About
Structural Analysis in Finite Element Analysis is actually NOT represented as linear model in many cases. There are some kind of non-linear term in the formulation such as Geometric Nonlinearity, Material Nonlinearity and Contact/B.C. Nonlinearity.
In this article, we are going to take a look at Geometric Nonlinearity.
Basic Formula
Let the displacement vector 𝑢, and think of how it moves from origin X.
\mathbf{u}(\mathbf{X}) = \mathbf{x}(\mathbf{X}) - \mathbf{X}\\
\mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \frac{\partial (\mathbf{X} + \mathbf{u})}{\partial \mathbf{X}} = \mathbf{I} + \frac{\partial \mathbf{u}}{\partial \mathbf{X}}The displacement gradient tensor is written as:
\nabla_{\! \mathbf{X}} \mathbf{u} = \left[ \frac{\partial u_i}{\partial X_j} \right]\\
\nabla \mathbf{u} =
\begin{bmatrix}
\frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2} & \frac{\partial u_1}{\partial x_3} \\
\frac{\partial u_2}{\partial x_1} & \frac{\partial u_2}{\partial x_2} & \frac{\partial u_2}{\partial x_3} \\
\frac{\partial u_3}{\partial x_1} & \frac{\partial u_3}{\partial x_2} & \frac{\partial u_3}{\partial x_3}
\end{bmatrix}
\in \mathbb{R}^{3 \times 3}* i = 1, 2, 3: spatial directions (x, y, z)
* j = 1, 2, 3: reference coordinate directions (X, Y, Z)
Cauchy Strain (Linear)
In the linear formulation, we use Cauchy Strain in the weak form formulation.
\begin{align}
\boldsymbol{\varepsilon} = \frac{1}{2} (\nabla \mathbf{u} + \nabla \mathbf{u}^\top)\\
\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial X_j} + \frac{\partial u_j}{\partial X_i} \right)
\end{align}Green-Lagrange Strain Tensor
The Green-Lagrange Strain tensor is described as
\begin{align}
\mathbf{E} = \frac{1}{2} (\mathbf{F}^\top \mathbf{F} - \mathbf{I})\\
E_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial X_j} + \frac{\partial u_j}{\partial X_i} + \sum_k \frac{\partial u_k}{\partial X_i} \frac{\partial u_k}{\partial X_j} \right)
\end{align}where F is deformation gradient, which is described as follows:
\mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \mathbf{I} + \nabla \mathbf{u}If you extend (3),
\mathbf{E} \approx \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^\top + (\nabla \mathbf{u})^\top \nabla \mathbf{u} \right)You would find that the last term in this extension is non-linear term.
Hencky Strain (logarithmic strain, true strain)
is commonly used in large deformation and plasticity theories. It is defined based on the continuous accumulation of infinitesimal stretches.
In 1 dimension,
\varepsilon_\text{Hencky} = \int_{l_0}^{l} \frac{1}{\tilde{l}} \, d\tilde{l} = \ln \left( \frac{l}{l_0} \right)- l_0: original length
- l: deformed length
\boldsymbol{\varepsilon}^\text{Hencky} = \ln \mathbf{V}
\quad \text{or} \quad
\boldsymbol{\varepsilon}^\text{Hencky} = \ln \mathbf{U}
\quad \text{or} \quad
\boldsymbol{\varepsilon}^\text{Hencky} = \frac{1}{2} \ln \mathbf{C}Where:
- \mathbf{F}: deformation gradient
- \mathbf{C} = \mathbf{F}^\top \mathbf{F}: right Cauchy–Green deformation tensor
- \mathbf{U}, \mathbf{V}: right and left stretch tensors from polar decomposition
\mathbf{F} = \mathbf{R} \mathbf{U} = \mathbf{V} \mathbf{R} - \ln: matrix logarithm (applied via eigenvalue decomposition)
