13.4 True Stress–Strain Determination from the Tension
Experiments Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
13.4.1 The Models for Estimation of Stress and Strains
in Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
13.4.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 203
13.4.3 Experimental and Numerical Procedure
of Construction a True Strain Curve According
to Experiment on High-Speed Tension . . . . . . . . . . . . 214
13.4.4 Testing the Procedure on Experiments . . . . . . . . . . . . . 216
13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
14 Flocking Rules Governing Swarm Robot as Tool to Describe
Continuum Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Ramiro dell’Erba
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
14.2 Tool Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
14.3 Results from Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . 227
14.4 Results for a Bending Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 230
14.5 Future Work and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 238
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15 Homogenization-Based Mechanical Behavior Modeling
of Composites Using Mean Green Operators for Infinite Inclusion
Patterns or Networks Possibly Co-continuous with a Matrix . . . . . 245
Patrick Franciosi and Mario Spagnuolo
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
15.2 Effective (Piece-Wise) Linear Elastic-like Properties
of Composites of the Matrix-Reinforced Type . . . . . . . . . . . . . 248
15.3 Elementary Axial and Planar Alignments of Axially
Symmetric or Fiber-like Elements . . . . . . . . . . . . . . . . . . . . . . 255
15.4 From Elementary Alignments to Bundles or Networks . . . . . . . 261
15.5 Other Potential Application Extension Directions . . . . . . . . . . . 266
15.5.1 From Linear to Nonlinear Behavior . . . . . . . . . . . . . . . 267
15.5.2 From Elasticity-Type to Coupled Piezo-Type
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
15.5.3 From Static to Dynamical Problems . . . . . . . . . . . . . . 270
15.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
16 Strain Gradient Models for Growing Solid Bodies . . . . . . . . . . . . . 281
Zineeddine Louna, Ibrahim Goda, and Jean-François Ganghoffer
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
16.2 Evolution Laws for Growing Solid Bodies with Strain
Gradient Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
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Contents
Experiments Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
13.4.1 The Models for Estimation of Stress and Strains
in Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
13.4.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 203
13.4.3 Experimental and Numerical Procedure
of Construction a True Strain Curve According
to Experiment on High-Speed Tension . . . . . . . . . . . . 214
13.4.4 Testing the Procedure on Experiments . . . . . . . . . . . . . 216
13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
14 Flocking Rules Governing Swarm Robot as Tool to Describe
Continuum Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Ramiro dell’Erba
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
14.2 Tool Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
14.3 Results from Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . 227
14.4 Results for a Bending Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 230
14.5 Future Work and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 238
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15 Homogenization-Based Mechanical Behavior Modeling
of Composites Using Mean Green Operators for Infinite Inclusion
Patterns or Networks Possibly Co-continuous with a Matrix . . . . . 245
Patrick Franciosi and Mario Spagnuolo
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
15.2 Effective (Piece-Wise) Linear Elastic-like Properties
of Composites of the Matrix-Reinforced Type . . . . . . . . . . . . . 248
15.3 Elementary Axial and Planar Alignments of Axially
Symmetric or Fiber-like Elements . . . . . . . . . . . . . . . . . . . . . . 255
15.4 From Elementary Alignments to Bundles or Networks . . . . . . . 261
15.5 Other Potential Application Extension Directions . . . . . . . . . . . 266
15.5.1 From Linear to Nonlinear Behavior . . . . . . . . . . . . . . . 267
15.5.2 From Elasticity-Type to Coupled Piezo-Type
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
15.5.3 From Static to Dynamical Problems . . . . . . . . . . . . . . 270
15.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
16 Strain Gradient Models for Growing Solid Bodies . . . . . . . . . . . . . 281
Zineeddine Louna, Ibrahim Goda, and Jean-François Ganghoffer
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
16.2 Evolution Laws for Growing Solid Bodies with Strain
Gradient Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
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Contents