Micromechanics
Encyclopedia
Micromechanics is the analysis of composite
or heterogeneous materials on the level of the individual constituents that constitute these materials.
, solid foam
s 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".
(1887) - Strains constant in composite, Rule of Mixtures for stiffness 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 visualized as each fiber having a vanishing diameter yet finite volume.
Composite Cylinder Assemblage (CCA) - Composite
composed of cylindrical fibers surrounded by cylindrical matrix, cylindrical elasticity
solution. Analogous method for composite
reinforced by spherical particles: Composite Sphere Assemblage (CSA)
Hashin-Shtrikman Bounds - Provide bounds on the elastic moduli
and tensor
s of transversally isotropic composites
(reinforced,e.g., by aligned continuous fibers) and isotropic composites
(reinforced, e.g., by randomly positioned particles).
Self-Consistent Scheme - Effective medium theory
based on Eshelby
elasticity
solution for inhomogeneity in infinite medium. Uses material properties of composite
for infinite medium.
Mori-Tanaka Method - Effective field theory based on Eshelby's
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.
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 can be used with commercial FEA codes, whereas analysis based on asymptotic
homogenization typically requires special-purpose codes.
In addition to studying periodic microstructures, embedding models and analysis using macro-homogeneous or mixed uniform boundary conditions 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), combines the merits of both asymptotic homogenization and FEA. A general-purpose micromechanics code, SwiftComp Micromechanics (formerly VAMUCH), accompanies this approach.
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
or heterogeneous materials on the level of the individual constituents that constitute these materials.
Aims of micromechanics of materials
Heterogeneous materials, such as compositesComposite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
, solid foam
Foam
-Definition:A foam is a substance that is formed by trapping gas in a liquid or solid in a divided form, i.e. by forming gas regions inside liquid regions, leading to different kinds of dispersed media...
s and polycrystals, consist of clearly distinguishable constituents (or phases) that show different mechanical and physical material properties
Material properties (thermodynamics)
The thermodynamic properties of materials are intensive thermodynamic parameters which are specific to a given material. Each is directly related to a second order differential of a thermodynamic potential...
.
Given the (linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
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
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
: 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
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...
. Such properties are often difficult to measure experimentally, but knowing what they are is a requirement, e.g. for structural analysis
Structural analysis
Structural analysis is the determination of the effects of loads on physical structures and their components. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, vehicles, machinery, furniture, attire, soil strata, prostheses and...
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
Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...
) 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
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
rather than on atomistic approaches such as molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
. In addition to the mechanical responses of inhomogeneous materials, their thermal conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....
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
VoigtWoldemar Voigt
Woldemar Voigt was a German physicist, who taught at the Georg August University of Göttingen. Voigt eventually went on to head the Mathematical Physics Department at Göttingen and was succeeded in 1914 by Peter Debye, who took charge of the theoretical department of the Physical Institute...
(1887) - Strains constant in composite, Rule of Mixtures for stiffness components.
Reuss (1929) - Stresses constant in composite, Rule of Mixtures for compliance components.
Strength of Materials (SOM) - Longitudinally: strains constant in composite
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
, stresses volume-additive. Transversely: stresses constant in composite
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
, strains volume-additive.
Vanishing Fiber Diameter (VFD) - Combination of average stress and strain assumptions visualized as each fiber having a vanishing diameter yet finite volume.
Composite Cylinder Assemblage (CCA) - Composite
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
composed of cylindrical fibers surrounded by cylindrical matrix, cylindrical elasticity
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
solution. Analogous method for composite
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
reinforced by spherical particles: Composite Sphere Assemblage (CSA)
Hashin-Shtrikman Bounds - Provide bounds on the elastic moduli
Elastic modulus
An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically when a force is applied to it...
and tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s of transversally isotropic composites
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
(reinforced,e.g., by aligned continuous fibers) and isotropic composites
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
(reinforced, e.g., by randomly positioned particles).
Self-Consistent Scheme - Effective medium theory
Effective Medium Approximations
Effective medium approximations or effective medium theory are physical models that describe the macroscopic properties of a medium based on the properties and the relative fractions of its components...
based on Eshelby
John D. Eshelby
John Douglas Eshelby was a scientist in micromechanics. His work has shaped the fields of defect mechanics and micromechanics of inhomogeneous solids for fifty years and provided the basis for the quantitative analysis of the controlling mechanisms of plastic deformation and fracture.Eshelby was...
elasticity
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
solution for inhomogeneity in infinite medium. Uses material properties of composite
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
for infinite medium.
Mori-Tanaka Method - Effective field theory based on Eshelby's
John D. Eshelby
John Douglas Eshelby was a scientist in micromechanics. His work has shaped the fields of defect mechanics and micromechanics of inhomogeneous solids for fifty years and provided the basis for the quantitative analysis of the controlling mechanisms of plastic deformation and fracture.Eshelby was...
elasticity
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
solution for inhomogeneity in infinite medium. Fourth-order tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
relates average inclusion strain
Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...
to average matrix strain
Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...
and approximately accounts for fiber interaction effects.
Numerical approaches to continuum micromechanics
Finite Element Analysis (FEA) based methods - Most such micromechanical methods use periodicPeriodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
homogenization, which approximates composites
Composite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
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 can be used with commercial FEA codes, whereas analysis based on asymptotic
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
homogenization typically requires special-purpose codes.
In addition to studying periodic microstructures, embedding models and analysis using macro-homogeneous or mixed uniform boundary conditions 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
Displacement field (mechanics)
A displacement field is an assignment of displacement vectors for all points in a region or body that is displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position...
in subcells and imposes traction and displacement
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...
continuity.
High-Fidelity GMC (HFGMC) - Like GMC, but considers a quadratic displacement field
Displacement field (mechanics)
A displacement field is an assignment of displacement vectors for all points in a region or body that is displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position...
in the subcells.
A recent approach, Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH), combines the merits of both asymptotic homogenization and FEA. A general-purpose micromechanics code, SwiftComp Micromechanics (formerly VAMUCH), accompanies this approach.
See also
- Composite materialComposite materialComposite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
- MetamaterialMetamaterialMetamaterials are artificial materials engineered to have properties that may not be found in nature. Metamaterials usually gain their properties from structure rather than composition, using small inhomogeneities to create effective macroscopic behavior....
- Negative index metamaterialsNegative index metamaterialsNegative index metamaterials or negative index materials are artificial structures where the refractive index has a negative value over some frequency range. This does not occur in any known natural materials, and thus is only achievable with engineered structures known as metamaterials...
- Micromechanics of Composites (Wikiversity learning project)