Application of the “Theory of Mixtures” to Temperature – Stress Equivalency in Nonlinear Creep of Thermoplastic/Agro-fibre Composites

Abstract

The viscoelastic characterization of agro-filler 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 fittings, 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.

Full text

©Rapra Technology, 2006
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.
455
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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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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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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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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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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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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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Polymers & Polymer Composites, Vol. 14, No. 5, 2006
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)
469
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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