associated flow rules

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Section 8.7

Solid Mechanics Part II Kelly234

8.7 Associated and Non-associated Flow Rules

Recall the Levy-Mises flow rule, Eqn. 8.4.3,

ij

p

ij

sd d λ ε = (8.7.1)

which gives the ratios between the plastic strains and stresses. 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

ij

p

ij Gd d λ ε = , (8.7.2)

where ijG is some function of the stresses and perhaps other quantities, for example

the hardening parameters. It is symmetric because 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

gd d

σ λ ε ∂

∂= . (8.7.3)

Here, g is a scalar function which when differentiated with respect to the stresses

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 whose plastic potential is the yield function,

f g = :

ij

p

ij

f d d

σ λ ε ∂

∂= . (8.7.4)

This flow rule is called an associated flow-rule, because 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, these are

taken to be coincident. The normal to the yield surface is in the directionij 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.



Section 8.7


Figure 8.7.1: Yield surface

The normality rule has been confirmed by many experiments on metals. However, it

is found to be seriously in error for soils and rocks, where, for example, itoverestimates plastic volume expansion. For these materials, one must use a non-

associative flow-rule.

Next, the Tresca and Von Mises yield criteria will be discussed. First note that, to

make the differentiation easier, the associated flow-rule 8.7.4 can be expressed in

terms of principal axes as

i

p

i

f d d

σ λ ε ∂

∂= . (8.7.5)

Tresca

Taking 321 σ σ σ >> , the Tresca yield criterion is

k f −−

=2

31 σ σ (8.7.6)

One has

21,0,

21

321

−=∂∂=∂∂+=∂∂ σ σ σ f f f (8.7.7)

so, from 8.7.5, the flow-rule associated with the Tresca criterion is

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

+

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

21

21

3

2

1

0λ

ε

ε

ε

d

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 (see Fig. 8.3.9). All plastic deformation occurs in the 31− plane. Note that

8.7.8 is independent of stress.

•

•

pd ε

pd 11, ε σ

pd 22 , ε σ

pd 33, ε σ

σd



Section 8.7


Figure 8.7.2: The plastic strain increment vector and the Tresca criterion in the

π-plane (for the associated flow-rule)

Von Mises

The Von Mises yield criterion is 02

2 =−= k J f . With

( ) ( ) ( )[ ] ( )⎥⎦⎤

⎢⎣

⎡+−=−+−+−

∂

∂=

∂

∂321

2

13

2

32

2

21

11

2

2

1

3

2

6

1σ σ σ σ σ σ σ σ σ

σ σ

J (8.7.9)

one has

( )( )( )( )( )( )⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−

+−

+−

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

2121

332

3121

232

3221

132

3

2

1

σ σ σ

σ σ σ

σ σ σ

λ

ε

ε

ε

d

d

d

d

p

p

p

. (8.7.10)

This are none other than the Levy-Mises flow rule 8.4.6.

Note that if one were to use the alternative but equivalent expression

02 =−= k J f , one would have a 22 / 1 J term common to all three principal

strain increments, which could be “absorbed” into the λ d giving the same flow-rule

8.7.10. Also, one can get 8.7.10 doing a more complicated calculation involving

arbitrary stress components (see the Appendix to this section).

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. However, it

1σ ′ 2σ ′

3σ ′

02

31=−−= k f σ σ

•

pd ε



Section 8.7


does agree with experimental observations 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 materials for which the associative flow-rule applies;

this is tied closely to the notion of stable and unstable materials.

8.7.2 Stable and Unstable Materials

Consider a material with stresses ijσ and strains ijε undergoing any type of

deformation (e.g. elastic or plastic). It is then acted upon by some external agency

which induces additional stresses and strainsijd σ and

ijd ε . The external agency

then releases the load. The following two statements then define a stable material:

(these statements are also known as Drucker’s postulate):

(1) Positive work is done by the external agency, i.e. 0>ijijd d ε σ , during the

application of the loads

(2) The net work performed by the external agency over the application/removal

cycle is nonnegative

Regarding (1), the one-dimensional situation is illustrated in Fig. 8.7.3a. A strain

hardening material is stable. The stress-strain curve in Fig. 8.7.3b is typical of an

unstable material. Strain softening materials, for example soils and certain steel

alloys at high temperature, exhibit this type of response.

Figure 8.7.3: Stable (a) and unstable (b) stress-strain curves

Note that the work discussed here is that done by the additional stresses. Note also

that the laws of thermodynamics insist that the total work done, ∫ ε σ d , is positive (or

zero) in a complete cycle.

Look now at (2) in the case of strain hardening (stable) plastically deforming

materials. Consider first the one dimensional case where a material is loaded from an

initial elastic state where the stress is *σ (point A in Fig. 8.7.4) up to the yield point

Y σ (point B) and then plastically (greatly exaggerated in the figure) up to a stress

ε

σ d

ε d

0>ε σ d d

σ d

ε d

0<ε σ d d

ε

)a( )b(



Section 8.7


σ σ d Y + (point C ). The material is then unloaded to D. The material is said to have

undergone a stress cycle. The work done (per unit volume) is the area under the

stress-strain curve:

( )( )∫ −−−

−= DC B A

d W ε σ ε σ * (8.7.11)

The elastic work is recovered, so

( )( )∫−

−=C B

pd W ε σ ε σ * (8.7.12)

This is the shaded work in Fig. 8.7.4. With σ d infinitesimal (or making

σ σ σ d Y >>− * ), this is ) p

Y d ε σ σ *− or, dropping the subscript,

) pd W ε σ σ *−= (8.7.13)

The requirement (2) of a stable material is then that this is non-negative,

) 0* ≥− pd ε σ σ (8.7.14)

Figure 8.7.4: Work W done during a loading/unloading cycle

The three dimensional case is illustrated in Fig. 8.7.5, for which one has

0* ≥− p

ijijijd ε σ σ (8.7.15)

ε

σ d

*σ A D

BY σ

C

pd ε

W



Section 8.7


Figure 8.7.5: Stresses during a loading/unloading cycle

8.7.3 Consequences of the Stability Postulate

The criteria that a material be stable have very interesting consequences.

Normality

In terms of vectors in stress (plastic strain increment) space, Fig. 8.7.6, Eqn. 8.7.15

reads

( ) 0* ≥⋅− p

d εσσ (8.7.16)

These vectors are shown with the solid lines in Fig. 8.7.6. Since the dot product isnon-negative, the angle between the vectors *

σσ − and pd ε (with their starting points

coincident) must be less than 90o. This implies that the plastic strain increment vector

must be normal to the yield surface since if it were not, an initial stress state *σ could

be found for which the angle was greater than 90o

(as with the dotted vectors in Fig.

1.7.4). Thus a consequence of a material satisfying the stability requirements is that

the normality rule holds.

Figure 8.7.6: Normality of the plastic strain increment vector

initial yield

surface

ijσ

•

•

•

*ijσ

new yield

surface

*σ

σ

•

•

pd ε

*σσ − •*

σ



Section 8.7


When the yield surface has sharp corners, as with the Tresca criterion, it can be shown

that the plastic strain increment vector must lie within the cone bounded by the

normals on either side of the corner, as illustrated in Fig. 8.7.7.

Figure 8.7.7: The plastic strain increment vector for sharp corners

Convexity

Using the same arguments, one cannot have a yield surface like the one shown in Fig.

8.7.8. In other words, the yield surface is convex: the entire elastic region lies to one

side of the tangent plane to the yield surface1.

Figure 8.7.8: A non-convex surface

The Principle of Maximum Plastic Dissipation

The rate form of Eqn. 8.7.16 is

( ) 0* ≥− p

ijijij ε σ σ & (8.7.17)

The quantity p

ijijε σ & is called the plastic dissipation, and is a measure of the rate at

which energy is being dissipated as deformation proceeds.

Eqn. 8.7.17 can be written as

1 note that when the plastic deformation affects the elastic response of the material, it can be shown that

the stability postulate again ensures normality, but that the convexity does not necessarily hold

•

pd ε*

σσ −•*

σ

tangent

plane convex

surface

•

pd ε



Section 8.7


p

ijij

p

ijijε σ ε σ &&

*≥ or p pεσεσ && ⋅≥⋅ * (8.7.18)

and in this form is known as the principle of maximum plastic dissipation. The

principle is illustrated graphically in Fig. 8.7.9: of all possible stress states *

ijσ (within

or on the yield surface), the one which arises is that which requires the maximumplastic work.

Another way of saying this is that, given a plastic strain increment vector pd ε , for a

convex yield locus, the stress point on the yield locus for which pd ε is normal to the

yield locus will provide the maximum value of pd εσ ⋅ .

Figure 8.7.9: Principle of Maximum Plastic Dissipation

Limitations

Note that the stability postulate leads to the principle of maximum plastic dissipation.

However, the stability postulate is only a sufficient condition, not a necessary one. In

other words, one can satisfy the principle of maximum dissipation and yet not satisfythe stability requirements. Indeed, the principle of maximum plastic dissipation is

often taken as a starting point in analyses with no mention of the stability postulate.

For convex yield surfaces, the principle of maximum plastic dissipation is a necessary

and sufficient condition for normality. This is true for both hardening materials and

softening materials (whereas a stable material requires strain hardening). Thus

softening (and perfectly-plastic) materials obey the principle of maximum plastic

dissipation, and so also satisfy normality and convexity.

Further, for many materials, hardening and softening, a non-associative flow rule is

required, as in 8.7.3. Here, the plastic strain increment is no longer normal to theyield surface and the principle of maximum plastic dissipation does not hold in

general. In this case, when there is hardening, i.e. the stress increment is directed out

from the yield surface, it is easy to see that one can have 0< p

ijij d d ε σ , Fig. 8.7.10,

contradicting the stability postulate (1),. With hardening, there is no obvious

instability, and so it could be argued that the use of the term “stability” in Drucker’s

postulate is inappropriate.

•

pd ε•

•*

σ

σ

yield locus

pd ε



Section 8.7


Figure 8.7.10: plastic strain increment vector not normal to the yield surface;

non-associated flow-rule

8.7.4 Problems

1. Derive the flow-rule associated with the Drucker-Prager yield criterion

k J I f −+= 21α

2. Derive the flow-rule associated with the Mohr-Coulomb yield criterion, i.e. with

321 σ σ σ >> ,

k =−

2

31 σ ασ

Here,

20,1

sin1

sin1 π φ

φ

φ α <<>

−

+=

Evaluate the volumetric plastic strain increment, that is

p p p p

d d d

V

V 321 ε ε ε ++=

Δ

Δ,

and hence show that the model predicts dilatancy (expansion).

3. Consider the plastic potential

k g −−

=2

31 σ βσ

Derive the non-associative flow-rule corresponding to this potential. Hence show

that compaction of material can be modelled by choosing an appropriate value of

β .

8.7.5 Appendix to §8.7

Differentiation to obtain the Von Mises Associated Flow-Rule

The Von Mises yield criterion is 02 =−= k J f where ijij ss J 21

2 = . Using the

product rule of differentiation, in index notation:

σ•

•

pd ε σd



Section 8.7


( )

( )

( )

( )

2

31

31

31

31

2

1

21

21

2

22

2

2

2

2

2

1

2

11

2

1

J

s

ss

s

ss

ss

ss

s

ss

s

ss

s

ss

ss

ss

ss

ss J

ij

pq pq

ij

pq pq

mmijij

mnijnjmi

pq pq

mn

mnkjkinjmi

pq pq

mn

ij

mnkk mn

pq pq

mn

ij

mn

mn

pq pq

ij

mnmn

pq pqij

mnmn

ij

==

−

=

−=

−=

∂

−∂=

∂

∂=

∂

∂=

∂

∂=

∂

∂

δ

δ δ δ δ

δ δ δ δ δ

σ

δ σ σ σ

σ σ σ

(8.7.19)

and in tensor notation:

( )

( )( )

( )

( )

2

31

31

31

21

21

21

2

2:2

:2

tr

::2

tr:

:2

::2

1

:

2

1

:

1

2

1:

J

J

s

ss

s

ss

Iss

IIss

s

σ

Iσσ

ss

s

σ

ss

ss

σ

ss

ssσ

ss

σ

==

−=

⊗−=

∂

−∂=

∂∂=

∂

∂=

∂

∂=

∂

∂

I

(8.7.20)

associated flow rules

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