A reusable route from eigenstrain and mechanical equilibrium to elastic energy, variational driving forces and computational solvers.
The physical starting point is the inclusion thought experiment used in micromechanics. Imagine cutting a region out of an unstressed matrix. Let that free region undergo its stress-free transformation strain. Apply tractions to restore the shape needed to fit the original hole, place it back, weld the coherent interface, and finally allow the entire body to relax elastically.
{% include eshelby-schematic.html %}This construction separates transformation strain from elastic strain. Eshelby showed how an ellipsoidal inclusion behaves inside an elastic medium. Mura developed eigenstrain as a general language for defects and inclusions. Khachaturyan then expressed arbitrary populations of coherent transformed regions through shape or phase fields and solved their collective elastic interactions in Fourier space.
Foundational sources: J. D. Eshelby, “The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems,” Proceedings of the Royal Society A 241 (1957), 376–396; T. Mura, Micromechanics of Defects in Solids, 2nd ed. (1987); A. G. Khachaturyan, Theory of Structural Transformations in Solids, Chapter 7.
Elastic stress effects run through our work on multivariant transformations, phase separation in ternary alloys, coherent precipitates and Ni-base superalloys, ferroic domains, grain-boundary migration and deformation in heterogeneous materials. The models range from homogeneous microelasticity to elastically anisotropic and inhomogeneous systems, and from phase-field evolution to FFT-based elasticity, dislocation dynamics and crystal plasticity.
The formulation below introduces the common physical chain. A local stress-free transformation strain creates elastic strain when compatibility and boundary conditions prevent unconstrained relaxation. Mechanical equilibrium determines the relaxed field, and its elastic energy alters the thermodynamic driving force for microstructural evolution.
A microstructure may require several independent composition fields \(c_1,\ldots,c_{N_c}\), several structural order parameters \(\eta_1,\ldots,\eta_{N_\eta}\), or additional fields such as polarization, orientation and plastic strain. The eigenstrain is a constitutive map from all these local fields to a symmetric second-rank tensor:
The interpolation functions \(\theta_p\) may depend linearly or nonlinearly on several fields. They encode phase fractions, variants or other local structural states; the tensors \(\epsilon^{0(p)}_{ij}\) encode their stress-free distortions. Cross-couplings are allowed, so a contribution may depend on more than one composition or order parameter.
This is one multivariant example, not the general definition. The square makes a variant’s strain contribution invariant under \(\eta_a\rightarrow-\eta_a\).
This formulation joins Mura's general eigenstrain method, the periodic Green-operator viewpoint of Michel, Moulinec and Suquet, and our CICP reference-medium formulation. Follow the assumptions in order: first define the inclusion, then regularize its interface if needed, then choose how stiffness and eigenstrain vary, and finally specify the macroscopic mechanical loading.
Let \(\chi_\Omega\) be one inside an inclusion and zero in the matrix. For a homogeneous medium,
Mura's Green-function solution treats the gradient of eigenstrain as the source of the elastic field. At a perfectly bonded sharp interface,
Displacement and traction are continuous. Total strain and individual stress components need not be. For tangent \(\mathbf s\) and normal \(\mathbf n\), \([\![\epsilon_{ss}]\!]=0\) and the traction components \(\sigma_{nn}\) and \(\sigma_{sn}\) are continuous. This is the sharp benchmark that a diffuse calculation should approach.
Current width:
Here \(d\) is a signed-distance-like coordinate and \(w\) controls the transition width. This tanh profile prescribes a smooth geometry. It is not the same decision as selecting a constitutive interpolation \(h(\eta)\) for an evolving order parameter.
At finite width there is no mathematical jump surface. Displacement, strain and stress vary through a layer while \(\nabla\!\cdot\!\sigma=0\). Integrating equilibrium across a shrinking layer recovers \([\![\sigma]\!]\mathbf n=0\) as \(w\to0\), provided that there is no sliding, surface elasticity or singular interfacial force.
| Field and physical meaning | First choice | Reason |
|---|---|---|
| Composition or a true phase fraction | Linear | Vegard or mixture interpretation |
| Evolving phase field, \(0\leq\eta\leq1\) | Cubic | Correct endpoints and zero endpoint slopes |
| Extra-flat pure-phase response | Quintic comparison | First and second derivatives vanish at both endpoints |
| Prescribed diffuse geometry | Spatial tanh | The width has a direct geometric meaning |
| Signed amplitude with equivalent \(+\eta\) and \(-\eta\) | \(\eta^2\) | Preserves the required sign symmetry |
For an evolving phase field, a useful starting model is
The modulus term and the eigenstrain term must both be retained. For several phases, the weights must form a deliberate partition or a deliberately softer grain-boundary state. Independently overlapping interpolation functions can otherwise double-count stiffness and eigenstrain.
In Eqs. (2.1)–(2.16), the paper writes \(C(\mathbf r)=C^0+C'(\mathbf r)\), permits an arbitrary eigenstrain distribution and solves equilibrium iteratively with the Green tensor of the homogeneous reference medium. In polarization form,
This is the appropriate extension when stiffness varies in space. A single homogeneous elastic-energy kernel \(B(\mathbf n)\) must not be used as though it solved a modulus-contrast problem. The CICP polycrystal example uses rotated contributions weighted by \(\eta_g^2\), and explicitly compares sharp and diffuse grain-boundary descriptions. When \(\sum_g\eta_g^2<1\) in a boundary, that choice can create a physically distinct softer boundary region whose width affects the effective properties.
Read the CICP paper and publication details →
Primary references for this section: T. Mura, Micromechanics of Defects in Solids, 2nd ed., Chapter 1, especially §§1, 3 and 6; H. Moulinec and P. Suquet's periodic FFT cell formulation and its later mixed-control extensions; S. Bhattacharyya, S. Heo, K. Chang and L.-Q. Chen, Communications in Computational Physics 11 (2012), 726–738.
The Green operator determines the nonzero modes. The macroscopic mechanical boundary condition determines the zero mode. For homogeneous stiffness,
The displacement fluctuation is periodic. The affine part need not be.
Prescribe the zero-mode strain \(\bar\epsilon=E^{\rm applied}\) and hold it fixed. At every grid point, \(\epsilon(\mathbf x)=E^{\rm applied}+\widetilde\epsilon(\mathbf x)\). Equilibrate the nonzero modes and calculate the resulting macroscopic stress:
The homogeneous strain is applied everywhere, but the total local strain and stress are generally nonuniform because the periodic fluctuation responds to eigenstrain and elastic contrast.
Prescribe only the average stress. Do not assign the same stress to every grid point. The homogeneous strain is now unknown and must be adjusted until the equilibrated local field has the required average:
For fixed linear elasticity, calculate the response to independent unit homogeneous strains once. These response columns give \(C^{\rm eff}\). If \(\Sigma^0\) is the residual average stress at \(\bar\epsilon=0\), then
residualStress = equilibrate(meanStrain = zero).averageStress
for each independent strain component J:
unitStrain[J] = 1
response = equilibrate(meanStrain = unitStrain).averageStress
effectiveStiffness[:, J] = response - residualStress
meanStrain = inverse(effectiveStiffness)
* (targetStress - residualStress)
solution = equilibrate(meanStrain)
check solution.averageStress against targetStress
check divergence of the local stressWith nonlinear constitutive behavior, update the homogeneous strain with the current consistent effective tangent and repeat. Mixed control fixes selected strain components while solving the remaining zero-mode components from their target average stresses.
Hold selected components of \(\bar\epsilon\) fixed and solve the other zero-mode components from prescribed components of \(\Sigma\). For example, epitaxial in-plane strain can be combined with zero average normal and transverse shear stresses. The controlled components must be stated explicitly; “periodic boundary conditions” alone do not specify this macroscopic loading.
For an epitaxial film, prescribe the in-plane homogeneous strains and use mixed control for the normal components:
A very soft vacuum gives \(\sigma\mathbf n\approx0\) at the exposed surface. Exactly zero vacuum stiffness makes the displacement in the vacuum undetermined and can make a displacement-based solver singular. Validate the result against vacuum stiffness, vacuum thickness and grid resolution. For in-plane stress control, do not let arbitrary vacuum thickness dilute the reported load. Use a solid-volume average or the membrane resultant \(N_{\alpha\beta}=\int_{\rm solid} \sigma_{\alpha\beta}\,dz\).
Vacuum supplies an approximately traction-free surface. It does not represent pressure, bottom-plane clamping, bending tractions, surface elasticity or a nonperiodic lateral edge. Those require an explicit surface load or a solver with the corresponding boundary conditions.
A smooth picture is not automatically a diffuse-interface model. The field definition, interpolation law, interface width, macroscopic boundary condition and averaging measure all change the physical problem. The calculation becomes interpretable only when each choice is stated and independently tested.
Explore eigenstrain shapes, Mohr’s circle, habit lines, Fourier elastic-energy kernels, two-phase interactions and soft or hard precipitates with diagrams and independently tested calculations.
Open the interactive elasticity laboratory →A coherent microstructure cannot generally realize every local eigenstrain independently. Elastic strain is the difference between compatible total strain and eigenstrain. The constitutive law gives stress, and mechanical equilibrium determines the displacement field.
Repeated spatial indices are summed. Eigenstrain and elastic strain are distinct: eigenstrain describes the local stress-free transformation, while elastic strain is the constrained response that produces stress.
The closed-form Fourier kernel introduced in the linked derivation assumes linear elasticity, homogeneous stiffness, coherent interfaces, periodic fields and zero applied mean stress. It is a foundation, not a universal restriction. Our broader formulations treat elastic anisotropy, spatially varying stiffness, polycrystals, applied loading and coupling with plastic deformation. Direct iterative and equivalent-eigenstrain solvers are used when elastic inhomogeneity makes the homogeneous kernel insufficient.
Understand the assumptions behind shape-independent elastic energy, uniform strain inside ellipsoidal inclusions, and the anisotropic patterns in the worked example.
Explore the theorems and their limits →Follow the homogeneous-modulus microelasticity developed in Chapter 7 of A. G. Khachaturyan’s Theory of Structural Transformations in Solids, especially §§7.2–7.3. The derivation produces the Fourier-space elastic-energy kernel and its composition and order-parameter derivatives, including the real-space chain rule.
Read the derivations →The total free energy combines chemical, interfacial and elastic contributions. Elastic energy therefore enters the variational driving forces for conserved composition fields and nonconserved structural fields. Mechanical equilibrium is solved at each required stage so that the evolving microstructure and mechanical fields remain coupled.
This chain connects transformation strains with morphology, orientation, interactions, coarsening, splitting, rafting and variant selection. It also makes clear where model assumptions enter and which solver is valid for a given material system.
This teaching presentation follows a connected micromechanical lineage. Eshelby’s inclusion construction and Mura’s eigenstrain method establish the physical problem. Khachaturyan’s Chapter 7 extends it to arbitrary coherent multiphase microstructures, homogeneous-modulus elastic energy and inclusion interactions. Our Communications in Computational Physics paper develops a spectral iterative perturbation method for spatially varying stiffness and effective properties. My thesis appendices provide one detailed multivariant derivation; the wider research program includes both homogeneous and inhomogeneous formulations.
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