THERMAL EFFECTS IN VISCOELASTIC MATERIALS
1
A Hussain, 2R J Greene and 3R A Tomlinson
University of Sheffield, Department of Mechanical Engineering
Sheffield, S1 3JD, UK
1
2
[email protected], [email protected], [email protected]
ABSTRACT
The thermal and thermoelastic effects of rubber are explored. A comparison is made between the theory of
thermoelasticity for elastic solids and that for rubber materials. The problems associated with using thermoelastic stress
analysis techniques for rubber are highlighted. Experimental results confirming the differing behaviour of aluminium and
rubber under cyclic load are presented.
Introduction
The science of thermoelasticity is a study of the coupling phenomenon between thermal energy and mechanical
deformation in elastic materials. The investigation of the non-contacting thermoelastic effect using infrared technology was
first reported in 1967 by Belgen [1,2]. This technology has developed to a stage where highly sensitive array detectors are
now widely used for thermoelastic stress analysis (TSA) and a variety of thermographic and thermal NDE studies [3].
Array systems such as the DeltaTherm system may be used to obtain full-field maps of the minute differential temperature
on the surface of a component due to the thermoelastic effect. In addition, these systems are also capable of providing
static thermal images of the absolute temperature of the component at the time the image is captured.
The majority of research using TSA has involved metals and composite materials, although early work on the
thermoelastic effect in 1805 was using rubber [4]. Little research using TSA has been performed on such viscoelastic
materials but the advanced technologies now available offer the opportunity to study in detailed both the temperature and
thermoelastic effects of such materials.
Rubbers behave differently than other materials in many ways, notably their high elastic properties and their thermoelastic
behaviour. A piece of rubber can extend more than 500% of its initial length without breaking, in contrast to metals which
will normally permanently deform, if not fracture, before even reaching 25% elongation. The coupling effects between
deformation and thermal energy of Indian rubber was realised and reported by Gough [4]. Gough’s observation on the
thermodynamics of rubber was confirmed by Joule [5] using vulcanised rubbers in 1859. Two effects that were
conclusively found were that rubber contracted upon heating (reversibly) when held in constant load and heated up
(reversibly) upon stretching. These effects were referred to as Gough-Joule effects.
Rubbers are described as entropic elasticity whereas most other solids behave as energetic elastic. The ‘entropic’ force in
rubber is a consequence of the changes in entropy, from a less probable state of lower entropy when extended to a more
probable state of higher entropy at its natural length. For metals and other solids on the other hand, the ‘energetic’
elasticity arises from positional perturbation of their molecules from equilibrium state hence changing their internal energy
state. [6,7,8] Since the relationship of deformation, energy and entropy is the subject of thermodynamics, many articles
have been written on the subject using the rubber as a medium for teaching thermodynamics [9,10,11,12,13]. An excellent
account on the development on the theory of rubber elasticity can be found in the work of Treloar [14].
In this paper the thermal and thermoelastic effects in rubber are investigated. First the comparison between the
thermoelasticity effects for elastic solids and rubber-like solids will be presented. Next the theory will be extended to
thermoelastic stress analysis. Then some experimental findings on the thermal and thermoelastic effects will be reported
and discussed.
Comparison between the thermoelasticity of elastic solids and rubber-like solids [15, 16]
The first law of thermodynamics, for a stationary closed system or a fixed mass can be expressed as
∆U = Q + W
(1)
where
∆U = the net change in the internal energy
Q = the net heat transfer to the system
W = the net work done on the system.
In the differential form and on a unit volume basis,
du = δq + δw
(2)
For homogeneous solid deformation, the work per unit volume is
δw = σ ij dε ij
(3)
The second law of thermodynamics gives the relationship between the entropy balance per unit volume s at constant
temperature, and is expressed by
ds ≥
∂q
,
T
(4)
where the equality holds for reversible processes and the inequality holds for irreversible processes.
Combining (2),(3) and (4) and assuming the reversible process, the internal energy change in the system is
du = Tds + σ ij dε ij
(5)
Introducing Helmholtz free energy function per unit volume
In total differential form, this gives
Substituting (5) into (7) yields
H = u − Ts
(6)
dH = du − Tds − sdT
(7)
dH = σ ij dε ij − sdT
(8)
H is a state function of strain and temperature, and using the relation for the total differential of H and comparing to (8)
therefore
 ∂H 

 ∂ε 
 ij T
σ ij = 
and
 ∂H 
s = −

 ∂T ε ij
(9)
Using the property of partial differential, this gives
 ∂s 
∂σ

 = − ij 
 ∂T 
 ∂ε 

ε ij
 ij T
(10)
Now, using the relation for the total differential of s, for s is a state function of strain and temperature, therefore
 ∂s 
 dε ij +  ∂s  dT
ds = 
 ∂ε 
 ∂T ε ij
 ij T
(11)
Combining (10) into (11) and substituting into (5), yields

 ∂σ  
 ∂s 
du = σ ij − T  ij   dε ij + T 
 dT
∂T ε ij
∂T ε pq 





(12)
But u is also a state function of strain and temperature, and using the relation for the total differential of u and equating to
(12), this gives
 ∂u 
∂σ

 = σ ij − T  ij 


 ∂ε 
 ∂T ε ij
 ij T
 ∂u 
 ∂s 

 = T

 ∂T ε ij
 ∂T ε ij
and
(13)
(14)
Each term in (14) describes another thermal property known as the heat capacity per unit volume of the solid at constant
strain Cε, which is equivalent to Cv. in the undeformed state,
 ∂u 
 ∂s 
Cε = 
 = T

 ∂T ε ij
 ∂T ε ij
(15)
For the elastic solids considering a general case in which the constitutive equation that relates the stress σij, strain εij and
temperature T can be expressed as
σ ij = 2µε ij + (λε kk − γ ∆T )δ ij
where
E
2(1 + ν )
Eα
γ=
1− 2ν
µ=
and
λ=
νE
(1 + ν )(1 − 2ν )
(i, j = 1,2,3),
(16)
are the Lamé constants,
E = the modulus of elasticity,
= Poisson’s ratio,
α = the coefficient of thermal expansion,
ν
δ ij
= Kronecker symbol = 1 if i = j and 0 if i ≠ j.
The components of the small strain tensor is expressed as
 ∂u
∂u j 

ε ij = 12  i +

∂
∂
x
x
j
i


(17)
noting that the higher orders have been ignored.
The change in the internal energy in the system can be obtained from (12). On the RHS the first bracketed term can be
solved from (16) and the partial derivative of the second term is simply (15), this yields
du = (σ ij + γTδ ij )dε ij + Cε dT
(18)
Integrating (18) using the limits (0 to εij) and (To to T) corresponding to the initial state and the deformed state yields
u −u o = 12 σ ijε ij +
Eα ε kk
(T + To ) + Cε (T − To )
2(1 − 2ν )
(19)
The entropy change in the system is obtained from (11) using equations (15), (10) and (16) and then integrating, yields
s −so =
Eα ε kk
T
+ Cε ln
(1 − 2ν )
To
(20)
Consider a simple uniaxial tensile case, choosing the direction of the load in such a way that the coordinate axes are
coincide with the principal directions, and assuming an isothermal process, then from (16) this gives
σ 11 = Eε11
and
ε 11 =
ε 11 + ε 22 + ε 33
1 − 2ν
(21)
Integrating (3) the work done on the system is
w = 12 Eε11
2
(22)
From (19), the change in the internal energy becomes
u −u o = 12 Eε 11 + To Eαε 11
2
(23)
From (20), the change in the entropy becomes
and the reversible heat generated is
s − s o = Eα ε11
(24)
q = To ( s − so ) = To Eα ε11
(25)
Now, substitute (23) (24) and (25) into (16), it is verified that the energy is conserved.
The constitutive equation most commonly used to describe the thermoelastic behaviour of rubber is obtained from the
statistical mechanics [15]. This equation cannot be deduced by coupling the thermodynamical and mechanical field
equations because no hypothesis on the structure of rubber molecules can be made [9].
The equation of state for rubber-like solids given by James and Guth [17] can be stated as
 L  Lo 2 
F = KT  −   
 Lo  L  
where
(26)
F is the force, K is a constant which depends on the composition of rubber network, L is the extended length, and Lo is the
unstretched length. The ratio of L to Lo is the extension ratio and is always kept the same for all temperatures [17]. In
addition, in deriving equation (26) restriction of incompressibility of the rubber is implied, which suggests that the
deformation takes place at constant volume.
Inserting F in place of σij, and the state variable L with εij (T remains) for the state functions u, s, and H, the equations
relating the energetic and entropic contributions accompanying a deformation of rubber can be derived.
In view of (3), the work done on the system in a small displacement taking PdV to be zero (constant volume condition) is
 ∂W 
∂W = FdL , hence F = 

 ∂L T
(27)
It has to be noted that even when the constant volume assumption is disregarded, the contribution of the term PdV at
atmospheric pressure is less than FdL by a factor of 10-3 or 10-4 [14].
Referring to (10), the entropy change per unit extension in terms of the measurable quantities is
F
 ∂S 
 ∂F 
  = −
 =−
T
 ∂L T
 ∂T  L
(28)
In view of (13), the internal energy change per unit extension yields
 ∂U 
 ∂F 

 = F −T
 =0
 ∂L T
 ∂T  L
(29)
The relationship between the internal energy, work and heat for elastic solids as given in (23), (24), and (25), can be seen
in Figure 1. Similar relationship but in terms of partial derivatives is shown in Figure 2 to depict the behaviour of rubber-like
materials whose constitutive equation is given by (26) and strictly within the assumptions as indicated.
(∂W/∂L)T
Force
Energy
∆U
W
T(∂S/∂L)T
Q
Strain
Extension (L/Lo)
Figure 1 Thermoelastic effects of elastic solids
Figure 2 Thermoelastic effects of rubber-like solids.
Note that (∂U/∂L)T=0
It can be seen from Figure 1 that all the energy contributions accompanying tensile deformation of elastic solids are
positive. In the sign convention used in deriving these relationships, the work done to the system and the heat absorbed
by the system are positive, indicating that the system cools down when stretched. For the case of compressive
deformation, the line of Q can be projected into the third quadrant. A negative value for Q signifies that the system gives
out heat, therefore the material heats up upon compression.
The thermoelastic response for the rubber-like solids is contrary to that of elastic solids. When a piece of rubber is
stretched, the material gives up heat. This effect is indicated by the line T(∂S/∂L)T or the heat change per unit extension as
plotted in Figure 2. When a rubber is subjected to rapid stretching where adiabatic conditions are assumed, then a
significant increase in temperature is attainable. This phenomenon was first described by Gough [4]. In this example the
contribution of internal energy is zero, since the constitutive equation (26) holds on the basis that rubbers are
incompressible materials. Justification of using this equation in this case is that the force-extension relation is established
from the basis of statistical theory in its elementary form [14]. Various models have been proposed which include the
terms to take account the compressibility of the rubber-like materials. In constant volume processes, the fractional
contribution of the internal energy to the force due to deformation as frequently found in the literature is expressed as the
ratio of fe/f. Experiments conducted in a constant volume environment are fairly complex, however fe/f of 0.2 has been
reported [7]. The fe/f in the present case is zero.
Extension to thermoelastic stress analysis, TSA
The application of the theory of thermoelasticity in the stress analysis of elastic solids has been well established. Detailed
account on the derivation of the equations involved has been given elsewhere [16]. The viability of the method in solving
various experimental stress studies has been appraised [18], and its practicality has been demonstrated in industrial
applications [19], and in the analysis of medical devices [20].
Briefly, the derivation of the equation for TSA follows from (20). On the condition where an adiabatic process is achieved
the basic equation for TSA can be written as
∆T = −
Eα ε kkTo
Cε (1 − 2ν )
(30)
noting that Cε is in unit volume. This equation valid for any elastic solid that is homogeneous, isotropic, whose behaviour
follows the constitutive equation (16) and the material properties are independent of temperature.
A long term objective of the present work is to establish a relationship for TSA that is applicable to rubber-like solids. The
behaviour of viscoelastic materials must be fully understood. Point of departure may involve the dependency of material
properties to temperature and contribution of heat generation inside the materials to temperature change.
Experiments on the thermoelastic effect
A test was carried out using aluminium 2024 and common fluoroelastomer rubber tensile specimens, having the thickness
of 1.6 mm for the aluminium and 2.1 mm for the rubber and the gauge area of 5.5 mm by 30 mm for both. The loading
machine was an MTS Table Top with a 5 KN capacity load cell. The machine was computer controlled and the same
system was used to record the period, displacement, and load, as defined by the operator. The driving frequency of 1 Hz
was chosen and 15 data points per cycle were recorded. Sample temperature during the deformation was measured using
DeltaTherm 1500 camera/detector using the standard 25 mm lens. A sequence of thermal images was captured by
utilising the ‘movie’ application available in DeltaVision software. The number of frames per cycle was equal to the number
of data points per cycle recorded by the MTS system, so that point to point synchronisation could be made during post
processing of the result. The Deltatherm camera was externally triggered, by manually starting the image recording and
the test machine simultaneously. The recording was done for 10 s. The camera was initially calibrated using a two-point
calibration with hot and cold temperatures. Two steel plates painted with black surface, one at room temperature and the
other heated slightly, and thermocouples were used to provide the reference temperatures. This type of calibration is a
standard procedure in TSA, and in doing so, the collected sample temperatures in thermoelastic unit can be converted into
Celsius unit if required. The aluminium sample was loaded in a load controlled mode with mean load of 350 N and the
amplitude of 344 N. The rubber sample was loaded in displacement controlled mode between in the range of 10 mm to 20
mm which resulting the load of 6 N to 30 N.
Results and Discussion
The loading and the temperature responses in the first three cycles for each sample are plotted as shown in Figure 3. The
load and the displacement were normalised with respect to the mean load and mean displacement, respectively. Two
contrasting thermoelastic behaviours can be seen although both aluminium and rubber samples were loaded in tension.
However both responded as predicted in the theory. In each cycle the temperature of rubber reaches maximum
corresponding to the maximum displacement. On the other hand the peak in aluminium temperature matches the valley of
the applied load.
Temp: Al
Temp: Rub
Force: Al
Disp: Rub
6.0
5.5
27.0
5.0
4.5
26.9
4.0
3.5
26.8
3.0
2.5
26.7
2.0
1.5
26.6
1.0
0.5
26.5
0.0
0
1
2
Applied load & disp (normalised)
Temperature (deg C)
27.1
3
Number of cycle
Figure 3 Thermoelastic effects accompanying deformations for aluminium and rubber
o
The average temperature difference of three cycles for aluminium and rubber measured from peak to peak were 0.082 C
and 0.088 oC, respectively. The temperature difference for aluminium is unusually large for the applied load and sample
dimensions. Cross checking with equation (30) using the material properties from [21], the temperature difference is in the
region of 0.02 oC for the applied load range. The significant temperature difference in rubber sample may be justified due
to its viscoelastic behaviour. In fact the mean temperature for rubber increases with every cycle as highlighted by the
horizontal line that has been drawn across the curve. A very slight increase in the mean temperature can also be observed
in the aluminium curve, but this trend was only temporary, only can be seen in the beginning stage of the cycle. The
possibility of the erroneous in the level of temperature difference displayed by DeltaTherm is largely due to in accuracy of
taking the reference temperature during the calibration of the camera/detector prior to the test. The thermocouple used
was accurate only to one tenth of a degree, lacking in providing correct minute temperature reading. Since a proper
calibration or temperature correction procedure is fairly difficult to accomplish therefore for quantitative analysis, stress or
strain calibration is recommended in TSA.
Conclusions
Using the quantitative equations and the laws of thermodynamics, the effects of elastic solids and rubber-like solids to
deformation have been shown. The reversible temperature-deformation responses under uniaxial cyclic load for aluminium
and rubber have been experimentally examined. Viscoelastic solids heat up while elastic solids cool down in tension and
vice versa as predicted by the theory.
Acknowledgments
This work is funded by the MARA Institute of technology, Malaysia. The rubber samples were supplied by MERL Ltd.
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