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1. INTRODUCTION AND BACKGROUND

To process agro-based thermoplastic composites,

40-60% w/w agro-particles are normally blended

with thermoplastics such as HDPE, PP, and PVC.

Products made out of these composites include

deck-boards, railings, railway ties etc. When in use,

a plastic or wood material deforms as a function

of stress and temperature1-3. Changes in stress lead

to nonlinear deformation in wood and plastics.

Similar behaviour would be expected in agro based

plastics composites. Temperature is known to

aggravate creep in both plastics and wood, and is

expected to do the same in the composites. When a

composite material is subjected to long-term stress in

a vacillating temperature (in weather, for example)

the creep deformation and non-linearity will be a

function of the temperature too. Since composites

are two-phase materials, a special approach is

needed to develop a predictive model that would

encompass temperature, stress and time in a single

expression.

Schapery’s work1 describes nonlinearity well for

plastics and composites but Schapery’s work is valid

for “single phase” materials. So the model is good

for calculating material creep constants, and as a

validation tool. Our approach has been to develop a

predictive model where the nonlinearity constants

are based on a “two phase” concept. The predicted

nonlinearity constants thereof will be validated

against the Schapery “single phase” experimental

constants, which deﬁ ne creep.

The effects of temperature on creep have been

investigated for many plastics2,4. In some ways

temperature and stress may act through a similar

mechanism. Crissman5 has pointed out that stress and

temperature dilations are responsible for the easier

movement of plastic macromolecules in amorphous

regions. The theory of temperature dilation led to

the famous Williams Landel Ferry (WLF) equation,

1This paper is dedicated to the memory of Amit Pramanick, who

passed away in December 2005.

2Professor, Wood Science and Technology, University of Toronto

Application of the “Theory of Mixtures” to Temperature – Stress

Equivalency in Nonlinear Creep of Thermoplastic/Agro-ﬁ bre

Composites

1A. Pramanick (deceased) and 2M. Sain

Centre for Biocomposites and Biomaterials Processing, Faculty of Forestry, 33 Willcocks Street,

University of Toronto, Toronto, Ontario, Canada M5S 3B3

Received: 14 April 2005 Accepted: 28 November 2005

SUMMARY

The viscoelastic characterization of agro-ﬁ ller based plastic composites is of paramount

importance for these materials’ long-term commercial success. To predict creep, it is

imperative to derive a relationship between deformation, time, temperature, and stress.

This work is the harbinger in modelling of the nonlinear creep behaviour of two-phase

materials, where an extended “theory of mixtures” has been used to describe all the creep

related parameters. The stress- and temperature-related shift factors were estimated in

terms of the activation energy of the constituents. 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. The model was validated under rigorous conditions

and is unique because it describes creep not through curve ﬁ ttings, but in terms of the

creep constants of the constituents. This constitutive model is not only a vanguard in the

prediction of long term creep of many biocomposites but also in the modelling of creep

under step loading of temperature.

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Polymers & Polymer Composites, Vol. 14, No. 5, 2006

Application of the “Theory Of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep…

the activation energy concept, and the TTSP (Time-

Temperature Superposition Principle) proposition.

The focus has always been to estimate the value of

the shift factor or the activation energy of a polymer

chain relaxation. The TTSP concept has been used

extensively in creep predictions of thermoplastics,

including HDPE. Elahi and Weitsman6,7 applied it

to chopped glass/urethane composites.

Although stress/temperature effects have been

investigated separately, the issue of stress-

temperature interactivity in creep has not been

resolved. So our goal here is to look at the interaction

between the two shift factors. At this moment

temperature shifts, like nonlinearity shifts, are

predicted only with the “single phase” approach.

These shift factors are currently determined

experimentally. We want to develop a “two phase”

stress/temperature model and also want to modify

Schapery’s expression (which does not involve a

temperature effect) extensively, to suit temperature

effects to the “single phase” approach.

The current modelling approaches themselves need

a revisit too. Current works deal with curve ﬁ tting

exercises where extensive analytical or numerical

solutions are proposed to solve the integrals and to

calculate the creep related constants8-13. Some articles

have focused solely on the pattern of deformation.

Sain et al.14 have shown that the creep behaviour of

wood-particles/plastic composites changes with an

alteration in the composition; Bledzki and Faruk15

have shown that the creep of PP/wood composites

changes with processing conditions; Xu et al.16

showed that the creep of wood particle/plastic

composite decreased with an increase in the ﬁ bre

content; and Bledzki and Gassan17 also asserted the

same for PP/cellulose composites.

Existing literature reports do not derive creep

expressions from the perspective of the constituents.

So these creep models are valid only for the

particular composite under consideration. Any

changes in its constituent materials would invalidate

these creep predictions. Our “two phase” generic

creep model, based on the ‘theory of mixtures’ in

the solid mechanics of composites2, will resolve

that problem. Both stress and temperature-related

shift factors and the interactions thereof would be

predicted with this theory. We have shown in our

previous work that deck-board materials made of

wood/plastics exhibit a Schapery type of nonlinear

behaviour18,19 under ambient conditions. We also

emphasize that our goal is to predict long term

performance, so all the shift factors will also be

validated for rigorous long-term experiments.

Broadly, our focuses are:

1) experimental determination of stress and

temperature related creep constants for rice

husk/HDPE composites with a modified

Schapery’s concept

2) a study of temperature and stress

interactivity

3) an estimation of the above creep constants

using the proposed “theory of mixtures”

4) a comparison of the predicted constants with

the experimental ones

5) the validation of the models for rigorous

long term experiments. A cursory look at the

moisture effect will also be taken.

2. THEORETICAL CONSIDERATIONS

2.1 The Theory of Flexural Deformation

In the four-point-bending mode (Figure 1), the

maximum tensile stress and strain occur at the

bottom surface of the beam. This stress can be

calculated with the formula:

σ = PLI1/(bd2 I2) (1)

where σ = stress at the bottom the beam, I1= bd3/12,

I2 = (bd3-b1d1

3)/12, P = load, L = span, b = breadth,

and d = depth of the material (Figure 2). Also note

that I1, I2 represent the moments of inertia of the cross

-sections of the solid and hollow bars respectively.

The strain can be calculated with the formula:

ε = 4.7dΔ/L2 (2)

where Δ is the deﬂ ection of the beam. Some

reports11,20,21 prefer to use compliance (= ε/σ) instead

of the absolute deformation.

2.2 The Theory of Creep

Creep is deﬁ ned as the total strain as a function of

time at the bottom surface of the beam. In order to

quantify the effect of stress on the material, creep

has been normalized:

D(t) = ε(t)/σD(t) = compliance (3)

456 Polymers & Polymer Composites, Vol. 14, No. 5, 2006

A. Pramanick and M. Sain

where σ represents a constant applied load and ε (t)

is the time dependent strain. The thermodynamic

theory permits us to express the nonlinear strain1

as follows:

εσ ψψ

σ

ττσ

() ( )tgD g D dg

dd

t

=+ −

′

∫

00 1

2

0

∆

(4)

In the above equations Do is the initial value of

the creep compliance, ΔD (ψ-ψ’) is the transient

component of the creep compliance, and ψ is the

reduced-time, calculated as follows:

ψ = ∫dt/aσ For (aσ > 0) (5)

ψ′ = ψ(τ) = ∫dt/aσ (6)

where go, g1, g2, and aσ are the material properties as a

function of stress. These stress-dependent properties

have speciﬁ c thermodynamic signiﬁ cance and go,

g1, and g2 reﬂ ects the dependence of the Gibbs free

energy on the applied stress22. These constants relate

to a reference point (ambient condition). Equation

(4) can be simpliﬁ ed into the following forms:

εσ ψσ

()tgD ggD=+

()

00 12

∆ (7)

εσ σ

σ

() logtgD ggD t

a

n

=+ ⎛

⎝

⎜⎞

⎠

⎟

00 121

(8)

The corresponding recovery equation is:

εσ

σ

ra

n

a

n

tgD t

att D tt() ( log log=+−

⎛

⎝

⎜⎞

⎠

⎟−−

()

21 1 ))

(9)

Figure 1. A typical four point bending setup

Figure 2. Cross sectional schematic view of the beam

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Polymers & Polymer Composites, Vol. 14, No. 5, 2006

Application of the “Theory Of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep…

2.2.1 Determination of Creep Coefﬁ cients

The nonlinear viscoelastic theory (Schapery’s

model) presents the constitutive behaviour of

polymeric materials in terms of Equation (7). From

the creep and recovery equations (Equation (9)) we

can obtain the creep constants, go, Do etc. According

to the Data Reduction Method proposed by11, the

constants can be determined by solving the said

equations. We have shown this in our previous

work18,19.

2.2.2 The Proposed Creep Model

Development Based on the ‘Theory of

Mixtures’

For discontinuous ﬁ bres with a low aspect ratio,

the composite stiffness can be expressed as follows,

where φ = volume fraction of the ﬁ bre, η= a factor

for shortness of the ﬁ bres, Ef = Young’s modulus

of the ﬁ bre, Em = modulus of the matrix, εo =

compliance2:

EEE E

fm o

=+− =111/( ( ) /

ηφ φ ε

and (10)

When under stress, the ﬁ bres and the matrix must

undergo the same strain as the whole composite does,

if the interface is not damaged at that level of stress.

Example of a hollow beam has been considered in

this study (Figure 3). Figures 4 and 5 show that the

interface was intact even after the material had been

subjected to 40 °C/ 30% RH/3.5 MPa for 1000 h. So

we propose the following creep equation:

Creep ==+

εσεε

c

n

tgt( ) / (log[ ])

01 (11)

where gε1 = 1/(η/gf

σ1ε1,f + 1/gm

σ2ε1,m) and gf

σ1 = gm

σ2 = g.

also, ó1 and ó2 are incremental stress distributions

in ﬁ brers and matrix, respectively.

If we incorporate the temperature effect:

Creep = εc(t,T)/σ = ε0 + Co/aT + gε1(log[t])n/aT (12)

Where gε1/aT = 1/(aT, f η/gfε1,f + aT, m /gmε1,m), aT = exp

(Ea (1/T-1/To)/R),

Co = creep at the end of one minute for 3.5 MPa

(ambient reference condition), and 1/aT =

temperature shift factor.

The value of η can be calculated as follows2:

η = 1 – (tanh(na)/na) (13)

where n = n2G/Eln(2R/d)

mf

=

where Gm = matrix shear stress, Ef = ﬁ bre modulus,

2R = distance between two adjacent ﬁ bres, and d =

diameter of each ﬁ bre and a = aspect ratio.

If we compare this creep expression with the

Schapery equation, we observe that:

Do go = εo and g1g2D1 = gε1

2.2.3 Temperature Factor

The temperature shift factor 1/aT can be incorporated

into the creep Equation (12). A thermally susceptible

material (such as a semi-crystalline plastic) exhibits

a vertical shift of the data plots along Y-axis that

needs to be considered23. We have incorporated that

part of the creep as Co/aT in Equation (10). The WLF

theory24 states that for an amorphous plastic this

shift factor can be calculated as follows:

log ()

acT T

cTT

T

g

g

=−−

+−

1

2 (14)

where c1 and c2 are material dependent constants

and Tg is the reference glass temperature.

The problem with this equation is that the value

of Tg is not very well deﬁ ned in semicrystalline

composites. The Arrhenius relationship is more

practical for this purpose: it deﬁ nes a shift factor

by

ln ( )aE

RT T

T

a

ref

=−

11

(15)

If temperature and stress act additively, we

should trust Equation (10). In the above, Ea is the

activation energy for chain relaxation and we have

values available for both plastic and wood. If the

temperature is changed during the course of the

creep experiment, the activity may be called the

“step loading of temperatures”. If at time t1 the

temperature is changed from T1 to T2, the following

relationships will hold:

458 Polymers & Polymer Composites, Vol. 14, No. 5, 2006

A. Pramanick and M. Sain

Figure 3. Creep set up according to ASTM standard

Figure 5. Fiber in the plastic; after creep

Figure 4. Fiber in the plastic; before creep

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Polymers & Polymer Composites, Vol. 14, No. 5, 2006

Application of the “Theory Of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep…

For t ≤ t1, when aT = aT, 1

εσε ε

cooT

T

n

tCag

at( ) / / (log )=+ +

1

1

1 (16)

For t > t1, when aT = aT2

εσεσ σε

ctCa

Tga

TaT

o

() / ( / / )

(l

=+ + −

011

1112

oog[ ]) (log[ ])

tngt

n

aT

+

σε

1

1 (17)

If the load is withdrawn at a time ta while the

temperature (= T) remains the same, the following

relationship should hold for the creep relaxation:

εεε

r

T

n

T

a

n

g

atg

att=−−

11

(log[ ]) (log[ ] )

(18)

3. EXPERIMENTAL

3.1 Materials

The materials were acquired from Extendex

Inc., Barrie, Ontario, Canada. They consisted of

contained 60% rice husk and 40% HDPE. A “two

step” procedure was followed to process them:

compounding and extrusion. First, the rice husk

was passed through a sieve of mesh size 16-80.

The average moisture content of the rice husk was

about 10%. The husks at the outlet of the dryer

achieved a moisture content of 1%. The dried husk

was then sent through a heated co-rotating twin-

screw extruder where the HDPE pellets were mixed

thoroughly and ejected as compounded pellets.

These pellets were passed through a conical proﬁ le

extruder. While the proﬁ le was being pulled out of

the extruder, a mist was used to cool the product

down. MAPE (maleated polyethylene) was used as

the coupling agent, which was mixed during the

pelletization in the extruder. The dimensions of

the cross-section of these rectangular proﬁ les were

600 mm x 40 mm x 40 mm whereas the thickness

was 5 mm.

3.2 Experimental Setup for Model

Validation and Constant Determination

A ﬂ exural creep testing rack was designed based

on ASTM D 611224. ASTM suggests the four-point

loading conﬁ guration (Figure 1) because plastic

lumber is relatively ductile and does not fail

within the maximum strain (3%) under three-

point loading. The span length was 600 mm (= L)

and the crosshead speed was 10 mm/minute. The

noses of the support and the loading beams were

conﬁ gured 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 was maintained

at each test specimen end. The deﬂ ection of the

specimen was measured at the midpoint of the load

span, at the bottom face of the specimen.

Both short-term and 1000-hour validation tests

(Figure 3) were performed at various percentages

(14 to 50%) of the maximum stress level. The

temperature was varied (20 °C-60 °C) to determine

the temperature shift factor of the material. The

reference temperature was 20 °C (68 °F) at a relative

humidity of 58%. Dead loads were placed in the

middle of the composite beams according to the

ASTM standard and the beams were allowed to

deform with respect to time. The creep tests were also

followed up with retraction of the load (recovery) in

some cases. Step loading was performed with both

load and temperature, where these two variables

were altered in a single experiment. A transducer

was placed at the bottom of the beams to monitor the

time dependent deﬂ ection. The whole set-up was

placed in a kiln room where the temperature and

humidity could be altered with a control panel.

4. RESULT AND DISCUSSION

4.1 Stress Effect

4.1.1 Nonlinearity in Creep of Plastic

Composites

The composite showed a significant stress

dependency, as is evident from Figure 6. The

compliance increased consistently with the stress

levels. So it was imperative to determine the stress

related constants (g1, g2, aσ etc.). This issue was taken

up in detail in our previous paper18,19 and will be

mentioned cursorily. It must be mentioned that all

the shift factors relate to a reference condition.

Calculation of Do, D1, g0, & g1, n, a0 and g2

Experimental Do and go

values are shown in Table 1.

It was assumed that the value of go was 1 at 14%

of the strength (25 MPa) level. The value of g1

represents the recoverable part of the creep after

the load was withdrawn. The value of g1 (Table 1)

decreased with increasing load. A high value of g1

460 Polymers & Polymer Composites, Vol. 14, No. 5, 2006

A. Pramanick and M. Sain

suggests quickness of creep recovery. However, the

value of g1 reached a plateau at a value of unity after

30% stress level. So a value of 1 was justiﬁ able at a

somewhat high stress level (> 30% ultimate stress

level). At 14% σu, it was assumed that the values

of aσ and g2 were unity. The D1 and n values were

determined using the standard Schapery equation

at 14% stress level (Equation (6)). (This value of n

is just an indicator of the shape of the curve. A far

more important value is that of the product D1g1g2

(= gε1) for creep prediction).

According to the theory of viscoelasticity, aσ

represents the horizontal shift. So a change in the

value of this constant signiﬁ es a change in the

viscosity caused by molecular movements and/or

stress dilation. We have also concluded that the

value of aσ was unity (19) and that the vertical

shifts (nonlinearity) of the plots were due to stress

(g1 x g2).

4.1.2 Model Validation for Stress Shift

At the ambient condition Equation (8), incorporating

the material values, assumes the following form for

this particular material:

εσ

co

tg gg t( ) / . . (log[ ]) .

=+0 04 0 0036

12

145

(19)

Equation (19) describes “single phase” material

creep behaviour. The creep constants in the above

equations were validated for several stresses

(Figures 7, 9 and 10). The above equation (19)

was also validated for a rigorous step loading

experiment (Figure 8), where 7.53 MPa stress was

followed by 5.59 MPa stress and an eventual stress

withdrawal19.

Our proposed “two phase” model (Equation (12))

was veriﬁ ed in Figure 11 and takes the following

form:

Figure 6. Stress dependency of the composite

Table 1. Stress related experimental creep coefﬁ cients for “single phase” composite

Stress level Compliance (MPa-1)g

0D0,%/MPa g1D1,%/(MPa.

minute^1.45)

n

14% 0.00035 1.45 0.00035 1.66 0.00036 1.45

22% 0.00040 1.14 0.00035 1.50 0.00036 1.45

27% 0.00054 1.55 0.00035 1.45 0.00036 1.45

30% 0.00056 1.60 0.00035 1.00 0.00036 1.45

50% 0.00045 1.30 0.00035 1.00 0.00036 1.45

40% 0.00054 1.54 0.00035 1.00 0.00036 1.45

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Polymers & Polymer Composites, Vol. 14, No. 5, 2006

Application of the “Theory Of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep…

Figure 7. Creep and recovery data for 14% stress level

Figure 8. Creep and recovery plot for stress step loading

Figure 9. Creep data for 27% of ultimate stress

Figure 10. Creep data for 40% of ultimate stress

462 Polymers & Polymer Composites, Vol. 14, No. 5, 2006

A. Pramanick and M. Sain

εσσ σ

ctgt( , ) . ( ) . (log[ ]) .

=+0 05 0 005 145

(20)

The constants in Equation (20) were derived from

pure HDPE (matrix) and wood (proxy for ﬁ bre) from

the literature3,9,26,27. The average stiffness modulus of

wood can be used to represent rice husk particles,

so we obtained a value of 2.5 GPa for the composite

stiffness using Equation (10). The relative wood-

creep is about 100% of the initial deformation in

12 years26. The estimated average creep compliance

for wood and HDPE9 ﬁ ts the predicted values

(Equation (11)) against the experimental values

(Figure 11) for 14% strength level (3.5 MPa). At

this stress level, the value of g was considered as

unity. As g is the increase in strain per unit stress

and the g values are stresses dependent, we have to

calculate the stress distribution in the particles and

matrix per unit MPa stress (Equation (12)):

σ = σ1 + σ2 = 2 MPa (for 50% volume of

particles)

gm

σ1 = (gf/η)σ2 = increase in length per 1 MPa stress

where gm = gf = 1.12 (Table 3) solving for g values,

we obtain: gm

σ1 = 1.21

For a stress transferring efﬁ ciency of η = 0.3, the g

value for the composite would be 1.21/1 MPa. This

yields a value of 1.8 (= 1.23.3) for the increment of

stress from 3.5 (14%) to 6.71 MPa (Table 2). The g

value based on prediction and the g (=g1g2) values

based on experiments are displayed in Table 3.

4.2 Temperature Effect

4.2.1 Temperature Stress Interaction

In order for Equation (12) to be valid, the stress

temperature effects should be additive (independent).

When the temperature was varied from 20 °C to 60 °C

a concomitant change in the stress levels (i.e. from

6.5 MPa to 3.5 MPa respectively) was also allowed.

Figure 12 depicts the temperature/stress effect for

creep. In order to segregate the temperature effect

Figure 11. Short term creep prediction at 14% stress and ambient condition (two phase)

Table 2. Experimental determination of g2 and ε1values, “single phase”

Stress %max

stress

ta,

minutes

Δεc/g1g2D1g1g2 = ε1

3.5 14% 1537 0.0190 1.00 0.0060

5.2 22% 1483 0.0270 1.37 0.0074

6.71 27% 2849 0.0427 1.90 0.0099

8.83 40% 252 0.0440 3.04 0.0100

11 50% 77 0.0410 4.44 0.0160

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Polymers & Polymer Composites, Vol. 14, No. 5, 2006

Application of the “Theory Of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep…

from the stress dilation, an extension of Equation

(6) was used:

log () log( / ) log

ε

σ

tgD

gg Dt a a

n

T

−=

()

−

00

12

∆

(21)

In Figure 12, log ((ε-go Do)/g1g2) values were plotted

along the Y-axis and log t values were plotted

along the X axis for the composite. The normalized

plots with respect to the stress factor g showed a

horizontal shift (temperature shift) along the time

axis. A point per point time-temperature shifting was

also attempted using the experimental compliance

data to obtain a smooth master curve in Figure 13.

This master curve conﬁ rmed that a creep test result

at 60 °C and 1000 minutes was equivalent to one

at 20 °C and 2 years. The master curve proves the

additivity of temperature and stress.

Activation energy (Ea) values show a shift of the

creep curves along the X-axis (Figure 14) due

to the changes in temperature (Equation 12).

Experimentally, a shift of 1.35 (=1/aT) per 10 °C is

equivalent to an Ea of 30 KJ/mol (Figure 14). From

various reports, it is clear that pure HDPE and

wood have Ea values of 90 kJ/mol28,29 and 20 kJ/mol

respectively. This conﬁ rms that the incorporation

of the rice husk particles reduces creep. The glass

transition temperature (Tg) of pure HDPE is –125 °C.

The Tg of the composite is difﬁ cult to establish.

We have attempted to calculate it with the W-L-F

equation where the universal constants c1 and c2

were assumed to be 17 and 52 respectively. The

values we obtained are given in Table 5. A high

Tg indicates a low sensitivity to temperature and

a low Ea.

4.2.2 Validation of the “Theory of Mixtures”

for Temperature–Stress Shifts

4.2.2.1 Temperature Shift

We have shown that the g values (stress shift) at

the ambient condition are compatible with the

experimental ones. This section will deal with the

validity of the theory in relation to stress-temperature

interaction and temperature shifts.

Incorporating the corresponding values for HDPE

and wood in Equation (16), we obtained the

composite shift factor aT as follows:

Table 3. Properties of the constituents

Compliance

(MPa-1)

Flexural

Modulus (E)

Shear modulus

(Gm)

ηn Creep modulus,

1/ε1,

gf, gm/unit

stress

Aspect

ratio (a)

Wood (for rice

husk)

8 GPa *N.A. 0.3 1.45 1/0.0008 1.12 2

HDPE 1.2 GPa 0.6 GPa N.A. 1.45 1/0.05 1.12 N.A.

Figure 12. Temperature and stress effects on creep

464 Polymers & Polymer Composites, Vol. 14, No. 5, 2006

A. Pramanick and M. Sain

Figure 13. Master curve for time-temperature-stress superposition

Figure 14. Energy of activation calculation

Table 4. Comparison of experimental and theoretical creep constants (stress related)

Methods Dogo= εo, %/MPa D1g1g2 = gε1 %/(MPa.minute^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

Table 5. Experimental temperature dependent creep coefﬁ cients, “single phase”

Material Model type 1/aTEaTg

Rice husk plastic comp Log Power Law 1.35/10 °C30kJ/mol -25 °C

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Polymers & Polymer Composites, Vol. 14, No. 5, 2006

Application of the “Theory Of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep…

Stress due to temperature shift on the composite =

σ = (1/aT -1) ε1 = 0.5(σf +σm), where σf and σm are the

stresses on the ﬁ bre and matrix respectively. But

strain on the composite = strain on the constituents

⇒ 1/a T, f -1=1/a T, m -1=1/ a T -1= σfεf = σmεm= σε1

(Table 6), where ε1 = 1/ (1/εf*η+εm)

Table 5 displays the calculated values where 1/aT

for the composite is 1.4. This theoretical value is

compatible with the experimental one (Table 6).

The predicted aT and g values were combined

in Equation (19) which describes the interaction

of these two shifts well and proves that they are

additive (Figures 15-17). Figure 18 shows the plots

for step loading of stress and temperature 4.1 MPa,

50 °C and 4.9 MPa, 60 °C. Here, predicted 1/aT and

g values were used from Table 6 and Table 4 and

in Equation (15). By comparing model equation

values with the experimental data in this ﬁ gure it is

obvious that the theory of additivity and the “theory

of mixture” hold well in this case too. It testiﬁ es

that the same g value of 1.2/MPa holds at 50 °C

as well as at 60 °C. The validity of this additivity

theory was also checked through a set of creep and

relaxation experiments. The data were plotted at

three temperatures and found to conform to the

Equation (14). This reinforces the assertion that

the creep stress related constants are independent

of temperature (Figure 25).

Table 6. Theoretical “two phase” aT value calculations

Material Ea

(activation energy)

(-1+1/aT)/10 °C 1/ε1

(=1/creep compliance)

σ,

MPa = stress distribution

HDPE 90 KJ/mol 2.36 20 (1/ε1,m) 47.2

Wood 20 KJ/mol 0.30 375 (1/ε1,f) 112.5

Composite 30 KJ/mol 0.41 197.5 (1/ε1)80

Figure 15. Validation of the two phase model at 20 °C, 6.5 MPa

Figure 16. Validation of the two phase model at 60 °C, 3.4 MPa

466 Polymers & Polymer Composites, Vol. 14, No. 5, 2006

A. Pramanick and M. Sain

Long-term tests (Figures 19-24) for a thousand

hours were also performed under 4.55 MPa and

3.37 MPa stresses and at various temperatures and

RH. It is evident that the model does very well at

all these conditions, except at very high humidity

(>80%) and at 23 °C. The problem with the latter

is probably an experimental one. At high RH, the

mechanism of creep is inﬂ uenced by the moisture

uptake. Due to the continuous moisture uptake, the

creep does not reach a plateau at long term, unlike

the situation at low RH. The creep rate at high RH

is actually constant thorough the creep period (just

like the diffusion behaviour itself).

Some plots at varying temperature and RH are shown

in Figure 2 and it is clear that the creep behaviour

changes radically at 80% RH/4.5 MPa/40 °C. This

also conﬁ rms that the creep actually remains

constant with time at high RH.

4.2.2.2 Inﬂ uence of Relative Humidity

The inﬂ uence of relative humidity on creep (RH) is

profound. RH affects the material creep constants

differently compared to temperature because of the

continuous diffusion throughout the creep process.

The impact of RH was not signiﬁ cant in between

21% and 60% RH (Figures 26-27). In this window

period the increase in creep, as we increased RH

from 30% to 60%, was insigniﬁ cant. The shift was

about 1.1 (= 1/aM) but the shift appeared to be so

high between 60% and 80% that the creep equation

actually changed from ΔD (t)∝ log (t) n to ΔD(t)

∝ t. In fact, this behaviour emulates the diffusion

behaviour itself (Figures 28-29). Figure 27 reveals

some good information, where the condition during

creep changes as follows:

At 10,000 minutes, the condition is changed from

30 °C/30%RH to 30 °C/60%; at 20,000 minutes it was

changed to 30 °C/ 80% RH; at 37,000 minutes, it was

Figure 17. Validation of the two phase model at 30 °C, 5.7 MPa

Figure 18. Effect of step loading of temperature and stress on the creep strain

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Application of the “Theory Of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep…

Figure 19. Creep prediction at ambient condition

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

Figure 21. Creep prediction at 40 °C, 51% RH

Figure 20. Creep prediction at 30 °C, 60% RH

468 Polymers & Polymer Composites, Vol. 14, No. 5, 2006

A. Pramanick and M. Sain

Figure 25. Creep and relaxation at three temperatures

Figure 24. Creep prediction at 55 °C, 21% RH

Figure 23. Validation for creep at 45 °C, 21% RH

changed to 40 °C/30% RH, and at 50,000 minutes,

it was changed to 40 °C/90% RH.

At the condition of 30 °C/80% RH the creep can be

described as ΔD (t) = 2.2 x 10-6 t whereas at 40 °C/90%

RH the same is ΔD (t) = 6.5 x 10-6. This yields a

1/aM value of about 3 for the concomitant jump of

10 °C and 10% RH. However, at this point it is not

clear if the jump is still 1.4 for 10 °C at this high

RH. We propose the following equation to predict

the behaviour of the last segment (37000 minutes

to 50,000 minutes) of creep as follows:

εc/s = .04+Co+.0052logt1.45 +2.2x10-6 (t-ta)-.0052 log

(t-ta) 1.45+1.4 log (t-tb) 1.45-2.2x10-6 (t-tb)

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Polymers & Polymer Composites, Vol. 14, No. 5, 2006

Application of the “Theory Of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep…

Figure 26. RH effect on creep in long-term situation

Figure 29. Diffusion pattern at 25 °C, 72% RH

Figure 28. Diffusion pattern at 25 °C, 93% RH

Figure 27. Long term step-loading depiction of model and experiment

470 Polymers & Polymer Composites, Vol. 14, No. 5, 2006

A. Pramanick and M. Sain

where ta = the time at which RH was changed to

80%, and tb = the temperature at which RH was

changed back to 30%. This equation complies

with the experimental data and proves that stress/

temperature additivity is valid even in a complex

step loading condition.

5. CONCLUSIONS

Creep models with “two phase” approach do not

exist, although a two-phase model would help us

to predict creep with the creep data relating to the

constituents A creep model was developed in this

work, to expresses creep in terms of wood and

HDPE creep properties. The model was validated

against the creep of rice husk based HDPE. The

creep performance was studied with an emphasis on

the temperature-stress equivalency. Some moisture

effects were also touched upon. Schapery’s model

was extensively modiﬁ ed to calculate the material

parameters such as temperature, stress shift

constants. The predictive parameters were validated

against the experimentally obtained parameters

for long term, and step loading processes. Despite

being a two-phase composite containing a plastic

and a ﬁ ller, this composite shows Time Temperature

Superposition behaviour.

It seems that the interaction between temperature

and stress is additive within the limited range of

temperature and stress, where the stress related

constants are unaffected by temperature. Preliminary

results show that at high relative humidity, the

model needs some changes. The following are the

achievements in short:

1. A “two-phase” creep model was developed

2. The temperature – stress interaction was

considered in the model

3. A protocol was developed to model step

loading of temperature in creep, which is

currently absent

4. The impact of humidity was explored for the ﬁ rst

time for the creep of agro based composites.

ACKNOWLEDGEMENT

We thank NSERC-Strategic Grant for ﬁ nancing this

project.

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