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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-linearity constitutive rule. Figure 1. shows an axial bar with distributed body force $$q$$. If we denote the axial force in the bar to be $$\sigma$$, 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$$ is the constitutive function which relates the gradient of the displacement with axial force, for a general three dimensional continua this relation will be replace by a non-linear tensorial relation. For the sack of concreteness, let us assume the $$\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.

Weak form of the non-linear axial bar
Having discussed the equations of the non-linear axial bar, we prosed to construct a finite element approximation for the discussed model. The first step in obtaining the finite element approximation is to construct the weak form. To obtain the weak form we first multiply the equilibrium equations equations discussed in the previous section with a test function (weighting function) and integrate over the whole domain $$[0,L]$$.

\int_{[0,L]}\frac{d}{dx}\left(AE\left(1+\frac{du}{dx}\right)\frac{du}{dx}\right) \varphi+q\varphi=0 $$ $$\varphi$$ is the test function or the weighting function. Now we apply integration by parts to shift a derivative from the first term to the test function.



\int_{[0,L]}\left(AE\left(1+\frac{du}{dx}\right)\frac{du}{dx}\right) \frac{d\varphi}{dx} + \left(AE\left(1+\frac{du}{dx}\right)\frac{du}{dx}\right)|_{x=0}^{x=L}=\int_{[0,L]}q\varphi $$

We consider a pure Dirichlet problem, hence we ignore the boundary term. The weak form for a pure Dirichlet problem for the non-linear axial bar is given by,



\int_{[0,L]}\left(AE\left(1+\frac{du}{dx}\right)\frac{du}{dx}\right) \frac{d\varphi}{dx}=\int_{[0,L]}q\varphi $$

Linearization of the weak form
Let $$u_0$$ denote an equilibrium configuration of the bar. The notion of Frechet derivative is used to obtain the linearized weak form.

Lf(u)= \frac{d}{d \epsilon}|_{\epsilon=0}f(u_0+\epsilon\hat{u}) $$

\int_{[0,L]} AE(u+\epsilon \hat{u})\cfrac{d(u+\epsilon \hat{u})}{dx}\cfrac{dw}{dx}~dx = \int_{[0,L]} axw dx $$

\int_{[0,L]} \left[AE(\hat{u})\cfrac{du_o}{dx}\cfrac{dw}{dx} + AE(u_0)\cfrac{d\hat{u}}{dx} \cfrac{dw}{dx} \right]~dx $$