- Email: [email protected]
Experimental study of cracks at interfaces with voids
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Available online www.sciencedirect.com Available online at at www.sciencedirect.com Structural Integrity Procedia 00 (2016) 000–000
ScienceDirect ScienceDirect
Procedia Structural (2016) 277–284 Structural IntegrityIntegrity Procedia200 (2016) 000–000
www.elsevier.com/locate/procedia
www.elsevier.com/locate/procedia
21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy
Experimental study of cracks at interfaces with voids
XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal
Michal K. Budzika*, Henrik M. Jensena
Thermo-mechanical modeling of a high pressure turbine blade of an Aarhus University, Department of Engineering, Inge Lehmanns Gade 10, Aarhus C, DK airplane gas turbine engine a a
Abstract
P. Brandãoa, V. Infanteb, A.M. Deusc*
Heterogeneities are inherent parts of adhesively bonded joints. In order to take the full advantage of the adhesive bonding, it is a Department Mechanical Engineering, Instituto Superior Técnico, de Flaws Lisboa, and Av. Rovisco 1, 1049-001 Lisboa, commonly acceptedofthat the bondline and the interface should be Universidade homogenous. voids Pais, present at surfaces of the Portugal adherents or trapped inside the bondline are expected to lower the resistance to fracture. Indeed, with a simple inspection of the b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, force vs. displacement curves, as obtained from mode I double Portugal cantilever beam experiments for assumed homogenous bond lines, c some fluctuations were observed. These fluctuations are due to theTécnico, aforementioned voids. A set of were designedLisboa, with CeFEMA, Department of Mechanical Engineering, Instituto Superior Universidade de Lisboa, Av.specimens Rovisco Pais, 1, 1049-001 Portugal direction. Specifically, we address the problem of crack strong/weak adhesion zones perpendicular to the crack propagation propagation along such interfaces with focus on the relation between the process zone size and the size of the void. In this paper, experimental results are presented followed by a fundamental analytical model. This is sufficient to gain phenomenological Abstract insight into the process of crack propagation along adhesively discontinuous interfaces. Copyright ©their 2016operation, The Authors.modern Published by Elsevier B.V.components This is an open article under the CC BY-NC-ND licenseoperating conditions, aircraft engine areaccess subjected to increasingly demanding © During 2016 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent Peer-review under responsibility of Scientific theAScientific Committee ofelement ECF21.method (FEM) was developed, in order to be able to predict Peer-review under responsibility the Committee of finite ECF21. degradation, one of which isofcreep. model using the the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation Keywords: adhesive bonding, crack propagation, heterogenities, interface fracture. company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D 1.rectangular Introduction block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be usefuland in theadhesive goal of predicting blade life, given a set of FDR Heterogeneities bondingturbine are inherently associated with data. each other. At the macroscopic joint
structure level, bonding is often used to bridge dissimilar (chemically or physically) materials. Once providing © 2016 The Authors. Published by Elsevier B.V. kinematic continuity, the stresses inside the materials are necessarily different with steep gradients expected near the Peer-review under responsibility of the Scientific Committee of PCF 2016. interfaces due to mismatch in elastic material parameters as first shown by Dundurs (1969). The adhesive (bondline) itself is also rarely homogenous. It is a common practice that different fillers are used in the constitution of the Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation. adhesive to reduce the costs or to increase mechanical or chemical parameters. Also, byproducts could be present in
* Corresponding author. Tel.: +45 4189 3217. E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.
* Corresponding Tel.: +351of 218419991. Peer-review underauthor. responsibility the Scientific Committee of ECF21. E-mail address: [email protected]
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016.
Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer review under responsibility of the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.036
Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284 Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000
278 2
the adhesive, curing kinetics can vary across it, air or different gases could be trapped etc. Finally, the interfaces are not always physically and mechanically homogenous or intact as desired. Being the most crucial in transferring the load, the interface may suffer from contaminations, lack of or poor surface treatment leading to lower adhesion forces or kissing bonds. Those last are very dangerous and have been given considerable attention over the past years, e.g. Brotherhood et al. (2003), due to the fact that the adhesion is severely limited but sufficient to transfer acoustic or ultrasonic waves, as such being hard to detect in a non-destructive manner. From the other side, patterning of the interface is attractive for some of the applications including microelectronic components, Tadepalli et al. (2008). In all cases, understanding the behavior of the joint with a degraded adhesion properties is crucial for reliable and robust design. While voids (or heterogeneities in general) inside the adhesive or in the bulk polymer received considerable attention, e.g. Bresson et al. (2013), it is only in recent years that interface heterogeneities have gained more attention. The perturbation theory of Gao and Rice (1989) and its variations [e.g. Willis (2012)] is most of the time used to predict final properties of the bonded joint with varying surface adhesion. The common geometry refers to the peel or double cantilever beam experiment, like in Patinet et al. (2013). Based on the contrast between strong and weak adhesion zones, crack front morphology is also explained. This is however often limited to the case when the interface consists of a strong/weak zone along the crack propagation direction and during steady-state propagation. Recently, an analytical model of a beam on an elastic foundation to analyze the effect of bonded/not-bonded pattern, when stacked perpendicularly to the direction of crack propagation, was proposed by Cuminatto et al. (2015). In the present paper, the focus is on experimental findings for double cantilever beam under mode I fracture mechanics loading when a constant rate of separation is applied to the adherents. The force vs. displacement data are collected for various systems including homogeneous and heterogeneous surface preparation. Nomenclature a b C Δ E Ea e F GI h I l λ λ-1 νa x, y, z
crack length width compliance (=Δ/F) applied displacement Young’s modulus of elasticity of the adherents Young’s modulus of elasticity of the adhesive thickness of the bondline applied transverse force the mode I energy release rate thickness of the adherent second moment of the area of the adherent length of the adherent bondline ‘wave number’ wave length ≡ process zone length Poisson’s ratio of the adhesive Cartesian coordinate system
2. Experimental Two PMMA plates of width, b = 25 mm, thickness, h = 5 mm and Young’s modulus of elasticity, E, of ca. 3.5 GPa, estimated from three point bending experiment, were bonded with an commercial acrylic adhesive (Bostik) to produce double cantilever beam specimens. Half of such a specimen is schematically shown in Fig. 1.
Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284 Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000
279 3
z
F, Δ, dΔ/dt
x y b w
a
Fig. 1. Schematic representation of half of the DCB specimen under mode I loading.
To introduce the pattern consisting of strong – weak – strong interfaces of length w - the weak zone, was masked with an anti-adhesive tape a prior to bonding. These weak zones are effectively voids with near 0 interface fracture energy. The regions of strong bonding result from the mechanical surface abrasion followed by soap cleaning, warm water rinsing and final rinsing in ethanol to remove the water. In the present paper, results will be shown for the weak zones of length w = 1, 10, 20 and 40 mm. The choice of w is related to the process zone length (λ-1 = 22 mm) as will be explained at later stage. 3. Analysis of steady-state fracture A simple analysis of the cantilever beam is made using the Bernoulli-Euler beam kinematics. At the present stage, it provides sufficient information about the most important parameters. 3.1. Force vs. displacement and equivalent R-curves Taking half of the specimen and neglecting transverse shear effects (h<
z ( x 0)
Fa3 3EI
(1) ���
, and E being the Young’s modulus of the adherent. The initial with I being second moment of the area, � � �� loading slope of the force, F, vs. displacement, Δ, curve be found from this boundary condition, once the initial crack position is known. This was used as a reference to study the effect of the finite stiffness of the bondline. Using the Irwin-Kies approach, the energy release rate – the driving force for crack propagation, can be expressed as:
GI
F 2 dC 2b da
(2)
with C being the compliance defined as, C = Δ/F. It is important to mention that a in Eq. (1) does not refer to the real position but rather to the apparent position of the crack i.e. an estimate. This value is overestimated to compensate for e.g. root rotation effects and finite rigidity of the bondline. However, assuming that there is no time-dependence in the bondline and/or in the adherent, the increase in a by δa is independent of the interpretation of a and δa/δt and thus da/dt refers always to the real crack speed. Using the boundary conditions from Eq.(1) we get:
4280
GI 3
Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000 Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284
F3 bh
3
F2 2bE
(3)
Introducing the steady-state crack propagation criterion GI = GIC with GIC as a constant, the following relation results: (4)
F 1 / 2
with α being a fracture energy dependent constant. Eq.(4) represents a trend curve for the expected steady-state fracture and can be used as a reference for comparison, as well as a method for direct estimation of GIc, by a fitting procedure from the raw experimental data. 3.2. Crack kinetics In the general loading case with the use of Eq.(1), we can write:
Eb (t ) a(t ) h 4 F (t )
3
(5)
Note, that this representation is adequate in a kinematical sense rather than the rheological. The crack speed can be found from:
da da da F dt d t dF t
(6)
After substitution and some manipulation we get: da h Eb dt 3 4 F
1/ 3
1 1 F t F t
(7)
By letting, F = α Δ-1/2 we get: da h Eb dt 3 4 F
1/ 3
1 / 2 1 / 2 t t
(7)
In the present case, since the loading conditions used for the experiments are a constant rate of separation, thus, ∂Δ/∂t = const. It can be seen that during the experiment crack speed, da/dt, decreases with increasing displacement. We consider the case of bi-valued energy release rate, viz. strong and weak interfaces. The crack speed is now dependent on a ‘constant’ α which in turn is dependent on the steady-state values of GIC. Since, in our case, the weak interface basically means no resistance to fracture, neglecting the existence of the process zone or complex in-plane shape of the crack at the time being, the crack speed is expected to instantaneously accelerate from the value characteristic for the strong interface (da/dt)strong to the one associated to the weak one (da/dt)weak once the boundary between them is achieved. In the opposite situation, though not of the interest in present paper, since GICstrong>> GICweak, the crack is expected to arrest.
Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284 Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000
281 5
3.3. Important length scale parameter – process zone For the homogeneous interfaces, during the steady-state propagation, the simple analysis from section 3.2 is suitable. However, once the homogeneity is affected, another length scale should be analyzed. Since we focus only on a one dimensional analysis presently, out-of-the plane aspects of the propagation the effect of the process zone should be investigated further. We follow a classical approach to account for the process zone as proposed by Kanninen (1973) and further extended by Penado (1993) which accounted for the bondline. From such analysis, a characteristic wave number, λ, can be defined as:
2 2
4
(8)
k EI
which is directly associated the process zone length, λ-1. In Eq.(8) k is the stiffness of the bondline k = m(bEa/e). The factor m can account for either plane stress (m = 1) or plane strain [m = f(νa) with νa being the Poisson’s ratio of the adhesive] conditions at the crack tip and inside the process zone. For the time being, we will limit analysis to this single length scale parameter, its relation to w, and its impact on stability of the crack growth during the experiment. A simple analysis yields that if the weak interface is at the distance > λ-1 then the crack kinetic should not be affected by the void. If the distance is = λ-1 the stability point is achieved. Finally, once the distance is < λ-1 then unstable crack growth is to be expected. Further refinement on the way to incorporate effects of the mismatch between λ-1 and the length of bonded zone can be made using the approach of Tadepalli et al. (2008) by letting the energy release rate take the form: G Ieff
Aeff A
(9)
GI
with Aeff being the effective bonding area, (λ-1 – w)b, where w is the length of the weak interface once the crack is propagating through the patterns and A = λ-1 b. We will focus more on non-dimensional λw parameter. For λw→0 crack speed should behave as da/dt ≈ da/dtstrong and the voids should lead to a slight oscillation over the average or reference values of α. For λw→∞, da/dt ≈ da/dtweak or, in the present case ∞. 4. Example of results and discussion
4.1. Homogeneous interfaces In Fig. 2 (a) crack propagating through a homogeneous interface is shown. w
10 mm
Fig. 2. Side view of the DCB specimen during crack propagation along (a) the bondline (with ‘strong’ interface surface preparation), (b) example of heterogeneous interfaces with a ‘void’ of w = 5 mm.
6 282
Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000 Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284
'Strong' interface
180 160 140
F (N)
120
F = 512
100
-0.5
80 60 40 20 0 0
20
40
60
80
(mm) Fig. 3. Experimental results of the DCB experiments with the crack propagating along homogeneous interface.
In Fig. 3 all (15) experimental results are shown for the case of crack propagating along homogeneous, strong, interfaces. All cohesive fractures inside the bondline were reported. In addition to experimental curves, the analytical curve following Eq.(7) is given. The bold line represents the average value with the dashed lines showing the lower and upper limits of α (for 95% confidence). Estimated, average, fracture energy, Eq. (3), is 1331 J/m2. The length of the process zone estimated from the experiments was λ-1 = 24 (±2) mm. 4.2. Effect of void In Fig. 4 (a) – (d) experimental results for different sizes of the weak zones/voids, w= 1, 10, 20 and 40 mm are presented together with the 95% confidence trend lines, as based on Eq.(7). Results for two experiments for each void are presented showing good reproducibility of the test results. Once more, the fracture was cohesive when the surfaces were prepared as strong with the crack locus changing to a pure interfacial above the void as depicted in Fig. 2 (b). Returning to Fig. 4, each void is clearly visible resulting in peaks on the curve. Even in the first case presented, Fig. 4 (a), for which the non-dimensional parameter wλ is barely equal 0.04 [and therefore the effect on the area carrying the load is low – Eq.(9)] the void is clearly detected with the results remaining within the confidence bars. While for wλ ≈ 0.08 the experimental curves still remain within the confidence bars, any higher values of wλ result in the confidence bar being passed. Without proposing any fine treatment at the time being, we assume that for the given material system, any void greater than 10% of the homogeneous process zone is potentially dangerous from the design perspective, overpassing the assumed confidence interval, and could lead to premature failure. Finally, in Fig. 5 crack propagation kinetics is shown for all the cases from Fig. 3 including the analytical solution of Eq. (3) – the bold continuous line. The continuous line is based on the measurement made with the frequency of 100 Hz, while the points correspond to every 3s of the test.
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Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000 Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284
180
180
1 mm 'weak'
140
140
120
F (N)
120
F (N)
10 mm 'weak'
160
160
100 80
100 80
60
60
40
40
20
20 0
0 0
20
40
60
80
0
20
40
20 mm 'weak'
180
160
160
140
140
120
120
100
100
F (N)
F (N)
180
60
80
(mm)
(mm)
80
80
60
60
40
40
20
20
0
40 mm 'weak'
0 0
20
40
60
80
0
20
40
(mm)
60
80
(mm)
Fig. 4. Experimental results of the DCB experiments with the crack propagating along homogeneous interface with different ‘void’ sizes, 1 to 40 mm from (a) – (d) respectively.
200 180 160
a (mm)
140 120 100
Analytical a (w = 1 mm) a (w = 10mm) a (w = 20mm) a (w = 40mm)
80 60 40 20 0 0
20
40
60
80
(mm) Fig. 5. Experimental results of the DCB experiments with the crack propagating along homogeneous interface.
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Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284 Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000
From the brief introduction given before, a sudden jump of the crack, viz. da/dΔ (in the present case ≡ da/dt → ∞) should be expected at the values of wλ = 1, and thus around w = 20 mm. The results suggest that this is, indeed, the case confirming that the process zone length could be used in the damage tolerant design or in deciding on acceptable sizes of the flaws inside the joints. 4. Conclusions
We have studied in an experimental manner, crack propagation along heterogeneous interfaces with sharp transitions between weak and strong adhesion zones (or bonds and voids) perpendicular to the crack propagation direction. We found the raw experimental force vs. displacement curves are sensitive to the void sizes. The behavior is mainly affected by two length scales – the size of the void, w, and the process zone length, λ-1, which could be gathered to a single non-dimensional parameter, wλ. In the present case, the voids resulting in wλ = 0.04 could be easily detected from the raw experimental curves. This, we believe, is an interesting and important finding showing good sensitivity of the experiment to the heterogeneities. We showed that the kinetics of the crack propagation is also affected by the aforementioned parameter. As expected, at the values of wλ close to 1, the crack becomes unstable, and as such it may be chosen as an additional design parameter following damage tolerance philosophy. References Bresson, G., Jumel, J., Shanahan, M.E.R., Serin, P., 2013. Statistical aspects of the mechanical behaviour a paste adhesive. International Journal of Adhesion and Adhesives 40, 70-79. Brotherhood, C.J., Drinkwater, B.W., Dixon, S., 2003. The detectability of kissing bonds in adhesive joints using ultrasonic techniques. Ultrasonics 41(7), 521-529. Cuminatto, C., Parry, G., Braccini, M., 2015. A model for patterned interfaces debonding – Application to adhesion tests. International Journal of Solids and Structures 75-76, 122-133. Dundurs, J., 1969. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. Transactions of the ASME Journal of Applied Mechanics 36, 650-652. Gao, H., Rice, J.R., 1989. A first order perturbation analysis of crack trapping by arrays of obstacles. Transactions of the ASME Journal of Applied Mechanics 56, 826-836. Kanninen, M.F., 1973. An augmented double cantilever beam model for studying crack propagation and arrest. International Journal of Fracture 9, 83-92. Patinet, S.,Alzate, L., Barthel, E., Dalmas, D., Vandembroucq, D., Lazarus,V., 2013. Finite size effects on crack front pinning at heterogeneous planar interfaces: experimental, finite elements and perturbation approaches. Journal of the Mechanics and Physics of Solids 61(2), 311-324. Penado, F.E., 1993. A closed form solution for the energy release rate of the double cantilever beam specimen with an adhesive layer. Journal of Composite Materials 27, 383-407. Tadepalli, R., Turner, K.T., Thompson, C.V., 2008. Effects of patterning on the interface toughness of wafer-level Cu-Cu bonds. Acta Materialia, 56, 438-447. Willis, J.R., 2012. Crack front perturbations revisited. International Journal of Fracture 184(1), 17-24.
ScienceDirect
Available online www.sciencedirect.com Available online at at www.sciencedirect.com Structural Integrity Procedia 00 (2016) 000–000
ScienceDirect ScienceDirect
Procedia Structural (2016) 277–284 Structural IntegrityIntegrity Procedia200 (2016) 000–000
www.elsevier.com/locate/procedia
www.elsevier.com/locate/procedia
21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy
Experimental study of cracks at interfaces with voids
XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal
Michal K. Budzika*, Henrik M. Jensena
Thermo-mechanical modeling of a high pressure turbine blade of an Aarhus University, Department of Engineering, Inge Lehmanns Gade 10, Aarhus C, DK airplane gas turbine engine a a
Abstract
P. Brandãoa, V. Infanteb, A.M. Deusc*
Heterogeneities are inherent parts of adhesively bonded joints. In order to take the full advantage of the adhesive bonding, it is a Department Mechanical Engineering, Instituto Superior Técnico, de Flaws Lisboa, and Av. Rovisco 1, 1049-001 Lisboa, commonly acceptedofthat the bondline and the interface should be Universidade homogenous. voids Pais, present at surfaces of the Portugal adherents or trapped inside the bondline are expected to lower the resistance to fracture. Indeed, with a simple inspection of the b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, force vs. displacement curves, as obtained from mode I double Portugal cantilever beam experiments for assumed homogenous bond lines, c some fluctuations were observed. These fluctuations are due to theTécnico, aforementioned voids. A set of were designedLisboa, with CeFEMA, Department of Mechanical Engineering, Instituto Superior Universidade de Lisboa, Av.specimens Rovisco Pais, 1, 1049-001 Portugal direction. Specifically, we address the problem of crack strong/weak adhesion zones perpendicular to the crack propagation propagation along such interfaces with focus on the relation between the process zone size and the size of the void. In this paper, experimental results are presented followed by a fundamental analytical model. This is sufficient to gain phenomenological Abstract insight into the process of crack propagation along adhesively discontinuous interfaces. Copyright ©their 2016operation, The Authors.modern Published by Elsevier B.V.components This is an open article under the CC BY-NC-ND licenseoperating conditions, aircraft engine areaccess subjected to increasingly demanding © During 2016 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent Peer-review under responsibility of Scientific theAScientific Committee ofelement ECF21.method (FEM) was developed, in order to be able to predict Peer-review under responsibility the Committee of finite ECF21. degradation, one of which isofcreep. model using the the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation Keywords: adhesive bonding, crack propagation, heterogenities, interface fracture. company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D 1.rectangular Introduction block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be usefuland in theadhesive goal of predicting blade life, given a set of FDR Heterogeneities bondingturbine are inherently associated with data. each other. At the macroscopic joint
structure level, bonding is often used to bridge dissimilar (chemically or physically) materials. Once providing © 2016 The Authors. Published by Elsevier B.V. kinematic continuity, the stresses inside the materials are necessarily different with steep gradients expected near the Peer-review under responsibility of the Scientific Committee of PCF 2016. interfaces due to mismatch in elastic material parameters as first shown by Dundurs (1969). The adhesive (bondline) itself is also rarely homogenous. It is a common practice that different fillers are used in the constitution of the Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation. adhesive to reduce the costs or to increase mechanical or chemical parameters. Also, byproducts could be present in
* Corresponding author. Tel.: +45 4189 3217. E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.
* Corresponding Tel.: +351of 218419991. Peer-review underauthor. responsibility the Scientific Committee of ECF21. E-mail address: [email protected]
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016.
Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer review under responsibility of the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.036
Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284 Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000
278 2
the adhesive, curing kinetics can vary across it, air or different gases could be trapped etc. Finally, the interfaces are not always physically and mechanically homogenous or intact as desired. Being the most crucial in transferring the load, the interface may suffer from contaminations, lack of or poor surface treatment leading to lower adhesion forces or kissing bonds. Those last are very dangerous and have been given considerable attention over the past years, e.g. Brotherhood et al. (2003), due to the fact that the adhesion is severely limited but sufficient to transfer acoustic or ultrasonic waves, as such being hard to detect in a non-destructive manner. From the other side, patterning of the interface is attractive for some of the applications including microelectronic components, Tadepalli et al. (2008). In all cases, understanding the behavior of the joint with a degraded adhesion properties is crucial for reliable and robust design. While voids (or heterogeneities in general) inside the adhesive or in the bulk polymer received considerable attention, e.g. Bresson et al. (2013), it is only in recent years that interface heterogeneities have gained more attention. The perturbation theory of Gao and Rice (1989) and its variations [e.g. Willis (2012)] is most of the time used to predict final properties of the bonded joint with varying surface adhesion. The common geometry refers to the peel or double cantilever beam experiment, like in Patinet et al. (2013). Based on the contrast between strong and weak adhesion zones, crack front morphology is also explained. This is however often limited to the case when the interface consists of a strong/weak zone along the crack propagation direction and during steady-state propagation. Recently, an analytical model of a beam on an elastic foundation to analyze the effect of bonded/not-bonded pattern, when stacked perpendicularly to the direction of crack propagation, was proposed by Cuminatto et al. (2015). In the present paper, the focus is on experimental findings for double cantilever beam under mode I fracture mechanics loading when a constant rate of separation is applied to the adherents. The force vs. displacement data are collected for various systems including homogeneous and heterogeneous surface preparation. Nomenclature a b C Δ E Ea e F GI h I l λ λ-1 νa x, y, z
crack length width compliance (=Δ/F) applied displacement Young’s modulus of elasticity of the adherents Young’s modulus of elasticity of the adhesive thickness of the bondline applied transverse force the mode I energy release rate thickness of the adherent second moment of the area of the adherent length of the adherent bondline ‘wave number’ wave length ≡ process zone length Poisson’s ratio of the adhesive Cartesian coordinate system
2. Experimental Two PMMA plates of width, b = 25 mm, thickness, h = 5 mm and Young’s modulus of elasticity, E, of ca. 3.5 GPa, estimated from three point bending experiment, were bonded with an commercial acrylic adhesive (Bostik) to produce double cantilever beam specimens. Half of such a specimen is schematically shown in Fig. 1.
Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284 Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000
279 3
z
F, Δ, dΔ/dt
x y b w
a
Fig. 1. Schematic representation of half of the DCB specimen under mode I loading.
To introduce the pattern consisting of strong – weak – strong interfaces of length w - the weak zone, was masked with an anti-adhesive tape a prior to bonding. These weak zones are effectively voids with near 0 interface fracture energy. The regions of strong bonding result from the mechanical surface abrasion followed by soap cleaning, warm water rinsing and final rinsing in ethanol to remove the water. In the present paper, results will be shown for the weak zones of length w = 1, 10, 20 and 40 mm. The choice of w is related to the process zone length (λ-1 = 22 mm) as will be explained at later stage. 3. Analysis of steady-state fracture A simple analysis of the cantilever beam is made using the Bernoulli-Euler beam kinematics. At the present stage, it provides sufficient information about the most important parameters. 3.1. Force vs. displacement and equivalent R-curves Taking half of the specimen and neglecting transverse shear effects (h<
z ( x 0)
Fa3 3EI
(1) ���
, and E being the Young’s modulus of the adherent. The initial with I being second moment of the area, � � �� loading slope of the force, F, vs. displacement, Δ, curve be found from this boundary condition, once the initial crack position is known. This was used as a reference to study the effect of the finite stiffness of the bondline. Using the Irwin-Kies approach, the energy release rate – the driving force for crack propagation, can be expressed as:
GI
F 2 dC 2b da
(2)
with C being the compliance defined as, C = Δ/F. It is important to mention that a in Eq. (1) does not refer to the real position but rather to the apparent position of the crack i.e. an estimate. This value is overestimated to compensate for e.g. root rotation effects and finite rigidity of the bondline. However, assuming that there is no time-dependence in the bondline and/or in the adherent, the increase in a by δa is independent of the interpretation of a and δa/δt and thus da/dt refers always to the real crack speed. Using the boundary conditions from Eq.(1) we get:
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GI 3
Budzik et al./ Structural Integrity Procedia 00 (2016) 000–000 Michal K. Budzik et al. / Procedia Structural Integrity 2 (2016) 277–284
F3 bh
3
F2 2bE
(3)
Introducing the steady-state crack propagation criterion GI = GIC with GIC as a constant, the following relation results: (4)
F 1 / 2
with α being a fracture energy dependent constant. Eq.(4) represents a trend curve for the expected steady-state fracture and can be used as a reference for comparison, as well as a method for direct estimation of GIc, by a fitting procedure from the raw experimental data. 3.2. Crack kinetics In the general loading case with the use of Eq.(1), we can write:
Eb (t ) a(t ) h 4 F (t )
3
(5)
Note, that this representation is adequate in a kinematical sense rather than the rheological. The crack speed can be found from:
da da da F dt d t dF t
(6)
After substitution and some manipulation we get: da h Eb dt 3 4 F
1/ 3
1 1 F t F t
(7)
By letting, F = α Δ-1/2 we get: da h Eb dt 3 4 F
1/ 3
1 / 2 1 / 2 t t
(7)
In the present case, since the loading conditions used for the experiments are a constant rate of separation, thus, ∂Δ/∂t = const. It can be seen that during the experiment crack speed, da/dt, decreases with increasing displacement. We consider the case of bi-valued energy release rate, viz. strong and weak interfaces. The crack speed is now dependent on a ‘constant’ α which in turn is dependent on the steady-state values of GIC. Since, in our case, the weak interface basically means no resistance to fracture, neglecting the existence of the process zone or complex in-plane shape of the crack at the time being, the crack speed is expected to instantaneously accelerate from the value characteristic for the strong interface (da/dt)strong to the one associated to the weak one (da/dt)weak once the boundary between them is achieved. In the opposite situation, though not of the interest in present paper, since GICstrong>> GICweak, the crack is expected to arrest.
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3.3. Important length scale parameter – process zone For the homogeneous interfaces, during the steady-state propagation, the simple analysis from section 3.2 is suitable. However, once the homogeneity is affected, another length scale should be analyzed. Since we focus only on a one dimensional analysis presently, out-of-the plane aspects of the propagation the effect of the process zone should be investigated further. We follow a classical approach to account for the process zone as proposed by Kanninen (1973) and further extended by Penado (1993) which accounted for the bondline. From such analysis, a characteristic wave number, λ, can be defined as:
2 2
4
(8)
k EI
which is directly associated the process zone length, λ-1. In Eq.(8) k is the stiffness of the bondline k = m(bEa/e). The factor m can account for either plane stress (m = 1) or plane strain [m = f(νa) with νa being the Poisson’s ratio of the adhesive] conditions at the crack tip and inside the process zone. For the time being, we will limit analysis to this single length scale parameter, its relation to w, and its impact on stability of the crack growth during the experiment. A simple analysis yields that if the weak interface is at the distance > λ-1 then the crack kinetic should not be affected by the void. If the distance is = λ-1 the stability point is achieved. Finally, once the distance is < λ-1 then unstable crack growth is to be expected. Further refinement on the way to incorporate effects of the mismatch between λ-1 and the length of bonded zone can be made using the approach of Tadepalli et al. (2008) by letting the energy release rate take the form: G Ieff
Aeff A
(9)
GI
with Aeff being the effective bonding area, (λ-1 – w)b, where w is the length of the weak interface once the crack is propagating through the patterns and A = λ-1 b. We will focus more on non-dimensional λw parameter. For λw→0 crack speed should behave as da/dt ≈ da/dtstrong and the voids should lead to a slight oscillation over the average or reference values of α. For λw→∞, da/dt ≈ da/dtweak or, in the present case ∞. 4. Example of results and discussion
4.1. Homogeneous interfaces In Fig. 2 (a) crack propagating through a homogeneous interface is shown. w
10 mm
Fig. 2. Side view of the DCB specimen during crack propagation along (a) the bondline (with ‘strong’ interface surface preparation), (b) example of heterogeneous interfaces with a ‘void’ of w = 5 mm.
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'Strong' interface
180 160 140
F (N)
120
F = 512
100
-0.5
80 60 40 20 0 0
20
40
60
80
(mm) Fig. 3. Experimental results of the DCB experiments with the crack propagating along homogeneous interface.
In Fig. 3 all (15) experimental results are shown for the case of crack propagating along homogeneous, strong, interfaces. All cohesive fractures inside the bondline were reported. In addition to experimental curves, the analytical curve following Eq.(7) is given. The bold line represents the average value with the dashed lines showing the lower and upper limits of α (for 95% confidence). Estimated, average, fracture energy, Eq. (3), is 1331 J/m2. The length of the process zone estimated from the experiments was λ-1 = 24 (±2) mm. 4.2. Effect of void In Fig. 4 (a) – (d) experimental results for different sizes of the weak zones/voids, w= 1, 10, 20 and 40 mm are presented together with the 95% confidence trend lines, as based on Eq.(7). Results for two experiments for each void are presented showing good reproducibility of the test results. Once more, the fracture was cohesive when the surfaces were prepared as strong with the crack locus changing to a pure interfacial above the void as depicted in Fig. 2 (b). Returning to Fig. 4, each void is clearly visible resulting in peaks on the curve. Even in the first case presented, Fig. 4 (a), for which the non-dimensional parameter wλ is barely equal 0.04 [and therefore the effect on the area carrying the load is low – Eq.(9)] the void is clearly detected with the results remaining within the confidence bars. While for wλ ≈ 0.08 the experimental curves still remain within the confidence bars, any higher values of wλ result in the confidence bar being passed. Without proposing any fine treatment at the time being, we assume that for the given material system, any void greater than 10% of the homogeneous process zone is potentially dangerous from the design perspective, overpassing the assumed confidence interval, and could lead to premature failure. Finally, in Fig. 5 crack propagation kinetics is shown for all the cases from Fig. 3 including the analytical solution of Eq. (3) – the bold continuous line. The continuous line is based on the measurement made with the frequency of 100 Hz, while the points correspond to every 3s of the test.
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180
180
1 mm 'weak'
140
140
120
F (N)
120
F (N)
10 mm 'weak'
160
160
100 80
100 80
60
60
40
40
20
20 0
0 0
20
40
60
80
0
20
40
20 mm 'weak'
180
160
160
140
140
120
120
100
100
F (N)
F (N)
180
60
80
(mm)
(mm)
80
80
60
60
40
40
20
20
0
40 mm 'weak'
0 0
20
40
60
80
0
20
40
(mm)
60
80
(mm)
Fig. 4. Experimental results of the DCB experiments with the crack propagating along homogeneous interface with different ‘void’ sizes, 1 to 40 mm from (a) – (d) respectively.
200 180 160
a (mm)
140 120 100
Analytical a (w = 1 mm) a (w = 10mm) a (w = 20mm) a (w = 40mm)
80 60 40 20 0 0
20
40
60
80
(mm) Fig. 5. Experimental results of the DCB experiments with the crack propagating along homogeneous interface.
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From the brief introduction given before, a sudden jump of the crack, viz. da/dΔ (in the present case ≡ da/dt → ∞) should be expected at the values of wλ = 1, and thus around w = 20 mm. The results suggest that this is, indeed, the case confirming that the process zone length could be used in the damage tolerant design or in deciding on acceptable sizes of the flaws inside the joints. 4. Conclusions
We have studied in an experimental manner, crack propagation along heterogeneous interfaces with sharp transitions between weak and strong adhesion zones (or bonds and voids) perpendicular to the crack propagation direction. We found the raw experimental force vs. displacement curves are sensitive to the void sizes. The behavior is mainly affected by two length scales – the size of the void, w, and the process zone length, λ-1, which could be gathered to a single non-dimensional parameter, wλ. In the present case, the voids resulting in wλ = 0.04 could be easily detected from the raw experimental curves. This, we believe, is an interesting and important finding showing good sensitivity of the experiment to the heterogeneities. We showed that the kinetics of the crack propagation is also affected by the aforementioned parameter. As expected, at the values of wλ close to 1, the crack becomes unstable, and as such it may be chosen as an additional design parameter following damage tolerance philosophy. References Bresson, G., Jumel, J., Shanahan, M.E.R., Serin, P., 2013. Statistical aspects of the mechanical behaviour a paste adhesive. International Journal of Adhesion and Adhesives 40, 70-79. Brotherhood, C.J., Drinkwater, B.W., Dixon, S., 2003. The detectability of kissing bonds in adhesive joints using ultrasonic techniques. Ultrasonics 41(7), 521-529. Cuminatto, C., Parry, G., Braccini, M., 2015. A model for patterned interfaces debonding – Application to adhesion tests. International Journal of Solids and Structures 75-76, 122-133. Dundurs, J., 1969. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. Transactions of the ASME Journal of Applied Mechanics 36, 650-652. Gao, H., Rice, J.R., 1989. A first order perturbation analysis of crack trapping by arrays of obstacles. Transactions of the ASME Journal of Applied Mechanics 56, 826-836. Kanninen, M.F., 1973. An augmented double cantilever beam model for studying crack propagation and arrest. International Journal of Fracture 9, 83-92. Patinet, S.,Alzate, L., Barthel, E., Dalmas, D., Vandembroucq, D., Lazarus,V., 2013. Finite size effects on crack front pinning at heterogeneous planar interfaces: experimental, finite elements and perturbation approaches. Journal of the Mechanics and Physics of Solids 61(2), 311-324. Penado, F.E., 1993. A closed form solution for the energy release rate of the double cantilever beam specimen with an adhesive layer. Journal of Composite Materials 27, 383-407. Tadepalli, R., Turner, K.T., Thompson, C.V., 2008. Effects of patterning on the interface toughness of wafer-level Cu-Cu bonds. Acta Materialia, 56, 438-447. Willis, J.R., 2012. Crack front perturbations revisited. International Journal of Fracture 184(1), 17-24.