Frontiers of Fracture Mechanics. Adhesion and Interfacial Fracture Contact Damage

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1 Frontiers of Fracture Mechanics Adhesion and Interfacial Fracture Contact Damage

2 Biology, Medicine & Dentistry The Next Frontiers For Mechanics One of the current challenges in materials & mechanics is how to make adequate connections to biology and medicine This requires new knowledge and teaming approaches beyond the boundaries of our current efforts medicine, dentistry biology, chemistry, physics materials and mechanics

3 The Benefits of The Mechanics Approach Theoretical/computational mechanics enable quantitative predictions of cause and effect However biological systems are complex Two approaches are often needed Experimental mechanics provides new insights and measurements Enables model validation Enables new clinical solutions

4 Background and Introduction This class presents selected examples at the frontiers of fracture mechanics These include: Problems involving adhesion and interfacial fracture Problems involving contact damage This focus is on applications of fracture mechanics in biomedical applications

5 Path to understanding the effects of multiple loadings on dental structure Tooth on tooth contact Hertzian contact on idealized Multilayer Structure P Ceramic Adhesive layer foundation

6 Basic FEA on idealized 3-layer structure Ceramic foundation Adhesive layer Ceramic Foundation

7 Sub-surface crack is the major clinical failure mode. FAILURES CAUSED BY HERTZIAN INDENTATION Cone crack Subsurface crack

8 Loading rate effects on 3-layer structure Plot of critical load Pm for radial cracking as function of loading rate P for coatings of soda-lime glass (filled symbols) and silicon (unfilled symbols) bonded to polycarbonate substrates. (Lee et al. 2002)

9 Contribution of Lawn s approach Experimental study demonstrates the existence of loading rate effect Silicon does not exhibit slow crack growth (SCG) but silicon tri-layer shows loading rate effect SCG model can only explain part of loading rate effect Material properties of the foundation and join layers may play important role

10 Approach of the Current Work Develop understanding of the constitutive behavior of individual layers Integrate the constitutive behavior into mechanics models Predict critical loads in multilayer structure Develop understanding of loading rate effects on deformation and cracking

11 Experiments Studied joins, foundations & multilayers with real dental materials Measured constitutive behavior for individual layers (monotonic compression tests) Studied cracking of multilayers (monotonic Hertzian contact tests) Compression Tests Hertzian contact Tests Indenter Capacitance Gage Indenter Capacitance Gage Specimen Specimen

12 Loading rate effect of critical load measurements Indenter Specimen Capacitance Gage Glass Dental cement RelyX Z100 dental restoration Loading rate 100 N/s

13 Critical load (N) Loading rate effect of critical load on dental multilayer Indenter Specimen Capacitance Gage Glass Dental cement RelyX 1 mm ~100 μm Z100 dental restoration 4 mm Loading Rate (N/s) 9 mm

14 Stress (MPa) Stress (MPa) Rate Dependent Young s Moduli of Cement & Foundation Layers Indenter Specimen Capacitance Gage 350 Dental Cement RelyX ARC (1 mm thick and 7 mm diameter) 350 Dental Restoration Z100 (4 mm thick and 9 mm diameter) 300 Strain rate (s -1 ) Strain rate (s -1 ) Strain Strain

15 Elasticity Analysis Coating/substrate bi-layer model P Ceramic ceramic a foundation Adhesive layer foundation P B and d are dimension coefficients B d 2 log E / E t f

16 Critical load (N) Bi-layer Slow Crack Growth Power law crack growth model da K v v0 dt KIC K is stress intensity factor = ya 1/2 N and v 0 are determined through four point bending tests Bi-layer SCG analytical solution (Lee. et al) N 1 N 1 Pm AN 1 P solution Experimental data Bi-layer SCG model analytical solution Best linear fit of experimental data Loading rate (N/s)

17 Indenter Specimen E Capacitance Gage Herztian contact Test data Glass Dental cement RelyX E Z100 dental restoration D Finite element analysis (FEA) K c c D N / 2 1 v N 1 N / 2 1 N / 2 Ic 0 f 0 N t SCG model for top glass layer 0 t R t dt D Glass Dental cement RelyX Z100 dental restoration Integration E Rupture time t R RDEASCG Model (Rate Dependent Elastically- Assisted SCG) P Critical load c P t R Pc

18 Critical load (N) Prediction of Loading rate effect of critical load with RDEASCG model Experimental data 3-layer RDEASCG model FEA results Bi-layer SCG model analytical solution Best linear fit of experimental data Loading Rate (N/s) Clinically relevant loading rates

19 Experimental Summary Loading Rate Effects -- studied constitutive behavior of individual layers -- studied rate-dependent critical loads in multilayer structure -- in situ study gives good observation of deformation and cracking in dental multilayer Modeling -- implemented rate-dependent Young s modulus into multilayer structure with real dental material -- developed analytical model for reasonable prediction of deformation and cracking in dental multilayer under monotonic loading

20 CYCLIC CONTACT EXPERIMENTS Model multilayers are subjected to cyclic Hertzian indentation for 10 6 cycles Peak load levels lower than observed crack initiation loads under monotonic loading 3 mm Model multilayer 8 mm Glass Epoxy Ceramic-filled Polymer Indenter Specimen Experimental set-up P Capacitance Gage

21 CYCLIC HERTZIAN INDENTATION 0.8 mm, P=60 N 3.18 mm, P=70 N 8 mm, P=90 N Surface deformation after crack nucleation 0.8 mm, P=60 N 3.18 mm, P=70 N 8 mm, P=90 N Hertzian cone crack (Soboyejo et al, 2001) Subsurface crack

22 MAXIMUM PRINCIPAL STRESS DISTRIBUTION Cement Ceramic high tensile stress area causes sub-surface crack 1 mm Polymer

23 DEFORMATION DURING CYCLIC LOADING P ~ N 200 mm Indenter 1 (120) ~ mm 200 (120) ~ mm (120) ~ mm Two Sources Time dependent response of epoxy layer Accumulation of plastic strain in epoxy layer

24 Deformation During Cyclic 2 0 Loading P -2 (m) N N N N 200 mm Indenter E E E E E+05 Time (sec) Sharp Change in sample compliance Sample loaded for million cycles at each load level at 5 Hz loading rate

25 P (N) Subsurface Pop-In Conditions 90 P After 2 X 10 6 Cycles After 3 X 10 6 Cycles After 10 6 Cycles (m) mm Indenter

26 Critical Load (N) Pop-In Criteria(Lawn, 2001) 1000 Cone Crack Radial Crack Cone Crack Critical Load [Lawn,2001] Radial Crack Critical Load [Lawn, 2001] Indenter Radius (mm)

27 POSSIBLE MECHANISMS OF SUB-SURFACE CRACKING Mechanics-driven crack growth due to flow of the cement into the crack Stress corrosion cracking Mechanical fatigue

28 HYDRAULIC FRACTURE TEST p glass glass oil a 2R crack c p hole Glass cracks under certain pressure. p =c/a

29 NORMALIZED STRESS INTENSITY FACTOR

30 CEMENT FLOWS INTO CRACK W 2p 0 H 0 2H 0 c W 2 a 0

31 CRACK GROWTH RATE da dt 2 2H p cos 3a sin 1 K p a K p a K p a p H c / 0 tan c / 2 0 c / da dt H p 0 cos 1 2 3K c p H a 0 If H= 1μm, p 0 =2 MPa, K c =0.5 MPa-s, μ=100 Pa-s, cos=0.005 N/m, then da/dt = 0.27 μm/s

32 Summary Cyclic Contact Pop-in conditions are strongly influenced by ball size in Hertzian indentation experiments Ball size effects explained using simple contact mechanics models Hydraulic fracture may be caused by water within cracks Mechanics model developed for hydraulic fracture modeling Mechanistically-based fatigue model needed

33 Modeling of Water Diffusion Cracking is shown in top ceramic layer after the dental multilayers immersed in water for some time. Major observations: 1. cracking occurs after some time, and becomes more extensive as time increases. 2. cracking first occurs near the edge of the sample and propagates parallel to the edge. 3. cracking initiates from the bottom surface of the top layer. after 6 days after 15 days after 70 days Courtesy of Prof. V.P. Thompson, NYU Dental School

34 Real Questions Is the stress induced by water diffusion high enough to cause the cracking? Can the crack pattern be understood by water diffusion induced SCG? Thermal diffusion analogy Water diffusion Water induced foundation expansion Heat transfer Thermal expansion

35 MAXIMUM PRINCIPAL STRESS DISTRIBUTION DURING WATER DIFFUSION Maximum Principal Stress (Pa) dental ceramic or glass dental cement dentin-like polymer Symmetry line Maximum tensile stress appears at the bottom surface of the top layer. Peak tensile stress first appears near the edge and is parallel to the edge.

36 Maximum Principal Stress Distributions in Dental Crown Multilayer E * V c s / t/ = center Location, x/r edge length scale 5 10 m ~ 289 days diffusivity m 2 / s E * c ~100GPa V s ~1% E V * c s / 3 ~ 300MPa

37 Slow Crack Growth Theory da dt v K / / 0 K IC Crack growth rate: N Stress intensity factor: 1/ 2 K ya Criteria to form subsurface crack: D t R 0 t K N N IC dt D N N / 21 N / 2 1v y a 0 i independent of time and load Driving force to form subsurface crack: t 0 t N dt 1/ N

38 SCG in Top Glass Layer Due to Water Diffusion 1/ N t N dt 0 1/ N * E V / 3 c s t/ = center Location, x/r edge Water diffusion is very important in determining the lifetime of dental multilayers.

39 Summary Modeling of Water Diffusion Model predicts the cracking in the top ceramic layer after the dental multilayers immersed in water for some time. Model also predicts the major observations: 1. cracking occurs after some time, and becomes more extensive as time increases (due to stresses associated with water distribution). 2. cracking first occurs near the edge of the sample and propagates parallel to the edge (consistent with stresses and slow crack growth). 3. cracking initiates from the bottom surface of the top layer (ditto). after 6 days after 15 days after 70 days Courtesy of Prof. V.P. Thompson, NYU Dental School

40 BIO-INSPIRATION - TOOTH STRUCTURE Enamel Dentin-Enamel-Junction (DEJ) Dentin

41 ELASTIC MODULUS DISTRIBUTION IN DENTIN-ENAMEL JUNCTION (DEJ) G.W. Marshall Jr., et al. J Biomed Mater Res 54, 87-95, 2001

42 MECHANICAL PROPERTIES OF DENTAL MATERIALS/MULTILAYERS Dental ceramic E: 50~200 GPa Dental cement E=5 GPa Dental restoration Dentin-like polymer E=20 GPa Enamel E=65 GPa Dentin-Enamel Junction (DEJ): Graded Real tooth Dentin E=20 GPa

43 DENTAL CROWN RESTORATION FGM DESIGN Dental ceramic E: GPa Functionally graded layer Dental cement E: 2-13 GPa Dentin-like polymer E: GPa

44 MAXIMUM PRINCIPAL STRESS DISTRIBUTION ceramic cement enamel DEJ Natural tooth dentin-like polymer 0.1 mm dentin Current crown Maximum Principal Stress (Pa) ceramic FGM cement FGM design dentin-like polymer

45 Maximum principal stress (MPa) Maximum Principal Stress ceramic polymer Mark II Glass Dicor Empress 2 Zicornia Dicor MGC FGM ceramic polymer 20 Natural Enamel Young s modulus of ceramic (GPa)

46 Young s modulus (GPa) Maximum principal stress (MPa) Effects of Different Distributions of Young s Modulus E=72 GPa E=20 GPa Ceramic FGM 60 I II III Cement thickness (μm) I II III Different Young s modulus distribution

47 Effects of FGM on Fracture Toughness E 1 Ceramic crack a E 2 Cement Stress intensity factor: K I Critical defect length: a a, f E c K c / 2 / E1, 1, 2

48 Critical Defect Length (m) Comparison of Critical Defect Lengths Zicornia Mark II Empress 2 FGM Enamel Dicor-MGC Dicor Glass Mark II Dicor-MGC Glass Dicor Empress 2 Zicornia Maximum Principal Stress in the Ceramic (MPa)

49 Brazil-nut Sandwich Sample Test in Determining Interfacial Toughness between a Dental Cement Composite and Glass Substrate Schematics of (a) teeth contact during chewing and (b) Brazil-nut sandwich samples under contact loading Schematic illustration of Brazil-nut sandwich sample and setup for fracture testing

50 Max. Force (N) INTERFACIAL TOUGHNESS (J/M2) Experimental Results of Brazil-nut Sandwich Sample Test Compression Angle Applied Force COMPRESSION ANGLE COMPRESSION ANGLE

51 Geometry of Brazil-nut Sandwich Sample in Numerical Simulation Interfacial crack Glass Substrate E (GPa) ν Epoxy Interlayer E (GPa) ν

52 Introduction to Cohesive Zone Models for Interfacial Failure Cohesive Zone Ahead of an Interfacial Crack Tip Cohesive Bond Structure Cohesive Bond Rupture Leads to Physical Crack Growth Various Kinds of Cohesive Zone Laws σ σ σ δ (a) Exponential Type (b) Bilinear Type (c) Parabolic Type δ δ

53 Interfacial Potential and Tractions Xu and Needleman (1994) proposed an interfacial potential of the following form 2 n n 1q r q n t ( ) n nexp {[1 r ] [ q ]exp 2 n n r1 r1n t to derive the normal and tangential traction as follows T n which gives n and T t t T n 2 2 n n n t 1 q n t exp { exp [ r ][1 exp ]} 2 2 n n n t r 1 n t T t 2 n n t 1 q n n t 2 { q }exp exp 2 n t t r 1n n t

54 Interfacial Properties Employed in Simulation COHESIVE ENERGY (J/M/M) NORMAL COHESIVE STRENGTH (MPA) TANGENTIAL COHESIVE STREGNTH (MPA) SET SET SET SET CRITICAL INTERFACIAL SEPARATION (micron) For simplicity, the cohesive energy and critical interfacial separation are assumed to be equal in both normal and tangential direction

55 MAX. FORCE (N) INTERFACIAL TOUGHNESS (J/M^2) Comparison between Simulation and Experimental Results COMPRESSION ANGLE Exp SET 1 SET 2 SET 3 SET EXP SET 1 SET 2 SET 3 SET COMPRESSION ANGLE

56 Interfacial Fracture Toughness Fracture Toughness of Glass/Cement and Zirconia/Cement Interface 40 Gc (J/m 2 ) Glass/Epoxy Zirconia/Epoxy 2 substrate 1 Adhesive substrate y

57 Interfacial Fracture Toughness Gc (J/m 2 ) Interfacial Fracture Toughness Crack Glass/Epoxy Interface Fracture Resistance: Zirconia/Cement Crack Crack F Models vs. Experiment f = 45, b = 170 f = 45, b = 175 f = 45, b = 180 Zone Model Experiments y H l L Interface 2 G tan y 1[ 0(1 tan 2 G0 1 tan G G 0 Gc (J/m 2 ) 2y y )( G / G 5 2d (,,, ) d (sin cos tany ) 2 2 cos (1 tan y ) 45,170 45,175 45,180 Zone Model Experiments Evans and Hutchinson (1989) Acta Metall. Mater., 37(3), [(sin cos tany )(sin( y ) cos( ) tany )] 2h 2 cos(1 tan y ) h 2 2 Glass/Cement 0 1) w 2 G G0 (1 tan y ) y

58 Microstructure Effect : FIB Finite elements simulations carried out to study the crack path selection, using the He & Hutchinson criteria, were not successful to fully capture the crack path selection. Knowledge of microstructure is needed for further improvement in model. In the following model clusters of zirconia clusters have been assumed to exist in the epoxy which affect the crack path selection. Glass Zirconia Clusters He MY, Hutchinson JW (1989), Int. J. Solids Structures, 25(9), Glass Epoxy

59 Focused Ion Beam Images of Interfacial Cracks in Ceramic/Cement Interface Interfacial Crack Glass Epoxy

60 Concluding Remarks This class presents an introduction to contact damage in dental multilayers Complex loading and geometry idealized to provide insights into mechanisms Rate-dependent slow crack growth model used to describe underlying physics Bio-inspired design concept presented for the design of robust interfaces Interfacial fracture mechanisms explored using a combination of models and experiments

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