671017.pdf

where휆푘(푖 = 1,2,3)are the coefficients. Suppose that the
principal directions ofd휀pand휎푖are the same, and we
substitute ( 14 )into( 9 ), giving the tensor expression in general
coordinate space as

d휀푝푖푗=

3

∑

푘=1

휆푘

휕Φ푘

휕휎푖푗

. (15)

Similarly,d휀pcan also be expressed in strain space. Let
Ψ 1 ,Ψ 2 ,Ψ 3 be three linearly independent potential functions
in principal strain space, and the following expression is
obtained:

d휀푝푖푗=

3

∑

푘=1

휇푘

휕Ψ푘

휕휀푖푗

. (16)

Define the plastic stress increment as

{d휎푝}=[퐷푒]{d휀푝}, (17)

where[퐷푒]is the elastic stiffness matrix, and then the
expressions in stress space and strain space are

d휎푝푖푗=

3

∑

푘=1

훽푘

휕퐺푘

휕휎푖푗

, (18)

d휎푝푖푗=

3

∑

푘=1

훼푘

휕퐹푘

휕휀푖푗

. (19)

For the total stress and the total strain, consider the three
principal stresses and the three principal strains as vectors in
three-dimensional space with the same principal directions,
and the following expressions are obtained:

휎푖푗=

3

∑

푘=1

휂푘

휕푊푘

휕휀푖푗

,

휀푖푗=

3

∑

푘=1

휒푘

휕Ω푘

휕휎푖푗

,

(20)

where푊푘,Ω푘(푖 = 1,2,3)are potential functions with lin-
early independent gradient vectors in strain space and stress
space, respectively.

4. Derivation of Constitutive Framework from

Tensor Theory

If퐴푖푗 and퐸푖푗 are symmetric second-order tensors with
the same principal directions, the following equations are
obtained according to tensor theory and vector fitting:

퐴푖푗=

3

∑

푘=1

휆푘

휕퐼퐸푘

휕퐸푖푗

, (21)

퐸푖푗=

3

∑

푘=1

훽푘

휕퐼퐴푘

휕퐴푖푗

, (22)

where퐼퐸푘(푘=1,2,3)are three independent invariants of퐸푖푗,

and퐼퐴푘(푘=1,2,3)are three independent invariants of퐴푖푗.

For example, for the stress tensor휎푖푗, the three independent

invariants can be퐼휎1 =휎푖푖,퐼휎2 = (1/2)휎푖푗휎푗푖,퐼휎3 =

(1/3)휎푖푘휎푘푛휎푛푚;퐼휎1=푝,퐼휎2=푞,퐼휎3=휃;or퐼휎1=휎 1 ,퐼휎2=휎 2 ,

퐼휎3=휎 3 ,where푝is the mean stress,푞is the deviatoric stress,

휃is the Lode’s angle, and휎 1 ,휎 2 ,and휎 3 are the three principal

stresses.

If퐴푖푗=d휀푝푖푗,퐸푖푗=휎푖푗,퐼휎푘(푘=1,2,3)are invariants of

휎푖푗,itcanbeobtainedfrom( 21 )that

d휀

푝

푖푗=

3

∑

푘=1

휆푘

휕퐼휎푘

휕휎푖푗

, (23)

which is equivalent to ( 15 )whenΦ푘=퐼휎푘.

If퐴푖푗=d휀푝푖푗,퐸푖푗=휀푖푗,퐼휀푘(푘=1,2,3)are invariants of휀푖푗,

then

d휀

푝

푖푗=

3

∑

푘=1

훼푘

휕퐼휀푘

휕휀푖푗

. (24)

If퐴푖푗=휀푖푗, 퐸푖푗=휎푖푗,then

휎푖푗=

3

∑

푘=1

훽푘

휕퐼휀푘

휕휀푖푗

,

휀푖푗=

3

∑

푘=1

휆푘

휕퐼휎푘

휕휎푖푗

.

(25)

Equation ( 25 ) corresponds to ( 20 ), respectively, when푊푘=

퐼휀푘,Ω푘=퐼휎푘.

Similarly, if퐴푖푗=Δ휀푖푗,퐸푖푗=Δ휎푖푗, then according to ( 21 ),

we have

Δ휀푖푗=

3

∑

푘=1

휆푘

휕퐼휎푘

휕(Δ휎푖푗)

. (26)

It should be noted that the derivations from vector field

theory and from tensor theory are actually coincident. The

potential functions in the derivation from vector field theory

are functions of invariants, which degenerate to tensor form

after composite derivation.

5. Elastoplastic Matrix in Stress Space

Without considering the effect of the Lode’s angle휃and

rotation of principal stress, the relations for the plastic strain

increment and stress increment are expressed as

d휀V푝=퐴d푝+퐵d푞, (27a)

d휀푝=퐶d푝+퐷d푞, (27b)

where퐴, 퐵, 퐶, 퐷are parameters.

The following equation is obtained from the potential

functionsΦ 1 =푝, Φ 2 =푞in ( 15 )withoutconsideringthe

effect of the Lode’s angle:

d휀푖푗푝=휆 1

휕푝

휕휎푖푗

+휆 2

휕푞

휕휎푖푗

. (28)