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. 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.