FraMCoS-2 Zurich (Switzerland), 1995 Proceedings

Aedificatio has kindly granted permission to upload proceedings (thanks to Prof. Witttmann). Proceedings were scanned by Prof. Landis.

IA-FraMCoS

Front Matter, Preface, Table of Contents

Volume 1

Chapter 1. Fracture Energy, Strain Softening, and Fracture Toughness
1-1-1
A Comparative Study of GF Test Results
D.R. Brokenshire and B.I.G. Barr
1-1-2 Determination of Tension Softening Diagrams of Various Kinds of Concrete by Means of Numerical Analysis
Y. Uchida, N. Kurihara, K. Rokugo and W. Koyanagi
1-1-3 Fracture Energy of Normal Strength Concrete, High Strength Concrete and Ultra High Strength Ductile Steel Fiber Reinforced Concrete
J.P. Ulfkjaer and R. Brincker
1-1-4 Effect of Model Fracture Law and Porosity on Tensile Softening of Concrete
A. Arslan, E. Schlangen and J.G.M. van Mier
1-1-5 Unified Test Procedure for Evaluating the Fracture Characteristics of Concrete
A.D. Jefferson and B.I.G. Barr
1-1-6 Influence of Notch Size, Eccentricity and Rotational Stiffness on Fracture Properties Determined in Tensile Tests
F.P. Zhou
1-1-7 The Effect of the Boundary Conditions on the Cylinder Splitting Strength
C. Rocco, G.V. Guinea, J. Planas and M. Elices
1-1-8 Toughness of Old Plain Concretes
M. Berra and A. Castellani
1-1-9 Rupture Modulus and Fracture Properties of Concrete
J. Planas, G.V. Guinea and M. Elices
1-1-10 The Effects of Mix Variables on Concrete Fracture Mechanics Parameters
Y.-C. Kan and S.E. Swartz
1-1-11 Effect of Concrete Materials on Fracture Performance
N.-Q. Feng, X.-H. Ji, Q.-F. and J.-T. Ding
1-1-12 Microstructural Effects on the Brittleness of High Strength Concretes
C. Tasedemir, M.A. Tasedemir, R. Grimm and G. König
1-1-13 Determination of KSIC and CTODC from Peak Loads and Relationship between Two-Parameter Fracture Model and Size Effect Model
T. Tang, S. Yang, C. Ouyang, W. Libardi and S.P. Shah
1-1-14 Mode I Fracture Behaviour of Recycled Concrete
S. Ishiguro, S.E. Stanzl-Tschegg, E.K. Tschegg and R. Travnicek
Chapter 2. Fracture Process Zone
1-2-1
Simulation and Observation of the Fracture Process Zone
T. Steiger, H. Sadouki and F.H. Wittmann
1-2-2 Studies of the Length of Concrete Micro Crack Zone
N.-Q. Feng, X.-H. Ji, Q.-F. Zhuang and J.-T. Ding
1-2-3 Further Studies of the Fracture Process Zone Associated with Mixed Mode Dynamic Fracture of Concrete
Z.K. Gou, M. Kosai, A.S. Kobayashi and N.M. Hawkins
1-2-4 Fracture Mechanics Characteristics at Crack Initiation and Propagation and their Dependence on Structure of Concrete
K. Kolver
1-2-5 Fracture Parameters for Concrete Based on Poly-linear Approximation Analysis of Tension Softening Diagram
Y. Kitsutaka
1-2-6 A Method to  Evaluate Tension Softening Model for Concrete
N. Nomura and H. Mihashi
1-2-7 A Method for Determining Fracture Properties of Concrete Through a Single Test
F.P. Zhou
1-2-8 Application of Electronic Speckle Pattern Interferometry (ESPI) to Observe the Fracture Process Zone
E. Hack, T. Steiger and H. Sadouki
1-2-9 Use of the Confocal Microscope to Study Pre-Existing Microcracks and Crack Growth in Concrete
H. Sunderland, A. Tolou, E. Denarié, L. Job and C. Huet
Chapter 3. Crack Formation Under Different Conditions
1-3-1
Transient Fluid Fracture Interaction
V. Slowik, V.E. Saouma and Y.-S. Roh
1-3-2 Tensile Hydraulic Fracture of Concrete and Rock
J.H.M. Visser and J.G.M. van Mier
1-3-3 Influence of Stress Corrosion on Fracture Energy of Cementitious Materials
A. Gerdes and F.H. Wittmann
1-3-4 Tensile Load-Bearing Behaviour of Concrete after Long-Term Static and Cyclic Tensile Preloading
F. Blaschke and G. Mehlhorn
1-3-5 Influence of the Long-Term and Repeated Loading on Fracture Mechanics of the Partially Prestressed Beams
C. Magureanu
1-3-6 Interface Fracture in Cement Based Materials
A. Vervuurt and J.G.M. van Mier
1-3-7 Interaction of Plural Cracks in Concrete During Flexural Failure
H. Koide, H. Akita and M. Tomon
1-3-8 Experimental Measurements of Microfracture in Cement Based Materials
E.N. Landis and S.P Shah
1-3-9 Experimental and Numerical Investigation of Failure of Quasi-Brittle Materials
J. Gao, M.L. Wang and H.L. Schreyer
1-3-10 On an Efficient New Numerical Method for Resolution of the Frictionless Contact Problems with a Variational Inequality Approach
A. Noune, R.M. Courtade and H. Benkhira
1-3-11 Simulations of Tensile Fracture in Concrete
A. Vervuurt, M.R.A. van Vliet, J.G.M. van Mier and E. Schlangen
1-3-12 Interaction of a Main Crack with Ordered Distributions of Microcracks
A. Brencich and A. Carpinteri
Chapter 4. Concrete Failure under Compression
1-4-1
Modelling Compressive Failure Using Rigid Particle Systems
Y.-Q. Liu, H. Hikosaka and J.E. Bolander Jr.
1-4-2 Softening Behaviour of Concrete under Uniaxial Compression
M.R.A. van Vliet and J.G.M van Mier
1-4-3 Experimental Observations of Concrete Behaviour under Uniaxial Compression
Y.-H. Lee, K. Willam and H.-D. Kang
1-4-4 Failure of Concrete with Oblique Weak Layer
H. Wang and X.-Z. Hu
1-4-5 Compressive Strength Variation of Concrete Specimens Due to Imperfection Sensitivity.
K. Ikeda and K. Maruyama
1-4-6 A Compressive Softening Model for Concrete
G. Markeset
1-4-7 Microcracking in Concrete under Compression: its Gradient Mechanisms and Reflection in Macro
I. Blechman
1-4-8 A Case of Compression Failure in Concrete Due to Stress Release
P. Stroeven
Chapter 5. Mixed Mode Fracture
1-5-1
A Simple Fracture Mechanics Model for Mixed-Mode Failure in Concrete
T. Olofsson, U. Ohlsson and M. Klisinski
1-5-2 Mixed-Mode Fracture in Concrete: a Non-Local Damage Approach
M. di Prisco and J. Mazars
1-5-3 Fracture Behavior of Concrete to Mixed Loading
S.-Y. Peng, N. Shirai and M. Irobe
1-5-4 Influence of Age of Loading on Mixed Mode Fracture Parameters of Concrete
B. Hu, G. Zhao and Y. Song
Chapter 6. Scaling Theories and Size Effect
1-6-1
Scaling Theories for Quasibrittle Fracture: Recent Advances and New Directions
Z.P. Bazant
1-6-2 Size Effect Mechanisms in Numerical Concrete Fracture
J.E. Bolander Jr. and Y. Kobashi
1-6-3 Size Effects in the Biaxial Tensile-Compressive Behaviour of Concrete: Physical Mechanisms and Modelling
P. Rossi and F.-J. Ulm
1-6-4 A Truncated Statistical Model for Analyzing the Size-Effect on Tensile Strength of Concrete Structures
A. Carpinteri, G. Ferro and S. Ivernizzi
1-6-5 Crack Surface Friction and Size Effect in Mode I Propagation for Mortar and Concrete
A. Turatsinze and A. Bascoul
1-6-6 Multifractal Scaling Law for the Fracture Energy Variation of Concrete Structures
A. Carpinteri and B. Chiaia
1-6-7 A Fractal Study of the Size Effect of Concrete Fracture Energy
N.-Q. Feng, X.-H. Ji, Q.-F. Zhuang and J.-T. Ding
1-6-8 Size Effect and Stability of Fracture
B. Trunk, H. Sadouki and F.H. Wittmann
1-6-9 Numerical Concrete Applied to Investigative Size Effect and Stability of Crack Propagation
H. Sadouki and F.H. Wittman
1-6-10 Representative Volumes of Cementitious Materials
Y. Xi
1-6-11 Size Effect on Shear Strength in Reinforced Concrete Beams with Shear Reinforcement
N. Shirai, K. Moriizumi and K. Ishii
1-6-12 Size Effect on Strength and Deformability of RC Beams Failing in Flexure
H. Adachi, N. Shirai, M. Nakanishi and K. Ogino
1-6-13 Size Effect in Concrete and Reinforced Concrete Structures
J. Ozbolt and R. Eligehausen
1-6-14 Large-Scale In-Situ Fracture of Ice
J.P. Dempsey, R.M. Adamson and S.V. Mulmule
1-6-15 Fracture Characteristics and Size Effect in Normal and High-Strength Concrete Beam
S.H. Eo, N.M. Hawkins and G.-S. Kwak
1-6-16 The Neural Network-Based Analysis of Size Effect in Concrete Fracture
A. Arslan and R. Ince
Chapter 7. Rate Effects
1-7-1
High Strain Rate Tensile Behaviour of Concrete: Significant Parameters
F. Toutlemonde, C. Boulay and P. Rossi
1-7-2
Effect of Temperature and Loading Rate on Fracture Behavior of Concrete Subjected to Uniaxial Tension
V. Mechtcherine, H. Garrecht and H.K. Hilsdorf
1-7-3
Dynamic Crack Propagation in PMMA
R. Jelinek, H.A. Dieterman and L.J. Sluys
Chapter 8. Fibre Reinforced Concrete
1-8-1
A Study on Approximation Method of Tension Softening Curve of Steel Fiber Reinforced Concrete
S. Matsuo, S. Matsuoka, A. Masuda and H. Yanagi
1-8-2 What is Interpreted from Fractured Surfaces in Concrete?
H. Mihashi, N. Nomura, H. Nakamura and T. Umeoka
1-8-3 Characterization of the Toughness of Fiber Reinforced Concrete Using the Load-CMOD Response
V.S. Gopalaratnam, R. Gettu, S. Carmona and D. Jamet
1-8-4 An R-Curve Approach for PUll-Out Fibres From a Matrix
C. Ouyang, A. Pacios and S.P. Shah
1-8-5 Fracture of Fibre-Reinforced Concrete Beams with Low Fibre Content
J.L. Vitek and P. Vitek
1-8-6 Effects of Fiber on Fracture Properties of Light Weight Concrete made with Fly-Ash Pelletized Aggregates
T.-P. Chang, C.-Y. Lin, C.-L. Hwang and M.M. Shieh
1-8-7 On Crack Propagation and Failure Modes in Fiber-Reinforced Concrete Slabs
R. Felicetti, P.G. Gambarova and N. Zanini

Volume 2

Chapter 9. Numerical Models
2-9-1
Fictitious Crack Models: A Classification Aproach
S. Weihe and B. Kröplin
2-9-2 New Developments in Microplane and Multicrack Models for Concrete
I. Carol and Z.P. Bazant
2-9-3 Enhanced Microplane Concrete Model
T. Hasegawa
2-9-4 Gradient-Enhanced Smeared Crack Models for Finite Element Analysis of Plain and Reinforced Concrete
J. Pamin and R. de Borst
2-9-5 Holonomic and Nonholonomic Simulations of Quasi-Brittle Fracture: a Comparative Study of Mathematical Programming Approaches
G. Bolzon, G. Maier and F. Tin-Loi
2-9-6 Embedded Cohesive Crack Models Based on Regularized Discontinuous Displacements
R. Larsson, K. Runesson and M. Akesson
2-9-7 Computational Aspects of Fracture Simulations with Lattice Models
E. Schlangen
2-9-8 A Finite Element Approach to 3D Crack Growth in Brittle Solids
G. Baker and C.A. Grummitt
2-9-9 An Adaptive Mesh Strategy Based on the Ale Formation to Transient Finite Element Analysis of Localisation
L. Bodé, G. Pijaudier-Cabot and A. Huerta
2-9-10 Particle Model for Fracture and Statistical Micro-Macro Correlation of Material Constants
M. Jirasek and Z.P. Bazant
2-9-11 2D and 3D-Modelling of Concrete as an Assemblage of Spheres Reevaluation of the Failure Criterion
W.J. Beranek and G.J. Hobbelman
Chapter 10. Damage Models
2-10-1
Damage in Discrete and Continuum Models
G. Pijaudier-Cabot, A. Delaplace and S. Roux
2-10-2 A Thermo-Mechanical Damage Model for Concrete at Elevated Temperatures
G. Baker and R. de Borst
2-10-3 Some Considerations on Explicit Damage Models Including Crack Closure Effects and Anisotropic Behaviour
A. Rouquand and C. Pontiroli
2-10-4 Gradient Damage and Reliability: Instability as Limit State Function
J. Carmeliet and R. de Borst
2-10-5 Additive Volumetric-Deviatoric Split of Finite Strain Tensor and its Implication for Cracking Models
Z.P. Bazant
2-10-6 Damage Evolution Laws for Concrete - a Comparative Study
M. Polanco-Loria and S.I. Sørensen
2-10-7 A Damage Mechanics Model for Concrete Subjected to Compression
S. Akyüz and M.A. Tasdemir
2-10-8 Failure Analysis of Pre-Damaged Concrete Structural Components
L. Bodé, J.L. Tailhan, G. Pijaudier-Cabit and Ch. La Bordierie
2-10-9
Poro-Fracture Analysis of Concrete Using a Damage Mechanics Models S.S. Bhattacharjee, F. Ghrib, R. Tinawi and P. Léger
Chapter 11. Fundamental Considerations
2-11-1
Localization and Mesh Sensitivity in Gradient Dependent Softening Plasticity
F. Meftah, J.M. Reynouard and O. Merabet
2-11-2
Strain Localization as Bifurcation Behaviour of Elasto-Plastic Softening Materials
H. Yoshikawa, R. Nagano and K. Willam
2-11-3
A Continuum Thermodynamics Approach for Studying Microstructural Effects on the Non-Linear Fracture Behaviour of Concrete Seen as a Multicracked Granular Composite Material
C. Huet
Chapter 12: Applications - 12.1 Reinforced Concrete and Concrete Structures
2-12-1
Approximate Fracture Mechanical Approach to the Prediction of Ultimate Shear Strength of RC Beams
B.L. Karihaloo
2-12-2 Hardening-Softening Behaviour and Minimum Reinforcement of RC Beams
K. Rokugo, N. Kurihara, T. Ito, Y. Uchida and W. Koyanagi
2-12-3 Crack Spacing in Reinforced Elements
L.J Sluys and M.A. Brioschi
2-12-4 Restrained Shrinkage - Its Impact on the Response of Reinforced Concrete Members
T.S. Ahn and V.S. Gopalaratnam
2-12-5 F.E. Analysis of SFRCT T-Girders Submitted to Shear: A Comparison with Experiment
P. Casanova, P. Rossi and A. Elouard
2-12-6 Fracture of Lightly Reinforced Concrete Beams: Theory and Experiments
J. Planas, G. Ruiz, M. Elices
2-12-7 Applications of Linear Elastic Fracture Mechanics to Reinforced and Prestressed Concrete Structures
L. Goffi
2-12-8 Dynamic Analysis of Reinforced Concrete Slabs: Comparison of Simplified Methods
F. Toutlemonde and P. Rossi
2-12-9 Reinforced Normal and High Strength Concrete Columns - Fracture Mechanics and Experiments
C. Claeson and K. Gyltoft
2-12-10 Shock Wave Loading on Reinforced Concrete Plates: Experimental Results and Comparisons with Explicit Damage Model Predictions
C. Pontiroli, A. Rouquand and E. Canton
2-12-11 Simulation of Punching Failure: Failure Mechanism and Size-Effect
Ph. Menétrey
2-12-12 Fracture of Concrete - A New Approach to Predict Crack Propagation
T. Corelius Hansen
2-12-13 Adaptive Mesh Refinement Procedure of Finite Element Method for Fracture Mechanics
L. Liang and Y.M. Lin
2-12-14 Fictitious Crack Modelling of Kiln Furniture Ceramics
M.A.J. van Gils, L.J.M.G.Dortmans and G. With
2-12-15 A Strategy for Analyzing the Mechanical Behaviour of Concrete Structures under Various Loadings: the LCPC Experience
P. Rossi and F.-J. Ulm
2-12-16 Discrete Crack Modeling in Concrete Structures
J. Cervenka and V.E. Saouma
2-12-17 Similarity Laws for Concrete Structures under Dynamic Loading
J. Mazars, J.F. Dubé and S. Ghavamian
2-12-18 Crack Propagation in Concrete Specimens Subjected to Sustained Loads
A. Carpinteri, S. Valente, F.P. Zhou, G. Ferrara and G. Melchiorri
2-12-19 Interfacial Fracture Toughness of Concrete Repair Materials
V.C. Li, Y.M. Lim and D.J. Foremsky
2-12-20 Application of Fracture Mechanics to Design of SFRC Tunnel Linings and Development of Strain-Hardening Cementitious Composites
H. Horii, P. Nanakorn and P. Kabele
2-12-21 Strength of a Concrete Element under the Action of Concentrated Tensile or Shear Force
V.I. Yagust and D.Z. Yankelevsky
2-12-22 Simulation of 3-D Concrete-Frame Collapses Due to Dynamic Loading
M. Hakuno
2-12-23 Linear or Nonlinear Fracture Mechanics of Concrete
G.A. Plizzari and V.E. Saouma
2-12-24 Mesh Sensitivity Effects in Smeared Finite Element Analysis of Concrete Fracture
V. Cervenka, R. Pukl, J. Ozbold and R. Eligehausen
2-12-25 Seismic Localization of Softening Cracking Damage in Concrete Frames
Z.P. Bazant and M. Jirasek
2-12-26 Application of Non-Linear Fracture Mechanics to the Seismic Assessment of Concrete Gravity Dams
M. Galli, G. Feltrin and H. Bachmann
12.2 Hygral and Thermal Gradients and Crack Formations
2-12-27
Crack Formation under Hygral or Thermal Gradients
A.M. Alvaredo
2-12-28 Modelling of Early Age Concrete Cracking Due to Thermo-Chemo-Mechanical Couplings
F.-J. Ulm, A. Elouard and P. Rossi
2-12-29 Bonded High Strength Concrete Overlays Exposed to Autogenous and Drying Shrinkage
T. Kanstad
2-12-30 Experimental and Numerical Study of Drying-Induced Warping of Concrete Slabs
A.M. Alvaredo, S. Gallo and F.H. Wittmann
2-12-31 Optimization of Mortars by Means of Fracture Mechanics
G. Martinola and F.H. Wittmann
2-12-32 An Insight in the Reduction of Drying Shrinkage of Concrete Due to Skin Micro-Cracking
L. Granger and P. Rossi
2-12-33 Water Movement within Mortar Analyzed as Capillary Flow
H. Akita and T. Fujiwara
2-12-34 Evaluation of Concrete Durability Based on Crack Growth Kinetics Data under influence of Elevated Temperatures
Y. Zaitsev, V. Shevchenko and T. Cherednichenko
2-12-35 Influence of Drying Induced Damage on the Hygral Diffusion Coefficient
F.H. Wittmann
12.3 Anchor Bolts
2-12-36
Anchor Pull-Out Tests - Influence of Boundary Conditions and Material Properties
V. Slowik, A.M. Alvaredo and F.H. Wittmann
2-12-37 Fracture Behaviour of Normal and High Strength Concrete in Anchor Pull-Out
R. Zeitler and J.-D. Wörner
2-12-38 Anchor Bolts in Concrete Structures  - Finite Element Calculations Based on Inner Softening Bands
U. Ohlsson and T. Olofsson
2-12-39 Fatigue Behaviour of Anchor Bolts in Concrete
E. Cadoni
2-12-40 FEM Analysis of Tensile Fracture Phenomena in Concrete Structures
A. Ali
2-12-41 Splitting of Concrete Covers - A Fracture Mechanics Approach
K. Noghabai
2-12-42 Computer Simulation of Fasteners in Concrete Under Fire
R. Pukl, M. Reick and R. Eligehausen

Volume 3

Front Matter, Preface, Table of Contents
Workshop 1. Numerical Modelling and Determination of Fracture Mechanics Parameters
3-1-1
Numerical Modelling and Determination of Fracture Mechanics Parameters for Concrete and Rock: Introduction
J.G.M. van Mier
3-1-2
Numerical Modelling and Determination of Fracture Mechanics Parameters. Hillerborg Type Models
M. Elices and J. Planas
3-1-3 On Fracture Mechanics in General and Discrete Cracks in Particular
V.E. Saouma
3-1-4 Softening, Damage and Higher-Order Continua
R. de Borst
3-1-5 Numerical Modelling and Determination of Fracture Mechanics Parameters for Concrete and Rock: Probabilistic Aspects
J. Carmeliet
3-1-6 The Need for Standard Test Models
B.I.G. Barr and S.E. Swartz
3-1-7 Numerical Modelling and Determination of Fracture Mechanics Parameters: Workshop Summary Report
E. Schlangen
Workshop 2. Fracture Mechanics and Design of Concrete Structures
3-2-1
Approximate Analytical Fracture Mechanics Approach to the Design of Concrete Structures
A. Carpinteri and R. Massabó
3-2-2 Applications of Fracture Mechanics to Anchors and Bond
L. Elfgren, K. Noghabai, U. Ohlsson and Th. Olofsson
3-2-3 Fracture Mechanics in the Design of Concrete Pipes
P.J. Gustafsson and O. Dahlblom
3-2-4 Using Fracture Mechanics Concepts to Predict the Shear Strength of Concrete Structures
S. Kono, N.M. Hawkins and A.S. Kobayashi
3-2-5 Application of Computational Fracture Mechanics to Repair of Large Concrete Structures
A. Ingraffea, B. Carter and P. Wawrzynek
3-2-6 Application of Fracture Mechanics to High Strength and Offshore Concrete Structures
M. Modéer
3-2-7 Fracture Mechanics and Design of Concrete Structures - Workshop Summary Report
B.L. Karihaloo
Workshop 3. Size Effects: Theoretical Concepts, Experimental Verifications, and Implications in Structural Design
3-3-1 Size Effect Aspects of Measurement of Fracture Characteristics of Quasibrittle Material
Z.P. Bazant
3-3-2 A Multifractal Approach to the Strength and Toughness Scaling of Concrete Structures
A. Carpinteri and B. Chiaia
3-3-3 Intrinsic Material Law for Predicting Size Effect in Concrete Structures
H. Mihashi and N. Nomura
3-3-4 Size Effects in Cracking of Concrete: Physical Explanations and Design Consequences
P. Rossi
3-3-5 Size Effect on Strength and Deformation of RC Beams Failing in Flexure
H. Adachi, N. Shirai, M. Nakanishi and K. Ogino
3-3-6 Size Effects: Theoretical Concepts, Experimental Verifications, and Implications in Structural Design - Conclusive Remarks
F.-J. Ulm
Workshop 4. Experimental Methods
3-4-1
Strain Softening of Concrete in Compression
S.P. Shah, S. Choi and D.C. Jansen
3-4-2 Effect of Confinement on the Fracture Behavior of Concrete under Compression
K.M. Nemati and P.J.M. Monteiro
3-4-3 An Experimental Study of the Microstructural Mechanism Influencing Crack Closure Stresses in a Fine-Sand Mortar
R. Dasgupta, J.C. Hay, J. McConell, K.W White and C. Vipulanandan
3-4-4 Uniaxial Tension
H.W. Reinhardt
Workshop 5. Smeared Versus Discrete Failure Analysis with Application to Pull-Out Problems
3-5-1
Discrete Versus Smeared Crack Analysis
K. Willam and I. Carol
3-5-2 Fastening Elements in Concrete Structures
R. Eligehausen and J. Oûbolt
3-5-3 Pull-Out Failure Mechanisms in Concrete
D.V. Phillips, D.R. Green, B.S. Zhang, C.J. Pearce and N. Bicanic
3-5-4 Application of the Finite Element Method to Anchoring Technology in Concrete
J. Nienstedt and Ch. Dietrich
3-5-5 Bridges Between Damage and Fracture Mechanics
J. Mazars and G. Pijaudier-Cabot
3-5-6 Discrete Crack Analysis of Anchor Bolt Pull-Out
J. Cervenka and V.E. Saouma
3-5-7 Smeared Versus Discrete Failure Analysis with Application to Pull-Out Problems - Concluding Remarks
J. Ozbolt and R. Pukl
Author Index