# What Is Eigenstrain?

Eigenstrains are internal mechanical deformations in a material not caused by external stress, such as thermal expansion, in continuum mechanics.

The term was introduced by Toshio Mura in the 1970s while working on mathematical generalization.

Non-uniform distribution of eigenstrains in a material, like a composite, results in corresponding eigenstresses that impact the material’s mechanical properties.

There are various internal material factors that cause eigenstrains, such as defects, thermal expansion, added phases, and prior plastic strains.

These are not due to external mechanical loads. Eigenstrains are also called “stress-free strains” or “inherent strains.”

When a material’s region has a different eigenstrain than its surroundings, it creates a residual stress state in both regions due to the restraining effect of the surroundings.

The objectives of eigenstrain theory are to either analyze residual stress distribution from known eigenstrain distribution or estimate total eigenstrain from partial data.

## What Are The Applications Of Eigenstrain?

1. Structural Engineering:

Residual stresses, caused by manufacturing processes or welding, represent the material’s eigenstrain state, which can impact the fatigue, wear, and corrosion of components. Eigenstrain analysis models these residual stresses.

1. Composite Materials:

Eigenstrains are significant in the study of composites, which have varying thermal and mechanical properties.

Local stresses and strains can cause decohesion or cracking, due to temperature changes, moisture, piezoelectric effects, or phase transformations.

Solutions and approximations for stress fields considering the periodic or statistical nature of the composite’s eigenstrain have been developed.

1. Strain Engineering:

Lattice misfit strains, caused by growing a crystal with different lattice parameters, are a type of eigenstrains. Controlling these strains can enhance the electronic properties of epitaxially grown semiconductors.

## What Is The Universality In Anisotropic Linear Anelasticity?

Linear elasticity refers to the concept where universal displacements, which can only be sustained by boundary tractions and can have arbitrary elastic constants, are specific to a given symmetry class.

Our previous work demonstrated that a bigger symmetry group leads to a larger space for universal displacements.

Now, we extend this idea to the field of linear anelasticity where total strain is separated into elastic strain and anelastic strain, commonly referred to as eigenstrain.

## How To Map Out Eigenstrains?

Measuring eigenstrains and their accompanying residual stresses is challenging for engineers, as they can only obtain limited information about the eigenstrain distribution.

Solving the inverse problem of eigenstrain, to fully determine eigenstrain, is a subject of ongoing research.

Knowing the complete residual stress state, through eigenstrain knowledge, plays a crucial role in design across many industries.

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