View source for Nonlinear finite elements/Nonlinear axially loaded bar

===Non-linear axial bar===
Consider an axial bar of length <math>L</math>, the displacement of the bar is denoted by <math>u</math>.  We assume the material of the bar to follow a non-linear constitutive rule. 
{|align="center" 
|[[image:DistAxialBar.png|thumb|400px|Figure 1. Distributed axial loading of a bar.]]
|}
Figure 1. shows the axial bar with distributed body force <math>q</math>. If we denote the axial force in the bar as <math>\sigma</math>, then <math>\sigma</math> depends on the deformation of the bar, this fact is consciously written as <math>\sigma(u)</math>. The equilibrium equation describing the axial bar is given as,
:<math>
\frac{d \sigma}{dx}+q=0
</math>
One should note that above equation is valid irrespective of the material behaviour of the axial bar. Taking into account that a rigid body motion do not produce axial force in the bar, on can conclude, the <math>\sigma</math> can depend on the gradient of the displacement but not the displacement as such.
:<math>
\sigma=f\left(\frac{du}{dx}\right)
</math>
In the above equation <math>f</math> is the constitutive function which relates the gradient of the displacement with axial force, for a general three dimensional continua this relation will be replace by a non-linear tensorial relation. For the sake of concreteness, let us assume the <math>\sigma</math> depends quadratically on the displacement gradient.
:<math>
\sigma=AE\left(1+\frac{du}{dx}\right)\frac{du}{dx}
</math>
<math>AE</math> is a constant, generally inferred as the linear axial stiffness. Using the above mentioned constitutive relation the equilibrium equation can now be written in terms of the displacements as,
:<math>
\frac{d}{dx}\left(AE\left(1+\frac{du}{dx}\right)\frac{du}{dx}\right)+q=0
</math>
The equilibrium equations must be supplemented with additional boundary conditions for the problem to be complete. 
The above equation admits two kinds of boundary conditions, 

1. Dirichlet boundary <math>u(x)=g</math>, <math>g</math> is a prescribed function defined only on the boundary <math>x\in\{0,L\}</math>

2. Newman boundary <math>\sigma|_x=AE\left(1+\frac{du}{dx}\right)\frac{du}{dx}|_x=n</math>, <math>n</math> describes the traction condition of the bar at the boundary.

Although the above discussed model for a materially non-linear axial bar is really simple, it contains most of the essential features of a small deformation materially non-linear solid continua.