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DoITPoMS Teaching & Learning Packages Mechanical Testing of Metals Overview of Plasticity and its Representation with Constitutive Laws

Overview of Plasticity and its Representation with Constitutive Laws

Plastic deformation of metals most commonly occurs as a result of the glide of dislocations, driven by shear stresses.  (In some cases, deformation twinning may contribute, but this also requires shear stresses in a similar way, and also involves no volume change.)  In a polycrystal (ie in most metallic samples), individual grains must deform in a cooperative way, so that each undergoes a relatively complex shape change (requiring the operation of multiple slip systems), consistent with those of its neighbours. 

The (deviatoric) stress needed to initiate global plasticity in a sample is termed the yield stress. In general, continuation of plastic deformation requires a progressively increasing level of applied stress. This effect is termed “work hardening” or “strain hardening”. It arises because, as more dislocations are created, and as they interact with each other (creating jogs and tangles), they tend to become less mobile, see Mechanisms of Plasticity .

The yield stress, and the work hardening characteristics, exhibit a complex dependence on crystal structure, grain size, crystallographic texture, composition, phase constitution, grain boundary structure, prior dislocation density, impurity levels etc.  Even for a given material, these plasticity characteristics can be dramatically changed by thermal or mechanical treatments or by exposure to various environments (chemical, irradiative etc). Accurate prediction of key mechanical properties of metallic alloys is virtually impossible, even if the microstructure has been carefully and comprehensively characterised.  Such properties must therefore be measured experimentally.  Since they are of great importance for many (industrial) purposes, the measurement techniques need to be fully understood.

The yield stress is usually taken to have a single value, but work hardening needs more complex definition.  This must be valid over an appreciable range of plastic strain  -  perhaps 50% or more in some cases.  Even metals that are relatively hard (and brittle) are normally required to have ductility levels (plastic strains to failure) of at least several % if they are to be used for engineering purposes.

Of course, there is no expectation that the work hardening curve will conform to any particular functional form.  However, in general, the work hardening rate (gradient of the true stress / true strain plot) tends to decrease progressively with increasing strain, perhaps eventually approaching a plateau.  This is a consequence of competition between the creation of new dislocations, and inhibition of their mobility (by forming tangles etc), and processes (such as climb and cross-slip) that will allow them to become more organised and to annihilate each other.  Initially, the former group of processes tends to dominate, but a balance may eventually be reached, so that the “flow stress” ceases to rise.  (With metals, it is very rare, except with single crystals, for the work hardening rate to rise with increasing strain, but this is quite common in certain types of polymer, as a consequence of molecular reorganisation  -  see the TLP on Crystallinity in Polymers).

Several analytical expressions have been proposed to characterise the work hardening of metals, but only two are in frequent use.  The first is the Ludwik-Hollomon equation

\[{\sigma _{}} = {\sigma _{\rm{Y}}} + K{\varepsilon ^n} \qquad \qquad \qquad (6)\]

where \( \sigma \) is the (von Mises) applied stress, \(\sigma _{\rm{Y}} \) is its value at yield, \( \varepsilon \) is the plastic (von Mises) strain, K is the work hardening coefficient and n is the work hardening exponent.  The second is the Voce equation

\[\sigma = {\sigma _{\rm{s}}} - ({\sigma _{\rm{s}}} - {\sigma _{\rm{Y}}})\exp \left( {\frac{{ - \varepsilon }}{{{\varepsilon _0}}}} \right) \qquad \qquad \qquad (7) \]

The stress \( \sigma \)s is a saturation level, while \( \varepsilon \)0 is a characteristic strain for the exponential approach of the stress towards this level. 

The simulation below can be used to explore their shapes.  In practice, the L-H is more common.  The L-H does allow linear work hardening (n = 1), which is sometimes observed, whereas Voce does not.

Simulation 3: Ludwik-Hollomon and Voce constitutive laws