# Nonlinear finite elements/Nonlinear axially loaded bar

### Non-linear axial bar

Consider an axial bar of length ${\displaystyle L}$, the displacement of the bar is denoted by ${\displaystyle u}$. We assume the material of the bar to follow a non-linear constitutive rule.

Figure 1. shows the axial bar with distributed body force ${\displaystyle q}$. If we denote the axial force in the bar as ${\displaystyle \sigma }$, then ${\displaystyle \sigma }$ depends on the deformation of the bar, this fact is consciously written as ${\displaystyle \sigma (u)}$. The equilibrium equation describing the axial bar is given as,

${\displaystyle {\frac {d\sigma }{dx}}+q=0}$

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 ${\displaystyle \sigma }$ can depend on the gradient of the displacement but not the displacement as such.

${\displaystyle \sigma =f\left({\frac {du}{dx}}\right)}$

In the above equation ${\displaystyle f}$ 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 ${\displaystyle \sigma }$ depends quadratically on the displacement gradient.

${\displaystyle \sigma =AE\left(1+{\frac {du}{dx}}\right){\frac {du}{dx}}}$

${\displaystyle AE}$ 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,

${\displaystyle {\frac {d}{dx}}\left(AE\left(1+{\frac {du}{dx}}\right){\frac {du}{dx}}\right)+q=0}$

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 ${\displaystyle u(x)=g}$, ${\displaystyle g}$ is a prescribed function defined only on the boundary ${\displaystyle x\in \{0,L\}}$

2. Newman boundary ${\displaystyle \sigma |_{x}=AE\left(1+{\frac {du}{dx}}\right){\frac {du}{dx}}|_{x}=n}$, ${\displaystyle n}$ 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.