Mechanics of deformable solids Definition

Infinitesimal strain theory

Builds on Deformation Strain tensor Green–Lagrange strain tensor

Definition

Let $u \colon B \to \R^3$ be the displacement of a deformation of an open set $B \subseteq \R^3$ . Infinitesimal strain theory is the approximation valid when $\norm{\nabla u(X)} \ll 1$ for all $X \in B$ , in which:

terms of order two and higher in $\nabla u$ are neglected;
the Green–Lagrange tensor is replaced by its linear part, the strain tensor $\varepsilon = \tfrac12 (\nabla u + (\nabla u)^T)$ ;
the reference and the deformed configurations are identified, so material and spatial coordinates are not distinguished.

Remarks

A uniform extension $u(X) = (10^{-3} X_1, 0, 0)$ satisfies the smallness condition. Its strain $\varepsilon = \operatorname{diag}(10^{-3}, 0, 0)$ differs from $E = \operatorname{diag}(10^{-3} + \tfrac12 10^{-6}, 0, 0)$ only in the neglected term. A rotation of the body by $90$ degrees does not satisfy the smallness condition, since $\nabla u = Q - I$ has norm of order $1$ . The linear theory would assign it a spurious strain. The smallness of the strain alone is not enough: rotations must also be small.

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Tag	`ZCHHTZ1FHS`
URI	`m2://ZCHHTZ1FHS`
Kind	definition
Writing	W0 stub
MSC	74A05, 74B05
Defines	Infinitesimal strain theory
Uses	Deformation, Strain tensor, Green–Lagrange strain tensor
Changed	3 November 2026

Infinitesimal strain theory

Definition

Remarks

Reading path

Further reading

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