Modeling size effects in nano indentation of polymers

MODELING SIZE EFFECTS IN NANO-INDENTATION OF POLYMERS POH LEONG HIEN NATIONAL UNIVERSITY OF SINGAPORE 2005 MODELING SIZE EFFECTS IN NANO-INDENTATION OF POLYMERS POH LEONG HIEN [B.Eng (Hons), NUS] A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 Acknowledgements The report was done with much advice and understanding from my project supervisors, Assoc Prof Swaddiwudhipong, Somsak and Assoc Prof Tam Chat Tim Their guidance and suggestions are greatly appreciated I am also grateful to Dr Hua Jun and Mr Tho Kee Kiat for their invaluable help throughout the study Last but not least, I would like to express my gratitude to the many academic, technical and administrative staffs for their assistance i Table of Contents Page Acknowledgements i Table of Contents ii List of Tables iv List of Figures v Abstract vii Introduction and Literature Review 1.1 Introduction 1.2 Literature Review 1.2.1 Constitutive model for polymeric materials 1.2.2 Size effects of crystalline materials 1.2.3 Size effects of polymeric materials 1.3 Scope of Investigation Theory 2.1 Constitutive models 9 2.1.1 Constitutive model for glassy polymers 2.1.2 Constitutive model for epoxy polymer 11 2.2 Strain gradient plasticity 13 2.2.1 Glassy polymers 13 2.2.2 Epoxy polymer 16 2.3 Constitutive models with strain gradient plasticity 17 ii 2.4 Obtaining effective strain gradient parameter 19 2.4.1 Effective strain gradient from indentation geometry 20 2.4.2 Effective strain gradient from nodal displacements 20 Numerical examples and discussion of results 3.1 Glassy polymers 24 24 3.1.1 Spherical indentation on polystyrene 25 3.1.2 Load controlled vs Displacement controlled method 28 3.1.3 Berkovich indentations at submicron level 29 3.2 Epoxy polymer 37 3.2.1 Uniaxial Compression 38 3.2.2 Berkovich indentations of epoxy polymer 40 Conclusions and Recommendations 46 4.1 Conclusions 46 4.2 Recommendations 48 References 49 iii List of Tables Table Page 3.1 Material parameters for polystyrene [van Melick et al 2003a] 26 3.2 Material parameters for polycarbonate [Govaert et al 2000] 30 3.3 Material parameters for epoxy 39 iv List of Figures Figure Page 1.1 Depth dependence of the hardness of PMMA [Balta Calleja et al 2004] 2.1 Uniaxial compression loading curve for epoxy [Xia et al 2003] 12 2.2 Compliance curves at different stress levels [Tervoort et al 1996] 18 2.3 Schematic diagram showing parameters in indentation tests 20 3.1 Finite element model of spherical indentation 27 3.2 Comparison of results from spherical indentations on polystyrene based on tests conducted by van Melick et al [2003a] 27 3.3 Comparison between load controlled (0.01 N/s) and displacement controlled numerical analysis 29 3.4 Finite element model of Berkovich indentation 31 3.5 Berkovich indentation numerical results for different domain sizes 32 3.6 Convergence study of finite element model simulating Berkovich indentation 32 3.7 Berkovich indentation on polycarbonate conducted by Chong and Lam [1999] at 100s loading time 34 3.8 Berkovich indentation on polycarbonate conducted by Lu et al [2003] at 158s loading time 34 3.9 Berkovich indentation on polycarbonate conducted by Bucaille et al [2002] at 10s loading time 35 3.10 Berkovich indentation on polycarbonate conducted by Bucaille et al [2002] at 1000s loading time 35 3.11 Depth dependence of the hardness of polycarbonate [Lam and Chong 1999] 36 3.12 Comparison between numerical analysis and experimental data [Lam and 39 Chong 2001] for uniaxial compression of epoxy v 3.13 Berkovich indentation numerical results for different domain sizes 41 3.14 Convergence study of finite element model simulating Berkovich indentation 41 3.15 Berkovich indentation conducted by Dutta et al [2004] at 10s loading time 42 3.16 Berkovich indentation conducted by Lam and Chong [2001] at 100s loading time 44 3.17 Strain gradient modulus Mg as a function of strain, ε0 is the uniaxial yield strain, n = 0.65 [Lam and Chong 2001] 44 3.18 Hardness of epoxy with depth [Lam and Chong 2001] 45 vi Abstract Modeling Size Effects in Nano-indentation of Polymers Conventional methods such as uniaxial tests used to determine mechanical properties are not feasible for those involving small volume of materials As such, in recent years, material characterization using indentation techniques at the micron and submicron levels have gained popularity It has been widely reported that when deformations are induced in the submicron level, the materials display strong size effects which alter the mechanical properties from their bulk characteristics Classical plasticity theory is unable to account for this phenomenon Strain gradient plasticity theory is proposed and it has been shown to successfully capture the size effects of various materials The present study adopts constitutive models for the viscoelastic-plastic deformation of glassy and epoxy polymers These models are then implemented in the commercial general purpose finite element package ABAQUS, via user subroutines It is demonstrated that strain gradient effects has to be considered into the adopted constitutive models in order to better describe the material response of glassy and epoxy polymers at submicron indentations The study also covers the two approaches of deriving the values of effective strain gradient via indentation geometry and directly from finite element nodal displacement parameters Comparison of results obtained from both approaches show good agreement with existing experimental values in all cases covered in the present study The latter, as expected, provides slightly more accurate solutions as vii compared to the test results than the former but at a marginally higher computing time and resources Keywords: Size effects; Nano-indentation; Glassy polymers; Epoxy; Strain gradient plasticity; Finite element method viii Parameter Value E 2200 MPa v 0.37 τ0 1.3 MPa α 0.1 γ0 1.8x10-18 s-1 GR 28 MPa Table 3.3 – Material parameters for epoxy 100 True stress (Mpa) 80 60 40 numerical analysis 20 experiment [Lam and Chong 2001] 0 0.02 0.04 0.06 0.08 True strain Figure 3.12 – Comparison between numerical analysis and experimental data [Lam and Chong 2001] for uniaxial compression of epoxy 39 3.2.2 Berkovich indentations of epoxy polymer Simulated Berkovich indentation on epoxy polymer is studied in this section Numerical analysis is carried out at the micron level and results compared with existing experimental data to demonstrate the applicability of the constitutive model in the indentation experiments Subsequently, simulations at submicron level with and without size effects are carried out and the results presented and discussed The finite element model for Berkovich indentation in section 3.1.3 is employed in this section Far field effect and convergence study are carried out using the epoxy polymer constitutive relations to ensure that the obtained numerical results are mesh independent and representative of semi-infinite target domain The finite element mesh for the target body employed in this section comprises 2842 second order C3D20/C3D27 solid elements over the domain sizes of 115 x 150 x 200 and 173 x 225 x 300 (µm)3 As demonstrated in Fig (3.13), the size of the domain used in the simulation is insensitive to the far-field effect for each target body consisting of 2842 elements For the domain size 115 x 150 x 200 (µm)3 adopted in the present study, numerical values obtained from analyses employing 1848 and 2842 solid elements demonstrate negligible differences as depicted in Fig (3.14), suggesting the convergence of the results The more refined mesh size is used to ascertain that only the converged values are presented in this study 40 2.50 115x150x200 173x225x300 Force (mN) 2.00 1.50 1.00 0.50 0.00 0.2 0.4 0.6 0.8 Indentation depth (µm) Figure 3.13 – Berkovich indentation numerical results for different domain sizes 3.00 2842 elements 1848 elements Force (mN) 2.50 2.00 1.50 1.00 0.50 0.00 0.2 0.4 0.6 0.8 Indentation depth (µm) Figure 3.14 – Convergence study of finite element model simulating Berkovich indentation 41 Berkovich indentation on epoxy polymer carried out earlier by Dutta et al [2004] is simulated The unloading portion of the load indentation depth curve is not available and hence not included As demonstrated in Fig (3.15), the numerical results agree reasonably well with experimental values This suggests that the adopted constitutive model is capable of capturing epoxy polymer behavior in indentation experiments at the micron level 100 experiment [Dutta et al 2004] Load (mN) 80 numerical analysis 60 40 20 0 Indentation Depth (µm) Figure 3.15 – Berkovich indentation conducted by Dutta et al [2004] at 10s loading time Simulation of Berkovich indentation at submicron level is carried out and results compared with experimental data from Lam and Chong [2001] as depicted in Fig (3.16) When strain gradient plasticity is not considered, the force at maximum indentation depth estimated from numerical analysis is lower than the experimental value This suggests 42 that the hardness of the target polymer obtained from finite element analysis is lower compared with that derived from indentation test The strain gradient plasticity discussed in section 2.2.2 is incorporated into the constitutive model Using Power law to model the stress strain behavior of epoxy, Lam and Chong [2001] obtained the value of 0.65 for parameter n As depicted in Fig (3.17), correlation with these experimental data based on the relation given earlier in Eq (2.25) yields the following relationship: M g = 93 exp( ε= − 10.25ε ε0 ) 0.65 ε ⋅ε (3.1) (3.2) where yield strain ε0 is assumed to be 0.035 and ε is taken as the effective strain As demonstrated in Fig (3.16), it is apparent that the incorporation of strain gradient plasticity in constitutive equations results in better agreement between the experimental and numerical results This suggests the existence of strain gradient effects in epoxy which is consistent with experimental findings from Lam and Chong [2001] shown in Fig (3.18), where size effects are dominant in epoxy polymer for indentations up to approximately 1.2 µm 43 3.5 experiment [Lam and Chong 2001] Load (mN) FEM without strain gradient 2.5 FEM with strain gradient 1.5 0.5 0 0.2 0.4 0.6 0.8 1.2 Indentation depth (µm) Figure 3.16 – Berkovich indentation conducted by Lam and Chong [2001] at 100s loading time 0.004 Mg=93exp (-10.25ε / ε0)0.65 Mg 0.0032 0.0024 0.0016 0.0008 0.99 1.01 1.02 1.03 1.04 1.05 (ε / ε0)0.65 Figure 3.17 – Strain gradient modulus Mg as a function of strain, ε0 is the uniaxial yield strain, n = 0.65 [Lam and Chong 2001] 44 350 330 Hardness (MPa) 310 290 270 250 230 210 190 170 0.5 1.5 2.5 3.5 Depth (µm) Figure 3.18 – Hardness of epoxy with depth [Lam and Chong 2001] As shown in the Fig (3.18), at 0.9µm, the hardness of epoxy is only marginally higher than that at large indentation displacement Thus, this may be a reason why there is only a slight improvement after incorporating the strain gradient plasticity depicted in Fig (3.16) since the indentation displacement of 0.9µm is near the threshold depth where hardness is independent of strain gradient effects 45 Conclusions and Recommendations 4.1 Conclusions Although material characterization using indentation techniques are gaining popularity in recent years, numerical studies of such experiments on polymers are lacking To further complicate the situation, from studies of crystalline materials, it has been widely reported that when deformations are induced in the submicron level, size effects are dominant which alter the mechanical properties from the bulk characteristics The first portion of the report investigates the size effects of glassy polymers A constitutive model for glassy polymers is adopted and implemented as user subroutines in ABAQUS, a finite element package Numerical analyses of indentations at micron and submicron levels are carried out to verify the applicability of the model in describing the viscoelastic-plastic behavior of polymers Substantial discrepancies between the obtained numerical results and experimental data are observed for indentations at submicron level when the strain gradient effects are not accounted for in the former However, the discrepancies diminish when the strain gradient effects are included in the constitutive model This provides a strong evidence of the presence of strain gradient effects in glassy polymers at submicron level and cannot be ignored without loss of significant accuracy This observation is also in agreement with various other experimental studies where the hardness of polymers is observed to increase with decreasing indentation depth 46 The effective strain gradient values are obtained from explicitly evaluating displacement shape functions from numerical analyses as well as approximated from indenter geometry and indentation depth and the two methods compared It is demonstrated that deriving effective strain gradients from nodal displacements provide better agreements with experimental data The trade-off of using the more accurate method, however, is the slightly longer computational time compared with that of the approximation method The second part of the study extends the investigation of size effects to epoxy polymer A constitutive model for epoxy polymer is adopted and implemented via the user subroutine in ABAQUS, a finite element package Numerical analyses of uniaxial compression and indentations in the micron range are carried out to verify the applicability of the adopted constitutive model in describing the behavior of epoxy polymer When implemented in the numerical analysis of submicron indentation, the incorporation of strain gradient effects into the adopted constitutive model gives rise to better agreement between numerical results and experimental data This suggests the presence of strain gradient effects in epoxy polymer 47 4.2 Recommendations The current investigation focuses on the size effects in submicron Berkovich indentations This may lead to speculations that the observed phenomena are indenter induced It would be appropriate to extend the study to include other indenter shapes such as the Vickers, Brinell, Boussinesq and Knoop This is so as to further confirm the dominance of strain gradient in submicron indentations The strain gradient effects can be extended to study the initiation of cracking in polymers This is especially useful since one of the major applications of polymers is to enhance the impact resistance of materials by requiring them to undergo large deformations before failure When used as a biomaterial, premature failure due to cracks may lead to dire consequences on the human body’s recovery Since the initiation of cracking is in the submicron range, strain gradient effects may be dominant This may help us to better understand polymeric 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95, 3655-3666 53 .. .MODELING SIZE EFFECTS IN NANO- INDENTATION OF POLYMERS POH LEONG HIEN [B.Eng (Hons), NUS] A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING NATIONAL... Finite element model of Berkovich indentation 31 3.5 Berkovich indentation numerical results for different domain sizes 32 3.6 Convergence study of finite element model simulating Berkovich indentation. .. experiments, coupled with the rising importance of polymers in several small volume applications lead to a surge in interest in indentations of polymers, in particular, glassy polymers [Dahl et al 1999;