Chapter 3 Element Analysis for Nonlinear Elastic Systems

Last updated: 01 Sep 2026

3.2 Stress and Strain Measures in Large Deformation

3.2.1 Deformation gradient

The description of the mapping from initial to current configuration is given by

where $\boldsymbol{X}$ and $\boldsymbol{x}$ are the position vectors in the reference and current configurations, respectively. The mapping $\phi$ is assumed to be smooth and invertible, and $\boldsymbol{u}$ is the displacement vector. The relationship between differential elements $\rm d\boldsymbol{X}$ and $\rm d\boldsymbol{x}$ is given by:

where $\boldsymbol{F} = \boldsymbol{I} + \nabla \boldsymbol{u}$ is the deformation gradient and is expressed as $F_{ij} = \partial x_i / \partial X_j$. The term $\nabla \boldsymbol{u}$ is the displacement gradient. The determinant of the deformation gradient, denoted as $J = \det(\boldsymbol{F})$, represents the local volume change during deformation and should satisfy $\text{det}(\boldsymbol{F}) > 0$ to make a valid mapping.

3.2.2 Lagrangian and Eulerian Strains

Lagrangian Strain

This strain measure is defined in the reference configuration. Consider the change in the squares of the lengths of different material line elements $\rm d\boldsymbol{X}$ and $\rm d\boldsymbol{x}$:

where $\boldsymbol{F}^\top \boldsymbol{F}-\boldsymbol{I}$ measures the change in squared lengths of line elements and $\boldsymbol{F}^\top \boldsymbol{F}$ is the right Cauchy-Green deformation tensor:

and the Green-Lagrange strain tensor is defined as:

where the factor of $1/2$ is introduced to make the strain measure consistent with the linearized strain measure for small deformations. The Green-Lagrange strain tensor can also be expressed in terms of the displacement gradient as:

which indicates that if the displacement gradient is small, the Green-Lagrange strain reduces to the infinitesimal strain tensor.

It should be noted that the infinitesimal strain tensor cannot be an exact measure of deformation because it does not remain zero in rigid body motions.

Eulerian Strain

The change in the squares of the lengths of different material line elements can also be expressed in terms of the deformed differential line element $\rm d\boldsymbol{x}$:

where the left Cauchy-Green deformation tensor is defined as:

and the Eulerian strain tensor can be defined as:

Similarly, $\boldsymbol{e}$ can also be expressed in terms of the displacement gradient as:

where $\nabla_x$ is the gradient operator at the current configuration. The Eulerian strain tensor also reduces to the infinitesimal strain tensor for small deformations.

3.2.3 Polar Decomposition

If the deformation gradient $\boldsymbol{F}$ is nonsingular, it can be decomposed into a unique orthogonal rotation tensor $\boldsymbol{R}$ and unique positive definite symmetric stretch tensors $\boldsymbol{U}$ and $\boldsymbol{V}$ such that

which means the deformation can be separated into a scaling part and a rotation part.

The right stretch tensor $\boldsymbol{U}$ and the left stretch tensor $\boldsymbol{V}$ have the same eigenvalues (principal stretches) but different eigenvectors (principal directions).

The relationship between $\boldsymbol{U}$ and $\boldsymbol{C}$ can be expressed as:

$\boldsymbol{C}=\boldsymbol{F}^\top \boldsymbol{F}=\boldsymbol{U}^\top \boldsymbol{R}^\top \boldsymbol{R} \boldsymbol{U} = \boldsymbol{U}^2$

so $U$ has the same eigenvectors as $C$ and its eigenvalues are the square roots of the eigenvalues of $C$. If $\lambda_i^2$ and $\boldsymbol{E}_i$ are the eigenvalues and eigenvectors of $\boldsymbol{C}$, then $\boldsymbol{C}$ can be expressed as

or equivalent matrix form

where $\boldsymbol{\Phi} = [\boldsymbol{E}_1, \boldsymbol{E}_2, \boldsymbol{E}_3]$ is the matrix of eigenvectors and $\boldsymbol{\Lambda} = \text{diag}(\lambda_1^2, \lambda_2^2, \lambda_3^2)$ is the diagonal matrix of eigenvalues. Then the right stretch tensor $\boldsymbol{U}$ can be expressed as

where $\sqrt{\boldsymbol{\Lambda}} = \text{diag}(\lambda_1, \lambda_2, \lambda_3)$.

3.2.4 Deformation of Surface and Volume

The infinitesimal volume element in the reference and current configurations can be expressed as

3.2.5 Cauchy and Piola-Kirchhoff Stresses

The stress vector in the current deformed configuration can be written as

where $\boldsymbol{\sigma}$ is the Cauchy stress tensor, also called the true stress.

A different stress vector can be defined using the same force $\Delta \boldsymbol{f}$ but with respect to the reference configuration:

where $\boldsymbol{P}$ is the first Piola-Kirchhoff stress tensor, also called the nominal stress. The first PK stress is not symmetric. Therefore, we have

With the relationship between the differential areas in the current and reference configurations, we have

By multiplying both sides by $\boldsymbol{F}^{-1}$, we can define the second Piola-Kirchhoff stress as

also

Another stress measure is the Kirchhoff stress defined as

With the above stress measures, we can define the internal virtual work as: