Micromechanics of composites

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Welcome to this learning project about Micromechanics of composites!

Learning Project Summary

• Project code:
• Suggested Prerequisites:
  • Introduction to Elasticity
  • Continuum mechanics
• Time investment: 6 months
• Assessment suggestions:
• Portal:Enegineering and Technology
• School:Engineering
• Department:Mechanical Engineering
• Stream:Applied Mechanics
• Level: Second year graduate

Content summary

This course is the micromechanics of composite materials. The purpose is to show you ways in which micromechanics may be used to determine the effective properties of composites.

Goals

This learning project aims to

• Show you some of the fundamental theorems in the micromechanics of composites.
• Give you a feel for how the theory can be used to determine the effective properties of composites.
• Give you an idea about numerical approaches based on the theory.

Contents

Learning materials

1. Review of some basic continuum mechanics
   1. Conservation of mass
   2. Balance of linear momentum
   3. Balance of angular momentum
   4. Balance of energy
2. Some basic ideas of micromechanics
   1. The RVE and governing equations
   2. Infinitesimal deformations
      1. Average strain in a RVE
      2. Average displacement in a RVE
      3. Average stress in a RVE
      4. Average stress power in a RVE
   3. Finite deformations
      1. Average deformation gradient in a RVE
      2. Average velocity gradient in a RVE
      3. Average stress in a RVE
      4. Average stress power in a RVE
3. Appendix: Some useful results and proofs
   1. Proof 1: Tensor-vector identity - 1
   2. Proof 2: Tensor-vector identity - 2
   3. Proof 3: Surface and volume integral relation - 1
   4. Proof 4: Integral of a cross product
   5. Proof 5: Surface and volume integral relation - 2
   6. Proof 6: Curl of a gradient - 1
   7. Proof 7: Curl of a gradient - 2
   8. Proof 8: Relation between axial vector and displacement
   9. Proof 9: Relation between axial vector and strain
   10. Proof 10: Rigid body motion
   11. Proof 11: More tensor identities
   12. Proof 12: Relation between volume averaged fields
   13. Proof 13: Average stress power identity - Cauchy stress
   14. Proof 14: Average stress power identity - 1st P-K stress

Readings and other resources

Primary texts

• S. Nemat-Nasser and M. Hori, 1993, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland.
• G. W. Milton, 2002, The Theory of Composites, Cambridge University Press.
• S. Torquato, 2002, Random Heterogeneous Materials, Springer.

Other reading materials

• T. Belytschko, W. K. Liu, and B. Moran. Nonlinear Finite Elements for Continua and Structures. John Wiley and Sons, Ltd., New York, 2000.
• J. Bonet and R. D. Wood. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 1997.
• F. Costanzo, G. L. Gray, and P. C. Andia. On the definitions of effective stress and deformation gradient for use in MD: Hill's macro-homogeneity and the virial theorem. Int. J. Engg. Sci., 43:533--555, 1985. http://dx.doi.org/10.1016/j.ijengsci.2004.12.002
• P. Chadwick. Continuum Mechanics: Concise Theory and Problems. George Allen and Unwin Ltd., London, 1976.
• M. E. Gurtin. The linear theory of elasticity. In C.~Truesdell, editor, Encyclopedia of Physics (Handbuch der Physik), volume VIa/2, pages 1--295. Springer-Verlag, Berlin, 1972.
• M. E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, New York, 1981.
• R. Hill. Elastic properties of reinforced solids : some theoretical principles. J. Mech. Phys. Solids, 11:357--372, 1963. http://dx.doi.org/10.1016/0022-5096(63)90036-X
• R. Hill. Theory of mechanical properties of fibre-strengthened materials: I. Elastic behavior. J. Mech. Phys. Solids, 12:199--212, 1964. http://dx.doi.org/10.1016/0022-5096(64)90019-5
• R. Hill. On constitutive macro-variables for heterogeneous solids at finite strain. Proc. Royal Soc. Lond. A, 326:131--147, 1972. http://dx.doi.org/10.1098/rspa.1972.0001
• R. Hill. On macroscopic effects of heterogeneity in elastoplastic media at finite strain. Math. Proc. Camb. Phil. Soc, 95:481--495, 1984. http://dx.doi.org/10.1017/S0305004100061818
• S. Nemat-Nasser. Averaging theorems in finite deformation plasticity. Mechanics of Materials, 31:493--523, 1999. http://dx.doi.org/10.1016/S0167-6636(98)00073-8
• S. Nemat-Nasser. Plasticity: A Treatise on Finite Deformation of Heteogeneous Inelastic Materials. Cambridge University Press, Cambridge, 2004.
• P. Perzyna. Constitutive equations for thermoinelasticity and instability phenomena in thermodynamic flow processes. In Stein E., editor, Progress in Computational Analysis of Inelastic Structures: CISM Courses and Lectures No. 321, pages 1--78. Springer-Verlag-Wien, New York, 1993.
• W. S. Slaughter. The Linearized Theory of Elasticity. Birhhauser, Boston, 2002.
• C. Truesdell and W. Noll. The Non-linear Field Theories of Mechanics. Springer-Verlag, New York, 1992.
• T. W. Wright. The Physics and Mathematics of Adiabatic Shear Bands. Cambridge University Press, Cambridge, UK, 2002.

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