Deformation Theory of Plasticity Revisited

CRNOGORSKA AKADEMIJA NAUKA I UMJETNOSTI

GLASNIK ODJELJENJA PRIRODNIH NAUKA, 13, 1999.

QERNOGORSKAYA AKADEMIYANAUK I ISSKUSTV

GLASNIK OTDELENIYA ESTESTVENNYH NAUK, 13, 1999.

THE MONTENEGRIN ACADEMY OF SCIENCES AND ARTS

GLASNIK OF THE SECTION OF NATURAL SCIENCES, 13, 1999

UDK 539.319

Vlado A. Lubarda

Deformation Theory of Plasticity Revisited

A b s t r a c t

Deformation theory of plasticity, originally intro duced for in nitesi-

mal strains, is extended to encompass the regime of nite deforma-

tions. The framework of nonlinear continuum mechanics with loga-

rithmic strain and its conjugate stress tensor is used to cast the formu-

lation. A connection between deformation and ow theory of metal

plasticity is discussed. Extension of theory to pressure-dep endent

plasticity is constructed, with an application to geomechanics. Deriva-

tions based on strain and stress decomp ositions are b oth given. Dual-

ity in constitutive structures of rate-typ e deformation and ow theory

for ssured ro cks is demonstrated.

Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA

Deformation Theory of Plasticity Revisited 2

OSVRTNADEFORMACIONU TEORIJU PLASTICNOSTI

I z v o d

U radu je data formulacija deformacione teorije plasticnosti ko ja obuh-

vata oblast konacnih deformacija. Meto di nelinearne mehanika kon-

tinuuma, logaritamska mjera deformacije i njen konjugovani tenzor

nap ona su adekvatno up otrebljeni u formulaciji teorije. Veza izmedju

deformacione i inkrementalne teorije plasticnosti je diskutovana na

primjeru p olikristalnih metala. Teorija je zatim prosirena na oblast

plasticnosti koja zavisi o d pritiska, sa primjenom u geomehanici. For-

mulacije na bazi dekomp ozicija tenzora deformacije i nap ona su p osebno

date. Dualnost konstitutivnih struktura deformacione i inkrementalne

teorije je demonstrirana na mo delu stijenskih masa.

INTRODUCTION

Commonly accepted theory used in most analytical and computa-

tional studies of plastic deformation of metals and geomaterials is

the so-called ow theory of plasticity (e.g., Hill, 1950,1978; Lubliner,

1992; Havner, 1992). Plastic deformation is a history dep endent phe-

nomenon, characterized by nonlinearity and irreversibility of under-

lining physical pro cesses (Bell, 1968). Consequently,in ow theory of

plasticity the rate of strain is expressed in terms of the rate of stress

and the variables describing the current state of material. The over-

all resp onse is determined incrementally byintegrating the rate-typ e

constitutive and eld equations along given path of loading or defor-

mation (Lubarda and Lee, 1981; Lubarda and Shih, 1994; Lubarda

and Kra jcinovic, 1995).

There has b een an early theory of plasticity suggested by Hencky

(1924) and Ilyushin (1947,1963), known as deformation theory of plas-

ticity,inwhich total strain is given as a function of total stress. Such

constitutive structure, typical for nonlinear elastic deformation, is in

Deformation Theory of Plasticity Revisited 3

general inappropriate for plastic deformation, since strain there de-

p ends on both stress and stress history, and is a functional rather

than a function of stress. However, deformation theory of plasticity

found its application in problems of prop ortional or simple loading,

in which all stress comp onents increase prop ortionally, or nearly so,

without elastic unloading ever o ccurring (Budiansky, 1959; Kachanov,

1971). The theory was particularly successful in bifurcation studies

and determination of necking and buckling loads (Hutchinson, 1974).

Deformation theory of plasticitywas originally prop osed for non-

linear but in nitesimally small plastic deformation. An extension to

nite strain range was discussed by Storen and Rice (1975). The

purp ose of this pap er is to provide a formulation of the rate-typ e

deformation theory for pressure-dep endent and pressure-indep endent

plasticity at arbitrary strains. After needed kinematic and kinetic

background is intro duced, the logarithmic strain and its conjugate

stress are conveniently utilized to cast the formulation. Relationship

between the rate-typ e deformation and ow theory of metal plasticity

is discussed. A pressure-dep endent deformation theory of plasticity

is constructed and compared with a non-asso ciative ow theory of

plasticity corresp onding to the Drucker-Prager yield criterion. Devel-

opments based on strain and stress decomp ositions are b oth given.

Duality in the constitutive structures of deformation and ow theory

for ssured ro cks is demonstrated.

1 KINEMATIC PRELIMINARIES

The lo cations of material points of a three-dimensional body in its

undeformed con guration are sp eci ed by vectors X. Their lo ca-

tions in deformed con guration at time t are sp eci ed by x, such

that x = x(X;t) is one-to-one deformation mapping, assumed twice

continuously di erentiable. The comp onents of X and x are mate-

rial and spatial co ordinates of the particle. An in nitesimal material

Deformation Theory of Plasticity Revisited 4

elementdX in the undeformed con guration b ecomes

@ x

dx = F dX; F = (1.1)

@ X

in the deformed con guration at time t. Physically p ossible deforma-

tion mappings have p ositivedetF,henceF is an invertible tensor; dX

can b e recovered from dx byinverse op eration dX = F dx.

By p olar decomp osition theorem, F is decomp osed into the pro d-

uct of a prop er orthogonal tensor and a p ositive-de nite symmetric

tensor, such that (Truesdell and Noll, 1965)

F = R U = V R: (1.2)

Here, U is the right stretch tensor, V is the left stretch tensor, and

, so that U R is the rotation tensor. Evidently, V = R U R

and V share the same eigenvalues (principal stretches ), while their

eigenvectors are related by n = R N . The right and left Cauchy-

i i

Green deformation tensors are

T 2 T 2

C = F F = U ; B = F F = V : (1.3)

If there are three distinct principal stretches, C and B have their

sp ectral representations (Marsden and Hughes, 1983)

3 3

X X

2 2

n n : (1.4) N N ; B = C =

i i i i

i i

i=1 i=1

2 STRAIN TENSORS

Various tensor measures of strain can b e intro duced. A fairly general

de nition of material strain measures is (Hill, 1978)

1 1

2n 0 2n

E = U I = 1 N N ; (2.1)

i i

(n)

2n 2n

i=1

where 2n is a p ositive or negative integer, and and N are the

i i

principal values and directions of U. The unit tensor in the initial

Deformation Theory of Plasticity Revisited 5

con guration is I . For n = 1, Eq. (2.1) gives the Lagrangian or

2 0

Green strain E = (U I )=2, for n = 1 the Almansi strain

(1)

0 2

E = (I U )=2, and for n = 1=2 the Biot strain E =

(1) (1=2)

(U I ). The logarithmic or Hencky strain is

E =lnU = ln N N : (2.2)

i i i

(0)

i=1

A family of spatial strain measures, corresp onding to material

strain measures of Eqs. (2.1) and (2.2), are

1 1

2n 2n

E = V I = 1 n n ; (2.3)

i i

(n)

2n 2n

i=1

E =lnV = ln n n : (2.4)

i i i

(0)

i=1

The unit tensor in the deformed con guration is I, and n are the

E E

principal directions of V . For example, E =(V I)=2, and E =

(1) (1)

(I V )=2, the latter b eing known as the Eulerian strain tensor.

2n T 2n

Since U = R V R, and n = R N , the material and

i i

spatial strain measures are related by

T T

E E

E = R EE R; E = R EE R; (2.5)

(n) (n) (0) (0)

i.e., the former are induced from the latter by the rotation R.

Consider a material line element dx in the deformed con guration

at time t. If the velo city eldisv = v (x;t), the velo cities of the end

p oints of dx di er by

dv = L dx; F F : (2.6)

The tensor L is called the velo city gradient. Its symmetric and an-

tisymmetric parts are the rate of deformation tensor and the spin

tensor

1 1

T T

D = L + L ; W = L L : (2.7)

2 2

Deformation Theory of Plasticity Revisited 6

3 CONJUGATE STRESS TENSORS

For any material strain E of Eq. (2.1), its work conjugate stress

(n)

T is de ned such that the stress p ower p er unit initial volume is

(n)

T : E = : D; (3.1)

(n) (n)

where = (det F) is the Kirchho stress. The Cauchy stress is

denoted by . For n =1,Eq. (3.1) gives

1 T 1 1

T = F F = U ^ U : (3.2)

(1)

The stress ^ = R R is induced from by the rotation R. Similarly,

T = F F = U ^ U: (3.3)

(1)

More involved is an expression for the stress conjugate to logarithmic

strain, although the approximation

T =^ + O E ^ (3.4)

(0)

(n)

may be acceptable at mo derate strains. If deformation is such that

principal directions of V and are parallel, the matrices E and T

(n) (n)

commute, and in that case T =^ exactly (Hill, 1978). If principal

(0)

directions of U remain xed during deformation,

_ _ ^

E = U U = D; T =^ : (3.5)

(0) (0)

The spatial strain tensors E in general do not have their conju-

(n)

T T E

gate stress tensors T suchthat T : E = T : E . However,

(n) (n) (n) (n) (n)

the spatial stress tensors conjugate to strain tensors E can be in-

(n)

tro duced by requiring that

T E

T : E = T : E ; (3.6)

(n) (n) (n) (n)

where ob jective, corotational rate of strain E is de ned by

(n)

E E E E

E = E ! EE + E ! ; ! = R R : (3.7)

(n) (n) (n) (n)

Deformation Theory of Plasticity Revisited 7

In view of the relationship E = R E R ,itfollows that

(n) (n)

T = R T R : (3.8)

(n) (n)

This is the conjugate stress to spatial strains E in the sense of Eq.

(n)

(3.6).

Note that R R is not the work conjugate to any strain measure,

since the material stress tensor T in Eq. (3.8) cannot be equal to

(n)

spatial stress tensor . Likewise, although ^ : D = : D, the stress

tensor ^ = R R is not the work conjugate to any strain measure,

b ecause D = R D R is not the rate of any strain. Of course,

itself is not the work conjugate to any strain, b ecause D is not the

rate of any strain, either.

4 DEFORMATION THEORY OF PLASTIC-

ITY

Simple plasticity theory has been suggested for prop ortional loading

and small deformation by Hencky(1924) and Ilyushin (1947,1963).

A large deformation version of this theory is here presented. It is

convenient to cast the formulation by using the logarithmic strain

E =lnU and its conjugate stress T . Assume that the loading is

(0) (0)

such that all stress comp onents increase prop ortionally, i.e.

T = c(t) T ; (4.1)

(0)

where T is the stress tensor at instant t ,andc(t) is monotonically

(0)

increasing function of t,withc(t )=1. Evidently, principal directions

of T in Eq. (4.1) remain xed during the deformation pro cess.

(0)

Since stress prop ortionally increases, with no elastic unloading

taking place, it seems reasonable to assume that elastoplastic resp onse

can b e describ ed macroscopically by the constitutive structure of non-

linear elasticity,inwhich total strain is a function of total stress. Thus,

decomp ose the total strain into its elastic and plastic parts,

E = E + E ; (4.2)

(0)

(0) (0)

Deformation Theory of Plasticity Revisited 8

and assume that

(0)

= E ; (4.3)

(0)

@ T

(0)

; (4.4) E = '

(0)

@ T

(0)

where is a complementary elastic strain energy per unit unde-

(0)

formed volume, a Legendre transform of elastic strain energy ,

(0)

T = T : E E : (4.5)

(0) (0) (0) (0) (0) (0)

Isotropic elastic behavior will b e assumed, so that = T

(0) (0) (0)

is an isotropic function of T . For plastically isotropic materials, i.e.

(0)

isotropic hardening, a function f = f T is also an isotropic

(0) (0) (0)

function of T . The scalar ' is an appropriate scalar function to

(0) (0)

be determined in accord with exp erimental data. Clearly, principal

directions of b oth elastic and plastic comp onents of strain are parallel

to those of T , as are the principal directions of total strain E .

(0) (0)

Consequently, E and U have their principal directions xed during

(0)

the deformation pro cess, the matrix U commutes with U, and by Eq.

(3.5)

1 T

_ _

E = U U ; T = R R: (4.6)

(0) (0)

The requirementfor xed principal directions of U severely restricts

the class of admissible deformations, precluding, for example, the case

of simple shear. This is not surprising b ecause the premise of defor-

mation theory { prop ortional stressing imp oses at the outset strong

restrictions on the analysis.

Intro ducing the spatial strain

E = R E R; (4.7)

(0) (0)

Eqs. (4.2)-(4.4) can b e rewritten as

E E E

E = E + E ; (4.8)

(0)

(0) (0)

Deformation Theory of Plasticity Revisited 9

(0)

E = ; (4.9)

(0)

: (4.10) E = '

(0)

Although deformation theory of plasticity is total strain theory,

the rate quantities are now intro duced for later comparison with the

ow theory of plasticity, and for application of the resulting rate-

typ e constitutive equations approximately b eyond prop ortional load-

ing. This is also needed whenever the b oundary value problem of

nite deformation is b eing solved in an incremental manner. Since

U U is symmetric, wehave

T 1

_ _

D = R E R ; W = R R ; (4.11)

(0)

and

T = R R; E = D: (4.12)

(0) (0)

By di erentiating (4.2)-(4.4), or by applying the Jaumann derivative

to (4.8)-(4.10), there follows

e p

D = D + D ; (4.13)

(0)

D = M : ; M = ; (4.14)

(0) (0)

@ @

@ f @f

(0) (0)

+ ' : : (4.15) D = '_

(0) (0)

@ @ @

Assume quadratic representation of the complementary energy

1 1

= I M :: ( ); M = I I ; (4.16)

(0) (0) (0)

2 2 2 +3

Deformation Theory of Plasticity Revisited 10

where and are the Lame elastic constants. Furthermore, let the

function f be de ned bythe second invariantof deviatoric part of

(0)

the Kirchho stress,

0 0

: : (4.17) f =

(0)

Substituting the last two expressions in Eq. (4.15) gives

p 0 0

D = '_ + ' : (4.18)

(0) (0)

The deviatoric and spherical parts of the total rate of deformation

tensor are accordingly

0 0 0

D = '_ + + ' ; (4.19)

(0) (0)

tr D = tr ; (4.20)

where = +(2=3) is the elastic bulk mo dulus.

Supp ose that a nonlinear relationship = ( )between the Kirch-

ho stress and the logarithmic strain is available from elastoplastic

pure shear test. Let the secant and tangent mo duli b e de ned by

; h = ; (4.21) h =

t s

and let

1=2 1=2

1 1

0 0 0 0

; (4.22) : T = : = T

(0) (0)

2 2

1=2 1=2

0 0

E E

= 2 E : E = 2 E : E : (4.23)

(0) (0)

Since from Eqs. (4.9) and (4.10)

E = + ' ; (4.24)

(0)

Deformation Theory of Plasticity Revisited 11

substitution into Eq. (4.23) provides an expression for

1 1

' = : (4.25)

(0)

2h 2

In order to derive an expression for the rate '_ , di erentiate Eqs.

(0)

(4.22) and (4.23) to obtain

_ _

= : ; =2E : D: (4.26)

(0)

In view of Eqs. (4.21), (4.24) and (4.25), this gives

0 0 0 0

: =2h h E : D = h : D : (4.27)

s t t

(0)

When Eq. (4.19) is incorp orated into Eq. (4.27), the rate is found to

1 1 : 1

: (4.28) '_ =

(0)

0 0

2 h h :

t s

Taking Eq. (4.28) into Eq. (4.19), the deviatoric part of the total rate

of deformation is

# "

0 0

( ): h 1

0 0

1 : (4.29) + D =

0 0

2h h :

s t

Eq. (4.29) can b e inverted to give

0 0

h ( ):D

0 0

: (4.30) =2h D 1

0 0

h :

During initial, purely elastic stages of deformation, h = h = . The

t s

onset of plasticity,beyond which Eqs. (4.29) and (4.30) apply,occurs

when , de ned by the second invariant of the deviatoric stress in Eq.

(4.22), reaches the initial yield stress in shear. The resulting theory

is referred to as the J deformation theory of plasticity. 2

Deformation Theory of Plasticity Revisited 12

5 RELATIONSHIP BETWEEN DEFORMA-

TION AND FLOW THEORY OF PLAS-

TICITY

For prop ortional loading de ned byEq. (4.1) the stress rates are

c_ c_

T ; = : (5.1) T =

(0) (0)

c c

Consequently, from Eq. (4.28) the plastic part of the rate of deforma-

tion tensor is

1 1 c_ 1

'_ = ; (5.2)

(0)

2 h h c

t s

while from Eq. (4.29)

0 1 1 c_ 1

p 0 e 0

D = D D = : (5.3)

2 h c

On the other hand, in the case of ow theory of plasticity,

_ _ _

E = E + E ; (5.4)

(0)

e 0

_ _ _

E = M : T ; E = _ T : (5.5)

(0) (0)

(0)

The yield surface is de ned by

1=2

p p

0 0 2

_ _

T dt; (5.6) : T 2 E : E k (#)=0; # =

(0) (0)

and the consistency condition gives (Lubarda, 1991,1994)

_ = ( : ): (5.7)

(0) p

4k h

Here, h = dk=d# designates the plastic tangent mo dulus. Since

T T

T = R R and E = R D R, the plastic part of the rate

(0) (0)

of deformation b ecomes

0 0 p

: : (5.8) D =

4k h t

Deformation Theory of Plasticity Revisited 13

In view of Eq. (5.1), this simpli es to

1 c_

p 0 0

D = _ = : (5.9)

(0) p

2h c

Constitutive structures (5.3) and (5.9) are in accord since

1 1 1

= : (5.10)

h h

The last expression holds b ecause in shear test k = , # = , and

1 d d 1

p e

= = ; = : (5.11)

d d

Also note that by (4.25), (5.2) and (5.9) there is a connection

_ '_ = ' : (5.12)

(0) (0) (0)

5.1 Application of Deformation Theory Beyond Pro-

p ortional Loading

If plastic secant and tangent mo duli are used, related to secant and

tangent mo duli with resp ect to total strain by

1 1 1 1 1

; (5.13) = =

p p

h h

t s

the plastic part of the rate of deformation can b e rewritten from Eq.

(4.29) as

0 0

1 1 1 ( ):

p 0

D = + : (5.14)

p p p

0 0

2h 2h 2h :

s s

Deformation theory agrees with ow theory of plasticity only under

prop ortional loading, since then sp eci cation of the nal state of stress

also sp eci es the stress history. For general (non-prop ortional) load-

ing, more accurate and physically appropriate is the ow theory of

plasticity, particularly with an accurate mo deling of the yield surface

Deformation Theory of Plasticity Revisited 14

and hardening b ehavior. Budiansky (1959), however, indicated that

deformation theory can be successfully used for certain nearly pro-

p ortional loading paths, as well. The rate in Eq. (5.14) do es not

then have to b e co directional with . The rst and third term (b oth

prop ortional to 1=2h )inEq. (5.14) do not cancel each other in this

case (as they do for prop ortional loading) , and the plastic part of the

rate of deformation dep ends on both comp onents of the stress rate

0 0

, one in the direction of and the other normal to it. In contrast,

according to ow theory with the von Mises smo oth yield surface, the

0 0

comp onent of the stress rate normal to do es not a ect the plastic

part of the rate of deformation. Physical theories of plasticity (e.g.,

Hill, 1967) indicate that yield surface of a p olycrystalline aggregate

develops a vertex at its loading stress p oint, so that in nitesimal in-

crements of stress in the direction normal to indeed cause further

plastic ow. Since the structure of the deformation theory of plas-

ticity under prop ortional loading do es not use a notion of the yield

surface, Eq. (5.14) can be adopted for an approximate description

of the resp onse in the case when the yield surface develops a vertex.

When Eq. (5.14) is rewritten in the form

# "

0 0 0 0

1 1 ( ): ( ):

p 0

+ ; (5.15) D =

p p

0 0 0 0

: :

2h 2h

the rst term on the right-hand side gives the resp onse to comp onent

of the stress increment normal to . The asso ciated plastic mo dulus

. The plastic mo dulus asso ciated with comp onent of the stress is h

increment in the direction of is h . Therefore, for continued plastic

ow with small deviations from prop ortional loading (so that all yield

segments whichintersect at the vertex are active { fully active load-

ing), Eq. (5.15) can b e used to approximately account for the e ects

of the yield vertex. The idea was used by Rudnicki and Rice (1975)

in mo deling inelastic b ehavior of ssured ro cks, as will be discussed

in section 7.1. For the full range of directions of stress increment,

the relationship b etween the rates of stress and plastic deformation is

Deformation Theory of Plasticity Revisited 15

not exp ected to b e necessarily linear, although it should b e homoge-

neous in these rates in the absence of time-dep endent (creep) e ects.

A corner theory that predicts continuous variation of the sti ness

and allows increasingly non-prop ortional increments of stress is for-

mulated by Chisto ersen and Hutchinson (1979). When applied to

the analysis of necking in thin sheets under biaxial stretching, the

results were in b etter agreement with exp erimental observations than

those obtained from the theory with smo oth yield characterization.

Similar conclusions were long known in the eld of elastoplastic buck-

ling. Deformation theory predicts the buckling loads b etter than the

ow theory with a smo oth yield surface (Hutchinson, 1974).

6 PRESSURE-DEPENDENT DEFORMATION

THEORY OF PLASTICITY

To include pressure dep endence and allow inelastic volume changes in

deformation theory of plasticity, assume that, in place of Eq. (4.4),

the plastic strain is related to stress by

" #

1=2

2 1

0 0 0 0

+ T E = ' ; (6.1) I : T T

(0)

(0) (0) (0)

(0)

3 2

where is a material parameter. It follows that the deviatoric and

spherical parts of the plastic rate of deformation tensor are

p 0 0 0

D = '_ + ' ; (6.2)

(0) (0)

1=2

tr D =2 J '_ + ' : (6.3)

(0) (0)

2 J

0 0

The invariant J =(1=2) : is the second invariant of deviatoric

part of the Kirchho stress.

Supp ose that a nonlinear relationship = ( ) between the

Kirchho stress and the plastic part of the logarithmic strain is avail-

able from the elastoplastic shear test (needed data for brittle ro cks is

Deformation Theory of Plasticity Revisited 16

commonly deduced from con ned compression tests; Lubarda, Mas-

tilovic and Knap, 1996a). Let the plastic secant and tangent mo duli

b e de ned by

; h = ; (6.4) h =

p p

and let in three-dimensional problems of overall compressive states of

stress

1=2

+ = J tr ; (6.5)

1=2

p p

p 0 0

E E

= 2 E : E =2' J : (6.6)

(0)

(0) (0)

The friction-typ e co ecient is denoted by . Note that from Eq.

(6.1), E = ' . By using the rst of Eq. (6.4), therefore,

(0)

' = : (6.7)

(0) p

1=2

In order to derive an expression for the rate '_ , di erentiate Eqs.

(0)

(6.5) and (6.6) to obtain

1 1

1=2

= J tr ; (6.8) ( : )+

2 3

1=2 1=2

=2 '_ J + ' J ( : ) : (6.9)

(0) (0)

2 2

Combining this with the second of Eq. (6.4) gives

1 : 1 tr 1 1 1

'_ = : (6.10) +

(0) p p p

1=2 1=2

2 h h 2 J 2h 3

t t

J J

2 2

Substituting Eqs. (6.7) and (6.10) into Eqs. (6.2) and (6.3) yields

0 0

1 1 1 ( ): 1

0 p 0

+ = D

p p p

1=2 1=2

2h 2 h h 2 J

s s

J J

2 2

(6.11)

1 1 tr

+ ;

1=2

2h 3

J 2

Deformation Theory of Plasticity Revisited 17

1 :

tr D = + tr : (6.12)

1=2

h 3

t 2 J

1=2

= J ,Eqs. (6.11) and (6.12) reduce to If = 0, i.e.

# "

0 0

1 ( ): h

p 0 0

D = 1 ; (6.13) +

p p

2 J

2h h

tr D = : (6.14)

1=2

6.1 Non-Coaxiality Factor

It is instructive to rewrite Eq. (6.11) in an alternative form as

! # "

0 0 0 0

1 1 1 : ( ):

p 0 0

D = tr + +

p p

1=2 1=2 1=2

3 2 J

2h 2h

J 2 J J

2 2 2

(6.15)

p 0 0

The rst part of D is coaxial with . The second part is in the

0 0

direction of the comp onentof the stress rate that is normal to .

There is no work done on this part of the plastic strain rate, i.e.

1 2

1=2

p 0 0

: D = J tr : (6.16) : +

Observe in passing that from Eqs. (6.12) and (6.16),

p 0

: D

; (6.17) tr D =

1=2

which o ers a simple physical interpretation of the parameter .

The co ecient

1 1 1 tr

1+ & = = (6.18)

p p

1=2 1=2

2h 2h

s s

J J

2 2

in Eq. (6.15) can b e interpreted as the stress-dep endent non-coaxiality

factor. Other de nitions of this factor app eared in the literature, e.g., Nemat-Nasser (1983).

Deformation Theory of Plasticity Revisited 18

6.2 Inverse Constitutive Relations

The deviatoric and volumetric part of the total rate of deformation

are obtained by adding to (6.11) and (6.12) the elastic contributions,

! !

0 0

( ): 1 1 1 1 1

0 0

+ + D =

p p p

1=2 1=2

2 2 2 J

2h h h

s s

J J

2 2

1 tr 1

;

1=2

(6.19)

1 1 :

tr D = + tr + : (6.20)

p p

1=2

3 h 2h

t t

The inverse relations are found to b e

" #

0 0 0

1 a ( ): D 1

0 0

=2 D tr D ; (6.21)

1=2

b bc 2 J c 2

# "

3 : D h

tr = : (6.22) tr D 1+

1=2

The intro duced parameters are

1+ ; (6.23) ; b =1+ a =1

p p p

1=2 1=2

h h h

s s

J J

2 2

and

c =1+ + : (6.24)

7 Relationship to Pressure-Dep endent Flow

Theory of Plasticity

For geomaterials like soils and ro cks, plastic deformation has its ori-

gin in pressure dep endent microscopic pro cesses and the yield condi-

tion dep ends on the hydrostatic comp onent of stress. Drucker and

Deformation Theory of Plasticity Revisited 19

Prager (1952) suggested that inelastic deformation commences when

the shear stress on o ctahedral planes overcomes cohesive and frictional

resistance to sliding. The resulting yield condition is

1=2

f = J + I k =0; (7.1)

with as the co ecient of internal friction, and k as the yield shear

strength. The rst invariant of the Kirchho stress is I =tr,andJ

1 2

is the second invariant of the deviatoric part of the Kirchho stress.

Constitutive equations in which plastic part of the rate of deforma-

tion is normal to lo cally smo oth yield surface in stress space are re-

ferred to as asso ciative ow rules. A sucient condition for this con-

stitutive structure is that material ob eys the Ilyushin's work postu-

late (Ilyushin, 1961). However, pressure-dep endent dilatant materials

with internal frictional e ects are not well describ ed by asso ciative

ow rules. For example, they largely overestimate inelastic volume

changes in geomaterials, and in certain high-strength steels exhibiting

the strength-di erential e ect (by which the yield strength is higher in

compression than in tension). For such materials, plastic part of the

rate of strain is taken to be normal to the plastic potential surface,

which is distinct from the yield surface. The resulting constitutive

structure is known as a non-asso ciative ow rule. For geomaterials

whose yield is governed by the Drucker-Prager yield condition, the

plastic p otential can b e taken as

1=2

= J I k =0: (7.2) +

The material parameter is in general di erent from in Eq. (7.1).

Thus,

1 @ 1

1=2

0 p

= _ J + I : (7.3) D = _

@ 2 3

The loading index _ is determined from the consistency condition.

Assuming known the relationship k = k (#) between the shear yield

Deformation Theory of Plasticity Revisited 20

stress and the generalized plastic shear strain

1=2

p 0 p 0

dt; (7.4) # = 2 D : D

the condition f = 0 gives

1 1 1

1=2

_ = J + I : : (7.5)

2 3

The plastic tangent mo dulus is h =dk=d#. Substituting Eq. (7.5)

into Eq. (7.3) results in

1 1 1 1 1

1=2 1=2

p 0 0

D = J + I J + I : : (7.6)

2 2

h 2 3 2 3

Aphysical interpretation of the parameter is obtained by observing

from Eq. (7.3) that

p 0

: D

1=2

p p 0 p 0

= ;_ tr D = ;_ (7.7) 2 D : D =

1=2

i.e.,

tr D

= : (7.8)

1=2

p 0 p 0

(2 D : D )

Thus, is the ratio of the volumetric and shear part of the plastic

strain rate, which is often called the dilatancy factor (Rudnicki and

Rice, 1975). Representative values of the friction co ecient and the

dilatancy factor for ssured ro cks indicate that = 0:3 1 and

= 0:1 0:5 (Lubarda, Mastilovic and Knap, 1996b). Frictional

parameter and inelastic dilatancy of material actually change with

progression of inelastic deformation, but are here treated as constants.

For more elab orate analysis, which accounts for their variation, the

pap er by Nemat-Nasser and Shokooh (1980) can b e consulted.

The deviatoric and spherical parts of the total rate of deformation

are

0 0 0

1 : 1

+ + tr ; (7.9) D =

1=2 1=2

2 2h 3

J 2 J

2 2

Deformation Theory of Plasticity Revisited 21

1 1 :

tr D = + tr + tr : (7.10)

1=2

3 h 3

t 2 J

These can b e inverted to give the deviatoric and spherical parts of the

stress rate

" #

0 0 0

1 1 ( ):D

0 0

=2 D tr D ; (7.11)

1=2

c 2 J c 2

" #

h : D 3

1+ tr D : (7.12) tr =

1=2

The parameter c is de ned in Eq. (6.24). The last expression is

identical to (6.22), as exp ected since (6.20) and (7.10) are in concert.

If the friction co ecient is equal to zero, Eqs. (7.11) and (7.12)

reduce to

0 0

( ):D 1

0 0

; (7.13) =2 D

1+h = 2 J

: D

tr =3 tr D : (7.14)

1=2

= 1+h

With vanishing dilatancy factor ( = 0), these coincide with the con-

stitutive equations of isotropic hardening pressure-indep endent metal

plasticity.

7.1 Relationship to Yield Vertex Mo del for Fissured

Ro cks

In a brittle ro ck, mo deled to contain a collection of randomly ori-

ented ssures, inelastic deformation results from frictional sliding on

the ssure surfaces. Individual yield surface may be asso ciated with

each ssure, so that the macroscopic yield surface is the envelop e of

individual yield surfaces for ssures of all orientations (Rudnicki and

Deformation Theory of Plasticity Revisited 22

Rice, 1975). Continued stressing in the same direction will cause con-

tinuing sliding on (already activated) favorably oriented ssures, and

will initiate sliding for a progressively greater numb er of orientations.

After certain amount of inelastic deformation, the macroscopic yield

envelop e develops a vertex at the loading point. The stress incre-

ment normal to the original stress direction will initiate or continue

sliding of ssure surfaces for some ssure orientations. In isotropic

hardening idealization with smo oth yield surface, however, a stress

increment tangential to the yield surface will cause only elastic de-

formation, overestimating the sti ness of the resp onse. In order to

takeinto account the e ect of the yield vertex in an approximate way,

Rudnicki and Rice, (op. cit.) intro duced a second plastic mo dulus h ,

which governs the resp onse to part of the stress increment directed

tangentially to what is taken to b e the smo oth yield surface through

the same stress p oint. Since no vertex formation is asso ciated with

hydrostatic stress increments, tangential stress increments are taken

to b e deviatoric, and thus

! !

0 0 0

1 : 1 1 :

p 0 0 0

D = + tr + :

1=2 1=2

3 2h 2 J

t J 2 J

2 2

(7.15)

The dilation induced by the small tangential stress increment is as-

sumed to b e negligible, i.e.,

1 :

+ tr : (7.16) tr D =

1=2

2 J

Comparing Eq. (7.15) with (6.15) of the pressure-dep endent deforma-

tion theory of plasticity, it is clear that the two constitutive structures

are equivalent, provided that identi cation is made

1=2

1 J

p p

= : (7.17) h = h

This derivation reconciles the di erences left in the literature in a de-

bate between Rudnicki (1982) and Nemat-Nasser (1982). It should

Deformation Theory of Plasticity Revisited 23

also b e noted that the constitutive structure in Eq. (7.15) is intended

to mo del the resp onse at a yield surface vertex for small deviations

from prop ortional loading . For increasingly non-prop ortional

stress increments, the relationship b etween the stress and plastic de-

formation rates is not exp ected to b e necessarily linear.

The expressions for the rate of stress in terms of the rate of defor-

mation are obtained by inversion of the expressions based on (7.10)

and (7.15). The results are given byEqs. (6.21) and (6.22), with the

parameters

a =1 ; b =1+ ; (7.18)

p p p

h h h

and with c given byEq. (6.24). In view of the connection (7.17), ex-

pressions in Eq. (7.18) are clearly in accord with (6.23). This demon-

strates a duality in the constitutive structures of deformation and ow

theory for the considered mo dels of pressure-dep endent plasticity.

8 DEFORMATION THEORY BASED ON STRESS

DECOMPOSITION

In the ow theory of plasticity the constitutive structure can b e built

by either decomp osing the rate of strain or the rate of stress into

elastic and plastic constituents (Hill, 1978; Lubarda, 1994,1999). It

is app ealing to formulate the deformation theory of plasticity in a

similar manner. Thus, instead of decomp osing the total strain, which

was done in section 4, decomp ose the stress tensor into its elastic and

plastic part,

T = T + T ; (8.1)

(0)

and assume that for isotropic pressure-indep endent plasticity

e 0

T =2 E + tr E I ; (8.2)

(0) (0)

(0)

T = E ; (8.3)

(0)

(0) (0)

Deformation Theory of Plasticity Revisited 24

where is an appropriate parameter. Note that T is a deviatoric

(0)

tensor, so that deviatoric part of the total stress is

0 0

T =(2 )E : (8.4)

(0) (0)

(0)

e 0 0

Since =2 D , from (8.4) by di erentiation,

p 0

= E D : (8.5)

(0) (0)

(0)

= ( ) between the Supp ose that a nonlinear relationship

logarithmic strain and plastic part of the conjugate stress is available

from elastoplastic pure shear test. Let the corresp onding secant and

tangent compliances b e de ned by

g = ; g ; (8.6) =

p p

and let

1=2 1=2

1 1

p p

p p p

= : T : T = ; (8.7)

(0) (0)

2 2

while is de ned as in Eq. (4.23). It follows that

= ; (8.8)

(0) p

and

: D E

1 1

(0)

=2 : (8.9)

p p (0)

0 0

g g E E

E : E

(0) (0)

Substituting Eq. (8.9) into Eq. (8.5), the plastic part of the Jaumann

rate of the Kirchho stress b ecomes

2 3

0 0

E E

E EE : D

(0) (0)

g 2

0 p

4 5

1 D + : (8.10) =

p p

0 0

E E

g g

E : E

(0) (0)

Deformation Theory of Plasticity Revisited 25

By adding the elastic contribution, the deviatoric part of the Jaumann

rate of stress is

2 3

0 0

E E

E EE : D

(0) (0)

1 1 1

0 0

4 5

=2 D : (8.11)

p p p

0 0

g g g E E

E : E

s s

(0) (0)

= This constitutive structure is in agreement with (4.30), b ecause

+ , and

1 1

h = ; h = : (8.12)

s t

p p

g g

It is also noted that the parameter is related to parameter '

(0) (0)

of section 4 by

(0)

: (8.13) 2' =

(0)

In the case of pressure-dep endent plasticity,we can take the plastic

part of the stress to b e related to strain according to

1=2

0 0 0 0

T = + E : E I ; (8.14) 2 E

(0)

(0) (0) (0)

(0)

where is a new material parameter. Furthermore, de ne

1=2

0 0

: E + = 2 E tr E ; (8.15)

(0)

(0) (0)

1=2

p p

p 0 0 0

= T : T ; (8.16)

(0) (0)

p 0

= ( ). The friction-typ e and assume known the relationship

co ecient is denoted by . It is easily veri ed that (6.5) and (8.15)

cannot lead to equivalent constitutive descriptions, if and are

b oth required to b e constant (although distinct) co ecients. Having

this in mind, and with the plastic secant and tangent compliances

de ned by

g = ; g = ; (8.17)

p 0 p 0

Deformation Theory of Plasticity Revisited 26

it follows that

; (8.18) =

p (0)

1=2

and

2 E : D

2 2 tr D 2 1

(0)

= + : (8.19)

(0) p p p

1=2 1=2

g g j g 3

4j j t t

2 2

0 0

E E

The notation is used j =2E : E . Consequently,

(0) (0)

0 0

E E

E 2 E : D

1 2

(0) (0)

p 0

+ tr D =

1=2 1=2

g 3

j j

2 2

3 2

(8.20)

0 0

E E

2 E EE : D

(0) (0)

5 4

: D

1=2

g j

: D 2 E

1 2

(0)

+ tr D : (8.21) tr =

1=2

g 3

t j

These give rise to dual, but not equivalent constitutive structures to

those asso ciated with Eqs. (6.12) and (6.15). Finally, it is noted that

p 0

2 E :

(0)

tr = ; (8.22)

1=2

which parallels Eq. (6.17).

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