Towards the Characterization of Viscoelastic MaterialsUsing Model-Based Reasoning Techniques

S.M.F.D . Syed Mustapha, T. N . Phillips and C. J . PriceUniversity of Wales, Aberystwyth, UK

sfs95,tnp,cjp@aber .ac.uk

Abstract

A rheological equation of state is required to sim-ulate the behaviour of a viscoelastic material inmany complicated industrial flow processes . Thegeneration of such a model is not a straightfor-ward operation . In this paper we show how thisprocess may be automated using model-basedreasoning techniques . Data from three simple ex-periments are required in our analysis in order tocompletely characterize a material and formulateits equation of state .This paper builds on the work in (Capelo, Ironi &Tentoni 1992 ; 1993) who have presented a systemwhich selects a mathematical model (the consti-tutive equation) for a given viscoelastic materialfrom its response to an externally applied forcein a so-called creep-test . This process is based ona qualitative abstraction of the plots generatedfrom this static experiment .We extend this work to fully characterize the ma-terial in terms of its rheological constants andfunctions . The viscosity is then determined froma steady simple shear experiment in which thedata is analyzed using a graph recognition tech-nique . The relaxation and retardation times aredetermined using data from a dynamic experi-ment . An optimization technique is used to de-terinine the number of relaxation and retardationtimes and the values of these parameters . Thissystem delivers to the user a constitutive relation-ship with material constants and functions whichmay be used to simulate the flow of the viscoelas-tic material under consideration .

IntroductionAn understanding of the behaviour and properties ofviscoelastic materials, whose ranks include materialssuch as paints, shampoo, polymers and lubricants, isimportant in many industrial processes . Many im-portant and industrially relevant questions remainunanswered . For example, in journal bearing lubri-cation, can normal stresses induced by viscoelasticitycompensate for the adverse effects of shear-thinning,

namely higher wear resulting from lower friction . Theformulation of a realistic viscoelastic model enablesthe behaviour of a given lubricant to be predicted bymeans of numerical simulation . The mathematical de-scription of a viscoelastic fluid is much more complexthan its Newtonian counterparts . In addition to theconservation equations of mass and momentum an ad-ditional equation, the constitutive equation or rheo-logical equation of state, is required which relates thestress to the deformation . For a viscoelastic liquid thisrelationship is nonlinear and it has no standard formwhich is universally valid for each fluid in every flowsituation . This situation is one of the reasons why thesubject. of viscoelasticity is so challenging .A constitutive equation is required in order to per-

form numerical simulations of viscoelastic flows. Al-though some information can be gleaned from experi-ments a complete description can only be obtained bysolving the full set of governing equations . This en-ables local characteristics such as information aboutthe stresses and the energy dissipation, for example,to be calculated . A constitutive model which is simpleenough to allow efficient numerical solution yet pos-sessing predictive capabilities needs to formulated .

In practice the constitutive equation is formulatedby appealing to rheometrical experiments . The be-haviour of the material is investigated in simple flowssuch as steady simple shear flow and small amplitudeoscillatory shear flow . Then a model, which is ableto simulate the behaviour of the fluid in these simpleflows, is chosen from among those that are available .The work in this paper automates much of this processgenerating a complete constitutive equation directlyfrom experimental observations .

In this paper we present an intelligent hybrid sys-tem which automates both the selection of the formof the constitutive equation and the determination ofthe material constants and functions which appearin this equation . In (Capelo, Ironi & Tentoni 1992;1993) a system is presented which selects a mathe-matical model (the constitutive equation) for a givenviscoelastic material from its response to an externallyapplied force in a so-called creep-test . This processis based on a qualitative abstraction of the graphicaloutput generated from this static experiment . Their

analysis shows that there are four physically feasibleclasses of models. Each class contains a family ofmod-els each member being identified by the number of re-laxation times of the material . At this stage the workof calculating the material functions and parametersof the constitutive equation (1) and also automatingthe process of determining the number of relaxationor retardation times (constant terms on the left andright. hand side, respectively) has not yet been done .

Therefore, in this paper we extend their work bybuilding an intelligent hybrid system which can fullycharacterize the material in terms of its rheologicalconstants and functions. A selection of rheometricalexperiments are necessary to provide data for this pro-cedure . Once the equivalence class for the materialhas been identified the material constants and func-tions may be determined . The viscosity is determinedfrom a steady simple shear (viscometric) experimentin which the data is analyzed using a graph recogni-tion technique developed by (Mustapha et al 1997) .The relaxation and retardation times are determinedusing data from a dynamic experiment . An optimiza-tion technique is used to determine the number of re-laxation and retardation times and the values of theseparameters . This system delivers to the user a con-stitutive relationship complete with values of the ma-terial constants and functions which may be used tosimulate the flow of the viscoelastic material underconsideration.

Formulation of a Rheological Modelfrom a General Constitutive EquationGood progress towards the formulation of a rheolog-ical model from given experimental data has beenmade by (Capelo, Ironi & Tentoni 1991 ; 1992 ; 1993 ;1995) . There are four basic processes which enablethem to formulate a general rheological equation ofstate . These are summarized in Figure 1 .A static creep test experiment is performed on a

given viscoelastic material and the strain responseis analyzed . This analysis is performed using graphrecognition techniques . In (Capelo, Ironi & Tentoni1993) this information is used to assign to this mate-rial one of four formal admissible constitutive equa-tions: n

aiD'o- = E OiD'-~,

(1)i=1 i=6

where n = m, m + 1, b = 0, 1, ai, i

7n, Oi,i =6,...,,n

are material functions and D is the timederivative operator . The function 01 is the viscosityand we will also denote this by ?7 . All the other ma-terial functions in (1) are assumed to be constant .

Process I shows a segmentation of the strain re-sponse based on several successive steps of stress im-posed on the material . Therefore the segmented com-ponents are quite clear. Excellent descriptions for

298

each of the components can be found in (Whorloa-1980). Using first and second order derivatives, thequalitative curve attributes can be obtained (Capelo,Ironi & Tentoni 1995) . . A full qualitative descrip-tion such as instantaneous, delayed elasticity and non-recoverable deformation can be obtained by applyingheuristic rules as shown in Process II . Consequently, inProcess III, if the existence of certain properties suchas instantaneous, delayed elasticity and full recovery(non-viscosity) are realised, then the value of the logi-cal triplet for this example is (T,T,F) . The other threepossibilities are (F,T,T), (F,T,F) or (T,T,T) . ProcessIV shows that a direct mapping to an equivalence classis a straightforward task once the logical triplet is ob-tained .The two major drawbacks in the procedure de-

scribed above are : i) the number of relaxation or re-tardation times are manually input from user and ii)the material parameters or functions such as the vis-cosity, relaxation and retardation times are unavail-able . To overcome these difficulties, we propose twofurther experiments to fully characterize the material .Thus, three experiments in all will suffice to determinea constitutive equation for a given viscoelastic mate-rial . The interaction between the output from thesethree experiments forms the basis of the intelligenthybrid system shown in figure 2. The additional ex-periments are:

1 . Dynamic experiment : Perform a small-amplitudeoscillatory shear experiment . The outputs from thisexperiment are the storage and loss moduli denotedby G' and G" respectively . The discrete relaxationspectra can be determined from this data .

2 . Viscometric experiment : If required, run a steadysimple shear experiment to determine the shear vis-cosity.

The details which show the output of the viscomet-ric data analyser (calculating viscosity) can be foundin (Mustapha 1995 ; Mustapha et al 1997) . Under theassumptions that the fluid is a linear viscoelastic liq-uid, the flow conditions are isothermal and extensionalviscosity is considered small, the system shown in fig-ure 2 is a complete one. Section 3 will explain in de-tail the calculation of a viscosity function from dataobtained from a simple steady shear experiment . Sec-tion 4 shows how the discrete relaxation spectrum ofa material may be calculated .

Determination of the Viscosity

Function

The most important material property which needsto be determined is the viscosity. The flow of thematerial in many situations can be predicted froma knowledge of this function . The viscosity can de-pend on variables such as shear rate, temperature

and pressure . However in many industrial and ev-ervdav situations it is the variation of viscosity withshear

_rate which is the most important. Therefore

in this paper we restrict ourselves to this situation.Several models are available for the purpose of fittingthe viscosity-shear rate data and each of them has itsown capabilities in characterising different shapes ofthe flow curve. In the system developed by (Mustapha1995 : hlustapha et al 1997), six models are used . Theprocess of determining several models on a piecewiserheogram is found in two stages :-

1 . Explanation process (qualitative technique) : It isimportant to explain to the user the meaning of thecurve in order to help choose the piecewise curve (upor down curve) and the range of the curve chosen(initial or endmost or whole part of the curve) . Themeaning of the mathematical models are also neces-sary since the user needs to know why and when touse them . In additional to these there are warningmessages for graph abnormalities and expert adviceto repeat. the experiment using the proper geome-try. Since only a qualitative description is requiredto fulfil these tasks, qualitative reasoning techniquesare the most appropriate.

2 . Curve-fitting process (quantitative technique) : Theobjective of this curve-fitting process is to find thebest model/s and also to minimise the number ofmodels used in characterising the selected curve.Frequently there will be no single model which is ca-pable of characterizing the material over the wholeshear rate range. The criteria for selecting the mod-els are as follows :-

(a) The model must cover the largest shear raterange. This is important in minimizing the num-ber of models used .

(b) The model must produce the highest correlationamong all other models or at least above 0.9975 .The correlation is obtained by Pearson's method(Weiss &!Hassett 1991) .

The six'iriodels'~~"ently used in the system arelisted in Table 1 where T is the shear stress and 7.is the shear rate . Figire 3 illustrates how severalmodels can.be; assigned to aflow curve in a piecewisesense.

Determination of the DiscreteRelaxation Spectrum

Suppose that following the creep test we have iden-tified which of the four admissable constitutive equa-tions describes a given viscoelastic material . In thisgeneral equation the number of relaxation times (thevalue m in (1)) needs to be determined in order to re-duce the number of plausible models further. Ideallywe would like to describe the rheological behaviour of

the material with as few parameters as possible fortwo important reasons. First of all to reduce the testof numerical simulations and secondly to avoid the ill-posed problems associated with the determination ofthe discrete relaxation spectra of a viscoelastic mate-rial. This is well-known to be an ill-posed problem inwhich the degree of ill-posedness increases as the num-ber of relaxation times increases (Honerkamp 1989) .The relaxation modulus 0(t), defined by

may be measured using dynamic mechanical methodsin which the material is strained sinusoidally at a fre-quency w. The measured stress response decomposesinto an in-phase and an out-of-phase component: thestorage modulus G' and the loss modulus G" . In os-cillatory shear we define a complex shear modulus G'through the equation

The complex shear modulus, G' (w), has the represen-tation

'~ .7(Lo) _

o(s) exp(-iws)ds

m

iwg~=

fo"o

F (1 + iwAi= i J) "

We write

where

b(t) = E gi exp(-t/.\j),

t > 0,j=i

G' (w) = G'(w) + iG"(w),

G'(w)

G" (w)

G"

m

a(t) = G*(L.,)-),'t)_

mg.,A?w2

E (1 + c.w2A?)'.i-im

9.iAjL';E (1 + w2A2)'j=1

Experimental data provides us with information atc.w = wi, i = 1, . . . , .'11 . Let us denote this by G; andG,', i = 1 . . . . . . 11, respectively. This results in thefollowing system of equations for g.t, Aj , j = 1' . . .'M:

m9jAjw2

(1 +w=A?)

m97Aiwi

(1 +w?A?)

for i = 1, . . . , M. This is a system of 2M equations for27n unknowns if m is known. However, the value of,in is not known priori and must be determined alongwith the other parameters to avoid unreliable numeri-cal simulations at a later stage. The nonlinear natureof this system of equations makes it extremely diffi-cult to solve. One has to resort to a nonlinear leastsquares method in order to fit the data .

We consider the minimization of the function2

m

1=

E

Gi -E hi,jgj

(0~)2i-1 j-1

=

2of m

1

G i,- E hiljgj

To,'i)2

,

i-1 j-1

with respect to rn, .1j and gj, j = 1, . . . , rn, where

,2hi,j

hij =

j-1 (1+,..iA~)~

The measured values Gt and G are assumed to beaffected by standard errors of size v; and a , respec-tively . The algebraic equations resulting from theleast squares process are solved using the Marquardt-Levenberg algorithm. This turns out to be a highlyefficient numerical algorithm for this problem. TheMarquardt-Levenberg algorithm is a standard opti-mization procedure although to our knowledge it hasnot been used previously in this context.The number of relaxation modes is increased gradu-

ally until there is no improvement in the minimizationof t . There are a number of criteria which may beapplied in order to terminate the procedure before theproblem becomes ill-posed . One termination criterionis to stop when negative values of the parameters ap-pear since these are physically infeasible . However,realising that this criterion is insufficient, one of thefuture tasks is to work out a more robust terminationcriterion .

Table 2 shows a sample 6-mode relaxation spec-trum generated for a polymer melt and a decrease ofstandard errors as the number of modes (i) increases.The corresponding result in Table 2 can be showngraphically in Figure 4. Given a discrete relaxationspectrum the corresponding discrete retardation spec-trum may be determined using the Laplace transform(Weiss & Hassett 1991) .

ConclusionsWe have built on the work in (Capelo, Ironi & Ten-toni 1991 ; 1992 ; 1993 ; 1995) and used data from ad-ditional experiments in order to fully characterize agiven viscoelastic material. The viscosity is deter-mined from asteady, simple.shear experiment in whichthe data is interpreted using qualitative and quanti-tative techniques . The discrete relaxation spectrum isdetermined from a small-amplitude oscillatory shearexperiment . The extraction of the relaxation spec-trum from this data is an ill-posed problem . A nonlin-ear least squares method is proposed for circumvent-ing the problems associated with ill-posedness. This

300

method also determines the appropriate number of re-laxation times for the material .One limitation of this work is that we have assumed

that viscosity is a function oL,shear rate only.practice it can also depend on pressure and in a non-isothermal setting, temperature.The methods described in this paper have provided

good results from rheological experiments with a va-riety of substances .

References

A. C. Capelo, L. Ironi, and S. Tentoni. A model-based system for the classification and analysis ofmaterials. Intelligent Systems Engineering Autumn,2(3) :145-158, 1993 .

A.C . Capelo, L. Ironi, and S. Tentoni. An algorithmfor the automated generation of rheological models .Applications of Artificial Intelligence in EngineeringVI, pages 963-979, 1991 .

A.C . Capelo, L. Ironi, and S . Tentoni. A qualitativesimulation algorithm for the rheological behaviour ofvisco-elastic materials. Applications of Artificial In-telligence in Engineering VII, pages 1117-1130, 1992 .

A.C . Capelo, L. Ironi, and S. Tentoni. Automaticselection of an accurate model of a visco-elastic ma-terial . Ninth International Workshop on Qualita-tive Reasoning about Physical Systems, 2(3) :145-158,1995 .

J. Honerkamp. Ill-posed problems in rheology . Rheol .Acta, 28:363-371, 1989 .

S . M . F . D . Syed h-lustapha . Rheometer advisory sys-tem. Master's thesis, University of Wales, Swansea,1995 .

S . M. F. D . Syed Mustapha, L . Moseley, T. E. R.Jones, T. N . Phillips, and C. J . Price. Viscometricflow interpretation using qualitative and quantitativetechniques . Technical report, University of Wales,1997 .

N.A . Weiss and M.J . Hassett . Introductory Statistics.Addison-Wesley Publishing Co ., New York, 1991 .

R. Whorlow. Rheological Techniques . Ellis Horwood,Chichester, 1980 .

Table l : Viscosity models

Table 2 : Relaxation spectrum for a polymer melt

Viscosity model Equations Meaning of model parameters

Newtonian model no = rjo -- viscosity at zero-shearBingham model rt = + rlp r)p - plastic viscosity

7v - yield stressSisko model 7 _ 77. + kyn'i rl - viscosity

- infinite-rate viscosityk - viscosity coefficient or consistencyn - rate index

Casson model rli/z -_ 'Y1/27'v/2 + ky k - consistency or viscosity coefficient7y - yield stress

Power Law 77 =W-1 k - consistency or viscosity coefficientn - Power Law index

Herschel-Bulklev rl = } + k~"-1 T, - yield stressk - consistency or viscosity coefficientn - Power Law index

i Ai gi U

1 28 .4075 4490 8 .7618e-012 3 .0371 7679 4 .5586e-013 0 .5184 18400 1 .9621e-014 0 .0911 38828 7 .9216e-025 0 .0164 74196 3 .3643e-026 0 .0023 167850 1 .8901e-02

Process 1

CurveRecognitionand Data

Segm en tation -slope and direction

Assignment of qualitative curveattributes by characterising

qualitative value of thequantity spaces

loosely concave andconcave at least one ofthesubdivision

NOT ((linear and growing ANDasymptotically-positive-horizontal) OR(weakly linear-and-growing ANDwymptoticaly-largely-positive-horizontal))

Figure 1 : Four major processes in analysing creep data and determining an equivalence class

vertical

Expenmxntal plots lm allplauside

rha~hrgical mlWelc

To find the bestrmtdeh furthe

electedshear raterare

Process 11

v i-Ay funcxinn4={!, u obtained fnrm

thebeat model

IkItrmInaUnnthe valorofm and the

discrete relaxationand r<tard.War

spectra

r

Ya,D'a =Y 0, WY

DCOMPLETE CONtinTUTIVE

FA)tATl0N

Process 111

lrt~ tea dataalver

Ifigure II

Ilrtxnnine which ofthe four admixsaMenxlilWivt equation

avdewrihe thegivenmaterial .

Specify n and b

ProcesstV

Behavioural class mapping tological triplet

ED',=ED'r r (T "T" F)

Figure 2 : An intelligent hybrid system for determining a complete rheological model for aviscoelastic material

StandardExperiment

vi-metric try"-k titansexperiment experiment experment

(Shady simple shear I (0,01.1-y shnrrl Icrrep tes0

shear stress

shear stress

Sisko model11

Power Law model

Casson model

shear rate

shear rate

Figure 3 : A flow curve which has models assigned to it

Figure 4 : A comparison of model data and measured data for polymer melt