# 5.6: Tensile Testing - Necking and Failure

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With a brittle material, tensile testing may give an approximately linear stress-strain plot, followed by fracture (at a stress that may be affected by the presence and size of flaws). However, most metals do not behave in this way and are likely to experience considerable plastic deformation before they fail. Initially, this is likely to be uniform throughout the gauge length.

Eventually, of course, it will fail (fracture). However, in most cases, failure will be preceded by at least some ** necking**. The formation of a neck is a type of instability, the formation of which is closely tied in with work hardening. It is clear that, once a neck starts to form, the (true) stress there will be higher than elsewhere, possibly leading to more straining there, further reducing the local sectional area and accelerating the effect.

In the complete absence of work hardening, the sample will be very susceptible to this effect and will be prone to necking from an early stage. (This will be even more likely in a real component under load, where the stress field is likely to be inhomogeneous from the start.) Work hardening, however, acts to suppress necking, since any local region experiencing higher strain will move up the stress-strain curve and require a higher local stress in order for straining to continue there. Generally, this is sufficient to ensure uniform straining and suppress early necking. However, since the work hardening rate often falls off with increasing strain (see earlier page), this balance is likely to shift and may eventually render the sample vulnerable to necking. Furthermore, some materials (with high yield stress and low work hardening rate) may indeed be susceptible to necking from the very start.

## Considère’s Construction

This situation was analysed originally by Armand Considère (1885), in the context of the stability of structures such as bridges. Instability (onset of necking) is expected to occur when an increase in the (local) strain produces no net increase in the load, *F*. This will happen when

\[\Delta F=0\]

This leads to

\[\begin{aligned}

&F=\sigma A, \quad \therefore \mathrm{d} F=A \mathrm{d} \sigma+\sigma \mathrm{d} A=0\\

&\therefore \frac{\mathrm{d} \sigma}{\sigma}=\frac{-\mathrm{d} A}{A}=\frac{\mathrm{d} L}{L}=\mathrm{d} \varepsilon\\

&\therefore \sigma=\frac{\mathrm{d} \sigma}{\mathrm{d} \varepsilon}

\end{aligned}\]

Necking is thus predicted to start when the slope of the true stress / true strain curve falls to a value equal to the true stress at that point. This construction can be explored using the simulation below, in which the true stress – true strain curve is represented by the L-H equation.

Simulation 4: Considère's construction (basic construction)

This condition is commonly expressed in terms of the nominal strain.

\[\begin{aligned}

&\therefore \frac{\mathrm{d} \sigma}{\mathrm{d} \varepsilon}=\frac{\mathrm{d} \sigma}{\mathrm{d} \varepsilon_{\mathrm{N}}} \frac{\mathrm{d} \varepsilon_{\mathrm{N}}}{\mathrm{d} \varepsilon}=\frac{\mathrm{d} \sigma}{\mathrm{d} \varepsilon_{\mathrm{N}}}\left(\frac{\mathrm{d} L / L_{0}}{\mathrm{d} L / L}\right)=\frac{\mathrm{d} \sigma}{\mathrm{d} \varepsilon_{\mathrm{N}}}\left(\frac{L}{L_{0}}\right)=\frac{\mathrm{d} \sigma}{\mathrm{d} \varepsilon_{\mathrm{N}}}\left(1+\varepsilon_{\mathrm{N}}\right)\\

&\therefore \sigma=\frac{\mathrm{d} \sigma}{\mathrm{d} \varepsilon_{\mathrm{N}}}\left(1+\varepsilon_{\mathrm{N}}\right)

\end{aligned}\]

The condition can therefore also be formulated in terms of a plot of true stress against nominal strain. On such a plot, necking will start where a line from the point εN = -1 forms a tangent to the curve. This construction can be explored using the simulation below.

Simulation 5: Considère's construction, based on a true stress-nominal strain plot

If the true stress – true strain relationship does conform in this way to the L-H equation, it follows that the necking criterion (Eqn.(9)) can be expressed as

\[\sigma_{\mathrm{Y}}+K \varepsilon^{n}=n K \varepsilon^{n-1}\]

which can be solved analytically.

## Ultimate Tensile Stress (UTS) and Ductility

It may be noted at this point that it is common during tensile testing to identify a “strength”, in the form of an “ultimate tensile stress” (** UTS**). This is usually taken to be the peak on the nominal stress v. nominal strain plot, which corresponds to the onset of necking. It should be understood that this value is not actually the true stress acting at failure. This is difficult to obtain in a simple way, since, once necking has started, the (changing) sectional area is unknown - although the behaviour can often be captured quite accurately via FEM modelling – see below. Also, the "ductility", often taken to be the nominal strain at failure (usually well beyond the strain at the onset of necking) does not correspond to the true strain in the neck when fracture occurs. UTS and ductility values therefore provide only rather loose indications of the strength and toughness of the material. Nevertheless, they are quite widely quoted.

## FEM Simulation of Tensile Tests

It is sometimes stated that the initiation of necking during tensile testing arises from (small) variations in sectional area along the gauge length of the sample. However, in practice, for a particular material, its onset does not depend on whether great care has been taken to avoid any such fluctuations. Furthermore, the introduction of such defects in an FEM model does not, in general, significantly affect the predicted onset. The (modelling) condition that does lead to necking is the assumption that, near the end of the gauge length, the sample is constrained from contracting laterally. In practice, due to the increasing sectional area in that region, and because the material beyond the gauge length will undergo little or no deformation, that condition is often a fairly realistic one.

In fact, for any true stress – true strain relationship, including an experimental one that cannot be expressed as an equation, FEM simulation can be used to predict the onset of necking. The simulation below shows, for two materials (with low and high work hardening rates), how the behaviour can be accurately captured and the stress and strain fields explored at any point during the test.(These two materials will also be used to illustrate some effects concerned with compression and indentation testing.) The L-H law is being used here, with best fit values for the 3 parameters in each case.

Simulation 6: Tensile test FEM simulation data, for two materials, together with the corresponding videos and experimental stress-strain curves.