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MJ16. Difference between associated and non-associated flow rules, plastic part of volumetric

deformation

Lecture 9, slides 27-36

Through experiments, it has been found that strain can be decomposed into two components: e

representing the elastic strains and p representing the plastic strains. It follows that:

The elastic component of the strain can be readily computed using linear Hooks law, knowing the

Youngs modulus:

While the elastic response can be determined analytically based on first principles, the plastic

response requires an empirical/experimental approach, whereby each material is tested in

uniaxial/multi-axial conditions and its stress-strain response is recorded. This data is then used to fit

curves (and create models) which describe the plastic response of the structure.

In trying to describe the plastic response of a material, two issues must be addressed. First, we must

be able to describe the instance where plastic response begins (i.e. yielding occurs). Second we must

be able to describe the relationship between stress and strain following yielding.

The first issue is dealt with by the concept of yield criterion/yield functions/yield

surface. A yield criterion is a way of organizing the experimental data based on a

predetermined model. It is in fact a curve fitting algorithm. For example the von

mises criterion, when applied to plane stress, allows for fitting a surface around

the experimental data which have this shape:

Yield functions ( f() ), based on a particular yield criterion, try to describe the onset of yielding

under different stress conditions. They are essentially the functions you would get once you have

fitted a particular criterion onto some specific data (i.e. whereas the criterion is a general

description, the yield function is specific for the material and derived from experimental data).

Under uniaxial stress (tension or compression), the yield function is simply equivalent to the material

yield strength (ft or fc).

Under biaxial stress, the yield function is more complex and depends on the relationship between

tensile and compressive stresses in either direction.

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By plotting all the individual yield stress combinations we will get the yield surface which basically

describes two regions the admissible region, enclosed by the yield function and the inadmissible

region/domain outside this region.

It is important to note that the yield stress (at least in perfect plasticity without consideration of

hardening or softening) is the ultimate stress state that can be attained at a point and thus the yield

function will take the form:

This means that only stress states below or at yield are admissible. In perfect elastoplasticity, the

stress states below yield stress (where the function is negative) represent the elastic stress states

and the stress states at yield (where the function is zero) represent yielding.

The relation becomes more complex when hardending/softening is taken into consideration (see

note 1 for description of strain hardening) because now the yield surface changes (expands or

contracts) based on the stress increment.

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So looking at the yield surface under biaxial stress, we get:

Having dealt with the first issue, namely how to describe the onset of plastic deformation, we can

now move on to try and describe the relation between stress and strain following yielding. The

fundamental question here is: with an increment of stress (d) beyond the yield stress (o), what is

the associated increment in plastic strain (dp)?

As you can see, the magnitude of the plastic strain increment is completely determined by the

hardening behavior of the solid. This is because during continued plastic flow, the stress must still be

on the yield surface at all times (recall from the image above that with hardening the yield surface

expands). It is obvious that for the perfectly plastic model, the response is infinite, meaning that

once we have reached yielding, any stress increment will result in infinite plastic strain. So we will

focus on models which take into account some kind of hardening/softening behaviour following

initial yielding.

Here we introduce the concept of flow rule. The flow rule describes the plastic relation between the

increment of stress and the associated plastic strain. It is in essence analogous to Hooks law, butwith the difference that it predicts behaviour at each stress increment based on the previous stress

state () and a small increment of strain (d). Thus it is often expressed as the rate of plastic strain (

):

Graphically this is represented by as the gradient of the yield function and is thus normal to the yield

surface (and represents the direction of fastest increase of the yield function).

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You will notice that there are two different flow rules (b and c). The non-associated flow rule (c) is

the general form of the flow rule in that it describes the rate of change of plastic strain ( ) due to

an increment of stress () as a function of the previous state (g()) times by the increment of strain

(d). This function g(), is known as the plastic flow potential and is itself a function of the deviatoric

stress (through its invariants J2 andJ3), strain history and temperature and must be determinedexperimentally.

For most material models (especially metals where plasticity is described more or less by dislocation

theory), the experimental data suggest that the yield function provides a close approximation of the

plastic flow potential and could be used to derive the flow law.

So after all this, what is the difference between associated and non-associated flow rules? A flow law

derived from the yield surface (and the yield function) is known as an associated plastic flow law.

Theories of plasticity that use a separate flow potential are known as non-associative plasticity

models.

The distinction between the two could be represented graphically as:

The lecture provides an example of how the plastic flow potential function using the Drucker-Prager

model differs from the yield function:

Thus just looking at the yield flow rule with respect to x the difference will be:

The only difference being the substitution of dilatancy coeff. for the internal friction coeff.

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Now the reasons as to why non-associative plasticity models are required for me are hard to

understand. I suspect that it is due to the fact that for some materials the yield function simply is not

close enough to the experimental results. However that could be completely wrong. So if someone

knows the reason, please just put a comment on the document.

Notes:

1) Experiments show that if you plastically deform a solid, then unload it, and then try to re-load it so as to induce further plastic flow, its resistance to plastic flow will have increased.

This is known as strain hardening. Obviously, we can model strain hardening by relating the

size and shape of the yield surface to plastic strain in some appropriate way. There are

many ways to do this, of course. Here we describe the simplest approaches.

Disclaimer:

This is my basic understanding of the principles of elastoplasticity. The lecture notes, while heavy on

equations, lack any real explanation of what we are trying to achieve. So this is my best attempt at

making sense of what we are trying to accomplish and fitting the equations in that description.

Please review the notes and suggest improvements.