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Chalasani, Prveen K.
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Nanoindentation of layered materials with a nonhomogeneous interface
h [electronic resource] /
by Praveen K. Chalasani.
260
[Tampa, Fla] :
b University of South Florida,
2006.
3 520
ABSTRACT: Indentation is used as a technique for mechanical characterization of materials for a long time. In the last few decades, new techniques of mechanical characterization at micro and nano level using indentation have been developed. Mechanical characterization of thin films has become an important area of research because of their crucial role in modern technological applications. Theoretical and computational models of indentation are less time consuming,cost effective, and flexible. Many researchers have investigated mechanical properties of thin films using theoretical and computational models. In this study, an indentation model for a thin layersubstrate geometry with the possibility of nonhomogeneous or homogeneous interface of finite thickness between layer and substrate has been developed. The layer and substrate can be nonhomogeneous or homogeneous. Three types of indenters are modeled: 1) Uniform pressure indenter 2) Flat indenter 3) Smooth indenter. Contact depth, maximum interfacial normal stress and maximum interfacial shear stress play an important role in design and mechanical characterization of thin films using indentation and the effect of modeling the interface as homogeneous and nonhomogeneous on these parameters is studied. A sensitivity analysis is also conducted to find the effect of indentation area, substrate to layer Young's modulus ratio, layer to interface thickness ratio on contact depth and critical interfacial stresses.
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Thesis (M.A.)University of South Florida, 2006.
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Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
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Title from PDF of title page.
Document formatted into pages; contains 41 pages.
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Adviser: Autar K. Kaw, Ph.D.
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Thin film.
Substrate.
Mechanical properties.
Shear stress.
Indenter.
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Dissertations, Academic
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x Mechanical Engineering
Masters.
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.1567
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Nanoindentation of Layered Material s with a Nonhomogeneous Interface by Praveen K. Chalasani A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Autar K. Kaw, Ph.D. Daniel Hess, Ph.D. Ashok Kumar, Ph.D. Date of Approval March 28, 2006 Keywords: thin film, substrate, mechanic al properties, shear stress, indenter Copyright 2006, Praveen K. Chalasani
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ACKNOWLEDGMENT I express my sincere thanks to my major professo r Autar K. Kaw who helped me at every step of my thesis and provide d me with much needed financial support. He is always ready to help students in all situations. I thank Dr. Ashok Kumar and Dr. Daniel Hess for their help as committee members. I want to thank my fellow gradua te student Cuong Nguyen for helping me with the Design of Experiments. I thank Depart ment of Mechanical Engineering for the financial support. I also thank Sue and othe r office staff for their support during the course of my study.
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TABLE OF CONTENTS LIST OF TABLES...iii LIST OF FIGURES..iv LIST OF NOMENCLATURE..vi ABSTRACT...viii CHAPTER 1 INTRODUCTION..1 1.1 Literature survey......1 1.2 Our study.........4 CHAPTER 2 FORMULATION...5 2.1 Geometry......................5 2.2 Stress and displacement field equations..6 2.3 Continuity conditions...9 2.4 Boundary conditions..11 2.5 Type of loads......11 2.6 Derivation of solution....12 CHAPTER 3 DESIGN OF EXPERIMENTS...14 3.1 Introduction...14 3.2 Factorial designs...14 3.3 Factorial design....15 k2 3.4 Factorial design....15 42 CHAPTER 4 RESULTS...19 4.1 Introduction 4.2 Contact depth.19 4.3 Normal stress......24 4.4 Shear stress. i
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4.5 Sensitivity analysis...33 4.6 Calculations..34 CHAPTER 5 CONCLUSIONS..................38 REFERENCES....................40 ii
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LIST OF TABLES Table 1: Notations for experimental combinations..16 Table 2: Contrast constants for design 42 Table 3: Values of two different levels of the factors......33 Table 4: Percentage contribution of factors to contact depth ratio......34 Table 5: Percentage contribution of factors to maximum normal stress ratio.35 Table 6: Percentage contribution of factors to maximum shear stress ratio iii
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LIST OF FIGURES Figure 1: Schematic diagram of the layersubstrate model.5 Figure 2: Normalized vertical displacement difference (u(y)u(0))(a 2 E 1 )/(h 1 L) as a function of distance along top surface from indentor axis for uniform pressure...21 Figure 3: Contact depth ratio as a function of Youngs modulus ratio between layer and substrate for uniform pressure..21 Figure 4: Normalized vertical displacement difference (u(y)u(0))(a 2 E 1 )/(h 1 L) as a function of distance along top surface from indentor axis for flat indentor...22 Figure 5: Contact depth ratio as a function of Youngs modulus ratio between layer and substrate for flat indentor.22 Figure 6: Normalized vertical displacement difference (u(y)u(0))(a 2 E 1 )/(h 1 L) as a function of distance along top surface from indentor axis for spherical indentor...23 Figure 7: Contact depth ratio as a function of Youngs modulus ratio between layer and substrate for spherical indentor Figure 8: Normalized interface normal stress, xx (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for uniform pressure.25 Figure 9: Maximum normal stress ratio as a function of Youngs modulus ratio between layer and substrate for uniform pressure..26 Figure 10: Normalized interface normal stress, xx (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for flat indentor.26 Figure 11: Maximum normal stress ratio as a function of Youngs modulus ratio between layer and substrate for flat indentor..27 iv
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Figure 12: Normalized interface normal stress, xx (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for spherical indentor Figure 13: Maximum normal stress ratio as a function of Youngs modulus ratio between layer and substrate for spherical indentor.....28 Figure 14: Normalized interface shear stress, xy (a 2 )/L at the layersubstrate as a function of distance along interface from indentor axis for uniform pressure...29 Figure 15: Maximum shear stress ratio as a function of Youngs modulus ratio between layer and substrate for uniform pressure......30 Figure 16: Normalized interface shear stress, xy (a 2 )/L at the layersubstrate as a function of distance along interface from indentor axis for flat indentor...30 Figure 17: Maximum shear stress ratio as a function of Youngs modulus ratio between layer and substrate for flat indentor..........31 Figure 18: Normalized interface shear stress, xy (a 2 )/L at the layersubstrate as a function of distance along interface from indentor axis for spherical indentor...31 Figure 19: Maximum shear stress ratio as a function of Youngs modulus ratio between layer and substrate for spherical indentor.....32 v
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LIST OF NOMENCLATURE a2 Indentor width. 1h Layer width. 2h Interface width. 1E Youngs modulus of layer. 2E Youngs modulus of interface. 3E Youngs modulus of halfplane. Poissons ratio of layer. 1 Poissons ratio of interface. 2 Poissons ratio of halfplane. 3 Displacement field along xdirection in layer. 1u Displacement field along xdirection in interface. 2u Displacement field along xdirection in halfplane. 3u Displacement field along ydirection in layer. 1 Displacement field along ydirection in interface. 2 Displacement field along ydirection in halfplane 3 Stress field along xdirection. xx vi
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Stress field along ydirection. yy Shear stress field. xy Shear modulus for halfplane. 3 L Applied Load. p Applied pressure distribution. d Contact depth vii
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NANOINDENTATION OF LATERED MATERIALS WITH A NONHOMOGENEOUS INTERFACE Praveen K. Chalasani ABSTRACT Indentation is used as a technique for mechanical characterization of materials for a long time. In the last few decades, new techniques of mechanical characterization at micro and nano level using indentation have been developed. Mechanical characterization of thin films has become an important area of research because of their crucial role in modern technological applications. Theoretical and computational models of indentation are less time consuming, cost effective, and flexible. Many researchers have investigated mechanical properties of thin films using theoretical and computational models. In this study, an indentation model for a thin layersubstrate geometry with the possibility of nonhomogeneous or homogeneous interface of finite thickness between layer and substrate has been developed. The layer and substrate can be nonhomogeneous or homogeneous. Three types of indenters are modeled: 1) Uniform pressure indenter 2) Flat indenter 3) Smooth indenter. Contact depth, maximum interfacial normal stress and maximum interfacial shear stress play an important role in design and mechanical characterization of thin films using indentation and the effect of modeling the interface as homogeneous and nonhomogeneous on these parameters is studied. A sensitivity analysis is also conducted to find the effect of indentation area, substrate to layer Youngs modulus ratio, layer to interface thickness ratio on contact depth and critical interfacial stresses. viii
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CHAPTER 1 INTRODUCTION Thin film coatings play an important role in modern technological applications. Thin films of micro and nano thickness are not uncommon in modern technological applications. So, mechanical characterization of thin films has become an important area of research. During early 1980s, it was found that load sensing indentation can be used to obtain mechanical properties of thin films and surfaces. Instruments that can produce submicron level indentations were developed. Since then extensive research has been done on depth sensing indentation and analysis of experimental data to obtain mechanical properties of materials. 1.1 Literature survey The procedure for depth sensing indentation is as follows. Load that varies linearly or in steps is applied to material while continuously measuring the indentation depth. Loading is followed by unloading and the data obtained is plotted to get loaddisplacement curve. Since it is time consuming and difficult to measure contact area of indentor by direct observation of hardness impressions, a simple and indirect method was developed by Oliver, Hutchings and Pethica (1986). Their method is based on loaddisplacement data and indenter area function (crosssectional area of indenter as a 1
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function of distance from its tip). The idea behind the method is that at peak load, the material conforms to the shape of indenter to some depth. If this depth can be known from load displacement data, then the projected area can be estimated from indenter shape function. So, the estimation of contact depth of indenter in material at peak load became prime focus of early depth sensing indentation research. Oliver et al (1986) found that final depth is a better estimate for contact depth than the depth at maximum load. Doerner and Nix (1986) observed that unloading curve is linear at peak load. They proposed a method of extrapolating the linear portion of unloading curve to zero load and using the extrapolated depth as contact depth. Experiments confirmed that extrapolated depth gives better estimation of contact depth when compared to either depth at peak load, h max or final depth, h f Oliver and Pharr (1992) showed that unloading curve is not linear for all materials even at initial stages and developed an analysis technique that accounts for the curvature in unloading data to estimate contact depth that accounts for the curvature of unloading curve. The analysis is based on analytical solutions to different indentor geometries. The technique provides a physically justifiable procedure for determining contact depth. The above mentioned methods are being used by researchers to obtain mechanical properties from loaddisplacement data. 2
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Many researchers have investigated the mechanical properties of nano range thin films using nanoindentation. Some developed theoretical and computational indentation models. Chen, Lei Liu, and Wang (2004) investigated the effects of thickness and different filmsubstrate combinations. They used aluminum and tungsten films on glass and silicon substrates so that they can have a combination of soft films on hard substrates and hard films on soft substrates. They reported the effect of substrate on measured film properties. They found that for a soft film on a hard substrate hardness decreases at small indentation depth, then remains constant, and increases with increasing indentation depth. For a hard film on a soft substrate, hardness increases at small indentation depth, then remains constant, and decreases with increasing indentation depth. Chudoba, Schwarzer and Ritcher (2000) studied elastic properties of thin films by indentation measurements with a spherical indenter. They used an analytical solution for the elastic deformation of substrate to simulate loaddisplacement data. From this solution they could determine Youngs modulus of thin films independent of substrate effects. Linss, Schwarzer, et al (2004) investigated the mechanical properties of graded thin films with varying Youngs modulus using theoretical modeling and nanoindentation. They showed that a graded coating can be distinguished from a homogeneous layer by elastic indentation using a variety of different spherical indentors. Chudoba et al (2004) derived the correct moduli at the lower and top most part of the graded coating using a mathematical model. Their theoretical values are in agreement with values obtained from experiments. 3
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1.2 Our study In this study we investigated the effect of nonhomogeneous interface between film and substrate and quantified the effect of various parameters like film thickness, type of indentor, elastic modulus ratio, and contact depth of indentor. For this purpose we have modeled a homogeneous or nonhomogeneous thin layer on a homogeneous halfplane (substrate) separated by a nonhomogeneous or homogeneous interface. We used three different type of indenter loads The advantage with the above model is mathematical formulation required is simple and readily available. Models for stress and displacement fields for a nonhomogeneous finite strip with exponential variation in Youngs modulus and Poissons ratio is available in literature (Delale and Erdogan, 1988; Kaw et al., 1992). Models for stress and displacement fields for a homogeneous halfplane are also present in literature (Delale and Erdogan, 1988; Kaw et al., 1992). Using the above mentioned mathematical models, the filmsubstrate model can be solved numerically with a high degree of accuracy. 4
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CHAPTER 2 FORMULATION 2.1 Geometry The geometry of the problem is shown in Fig 1. The model consists of two nonhomogeneous layers of infinite length and finite width and respectively deposited on a homogeneous halfplane. Loads of different distributions are applied symmetrically about axis over a length of on the top layer. Youngs modulus and Poissons ratio vary exponentially along the width of nonhomogeneous layers where as, they are constant in the homogeneous halfplane. This model can be solved mathematically for displacements and stresses. 1h 2h x a2 y1x 2x 3x a2)(11xE)(11x )(22xE)(22x 2h3E3 ondistributiloaduniform 1h Layer 1 Layer 2 Halfp lan e Figure 1: Schem a tic diagram of the layersubstrate m odel 5
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2.2 Stress and displacement field equations For i th nonhomogeneous layer, Youngs modulus, and Poissons ratio, vary exponentially through the width as )(iixE )(iix (1) iixiiieExE0)( (2) iixiiiiiexbax)()(00 Where ,,and are found using the Youngs modulus and Poissons ratios at the edges of the i ia0 ib0 i iE0 th strip iiihxx ,0. The equations for stresses and displacement fields for i th layer are given as (Delale and Erdogan, 1988; Kaw et al., 1992) The displacement along the xdirection is given by iixmiiiiiiiiiiemcmcmxcEyxu22122222020)()()()(22),( iixmiiiiiiiemcmcmxc11321414)()()()( iiiiiiiiiicbcaxmxmcb102012120 2)()()(12010110iixmiiiiiiiecacbmca ))()(()(304022240iiiiiiiiiicbcaxmxmcb dyecacbmcaiixmiiiiiii)cos(2)()()(24030230 (3) The displacement along ydirection is given by 6
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))()(()((12),(212100iiiiiiixccmEyx ))()(()(())(24322212iiiixmiixccmecmii + iixmiiecm1422 iixmiiiiiiexccxba12100 dyexcciixmiii2)sin())()((243 (4) The stress field is given by dyexccexccyxiiiixmiiixmiiiiixxcos)()()()(22),(21432102 (5) iixmiiiiiiiiyyecmxccmyx1)(2)()()(212),(2121210 dyecmxccmiixmiiiiiicos)(2)()()(2424322 (6) iixmiiiiiiixyecxccmyx1)()()(22),(22110 dyecxccmiixmiiiiisin)()()(24432 (7) where, 2122142iiim and (8) and 2122242iiim (9) The constants and in equations(19) are obtained from the Youngs modulus and Poissons ratio at the two edges i ia0 ib0 iiihxx ,0 of the i th strip. The mathematical 7
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equations for the constants when Youngs modulus and Poissons ratio are varying exponentially across the width are then given by iiiihEE01ln (10) (11) iia00 ihiiiaehbii010 (12) where = Width of i ih th layer = Youngs modulus of the i iE0 th layer at 0 ix = Youngs modulus of the i iE0 th layer at iihx = Poissons ratio of the i i0 th layer at 0 ix = Poissons ratio of the i i1 th layer at iihx The equations for stress and displacement fields for the homogeneous half plane are given by (Gupta, 1973) The displacement in ydirection is given by dyexcccyxuxcos2112),(30323233133 (13) The displacement in ydirection is given by 8
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dyexcccyxxsin2112),(30323233133 (14) The stress field is given by ydecxcyxxxxcos22),(3032331333 (15) ydecxcyxxyycos)2(22),(3032331333 (16) ydecxcyxxxysin)1(22),(3032331333 (17) where and shear modulus 3 3 are given by 3343 33312E = Poissons ratio of halfplane, 3 = Youngs modulus of halfplane. 3E 2.3 Continuity conditions When we use equations (3)(8) to layer 1 and layer 2 ,we have eight unknown functions, , )(11c )(12c )(13c )(14c 21c 22c 23c ,and From Equations (13)(17) for halfplane we have two more unknown functionsand To find stresses and displacements we need to solve for ten unknown functions present in the equations (38) and (1317) ,using continuity and boundary conditions. 24c )(31c )(32c 9
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Continuity conditions are applied to stress and displacement equations (110) to get relationships between ten unknown functions , , )(11c )(12c )(13c )(14c 21c , ,)and 22c 23c 24c (31c )(32c The continuity conditions at the interface 11hx or 02 x of layer 1 and layer 2 are given by (18) ),0(),(211yyhxxxx (19) ),0(),(211yyhxyxy (20) ),0(),(211yuyhu (21) ),0(),(211yyh The above continuity conditions provide four equations relating the variables. We get another four equations by applying continuity conditions at the interface between layer 2 and homogeneous halfplane, and are given by (22) ),0(),(322yyhxxxx (23) ),0(),(322yyhxyxy (24) ),0(),(322yuyhu (25) ),0(),(322yyh Finally, we get two more equations relating unknown functions by applying boundary conditions to the stress and displacement field equations. 10
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2.4 Boundary conditions The boundary conditions on the top surface of the model are given by (26) 0),0(1yxy )(),0(1ypyxx aya (27) 0),0(1yxx ay where, is the applied pressure distribution, )(yp is the length over which the load is applied. a2 The two boundary conditions (2627) along with eight continuity conditions (1825) can be used to solve for ten unknown functions , , )(11c )(12c )(13c )(14c 21c 22c , ,)and 23c 24c (31c )(32c 2.5 Types of loads In this analysis we have used three different types of load distributions representing different types of indentors. Flat Indentor: The pressure distribution for a flat indentor is given by )(yp 11
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ayayaLyp,)(22 (28) where, = the applied load, L = the loading length. a2 Smooth Indentor: The pressure distribution for a smooth indentor is given by )(yp ayayaaLyp ,2)(222 (29) where, is the applied load, L is the loading length. a2 Uniform pressure: The uniform pressure distribution is given by ayaaLyp,2)( (30) where, = the applied load, L = the loading length. a2 2.6 Derivation of solution The ten equations (1827) resulting from continuity and boundary conditions can be arranged in the matrix form as 12
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CBA where, is the coefficient matrix, A is the matrix of unknown functions, B is the right hand side array. C The above matrix form can be solved numerically for unknown functions. These functions are substituted in the stress and displacement field equations (38) and (1317), and these equations are solved numerically to get stresses and displacements at any given point in the model. 13
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CHAPTER 3 DESIGN OF EXPERIMENTS 3.1 Introduction Design of experiments is a scientific way of planning the experiments involving more than one factor so that appropriate data that can be analyzed using statistical techniques is collected. Statistical analysis of collected data is important to reach valid conclusions. Since any valid scientific research involves experiments and statistical analysis of data collected from experiments, design of experiments involving multiple factors is an integral part of scientific study. In this chapter we discuss 2 k factorial design (Montgomery, 2001) which we used in our study. 3.2 Factorial designs Factorial designs are used to study the combined effects of several factors on a response. Most experiments have two or more factors involved. Factorial designs are more useful for experiments involving more than two factors. There are several special cases of general factorial design that are used in research work because they form the basis for other designs of considerable practical importance. The most important special case of general factorial design is that of k factors, each at two different levels. This special case requires observations and is calledfactorial design. k22......222 k2 14
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3.3 Factorial design k2 Factorial design is a design with k factors each at two different levels. The model includes k main effects, twofactor interactions, three factor interactions, and one factor interaction. There are k2 2k 3k k 12 k total number of effects in a factorial design. For example, a factorial design has k2 32 123 total number of effects. In our study we have four important factors, so factorial design is appropriate for our case. The following section discusses factorial design in detail. 42 42 3.4 Factorial design 42 Let A, B, C, D are four main factors involved in the experiment. The total number of observations or runs required is given by. Each factor has two different levels indicated by and 1624 One level is indicated by where as represents other level. There are total number of effects in factorial design, they are 124 Main effects: A, B, C, D Two factor interactions: AB, AC, AD, BC, BD, and CD Three factor interactions: ABC, ABD, ACD, and BCD Four factor interaction: ABCD. Table 1 shows the 16 runs or observations required for design. 42 15
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Table 1: Notations for experimental combinations Run No A B C D Run label Response 1 (1) Data1 2 + a Data2 3 + b Data3 4 + + ab Data4 5 + c Data5 6 + + ac Data6 7 + + bc Data7 8 + + + abc Data8 9 + d Data9 10 + + ad Data10 11 + + bd Data11 12 + + + abd Data12 13 + + cd Data13 14 + + + acd Data14 15 + + + bcd Data15 16 + + + + abcd Data16 Table 2 below shows the contrast constants for the design. 42 Table 2: Contrast constants for design 42 Run label A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD (1) + + + + + + + a + + + + + + + b + + + + + + + ab + + + + + + + c + + + + + + + ac + + + + + + + 16
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Table 2: Continued bc + + + + + + + abc + + + + + + + d + + + + + + + ad + + + + + + + bd + + + + + + + abd + + + + + + + cd + + + + + + + acd + + + + + + + bcd + + + + + + + abcd + + + + + + + + + + + + + + + The contrast constants for interaction effects shown are obtained by multiplying the contrast constants of individual effects. For example, the contrast constant for interaction effect BC for run (1) is + because both B and C has contrast constant for run (1). Contrast: The next step is to find contrasts from contrast constants. Contrast for each effect is obtained by multiplying the contrast column of each effect with the response column in Table 1 and then taking the sum of the elements of the resulting column. For example, for factor A contrast is obtained by multiplying column A in Table 2 with response column in Table 2 and adding all the elements of resulting column. So contrast for A is given by Contrast A = (Data1+Data2Data3+Data4Data5+Data6Data7+Data8Data9+Data10Data11+Data12Data13+Data14Data15+Data16) 17
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Similarly we can find Contrasts for all the 124 effects. Sum of squares: The sum of squares for the effects are calculated using the following formulae. nContrastSS422 where, n = number of runs. Total sum of squares: The total sum of squares is obtained by adding the individual sum of squares of effects. ABCDCBATSSSSSSSSSS .......... The percentage contribution of each of the effects is obtained by taking the ratio of sum of squares of effect to total sum of squares and then multiplying the result with 100. For example, the percentage contribution of effect A is given by Percentage contribution of A 100TASSSS The factor having the highest percentage contribution is said to have the most effect on the experiment. 18
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CHAPTER 4 RESULTS 4.1 Introduction In this chapter we present the results obtained from FORTRAN program and statistical analysis. The layersubstrate indentation model discussed in chapter 2 can be solved mathematically for stresses and displacements using the formulation described in chapter 2. Our objective is to study the effect of nonhomogeneous interface between homogeneous layer and homogeneous substrate on contact depth at the surface, maximum normal stress and maximum shear stress at the interface and to quantify the effect of various parameters on the results using statistical analysis. To achieve the above purpose we have used the formulation presented in chapter 2. Layer 1 in the model described in chapter 2 is our homogeneous layer and Layer 2 can be used as either nonhomogeneous or homogeneous interface between homogeneous layer and homogeneous substrate represented by homogeneous halfplane. 4.2 Contact depth Contact depth, is the depth through which the indentor is in contact with the material. It is obtained from the model by taking the difference of displacements at at and. d 0x 0y ay 19
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)0,0(),0(uaud We have obtained results for contact depth for both homogeneous and d nonhomogenous interface for all the three types of indentors, for different Youngs modulus ratio and for different indentor widthlayer thickness 1ha ratio. Below are the values we have chosen for each parameter. Youngs modulus ratio, 20015:31EE 12015 6015 1, 1560 15120 15200 Indentor widthlayer thickness ratio ,61:1ha ,31 21 We had to choose Youngs modulus ratio between 20015 and 15200 to keep the exponentially varying Poissons ratio below 0.5. Fig 2, Fig 4, and Fig 6 show how normalized vertical displacement difference LhEauyu112)0()( varies with distance from indentor axis along surface for uniform indentor, flat indentor and spherical indentor, respectively. The figures contain plots for both nonhomogeneous and homogeneous interface. Fig 3, Fig 5, and Fig 7 below are contact depth ratio hnhdd as a function of Youngs modulus ratio plots for uniform indentor, flat indentor, and spherical indentor, respectively. 20
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0.080.060.040.02000.040.080.12distance from indentor axis, y (u(y)u(0))(a2E1)/(h1L)h nh Figure 2: Normalized vertical displacement difference (u(y)u(0))(a 2 E 1 )/(h 1 L) as a function of distance along top surface from indentor axis for uniform pressure 0.9981.0001.0021.0041.0061.00802468101214 a/h=0.167 a/h=0.34 a/h=0.5 Y oun g 's modulus ratio, E 1/E3Contact depth ratio, dnh/dh Figure 3: Contact depth ratio as a function of Youngs modulus ratio between layer and substrate for uniform pressure 21
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0.0080.00400.0040.0080.01200.050.10.15Distance from indentor axis, y (u(y)u(0))(a2E1)/(h1L)nh h Figure 4: Normalized vertical displacement difference, (u(y)u(0))(a 2 E 1 )/(h 1 L) as a function of distance along top surface from indentor axis for flat indentor 0.0000.5001.0001.5002.000024681012 14 Contact depth ratio, dnh/dh Y oun g 's modulus ratio, E 1/E3 a/h=0.334a/h=0.167 a/h=0.5 Figure 5: Contact depth ratio as a function of Youngs modulus ratio between layer and substrate for flat indentor 22
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0.080.060.040.02000.050.10.15Distance from indentor axis, y (u(y)u(0))(a2E1)/(h1L)nh h Figure 6: Normalized vertical displacement difference (u(y)u(0))(a 2 E 1 )/(h 1 L), as a function of distance along top surface from indentor axis for spherical indentor 1.0001.0021.0041.0061.00802468101214 a/h=0.167a/h=0.334 a/h=0.5 Y oun g 's modulus ratio, E 1/E3Contact depth ratio, dnh/dh Figure 7: Contact depth ratio as a function of Youngs modulus ratio between layer and substrate for spherical indentor 23
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We can see from Fig 3, Fig 5, and Fig 7 the contact depth ratio homddnon is close to1.0 except for flat indentor where the ratio is close to 2.0 for values of 31EE close to zero and the ratio increases with increase in 1ha ratio. So there is no significant difference between contact depth values for homogeneous and nonhomogeneous interface either for soft layer on hard substrate, 0.131EE or for hard layer on soft substrate, 0.131EE Also, from Fig 2, Fig 4 and Fig 6 one can notice that contact depth curves for homogeneous and nonhomogeneous interface are overlapping each other. So, we can conclude that for 33.13031EE contact depth results are not effected by nonhomogeneous nature of interface. 4.3 Normal stress The maximum normal stress maxxx at the interface should be maintained below a critical value to avoid debonding between layer and substrate. We have obtained results for maximum normal stress at the layersubstrate interface maxxx for both homogeneous and nonhomogenous interface for all the three types of indentors, for different Youngs modulus ratio and for different 1ha ratio. Below are the values we have chosen for each parameter. 24
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Youngs modulus ratio, 20015:31EE 12015 6015 1, 1560 15120 15200 Indentor widthlayer thickness ratio, ,61:1ha ,31 21 Fig 8, Fig 10, and Fig 12 show how normalized interface normal stress xx (a 2 )/L changes with distance from indentor axis along interface for uniform pressure, flat indentor, and spherical indentor, respectively. The curves represent normal stress as a function of distance for homogeneous and nonhomogeneous interface. Fig 9, Fig 11, and Fig 13 below are plots for maximum normal stress ratio as a function of Youngs modulus ratio for uniform indentor, flat indentor, and spherical indentor respectively. The three different curves represent three different ratios between indentor width and layer thickness. 0.120.10.080.060.040.0200.050.10.150.20.25Distance from indentor axis, yNormalized interface normal stress, xx(a2)/Lnh h Figure 8: Normalized interface normal stress, xx (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for uniform pressure 25
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0.991.001.011.02051015 Maximum normal stress ratio, xx(nh)/xx(h) Y oun g 's modulus ratio, E 1/E3a/h=0.17 a/h=0.34 a/h=0.5 Figure 9: Maximum normal stress ratio along interface as a function of Youngs modulus between layer and substrate for uniform pressure 0.10.080.060.040.0200.050.10.150.20.25Distance from indentor axis, yNormalized interface normal stress, xx(a2)/Lnh h Figure 10: Normalized interface peel stress, xx (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for flat indentor 26
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0.9911.011.02051015 Maximum normal stress ratio, xx(nh)/xx(h) Y oun g 's modulus ratio, E 1/E3a/h=0.17 a/h=0.34a/h=0.5 Figure 11: Maximum normal stress ratio along interface as a function of Youngs modulus between layer and substrate for flat indentor 0.120.10.080.060.040.02000.050.10.150.20.25Distance from indentor axisNormalized interface normal stress, xx(a2)/Lh nh Figure 12: Normalized interface peel stress, xx (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for spherical indentor 27
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0.9911.011.020510 15 Maximum normal stress ratio, xx(nh)/xx(h)Young's modulus ratio,E1/E3a/h=0.17 a/h=0.34 a/h=0.5 Figure 13: Maximum normal stress ratio along interface as a function of Youngs modulus between layer and substrate for spherical indentor W e can see from the above plots the maximum normal stress ratio )()(hnhxxxx is close to 1.0 and the ratio increases with increase in 1ha ratio. So there is no significant difference between maximum normal stress ratio values for both homogeneous and nonhomogeneous interface either for soft layer on hard substrate 0.113EE or for hard layer on soft substrate 0.113EE Again for normal stress as a function of distance curves for homogeneous and nonhomogeneous interface are overlapping each other as shown in Fig 8 Fig 10, and Fig 12 So, we can conclude that for 33.13031EE nonhomogeneous interface does not have significant effect on normal stress results. 28
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4.4 Shear stress Shear stress at the interface should be maintained below a critical value to prevent debonding between layer and substrate. We have obtained results for maximum shear stress at the layersubstrate interface xy for both homogeneous and non homogenous interface for all the three types of indentors, for different Youngs modulus ratio and for different 1ha ratio. Below are the values we have chosen for each parameter. 15200,15120,1560,1,6015,12015,20015:13EE 21,31,61:1ha Fig 14, Fig 16, and Fig 18 are plots showing how normalized interface shear stress, xy (a 2 )/L changes with distance from indentor axis along interface. Fig 15, Fig 17, and Fig 19 below are maximum shear stress ratio as a function of Youngs modulus ratio plots for uniform indentor, flat indentor, and spherical indentor, respectively. 0.0240.0200.0160.0120.0080.0040.00000.050.10.150.20.25Distance from indentor axis, yNormalized interface shear stress, xy(a2)/Lh nh Figure 14: Normalized interface shear stress, xy (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for uniform pressure 29
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0.9811.021.041.0602468101214 Maximum shear stress ratio, xy(nh)/xy(h) Y oun g 's modulus ratio, E 1/E3a/h=0.17 a/h=0.34 a/h=0.5 Figure 15: Maximum shear stress ratio along interface as a function of Youngs modulus between layer and substrate for uniform pressure 0.0240.0200.0160.0120.0080.0040.00000.050.10.150.20.25Distance from indentor axis, yNormalized interface shear stress, xy(a2)/Lh nh Figure 16: Normalized interface shear stress, xy (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for flat indentor 30
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0.9811.021.041.0602468101214 Maximum shear stress ratio, xy(nh)/xy(h) Y oun g 's modulus ratio, E 1/E3a/h=0.17 a/h=0.34 a/h=0.5 Figure 17: Maximum shear stress ratio along interface as a function of Youngs modulus between layer and substrate for flat indentor 0.0250.0200.0150.0100.0050.00000.050.10.150.20.25Distance from indentor axis, yNormalized interface shear stress, xy(a2)/Lh nh Figure 18: Normalized interface shear stress, xx (a 2 )/L at the layersubstrate interface as a function of distance along interface from indentor axis for spherical indentor 31
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0.9811.021.041.06051015 Maximum shear stress ratio, xy(nh)/xy(h) Y oun g 's modulus ratio, E 1/E3a/h=0.17 a/h=0.34 a/h=0.5 Figure 19: Maximum shear stress ratio along interface as a function of Youngs modulus between layer and substrate for spherical indentor We can see from the above plots the maximum shear stress ratio hnhxyxy is close to 1.0 and the ratio increases with increase in 1ha ratio. So there is no significant difference between maximum shear stress values for homogeneous and nonhomogeneous interfaces either for soft layer on hard substrate 0.113EE or for hard layer on soft substrate 0.113EE We can also see from Fig 14, Fig 16, and Fig 18 that there is no significant difference between homogeneous and nonhomogeneous interface curves. So, we can conclude for 33.13031EE nonhomogeneous interface does not effect shear stress results at the interface significantly. 32
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4.5 Sensitivity analysis The second part of study deals with using statistical analysis to quantify the effect of various factors on contact depth ratio, maximum normal stress ratio, and maximum stress ratio. For this purpose we have designed experiments using factorial design discussed in previous chapter. 42 The four factors we have chosen for factorial design in our study are 42 A) Type of indentor, B) 12hh ratio, C) 1ha D) Youngs modulus ratio 31EE The above factors are chosen at two different levels. Table 3 shows the two levels chosen for each factor to execute factorial design. 42 Table 3: Values of two different levels of the factors Factor Symbol Level 1 Level 2 Type of indentor A Flat indentor Spherical indentor 12hh B 05.0 15.0 1ha C 1667.0 5.0 31EE D 20015 15200 33
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4.6 Calculations Using the above levels for factors as input we have calculated using the method described in previous chapter the percentage contribution of each factor by itself and together with other factors to a response. Table 4, Table 5, and Table 6 below present percentage contribution of factors AType of indentor, Bratio between interface width and layer width, Cratio between indentor width and layer width, DYoungs modulus ratio and their combinations to contact depth ratio, maximum normal stress ratio, and maximum shear stress ratio respectively. Table 4: Percentage contribution of factors to contact depth ratio Factor Effect Estimate Sum of Squares Percentage Contribution A 0.014160791 0.01203168 16.66 B 0.002276597 0.000310974 0.43 C 0.014212838 0.012120286 16.78 D 0.013699431 0.011260465 15.59 AB 0.002096113 0.000263621 0.36 AC 0.013957742 0.011689113 16.18 AD 0.013795987 0.011419756 15.81 BC 0.002096113 0.000263621 0.36 BD 0.001850729 0.000205512 0.28 CD 0.002096113 0.000263621 0.36 ABC 0.001994151 0.000238598 0.33 ABD 0.001901632 0.000216972 0.30 ACD 0.01383597 0.011486044 15.90 BCD 0.001890864 0.000214522 0.29 ABCD 0.00192271 0.000221809 0.30 34
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Table 5: Percentage contribution of factors to maximum normal stress ratio Factor Effect Estimate Sum of squares Percentage Contribution A 0.0000823695 0.0000004071 0.39 B 0.0006275050 0.0000236258 22.71 C 0.0002036027 0.0000024872 2.39 D 0.0009689179 0.0000563281 54.15 AB 0.0000458643 0.0000001262 0.12 AC 0.0000606182 0.0000002205 0.21 AD 0.0000648704 0.0000002525 0.24 BC 0.0000458643 0.0000001262 0.12 BD 0.0005713832 0.0000195887 18.83 CD 0.0000458643 0.0000001262 0.12 ABC 0.0000345889 0.0000000718 0.069 ABD 0.0000341496 0.0000000700 0.067 ACD 0.0000477291 0.0000001367 0.13 BCD 0.0000830774 0.0000004141 0.39 ABCD 0.0000260699 0.0000000408 0.039 Table 6: Percentage contribution of factors to maximum shear stress ratio Factor Effect Estimate Sum of squares Percentage Contribution A 0.00002778 0.0000000463 0.021 B 0.0013148942 0.0001037368 47.06 C 0.0000566192 0.0000001923 0.087 D 0.0010163401 0.0000619768 28.11 AB 0.0000916392 0.0000005039 0.22 AC 0.0000441751 0.0000001171 0.053 AD 0.0000467626 0.0000001312 0.059 BC 0.0000916392 0.0000005039 0.22 BD 0.0008835317 0.0000468377 21.24 35
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Table 6: Continued CD 0.0000916392 0.0000005039 0.22 ABC 0.0000751591 0.0000003389 0.15 ABD 0.0000674468 0.0000002729 0.12 ACD 0.0000233928 0.0000000328 0.014 BCD 0.0002831228 0.0000048095 2.18 ABCD 0.0000827806 0.0000004112 0.18 From Table 5 we can see that the contribution of individual factors A (Type of load), C (Indentor width to layer width ratio,a/h 1 ) and D (Youngs modulus ratio E 3 /E 1 ) to contact depth ratio response is significant and almost equal (close to 15%) where as contribution from B ( Interface width to layer width ratio, h 2 /h 1 ) is minimum(0.43%). So, we can conclude that contact depth ratio response equally depends on A (Type of indentor), C (Indentor width to layer width ratio, a/h 1 ), D (Youngs modulus ratio, E 3 /E 1 ) and the influence of B (Interface width to layer width ratio, h 2 /h 1 ) on contact depth ratio is minimum. From Table 6 we can see that maximum normal stress ratio response has largest contribution from D54.15% (Youngs modulus ratio, E 3 /E 1 ) followed by B22.71% (Interface width to layer width ratio, h 2 /h 1 ).The contributions of A (Type of load) and C (Indentor width to layer width ratio, a/h 1 ) are 0.39% and 2.39%, respectively. So, we can conclude that D (Youngs modulus ratio, E 3 /E 1 ) is the most significant factor and A (Type of load) is the least significant factor for maximum normal stress ratio. From Table 7 we can see that B (Interface width to layer width ratio, h 2 /h 1 ) followed by D (Youngs modulus ratio, E 3 /E 1 ) with contributions 47.06% and 28.11%, 36
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respectively are significant factors for maximum shear stress response and contributions from A (Type of load) and C (Indentor width to layer width ratio, a/h 1 ) are 0.027% and 0.087%, respectively. So, we can conclude that B (Interface width to layer width ratio, h 2 /h 1 ) is the most significant factor and A (Type of load) is the least significant factor for normal shear stress response. 37
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CHAPTER 5 CONCLUSIONS The objective of this study is to study the effect of various parameters involved in layersubstrate indentation experiments on final results. We have shown that either for soft layer on hard substrate or for hard layer on soft substrate the results for contact depth, maximum normal stress and maximum shear stress are almost same for homogeneous change of Youngs modulus at the interface and nonhomogeneous change of Youngs modulus at the interface. So, we cannot conclude whether the interface is homogeneous or nonhomogeneous using contact depth results from layersubstrate indentation experiments for Youngs modulus ratio less than 13.33 33.1331EE Also, using statistical analysis we have quantitatively found the effect of various factors on contact depth ratio, maximum normal stress ratio and maximum shear stress ratio. Youngs modulus ratio followed by ratio between interface thickness and layer thickness have major impact on both maximum normal stress ratio and maximum normal stress ratio where as the impact of type of load and indentor widthlayer thickness ratio is insignificant. Also, Youngs modulus ratio, indentor widthlayer thickness ratio and type of load have equal and significant impact on contact depth ratio where as ratio between interface thickness and layer thickness has no significant impact on contact depth ratio. 38
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Although, conclusions on the effect of nonhomogeneous interface are based on restricted choice of Youngs modulus ratio, our study is a good start in characterization of interface which is important in design and deposition of thin films on substrates. In the next part of our study we propose to overcome the limitations on Youngs modulus ratio by modeling the interface as multiple nonhomogeneous thin sub strips with Youngs modulus and Poissons ratio varying exponentially across each sub strip thickness. Further, sensitivity analysis provides valuable information about the impact of various parameters on indentation results which can be used in design and mechanical characterization of layersubstrate combinations. 39
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REFERENCES Chudoba, T., Schwarzer, N. and Richter, F. Determination of elastic properties of thin films by indentation measurements with a spherical indenter. Surface and Coatings Technology 127 (2000), P. 917. Chudoba, T., Schwarzer, N., Linss, V., Ritcher, F. Determination of mechanical properties of graded coatings using nanoindentation. Thin Solid Films 469470 (2004), P. 239247. Delale, F. and Erdogan, F. On the mechanical modeling of the interfacial region in bonded halfplanes. ASME Journal of Applied Mechanics 55 (1988), P. 317324. Doerner, M.F. and Nix, W.D. A method for interpreting the data from depth sensing indentation instruments. Journal of Materials Research 1 (1986), P. 601609. Gupta, G.D. A layered composite with a broken laminate. International Journal of Solids and Structures 9 (1973) 11411154. Kaw, A.K., Selvarathinam, A.S. and Besterfield, G.H. Comparison of interphase models in a fracture problem in fiber reinforced composites. Journal of Theoretical and Applied Mechanics 17 (1992), P. 133147. Linss, V., Schwarzer, N., Chudoba, T., Karniychuk, M., Richter F. (2004). Mechanical properties of a graded BCN sputtered coating with varying Youngs modulus: deposition, theoretical modeling and nanoindentation. Surface and Coatings Technology 195 (2005), P. 287297. Montgomery, D.C. 2001, Design and analysis of experiments, John Wiley & Sons, Inc. New York. Oliver, W.C., Hutchings, R., and Pethica, J.B. Microindentation Techniques in materials science and engineering. ASTM STP 889, edited by P.J. Blau and B.R. Lawn. American Society for Testing and Materials (1986), P. 90108. Oliver, W.C and Pharr G.M. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal Materials Research 7 (1992), P. 15641583. 40
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Shaohua, Lei, Tzuchiang. Investigation of mechanical properties of thin films by nanoindentation, considering the effects of thickness and different coatingsubstrate combinations. Surface and Coatings Technology. 191 (2005), P. 2532. 41
