Pertanika 2(2), 111-117 (1979)

Theoretical Considerations for ViscoelasticCharacterization of Biomaterials

NARESH CHANDRA SAXENADepartment of Engineering Sciences, Faculty of Agricultural Engineering, Universiti Pertanian Malaysia

Key words: Viscoelasticity; Biomaterials; Characterization.

RINGKASAN

Untuk sesuatu rekaan, pembinaan serta ujian jentera-jmtera memproses dan mesin-mesin pengangkutanbahan-bahan biological, sifat-sifat mekanikal bahan-bahan tersebut perlu diketahui terlebeh dahulu. Olehkerana percubaan-percubaan yang lampau dapat mempastikan bahawa bahan-bahan biological bersifat visco­elastic, perlulah ianya dinyatakan supaya dapat dipastikan tindak balas sifat-sifat tersebut dalam kerja-kerjamengangkut dan memproses. Untuh bahan-bahan kejuruteraan, penggunaan teori viscoelasticity adalahterbukti dan nyata, tetapi bagi bahan-bahan biological perkara ini tidaklah sedemikian. Untuk bahan-bahanbiological, teori asas viscoelasticity perlu diubah untuk penyesuaian dengan sifat-sifat biological. Oleh itu,teori-teori yang perlu adalah dibincangkan dalam kajian ini. Ianya juga memperkenalkan istilah-istilahbio-engineering dan menghuraikan gambaran viscoelastic terutamanya dibidang bio-creep dan relaxation.

Ka}ian ini berusaha untuk menunjukkan persamaan sxfat tersebut dengan model-model mekanikal danpersamaan mereka secara am. Ringkasan barrel-effect untuk bahan viscoelastic yang berunsU1' biologicaljuga diberikan. Peranan shift-factors sebagai sifat bahan juga diliputi dalam perbincangan ini.

SUMMARY

The mechanical properties of biomaterials need to be known and standardised before design, constructionand testing of processing and handling machinery can be undertaken. Since experiments in the past haveproved the viscoelastic nature of biomaterials, it is necessary to characterize them in order to ascertain theirresponse to handling and processing. For engineering materials, the application of the theory of viscoelasticityis now quite well known,. but this is not the case for biomaterials. In the characterization of biomaterialsit will be necessary to modify the basic theory in the context of the bio-effects. The theoretical considerationsare discussed in this study. The various bio-engineering terms are defined and viscoelastic representation withparticular emphasis on bio-creep and l·ela."<ation is described.

An attempt is made to show analogy with the mechanical models and their generalization. Barrel­effect for viscoelastic materials of bio-origin is outlined. The shift-factors and their significance as a materialproperty are also included.

INTRODUCTION

One of the basic needs in the food industryis knowledge of mechanical properties of theproducts requiring processing and handling. Thestructural complexity of food products presents achallenge in measuring the rheological properties.In order to provide food of higher quality it isnecessary to understand the physical laws govern­ing the response of biological materials to handlingand processing. Damage and spoilage have to becontrolled to increase the efficiency of harvesting,

Key to author's name: Saxena, N. C.

111

handling and storage facilities. The importanceof knowledge of these basic engineering para­meters for food products and the application ofthis information to an engineering analysis isobvious. For example, the response of specificfood products to load or deformation at varioustemperature conditions is required. Only recently,food and agricultural engineers have begun toapply the basic theories of engineering to thebehaviour of food and agricultural products.The scope of Biomaterials Science coming underBiomechanics or Bioengineering seems unlimited.

N. C. SAXENA

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D£F'ORMATJON -

BJO YIEU- POUJT"

'"

tBased on experimental evidences attributed

to Zoerb and Hall (1960), Mohsenin (1963),Timbers (1964) and Morrow (1965), agriculturalproducts are viscoelastic. From the very limiteddata available in this area it would appear thatviscoelastic behaviour is non-linear. In anattempt to explain the rheological behaviour ofagricultural products, simplified assumptions havebeen made and the theories of linear viscoelasticityapplied.

Biomaterials are, in fact, plant and animalmaterials. They are the principal raw materialsfor food and agricultural industries. They alsoinclude the final products after primary, secondaryor tertiary processing; and the agricultural andfood engineers and scientists are concerned withtheir physical behaviour. The nature of bio­materials, e.g. specimen for testing (size, shape,non-availability of prepared specimen, ripeness,type, moisture, temperature, variety of the samekind, harvesting time effect, soil and irrigationeffect, fertilizer and pesticide effect) presentnumerous obstacles in actual experimentation.It is, therefore, necessary for these problems andconditions to be specified for each testing of bio­materials. Since the citing of examples of evena single biomaterial will require mention anddiscussion of the whole spectrum under whichtesting is done and results obtained, no attempthas been made to analyse any example in thisstudy.

BIOENGINEERING TERMINOLOGY

Fl"!<lo)

Fig. 1. Illustration of Terms.

The following terms and definitions are usedin this study. These are illustrated in Fig. 1.

Bioyield point is that point on the force­deformation curve at which an increase in defor­mation occurs with a decrease or no change of theapplied force. In many agricultural products,the presence of this bioyield point is an indicationof initial cell rupture in the cellular structure ofthe material.

Rupture point is that point on the force­deformation curve at which the crack is visibleto the unaided eye. It indicates a failure in themacrostructure while bioyield point indicates afailure in microstructure of the specimen.

The degree of elasticity, D, is defined as theratio of elastic deformation to the sum of elasticand plastic deformation when a material is loadedand then unloaded to zero load.

The hysteresis is defined as the energyabsorbed by the material in a cycle of loadingand unloading and is evaluated as the area betweenloading and unloading curves.

Viscoelasticity, in general, is a combinedsolid-like and liquid-like behaviour in which thestress-strain relationship is time dependent.

Linear viscoelasticity is defined as a visco­elastic behaviour in which the ratio of stress tostrain is a function of time alone and not of thestress magnitude.

Stress relaxation is the decay of stress withtime after the material is suddenly deformed toa given deformation or constant strain.

Creep is the deformation as a function oftime when the material is suddenly subjected toa constant load.

112

THEORETICAL CONSIDERATIONS FOR VISCOELASTIC CHARACTERIZATION OF BIOMATERIALS

Relaxation time is the time required for thestress in the Maxwell model, to decay to lie orapproximately 37 per cent of its original value.

Retardation time is the time required for theKelvin model to deform to I - lie or approxi­mately 63 per cent of its total deformation.

VISCOELASTIC REPRESENTATION

The mechanism of response of an engineeringmaterial may be studied on any of three. basiclevels: the molecular, structural, and the pheno­menologicallevel. At the first level, the responseof the material is inferred from the properties ofits microscopic elementary particles; at the second,the material is considered as being made up ofnonhomogeneous, visible units whose interactionproduces the observed behaviour; at the third,the material is assumed to be macroscopicallyhomogeneous and isotropic such that the behaviourof any part of it, in any direction, is the same asthat of the whole.

The behaviour of all real materials will fallsomewhere between the two extremes of theEuclid-solid and Pascalian-liquid. The Euclid­solid is a completely rigid body, which is in­compressible, while the Pascalian-liquid is alsoincompressible but offers absolutely no resistanceto deformation. Next to the Euclid-solid inhierarchy is the Hooke-solid, for which stress isdirectly proportional to strain, while next to thePascalian-liquid is the Newtonian-liquid, forwhich stress is directly proportional to the time­rate of strain. Another type of idealized behaviouris that of the St. Venant-solid, which, up to acertain stress, called the yield stress, acts like aHooke-solid, but once that stress is reached, itdeforms plastically at constant stress.

In general, the response of all real materialsis made up of a complex combination of themany idealized responses mentioned above. Infact, any quantitative representation of this com­plex response will in itself be idealized, the degreeof idealization depending on the accuracy desiredor achieved.

The fundamental problem in a study ofmaterial response is the determination of thefunctional relationship between the strain, andthe stress, and their time derivatives. This isthe rheological equation of state of the material.

113

.One of ~he fu~dament~l methods of repre­sentmg the ltnear vIscoelastIC materials is that ofthe Linear operator equation,

Pd'ij = Q E if

Here the linear operators P and Q are defined as,m lIm

P = r.l.. m -o otm

o 00Q = r. J3n -

o Otn

whereo£ and J3 are non-zero coefficients.

From the superposition principle creep ISexpressed mathematically as,

<ttl ~ 6 r~ H (tl]and relaxation as,

6 (t) =e [E + 1> (t)]

where If and 1> are creep and relaxation functionsrespectively.

The phenomena of creep and relaxation,according to Gross (1953), are represented by afinite series of exponential functions with timedistributed over a finite interval. This leads totwo distribution functions of time constants i.e.distribution functions of both retardation andrelaxation times.

Despite their mathematical elegance thepractical application of the distribution functions,is difficult to quantify. It is suggested, therefore,that experimental data should not be expressedin terms of distributions, but rather in experi­mental parameters which are read or calculateddirectly from the experimental data.

Another fundamental way and one of themost widely used and easily interpreted methodsof representing the behaviour of a viscoelasticmaterial is to seek an analogy with mechanicalmodels made up of various combinations ofHookean springs and Newtonian dashpots. Theseare the two primary elements of a mechanicalmodel.

MECHANICAL MODELS

The combination of elastic and VICOUSSelements in series forms a Maxwell model asshown in Fig. 2(a) and the relationship betweenstress and strain is,

N. C. SAXENA

0- (t) k E1 H(t) + k 112 <5 (t) +n t

k I Ei exp (-Ei -) H(t)E 1=3 11i

dwhere 8 (t) = H(t)

dt

CP-) MAXWUL MOllEL (bl J<ELVIN-VOlliT MOll&L The force response to a unit extension is definedby Bland (1960) as the relaxation function. It istherefore,

n t</>(t) = k Ei exp(-Ei -) H(t)

i =3 11i

Stress relxation in materials can be repre­sented by a generalized Maxwell model andcreep can be represented by Kelvin chains. Thisis the limitation of the rheological models inthat one model may adequately represent relax­ation while creep may be extremely difficult torepresent.

(j:) GEHEtRALlZ£D MAXWELl- MOllEI.

Fig. 2. Mechanical Models.

U (T

+ EE 11

In a Kelvin-Voigt model, the two primaryelements are connected in parallel as shown inFig. 2(b), and the stress-strain relationship is

CT = EE + 11 E

There are two ways of systematically buildingup more complicated models, the Kelvin chain(or generalized Kelvi n-Voigt model) and thegeneralized Maxwell model. In the former anarbitrary number of Kelvin units are in series.In the generalized Maxwell model, Maxwellunits are in parallel as shown in Fig. 2(c).

In the generalized Maxwell model, if it isgiven a sudden deformation,

€ = k H(t)

where Heaviside function H (t) is

H (t) = 0, t < 0 and H (t) = 1, t ~ 0

The relaxation behaviour of a linear viscoelasticmaterial can be represented as

It can therefore be postulated that therelaxation behaviour of a biomaterial is repre­sented by generalized Maxwell model and thenumber of Maxwell units can be determinedfrom the experiment.

If the stress in a material falls to zero forlarge values of time, then there should be nospring in parallel with the other elements when ag~neralizcd Maxwell model is postulated tosimulate the behaviour of the biomaterials. Ifon the ot.her hand, the stress does not approachze:o as tIme approaches infinity, then obviouslythIs type of behaviour should be represented byan elastic clement in parallel with the generalizedmodel.

PRONY-DIRICHLET SERIES

In a generalized Maxwell model, the step in:putin strain gives

t

CT(t) = [

where the constant term Eo has been added toallow for an elastic response. This model givesrelaxation function (stress divided by constantstrain) as>,

-tn

</> (t) = Eo + k E i e tf, I

1-1

114

THEORETICAL CONSIDERATIONS FOR VISCOELASTIC CHARACfERIZATION OF BIOMATERIALS

8

LOAJI

(c) Make use of the fact that at a sufficientlong time,

(e) Get a set of algebraic equations for thecoefficients Ei.

~ (00) = Eo

The exponential nature of these functionsmakes it convenient when using Laplace Trans­forms for analytical manipulations. The aboveexponential series is called Prony-Dirichlet series.Under certain conditions it is mathematicallycomplete. To determine the coefficients of thi'sseries, Brisbane (1966) has outlined the followingsteps;

(a) Choose 'l) decades of time over the timeinterval of interest from the relaxation functioncurve. For example 1, 10, 103, 102 seconds.

(d) Equate the series value of E (t) to theexperimental values at (n-1) points.

(b) Choose values of i such that one fallswithin each decade on the curve.

The resulting set of equations for the co­efficients E i can be solved on a computer.

BARREL EFFECT Fig. 3. Barrel effect under compression.

Bartenev and Zuyev (1968) proposed that undercompressive loads the viscoelastic material failsusually either by shear or by rupture and not by acombination of both shear and rupture providedthe compression proceeds without slipping orsliding on bearing surfaces. They furthersuggested the existence of the 'barrel' effect asshown in Fig. 3. Under compression of thespecimen the points A and B are in tension asthe free side surfaces bulge in the form of a barrel.Along the line AB of the barrel a growth of smalltears takes place from the surface into the depthof the material.

SHIFT FACTORS

The temperature and moisture shift factorsare used to describe the effects of temperatureand moisture on any viscoelastic function. Theshift factor is an inherent property of the material.

The thermo-rheologically simple nature of amaterial indicates that an increase in temperaturecorresponds to an increase in time. The samereasoning is used in a hydro-rheologically simplematerial so that an increase in moisture contentwould correspond to an increase in either tempera­ture or time.

Temperature shift factor aT(T) is defined asaT = ef(T)

f(T) is a shift function of temperature and shiftis positive as temperature increases when thesame viscoelastic function. is plotted againstlogarithm of time, as shown in Fig. 4.

i

lo/t-Fig. 4. Shift-Principle.

115

N. C. SAXENA

Let !is (t, To) be the relaxation function atreference temperature To and 1> (t, T) at anytemperature T.

Let us change independent variable such that

1> (t, To) = L 1 (log t)

In other words, relaxation function is plottedagainst logarithm of time.

For thermo-rheologically simple materials,viscoelastic functions when plotted against logof time exhibit a shift but no change in shapewhen temperature is changed. The relationshipis, therefore,

1> (t, T) = L 1 (log t + f(T))

Christensen (1971) described the methodfor determining the shift factors which are definedin this study. The viscoelastic mechanicalproperty-relaxation function, creep function orcomplex moduli, when plotted against thelogarithm of time can be superimposed to form asingle curve merely by shifting the various curvesat different temperatures along the logarithmof time axis. If the curves do coincide withinexperimental error the basic postulate of thermo­rheologically simple material is verified. Hefurther claimed that there are no general inclusiveguide lines that can be given to answer thequestion whether a given material can be expectedto exhibit the thermo-rheologically simple typeof behaviour. The only safe and certain answerlies in experimentally verifying or invalidatingthe shifting procedure for every material studied.

where

f(T o)

df(T)o and > 0

dt

It can, therefore, be postulated that a bio­material is both thermo and hydro-rheologicallysimple. That is, the material has only one time­temperature shift factor and one time-moistureshift factor.

Substituting the log shift-factor In place ofthe shift-function, we obtain

Based on the temperature shift factor, ananalogy is established for a moisture shift factor.It is also defined as

REFERENCES

Biomaterials are viscoelastic and needviscoelastic characterization.

The Theory of Viscoelasticity as appliedto engineering materials is standardised,but it should be modified when biomaterialsare to be characterised.

Viscoelastic behaviour can be representedmathematically by the linear operatorequation and/or by an analogy of mechanicalmodels.

Stress-relaxation of biomaterials can berepresented by a generalized Maxwell modelwhich is expressed as a PRONY-DIRICH­LET series. The coefficients of this seriesare found by BRISBANE's method usingcomputer.

"Barrel-effect" is observed in viscoelasticmaterials.

The biomaterials can be postulated to beboth thermo and hydro-rheologically simple.

CONCLUSIONS

(i)

(v)

(ii)

(vi)

(iv)

(iii)

L 1 (log t + log aT)

L1 (log t . aT)

L 1 (log n1> (t, T)

where f(M) = shift function of moisture.

Hydro-rheologically simple materials aredefined as those materials which have all materialfunctions shifting in the same way and in thesame amount.

Thus the relaxation function at any tempera­ture can be directly obtained from the relaxationfunction at base temperature Toby replacing twith ~.

Developing the relaxation function at anymoisture content in the same way it is given that

BARTENEV, G. M. and ZUYEV, Y. S. (1968): Strengthand Failure of Viscoelastic Materials. New York.Pergamon Press.

<p (t, M o)

<p (t, M)

L 2 (log t) and

L 2 ( logt. an)

BRISBANE, J. J. (1966): Characterization of LinearViscoelastic Materials. Huntsville, Alahama.Rohm and Haas Company, Redstone ArsenalResearch Division.

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THEORETICAL CONSIDERATIONS FOR VISCOELASTIC CHARACTERIZATION OF BIOMATERIALS

CHRISTENSEN, R. M. (1971): Theory of Viscoelasticity:An Introduction. New York, Academic Press.

GROSS, B. (1953): Mathematical Structure of theTheories of Viscoelasticity. Paris. Hermann andCie.

MOHSENIN, N. N. (1963): Engineering Approach toEvaluating Textural Factors in Fruits and Vege­tables. Transactions of the A.S.A.E., 6(2). 92,85-88.

MORROW, C. T. (1965): Viscoelasticity in a SelectedAgricultural Product. M.S. Thesis. ThePennsylvania State University.

TIMBERS, G. E. (1964): Some Mechanical and Rheo­logical Properties of the Netted Gem Potato.Master's Thesis. University of British Columbia.

ZOERB, G. C. and HALL, C. W. (1960): Some Mechani­cal and Rheological Properties of Grains. J. agri.Engng Res. 5: 1,83-93.

(Received 17 February 1979)

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