In solid mechanics, we often encounter nonelastic strains. Examples of nonelastic strains include:
Thermal expansion: Reversible changes in volume or shape driven by an increase or decrease in temperature.
Plastic strains: Permanent, irreversible deformations that remain after yield-inducing loads are removed (often due to the motion of a type of one-dimensional defect, called dislocations).
Initial strains:
Initial strains represent the deformation state that already exists in a material before any new external loads or current analyses are applied. We typically encounter initial strains as a result of a material's manufacturing or processing history. Rather than simulating the entire history of how a part was made, engineers treat the leftover deformation as a baseline "initial strain."Misfit strains:
Misfit strains occur at the microscale when a foreign particle or a new phase forms inside a host material, but its natural geometric size or lattice spacing does not perfectly match the surrounding atomic lattice. Because the two materials are bonded together, they are forced to stretch or compress to fit each other.- Example: In precipitation-hardened aluminum (like the aluminum used in aircraft), copper atoms cluster together to form tiny "precipitates" within the aluminum lattice. The natural crystal spacing of these copper-rich precipitates is slightly different from the surrounding aluminum matrix. The purely geometric difference between the precipitate's natural size and the "hole" it occupies in the aluminum matrix is the misfit strain. Another classic example is doping in semiconductors, where substituting a larger atom (like Phosphorus) into a Silicon lattice creates a local misfit strain.
Toshio Mura generally referred to these nonelastic strains as eigenstrains. J.D. Eshelby (1957) originally referred to them as "stress-free strains." Some researchers also call them "intrinsic strains." In this text, we will use Mura's terminology and denote the eigenstrain tensor by \epsilon_{ij}^*.
When localized eigenstrains develop within a body, the principle of continuum compatibility dictates that the material cannot physically tear or overlap. As a result, the surrounding material is forced to stretch, compress, or bend to accommodate the deformation and ensure a seamless fit. The internal stresses required to maintain this forced compatibility are called eigenstresses. Because no external forces are applied, this internal stress state must perfectly balance out across the part, making it self-equilibrating. In engineering practice, these self-equilibrated internal stresses, usually resulting from manufacturing processes or plastic yielding, are more commonly known as residual stresses.
(Note: The prefix "eigen" comes from German, meaning "proper" or "inherent." It is important to clarify that eigenstrains and eigenstresses have absolutely nothing to do with the mathematical eigenvalues of the strain or stress tensors. As we have discussed, those eigenvalues represent the principal strains and principal stresses, which are the normal strains or stresses in a specific coordinate system where the shear components are zero. Notice that in anisotropic materials, these two coordinate systems do not necessarily align.)
Example of an Eigenstrain
Suppose the temperature of an inhomogeneity (region Ω) enclosed by a larger, unconstrained material block increases by ΔT. The inhomogeneity "wants" to expand. If it were completely free and unattached, it would experience a purely stress-free thermal strain. For an isotropic material, this eigenstrain is written as:
\epsilon_{ij}^*=\alpha \delta_{ij} \Delta T,where α is the coefficient of thermal expansion and δij is the Kronecker delta (which equals 1 if i = j, and 0 if i ≠ j). This assumes the expansion is uniform in all normal directions. If the material is anisotropic, the expansion depends on orientation, and αδij is replaced by a general thermal expansion tensor αij.
Fictitious Eigenstrains (The Equivalent Inclusion Method)
It is important to note that eigenstrains are not always literal, physical deformations; they can also be employed as a highly effective mathematical tool. In micromechanics, engineers frequently analyze inhomogeneities, regions within a material that possess a different elastic stiffness than the surrounding matrix. Calculating the stress field around these mismatched materials under an applied external load is mathematically cumbersome. However, J.D. Eshelby (1957) demonstrated a brilliant workaround: we can mathematically replace the "foreign" particle with the original host material, provided we introduce a purely theoretical, fictitious eigenstrain (often called an equivalent eigenstrain) into that region. This fictitious eigenstrain is precisely calculated so that the resulting stress and strain fields perfectly match the reality of the stiff particle. This conceptual leap, known as the Equivalent Inclusion[1] Method, is a cornerstone of composite mechanics because it allows researchers to solve complex, multi-material problems using the vastly simpler equations of a uniform, homogeneous material.
The Decomposition of Strain
When dealing with infinitesimal deformations, the total strain 𝜖ij can be decomposed additively into the elastic strain \epsilon_{ij}^{\rm pl} and the eigenstrain \epsilon_{ij}^*:
\epsilon_{ij}=\epsilon_{ij}^{\rm el}+\epsilon_{ij}^*.It is crucial to distinguish between these three strains:
Total Strain (𝜖ij): This is the actual, physical geometric deformation of the material. Because the material must remain continuous (no tearing or overlapping), the total strain must be compatible. This means it is directly derived from a continuous displacement field ui:
\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial u_j}+\frac{\partial u_j}{\partial x_i}\right)Eigenstrain (𝜖*ij): The inherent, stress-free shape change the material wants to undergo.
Elastic Strain (𝜖ijel): The strain due to applied forces or internal stresses.
Hooke's Law with Eigenstrains
In a linear elastic material, the stress tensor is a linear function of elastic strain. Therefore, we have
\sigma_{ij}=\sum_{k=1}^3\sum_{l=1}^3 C_{ijkl}\epsilon_{ij}^{\rm el},where Cijkl is the fourth-order stiffness tensor.
Substituting the strain decomposition into this equation gives us the relationship between stress, total strain, and eigenstrain:
\sigma_{ij}=\sum_{k=1}^3\sum_{l=1}^3 C_{ijkl}(\epsilon_{ij}-\epsilon_{ij}^{*}).- Using index notation where summation over the repeated indices k and l is implied, the above equations in solid mechanics are written as:
References
Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 241(1226), 376–396. https://doi.org/10.1098/rspa.1957.0133
Korsunsky, A. M. (2017). A teaching essay on residual stresses and eigenstrains. Butterworth-Heinemann.
Mura, T. (1987). Micromechanics of defects in solids (2nd rev. ed.). Springer.
