Thermo-viscoplasticity of carbon black-reinforced thermoplastic elastomers

Published Date
International Journal of Solids and Structures

1 June 2009, Vol.46(11):2298–2308, doi:10.1016/j.ijsolstr.2009.01.015
Open Archive, Elsevier user license

Author 

• A.D. Drozdov ^,
• J. Christiansen

• Danish Technological Institute, Gregersensvej 1, DK-2630 Taastrup, Denmark

Received 28 March 2008. Revised 15 January 2009. Available online 28 January 2009. 

Abstract

Observations are reported on carbon black–filled thermoplastic elastomer (TPE) in uniaxial loading–unloading tensile tests with various strain rates (ranging from  to  ) at temperatures in the interval from 25 to 90 °C. A constitutive model is derived for the viscoplastic response of a TPE composite at three-dimensional deformations with finite strains. The stress–strain relations involve six adjustable parameters that are found by fitting the experimental data. It is shown that the model correctly describes the observations, and its parameters are affected by temperature and strain rate in a physically plausible way.

Keywords

Thermoplastic elastomers

Polymer composite

Viscoplasticity

Finite strains

Constitutive equations

---

1 Introduction

This paper is concerned with the experimental and theoretical analysis of the mechanical response of a carbon black-reinforced thermoplastic elastomer (TPE) at quasi-static deformations with finite strains.

Thermoplastic elastomers are an important class of polymers that combine mechanical properties of rubbers with high-speed processability and recyclability of thermoplastics (Holden et al., 2004 and Grady and Cooper, 2005). The experimental investigation focuses on the viscoplastic behavior of Thermoplast K reinforced with a relatively large amount (about 60–70 wt.%) of carbon black particles. The polymeric matrix of this composite consists of a hydrogenated styrene block copolymer (HSBC)-based thermoplastic elastomer with addition of polypropylene segments. A rather complicated chemical structure of the TPE composite and high concentration of filler ensures its strong thermal and chemical resistance. Due to these properties, Thermoplast K may be employed as a sealing material for low-temperature proton exchange membrane fuel cells (PEM FC) with operating temperatures up to 100 °C. From the standpoint of applications, the objectives of this work are (i) to study the mechanical response of Thermoplast K at various temperatures, and (ii) to develop a simple constitutive model for its thermo-viscoplastic response to be used in finite-element analysis of a fuel cell.

Mechanical properties of thermoplastic elastomers has attracted substantial attention in the past decade (Boyce et al., 2001a, Boyce et al., 2001b, Boyce et al., 2001c, Blundell et al., 2002, Blundell et al., 2004, Le et al., 2003, Krijgsman and Gaymans, 2004, Baeurle et al., 2005, Indukuri and Lesser, 2005, Qi and Boyce, 2005, Drozdov, 2006, Drozdov, 2007, Drozdov and Christiansen, 2006, Drozdov and Christiansen, 2007, Laity et al., 2006, Long and Sotta, 2006, Yi et al., 2006, Roland et al., 2007, Sarva et al., 2007 and Eceiza et al., 2008). Most of these studies, however, concentrated on the time- and rate-dependent behavior of unfilled TPEs at ambient temperature. Only a few works dealt with the viscoelastic and viscoplastic responses of thermoplastic elastomers reinforced with carbon black (Kurian et al., 1995, Datta et al., 1996, Katbab et al., 2000 and Yamauchi et al., 2005), and none of them investigated mechanical properties at elevated temperatures. From the viewpoint of fundamental research, the objectives of the present study are (i) to report observations in uniaxial tensile tests with finite strains on a carbon black-reinforced TPE in a relatively large range of temperatures, and (ii) to analyze the effects of temperature and strain rate on adjustable parameters in the stress–strain relations.

Derivation of the constitutive model is based on a homogenization concept, according to which a complicated morphology of a TPE composite is replaced with a simple structure that captures essential features of its mechanical response. Following common practice (Tomita et al., 2007), a thermoplastic elastomer is thought of as a permanent non-affine network of chains bridged by junctions. The assumption that the network is permanent means that detachment of active chains from their junctions and merging of dangling chains with the network are neglected. With reference to Green and Tobolsky, 1946 and Tanaka and Edwards, 1992, the latter is tantamount to the neglect of viscoelasticity of the equivalent network. Non-affinity presumes that junctions slide with respect to their reference positions under deformation, which implies that the viscoplastic response is associated with sliding (plastic flow) of junctions. To simplify the analysis, it is postulated that plastic flow is volume-preserving, the strain energy of a chain is described by the neo-Hookean formula, and the energy of interaction between chains is accounted for by the incompressibility condition for the equivalent network.

The exposition is organized as follows. Observations in uniaxial tensile loading–unloading tests and relaxation tests at various temperatures are reported in Section 2. A constitutive model in finite viscoplasticity of TPE composites is developed in Section 3. In Section 4, the stress–strain relations are simplified for uniaxial tension of an incompressible medium. Adjustable parameters in the governing equations are found in Section 5 by fitting the experimental data. Results of numerical simulation for simple shear of a TPE composite are reported in Section 6. Some concluding remarks are formulated in Section 7.

2 Experimental results

Carbon black-reinforced thermoplastic elastomer Thermoplast K TV5LVZ [density , melt flow index 12 g/10 min at 230 °C, elongation at break 520%, hardness (shore A) 50] was purchased from Kraiburg TPE GmgH (Germany). Dumbbell specimens for uniaxial tensile tests (ASTM standard D638) with cross-sectional area 10.2 mm × 3.4 mm were molded by using injection-molding machine Arburg 320C.

Mechanical tests were conducted with the help of universal testing machine Instron-5569 equipped with a thermal chamber and an electro-mechanical sensor for control of longitudinal strains in the active zone of samples. The tensile force was measured by a standard load cell. The engineering stress σ was determined as the ratio of the axial force to the cross-sectional area of specimens in the stress-free state.

The experimental program involved two series of tests. The first consisted of tensile loading–unloading tests (1 cycle) with the maximum engineering strain  conducted at the temperatures T = 25, 50, 70, and 90 °C with cross-head speeds of 1, 5, 10, 20, 100, and 150 mm/min. These cross-head speeds corresponded to the strain rates of  ,  ,  ,  ,  , and , respectively. In a test, a sample was stretched up to the maximum strain ϵ_max with a constant strain rate   and unloaded down to the zero stress with the strain rate  . Each test was performed on a new specimen. In all tests at elevated temperatures, samples were pre-heated in a thermal chamber at the required temperature for at least half an hour, and, afterwards, equilibrated in the testing machine for 10 min before deformation.

To assess repeatability of observations, several tests were carried out on three different specimens (appropriate data are not presented). The maximum difference between the engineering stresses measured on different samples did not exceed 4%. This distinguishes the mechanical response of carbon black-reinforced TPEs from that of filled rubbers, as cyclic preloading of specimens is necessary in the latter case to reach an acceptable level of repeatability of measurements (Haupt and Sedlan, 2001).

The range of temperatures was chosen to cover the entire interval of operating temperatures of a low-temperature PEM FC. The interval of strain rates was restricted by the requirement that the duration of a test did not exceed 40 min (from below) and by the maximal cross-head speed of the testing machine (from above). The maximum engineering strain in tensile tests ϵ_max was limited by the range of admissible strains for the extensometer (less than 100%).

Observations in cyclic tensile tests at various temperatures T are reported in Fig. 1, Fig. 2, Fig. 3 and Fig. 4, where the engineering tensile stress σ is plotted versus tensile strain ϵ. The following conclusions are drawn:

• 1.
  The stress–strain diagrams at loading are strongly affected by cross-head speed. The engineering tensile stress grows with strain rate  . The maximum increase in stress is about 27% at room temperature, 35% at T = 50, 40% at T = 70, and 65% at T = 90 °C, which means that the effect of strain rate on σ becomes more pronounced at elevated temperatures.
• 2.
  At room temperature, the stress–strain diagrams at unloading are practically independent of cross-head speed. With the growth of temperature, the influence of strain rate on σ at retraction becomes stronger. At the maximum temperature T = 90 °C, the tensile stress at unloading noticeably increases with  .
• 3.
  The residual engineering strain ϵ (the strain at which the tensile stress vanishes at unloading) is practically independent of strain rate and temperature. In all tests, this strain belongs to the interval between 10 and 15%.
• 4.
  Given a strain rate  , the engineering tensile stress is strongly affected by temperature. At the highest strain rate of  , the maximum tensile stress (reached at the maximum strain ϵ_max) equals 1.50, 1.13, 0.68, and 0.38 MPa at the temperatures T = 25, 50, 70, and 90 °C, respectively. This means that the maximum stress decreases by a factor of 4 with growth of temperature in the interval under consideration.

Fig. 1. Engineering stress σ versus tensile strain ϵ. Symbols: experimental data in loading–unloading tests with various cross-head speeds   mm/min at room temperature. Solid lines: results of numerical simulation.

Fig. 2. Engineering stress σ versus tensile strain ϵ. Symbols: experimental data in tensile loading–unloading tests with various cross-head speeds   mm/min at T = 50 °C. Solid lines: results of numerical simulation.

Fig. 3. Engineering stress σ versus tensile strain ϵ. Symbols: experimental data in tensile loading–unloading tests with various cross-head speeds   mm/min at T = 70 °C. Solid lines: results of numerical simulation.

Fig. 4. Engineering stress σ versus tensile strain ϵ. Symbols: experimental data in tensile loading–unloading tests with various cross-head speeds   mm/min at T = 90 °C. Solid lines: results of numerical simulation.

The other series of tests involved uniaxial tensile relaxation tests with the engineering strain   conducted at the temperatures T = 25, 50, 70, 90, and 110 °C. In each test, a specimen was loaded with a constant cross-head speed of 100 mm/min up to the required strain ϵ. Afterwards, a decrease in stress was measured as a function of time while the strain was preserved constant. Following the ASTM protocol for short-term relaxation tests (ASTM STP 676), the duration of relaxation tests  was chosen.

The experimental data in relaxation tests are reported in Fig. 5, where the engineering stress σ is plotted versus relaxation time  , where   is the instant when the relaxation process starts. Following common practice, the semi-logarithmic plots are used with  . The following conclusions are drawn:

• 1.
  The relaxation curves are typical of elastomers. They demonstrate a practically linear decay in tensile stress with logarithm of relaxation time.
• 2.
  The tensile stress strongly decreases with temperature at all relaxation times  . The slopes of the curves   decay with temperature.
• 3.
  The amounts of stress relaxing within the time   weakly decrease with temperature T and equal 34.6, 33.5, 31.8, 28.3, and 22.8% at the temperatures T = 25, 50, 70, 90, and 110 °C, respectively.

Fig. 5. Engineering stress σ versus relaxation time   in tensile relaxation tests with the engineering strain  . Symbols: experimental data at various temperatures T °C. Solid lines: guides for the eye.

3 Constitutive model

With reference to the homogenization concept (Bergstrom et al., 2002), a carbon black-filled thermoplastic elastomer (polymer composite with a complicated micro-structure) is thought of as an equivalent one-phase continuum. A non-affine network of flexible chains bridged by permanent junctions is chosen as the equivalent structure. To simplify derivations, we presume macro-deformation of the equivalent network to be volume-preserving. The incompressibility condition is in good agreement with available experimental data for the Poisson ratio (close to 0.5) of thermoplastic elastomers (Kojima et al., 2004).

3.1 Kinematic equations for sliding of junctions

Under deformation, junctions in a non-affine polymer network slide with respect to their reference positions. Denote by   the deformation gradient for macro-deformation and by   the deformation gradient for sliding (plastic flow) of junctions. According to the multiplicative decomposition of the deformation gradient, the deformation gradient for elastic deformation   reads

equation1

where the dot stands for inner product. The incompressibility conditions for macro-deformation and plastic flow together with Eq. (1) imply that the elastic deformation is volume-preserving.

Differentiating Eq. (1) with respect to time and introducing the velocity gradients by the conventional formulas

equation2

we find that

equation3

where the velocity gradient for elastic deformation   is given by

equation4

The left and right Cauchy–Green tensors for elastic deformation are determined by

equation5

where   stands for transpose. It follows from the incompressibility condition for elastic deformation that the third principal invariants of these tensors equal unity, whereas the first and second invariants read

equation6

where the colon stands for convolution, and I is the unit tensor.

Differentiation of the last equality in Eq. (5) with respect to time and use of Eq. (3)imply that

equation7

where

equation8

is the rate-of-strain tensor and

equation9

is the vorticity tensor for elastic deformation. Combination of Eqs. (5), (6) and (7)results in the formula

equation10

To describe plastic flow of junctions, it is postulated that

• 1.
  The rate-of-strain tensor for elastic deformation is connected with the rate-of-strain tensor for macro-deformation
  equation11
  by the linear relation
  equation12
  where ϕ is a scalar function to be determined in what follows,
• 2.
  The vorticity tensor for elastic deformation vanishes
  equation13

It follows from Eqs. (8), (9), (12) and (13) that

equation14

Insertion of Eq. (14) into Eq. (4) results in the expression

equation15

Substitution of Eqs. (1) and (2) into Eq. (15) implies the differential equation

equation16

To deduce another form of Eq. (16), we introduce the left and right Cauchy–Green tensors for macro-deformation

equation17

By analogy with Eq. (7), we find that

equation18

Combination of Eqs. (17) and (18) implies that

It follows from Eq. (2) that

Insertion of these expressions into Eq. (16) results in the differential equation

equation19

which expresses the deformation gradient for sliding of junctions by means of the deformation gradient for macro-deformation.

The approach based on Eqs. (12) and (13) slightly differs from the standard hypotheses in finite viscoplasticity of solid polymers. As there is no physically reasonable way to distinguish between elastic and plastic vorticity tensors, the conventional theories (i) postulate that the vorticity tensor for plastic deformation vanishes [which allows the vorticity tensor for elastic deformation   to be found from Eqs. (4) and (9)] and (ii) introduce algebraic relations between the rate-of-strain tensor for plastic deformation   and the rate-of-strain tensor for macro-deformation  and the Cauchy stress tensor Σ. Derivation of the latter equations leads, however, to some complications, as the tensor   on the one hand and the tensors   and Σ on the other are defined in different bases.

Within the present approach, kinematic equations are formulated for components  of the velocity gradient for elastic deformations  . These tensors are defined in the same basis as the tensor  , which implies that Eq. (12) is objective. It follows from Eqs. (12) and (13) that the deformation gradient   can be found from Eq. (19), which means that plastic flow in the equivalent network is entirely determined by history of macro-deformation.

3.2 Stress–strain relations

The strain energy of a polymer chain is determined by the conventional formula for a neo-Hookean medium

equation20

where μ stands for rigidity per chain. The strain energy per unit volume of a network equals the sum of strain energies of individual chains

equation21

where  , and N stands for the number of chains per unit volume. Eq. (21) is grounded on the assumption that the energy of inter-chain interactions is accounted for by means of the incompressibility condition only.

Eq. (20) coincides with an expression for the strain energy of a Gaussian chain developed within the statistical theory of rubber elasticity (in the latter case, the coefficient   is proportional to absolute temperature T). It is worth noting that Eq. (20)is adopted for its simplicity only, which means that no proportionality is presumed between rigidity of a chain and temperature of the network. Although a decrease in the elastic modulus μ with T may be explained within the concept of polymer networks with constrained junctions (Drozdov et al., 2008), we do not dwell on this issue in the present study.

Differentiating Eq. (21) with respect to time and using Eqs. (10) and (12), we obtain

equation22

The Clausius–Duhem inequality for isothermal deformation of an incompressible medium reads

equation23

where Q stands for internal dissipation per unit volume and unit time, and   denotes the deviatoric component of the Cauchy stress tensor Σ. Inserting Eq. (22) into Eq. (23) and assuming internal dissipation to vanish, we arrive at the stress–strain relation

equation24

where p denotes an unknown pressure.

3.3 Kinetics of plastic flow

To describe changes in the coefficient ϕ in Eq. (24) at active loading and unloading, we presume evolution of ϕ with time to be governed by strain energy of the equivalent network W. The growth of ϕ with time at active loading is determined by the differential equation

equation25

where   are adjustable parameters. Eq. (25) together with the initial condition   implies that the function   monotonically increases with time and reaches its ultimate value Φ at relatively large deformations. With reference to Eq. (12), this means that the rate-of-strain tensor for elastic deformation   coincides with the rate-of-strain tensor for macro-deformation   at the initial instant   (no sliding of junctions in the reference state), and it becomes proportional to   (with the coefficient of proportionality  ) at the steady regime of plastic flow. According to Eq. (21), the rate of increase in ϕ is proportional to the dimensionless strain energy , whereas the coefficient of proportionality decays due to sliding of junctions and vanishes when ϕ equals its maximum value Φ.

Eq. (25) may be treated as a kinetic equation for a chemical reaction of the second order. Although some justification for this order can be presented (sliding of junctions in the polymer matrix induces sliding of aggregates of filler, which, in turn, accelerates plastic flow in the rubbery phase), Eq. (25) is treated as a merely phenomenological relation.

Evolution of the coefficient ϕ at unloading is described by the differential equation similar to Eq. (25),

equation26

where  ,  , and   are adjustable parameters. The initial condition for Eq. (26) expresses continuity of ϕ at the instant when transition occurs from active loading to unloading.

Eq. (26) implies that the function ϕ increases with time at unloading. The first term in the right-hand side of Eq. (26) coincides formally with an appropriate term in Eq. (25), whereas the last term characterizes self-acceleration of plastic flow at retraction. This term is absent in Eq. (25) for deformation of a virgin material, but a similar term (with a negative coefficient) should appear in an analog of Eq. (25) for reloading of a thermoplastic elastomer under cyclic deformation. We do not dwell on this refinement as the present study deals with the first cycle of deformation only.

3.4 Adjustable parameters

Eqs. (24), (25) and (26) provide a set of stress–strain relations for an arbitrary three-dimensional deformation of a particulate composite with TPE matrix at finite strains. These equations involve six adjustable parameters:

• 1.
  The coefficient μ characterizes elasticity of the equivalent network.
• 2.
  The coefficients a, α, and β describe evolution of the rate of plastic flow at active loading and unloading.
• 3.
  The quantities   determine the rate of sliding of junctions in the steady regime of plastic flow under active deformation and retraction.

In general, these quantities depend on strain rate and temperature. To avoid formulation of an overly complicated constitutive model with 6 unknown functions of two variables, the following simplifications are introduced:

• (A)
  The elastic modulus μ is independent of strain rate. The effect of temperature T on μ is described by the linear equation
  equation27
  where   are positive coefficients.
• (B)
  The steady rate of plastic flow at loading Φ is independent of temperature T and weakly (logarithmically) decreases with strain rate
  equation28
  where   are positive coefficients, and
  equation29
  stands for the equivalent strain rate for macro-deformation.
• (C)
  The steady rate of plastic flow at unloading Ψ is independent of strain rate, and linearly decreases with temperature,
  equation30
  where   are positive coefficients.
• (D)
  The coefficients a, α, and β are independent of temperature and proportional to the equivalent strain rate
  equation31
  where  ,  , and   are positive constants.

Our aim now is to examine validity of Eqs. (27), (28), (30) and (31) by fitting the experimental data depicted in Fig. 1, Fig. 2, Fig. 3 and Fig. 4.

4 Uniaxial deformation

Uniaxial tension of an incompressible specimen is described by the formulas

equation32

where   are Cartesian coordinates in the initial and actual states, respectively, and   stands for elongation ratio. The deformation gradient for macro-deformation reads

equation33

where   are basic vector of the Cartesian frame in the refeence state. Insertion of Eq. (33) into Eqs. (2) and (11) implies that

equation34

where

equation35

Transition from the initial state into the intermediate (stress-free) state is described by the equations similar Eq. (32),

where   are Cartesian coordinates in the stress-free state, and  is a function to be found. By analogy with Eq. (33), one can write

equation36

Substitution of Eqs. (33) and (36) into Eq. (1) yields

equation37

Combination of Eqs. (3) and (37) results in

equation38

According to Eq. (38), Eq. (13) is satisfied identically. Insertion of Eqs. (34) and (38)into Eq. (12) implies the differential equation

equation39

Substituting Eqs. (5) and (37) into Eq. (24), we find that

where

Excluding the unknown pressure from the boundary condition   and introducing the engineering tensile stress  , we arrive at the formula

equation40

Eqs. (39) and (40) together with Eqs. (25) and (26), where

equation41

describe uniaxial deformation of a TPE composite at tension–compression.

5 Fitting of observations

Adjustable parameters in the governing equations are found by fitting the observations reported in Fig. 1, Fig. 2, Fig. 3 and Fig. 4. Each set of experimental data is matched separately.

5.1 Loading

We begin with approximation of loading paths of the stress–strain curves, when the elongation ratio k obeys the differential equation

To determine the quantities μ, a and Φ, we fix some intervals  , where the best-fit parameters a and Φ are assumed to be located, and divide these intervals into   sub-intervals by the points   with  , . For each pair  , the stress–strain relations are integrated numerically from   to   with  . The governing equations are presented in the form

equation42

Integration of Eq. (42) is performed by the Runge–Kutta method with the step . The modulus μ is found by the least-squares technique from the condition of minimum of the function

equation43

where summation is performed over all elongation ratios   at which observations are reported,   is the engineering tensile stress measured in a test, and   is given by Eq. (40). The best-fit quantities a and Φ are found from the condition of minimum of function (43). Afterwards, the initial intervals are replaced with the new intervals ,  , and the calculations are repeated three times. The best-fit material parameters μ, a, and Φ are plotted versus strain rate   in Fig. 6, Fig. 7and Fig. 8.

Fig. 6. Elastic modulus μ versus strain rate  . Symbols: treatment of observations in loading–unloading tests at various temperatures T °C. Solid lines: their approximations by constants.

Fig. 7. Parameter Φ versus strain rate  . Symbols: treatment of observations in loading–unloading tests at various temperatures T °C. Solid line: their approximation by Eq. (28).

Fig. 8. Parameter a versus strain rate  . Symbols: treatment of observations in loading–unloading tests at various temperatures T °C. Solid line: their approximation by Eq. (31).

In accord with assumption (A), the dependencies of μ on   (found by matching observations at each temperature T separately) are approximated in Fig. 6 by constants. These constants are plotted versus temperature T in Fig. 12, where the experimental data are matched by Eq. (27). The coefficients   in Eq. (27)are calculated by the least-squares method and are listed in Table 1.

Table 1. Adjustable parameters for TPE composite.

Parameter	Dimension	Value
	MPa	2.19
	MPa K−1
	s−1
	s−1
	s−1
	K−1

Following assumption (B), the dependencies of Φ on   (determined at all temperatures T under consideration) are approximated by Eq. (28), where   is replaced with   and the coefficients   are found by the least-squares method. These coefficients are collected in Table 1.

With reference to assumption (D), the dependencies of a on   (found at all temperatures T) are approximated by the equation

where the coefficient   is calculated by means of the least-squares algorithm (its best-fit value is reported in Table 1).

5.2 Unloading

We proceed with fitting unloading paths of the stress–strain diagrams when the elongation ratio k is governed by the differential equation

For each temperature and strain rate, Eq. (42) are first integrated with the adjustable parameters collected in Table 1. Afterwards, we fix some intervals  ,  , and , where the parameters α, β, and Ψ are assumed to be located, and divide these intervals into   sub-intervals by the points  ,  , and  with  ,  ,    . For each triplet  , the stress–strain relations are integrated numerically from  to  . The governing equations are presented in the form

equation44

Integration is carried out by the Runge–Kutta method with the step  . The best-fit parameters α, β, and Ψ are determined from the condition of minimum of function (43), where   is given by Eq. (40). When these quantities are found, the initial intervals are replaced with the new intervals  ,  , , and the calculations are repeated three times.

The best-fit material parameters α, β, and Ψ are depicted versus strain rate   in Fig. 9, Fig. 10 and Fig. 11. With reference to assumption (D), the experimental dependencies   are approximated by the relations

equation45

where   are determined by the least-squares technique. Eq. (45) coincide with Eq. (31), where   is replaced with  . The best-fit values of   are reported in Table 1.

Fig. 9. Parameter α versus strain rate  . Symbols: treatment of observations in loading–unloading tests at various temperatures T °C. Solid line: their approximation by Eq. (31).

Fig. 10. Parameter β versus strain rate  . Symbols: treatment of observations in loading–unloading tests at various temperatures T °C. Solid line: their approximation by Eq. (31).

Fig. 11. Parameter Ψ versus strain rate  . Symbols: treatment of observations in loading–unloading tests at various temperatures T °C. Solid lines: their approximations by constants.

Fig. 11 shows that the parameter Ψ is practically independent of strain rate, in accord with hypothesis (C). The experimental data depicted in this figure are approximated by constants. The dependence of these constants on temperature is illustrated in Fig. 12, where the data are matched by Eq. (30). The coefficients   in Eq. (30)are determined by the least-squares algorithm and are listed in Table 1.

Fig. 12. Elastic modulus μ and parameter Ψ versus temperature T. Circles: treatment of observations in loading–unloading tests. Solid lines: their approximation by Eqs. (27) and (30).

5.3 Discussion

Fig. 1, Fig. 2, Fig. 3 and Fig. 4 demonstrate good agreement between the observations in tensile loading–unloading tests at various temperatures and the results of numerical simulation.

Despite some scatter of the data depicted in Fig. 5 and Fig. 11, these figures confirm that the parameters   are independent of strain rate. Fig. 12 shows that the effect of temperature on their average values (over tests with various strain rates at fixed temperatures) is adequately described by Eqs. (27) and (30).

Although Fig. 7 and Fig. 8 reveal an acceptable agreement between the experimental data and their approximations by Eqs. (28) and (31), the scatter of observations is noticeable. It may be explained by the fact that the loading paths of the stress–strain diagrams are rather simple curves that do not require three adjustable parameters for their fitting. Deviations of the data depicted in Fig. 9 and Fig. 10 from their approximations by Eq. (45) are noticeably lower, which means that kinetic Eq. (26)provides an adequate model for viscoplastic flow of junctions at unloading.

6 Numerical simulation

To assess ability of the constitutive equations to predict mechanical properties of TPE composites, we concentrate of their thermo-viscoplastic response at simple shear. The importance of our analysis for applications is grounded on the assumption that shear provides the main mode of deformation of sealing materials in fuel cells (Wang and Wang, 2005 and Malzbender et al., 2007). From the standpoint of fundamental research, the study of simple shear allows basic hypotheses of the model to be examined for macro-deformations with non-vanishing vorticity tensors.

Simple shear of an infinite layer is described by the equations

where   stands for shear. The deformation gradient for macro-deformation reads

equation46

Insertion of Eq. (46) into Eqs. (2) and (11) yields

equation47

where   is given by Eq. (35). Combination of Eqs. (29) and (47) implies that

equation48

Differentiating the first equality in Eq. (5) with respect to time and using Eq. (3), we obtain

Combination of this relation with Eq. (14) results in the differential equation

equation49

The initial condition for Eq. (49) reads  . We search the tensor   in the form

equation50

where   are unknown functions of time that obey the incompressibility condition

equation51

Substitution of Eq. (50) into Eq. (49) results in the differential equations

equation52

It follows from the third equality in Eq. (52) that

equation53

The first two equalities in Eq. (52) imply that the functions   coincide. This conclusion together with Eqs. (51) and (52) yields

equation54

Combination of the last equality in Eq. (52) with Eq. (54) results in the differential equation

equation55

Substitution of Eq. (49) into Eq. (24) implies that

where the shear stress Σ reads

equation56

Bearing in mind Eqs. (50), (53) and (54), we find that

It follows from Eqs. (25), (26), (31), (48) and (55) that the constitutive equations for active loading with a constant shear rate   read

equation57

and the governing equations for unloading with the shear rate   are given by

equation58

To assess intensity of stresses at cyclic loading of the TPE composite, numerical simulation is performed of Eqs. (56), (57) and (58) for simple shear with the strain rates   and the maximum shears  , 2.0, and 2.5 at the temperatures T = 60, 80, and 100 °C. Integration of the governing equations is carried out by the Runge–Kutta method with the step  . The stress Σ is plotted versus shear k in Fig. 13, Fig. 14 and Fig. 15. The following conclusions are drawn:

• 1.
  The shear stress Σ monotonically increases with k at active loading and decreases at unloading. The functions   are strongly nonlinear both at loading and unloading.
• 2.
  The stress–strain diagrams are substantially affected by temperature. The maximum stress noticeably decreases with T. For example, the shear stress Σ at  equals 4.31, 2.96, and 1.56 MPa at the temperatures T = 60, 80, and 100 °C, respectively.
• 3.
  Given a temperature T, the shear stress increases with strain rate, but this growth is relatively weak. When the strain rate increases by 5 orders of magnitude, the maximum stress (at  ) grows by 90% at all temperatures under consideration.
• 4.
  The residual shear k (at which Σ vanishes under retraction) strongly increases with maximum shear  . This quantity is practically independent of temperature, and it equals 0.53, 1.26, and 1.90 for cyclic deformation with the strain rate  , and 0.62, 1.34, and 1.98 for loading–unloading with the strain rate  .

Fig. 13. Shear stress Σ versus shear k in loading–unloading tests at T = 60 °C. Solid lines: results of numerical simulation for shear with the strain rates   (upper curves) and  (lower curves) and various maximum shears  .

Fig. 14. Shear stress Σ versus shear k in loading–unloading tests at T = 80 °C. Solid lines: results of numerical simulation for shear with the strain rates   (upper curves) and  (lower curves) and various maximum shears  .

Fig. 15. Shear stress Σ versus shear k in loading–unloading tests at T = 100 °C. Solid lines: results of numerical simulation for shear with the strain rates   (upper curves) and  (lower curves) and various maximum shears  .

Although the above results seem plausible, they should be treated with caution for two reasons: (i) observations were obtained in uniaxial tensile tests with the maximum elongation ratio  , whereas numerical simulation was performed for shear tests with the larger maximum shear  , and (ii) adjustable parameters for unloading α, β, and Ψ may, in general, depend on  , but the effect of maximum deformation on these quantities was not investigated experimentally.

7 Concluding remarks

Observations have been reported in uniaxial tensile tests on carbon black–filled thermoplastic elastomer Thermoplast K at various temperatures in the range from ambient temperature to 90 °C. The experimental data reveal an interesting phenomenon: (i) the unloading curves measured at various strain rates practically coincide at room temperature, (ii) the difference between these curves grows with temperature, and (iii) it becomes significant at the highest temperature under consideration.

Constitutive equations have been developed for the viscoplastic response of TPE composites. With reference to the homogenization concept, a composite is treated as an equivalent non-affine network of flexible chains linked by permanent junctions. Non-affinity means that macro-deformation induces sliding of junctions with respect to their reference positions. The effect of strain on the rate of sliding is described by Eqs. (25) and (26).

Modeling a carbon black-reinforced composite as an equivalent one-phase continuum noticeably simplifies derivations and allows the number of adjustable parameters to be reduced substantially. It leads, however, to the neglect of some effects driven by deformation-induced evolution in distribution of filler (Klüppel, 2003). In particular, this approach disregards mechanically induced anisotropy of filled elastomers under cyclic loading. The latter can be described by means of the constitutive models developed by Ihlemann, 2005, Diani et al., 2006a, Diani et al., 2006b and Itskov et al., 2006.

Stress–strain relations for an arbitrary three-dimensional deformation with finite strains are derived by using the laws of thermodynamics. These equations involve 6 material parameters that are found by matching experimental data in tensile loading–unloading tests with various strain rates. Good agreement is demonstrated between the observations and the results of numerical simulation. An advantage of the model is that loading and unloading paths of the stress–strain diagrams are described by similar governing equations (25) and (26).

Phenomenological equations (27), (28), (30) and (31) are proposed to describe the effect of temperature and strain rate on the adjustable parameters. Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11 and Fig. 12 reveal that these relations are satisfied with an acceptable level of accuracy.

The constitutive equations have been applied to predict the effect of temperature and strain rate on the shear stress at simple shear of a TPE composite. The results of numerical simulation show that the shear stress noticeably grows with strain rate and decreases with temperature. It is revealed that the residual strain after unloading to the zero stress strongly increases with maximum strain under loading. The curves presented in Fig. 13, Fig. 14 and Fig. 15 appear to be physically plausible, which may serve as a justification of assumptions (12) and (13) that slightly differ from the conventional hypotheses in finite viscoplasticity of solid polymers.

As our purpose is to develop simple stress–strain relations to be employed in finite-element analysis, a number of assumptions have been made in the derivation of constitutive equations. The most important among them are that (i) deformation of the equivalent polymer network is volume-preserving [which restricts the maximum intensity of macro-deformation as carbon black-reinforced rubber-like materials violate the incompressibility condition at very large deformations due to debonding of filler particles from the matrix (Gurvich and Fleischman, 2003)], and (ii) junctions in the equivalent network are permanent (with reference to Green and Tobolsky, 1946and Tanaka and Edwards, 1992, this means that the viscoelastic response of the polymer composite is disregarded). Observations depicted in Fig. 5 show that the latter assumption is acceptable from the engineering standpoint for macro-deformations with strain rates of order of   and higher (when relaxation of stresses in a test does not exceed 5%).

Acknowledgments

This work was partially supported by The Danish Energy Authority through project ENS–33033–0096.

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