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Finite deformation tensors

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.

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

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 defomed 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 and available on my website www.math.cmu.edu/~wn0g/noll

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 an 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

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

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.

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.

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

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

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).

Examples

Uniaxial extension of an incompressible material

This 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:

$\mathbf{F}=\begin{bmatrix} \alpha & 0 & 0 \\ 0 & \alpha^{-0.5} & 0 \\ 0 & 0 & \alpha^{-0.5} \end{bmatrix}$

$\mathbf{B}=\mathbf{C}=\begin{bmatrix} \alpha^2 & 0 & 0 \\ 0 & \alpha^{-1} & 0 \\ 0 & 0 & \alpha^{-1} \end{bmatrix}$

Simple shear

$\mathbf{F}=\begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

$\mathbf{B}=\begin{bmatrix} 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

$\mathbf{C}=\begin{bmatrix} 1 & \gamma & 0 \\ \gamma & 1+\gamma^2 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Rigid body rotation

$\mathbf{F}=\begin{bmatrix} \cos \theta & \sin \theta & 0 \\ - \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$

$\mathbf{B}=\mathbf{C}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \mathbf{1}$

References

C. W. Macosko Rheology: principles, measurement and applications, VCH Publishers, 1994, ISBN 1-56081-579-5

Categories: Continuum mechanics | Non-Newtonian fluids

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Finite_deformation_tensors". A list of authors is available in Wikipedia.