On mathematical forms for the material functions in nonlinear viscoelasticity

.I. Mech. Phys. Solids, 1969, Vol. 17, pp. 339 to 358. Pergan~on Press. Printed in Great Britain.

ON

RIATHENATICAL

FIJNCTIOXS

IN

FOEIRIS FOR

NONLINEAR

THE

RiATEHIAL

VISCXXtLASTICITY

of h’unwrical and Applied ~~at~ler!~at~~, National Physical Labor&orb Ted(iin~oI~, Middlesex

Division

SUMMARY

‘I’rrm~ special fortus of material functions are introduced into the Green-Kivlin constitutivc Two of these reduce the third-order equations to single equations of nonlinear viscoelasticity. integral forms, rationally and e~~erill~ental~y derived by others. Pub~sl~ed creep and stressrelaxation data are analysed using bot.h tbc proposed and previously published constitutire equations, particular attention being given to multi-step data. Specific results &ow the 1inGt.sof

applicability of the third-order constitutivc relations for the Inaterials considered.

IlGi{lX<: the last decade

a number

of general

f~~r~r~~~lat~d to represent the mechanical In particular, as those

the rational

derived

by

capable of representing

behaviour

constitutivc

relations

have

of nonlinear viscoelastic

theories listed by TRUESIBLL and I?OX,L (1~~5),

SCXIABXY

(1966)

from

irreversible

been

materials. as well

thermodynamics,

are

it large variety of real materials under quite general deforma-

tions. However, these gcncral theories are too complicated for practical applicatious; even the determination of material parameters may be beyond present testing procedures.

Thus,

~~pplicabilit~.

simplification

A number

or specialization

of writers haw

IICTANGand LEE (1966) have developed PIIWN mental

is a prerequisite

given some attention forms applicable

to practical

to simpli~catiol~s.

to very short times, and

and Rot:m~ (1968) developed an integral series arranged so that experdata may be incorporated directly. The present paper is intended as an

additional reducillg

relaxation

step in the area of simplifications. the

evaluation

tests.

of it eonstitutive

of material

Additional equation

functions

tests arc considered

Particular

the isothermal

is directed

to

creep or stress-

which partly determine

to predict general deformations

We shall consider primarily

attention

to simple (one-step)

the ability

of a specific material.

theory of (initially)

isotropic

simple

materials in the explicit form of a third-order integral polynomial. This formulation is chosen because it in&ides the effects of physical and geometric nonlinearity in a general fashion, and is the lowest order nonlinear theory applicable to general Recently these integral polynomial equations have receivccl condeformations. siderable attention. LOC~ETT (1965) has shown that in principle all twelve material functions can be determined by experiment, but LIFSHXTZ aud KOLSKY (1966) have found these tests exceedingly difftcult to perform accurately. a39

forms for the material functions in nonlinear viscoelasticity

On mathematical

M,

= P (T&

dot denotes

$1 and $3 are functions

Tab = tr (M, MB),

T, = tr (Ma),

C/T(N) = &I d~2 . . . dTN,

Tafi7, = tr (M, elf, J&,), where the superposed

:I41

a material

of one variable

time derivative.

Material

sl; #3, . . . , $6 are functions

functions

of ~1, and sg,

$7, . . * 9 412 arc functions of ~1, s2 and 83; where sa = t - TV. It has been shown by LOCI~TT (1965) that the following symmetries may be assunled without loss of generality

: *3, *4, $6, $9, $10 = s (1,2), $7 = 80, %3), ('2.4) $8, $11

where s (i, j) denotes

=

3),

sp,

symmetry

412 =

s (1,

in the ith and jth

3),

I

arguments.

is given by defining P = I and Q = E; and a stress given by setting P = E and Q = I:. In a one-dimensional and the displacement to

Q =

I

tJ (.q)

Zjl d

T(~)

deformation

which depends

formulation

on one spatial

vector is in the CT-direction, the constitutive

is

coordinate

equation

X,

reduces

h' (81,s2)kl p2 &2)

+

--m

t

-cc

+ Note that Lagraugian

A creep formulation

relaxation

and classical

us

L (Sl, S2,S3)p1 p2 p3 +3,.

(2.5)

-cc

strain

arc related

by

E = E + i-G’, and either may be used in (2.5); of course, the material functions are different each case, and a third-order theory in E is not identical with one in E. In addition developed

to (2.5)

we will use one other

by PIPKIN (1964). By employing

change, he developed a four-function sible materials. The two alternative

simplification

a geometrical

stress-relaxation forms arc

of (2.3),

condition

formulation

in

originally

of zero volume for incompres-

RT aR FT aF

$2 M1

&l)

+

$6 M1 Mz d7<2)

/i

--m

+

t

sss

(#loT12 M3 -I-$12

MI

Mr M3)d7(3).

(2.6)

--00

The material

functions

are different

for each form, and

P = E, where s and p denote deviatoric

a = S + PI,

stress and hydrostatic

pressure.

In a recent paper, LOCKETT and STAFFORD (1969) have shown that an equation similar to (2.6) could be derived for the creep formulation, and the conditions of plane strain and plane stress reduced these constitutive equations to relations involving only three material functions. Experimental programs for evaluating the ’ plane ’ equations were described: however, the number of tests required

01~r~lathw~aticalfwns for 11~nlntcrialfunctions ill nollli1learviscoelasticitg

$4 (cr.v) =fi (4 H (!/ - 4 +.fi (y) IZ(2’ *j

y),

(X) II (!/ - X) II (Z -- X) 4-5 (r/) Ii (X -

(J’, ,//, 2) =.fi

,/,/)ZZ (2 - 9)

-tf,(z)zz(m-X)ZZ(?J-z). Ikpm’kwr

OII

J.

y.

. . .

I

(3.2)

has I~cn given the same functioual form in order that the

v’,lsl~o~rltlIw continuous at ,7’= jj = 2 r forms.

:1.1:1

. . . ; this rcsldts in completely symmetric

ZZ dcuotcs the Hea\-isidc step function

J
superposition

theory

FIKIIII.ET and LAI (1067) and LAI and FINDLEY (1968) a type of nonlinear

c.splicitl!~ tlclinc the material functions

superposition.

as in (XT),

Although

they did not

their resultant equations

are

c.q~~i~*nlcnt. .\ special form of (3.L’), which is related to strain-dependent

modulus theory,

is ol)t:rined by assuming that certain functions fi are related.

For compressible

Irlatwinls. *,li $, (z),

i _= 3, 4, 7, 8,

.fi (L?)mmzdi $2(X), i -mlli, 9, IO, 1

A,&- (x),

,i =

11,

wltere the constants Ai arc found from expcrimcnt. ~)mc simplification

is obtained from the introduction

(3.3) 1

For iucompressible materials, of the condition of no volume

change (see LOCKETT and STAFFORT),1969).

:s.:s z’wtluct

rrollzilrcul.it~/

Finally, we discuss one form which may be considcrcd to be an analytic approxinrat,ion to the material functions.

In view of the one-step test requirement,

we

sl~ppose that second- and higher-order material functions are separable and exprcssibl~ as the products

whew flmct,ions f,(j) are related by the symmetry c+c*. IAI

conditions (‘2.4), e.g. &(l) = f6(2),

and FIXDLEY (1965) and FINBLKY and ONARAN (1968) have proposed

~‘or~rlssimilar to (XLI). However, they considered only completel>. symmetric forms, an unnecessary limitation in three dimensions. The effect of this approximation cannot be evaluated quantitatively, as none of the second- or higher-order terms have ever been completely determined. Qualitati\-cly, there is experimental evidence to support the assumption that # can be represented by a series. Then (3..L) is an approximation to # by a product of wricbs. and this is acceptable only when I/ contains a wry small number of terms,

On mathematical

forms for the material functions in nonlinear viscoelasticity

345

A4 linear recovery may be acceptable for special materials or small inputs, but, generally, nonlinear materials will ha\-e a nonlinear recovery. Although one-step tests may be sufficient for determining the material functions, it is well known that such tests are incapable of assessing the degree of applicability of a specific constitutive equation to general deformations. This is readily illustrated by result (3.8), as one may use the functions determined therefrom in any of the three constitutive equations (3.5), (3.6) or (3.7). H ence, the selection of a constitutivc equation must be guided by additional information, and WC believe that a nndtiple-step test cm provide criteria for selecting a constitutive equation. PIPKIX and ROGERS (1068) observed that multi-step inputs are a more critical test of a constitutive equation’s ability to describe the response, particularly in comparison with constant rate inputs. A brief inspection of the limited multi-step data available (PIPKIN and ROWERS,1963; MCGUIRTand LIANIS, 1967; ZAPAS and CHAFT, 1965; and FIXULEY and LAI, 1966) shows that in general both creep and stress-relaxation equations predict a lower response than measured. This seems particularly marked in unloading tests which are shown in Figs. 1 and 2. Hence, it is desirable to be able to predict, from a knowledge of the material functions, which type of constituti\-e equation will give the better (and usually larger) response prediction. Since we shall test the constitutive equations using step inputs, these are applied to the one-dimensional equations to obtain algebraic equations from which inequalities may be readily established. Let PRODi, SUPi and LINi denote respectively the product nonlinear form (3.7), the superposition form (3.6) and physically linear form (3.5), where the subscript i = 2, 3 denotes a second- or third-order term. The comparison of SUP with PROD and LIN gives the same result for both second and third-order terms and is independent of the sign of the material function:

(SUPa - PROD%) (SUP* - LINf) The comparison of PROD with LIN varies with order:

Here, sgn (P,+i -- I’%) is the sigrl of the difference in two steps. The statement (3.10) is not completely independent of ft, but holds (for all materials investigated) provided that IPn+il > 6.63 IPnl. It should b e remembered that the fully nonlinear equations (2.5) will (in principle) follow the first three steps exactly, and thereafter may be expected to produce a better prediction than any of the approximations. This conclusion was partially confirmed by NEIS aud SACK&IAN (1967) in the onedimensional seven-step creep tests on polyethylene. The term sgn (I’,+1 - Plz) makes the inequalities dependent on the input. This implies that if one approximation is best for decreasing loads another may be best for increasing loads. Unfortunately, there are insufficient data to test this conclusion. Hence, these inequalities can only serve as a guide; however, it seems that a large step followed by several unloading steps is a more critical test than several loading steps.

On mathematical To

provide

347

forms for the material functions in nonlinear viscoclasticity

an additional

comparison

tieri\-ed from the thermodynamics

we include

of irreversible

a one-dimensional

processes.

Following

equation

M. A. Biot’s

linear analysis. SCIIAPERY (1966) has developed explicit equations for nonlinear mntcrials wherein the physical nonlinearity is contained in a ‘ reduced time.’ whcrc this reduction is an implicit function of strain. In a recent paper, Scrr.4rElt\i (19GH)

has csonsidcred a more general

depend

on flmctions

strain.

Hence,

entropy

production

of stress as well as (his previously

for a stress-relaxation

formulat5on

inequality considered)

which

ma>

functions

one may choose

timr: to be a function of strain. and for the creep formulation choose a function stress. Schaprry’s oike-dimensional c73~p formulntious may be written as

whcrc U is the viscoelastic vcnicnt

of

t

t

that creep and recorery

of

the rcduccd

component of the linear creep compliancc~. It is probable data for several stress levels and times are the most co11-

data for c\-nlllatir\g functions

11~;. AD and f(u)

as well as CL(u) and 0.

Howe\-er, to enable cbomparison with the previous results, a special form of (4.8) is chosen which can bc evaluated from onc-step tests. The response to step input o = ergZZ (t) is E =

(7 (00)

+f(uO)

n

[t

,lG

(go)/--ID

(WI)]

gzio)

(.L!J)

.

(4.10) IGluation

(4.8) can take

’ linear ’ equation equation

like (3.6).

most successful nonlinearity

011

a number

of forms:

setting

do

like (3.5), and f = 1 with ilc: :-= AD produces WC are interested

in forms employing

form (for the data investigated)

in the reduced

time.

Thus, letJ(u)

2: _I(; L-m1 yields

;L

a ‘ superposition

reduced

is that which includes

all of the

= .Jc (u) = 1; hence, (1.11)

l’lic rcsponsc

to multi-step

loading

’

time, and the

(4.4) is

When (4.12) is unsatisfactory, the next level of approximation is to include citherf (u) or rlc: (u) ; the latter choice gives somewhat better results. To be specific, let one-step tests define the relationship between Jo and &, and then express _4~ as a poIynomia1. viz. 1 -;- 1;~ CT-I- liy u2 / . . . . IIencc, :I linear _-I(: is tlcfinctl I);\,one two-step test, a quadratic . la by out: three-step test. :u1d so forth. Since this

-

--

fSffP

_

1.“

0.1

*i

- ---

.____ -----_ t

-.-.

__ ___...

‘PROD

_ _..._ _._I._-.__.._._

. . _._ _-

t.0

0.5

Time

(hours

.__----_.

-.

1.5

)

-- .-

-.----.

:MJ

On ntatlkcmaticalforms for tile matcrinl functions in nonlincnr viscoelasticity

The predictions

for the FOAM

rtf t.he constitutire

equations

material

follow

are shown in Fig. I, and clearly none

the data

in a satisfactory

manner.

From

inequalities (4.18) the superposition theory is superior to the others, and this is readily observed. The linear form is not shown as it was almost uniformly 98 per ceut of the product

form.

The reduced-time

equation

(4.12) gave a good prediction

of the second step.

However, the error increased T’ery rapidly to 04 per cent strain Evidently the function & is needed to characterize this step.

at the fourth mnterial; a quadratic

_,Ic, was found to give rxcellent

results.

(Yostrain 049 0.87

-__ ____-

hrs

-._.

2cte3.5

~~ ____

-__L^___--”

0.25 T‘

_.~_.____

0

0.5

lo-

PIG. 3. Creep of polyvinylcllloride

I.5

(VINDLIZY

amI

Time

LAI, 1066).

(hours)

Y

b

On nmthcmatic:d

fornrs for 11x2 nletwial

functions

in nonlinear

351

riscoclssticity

This result can be used for comparison with an equation like (3.7) is obtained. Ii\-c tliflcrcnt two-step tests rclportcd l)y T,ifshitz am1 Kolsky; typical results are ~11c~wnin Fig. A

/

Data: A-+-

gc507.2

H(t)+ZbO*SH

B-A-

a-353.5

H (t1+414

(t-l

1

H (t-60 1

Predictions: A

b

Product form Superposition

-.-

-.-‘I

50

where

approximation

second-step,

t(min1

the data exactly except at very long times, When the initial step was increased to

were significantly

this error was a constant

and shifting

I

I50

to appear.

507.2 psi, all of the predictions Fig. 4. However,

I

I

of the data which has an initial step of 353.5 psi.

followed

a small error begins

I

100

Curve U of Fig. 4 is typical The product

I

I

I

0

the curves upward

low, as shown

by curve A of

0.74 per cent strain along the entire yields very good agreement.

Hence,

this error seems to be of an elastic nature, and therefore it does not inralidatc

the

assumption

the

product

of product

for the viscoelastic

response.

Evidently,

form is valid for some materials over a fairly wide range of time.

The question is partly yields

separability

of which approximation

answered

by the previously

Second-order SUP > PROD

terms > LIN

will give the best fit to multi-step

derived

inequalities.

Third-order PROD

data

Using Fig. 3 in (4.13)

terms

> LIN > SUP

(.E.lO)

Thus, we know a pGo~i that only the superposition form can improve 011 the However, the third-order term dominates, particularly at high product form. strcsscs. and the sllperposition prediction is observed to have a significant error.

‘/I

“BKZ

.\”

1 “,I

,\

.v

1,

I1 (A. t)

!,::

.v_!‘,\:.

7. (I) (A’

i,:x

1)

.\.:,‘A

/-: (!)/A : ;.’ (I).

(.-I.1

1

(.i.(i /

On matlxn7at~ical forms for the nlatcrial functions in nonlinear viscoclasticity

u

-------

h2---J/h

++$lo(tf,f)[(A” - J)2 +2 (;-- 1)2]

&I/,~@) + t z&(t,t)

x4 -

SP

+

x +

R -

;

+

;

.

(5.8)

i Since

(5.8)

is restricted

to small finite strains, we write h==l+E,

and expand

to coincide

the terms in the equations

in powers of F. Equation

with (5.5) and (5.6) to third order in E by choosing

(5.8) can be made

the following

defini-

tions for the *’ s:

MCGUIRT and LIAXIS (1967)

t $2 i

240

The error induced

241

B+Y

2GL- /3 - 5/6(~(- ,fI)

‘Go

5/3(40 + 2@0) - 213 (240 + 7Go)

B $10 B $12

for one-step

+ -

$2

ZAPAS and CRAFT (1965)

by ignorillg

fourth

2/3(~ - B)

and higher terms can be readily

estimated

tests : [O -

CWILI< 12.7 40~~;

(-

JP ~~1,

(5.9a)

(5.9b) The question of multi-&p loads is an open one. Neither of the papers presented more than two-step data, none of which was in the range of strains supposedly applicableto

(.5.7). ~q~~ations(3.9)a~~d(.3.10)mayhe~~sedtocleterminewhichapprolrimatio~~

c 0 _ I 0lU

.5 I 393 j

I 0 0

lb

>4i

EC Loading

130 program

rho

On nmt~lttrenratical forms for the material functions in nonlinear viscoelasticity

2,8 2.4 2.0 n

-

sup

----

Lin

----

Prod

--A---

Z.rC!

I.4 I.3 /

200
I.2

<2so

F’i-,

l,Ol 50

0

too

Loading

(minutes) FIO 0. Stress-relasntion

of 1’IR

(ZAPAS

and

CRAIT,

1905).

150

program

200 min

On mathematical

forms for the material functions in nonlinear viscoelasticity

357

It has been noted by Locr~~r and STAFFORU (1969) that when the material functions arc completely symmetric only uniaxial tests are required. Thus, only the superposition form can be determined from uniaxial tests, while the product and linear forms require biaxial tests. In a recent paper, PIPKIN and ROGERS (1968) proposed an integral series representation of a different form. The first term represents exactly the response to single-step inputs, the second term generalizes the exact representation to twostep inputs, and so forth. By noting that some multi-step data (see Pig. 2) was well represented by the first term of their series, they concluded that their form was preferable to (2.3); their first term being

El(t)

a, Cl

=

{u(T),

t -

T>,

(‘3.1)

0 where d, C s (~C/JU) ti (T) C/Tfor continuous loading histories. Although the term (6.1) itself is not exactly a superposition form, nearly all published one-step data will force it to become one, as the usual experimental result seems to be a separable function of time and input, like (4.1), (5.5) and (5.6). Hence, the superposition curves in Figs. 1 and 2 are identical to the results reported by Pipkin and Rogers. Their second-order function C2 (~1, t - tl, ~2, t - tz) requires the evaluation of a function of four variables. However, LIFSHITZ and KOI,SI(Y (1966) were unable to evaluate a function of two variables K (tl, tg); hence, the complete evaluation of Cz appears impractical. Also, Lifshitz and Kolsky’s results indicate that the two-step response is a separable function of time(s) and strain(s). If this separability were to hold for all step-tests, the constitutive equation would again reduce to a superposition form. The demonstration that the incompressible third order theory, equation (2.6), can fit the forms derived by Lianis, and by Zapas and Craft, is not particularly meaningful. It may be argued that almost any relation will fit one- or two-step data for sufficiently small stresses and strains. Rather, this should be viewed as providing some justification for the use of 4-function and 3-function theories of LOCKETTand STAFFORD(1969) to describe incompressible three- and two-dimensional deformations. The final criteria of applicability should include multi-step tests, and on the basis of Section 5 one may expect the superposition form to yield more accurate predictions. An important question is how to proceed when none of the equations derivable from one-step tests will follow multi-step data. No complete answer can be proposed, but it appears that the next level of sophistication would be the rcducedtime equations like (4.8) developed by Schapery. Here, evaluation of, say, AG, requires only one two-step test, one three-step test, etc., while an integral polynomial theory requires at least YLtwo-step tests, 44 three-step tests, etc. For example, we found that choosing & to be linear in u gave excellent results for PVC. However, it must be remembered that reduced-time equations will only follow data in which the functional form of the response is unchanged by additional steps. If it does change, one must resort to the more general integral polynomial theories. Although the applicability of the various one-dimensional constitutive relations has been discussed in detail, there remain open a number of questions in the general

I !I(; I 1 !)I;.; I !Ifi(i

I !JCii I !)lih

t !I(iii

I !I57 I!lfili t !lliS

1 !lliT

1!J(i.L

%Al~‘\S, I,. .T. nlltl I’ll \l”I’.

‘I’.

1 !lli.i