671017.pdf

2. Classical Elastoplastic Theory of

Geomaterials

The incremental form of a stress-strain relationship in tradi-
tional geomechanics is generally expressed as

d휎푖푗=퐷푖푗푘푙d휀푘푙. (1)

Determining퐷푖푗푘푙is the major topic for constitutive models
of geomaterials. Obviously,퐷푖푗푘푙can be obtained by fitting
experimental data, given that experiments on stress and
strain tensors are conducted. However, it is extremely difficult
to do so. Therefore, experiments on the stress-strain rela-
tionship of geomaterials are usually conducted in principal
stress/strain space; that is, only the relationship between
the principal stress휎푖(푖 = 1,2,3)and the principal strain
휀푖(푖 = 1,2,3),isobtained.Toobtain퐷푖푗푘푙, the constitutive
tensor in general coordinate space should be derived from
the stress-strain relationship in principal stress/strain space.
From a mathematical point of view, these can be treated as
the problems of coordinate transformation [ 34 – 36 ].
The relationship between the plastic strain increment and
stress increment in principal stress/strain space is defined as

{d휀푝푖}3×1=[퐴]3×3{d휎푖}3×1, (2a)

[퐴]3×3=[

[

푎 11 푎 12 푎 13

푎 21 푎 22 푎 23

푎 31 푎 32 푎 33

]

]

, (2b)

where푎푖푗are the functions of total stress휎푖, total strain휀푖(푖 =
1,2,3)or stress path.
When the matrix rank of[퐴]is 1, or|퐴| = 0, there exists a
vector{훼 1 훼 2 훼 3 }푇and coefficients훽 1 ,훽 2 ,훽 3 to express[퐴]
as

[퐴]={훽 1 훽 2 훽 3 }

푇

{훼 1 훼 2 훼 3 }, (3)

Therefore, substituting ( 3 )into(2a)and(2b)gives

{d휀푝푖}={훽 1 훽 2 훽 3 }푇{훼 1 훼 2 훼 3 }{d휎푖}, (4a)

that is,

{d휀

푝

푖}=d휆{훽^1 훽^2 훽^3 }

푇

, (4b)

where

d휆=

3

∑

푖=1

훼푖d휎푖. (5)

According to (4b),

d휀푝 1 :d휀푝 2 :d휀푝 3 =훽 1 :훽 2 :훽 3. (6)

훽푖(푖 = 1,2,3)is a function of휎푖or휀푖.When훽={훽 1 훽 2 훽 3 }푇
is of a field with potential, there is a potential function푄such
that

훽푖=

휕푄

휕휎푖

. (7)

Substituting ( 7 )into(4b), we have

d휀푝푖 =d휆

휕푄

휕휎푖

. (8)

If we assume that d휀푖푝 and휎푖 have the same principal

directions, the coordinate transformation can be expressed as

follows:

d휀

푝

푖푗=d휀

푝

푖

휕휎푖

휕휎푖푗

. (9)

Substituting ( 9 )into( 8 )gives

d휀

푝

푖푗=d휆

휕푄

휕휎푖푗

. (10)

Similarly, from elastic potentials theory, there is a poten-

tial function푊in principal stress space, and휀푖is defined as

휀푖=

휕푊

휕휎푖

. (11)

If we assume that휎푖and휀푖havethesameprincipaldirection,

the coordinate transformation can be expressed as follows:

휀푖푗=휀푖

휕휎푖

휕휎푖푗

. (12)

Substituting ( 11 )into( 12 ),

휀푖푗=

휕푊

휕휎푖푗

. (13)

In conclusion, traditional plastic potential theory corre-

sponds to the case that the matrix[퐴]in (2a)and(2b)has

rank 1, and훽can be expressed as the gradient vector of a

potentialfunction.Basedonmathematicalprinciples,amore

general potential function-based constitutive framework can

be established according to vector field theory and tensor

theory as described below.

3. Derivation of Constitutive Framework from

Vector Field Theory

Obviously, when the three principal components of the

plastic strain increment, d휀푖푝(푖 = 1,2,3), are considered to

be components of a vectord휀p,theprincipalcomponentscan

be expressed as three linearly independent 3D vectors by a

vector fitting method. The gradient vectors of three linearly

independent potential functions are selected as the linearly

independent vectors.

Whend휀pis expressed in principal stress space, the

coordinate orientations ofd휀pand휎푖arethesame,and

Φ 1 ,Φ 2 ,Φ 3 are three linearly independent potential func-

tions in principal stress space, and then the following expres-

sion is obtained:

d휀

푝

푖 =

3

∑

푘=1

휆푘

휕Φ푘

휕휎푖

, (14)