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Nonlinear Axial Bar[edit]

We describe the deformation of a nonlinear elastic bar. The non-linearity to be discussed can be classified as material non-linearity.

Figure 1. Distributed axial loading of a bar.

In our previous discussion we had assumed that the bar had a constant Young's modulus of , that is, the constitutive model of the bar was linear elastic.

Let us now assume that the stress-strain relationship is nonlinear, and of the form

where is a material constant which can be interpreted as the initial Young's modulus. Figure 2 shows how the stress-strain relationship for this material looks.

Figure 2. Stress-strain relationship for the nonlinear bar.

The governing differential equation for the bar is

If we plug in the stress-strain relation into the governing equation, we get

Notice that we have not included the expression for in the above equation. The reason is that we want to maintain the symmetry of the stiffness matrix.

Effect of including expression for [edit]

You can see what happens when we include the expression for in the steps below. We get,

or,

Now, when we try to derive the weak form, we get

The first integral is the same as before, and we get

The third integral remains the same. However, the second integral becomes

Rearranging, we get

After collecting all the terms, the weak form becomes

Separating the terms corresponding to the internal and external forces, we get

If we choose trial and weighting functions

and plug them into the weak form, we get

Taking the constants out of the integrals, we get (regarding this step, read the Discuss page of this article)

Therefore, the coefficients of the stiffness matrix are given by

This stiffness matrix is not symmetric because of the second term inside the integral. That is, . Also, we have not been able to remove the dependence of the stiffness matrix on the displacement.