Finite deformation tensors

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In continuum mechanics, finite deformation tensors are used when the deformation of a body is sufficiently large to invalidate the assumptions inherent in small strain theory. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue.

1 Deformation gradient tensor
- 1.1 Polar Decomposition
2 Rotation-Independent Deformation Measures
3 Examples
4 See also
5 References

[edit] Deformation gradient tensor

The position (vector) of a particle in the initial, undeformed state of a body is denoted $\mathbf {X}$ relative to some coordinate basis. The position of the same particle in the deformed state is denoted $\mathbf {x}$ . If $d \mathbf {X}$ is a line segment joining two nearby particles in the undeformed state and $d \mathbf {x}$ is the line segment joining the same two particles in the deformed state, the linear transformation between the two line segments is given by

$d\mathbf{x} = \mathbf{F} d\mathbf{X}$

The quantity $\mathbf{F}$ is called the deformation gradient and is given by:

$\mathbf{F} = \nabla_{\!X} \mathbf {x} =\frac {\partial \mathbf{x}} {\partial \mathbf {X}}$

or, in index notation:

$F_{ij} = \frac {\partial x_i} {\partial X_j}$

It is assumed that $\mathbf{x}$ is a differentiable function of $\mathbf {X}$ and time t, i.e, that cracks and voids do not open or close during the deformation.

$\mathbf{F}$ is a second-order tensor and contains information about both the stretch and rotation of the body.

Note: The notation and terminology used here was introduced in the "Non-Linear Field Theories of Mechanics” by C.Truesdell and myself (Walter Noll), published in 1965. I invented much of this notation and terminology, but I now realize that some of it is misleading and should be changed. For example “Deformation Gradient” should be replaced by “Transplacement Gradient”. A modern, frame-free and coordinate-free analysis of the mathematical concept of deformation can be found in the first four parts of my "Five Contributions to Natural Philosophy", published in 2005 on my website www.math.cmu.edu/~wn0g/noll

[edit] Polar Decomposition

The deformation gradient $\mathbf{F}$ can be decomposed using the polar decomposition theorem into a product of two second-order tensors:

$\mathbf{F} = \mathbf{R}\mathbf{U} = \mathbf{V} \mathbf{R}$

where $\mathbf{R}$ is a proper orthogonal tensor, and $\mathbf{U}$ and $\mathbf{V}$ are both positive definite symmetric tensors of second order.

The tensor $\mathbf{R}$ represents a rotation. The tensors $\mathbf{U}$ and $\mathbf{V}$ represent stretches. $\mathbf{U}$ is called the right stretch tensor. $\mathbf{V}$ is called the left stretch tensor.

The spectral decompositions of $\mathbf{U}$ and $\mathbf{V}$ are

$\mathbf{U} = \sum_{i=1..3} \lambda_i \mathbf{N}_i \otimes \mathbf{N}_i$

and

$\mathbf{V} = \sum_{i=1..3} \lambda_i \mathbf{n}_i \otimes \mathbf{n}_i$

where

$λ i$ are the principal stretches, and $\mathbf{N}_i$ , $\mathbf{n}_i$ are the directions of the principal stretches (principal directions).

The principal directions are related by

$\mathbf{n}_i = \mathbf{R} \mathbf{N}_i$

[edit] Rotation-Independent Deformation Measures

Since a pure rotation should not induce any stresses in a deformable body, it is often convenient to use rotation-independent measures of the deformation in continuum mechanics.

As a rotation followed its inverse rotation leads to no change ( $\mathbf{R}\mathbf{R}^T=\mathbf{R}^T\mathbf{R}=\mathbf{1}$ ) we can exclude the rotation by multiplying $\mathbf{F}$ by its transpose.

[edit] The Right Cauchy-Green deformation tensor

The right Cauchy-Green deformation tensor (named after Augustin Louis Cauchy and George Green) is defined as::

$\mathbf{C}=\mathbf{F^T}\mathbf{F}=\mathbf{U}^T\mathbf{U}=\mathbf{U}^2$

$C_{ij}=\sum_{k=1..3}\frac {\partial x_k} {\partial X_i} \frac {\partial x_k} {\partial X_j}$

The spectral decomposition of $\mathbf{C}$ is

$\mathbf{C} = \sum_{i=1..3} \lambda_i^2 \mathbf{N}_i \otimes \mathbf{N}_i$

Physically, the Cauchy-Green tensor gives us the square of local change in distances due to deformation.

[edit] The Left Cauchy-Green deformation tensor

Reversing the order of multiplication in the formula for the Finger tensor leads to the left Cauchy-Green deformation tensor which is defined as:

$\mathbf{B}=\mathbf{F}\mathbf{F^T}=\mathbf{V}\mathbf{V^T}=\mathbf{V}^2$

In index notation:

$B_{ij}=\sum_{k=1..3}\frac {\partial x_i} {\partial X_k} \frac {\partial x_j} {\partial X_k}$

The spectral decomposition of $\mathbf{B}$ is

$\mathbf{B} = \sum_{i=1..3} \lambda_i^2 \mathbf{n}_i \otimes \mathbf{n}_i$

[edit] The Finger deformation tensor

The inverse of the left Cauchy-Green tensor is often called the Finger tensor. This tensor is named after Josef Finger (1894).

[edit] Examples

[edit] Uniaxial extension of an incompressible material

This is the case where a specimen is stretched in 1-direction with a stretch ratio of $\mathbf{\alpha=\alpha_1}$ . If the volume remains constant, the contraction in the other two directions is such that $\mathbf{\alpha_1 \alpha_2 \alpha_3 =1}$ or $\mathbf{\alpha_2=\alpha_3=\alpha^{-0.5}}$ . Then: