micromechanics

{{Unreferenced section|date=May 2010}}

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. While the constituents can often be modeled as having isotropic behaviour, the microstructure characteristics (shape, orientation, varying volume fraction, ..) of heterogeneous materials often leads to an anisotropic behaviour.

Anisotropic material models are available for linear elasticity. In the nonlinear regime, the modeling is often restricted to orthotropic material models which do not capture the physics for all heterogeneous materials. An important goal of micromechanics is predicting the anisotropic response of the heterogeneous material on the basis of the geometries and properties of the individual phases, a task known as homogenization.S. Nemat-Nasser and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, Second Edition, North-Holland, 1999, {{ISBN|0444500847}}.

Micromechanics allows predicting multi-axial responses that are often difficult to measure experimentally. A typical example is the out-of-plane properties for unidirectional composites.

The main advantage of micromechanics is to perform virtual testing in order to reduce the cost of an experimental campaign. Indeed, an experimental campaign of heterogeneous material is often expensive and involves a larger number of permutations: constituent material combinations; fiber and particle volume fractions; fiber and particle arrangements; and processing histories). Once the constituents properties are known, all these permutations can be simulated through virtual testing using micromechanics.

There are several ways to obtain the material properties of each constituent: by identifying the behaviour based on molecular dynamics simulation results; by identifying the behaviour through an experimental campaign on each constituent; by reverse engineering the properties through a reduced experimental campaign on the heterogeneous material. The latter option is typically used since some constituents are difficult to test, there are always some uncertainties on the real microstructure and it allows to take into account the weakness of the micromechanics approach into the constituents material properties. The obtained material models need to be validated through comparison with a different set of experimental data than the one use for the reverse engineering.

A key point of micromechanics of materials is the 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".

Voigt{{cite journal|author=Voigt, W.|title=Theoretische Studien über die Elasticitätsverhältnisse der Krystalle|journal=Abh. KGL. Ges. Wiss. Göttingen, Math. Kl.|volume=34|pages=3–51|year=1887}} (1887) - Strains constant in composite, rule of mixtures for stiffness components.

Reuss (1929){{cite journal|author=Reuss, A.|title=Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle|journal=Journal of Applied Mathematics and Mechanics|volume=9|issue=1|pages=49–58|year=1929|doi=10.1002/zamm.19290090104|bibcode=1929ZaMM....9...49R}} - 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){{cite journal|doi=10.1115/1.3162088|author=Dvorak, G.J., Bahei-el-Din, Y.A.|title=Plasticity Analysis of Fibrous Composites|journal=Journal of Applied Mechanics|volume=49|issue=2|pages=327–335|year=1982|bibcode = 1982JAM....49..327D }} - 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){{cite journal|author=Hashin, Z.|title=On Elastic Behavior of Fibre Reinforced Materials of Arbitrary Transverse Phase Geometry|journal=J. Mech. Phys. Sol.|volume=13|issue=3|pages=119–134|year=1965|doi=10.1016/0022-5096(65)90015-3|bibcode = 1965JMPSo..13..119H }} - Composite composed of cylindrical fibers surrounded by cylindrical matrix layer, cylindrical elasticity solution. Analogous method for macroscopically isotropic inhomogeneous materials: Composite Sphere Assemblage (CSA){{cite journal|author=Hashin, Z.|title=The Elastic Moduli of Heterogeneous Materials|journal=Journal of Applied Mechanics|volume=29|issue=1|pages=143–150|year=1962|doi=10.1115/1.3636446|url=http://www.dtic.mil/get-tr-doc/pdf?AD=AD0245469|archive-url=https://web.archive.org/web/20170924114434/http://www.dtic.mil/get-tr-doc/pdf?AD=AD0245469|url-status=dead|archive-date=September 24, 2017|bibcode=1962JAM....29..143H}}

Hashin-Shtrikman Bounds - Provide bounds on the elastic moduli and tensors of transversally isotropic composites{{cite journal|author=Hashin, Z., Shtrikman, S.|title=A Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials|journal=J. Mech. Phys. Sol.|volume=11|issue=4|pages=127–140|year=1963|doi=10.1016/0022-5096(62)90005-4|bibcode = 1962JMPSo..10..343H }} (reinforced, e.g., by aligned continuous fibers) and isotropic composites{{cite journal|author=Hashin, Z., Shtrikman, S.|title=Note on a Variational Approach to the Theory of Composite Elastic Materials|journal=J. Franklin Inst.|volume=271|issue=4|pages=336–341|year=1961|doi=10.1016/0016-0032(61)90032-1}} (reinforced, e.g., by randomly positioned particles).

Self-Consistent Schemes{{cite journal|doi=10.1016/0022-5096(65)90010-4|author=Hill, R.|title=A Self-Consistent Mechanics of Composite Materials|journal=J. Mech. Phys. Sol.|volume=13|issue=4|pages=213–222|year=1965|bibcode = 1965JMPSo..13..213H |url=https://hal.archives-ouvertes.fr/hal-03619975/file/Hill1965.pdf }} - Effective medium approximations based on Eshelby's{{cite journal|author=Eshelby, J.D.|title=The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems|journal=Proceedings of the Royal Society|volume=A241|issue=1226|pages=376–396|year=1957|doi=10.1098/rspa.1957.0133|jstor=100095|bibcode=1957RSPSA.241..376E|s2cid=122550488|url=https://hal.archives-ouvertes.fr/hal-03619957/file/Eshelby1957.pdf }} elasticity solution for an inhomogeneity embedded in an infinite medium. Uses the material properties of the composite for the infinite medium.

Mori-Tanaka Method{{cite journal|author=Mori, T., Tanaka, K.|title=Average Stress in the Matrix and Average Elastic Energy of Materials with Misfitting Inclusions|journal=Acta Metall.|volume=21|issue=5|pages=571–574|year=1973|doi=10.1016/0001-6160(73)90064-3}}{{cite journal|author=Benveniste Y.|title=A New Approach to the Application of Mori-Tanaka's Theory in Composite Materials|journal=Mech. Mater.|volume=6|issue=2|pages=147–157|year=1987|doi=10.1016/0167-6636(87)90005-6}} - Effective field approximation based on Eshelby's 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.

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{{cite journal|author=Michel, J.C., Moulinec, H., Suquet, P.|title=Effective Properties of Composite Materials with Periodic Microstructure: A Computational Approach|journal=Comput. Meth. Appl. Mech. Eng.|volume=172|issue=1–4|pages=109–143|year=1999|doi=10.1016/S0045-7825(98)00227-8|bibcode=1999CMAME.172..109M}} can be used with commercial FE codes, whereas analysis based on asymptotic homogenization{{cite conference|author=Suquet, P.|title=Elements of Homogenization for Inelastic Solid Mechanics|book-title=Homogenization Techniques in Composite Media|editor1=Sanchez-Palencia E. |editor2=Zaoui A. |pages=194–278|publisher=Springer-Verlag|location=Berlin|year=1987|isbn=0387176160}} typically requires special-purpose codes.

The Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH){{cite journal|author=Yu, W., Tang, T.|title=Variational Asymptotic Method for Unit Cell Homogenization of Periodically Heterogeneous Materi-als|journal=International Journal of Solids and Structures|volume=44|issue=11–12|pages=3738–3755|doi=10.1016/j.ijsolstr.2006.10.020|year=2007}} and its development, Mechanics of Structural Genome (see below), are recent Finite Element based approaches for periodic homogenization. A general introduction to Computational Micromechanics can be found in Zohdi and Wriggers (2005).

In addition to studying periodic microstructures, embedding models{{cite journal|author1=González C. |author2=LLorca J. |title=Virtual Fracture Testing of Composites: A Computational Micromechanics Approach|journal=Eng. Fract. Mech.|volume=74|issue=7 |pages=1126–1138|year=2007|doi=10.1016/j.engfracmech.2006.12.013}} and analysis using macro-homogeneous or mixed uniform boundary conditions{{cite journal|author1=Pahr D.H. |author2=Böhm H.J. |title=Assessment of Mixed Uniform Boundary Conditions for Predicting the Mechanical Behavior of Elastic and Inelastic Discontinuously Reinforced Composites|journal=Computer Modeling in Engineering & Sciences|volume=34|pages=117–136|year=2008|doi=10.3970/cmes.2008.034.117}} 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, allowing, e.g., the handling of viscoelastic, elastoplastic and damage behavior.

A unified theory called mechanics of structure genome (MSG) has been introduced to treat structural modeling of anisotropic heterogeneous structures as special applications of micromechanics.{{cite journal|author=Yu W.|title=A Unified Theory for Constitutive Modeling of Composites|journal=Journal of Mechanics of Materials and Structures|volume=11|issue=4|pages=379–411|year=2016|doi=10.2140/jomms.2016.11.379|doi-access=free}} Using MSG, it is possible to directly compute structural properties of a beam, plate, shell or 3D solid in terms of its microstructural details.{{cite journal|author=Liu X., Yu W.|title=A Novel Approach to Analyze Beam-like Composite Structures Using Mechanics of Structure Genome|journal=Advances in Engineering Software|volume=100|pages=238–251|year=2016|doi=10.1016/j.advengsoft.2016.08.003}}

{{cite journal|author=Peng B., Goodsell J., Pipes R. B., Yu W.|title=Generalized Free-Edge Stress Analysis Using Mechanics of Structure Genome|journal=Journal of Applied Mechanics|volume=83|issue=10|pages=101013|year=2016|doi=10.1115/1.4034389|bibcode=2016JAM....83j1013P}}

{{cite journal|author=Liu X., Rouf K., Peng B., Yu W.|title=Two-Step Homogenization of Textile Composites Using Mechanics of Structure Genome|journal=Composite Structures|volume=171|pages=252–262|year=2017|doi=10.1016/j.compstruct.2017.03.029}}

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.{{cite journal|author1=Moulinec H. |author2=Suquet P. |title=A Numerical Method for Computing the Overall Response of Nonlinear Composites with Complex Microstructure|journal=Comput. Meth. Appl. Mech. Eng.|volume=157|issue=1–2 |pages=69–94|year=1997|doi=10.1016/S0045-7825(97)00218-1|bibcode = 1998CMAME.157...69M |arxiv=2012.08962|s2cid=120640232 }} 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 typically 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.{{cite journal|author1=Kanit T. |author2=Forest S. |author3=Galliet I. |author4=Mounoury V. |author5=Jeulin D. |title=Determination of the Size of the Representative Volume Element for Random Composites: Statistical and Numerical Approach|journal=Int. J. Sol. Struct. | volume=40 |issue=13–14 | pages=3647–3679 | year=2003 | doi=10.1016/S0020-7683(03)00143-4}}

• Micromechanics of Failure
• Eshelby's inclusion
• Representative elementary volume
• Composite material
• Metamaterial
• Negative index metamaterials
• John Eshelby
• Rodney Hill
• Zvi Hashin

{{Reflist}}

{{refbegin}}

• {{cite book|author=Mura, T.|year=1987|title=Micromechanics of Defects in Solids|publisher=Martinus Nijhoff|location=Dordrecht|isbn=978-90-247-3256-2}}
• {{cite book|author=Aboudi, J.|year=1991|title=Mechanics of Composite Materials|publisher=Elsevier|location=Amsterdam|isbn=0-444-88452-1}}
• {{cite book|author1=Nemat-Nasser S. |author2=Hori M. |title=Micromechanics: Overall Properties of Heterogeneous Solids|year=1993|location=Amsterdam|isbn=978-0-444-50084-7|publisher=North-Holland}}
• {{cite book|author=Torquato, S.|year=2002|title=Random Heterogeneous Materials|publisher=Springer-Verlag|location=New York|isbn=978-0-387-95167-6}}
• {{cite book|author=Nomura, Seiichi|year=2016|title=Micromechanics with Mathematica|location=Hoboken|publisher=Wiley|isbn=978-1-119-94503-1}}
• {{cite book|author=Zohdi, T. and Wriggers, P. |year=2005|title=Introduction to Computational Micromechanics|location=Heidelberg|publisher=Springer-Verlag|isbn=978-3-540-32360-0}}

{{refend}}

• Micromechanics of Composites (Wikiversity learning project)

Category:Composite materials