# Micromechanics of composites

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

* Review of some basic continuum mechanics
** Conservation of mass
** Balance of linear momentum
** Balance of angular momentum
** Balance of energy
* Some basic ideas of micromechanics
** The RVE and governing equations
** Infinitesimal deformations
*** Average strain in a RVE
*** Average displacement in a RVE
*** Average stress in a RVE
*** Average stress power in a RVE
** Finite deformations
*** Average deformation gradient in a RVE
*** Average velocity gradient in a RVE
*** Average stress in a RVE
*** Average stress power in a RVE
* 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
#### 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.
* Learning materials
* Learning materials and learning projects are located in the main Wikiversity namespace. Simply make a link to the name of the lesson (lessons are independent pages in the main namespace) and start writing!
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