About Eigenstrains

Residual stresses in mechanical parts are stresses that exist in the absence of externally applied loads. Almost all manufacturing processes, including additive manufacturing, introduce residual stresses into mechanical parts. Residual stresses are sometimes intentionally introduced to improve the in-service response; for example, prestressed concrete slabs used in bridge construction. However, manufacturers often try to minimize residual stresses since they can cause fracture during the manufacturing process, lead to unwanted distortions, and significantly impact fatigue behavior. Three primary classes of manufacturing effects that can lead to residual stresses are as follows:

Mechanical (for example, inelastic deformation)
Thermal (for example, nonuniform thermal expansion)
Changes in material microstructure (for example, phase transformations)

Eigenstrains ( $ε_{s t r a i n}$ ), which are often referred to as inherent strains, have long been used in simulating residual stresses from welding operations. Thermal strains are a subset of eigenstrains.

In a linear elastic deformation, the stress induced by an eigenstrain can be represented as

σ = C \cdot (ε_{t o t a l} - ε_{e i g e n}) = C \cdot ε_{e l a s t i c},

where

$σ$ is the Cauchy stress,
$C$ is the elasticity matrix,
$ε_{t o t a l}$ is the total strain,
$ε_{e i g e n}$ is the eigenstrain, and
$ε_{e l a s t i c}$ is the elastic strain.

Using constitutive equations (such as the one shown above) eigenstrains can be used to compute residual stresses coming from mechanical, thermal, and microstructural sources.

An eigenstrain in 3D is represented as a standard strain tensor with six components:

ε_{e i g e n} = [\begin{matrix} \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{11} \\ \begin{matrix} ε_{22} \\ ε_{33} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} ε_{12} \end{matrix} \\ \begin{matrix} ε_{13} \\ ε_{23} \end{matrix} \end{matrix} \end{matrix}] .

The components of the eigenstrain are functions of many factors, including material properties, manufacturing processes, and thermal history. Various methods can be used to determine appropriate eigenstrains:

Destructive and nondestructive tests of manufactured parts
Numerical simulation
Analytical formulas for simple scenarios

Once an appropriate eigenstrain field has been determined, it can be applied in a layer-by-layer simulation to predict the residual stresses and distortions in a 3D printed component.