1 Introduction to Strain Theory and Kinematics

In continuum mechanics, strain is a measure of the deformation of a material body. Unlike displacement, which describes the change in position of a point, strain describes the change in relative distance between material points (stretching) and the change in angle between material lines (distortion).

1.1 Configurations and Coordinates

In continuum mechanics, we distinguish between the material body as it exists originally and how it exists after deformation. This distinction is crucial for defining the correct derivatives for strain calculations.

To describe motion, we define two specific configurations:

  1. Reference Configuration (\(\Omega_0\)): The state of the body at \(t=0\) (Original).

    • Coordinates: \(\mathbf{X}\) (Material coordinates).

    • These coordinates are “fixed” to the material particle. As the body deforms, a specific particle keeps its same label \(\mathbf{X}\).

  2. Current Configuration (\(\Omega_t\)): The state of the body at time \(t\) (Deformed).

    • Coordinates: \(\mathbf{x}\) (Spatial coordinates).

    • This is the new physical position of the particle in space.

1.1.1 The Motion Map

The motion is described by a function \(\boldsymbol{\chi}\) that maps a material point \(\mathbf{X}\) to its new position \(\mathbf{x}\):

\[\mathbf{x} = \boldsymbol{\chi}(\mathbf{X}, t)\]

The Displacement Vector \(\mathbf{u}\) connects the original position to the new position:

\[\mathbf{u}(\mathbf{X}, t) = \mathbf{x} - \mathbf{X}\]

1.1.2 The Deformation Gradient Tensor (\(\mathbf{F}\))

The fundamental measure of deformation is the Deformation Gradient, \(\mathbf{F}\). It maps infinitesimal line segments \(d\mathbf{X}\) from the reference configuration to \(d\mathbf{x}\) in the current configuration.

\[d\mathbf{x} = \mathbf{F} d\mathbf{X}\]

It is defined as the partial derivative of the current position with respect to the reference position:

\[\mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}}\]

Substituting \(\mathbf{x} = \mathbf{X} + \mathbf{u}\), we can express \(\mathbf{F}\) in terms of the displacement gradient:

\[\mathbf{F} = \frac{\partial (\mathbf{X} + \mathbf{u})}{\partial \mathbf{X}} = \frac{\partial \mathbf{X}}{\partial \mathbf{X}} + \frac{\partial \mathbf{u}}{\partial \mathbf{X}} = \mathbf{I} + \nabla_{\mathbf{X}} \mathbf{u}\]

Where \(\mathbf{I}\) is the identity tensor and \(\nabla \mathbf{u}\) is the displacement gradient tensor.

Note

Be careful not to confuse \(\mathbf{F}\) with its inverse. The derivative \(\partial \mathbf{X} / \partial \mathbf{x}\) represents \(\mathbf{F}^{-1}\), which maps spatial vectors back to the reference configuration.

Note

Additionally:

  • \(J = \det(\mathbf{F})\) represents the volume change (Jacobian).

  • \(\mathbf{F}\) contains information about both rotation and stretch.

1.1.3 Polar Decomposition

Since rigid body rotation should not induce strain, we often decompose \(\mathbf{F}\) into a rotation tensor (\(\mathbf{R}\)) and a stretch tensor (\(\mathbf{U}\) or \(\mathbf{V}\)):

\[\mathbf{F} = \mathbf{R}\mathbf{U} = \mathbf{V}\mathbf{R}\]

  • R: Orthogonal rotation tensor.

  • U: Right stretch tensor (defined in the reference configuration).

  • V: Left stretch tensor (defined in the current configuration).

1.2 Strain Tensors

Strain tensors are symmetric tensors derived from \(\mathbf{F}\) that remove the rigid body rotation component.

1.2.1 Right Cauchy-Green Deformation Tensor (\(\mathbf{C}\))

This tensor operates entirely in the reference configuration. It provides a measure of the square of local line stretching.

\[\mathbf{C} = \mathbf{F}^T \mathbf{F} = (\mathbf{R}\mathbf{U})^T (\mathbf{R}\mathbf{U}) = \mathbf{U}^T \mathbf{R}^T \mathbf{R} \mathbf{U} = \mathbf{U}^2\]

Since \(\mathbf{R}^T \mathbf{R} = \mathbf{I}\), the rotation is effectively removed. \(\mathbf{C}\) is symmetric and positive definite.

1.2.2 Green-Lagrange Strain Tensor (\(\mathbf{E}\))

The Green-Lagrange strain tensor measures the difference in squared lengths between the deformed and undeformed configurations. It is the most common measure for large deformations in Finite Element Analysis (FEA).

\[\mathbf{E} = \frac{1}{2}(\mathbf{C} - \mathbf{I}) = \frac{1}{2}(\mathbf{F}^T \mathbf{F} - \mathbf{I})\]

  • Usage: Hyperelastic materials, geometric non-linearities.

  • Characteristics: Zero when there is no deformation. Non-linear with respect to displacement.

1.2.3 Infinitesimal (Cauchy) Strain Tensor (\(\boldsymbol{\varepsilon}\))

If deformations are small (\(\nabla \mathbf{u} \ll 1\)), we can ignore the higher-order terms in \(\mathbf{E}\). This linearizes the theory.

\[\boldsymbol{\varepsilon} \approx \frac{1}{2}(\nabla \mathbf{u} + (\nabla \mathbf{u})^T)\]

  • Usage: Standard structural analysis, metals below yield stress.

  • Limitation: Inaccurate if the body undergoes large rotations, as it predicts non-zero strain for pure rigid rotation.

1.2.4 Hencky (Logarithmic) Strain (\(\mathbf{E}_{ln}\))

Defined as the natural logarithm of the principal stretches. It is additive, meaning the strain of two sequential deformations is the sum of the individual strains.

\[\mathbf{E}_{ln} = \ln(\mathbf{U}) = \frac{1}{2}\ln(\mathbf{C})\]

  • Usage: Plasticity, metal forming, polymer testing.

1.3 Summary of Relationships

Comparison of Strain Measures

Tensor

Symbol

Relation to F

Deformation Gradient

\(\mathbf{F}\)

\(\mathbf{I} + \nabla \mathbf{u}\)

Right Cauchy-Green

\(\mathbf{C}\)

\(\mathbf{F}^T \mathbf{F}\)

Green-Lagrange

\(\mathbf{E}\)

\(\frac{1}{2}(\mathbf{F}^T \mathbf{F} - \mathbf{I})\)

Infinitesimal

\(\boldsymbol{\varepsilon}\)

\(\frac{1}{2}(\mathbf{F} + \mathbf{F}^T) - \mathbf{I}\) (Linearized)

1.4 Strain formulations

There are several ways to formulate the strain tensor. The most common are:

  1. Cauchy Strain Cauchy Strain (Infinitesimal)

  2. Green-Lagrange Strain Green-Lagrange Strain

  3. Logarithmic/Hencky Strain Hencky (Logarithmic) Strain

1.5 References

  1. Textbooks:

    • Continuum Mechanics, A.J.M. Spencer, Dover Publications.

    • Nonlinear Solid Mechanics: A Continuum Approach for Engineering, Gerhard A. Holzapfel.

    • Introduction to the Mechanics of a Continuous Medium, L.E. Malvern.

  2. Online Resources:
  3. YouTube: