Master of Science in Aerospace Engineering School of Mechanical

German Institute of Science and Technology – TUM Asia Pte. Ltd.

Master of Science in Aerospace Engineering School of Mechanical and Aerospace Engineering

Internship Report on Characterization of Interlaminar Fracture Toughness of Glass prepreg-Aluminium Fiber Metal Laminates subjected to different Adhesion nature between them

Submitted by: VISHAKH RAJENDRAN G1403000A

Internship Period: October 2015 – January 2016 Internship work carried out at School of Mechanical and Aerospace Engineering Nanyang Technological University Singapore 639798 Under the Guidance of Dr. Chai Gin Boay , Associate Professor, NTU Dr. Manikandan Periyasamy , PHd Student , NTU 1

DECLARATION

I hereby declare that the entire work embodied in this Internship is an original work and has been carried out by me. Experiments were carried out at the Structures Laboratory at the school of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798. No part of it has been submitted for any degree or diploma of any institution previously.

Date:

Signature of the student

Singapore

Vishakh.R

2

Date: 21 / 6 / 2016

CERTIFICATE

This is to certify that the Internship report titled, “Characterization of

Interlaminar Fracture Toughness of Glass prepreg-Aluminium Fiber Metal Laminates subjected to different Adhesion nature between them”

submitted by VISHAKH RAJENDRAN is a Bonafide record of the work carried out at School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore from October 2015 till January 2016 under my supervision and guidance.

Signature of Guide Dr. Chai Gin Boay Assistant Professor School of Mechanical and Aerospace Engineering Nanyang Technological University Singapore

3

ACKNOWLEDGEMENTS

I take this opportunity to express my profound gratitude and deep regards to Dr. Chai Gin Boay, Assistant Professor, School of Mechanical and Aerospace Engineering, Nanyang Technological University for having given me this Internship offer and this project in particular. I would like to thank him for the time that he took to respond to emails and discussions amidst his busy schedule.

I am highly indebted to Dr. Manikandan Periyasamy, PHd student, School of Mechanical and Aerospace Engineering, Nanyang Technological University for his exemplary guidance, constant monitoring and valuable feedback. He had provided great quantum of advice and knowledge during the course of this Internship.

4

Abstract Contents Chapter 1 – Introduction 1.1

Introduction to Fiber Metal Laminate (FML) .................................................................14

1.2

Fracture Mechanics ......................................................................................................15

1.3

Stress Intensity Factor ..................................................................................................15

1.4

Fracture Toughness ......................................................................................................16

1.5

Strain Energy Release Rate ...........................................................................................17

1.6

Modes of Fracture ........................................................................................................18

1.7

Measurement of Mode 1 Fracture Toughness...............................................................19

1.8

Double Cantilever Beam Test ........................................................................................19

1.9

Fiber Bridging Phenomenon..........................................................................................21

1.10

Specimen Parameters ...................................................................................................22

1.11

Test Procedure .............................................................................................................23

1.12

Approaches to Determine Mode 1 Fracture Toughness.................................................25

1.13

Beam Theory ................................................................................................................25

1.14

Modified Beam Theory .................................................................................................26

1.15

Compliance Calibration Method ...................................................................................27

1.16

Modified Compliance Calibration Method ....................................................................28

1.17

Compliance Method .....................................................................................................30

1.18

Types of Possible Fracture modes in the adhesive.........................................................30

Chapter 2 - Review of Literature 2.1

Citation Number 1 ........................................................................................................33

2.2

Citation Number 2 ........................................................................................................35

2.3

Citation Number 3 ........................................................................................................37

2.4

Citation Number 4 ........................................................................................................39

Chapter 3 - Experimentation 3.1

Experimental results for Specimen 1 .............................................................................41

3.2


3.3


3.4

Experimental results for Specimen 4 .............................................................................45 5

3.5


3.6

Comparative Plot for Batch 1 Specimens (Specimens with adhesive film)......................47

3.7


3.8


3.9


3.10 Comparative Plot of Load vs. Displacement for Batch 2 specimens (Specimens without adhesive film)...........................................................................................................................52 Chapter 4 - Determination of Fracture Toughness by Compliance method 4.1 Determination of Fracture Energy by Compliance Method (Specimens with Adhesive Film) (Batch 1 Specimens) ........................................................................................................53 4.2 Determination of Fracture Energy by Compliance Method (Specimens without Adhesive Film) (Batch 2 Specimens) ........................................................................................................58 Chapter 5 - Determination of Fracture Toughness by Beam Theory and it's modifications 5.2 Approach Number 1: Modified Beam Theory Approach for specimens with adhesive film (Batch 1 specimens) .................................................................................................................62 5.3 Approach Number 2: Compliance Calibration Approach to Beam Theory for specimens with adhesive film ....................................................................................................................65 5.4 Approach Number 3: Modified Compliance Calibration Approach for specimens with adhesive film ............................................................................................................................68 5.5 Comparison of Fracture Energy by the Various Beam Theory Approaches for specimens with adhesive film (Comparison of Batch 1 specimens) ............................................................70 5.6

Beam Theory for Specimens without Adhesive Film (Batch 2 Specimens) ......................71

5.7

Modified Beam Theory Approach for Specimens without adhesive film ........................72

5.8

Compliance Calibration Approach to Beam Theory for specimens without adhesive film 73

5.9 Modified Compliance Calibration Approach to Beam Theory for specimens without adhesive film ............................................................................................................................74 5.10 Comparison of Fracture Energy by the Various Beam Theory Approaches for specimens without adhesive film (Batch 2 specimens) ...............................................................................75 Chapter 6 - Determination of Fracture Toughness by Area Method Chapter 7 - Results , Discussions and Conclusions 7.1

Results for Specimens with adhesive film (Batch 1 Specimens) .....................................81

7.2

Results for specimens without adhesive film (Batch 2 Specimens) ................................82

7.3

Discussions ...................................................................................................................83 6

7.4

Conclusions ..................................................................................................................87

References Appendix A Appendix B

List of Figures Figure 1 View of a Fiber Metal Laminate...................................................................................14 Figure 2 Stress Intensity Factor for through thickness crack in an infinite plate under tension ..16 Figure 3 : The three Fracture modes .........................................................................................18 Figure 4 View of a DCB test specimen.......................................................................................20 Figure 5 Depiction of various methods of load application in DCB test......................................20 Figure 6 Depiction of Bridged fibers observed during DCB test .................................................21 Figure 7 Closer View of the Bridged Fibers................................................................................22 Figure 8 Composition of Fiber Metal Laminate .........................................................................22 Figure 9 Plot of Load vs. Cross head displacement showing the Delamination Initiation point for a sample specimen ...................................................................................................................24 Figure 10 Figure depicting the observance of curvature in the part with the Aluminium sheet .25 Figure 11 Plot of Cube root of Compliance vs. Crack Length depicting X - intercept ..................27 Figure 12 Plot of Log of Compliance vs. Log of Crack Length .....................................................28 Figure 13 Plot of delamination length normalized by thickness vs. cube root of compliance .....28 Figure 14 Depiction of Equal crack lengths spacing in Area Method ..........................................29 Figure 15 Division of Load vs. Displacement plot into partitions for area method .....................30 Figure 16 Depiction of Adhesion Fracture .................................................................................31 Figure 17 Depiction of Cohesive Fracture .................................................................................31 Figure 18 Depiction of Substrate Fracture ................................................................................32 Figure 19 Depiction of Hybrid Fracture .....................................................................................32 Figure 20 FML sequence in Citation number 1 ..........................................................................33 Figure 21 Sequence of specimens used in Citation number three .............................................38 Figure 22 Specimen used in Citation number 4 .........................................................................39 Figure 23 Islands of adhesives seen on top and bottom adherends ..........................................40 Figure 24 Plot of Load vs. Cross Head displacement for specimen 1 ..........................................41 Figure 25 Plot of Load vs. Cross Head displacement for specimen 2 ..........................................43 Figure 26 Plot of Load vs. Cross Head displacement for specimen 3..........................................44 Figure 27 Plot of Load vs. Cross Head displacement for specimen 4 ..........................................45 Figure 28 Plot of Load vs. Cross Head displacement for specimen 5 ..........................................46 Figure 29 Comparative plot of Load vs. Displacement for batch 1 specimens (With adhesive film) .........................................................................................................................................48 Figure 30 Plot of Load vs. Cross Head displacement for specimen 6 ..........................................48 Figure 31 Plot of Load vs. Cross Head displacement for specimen 7 ..........................................49 7

Figure 32 Plot of Load vs. Cross Head displacement for specimen 8..........................................50 Figure 33 Stabilization of Force beyond certain displacement...................................................52 Figure 34 Comparative plot of Load vs. Cross head displacement for batch two specimens ......52 Figure 35 Second step in Compliance method: Construction of Plot of Compliance C vs. Crack Length (a).................................................................................................................................54 Figure 36 Comparison of Compliance for all specimens with Adhesive film (Batch 1 specimens) ................................................................................................................................................56 Figure 37 Comparison of Fracture Energies with respect to Crack length for specimens with adhesive film (Batch 1 Specimens) ...........................................................................................57 Figure 38 Comparison of Fracture Energies with Error Bars for Specimens with Adhesive film (Batch 1 Specimens) .................................................................................................................58 Figure 39 Comparison of Compliance for Specimens without Adhesive film (Batch 2 Specimens) ................................................................................................................................................58 Figure 40 Comparison of Fracture Energies with respect to Crack Length by Compliance method for specimens without Adhesive film (Batch 2 specimens) ........................................................59 Figure 41 Comparison of Fracture Energies with Error Bars for Specimens with no Adhesive film by Compliance method ............................................................................................................59 Figure 42 Comparison of Fracture Energies with respect to Crack Length by Beam Theory for specimens with Adhesive film (Batch 1 specimens) ..................................................................61 Figure 43 Comparison of Fracture Energies with Error Bars for Specimens with Adhesive film by Beam theory ............................................................................................................................62 Figure 44 Plot of Cube root of Compliance vs. Crack Length for specimen 1 ..............................63 Figure 45 Depiction of X-Intercept (Δ) for Specimen 1 by Modified Beam Theory .....................63 Figure 46 Comparison of Fracture Energies with respect to Crack Length by Modified Beam approach for specimens with Adhesive film..............................................................................65 Figure 47 Plot of Log C vs. Log a for Compliance Calibration Approach......................................66 Figure 48 Comparison of Fracture Energies with respect to Crack Length by Compliance Calibration approach for specimens with Adhesive film ............................................................67 Figure 49 Plot of a/h vs. c ^ (1/3) for Modified Compliance Calibration approach .....................68 Figure 50 Comparison of Fracture Energies with respect to Crack Length by Modified Compliance Calibration approach for specimens with Adhesive film .........................................69 Figure 51 Figure Comparison of Fracture Energy by Beam theory method and its modification approaches for specimens with adhesive film (Batch 1) ............................................................70 Figure 52 Comparison of Fracture Energies with respect to Crack Length by Beam Theory for specimens without adhesive film (Batch 2 specimens) .............................................................71 Figure 53 Comparison of Fracture Energies with respect to Crack Length by Modified Beam Theory approach for specimens without Adhesive film.............................................................72 Figure 54 Comparison of Fracture Energies with respect to Crack Length by Compliance Calibration approach for specimens without Adhesive film ......................................................73 Figure 55 Comparison of Fracture Energies with respect to Crack Length by Modified Compliance Calibration approach for specimens without Adhesive film ...................................74 Figure 56 Comparison of Fracture Energy by Beam theory method and its modification approaches for specimens without adhesive film (Batch 2) ......................................................75 8

Figure 57 Area method to determine Fracture Energy ..............................................................76 Figure 58 Comparison of Fracture energies vs. Crack length for Specimens with adhesive film by Area method ............................................................................................................................78 Figure 59 Comparison of Fracture Energies with Error Bars for Specimens with Adhesive film by Area Method ............................................................................................................................78 Figure 60 Comparison of Fracture Energies with Error Bars for Specimens without Adhesive film by Area Method (Batch 2) ........................................................................................................79 Figure 61 Comparison of Fracture Energies with Error Bars for Specimens without Adhesive film by Area Method .......................................................................................................................80 Figure 62(a) Surface of Aluminium plate post fracture..............................................................83 Figure 63 (a) Surface of Composite post fracture ......................................................................85 Figure 64 Depiction of Cohesive fracture ..................................................................................87

9

List of Tables Table 1Meaning of Variables used in Beam theory expression ..................................................26 Table 2 Meaning of variables used in the Area method expression ...........................................29 Table 3 Data regarding Load and Cross head displacements at different crack lengths for Specimen 1 ..............................................................................................................................42 Table 4 Data regarding Load and Cross head displacements at different crack lengths for Specimen 2 ..............................................................................................................................43 Table 5 Data regarding Load and Cross head displacements at different crack lengths for Specimen 3 ..............................................................................................................................44 Table 6 Data regarding Load and Cross head displacements at different crack lengths for Specimen 4 ..............................................................................................................................45 Table 7 Data regarding Load and Cross head displacements at different crack lengths for Specimen 5 ..............................................................................................................................47 Table 8 Data regarding Load and Cross head displacements at different crack lengths for Specimen 6 ..............................................................................................................................48 Table 9 Data regarding Load and Cross head displacements at different crack lengths for Specimen 7 ..............................................................................................................................50 Table 10 Data regarding Load and Cross head displacements at different crack lengths for Specimen 8 ..............................................................................................................................51 Table 11 : First Step in Compliance method: Determining Compliance C ...................................53 Table 12 : Third step in Compliance Method: Determination of slope and Fracture energy .......55 Table 13 Depiction of Average Fracture Energy and Standard Deviation for the Specimens with Adhesive film (Batch 1 specimens) ...........................................................................................57 Table 14 Depiction of Average Fracture Energy and Standard Deviation for Specimens without Adhesive film (Batch 2 Specimens) ...........................................................................................59 Table 15 List of Specimens with adhesive film with their Average Fracture Energies and Standard Deviation by Beam Theory.........................................................................................61 Table 16 Calculation of C^1/3 and Fracture energy by Modified Beam Theory for Specimen 1..64 Table 17 List of Batch 1Specimens with their Average Fracture Energies and Standard Deviation by Modified Beam Theory ........................................................................................................65 Table 18 Depiction of Calculation of Fracture Energy by Compliance Calibration approach to Beam theory ............................................................................................................................66 Table 19 List of Batch 1 Specimens with their Average Fracture Energies and Standard Deviation by Compliance Calibration Approach ........................................................................................67 Table 20 Determination of Fracture energy by Modified Calibration approach .........................68 Table 21 List of Batch 1Specimens with their Average Fracture Energies and Standard Deviation by Modified Compliance Calibration Approach .........................................................................69 Table 22 List of Batch 2 specimens with their average fracture energy and standard deviation by Beam theory ............................................................................................................................71 Table 23 List of Batch 2 Specimens with their Average Fracture Energies and Standard Deviation by Modified Beam Theory Approach ........................................................................................72

10

Table 24 List of Batch 2 Specimens with their Average Fracture Energies and Standard Deviation by Compliance Calibration Approach ........................................................................................73 Table 25 List of Batch 2 Specimens with their Average Fracture Energies and Standard Deviation by Modified Compliance Calibration Approach .........................................................................74 Table 26 Determination of Fracture energy by Area method ....................................................77 Table 27 List of Batch 1 specimens with their fracture energies by Area method ......................79 Table 28 List of Batch 2 specimens with their fracture energies by Area method ......................80 Table 29 Results showing values of Fracture energies obtained by the various methods for specimens with adhesive film (Batch 1) ....................................................................................81 Table 30 Maximum, Minimum and Average fracture energies for the Batch 1 specimens .........82 Table 31 Results showing values of Fracture energies obtained by the various methods for specimens without adhesive film (Batch 2)...............................................................................82 Table 32Maximum, Minimum and Average fracture energies for batch 2 specimens ................83

11

ABSTRACT Fiber metal laminates (FML’s) are believed to have better properties in comparison to their metallic and composite counterparts. A FML consists of several layers of metals bonded to several layers of composites in which the stacking sequence may differ. A FML shows improvement in properties such as strength but most importantly in damage tolerance. Though content exists in literature on the Mode 1 damage tolerance of composites, very few is found in the domain of FML. The current Internship work aims to manufacture a GFRP and characterize its behaviour (Gíc properties) when subjected to Mode -1 opening under different adhesion nature between them. Two batch of FML’s are manufactured (one without adhesive film and another with a Redux 335K adhesive film in between the metal layer and one of the composites). The fiber metal laminate considered for this internship study is a 0.6mm Aluminium metal sheet sandwiched between two GFRP structures by means of adhesives. Crack is pre-initiated in the FML by help of a small thickness paper. The location of the precrack is between the aluminium metal and one of the composite blocks. Fabrication is done in an Autoclave by placing the metal sandwiched by the prepregs setup and the necessary attachments in an Autoclave. The required temperature and pressure was applied and an optimum cycle time was provided for formation of FML. The manufactured specimens were cut to a length of 160mm and breadth of 20mm. It was then subjected to tensile loads in the UTM with a loading rate of close to 1mm / min. Load (P) and Cross head displacement (δ) were measured at different crack lengths. Different methods were used to obtain the value of fracture energy from the available data of Load, displacement and crack length. Among the various types available in

12

literature, those used in this Internship are the Compliance method, Beam theory, Modifications to beam theory and the Area method. Resistance curve (A curve between Gic and Crack length) is plotted to study Mode 1 Interlaminar Fracture Toughness behaviour. Some specimens were loaded till fracture and the surfaces of the adherends post fracture were studied under a microscope. The type of fracture was found out to be Cohesive fracture as fracture was found between the adhesive. No fracture was found in the adherends. From the load displacement curve, the type of failure is found to be progressive failure rather than catastrophic failure.

13

CHAPTER 1 – INTRODUCTION 1.1 Introduction to Fiber Metal Laminate (FML) A Fiber Metal laminate (FML) consists of layers of metals bonded with layers of composite. The resultant material formed has a wide range of specific advantages in comparison to their purely metallic and purely composite counterparts. Some of the properties where a Fiber Metal laminate exhibits considerable advantage are in Impact, Corrosion resistance and Fatigue. An example of a fiber metal laminate is the famous GLARE which is composed of several thin layers of Aluminium metal interspersed with pre-pegs of glass fibers. FML’s are known for their better damage tolerance behaviour as the elastic strain is larger than any other metal laminate during impact. Another advantage of FML is the fact that it can be tailored according to requirement.

Figure 1 View of a Fiber Metal Laminate

As these FML’s gain popularity, of considerable consideration is their response to the onset of Delaminations. Delaminations between the metal and composite layer is of great concern in FML’s as it is established that FML’s are very sensitive to the presence of discontinuities. Since, there is an absence of through the thickness reinforcements, susceptibility to failure caused by inter-laminar crack initiation and propagation is high. Thus, the study about the resistance to Inter-Laminar fracture of FML gains importance.

14

Knowledge of a material’s response to Inter-laminar fracture is helpful for material selection and product development. To date, there is not much knowledge of the response of monotonic loading on FML’s

[1].

The fracture toughness characterization of

Fiber Metal Laminates is still in the phases of development.

1.2 Fracture Mechanics Fracture is a form of failure that results in the fragmentation of a solid body into two or more parts under the action of stress. Displacement discontinuity surfaces develop within the solid and this ultimately leads to fracture. The branch of Structural Mechanics that deals with the study of structures containing cracks/voids and Delaminations in called as Fracture Mechanics. It is broadly based on the works of A.A.Griffith. A material is assumed to be suitable for the designed purpose if the material strength is greater than the expected stress that is to be applied on it. Thus, yield and applied stress becomes the two important variables for this approach. However, the application of the above said approach to structures with cracks and voids is not beneficial. The reason can be attributed to the presence of stress intensity at the crack tip

[2] [3]

. To overcome this, a

factor which considers three important variables like applied load, flaw size and fracture toughness is introduced. This factor is christened as “Stress Intensity Factor”.

1.3 Stress Intensity Factor A factor which predicts the stress state at crack tips is the stress intensity factor. On the basis of Linear theory, stresses at crack tip tends to become infinity but it is not so in actual. The stresses at crack tip in actual take a finite value as there is a plastic zone at the edge of the crack tip that limits the stress values to finite limits. Whether a crack 15

grows or not can be determined from the magnitude of the Stress intensity factor. The Stress intensity factor for a through the thickness crack in an infinite plate under tension is given as-

Figure 2 Stress Intensity Factor for through thickness crack in an infinite plate under tension

An engineering approach to determine whether a crack is growing or not is to perform a series of experiments and determine the critical stress intensity factor. This critical stress intensity factor Kc is called as Fracture Toughness. Crack does not grow when K < Kc .

1.4 Fracture Toughness It is the ability of a crack tip to resist fracture. From the design perspective, it is one of the most important parameters. In other words, it is a quantitative method of expressing resistance to fracture when there is a crack. It may also be defined as the amount of stress needed to propagate a pre-existing flaw. High fracture toughness implies ductile fracture and low fracture toughness implies brittle fracture. Metals have high fracture toughness indicating they can easily undergo ductile fracture and a composite has low fracture toughness indicating they can undergo brittle fracture. The present internship 16

considers a fiber metal laminates which is a combination of both of the above. The huge variation in fracture toughness and the difference in the fracture method between the constituents of a FML make it interesting. Fracture Toughness is represented by symbol K and a roman numeral gives the mode of fracture.

1.5 Strain Energy Release Rate This quantity is measureable in experiments and is based on energy considerations. The fall in potential energy for a unit increase in crack area is called as Strain Energy release rate. This energy approach was originally postulated by Griffith [4]. He stated that when energy available for the growth of crack is sufficient to overcome the resistance offered by the material, crack extension occurs. The resistance offered by the material may include the work of plasticity, surface energy or the dissipation of energy during the propagation of a crack. The strain energy release rate is denoted by U. π=U–F

(Equation 1)

Where, π denotes Total Potential energy and F denotes work done by the external forces G = -dπ/dA

(Equation 2)

For a structure to be load controlled (F= 2U), G takes the form of equation three and for a structure to be displacement controlled (F=0), G takes the form of equation 4. G = dU/dA

(Equation 3)

G = -dU/dA

(Equation 4)

17

For a specimen width of b, G = - 1 dU b

(Equation 5)

dA

Where, G is referred to as Strain Energy Release Rate and is always positive. In the case of mixed loading at the crack tip, the total strain energy release rate is the sum of the strain energy release rate in each of the individual pure modes. G = Gí + Gíí + Gííí

(Equation 6)

Thus, for a pure mode loading, when the strain energy release rate at the crack tip exceeds a critical value in that corresponding mode, crack growth occurs.

1.6 Modes of Fracture The process of fracture consists of two processes, crack initiation followed by crack propagation. In general, a crack can be loaded in three different modes namely Crack Opening mode, Crack Sliding mode and Crack Tearing mode.

Figure 3 : The three Fracture modes

Mode 1 – Crack Opening Mode - A fracture mode where the delamination faces open away from the other and there is absence of relative crack face sliding.

18

Mode 2 – Crack Sliding Mode – A fracture mode where the delamination faces slide over each other in the direction of delamination growth and there is absence of relative crack face opening. Mode 3 – Crack Tearing Mode – A fracture mode where the delamination faces tear off literally. Gíc , Gííc and Gíííc are the critical stress intensity factors or fracture toughness values in the three different fracture modes. Fracture Toughness measurement requires the presence of an initial delamination in the specimen and this is achieved by using a non-adhesive insert.

1.7 Measurement of Mode 1 Fracture Toughness Knowledge of a material’s resistance to Inter-laminar fracture is useful in product development and material selection. A measurement of Mode 1 Inter-Laminar Fracture Toughness is useful for the establishment of design allowables to be used in damage tolerance studies. ASTM standard D5528

[5]

is followed in this Internship for the process

of testing and calculation of Fracture Toughness under Mode 1 Loading. A double cantilever beam (DCB) test is followed to determine the Mode 1 fracture toughness.

1.8 Double Cantilever Beam Test This popular test method is used in this current Internship to determine the fracture toughness of FML. Testing is carried out in a Universal Testing Machine with a force and cross head displacement value recorder. Two aluminium blocks are attached to the FML by means of very strong glue (in this case cyanoacrylate). A crack is pre initiated 19

in the specimen between the composite and the metal layers by means of an insert. The insert is introduced in the specimen during the manufacturing stages itself. During the test, a vertical force is applied on the specimen. A vertical Tensile loading is applied on the two aluminium blocks and in turn on the specimen. The load is applied at a certain rate (typically around 1mm / min). The crack begins to propagate with loading and the values of load and cross head displacement are recorded.

Figure 4 View of a DCB test specimen

The load in the DCB test may either be applied by means of piano hinges or loading blocks. The latter approach was followed in the current Internship.

Figure 5 Depiction of various methods of load application in DCB test

20

1.9 Fiber Bridging Phenomenon Fiber bridging occurs as a result of the interaction between delamination cracks and inclined or misaligned fibers. Fiber bridging leads to an increase in the resistance as the crack grows. With the growth of the delamination from the tip of the pre initiated crack, the development of resistance type fracture behaviour is observed. Gíc increases monotonically with increase in the cross head displacement and then stabilize with further growth in delamination. Fiber bridging is the reason for the above observed phenomenon. Thus, the significance of Gíc values calculated beyond the end of the voluntarily pre-initiated crack is questionable and values of Gíc which are measured from the implanted insert is preferred. The figure below shows the specimen under loading in the UTM and the highlighted box shows the bridged fibers observed during the DCB test carried out in this Internship.

Figure 6 Depiction of Bridged fibers observed during DCB test

21

Figure 7 Closer View of the Bridged Fibers

1.10

Specimen Parameters

Specimen Length

:

160mm

FML Thickness pattern

: 18.75mm Composite layer followed by 6mm Al sheet followed by 11.25mm Composite.

Composite Aluminium Sheet

Composite Pre – Initiated crack of 50mm Figure 8 Composition of Fiber Metal Laminate 22

Number of Plies

:

10 plies in one part and 6 plies in another

(ASTM suggests that the number of plies be even, see [5]) Thickness of Aluminium sheet

:

Average Thickness of specimen

: 36mm (6mm Aluminium + 2 Composite layers)

Thickness of Composite Layers

:

Specimen width

:

20mm

Initial delamination Length

:

45mm

Nature of Composite

:

Prepregs used

:

Adhesive film used

:

1.11

6mm

18.75mm and 11.25mm

Woven Fabric Composite GFRP fibers Redux 35K adhesive film

Test Procedure

Suitable arrangements are made in the UTM to carry out the test. The test specimen along with the end blocks are placed in the UTM. The readings from the load cell and the cross head displacement are continuously recorded in a digital computer. A loading rate of around 1mm / minute is applied on the specimen. Load increases linearly with the applied displacement till the onset of delamination growth. A steady decrease in load is seen after this onset.

23

Initiation of Delamination / Crack Growth

Linear Variation of Load with Displacement up to the Delamination Initiation point

Figure 9 Plot of Load vs. Cross head displacement showing the Delamination Initiation point for a sample specimen

The experiment is carried on up till the two parts separate completely from each other. One part consists of the top composite layer and the other part consists of the Aluminium sheet and the bottom Composite layer. On complete separation, the top composite layer (which is only composite and no Aluminium sheet with it) appears straight without any curvature in it. But the bottom composite layer which has the Aluminium sheet with it is curved. The above phenomenon is depicted in the following figure where the lower arm contains the Aluminium sheet. It can be seen from the picture that the lower arm has a curvature in comparison to the upper arm.

24

Figure 10 Figure depicting the observance of curvature in the part with the Aluminium sheet

1.12

Approaches to Determine Mode 1 Fracture Toughness

According to ASTM D5528, three different methods can be adopted to calculate the Fracture Toughness Gíc – Modified Beam Theory, Compliance Calibration method and Modified Compliance Calibration method. However, none of these is better compared to the other because all these theories have their own assumptions and are found to yield results that do not differ by more than 3%. However, the modified beam theory is widely accepted to have given the most conservative results. Thus, the modified beam theory is widely used. In addition to these three theories, the area method may also be used to determine the Fracture Toughness.

1.13

Beam Theory

The expression for the strain energy release rate from the beam theory expression is

(Equation 7) 25

Where, Table 1Meaning of Variables used in Beam theory expression

Notation

Expansion

P

Load

δ

Load point displacement

b

Width of specimen

a

Delamination Length

The beam theory is valid for a perfectly built in double cantilever beam .i.e. clamped at the delamination front. However, since the beam is not perfectly built in (that is, rotation may occur at the delamination front), this expression will tend to overestimate the values for Gí. However the above drawback is corrected by three commonly methods according to ASTM D5528 [5].

1.14

Modified Beam Theory

To account for the rotation at the delamination front, we may consider a slightly longer delamination, i.e., a + |Δ|

(Equation 8)

Δ is determined experimentally. Δ is the X-Intercept of the linear curve – fit plot between the cube root of compliance (C1/3) and delamination length (a). Compliance is the ratio between the displacements to the applied load.

26

20

C^(1/3)

15 10 5 0 -2

0

Δ -5

2

4

6

8

10

a (in mm)

Figure 11 Plot of Cube root of Compliance vs. Crack Length depicting X - intercept

1.15

Compliance Calibration Method

In this method, a plot of log(C) vs. log (a) is created where C is the Compliance. A least error square linear fit is constructed and its slope (n) is determined. The value of ‘n’ is used in the beam theory to improve its predictions.

(Equation 9)

Where, n is the slope of the plot constructed between Logarithm of Compliance vs. The Logarithm of the crack length.

27

20 15

Log (C) 10 5

n = Δy / Δx

0 0

2

4

6

8

10

Log (a) Figure 12 Plot of Log of Compliance vs. Log of Crack Length

1.16

Modified Compliance Calibration Method

A plot of delamination length normalized by specimen thickness versus cube root of compliance is constructed. A least error-square linear fit is drawn and the slope (A1) is determined.

a/h

(Equation 10)

20 18 16 14 12 10 8 6 4 2 0

A1 = Δy / Δx

0

2

4

6

8

10

C^ (1/3)

Figure 13 Plot of delamination length normalized by thickness vs. cube root of compliance

28

Area Method

By this method,

(Equation 11)

Δa is the increment in crack length from a1 to a2, a2 to a3 and so on. a1, a2, a3, are equally spaced as shown in the following figure.

Figure 14 Depiction of Equal crack lengths spacing in Area Method

The area under the load displacement curve is calculated by –

ΔU1 = U2 – U1 = 0.5 * (P1 δ2 – P2 δ1)

(Equation 12)

Table 2 Meaning of variables used in the Area method expression

Notation

Expansion

ΔU

Area under the Load displacement curve

Δa

Increase in crack length

B

Specimen width

P1 , P2

Load at crack length a1 and a2

δ1 , δ2

Displacement at crack length a 1 and a2

29

Figure 15 Division of Load vs. Displacement plot into partitions for area method

1.17

Compliance Method

(Equation 13)

C=δ/P

Where,

1.18

Types of Possible Fracture modes in the adhesive

The types of possible fracture modes that might occur are – 

Substrate Fracture



Adhesion Fracture



Cohesion Fracture



Hybrid Fracture 30

Adhesion Fracture Fracture that occurs in the region between the substrate and the adhesive is Adhesion fracture. There is occurrence of partial or total separation of the adhesive from the surface of the substrates. Possible causes of this type of fracture may be improper preparation of the substrate surface before adhesive application, selection of substrate and adhesive which do not bond together well, inadequate preparation time and inadequate thickness.

Figure 16 Depiction of Adhesion Fracture

Cohesion Fracture Occurrence of fracture is observed in the adhesive material itself. Post fracture, there is presence of adhesive traces on the surface of both the adherents. Possible causes for this type of fracture may be improper adhesive curing time, defects and bubbles in the adhesive.

Figure 17 Depiction of Cohesive Fracture 31

Substrate Fracture Breaking of the adhesive bonding into one or many substrates. Possible causes for this type of fracture may include improper design of the substrate to withstand the applied load and that the resistance of the substrate is less than the adhesive bond strength. There is deterioration of the mechanical properties of the substrate.

Figure 18 Depiction of Substrate Fracture

Hybrid Fracture It is a type where there is combination of two or more of the above mentioned types.

Figure 19 Depiction of Hybrid Fracture

32

CHAPTER 2 – REVIEW OF LITERATURE 2.1 Citation Number 1 Title of Paper – The effect of adhesive bonding between Aluminium and Composite Prepreg on the Mechanical Properties of Carbon Fiber Reinforced Metal Laminates. [6] Authors – Glyn Lawcock, Lin Ye, Yiu Wing Mai and Chin Teh Sun Objective of Paper- To investigate the role of adhesion between fiber composite prepregs and aluminium sheets on Inter laminar fracture energy.

Different surface

treatments were applied to the metal to alter the adhesive response between metal and composite. Important Properties- Carbon fiber reinforced metal laminate with 2024 – T3 aluminium sheets and one layer of composite (Fiberite T300). One method of surface treatment is hand abrasion with acetone solvent wipe and the second method is degreasing with methyl ethyl ketone. Length, Width = 160mm and 20mm. Thickness = 0.38mm Aluminium alloy sheets and 0.30mm composite.

Figure 20 FML sequence in Citation number 1 33

Procedure Used- Two different adhesion natures were created using different surface treatments for Aluminium. Double Cantilever beam test is conducted to determine the interfacial fracture toughness. Dogbone tensile specimens were made in order to test the mechanical properties. For testing of interlaminar fracture toughness, specimens were loaded at 0.1mm/min. The load displacement curves were recorded continuously for different crack lengths. A tensile test was conducted to determine the mechanical properties. Results and Conclusions made- Not much difference was found in the mechanical properties between the two specimens. Interlaminar shear strength decreased by 10% in the specimen with poor interfacial adhesion. Residual strength is independent of the adhesion nature between the composite and the metal. Bridging fibers in crack wake helps in increasing the fatigue resistance. Stable crack growths were observed for both the specimens. Discrete load steps occur for rapid crack growth. However the two specimens differ in the critical load required to cause crack growth. Initiation fracture energy varies considerably between the two specimens, indicating its sensitivity to the nature of the adhesion between Laminate and Aluminium. This difference is observed due to the different crack propagation paths. This difference in path was observed due to the difference in adhesion nature. The specimen with poor adhesion resulted in crack path along the interface between the composite and the metal. Crack propagation was observed within the composite layer for the specimen with strong adhesion.

34

2.2 Citation Number 2 Title of Paper – Mode 1 Delamination Characteristics of Cross – plies Fiber Metal Laminates [7] Authors – J.Laliberte , P.V.Straznicky and C.Poon Objective of Paper- To predict the dependence of Mode – 1 energy release rate upon the layup of the FML. Although some data exists in literature regarding delamination behaviour of unidirectional laminate, there is very few regarding the behaviour of cross ply FMLS’s. Important Parameters- Three variants of GLARE were tested. 1. GLARE-3-2/1 [AL, (0/90), AL], 2. GLARE-4-2/1 [AL , 0/90/0 , AL] and 3. GLARE-5-2/1 [AL, 0/90/90/0, AL] Procedure Used- Series of delamination characterization testes was employed to improve the accuracy of low velocity impact finite element simulations. A glass reinforced aluminium laminate was tested. A double cantilever test was used for the experimentation. Finite element modelling of the low velocity impact damage was done using LS-DYNA explicit. A modified data scheme which accounts for the deformation of adherends was employed to convert load data to fracture energy values. Specimens were loaded quasi-statically at rate of 5mm/min. Load and displacement at different crack lengths was measured.

35

Modified area method contains corrections for adherends bending. Area method determines higher Gi value because it does not consider the flexibility of the specimen. And the displacement system takes into account the crack opening displacement. Results and Conclusions- Mode 1 energy release rate has limited dependence upon the stack up sequence. Some of the energy is stored as elastic strain energy. One of the concerns for simulating this test is the phenomenon of fiber bridging. Since fiber bridging is also observed in the impact simulations, the Gi data could still be applied to the simulations. Simulation shows that Mode 1 behaviour dominates the delamination propagation in low velocity impact of GLARE. In the simulation, mode 2 was found to dominate during impact onset and mode 1 during the rebound phase of impact. Thus, for initiation, mode 2 properties are important and the properties of mode 1 for propagation of delamination. Delaminations were observed in the prepreg layers in which the fibers were oriented parallel to the direction of delamination. Each of the different methods used for determination of Gi gives different values due to the fact that different methods have different assumptions. Flexibility of the adherends also affects the end results.

36

2.3 Citation Number 3 Title of Paper – Fracture Toughness of Fiber Metal Laminates (Ti – Polymer Matrix composite interface). [8] Authors – Hieu Truong, Dimitris C.Lagoudas , Ozden Ochoa and Khalid Lafdi Objective of Paper- To process and characterize the fracture toughness of carbon fiber reinforced polymer matrix and carbon nanotube modified interface between composite and titanium. To use data obtained from study of surface of adherends. Important Parameters- T300 plain weave carbon fabric with epon 862 (Epicure curing agent) and plain Titanium foil. Length of FML =125mm and width = 25.4mm. An initial crack length of 50mm was used. Cross head displacement = 3mm/min. Procedure used- Vacuum assisted resin transfer moulding is used for fabrication and double cantilever beam test is used for mode 1 interlaminar fracture toughness characterization. Surface composition plays important role in bonding of FML. Thus, different materials may be considered at the interface of FML to improve fracture toughness properties. These CNT’s create micro and nano scale roughness on the surface to enable bonding. Different specimens are considered (one with crack initiated at interface of fuzzy T300 fabric and Titanium and the other at the interface of fuzzy Ti fabric and fuzzy T300 fabric).ASTM standard D5528-01 is used for the testing. Piano

37

hinges are used to apply the loads. A uniform crack front is assumed to exist across the width of the specimen.

Figure 21 Sequence of specimens used in Citation number three

Results and Conclusions- It is observed that fracture toughness is higher at fTi interface compared to plain Ti interface. In the case of crack growth jumping from one region to the other (adhesive fracture), the value of fracture energy varies significantly. Thus adhesive fracture is not desirable. However if cohesive fracture was observed, there is not much jump observed on the values of fracture toughness. Thus, characterization of the nature of fracture is important. During the experimentation phase, it was observed that crack propagation did not occur in the same plane but jumped planes. A feature called scarp is observed on the crack surface. A scarp is a region where adjacent crack planes converge. A scarp is an indication of mode 1 fracture. On the other hand, cusps indicate mode 2 fracture. A study on the fracture surface helps to predict the dominant type of fracture. Thus, characterization of fracture and the surface after fracture are found to give vital information regarding the values of fracture energy and the dominant type of fracture.

38

2.4 Citation Number 4 Title of Paper- A practical test method for Mode 1 fracture toughness of adhesive joints with dissimilar substrates. [9] Authors- Raymond G.Boeman , Donald Erdman , Lynn Klett and Ronny Lomax Objective of Work- To discuss a practical test for determination of mode 1 fracture toughness in presence of adhesive joints between dissimilar materials. The two adherends used differ in their flexural rigidity .Thus results obtained from this test can be extended to FML’s. And to discuss methods to achieve symmetric loading conditions. Backing beams made of steel can be bonded to the substrates to prevent substrate failure. Important parameters- Samples were prepared by bonding SRIM composite and ecoat steel with Teflon film at one end. Size of bonded panel is 100mm by 250mm. The length of pre crack is accurately 25mm.

Figure 22 Specimen used in Citation number 4

Procedure Used- The test is based on the popular DCB test. The test is done on a uniform rectangular specimen with load introduction though means of piano hinges or end blocks with clevis holes. The composite was made from a continuous-strand mat preform infiltrated with an isocyanurate (Dow MM364) resin by the structural reaction injection moulding (SRIM) process. A loading rate of 1.27mm/min was applied. An

39

experimental compliance method using a third order compliance fit was chosen for this method. Compliance was determined at different crack lengths and is plotted with respect to crack length. The slope of compliance vs. Crack length curve dc/da is obtained by differentiating the polynomial fit. The point at which 5% deviation from linearity was observed for the load-displacement curves was taken as the critical load. DCB specimens with dissimilar substrates are inherently mixed mode loading specimens due to nonsymmetric flexure of the unbonded (i.e., cracked) portion of the substrates. From a physical standpoint it is argued here that if the heights of the two backing beams are chosen such that symmetric bending is achieved, then conditions for mode I are established. Backing beam was used to ensure symmetric loading conditions. Results and Discussions- Controlled stable crack growth was observed. The behaviour was found to be similar to other experiments where identical adherends were used (both adherends are composites).

Figure 23 Islands of adhesives seen on top and bottom adherends

From the above figure, it is visible that there are islands of adhesives on the top steel substrate whereas the remaining adhesives are on the bottom composite substrate. Fracture toughness was found to constant with respect to crack length as fiber bridging was not significant. 40

CHAPTER THREE – EXPERIMENTATION Two different batches of specimens were tested in this Internship. The two batches were manufactured and tested separately at different points of time. One batch of specimens (Batch 1) is manufactured with Redux 335K adhesive film in between metal and composite. One batch of specimens (Batch 2) is manufactured without any adhesive film in between. Results of five specimens are presented from Batch 1. But in the second batch, data from only three specimens are presented because some specimens did not bond well to the end blocks despite multiple attempts. During experimentation, the end block disbonded for this specimen. As mentioned before, the values of the load and the corresponding crack opening displacement is recorded. The plot of the applied load versus the crack opening extension is given below for the different specimens with the tabular column depicting load and the cross head displacements observed at different crack lengths. Specimens 1 to 5 belong to batch 1 (with an adhesive film) and 6 to 8 belong to batch 2 (without adhesive film).

Load (N)

3.1 Experimental results for Specimen 1 40 35 30 25 20 15 10 5 0 0

20

40

60

80

100

120

140

Tensile Extension (mm) Figure 24 Plot of Load vs. Cross Head displacement for specimen 1

41

Table 3 Data regarding Load and Cross head displacements at different crack lengths for Specimen 1

Load P (N)

Crack Length a

Crosshead Displacement

36.32 36.72 35.63 36.29 36.02 36.98 36.57 34.74 35.44 31.82 32.71 34.31 27.13 27.21 27.21 27.5 15.43 17.77 19.14 20.96 20.97 22.3 22.37 22.08 22.74 22.89

46.5 (mm) 49 50 51 52 54 56 59 60.2 62 63 66 74 76 77 79 89 91 92.5 93 94 94.5 95 97 99.5 100

11.93 (mm) 13.9 15.35 16.33 17.39 18.63 20.45 22.57 25.51 27.93 30.19 33.85 35.48 36.03 36.57 37.46 38.47 41.46 43.75 47.66 50.27 54.97 57.35 59.44 62.13 63.39

42

3.2 Experimental results for Specimen 2 50

Load (N)

40 30 20 10 0 0

20

40

60

80

100

Tensile Extension (mm) Figure 25 Plot of Load vs. Cross Head displacement for specimen 2 Table 4 Data regarding Load and Cross head displacements at different crack lengths for Specimen 2

Load P

Crack Length a (mm)

(N) 36.01 38.07 39.24 37.97 33.79 32.38 28.07 27.88 28.71 28.3 26.2 26.12 25.2 25.29 24.37 24.58 23.67 23.67 23.61 23.82 23.61 23.1 22.73

Crosshead Displacement (mm)

45.5 47.8 49 51 54 57 60 61 63 66 68 69 70 72 73.5 76 78 79 80 81 84 85 88.5

12.96 14.99 16.58 18.68 20.19 22.36 24.31 25.73 27.18 28.75 30.2 31.59 32.8 34.6 36.51 37.96 39.54 41.35 43.29 45.46 48.22 49.3 52.54 43

22.91 22.66 21.42 22 23.17 23.1 24.54 25.96 25.34 25.26

89.3 90.5 93 94 96 98.5 99 99 100 101

53.93 55.98 57.95 59.93 63.15 65.54 69.9 75.17 77.71 79.66


Load (N)

40 30 20 10 0 0

20

40

60

80

100

120

140

160



Load P (N)

Crack Length a (mm)


42.45 46.53 47.11 46.4 43.75 42.5 40 38.87 38.91

46 47 49 50.5 53 54.5 58 60 61

13.55 16.27 17.87 19.3 22.06 23.73 25.98 28.31 30.29 44

38.21 37.46 34.97 35.13 33.94 33.88 34.49 31.94 32.66 32.94 32.02 31.48 32.39 28.03 27.01 24.12 21.84 22.21 22.79 23.21

62.5 64 67 68.7 70 73 74 76 78 79 81 82 84.3 86.5 88 90 93 95 98.5 100

32.26 33.82 36.05 38.16 40.19 43.13 45.11 47.45 50.02 51.86 54.4 57.12 59.88 62.78 65.02 66.83 68.83 70.26 74.71 77.07


Load (N)

50

40 30 20 10 0 0

20

40

60

80

100

120



Load P (N)

Crack Length a (mm)

45


49.86 51.12 47.19 46.19 47.9 41.47 40.84 38.56 36.81 36.3 36.88 36.94 37.89 40.06 38.21 33.67 34.87 34.96 34.21 29.22 27.9 25.98 22.37 23.51 24.49 23.5

45.5 49 49.5 51 54 57 59 62 64.5 67 68 69 70 71 72 75 76 77 79 83 86 87 91 93 94 95.5

16.27 18.96 21.51 23.77 26.04 28.81 30.72 33.35 35.4 37.62 40.11 43.02 45.46 49.05 50.45 52.39 54.68 56.76 58.53 60.49 63.18 64.76 65.9 67.46 69.22 70.99

3.5 Experimental results for Specimen 5 40 35

Load (N)

30 25 20 15

10 5 0 0

20

40

60

80

100

120

Tensile Extension (mm) Figure 28 Plot of Load vs. Cross Head displacement for specimen 5 46


Load P (N)

Crack Length a (mm)

Cross Head displacement (mm)

33.09 35.35 35.6 33.69 35.51 36.71 37.38 28.7 28.03 26.54 25.56 28.28 29.89 12.52 14.56 16.54 18.27 19.62 20.35 21.91 22.29 23.27 24.23 24.01

45 46 47 51 53 54 55 60 63 64 66 67 68 89 91 92 93 93 94 95 96 96.5 97 98

11.8 14.3 15.74 17.54 20.26 22.6 24.84 27.1 27.91 29.31 31.03 32.71 34.84 35.78 37.88 40.41 44.21 47.47 49.1 55.91 58.71 60.89 63.56 68.28

3.6 Comparative Plot for Batch 1 Specimens (Specimens with adhesive film)

47

60

Specimen 1

Load (N)

50

Specimen 2

40 30

Specimen 3

20

Specimen 4

10

Specimen 5

0 0

20

40

60

80

100

120

Tensile Extension (mm) Figure 29 Comparative plot of Load vs. Displacement for batch 1 specimens (With adhesive film)

Second Batch (Specimens without adhesive film)


Load (N)

30 25 20 15 10

5 0 0

10

20

30

40

50



Load P (N)

Crack Length a (mm)


26.61

45

11.12 48

26.58 26.31 25.45 24.39 23.75 23.65 23.38 24.02 23.73 23.73 23.8 24.21 24.15 24.57 24.08 23.16 23.12 21.77 19.46 19.14 18.34 18.44 18.54 18.57

47.5 49 50.5 52 53.5 54 56.8 58 59 60.5 62 63.5 65 66.8 68 70.9 73 77 80 81.8 83.3 84.5 85.7 87.5

12.6 13.83 15 15.84 16.86 17.8 18.8 20.2 21.29 22.43 23.96 25.51 26.95 28.27 30.9 32.8 35.22 38.4 39.77 40.46 41.49 42.84 44.47 46.73


Load (N)

30 25 20 15 10 5 0 0

10

20

30

40


49

50


Load P

Crack Length a

(N)

(mm)

27.81 29.88 30.6 30.57 29.57 29.38 28.65 27.95 28.15 26.12 23.09 20.76 21.73 22.51 20.25 18.47 16.74 15.07 15.16

46 47 51 52 55 57 59.5 62.5 66 72 75 78.2 78.5 81 83 88 90 96.5 100


7.236 8.745 10.36 11.24 12.35 14 15.89 18.17 20.97 23.51 24.11 26.58 28.98 31.8 34.29 36.92 38.87 40.91 42.87

Load (N)

3.9 Experimental results for Specimen 8 40 35 30 25 20 15 10 5 0

0

20

40

60

80

100


50


Load P (N)

Crack Length a (mm)


32.39 35.1 36.15 37.49 36.99 34.15 33.56 30.97 30.49 28.72 28.05 27.45 25.67 23.1 22.45 22.52 23.05 23.32 22.76 22.82 23.09 23.06 23.33

45 46 47 49.5 54 57 59 63.5 66 69.5 71.5 75 78.5 82 85 89 91 93 96 97.5 101.5 103 104

10.5 12.5 14.07 15.65 18.25 22.08 23.95 27.91 30.79 33.48 37.11 40.44 44.19 47.91 50.43 53.17 56.4 62.44 65.61 69.46 74.21 77.05 77.05

After a particular distance from the pre initiated crack front, the load tends to stabilize with respect to the Cross head displacement. This can be seen from figure that is depicted below.

51

Figure 33 Stabilization of Force beyond certain displacement

3.10 Comparative Plot of Load vs. Displacement for Batch 2 specimens (Specimens without adhesive film) 40 35

Load (N)

30

Specimen 6 Specimen 7

25 20 15 10 5 0 0

20

40

60

80

100

Tensile Extension (mm) Figure 34 Comparative plot of Load vs. Cross head displacement for batch two specimens

52

Chapter 4– Compliance Method The following set of data gives the determination of Fracture energy by Compliance method for Specimens with adhesive film. A total of five specimens were manufactured and tested in this set. The adhesive film used is Redux 335K. The initial crack length is 45mm and in the crack propagation region, we consider a distance of 55mm. In total, let us record the values of Load, Displacement and the Crack distances for 100mm (45mm lies in the region of the pre-initiated crack and another 55mm beyond the region of preinitiated crack). This is done as it is known from Literature that Gí tends to stabilize with further increase in delaminations beyond a particular point. Also the effect of fiber bridging is predominant as we progress further with crack propagation. Determination of the values within this distance of 100mm provides a good estimate of the Fracture Energy.

4.1 Determination of Fracture Energy by Compliance Method (Specimens with Adhesive Film) (Batch 1 Specimens) The first step in this method is to determine the Compliance (in mm/N) with the data available regarding Load (P) and Cross head displacement (δ). Compliance is calculated as C = δ / P. The determination of Compliance for Specimen 1 is presented below. Table 11 : First Step in Compliance method: Determining Compliance C

Load P (In N)

Crack Length a (In mm)

36.32 36.72 35.63 36.29 36.02 36.98 36.57

46.5 49 50 51 52 54 56

Cross Head Displacement (In mm) 11.93 13.9 15.35 16.33 17.39 18.63 20.45 53

Compliance C (mm/N) 0.3285 0.3785 0.4308 0.4500 0.4828 0.5038 0.5592

34.74 59 22.57 0.6497 35.44 60.2 25.51 0.7198 31.82 62 27.93 0.8777 32.71 63 30.19 0.9230 34.31 66 33.85 0.9866 27.13 74 35.48 1.3078 27.21 76 36.03 1.3241 27.21 77 36.57 1.3440 27.5 79 37.46 1.3622 15.43 89 38.47 2.4932 17.77 91 41.46 2.3331 19.14 92.5 43.75 2.2858 20.96 93 47.66 2.2739 20.97 94 50.27 2.3972 22.3 94.5 54.97 2.4650 22.37 95 57.35 2.5637 22.08 97 59.44 2.6920 22.74 99.5 62.13 2.7322 22.89 100 63.39 2.7693 Now that the compliance is determined, a plot is constructed between the Compliance (On Y axis) and the Crack length (On X axis) to determine the slope dc/da. The relationship between compliance and crack length is determined from the constructed graph. This is the second step in this method. 3.0

y = 0.0004x2 - 0.018x + 0.2073 R² = 0.9876

Compliance C

2.5

2.0 1.5 1.0

0.5 0.0 45

55

65

75

85

95

105

Crack Length (a) Figure 35 Second step in Compliance method: Construction of Plot of Compliance C vs. Crack Length (a)

It can be seen from the graph that, y = 0.0004x2 – 0.018x + 0.2073

54

C = 0.0004a2 – 0.018a + 0.2073 dc / da = 0.0008 – 0.018. The value of dc/da is used in the Equation 13 to determine the Fracture Toughness. This forms the third step in this method. Table 12 : Third step in Compliance Method: Determination of slope and Fracture energy

Crack Length a (In Compliance mm) C (mm/N) 49 0.3785 50 0.4308 51 0.4500 52 0.4828 54 0.5038 56 0.5592 59 0.6497 60.2 0.7198 62 0.8777 63 0.9230 66 0.9866 74 1.3078 76 1.3241 77 1.3440 79 1.3622 89 2.4932 91 2.3331 92.5 2.2858 93 2.2739 94 2.3972 94.5 2.4650 95 2.5637 97 2.6920 99.5 2.7322 100 2.7693

Slope dc/da 0.0212 0.0220 0.0228 0.0236 0.0252 0.0268 0.0292 0.0302 0.0316 0.0324 0.0348 0.0412 0.0428 0.0436 0.0452 0.0532 0.0548 0.0560 0.0564 0.0572 0.0576 0.0580 0.0596 0.0616 0.0620

Fracture Energy Gic (N/mm) 0.7146 0.6982 0.7507 0.7655 0.8615 0.8960 0.8810 0.9470 0.7999 0.8667 1.0241 0.7581 0.7922 0.8070 0.8546 0.3167 0.4326 0.5129 0.6194 0.6288 0.7161 0.7256 0.7264 0.7963 0.8121

From the values of the fracture energy, the Average and the Standard Deviation is determined.

Average = 0.7437 N/mm Standard Deviation = 0.154. 55

Similar approaches were followed to determine the Fracture energies for the other specimens by this Compliance method. The tabular column for the determination of fracture energies for the other specimens of Batch 1 is shown in Appendix A and Batch 2 specimens are shown in Appendix B. The comparison plot between Crack Length (a) and the Compliance (C) for the first batch of specimens is given below.

Compliance C (mm/N)

3.5

Specimen 1

3

Specimen 2 2.5

Specimen 3

2

Specimen 4 Specimen 5

1.5 1 0.5 0

45

55

65

75

85

95

105

Crack Length a (mm) Figure 36 Comparison of Compliance for all specimens with Adhesive film (Batch 1 specimens)

56

Fracture Energy Gic (N/mm)

4 3.5

Specimen 1

3

Specimen 2

2.5 2

Specimen 3

1.5 1

Specimen 4

0.5 0

45

55

65

75

85

95

105

Specimen 5

Crack Length a (mm) Figure 37 Comparison of Fracture Energies with respect to Crack length for specimens with adhesive film (Batch 1 Specimens)

The comparison of the Fracture Energies for the various specimens with their Error Bars is given below. The value of the Average Fracture Energy and Standard Deviation is given in the tabular column below. Table 13 Depiction of Average Fracture Energy and Standard Deviation for the Specimens with Adhesive film (Batch 1 specimens)

Specimen Specimen 1 Specimen 2 Specimen 3 Specimen 4 Specimen 5

Average Fracture Energy (In0.7437 N/mm) 0.8137 (N 1.5000 1.5800 (N/m\llllllllllllllllllllllll(N/mm 0.7721

57

Standard Deviation 0.1547 0.1527 0.3567 0.3319 0.2607

Fracture Energy (Gi)

2.5

2.0 1.5 1.0 0.5 0.0 1

2

3

4

Specimens

5

Figure 38 Comparison of Fracture Energies with Error Bars for Specimens with Adhesive film (Batch 1 Specimens)

4.2 Determination of Fracture Energy by Compliance Method (Specimens without Adhesive Film) (Batch 2 Specimens) The procedure for Compliance method was shown in section 4.1. The tabular column for the determination of Compliance and Fracture energy by this method is shown in Appendix.

3.0

Compliance C

2.5 Specimen 6

2.0 1.5

Specimen 7

1.0

Specimen 8

0.5 0.0 45

55

65

75

85

95

Crack Length (a) Figure 39 Comparison of Compliance for Specimens without Adhesive film (Batch 2 Specimens)

58

Fracture Energy (Gí)

2.0

1.8 1.6

Specimen 6

1.4 1.2 1.0

Specimen 7

0.8 0.6 0.4

Specimen 8

0.2

0.0 50

60

70

80

90

100

Crack Length (a) Figure 40 Comparison of Fracture Energies with respect to Crack Length by Compliance method for specimens without Adhesive film (Batch 2 specimens)

Table 14 Depiction of Average Fracture Energy and Standard Deviation for Specimens without Adhesive film (Batch 2 Specimens)


Specimen Specimen 6 Specimen 7 Specimen 8

Average 0.6496 0.544 0.8698

Standard Deviation 0.12366 0.19283 0.12204

1.2 1 0.8

0.6 0.4

0.2 0 6

7

8

Specimens Figure 41 Comparison of Fracture Energies with Error Bars for Specimens with no Adhesive film by Compliance method 59

CHAPTER 5 – CALCULATIONS BY BEAM THEORY AND ITS MODIFICATIONS A calculation by the Beam theory is a straight forward approach where the values of Load, Crack Length and Cross head displacement are substituted in equation (7) to obtain the Fracture energy. It is to be noted that the width of the specimen is 20mm. 5.1 Beam Theory for Specimens with Adhesive Film (Batch 1 Specimens)

The following tabular column is presented for specimen 1. Load P (N)

Crack Length including Pre-inserted crack a (mm)

Cross Head Displacement (mm)

Fracture Energy Gi by Beam Theory

36.32 36.72 35.63 36.29 36.02 36.98 36.57 34.74 35.44 31.82 32.71 34.31 27.13 27.21 27.21 27.5 15.43 17.77 19.14 20.96 20.97 22.3 22.37 22.08 22.74 22.89

46.5 49 50 51 52 54 56 59 60.2 62 63 66 74 76 77 79 89 91 92.5 93 94 94.5 95 97 99.5 100

11.93 13.9 15.35 16.33 17.39 18.63 20.45 22.57 25.51 27.93 30.19 33.85 35.48 36.03 36.57 37.46 38.47 41.46 43.75 47.66 50.27 54.97 57.35 59.44 62.13 63.39

0.6989 0.7812 0.8204 0.8715 0.9034 0.9569 1.0016 0.9967 1.1263 1.0751 1.1756 1.3198 0.9756 0.9675 0.9692 0.9780 0.5002 0.6072 0.6790 0.8056 0.8411 0.9729 1.0128 1.0148 1.0650 1.0882

60

Average Fracture Energy = 0.9309N/mm Standard Deviation = 0.18N/mm Similar approaches have been followed to determine the Fracture energy using Beam theory for other specimens. The values of fracture energies determined by beam theory for the other specimens are shown in Appendix. The Comparison between the fracture energies with respect to crack length obtained for the specimens with adhesive film (batch 1) by Beam theory is given below.


4.0 3.5

Specimen 1

3.0

Specimen 2

2.5 2.0

Specimen 3

1.5 1.0

Specimen 4

0.5 0.0 45

55

65

75

85

95

105

Specimen 5

Crack Length a (mm) Figure 42 Comparison of Fracture Energies with respect to Crack Length by Beam Theory for specimens with Adhesive film (Batch 1 specimens)

Table 15 List of Specimens with adhesive film with their Average Fracture Energies and Standard Deviation by Beam Theory

Specimen

Average

St .Dev

Specimen 1 Specimen 2 Specimen 3 Specimen 4 Specimen 5

0.9309 1.0181 1.4201 1.55 0.9037

0.1806 0.1791 0.1615 0.2292 0.2403

61

Fracture Energy (N/mm)

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1

2

3

4

5

Specimens Figure 43 Comparison of Fracture Energies with Error Bars for Specimens with Adhesive film by Beam theory

As we had seen in the Introduction chapter, Beam theory tends to over-estimate the values of Fracture energy. Thus the following approaches were made to the Beam theory.

5.2 Approach Number 1: Modified Beam Theory Approach for specimens with adhesive film (Batch 1 specimens) In this method, a plot is constructed between the cube root of compliance and the crack length. The x- intercept of the curve is taken and is represented as Δ. This value is substituted in equation (8) to obtain the fracture energy. The value Δ is used to account for the rotation that might have occurred at delamination front. Addition of Δ to the crack length leads to the consideration of a longer length of delamination. A sample plot between the cube root of Compliance and the crack length for specimen 1 is shown

62

below. 1.6 1.4

y = 0.0134x + 0.0842 R² = 0.9906

C^ (1/3)

1.2

1.0 0.8 0.6 0.4 0.2 0.0 45

55

65

75

85

95

105

Crack Length (a) Figure 44 Plot of Cube root of Compliance vs. Crack Length for specimen 1

Now the relation between C1/3 and Crack length (a) is given as Y = 0.0134x + 0.0842 When y tends to 0, then 0 = 0.0134x + 0.0842, thus the X intercept is -6.28

0.2

C ^ (1/3)

0.2

y = 0.0134x + 0.0843

0.1 0.1 0.0 -0.1

X Intercept X intercept (Δ)

Crack Length (a)

Figure 45 Depiction of X-Intercept (Δ) for Specimen 1 by Modified Beam Theory

Thus, the value of this Δ is now used to determine the Fracture Energy.

63

Table 16 Calculation of C^1/3 and Fracture energy by Modified Beam Theory for Specimen 1

Crack Length a

Cross Head

Compliance C

(mm)

Displacement (δ)

(N/mm)

C 1/3

Fracture Energy Gi (N/mm)

46.5 49 50 51 52 54 56 59 60.2 62 63 66 74 76 77 79 89 91 92.5 93 94 94.5 95 97 99.5 100

11.93 13.9 15.35 16.33 17.39 18.63 20.45 22.57 25.51 27.93 30.19 33.85 35.48 36.03 36.57 37.46 38.47 41.46 43.75 47.66 50.27 54.97 57.35 59.44 62.13 63.39

0.3285 0.3785 0.4308 0.4500 0.4828 0.5038 0.5592 0.6497 0.7198 0.8777 0.9230 0.9866 1.3078 1.3241 1.3440 1.3622 2.4932 2.3331 2.2858 2.2739 2.3972 2.4650 2.5637 2.6920 2.7322 2.7693

0.6900 0.7234 0.7553 0.7663 0.7845 0.7957 0.8239 0.8661 0.8962 0.9575 0.9736 0.9955 1.0936 1.0981 1.1036 1.1085 1.3560 1.3263 1.3173 1.3150 1.3384 1.3508 1.3686 1.3911 1.3980 1.4043

0.6157 0.6924 0.7288 0.7759 0.8061 0.8571 0.9006 0.9008 1.0199 0.9762 1.0690 1.2050 0.8992 0.8936 0.8961 0.9059 0.4672 0.5680 0.6358 0.7546 0.7884 0.9122 0.9500 0.9530 1.0017 1.0239

Similar approaches have been followed for the other specimens to determine the Fracture energy by Modified Beam Theory.

64


4

3.5

Specimen 1

3

Specimen 2

2.5 2

Specimen 3

1.5

1

Specimen 4

0.5 0 45

55

65

75

85

95

105

Specimen 5

Crack Length a (mm) Figure 46 Comparison of Fracture Energies with respect to Crack Length by Modified Beam approach for specimens with Adhesive film

Table 17 List of Batch 1Specimens with their Average Fracture Energies and Standard Deviation by Modified Beam Theory


Average Fracture Energy Standard Deviation 0.1670 (In0.8537 N/mm) 0.9043 0.1779 1.3815 0.1609 (N 1.45 0.3708 0.7537 0.2035 (N/m\llllllllllllllllllllllll(N/mm

5.3 Approach Number 2: Compliance Calibration Approach to Beam Theory for specimens with adhesive film Another method to correct the overestimation of fracture energy is by the Compliance calibration approach. In this method, a plot of the log of Compliance versus log of Crack length is constructed. The slope of the plot is used in equation to determine the fracture energy.

65

0.6

Logarithm C

0.4

y = -1.0874x2 + 6.7879x - 8.774

0.2 0.0 1.6

1.7

1.7

1.8

1.8

1.9

1.9

2.0

2.0

2.1

-0.2 -0.4 -0.6

Logarithm a Figure 47 Plot of Log C vs. Log a for Compliance Calibration Approach

The relation between log C and log a is given as y = -1.0874x2 + 6.7879x – 8.774 Log C = -1.0874 (loga) 2 + 6.7879 (loga) – 8.774 This equation is differentiated to determine the slope as slope = -2.1748x + 6.7879 Table 18 Depiction of Calculation of Fracture Energy by Compliance Calibration approach to Beam theory Load P (N) 36.32 36.72 35.63 36.29 36.02 36.98 36.57 34.74 35.44 31.82 32.71 34.31 27.13 27.21 27.21

Crack Length (a) 46.5 49 50 51 52 54 56 59 60.2 62 63 66 74 76 77

Displacement (δ)

Compliance (mm/N)

Log C

Log a

Slope (n)

11.93 13.9 15.35 16.33 17.39 18.63 20.45 22.57 25.51 27.93 30.19 33.85 35.48 36.03 36.57

0.3285 0.3785 0.4308 0.4500 0.4828 0.5038 0.5592 0.6497 0.7198 0.8777 0.9230 0.9866 1.3078 1.3241 1.3440

-0.4835 -0.4219 -0.3657 -0.3468 -0.3162 -0.2978 -0.2524 -0.1873 -0.1428 -0.0566 -0.0348 -0.0059 0.1165 0.1219 0.1284

1.6675 1.6902 1.6990 1.7076 1.7160 1.7324 1.7482 1.7709 1.7796 1.7924 1.7993 1.8195 1.8692 1.8808 1.8865

3.1615 3.1121 3.0930 3.0743 3.0559 3.0203 2.9859 2.9367 2.9176 2.8898 2.8747 2.8308 2.7227 2.6975 2.6852

66

Fracture Energy Gi (N/mm) 0.7365 0.8104 0.8458 0.8931 0.9203 0.9633 0.9969 0.9757 1.0954 1.0356 1.1265 1.2453 0.8854 0.8699 0.8675

27.5 15.43 17.77 19.14 20.96 20.97 22.3 22.37 22.08 22.74 22.89

79 89 91 92.5 93 94 94.5 95 97 99.5 100

37.46 38.47 41.46 43.75 47.66 50.27 54.97 57.35 59.44 62.13 63.39

1.3622 2.4932 2.3331 2.2858 2.2739 2.3972 2.4650 2.5637 2.6920 2.7322 2.7693

0.1342 0.3968 0.3679 0.3590 0.3568 0.3797 0.3918 0.4089 0.4301 0.4365 0.4424

1.8976 1.9494 1.9590 1.9661 1.9685 1.9731 1.9754 1.9777 1.9868 1.9978 2.0000

2.6609 2.5484 2.5274 2.5119 2.5068 2.4967 2.4917 2.4867 2.4671 2.4430 2.4383

0.8675 0.4249 0.5115 0.5685 0.6732 0.7000 0.8081 0.8396 0.8345 0.8672 0.8845

Similar approaches have been followed for the other specimens to determine the Fracture energy by Compliance calibration approach to Beam Theory.


4 3.5

Specimen 1

3

Specimen 2

2.5 2

Specimen 3

1.5 1

Specimen 4

0.5 0 45

55

65

75

85

95

105

Specimen 5

Crack Length a (mm) Figure 48 Comparison of Fracture Energies with respect to Crack Length by Compliance Calibration approach for specimens with Adhesive film

Table 19 List of Batch 1 Specimens with their Average Fracture Energies and Standard Deviation by Compliance Calibration Approach

Specimen

Average Fracture Energy (In N/mm)

Standard Deviation

Specimen 1 Specimen 2 Specimen 3 Specimen 4

0.8557 (N 0.8724 1.4601 (N/m\llllllllllllllllllllllll(N/mm 1.6

0.1824 0.1064 0.1838 0.2307

67

Specimen 5

0.7964

0.2152

5.4 Approach Number 3: Modified Compliance Calibration Approach for specimens with adhesive film

a/h

Another method to correct the overestimation of fracture energy is by the Modified Compliance calibration approach. In this method, a plot of a/h vs. C 1/3 is constructed. 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0

y = 0.2799x2 + 1.4482x + 0.1502 R² = 0.9912

0.5

0.7

0.9

1.1

1.3

1.5

c ^ (1/3) Figure 49 Plot of a/h vs. c ^ (1/3) for Modified Compliance Calibration approach

Table 20 Determination of Fracture energy by Modified Calibration approach

Crack Length a (mm)

Compliance C (mm/N)

C1/3

a/h

Slope (A1)

C2/3


46.5 49 50 51 52 54 56 59 60.2 62 63 66 74

0.3285 0.3785 0.4308 0.4500 0.4828 0.5038 0.5592 0.6497 0.7198 0.8777 0.9230 0.9866 1.3078

0.6900 0.7234 0.7553 0.7663 0.7845 0.7957 0.8239 0.8661 0.8962 0.9575 0.9736 0.9955 1.0936

1.2917 1.3611 1.3889 1.4167 1.4444 1.5000 1.5556 1.6389 1.6722 1.7222 1.7500 1.8333 2.0556

1.8346 1.8533 1.8711 1.8773 1.8875 1.8938 1.9096 1.9332 1.9501 1.9844 1.9934 2.0057 2.0606

0.4761 0.5233 0.5704 0.5872 0.6154 0.6331 0.6788 0.7501 0.8032 0.9167 0.9480 0.9910 1.1959

0.7131 0.7932 0.8063 0.8582 0.8813 0.9525 0.9903 0.9756 1.0777 0.9745 1.0600 1.2118 0.8899

68

76 77 79 89 91 92.5 93 94 94.5 95 97 99.5 100

1.3241 1.3440 1.3622 2.4932 2.3331 2.2858 2.2739 2.3972 2.4650 2.5637 2.6920 2.7322 2.7693

1.0981 1.1036 1.1085 1.3560 1.3263 1.3173 1.3150 1.3384 1.3508 1.3686 1.3911 1.3980 1.4043

2.1111 2.1389 2.1944 2.4722 2.5278 2.5694 2.5833 2.6111 2.6250 2.6389 2.6944 2.7639 2.7778

2.0631 2.0662 2.0690 2.2075 2.1909 2.1859 2.1846 2.1977 2.2047 2.2146 2.2272 2.2311 2.2346

1.2058 1.2179 1.2288 1.8387 1.7591 1.7352 1.7292 1.7912 1.8248 1.8732 1.9352 1.9544 1.9720

0.9015 0.9092 0.9357 0.4131 0.5282 0.6059 0.7245 0.7467 0.8575 0.8818 0.8825 0.9437 0.9633


4 3.5 3 2.5

Specimen 1

2

Specimen 2

1.5

Specimen 3

1

Specimen 4

0.5

Specimen 5

0 45

55

65

75

85

95

105

Crack Length a (mm) Figure 50 Comparison of Fracture Energies with respect to Crack Length by Modified Compliance Calibration approach for specimens with Adhesive film

Table 21 List of Batch 1Specimens with their Average Fracture Energies and Standard Deviation by Modified Compliance Calibration Approach


Average Fracture Energy (In N/mm) 0.8645 0.89 (N 1.4 1.58 (N/m\llllllllllllllllllllllll(N/mm 0.7795 69

Standard Deviation 0.1705 0.1362 0.1657 0.3259 0.2241

5.5 Comparison of Fracture Energy by the Various Beam Theory Approaches for specimens with adhesive film (Comparison of Batch 1 specimens) The figure depicted below represents the Fracture Energy obtained by Beam theory and the various modifications to beam theory. As can be seen, the results obtained from beam theory (Blue columns) tend to overestimate the fracture energy and is minimized by Modified beam theory (Red columns) and by the Compliance calibration approach (Green columns) and by the modified compliance calibration approach (Violet columns). . In most of the cases, modified beam theory and the compliance calibration approach has given values lesser than the Beam theory except for certain cases.

2.5


Beam Theory 2


1.5

1

Beam Theory by Compliance Calibration Approach

0.5 Modified Compliance Calibration 0 1

2

3

Specimens

4

5

Figure 51 Figure Comparison of Fracture Energy by Beam theory method and its modification approaches for specimens with adhesive film (Batch 1)

70

5.6 Beam Theory for Specimens without Adhesive Film (Batch 2 Specimens) Similar to the procedures described in the initial phases of chapter 5, Beam theory and


it’s modifications are used for Batch 2 specimens. 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

Specime n6 Specime n7 Specime n8 50

60

70

80

90

100

Crack Length a (mm) Figure 52 Comparison of Fracture Energies with respect to Crack Length by Beam Theory for specimens without adhesive film (Batch 2 specimens)

Table 22 List of Batch 2 specimens with their average fracture energy and standard deviation by Beam theory

Specimen


Standard Deviation

Specimen

0.6749

0.100456

Specimen 7

(N/m\llllllllllllllllllllllll(N/mm 0.5425

0.087306

Specimen 8

0.999

0.156669

71

5.7 Modified Beam Theory Approach for Specimens without adhesive film


2.0 1.8 1.6

Specimen 6

1.4 1.2 1.0

Specimen 7

0.8

0.6 Specimen 8

0.4 0.2

0.0 50

60

70

80

90

100

Crack Length (a) Figure 53 Comparison of Fracture Energies with respect to Crack Length by Modified Beam Theory approach for specimens without Adhesive film

Table 23 List of Batch 2 Specimens with their Average Fracture Energies and Standard Deviation by Modified Beam Theory Approach

Specimen Specimen Specimen 7 Specimen 8

Average Fracture Energy (In N/mm) 0.6002 0.5099 (N 0.9787 (N/m\llllllllllllllllllllllll(N/mm

72


5.8 Compliance Calibration Approach to Beam Theory for specimens without adhesive film


2 1.8 1.6

Specimen 6

1.4 1.2 1

Specimen 7

0.8 0.6

Specimen 8

0.4

0.2 0 45

65

85

105

Crack Length a (mm) Figure 54 Comparison of Fracture Energies with respect to Crack Length by Compliance Calibration approach for specimens without Adhesive film

Table 24 List of Batch 2 Specimens with their Average Fracture Energies and Standard Deviation by Compliance Calibration Approach

Specimen


Standard Deviation

Specimen Specimen 7 Specimen 8

0.6 (N 0.58 0.9646 (N/m\llllllllllllllllllllllll(N/mm

0.0896 0.1284 0.1091

73

5.9 Modified Compliance Calibration Approach to Beam Theory for specimens without adhesive film Fracture Energy (Gi)

2.0 1.8 1.6

Specime n6 Specime n7

1.4

1.2 1.0 0.8 0.6 0.4 0.2 0.0 50

60

70

80

90

100

Crack Length (a) Figure 55 Comparison of Fracture Energies with respect to Crack Length by Modified Compliance Calibration approach for specimens without Adhesive film

Table 25 List of Batch 2 Specimens with their Average Fracture Energies and Standard Deviation by Modified Compliance Calibration Approach

Specimen Specimen Specimen 7 Specimen 8

Average Fracture Energy (In N/mm) 0.6018 0.5865 (N 0.9568 (N/m\llllllllllllllllllllllll(N/mm

74


5.10 Comparison of Fracture Energy by the Various Beam Theory Approaches for specimens without adhesive film (Batch 2 specimens)


1.4

Beam Theory

1.2 1


0.8 0.6

0.4 0.2 0 6

7

8

Compliance Calibration Approach to Beam Theory Modified Comliance Calibration Theory

Specimens Figure 56 Comparison of Fracture Energy by Beam theory method and its modification approaches for specimens without adhesive film (Batch 2)

75

CHAPTER 6: DETERMINATION OF FRACTURE ENERGY BY AREA METHOD In this method, the Load – Displacement curve is partitioned with respect to crack length. The area under each partition is determined (ΔU). The crack lengths at the boundaries of the partition is determined and the incremental change in crack length (Δa) is given as Δa = a2 – a1. With information about ΔU and Δa , the Fracture energy is defined by equation (11). A sample Load vs. displacement plot is shown below. This corresponds to specimen 2 (with adhesive film). 40

a2

a1

35

a3

Force P (N)

30

a4

a5

a6

25 20

a7 15 10 5 0 0

10

20

30

40

50

60

Cross Head Displacement δ (mm) Figure 57 Area method to determine Fracture Energy

Various techniques can be used to determine the change in area (Δa). Δa = 0.5 * (P1 δ2 – P2 δ1)is used to determine the change in area in this internship.

76

70

A sample tabular column is shown below to determine the fracture energy by this method. The tabular column shown below is for specimen 1. Table 26 Determination of Fracture energy by Area method

Load P

Crack

Cross Head

Change in

Change in Area

Fracture

(N)

Length a

Displacement

Crack

(ΔU)

Energy Gi

36.32 36.72 35.63 36.29 36.02 36.98 36.57 34.74 35.44 31.82 32.71 34.31 27.13 27.21 27.21 27.5 15.43 17.77 19.14 20.96 20.97 22.3 22.37 22.08 22.74 22.89

(mm) 46.5

δ11.93 (mm)

Length 2.5 (Δa)

(N/mm) 0.6678

49 50 51 52 54 56 59 60.2 62 63 66 74 76 77 79 89 91 92.5 93 94 94.5 95 97 99.5 100

13.9 15.35 16.33 17.39 18.63 20.45 22.57 25.51 27.93 30.19 33.85 35.48 36.03 36.57 37.46 38.47 41.46 43.75 47.66 50.27 54.97 57.35 59.44 62.13 63.39

1.0 1.0 1.0 2.0 2.0 3.0 1.2 1.8 1.0 3.0 8.0 2.0 1.0 2.0 10.0 2.0 1.5 0.5 1.0 0.5 0.5 2.0 2.5 0.5 4.0

33.3892 34.1975 12.3932 21.4383 13.9852 37.4710 57.4760 43.1683 89.0555 23.5278 35.7073 149.4842 6.0415 7.3467 6.8058 239.9586 -21.9421 -8.0535 -2.3938 27.1145 15.8499 24.6130 31.6924 10.0824 9.6664 132.8013

77

1.7099 0.6197 1.0719 0.3496 0.9368 0.9579 1.7987 2.4738 1.1764 0.5951 0.9343 0.1510 0.3673 0.1701 1.1998 -0.5486 -0.2684 -0.2394 1.3557 1.5850 2.4613 0.7923 0.2016 0.9666 1.6600


7.0 6.0

Specimen 1

5.0 4.0

Specimen 2

3.0

2.0

Specimen 3

1.0 0.0 -1.0

45

55

65

75

85

95

105

Specimen 4

Specimen 5

-2.0

Crack Length a (mm) Figure 58 Comparison of Fracture energies vs. Crack length for Specimens with adhesive film by Area method

Since the load and displacements were measured at irregular crack lengths, the area method tends to give large standard deviations. This can be seen from the figure above also in which some points on the curve touch fracture energies of even 4N/mm. Thus, in

Fracture Energy (N/mm)

the error bar graphs for area method, standard deviations are not used.

1.8 1.6

1.4 1.2 1 0.8 0.6

0.4 0.2 0 1

2

3

4

5

Specimens Figure 59 Comparison of Fracture Energies with Error Bars for Specimens with Adhesive film by Area Method

78

Table 27 List of Batch 1 specimens with their fracture energies by Area method


Average Fracture Energy 0.89 0.853 1.446 1.567 0.707


4.0 3.5 Specimen 6

3.0 2.5 2.0

Specimen 7

1.5 1.0

Specimen 8

0.5 0.0 50

60

70

80

90

100

Crack Length (a) Figure 60 Comparison of Fracture Energies with Error Bars for Specimens without Adhesive film by Area Method (Batch 2)

79


1.2 1 0.8

0.6 0.4 0.2 0 1

2

3

Specimens Figure 61 Comparison of Fracture Energies with Error Bars for Specimens without Adhesive film by Area Method

Table 28 List of Batch 2 specimens with their fracture energies by Area method

Specimen Specimen 1 Specimen 2 Specimen 3

Average Fracture Energy 0.6432 0.6892 0.99

80

CHAPTER 7: RESULTS AND DISCUSSIONS

7.1 Results for Specimens with adhesive film (Batch 1 Specimens) Thus, the value of Fracture Toughness has been determined by Compliance Method, Beam theory, Modified Beam theory, Compliance Calibration approach to Beam theory, Modified Compliance Calibration approach to Beam theory and Area Method. The comparisons between the values of Fracture Energy obtained by Beam theory and its modifications have already been discussed in chapter 6. The following table gives the values of Fracture energy obtained by the different methods for the various specimens. Table 29 Results showing values of Fracture energies obtained by the various methods for specimens with adhesive film (Batch 1)

Specimen


Average Value of Fracture Energy (Gi) Complianc Beam Modified Compliance Modified e Method Theory (or Beam Calibration Compliance Displacem Theory Approach Calibration ent to Beam Approach Method) Theory to Beam Theory

Area Method

0.7437

0.9309

0.8537

0.8557

0.8645

0.89

0.8137

1.0181

0.9043

0.8724

0.89

0.853

1.5000

1.4201

1.3815

1.4601

1.4

1.446

1.5800

1.55

1.45

1.6

1.58

1.567

0.7721

0.9037

0.7537

0.7964

0.7795

0.707

81

Table 30 Maximum, Minimum and Average fracture energies for the Batch 1 specimens

Specimen


Average of Fracture Energy from all methods 0.8564

Maximum Minimum value of Standard Calculated value Fracture Energy Deviation of Fracture calculated Energy 0.9309

0.7437

0.0623

0.8919

1.0181

0.8137

0.0694

1.42

1.5

1.3815

0.0430

1.54

1.58

1.45

0.0537

0.7854

0.9037

0.707

0.0655

7.2 Results for specimens without adhesive film (Batch 2 Specimens) Table 31 Results showing values of Fracture energies obtained by the various methods for specimens without adhesive film (Batch 2)

Specimen

Specimen 6 Specimen 7 Specimen 8

Average Value of Fracture Energy (Gi) Complianc Beam Modified Compliance Modified e Method Theory (or Beam Calibration Compliance Displacem Theory Approach Calibration ent to Beam Approach Method) Theory to Beam Theory

Area Method

0.6496

0.6749

0.6002

0.6

0.6018

0.6432

0.544

0.5425

0.5099

0.58

0.5865

0.5892

0.8698

0.999

0.9787

0.9646

0.9568

0.99

82

Table 32Maximum, Minimum and Average fracture energies for batch 2 specimens

Specimen

Specimen 6 Specimen 7 Specimen 8

Average of Maximum Minimum value of Standard Fracture Calculated value Fracture Energy Deviation Energy of Fracture calculated from all Energy methods 0.628 0.6749 0.6 0.0320 0.56

0.5892

0.5099

0.0623

0.95

0.999

0.8698

0.0467

7.3 Discussions Some specimens were loaded till fracture and the surface of the specimens were observed under a microscope. The following images are the images depicting the surface of the aluminium plate (i.e. the part of the FML containing the Aluminium plate adhered to a part of the Composite) at different portions on the surface.

Figure 62(a) Surface of Aluminium plate post fracture

83

Figure 62 (b) Surface of Aluminium plate post fracture

Figure 62(c) Surface of Aluminium plate post fracture

84

The following images are the images depicting the surface of the Composite (.i.e., the part of the FML containing only composite).

Figure 63 (a) Surface of Composite post fracture

Figure 65 (b) Surface of Composite post fracture 85

Figure 65 (c) Surface of Composite post fracture

The photographs above indicate that there are patches of adhesive on both the Aluminium and Composite surface. Since we are able to see traces of the adhesive on the surface of both adherents, this is an indication of Cohesive Fracture. Thus, the fracture has occurred in the adhesive material itself. The resistance of the adhesive is less than the substrates. Literature suggests suitable methods to improve the performance of the adhesive, the study of which is beyond the scope of this Internship. Some of the techniques would be to respect the adhesive curing time, minimizing the presence of voids and defects in the adhesive and the prevention of bubbles and pores in the adhesive.

86

Adherent (Metal + Composite)

Adherent (Only Composite) Figure 64 Depiction of Cohesive fracture

Fracture has occurred through the Adhesive

7.4 Conclusions The values of average fracture toughness and variation of fracture toughness with crack length is determined for the two batches of specimens (one with an adhesive film and other without adhesive film) by making use of the various methods available in literature. Interlaminar Fracture Toughness is an important aspect of damage tolerance studies which is an essential part for composites certification. Using information from the strain energy release rate and the fracture toughness contours, the designer can decide if composite part is safe from growth of the delamination. Modified beam theory was found to give the least values for fracture toughness. And beam theory was found to give the highest values for fracture toughness. The nature of fracture was found out to be Cohesive fracture.

87

REFERENCES [1] Glyn Lawcock , Lin Ye , Yiy Wing Mai and Chin Teh Sun, The effect of adhesive bonding between Aluminium and Composite Prepreg on the mechanical properties of Carbon Fiber Reinforced metal laminates, Elsevier, edition: 1996, pages [2] H.M.Westergaard. Bearing pressure and cracks, Journal of Applied Mechanics, Vol.6, A49-A53, 1939. [3] G.C.Sih , P.C.Paris and G.R.Irwin , Cracks in rectilinearly anisotropic bodies , International Journal of Fracture Mechanics , Vol.1 , No.3. pp 189 – 203, 1965. [4] A.A.Griffith , The phenomenon of rupture and flow in solids , Philosophical Transactions of the Royal Society , Series A , Vol 221 , pp 163 – 198 , 1920. [5] ASTM 5528, Standard Test Method for Mode 1 Inter laminar Fracture Toughness of Uni-directional Fiber Reinforced Polymer Matrix Composites,2007. [6] Glyn Lawcock , Lin Ye , Yiu Wing Mai and Chin Teh Sun , The effect of adhesive bonding between Aluminium and Composite Prepreg on the mechanical properties of Carbon fiber reinforced metal laminates , Elsevier composite science and technology , 1997. [7] J.Laliberte , P.V.Straznicky and C.Poon , Mode 1 delamination characteristics of Cross ply fiber metal laminates , Poster reference : ICD1003810R , Carleton University. [8] Hieu Truong, Dimitis Lagoudas , Ozden O Ochoa and Khalid Lafdi , Fracture Toughness of fiber metal laminates , Journal of composite materials ,

88

[9] Raymond G.Boeman, Donald Erdman , Lynn Klett and Ronny Lomax , A Practical test method for mode 1 fracture toughness of adhesive joints with dissimilar substrates , SAMPE automotive conference.

89

Appendix A Appendix A shows the values of fracture energies obtained by all the methods at different crack lengths for first batch of specimens (with adhesive film) Specimen 2 Crack Length a (mm)

Compliance Method

Beam Theory


Compliance Calibration Approach

Area Method

0.8762

Modified Compliance Calibration approach 0.9378

45.5

0.9434

0.7693

0.6376

47.8

1.1044

0.8954

0.7483

0.9958

1.0887

0.8957

49

1.2010

0.9958

0.8355

1.0940

1.1914

1.2933

51

1.1678

1.0431

0.8807

1.1232

1.1887

1.1285

54

0.9762

0.9475

0.8070

0.9908

1.0203

0.8483

57

0.9436

0.9527

0.8178

0.9680

0.9943

1.3293

60

0.7446

0.8530

0.7374

0.8429

0.8193

1.1120

61

0.7462

0.8820

0.7642

0.8636

0.8293

0.2384

63

0.8160

0.9290

0.8084

0.8933

0.8885

0.4685

66

0.8289

0.9246

0.8093

0.8656

0.8882

1.2676

68

0.7311

0.8727

0.7667

0.8028

0.8008

0.9709

69

0.7368

0.8969

0.7893

0.8179

0.8113

1.5167

70

0.6954

0.8856

0.7808

0.8007

0.7776

0.5301

72

0.7195

0.9115

0.8062

0.8101

0.7988

1.3356

73.5

0.6815

0.9079

0.8050

0.7967

0.7689

0.2767

76

0.7159

0.9208

0.8194

0.7912

0.7916

0.9172

78

0.6807

0.8999

0.8031

0.7606

0.7572

1.0711

79

0.6891

0.9292

0.8304

0.7788

0.7706

1.2100

80

0.6940

0.9582

0.8574

0.7966

0.7814

1.0536

81

0.7149

1.0026

0.8984

0.8267

0.8079

0.6274

84

0.7274

1.0165

0.9142

0.8180

0.8150

1.2523

85

0.7044

1.0049

0.9048

0.8022

0.7937

0.6649

88.5

0.7091

1.0121

0.9149

0.7856

0.7927

0.6918

89.3

0.7267

1.0377

0.9389

0.8004

0.8110

1.2593

90.5

0.7201

1.0512

0.9523

0.8033

0.8084

1.1406

93

0.6607

1.0010

0.9091

0.7500

0.7482

0.2200

94

0.7042

1.0520

0.9563

0.7820

0.7913

0.0090

96

0.7972

1.1431

1.0412

0.8367

0.8779

0.5980

98.5

0.8124

1.1528

1.0523

0.8276

0.8862

0.3169

99

0.9214

1.2995

1.1868

0.9293

1.0017

0.6000

99

1.0311

1.4783

1.3501

1.0572

1.1281

2.8136

90

0.5044

Specimen 3 Crack Length a (mm)

Compliance Method

Beam Theory



Area Method

0.9132


46

0.5902

0.9378

0.8999

47

0.8065

1.2080

1.1602

1.1795

1.2283

0.8126

1.5045

49

1.0265

1.2886

1.2395

1.2649

1.3278

1.3343

50.5

1.1411

1.3300

1.2808

1.3106

1.3678

1.7921

53

1.2298

1.3657

1.3175

1.3541

1.3784

1.6773

54.5

1.2824

1.3879

1.3402

1.3809

1.3899

1.1068

58

1.3880

1.3438

1.3003

1.3476

1.3583

1.5320

60

1.4467

1.3755

1.3324

1.3852

1.3817

1.8958

61

1.5178

1.4491

1.4044

1.4623

1.4456

1.6309

62.5

1.5622

1.4792

1.4347

1.4972

1.4694

1.3967

64

1.5962

1.4846

1.4410

1.5072

1.4749

1.3979

67

1.5561

1.4112

1.3715

1.4407

1.4006

1.0003

68.7

1.6648

1.4635

1.4233

1.4987

1.4620

2.2447

70

1.6213

1.4615

1.4221

1.5001

1.4429

0.8516

73

1.7706

1.5013

1.4624

1.5489

1.5066

1.0193

74

1.8884

1.5769

1.5366

1.6296

1.5888

2.4467

76

1.7113

1.4956

1.4584

1.5506

1.4792

0.5990

78

1.8853

1.5708

1.5327

1.6337

1.5772

1.1522

79

1.9666

1.6218

1.5829

1.6893

1.6332

1.6422

81

1.9506

1.6129

1.5751

1.6851

1.6207

2.9118

82

1.9300

1.6446

1.6066

1.7208

1.6342

0.3794

84.3

2.1517

1.7255

1.6867

1.8114

1.7510

4.0342

86.5

1.6892

1.5258

1.4923

1.6067

1.4832

2.1137

88

1.6178

1.4968

1.4645

1.5793

1.4424

2.9599

90

1.3424

1.3433

1.3149

1.4212

1.2586

1.6718

93

1.1650

1.2123

1.1875

1.2876

1.1204

0.0721

95

1.2492

1.2320

1.2073

1.3118

1.1614

0.4149

98.5

1.3971

1.2964

1.2714

1.3863

1.2510

0.3734

100

1.4855

1.3416

1.3161

1.4372

1.3082

-0.0123


Compliance Method

Beam Theory



Area Method

1.3270

Complianc e Calibration Approach 1.3548

45.5

1.3176

1.3372

49

1.7509

1.4835

1.4730

1.5010

1.5264

10.2434

91

0.8116

49.5

1.5366

1.5380

1.5272

1.5558

1.4984

2.1360

51

1.6001

1.6146

1.6036

1.6324

1.5604

0.5350

54

1.9961

1.7324

1.7212

1.7495

1.7426

2.5010

57

1.7026

1.5720

1.5624

1.5859

1.5467

1.2170

59

1.7847

1.5948

1.5854

1.6079

1.5847

1.4788

62

1.7694

1.5556

1.5469

1.5668

1.5559

1.3741

64.5

1.7479

1.5152

1.5070

1.5250

1.5259

0.9977

67

1.8316

1.5287

1.5207

1.5374

1.5629

1.7142

68

1.9450

1.6315

1.6232

1.6404

1.6682

2.6229

69

2.0059

1.7273

1.7186

1.7362

1.7551

1.2316

70

2.1678

1.8455

1.8363

1.8545

1.8851

0.9344

71

2.4874

2.0756

2.0655

2.0851

2.1372

3.6707

72

2.3214

2.0080

1.9983

2.0167

2.0492

2.5264

75

1.9386

1.7640

1.7558

1.7702

1.7841

0.3559

76

2.1279

1.8816

1.8730

1.8877

1.9240

1.6902

77

2.1877

1.9328

1.9240

1.9386

1.9814

1.3056

79

2.1885

1.9009

1.8925

1.9057

1.9681

2.2445

83

1.7332

1.5972

1.5904

1.5996

1.6409

1.3204

86

1.6736

1.5373

1.5310

1.5386

1.5933

4.1347

87

1.4782

1.4504

1.4446

1.4513

1.4780

1.6463

91

1.1760

1.2150

1.2103

1.2147

1.2328

-0.5029

93

1.3431

1.2790

1.2742

1.2782

1.3366

-0.6183

94

1.4814

1.3526

1.3475

1.3514

1.4350

1.8646

95.5

1.3972

1.3102

1.3054

1.3086

1.3849

1.2282

Area Method


Compliance Method

Beam Theory



45

0.8787

0.6508

0.4991

0.5280


46

1.0153

0.8242

0.6353

0.6715

0.6236

1.1832

47

1.0424

0.8942

0.6926

0.7315

0.6739

0.5884

51

0.9790

0.8690

0.6852

0.7220

0.6787

0.7464

53

1.1128

1.0181

0.8092

0.8519

0.8054

1.4695

54

1.2028

1.1523

0.9194

0.9676

0.9094

1.6772

55

1.2610

1.2662

1.0140

1.0668

0.9965

1.5005

60

0.7846

0.9722

0.7917

0.8323

0.7589

0.3450

63

0.7719

0.9313

0.7652

0.8044

0.7526

2.0207

64

0.6991

0.9116

0.7510

0.7896

0.7282

0.9297

66

0.6615

0.9013

0.7465

0.7850

0.7244

-1.0365

67

0.8178

1.0355

0.8599

0.9043

0.8555

0.1893

92

1.4014

68

0.9224

1.1486

0.9562

1.0057

0.9612

0.7539

89

0.2676

0.3775

0.3272

0.3464

0.3318

-0.5837

91

0.3481

0.4546

0.3952

0.4187

0.4174

-0.9541

92

0.4281

0.5449

0.4743

0.5028

0.5133

-0.1764

93

0.4937

0.6514

0.5679

0.6021

0.6218

-0.1700

93

0.5353

0.7511

0.6548

0.6943

0.7170

-0.0668

94

0.6253

0.7972

0.6959

0.7383

0.7697

1.5497

95

0.6521

0.9671

0.8454

0.8972

0.9313

1.0026

96

0.7134

1.0224

0.8949

0.9501

0.9877

-0.4472

96.5

0.7764

1.1012

0.9645

1.0243

1.0710

0.1838

97

0.7682

1.1908

1.0436

1.1086

1.1634

3.2087

98

0.7000

1.25464 5

1.10096

1.170022

1.215715

0.781056

93

Appendix B Appendix B shows the values of fracture energies obtained by all the methods at different crack lengths for second batch of specimens (without adhesive film)


Complianc e Method

Beam Theory



Area Method

0.4378


45

0.4231

0.4932

0.4184

47.5

0.4840

0.5288

0.4523

0.4695

0.4674

0.6016

49

0.5105

0.5569

0.4784

0.4945

0.4900

0.7113

50.5 52

0.5117

0.5670

0.4891

0.5035

0.4941

0.6213

0.5012

0.5572

0.4826

0.4949

0.4835

0.5836

53.5

0.5048

0.5613

0.4880

0.4986

0.4859

1.2006

54

0.5104

0.5847

0.5089

0.5194

0.5006

0.2541

56.8

0.5524

0.5804

0.5084

0.5157

0.5108

0.4312

58

0.6073

0.6274

0.5510

0.5575

0.5551

0.8010

59

0.6124

0.6422

0.5652

0.5707

0.5652

0.4509

60.5

0.6419

0.6598

0.5824

0.5864

0.5847

0.5789

62

0.6755

0.6898

0.6106

0.6131

0.6128

0.4511

63.5

0.7297

0.7294

0.6475

0.6484

0.6532

0.6066

65

0.7567

0.7510

0.6683

0.6676

0.6747

0.2855

66.8

0.8213

0.7799

0.6961

0.6934

0.7124

1.6348

68

0.8132

0.8207

0.7339

0.7297

0.7346

0.6395

70.9

0.8067

0.8036

0.7217

0.7147

0.7245

0.6828

73

0.8432

0.8366

0.7536

0.7441

0.7570

0.7567

77

0.8140

0.8143

0.7373

0.7244

0.7381

0.9877

80

0.6902

0.7256

0.6593

0.6456

0.6488

0.3632

81.8

0.6907

0.7100

0.6465

0.6319

0.6415

0.8680

83.3

0.6519

0.6851

0.6248

0.6097

0.6155

0.4294

84.5

0.6733

0.7012

0.6402

0.6240

0.6330

0.5369

85.7

0.6950

0.7215

0.6596

0.6422

0.6532

0.5634

87.5

0.7190

0.7438

0.6812

0.6621

0.6759

94

0.3972


Compliance Method

Beam Theory



Area Method

0.2833


46

0.1160

0.3281

0.2997

47

0.1696

0.4170

0.3816

0.3645

0.3778

0.2622

51

0.3277

0.4662

0.4295

0.4263

0.4428

0.6810

52

0.3645

0.4956

0.4572

0.4579

0.4708

0.3764

55

0.4459

0.4980

0.4614

0.4739

0.4860

0.6392

57

0.5093

0.5412

0.5027

0.5246

0.5316

0.6575

59.5

0.5664

0.5738

0.5346

0.5684

0.5695

0.6370

62.5

0.6328

0.6094

0.5697

0.6185

0.6148

0.5330

66

0.7528

0.6708

0.6292

0.6989

0.6957

0.4753

72

0.8119

0.6397

0.6031

0.6939

0.6987

0.7242

75

0.6984

0.5567

0.5261

0.6152

0.6173

0.8844

78.2

0.6197

0.5292

0.5013

0.5957

0.5924

2.0035

78.5

0.6847

0.6017

0.5700

0.6784

0.6720

0.3867

81

0.7854

0.6628

0.6289

0.7576

0.7577

1.5990

83

0.6684

0.6274

0.5961

0.7248

0.7178

0.5715

88

0.6243

0.5812

0.5537

0.6881

0.6922

1.2486

90

0.5352

0.5422

0.5172

0.6480

0.6518

0.3810

96.5

0.4928

0.4792

0.4584

0.5892

0.6112

0.1847

100

0.5309

0.4874

0.4671

0.6079

0.6430

0.1765

0.6747


Compliance Method

Beam Theory



Modified Compliance Calibration approach

Area Method

45 46 47 49.5 54 57 59 63.5 66 69.5 71.5 75

0.5298 0.6530 0.7253 0.8679 0.9988 0.9388 0.9630 0.9280 0.9575 0.9218 0.9186 0.9456

0.5668 0.7154 0.8116 0.8890 0.9376 0.9921 1.0217 1.0209 1.0668 1.0376 1.0919 1.1101

0.5169 0.6536 0.7429 0.8172 0.8677 0.9218 0.9516 0.9555 1.0008 0.9765 1.0293 1.0492

0.6105 0.7660 0.8641 0.9332 0.9609 1.0015 1.0213 0.9990 1.0321 0.9886 1.0314 1.0334

0.6688 0.8247 0.9159 1.0213 1.0829 1.0582 1.0737 1.0265 1.0521 1.0012 1.0153 1.0239

0.9081 1.0496 0.3826 0.5850 1.6125 0.9611 1.0829 1.0259 0.9751 1.5836 0.8262 1.2494

95

78.5 82 85 89 91 93 96 97.5

0.8846 0.7631 0.7585 0.8140 0.8793 0.9272 0.9221 0.9465

1.0838 1.0122 0.9990 1.0090 1.0714 1.1743 1.1666 1.2193

1.0269 0.9613 0.9503 0.9620 1.0226 1.1218 1.1161 1.1672

96

0.9948 0.9165 0.8942 0.8900 0.9382 1.0210 1.0037 1.0436

0.9644 0.8528 0.8362 0.8612 0.9174 0.9791 0.9650 0.9947

1.4933 0.7446 0.3624 0.5570 1.5499 0.9074 1.3948 0.5603