**Non Linear Behaviour**

**Introduction**

Two main types of non-linear need to be considered. Both add considerably to the complication
of the analysis and the time needed to carry it out.

Significantly more expertise is needed by the user of the software to achieve good results than for linear analysis.

**Geometrical effects**

Looking at Figure 1, below, there are a number of issues to address.

This is a shallow depth cantilever beam, length L1, with a transverse load, F, attached to the free end.

This is assumed to result in the cantilever bending into the curve shown. As the load
is attached to the end, the bending moment along the beam is reduced as L1 reduces to L2.

However as the beam bends, the load F ceases to be purely transverse and starts to generate
a tensile component within the beam. To obtain accurate results from analysis, these effects need to
be considered.

This problem is normally tackled by splitting the load F into several intervals, perhaps as many as 10
and applying it in increments.

After each load increment is applied, the deflection of the beam is checked, L determined and the bending
moment along the beam re-calculated. Due to the orientation of the beam the element stiffnesses may have
changed and it may be appropriate for these to be computed again and re-assembled. The varying orientation along the beam with
respect to the load may be used to determine any tensile effects, but for most applications involving
slender beams the tensile effects are likely to be negligible compared to the effects of bending.

In such an analysis, normally the only action required by the user of the software is to specify the number / size
of the load increments.

**Material Non Linearity**

In many engineering applications involving metals, linear elastic behaviour can be assumed and will give results which are sufficiently accurate. For components constructed totally or in part from polymers, eg tyres, the non linear behaviour of the matertial must be considered in all analysis. This analysis is often further complicated by the fact that mechanical properties are often strongly dependent upon the rate of loading and temperature.

Modern efficient design sometimes requires components to be stressed locally beyond the yield point and because
almost all metals and alloys work harden, there may still be a safety margin when this is done.

In metal forming blanks undergo considerable bulk plastic deformation and analysisng the metal flow
can keep tooling costs to a minimum.

When analysing the behaviour of metals the stress v strain curve is often split into linear elastic behaviour up to the yield point and some type of post yielding behaviour which may or may not be linear.

One common model assumes linear elasticity initially where the gradient of the stress v strain line is equal
to the Young's modulus, E and after yielding by a second straight portion of stress v strain curve normally with
a much shallower gradient, see diagram below:

During FEA, the von Mises stress (assuming ductile behaviour) in every element is checked at every load increment
against the yield stress (sigma_{y}) and depending whether it is less than or greater than sigma_{y},
the appropriate modulus is used in computing each element stiffness. In applications where there is only local yielding, there
will be no large changes in the geometry of the overall structure or the individual elements. However if extensive
yielding has occured which leads to significant deformation, it will be necessary to re-calculate the geometry of the elements that
have yielded and take this change into account as well as the modulus value when recalculating individual element stiffnesses
to solve subsequent load increments.

To provide closer matching of the stress - strain curve to experimental results, the post yield stress - strain curve
may be split up into more than one segment, see below:

Elastic, power hardening is shown in the schematics below:

The schematic on the left has linear axes and the
equivalent diagram on the right has logarithmic axes, with the log decades having the same length in both directions.
On the log axes, in the elastic region, the equation: sigma = E e, has a gradient of 1. The exponent (slope) of the plastic region
on the log axes, n, is typically between 0.05 and 0.4 for most metals where this equation fits well.

Sophisticated software will offer a range of post yielding criteria and the user will need experience and possibly experimental data if accurate results are to be obtained.

David Grieve, 28th September 2012.