On the correct form of rate-type constitutive equations for elastic behaviour

(1)

On the correct form of rate-type constitutive equations for

elastic behaviour

Citation for published version (APA):

Wijngaarden, van, H., & Veldpaus, F. E. (1986). On the correct form of rate-type constitutive equations for elastic behaviour. (DCT rapporten; Vol. 1986.038). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1986

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I

I-I

I

I. I

I

ELASTIC BEHAVIOUR H. Van Wij~gaarden

Philips Research Laboratories

5600 MD Eindhoven, The Netherlands

and

F.E. Veldpaus

Technical University of Eindhoven

5600 MB Eindhoven, The Netherlands

'WFW 86.038 13.279

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I

1_

I

I. I

I

-1-MS. 13.279

ON THE CORRECT FORM OF RATE-TYPE CONSTITUTIVE EQUATIONS FOR

ELASTIC BEHAVIOUR

H. van Wijngaarden

Philips Research Laboratories

5600 MD Eindhoven, The Netherlands

and

F.E. Veldpaus

Technical University of Eindhoven

5600 MB Eindhoven, The Netherlands

SUMMARY

Deformations are elastic if the stresses can be expressed in

terms of the total deformation and do not depend on the way

this deformation has been reached. Using this concept a class

of rate-type constitutive models is developed for

deforma-tions of this kind. It is proved that many earlier proposed

rate type models, like the Jaumann and the Dienes rate type,

(4)

I

-2-1. Introduction

In this paper attention is focussed mainly on the

des-I

I

cription of elastic behaviour,

stitutive equation of the type

v

'I

w=.rL:iD

starting from a rate type

con-( 1. 1)

I

I. I

I

V

Here, a; is some objective rate of the Cauchy stress tensor

ar,

1D

is the de forma t ion rate tensor and 4(C is a four

th-order tensor, the so-called elasticity tensor. This tensor

does not depend on

J-

or

j )

but can be a function of ([J •

In the geometrical linear theory the deformations and

rota-V

tions are very small and the objective rate ~ may be

re-.

placed by the material rate ~ • For a given state at time to

and a given deformation path in the time interval [to, t ] the

current stress tensorQr(t) can then be determined by

integra-tion of 4~:Dover that interval. In the geometrical

non--linear theory, however, i t is not allowed to replace the

ob-jective rate in the constitutive equation by the material

rate. In the literature many objective rates are used. Some

examples are the Jaumann-Zaremba rate [11, the Truesdell rate

[2], the Cotter-Rivlin r a t e ( 3 ) and the Dienes rate [4).

The reason to choose (1.1) as the starting point for the

description of elastic behaviour is that this form is very

convenient for the derivation of constitutive equations for

elastic-plastic behaviour. This is shortly pointed out in

Section 4 and results in a similar equatio~as (1.1), where

4~ has to be replaced by the so-called elastic-plastic

material tensor 4~.

Nagtegaal and de Jong [5) showed that the constitutive

17

equation (1.1) with the Jaumann rate for ~ yields

unaccepta-ble results in the simple shear test. Other authors [6,7,8]

tried to get acceptable results by using other objective

stress rates. Lee, Mallett and Wertheimer (6) use a

modifica-tion of the Jaumann rate in their analysis of the simple

shear test for the kinematic hardening of elastic-plastic

materials. However, a generalization of their procedure to

(5)

I

tt

I

I. I

I

-3-(a1

also studied the simple shear test. He uses symmetrical

and non-symmetrical objective rat~s that can be written as

the Jaumann rate plus an objective function of the tensor

([). fi) ..

ID.Gr.

His analysis of the effect of the chosen function

on the stresses in the simple shear test results among other

things in the conclusion that the Truesdell rate and the

Cotter-Rivlin rate lead to acceptable results. His conclusion

that other objective rates, like the Dienes rate, are

super-fluous, is not really proved.

In this paper a class of objective rates is studied and

i t is examined which of these rates can result in a correct

description of elastic behaviour. Each of the earlier

men-tioned objective rates (Jaumann etc.) belongs to this class.

I t will be shown in Section 3 that for each objective rate in

fA V /.., /i. II\-T

t his c I ass the r e ex i s t s a t ens0 r Ir"\ s u c h t hat<rr

=

IA

•

ar . In

where the tensor

C-

=-14

.<IT.

#\T

is invariant under

Galilei transformations. Using this tensor the constitutive

equation (1.1) can be written as

( 1.2 )

A ~

Because

err

is the material rate of (JT integration of (1.2)

""

yields the invariant stress tensor «r , whereupon ~ can be

determined. When 4iLis replaced by 4~ this procedure to

de-termine (/J" yields one of the possibilities that are used in

Section 5 to investigate which objective derivatives can

re-sult in a correct description of some kind of elastic

be-haviour. For isotropic fourth-order elasticity tensors i t

will be proved that this is not the case for the Jaumann rate

and the Dienes rate. The discussion in Section 5 leads to the

introduction of a special class of objective rates, each of

which can result in a correct description of one kind of

elastic behaviour. The Truesdell rate and the Cotter-Rivlin

rate belong to this class. In Section 6 the objective rates

of this class are used in the analysis of the torsion of

(6)

I

-4-Similarly, left polar decomposition of ~ leads to the

intro-duction of the left Cauchy strain tensor

18

and the left

stretch tensor

\V:

( 2 • 1 ) ( 2 • 2 ) J~o

J

=

de~(fFJ

=

dV.

cI\I. J o

Right polar decomposition of

/iF

leads to the introduction of

the right Cauchy strain tensor~, of the right stretch

tensor

tLl

and of the so-called principal strain rotation

tensor

fR :

2. Some kinematic notions

Let ~ be the deformation tensor of the current

configu-ration C(t) of a body with respect to a reference

configura-tion CO=C(tO). The determinant of ~, i . e . det(~), is

positive and equal to the ratio J of the current volume dV

of an infinitesimally small material element and the

reference volume _{dVO of that element}

I

Since

\V

is symmetric and positive definite the eigenvalues

A ,

~ 2 and).. 3 are real and positive. If the corresponding,

I

mutually orthogonal unit eigenvectors of

\V

are denoted by

n1,

02 and n3 the tensors

\V

and

18

are given in spectral

re-presentation by

I

I. I

I

fF;;

\V.

fR

.I ( 2 • 3 )

The principal logarithmic strains

E

l '

E

2 and C3 are defined

by

I

J ( 2 .4 )

.

Let

~

be the material rate of

~.

Then the deformation

rate tensor

~

and the spin

tensor~

follow from

I

(7)

( 2 .9) ( 2 .8) ( 2 • 6 ) (2.7 ) (2.10) (2.13) (2.11) (2.12) J

easily seen that

i= 1,2,3 T ..J'l-.: -.JZL

IN.

n.

,

.

n.: .::

3

ID:.

L

(i ..

n.. n.)

+ .!.

(\'vJ.

(B

+

lB.

\'yj

T)

"=,

. "

~

IN:.

-9l. .,.

~

( \\,./.

1[3 _

1l3. \\,./

T)

Furthermore i t is seen that the trace of

ll),

tr(ID), is equal

to

where

\~

is some skew-symmetrical tensor. Because

~

is

symme-trical while

#V

and~ are skew-symmetrical i t is found that

Comparison of this result with (2.7) yields that lA/has to

sa-tisfy

.

Therefore i t follows from (2.4) that

IB

can also be written as

-5-Using

lB

=

!F.

fFT and

i

=CiDf

~).

rr

i t is

7

IN:; -

IN

j

.

JB

=

(II)

+

..9L).

fij

+

fi3.

(rD

+

..9'1..)'

.

A second relation for

/B

can be derived from (2.4). Supposing

that the mutually orthogonal unit eigenvectors n1,

n2

and n3

are differentiable functions of t i t can be shown that there

exists a skew-symmetrical tensor

IN

such that

I

'I

I

,_

I

I. I

I

(8)

I

-6-I

I

A motion of the body is called an objective equivalent

motion if i t is generated from the real motion by a rigid

bo-dy translation and a rigid body rotation

<It,

where

CQ

is a

ro-tation tensor

I

Jet

((4»)

=. I (2.14)

I

1-I

The transformation of the real motion to an objective

equivalent motion is called an objective transformation.

Because rigid body translations are irrelevant in the sequel

they are left out of consideration. On that account,

objective transformations are determined by the rigid body

rotat ion tensor

CCi

and can be denoted by T(CQ). Quan t i t ie s

that are related to an objective equivalent motion are

labelled with an asterisk. If a quantity remains unchanged

for every T(~) i t is called invariant. A scalar quantity

a vectorial quantity

l'

and a second-order tensorial

-J

~

=([{.p

~,

(2.15)

.J

are called objective if for every T(~):

quantity

#

I

A fourth-order tensorial quantity 4#iS called objective if

for every T«Q) and every

IN

I. I

If#

f ;

1M ::

CQ. [

"cfI>

(2.16)

Hence, each invariant scalar quantity is also objective. The

unit tensor

I

is the only second-order tensor that is both

objective and invariant. However, there are three

fourth-order tensors that are both objective and invariant. These

are denoted by

4JI ,

4][ T and

Jl.ll

and are defined by the

requirement that for every

1M

I

"I·

1/'1::

/11

oJ

'IJ[

T:

1M

=

11'1

T .

"

JJ.

jJ :

1/1::.

1:-... (

//'1) .

JL

(2.17)

I

(9)

-7-invariant nor objective :

From their definitions i t follows that J and

UJ.

are invariant,

that

ID

and

\V

are objective and that

W,

fR

and.7J.. are neither

(2.18) (2.19) (2.20)

in the reference state.

I t is t a c i t l y assumed here that ([)=

If

I

I_

I

I. I

I

(10)

I

-8-3. Objective and invariant stress quantities

Leta} be the Cauchy stress tensor. The mechanical power

supplied in the current state to an infinitesimally small

material element with current volume dV is equal to

v:~dV=tr(~.~)dV. The mechanical power per unit of reference

volume, ~ , is given by

I

1t=

].q;r:])

( 3. 1 )

I

.-The principle of objectivity states that ~ is invariant,

i . e . that 7r¥ = 1r for every T(Q). Because]I) is objective and

J is invariant i t is seen that the Cauchy stress tensor is

objective

I

for every T(~) ( 3 .2)

I

However, the material rate ~of aT is not objective because

I

; F

0 . (3.3)

and this result shows that the tensoraT-

-SL.ar-qr.11l

V

called Jaumann or Zaremba rate qj) of

err,

, the

so-I

I. I

Using (2.20) ([; can be eliminated, yielding

.J ( 3 .4 )

I

::. «1- -

.3l.. • rtr - err. .!hT ( 3 • 5 )

I

V

is objective. Furthermore, i t follows that every rate

«r

of

the form

V

CD""::.

<Uj -

1/1

(

3 • 6 )

is objective if

If?

is objective. I t is not necessary for /~

to be symmetrical. Atluri [8] uses non-symmetrical tensors

but i t is not clear whether the resulting non-symmetrical

V

objective rates U offer any advantages in the process of

formulating rate-type constitutive equations. Therefore, only

(11)

tion of

(3.12)

(3.13)

( 3 .9)

}

for every T (

a»

-9-for every T(~) ;

of the type

( fA

lI•

(J)) :: - (

fAl<. ~).

(.9L

+ 11-1) for eve r y T (~)

IA"

=-

fA.

CQ

T

Because

ar

is objective the tensor ~, defined by

and this implies that

i t is seen that

This tensor is not objective. However, using (2.20) and (3.9)

G

:=.

fA .

<IT.

fA

T ( 3 • 1 4 )

is invariant. In the sequel i t is assumed that fA is regular

for all t 7tO_ From (3.11) and (3.10) i t then follows that

!A

.=. -

/A.(

-S7l..+

11-1)

C/.I

IH::.

11-/

+ aT. $ (3 • 8 )

combining (3.7) and (3.8) with (3.5) and (3.6) yields

;;. =.

cU- -

(-S1l..

+

II-I).

err -

crT.

(...9l-

+

/1-1)

T (3 • 10 )

Hence, each objective tensor IHresults in an objective rate

and (3.10) specifies a class of objective rates. An important

property of these rates is that they are associated with the

material rate of invariant stress tensors. Let~ be the

solu-In e,;,T

IN

=-

IH.

a:r-

+ <!r.

IJ.-j

(

3 .7)

with an objective tensor

IH

and a skew-symmetrical tensor $

are considered. To guarantee objectivity of IN i t is necessary

V)

and sufficient that/H can be written as

only tensors

I

-I

I

I. I

I

(12)

I

-10-I

I

From the foregoing i t is

(3.15)

clear that there are two

stra-tegies for arriving at objective rates of the earlier

men-tioned class. The direct method is based on the choice of an

objective tensor

11-/,

while the indirect method is based on the

choice of a regular tensor~ that satisfies (3.13). In the

last method the corresponding objective rate follows from

(3.11) and (3.10). To illustrate the direct method/~1is

I

Ie

I

chosen as

.

//-1

=- -

r .

t ...

(if)) .

1[

= -

't

1 JL

where

Q

is a constant. Using (3.10) this results in

(3.16) solution of (3.17) (3.20 ) (3.13) because i t follows that strain rotation

?

~

J-

L (

(J2.1'.

ezr)

) (3.18)

if

¥

=

0, i . e . the Jaumann rate is

ob-From (3.11) i t is seen that

fA

is given in

where

r

is a constant and ~ is the principal

tensor. This tensor is regular and satisfies

fTt:rz;.o..for

every T(q). Using (3.14) and (3.15)

V

the associated objective rate aT is given by

where~,

the so-called Jaumann rotation tensor, is the

Other choices for the tensor ~/have been given e.g. by Lee

cs. [6] and Atluri [8J.

To illustrate the indirect method ~ is chosen as

this case by

v

iT Of cour seq)

=

Cl7

J

t a i ned if

Ii-I

=

(1).

I. I

I

(13)

I

11

-In an analogous way i t is seen that the choice

v

J-:=.

:J

-1.4' (

J~4'.

CIT)

D

I

V wher e

<DO

V

CO-o

-v

=

Q)D

is the Dienes rate of ~

(fR.

fi<').

err -

crY.

erR.

(RT ) '

( 3 .21 )

<3.22)

Fin a 11 y, i f

fA

i s c h0 sen as

V

leads to the Cotter-Rivlin rate ~R

(3.24)

(3.25) (3.23)

y

-~«

=

cO- -

(.51-

jj)) .

ar -

<IT.

(.n.-

iD)

I •

J

y

r

fA

=

J .

IF

I

e

I

the Truesdell or Green rate ~RV is found:

""

From the' definition (3.14) of<IJit follows that the invariant

stress tensor associated with this choice for

A

is equal to

the second Piola-Kirchhoff stress tensor if

r

=

f

In the last three examples, resulting in the Dienes, the

Cotter-Rivlin and the Truesdell rates, the tensor ~ at time t

is completely determined by the deformation tensor

~

of the

current con~igurationC(t) with respect to the reference

configuration CO=C(tO)' This is not the case in the first

example because the

solution~(t)

of (3.19) will depend

I. I

I

J (3.26)

I

(14)

I

-I

I

I. I

I

-12-on the path of.fl..=~('L) for to::!" s t . As a consequence, i t is

not po s sib l e i n t h i s cas e to in t e r pret

IA

=

J

If if>T i n

kinematic terms. The given examples show that the most widely

used objective rates belong to the class of rates that is

(15)

I

-13-4. Some comme~ts on rate-type constitutive equations

An essential step in the procedure generally used to

derive a constitutive equation for elastic-plastic behaviour

is the decomposition of the total deformation rate

ID

into an

elastic part (])e and a plastic part '[])P,

where the elastic part

IDe

is defined by

\7

Here

err

is some specified objective stress rate while 4/f1 is a

specified objective fourth-order tensor that may depend on

(current) stress and deformation parameters like a:r and fFbut

po fl) . ~(l.

not on rate quantities like err and . S~nce u...J and cr are

sym-metrical 4/Nmust be left-symmetrical and may be chosen

right-I t is assumed that there exists an objective left- and

right-symmetrical fourth-order tensor 4~, the so-called

elas-ticity tensor, such that

for every ~

I

1-I

I

I. I

-symmetrical, i . e .

l{/t1 :

-z

=-

'1

1M:

~

T

z ;

"l11

_=-

~

_':

~t7

_for _every

Z

( 4 • 1 ) ( 4 .2 ) ( 4 .3) ( 4 • 4 )

I

From (4.1) and (4.2) i t then follows that

v

I.( 'I P

crY=-

Cf:7D -

CC:

II>

( 4 • 5 )

I

I t remains to relate lDPto

ID

or qT. For time-independent

plas-ticity this relation can be derived in the usual way by the

introduction of a yield criterion, a flow rule and a

harden-ing model (Lehmann,

(91).

This finally results in

I

(16)

I

-14-Combining this with (4.6) yields a relation for the material

'"

rate aT. Integration of that relation results in

(4.8) ( 4 • 7 ) i7 <!Y"-t

f

(

V T

IA .

!L:

TD).

/A cI7:

where the elastic-plastic tensor 4LLis left- and

right-V

symmetrical, objective and independent of <t:randJI).

V

The objective rate ~ is not the material rate of (f) and,

as a consequence, straightforward integration of (4.6) to

determine ~is not possible. However, each objective rate

considered in this paper is associated with a regular tensor

'"

/Aand an invariant stress tensor a ; , such that

I

I_

I

Her e

a;. (

to)

=

cr::r

(t 0) sin c e

/A

=][

for t

=

to' S tar tin g fro m

(4.6) the Cauchy stress tensor(Y(t) is therefore given by

( 4 .9)

Similar expressions forCIr(t) were used by Nagtegaal and

Veldpaus [11] for the numerical integration of the

constitu-tive equations for isotropic hardening elastic-plastic

be-haviour. However, they only considered the Jaumann rate,

whereas (4.9) and (4.10) are applicable to any objective rate

that belongs to the class defined by (3.10).

If the outlined procedure to determine aT is used in a

general solutionprocess for elastic-plastic problems i t must

be applicable for elastic problems. I t will be shown in the

next Section that this requirement restricts the allowable

combinations of objective stress rates and elasticity

I

I. I

I

where the symmetrical tensor

11-<

is defined by

i-lK

(t)::

/A-I

(f). [

r

IA.

('f/L :

TD).

fAT cI 1:" ]

to

(4.10)

I

tensors.

(17)

I

-15-5. Elastic behaviour

Before analyzing rate-type constitutive equations for

the description of elastic behaviour some definitions are

given. I t is assumed that the material is stress-free in the

reference configuration CO.

A deformation from Co to the current configuration C(t)

is called Cauchy elastic ifV(t)

is determined completely by

~e).In

this case the stress-strain relation must be of the

form

(Hunter

[11])

I

1_

( 5 • 1 )

I

An elastic material is isotropic if

q;.::

a;

([JiS isotropic.

Then the stress-strain relation becomes (Hunter,l111)

( 5 • 2 )

where C(O,

d

_{1 and}

d

_{2 are} scalar functions of the invariants of

~. Using the spectral representation (2.4) of

13

i t follows

that, for an isotropic elastic material,

ar

is given by

A material is Green-elastic if the stresses can be derived

from an elastic potential~. For an isotropic Green-elastic

material <ZIcan also be given by (5.3), but in this case the

principal Cauchy stresses 0""1,0"2 and 0'""3 follow from

I. I

I

112 ,3 ( 5 • 3 )

(18)

I

-16-Now suppose that the procedure of the preceding section

is used for the analysis of an elastic deformation from the

stress-free reference configuration Co to the current

configuration C(t). Then, the rate-type constitutive equation

(4.5) reduces to

( 5 • 5 ) ( 5 .4 )

logarithmic strains c_{1 ,} [2 and

)

~-J

where the principal

defined in (2.5).

I

I-quantities ~1 and ~2 may be functions of the invariants of

m.

The constitutive equation (5.5) then becomes

V

If both QT and 4([. are specified,CIT(t) follows from (4.9) and

(4.10) if 4jL is replaced by 4(1:. Because <II = CD for _{t=t o this}

yields

This relation represents some kind of elastic behaviour

if !I«t) is independent of the deformation path and hence is

completely determined by the deformation tensor f ( t ) of C(t)

with respect to CO' A few examples are considered here. First

of all i t is assumed that 4~ is an isotropic fourth-order

tensor with the earlier required symmetry properties:

( 5 .6)

( 5.8 )

( 5 .7)

defined in (2.17). The scalar

4][ ,41l. T

.11.-.JI.-where /I and are

If([..::

i

r,

('1][

+

41[")

7-

~2

JL .

JL

I:

([j:=.

JJ~::

fA-

J

[f

IA.

('fa:::

Ii)).

IA

T

cI

't

J.

fA-T to

I

I. I

I

(19)

I

-17-and this tensor

lK

is independent of the deformation path if

both integrals are path-independent. For the second integral

this is true if there exists a function

oJ.

= ~ (J) such that

V

If the Dienes rate is chosen for

v

the tensor

/A

is given by

fD

lInT ILl" -. - I ' j)

(3.20). Using

=i ,,,.(

.lL.l +u.t. .1Ll).I" i t follows from

(5.9) that (5.11) (5.10) ( 5.9 ) eft.il

J

t:

1'/

Y\:;.

J-2¥ [

.

In

I~•

J,..e.c..

....!

J"l.

1 ( ·lU.\U.-,~1J.L.lLL)el't.I,",- , . (j)/+

f_o

2-11<

=

/A-

J

However, by choosing different deformation paths, i t can be

shown that the first integral in (5.10) is path-dependent if

~1fo. This implies that the Dienes rate, combined with the

constitutive equation (5.8), cannot result in a correct

des-cription of the material behaviour in general elastic

defor-mations. The same conclusion is found if the stress rate in

V

(5.8) is replaced by the Jaumann rate. R~placing crby the

Truesdell rate, i . e . choosing

#1.=

"')YIF-J. , i t can be shown that

IK

is path-independent if the scalar quantities ~1 and ~2

sa-tisfy the differential equation

and i t is easily seen that

I

I_

I

I. I

where

~

denotes the material rate. Since

U,

and

~2

only

de-e.l

L-pen don the in v a ria n t s 0 fll3 i t can be s how n t h a t (5. 1 2) can

be satisfied for every deformation path only if

u,

and ~2 are

a function of J dec (11=) and if there exists a function

!

= g(J) with

~

(1)=0, such that

I

~

(

. J

1.

0 )

+-de

~

• 1.

0

2 j ~1

J

::

0

J

( 5. 12 )

I

- 2

1

+ :1

Go

J (5.13)

I

(20)

I

-18-I

where GO is a constant. In that case c;v- and

II.<

are given by

I

(IT

=

Ik

G

_o

(Jt>

-]I)

J

(5.14)

I

Ie

These results show that the combination of the Truesdell rate

with the elasticity tensor 4tr:as specified by (5.7) results

in a correct description of some kind of elastic behaviour i f

(5.13) is satisfied. The stress-strain relation associated

with this combination is given by (5.14).

If the Cotter-Rivlin rate is used in (5.8), i.e. if

fA

=

J.fFT , i t is found that

IK

is path-independent if there exist

a constant GO and a function

!

= ~(J) with a(1)=0, such

that

The associated elastic stress-strain relation is then given

by

(5.16) ( 5 . 15 )

The same conclusions about these objective rates can be

derived in a different way. In the approach outlined above i t

Q

is assumed that both 4<1:" and qr are specified. The first

question in this approach is whether or not this combination

yields a correct description of some kind of elastic

beha-viour. The second question concerns the associated

stress--strain relation. In the alternative approach i t is assumed

that 4~ and the elastic stress-strain relation (5.1) are

V

given. Then starting from

rr:r

=

4(:lD

i t must be investigated

which objective stress rate o£ the type (3.10) will result in

I

I. I

I

mustthe givenbe determined such thatstress-strain relation. This means that a tensor

I

err

.

( $l. +

/I-I).

([T - (fJ.

(§l..

T-

I}-/)

T

=-

l(

a::.

(5.17)

I

In any vector basis this symmetrical tensor equation yields a

(21)

ba-I

I

Ie

I

-19-sis. Hence, i t is probable that these components are not

de-termined uniquely by (5.17). This is in agreement with

Sec-tion 3, where i t was shown that an objective stress rate of

the type (3.10) does not change if IH is replaced by IH+ ar.J

provided that $ is skew-symmetrical. In general i t is

possi-ble to determine a class of tensors

IH

such that (5.17) is

sa-tisfied for the given stress-strain relation and the given

elasticity tensor. In practice this possibility is not of

great importance, but the tensor equation can be used as a

starting point for a further investigation of the commonly

used objective stress rates like the Jaumann, the Dienes, the

Truesdell and Cotter-Rivlin rates. For simplicity only

isotropic elastic behaviour is considered and i t is assumed

again that 4([is given by (5.7) with as yet unspecified

scalar quantities _Pl and P2· By taking the material rate of

(5.4), using (2.13) _{for ni and (2.11)} for

rD,

i t is seen that

the tensor equation (5.17) for

JI-J

transforms into

I

[JH -

i

(lIN.

II) -

lB.

\\.JT)J .

qr + rrr. T

[ 11-1 -

i

(\\,y'./5 -

lB.

\\v'I)J ::

I

,}

=

,~

(cTz / '

it

-/,l

t"

riD))

n.v;.

,

For an isotropic material the principal stresses ell ,v2 _{and<T 3}

are functions of the principal logarithmic strains Co 1,

e

2 and

c3 only and therefore ()i (i=1,2,3) is proportional to

el' c

₂

and eo_{3 •} Furthermore, according to ( 2 . 1 4 ) , t ...(ID) is equal _{toE 1}

Let Hij(i,j = 1,2,3) be the elements of the matrix

represen-tation H ofU~ with respect to the vector basis, spanned by

the e i g en v e c tor s n l '

n

2 and

n

3 0 f

18 ,

i .e. Hi j Iii

.lH.

n

j for

i , j = 1,2,3. From (5.18) i t then follows that the diagonal

elements H11, H22 and H33 of ~ have to satisfy

I. I

I

2 1-1 .. .

0". .::

"

(.

.

+t'2 +t: _{3 .} Hence (5.19) can be written as

(5.19)

I

-..u.. [.. _# ' )

i.

/ - 1 tJ ./ ~ J for i 1 , 2 ,3, (5.20)

(22)

I

-20-and because this must hold for all strain rates Hii must be a

linear function of these strain rates. This is true for each

of the earlier mentioned, commonly used objective stress

rates since the tensor

lH

for these stress rates is given by

I

JI-/

=

~.

ID

~.

t

~

(

iD)

1L

+

11-1

- - T

11-1 :: - /I-I

( 5 • 21)

For the Jaumann rate B=O and W=~, for the Truesdell rate 3=+1

and IH=<D, for the Cotter-Rivlin rate B=-1 and 11-/= 0 while for

the Dienes rate B=O, and I~ follows from

I

1_

H '. ::.

..!.( .\.

~

- ).. .)"

V ..

L~ 2 . I ' Lj

'v./..

, 1

= "' ..

L

\V.

n

J.

(5.22)

I

considered.In the Thesequelassociated objectiveonly tensors

IH

of the typestress rates,(5.21) are

I

I. I

I

are called B-type objective stress rates. They belong to a

subclass of the class of objective rates as defined by

(3.10). As a consequence of this restriction i t is not t r i

-vial that the rate-type constitutive equation (5.8) or the

equivalent form (5.18) can describe any kind of i~otropic

elastic behaviour at all. This can be seen in the following

way. The components of ~ have to satisfy for every

deforma-tion path the six differ~ntial equations that can be derived

from (5.18). This is impossible unless special requirements

are fulfilled. An evaluation of these requirements is given

in the remainder of this section. For simplicity only

con-stant factors B and ~ are considered. . .

Fr0m (5. 2 1) i t i s s e e n t h a t Hi i = B € i -

Y(

c1 + £ 2 + C 3) for

i=1,2,3. These main diagonal components of H must satisfy

(5.20) for every strain rate and therefore

Jcr;;

=(

/oA,+

If\cr)S ..

+.,A- -2\1'0": (5.24)

() ( . / , - L ,

J

/"

2 0 L

1

must hold for i , j = 1,2,3. Using J = det(lF)

A

₁

·A

₂

·A

₃ and

(23)

I

-21-I

I

~o ...

c'j=

',2,3 . (5.25)

For 3=+1 and 3=-1 the earlier derived stress-strain relations

(5.28) (5.27) (5.29) -1. (\ '1

G;,

A. . 6..

.. ~J ( j1..(" \

-2\\)

.0-;

=

~{Ac:

-1.~

·1)t-;:Jr:. .I

If 3 -+ 0, as is the case for the Jaumann and Dienes

rates, the solution for ~ is given by

Rivlin rate, respectively, are found again. The trivial case

GO=O is not considered in the remaining part of this section.

A preliminary conclusion, based on these results, is that

17

the rate-type constitutive equation <lJ = )11ll:>+ )12tr(lD)JI"

V

with a 3-type objective stress rate for ~ may result in a

correct description of isotropic elastic behaviour only if )11

and )12 satisfy (5.26). This condition, however, might not be

sufficient because up to now only three of the equations that

can be derived from (5.18) have been taken into account since

only the diagonal components of H have been considered. For a

correct description of isotropic elastic behaviour the

re-maining three equations in (5.18) must also be satisfied.

Using (2.11) for [) and (5.29) or (5.30) for ar this

I t is easily seen that the solution of (5.27) is given by

scalar _{quantities }l1 and )12 only depend on J and} j f there

exists a constant GO and a function

S

₌

i

(J ) with J(1)=0,

such that

1.~

~

G

)'2~

::.

J

ell

(5.26)

~,

j

= -

2 + :2 _." _~z

01]

The differential equations (5.25) then become

(5.14) and (5.16) for the Truesdell rate and the

Cotter-This set has a solution if and only if the as yet unspecified

I

1_

I

I. I

I

(24)

I

-22-I

I

results in three equations for the components

H

_{i j} (i,j=1,2,3;

-

-,.

j ";> i) 0 f the mat r i x rep res e n tat ion 0 f IH = - IH i n t h e

chosen vector basis:

Green elasticity, however, the principal stresses cr"C'2 and

03

must be derivable from an elastic potential ~ as

specified by (5.4). For a stress-strain relation of the type

(5.28) this is true only if ){ =

i .

This result can be

derived by differentiating (5.4) with respect to

E

j and

substituting in (5.24).

This must hold for every Wij. This is neither the case for

the Jaumann rate (where Hij = 0) nor for the Dienes rate

(where Hij is given by (5.22», so these rates do not result

in a correct description of isotropic elastic behaviour.

How-ever, i t is easily seen that the factor enclosed by square

brackets in (5.31) is zero if 8 2 =1. Then Hij=O for i,j=1,2,3

and according to (5.21)

IH

is equal to 8 [j) -

&'

t ..

r1t>JiI.

This

con-firms the earlier derived conclusion that the Truesdell rate

(a=+1) and the Cotter-Rivlin rate (B=-1) with the

constituti-ve equation (5.8) and ~, and ~2 according to (5.26) yield a

correct description of isotropic elastic behaviour. The

cor-responding stress-strain relation is given by (5.28). If B2

f,

and a stress-strain relation of the type (5.28) must be

re-pre sen ted cor r e c t 1Y by a rat e - t ypeequa t ion

cC-

=

~

1

TD

+);12

f ... (W)

JL ,

then a B-type objective stress rate (5.23) must be used with

a skew-symmetrical tensor

/H

as specified by (5.30).

(5.30)

For

have been

J.v..

'J

Up to now no statements on the value of ~

made. For Cauchy elasticity

0

may be arbitrary.

I

Ie

I

I. I

I

(25)

I

-23-I

6. Tension-torsion of cylindrical bars

I

To evaluate a proposed rate-type constitutive equation

physical and numerical tests have to be done. Quite popular

nowadays is the so-called simple shear test

([4),[51,[6],[8]

e t c . ) . However, this is a theoretical test and no

experimen-tal data are available for large deformations. An alternative

is the torsion test of solid circular bars or the combined

I

Ie

I

tension-torsion test, which is much easier to realize than

the simple shear test. As will be shown, the same phenomena,

like the oscillations of stresses for the Jaumann rate, also

occur in the tension-torsion test.

Let r , ~ and z be the current cylindrical coordinates

of a material point with cylindrical coordinates rO' ~O and

Zo in the reference configuration. Then the position vectors

i and Xo of this material point in the current configuration

C(t} and in the reference configuration Co are given by

I

x ( 6 • 1 )

I

I. I

where _{e z is an axial unit vector and e r} (f) is the radial

unit vector for _a point with circumferential coordinate

If •

I t is assumed that the strain field in the bar is

aXi-symme-t r i c , independent of the axial coordinate zo and symmetric

with respect to the midplane zO=O of the bar. For the current

coordinates r ,

f

and z this results in

( 6 • 3 ) ( 6.2 )

n=

of C(t) with respect to

e

+ 1'0

fF

_::.

CJr

_e

r

0'0

Co is given by

and hence the deformation tensor

Here,

elf

=

eY'

('f) is the unit vector in the circumferential

direction, i . e .

e y

= ezxer. For brevity, the vectors er(fo}

and

e,(fo}

are denoted in (6.3) by e~ and e~

,

respective-1 y. Fr0 m (6. 3) i t i s s e e n t h a t the de t e r min ant 0 f

11=

is

I

(26)

I

-24-I

equal to

I

( 6.4 )

I

T"

The left Cauchy strain tensor

1[3

follows from

In

=

TF.

iF •

After

some elementary calculations the unit eigenvectors n1' n2 and

\2 \2 t

n3 and the corresponding eigenvalues /\ 1 ' /\ 2 and )\ 3 can be

determined. This yields

( 6 •5 )

I

A -

~-where the angle ~ between e~ and n2

( 6.6 ) ( 6. 7 ) follows from C.oS 2.

t

=

J

(11'£:)."-.

d.

)1.2._)..2. 2 J

I

I'

I

I. I

I

For an isotropic elastic material the principal Cauchy

stresses ()1' <:5""2 and

03

are completely determined by

A

₁ '

>.

2

and ->-3 while the eigenvectors of the Cauchy stress tensor

coincide with the eigenvectors n1' n2 and n3 of

£B.

Using the

spectral representation (5.3) for

rr:r

and the relations (6.6)

for these eigenvectors the componentso-ij ( i , j = r , f ' ,z) of

the matrix representation of~ with respect to the vector

basis, spanned by

e

_{r ,} e~ and

e

_{z ,} can be determined. This

(27)

I

-25-I

These stresses depend on rO and t but not on zo and ~O.

Therefore, the only relevant equilibrium equation and the

corresponding boundary condition are given by

I

I·

I

( 6 • 8 ) (6.10) ( 6 • 9 )

these end planes.

in

tor

c:r::-

....

=

0

I=.

J J

R

~1'fr~rclr

o

where R is the current radius of the bar

where RO is the radius of the bar in the reference

configura-tion. The axial force N and the torque T in the end planes

zO=-IO and zQ=+IO of the bar follows from

N:=.

I

I. I

I

The given set of equations for the tension-torsion test

must be completed by the constitutive equations. Only

rate-p

-type constitutive equations of the type (5.8), i . e .

~= ~,DD

+

~2tr(ID)JI,

with a a-type objective stress rate for

~

and

with ~1 and ~2 according to (5.26), are considered. First of

all i t is assumed that the skew-symmetrical tensor

lH

in the

definition (5.23) of a-type objective stress rates satisfies

the conditions (5.31). This excludes the Jaumann rate (where

a=0 and

IH

=

a»

and the Die n e s rat e (w her e a = 0 and IJ-I f0 I low s

from (5.22». From Section 5 it follows that the

stress-strain relation is given in this case by (5.28), so

(28)

I

-26-Here, ~ =

i

(J) must be a known function of J with ~ (1) = O.

Using (6.5), (6.7), (6.8) and (6.11) i t is possible now to

derive from (6.9) and (6.10) a set of three equations for

r=r(ro,t), t.= E(t), ~= o ( t ) , N = N(t) and T=T(t). If eitherc

or N and ~or T are prescribed as functions of time the

re-maining unknowns can be solved, at least in principle.

How-ever, the equations are highly non-linear. For that reason

the equations are simplified by assuming incompressibility.

From J=1 and (6.4) i t is easily seen that the current radius

r of a material point and the radius rO of that point in the

reference configuration are related by

I

I-I

cr

_l

_{=- ]}

-7..'1 [

_~

+ (6.11)

I

r.:

(6.12)

The relations (6.5) up to (6.10) remain valid but the

stress--strain relations (6.11) have to be adjusted. I t can be shown

(Hunter [11]) that these relations must be replaced by

In Fig. 1 this stress is plotted as a function of 3 and

E

for

-2~3 ~ 2 and

-i

~ E ~ 1. It is noted again that 3=+1 and 3=-1

correspond to the Truesdell and the Cotter-Rivlin rate,

res-pectively. (6.13) (6.14) 1,2,3, for i

o:.=-p+

where the unknown function p=p(ro,t) has to be determined

from the equilibrium equation. If 3=0 i t can be proved that p

equals the hydrostatic pressure.

Two special cases are considerd in more detail. The

first case concerns pure tension of the bar. Then ~(t) = 0

and i t follows that all stresses except~z are equal to

zero. The stress O'zz turns out to be independent of rO and

is given by

I

I. I

I

(29)

I

-27-I

I

The second case concerns torsion without axial loading,

i.e.of= ~(t) is prescribed and N(t)=O. Some of the results

for this case are plotted in Fig. 2. In Fig. 2.a the relative

axial elongation is represented as a function of

a

and~for

-2:!u~1 and 1£01:!10. Fig. 2.b gives a plot of the

dimension-3

less torque T/(2rr

R

_{o GO)} as a function of the same arguments.

If experimental data for these two cases are available a

choice of the value of

a

can be made by using these figures.

In section 5 i t is concluded that neither the Jaumann

mations.

cribe elastic behaviour correctly in the case of large

defor-rate nor the Dienes rate can result in a correct description

of elastic behaviour. To illustrate this these objective

stress rates are also applied to analyse the tension-torsion

test. The derivations for the case of incompressibility are

given in Appendix A. To show that the results depend not only

and the Dienes rate cannot

des-again shows that the Jaumann

on the deformation but also on the deformation path two

cal-culations are made. In the first calculation axial

displace-ments of the end planes of the bar are suppressed (E=O) while

the bar i s t w i s ted up tool

=

eL,.

I n t h e sec0 n d c a I cuI at ion the

bar is twisted up to

0I=~,

while the axial force

N

is kept

zero. For both the Jaumann and the Dienes rate this results

in an elongation of the bar. After

01

=

C/.

has been reached the

b«r is pushed back to its original length. The final

deforma-tion is therefore character ized by

t.

= 0 and

01 =01

1 , as is the

case in the f i r s t calculation. However, the deformation paths

are different. In Fig. 3.a and Fig. 3.b the dimensionless

tor que TI (_{2 17 R.30} GO) is plot ted a s a fun c t ion 0 f

of

₁ for the

Jaumann rate and the Dienes rate, respectively. The solid

lines give the results for the first calculation while the

dotted lines represent the second calculation. Only for small

values of d,do the solid and dotted lines coincide and this

I

I. I

I

It is noted that pure torsion of the bar ( t . =0, the

f i r s t calculation) is very similar to the simple shear test.

It can be shown from the results in AppendiX A that the shear

stress cr~z in pure torsion of the bar is given by

(30)

I

Ie

I

I. I

I

-28-and the torque T are oscillating as can be seen also from

Fig. 3.a. For the Dienes rate an analytic expression can also

be derived for ~ • This expression, however, is rather

com-plicated, but very similar to the expression of the shear

(31)

I

I. I

I

-29-7. Conclusions

In this paper attention has been focussed On

constitu-\7

tive equations of the rate type (1.1), i.e.<V= 4~:lD. It

has been shown that the commonly used objective stress rates

( 7 . ) T

bel0 n g t o t h e c las s de fin e d by (3. 10 ), i .e.

err

=

0- - (.A .,.IH .,tfT - C11". (..a·d~/)

where Il-l.wt w.H-IT is an objective tensor. For a chosen

objec-tive rate IH is not uniquely determined by (3.10): if a given

tensor

IH

results in the chosen objective stress rate, then

e a c h ten s0 r

I

~I + (fT. S wit h s k e w- s y mm e t ric a1 S will res u1t i n t hat rat e. I f

/1-1

i s a n 0b j e c t i v e t ens0r, the n for eve r y

11-/

a

tensor ~ can be found which is related to an invariant stress

tensor

crT

such that

~ =

IA.

([i •

fAT

and

efT

IA.

J....

lAT. As

out-lined in Section 4, these equations can be used as the

start-ing point for the numerical integration of the rate-type

con-stitutive equation (1.1).

For special purposes, for example in elastic-plastic

problems, i t is convenient that elastic behaviour can be

cha-racterized not by a stress-strain relation like (5.1) but by

v

a rate-type equation like (]j

=

4a:::: ii), where 4([. is the

fourth-order elasticity tensor. For specified 4a::. and chosen

objec-tive stress rate the Cauchy stress tensor <CT can be determined

as soon as ~ is given as a function of time. The resulting

stress-strain relations must be independent of the

deforma-tion path. It was proved in Section 5 that this is not the

V

case if the Jaumann or the Dienes rate is chosen for a:J •

Hence, these rates do not result in a correct description of

elastic behaviour.

To describe isotropic elastic behaviour the rate-type

V

equation qy- =~1lD + ~2t,"('D)][ is used with scalar quantities

P1 and P2 that can be functions of the invariants of the left

Cauchy strain tensor 173

=

It-. {FT. From the requirement that

this rate-type equation must lead to a correct description of

isotropic elastic behaviour i t was seen that the diagonal

components of the matrix representation H of IH in the

eigen-vector space of ~ must be linear functions of the logarithmic

strain rates i . e . in the main diagonal components of the

(32)

introduc-I

I

I. I

I

-30-tion of a-type objective stress rates, i . e . stress rates of

the type (3.10) with /J-I=aID-tfr(lD )if+IH, where

iR

is a

skew-symmetrical tensor. It was shown that each of these

rates results in a correct description of one kind of

isotropic elastic behaviour, provided that

IH

satisfies some

special requirements. These requirements turned out to be

fulfilled for the Truesdell rate and the Cotter-Rivlin rate.

Furthermore i t was seen that for an isotropic Green-elastic

material the factor ~ in the definition of a-type objective

stress rates must be equal to

i .

If a a-type objective rate is used for the description

of a given isotropic elastic material the value of B must be

determined from data of large deformation experiments.

Tension-torsion tests on cylindrical bars seem to be very

(33)

I

I-I

I

I. I

I

-31-8. Final remark

The applicability of a rate-type constitutive equation

for the description of elastic behaviour is investigated in

thi.paper by requiring that this equation results in a

stress-strain relation such that the stresses depend on the

deformation but not on the deformation path. As stated in

Section 4, the constitutive equation for elastic-plastic

behaviour is of the rate type (1.1) as well. However, i f

kinematic hardening has to be taken into account a second

rate-type equation appears, namely

(8. 1)

where

a:J

is the shift or back stress tensor and

lD

P is the

plastic part of the deformation rate tensor. In order to

decide which objective rate has to be used in (8.1) a similar

physical statement as in the case of elasticity would be very

helpful. As long as this is lacking one has to tryout

different rates to find out which rate results in the best

f i t to experimental data. However, the objective rate to be

used in (8.1) will probably be a different rate from the one

(34)

I

-32-Literature 9. Lehmann, Th.

"Some theoretical considerations and experimental

re-sults concerning elastic-plastic stress-strain

rela-tions", Ingenieur-Archiv 52 (1982).

I

I. I

I

1 • 2. 3 • 4. 5 • 6. 7 • 8. Jaumann, G.

"Geschlossenes System physikalischer und chemischer

Differentialgesetze", Sitzungsber. Akad. Wiss. Wien (2a)

120 (1911).

Truesdell, C.

"The simplest rate theory of pure elasticity", Comm.

Pure Appl. Math. 8 (1955).

Cotter, B.A. and Rivlin, R.S.

"Tensors associated with time-dependent stress", Quart.

Appl. Math. 13 (1955).

Dienes, J.K.

"On the analysis of rotation and stress rate in

deform-ing bodies", Acta Mech. 32 (1979).

Nagtegaal, J.C. and de Jong, J.E.

"Some aspects of non-isotropic workhardening in finite

strain plasticity", in: Lee, E.H. and Mallett, R.L.,

eds., Plasticity of Metals at Finite Strain: Theory,

Ex-periment and Computation, Stanford, 1982.

Lee, E.H.,Mallett, R.L. and Wertheimer, T.B.

"Stress Analysis for Anisotropic Hardening in

Finite--Deformation Plasticity", J.O. Appl. Mech. 50 (1983).

Lee, E. H •

"Finite deformation effects in plasticity theory",

in: Developments in Theoretical and Applied Mechanics,

Huntsville, (1982).

Atluri, S.N.

"On constitutive relations at finite strain

hypo-elas-ticity and elasto-plashypo-elas-ticity with isotropic or kinematic

hardening", Compo Meth. in Appl. Mech. and Engng. 43

(35)

I

I-I

I

I. I

I

10 1 1 •

-33-Nagtegaal, J.C. and Veldpaus, F.E.

"Numerical analyses of forming processes"; ed. Pittman

c . s . , John Wiley and Sons, Chichester, (1984).

Hunter, S.C.

"Mechanics of continuous media", 2 nd edition, Ellis

(36)

I

-34-I

Appendix A

I

For an incompressible, isotropic elastic material the

Jaumann rate form of the constitutive equation is given by

( A. 1 )

where aJd is the deviatoric part of the Cauchy stress tensor

and

e

_z are given by

The hydrostatic pressure p is not determined by the

constitu-tive equations but has to be determined from the equilibrium

equations. The deformation rate tensor

ID

and the spin tensor

..n-

follow from their definition (2.6) and from (6.3).

( A • 2 )

p

-J err

= -

pH

~

(JJd

Taking into account that the current radius r and the

ori-ginal radius rO are related by (6.12) in the case of

incom-pressibility, i t follows that the matrix representation D and

..n

of

jj)

and..9l. , respectively in the vector basis spanned by

I

I-I

I

I. I

I

,

.

0

- 0<

2.., 0

--e

₂ 0 0 Q.

.

D=

₀ _!.

i

-y.

..S1..- ~20 0 -If) l Q _J J

.

₀

_'1-)

0 0

- ¥-]

l ~ where E a and

tf

J are defined by ( A. 3 ) (A.4 ) some and after ( A. 1 )

cl

"t

if

=-

§ ] :1

Eo..

=

]2." (

'T

E)

J t

r

~

{/+£)-Y.z to

After substitution of these results in

I

elementary calculations i t is seen that the components of the

matrix representation~d of ~d in the earlier chosen vector

basis have to satisfy

(37)

I

-35-This is a set of four differential equations for

cJ cI d 0-'"

o r r '

Off

,O'"zz and I"'z

in

. d

O;-r 0 0 ₀ ₀

0

or,.

• d

orr.

• c1 ₊ cl ,j '"

_::::2&D.

tJ

0 _l

_"';.2

_{0;1. -}

csr,.

0 -• c1 _' _q o-c1_o-ti

'"

(A.5 ) 0 o;;~ ~ 0 - 2.

";l

H "'f' J

The associated initial condition isg:-= Q for t=to, i.e.

the reference configuration. Further elaboration yields

I

I-I

I

( A. 7 ) ( A. 8 ) ( A. 6 ) o o J o c.os

'f

0 c.rr> given by

G

Eo - cs:c1 o Q. l'l. • d

O:z

=

2.

y..)

"Pz.

+ 1~ E~ cJ

0;.,..

-• eI

a:r

with an angle

Using the Dienes rate the constitutive equation for an

incompressible, isotropic elastic material is given by

where the rotation tensor

fR

follows from

If-

\V.

fR

and

\V=ViiS'.

With (2.4) and the results of Section 6

\V

and

fi<.

can

be determined. For the matrix representation R of

lK

in the

vector basis, _{spanned by e r ,} e~ and

e

_{z ,} this results in

I

I. I

I

( A. 9 )

(38)

I

-36-consequence, oscillations can appear in the stresses if the

Jaumann rate is used but not if the Dienes rate is used.

Further elaboration of (A.7), using (A.B), again yields a set

cl eI

o f e qua t ion s for the non - t r i v i a1 s t r E;sse s 0-r r ' ~l" ' C}z z a n d

~z • The resulting set is given by (A.6) if ~J is replaced

by

tD .

However, there is a very significant difference

between the solutions of these

I

I-I

I

I. I

I

that ~D i s b0un de d (-11t ~,I- !.!1T ) 2 TO 2 sets. while From (A.9) tI. is not. T) i t is As a seen

(39)

.- 31"

..OJ»

•

f

• ••

_•

lIa

1.0 ~,q.

,.

'on of

~ ~nd

/

6:

as a funct.l,. st<e ss 0;. • nsvt"t ,.e

'"'e n,._t:a.t. e t.,!pe CO

E. tot: to··

,-O!>

, l '" a)C.i.a1 d ...,ens).on e .. 'tn_e l,.'~ ' ' I e e1.onQa.t.i.on tone r e 1. a t.l,. , "e toens).on..