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Composite Materials

Journal of Thermoplastic

http://jtc.sagepub.com/content/19/1/35

The online version of this article can be found at:

DOI: 10.1177/0892705706055443 2006 19: 35Journal of Thermoplastic Composite Materials

A. Pramanick and M. Sain

Characterization of Thermoplastic/Agro-fiber Composites

Temperature-Stress Equivalency in Nonlinear Viscoelastic Creep

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Temperature–Stress Equivalency in

Nonlinear Viscoelastic Creep

Characterization of

Thermoplastic/Agro-fiber Composites

A. PRAMANICK AND M. SAIN*

Faculty of Forestry, 33 Willcocks Street

University of Toronto, Toronto, Ontario

M5S 3B3, Canada

ABSTRACT: The viscoelastic characterization of agro-filler based plastic compo-

sites is of paramount importance for the material’s long-term commercial success.

To predict creep, it is important to derive a relationship between deformation, time,

temperature, relative humidity, and stress. Since temperature shift can interfere with

stress shift in creep, the predictive model should incorporate the relationship between

these two shifts. Rice husk–HDPE beams were subjected to creep and recovery in the

flexural mode and stress/time/temperature-related creep behavior of the same was

studied. Temperature-related creep constants and shift factors were determined

for the material and the constants were compared against theoretical two-phase

constants. The combined effect of temperature and stress on creep strain was

accommodated in a single analytical function where the interaction was shown to

be additive. This means that the stress equivalency of temperature is possible. This

constitutive equation can predict creep in the long run. Although stress dependency

is nonlinear, temperature dependency is linear and thermorheologically complex.

The ‘single-phase’ material behavior (creep constants) was also compared with a

‘two-phase’ predictive model, where the creep constants were estimated with the

‘theory of mixtures’.

KEY WORDS: HDPE–rice husk composites, creep, viscoelastic, temperature–stress,

nonlinear, two phase.

Journal of THERMOPLASTIC COMPOSITE MATERIALS, Vol. 19—January 2006 35

0892-7057/06/01 0035–26 $10.00/0 DOI: 10.1177/0892705706055443

ß2006 SAGE Publications

*Author to whom correspondence should be addressed. E-mail: m.sain@utoronto.ca

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INTRODUCTION

AGRO-BASED FIBERS AND particles are being considered as fillers for

thermoplastic composites. In these composites, typically 40–60%

agro-particles are blended with thermoplastics such as HDPE, PP, PVC,

etc. These composites have various uses: as deck-boards, railings, railway

ties, automotive parts, etc. In future, this material may replace both

wood and plastics. When used in building applications, plastic composite

can creep in the long run. Since creep is affected both by load and

temperature, any predictive model should consider the interactive effects

of these two factors among others. A study of the stress/temperature inter-

action is imperative for the prediction of creep under changing temperature

and load. ASTM standard defines some procedures to standardize the

performance of wood–plastic lumber; according to this standard, compli-

ance (stiffness/stress) of wood–plastic lumber should be used as one of the

creep-related performance parameters. Incidentally, compliance has been

used as the measure of creep strain of plastic materials by many authors,

e.g., Woo [1]. ASTM D 6112 [2] also suggests that wood–plastic lumber,

when used as load bearing material, should be tested in four-point flexural/

bending mode rather than tensile mode for their creep properties. All

plastic-based materials and wood exhibit viscoelastic behavior and creep

under stress. In many plastic materials, creep is nonlinear with respect to

stress in the sense that compliance is a function of stress. Temperature and

moisture [3] can also induce nonlinearity. An issue with temperature is its

influence on stress, i.e., whether the combined stress–temperature effect

could be additive or interactive.

Individual temperature and stress-related creep issues have been dealt

with in the field of thermosetting-based composites [1,4]. It is imperative

to deal with the effect of these factors, when they occur concomitantly. We

have shown in our earlier work that creep of HDPE–rice husk deck-boards

show nonlinear behavior with respect to stress [5] and follow the power

law. Authors have proven that pure HDPE creep exhibits nonlinear

behavior under stress [6]. Xu et al. [7] showed that with an increase in fiber

content, the creep of wood particles-filled plastic composite decreases. This

proves the influence of natural fiber on creep of plastic, but it has not been

clarified whether this influence is due to its reduction in plastic content or

due to the rigidity of the filler as there is a question of interfacial bond

energy. Martinez-Guerrero [8] suggests that stress influences compliance

in creep of wood–plastic lumber. Knauss and Emory [9] attributed stress-

related nonlinearity to changes in the free volume during deformation.

Rangaraj and Smith [10,11] ascribed nonlinearity to micro-damages

from deformation, and used power law to link damage with nonlinearity.

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Many authors have used the power law to express creep behavior of

plastic-based materials [10–12]. Schapery showed that a thermodynamic

approach to a nonlinear viscoelastic model is similar to the Boltzman

superposition principle, where the power law concept can fit in [13].

Another popular model to be considered is the Kelvin–Maxwell model

[14–16]. Kelvin–Maxwell, in conjunction with the Boltzman superposition

principle yields an expression that actually resembles Schapery’s expression.

Martinez-Guerrero concludes that plastic lumber does not follow the

Kelvin–Maxwell model [8]. However, Pooler [17] suggests that a modifica-

tion of this model, which is the Prony series, fits the creep behavior of wood

particle-filled plastic well. Pooler, however, did not try to explore Schapery’s

model analytically. The Prony series model application calls for numerical

calculations and is very material specific. All of the above models predict

creep through the determination of creep constants (and stress shift factors),

which treat the materials as ‘single-phase’ ones.

Temperature may influence compliance in a similar way as stress.

Crissman [20] has pointed out that stress and temperature dilations are

responsible for easier movement of the plastic macromolecules in

amorphous regions. The theory of temperature dilation has led to the

famous WLF (Williams–Landel–Ferry) equation, the activation energy

concept, and the TTSP (time–temperature superposition principle)

proposition. According to these assertions, temperature effects may be

described by altering the timescale of the viscoelastic response. That means,

if Dis creep strain (per unit stress) at a temperature of T

1

and time t, the

creep at a temperature of T

2

can be described as follows:

Dðt,T1Þ¼Dt

aT

,T2

where a

T

is a shift factor.

For thermorheologically complex material such as semicrystalline

plastics, a vertical shift of the data plots along the Y-axis should be

considered [17]. The famous WLF [21] assertion states that for an

amorphous plastic, the shift factor can predict the change in creep strain

corresponding to a temperature:

log aT¼c1ðTTgÞ

c2þTTg

:

An Arrhenius relationship is also typically used [16]:

ln aT¼Ea

R

1

T1

Tref

:

Viscoelastic Characterization of Agro-based Plastic Composites 37

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In this equation E

a

is the activation energy of chain relaxation, and T

ref

is

the reference temperature. Both of these equations are valid when there is a

linear temperature shift in creep strain due to a change in the temperature.

Elahi and Weitsman [18,19] used the concept of TTSP in their chopped

glass/urethane composites, which implies the applicability and universality

of this concept to composites. Many authors used the concept of vertical

shift in semicrystalline plastics in conjunction with horizontal shift (TTSP),

but the real cause for the vertical shift is not very clear [22]. While the

vertical shift factor had been thought to be a representation of the change

in crystallinity, crystallinity may not change at such a low temperature as

around 60C [24]. However, both vertical and horizontal shift factors are

needed to model viscoelastic behavior of HDPE and composites. In the

studies involving temperature effect on creep, the prediction of creep is done

through the calculations of shift factors. Like the current stress models,

temperature models are also used to calculate shift factors with a ‘single-

phase’ approach.

Through the perusal of the literature, one would notice that the power law

and Schapery models have been used to describe stress-related nonlinearity

in thermosetting composites and pure thermoplastics. The studies on

temperature shifts never looked at the interactive shift factors due to

concomitant changes in temperature and stress. We have proven that

Schapery models can be used to describe the nonlinear creep of rice-based

HDPE composite creep [25]. Here, we would like to incorporate the

temperature shift factor into the power law/Schapery equation and generate

constitutive equations that define, characterize, and predict long-term creep

of the material. So with a single analytical function, which is nonexisting at

the moment, we will be able to encompass both stress- and temperature-

related shift factors. The constants, thus determined, will be validated

through rigorous step loading and long-term creep experiments. The

emphasis here is on the temperature effect, but because the temperature

and stress are both acting on creep, a cursory look at the stress effect also

will be taken. This work is also the vanguard to use power law concept in

step loading of temperature/stress.

We follow a two-step approach in this study. In the first step, which is a

‘single-phase’ approach, we characterize the material in hand (rice husk/

HDPE composite) for creep, and develop creep prediction equations

thereof. Here we determine creep constants, i.e., stress–temperature shift

factors for the experimental material. Extensively modified Schapery’s

concept is adhered to, because this concept is a blend of several creep

concepts, to develop the equation describing the validation material. The

predictive equation in the first step is actually a validation for the theory

developed in the second step. In the second step we strive to predict the said

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constants with those of the constituent materials (wood particle and HDPE)

from the literature. The second step is a generic ‘two-phase’ model, but in

this article we only emphasize the first step. However, a discussion about the

‘two-phase’ model is also included in the text. We must also emphasize that

no article exists to date on the ‘two-phase’ creep approach in composites.

For the ‘two-phase’ model, the HDPE and wood rheological data are

obtained from literature [26–29]. In the literature, HDPE and wood

behaviors were studied at different conditions, and shifts were calculated.

Wood also creeps in a nonlinear fashion and exhibits temperature shift.

There is a conspicuous absence of rice husk creep data. However, since

wood particles and rice husk are both lignocellulosic materials with similar

adhesion properties against HDPE, we propose to use the shift factor values

of wood as a substitute for rice husk.

THEORETICAL CONSIDERATIONS

Theory of Flexural Deformation

When a beam is loaded in four-point bending mode (Figure 1), maximum

tensile stress occurs at the bottom surface of the beam, whereas, compressive

stress occurs at the top. The ultimate tensile stress occurring at the bottom

can be calculated using the formula:

¼PLI1

bd 2I2

,

where is the stress at the bottom of the beam; I

1

¼bd

3

/12, I

2

¼ðbd 3

b1d3

1Þ=12;P, load; L, span; b, breadth; and d, depth of the material. Also

note that I

1

,I

2

represent the moments of inertia of cross sections of solid and

Figure 1. A typical four-point bending setup.

Viscoelastic Characterization of Agro-based Plastic Composites 39

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hollow bars, respectively (Figure 2). The corresponding strain in the bottom

surface of the material can be calculated using the formula [2]:

"¼4:7d

L2,

where is the deflection of the beam.

Many studies [7,13,15] prefer to use normalized deformation, creep

or instantaneous, over absolute deformation, where normalized defor-

mation ¼"/. This parameter, known as compliance, is useful in the study of

changes due to stress variation.

Theory of Creep

STRESS FACTOR

In the present work, creep is defined as the total strain at the bottom

surface of the beam. In order to quantify the effect of stress on the material,

throughout this article, creep has been normalized:

DðtÞ¼"ðtÞ

¼compliance ð1Þ

where represents a constant applied load and "(t) is the time-dependent

strain. The thermodynamic theory permits us to express the nonlinear

material properties in strain [13] as follows:

"ðtÞ¼g0D0þg1Zt

0

Dð 0Þdg2

dd ð2Þ

where

D0Dð0Þand Dð 0ÞDð 0ÞD0ð3Þ

d, b

b1, d1

Figure 2. Cross-sectional schematic view of the beam.

40 A. PRAMANICK AND M. SAIN

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In Equation (3), D

0

is the initial value of the creep compliance, Dð 0Þ

is the transient component of the creep compliance, and (at a constant

temperature) is the reduced time calculated as follows:

¼Zdt

a

for a>0ð4Þ

0¼ ðÞ¼Zdt

a

ð5Þ

In the above equations, g

0

,g

1

,g

2

, and a

are the material properties as a

function of stress. In general, these stress-dependent properties have specific

thermodynamic significance and the changes in g

0

,g

1

,andg

2

reflect third-

and higher-order dependence of the Gibb’s free energy on the applied stress

[13]. Equation (2) can be simplified in a single-step load, where the value of

is assumed to be constant, to the following form:

"ðtÞ¼g0D0þg1g2D ðÞð6Þ

By substituting a constant stress into Equation (2), dg2=d¼0 (except

when ¼0, where d¼). Equation (2) morphs into:

"ðtÞ¼g0D0þg1g2Dt

a

n

ð7Þ

For nonlinear creep Equation (7) shows that the initial elastic response is

particularly linear even though the creep is strongly nonlinear and the

transient component of the creep D( ) is modeled by the log power law:

Dt

a

¼D1log t

a

n

ð8Þ

Determination of Stress-related Creep Coefficients

A full nonlinear viscoelastic theory presents a constitutive behavior,

a stress–strain relation, of polymeric materials (Figure 3) through the

following equations:

"ðtÞ¼g0D0þg1g2D1log t

a

n

þ"pðtÞð9Þ

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where a

is a timescale shift factor. It mathematically (and horizontally)

shifts the creep data parallel to the time axis relative to a master curve for

creep strain versus time, nis the index that determines the shape of the creep

curve, and "

p

is the plasticity that occurs during creep. It should be noted

that for linear viscoelastic strain, Equation (9) reduces into the following

equation (g

1

,g

2

, and a

are equal to unity in the linear region):

"ðtÞ¼DðtÞ¼ðD0þD1ðtÞðlog tÞnÞð10Þ

Equation (9) relates stress with strain through material constants; so in

order to predict creep for a given level of stress we estimate the creep

coefficients as a function of stress. According to the data reduction method

proposed by Papanicolaou et al. [12] for ‘carbon–epoxy resin’ composite,

the constants can be calculated through solving Equations (9) and (10) along

with the following equation for recovery (Figure 3):

"rðtÞ¼g2D1log t

a

þtta

n

D1logðtta

ðÞÞ

n

þ"pðtaÞð11Þ

Equation (11) for recovery assumes that at zero stress the a

,g

0

,g

1

,g

2

are

all unity.

Calculation of the Basic Creep Constants

The value D

0

can be determined from the instantaneous deflection data of

the creep/time curve in the linear range. According to Equation (2), g

0

could

be calculated from the creep plot when tis equal to zero. The following

formula may be employed to calculate compliance:

"ð0Þ¼Dð0Þ¼D0g0)g0¼"ð0Þ

D0

ð12Þ

εor

Strain

εoc

εc (t)

0

0

Time

εr (t)

∆εc

Figure 3. A typical creep diagram depicting strain–time relationship.

42 A. PRAMANICK AND M. SAIN

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The nand D

1

values can be calculated through curve fitting of Equation (9)

in the linear region. We may calculate g

1

with the following equation:

g

1

¼("

c

(t

a

)g

0

D

0

"

p

)/"

r

(t

a

). In order to calculate a

, we can use the

recovery equation (11) to obtain a fit; g

2

can be calculated from the

following equation where the assumption is that g

2

¼1 in the linear region:

"cðtaÞ"p

g1lognðtaÞ

nl

g1lognðtaÞ

"cðtaÞ"pðtaÞ

l

¼g2ð13Þ

where the subscript l means linear and nl means nonlinear.

Validation of the Model with a ‘Two-step’ Loading

When stress is applied stepwise in time with the following conditions:

¼a, for 0 <t<ta

¼b, for ta<t<tb

¼b¼c

ð14Þ

where the superscripts refer to the properties associated with the corre-

sponding stress levels. Equation (2) morphs into the following:

"cðtÞ¼bgb

0D0þagb

1gb

2D1log ta

aa

þtta

ab

n

þðgb

2bga

2aÞD1log tta

ab

nð15Þ

"rðtÞ¼agb

1gb

2D1log ta

aa

þtbta

ab

þttb

ac

n

þðgb

2bga

2aÞ

D1log tbta

ab

þttb

ac

nbD1gb

2log ttb

ac

n

ð16Þ

TEMPERATURE FACTOR

We assumed that temperature and stress act additively. So the tem-

perature acts upon time factor of the constitutive equation (6) as follows:

"ðtÞ¼ðg0D0ÞFðTÞþg1g2Dt

aaT

n

þ"pð17Þ

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or

"ðtÞ¼ðg0D0ÞFðTÞþg1g2Dt

a

nEa

RT þ"pð18Þ

where E

a

is the activation energy of chain relaxation. In the semicrystalline

materials, there could be an instantaneous deformation due to the

temperature change, which is known as the vertical shift factor. In

the above equation, F(T) denotes the vertical shift factor. If temperature

is changed during the course of creep, we may call the activity as the step

loading of temperatures. If at the time t

1

the temperature is changed from T

1

to T

2

, the following relationships will hold:

For tt

1

, when a

T

¼a

1T

"cðtÞ"p¼ðg0D0ÞFðT1Þþg1g2D1log t

a1T

n

ð19Þ

For t>t

1

, when a

T

¼a

2T

"cðtÞ¼ðg0D0ÞFðT2Þþg1g2D1log t1

aa1T

þtt1

aa2T

n

þ"pð20Þ

If the load is withdrawn at time t

a

while the temperature remains same,

the following relationship should hold:

"r¼D1log ta

aaT

þt

aT

ta

aT

n

D1log t

aT

ta

aT

n

g2þ"pðtaÞ

ð21Þ

‘TWO-PHASE’ APPROACH WITH ‘THE THEORY OF MIXTURE’

In this approach the number of parameters needs to be limited, or else the

calculations will be cumbersome. So Equation (6) needs simplification, where

we assume that D

0

g

0

¼"

0

and g

1

g

2

D

1

¼g"

1

. Now, for discontinuous fibers

with a low aspect ratio, the composite stiffness can be expressed in terms of

the following equation where is the volume fraction of the fiber, is the

factor for shortness of the fibers, E

f

is the modulus of the fiber, E

m

is the

modulus of the matrix, "

0

is the compliance [16]:

E¼1

EfþEmð1Þand "0¼1

Eð22Þ

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The assumption for Equation (22) is that when under stress the fibers

and the matrix must undergo the same strain as the whole composite does

at low stress. Thus, we propose the following creep equation (based on

Equation (6)):

Creep ¼"cðtÞ

¼"0þg"1log½tnð23Þ

where g"1¼1=ð=g1

f"1, f þ1=g2

m"1, mÞand g1

f¼g2

m¼g:Also,

1

and

2

are incremental stress distribution in fibers and matrix, respectively.

If we incorporate temperature shift and temperature stress shifts are

additive:

Creep ¼"cðtÞ

¼"0a0þg"1log½tn

a

ð24Þ

where g"1=a¼1=ða,f=g1

f"1, f þa,m=g2

m"1, mÞ,a

¼exp(E

a

(1/T1/T

0

)/R).

The difference between a

and a

0

is that the latter (vertical shift) is measured

with respect to "

0

, not "

1

. However, these two values depict the same shifts.

The value of can be calculated as follows [16]:

¼1tanh na

na

where n¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2Gm=Eflnð2R=dÞ

p,G

m

is the matrix shear stress, E

f

is the fiber

modulus, 2Ris the distance between two adjacent fibers, dis the diameter of

each fiber, and ais the aspect ratio.

Using Equation (23), the stress shift per unit stress is calculated as follows:

¼1þ2¼2 MPa ðfor 50%volume of particlesÞ

g1

m¼gf

2

¼increase in length per 1 MPa stress

where g

m

¼g

f

¼1.12 (Table 4). Solving for gvalues, we obtain g1

m¼1:21:

The temperature shift (Equation (24)) can be calculated as follows:

Stress generated due to temperature shift on the composite ¼¼

(1/a

1)"

1

¼0.5(

f

þ

m

), where

f

and

m

are the stresses on the fiber

and matrix respectively. But the strain on the composite ¼strain on the

constituents )1/a

,f

1¼1/a

,m

1¼1/a

1¼

f

"

f

¼

m

"

m

¼"

1

(Table

6), where "1¼1=ð1="

fþ"mÞ:

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METHODOLOGY

Materials

Material for this experiment was acquired from Extendex Inc., Barrie,

Canada. This material is commercially marketed as deck-boards and railings

(Figure 4). The railings are our focus and they contain 60% rice husk

and 40% HDPE. A ‘two-step’ procedure was followed to process these

materials – compounding and extrusion. In the first step, rice husk goes

through a sieve of mesh size 16–80 with the moisture content of 10%. The

husks at the outlet of the drier achieve a moisture content of 1%. The dried

husk is sent through a heated co-rotating twin-screw extruder, where HDPE

pellets are mixed thoroughly and are ejected as compounded pellets. These

pellets are subsequently passed through a conical profile extruder. While

the profile is pulled out of the extruder, a mist is used to cool the product

down. MAPE (maleated polyethylene) is used as the coupling agent, which

is mixed during the pelletization in the twin-extruder. The dimension of the

cross-section of these rails is 600 40 40 mm

3

, whereas the thickness of the

same is 5 mm.

Experimental Setup

Two types of tests (creep and instantaneous) were carried out and both

were done in the flexural mode. A flexural creep testing rack was designed

based on ASTM D 6112. ASTM uses the four-point loading configuration

(Figure 1) because plastic lumbers are relatively ductile and do not fail by

the maximum strain (3%) under the three-point loading. The span length

for the test was 600 mm (¼L) and the crosshead speed of 10 mm/min. The

noses of both the support and loading beams were configured with

cylindrical surfaces with a radius of 1.27 mm in order to avoid excessive

indentation of the specimen. In order to allow for overhanging, at least

10% of the support span were maintained at each test specimen ends.

Figure 4. Composite railings and deck-board.

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The deflection of the specimen was measured at the midpoint of the load

span at the bottom face of the specimen.

The instantaneous failure test (four-point flexural) was done by a Zwick

‘strength testing’ machine. A load was applied to the object in the middle of

the span and stress/strain diagram was plotted until the material failed. The

purpose of this test was to determine the strength of the material. The peak

stress was determined from the ensuing stress–strain curve.

Both short-term and 1000-h tests (Figure 5) were performed at various

stress levels (14–50%) of the maximum stress level (ultimate stress,

u

).

The temperature was also varied for the purpose of determining the

temperature shift of the plots (20–60C). The creep tests were also followed

up with retraction of the load when full or a part of the strain/creep was

recovered. Step loading was carried out by adding an extra load during the

process and with respect to temperature in the sense that temperature was

varied in some cases. A transducer was placed at the bottom of the beams

(Figure 5) to note the voltage of the transducer with respect to the creep

level. The whole setup was ensconced in a kiln room where the temperature

and humidity could be altered with a control panel.

RESULTS AND DISCUSSION

Stress Effect on Creep

The composite beams tested for strength did not show extreme variations

in the stress/strain properties (Figure 6). This lack of variation proves

uniformity of its strength and stiffness properties amongst specimens. The

maximum force level to break the beams was about 1900 N. At this force,

the ultimate stress (

u

) is in between 25 and 20 MPa.

NONLINEARITY IN CREEP

The composite creep showed a significant stress dependency, as is evident

from Figure 7. The compliance went up consistently with the applied

Figure 5. Creep setup according to ASTM standards.

Viscoelastic Characterization of Agro-based Plastic Composites 47

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stress level. The composite also showed a low plasticity, about 2–5% of the

total strain. The stress-related constants were determined to develop a basic

empirical equation for the ambient conditions. D

0

and g

0

values were

calculated according to Equation (12) and are shown in Table 1. It was

assumed that the value of g

0

was 1 at 14% stress (base line creep). The g

0

value in general increases with stress. However, the increase is prominent

only from 14 to 27%. Beyond that, the average g

0

value hovers at

around 1.50. We propose to use this value for the stress level above 27%.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 200 400 600 800 1000 1200 1400 1600

Time

(

min

)

Strain (%/MPa)

5.2 MPa 11 MPa

3.39 MPa 8.39 MPa

Figure 7. Stress dependency of the composite.

Figure 6. Stress/strain diagram.

Table 1. Stress-related creep coefficients (single phase).

Stress

level (%)

Compliance

(MPa

1

)g

0

D

0

g

1

D

1

10

2

n

14 0.00035 1.00 0.00035 1.66 0.0036 1.45

22 0.00040 1.14 0.00035 1.50 0.0036 1.45

27 0.00054 1.55 0.00035 1.45 0.0036 1.45

30 0.00056 1.60 0.00035 1.00 0.0036 1.45

50 0.00045 1.30 0.00035 1.00 0.0036 1.45

40 0.00054 1.54 0.00035 1.00 0.0036 1.45

48 A. PRAMANICK AND M. SAIN

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No error was introduced because our long-term tests were based on low

stress where g

0

was close to 1. In the cases where high stresses were used,

the exact values were chosen for g

0

from Table 1.

The g

1

values represent the part of creep that is recoverable after the load

is withdrawn. This value is important only if we are interested in the recovery

part of the creep process. It is observed that the values of g

1

(Table 1)

decrease with the increase in load. A high value of g

1

suggests quickness

in recovery. This implies that at low stress (and low strain), as the load is

retracted, the recovery may behave like elastic recovery. But g

1

plateaus

after 30% stress levels off to a value of 1 (Figure 8). Therefore, for practical

purposes, a value of 1 is justifiable at high stress level (>30% ultimate stress

level). Below this we must use the appropriate values.

Using Equation (9), the value of nwas estimated, as we obtained a

straight line between log "(t) and log(log(t)). So the slope of this plot is

nand it does not change with time because of this straight line relationship.

The values of nand D

1

are presented in Table 1. Equation (11) was validated

with a

value as unity and that means the stress-related nonlinearity is a

function of g

2

only [9,30]. The g

2

values were calculated using Equation (13)

(Table 2). The values of g

2

go up with the stress level. The g

2

values are also

0

0.5

1

1.5

2

0 204060

Stress level (%)

g1 values

Figure 8. Stress dependency of g

1

values.

Table 2. Estimation of g

2

values.

Stress

% max

stress

t

a

(min) "

c

/g

1

g

2

3.5 14 1537 0.01988 1

5.2 22 1483 0.027333 1.376317

6.71 27 2849 0.042727 1.901831

11 50 77 0.041818 4.448346

8.83 40 252 0.044545 3.043111

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linearly related with the stress level (Figure 9). This makes it simple to make

prediction based on the creep level.

MODEL VALIDATION FOR STRESS NONLINEARITY

At ambient conditions, the final equation assumes the following form:

ð"cðtÞ"pÞ= ¼g0ð0:035Þþg1g2ð0:0036 log½t1:45Þ%MPa:

For some stresses log power law models were verified through Figures 10–12,

where creep and recovery were studied. It is evident from Figures 11 and 12

that the model works very well for long-term creep. In this case, a linear

model was adopted. In the case of step loading, which is followed up with the

load retraction, the log power law model fit excellently well (Figure 13).

Temperature Effect

TEMPERATURE STRESS INTERACTION

In one set of experiments, temperature was varied from 20 to 60C and

the stress level was also concomitantly varied, but in the reverse order

0

1

2

3

4

5

0 2 4 6 8 10 12

Stress (MPa)

g2

Log based Linear (log based)

Figure 9. The g

2

–stress relationship.

0

0.02

0.04

0.06

0.08

0 500 1000 1500 2000 2500 3000 3500

Time (min)

Strain (%/MPa)

Lo

g

power law Expt.

Figure 10. Creep and recovery data for 14% stress level.

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0

0.02

0.04

0.06

0.08

0.1

0.12

1 10 100 1000 10,000

Time (min)

Strain (%/MPa)

Experimental Log power law

Figure 11. Creep data for 27% ultimate stress.

0.01

0.1

1

1 10 100 100

0

Time (min)

Strain (%/MPa)

Expt. Lo

g

power law

Figure 12. Creep data for 40% ultimate stress.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1000 2000 3000 4000 5000 6000

Time (min)

Strain (%)

Experimental

Predicted

Figure 13. Creep and recovery plot for step loading.

Viscoelastic Characterization of Agro-based Plastic Composites 51

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i.e., from 6.5 to 3.4 MPa respectively. Figure 14 depicts the temperature/

stress effect on creep. In order to segregate the temperature effect from the

stress dilation an extension of Equation (17) was used:

log ð"ðtÞðg0D0ÞFðTÞ"pÞ

g1g2

¼log Dt

aaT

n

ð25Þ

In Figure 14, log((""

0

"

p

)/(g

1

g

2

)) values were plotted along the Y-axis

and log tvalues were plotted along the X-axis for the composite. In this

particular experiment the total creep increased with increase in temperature,

although at higher temperatures applied stresses were lower. Although the

creep strain was affected by both temperature and stress, the normalized

plots with respect to stress factors (g

1

and g

2

) showed a horizontal shift

along the time axis. The shift factor a

T

, that depicts horizontal shift, seems

to increase uniformly with temperature. A point per point time–temperature

shifting too was attempted on the experimental compliance data. The creep

compliance curve of Figure 14 was shifted point by point to obtain a smooth

master curve in Figure 15. This master curve confirms that a creep test result

at 60C and 1000 min is equivalent to that at 20C and 2 years.

It has already been mentioned that activation energy (E

a

) indicates a shift

of the creep curves along the X-axis due to the changes in temperature.

As described by Equation (18), a plot of 1/Tversus log(""

0

)/(g

1

g

2

)at

371 min of creep yields us a straight line (Figure 16) for the composite,

where the line yields a slope of about –3200. After we equated this slope

value with E

a

/R, we obtained a value of 30 kJ/mol for the E

a

. The value

of E

a

can be converted into a

T

through Equations (17) and (18). Thus we

obtain a value of 0.15 for a

T

for an increase of 10C. Using this value we

obtain Figure 17 where the experimental values show conformation with

-3

-2.75

-2.5

-2.25

-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0 0.5 1 1.5 2 2.5 3 3.5 4

Log (time, min)

Log (normalized strain, %/MPa)

20°C, 6.5 MPa 30°C, 5.7 MPa 40°C, 4.93 MPa 50°C, 4.15 MPa 60°C, 3.4 MPa

Figure 14. Effects of temperature and stress on creep.

52 A. PRAMANICK AND M. SAIN

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y = -3297.7 x +6.9837

R2 = 0.9946

-5

-4

-3

-2

-1

0

0.0029 0.003 0.0031 0.0032 0.0033 0.0034 0.003

5

1/T (K)

Log(e-e0)/g1/g2

Figure 16. Energy of activation.

-3

-2.5

-2

-1.5

-1

-0.5

0

0.1 11

0

Time (log(t/aT), min)

Strain (%/MPa,

normalized)

20°C30°C40°C50°C60°C

Figure 15. Master curve for time–temperature–stress superposition.

0

0.05

0.1

0.15

0.2

0.25

0 1000 2000 3000 4000 5000 600

0

Time (min)

Strain (%/MPa)

20°C 30°C 40°C 50°C

Figure 17. Constitutive equation and experimental data.

Viscoelastic Characterization of Agro-based Plastic Composites 53

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the model (Equation (17)). From various literatures, it is clear that a pure

HDPE has an E

a

value of 90 kJ/mol [22,24]. A high value of E

a

signifies a

higher sensitivity to temperature, so it is no surprise that the value of this

composite turns out to be only 30 kJ/mol. This attests to the fact that the

incorporation of the rice husk particles has reduced the creep. The T

g

value

of the composite was also estimated through the WLF equation, where the

universal constants c

1

and c

2

were assumed to be 17 and 52 respectively. For

pure HDPE the T

g

value is close to 125C. The value we obtained for the

composite is given in Table 3. This value indicates that the effective T

g

of

the composite is way higher than pure HDPE.

A vertical shift factor was also observed for these materials, as the matrix

is made of a semicrystalline material. The instantaneous deformation

was actually linear with respect to temperature with the formula: F(T)¼

(0.0015T0.4K)/0.04 (Table 3, Figure 18).

VALIDATION OF THE CREEP MODEL FOR THE

MATERIAL AS ‘SINGLE PHASE’

Figure 17 shows a comparison of experimental data and Equation (17),

where the temperature was varied from 20 to 50C and stress was varied

from 6.5 to 4.15 MPa respectively for the composite. The a

T

and gvalues as

per the described in the theory and in Tables 1–3 were incorporated into

Equation (17) and this equation can describe the related creep behavior.

y = 0.0015 x -0.4052

R2 = 0.9919

0

0.02

0.04

0.06

0.08

0.1

0.12

290 300 310 320 330 340

Temperature (K)

Momentary strain (%/MPa)

Figure 18. Vertical shifts at several temperatures.

Table 3. Temperature-dependent creep coefficients (single phase).

Material Model type a

T

F(T)E

a

T

g

Rice husk

plastic composites

Log power law 0.15/10C (0.0015T0.4)/0.04 30 kJ/mol 25C

54 A. PRAMANICK AND M. SAIN

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Figure 19 shows the plots for two stress levels, 4.1 and 4.9 MPa. With

respect to an ambient condition of T¼20CandRH¼58%, the stress-

related vertical shift factor g

2

should be around 1.2. By comparing model

equation values (Equation (20)) and experimental data in this figure, it is

obvious that the theory of additivity holds well. Figure 19 also testifies that

the same g

2

value holds at 50 and 60C as well regarding the stress level.

Figure 19 also represents a step-temperature loading experiment where the

temperature was elevated to 60C from 50C during the creep at 60 min

(while the stress level was also changed). As expected from Equation (20),

the creep went up followed by an increase in the instantaneous deformation.

The increase in instantaneous deformation at the start of 60C suggests

an increase in the vertical shift factor. But due to the slow increase in

temperature, the instantaneous creep increase does not take place abruptly.

However, the overall creep in the long run can be predicted well with

Equation (20).

A three-step loading of temperature, keeping the stress level same depicts

a slightly different picture where temperature was maintained as follows:

60C for 1600 min, 40C for 1600–6000 min, 60C for 6000–8000 min.

The three-step temperature loading (Figure 20) experiment shows that

Equation (20) is valid for a general prediction of creep at varying

temperatures. It is valid only when the vertical shift factor is considered

as F(T

1

) rather than as F(T

2

) for a decrease in the temperature. It is not valid

if the instantaneous drop in modulus due to temperature drop (from 60 to

40C) is considered in the equation. That is probably due to the fact that

at lower temperature the recovery is slowed down just like creep. So it is

difficult for the beam to recover to the full potential. Due to the fact that not

many articles exist about this aspect, this aspect may be investigated in

greater detail.

The validity of this additivity theory is also checked through a set of creep

and relaxation experiments. Plotted data at three temperatures conform

0

0.05

0.1

0.15

0.2

0.25

0 20406080100120

Time (min)

Strain (%/MPa)

4.1 MPa, 50°C, %RH 4.9 MPa, 60°C, %RH Model

Figure 19. Effect of step loading of temperature and stress on the creep strain.

Viscoelastic Characterization of Agro-based Plastic Composites 55

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to Equation (21). This reinforces the assertion that the creep stress-related

constants are independent of temperature (Figure 21).

Long-term tests (Figure 23) for the composite, according to the ASTM

standard for a thousand hours, were performed with 4.55 MPa and at

40C and variable RH. A model plot for a temperature of 40C/60% RH/

4.55 MPa is also plotted with the experimental data in the background

in Figure 23. The model data points are ensconced in between the data of

72 and 51% showing the validity of the model for 1000 h test:

"cðtÞ"pðtÞ

¼g0ð0:035ÞFðTÞþg1g2ð0:0036Þlog t

aT

1:45

%

Since the creep rate is extremely low and the plasticity of the material is only

2–5%, it is expected that the material viscoelastic property will not change

much for moderate conditions like this. So a 1000 h test should suffice for

moderate conditions. At least it shows that the shape of the model curve is

0

0.05

0.1

0.15

0.2

0.25

0 2000 4000 6000 8000

Time (min)

Strain (%/MPa)

Expt. Model

Figure 20. A three-step temperature loading experiment.

0

0.05

0.1

0.15

0.2

0.25

0 20 40 60 80 100 120 140

Time (min)

Strain (%/MPa)

40°C, %RH 50°C, %RH 60°C, %RH model model model

Figure 21. Creep and relaxation at several temperatures.

56 A. PRAMANICK AND M. SAIN

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similar to that of the experimental plots and these curves plateau after

certain hours of creep. So the model is definitely valid in the long run.

In fact, it is better suited for long-term test because of some inaccuracies in

the first few minutes of data collection.

VALIDATION OF THE CREEP MODEL

FOR ‘TWO-PHASE’ MATERIALS

In this model both gand a

values are culled from literatures for wood

and HDPE creep experiments. The a

0

value is related to a

through the

ratio of "

1

and "

0

. Tables 4–6 display the predicted calculated shift values for

the composites and the constituents. Based on Equation (24) and Tables 4–6

we construct the following equations to describe ‘two-phase’ creep behavior

under 40C/72% RH/4.5 MPa:

Creep ¼"cðtÞ

¼"01:22þ1:21:05"1log½t1:451:42

Table 4. Creep properties of the constituents.

Compliance

(MPa

1

)

Modulus (E)

(GPa)

Shear

modulus

(Gm) n

Creep

modulus,

1/"

1

g

f

,g

m

/unit

stress

Aspect

ratio (a)

Wood

(for rice husk)

8 NA 0.3 1.45 1/0.0008 1.12 2

HDPE 1.2 0.6 GPa NA 1.45 1/0.05 1.12 NA

Table 6. Theoretical ‘two-phase’ a

value calculations.

Material

E

a

(activation

energy)

(kJ/mol)

(1þ1/a

)/

10C

1/"

1

(¼1/creep

compliance)

(MPa) ¼stress

distribution

HDPE 90 2.36 20 (1/"

1

,m) 47.2

Wood 20 0.30 375 (1/"

1

,f) 112.5

Composite 30 0.41 197.5 (1/"

1

)80

Table 5. Comparison of experimental and theoretical creep constants

(stress related).

Methods

D

0

g

0

¼"

0

(%/MPa)

D

1

g

1

g

2

¼g"

1

(%/(MPa min

1.45

))

g/unit

stress

Theory of mixture 0.05 0.0050 1.20

Experimental 0.052 (average) 0.0060 (for 14% stress) 1.25

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The above equation is validated in Figure 22 where 1/a

value at 400Cis

1.4

2

and gvalue is 1.2. Since RH was 72% and the moisture shift is about

1.2/20% RH change (Figure 23), we had to include a 1.05 shift in the above

final expression. A cursory look at Figure 23 testifies that the shift due to the

changes in RH is uniform at low RH (21–72%). But it changes dramatically

at 90% RH. So the 1.2 value per 20% change in RH is valid only in the

limited range. We must recall that all the shifts are based on 20C/58% RH/

3.5 MPa.

CONCLUSION

We characterized and modeled creep behavior of agro-based plastic

composites with respect to stress and temperature. We selected a rice-based

HDPE composite for that purpose. Since this material is a ‘two-phase’ one,

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10,000 20,000 30,000 40,000 50,000 60,000

Time (min)

Strain (%/MPa)

23.5°C/58% RH 40°C, 21% RH 40°C, 72% RH

40°C, 51% RH 40°C, 93% RH Theoretical plot

Figure 23. Long-term depiction of model and experiment.

0

0.05

0.1

0.15

0.2

0 10,000 20,000 30,000 40,000

Time (min)

Compliance (%/MPa)

Expt. data, 40°C, 72% RH

Prediciton

Figure 22. Validation for creep at 40C, 72% RH.

58 A. PRAMANICK AND M. SAIN

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we took a two-step approach. In the first, the temperature–stress equivalence

was studied for the material, where the material was considered as ‘single

phase’. The material constants were determined with the help of extensively

modified Schapery’s model, where the temperature shift factor was also

incorporated. The constants in the model were validated against step-loaded

(temperature and stress) creep and recovery data. In this so-called ‘single-

phase’ characterization, the temperature effect was found to be linear but was

thermorheologically complex exhibiting vertical shifts. The activation energy

of creep chain relaxation is lower than the literature E

a

value of HDPE in

general. Despite being a two-phase composite material this composite shows

time–temperature superposition behavior. The interaction between tempera-

ture and stress is additive within a limited range of temperature and stress.

A cursory look at the moisture effect also was taken.

The second step of creep prediction consisted of applying ‘the theory of

mixture’ to the predictive model. In this case the material is considered as

‘two phase’ and the model has a universal application. We have touched

upon the fact that creep data for HDPE and wood from literature can be

used to predict the behavior of this composite when put into the said theory.

The experimental constants determined with the modified Schapery model

actually conforms well to the theoretically predicted constants even for

long-term creep.

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