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.

Deformation gradient tensor

The position (vector) of a particle in the initial, undeformed state of a body is denoted <math> \mathbf {x^\prime} </math> relative to some coordinate basis. The position of the same particle in the deformed state is denoted <math> \mathbf {x} </math>. Then <math>d \mathbf {x ^\prime} </math> is a line segment joining two nearby particles in the undeformed state and <math>d \mathbf {x } </math> is the line segment joining the same two particles in the defomed state. The deformation gradient F is defined as:

<math> \mathbf { F } = \nabla \mathbf {x} =\frac {\partial \mathbf{x}} {\partial \mathbf {x^\prime}} </math>

or, in subscript notation:

<math>F_{i,j} = \frac {\partial x_i} {\partial x_j^\prime}</math>

It is assumed that <math> \mathbf {x} </math> is a differentiable function of <math> \mathbf {x^ \prime} </math> and time t, i.e that cracks and voids do not open or close during the deformation.

F is a second-order tensor and contains information about both the stretch and rotation of the body. If inertia terms are small, rotation does not induce stress and so can be removed from the analysis.

Finger tensor (The Left Cauchy-Green deformation tensor)

The deformation gradient F can be decomposed using the polar decomposition theorem into:

<math>\mathbf{F}=\mathbf{V} \mathbf{R}</math>

where V is a symmetric tensor and R is an orthogonal tensor. R then represents the rotation and so V represents the stretch. As the addition of a rotation and its inverse rotation leads to no change (<math>\mathbf{R}\mathbf{R^T}=\mathbf{1}</math>) we can exclude the rotation by multiplying F by its transpose:

<math>\mathbf{B}=\mathbf{F}\mathbf{F^T}=\mathbf{V}\mathbf{V^T}</math>

This tensor is named the Finger tensor, after Josef Finger (1894).

In subscript notation:

<math>B_{ij}=\sum_{k=1..3}\frac {\partial x_i} {\partial x_k^\prime} \frac {\partial x_j} {\partial x_k^\prime}</math>

Physically, the Finger tensor tensor gives us the local changes in area within a sample:

<math>\mu^2=\mathbf{n} \mathbf{B} \mathbf{n} </math>,

where <math>\mu</math> is the ratio of undeformed surface to the deformed surface and <math>\mathbf{n}</math> is the normal vector to the surface.

Cauchy-Green tensor (The right Cauchy-Green deformation tensor)

Rversing the order of multiplication in the formula for the Finger tensor leads to the Cauchy-Green tensor:

<math>\mathbf{C}=\mathbf{F^T}\mathbf{F}</math>

or

<math>C_{ij}=\sum_{k=1..3}\frac {\partial x_k} {\partial x_i^\prime} \frac {\partial x_k} {\partial x_j^\prime}</math>

named after Augustin Louis Cauchy and George Green.

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

<math>\alpha^2=\mathbf{n^\prime}\mathbf{C}\mathbf{n^\prime}</math>

where <math>\alpha</math> is the ratio of lengths of a vector in deformed and undeformed states and <math>\mathbf{n^\prime}</math> is the direction of the vector in undeformed state.

Examples

Uniaxial extension of an incompressible material

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

<math>\mathbf{F}=\begin{bmatrix} \alpha & 0 & 0 \\

0 & \alpha^{-0.5} & 0 \\ 0 & 0 & \alpha^{-0.5} \end{bmatrix}</math>

<math>\mathbf{B}=\mathbf{C}=\begin{bmatrix} \alpha^2 & 0 & 0 \\

0 & \alpha^{-1} & 0 \\ 0 & 0 & \alpha^{-1} \end{bmatrix}</math>

Simple shear

<math>\mathbf{F}=\begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}</math>

<math>\mathbf{B}=\begin{bmatrix} 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}</math>

<math>\mathbf{C}=\begin{bmatrix} 1 & \gamma & 0 \\ \gamma & 1+\gamma^2 & 0 \\ 0 & 0 & 1 \end{bmatrix}</math>

Solid body rotation

<math>\mathbf{F}=\begin{bmatrix} \cos \theta & \sin \theta & 0 \\ - \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}</math>

<math>\mathbf{B}=\mathbf{C}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \mathbf{1}</math>

See also

• Piola-Kirchhoff stress tensor, the stress tensor for finite deformations.

Source

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

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Finite deformation tensors".