Introduction to Plasticity - LabMeCComputational Plasticity 3 Some basics first. Mathematical theory of plasticity The mathematical theory of plasticity attempts to describe mathematically

Computational Plasticity 1

Introduction to Plasticity

Topics: •  Phenomenological aspects

•  Simple 1-D plasticity model

•  von Mises model

• The associated initial boundary value problem (IBVP)


Some basics first. What is plasticity? Plasticity is the property, displayed by many common solid materials, of sustaining permanent (or plastic) deformation after being subjected to certain loading programmes. Typical materials that may behave as plastic include: metals, concrete, rock, soils, biological tissues, etc. The term plasticity is usually used to describe the behaviour of materials for which plastic deformations do not depend on the rate of loading. This type of behaviour is also referred to as rate-independent plasticity. Materials for which plastic deformations depend on rate of loading are usually termed rate-dependent plastic or visco-plastic materials. Note that whether a material behaves as plastic or visco-plastic depends on a number of factors such as: rate of loading, temperature, moisture, etc, etc.


Some basics first. Mathematical theory of plasticity The mathematical theory of plasticity attempts to describe mathematically the behaviour of materials classed as plastic. Plasticity theory encompasses a wide class of solid constitutive models. What is a solid constitutive model? A set of equations that describe the stress at any instant t as a functional of the history of strains up to instant t:

76 Continuum Mechanics and Thermodynamics Ch. 3

a rather general class of constitutive models of continua. The use of in-ternal variables to formulate constitutive models of dissipative materials isthen addressed in Section 3.5.2. We remark that all dissipative constitutivemodels discussed in Parts II and III of this book are based on the internalvariable approach. Section 3.5.4 summarises a generic purely mechanical in-ternal variable model. The discussion on constitutive theory ends in Section3.5.5 where the fundamental constitutive initial value problems are stated.

3.5.1 Constitutive axioms

In the present context, the axioms stated in this section must be satisfiedfor any constitutive model. Before going further, it is convenient to intro-duce the definitions of thermokinetic and calorodynamic processes (Trues-dell, 1969). A thermokinetic process of B is a pair of fields

!(p, t) and !(p, t).

A set{"(p, t), e(p, t), s(p, t), r(p, t), b(p, t), q(p, t)}

of fields satisfying the balance of momentum, the first and the second prin-ciples of thermodynamics is called a calorodynamic process of B .

Thermodynamic determinism

The basic axiom underlying the constitutive theory discussed here is theprinciple of thermodynamically compatible determinism (Truesdell, 1969).It postulates that “the history of the thermokinetic process to which aneighbourhood of a point p of B has been subjected determines a caloro-dynamic process for B at p”. In particular, we shall be concerned withso-called simple materials, for which the local history (history at point ponly) of F, ! and g su!ces to determine the history of the thermokineticprocess for constitutive purposes. In this case, regarding the body force band heat supply r as delivered, respectively, by the linear momentum bal-ance (3.132)1 and conservation of energy (3.136) and introducing the specificfree energy (3.139), the principle of thermodynamic determinism implies theexistence of constitutive functionals F, G, H and I of the hystories of F, !and g such that, for a point p,

"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)





!(p, t) and !(p, t).





"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)





!(p, t) and !(p, t).





"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)

Sec. 3.5 Constitutive Theory 83

underlying microscopic dissipation mechanisms.

The phenomenological approach to irreversible thermodynamics hasbeen particularly successful in the field of solid mechanics. Numerous wellestablished models of solids, such as classical isotropic elastoplasticity andviscoplasticity, discussed in Parts II and III of this book, have been devel-oped on a purely phenomenological basis providing evidence of how powerfulsuch an approach to irreversible thermodynamics can be when the majorconcern is the description of the essentially macroscopic behaviour. In someinstances, however, the inclusion of microscopic information becomes essen-tial and a purely phenomenological methodology is unlikely to describe thebehaviour of the material with su!cient accuracy. One such case is illus-trated in Chapter 16, where a microscopically-based continuum model ofductile metallic crystals is described.

3.5.4 The purely mechanical theory

Thermal e"ects are ignored in the constitutive theories addressed in Parts IIand III of this book. It is, therefore, convenient at this point to summarisethe general internal variable-based constitutive equations in the purely me-chanical case. By removing the thermally-related terms of the above theory,we end up with the following set of mechanical constitutive equations:

!

""""#

""""$

! = !(F, !)

P = "#!

#F

! = f(F, !).

(3.164)

The infinitesimal strain case

In the infinitesimal strain case, the infinitesimal strain tensor, ", replacesthe deformation gradient and the stress tensor # of the infinitesimal theoryreplaces the first Piola-Kirchho" stress. We then have the general constitu-tive law

!

""""#

""""$

! = !(", !)

# = "#!

#"

! = f(", !).

(3.165)

3.5.5 The constitutive initial value problem

Our basic constitutive problem is defined as follows: “Given the history ofthe deformation gradient (and the history of temperature and temperature


Phenomenological Aspects of Plasticity


Typical tensile test (ductile metals)

Features to note: •  Elastic domain/Yield limit

•  Plastic straining above yield limit •  Evolution of the yield stress (hardening)

Sec. 6.1 Phenomenological Aspects 157

!

"

!0

" pO0 O1

Y1Y0

Z 0

Z 1!1

Figure 6.1: Uniaxial tension experiment with ductile metals

(or strain) loading is reversed at, say, point Z0, the bar returns to an un-stressed state via path Z0O1. The new unstressed state, O1, di!ers fromthe initial unstressed state, O0, in that a permanent change in the shape ofthe bar is observed. This shape change is represented in the graph by thepermanent (or plastic) axial strain !p. Monotonic reloading of the bar to astress level "1 will follow the path O1Y1Z1. Similarly to the initial elasticsegment O0Y0, the portion O1Y1 is also virtually straight and unloadingfrom Y1 (or before Y1 is reached) will bring the stress-strain state back tothe unstressed configuration O1, with no further plastic straining of the bar.So, the behaviour of the material in the segment O1Y1 may also be regardedas linear elastic. Here, it is important to emphasise that even though somediscrepancy between unloading and reloading curves (such as lines Z0O1

and O1Y1) is observed in typical experiments, the actual di!erence betweenthem is in fact much smaller than that shown in the diagram of Figure 6.1.Again, loading beyond an elastic limit (point Y1 in this case) will causefurther increase in plastic deformation.

Some important phenomenological properties can be identified in theabove described uniaxial test. They are enumerated below:

1. The existence of an elastic domain, i.e., a range of stresses withinwhich the behaviour of the material can be considered as purely elas-tic, without evolution of permanent (plastic) strains. The elastic do-main is delimited by the so-called yield stress. In Figure 6.1, segmentsO0Y0 and O1Y1 define the elastic domain at two di!erent states. The


Simple 1-D Plasticity Model


First modelling step: Mathematical model of observed stress-strain curve

Sec. 6.1 Phenomenological Aspects 157

!

"

!0

" pO0 O1

Y1Y0

Z 0

Z 1!1

Figure 6.1: Uniaxial tension experiment with ductile metals

(or strain) loading is reversed at, say, point Z0, the bar returns to an un-stressed state via path Z0O1. The new unstressed state, O1, di!ers fromthe initial unstressed state, O0, in that a permanent change in the shape ofthe bar is observed. This shape change is represented in the graph by thepermanent (or plastic) axial strain !p. Monotonic reloading of the bar to astress level "1 will follow the path O1Y1Z1. Similarly to the initial elasticsegment O0Y0, the portion O1Y1 is also virtually straight and unloadingfrom Y1 (or before Y1 is reached) will bring the stress-strain state back tothe unstressed configuration O1, with no further plastic straining of the bar.So, the behaviour of the material in the segment O1Y1 may also be regardedas linear elastic. Here, it is important to emphasise that even though somediscrepancy between unloading and reloading curves (such as lines Z0O1

and O1Y1) is observed in typical experiments, the actual di!erence betweenthem is in fact much smaller than that shown in the diagram of Figure 6.1.Again, loading beyond an elastic limit (point Y1 in this case) will causefurther increase in plastic deformation.

Some important phenomenological properties can be identified in theabove described uniaxial test. They are enumerated below:

1. The existence of an elastic domain, i.e., a range of stresses withinwhich the behaviour of the material can be considered as purely elas-tic, without evolution of permanent (plastic) strains. The elastic do-main is delimited by the so-called yield stress. In Figure 6.1, segmentsO0Y0 and O1Y1 define the elastic domain at two di!erent states. The

Sec. 6.2 One-dimensional Constitutive Model 159

!

"

!O0

O1

Y1 Z 0

Y0

"0

"

! p

Eslope

Eslope ep

Z 1

Figure 6.2: Uniaxial tension experiment. Mathematical model

stress-strain curve always follows the path defined by O0Y0Y1Z1. This pathis normally referred to as the virgin curve and is obtained by a continuousmonotonic loading from the initial unstressed state O0.

Under the above assumptions, after being monotonically loaded fromthe initial unstressed state to the stress level !0, the behaviour of the barbetween states O1 and Y1 is considered linear elastic, with constant plasticstrain, "p, and yield limit, !0. Thus, within the segment O1Y1, the uniaxialstress corresponding to a configuration with total strain " is given by

! = E (" ! "p), (6.1)

where E denotes the Young’s modulus of the material of the bar. Note thatthe di!erence between the total strain and the current plastic strain, "!"p,is fully reversible. That is, upon complete unloading of the bar, "!"p is fullyrecovered without further evolution of plastic strains. This motivates theadditive decomposition of the axial strain described in the following section.

6.2.1 Elasto-plastic decomposition of the axial strain

One of the chief hypothesis underlying the small strain theory of plasticityis the decomposition of the total strain, ", into the sum of an elastic (or


Mathematical modelling (contd.)


!

"

!O0

O1

Y1 Z 0

Y0

"0

"

! p

Eslope

Eslope ep

Z 1




! = E (" ! "p), (6.1)




1. Elastoplastic decomposition of axial strain 2. Uniaxial elastic uniaxial constitutive law

160 The Mathematical Theory of Plasticity Ch. 6

reversible) component, !e, and a plastic (or permanent) component, !p,

! = !e + !p, (6.2)

where the elastic strain has been defined as

!e = ! ! !p. (6.3)

6.2.2 The elastic uniaxial constitutive law

Following the above definition of the elastic axial strain, the constitutivelaw for the axial stress can be expressed as

" = E !e. (6.4)

The next step in the definition of the uniaxial constitutive model is toderive formulae that express mathematically the fundamental phenomeno-logical properties enumerated in Section 6.1. The items 1 and 2 of Sec-tion 6.1 are associated with the formulation of a yield criterion and a plas-tic flow rule, whereas item 3 requires the formulation of a hardening law .These are described in the following.

6.2.3 The yield function and the yield criterion

The existence of an elastic domain delimited by a yield stress has beenpointed out in item 1 of Section 6.1. With the introduction of a yieldfunction, !, of the form

!(", "y) = |"|! "y , (6.5)

the elastic domain at a state with uniaxial yield stress "y can be defined inthe one-dimensional plasticity model as the set

E = { " | !(", "y)<0 }, (6.6)

or, equivalently, the elastic domain is the set of stresses " that satisfy

|"| < "y . (6.7)

Generalising the results of the uniaxial tension test discussed, it has beenassumed in the above that the yield stress in compression is identical tothat in tension. The corresponding idealised elastic domain is illustrated inFigure 6.3.

It should be noted that, at any stage, no stress level is allowed abovethe current yield stress, i.e., plastically admissible stresses lie either in the



! = !e + !p, (6.2)


!e = ! ! !p. (6.3)



" = E !e. (6.4)




!(", "y) = |"|! "y , (6.5)


E = { " | !(", "y)<0 }, (6.6)


|"| < "y . (6.7)





! = !e + !p, (6.2)


!e = ! ! !p. (6.3)



" = E !e. (6.4)




!(", "y) = |"|! "y , (6.5)


E = { " | !(", "y)<0 }, (6.6)


|"| < "y . (6.7)




!

"

!O0

O1

Y1 Z 0

Y0

"0

"

! p

Eslope

Eslope ep

Z 1




! = E (" ! "p), (6.1)





Mathematical modelling (contd.) Sec. 6.2 One-dimensional Constitutive Model 159

!

"

!O0

O1

Y1 Z 0

Y0

"0

"

! p

Eslope

Eslope ep

Z 1




! = E (" ! "p), (6.1)




3. Yield function and yield criterion definition Elastic domain



! = !e + !p, (6.2)


!e = ! ! !p. (6.3)



" = E !e. (6.4)




!(", "y) = |"|! "y , (6.5)


E = { " | !(", "y)<0 }, (6.6)


|"| < "y . (6.7)





! = !e + !p, (6.2)


!e = ! ! !p. (6.3)



" = E !e. (6.4)




!(", "y) = |"|! "y , (6.5)


E = { " | !(", "y)<0 }, (6.6)


|"| < "y . (6.7)




!

"0

tension

!y

# !y

compression

elasticdomain

Figure 6.3: Uniaxial model. Elastic domain

elastic domain or on its boundary (the yield limit). Thus, any admissiblestress must satisfy the restriction

!(!, !y) ! 0. (6.8)

For stress levels within the elastic domain, only elastic straining may occur,whereas on its boundary (at the yield stress), either elastic unloading orplastic yielding (or plastic loading) takes place. This yield criterion can beexpressed by

If !(!, !y)<0 =" "p =0,

If !(!, !y)=0 ="!

"p =0 for elastic unloading,

"p #=0 for plastic loading.

(6.9)

6.2.4 The plastic flow rule. Loading/unloading conditions

Expressions (6.9) above have defined a criterion for plastic yielding, i.e.,they have set the conditions under which plastic straining may occur. Bynoting in Figure 6.3 that, upon plastic loading, the plastic strain rate "p ispositive (stretching) under tension (positive !) and negative (compressive)under compression (negative !), the plastic flow rule for the uniaxial modelcan be formally established as

"p = # sign(!), (6.10)


Mathematical modelling (contd.)


!

"

!O0

O1

Y1 Z 0

Y0

"0

"

! p

Eslope

Eslope ep

Z 1




! = E (" ! "p), (6.1)




Yield criterion


!

"0

tension

!y

# !y

compression

elasticdomain



!(!, !y) ! 0. (6.8)


If !(!, !y)<0 =" "p =0,

If !(!, !y)=0 ="!



(6.9)



"p = # sign(!), (6.10)



!

"

!O0

O1

Y1 Z 0

Y0

"0

"

! p

Eslope

Eslope ep

Z 1




! = E (" ! "p), (6.1)




4.  Plastic flow rule

where and the plastic multiplier satisfies the complementarity condition


where sign is the signum function defined as

sign(a) =

!

+1 if a ! 0

"1 if a < 0(6.11)

for any scalar a and the scalar ! is termed the plastic multiplier . The plasticmultiplier is non-negative,

! ! 0, (6.12)

and satisfies the complementarity condition

! ! = 0. (6.13)

The constitutive equations (6.10)–(6.13) imply that, as stated in the yieldcriterion (6.9), the plastic strain rate vanishes within the elastic domain,i.e.,

! < 0 =# ! = 0 =# "p = 0, (6.14)

and plastic flow ("p $= 0) may occur only when the stress level # coincideswith the current yield stress

|#| = #y =# ! = 0 =# ! ! 0. (6.15)

Expressions (6.8), (6.12) and (6.13) define the so-called loading/unloadingconditions of the elastic-plastic model. That is, the constraints

! % 0 ! ! 0 !! = 0, (6.16)

establish when plastic flow may occur.

6.2.5 The hardening law

Finally, the complete characterisation of the uniaxial model is achieved withthe introduction of the hardening law . As remarked in item 3 of Section6.1, an evolution of the yield stress accompanies the evolution of the plasticstrain. This phenomenon, known as hardening, can be incorporated intothe uniaxial model by simply assuming that, in the definition (6.5) of !,the yield stress #y is a given function

#y = #y("p) (6.17)

of the accumulated axial plastic strain, "p. The accumulated axial plasticstrain is defined as

"p &" t

0|"p| dt, (6.18)


!

"0

tension

!y

# !y

compression

elasticdomain



!(!, !y) ! 0. (6.8)


If !(!, !y)<0 =" "p =0,

If !(!, !y)=0 ="!



(6.9)



"p = # sign(!), (6.10)



sign(a) =

!

+1 if a ! 0

"1 if a < 0(6.11)


! ! 0, (6.12)


! ! = 0. (6.13)


! < 0 =# ! = 0 =# "p = 0, (6.14)


|#| = #y =# ! = 0 =# ! ! 0. (6.15)


! % 0 ! ! 0 !! = 0, (6.16)




#y = #y("p) (6.17)


"p &" t

0|"p| dt, (6.18)



sign(a) =

!

+1 if a ! 0

"1 if a < 0(6.11)


! ! 0, (6.12)


! ! = 0. (6.13)


! < 0 =# ! = 0 =# "p = 0, (6.14)


|#| = #y =# ! = 0 =# ! ! 0. (6.15)


! % 0 ! ! 0 !! = 0, (6.16)




#y = #y("p) (6.17)


"p &" t

0|"p| dt, (6.18)

Note that



sign(a) =

!

+1 if a ! 0

"1 if a < 0(6.11)


! ! 0, (6.12)


! ! = 0. (6.13)


! < 0 =# ! = 0 =# "p = 0, (6.14)


|#| = #y =# ! = 0 =# ! ! 0. (6.15)


! % 0 ! ! 0 !! = 0, (6.16)




#y = #y("p) (6.17)


"p &" t

0|"p| dt, (6.18)



sign(a) =

!

+1 if a ! 0

"1 if a < 0(6.11)


! ! 0, (6.12)


! ! = 0. (6.13)


! < 0 =# ! = 0 =# "p = 0, (6.14)


|#| = #y =# ! = 0 =# ! ! 0. (6.15)


! % 0 ! ! 0 !! = 0, (6.16)




#y = #y("p) (6.17)


"p &" t

0|"p| dt, (6.18)



sign(a) =

!

+1 if a ! 0

"1 if a < 0(6.11)


! ! 0, (6.12)


! ! = 0. (6.13)


! < 0 =# ! = 0 =# "p = 0, (6.14)


|#| = #y =# ! = 0 =# ! ! 0. (6.15)


! % 0 ! ! 0 !! = 0, (6.16)




#y = #y("p) (6.17)


"p &" t

0|"p| dt, (6.18)

Loading/Unloading criterion



!

"

!O0

O1

Y1 Z 0

Y0

"0

"

! p

Eslope

Eslope ep

Z 1




! = E (" ! "p), (6.1)




5. Hardening law

where is the accumulated plastic strain. Its evolution equation reads or, equivalently,



sign(a) =

!

+1 if a ! 0

"1 if a < 0(6.11)


! ! 0, (6.12)


! ! = 0. (6.13)


! < 0 =# ! = 0 =# "p = 0, (6.14)


|#| = #y =# ! = 0 =# ! ! 0. (6.15)


! % 0 ! ! 0 !! = 0, (6.16)




#y = #y("p) (6.17)


"p &" t

0|"p| dt, (6.18)



sign(a) =

!

+1 if a ! 0

"1 if a < 0(6.11)


! ! 0, (6.12)


! ! = 0. (6.13)


! < 0 =# ! = 0 =# "p = 0, (6.14)


|#| = #y =# ! = 0 =# ! ! 0. (6.15)


! % 0 ! ! 0 !! = 0, (6.16)




#y = #y("p) (6.17)


"p &" t

0|"p| dt, (6.18)


!A

0

!y

"p

!y ( )"p

hardening slope, H

Figure 6.4: One-dimensional model. Hardening curve

thus ensuring that both tensile and compressive plastic straining contributeto !p. Clearly, in a monotonic tensile test one has

!p = !p, (6.19)

whereas in a monotonic compressive uniaxial test,

!p = !!p. (6.20)

The curve defined by the hardening function "y(!p) is usually referred to asthe hardening curve (Figure 6.4).

From the definition of !p, it follows that its evolution law is given by

˙!p = |!p|, (6.21)

which, in view of the plastic flow rule, is equivalent to

˙!p = #. (6.22)

6.2.6 Summary of the model

The overall one-dimensional plasticity model is defined by the constitutiveequations (6.2), (6.4), (6.5), (6.10), (6.16) (6.17) and (6.22). The model issummarised in Box 6.1.


!A

0

!y

"p

!y ( )"p

hardening slope, H

Figure 6.4: One-dimensional model. Hardening curve

thus ensuring that both tensile and compressive plastic straining contributeto !p. Clearly, in a monotonic tensile test one has

!p = !p, (6.19)

whereas in a monotonic compressive uniaxial test,

!p = !!p. (6.20)

The curve defined by the hardening function "y(!p) is usually referred to asthe hardening curve (Figure 6.4).

From the definition of !p, it follows that its evolution law is given by

˙!p = |!p|, (6.21)

which, in view of the plastic flow rule, is equivalent to

˙!p = #. (6.22)

6.2.6 Summary of the model

The overall one-dimensional plasticity model is defined by the constitutiveequations (6.2), (6.4), (6.5), (6.10), (6.16) (6.17) and (6.22). The model issummarised in Box 6.1.


Mathematical modelling… the constitutive model, at last !!


Box 6.1: One-dimensional elasto-plastic constitutive model

1. Elasto-plastic split of the axial strain

! = !e + !p

2. Uniaxial elastic law" = E !e

3. Yield function!(", "y) = |"|! "y

4. Plastic flow rule!p = # sign(")

5. Hardening law"y = "y(!p)

˙!p = #

6. Loading/unloading criterion

! " 0 # # 0 #! = 0

6.2.7 Determination of the plastic multiplier

So far, in the uniaxial plasticity model introduced above, the plastic multi-plier , #, was left indeterminate during plastic yielding. Indeed, expressions(6.12) and (6.13) just tell us that # vanishes during elastic straining but mayassume any non-negative value during plastic flow. In order to eliminatethis indetermination, one should note firstly that, during plastic flow , thevalue of the yield function remains constant

! = 0, (6.23)

since the absolute value of the current stress always coincides with the cur-rent yield stress. Therefore, the following additional complementarity con-dition may be established:

! # = 0 (6.24)

which implies that, the rate of ! vanishes whenever plastic yielding occurs(# $=0),

! = 0, (6.25)





!(p, t) and !(p, t).





"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)





!(p, t) and !(p, t).





"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)





!(p, t) and !(p, t).





"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)






!

""""#

""""$

! = !(F, !)

P = "#!

#F

! = f(F, !).

(3.164)



!

""""#

""""$

! = !(", !)

# = "#!

#"

! = f(", !).

(3.165)




Determination of the plastic multiplier During plastic flow, we have so that the following additional complementarity condition can be established The derivative of gives so that we get In addition, the elastic law gives By combining the above, we find that the plastic multiplier is uniquely determined as




! = !e + !p





˙!p = #


! " 0 # # 0 #! = 0



! = 0, (6.23)


! # = 0 (6.24)


! = 0, (6.25)




! = !e + !p





˙!p = #


! " 0 # # 0 #! = 0



! = 0, (6.23)


! # = 0 (6.24)


! = 0, (6.25)




! = !e + !p





˙!p = #


! " 0 # # 0 #! = 0



! = 0, (6.23)


! # = 0 (6.24)


! = 0, (6.25)


and, during elastic straining, (! = 0), ! may assume any value. Equation(6.25) is called the consistency condition By taking the time derivative ofthe yield function (6.5), one obtains

! = sign(") " ! H ˙#p, (6.26)

where H is called the hardening modulus, or hardening slope, and is definedas (refer to Figure 6.4)

H = H(#p) =d"y

d#p. (6.27)

Under plastic yielding, equation (6.25) holds so that one has the followingexpression for the stress rate

sign(") " = H ˙#p. (6.28)

From the elastic law, it follows that

" = E(# ! #p). (6.29)

Finally, by combining the above expression with (6.22), (6.28) and (6.10),the plastic multiplier, !, is uniquely determined during plastic yielding as

! =E

H + Esign(") # =

E

H + E|#|. (6.30)

6.2.8 The elasto-plastic tangent modulus

Let us now return to the stress-strain curve of Figure 6.2. Plastic flow ata generic yield limit produces the following tangent relation between strainand stress

" = Eep #, (6.31)

where Eep is called the elasto-plastic tangent modulus. By combining ex-pressions (6.31), (6.29), the flow rule (6.10) and (6.30) the following expres-sion is obtained for the elasto-plastic tangent modulus

Eep =E H

E + H. (6.32)

Equivalently, the hardening modulus, H , can be expressed in terms of theelastic modulus and the elasto-plastic modulus as

H =Eep

1 ! Eep/E. (6.33)



! = sign(") " ! H ˙#p, (6.26)


H = H(#p) =d"y

d#p. (6.27)


sign(") " = H ˙#p. (6.28)


" = E(# ! #p). (6.29)


! =E

H + Esign(") # =

E

H + E|#|. (6.30)



" = Eep #, (6.31)


Eep =E H

E + H. (6.32)


H =Eep

1 ! Eep/E. (6.33)



! = sign(") " ! H ˙#p, (6.26)


H = H(#p) =d"y

d#p. (6.27)


sign(") " = H ˙#p. (6.28)


" = E(# ! #p). (6.29)


! =E

H + Esign(") # =

E

H + E|#|. (6.30)



" = Eep #, (6.31)


Eep =E H

E + H. (6.32)


H =Eep

1 ! Eep/E. (6.33)



! = sign(") " ! H ˙#p, (6.26)


H = H(#p) =d"y

d#p. (6.27)


sign(") " = H ˙#p. (6.28)


" = E(# ! #p). (6.29)


! =E

H + Esign(") # =

E

H + E|#|. (6.30)



" = Eep #, (6.31)


Eep =E H

E + H. (6.32)


H =Eep

1 ! Eep/E. (6.33)



! = sign(") " ! H ˙#p, (6.26)


H = H(#p) =d"y

d#p. (6.27)


sign(") " = H ˙#p. (6.28)


" = E(# ! #p). (6.29)


! =E

H + Esign(") # =

E

H + E|#|. (6.30)



" = Eep #, (6.31)


Eep =E H

E + H. (6.32)


H =Eep

1 ! Eep/E. (6.33)

The tangential relation between stress and strain reads By combining the previous relations we obtain Equivalently, we have


The elasto-plastic tangent modulus Sec. 6.2 One-dimensional Constitutive Model 159

!

"

!O0

O1

Y1 Z 0

Y0

"0

"

! p

Eslope

Eslope ep

Z 1




! = E (" ! "p), (6.1)






! = sign(") " ! H ˙#p, (6.26)


H = H(#p) =d"y

d#p. (6.27)


sign(") " = H ˙#p. (6.28)


" = E(# ! #p). (6.29)


! =E

H + Esign(") # =

E

H + E|#|. (6.30)



" = Eep #, (6.31)


Eep =E H

E + H. (6.32)


H =Eep

1 ! Eep/E. (6.33)



! = sign(") " ! H ˙#p, (6.26)


H = H(#p) =d"y

d#p. (6.27)


sign(") " = H ˙#p. (6.28)


" = E(# ! #p). (6.29)


! =E

H + Esign(") # =

E

H + E|#|. (6.30)



" = Eep #, (6.31)


Eep =E H

E + H. (6.32)


H =Eep

1 ! Eep/E. (6.33)



! = sign(") " ! H ˙#p, (6.26)


H = H(#p) =d"y

d#p. (6.27)


sign(") " = H ˙#p. (6.28)


" = E(# ! #p). (6.29)


! =E

H + Esign(") # =

E

H + E|#|. (6.30)



" = Eep #, (6.31)


Eep =E H

E + H. (6.32)


H =Eep

1 ! Eep/E. (6.33)



! = sign(") " ! H ˙#p, (6.26)


H = H(#p) =d"y

d#p. (6.27)


sign(") " = H ˙#p. (6.28)


" = E(# ! #p). (6.29)


! =E

H + Esign(") # =

E

H + E|#|. (6.30)



" = Eep #, (6.31)


Eep =E H

E + H. (6.32)


H =Eep

1 ! Eep/E. (6.33)


von Mises Plasticity Model


The von Mises yield criterion “Plastic flow begins when the stress deviator invariant reaches a critical value:” This is equivalent to say that plastic yield begins when the stored distortional elastic strain energy reaches a critical value. Note that the stored elastic energy can be decomposed as a sum of distortional and volumetric contributions where Hence, the critical value of distortional strain energy at which plastic yield starts is


where J3 is the third principal invariant of stress deviator§

J3 ! I3(s) ! det s = 13 tr(s)3. (6.98)

The Lode angle is the angle, on the deviatoric plane, between s and thenearest pure shear line (a pure shear line is graphically represented in Figure6.11). It satisfies

"!

6# " # !

6. (6.99)

Despite being used often in computational plasticity, the above invariantrepresentation results in rather cumbersome algorithms for integration of theevolution equations of the Tresca model. This is essentially due to the highdegree of non-linearity introduced by the trigonometric function involved inthe definition of the Lode angle. The multi-surface representation, on theother hand, is found by the authors to provide an optimal parametrisationof the Tresca surface which results in a simpler numerical algorithm and willbe adopted in the computational implementation of the model addressed inChapter 8.

6.4.2 The von Mises yield criterion

Also appropriate to describe plastic yielding in metals, this criterion wasproposed by von Mises (1913). According to the von Mises criterion, plasticyielding begins when the J2 stress deviator invariant reaches a critical value.This condition is mathematically represented by the equation

J2 = R(#), (6.100)

where R is the critical value, here assumed to be a function of a hardeninginternal variable, #, to be defined later.

The physical interpretation of the von Mises criterion is given in thefollowing. Since the elastic behaviour of the materials described in thischapter is assumed linear elastic, the stored elastic strain-energy at thegeneric state defined by the stress $ can be decomposed as the sum

%e = %ed + %e

v, (6.101)

of a distortional contribution,

& %ed =

1

2 Gs : s =

1

GJ2, (6.102)

§The equivalence between the the two rightmost terms in (6.98) is established bysumming the characteristic equation (2.73) [page 28] for i=1, 2, 3 and taking into accountthe fact that I1(s) = 0 (s is a traceless tensor) and that tr(S )3 =

!

is3i for any

symmetric tensor S.



J3 ! I3(s) ! det s = 13 tr(s)3. (6.98)


"!

6# " # !

6. (6.99)




J2 = R(#), (6.100)



%e = %ed + %e

v, (6.101)


& %ed =

1

2 Gs : s =

1

GJ2, (6.102)


!

is3i for any

symmetric tensor S.



J3 ! I3(s) ! det s = 13 tr(s)3. (6.98)


"!

6# " # !

6. (6.99)




J2 = R(#), (6.100)



%e = %ed + %e

v, (6.101)


& %ed =

1

2 Gs : s =

1

GJ2, (6.102)


!

is3i for any

symmetric tensor S.



J3 ! I3(s) ! det s = 13 tr(s)3. (6.98)


"!

6# " # !

6. (6.99)




J2 = R(#), (6.100)



%e = %ed + %e

v, (6.101)


& %ed =

1

2 Gs : s =

1

GJ2, (6.102)


!

is3i for any

symmetric tensor S.

Sec. 6.4 Classical Yield Criteria 183

and a volumetric contribution,

! "ev =

1

Kp2, (6.103)

where G and K are, respectively, the shear and bulk modulus. In view of(6.102), the von Mises criterion is equivalent to stating that plastic yieldingbegins when the distortional elastic strain-energy reaches a critical value.The corresponding critical value of the distortional energy is

1

GR.

It should be noted that, as in the Tresca criterion, the pressure componentof the stress tensor does not take part in the definition of the von Misescriterion and only the deviatoric stress can influence plastic yielding. Thus,the von Mises criterion is also pressure-insensitive.

In a state of pure shear , i.e., a state with stress tensor

[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)

Thus, a yield function for the von Mises criterion can be defined as

!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#

R is the shear yield stress. Let us now consider a state ofuniaxial stress :

[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)

In this case, we have

[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)

The above expression for the J2-invariant suggests the following alternativedefinition of the von Mises yield function:

!(!) = q(!) ! $y , (6.110)



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)


…pure shear, and a von Mises yield function can be defined as …uniaxial stress, and the von Mises yield function can be defined as



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)



! "ev =

1

Kp2, (6.103)


1

GR.



[!] =

!

"

0 # 0# 0 00 0 0

#

$ , (6.104)

we have, s = ! andJ2 = #2. (6.105)


!(!) =%

J2(s(!)) ! #y, (6.106)

where #y "#


[!] =

!

"

$ 0 00 0 00 0 0

#

$ . (6.107)


[s] =

!

"

23$ 0 00 ! 1

3$ 00 0 ! 1

3$

#

$ (6.108)

andJ2 = 1

3$2. (6.109)


!(!) = q(!) ! $y , (6.110)


where !y !"

3R is the uniaxial yield stress and

q(!) !!

3 J2(s(!)) (6.111)

is termed the von Mises e!ective or equivalent stress . The uniaxial andshear yield stresses for the von Mises criterion are related by

!y ="

3 "y. (6.112)

Note that this relation di!ers from that of the Tresca criterion given by(6.85). Obviously, due to its definition in terms of an invariant of the stresstensor, the von Mises yield function is an isotropic function of !.

The von Mises and Tresca criteria may be set to agree with one anotherin either uniaxial stress or pure shear states. If they are set by using theyield functions (6.84) and (6.110) so that both predict the same uniaxialyield stress !y, then, under pure shear, the von Mises criterion will predicta yield stress 2/

"3 (# 1.155) times that given by the Tresca criterion.

On the other hand, if both criteria are made to coincide under pure shear(expressions (6.83) and (6.106) with the same "y), then, in uniaxial stressstates, the von Mises criterion will predict yielding at a stress level

"3/2

(#0.866) times the level predicted by Tresca’s law.

The yield surface (" = 0) associated with the von Mises criterion isrepresented, in the space of principal stresses, by the surface of an infinitecircular cylinder the axis of which coincides with the hydrostatic axis. Thevon Mises surface is illustrated in Figure 6.8 where it has been set to matchthe Tresca surface (shown in the same figure) under uniaxial stress. Thecorresponding #-plane representation is shown in Figure 6.9(b). The vonMises circle intersects the vertices of the Tresca hexagon. The yield surfacesfor the von Mises and Tresca criteria set to coincide in shear is shown inFigure 6.11. In this case, the von Mises circle is tangent to the sides ofthe Tresca hexagon. It is remarked that, for many metals, experimentallydetermined yield surfaces fall between the von Mises and Tresca surfaces.A more general model which includes both the Tresca and the von Misesyield surfaces as particular cases (and, in addition, allow for anisotropy ofthe yield surface) is described in Section 10.3.4 (starting page 482).

6.4.3 The Mohr-Coulomb yield criterion

The criteria presented so far are pressure-insensitive and adequate to de-scribe metals. For materials such as soils, rocks and concrete, whose be-haviour is generally characterised by a strong dependence of the yield limiton the hydrostatic pressure, appropriate description of plastic yielding re-quires the introduction of pressure-sensitivity. A classical example of a


where !y !"

3R is the uniaxial yield stress and

q(!) !!

3 J2(s(!)) (6.111)

is termed the von Mises e!ective or equivalent stress . The uniaxial andshear yield stresses for the von Mises criterion are related by

!y ="

3 "y. (6.112)

Note that this relation di!ers from that of the Tresca criterion given by(6.85). Obviously, due to its definition in terms of an invariant of the stresstensor, the von Mises yield function is an isotropic function of !.

The von Mises and Tresca criteria may be set to agree with one anotherin either uniaxial stress or pure shear states. If they are set by using theyield functions (6.84) and (6.110) so that both predict the same uniaxialyield stress !y, then, under pure shear, the von Mises criterion will predicta yield stress 2/

"3 (# 1.155) times that given by the Tresca criterion.

On the other hand, if both criteria are made to coincide under pure shear(expressions (6.83) and (6.106) with the same "y), then, in uniaxial stressstates, the von Mises criterion will predict yielding at a stress level

"3/2

(#0.866) times the level predicted by Tresca’s law.

The yield surface (" = 0) associated with the von Mises criterion isrepresented, in the space of principal stresses, by the surface of an infinitecircular cylinder the axis of which coincides with the hydrostatic axis. Thevon Mises surface is illustrated in Figure 6.8 where it has been set to matchthe Tresca surface (shown in the same figure) under uniaxial stress. Thecorresponding #-plane representation is shown in Figure 6.9(b). The vonMises circle intersects the vertices of the Tresca hexagon. The yield surfacesfor the von Mises and Tresca criteria set to coincide in shear is shown inFigure 6.11. In this case, the von Mises circle is tangent to the sides ofthe Tresca hexagon. It is remarked that, for many metals, experimentallydetermined yield surfaces fall between the von Mises and Tresca surfaces.A more general model which includes both the Tresca and the von Misesyield surfaces as particular cases (and, in addition, allow for anisotropy ofthe yield surface) is described in Section 10.3.4 (starting page 482).

6.4.3 The Mohr-Coulomb yield criterion

The criteria presented so far are pressure-insensitive and adequate to de-scribe metals. For materials such as soils, rocks and concrete, whose be-haviour is generally characterised by a strong dependence of the yield limiton the hydrostatic pressure, appropriate description of plastic yielding re-quires the introduction of pressure-sensitivity. A classical example of a



(a) (b)

!2

"!plane

0

!3

!1

hydrostatic

axis

von Mises

Tresca

!1 !2

!3

Figure 6.9: (a) The !-plane in principal stress space and, (b) The !-plane repre-sentation of the Tresca and von Mises yield surfaces.

Multi-surface representation

Equivalently to the above representation, the Tresca yield criterion can beexpressed by means of the following six yield functions

!1(!, !y) = !1 ! !3 ! !y

!2(!, !y) = !2 ! !3 ! !y

!3(!, !y) = !2 ! !1 ! !y

!4(!, !y) = !3 ! !1 ! !y

!5(!, !y) = !3 ! !2 ! !y

!6(!, !y) = !1 ! !2 ! !y,

(6.91)

so that, for fixed !y, the equation

!i(!, !y) = 0 (6.92)

corresponds to a plane in the space of principal stresses for each i=1, ..., 6(see Figure 6.10).

In the multi-surface representation, the elastic domain for a given !y

can be defined as

E = { ! | !i(!, !y) < 0 i = 1, ..., 6 }. (6.93)

Definitions (6.87) and (6.93) are completely equivalent. The yield surface –the boundary of E – is defined in this case as the set of stresses for which!i(!, !y)=0 for at least one i with !j(!, !y)"0 for j #= i.


!3 " p

!1

!2

Tresca

von Mises

#$3

Figure 6.8: The Tresca and von Mises yield surfaces in principal stress space.

represented in a particularly simple and useful format as a three-dimensionalsurface in the space of principal stresses.

In principal stress space, the Tresca yield surface, i.e., the set of stressesfor which ! = 0, is graphically represented by the surface of an infinitehexagonal prism with axis coinciding with the hydrostatic line (also knownas the space diagonal), defined by !1 =!2 =!3. This is illustrated in Figure6.8. The elastic domain (for which ! < 0) corresponds to the interior of theprism. Due to the assumed insensitivity to pressure, a further simplificationin the representation of the yield surface is possible in this case. The Trescayield surface may be represented, without loss of generality, by its projectionon the subspace of stresses with zero hydrostatic pressure component (!1+!2+!3 = 0). This subspace is called the deviatoric plane, also referred toas the "-plane. It is graphically illustrated in Figure 6.9(a). Figure 6.9(b)shows the "-plane projection of the Tresca yield surface.

Graphical representation

principal stress space deviatoric plane

( -plane)


!3 " p

!1

!2

Tresca

von Mises

#$3

Figure 6.8: The Tresca and von Mises yield surfaces in principal stress space.

represented in a particularly simple and useful format as a three-dimensionalsurface in the space of principal stresses.

In principal stress space, the Tresca yield surface, i.e., the set of stressesfor which ! = 0, is graphically represented by the surface of an infinitehexagonal prism with axis coinciding with the hydrostatic line (also knownas the space diagonal), defined by !1 =!2 =!3. This is illustrated in Figure6.8. The elastic domain (for which ! < 0) corresponds to the interior of theprism. Due to the assumed insensitivity to pressure, a further simplificationin the representation of the yield surface is possible in this case. The Trescayield surface may be represented, without loss of generality, by its projectionon the subspace of stresses with zero hydrostatic pressure component (!1+!2+!3 = 0). This subspace is called the deviatoric plane, also referred toas the "-plane. It is graphically illustrated in Figure 6.9(a). Figure 6.9(b)shows the "-plane projection of the Tresca yield surface.


Sec. 6.5 Plastic Flow Rules 193

!1

!2

!!3

N

"#3p

Figure 6.17: The Prandtl-Reuss flow vector.

vector is given by

N ! !!

!!=

!

!"

!"

3 J2(s)#

=$

32

s

||s||, (6.136)

and the flow rule results in

"p = #$

32

s

||s||. (6.137)

Here, it should be noted that the Prandtl-Reuss flow vector is the deriva-tive of an isotropic scalar function of a symmetric tensor – the von Misesyield function. Thus (refer to Section A.1.2, page 855, where the deriva-tive of isotropic functions of this type is discussed), N and ! are coaxial,i.e., the principal directions of N coincide with those of !. Due to thepressure-insensitivity of the von Mises yield function, the plastic flow vectoris deviatoric. The Prandtl-Reuss flow vector is a tensor parallel to the de-viatoric projection s of the stress tensor. Its principal stress representationis depicted in Figure 6.17. The Prandtl-Reuss rule is usually employed inconjunction with the von Mises criterion and the resulting plasticity modelis referred to as the von Mises associative model or, simply, the von Misesmodel.

Associative Tresca

The associative Tresca flow rule utilises the yield function (6.84) as the flowpotential. Since ! here is also an isotropic function of !, the rate of plasticstrain has the same principal directions as !. The Tresca yield function is

The Prandtl-Reuss associative plastic flow rule


Drucker-Prageruniaxial fit

Mohr-Coulomb1!

2!

ft’

ft’

fc’

fc’

fbt’

fbc’

Drucker-Pragerbiaxial fit

Figure 6.16: Plane stress. Drucker-Prager approximation matching the Mohr-Coulomb surface in uniaxial tension and uniaxial compression.

6.5 Plastic flow rules

6.5.1 Associative and non-associative plasticity

It has already been said that a plasticity model is classed as associative ifthe yield function, !, is taken as the flow potential, i.e.,

" = !. (6.128)

Any other choice of flow potential characterises a non-associative (or non-associated) plasticity model.

In associative models, the evolution equations for the plastic strain andhardening variables are given by

!p = !"!

"", (6.129)

and

# = !!"!

"A. (6.130)

Associativity implies that the plastic strain rate is a tensor normal to theyield surface in the space of stresses. In the generalised case of non-smoothyield surfaces, the flow vector is a subgradient of the yield function, i.e, wehave

!p = !N ; N " "!!. (6.131)

In non-associative models, the plastic strain rate is not normal to the yieldsurface in general.


!1

!2

!!3

N

"#3p


vector is given by

N ! !!

!!=

!

!"

!"

3 J2(s)#

=$

32

s

||s||, (6.136)


"p = #$

32

s

||s||. (6.137)


Associative Tresca



!1

!2

!!3

N

"#3p


vector is given by

N ! !!

!!=

!

!"

!"

3 J2(s)#

=$

32

s

||s||, (6.136)


"p = #$

32

s

||s||. (6.137)


Associative Tresca


IMPORTANT: Plastic flow is isochoric

Isotropic hardening law We assume where the accumulated plastic strain (the hardening internal variable) is defined as or In view of the Prandtl-Reuss flow rule, this is equivalent to



in the crystallographic micro-structure which cause an isotropic increase inresistance to plastic flow. In the constitutive description of isotropic harden-ing, the set ! normally contains a single scalar variable, which determinesthe size of the yield surface. Two approaches: strain hardening and workhardening are particularly popular in the treatment of isotropic hardeningand are suitable to model the behaviour of a wide range of materials. Theseare described below.

Strain hardening

In this case the hardening internal state variable is some suitably chosenscalar measure of strain. A typical example is the von Mises e!ective plasticstrain, also referred to as the von Mises equivalent or accumulated plasticstrain, defined as

! p !! t

0

"

23 "p : "p dt =

! t

0

"

23 ||"p|| dt. (6.167)

The above definition generalises the accumulated axial plastic strain (6.18)[page 162] of the one-dimensional model to the multiaxially strained case.Its rate evolution equation reads

˙! p ="

23 "p : "p =

"

23 ||"p||, (6.168)

or, equivalently, in view of the Prandtl-Reuss flow equation (6.137),

˙! p = ". (6.169)

Accordingly, a von Mises isotropic strain hardening model is obtained byletting the uniaxial yield stress be a function of the accumulated plasticstrain:

#y = #y(!p). (6.170)

This function defines the strain hardening curve (or strain hardening func-tion) which can be obtained, for instance, from a uniaxial tensile test.

Behaviour under uniaxial stress conditions Under uniaxial stress condi-tions the von Mises model with isotropic strain hardening reproduces thebehaviour of the one-dimensional plasticity model discussed in Section 6.2and summarised in Box 6.1 (page 164). This is demonstrated in the follow-ing. Let us assume that both models share the same Young’s modulus, E,and hardening function #y = #y(!p). Clearly, the two models have identi-cal uniaxial elastic behaviour and initial yield stress. Hence, we only needto show next that their behaviour under plastic yielding is also identical.Under a uniaxial stress state with axial stress # and axial stress rate # in



Strain hardening


! p !! t

0

"

23 "p : "p dt =

! t

0

"

23 ||"p|| dt. (6.167)


˙! p ="

23 "p : "p =

"

23 ||"p||, (6.168)


˙! p = ". (6.169)


#y = #y(!p). (6.170)





Strain hardening


! p !! t

0

"

23 "p : "p dt =

! t

0

"

23 ||"p|| dt. (6.167)


˙! p ="

23 "p : "p =

"

23 ||"p||, (6.168)


˙! p = ". (6.169)


#y = #y(!p). (6.170)





Strain hardening


! p !! t

0

"

23 "p : "p dt =

! t

0

"

23 ||"p|| dt. (6.167)


˙! p ="

23 "p : "p =

"

23 ||"p||, (6.168)


˙! p = ". (6.169)


#y = #y(!p). (6.170)



recall 1-D model…

Sec. 6.6 Hardening Laws 203

the direction of the base vector e1, the matrix representations of the stresstensor and the stress rate tensor in the three-dimensional model are givenby

[!] = !

!

"

1 0 00 0 00 0 0

#

$ , [!] = !

!

"

1 0 00 0 00 0 0

#

$ . (6.171)

The corresponding stress deviator reads

[s] = 23!

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ . (6.172)

In this case, the Prandtl-Reuss flow equation (6.137) gives

["p] = "p

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ , (6.173)

where

"p = # sign(!) (6.174)

is the axial plastic strain rate. Note that the above expression coincides withthe one-dimensional plastic flow rule (6.10). Now, we recall the consistencycondition (6.60), which must be satisfied under plastic flow. In the presentcase, by taking the derivatives of the von Mises yield function (6.110), with!y defined by (6.170), we obtain

! = N : ! ! H ˙"p

= 0, (6.175)

where N " $!/$! is the Prandtl-Reuss flow vector (6.136) and H = H("p)is hardening modulus defined in (6.27). To conclude the demonstration, wecombine (6.175) with (6.136), (6.171)2 and (6.172) to recover (6.28) and,then, following the same arguments as in the one-dimensional case we findthat, under uniaxial stress conditions, the isotropic strain hardening vonMises model predicts the tangential axial stress-strain relation

! =EH

E + H", (6.176)

which is identical to equation (6.31) of the one-dimensional model.


…back to the uniaxial stress case



[!] = !

!

"

1 0 00 0 00 0 0

#

$ , [!] = !

!

"

1 0 00 0 00 0 0

#

$ . (6.171)


[s] = 23!

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ . (6.172)


["p] = "p

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ , (6.173)

where

"p = # sign(!) (6.174)


! = N : ! ! H ˙"p

= 0, (6.175)


! =EH

E + H", (6.176)




[!] = !

!

"

1 0 00 0 00 0 0

#

$ , [!] = !

!

"

1 0 00 0 00 0 0

#

$ . (6.171)


[s] = 23!

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ . (6.172)


["p] = "p

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ , (6.173)

where

"p = # sign(!) (6.174)


! = N : ! ! H ˙"p

= 0, (6.175)


! =EH

E + H", (6.176)



!1

!2

!!3

N

"#3p


vector is given by

N ! !!

!!=

!

!"

!"

3 J2(s)#

=$

32

s

||s||, (6.136)


"p = #$

32

s

||s||. (6.137)


Associative Tresca




[!] = !

!

"

1 0 00 0 00 0 0

#

$ , [!] = !

!

"

1 0 00 0 00 0 0

#

$ . (6.171)


[s] = 23!

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ . (6.172)


["p] = "p

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ , (6.173)

where

"p = # sign(!) (6.174)


! = N : ! ! H ˙"p

= 0, (6.175)


! =EH

E + H", (6.176)




[!] = !

!

"

1 0 00 0 00 0 0

#

$ , [!] = !

!

"

1 0 00 0 00 0 0

#

$ . (6.171)


[s] = 23!

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ . (6.172)


["p] = "p

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ , (6.173)

where

"p = # sign(!) (6.174)


! = N : ! ! H ˙"p

= 0, (6.175)


! =EH

E + H", (6.176)




[!] = !

!

"

1 0 00 0 00 0 0

#

$ , [!] = !

!

"

1 0 00 0 00 0 0

#

$ . (6.171)


[s] = 23!

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ . (6.172)


["p] = "p

!

"

1 0 00 ! 1

2 00 0 ! 1

2

#

$ , (6.173)

where

"p = # sign(!) (6.174)


! = N : ! ! H ˙"p

= 0, (6.175)


! =EH

E + H", (6.176)



Summary of the von Mises model Sec. 7.3 Integration Algorithm for the von Mises Model 247

1. A linear elastic law! = De : "e , (7.73)

where De is the standard isotropic elasticity tensor;

2. A yield function of the form

!(!, !y) =!

3 J2(s(!)) ! !y , (7.74)

where!y = !y("p) . (7.75)

is the uniaxial yield stress – a function of the accumulated plasticstrain, "p.

3. A standard associative flow rule

"p = # N = #$!

$!, (7.76)

with the (Prandtl-Reuss) flow vector, N, explicitly given by

N " $!

$!=

"

3

2

s

#s#. (7.77)

4. An associative hardening rule, with the evolution equation for thehardening internal variable given by

˙" p =#

23 #"p# = #. (7.78)

7.3.2 The implicit elastic predictor/return mapping scheme

Given the increment of strain

"" = "n+1 ! "n , (7.79)

corresponding to a typical (pseudo-) time increment [tn, tn+1], and given thestate variables {"e

n, "pn} at tn, the elastic trial strain and trial accumulated

plastic strain are given by

"e trialn+1 = "e

n + ""

"p trialn+1 = "p

n .(7.80)

The corresponding trial stress is computed as

!trialn+1 = De : "e trial

n+1 , (7.81)

Sec. 7.3 Integration Algorithm for the von Mises Model 247




!(!, !y) =!

3 J2(s(!)) ! !y , (7.74)

where!y = !y("p) . (7.75)



"p = # N = #$!

$!, (7.76)


N " $!

$!=

"

3

2

s

#s#. (7.77)


˙" p =#

23 #"p# = #. (7.78)



"" = "n+1 ! "n , (7.79)




"e trialn+1 = "e

n + ""

"p trialn+1 = "p

n .(7.80)



n+1 , (7.81)





!(!, !y) =!

3 J2(s(!)) ! !y , (7.74)

where!y = !y("p) . (7.75)



"p = # N = #$!

$!, (7.76)


N " $!

$!=

"

3

2

s

#s#. (7.77)


˙" p =#

23 #"p# = #. (7.78)



"" = "n+1 ! "n , (7.79)




"e trialn+1 = "e

n + ""

"p trialn+1 = "p

n .(7.80)



n+1 , (7.81)





!(!, !y) =!

3 J2(s(!)) ! !y , (7.74)

where!y = !y("p) . (7.75)



"p = # N = #$!

$!, (7.76)


N " $!

$!=

"

3

2

s

#s#. (7.77)


˙" p =#

23 #"p# = #. (7.78)



"" = "n+1 ! "n , (7.79)




"e trialn+1 = "e

n + ""

"p trialn+1 = "p

n .(7.80)



n+1 , (7.81)



! = !e + !p, (6.2)


!e = ! ! !p. (6.3)



" = E !e. (6.4)




!(", "y) = |"|! "y , (6.5)


E = { " | !(", "y)<0 }, (6.6)


|"| < "y . (6.7)







!(!, !y) =!

3 J2(s(!)) ! !y , (7.74)

where!y = !y("p) . (7.75)



"p = # N = #$!

$!, (7.76)


N " $!

$!=

"

3

2

s

#s#. (7.77)


˙" p =#

23 #"p# = #. (7.78)



"" = "n+1 ! "n , (7.79)




"e trialn+1 = "e

n + ""

"p trialn+1 = "p

n .(7.80)



n+1 , (7.81)





!(!, !y) =!

3 J2(s(!)) ! !y , (7.74)

where!y = !y("p) . (7.75)



"p = # N = #$!

$!, (7.76)


N " $!

$!=

"

3

2

s

#s#. (7.77)


˙" p =#

23 #"p# = #. (7.78)



"" = "n+1 ! "n , (7.79)




"e trialn+1 = "e

n + ""

"p trialn+1 = "p

n .(7.80)



n+1 , (7.81)





!(p, t) and !(p, t).





"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)





!(p, t) and !(p, t).





"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)





!(p, t) and !(p, t).





"(t) = F (F t, !t, gt)

"(t) = G(F t, !t, gt)

s(t) = H(F t, !t, gt)

q(t) = I (F t, !t, gt)

(3.143)






!

""""#

""""$

! = !(F, !)

P = "#!

#F

! = f(F, !).

(3.164)



!

""""#

""""$

! = !(", !)

# = "#!

#"

! = f(", !).

(3.165)




Box 6.2: A general elasto-plastic constitutive model

1. Additive decomposition of the strain tensor

! = !e + !p

or! = !e + !p !(t0) = !e(t0) + !p(t0)

2. Free-energy function

" = "(!e, ")

where " is a set of hardening internal variables.

3. Constitutive equation for # and hardening thermodynamicforces A

# = #$"

$!eA = #

$"

$"

4. Yield function! = !(#, A)

5. Plastic flow rule and hardening law

!p = % N(#, A)

" = % H(#, A)


! ! 0 % " 0 %! = 0

Associative flow rule

As we shall see later, many plasticity models, particularly for ductile metals,have their yield function, !, as a flow potential, i.e.,

" # !. (6.58)

Such models are called associative (or associated) plasticity models. Theissue of associativity will be further discussed in Section 6.5.1.


Sec. 7.2 General Elasto-Plastic Integration Algorithm 219

presented here is later specialised and applied to the von Mises model inSection 7.3. Specialisation to the other basic plasticity models of Chapter6, i.e., the Tresca, Mohr-Coulomb and Drucker-Prager models, is presentedin Chapter 8 and its plane stress implementation is addressed in Chapter9. Further applications of the algorithms described in the present chapterare made in Chapter 10, in the context of advanced plasticity models, andin Chapter 12, where the numerical implementation of damage mechanicsmodels is discussed.

7.2.1 The elasto-plastic constitutive initial value problem

Consider a point p of a body B with constitutive behaviour described bythe general elasto-plastic model of Box 6.2 (page 170). Assume that at agiven (pseudo-)time t0 the elastic strain, !e(t0), the plastic strain tensor,!p(t0), and all elements of the set "(t0) of hardening internal variables areknown at point p. Furthermore, let the motion of B be prescribed betweent0 and a subsequent instant, T . Clearly, the prescribed motion defines thehistory of the strain tensor, !(t), at the material point of interest betweeninstants t0 and T . The basic elasto-plastic constitutive initial value problemat point p is stated in the following.

Problem 7.1 (The elasto-plastic constitutive initial value problem)Given the initial values !e(t0) and "(t0) and given the history of the straintensor, !(t), t ! [t0, T ], find the functions !e(t), "(t) and !(t) for the elas-tic strain, hardening internal variables set and plastic multiplier that satisfythe reduced general elasto-plastic constitutive equations

!e(t) = !(t) " !(t) N(#(t), A(t))

"(t) = !(t) H(#(t), A(t))(7.6)

!(t) # 0 !(#(t), A(t)) $ 0 !(t)!(#(t), A(t)) = 0 (7.7)

for each instant t ! [t0, T ], with

#(t) = "#$

#!e

!!!!t

A(t) = "#$

#%

!!!!t

. (7.8)

Remark 7.1 We refer to the system of di"erential equations (7.6) as re-duced in that it is obtained from the model of Box 6.2 by incorporating theplastic flow equation into the additive strain rate decomposition. In thisway, the plastic strain does not appear explicitly in the system and the onlyunknowns are the elastic strain, the set of internal variables and the plasticmultiplier. Note that with the solution !e(t) of Problem 7.1 at hand, thehistory of the plastic strain tensor is obtained from the trivial relation

!p(t) = !(t) " !e(t), (7.9)


The Initial Boundary Value Problem



( , )t

!( , )

"u

"u

u .

t

!( , )t"

"

"t

!( , )"t t

t ( , ) tx

b( , )tx

Figure 3.16: The initial boundary value problem. Schematic illustration.

(ii) The essential boundary condition. The motion is a prescribed functionon the part of the boundary of B that occupies the region !!u in thereference configuration

!(p, t) = p + u(p, t) t ! [t0, T ], p ! !!u,

where u is the corresponding prescribed boundary displacement field.For simplicity, it is assumed here that !!u

!

!!t = ". We define theset of kinematically admissible displacements of B as the set of allsu"ciently regular displacement functions that satisfy the kinematicconstraint (the essential boundary condition)

K = {u : ! # R $ U | u(p, t) = u(p, t), t ! [t0, T ], p ! !!u}.(3.177)

The body B is assumed to be made from a generic material modelled bythe internal variable-based constitutive equations associated with Problem3.1 and the internal variable field, ", is known at the initial time t0, i.e.,

"(p, t0) = "0(p). (3.178)

The fundamental quasi-static initial boundary value problem is stated in itsspatial version in the following.

Problem 3.3 (The spatial quasi-static initial boundary value prob-lem) Find a kinematically admissible displacement function, u ! K , suchthat, for all t ! [t0, T ], the virtual work equation is satisfied"

!(!,t)[#(t) : %x$&b(t)·$] dv&

"

!("!t,t)t(t)·$ da = 0 '$ ! Vt. (3.179)


( , )t

!( , )

"u

"u

u .

t

!( , )t"

"

"t

!( , )"t t

t ( , ) tx

b( , )tx

Figure 3.16: The initial boundary value problem. Schematic illustration.

(ii) The essential boundary condition. The motion is a prescribed functionon the part of the boundary of B that occupies the region !!u in thereference configuration

!(p, t) = p + u(p, t) t ! [t0, T ], p ! !!u,

where u is the corresponding prescribed boundary displacement field.For simplicity, it is assumed here that !!u

!

!!t = ". We define theset of kinematically admissible displacements of B as the set of allsu"ciently regular displacement functions that satisfy the kinematicconstraint (the essential boundary condition)

K = {u : ! # R $ U | u(p, t) = u(p, t), t ! [t0, T ], p ! !!u}.(3.177)

The body B is assumed to be made from a generic material modelled bythe internal variable-based constitutive equations associated with Problem3.1 and the internal variable field, ", is known at the initial time t0, i.e.,

"(p, t0) = "0(p). (3.178)

The fundamental quasi-static initial boundary value problem is stated in itsspatial version in the following.

Problem 3.3 (The spatial quasi-static initial boundary value prob-lem) Find a kinematically admissible displacement function, u ! K , suchthat, for all t ! [t0, T ], the virtual work equation is satisfied"

!(!,t)[#(t) : %x$&b(t)·$] dv&

"

!("!t,t)t(t)·$ da = 0 '$ ! Vt. (3.179)

The initial boundary value problem

Kinematically admissible displacements


Sec. 3.7 The Initial Boundary Value Problem 89

The space of virtual displacements at time t is defined by

Vt = {! : "(!, t) ! U | ! = 0 on "(!!u, t)} (3.180)

and, at each point of B, the Cauchy stress is given by

#(t) = P(t)F(t)T /J(t), (3.181)

where P(t) is the solution of constitutive initial value problem 3.1 (page 84)with prescribed deformation gradient

F(t) = "p"(p, t) = I + "pu(p, t). (3.182)

The problem can be equivalently formulated in the reference configu-ration of B in terms of the material version of the principle of virtual work(3.173). For completeness, we state the material version of the fundamentalinitial boundary value problem in the following.

Problem 3.4 (The material quasi-static initial boundary value prob-lem) Find a kinematically admissible displacement function, u # K , suchthat, for all t # [t0, T ],

!

![P(t) : "p! $ b(t) · !] dv $

!

!!t

t(t) · ! da = 0 %! # V, (3.183)

whereV = {! : ! ! U | ! = 0 on !!u} (3.184)

and the Piola-Kirchho! stress, P(t), is the solution of initial value problem3.1 with prescribed deformation gradient (3.182).

3.7.2 The infinitesimal problem

Under infinitesimal deformations, the quasi-static initial boundary valueproblem is based on the weak form (3.176). It is stated in the following.

Problem 3.5 (The infinitesimal quasi-static initial boundary valueproblem) Find a kinematically admissible displacement, u # K , such that,for t # [t0, T ],

!

![#(t) : "! $ b(t) · !] dv $

!

!!t

t(t) · ! da = 0 %! # V, (3.185)

whereV = {! : ! ! U | ! = 0 on !!u}. (3.186)

and, at each point p, #(t) is the solution of the constitutive initial valueproblem 3.2 (page 84) with prescribed strain

$(t) = "su(p, t). (3.187)

Computational Plasticity is about solving problems of this type !!

Introduction to Plasticity - LabMeCComputational Plasticity 3 Some basics first. Mathematical theory of plasticity The mathematical theory of plasticity attempts to describe mathematically

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