Nonlinear finite elements/Nonlinear axially loaded bar

Non-linear axial bar

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

Figure 1. Distributed axial loading of a bar.

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

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

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

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

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

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

$AE$ 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,

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

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

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