Micromechanics (or, more precisely, micromechanics of materials) is the analysis of
composite or
heterogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
materials on the level of the individual constituents that constitute these materials.
Aims of micromechanics of materials
Heterogeneous materials, such as
composites, solid
foam
Foams are materials formed by trapping pockets of gas in a liquid or solid.
A bath sponge and the head on a glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the ...
s,
polycrystals, or
bone
A bone is a rigid organ that constitutes part of the skeleton in most vertebrate animals. Bones protect the various other organs of the body, produce red and white blood cells, store minerals, provide structure and support for the body, an ...
, 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
Microstructure is the very small scale structure of a material, defined as the structure of a prepared surface of material as revealed by an optical microscope above 25× magnification. The microstructure of a material (such as metals, polymers ...
characteristics (shape, orientation, varying volume fraction, ..) of heterogeneous materials often leads to an
anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...
behaviour.
Anisotropic material models are available for
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
elasticity. In the
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
regime, the modeling is often restricted to
orthotropic material
In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength ca ...
models which does not capture the physics for all heterogeneous materials. Micromechanics goal is to predict 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, .]
Micromechanics allows to predicting multi-axial properties 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 involve 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 constituents: by identifying the behaviour based on
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
simulation results; by identifying the behaviour through an experimental campaign on each constituents; 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 is 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.
Generality on micromechanics
The 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
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such ...
rather than on atomistic approaches such as
nanomechanics
Nanomechanics is a branch of ''nanoscience'' studying fundamental ''mechanical'' (elastic, thermal and kinetic) properties of physical systems at the nanometer scale. Nanomechanics has emerged on the crossroads of biophysics, classical mechanics, s ...
or
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
. In addition to the mechanical responses of inhomogeneous materials, their
thermal conduction
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted .
Heat spontaneously flows along a te ...
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 Voigt (mainly written Vogt, also Voight) is a German surname, and may refer to:
*Alexander Voigt, German football player
*Angela Voigt, East German long jumper
*Christian August Voigt (1808–1890), Austrian anatomist
*Cynthia Voigt, author of bo ...
(1887) - Strains constant in composite,
rule of mixtures for
stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
components.
Reuss (1929)
- 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)
- 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)
-
Composite composed of cylindrical fibers surrounded by cylindrical matrix layer, cylindrical
elasticity solution. Analogous method for macroscopically
isotropic inhomogeneous materials: Composite Sphere Assemblage (CSA)
Hashin-Shtrikman Bounds - Provide
bounds on the
elastic moduli
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
and
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
s of transversally isotropic
composites (reinforced, e.g., by aligned continuous
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
s) and
isotropic composites (reinforced, e.g., by randomly positioned particles).
Self-Consistent Schemes
-
Effective medium approximations based on
Eshelby's elasticity solution for an inhomogeneity embedded in an infinite medium. Uses the material properties of the
composite for the infinite medium.
Mori-Tanaka Method
- 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
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
s 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
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...
(FEA)
Most such micromechanical methods use
periodic homogenization
Homogeneity is a sameness of constituent structure.
Homogeneity, homogeneous, or homogenization may also refer to:
In mathematics
* Transcendental law of homogeneity of Leibniz
* Homogeneous space for a Lie group G, or more general transformat ...
, which approximates
composites by periodic phase arrangements. A single repeating volume element is studied, appropriate
boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
being applied to extract the composite's macroscopic properties or responses. The Method of Macroscopic Degrees of Freedom
can be used with commercial
FE codes, whereas analysis based on
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
homogenization
Homogeneity is a sameness of constituent structure.
Homogeneity, homogeneous, or homogenization may also refer to:
In mathematics
* Transcendental law of homogeneity of Leibniz
* Homogeneous space for a Lie group G, or more general transformat ...
typically requires special-purpose codes.
The Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH)
and its development, Mechanics of Structural Genome (see below), are recent Finite Element based approaches for periodic homogenization.
In addition to studying periodic
microstructures
Microstructure is the very small scale structure of a material, defined as the structure of a prepared surface of material as revealed by an optical microscope above 25× magnification. The microstructure of a material (such as metals, polymers ...
, embedding models
and analysis using macro-homogeneous or mixed uniform boundary conditions
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
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linear ...
,
elastoplastic and
damage behavior.
Mechanics of Structure Genome (MSG)
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.
Using MSG, it is possible to directly compute structural properties of a beam, plate, shell or 3D solid in terms of its microstructural details.
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
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
continuity. It was developed into the High-Fidelity GMC (HFGMC), which uses quadratic approximation for the
displacement fields in the subcells.
Fast Fourier Transforms (FFT)
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.
FFT-based methods at present appear to provide the numerically most efficient approach to periodic homogenization of elastic materials.
Volume Elements
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
In machine learning, particularly in the creation of artificial neural networks, ensemble averaging is the process of creating multiple models and combining them to produce a desired output, as opposed to creating just one model. Frequently an ens ...
over a number of SVEs may be used for improving the approximations to the macroscopic responses.
See also
*
Micromechanics of Failure
*
Eshelby's inclusion
*
Representative elementary volume
*
Composite material
A composite material (also called a composition material or shortened to composite, which is the common name) is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or ...
*
Metamaterial
A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...
*
Negative index metamaterials
*
John Eshelby
*
Rodney Hill
*
Zvi Hashin
Zvi Hashin (1929–29 October 2017) was an Israeli mechanical engineer. He was a professor for engineering sciences at Tel Aviv University. In 2012, he won the Benjamin Franklin Medal in Mechanical Engineering, for his research on micro-mechani ...
References
External links
*
Micromechanics of Composites (Wikiversity learning project)
Further reading
*
*
*
*
*
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Composite materials