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, polycrystals, or bone, 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.[1] 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.

Because most heterogeneous materials show a statistical rather than a deterministic arrangement of the constituents, the methods of micromechanics are typically based on the concept of the representative volume element (RVE). An RVE is understood to be a sub-volume of an inhomogeneous medium that is of sufficient size for providing all geometrical information necessary for obtaining an appropriate homogenized behavior.

Most methods in micromechanics of materials are based on continuum mechanics rather than on atomistic approaches such as nanomechanics or 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[2] (1887) - Strains constant in composite, rule of mixtures for stiffness components.

Reuss (1929)[3] - 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)[4] - Combination of average stress and strain assumptions that can be visualized as each fiber having a vanishing diameter yet finite volume.

Composite Cylinder Assemblage (CCA)[5] - Composite composed of cylindrical fibers surrounded by cylindrical matrix layer, cylindrical elasticity solution. Analogous method for macroscopically isotropic inhomogeneous materials: Composite Sphere Assemblage (CSA)[6]

Hashin-Shtrikman Bounds - Provide bounds on the elastic moduli and tensors of transversally isotropic composites[7] (reinforced, e.g., by aligned continuous fibers) and isotropic composites[8] (reinforced, e.g., by randomly positioned particles).

Self-Consistent Schemes[9] - Effective medium approximations based on Eshelby's[10] elasticity solution for an inhomogeneity embedded in an infinite medium. Uses the material properties of the composite for the infinite medium.

Mori-Tanaka Method[11][12] - Effective field approximation based on Eshelby's[10] elasticity solution for inhomogeneity in infinite medium. As is typical for mean field micromechanics models, fourth-order concentration tensors relate the average stress or average strain tensors in inhomogeneities and matrix to the average macroscopic stress or strain tensor, respectively; inhomogeneity "feels" effective matrix fields, accounting for phase interaction effects in a collective, approximate way.

Numerical approaches to continuum micromechanics

Methods based on Finite Element Analysis (FEA)

Most such micromechanical methods use periodic homogenization, which approximates composites by periodic phase arrangements. A single repeating volume element is studied, appropriate boundary conditions being applied to extract the composite's macroscopic properties or responses. The Method of Macroscopic Degrees of Freedom[13] can be used with commercial FEA codes, whereas analysis based on asymptotic homogenization[14] typically requires special-purpose codes.

The Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH)[15] is a recent Finite Element (FE) based approach for periodic homogenization.

In addition to studying periodic microstructures, embedding models[16] and analysis using macro-homogeneous or mixed uniform boundary conditions[17] can be carried out on the basis of FE models. Due to its high flexibility and efficiency, 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. It was developed into the High-Fidelity GMC (HFGMC), which uses quadratic approximation for the displacement fields in the subcells.

A further group of periodic homogenization models make use of Fast Fourier Transforms (FFT), e.g., for solving an equivalent to the Lippmann–Schwinger equation.[18] FFT-based methods at present appear to provide the numerically most efficient approach to periodic homogenization of elastic materials.

Ideally, the volume elements used in numerical approaches to continuum micromechanics should be sufficiently big to fully describe the statistics of the phase arrangement of the material considered, i.e., they should be Representative Volume Elements (RVEs). In practice, smaller volume elements must be used due to limitations in available computational power. Such volume elements are often referred to as Statistical Volume Elements (SVEs). Ensemble averaging over a number of SVEs may be used for improving the approximations to the macroscopic responses.[19] Parallel finite element programs such as ParaFEM have enabled researchers to study full size Representative Volume Elements (RVEs) for meshes derived from 3D tomography data.[20]

See also

External links

References

  1. ↑ S. Nemat-Nasser and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, Second Edition, North-Holland, 1999, ISBN 0444500847.
  2. ↑ Voigt, W. (1887). "Theoretische Studien über die Elasticitätsverhältnisse der Krystalle". Abh. Kgl. Ges. Wiss. Göttingen, Math. Kl. 34: 3–51.
  3. ↑ Reuss, A. (1929). "Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle". Journal of Applied Mathematics and Mechanics. 9: 49–58. doi:10.1002/zamm.19290090104.
  4. ↑ Dvorak, G.J., Bahei-el-Din, Y.A. (1982). "Plasticity Analysis of Fibrous Composites". J. Appl. Mech. 49: 327–335. Bibcode:1982JAM....49..327D. doi:10.1115/1.3162088.
  5. ↑ Hashin, Z. (1965). "On Elastic Behavior of Fibre Reinforced Materials of Arbitrary Transverse Phase Geometry". J. Mech. Phys. Sol. 13: 119–134. Bibcode:1965JMPSo..13..119H. doi:10.1016/0022-5096(65)90015-3.
  6. ↑ Hashin, Z. (1962). "The Elastic Moduli of Heterogeneous Materials". J. Appl. Mech. 29: 143–150. doi:10.1115/1.3636446.
  7. ↑ 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. Bibcode:1962JMPSo..10..343H. doi:10.1016/0022-5096(62)90005-4.
  8. ↑ 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.
  9. ↑ Hill, R. (1965). "A Self-Consistent Mechanics of Composite Materials". J. Mech. Phys. Sol. 13: 213–222. Bibcode:1965JMPSo..13..213H. doi:10.1016/0022-5096(65)90010-4.
  10. 1 2 Eshelby, J.D. (1957). "The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems". Proceedings of the Royal Society. A241: 376–396. JSTOR 100095.
  11. ↑ 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.
  12. ↑ 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.
  13. ↑ Michel, J.C., Moulinec, H., Suquet, P. (1999). "Effective Properties of Composite Materials with Periodic Microstructure: A Computational Approach". Comput. Meth. Appl. Mech. Eng. 172: 109–143. Bibcode:1999CMAME.172..109M. doi:10.1016/S0045-7825(98)00227-8.
  14. ↑ 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. ISBN 0387176160.
  15. ↑ 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. ISBN 978-90-481-3466-3.
  16. ↑ González C.; LLorca J. (2007). "Virtual Fracture Testing of Composites: A Computational Micromechanics Approach". Eng. Fract. Mech. 74: 1126–1138. doi:10.1016/j.engfracmech.2006.12.013.
  17. ↑ 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. Eng. Sci. 34: 117–136. doi:10.3970/cmes.2008.034.117.
  18. ↑ Moulinec H.; Suquet P. (1997). "A Numerical Method for Computing the Overall Response of Nonlinear Composites with Complex Microstructure". Comput. Meth. Appl. Mech. Eng. 157: 69–94. Bibcode:1998CMAME.157...69M. doi:10.1016/S0045-7825(97)00218-1.
  19. ↑ Kanit T.; Forest S.; Galliet I.; Mounoury V.; Jeulin D. (2003). "Determination of the Size of the Representative Volume Element for Random Composites: Statistical and Numerical Approach". Int. J. Sol. Struct. 40: 3647–3679. doi:10.1016/S0020-7683(03)00143-4.
  20. ↑ Levrero, F.; Margetts, L.; et al. (2016). "Evaluating the macroscopic yield behaviour of trabecular bone using a nonlinear homogenisation approach". Journal of the Mechanical Behavior of Biomedical Materials. 61: 384–396. doi:10.1016/j.jmbbm.2016.04.008.

Further reading

  • Aboudi, J., Arnold, S.M., Bednarcyk, B.A. (2013). Micromechanics of Composite Materials A Generalized Multiscale Analysis Approach. Amsterdam: Elsevier. ISBN 978-0-12-397035-0. 
  • 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 978-0-444-50084-7. 
  • Torquato, S. (2002). Random Heterogeneous Materials. New York: Springer-Verlag. ISBN 978-0-387-95167-6. 
  • Nomura, Seiichi (2016). Micromechanics with Mathematica. Hoboken: Wiley. ISBN 978-1-119-94503-1. 
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