Numerical methods to estimate material properties usually involve analysis of a representative volume element (RVE) or unit cell (UC). The representative volume element (RVE) or unit cell (UC) is the smallest volume over which a measurement can be made that will yield a value representative of the whole. RVEs and UCs are widely used in the characterisation of materials with multiscale architectures such as composites. However, finite element (FE) software packages such as Abaqus and Comsol MultiPhysics do not offer the capability for RVE and UC modelling directly on their own. To apply them to analyse RVEs and UCs, the generation of the FE models for them, the imposition of boundary conditions, and the extraction of directly relevant results are essentially the responsibility of the user. These have tended to be incorrectly implemented by users! For the first time, this book will provide a comprehensive account on correct modelling of RVEs and UCs, which will eliminate any uncertainties and ambiguities.The book offers a complete and thorough review on the subject of RVEs and UCs, establishing a framework on a rigorous mathematical and mechanical basis to ensure that basic concepts, such as symmetry and free body diagrams, are applied correctly and consistently. It also demonstrates to readers that rigorous applications of mathematics and mechanics are meant to make things clear, consistent, thorough and, most of all, simple and easy to follow, rather than the opposite as many perceive. As a result, the book shows that the appropriate use of RVEs and UCs can deliver an effective and reliable means of material characterisation. It not only provides a much needed comprehensive account on material characterisation but, more importantly, explains how such characterisation can be conducted in a consistent and systematic manner. It also includes a ready-to-use open source code for UCs that can be downloaded from a companion site for potential users to utilise, adapt and expand as they wish.
1. Chapter 1. Introduction — background, objectives and basic concepts - p. 3 2. Chapter 2. Symmetry, symmetry transformations and symmetry conditions - p. 11 3. Chapter 3. Material categorisation and material characterisation - p. 43 4. Chapter 4. Representative volume elements and unit cells - p. 67 5. Chapter 5. Common erroneous treatments and their conceptual sources of errors - p. 79 6. Part Two. Consistent formulation of unit cells and representative volume elements - p. 104 7. Chapter 6. Formulation of unit cells - p. 107 8. 6.1. Introduction - p. 107 9. 6.2. Relative displacement field and rigid body rotations - p. 108 10. 6.3. Relative displacement boundary conditions for unit cells - p. 114 11. 6.4. Typical unit cells and their boundary conditions in terms of relative displacements - p. 115 12. 6.5. Requirements on meshing - p. 176 13. 6.6. Key degrees of freedom and average strains - p. 177 14. 6.7. Average stresses and effective material properties - p. 179 15. 6.8. Thermal expansion coefficients - p. 182 16. 6.9. “Sanity checks” as basic verifications - p. 183 17. 6.10. Concluding remarks - p. 185 18. Chapter 7. Periodic traction boundary conditions and the key degrees of freedom for unit cells - p. 189 19. 7.1. Introduction - p. 189 20. 7.2. Boundaries and boundary conditions for unit cells resulting from translational symmetries - p. 193 21. 7.3. Total potential energy and variational principle for unit cells under prescribed average strains - p. 197 22. 7.4. Periodic traction boundary conditions as the natural boundary conditions for unit cells - p. 198 23. 7.5. The nature of the reactions at the prescribed key degrees of freedom - p. 202 24. 7.6. Prescribed concentrated ‘forces’ at the key degrees of freedom - p. 209 25. 7.7. Examples - p. 211 26. 7.8. Conclusions - p. 219 27. Chapter 8. Further symmetries within a UC - p. 223 28. 8.1. Introduction - p. 223 29. 8.2. Further reflectional symmetries to existing translational symmetries - p. 225 30. 8.3. Further rotational symmetries to existing translational symmetries - p. 251 31. 8.4. Examples of mixed reflectional and rotational symmetries - p. 291 32. 8.5. Centrally reflectional symmetry - p. 303 33. 8.6. Guidance to the sequence of exploiting existing symmetries - p. 314 34. 8.7. Concluding statement - p. 315 35. Chapter 9. RVE for media with randomly distributed inclusions - p. 319 36. Chapter 10. The diffusion problem - p. 347 37. Chapter 11. Boundaries of applicability of representative volume elements and unit cells - p. 361 38. Part Three. Further developments - p. 367 39. Chapter 12. Applications to textile composites - p. 371 40. Chapter 13. Application of unit cells to problems of finite deformation - p. 417 41. Chapter 14. Automated implementation: UnitCells© composites characterisation code - p. 439 42. Index - p. 459
Numerical methods to estimate material properties usually involve analysis of a representative volume element (RVE) or unit cell (UC). The representative volume element (RVE) or unit cell (UC) is the smallest volume over which a measurement can be made that will yield a value representative of the whole. RVEs and UCs are widely used in the characterisation of materials with multiscale architectures such as composites. However, finite element (FE) software packages such as Abaqus and Comsol MultiPhysics do not offer the capability for RVE and UC modelling directly on their own. To apply them to analyse RVEs and UCs, the generation of the FE models for them, the imposition of boundary conditions, and the extraction of directly relevant results are essentially the responsibility of the user. These have tended to be incorrectly implemented by users! For the first time, this book will provide a comprehensive account on correct modelling of RVEs and UCs, which will eliminate any uncertainties and ambiguities.The book offers a complete and thorough review on the subject of RVEs and UCs, establishing a framework on a rigorous mathematical and mechanical basis to ensure that basic concepts, such as symmetry and free body diagrams, are applied correctly and consistently. It also demonstrates to readers that rigorous applications of mathematics and mechanics are meant to make things clear, consistent, thorough and, most of all, simple and easy to follow, rather than the opposite as many perceive. As a result, the book shows that the appropriate use of RVEs and UCs can deliver an effective and reliable means of material characterisation. It not only provides a much needed comprehensive account on material characterisation but, more importantly, explains how such characterisation can be conducted in a consistent and systematic manner. It also includes a ready-to-use open source code for UCs that can be downloaded from a companion site for potential users to utilise, adapt and expand as they wish.
1. Chapter 1. Introduction — background, objectives and basic concepts - p. 3 2. Chapter 2. Symmetry, symmetry transformations and symmetry conditions - p. 11 3. Chapter 3. Material categorisation and material characterisation - p. 43 4. Chapter 4. Representative volume elements and unit cells - p. 67 5. Chapter 5. Common erroneous treatments and their conceptual sources of errors - p. 79 6. Part Two. Consistent formulation of unit cells and representative volume elements - p. 104 7. Chapter 6. Formulation of unit cells - p. 107 8. 6.1. Introduction - p. 107 9. 6.2. Relative displacement field and rigid body rotations - p. 108 10. 6.3. Relative displacement boundary conditions for unit cells - p. 114 11. 6.4. Typical unit cells and their boundary conditions in terms of relative displacements - p. 115 12. 6.5. Requirements on meshing - p. 176 13. 6.6. Key degrees of freedom and average strains - p. 177 14. 6.7. Average stresses and effective material properties - p. 179 15. 6.8. Thermal expansion coefficients - p. 182 16. 6.9. “Sanity checks” as basic verifications - p. 183 17. 6.10. Concluding remarks - p. 185 18. Chapter 7. Periodic traction boundary conditions and the key degrees of freedom for unit cells - p. 189 19. 7.1. Introduction - p. 189 20. 7.2. Boundaries and boundary conditions for unit cells resulting from translational symmetries - p. 193 21. 7.3. Total potential energy and variational principle for unit cells under prescribed average strains - p. 197 22. 7.4. Periodic traction boundary conditions as the natural boundary conditions for unit cells - p. 198 23. 7.5. The nature of the reactions at the prescribed key degrees of freedom - p. 202 24. 7.6. Prescribed concentrated ‘forces’ at the key degrees of freedom - p. 209 25. 7.7. Examples - p. 211 26. 7.8. Conclusions - p. 219 27. Chapter 8. Further symmetries within a UC - p. 223 28. 8.1. Introduction - p. 223 29. 8.2. Further reflectional symmetries to existing translational symmetries - p. 225 30. 8.3. Further rotational symmetries to existing translational symmetries - p. 251 31. 8.4. Examples of mixed reflectional and rotational symmetries - p. 291 32. 8.5. Centrally reflectional symmetry - p. 303 33. 8.6. Guidance to the sequence of exploiting existing symmetries - p. 314 34. 8.7. Concluding statement - p. 315 35. Chapter 9. RVE for media with randomly distributed inclusions - p. 319 36. Chapter 10. The diffusion problem - p. 347 37. Chapter 11. Boundaries of applicability of representative volume elements and unit cells - p. 361 38. Part Three. Further developments - p. 367 39. Chapter 12. Applications to textile composites - p. 371 40. Chapter 13. Application of unit cells to problems of finite deformation - p. 417 41. Chapter 14. Automated implementation: UnitCells© composites characterisation code - p. 439 42. Index - p. 459
