Micromechanics

Micromechanics (or, more precisely, micromechanics of materials) is the analysis of composite or heterogeneous materials on the level of the individual constituents that constitute these materials.

Aims of micromechanics of materials

Heterogeneous materials, such as composites, solid foams and polycrystals, consist of clearly distinguishable constituents (or phases) that show different mechanical and physical material properties.

Given the (linear and/or nonlinear) material properties of the constituents, one important goal of micromechanics of materials consists of predicting the response of the heterogeneous material on the basis of the geometries and properties of the individual phases, a task known as homogenization. The benefit of homogenization is that the behavior of a heterogeneous material can be determined without resorting to testing it. Such tests may be expensive and involve a large number of permutations (e.g., in the case of composites: constituent material combinations; fiber and particle volume fractions; fiber and particle arrangements; and processing histories). Furthermore, continuum micromechanics can predict the full multi-axial properties and responses of inhomogeneous materials, which are often anisotropic. Such properties are often difficult to measure experimentally, but knowing what they are is a requirement, e.g. for structural analysis involving composites. To rely on micromechanics, the particular micromechanics theory must be validated through comparison to experimental data.

The second main task of micromechanics of materials is localization, which aims at evaluating the local (stress and strain) fields in the phases for given macroscopic load states, phase properties, and phase geometries. Such knowledge is especially important in understanding and describing material damage and failure.

Most methods in micromechanics of materials are based on continuum mechanics rather than on atomistic approaches such as molecular dynamics. In addition to the mechanical responses of inhomogeneous materials, their thermal conduction behavior and related problems can be studied with analytical and numerical continuum methods. All these approaches may be subsumed under the name of "continuum micromechanics".

Analytical methods of continuum micromechanics

Voigt^[1] (1887) - Strains constant in composite, Rule of Mixtures for stiffness components.

Reuss (1929)^[2] - Stresses constant in composite, Rule of Mixtures for compliance components.

Strength of Materials (SOM) - Longitudinally: strains constant in composite, stresses volume-additive. Transversely: stresses constant in composite, strains volume-additive.

Vanishing Fiber Diameter (VFD)^[3] - Combination of average stress and strain assumptions visualized as each fiber having a vanishing diameter yet finite volume.

Composite Cylinder Assemblage (CCA)^[4] - Composite composed of cylindrical fibers surrounded by cylindrical matrix, cylindrical elasticity solution. Analogous method for composite reinforced by spherical particles: Composite Sphere Assemblage (CSA)^[5]

Hashin-Shtrikman Bounds - Provide bounds on the elastic moduli and tensors of transversally isotropic composites^[6] (reinforced,e.g., by aligned continuous fibers) and isotropic composites^[7] (reinforced, e.g., by randomly positioned particles).

Self-Consistent Scheme^[8] - Effective medium theory based on Eshelby^[9] elasticity solution for inhomogeneity in infinite medium. Uses material properties of composite for infinite medium.

Mori-Tanaka Method^[10][11] - Effective field theory based on Eshelby's^[9] elasticity solution for inhomogeneity in infinite medium. Fourth-order tensor relates average inclusion strain to average matrix strain and approximately accounts for fiber interaction effects.

Numerical approaches to continuum micromechanics

Finite Element Analysis (FEA) based methods - Most such micromechanical methods use periodic homogenization, which approximates composites by periodic phase arrangements, explicitly models a repeating unit cell, and applies appropriate boundary conditions to extract the composite's properties or response. The Method of Macroscopic Degrees of Freedom^[12] can be used with commercial FEA codes, whereas analysis based on asymptotic homogenization^[13] typically requires special-purpose codes.

In addition to studying periodic microstructures, embedding models^[14] and analysis using macro-homogeneous or mixed uniform boundary conditions^[15] can be carried out on the basis of Finite Element models. Due to its high flexibility and efficiency, the FEA at present is the most widely used numerical tool in continuum micromechanics.

Generalized Method of Cells (GMC) - Explicitly considers fiber and matrix subcells from periodic repeating unit cell. Assumes 1st-order displacement field in subcells and imposes traction and displacement continuity.

High-Fidelity GMC (HFGMC) - Like GMC, but considers a quadratic displacement field in the subcells.

A recent approach, Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH),^[16] combines the merits of both asymptotic homogenization and FEA. A general-purpose micromechanics code, SwiftComp Micromechanics (formerly VAMUCH), accompanies this approach.^[17]

See also

• Composite material
• Metamaterial
• Negative index metamaterials
• Micromechanics of Composites (Wikiversity learning project)

References

1. ^ Voigt, W. (1887). "Theoretische Studien über die Elasticitätsverhältnisse der Krystalle". Abh.Kgl.Ges.Wiss.Göttingen, Math.Kl. 34: 3–51. 
2. ^ Reuss, A. (1929). "Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle". Journal of Applied Mathematics and Mechanics 9: 49–58. 
3. ^ Dvorak, G.J., Bahei-el-Din, Y.A. (1982). "Plasticity Analysis of Fibrous Composites". J.Appl.Mech. 49: 327–335. doi:10.1115/1.3162088. 
4. ^ Hashin, Z. (1965). "On Elastic Behavior of Fibre Reinforced Materials of Arbitrary Transverse Phase Geometry". J.Mech.Phys.Sol. 13: 119–134. doi:10.1016/0022-5096(65)90015-3. 
5. ^ Hashin, Z. (1962). "The Elastic Moduli of Heterogeneous Materials". J.Appl.Mech. 29: 143–150. 
6. ^ Hashin, Z., Shtrikman, S. (1963). "A Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials". J.Mech.Phys.Sol. 11: 127–140. doi:10.1016/0022-5096(62)90005-4. 
7. ^ Hashin, Z., Shtrikman, S. (1961). "Note on a Variational Approach to the Theory of Composite Elastic Materials". J.Franklin Inst. 271: 336–341. doi:10.1016/0016-0032(61)90032-1. 
8. ^ Hill, R. (1965). "A Self-Consistent Mechanics of Composite Materials". J.Mech.Phys.Sol. 13: 213–222. doi:10.1016/0022-5096(65)90010-4. 
9. ^ ^{a b} Eshelby, J.D. (1957). "The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems". Proc.Roy.Soc.London A241: 376–396. JSTOR 100095. 
10. ^ Mori, T., Tanaka, K. (1973). "Average Stress in the Matrix and Average Elastic Energy of Materials with Misfitting Inclusions". Acta metall. 21: 571–574. doi:10.1016/0001-6160(73)90064-3. 
11. ^ Benveniste Y. (1987). "A New Approach to the Application of Mori-Tanaka's Theory in Composite Materials". Mech.Mater. 6: 147–157. doi:10.1016/0167-6636(87)90005-6. 
12. ^ Michel, J.C., Moulinec, H., Suquet, P. (1999). "Effective Properties of Composite Materials with Periodic Microstructure: A Computational Approach". Comput.Meth.Appl.Mech.Engng. 172: 109–143. Bibcode 1999CMAME.172..109M. doi:10.1016/S0045-7825(98)00227-8. 
13. ^ Suquet, P. (1987). "Elements of Homogenization for Inelastic Solid Mechanics". In Sanchez-Palencia E., Zaoui A.. Homogenization Techniques in Composite Media. Berlin: Springer-Verlag. pp. 194–278. 
14. ^ González C., LLorca J. (2007). "Virtual Fracture Testing of Composites: A Computational Micromechanics Approach". Engng.Fract.Mech. 74: 1126–1138. doi:10.1016/j.engfracmech.2006.12.013. 
15. ^ Pahr D.H., Böhm H.J. (2008). "Assessment of Mixed Uniform Boundary Conditions for Predicting the Mechanical Behavior of Elastic and Inelastic Discontinuously Reinforced Composites". Comput.Model.Engng.Sci. 34: 117–136. 
16. ^ Yu, W., Tang, T. (2010). "Variational Asymptotic Method for Unit Cell Homogenization". In Gilat R., Banks-Sills L.. Advances in Mathematical Modeling and Experimental Methods for Materials and Structures: Solid Mechanics and Its Applications. Springer-Verlag. pp. 117–130. 
17. ^ Mendelkow, Jacoba (May 11, 2011). "USU Spinout Company Holds the Key to Future of Modeling Composite Materials". Commercialization and Regional Development at Utah State University. http://crd.usu.edu/htm/in-the-news/articleID=12786. Retrieved 27 July 2011. 

Further reading

• Mura, T. (1987). Micromechanics of Defects in Solids. Dordrecht: Martinus Nijhoff. ISBN 978-90-247-3256-2. 
• Aboudi, J. (1991). Mechanics of Composite Materials. Amsterdam: Elsevier. ISBN 0-444-88452-1. 
• Nemat-Nasser S., Hori M. (1993). Micromechanics: Overall Properties of Heterogeneous Solids. Amsterdam: North-Holland. ISBN 978-0444500847. 
• Torquato, S. (2002). Random Heterogeneous Media. New York: Springer-Verlag. ISBN 978-0387951676.