In
continuum mechanics an eigenstrain is any mechanical
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
in a material that is not caused by an external mechanical stress, with
thermal expansion
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kinetic ...
often given as a familiar example. The term was coined in the 1970s by
Toshio Mura, who worked extensively on generalizing their mathematical treatment.
A non-uniform distribution of eigenstrains in a material (e.g., in a
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 ...
) leads to corresponding eigenstresses, which affect the mechanical properties of the material.
Overview
Many distinct physical causes for eigenstrains exist, such as
crystallographic defects
A crystallographic defect is an interruption of the regular patterns of arrangement of atoms or molecules in crystalline solids. The positions and orientations of particles, which are repeating at fixed distances determined by the unit cell param ...
, thermal expansion, the inclusion of additional phases in a material, and previous plastic strains.
All of these result from internal material characteristics, not from the application of an external mechanical load. As such, eigenstrains have also been referred to as “stress-free strains”
and “inherent strains”.
When one region of material experiences a different eigenstrain than its surroundings, the restraining effect of the surroundings leads to a stress state on both regions.
Analyzing the distribution of this
residual stress
In materials science and solid mechanics, residual stresses are stresses that remain in a solid material after the original cause of the stresses has been removed. Residual stress may be desirable or undesirable. For example, laser peening i ...
for a known eigenstrain distribution or inferring the total eigenstrain distribution from a partial data set are both two broad goals of eigenstrain theory.
Analysis of eigenstrains and eigenstresses
Eigenstrain analysis usually relies on the assumption of
linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
, such that different contributions to the total strain
are additive. In this case, the total strain of a material is divided into the elastic strain e and the inelastic eigenstrain
:
:
where
and
indicate the directional components in 3 dimensions in
Einstein notation.
Another assumption of linear elasticity is that the stress
can be linearly related to the elastic strain
and the stiffness
by
Hooke’s Law:
:
In this form, the eigenstrain is not in the equation for stress, hence the term "stress-free strain". However, a non-uniform distribution of eigenstrain alone will cause elastic strains to form in response, and therefore a corresponding elastic stress. When performing these calculations, closed-form expressions for
(and thus, the total stress and strain fields) can only be found for specific geometries of the distribution of
.
Ellipsoidal inclusion in an infinite medium
One of the earliest examples providing such a closed-form solution analyzed a ellipsoidal inclusion of material
with a uniform eigenstrain, constrained by an infinite medium
with the same elastic properties.
This can be imagined with the figure on the right. The inner ellipse represents the region
. The outer region represents the extent of
if it fully expanded to the eigenstrain without being constrained by the surrounding
. Because the total strain, shown by the solid outlined ellipse, is the sum of the elastic and eigenstrains, it follows that in this example the elastic strain in the region
is negative, corresponding to a compression by
on the region
.
The solutions for the total stress and strain within
are given by:
:
:
Where
is the Eshelby Tensor, whose value for each component is determined only by the geometry of the ellipsoid. The solution demonstrates that the total strain and stress state within the inclusion
are uniform. Outside of
, the stress decays towards zero with increasing distance away from the inclusion. In the general case, the resulting stresses and strains may be asymmetric, and due to the asymmetry of
, the eigenstrain may not be coaxial with the total strain.
Inverse problem
Eigenstrains and the residual stresses that accompany them are difficult to measure (see:
Residual stress
In materials science and solid mechanics, residual stresses are stresses that remain in a solid material after the original cause of the stresses has been removed. Residual stress may be desirable or undesirable. For example, laser peening i ...
). Engineers can usually only acquire partial information about the eigenstrain distribution in a material. Methods to fully map out the eigenstrain, called the inverse problem of eigenstrain, are an active area of research.
Understanding the total residual stress state, based on knowledge of the eigenstrains, informs the design process in many fields.
Applications
Structural engineering
Residual stresses, e.g. introduced by manufacturing processes or by welding of structural members, reflect the eigenstrain state of the material.
This can be unintentional or by design, e.g.
shot peening
Shot peening is a cold working process used to produce a compressive residual stress layer and modify the mechanical properties of metals and composites. It entails striking a surface with shot (round metallic, glass, or ceramic particles) with ...
. In either case, the final stress state can affect the fatigue, wear, and corrosion behavior of structural components.
Eigenstrain analysis is one way to model these residual stresses.
Composite materials
Since composite materials have large variations in the thermal and mechanical properties of their components, eigenstrains are particularly relevant to their study. Local stresses and strains can cause decohesion between composite phases or cracking in the matrix. These may be driven by changes in temperature, moisture content, piezoelectric effects, or phase transformations. Particular solutions and approximations to the stress fields taking into account the periodic or statistical character of the composite material's eigenstrain have been developed.
Strain engineering
Lattice misfit strains are also a class of eigenstrains, caused by growing a crystal of one lattice parameter on top of a crystal with a different lattice parameter.
Controlling these strains can improve the electronic properties of an epitaxially grown semiconductor.
See:
strain engineering.
See also
Residual stress
In materials science and solid mechanics, residual stresses are stresses that remain in a solid material after the original cause of the stresses has been removed. Residual stress may be desirable or undesirable. For example, laser peening i ...
References
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Continuum mechanics