# Micromechanics of composites

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Welcome to this learning project about **Micromechanics of composites**!

## Learning Project Summary[edit | edit source]

**Project code:****Suggested Prerequisites:****Time investment:**6 months**Assessment suggestions:****Portal:Enegineering and Technology****School:Engineering****Department:Mechanical Engineering****Stream:Applied Mechanics****Level:**Second year graduate

## Content summary[edit | edit source]

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[edit | edit source]

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[edit | edit source]

### Learning materials[edit | edit source]

- Review of some basic continuum mechanics
- Some basic ideas of micromechanics
- Appendix: Some useful results and proofs
- Proof 1: Tensor-vector identity - 1
- Proof 2: Tensor-vector identity - 2
- Proof 3: Surface and volume integral relation - 1
- Proof 4: Integral of a cross product
- Proof 5: Surface and volume integral relation - 2
- Proof 6: Curl of a gradient - 1
- Proof 7: Curl of a gradient - 2
- Proof 8: Relation between axial vector and displacement
- Proof 9: Relation between axial vector and strain
- Proof 10: Rigid body motion
- Proof 11: More tensor identities
- Proof 12: Relation between volume averaged fields
- Proof 13: Average stress power identity - Cauchy stress
- Proof 14: Average stress power identity - 1st P-K stress

### Readings and other resources[edit | edit source]

#### Primary texts[edit | edit source]

- 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[edit | edit source]

- 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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