8.7 Associated and Non-associated Flow Rules
Recall the Levy-Mis flow rule, Eqn. 8.4.3,
ij p ij s d d λε= (8.7.1)
The plastic multiplier can be determined from the hardening rule. Given the
hardening rule one can more generally, instead of the particular flow rule 8.7.1, write
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ij p ij G d d λε=, (8.7.2)
where ij G is some function of the stress and perhaps other quantities, for example the hardening parameters. It is symmetric becau the strains are symmetric.
A wide class of material behaviour (perhaps all that one would realistically be interested in) can be modelled using the general form
ij
p ij g
d d σλε∂∂=. (8.7.3)
Here, g is a scalar function which, when differentiated with respect to the stress, gives the plastic strains. It is called the plastic potential . The flow rule 8.7.3 is called a non-associated flow rule .
Consider now the sub-class of materials who plastic potential is the yield function, f g =:
ij
p ij f
d d σλε∂∂=. (8.7.4)
This flow rule is called an associated flow-rule , becau the flow rule is associated with a particular yield criterion.
8.7.1 Associated Flow Rules
The yield surface ()0=ij f σ is displayed in Fig 8.7.1. The axes of principal stress
and principal plastic strain are also shown; the material being isotropic, the are taken to be coincid
ent. The normal to the yield surface is in the direction ij f σ/∂ and
so the associated flow rule 8.7.4 can be interpreted as saying that the plastic strain increment vector is normal to the yield surface , as indicated in the figure. This is called the normality rule .
Figure 8.7.1: Yield surface
The normality rule has been confirmed by many experiments on metals. However, it is found to be riously in error for soils and rocks, where, for example, it
overestimates plastic volume expansion. For the materials, one must u a non-associative flow-rule.
Next, the Tresca and Von Mis yield criteria will be discusd. First note that, to make the differentiation easier, the associated flow-rule 8.7.4 can be expresd in terms of principal stress as
vicentei
p i f
d d σλε∂∂=. (8.7.5)
Tresca
Taking 321σσσ>>, the Tresca yield criterion is
k f −−=2
31σσ (8.7.6)
fancy free
One has
如何锻炼心理素质21,0,213
21−=∂∂=∂∂+=∂∂σσσf
f f (8.7.7)
so, from 8.7.5, the flow-rule associated with the Tresca criterion is
⎥⎥
雅思词汇书⎥⎦⎤⎢⎢⎢⎣⎡−+=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡1213210λεεεd d d d p p p . (8.7.8)
This is the flow-rule of Eqns. 8.4.33. The plastic strain increment is illustrated in Fig. 8.7.2 (e Fig. 8.3.9). All plastic deformation occurs in the 31− plane. Note that 8.7.8 is independent of stress.
σ
Figure 8.7.2: The plastic strain increment vector and the Tresca criterion in the
π-plane (for the associated flow-rule)
Von Mis
The Von Mis yield criterion is 022=−=k J f . With
()()()[]
()⎥⎦
⎤
⎢⎣⎡+−=−+−+−∂∂=∂∂321213232221112213261σσσσσσσσσσσJ (8.7.9)
one has
()()()()()()⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+−+−+−=⎥⎥⎥⎦
⎤⎢⎢⎢⎣⎡212133231
212323221132321σσσσσσσσσλεεεd d d d p p p . (8.7.10)
This are none other than the Levy-Mis flow rule 8.4.61.
The associative flow-rule is very appealing, connecting as it does the yield surface to the flow-rule. Many attempts have been made over the years to justify this rule, both mathematically and physically. However, it should be noted that the associative flow-rule is not a law of nature by any means. It is simply very convenient. That said, it
1
note that if one were to u the alternative but equivalent expression 02=−=
k J f , one would
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have a 22/1J term common to all three principal strain increments, which could be “absorbed” into the λd giving the same flow-rule 8.7.10
2
′2
3
1−−=
k f σσrsl
does agree with experimental obrvations of many plastically deforming materials, particularly metals.
In order to put the notion of associative flow-rules on a sounder footing, one can define more clearly the type of material for which the associative flow-rule applies; this is tied cloly to the notion of stable and unstable materials.
8.7.2 Drucker’s Postulate
Stress Cycles
First, consider the one-dimensional loading of a hardening material. The material may have undergone any type of deformation (e.g. elastic or plastic) and is now
subjected to the stress *σ, point A in Fig. 8.7.3. An additional load is now applied to the material, bringing it to the current yield stress σ at point B (if *σ is below the yield stress) and then plastically (greatly exaggerated in the figure) through the infinitesimal increment σd to point C. It is conventional to call the additional loads the external agency . The external agency is then removed, bringing the stress back to *σ and point D. The material is said to have undergone a stress cycle .
haydenFigure 8.7.3: A stress cycle for a hardening material
Consider now a softening material, Fig. 8.7.4. The external agency first brings the material to the current yield stress σ at point B. To reach point C, the loads must be reduced. This cannot be achieved with a stress (force) control experiment, since a reduction in stress at B will induce elastic unloading towards A. A strain
(displacement) control must be ud, in which ca the stress required to induce the (plastic) strain will be en to drop to σσd + (0<σd ) at C. The stress cycle is completed by unloading from C to D.
ε
σσ
Figure 8.7.4: A stress cycle for a softening material
Suppo now that *σσ=, so the material is at point B, on the yield surface, before action by the external agency. It is now not possible for the material to undergo a stress cycle, since the stress cannot be incread. This provides a means of distinguishing between strain hardening and softening materials: Strain-hardening … Material can always undergo a stress-cycle Strain-softening … Material cannot always undergo a stress-cycle
Drucker’s Postulate
The following statements define a stable material : (the statements are also known as Drucker’s postulate ):
(1) Positive work is done by the external agency during the application of the loads (2) The net work performed by the external agency over a stress cycle is
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nonnegative
By this definition, it is clear that a strain hardening material is stable (and satisfies Drucker’s postulates). For example, considering plastic deformation (*σσ= in the above), the work done during
an increment in stress is εσd d . The work done by the external agency is the area shaded in Fig. 8.7.5a and is clearly positive (note that the work referred to here is not the total work, ∫
+εεε
εσd d , but only that part which is done
by the external agency 2). Similarly, the net work over a stress cycle will be positive.
On the other hand, note that plastic loading of a softening (or perfectly plastic) material results in a non-positive work, Fig. 8.7.5b.
2
the laws of thermodynamics insist that the total work is positive (or zero) in a complete cycle.
goddes
ε
σσ
