Eshelby stress tensor

In continuum mechanics, the Eshelby stress tensor is a tensor field quantity that describes the energy changes and forces acting on defects, cracks, or dislocations inside a material reference lattice.

While a standard Cauchy stress tensor $𝝈$ describes the forces acting on the physical atoms or mass particles themselves, the Eshelby tensor looks at the material from a thermodynamic and structural standpoint. It describes forces associated with changes in the material configuration rather than ordinary spatial forces. The Eshelby tensor represents the spatial (or material) part of the broader relativistic stress-energy tensor mapped into classical solid mechanics.

The tensor is named after John Douglas Eshelby (1916–1981), a pioneering British materials scientist and physicist.

Definition

In the context of the $J$ -integral, the Eshelby tensor $𝐇$ (in index notation, $H_{k j}$ ) is defined as:

H_{k j} = W δ_{k j} - σ_{i j} \frac{\partial u_{i}}{\partial x_{k}}

where:

$W$ is the strain energy density of the material.
$δ_{k j}$ is the Kronecker delta (the identity matrix).
$σ_{i j}$ is the Cauchy stress tensor.
$\frac{\partial u_{i}}{\partial x_{k}}$ is the displacement gradient tensor.

The Eshelby tensor can also be defined via the Piola–Kirchhoff stress tensors:

H = W I - (\nabla u)^{T} P

where:

$W$ is the strain energy density in the original configuration,
$I$ is the identity matrix,
$\nabla u$ is the displacement gradient, and
$P$ is the first Piola-Kirchhoff stress tensor.

Connection to the J-integral

In a two-dimensional quasi-static system, the global material force vector $F_{k}$ acting on a defect enclosed by a contour $Γ$ is defined by the line integral of the Eshelby tensor:

F_{k} = \oint_{Γ} H_{k j} n_{j} d Γ

where $n_{j}$ is the outward unit normal vector to the contour.

Generalized Vector Form

The J-integral can be generalized into a configurational force vector $J_{i}$ representing the components of the energy release rate along any arbitrary coordinate axis $x_{i}$ . In terms of the Eshelby tensor (and in index notation), this is written as:

J_{i} := lim_{ε \to 0} \int_{Γ_{ε}} H_{i j} n_{j} d Γ = lim_{ε \to 0} \int_{Γ_{ε}} (W n_{i} - n_{j} σ_{j k} \frac{\partial u_{k}}{\partial x_{i}}) d Γ

where:

$Γ_{ε}$ is a shrinking contour of radius $ε$ immediately surrounding the crack tip.
$H_{i j}$ is the components of the Eshelby stress tensor.
All other field variables ( $W$ , $σ$ , $u$ ) are evaluated along the contour path as a function of spatial coordinates.

References

↑ Kienzler, R.; Herrmann, G. (2000). Properties of the Eshelby Tensor. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-57010-0_4. ISBN 978-3-642-63121-4. {{cite book}}: Unknown parameter |book-title= ignored (help)
↑ Desmorat, Rodrigue; Gravouil, Anthony; Kolev, Boris (2026). "Classical and relativistic balance of configurational forces". European Journal of Mechanics - A/Solids. 118: 106111. doi:10.1016/j.euromechsol.2026.106111.{{cite journal}}: CS1 maint: article number as page number (link)
↑ "Eshelby Tensor". MOOSE Framework. Idaho National Laboratory. Retrieved November 10, 2026.
↑ "J-integral Theory". COMSOL Documentation. COMSOL, Inc. Archived from the original on 2 May 2025. Retrieved 2 May 2025.{{cite web}}: CS1 maint: bot: original URL status unknown (link)

[1] Kienzler, R.; Herrmann, G. (2000). Properties of the Eshelby Tensor. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-57010-0_4. ISBN 978-3-642-63121-4. {{cite book}}: Unknown parameter |book-title= ignored (help)

[2] Desmorat, Rodrigue; Gravouil, Anthony; Kolev, Boris (2026). "Classical and relativistic balance of configurational forces". European Journal of Mechanics - A/Solids. 118: 106111. doi:10.1016/j.euromechsol.2026.106111.{{cite journal}}: CS1 maint: article number as page number (link)

[3] "Eshelby Tensor". MOOSE Framework. Idaho National Laboratory. Retrieved November 10, 2026.

[COMSOL_Doc-4] "J-integral Theory". COMSOL Documentation. COMSOL, Inc. Archived from the original on 2 May 2025. Retrieved 2 May 2025.{{cite web}}: CS1 maint: bot: original URL status unknown (link)

Eshelby stress tensor

Definition

Connection to the J-integral

Generalized Vector Form

See also

References