# Micromechanics of composites

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

## Learning Project Summary edit

**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

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

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

### Learning materials edit

- 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

#### Primary texts edit

- 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

- 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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