# Infinitesimal strain theory

**Infinitesimal strain theory**, also called small strain theory or small deformation theory, is the branch of continuum mechanics that describes the deformation of a solid when displacement gradients are small compared with unity, so that strain can be measured by a tensor that is linear in the displacement gradients.<sup>[1](https://www.ldeo.columbia.edu/~benh/courses/CD_2016spring/3_readings/Mase1970_Ch3n2.pdf)</sup> It is the kinematic foundation of linear elasticity and is used wherever deformations are a small fraction of body dimensions, as in most structural, geophysical, and microfabrication problems.<sup>[2](https://www.iue.tuwien.ac.at/phd/singulani/disssu4.html)</sup>

| Key fact | Detail |
|---|---|
| Defining tensor | \( \boldsymbol{\epsilon} = \tfrac{1}{2}(\mathbf{F} + \mathbf{F}^T) - \mathbf{I} \), equivalently \( \epsilon_{ij} = \tfrac{1}{2}(u_{i,j} + u_{j,i}) \)<sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup> |
| Validity condition | Displacement gradients small compared with unity; the practical limitation is small rotations, not small strains<sup>[1](https://www.ldeo.columbia.edu/~benh/courses/CD_2016spring/3_readings/Mase1970_Ch3n2.pdf)</sup><sup> • </sup><sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup>  |
| Independent components | Six: three normal strains and three shear strains, since the tensor is symmetric<sup>[2](https://www.iue.tuwien.ac.at/phd/singulani/disssu4.html)</sup> |
| Shear convention | Tensor shear terms are half the engineering shear strains: \( \gamma_{ij} = 2\epsilon_{ij} \)<sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup> |
| Volume change | Dilatation \( e = \mathrm{tr}(\boldsymbol{\epsilon}) = \nabla \cdot \mathbf{u} \), exact only to first order<sup>[4](https://biomechanics.stanford.edu/me338_09/me338_n06.pdf)</sup> |
| Compatibility | Six Saint-Venant equations, \( \nabla \times (\nabla \times \boldsymbol{\epsilon}) = \mathbf{O} \), ensure a single-valued displacement field<sup>[5](https://archive.nptel.ac.in/content/storage2/courses/112103167/module5/lec37.pdf)</sup> |
| Typical scale | Elastic-wave strains in the Earth are on the order of \( 10^{-4} \) to \( 10^{-5} \)<sup>[6](https://web.ics.purdue.edu/~nowack/geos557/lecture5-dir/Lecture5.htm)</sup> |

## The strain tensor and its derivation

The theory starts from a displacement field \( \mathbf{u} \), which accounts for every movement of material points, including rigid-body translation and rotation that do not constitute deformation.<sup>[2](https://www.iue.tuwien.ac.at/phd/singulani/disssu4.html)</sup> The deformation gradient \( \mathbf{F} \) contains both stretch and rotation. When all displacement gradients are small, the nonlinear (quadratic) terms in the exact finite-strain measures can be neglected, and the Green–Lagrange strain tensor linearizes to the infinitesimal strain tensor<sup>[7](https://courses.washington.edu/mengr503/Chapter_3.pdf)</sup>

\[ \boldsymbol{\epsilon} = \frac{1}{2}(\mathbf{F} + \mathbf{F}^T) - \mathbf{I} = \frac{1}{2}\left(\nabla \mathbf{u} + (\nabla \mathbf{u})^T\right). \]

This definition reproduces the familiar engineering measures: normal strain \( \epsilon = \Delta L / L_0 \), the fractional length change, and shear strain \( \gamma = D/T \), the change in right angle between two originally perpendicular line segments.<sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup> Normal strain is dimensionless; a rope that stretches 0.015 m from an initial 1 m, or 0.15 m from 10 m, has the same strain of 0.015, or 1.5%, which is what makes the definition useful.<sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup>

In Cartesian components the tensor reads

\[ \boldsymbol{\epsilon} = \begin{bmatrix} \dfrac{\partial u_x}{\partial X} & \tfrac{1}{2}\left(\dfrac{\partial u_x}{\partial Y} + \dfrac{\partial u_y}{\partial X}\right) & \tfrac{1}{2}\left(\dfrac{\partial u_x}{\partial Z} + \dfrac{\partial u_z}{\partial X}\right) \\ & \dfrac{\partial u_y}{\partial Y} & \tfrac{1}{2}\left(\dfrac{\partial u_y}{\partial Z} + \dfrac{\partial u_z}{\partial Y}\right) \\ \text{sym} & & \dfrac{\partial u_z}{\partial Z} \end{bmatrix}. \]

The diagonal entries \( \epsilon_{xx}, \epsilon_{yy}, \epsilon_{zz} \) are the normal strains, measuring relative extension or contraction along each axis; the off-diagonal entries measure local shearing, the change in angle between originally perpendicular directions.<sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup><sup> • </sup><sup>[6](https://web.ics.purdue.edu/~nowack/geos557/lecture5-dir/Lecture5.htm)</sup>  A deformation is thus represented by three normal and three shear components, one shear for each coordinate plane.<sup>[2](https://www.iue.tuwien.ac.at/phd/singulani/disssu4.html)</sup>

## Engineering and tensorial shear strain

A persistent source of error is the factor of two between conventions. The engineering shear strain \( \gamma_{xy} = D/T \) is the total angular change, while the tensor component \( \epsilon_{xy} \) is half of it, so that \( \gamma_{ij} = 2\epsilon_{ij} \). If \( \gamma_{xy} = 0.10 \), the strain tensor contains 0.05 in the corresponding off-diagonal slots.<sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup> The half factor is necessary so that coordinate transformations can be performed with the ordinary second-rank tensor rule \( \boldsymbol{\epsilon}' = \mathbf{Q} \cdot \boldsymbol{\epsilon} \cdot \mathbf{Q}^T \), the same rule used for the stress tensor.<sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup><sup> • </sup><sup>[8](https://www.continuummechanics.org/principalstrain.html)</sup>

## What the linearization buys, and what it costs

Because the quadratic terms are dropped, no distinction is made between material coordinates \( \mathbf{X} \) and spatial coordinates \( \mathbf{x} \), and the Green–Lagrange and Eulerian (Almansi) strain tensors coincide with \( \boldsymbol{\epsilon} \) in the small-strain regime.<sup>[7](https://courses.washington.edu/mengr503/Chapter_3.pdf)</sup><sup> • </sup><sup>[5](https://archive.nptel.ac.in/content/storage2/courses/112103167/module5/lec37.pdf)</sup>

The key limitation is frequently misstated. The small strain tensor is limited by small rotations, not by small strains: computed strains lose accuracy as rigid-body rotations become finite, while the theory is in fact not limited to small strains at all in that respect.<sup>[3](https://www.continuummechanics.org/smallstrain.html)</sup>

Under the small-strain assumption, an infinitesimal rigid motion has the form \( \mathbf{u} = \mathbf{W}\mathbf{X} + \mathbf{c} \), with \( \mathbf{W} \) a constant skew-symmetric tensor and \( \mathbf{c} \) a constant vector, equivalently \( \mathbf{u} = \boldsymbol{\omega} \times \mathbf{X} + \mathbf{c} \).<sup>[5](https://archive.nptel.ac.in/content/storage2/courses/112103167/module5/lec37.pdf)</sup>

## Volumetric and deviatoric decomposition

The strain tensor splits into a volume-changing part and a shape-changing part. The volumetric strain (dilatation) is

\[ e = \frac{dv - dV}{dV} = \epsilon_{11} + \epsilon_{22} + \epsilon_{33} + O(\epsilon_{ij}^2) = \mathrm{tr}(\boldsymbol{\epsilon}) = \nabla \cdot \mathbf{u}, \]

The volumetric part \( \boldsymbol{\epsilon}^{\mathrm{vol}} = \tfrac{1}{3} e \, \mathbf{I} \) is a spherical tensor carrying the volume change, while the deviatoric part \( \boldsymbol{\epsilon}^{\mathrm{dev}} = \boldsymbol{\epsilon} - \boldsymbol{\epsilon}^{\mathrm{vol}} \) is traceless and carries the shape change at constant volume; incompressibility corresponds to \( \nabla \cdot \mathbf{u} = 0 \).<sup>[4](https://biomechanics.stanford.edu/me338_09/me338_n06.pdf)</sup>

## Principal strains

Because \( \boldsymbol{\epsilon} \) is symmetric, it has three real eigenvalues, the principal strains \( \lambda^{\epsilon}_1, \lambda^{\epsilon}_2, \lambda^{\epsilon}_3 \), obtained from \( \det(\boldsymbol{\epsilon} - \lambda \mathbf{I}) = 0 \), with mutually orthogonal principal directions \( \mathbf{n}^{\epsilon}_i \). In these directions the strain is purely normal and the shear contribution vanishes.<sup>[4](https://biomechanics.stanford.edu/me338_09/me338_n06.pdf)</sup> The eigenvalues are the roots of the characteristic equation

\[ (\lambda^{\epsilon})^3 - I_{\epsilon}(\lambda^{\epsilon})^2 + II_{\epsilon}\,\lambda^{\epsilon} - III_{\epsilon} = 0, \]

with invariants \( I_{\epsilon} = \mathrm{tr}(\boldsymbol{\epsilon}) \), \( II_{\epsilon} = \tfrac{1}{2}[\mathrm{tr}^2(\boldsymbol{\epsilon}) - \mathrm{tr}(\boldsymbol{\epsilon}^2)] \), and \( III_{\epsilon} = \det(\boldsymbol{\epsilon}) \).<sup>[4](https://biomechanics.stanford.edu/me338_09/me338_n06.pdf)</sup> The principal strains are unchanged by any coordinate transformation of the tensor.<sup>[8](https://www.continuummechanics.org/principalstrain.html)</sup>

## Compatibility of strain fields

Six independent strain components are built from only three displacement components, so a strain field specified arbitrarily cannot in general come from any continuous displacement field: the strain–displacement relations would over-determine the three unknown displacements.<sup>[7](https://courses.washington.edu/mengr503/Chapter_3.pdf)</sup> The conditions that a strain field must satisfy to guarantee a single-valued displacement field are the Saint-Venant [compatibility equations](https://www.edgepedia.org/compatibility-equations), written compactly as

\[ \nabla \times (\nabla \times \boldsymbol{\epsilon}) = \mathbf{O}, \]

which give six independent equations for the symmetric tensor.<sup>[5](https://archive.nptel.ac.in/content/storage2/courses/112103167/module5/lec37.pdf)</sup> Physically, if compatibility is violated, pieces of the deformed body cannot be reassembled without gaps or overlaps, an impossible state for a simply connected continuum.<sup>[9](https://mechcodex.com/learn/strength-of-materials/strain-tensor)</sup> Every strain field derived from a displacement field automatically satisfies these conditions; they matter when strains are prescribed directly, as in dislocation theory, where incompatible elastic strain reflects the underlying defect structure.<sup>[5](https://archive.nptel.ac.in/content/storage2/courses/112103167/module5/lec37.pdf)</sup><sup> • </sup><sup>[4](https://biomechanics.stanford.edu/me338_09/me338_n06.pdf)</sup>  Saint-Venant's condition also has a finite-strain counterpart for the Eulerian left Cauchy–Green deformation tensor, which reduces to the linear compatibility condition in the small-strain regime.<sup>[10](https://onlinelibrary.wiley.com/doi/full/10.1002/zamm.202100475)</sup>

## Use and typical magnitudes

The theory underlies linear elasticity, whose simplicity makes it the standard first model for predicting solid behavior despite its strong assumptions.<sup>[5](https://archive.nptel.ac.in/content/storage2/courses/112103167/module5/lec37.pdf)</sup> Its accuracy is adequate whenever strains and, critically, rotations are small: elastic strains in metals are typically below about one percent, and elastic-wave strains in the Earth are on the order of \( 10^{-4} \) to \( 10^{-5} \).<sup>[6](https://web.ics.purdue.edu/~nowack/geos557/lecture5-dir/Lecture5.htm)</sup>

## References

1. [Deformation and Strain, Mase (1970), Ch. 3, Columbia University course reading](https://www.ldeo.columbia.edu/~benh/courses/CD_2016spring/3_readings/Mase1970_Ch3n2.pdf)
2. [Infinitesimal Strain Theory, PhD dissertation chapter, TU Wien](https://www.iue.tuwien.ac.at/phd/singulani/disssu4.html)
3. [Small Strains, Continuum Mechanics (Bob McGinty)](https://www.continuummechanics.org/smallstrain.html)
4. [ME338 Kinematics lecture notes, Stanford University](https://biomechanics.stanford.edu/me338_09/me338_n06.pdf)
5. [Lecture 37: Linear Elasticity, NPTEL Continuum Mechanics (IITs/IISc)](https://archive.nptel.ac.in/content/storage2/courses/112103167/module5/lec37.pdf)
6. [Lecture 5: Strain and Rotation, Purdue University GEOS 557](https://web.ics.purdue.edu/~nowack/geos557/lecture5-dir/Lecture5.htm)
7. [An Introduction to Continuum Mechanics, 2nd ed., Ch. 3 (J.N. Reddy), University of Washington course text](https://courses.washington.edu/mengr503/Chapter_3.pdf)
8. [Principal Strains & Invariants, Continuum Mechanics (Bob McGinty)](https://www.continuummechanics.org/principalstrain.html)
9. [Strain Tensor, MechCODEX](https://mechcodex.com/learn/strength-of-materials/strain-tensor)
10. [Fülöp (2023), Explicit compatibility condition formula for the Eulerian left Cauchy–Green deformation tensor field, ZAMM](https://onlinelibrary.wiley.com/doi/full/10.1002/zamm.202100475)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Continuum, solid, and fluid mechanics › Solid mechanics › Deformation and shear modes › Strain and deformation measures*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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