A Miniaturized Mixed ModeBending apparatus for in-situ

characterization of bi-material interfacedelamination

Master thesisby

M.H.L. ThissenMT 08.08

March, 2008Committeeprof.dr.ir. M.G.D. Geersdr.ir. J.A.W. van Dommelendr.ir. J.P.M. Hoefnagelsdr.ir. F.G.A. Homburgdr.ir. O. van der Sluis

Eindhoven University of TechnologyDepartment of Mechanical EngineeringSection Mechanics of Materials

Abstract

A new miniature mixed mode bending (MMMB) setup for in-situ charac-terization of interface delamination in miniature multi-layer structures wasdesigned and realized. This set up consists of a novel test configuration toaccomplish the full range of mode mixities and was specially designed withsufficiently small dimensions to fit in the chamber of a scanning electronmicroscope or under an optical microscope for detailed real-time fractureanalysis during delamination. Special care was taken to minimize the ef-fects of friction, the influence of gravity, and the non-linearities due to thegeometry of the setup. The performance of the setup was verified usingspecially-designed calibration samples together with an extensive finite el-ement method analysis. Delamination experiments conducted on homoge-neous bilayer samples in mode I and mixed mode loading were visualizedwith a scanning electron microscope and showed the formation of smallmicro cracks ahead of the crack tip along the interface followed by theirbridging into a full crack demonstrating the advantages of in-situ testing toreveal the microscopic delamination mechanism. Energy release rate calcu-lations of this interface showed a constant interface toughness for differentcrack lengths when these were determined visually with the scanning elec-tron microscope, whereas, crack lengths determined with stiffness lines gavea variable energy release rate.

1

2

Contents

Preface 5

1 Miniature Mixed Mode Bending apparatus 7

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Design of the Miniaturized Mixed Mode Bending (MMMB)apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Novel test configuration for mixed mode loading . . . 9

1.2.2 Design of the test apparatus . . . . . . . . . . . . . . . 13

1.2.3 Numerical analysis . . . . . . . . . . . . . . . . . . . . 16

1.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Proof of principle . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Theoretical aspects 27

2.1 Fracture of bi-material interfaces . . . . . . . . . . . . . . . . 27

2.2 Defining mode mixity . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Finite Element simulations of sample deformation 31

3.1 Results of numerical analysis . . . . . . . . . . . . . . . . . . 32

3.1.1 Finite Element model of a sample . . . . . . . . . . . 32

3.1.2 Thickness . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.3 Sandwiching . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Overall conclusion 39

4.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . 39

3

A Elastic hinges 43

B Flexure pin design 45

C Variation of sample thickness and alignment of the device 47

D Connector alignment 49

E Protective plates 51

F Mode angle 53

G Hysteresis 55

H Clearance in connectors 57

4

Preface

The main part of this report is reproduced from a paper in preparation,titled, ’A Miniaturized Mixed Mode Bending apparatus for in-situ char-acterization of bi-material interface delamination’. Footnotes in this paperrefer to more detailed information given in the appendices. In chapter 2 and3 the theory is discussed which is used in the paper and the finite elementsimulations, that are used to come to the eventual design of the new mixedmode bending setup are described in more detail. The overall conclusion atthe end of this report will have a great resemblance with the conclusion inthe paper. The conclusion is followed by the recommendations. A discussionis only given in the paper, in the section ’Proof of principle’.

5

6

Chapter 1

Miniature Mixed ModeBending apparatus

This chapter is based on the paper, A Mixed Mode Bending apparatus forin-situ characterization of bi-material interface delamination, which is to besubmitted.

1.1 Introduction

The demands by the semiconductors industry for high levels of integra-tion, lower costs, and a growing need for complete system solutions has ledto the emergence of ”System In Package” (SIP) solutions in which ”thepackage contains the system”. Since SIP-microsystems have multiple thinand stacked layers manufactured using different processes and various ma-terials with different stiffness, internal (intrinsic and/or thermal) mismatchstresses are inevitably present, making interface delamination a primary fail-ure mechanism [1]. It is necessary to characterize interfaces in these systemsover a complete range of mode angles, which are related to the ratio of theshear stresses to the normal stresses at the crack tip, since the interfacefracture toughness varies with the mode angle [2]. Currently, the industry isstill heavily depending on trial-and-error methods for product and processdevelopment. Consequently, a strong demand exists for a generic and ac-curate mixed-mode bending (MMB) delamination setup that enables thecharacterization of interface properties over the full range of mode mixities.

A number of experimental techniques have been developed to measure spe-cific interfacial properties, of which the fracture toughness is the most easilyobtainable. Techniques reported in the literature to measure fracture tough-ness include the well-known double cantilever beam test for pure mode-Iloading [3] and end notch flexure test for applying mode-II loading [4], aswell as the mixed-mode bending (MMB) setups [5], [6] and [7], which covera range of mode mixities, see figure 1.1. The primary difficulty in many of

7

the existing delamination experiments is identification of the crack tip lo-cation in order to track the crack length, which is needed to calculate thefracture toughness. Stiffness lines for different crack lengths are generatedwith finite element models to find the crack length or optical magnificationlens systems have been employed in order to track the crack. Both methodsproved to be inefficient. Therefore, detailed in-situ characterization of thedelamination experiment is crucial to pin-point the crack tip location, tomeasure additional delamination characteristics such as the crack openingprofile and the crack growth rate, and to obtain more insight in the fractureprocess occurring along the interface.

P

P

(a) DCB

P

(b) ENF

PP

A

B

(c) Mixed Mode

Figure 1.1: Different loading configurations for interface delamination

In-situ characterization, however, puts serious constraints on the overall sizeof the delamination set up. Evaluation of existing MMB setups elucidatesthe difficulties to use them for in-situ testing. In case of the setup of Reederand Crews, [5], shown in figure 1.2a, restricted dimensions of the designspace prevent the lever from having sufficient length to cover the completerange of mode mixities. Furthermore, this test is difficult to perform in ahorizontal plane (corresponding to a direction of load application that liesin the horizontal place) which is necessary to enable the use of standardmicroscopes to follow the crack tip movement during in-situ delaminationtesting. This is because the viewing axis of, for example, a scanning electronmicroscope (SEM) is, in general, in vertical direction. Merrill and Ho’s setup,[6], shown in figure 1.2b, was also constructed such that the loading directionis vertically, leading to similar limitations. Furthermore, gravity acts on bothof the above mentioned setups, resulting in additional non-linear forces onthe sample. In Thijsse et al. [7], a counter balance was added to the loadingconfiguration of Reeder and Crews to minimize the influence of gravity.However, a consequence of gravity that remains is a strong dependence of themode angle on the crack length. Finally, all currently available methods arehampered by non-linearities caused by friction in hinges that are attachedto the sample and rotation points/joints in the loading frame.

The present work focuses on the design and realization of a miniaturizedmixed mode bending setup which enables in-situ delamination testing forfull range of mode mixities. This setup will be designed such that effects offriction, gravity, and geometrical non-linearities are minimized.

8

F1

F2

F

(a)

F

Adjustable

F2

F1

S1S2

S3S4

Adjustable

Adjustable

(b)

Figure 1.2: The MMB setup configuration of (a) Reeder and Crews [5] and(b) Merrill and Ho [6] that enable mixed mode loading.

1.2 Design of the Miniaturized Mixed ModeBending (MMMB) apparatus

1.2.1 Novel test configuration for mixed mode loading

The key constraint in the design of the new setup is its size, which should besmall enough (i) to handle multi-layer structures, representative for stackedlayers present in SIPs and (ii) to fit in a micro tensile stage (with the availabledesign space of 55 x 47 x 29 mm) which in turn fits in the chamber of aSEM for in-situ delamination testing. Simple down scaling of the existingMMB setups is not feasible because of their load frame configuration and thesample orientation that prevents in-situ microscopic observation. Thereforea new test configuration was developed that meets the above mentionedrequirements and still is able to apply MMB loading comparable to theloading conditions of the Reeder and Crews configuration [5], see figure 1.1c,which is preferred because it was standardized by ASTM (ASTM D6671-01)[8] and generally accepted for characterization of interfacial delamination.A schematic representation of the new loading geometry of the MMMBapparatus is depicted in figure 1.3.

The setup consists of four rigid parts (a to d in figure 1.3) connected withhinges. Advantages of the present design are its working mechanism thatallows to access the maximum range of loading modes, from double cantileverbending (pure Mode I delamination), to pure Mode II delamination, to endnotch flexure in a single setup and the compact geometry (allowing it tobe used in the chamber of an electron microscope). This was realized by aninnovative new lever mechanism. Frame ’c’ is pinned to the outside worldallowing it only to rotate in the test plane. By the application of forcePMMMB on a certain position on part ’b’, part ’d’ moves downward and part’a’ moves upward generating two oppositely directed forces PA and PB. Theratio of the forces PA and PB depends on the position of the loading point onpart ’b’, triggering different loading modes as discussed in more detail below.Another benefit of the design is the insensitivity of force measurements to

9

its self weight, because loading of the sample is done in the horizontal planewhich also allows the microscope to be able to track the crack tip duringdelamination.

ENF Pure Mode II PMMMBMode I (DCB)

Hinges

CrackRigid

MMB loading

a

c

d

Fulcrum

b

PAP

PB

PB PA

Figure 1.3: Schematic of the new loading geometry for mixedmode bending.

An analysis of the loads applied to the sample in the new test setup is per-formed. The loads experienced by the sample (PA, PB, PC , PD) are depictedin the left hand side of figure 1.4 and can be written as:

PA = PMMMB

(H

γ

)(1.1a)

PB = PMMMB

(α

β

(1− H

γ

))(1.1b)

PC =PB

2(1.1c)

PD = PA +PB

2. (1.1d)

Where, PMMMB is the applied mixed mode bending load, H is the positionof load application, and α, β, γ are the dimensions of the loading mechanism.

All of the loads on the sample can be decomposed into pure mode I andpure mode II loads, PI and PII , respectively, which are defined by the loadconfigurations depicted in the top and bottom right hand side of figure 1.4.The corresponding expressions for Mode I and pure Mode II loads are givenby:

PI =PA − PD

2= PMMMB

(H

γ− α

4β

(1− H

γ

))(1.2)

10

H

L/2

PH

L/2

MMMB

PAPB

PCPD

PI

PI

II

PII

P

4

IIP

4

IIP

2

a b

g

PE =

PF =PG =

Figure 1.4: Schematic of the mixed mode bending setup showingthe loads applied to the sample (left) and a decomposition ofthese loads into pure mode I and pure Mode II loading (right)

PII = PB = PMMMBα

β

(1− H

γ

). (1.3)

The pure Mode II loading is chosen such that the two loads PE and PF

acting on top and bottom arms of the specimen in the pre-cracked region,see bottom right of figure 1.4, are equal in order to have a zero Mode Icomponent on the interface. It is clear from the above analysis that whenthe load is applied at the right hand side of the upper lever (part ’b’), i.e.when H = γ, the applied loading corresponds to pure mode I loading:

PI = PA = PMMMB and PII = 0 for H = γ. (1.4)

On the other extreme when the load is applied at the left end side of theupper lever (part ’b’), i.e. when H = 0, then the applied load resemblesconventional ENF loading, i.e.

PII = PB, but PI = −PMMMB(α

4β). for H = 0 (1.5)

Since in this case PI 6= 0, this ENF test can be considered as a combinedmode II and compressive mode I test. Therefore, it is concluded that the EndNotch Flexure (ENF) test, most generally used for mode II fracture analy-sis, does not represent a pure mode II test due to a compressive loading ofthe fractured interface, which promotes friction between the two contactinglayers. Friction leads to an energy release rate measurement which overesti-mates the interface toughness in the experiments. In addition, a position canbe identified in the new loading geometry where the mode I component is

11

fully zero and consequently a pure mode II loading is obtained. This positionis given by:

H =α4β γ

(1 + α4β )

. (1.6)

12

1.2.2 Design of the test apparatus

In this section, several aspects of the loading configuration will be discussed.The device consists of a mechanism of several moving parts. The hinges, thatallow these parts to rotate, introduce non-linearities in the force measure-ments, making it difficult to find representative energy release rate valuesfor the interface due to additional energy dissipation through friction at ro-tating hinges. In the present design, however, elastic hinges are used in orderto exclude the undesired influence of friction and clearance 1. Figure 1.5ashows a schematic of a standard elastic hinge, while figure 1.5b shows themodified elastic hinge design that is used in the current setup. The designof elastic hinges is, in general, based on the principle of elastic bending of athin beam. The torsional stiffness, kΨΨ, of the flexure hinge which is deter-mined by the geometry (t, h and D as shown in figure 1.5) and the Young’sModulus, E, is given by [9]:

kΨΨ = 0.093Eth2

√h

D. (1.7)

The torsional stiffness combined with the achievable maximum rotation, Ψ,determines the maximum moment, T, given by [9]:

T = kΨΨ Ψ. (1.8)

The geometry (t, h, D) and material has been chosen such that the elastichinges can undergo angular rotation, Ψ, that provides displacements largeenough for delamination experiments in the MMMB setup. Another impor-tant design criterion is the maximum bending stress, σ, whose maximumvalue at any time during the experiment should remain below the yieldstress, σy, of the chosen material to avoid permanent deformation of thehinge. This bending stress is given by [9]:

σ = 0.58ΨE

√h

D. (1.9)

From this analysis, it is clear that the material to be selected should havea low Young’s modulus, E, and a high yield strength, σy, to be able toachieve a favorable combination of loads and displacements in the setup. AnTi-6Al-4V alloy was chosen for the loading frame of the setup because of itscombination of yield strength and Young’s modulus.

Because of the limited space, the alternative hinge geometry visualized infigure 1.5b is used. With this geometry, almost half of the installation spacerequired with the conventional flexure hinge shown in figure 1.5a is needed.

1See appendix A.1

13

Dh

t

Groove

(a)

y

(b)

Figure 1.5: (a) Conventional elastic hinge with t: thickness, D: diameter andh: bridge height and (b) modified hinge geometry with D = 5 mm, h = 0.05mm and t = 2 mm. Ψ indicates the rotation angle.

A groove was introduced in the hinge design (shown in figure 1.5b) such thatit closes well before the maximum equivalent Von Mises stress reaches theyield stress of the material to prevent the hinge from undergoing permanentdeformation. This groove thickness is found with a finite element model ofthis hinge, where in the model an axial force and a desired rotation are sim-ulated. In total, the set up consists of 8 elastic hinges, each with dimensionsof D = 5 mm, h = 0.05 mm and t = 2 mm and a maximum rotation of 5.2degrees which was evaluated from the finite element simulations.

The main parts of the setup are depicted in figure 1.6. The ’Main LoadingMechanism’, MLM, is connected to the ’Position bar’ with the so called’Mode selector’. To change the applied mode angle, the Mode selector can beplaced at various discrete positions on the MLM. The arrow on the ’positionbar’ indicates the fixed direction and position of the externally applied force.Two ’connectors’ are attached on both sides of the sample at the end where apre-crack is located. The main loading mechanism and the support hinge areconnected to the sample by the connector parts. The support hinge connectsthe sample to the real world leaving only one rotational degree of freedom.Figure 1.7 shows a picture of the whole setup placed inside a micro tensilestage (Kammrath und Weiss). The range of sample dimensions which can fitin the bending device are (Length × Width × Height): 35 × 1-7.5 × 0.5-6mm.

Other elements in the setup design include 1) an adjustment mechanism toadjust the setup to a certain sample height, which is done with the ’Sampleheight adjuster’, 2) A special screw mechanism (parts 9 and 10 in figure1.7) to adjust the alignment of the sample, and 3) different measures toovercome out-of-plane deflections in the device and to avoid friction betweenthe moving parts and the rigid bottom plate. The latter is achieved by twoelastic supporting pins, positioned in vertical direction, which support thesystem and increase its stability by moving along in horizontal direction (bymeans of elastic bending) with the moving frames of the test device during

14

experiments, thereby keeping the MLM in-plane 2.

1

2

34

5

6

36.5 mm

20

mm

7

Figure 1.6: Design of the new MMMB device: 1: Main Load-ing Mechanism (MLM), 2: Mode selector, 3: Sample, 4: Sup-port, 5: Support hinge, 6: Connector, 7: Position bar.

1

2 35

7

8

910

12

11

35 mm

Figure 1.7: Total MMMB device, placed in a micro tensile stage, with: 1:Main Loading Mechanism (MLM), 2: Mode selector, 3: Sample, 5: Supporthinge, 7: Position bar, 8: Sample height adjuster, 9: Setscrew for sampleheight, 10: Setscrew for alignment, 11: Micro tensile stage, 12: Load cell.

2See appendix B.1

15

1.2.3 Numerical analysis

The MMMB setup together with a homogeneous bilayer sample is modeledin a finite element program (MSC. Marc/Mentat) as shown in Figure 1.8with the boundary conditions applied. Simulations were performed by as-suming plane strain with linear elastic material behavior to understand thebehavior of the device. The bilayer sample, containing a predefined crack,is created using 8-point quadrilateral elements. The crack tip mesh has arosette shape, which contains Transition Elements (TE) and Quarter-PointElements (QPE). Contact conditions in the crack are simulated without fric-tion. In total the sample consists of approximately 30.000 elements and theMLM exist of 100.000 eight-point quadrilateral elements. The material prop-erties of the sample and the MLM are depicted in table 1.1. Results from thesimulations elucidate the behavior of the total MMMB device when the loadis applied at different loading positions. In mode I loading, the maximumstroke that the MLM device can undergo at the cross head of the tensilestage when one of the hinges reaches its maximum deflection is 1.7 mm. ForENF loading, the maximum stroke is 0.5 mm.

Figure 1.8: FE model, with boundary conditions, of theMLM.

Table 1.1: Material properties of Ti-6Al-4V and Brass.Brass (sample)a Ti-6Al-4V (MLM)b

Young’s modulus 112 GPa 113.8 GPaPoissons ratio 0.346 0.342Yield stress 204 MPa 880 MPa

aDetermined from uni-axial tensile experimentsbSee [10]

16

Mode angles 3 from the simulations are calculated from the normal andshear stress profiles ahead of the crack tip according to [1]:

ψ = arctan(

σ12

σ22

), (1.10)

with σ22 the normal stress and σ12 the shear stress. All loading positionsare simulated and the calculated mode angles are shown in figure 1.9a. Itcan be observed that for the first 3 load positions, the mode angle staysalmost constant at approximately 90◦. Then the mode angle drops down to0◦ at the position 13, which corresponds to loading at the right hand sideof the device. This behavior can be explained by the analysis of the appliedMMMB loading performed in the previous section where a special position(H = γ/5 in case of the present setup) was identified between position 3 and4 at which pure mode II loading can be applied. The simulated mode anglefor this position is shown in figure 1.9a as well. When the load is applied atpositions 1, 2, 3 (i.e. H < γ/5) an additional compressive mode I componentexists (Eq. 1.1b) in addition to the pure mode II component acting at thecrack tip. However, the presence of the compressive mode I component actingon the end of the specimen does not influence the mode angle obtained fromthe stress field ahead of the crack tip. It is worth mentioning again thatthe delamination tests performed at those positions (1, 2 and 3) are notcompletely representative of pure mode II tests because of friction in thefractured interface behind the crack tip due to this compressive mode Icomponent 4. This also applies to the conventional ENF test which is mostwidely used as a Mode II delamination test. Implications of these differencesare discussed in the results and discussion section.

1 2 3 4 5 6 7 8 9 10 11 12 130

10

20

30

40

50

60

70

80

90

100

Position [−]

Mod

e an

gle

[Deg

rees

]

(a)

3 6 9 12 150

10

20

30

40

50

60

70

80

90

Crack length [mm]

Mo

de

an

gle

[D

eg

ree

s]

Position 4

Position 7

(b)

Figure 1.9: Mode angle obtained by FEM analysis as a function of (a) loadingposition for a crack length of 15 mm and (b) crack length for two loadingpositions.

The FE model is also used to check if the mode angle changes with cracklength. Figure 1.9b shows the mode angle dependence as function of crack the

3See appendix F and chapter 24See appendix F

17

length for two different loading positions. The mode changes approximately0.1% from the mean mode angle in the middle regime of crack lengths.However for the shortest and the longest crack (equal to half the samplelength), the deviation of the mode angle increases to a maximum of 2.9%.

In order to see the influence of flexure hinges on force measurements, twodifferent simulations are compared in figure 1.10. One is the previously de-scribed FE model with real elastic hinges and the other has the same geom-etry, however, the flexure hinges are replaced by ideal, infinitely complianthinges. It can be seen from the figure that the elastic hinges slightly raisethe actuation force acting on the system by approximately 0.36%, however,this can easily be accounted for.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

Displacement [mm]

Forc

e [N

]

0 0.2 0.4 0.6 0.8 10,3

0,32

0,34

0,36

0,38

0,40

Rela

tive d

iffe

rence [%

]

Elastic hinges

Frictionless hinges

Relative difference

Figure 1.10: Effect of elastic hinges on the obtained actuationforce. The difference is defined as the force increase due tostiffness of the hinges relative to infinity compliant hinges.

18

1.3 Calibration

In addition to the numerical analysis, calibration measurements with theactual setup are conducted to determine any inaccuracies of the obtainedresults coming from the geometry, machine compliance and any other possi-ble factors like clearance at connectors, etc. Calibration is done with speciallydesigned calibration test samples (shown in figure 1.11) suitable for loadingfrom position 1 to position 13. These are homogeneous, single layer samples(i.e. without an interface), but have a well defined notch, with a height of 30µm, representing an existing crack of a fixed length. Figure 1.11 shows twodifferent types of test samples. One type is a solid piece of brass, dimensions35 × 2.5 × 1 mm, with 5 different notch lengths. These samples are usedto perform tests for all positions for which the mode I component is largeenough to keep the notch from closing anywhere along the notch. The sec-ond type is used for all other positions, i.e., from pure mode II to ENF andcontains an elastic beam at the end of the notch to prevent contact betweenthe two arms of the notched portion.

(a) (b)

Figure 1.11: Calibration samples: (a) End parts of a Mode I/mixed modesample and a mode II sample. (b) A detailed image of the elastic hinge ofthe mode II calibration sample.

Figure 1.12 shows a graph with the results of five samples loaded with themode selector in position 7. Three hysteresis loops for every sample areplotted. From these curves the error, defined as the dissipated energy duringa loading-unloading cycle, relative to the energy supplied during loading canbe calculated.

All combinations of crack length and loading position have been tested. Themaximum relative hysteresis was found to be 7% at position 2. The meanvalue was 3% with a minimum hysteresis of 1%. Positions 1, 2 and 3 areidentified to give the largest error 5.

From the calibration results, it can be remarked that the curves in figure 1.12are not completely linear, whereas finite element simulations give a linearresponse during loading and subsequent unloading. From this it is clear that

5See appendix G

19

0 0.05 0.1 0.150

2

4

6

8

10

Displacement [mm]F

[N

]

3 mm 6 mm

9 mm

12 mm

15 mm

Figure 1.12: Hysteresis loops for loading position 7 for fivecrack lengths. The numbers indicate the crack length.

the geometry of the MMMB device is not the cause of this non-linear behav-ior. The source of deviation, which must lie in aspects that are not includedin simulations, such as the flexure pins and the connectors. Still from exper-iments it can be concluded that the flexure pins move with the system, sothat no slip occurs. The source of deviation was found to be the connectors,which attach the sample to the MLM and the support hinge. All connectionsare achieved with so-called ’T-shaped swallowtails’. Although they fit nearlyperfectly, still a clearance with a maximum of 15-20 µm is present 6. Figure1.13 shows a measurement of a calibration sample having a crack of 3 mm,loaded in Mode I. A kink appears at the beginning of the load-displacementcurve at a certain force level, both during loading and unloading. The rest ofthe curve is also slightly bent. Moreover, a corrected curve is plotted in thisfigure. To realize this correction, the vertical opening displacement of thesample arms is measured directly using digital image correlation. This dis-placement is then plotted against the measured forces. The corrected curveshows a better correspondence with the simulated results although a smalldeviation still exists. A reason for this can be the slight angle rotation of theconnectors due to clearance, which is not taken into account in the verticalopening displacement of the arms measured with DIC. Also the clearancedue to the mode selector is not taken into account, as the clearance betweenthe connector and the MLM and the connector and the support hinge ismore significant.

6See appendix H

20

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

2

4

6

8

10

12

14

16

Displacement [mm]

For

ce [N

]

Measurement position 13Corrected for slackResult of simulation

Figure 1.13: Correction for clearance in the T-shaped swal-lowtail connectors as measured using digital image correla-tion on a calibration sample with a crack of 3 mm, which isloaded in position 13.

1.4 Proof of principle

A batch of bilayer samples, consisting of two brass layers, with a thicknessof 0.5 mm each, glued together (with a glue (Araldite 2020) thickness ofapproximately 4 µm) is tested with this newly designed setup to find theinterface strength. The measurements performed in the SEM revealed howthe crack is propagating in the interface. In figure 1.14 an interface with thecrack coming from the right, is shown. In front of this crack, small pre-cracksappear up to 50 - 100 µm in front of the crack tip. These small pre-cracksgrow, connecting each other, which results in propagation of the main crack.

20mm

Glue

Pores

(a)

20mm

Cracks

(b)

Figure 1.14: The ellipses indicate areas (a) before crack evolution, show-ing the initial interface and (b) showing micro cracks after loading of theinterface.

In the SEM, first, the same magnification as previously obtained in a ded-icated optical camera system was used to locate the crack tip. After this

21

location was pin pointed, a higher magnification showed that the actualcrack tip was not located at this position, but was propagated much furtheralong the interface. This illustrates the benefits of using a SEM for deter-mining the location of the crack tip. Also, an advantage of a SEM over anoptical microscope is its much larger depth of view, which makes it possi-ble to track the crack tip location realtime by keeping the crack in focus,even when there are small out-of-plane movements; a feat that is practicallyimpossible with an optical microscope at high magnification. Tracking this’crack tip’ makes it possible to find the precise crack length within 20 µm,evolving with time. This enables to accurately extract the Energy ReleaseRate (ERR) without the use of stiffness lines that need to be generatedwith finite element simulations, as has been done in some other studies andalways include many assumptions.

Figure 1.15 shows the result of a mixed mode loading experiment conductedin a SEM, where the sample is loaded with the mode selector in position 11.In this graph numerically generated stiffness lines and stiffness lines fittedto the unloading loops are plotted. The origin of the numerically generatedstiffness lines is determined from the initial loading slope. This origin is usedfor all numerically generated stiffness lines, following the standard procedureof determining the ERR, as discussed next. Each stiffness line represents asample with a certain fixed crack length, which is also indicated in figures1.15 and 1.16a for all stiffness lines. The area between two successive stiffnesslines and the load-displacement curve represents the energy consumed by acrack growth of da, which yields the energy release rate after dividing thisarea by w · da, where w is the width of the sample.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

1

2

3

4

5

6

7

8

Displacement [mm]

Fo

rce

[N

]

Measured data

Fitted data

Stiffness lines measurements

Siffness lines FEM

4

5

1.5

3.34

4.76

6.38

8.8

Figure 1.15: Mixed mode loading of a bilayer brass sampleon position 11, conducted in a SEM. The stiffness lines areused to calculate the ERR. The numbers indicate the cracklength corresponding to the stiffness lines.

Because this experiment is conducted in a SEM, it was possible to visually

22

measure the crack lengths for every position where unloading occurs in figure1.15. The ERR found with the numerically obtained stiffness and the ERRfound by visual measurements of the crack length are plotted in figure 1.16.Using the numerically obtained stiffness K(a), a continuous description ofthe energy release rate as a function of the crack length is obtained from

dU

da=

dU

dx

dx

da, (1.11)

where x is the displacement and dUdx is obtained from the fitted line shown

in figure 1.15. From figure 1.15 it can be observed that the numericallygenerated stiffness lines can certainly not be used to indicate the lengthof a crack. Figure 1.16a shows the stiffness of a sample plotted as a func-tion of crack length. This figure shows that the stiffness of a sample witha certain crack length as measured using the SEM is lower than the finiteelement model with equal crack length predicts. This may be due to theinterface ahead of the crack tip, which may open slightly, thereby decreas-ing the stiffness, without increasing the crack length. From figure 1.16b, itcan be remarked that the ERR given by the numerically generated stiff-nesses is not constant, whereas the ERR found by visual measurements ofthe crack tip is constant. In this case, only the ERR obtained with FEMsimulations in the area for crack lengths longer than 6 mm may be usedto estimate the interface toughness. As the boundaries of this area are notknown a-priori, the large variations in crack length seen for the FEM sim-ulations make the methodology of determining the ERR through stiffnesslines from FEM simulations unreliable, thereby demonstrating the necessityof conducting an accurate experimental observation of the crack length. Thecalibration results and the experiments in the SEM are, despite of the smallnon-linearity, show good results. Low hysteresis is measured and the firstinterface measurements give a stable interface toughness as a function ofcrack length.

The interface toughness of the brass bilayer samples is shown in figure 1.17for five different loading positions. Mode I loading, position 13, gives thelowest interface toughness, whereas the ENF loading, position 1, gives thehighest interface toughness. From figure 1.9a it is known that a mode angleof 90 degrees is obtained for position 1 and 3. However the ERR for thesetwo positions is not equal in figure 1.17. This is due to the higher frictionraised by the higher negative mode I component in position 1.

23

0 2 4 6 8 10 1210

0

101

102

103

Crack length [mm]

K [N

/mm

]

Stiffness K from FEM simulationsStiffness K from measurements

(a)

1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

80

90

Crack length [mm]

Ene

rgy

Rel

ease

Rat

e [J

/m2 ]

simulations for crack lengthSEM measured crack lenghts

(b)

Figure 1.16: (a) Stiffness determined with simulations and experimentallydetermined stiffness as a function of crack length. (b) ERR as a function ofthe crack length determined with finite element models and visually deter-mined crack lengths using a SEM.

0 1 2 3 4 5 6 7 8 9 10 11 12 130

100

200

300

400

500

600

700

Position [-]

En

erg

y R

ele

ase

Ra

te [

J/m

2]

90 90 18.5 8 0

Mode angle [degrees]

Figure 1.17: Interface toughness of a bilayer brass sample,with a layer thickness of 0.5 mm, as a function of loadingposition and mode angle.

1.5 Conclusion

A new test configuration capable of applying a mixed mode bending load to adouble layer delamination sample with pre-crack was successfully introducedand designed with elastic hinges to overcome non-linearities due to friction.An analysis of the setup proved the capability of the new setup to achievea full range of mode mixities, with a constant mode angle as a function ofthe crack length. The analysis also revealed the fact that the conventionalENF test is not a true representation of mode II delamination, but that

24

both ENF and pure mode II loading is achievable with the new setup. Thedifference in ERR for an ENF and a pure mode II configuration was foundfor bilayer samples. Finally, an in-situ test done in a SEM showed a high-resolution observation of the delamination mechanism. It is also found thatthe crack length can be determined much more precisely. Energy release ratecalculations of this interface showed a constant interface toughness for cracklengths determined visually with the scanning electron microscope, whereas,crack lengths determined from a comparison of the sample stiffness withnumerical simulations gave a variable energy release rate. This is due toan overestimation of the sample stiffness by FEM. The interface toughnessmeasured on five different loading positions of the miniaturized mixed modebending device of a bilayer brass sample showed good results.

25

26

Chapter 2

Theoretical aspects

This chapter introduces how the interface strength between two layers iscalculated using the Griffith energy balance and discussed the conditionsleading to propagation of the crack. To define the loading mode, a crack tipexperiences, a definition is presented which can handle homogeneous andheterogeneous samples.

2.1 Fracture of bi-material interfaces

Linear elastic fracture mechanics (LEFM) assumes the material used to be-have purely linear elastic [11]. LEFM models make use of predefined cracksin the models. The aim of LEFM is to predict if and how the initially in-duced crack will propagate, by making use of the Griffith energy balance. Inthis approach, the criteria for crack propagation consider for given loadingconditions the energy balance in the material. It is supposed that energy dis-sipation is only due to crack propagation. The other energies are the storedelastic energy in the material and the externally supplied mechanical energy.The change in kinetic energy can be neglected, because of the slow speedat which the crack is propagating. Thus, the Griffith energy balance, whichregards the energy per unit of newly-created fracture surface, or per unit ofcrack length a if the width of the material is constant, is given by:

dUe

da− dUi

da=

dUa

da, (2.1)

with Ue the mechanically supplied energy, Ui the elastically stored energy inthe material and Ua the dissipated energy due to crack propagation. Dividingthe left hand side of equation (2.1) by the material width B gives the energyrelease rate, G.

G =1B

(dUe

da− dUi

da

)[J/m2] (2.2)

27

Crack growth will occur, if the energy release rate reaches a critical valueG = Gc. This value can be defined as the interface strength.

2.2 Defining mode mixity

There are three principal methods to load a crack: the opening mode (modeI, 0 degrees), the sliding/shearing mode (mode II, 90 degrees) and the tearingmode (mode III), see figure 2.1. Only Mode I and II are considered in thepresent study. In general, the interface strength under mode I loading willbe less than under mode II loading, depicted schematically in figure 2.2. InLEFM, the mode mixity (i.e. a measure for the ratio of mode I and mode IIloading) is defined by the local stress field at the crack tip.

(a) Mode I (b) Mode II (c) Mode III

Figure 2.1: Different loading modes.

-90o

90o

0o

Gc

y

Figure 2.2: Mode dependency of Gc.

In LEFM the crack is assumed to have an infinitely sharp crack tip and tobehave purely linear elastic under deformation. These assumptions resultin a singularity in the stress field as the distance from the crack tip, r,see figure 2.3 approaches zero. Because only mode I and mode II loadingare discussed, the shape of the singularity will be described with the stressintensity factors KI and KII . The complex stress intensity factor is definedas K = KI + iKII . The stresses at a distance r ahead of the crack tip andfor θ = 0 (i.e. at the interface) is given by [1]:

(σ22 + iσ12)θ=0 =KI + iKII√

2πrriε, (2.3)

28

(a) (b)

Figure 2.3: (a) Coordinate system defining r and θ and (b) schematic of theoscillating stress field close to the crack tip with θ = 0.

which describes a stress field that oscillates with the distance r from thecrack tip. The normal and shear stress are defined by σ22 and σ12. Theoscillatory index ε represents the elastic mismatch between two materialsand is defined by one of Dunders’ parameters, β [12]:

ε =12π

ln(

1− β

1 + β

)(2.4)

β =µ1(κ2 − 1)− µ2(κ1 − 1)µ1(κ2 + 1) + µ2(κ1 + 1)

, (2.5)

with µ = E2(1+ν) and κ = 3− 4ν for plane strain conditions, where E is the

Young’s modulus and ν the Poisson’s ratio. In equation (2.5), µ1 and µ2 rep-resent the shear modulus of the two different materials. For a homogeneoussample, consisting of two equal material layers, the oscillatory index, ε, is 0,resulting in a stress field without oscillations. The mode angle, ψ, is definedby the orientation of the stress field. Because the stress field oscillates forheterogeneous samples, ψ needs to be defined at a certain reference lengthr = D:

ψ = arctan(

σ12

σ22

)

r=D

= arctan[Im(KDiε)Re(KDiε)

]. (2.6)

Infinite stresses at the crack tip can not exist in reality. A zone will existnear the crack tip where plastic deformation takes place, before the crackpropagates. The size of this proces zone is equal to the distance to the cracktip of the position where the linear elastic stress field equals the yield stressof the weakest material [13]. The size of the proces zone is used to definethe reference length D. If this reference changes from D1 to D2, the changein ψ is given by:

ψD2 − ψD1 = ε ln(

D2

D1

). (2.7)

29

The phase index ε∗, is defined as the change of ψ corresponding to a changein D of one decade:

ε∗ = ε ln (10). (2.8)

This phase index is material and interface dependent.

2.3 Conclusion

The interface toughness is described as Gc and can be expressed as a functionof different mixed mode loading conditions ranging from 0 to 90 degrees. Themode mixity is the ratio between mode I and mode II loading. The modeangle, ψ, has to be calculated for a given reference length, because of theoscillating stress field, due to material mismatch, in heterogeneous bilayersamples. The MMMB device makes it possible to measure the interfacetoughness for the range of mode angles from 0 to 90 degrees.

30

Chapter 3

Finite Element simulations ofsample deformation

The behavior of the sample is important for the design of the MMMB de-vice. With finite element simulations, the deflections required to produce anEnergy Release Rate, which is high enough for the crack to propagate ac-cording to an assumed critical value, are calculated. From these simulationsthe corresponding load ranges are obtained. These parameters are used inthe design of the MMMB setup.

In chapter 1 the loading method of Reeder and Crews was chosen (figure3.1), due to the ASTM standard, and the world wide acceptance of thismethod for its use for interface toughness measurement. The behavior ofthe FE samples, when applying this loading method, is analyzed in thischapter. Simulations on homogeneous bilayer samples give more insight inthe behavior of the loading method of the new setup. Furthermore, the effectof the thickness of the samples on the mode angle ahead of the crack tip isinvestigated. Finally, a sandwiching method is discussed, which can be usedto obtain the interface strength of very thin or compliant samples.

P1P2

Figure 3.1: Mixed Mode Loading method definedby Reeder and Crews [4] and used in the newsetup.

31

3.1 Results of numerical analysis

3.1.1 Finite Element model of a sample

In Marc/Mentat a Finite Element model of a bilayer sample is created using8-point quadrilateral elements. The samples are described with 2D plane-strain models containing a predefined crack, with a crack tip mesh. Thismesh, which contains Transition Elements (TE) and Quarter-Point Elements(QPE), is shown in figure 3.2 [14]. The dimensions of the sample are 35 ×2.5 × 1 mm, (Length × Width × Height), and the material properties areassumed to be linear elastic. In this case the properties of Brass are used,see table 1.1.

Crack

Figure 3.2: The finite element model of a sample with a detailedview of the crack tip mesh. The sample dimensions are: Length:35, Width 2.5 and Height (thickness): 0.5-6 mm.

3.1.2 Thickness

From chapter 2 it is known that a homogeneous sample, consisting of twolayers of equal material, has a constant mode angle. This is not the case forthin samples as can be seen in figure 3.3 which shows the mode angle aheadof the crack tip for homogeneous samples with different thicknesses underMode I loading 1.

The mode angles are not constant for the thinner samples. To examinethis in more detail, the normal stress and the shear stress components, σ22

and σ12, are plotted in figure 3.4a for a sample of 1.9 mm, while figure3.4b shows the behavior of these stresses for a sample of 0.6 mm. The thinsample experiences negative normal stresses, the other sample stays in thepositive regime of the normal stress. From the definition of the mode angle,ψ, equation (2.6), it can be seen that the change in polarity of σ22 which ismore than a factor of 10 higher than σ12, causes peaks in the mode angle,depicted in figure 3.3 for a sample with a thickness of 0.84 mm.

1See appendix F

32

0 0.1 0.2 0.3-10

-8

-6

-4

-2

0

2

4

6

length ahead of crack tip [mm]

Mo

de

an

gle

[D

eg

ree

s] 1.4 mm

1.12 mm

0.84 mm

1.96 mm

Figure 3.3: The effect on the mode angle of vari-ous sample thicknesses under Mode I loading.

0 0.1 0.2 0.3 0.4 0.5 0.6-1

0

1

2

3

4

5x 10

9

length ahead of crack tip [mm]

Str

ess [

Pa

]

Normal stress σ22

Shear stress σ12

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

8

length ahead of crack tip [mm]

Str

ess [

Pa

]

Normal stress σ22

Shear stress σ12

(b)

Figure 3.4: The normal and shear stress as a function of the distance aheadof the crack tip, for mode I loading of a sample with a thickness of (a) 1.9mm and (b) 0.6 mm.

For the loading configuration of the MMMB device, the figures 3.5, 3.6 and3.7 are generated for a sample with a thickness of 1.9 mm.

The figure shows a rather constant behavior of the mode angle under differ-ent loading modes. The mode I loading, figure 3.5, shows as expected a modeangle of 0 degrees. The mode II loading, figure 3.6, shows a sign change of90 to -90 degrees. This is due to the behavior of the normal stress at thecrack tip. The mixed mode loading in figure 3.7 shows a slight increase of themode angle. Therefore, it was chosen to use a reference length, D = 50 µm,to calculate the mode angle, ψ, equation (2.6).

33

0 0.1 0.2 0.3 0.4 0.5 0.6-1

0

1

2

3

4

5x 10

9

length ahead of crack tip [mm]

Str

ess [

Pa

]

Normal stress σ22

Shear stress σ12

(a) Stresses

0 0.1 0.2 0.3 0.4 0.5 0.6−10

0

10

20

30

40

50

60

70

80

90

length ahead of crack tip [mm]

Mod

e an

gle

[Deg

rees

]

(b) Mode angle

Figure 3.5: Mode I Loading, sample thickness is 1.9 mm.

Normal stress σ22

Shear stress σ12

(a) Stresses (b) Mode angle

Figure 3.6: Mode II Loading, sample thickness is 1.9 mm.

Normal stress σ22

Shear stress σ12

(a) Stresses (b) Mode angle

Figure 3.7: Mixed Mode Loading, sample thickness is 1.9 mm, ratio σ22/σ12

= 1.

34

3.1.3 Sandwiching

In some cases the compliance of the material layers in the sample are suchthat the deflection which is needed to induce propagation of the crack cannot be achieved by the setup, since the maximum rotation of the hingesis too small. In these cases, a higher stiffness of the samples is desired toachieve delamination.

Producing samples with thicker layers is an option, but often this is notfeasible due to the fact that it is not possible to make them. However, if itis necessary to add more stiffness to the samples, the so-called sandwich-ing technique, which is shown in figure 3.8a, can be helpful. This methodmakes use of substrates, consisting of a different material than the layers ofthe sample, to increase the stiffness of both layers. The substrates are at-tached on both sides of the samples. The increase in overal stiffness resultsin smaller deflections of the sample needed to propagate the crack. Sand-wiching therefore increases the range of materials and dimensions that canbe tested with the MMMB device [15]. Another advantage of this methodis the fact that brittle or fiber reinforced samples can be tested as wel. Thesubstrates decrease the deflection of the layers, prohibiting failure in thebeams at the vicinity of the crack tip, due to compression, see figure 3.8b.

The influence of substrates on the mode angle of heterogeneous samplesis shown in figures 3.9 and 3.10. Table 3.1 gives the material propertiesof the heterogeneous samples and the substrates. With increasing Young’smodulus of the substrates, the mode angle becomes less unstable, figure 3.9.Increasing the thickness of the layers in heterogeneous samples, results inshifting of the mode angle. This also happens, but then with a larger effect,if the thickness of the Silicon substrate changes.

Substrate

SubstrateLayer 1

Layer 2

(a)

P

P

Compression failure

(b)

Figure 3.8: (a) Sandwiching method. The sample is glued between two sub-strate layers to increase the stiffness. (b) Compression failure in a brittlesample.

35

Table 3.1: Material properties of the substrates and the heterogeneous sam-ples

Molding Compound Epoxy (MCE) Lead Frame (LF) Silicon (Si)Young’s modulus 21.6 GPa 123 GPa 165 GPaPoisson’s ratio 0.25 0.33 0.27

0 0.1 0.2 0.3-100

-50

0

50

100

Distance ahead of crack tip [mm]

Mode a

ngle

[D

egre

es]

0.28 LF 0.28 MCE

Pa

Figure 3.9: Effect of sandwiching with substrates with differ-ent Young’s moduli on the mode angle. The numbers repre-sent the thickness of the layers in mm.

0 0.1 0.2 0.3-100

-50

0

50

100

Distance ahead of crack tip [mm]

Mo

de

an

gle

[D

eg

ree

s]

. 2x0.28 Si

0.28 LF 0.28 MCE

0.56 LF 0.56 MCE

0.7 LF 0.7 MCE

0.28 LF 0.28 MCE, substr

0.28 LF 0.28 MCE, substr. 2x0.42 Si

Figure 3.10: Influence of increasing layer thickness of a heteroge-neous sample and the effect of different Silicon substrate thick-nesses on the mode angle. The numbers represent the thicknessof the layers in mm.

36

3.2 Conclusion

This chapter revealed that it is in some cases necessary to use a referencelength ’D’ to calculate the mode angle for homogeneous samples. The modeangle of these homogeneous layers is not constant anymore, because of thesmall layer thickness. Simulations including heterogeneous samples showthat the mode angle changes especially in the first 50 µm ahead of the cracktip. Sandwiching these samples between stiffer substrates does not changethis effect, but the overal behavior is less unstable. The use of these sub-strates makes it possible to measure a high variety of samples in the setup.Because the deflections of a sample can be controlled by the right choice ofsubstrates, the maximum displacements necessary to propagate the crack ina sample can be decreased.

37

38

Chapter 4

Overall conclusion

A new test configuration capable of applying a mixed mode bending loadto a double layer delamination sample with a pre-crack was successfully in-troduced and designed with elastic hinges to overcome non-linearities dueto friction. An analysis of the setup proved the capability of the new setupto achieve a full range of mode mixities, with a constant mode angle as afunction of the crack length. The analysis also revealed the fact that theconventional ENF test is not a true representation of mode II delamination,but that both ENF and pure mode II loading is achievable with the newsetup. The difference in ERR for an ENF and a pure mode II configurationwas found for bilayer samples. A high number of different sample materialscan be tested in the MMMB device, especially when a sample is sandwichedbetween two substrates, which enables smaller displacements, therefore alsosmaller rotations of the elastic hinges, necessary to propagate a crack. Fi-nally, an in-situ test done in a SEM showed a high-resolution observationof the delamination mechanism. It was also found that the crack length canbe determined much more precisely. Energy release rate calculations of aninterface showed a constant interface toughness for crack lengths determinedvisually with the scanning electron microscope, whereas, crack lengths deter-mined from a comparison of the sample stiffness with numerical simulationsgave a variable energy release rate. It turned out that FE simulations pre-dict a higher stiffness of a sample as obtained from the force-displacementcurve. Therefore the stiffness lines generated with FEM can not be usedto find the actual crack length. The interface toughness measured on fivedifferent loading positions of the miniaturized mixed mode bending deviceof a bilayer brass sample showed good results.

4.1 Recommendations

In this section, some recommendations, to solve some experienced difficul-ties, are given. If these difficulties are improved, the functionality of thesetup will improve.

39

First of all, the clearance experienced in the connectors and in the Modeselector should be solved. A mechanism which can set pre-tension in theconnections should be designed. The force-displacement curve will be (more)linear if it is possible to solve this clearance.

The protective plate on top of the MLM disturbs the signal which is mea-sured by the detector in the SEM. As a consequence the resolution is low,and it is difficult to track the crack. Without this plate, the resolution ismore than high enough to obtain the proper information. If the geometryof this protective plate is altered, or if a conductive material is chosen, thedisturbance of the signal should be solved.

A simple modification of the MMMB device makes it possible to measuresamples with a length of 15 mm. A benefit of this modification is the relativehigher displacements which can be achieved compared to the sample. Toachieve this modification, the fulcrum and cantilever ’d’ should be moved15 mm closer to cantilever ’a’, figure 1.3.

To track the crack tip during an experiment conducted in the SEM, themicro tensile stage has to be shifted in order to keep the tip in the field ofview. This shifting is controlled by hand and makes it hard to get properinformation about the crack length. Software to track the crack tip and shiftthe stage to keep it in the field of view should be developed.

The SEM enables a high resolution magnification of the crack tip, whichmakes it possible to conduct digital image correlation on the area aroundthe crack tip. This information can give more information about the cracktip and the crack mechanism.

40

Bibliography

[1] J.W. Hutchinson and T.Y. Suo. Mixed Mode cracking in layered ma-terials, Advances in Applied Mechanics, volume 29. Academic Press,inc., 1992.

[2] ASTM D 5528-01. Standard test method for mode I interlaminar frac-ture toughness of unidirectional fiber-reinforced polymer matrix com-posites. Annual Book of ASTM Standards, 15.03, 2001.

[3] L.A. Carlsson, J.W. Gillespie, and JR R.B. Pipes. On the analysis anddesign of the end notched flexure (enf) specimen for mode II testing.Jl. Comp. Mater, 20, 1986.

[4] J.R. Reeder and J.R. Crews. Mixed-mode bending method for delami-nation testing. AIAA Journal, 28(7):1270–1276, 1990.

[5] J.H. Crews Jr. and J.R. Reeder. A mixed-mode bending apparatus fordelamination testing. Technical Memorandum 100662, NASA, august1988.

[6] C.C. Merrill and P.S. Ho. Effect of mode-mixity and porosity on in-terfacial fracture of low-k dielectrics. In Materials Research SocietySymposium Proceedings, volume 812. Materials Research Society, 2004.

[7] J. Thijsse, O van der Sluis, J.A.W. van Dommelen, W.D. van Driel,and Geers M.G.D. Interfacial adhesion method for semiconductor ap-plications covering the full mode mixity. Microelectronics Reliability 48,pages 401–407, 2008.

[8] ASTM D 6671-01. Standard test method for mixed mode I mode II in-terlaminar fracture toughness of unidirectional fiber reinforced polymermatrix composites. Annual Book of ASTM Standards, 15.03, 2001.

[9] M.P. Koster. Constructieprincipes, voor het nauwkeurig bewegen enpositioneren, volume 5. HB uitgevers.

[10] R. Boyer, G. Welsch, and E.W. Collings. AMaterials Properties Hand-book: Titanium Alloys. eds. ASM International, 1994.

[11] M.F. Kanninen and C.H. Popelar. Advanced fracture mechanics. OxfordClarendon Press, 1985.

41

[12] J. Dundurs. Edge-bonded dissimilar orthogonal elastic wedges undernormal and shear loading. Journal of Applied Mechanics, Transactionsof the ASME, 37:650–652, 1969.

[13] C.F. Shih. Cracks on bimaterial interfaces: elasticity and plasticityaspects. Materials science and engineering, A143:77–90, 1991.

[14] P.J.G Schreurs. Breukmechanica, syllabus bij de cursus breukmechan-ica. Reader, Technische Universiteit Eindhoven, 1996.

[15] J.R. Reeder, K Demarco, and S.W. Whitley. The use of doublers indelamination toughness testing. Proceedings of the American Societyfor composites 17th technical conference.

[16] R.T. Fenner. Mechanics of solids. 1999.

42

Appendix A

Elastic hinges

A Finite Element model of a elastic hinge is created in MSC Marc/mentat, inorder to check the analytical formulas that describe the mechanical behaviorof elastic hinges in chapter 1. Figure A.1 shows the FEM model. The linearbehavior of this hinge is depicted in figure A.2. This behavior is measuredby rotating the hinge, which is achieved by moving the top of the hinge inthe direction of the vector, indicated in figure A.1. The moment needed torotate the hinge has a maximum of 0.68 Nmm at the maximum rotation.Figure A.1 shows the equivalent Von Mises stresses present in the flexurehinge due to rotation together with an applied tensile load of 20 N. Themaximum Von Mises stress is 7.42 MPa due to this loading. This is still 138MPa lower than the Yield stress of Ti-6Al-4V, the material from which thehinge is made, see table 1.1. With a maximum rotation of 5.2 degrees, thistype of hinge is suited to use in the setup.

Figure A.1: FE model of the type of elastic hinges used inthe MMMB setup showing the equivalent Von Mises stressesin Pa occurring by maximum rotation.

43

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Rotation angle [Degrees]

Mom

ent [

Nm

m]

Figure A.2: Linear behavior of the elastic hinges.

In the design of the MMMB mechanism, the bottom left three hinges areplaced on one line, see figure 1.3. This configuration leads to the smallest hor-izontal (unwanted) displacement, dL, with a maximum vertical movement.Figure A.3 shows this behavior. One advantage of a triangle configuration,i.e. the middle hinge being out of line, is the increase of the maximum verti-cal displacement by the same rotation angle. Still, the unwanted horizontaldisplacement increases relatively more.

40 graden

6,80 mm 12,95 mmdL

Rotation point

5 graden5 degrees

l

(a)

dL

5 degrees

lRotation point

(b)

Figure A.3: The configuration presented in (a) shows that placing all 3 hingesin line results in a small (unwanted) horizontal displacement dL, whereas(b) shows that a triangle geometry results in a larger (unwanted) horizontaldisplacement. In both figures λ is equal and 5 degrees indicates the maximumrotation of the elastic hinges.

44

Appendix B

Flexure pin design

The whole MLM mechanism is only supported at three locations. One is theconnection to the rigid structure. This is the rigid rotation point (fulcrum),depicted in 1.3. The other two are vertical supporting pins. These flexurepins are positioned at the locations where the MLM exerts force on thesample. The main focus is to keep the MLM mechanism in-plane, thus tocarry the mass of the MLM and the mass of the sample. This gives morestability, but especially the alignment is improved. These flexure pins aredepicted in figure B.1.

Flexure pins

(a)

Flexure pin

(b)

Figure B.1: Placement of the flexure pins. (a) A 3D view, indicating thelocation of the flexure pins and (b) a cross-section of the device showinghow the flexure pins are attached to and constructed in the setup.

These flexure pins have the same advantages as the elastic hinges. No frictionor stick-slip occurs while these pins move. The MLM is situated on top ofthese flexure pins, thus the friction between the pins and the surface of theMLM should be high enough such that the pins move along with the device.Slip would lead to energy loss. Two important design principles are usedto come to this design. It is desired to give the pins the maximum lengthas possible, such that they just fit in the vertical direction of the setup, inorder to reach the desired horizontal deflection of the top of 2 mm. Also the

45

pin should not fail (kink) under a loading of 200 g. The axial force which aflexure pin can withstand is calculated from [16]:

Pk =π2EI

l2, (B.1)

where E is the Young’s modulus of the material, l the length of the flexurepin, and I the moment of inertia given by:

I =πD4

64, (B.2)

whit D the diameter of the flexure pin.

The moment needed to deflect the pin by a distance, dD, is calculated by:

M0 =2 dDEI

l2. (B.3)

Substitution of equation B.1 in B.3 yields:

M0 =2dDPk

π2. (B.4)

The bending stress can be obtained as:

σmax =23

σyield ≥ (D/2)M0

I, (B.5)

with 23 representing a safety factor.

With a length of the pins of 14 mm, and using material, Ti-6Al-4V see table1.1, the diameter of the pins is determined with the above equations. Adiameter of 0.25 mm is calculated such that it gives the strength needed forthe pin to withstand buckling.

46

Appendix C

Variation of sample thicknessand alignment of the device

The goal is to develop a robust interface testing device which can handledifferent geometries of samples. In this setup, the thickness and width ofthe samples can cover the range 0.5-6 and 1-7.5 mm respectively. To tunethe setup for a given sample thickness, the Setscrew sample thickness (#9)is used, see figure C.1a. By rotating this setscrew in clockwise direction theSample thickness adjuster moves in the direction indicated with vector V2.To fix the Sample thickness adjuster when the preferred sample thickness isset, the bolts indicated by #13 have to be tightened.

10

9

8

13V2

14

f

Fixed

15

16

(a)

f

(b)

Figure C.1: (a) Thickness control system. 8: Sample thickness adjuster, 9:Setscrew sample thickness, 10: Setscrew sample angle rotation, 14: Align-ment device. (b) Design of the Alignment device. The symbol φ indicatesthe angle rotation if the setscrew is rotated in clockwise direction.

To align this thickness control system, such that the sample is always par-allel with the MLM, an alignment device indicated with #14 and shown in

47

figure C.1b is installed. The spring (#15) which presses the Sample thick-ness adjuster on the supports of the Alignment device, and the Alignmentdevice itself are fixed in the setup with the bolt indicated by #16. The an-gle φ, indicated in figure C.1b is changed by rotating the Setscrew sampleangle rotation (#10). A rotation of the setscrew of 5 degrees results in anangle rotation of 0.012 degrees of the Alignment device. This rotation istransferred to the Sample thickness adjuster. When the sample is alignedthe bolts (#13) have to be tightened.

48

Appendix D

Connector alignment

For the setup to function correctly and to give reproducible results, it isimportant that the points of application of the forces on the sample are welldefined. To position the sample accurately in the MMMB setup, two attach-ment parts are glued with instant glue onto the sample. To do this correctlyand in a reproducible manner, a simple alignment tool was designed. In fig-ure D.1, the tool is presented. The two steps needed to glue the connectorsto the sample are shown in figure D.1a and b.

(a)

Connector

Sample

(b)

Figure D.1: The design of the alignment tool and use of this tool with, (a)step 1: top side gluing, and (b) step 2: bottom side gluing.

49

50

Appendix E

Protective plates

The MLM is rather fragile if it is loaded out of plane. This may happenwhen the sample is removed into or out of the setup. The elastic hinges arevulnerable to the out-of-plane loading and tend to go to plastic yielding.

This problem, which can lead to permanent deformation, is solved by fix-ing a protective plate on top of the parts, which contain elastic hinges,see figure E.1. These protective plates are made of polymethylmethacrylate(PMMA) and float 0.1 mm above the mechanisms. This orientation pro-hibits a too large out-of-plane deflection of the elastic hinges when takingout the sample, i.e. only elastic deformation can occur. During experiments,the measurements are not affected by the plates.

Protective

plates

Figure E.1: The design of the MMMB setup with protectiveplates.

51

52

Appendix F

Mode angle

For mode angle calculations the stress field ahead of the crack tip is usedin this report. Figure 1.9 shows for a pure mode II loading a mode angle of93 degrees. In theorie this should be 90 degrees. Therefore the mode angleof a pure mode II loading is also calculated with the Stress intensity factorsgiven by the simulations according to equation (2.6), resulting in a modeangle of 94 degrees. Again a higher value than the expected 90 degrees. A90 degrees mode angle is only obtained, when the sample is loaded in shearwithout bending. In this case (1), one side of the sample is fixed in the xand y direction with boundary conditions, whereas the other side is fixed iny direction with a small x displacement, see figure F.1. The layer thicknessof this sample is 1 mm. The behavior of the normal stress on the interfaceand ahead of the crack tip is shown in figure F.2.

x

y

Figure F.1: Sample with crack, the arrows indicate the x andy directions.

In case (2), the same sample is loaded in shear with the difference, withrespect to case (1), that the side were the small x displacement is appliedis not fixed in y direction. In figure F.2, also the behavior of the normalstress due to this kind of loading is plotted. This plot shows that the normalstresses are rather high and negative at the reference length of 50 µm aheadof the crack tip. According to equation (2.6), this indeed gives a mode anglehigher than 90 degrees.

For the simulation of the sample in case (1), the normal stresses return tozero degrees ahead of the crack tip after a disturbance in the vicinity of thecrack tip.

53

0 5 10 15 20 25 30-300

-200

-100

0

100

200

300

Distance along the interface [mm]

Norm

al str

ess [M

Pa]

Case 2

Case 1

Figure F.2: The normal stress distribution in a sample de-scribed by case 1 and case 2. The crack is from 0 to 15 mm.

To explain why the mode angle stays constant from pure mode II to ENFin figure 1.9, figure F.3 is shown. This figure shows the normal stress in thesame sample under ENF loading. At the crack tip (15 mm) the normal stressbecomes negative. Again leading to a mode angle higher than 90 degrees.Because the ratio between the normal and shear is the same in every FEmodel, the mode angle will be equal.

0 5 10 15 20 25 30-300

-200

-100

0

100

200

300

Distance along the interface [mm]

No

rma

l str

ess [

MP

a]

Figure F.3: The normal stress distribution in an ENF test.The crack is from 0 to 15 mm.

54

Appendix G

Hysteresis

To find the inaccuracies in the interface measurements, first, hysteresis testswere performed, making use of specially-designed calibration samples. Withthese calibration samples, with a crack length of 3, 6, 9, 12 and 15 mm, allexisting 13 positions of the setup were tested . The results of these measure-ments are combined in figure G.1. The hysteresis is defined as the dissipatedenergy during a loading-unloading cycle, relative to the energy supplied tothe system during loading. The mean hysteresis values of all calibration ex-periments are depicted in figures G.1. Figure G.1a gives information aboutthe influence of the loading positions on the measured error. The contribu-tion of the crack length is shown in figure G.1b. Position 1, 2, 3 clearly showa higher hysteresis value than the other positions. One reason can be the factthat these are mode II dominant tests with a negative mode I componentcausing friction in the fractured interface. An other reason can be the use ofa differently shaped calibration sample. Also position 12 and 13 experiencea higher mean error. This may be due of the higher sensitivity for clearancein these positions.

1 2 3 4 5 6 7 8 9 10 11 12 130

1

2

3

4

5

6

7

Position [-]

Mean h

yste

resis

[%

]

(a)

3 6 9 12 150

1

2

3

4

5

6

Crack length [mm]

Mean h

yste

resis

[%

]

(b)

Figure G.1: Hysteresis measurements, as a function of (a) the influence ofposition of load application, averaged over the different crack lengths and (b)the influence of crack length averaged over the different loading positions.

55

Figure G.1b shows higher hysteresis values for the calibration samples withcrack lengths of 3 and 15 mm. These are the minimum and maximum cracklengths the setup can handle. Therefore, the range of crack lengths from 6to 12 mm is best to use, for calculating of the energy release rate.

56

Appendix H

Clearance in connectors

From the calibration results it can be observed that the curves in the graphsare not completely linear, whereas finite element simulations, where theMLM together with the sample is modeled, give a linear behavior. From thisit is clear that the geometry of the MLM is not the cause of this non-linearbehavior. The error should be found in aspects that were not included inthe simulation. These aspects are the connectors, which attach the sampleto the MMMB part and the support hinge, and the ’Mode selector’. Allconnections are achieved with T-shaped swallowtails. In these connections,clearance can occur if the geometries do not fit perfect, see figure H.1.

slackPlay

Figure H.1: Clearance between the T-shaped swallowtail con-nector and the MLM.

In total the setup has 5 places were clearance can occur. Digital Image Cor-relation (DIC) is used to find the relative displacements between connectedparts. The clearance between the connectors and the MLM is measured anddepicted in figure H.2a. The maximum clearance in these connectors is ap-proximately 20 µm. This magnitude can vary with the alignment of theconnectors to the sample. Figure H.2b depicts the displacement of the microtensile stage measured by a Linear Variable Differential Transformer (LVDT)

57

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−0.005

0

0.005

0.01

0.015

0.02

0.025

Displacement y [mm]

Cle

aran

ce [m

m]

(a)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.02

0.04

0.06

0.08

0.1

0.12

Displacement y [mm]

Effe

ctiv

e di

spla

cem

ent [

mm

]

(b)

Figure H.2: Loading of a sample with a crack length of 3 mm on position 13showing, (a) the measured clearance between the connector and the MLM asmeasured with digital image correlation and (b) the effective displacementplotted against the measured displacement from the LVDT.

(x-axis) plotted against the displacement of the connector, measured withdigital image correlation. The same kink in the curve of this figure is visible swas found in figure 1.12, indicating that clearance contributes to the curvedgraphs in the calibration measurements.

Figure H.3 shows the influence on the measurements of a calibration sampleif the clearance is reduced. To decrease the clearance, foils of 10 µm areinserted in the gaps between the connectors and the MLM/support hinge.The clearance in the mode selector is not reduced, because it contributesvery little. This measurement resulted in a graph with less curvature thana normal measurement (without a foil). This experiment suggests that themeasurements can be linear in future, if clearance can be avoided.

0 0.05 0.1 0.150

2

4

6

8

10

12

14

Displacement [mm]

Forc

e [N

]

Normal clearance

Decreased clearance

Figure H.3: Decreased clearance results in less curvature ofthe graph.

To solve the clearance between connected parts, the method to attach theseseparate parts to each other should be changed. The mechanism should

58

incorporate pre-tension, such that clearance is avoided.

59