BJS (Al-CFRP) AJS-45° (Al-CFRP)
10,000 100,000 104 105 Cycles to failure Nf
1,000,000 106
Figure 15. Comparison of nominal shear and tensile stress S-N curves for TASS, BJS and AJS-45°, pulsating (R = 0.1) load conditions. The results of the LWF-KS2 validation tests are evaluated due to inhomogeneous stress fields in the adhesive layer in form of the applied forces and displacements measured locally near the adhesive layer (cf. Figure 16). Analogous to basic tests with stiff test specimens, an increasing stiffness with increasing tensile component can be recorded. Nevertheless, in contrast to basic tests, the larger deformation of adherends and the resulting inhomogeneity of the stress in the adhesive layer at 90° load compared to 0°
load leads to a decrease in the quasi-static ultimate load (cf. Figure 16, left). Due to the preventive steel adherends with higher stiffness (avoidance of joining part failure in notch radius) of the 90°-loaded LWF-KS2 specimen in the fatigue tests, the endurable load amplitudes of the 90° test series are higher than those of the 0° test series (cf. Figure 16, right). Furthermore, the slope of the F-N curves for all load conditions are of a similar order of magnitude.
Figure 16. Force-displacement curves (nominal strain rate 0.05 s-1) and F-N curves for 0°, 45° and 90° loaded LWF-KS2 specimens, pulsating (R = 0.1) load conditions. Stress based approaches for multiaxial failure hypothesis In following a criterion for upper stress-based lifetime prognosis of multiaxial stressed adhesive joints with hyperelastic material behaviour is shown initially under several simplifying assumptions in idealised nominal stress space. These findings as well as numerical simulations of the basic tests lead to a local stress analysis.
Nominal stress analysis The failure-based S-N curves in Figure 15 in combination with the procedure to determine the corresponding numbers of cycles to crack initiation shown in Figure 11 are used to describe the multiaxial fatigue strength in form of a failure- and crack-based
master S-N curve. In order to idealize the stress states, the following simplifying assumptions are made analogous to structural adhesive joints: (1) In first approximation, a homogeneous stress field in the adhesive layer is assumed due to stiff adherends. (2) Failure hypotheses are considered, which describe the stress multiaxiality by means of coordinate independent invariants of the stress tensor. It is assumed that a quasi-static strength criteria is also suitable to describe the fatigue behaviour. (3) The stress state in the adhesive layer is idealized assuming the validity of linear elasticity theory. With BJS and AJS-45°, in addition to the externally applied load stresses , additional transverse stresses (m and n ) occur as a result of the lateral strain obstruction of adhesive layers. These are taken in to account with a poisson ratio of o 0.488 and linear elasticity equations: p
m n =1p
(20)
The considered failure hypothesis is a combination of the second invariant of the deviatoric part of the Cauchy stress J2 (cf. equation (23)) with the first invariant of the Cauchy stress I1 (cf. equation (22)). The second invariant of the stress deviator represents the distortion energy and can be equated with the von Mises criterion. An extension by the hydrostatic part I1 extends this to a pressure-dependent model, which can be found for example as generalized Mohr-Coulomb criterion in linear form in the Drucker-Prager-model. Under linear elastic aspects, the hydrostatic weighting factor r can also be understood as lateral strain parameter, so that equation (21) can be transferred to the linear elastic strain energy density and thus to the failure criteria according to Beltrami. An extension of this approach in form of a yield function as well
as equivalent stress st for fatigue loading with plastically compressible adhesive joints can be found in [13] and [9].
=
st uB; < rv=;
(21)
v= w < m < n
(22)
;
;
; ; ; B; yzw m { < zm n { < 6n w 8; | < wm < mn < wn x
(23)
The creation of the nominal upper stress based master S-N curve is carried out in a regression analysis. For this purpose, the data points (upper nominal stress, number of cycles to failure) of different test series (TASS, BJS, AJS-45°) converted into an upper equivalent stress by means of equation (21) with start parameter c0 and interpreted as the result of a common S-N curve with scattering span kl , the associated standard deviation and RMSE in an Excel spreadsheet. The optimization result (failure and initial crack-based
master
rf = 0.377 and ri = 0.406,
S-N slope
curve)
with
parameter
hydrostatic
weighting
.f = 9.0 and .i = 9.3
with
factor minimum
scattering span of kNf = 1:3.1 and kNi = 1:4.7 is shown in Figure 17, left. Equation (21) can also be interpreted as isochronous elliptical equation that can be used to represent fatigue strength by ellipses with constant number of cycles (cf. Figure 17, right). Based on previously made assumptions, the strengths of the TASS are on the ordinate axis with vanishing hydrostatic component. As the tensile component increases, the hydrostatic component increases as result of lateral strain obstruction.
10
Equivalent upper nominal stress [MPa]
10 ~
TASS (Al-CFRP) AJS-45° (Al-CFRP) BJS (Al-CFRP) Crack initiation
1
Probability of failure
;
v=O stO
;
1
6
TASS (Al-CFRP) AJS-45° (Al-CFRP) BJS (Al-CFRP) Nf=10^3 Nf=10^4 Nf=10^5 Nf=10^6 Nf=10^7
4
2
10% 50% 90%
k = 9.0 TN = 1 : 3.1
10,000 104
B;O
stO
8
(J2u)1/2 [MPa]
stO B;O < r v=O
~
;
0 100,000 105
1,000,000 106
Cycles to failure Nf
10,000,000 107
0
2
4 6 I1u [MPa]
8
10
Figure 17. Upper nominal stress-based master S-N curve for pulsating (R = 0.1) load conditions and stress invariant representation of isochronous failure curve for ta = 2 mm. Local stress concept Adhesive layers of real components with flexible adherends are characterized by inhomogeneous stress fields due to elastic and plastic deformations of the parts to be joined. For this reason, a local master S-N curve will be introduced below based on the local stress concept (peak stress or hot-spot concept). For the evaluation of the local adhesive layer stresses in the basic tests, the geometries in Figure 2 are modelled using finite elements in Abaqus/CAE (version R2017x). The discretization of the adherends, the CFRP coupon and the auxiliary adhesive layer is carried out with continuum elements of form C3D8. A hybrid C3D8H formulation with an element edge length of 0.25 mm (8 elements above adhesive layer thickness) is used for the adhesive layer. The parts are joined by tied contacts. According to experimental investigations, the force introduction is realized by means of kinematic coupling via reference points. A detailed view of the assembled models is shown in Figure 18. The adherends and the auxiliary adhesive layer are isotropic and the CFRP coupon orthotropically linear elastic. The hyperelasticity of the adhesive is
modeled with the Marlow model [14] implemented in Abaqus/Standard (version R2017x), neglecting the quasi-static failure. Only the nominal stress-strain curve of the tensile tests (cf. Figure 3) with a poisson ratio of 0.488 (residual compressibility) serve as input variables. The cyclic stress softening with permanent residual deformation can be considered by coupling the Marlow model with a fictive von Mises plasticity and a damage approach according to Ogden-Roxburgh [15] based on [16] and [17].
Figure 18. Simulation models of basic test specimens. Based on the assumption that the cohesive failure occurs in middle of the adhesive layer, the stress distribution is evaluated in middle of the adhesive layer. The curves of the stress invariants v= and B; are compared in Figure 19 as examples of load levels, which lead with 50 % probability of failure to 105 load cycles to failure in experiments.
25
25
TASS BJS AJS-45°
FBJS = 771 N
20
TASS BJS AJS-45°
20
FTASS = 1480 N
FAJS−45° = 1548 N
15
I1 [MPa]
J2 [MPa²]
15
10
10
FAJS−45° = 1548 N
5
5
FTASS = 1480 N
FBJS = 771 N 0
0 0.0
0.2
0.4 0.6 x / lo [-]
0.8
1.0
0.0
0.2
0.4 0.6 x / lo [-]
0.8
1.0
Figure 19. Distribution of second deviatoric invariant J2 (left) and first invariant of CAUCHY stress (right) in symmetric plane and middle of the adhesive layer over the normalized overlap length lo for TASS, BJS and AJS-45° with load equivalence of 105 . In order to consider these stress inhomogeneities, the previously introduced method for creating a master S-N curve is transferred to local stress conditions. For this purpose, the experimental tested upper forces (TASS, BJS and AJS-45°) are defined as loads in the corresponding FE models. The failure critical location in the adhesive layer is determined by the local maximum of B; (shear dominating) for TASS and by v= (hydrostatic dominating) for BJS. The value pairs of B; and v= in highest stressed elements are assigned to the respective number of cycles to failure and crack-initiation. The identification of the master S-N curve parameter is carried out in two steps in Excel and initially only using the test series of the TASS and BJS. The following application of equation (21) with first parameter set to the FE results in Abaqus/CAE (version R2017x) confirms the assumption of the failure critical location for the TASS and BJS. Furthermore, by evaluating the scalar failure criterion at the AJS-45°, unique locations of maximum stress are identified. The value pairs of B; and v= in highest stressed elements allow a second parameter optimization with inclusion of experimental results of AJS-45°. This procedure results in the local master S-N curve shown in Figure 20
with
hydrostatic
weighting
factor
rf = 0.038 and ri = 0.039,
slope
parameter
.f = 8.1 and .i = 8.3 and minimum scattering span of kNf = 1:3.3 and kNi = 1:4.9, analogous to Figure 17. 10
10 ~
TASS (Al-CFRP) AJS-45° (Al-CFRP) BJS (Al-CFRP) Crack initiation
1
Probability of failure
;
v < r =O stO
;
1
6
TASS (Al-CFRP) AJS-45° (Al-CFRP) BJS (Al-CFRP) Nf=10^3 Nf=10^4 Nf=10^5 Nf=10^6 Nf=10^7
4
2
10% 50% 90%
k = 8.1 TN = 1 : 3.3
10,000 104
B;O
stO
8
(J2u)1/2 [MPa]
Equivalent upper stress [MPa]
stO B;O < r v=O
~
;
0 100,000 105
1,000,000 106
Cycles to failure Nf
10,000,000 107
0
10
20 I1u [MPa]
30
40
Figure 20. Upper local stress-based master S-N curve for pulsating (R = 0.1) load conditions and stress invariant representation of isochronous failure curve for ta = 2 mm. In order to verify the failure criterion and the resulting methodology of predicting the fatigue lifetime, the previously generated FE models are simulated quasistatically at different forces. The projection of the results against the master S-N curve provides the 50 % S-N curve of the respective specimen. Furthermore, the sensitivity of the mesh size on the fatigue life to be predicted is investigated. The successive coarsening of the adhesive layer discretization from 8 elements up to 2 elements above the adhesive layer thickness leads to comparable results with the identical parameter r. For example of the AJS-45°, a doubling of the element edge length from 8 to 4 elements over the adhesive layer thickness only leads to an increase of the predicted lifetime by about 4 % and a coarsening from 8 to 2 elements only to an increase of 9 %.
Validation The validation of the stress-based lifetime prediction concept is carried out in the following on basis of the CFRP laminated LWF-KS2 specimens (cf. Figure 16). In contrast to basic test specimens, the stress inhomogeneity in the adhesive layer is caused by load-induced adherend deformation (elastic/ plastic) in addition to the adhesive necking at overlap ends with increasing lateral strain obstruction in middle of the specimen. The further procedure provides a quasi-static simulation (with expected upper forces) of the geometry to be prognosticated with subsequent evaluation of the stress states and projection against an S-N curve, which is usual in the application-oriented lifetime prognosis. Therefore the LWF-KS2 specimen shown in Figure 2 is modelled analogous to the basic test specimens. The discretization of metallic parts to be joined is performed with shell elements of the form S4 with an assigned thickness of 1.5 mm. In order to consider possible plastifications of sheet metals, an elasto-plastic material behaviour with isotropic hardening von Mises plasticity is considered according to a corresponding yield curve. The element form and material definition of the CFRP coupon, the auxiliary (EP) and investigation (PU) adhesive layer and further boundary conditions correspond to the models of the basic test specimens. A detailed view of the assembled models is shown in Figure 21.
Figure 21. Simulation model of LWF-KS2 specimen.
The resulting stress components in the adhesive layer elements for different load levels (analogous to tested load horizons) are converted with equation (21) into failurerelevant equivalent stress with hydrostatic weighting factor rf = 0.038 / ri = 0.039 in post-processing. The distribution of the scalar failure criterion in middle of the adhesive layer are compared in Figure 22 as examples for a load level which leads with 50 % probability of occurrence to a prediction of 105 load cycles to failure. The projection of the exemplary marked maxima of the curves with the master S-N curve in Figure 20 leads to the endurable number of failure cycles.
Figure 22. Distribution of equivalent upper stresses in middle of adhesive layer vs. normalized overlap length lo and width wo for LWF-KS2 specimen with 105 load cycles. The predicted lifetimes (cycles to crack initiation and adhesive failure) of the simulated load stages are compared to the experimental S-N curves in Figure 23. The resulting prediction accuracies for the highest and lowest experimentally tested load horizons (a-max and a-min) are illustrated in Table 1 in form of the ,pre ⁄,exp ratio. ,i-exp represents the number of cycles to crack initiation and ,f-exp the number of cycles to adhesive failure with 50 % probability of occurrence. Based on the basic experiments the amplitude compliance is fit to the creep model (8) to identify the fatigue initiation life. The transition point is assessed using the recursive adjustment method shown in
Figure 11 with Matlab (R2018b). ,i-pre and ,f-pre represent the predicted quantities. ,pre ⁄,exp 1 is thus an underestimation (conservative design) and ,pre ⁄,exp 1 an overestimation of the expected lifetime. The predicted number of cycles to crack initiation and failure deviates from the F-N curves by up to factor 2.5 (conservative), which is due to scatter in fatigue tests and the indirect determination of crack initiation by means of stiffness evaluation, especially for relatively small deformations.
Figure 23. Comparison of experimental and numerical predicted F-N curves (crack initiation and failure) for 0°, 45° and 90° loaded LWF-KS2 specimen, R = 0.1. Table 1. Comparison of experimental cycles to crack initiation/ failure with numerical predicted lifetimes for 0°, 45° and 90° loaded LWF-KS2 specimen. KS2-0°
KS2-45°
KS2-90°
Ni-pre ⁄Ni-exp Nf-pre ⁄Nf-exp Ni-pre ⁄Ni-exp Nf-pre ⁄Nf-exp Ni-pre ⁄Ni-exp Nf-pre ⁄Nf-exp Fa-max ax 0.77
0.43
0.59
0.80
1.42
0.58
Fa-min 0.71
0.42
2.29
1.85
2.16
0.40
The results confirm the applicability of the shown method for lifetime prognosis to inhomogeneous stressed adhesive layers and thus to industrially manufactured and hyperelastically bonded sheet metal and CFRP components.
Conclusions Use of hyperelastic adhesives for structural applications e.g. in transportation industry requires robust and reliable design methods. Especially lifetime prognosis is of strong demand. The deformation of semi-structural polyurethane adhesives can be characterized with uniaxial tensile tests and modelled in according to Marlow with sufficient accuracy. For fatigue investigations of hyperelastic adhesives on CFRP adherends, geometrically adapted (adhesive layer thickness, overlapping length, adhesive layer diameter) and with CFRP laminated test specimens in form of the Thick Adherend Shear, Butt Joint as wells as Angle Joint Specimen can be used. In contrast to structural adhesives with comparatively small adhesive layer thicknesses, flexible adhesive systems do not have a homogeneous stress state due to the adhesive layer necking in a thicker layer. Accordingly, a numerical evaluation of the local stress distribution and strength evaluation appears inevitably useful. The combination of the investigated adhesive and material reacts under quasi-static load with constant strength and decreasing rupture strain. There is no influence on the fatigue strength. The influence of stress multiaxiality on fatigue strength are described with an isotropic and pressure dependent equivalent stress hypothesis. The obligatory parameter
identification and optimization in combination leads to the local master S-N curve and thus enables the lifetime prognosis of multi-axially stressed adhesive joints. This concept is based on the assumption of absence of crack initiation and propagation during tests. However, it is shown by means of image acquisition and evaluation that cracks already occur in early stages of the tests and change the nature of the test into a fracture mechanics problem. Use of cycles to failure is then technically speaking only valid for the individual tests and sample types. Since image acquisition during tests is rather extensive a method is proposed to estimate the first occurrence of cracks from the results of load/ displacement curves. Consequently all test results are reevaluated and compared to former evaluation. As result S-N curves are shifted to less cycle numbers without changing the slope. Using numbers of cycles to crack initiation result in more generality of the method. The local stress analysis on the LWF-KS2 specimen under consideration of the experimental boundary conditions from the fatigue tests reflects a stress inhomogeneity due to the joint part deformation. This effect supports the suitability of the specimen for the validation of predicted results. The shown validation process confirms the prognostic capability of the published computationally efficient method for stress-based lifetime prediction of adhesively bonded joints with hyperelastic material behaviour under fatigue loading.
Acknowledgements All shown research results are taken from IGF research project 19187 BG by Research Association for Automotive Technology (FAT) and Research Association on Welding and Allied Processes (DVS), which have been founded by the AiF under the program for promotion of industrial research (IGF) by the Federal Ministry of Economics and Energy based on a decision of the German Bundestag.
References [1] Lutz A, Droste A, Brändli C. Strukturkleben im Fahrzeugleichtbau: Eigenschaften und Haftungsspektrum moderner Strukturklebstoffe, Simulation und Anwendung im Karosserierohbau und in der Montage. Landsberg: Verl. Moderne Industrie; 2013. [2] Costa M, Viana G, da Silva LFM, Campilho RDSG. Environmental effect on the fatigue degradation of adhesive joints: A review. The Journal of Adhesion 2016;93(1-2):127–46. https://doi.org/10.1080/00218464.2016.1179117. [3] Razavi SMJ, Ayatollahi, Samari M, da Silva LFM. Effect of interface non-flatness on the fatigue behavior of adhesively bonded single lap joints. Proceedings of the IMechE 2019;233(7):1277–86. https://doi.org/10.1177/1464420717739551. [4] Pereira AM, Reis PNB, Ferreira JAM. Effect of the mean stress on the fatigue behaviour of single lap joints. The Journal of Adhesion 2017;93(6):504–13. https://doi.org/10.1080/00218464.2015.1110699. [5] Shenoy V, Ashcroft IA, Critchlow GW, Crocombe AD, Abdel Wahab MM. An evaluation of strength wearout models for the lifetime prediction of adhesive joints subjected to variable amplitude fatigue. International Journal of Adhesion and Adhesives 2009;29(6):639–49. https://doi.org/10.1016/j.ijadhadh.2009.02.008. [6] Hennemann O-D, Hahn O, Schlimmer M et al. Methodenentwicklung zur Berechnung und Auslegung geklebter Stahlbauteile im Fahrzeugbau bei schwingender Beanspruchung: Schriftenreihe Forschung für die Praxis P 653. Düsseldorf: Verlag und Vertriebsgesellschaft mbH; 2012. [7] Matzenmiller A, Hanselka H, Mayer B et al. Schwingfestigkeitsauslegung von geklebten Stahlbauteilen des Fahrzeugbaus unter Belastung mit variablen
Amplituden: Schriftenreihe Forschung für die Praxis P 796. Düsseldorf: Verlag und Vertriebsgesellschaft mbH; 2012. [8] Schmidt H. Schwingfestigkeitsanalyse struktureller Klebverbindungen unter Belastung mit variablen Amplituden. Zugl.: Darmstadt, Techn. Univ., Diss., 2014. Stuttgart: Fraunhofer-Verl; 2014. [9] Mayer B, Meschut G, Matzenmiller A, Hanselka H et al. Analyse der Schwingfestigkeit geklebter Stahlverbindungen unter mehrkanaliger Belastung: Schriftenreihe Forschung für die Praxis P 1028. Düsseldorf: Verlag und Vertriebsgesellschaft mbH; Forthcoming 2020. [10] Abdel Wahab MM. Fatigue in Adhesively Bonded Joints: A Review. ISRN Materials Science 2012;2012(3):1–25. http://dx.doi.org/10.5402/2012/746308. [11] Meschut G, Teutenberg D, Wünsche M. Prüfkonzept für geklebte Stahl/CFKStrukturen. Adhaes Kleb Dicht 2015;59(3):16–21. https://doi.org/10.1007/s35145015-0513-6. [12] DVS - Deutscher Verband für Schweissen und verwandte Verfahren e.V., EFB Europäische Forschungsgesellschaft für Blechverarbeitung e.V. Testing of Properties of Joints: Testing of Properties of mechanical and hybrid (mechanical/bonded) joints (DVS/EFB 3480-1). Düsseldorf: Verlag für Schweißen und verwandte Verfahren DVS-Verlag GmbH; 2007. [13] Schlimmer M. Anstrengungshypothese für Metallklebverbindungen. Mat.-wiss. u. Werkstofftech. 1982;13(6):215–21. https://doi.org/10.1002/mawe.19820130606. [14] Marlow RS. A general first-invariant hyperelastic constitutive model. In: Busfield J, editor. Constitutive models for rubber III: Proceedings of the third European Conference on Constitutive Models for Rubber 2003, London, UK. Lisse: Balkema; 2003, p. 157–160.
[15] Ogden RW, Roxburgh DG. A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Sciences
1999;455(1988):2861–77.
https://doi.org/10.1098/rspa.1999.0431. [16] Govindarajan SM, Hurtado JA, Mars WV (eds.). Simulation of Mullins effect and permanent set in filled elastomers using multiplicative decomposition: Constitutive models for rubber V. Proceedings of the 5th European Conference on Constitutive Models for Rubber, ECCMR 2007, Paris, France. London: Taylor & Francis; 2008. [17] Kassner M. Entwicklung einer Methodik zur Simulation der Lebensdauer von Klebverbindungen mit Polyurethan-Klebstoffen. Zugl.: Paderborn, Univ., Diss., 2015. Aachen: Shaker Verlag; 2016.
IJAA - Joint design AB19_67: H.L Wan, J.Y Min, J. Zhang, J.P Lin, C.C Sun, Effect of adherend deflection on lap-shear tensile strength of laser-treated adhesive-bonded joints AB19_163: J Weiland, MZ Sadeghi, A Schiebahn, U Reisgen, KU Schroeder, Analysis of back-face strain measurement for adhesively bonded single lap joints using Strain Gauge, Digital Image Correlation and Finite Element Method AB19_158: MZ Sadeghi, J Weiland, A Preisler, J Zimmermann, A Schiebahn, U Reisgen, KU Schroeder, Damage detection in adhesively bonded single lap joints by using backface strain: proposing a new position for backface strain gauges AB19_38: A Barroso, JC Marín, V Mantič, F París, Premature failures in standard test specimens with composite materials induced by stress singularities in adhesive joints AB19_45: Ioannis Katsivalis, Ole Thybo Thomsen, Stefanie Feih, Mithila Achinth, Development of Cohesive Zone Models for the Prediction of Damage and Failure of Glass/Steel Adhesive Joints AB19_26: K Sekmen, E Paroissien, F Lachaud, Simplified stress analysis of multilayered adhesively bonded structures AB19_27: B Ordonneau, E Paroissien, M Salaün, J Malrieu, A Guigue, S Schwartz, A methodology for the computation of the macro-element stiffness matrix for the stress analysis of a lap joint with functionally graded adhesive properties AB19_52: Mohamed Nasr Saleh, Milad Saeedifar, Dimitrios Zarouchas, Sofia Teixeira De Freitas, Stress analysis of double-lap bi-material joints bonded with thick adhesive AB19_159: MZ Sadeghi, J Zimmermann, J Weiland, A Gabener, U Reisgen, KU Schroeder, Failure load prediction of adhesively bonded single lap joints by using various FEM techniques AB19_13: M. A. Morgado, R.J.C Carbas, D.G. dos Santos, L.F.M da Silva, Strength of CFRP joints reinforced with adhesive layers AB19_146: H Grefe, MW Kandula, K Dilger, Influence of the fibre orientation on the lap shear strength and fracture behaviour of adhesively bonded composite metal joints at high strain rates AB19_160: JPA Valente, RDSG Campilho, EAS Marques, JJM Machado, LFM da Silva, Geometrical optimization of adhesive joints under tensile impact loads using cohesive zone modelling AB19_166: T Tannert, A Gerber, T Vallée, Hybrid adhesively bonded timber-concrete-composite floors AB19_189: A. Rudawska, J.W. Sikora, M. Müller, P. Valasek: Effect of Environmental Ageing at Lower and Sub-zero Temperatures on Adhesive Joint Strength AB19_36: RCM Sales, AF de Sousa, CBG Brito, JLS Sena, NNA Silveira, GM Cândido, MV Donadon, Analysis of hygrothermal effects on mixed mode I/II interlaminar fracture toughness of carbon composite joints AB19_197: Y Tan, X Li, G Chen, Q Gao, G-Q Lu, X Chen, Effects of thermal aging on long-term reliability and failure modes of sintered-silver lap-shear joint AB19_33: K.V. Machalická, M. Vokáč, P. Pokorný, M. Pavlíková, Effect of various artificial ageing procedures on adhesive joints for civil engineering applications AB19_135: J Zimmermann, MZ Sadeghi, KU Schroeder, Exposure of structural epoxy adhesive to combination of tensile stress and γ-radiation
AB19_103: C Thongchom, A Lenwari, RS Aboutaha, Bond properties between CFRP plate and firedamaged concrete AB19_75: JR Ni, JY Min, JP Lin, S Wang, HL Wan, Effect of adhesive type on mechanical properties of galvanized steel/CFRP adhesive-bonded joints AB19_176: Barbora Nečasová, Pavel Liška, Jiří Šlanhof, Barbora Kovářová, Effect of Adhesive Joint Stiffness on Maximal Size of Large-format Cladding Comparison of Artificial and Real Environment AB19_83: S Çavdar, D Teutenberg, G Meschut, A Wulf, O Hesebeck, M Brede, B Mayer, Stressbased fatigue life prediction of adhesively bonded hybrid hyperelastic joints under multiaxial stress conditions AB19_94: F Moroni, F Musiari, C Favi, Effect of the surface morphology over the fatigue performance of metallic single lap-shear joints AB19_7: D. V. Srinivasan, A. Aggarwal, S. Idapalapati, Temperature-dependent Peel Performance of Adhesively Rebonded Hybrid Joints AB19_154: S Psarras, G Sotiriadis, T Loutas, V Kostopoulos, Evaluating experimentally and numerically different scarf-repair methodologies