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Mullins’ effect in thermoplastic elastomers: Experiments and modeling
Mechanics Research Communications 36 (2009) 437–443
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Mullins’ effect in thermoplastic elastomers: Experiments and modeling A.D. Drozdov * Danish Technological Institute, Plastics Technology, Gregersensvej 1, DK–2630 Taastrup, Denmark
a r t i c l e
i n f o
Article history: Received 25 September 2008 Received in revised form 2 December 2008 Available online 25 December 2008
Keywords: Thermoplastic elastomer Viscoplasticity Damage Mullins’ effect
a b s t r a c t Observations are reported on carbon black-reinforced thermoplastic elastomer in uniaxial cyclic tensile tests with various maximum strains at room temperature. A constitutive model is derived for a polymer composite that accounts for its viscoplastic response and damage. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. It is shown that the model correctly reproduce characteristic features of the Mullins effect. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction This paper deals with experimental investigation and numerical simulation of Mullins’ effects for a carbon black-filled thermoplastic elastomer (TPE). Observations in cyclic tensile tests on particle-reinforced rubbers reveal a number of phenomena that are conventionally referred to as the Mullins effect (Mullins, 1947): (i) unloading and reloading paths of stress–strain diagrams differ substantially (hysteresis of energy), whereas the residual strain (measured when the stress vanishes) remains rather small, (ii) under deformation with a strain-controlled program (the maximum strain per cycle is fixed), the stress monotonically decreases with number of cycles (strain-softening), and (iii) when stretching of a sample proceeds after several cycles of loading– unloading, the stress–strain curve rapidly reaches that for a virgin specimen under the same loading conditions (strainhardening). Despite an agreement that interactions between a rubbery matrix and aggregates of filler provide a physical ground for the Mullins effect (Bueche, 1961), there is no constitutive model that adequately describes the entire set of phenomena. This may be explained by the fact that (i) some effects contradict one another (softening versus hardening), (ii) the account for rubber–filler interactions requires modeling of deformation in the matrix and filler network separately which leads to a substantial increase in the number of material constants, and (iii) Mullins’ phenomena are accompanied by (a) stress relaxation, (b) material anisotropy, and (c) formation, growth, and coalescence of micro-voids and micro-cracks, whose role remains a subject of debate. To develop constitutive equations to be employed in numerical simulation, the attention is conventionally focused on one aspect of observations only. Hysteresis of energy under cyclic loading is adequately described by taking into account alteration (breakage and restoration) of links between polymer chains and between the matrix and aggregates of filler (Chagnon et al., 2006; De Tommasi et al., 2006). This approach is similar to the concept of transient networks in viscoelasticity of polymers (Drozdov, 1997;
* Tel.: +45 72 20 31 42; fax: +45 72 20 31 12. E-mail address: [email protected] 0093-6413/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.12.007
438
A.D. Drozdov / Mechanics Research Communications 36 (2009) 437–443
Tanaka and Edwards, 1992). A variant of this theory was proposed in Drozdov and Dorfmann (2001), where detachment– attachment of links was treated as transition of chains from flexed to extended conformation and vice versa. The latter is closely connected to the two-phase theory of composites (Johnson and Beatty, 1993). Strain-softening of specimens under cyclic deformation is conventionally predicted by accounting for damage (Govindjee and Simo, 1992). In the simplest (pseudo-elastic) model, a filled rubber is treated as an elastic continuum subjected to damage, whose evolution is described by the only internal variable (Ogden and Roxburgh, 1999; Horgan et al., 2004). More sophisticated models accounting for combined effects of damage, viscoelasticity, and viscoplasticity of particle-reinforced rubbers were developed by Chagnon et al. (2004), Diani et al. (2006), and Goktepe and Miehe (2005), to mention a few. In some sense, these constitutive equations may be thought of as variants of the concept of breakage–restoration of links, where breakage (fracture) of chains is characterized by one or several damage variables at the phenomenological level. The objective of this work is three-fold: (i) to study experimentally cyclic deformation of a TPE composite, (ii) to develop a constitutive model and to find adjustable parameters in the stress–strain relations by fitting the observations, and (iii) to perform numerical simulation of the mechanical response and to demonstrate that the model adequately predicts characteristic features of the Mullins effect. 2. Experimental procedure Thermoplastic elastomer Thermoplast K TV5LVZ (a hydrogenated styrene block copolymer–based TPE with addition of polypropylene segments reinforced with 60 to 70 wt.% of carbon black) was purchased from Kraiburg TPE GmgH (Germany). Dumbbell specimens for uniaxial tensile tests (ASTM standard D638) with the cross-sectional area 10.2 3.4 mm were molded by using injection-molding machine Arburg 320C. Mechanical tests were performed at ambient temperature by using a universal testing machine Instron–5569 equipped with 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 r was determined as the ratio of axial force to cross-sectional area of specimens in the stress-free state. The experimental program involved 7 uniaxial tensile tests conducted with a constant strain rate _ ¼ 6:67 102 s1. In the first test, a specimen was loaded up to the maximum strain max ¼ 1:0. In other tests, specimens were stretched up to strains max 1 ¼ 0:3, 0.4, 0.5, 0.6, 0.7, 0.8 with the strain rate _ , unloaded down to the zero stress with the strain rate _ (to avoid buckling of samples, retraction was performed down to the engineering stress rmin ¼ 0:01 MPa), reloaded up to the maximum strain max 2 ¼ 0:9 with the strain rate _ , and unloaded down to the zero stress with the strain rate _ . Each test was conducted on a new specimen. Observations are reported in Fig. 1, where the engineering stress r is plotted versus engineering strain . To avoid overlapping of data, observations for the last unloading are omitted in Fig. 1A (as an example, they are presented in Fig. 1B). The following conclusions are drawn: (i) the stress–strain diagrams at loading and retraction are strongly nonlinear, (ii) the stress–strain curves at first loading and reloading differ substantially, (iii) the residual strain (at which the stress vanishes at unloading) monotonically grows with max 1 , and (iv) the hysteresis energy (area between unloading and subsequent reloading paths) strongly increases with max 1 . 3. Constitutive model With reference to the homogenization concept (Bergstrom et al., 2002), a particle-reinforced TPE is thought of as an equivalent continuum whose response coincides with that of the composite. An incompressible, permanent, non-affine
Fig. 1. Engineering stress r versus tensile strain . (A) Symbols: experimental data in tensile tests with various maximum strains max 1 (solid line – loading, – max 1 ¼ 0:3, – max 1 ¼ 0:4, I – max 1 ¼ 0:5, } – max 1 ¼ 0:6, – max 1 ¼ 0:7, – max 1 ¼ 0:8). (B) Circles: experimental data in a cyclic tensile test with max 1 ¼ 0:3, max 2 ¼ 1:0. Solid line: results of numerical simulation.
A.D. Drozdov / Mechanics Research Communications 36 (2009) 437–443
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network of chains is chosen as the equivalent medium. Sliding (plastic flow) of junctions between chains under deformation of the non-affine network reflects its viscoplastic behavior. Unlike the standard approach, where damage characterizes a decrease in the elastic moduli (due to breakage of chains), damage serves in the present model as a measure of acceleration of plastic flow of junctions (induced by growth of free volume between chains). Elastic deformation of a chain in an equivalent network is determined by means of the multiplicative decomposition formula
Fe ¼ F F1 p ;
ð1Þ
where F, Fp , Fe denote deformation gradients for macro-deformation, sliding of junctions, and elastic deformation, respece F1 tively, and the dot stands for inner product. The velocity gradient for elastic deformation L e ¼ dF e reads dt
Le ¼ L Fe lp F1 e ;
L¼
dF 1 F ; dt
lp ¼
dFp 1 F : dt p
The rate-of-strain and vorticity tensors for macro-deformation and elastic deformation, respectively, are introduced by the formulas
D¼
1 ðL þ L> Þ; 2
W¼
1 ðL L> Þ; 2
De ¼
1 ðLe þ L>e Þ; 2
W
e
¼
1 ðLe L>e Þ; 2
where > stands for transpose. To describe sliding of junctions, it is postulated that (i) the rate-of-strain tensor for elastic deformation is proportional to the rate-of-strain tensor for macro-deformation
De ¼ ð1 /ÞD;
ð2Þ
where / is a scalar function, and (ii) the vorticity tensor for elastic deformation vanishes, We ¼ 0. The left and right Cauchy– Green tensors for elastic deformation are given by
Be ¼ Fe F>e ;
Ce ¼ F>e Fe :
ð3Þ
The incompressibility condition for elastic deformation implies that the third principal invariants of these tensors equal unity. Their first and second principal invariants read J e1 ¼ Ce : I and J e2 ¼ C1 e : I, where I is the unit tensor, and the colon stands for convolution. Differentiating these relations with respect to time and using Eqs. (2) and (3), we find that
dJ e1 ¼ 2ð1 /ÞBe : D; dt
dJ e2 ¼ 2ð1 /ÞB1 e : D: dt
ð4Þ
Each chain in the equivalent network is modeled as a flexible chain with constrained junctions. Two types of constraints are introduced (Erman and Flory, 1978; Erman and Monnerie, 1989): (i) positions of a chain ends (determined by vectors X1 and X2 ) do not coincide with positions of the junctions (characterized by vectors Q 1 and Q 2 ); the vectors xm ¼ Xm Q m (m ¼ 1; 2) are independent Gaussian random vectors with the zero mean and a given standard deviation, and (ii) the distance between the center of mass of a chain and the middle point of the vector between junctions Q ¼ Q 2 Q 1 is fixed. Under stands for an analog ½ðJ e1 3Þ þ Að3J 2e1 4J e2 15Þ, where l these conditions, the strain energy of a chain reads w ¼ 12 l of the chain rigidity, and A is a dimensionless positive constant (Drozdov and Christiansen, 2007a). Neglecting the energy of inter-chain interaction (this energy is accounted for by means of the incompressibility condition), we calculate the strain energy density per unit volume of the equivalent network as the sum of strain energies of individual chains
W¼
l 2
½ðJ e1 3Þ þ Að3J 2e1 4J
e2
15Þ;
ð5Þ
N, and N stands for the number of chains per unit volume. Inserting Eq. (5) into the Clausius–Duhem inequality where l ¼ l þ R0 : D P 0, where Q stands for internal dissipation per for isothermal deformation of an incompressible medium Q ¼ dW dt 0 unit volume and unit time, and R denotes the deviatoric component of the Cauchy stress tensor R, using Eq. (4), and assuming internal dissipation to vanish, we arrive at the stress–strain relation
R ¼ pI þ lð1 /ÞðW 1 Be W 2 B1 e Þ;
W 1 ¼ 1 þ 6AJ e1 ;
W 2 ¼ 4A;
ð6Þ
where p stands for an unknown pressure. To describe evolution of the coefficient / at cyclic deformation, we distinguish (i) the first loading of a virgin material, (ii) its unloadings, and (iii) subsequent reloadings. Under deformation of a virgin material, / obeys the equation
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / ¼ U0 ½1 expða Je1 3Þ;
ð7Þ
where a and U0 are adjustable parameters. Eq. (7) means that / vanishes in the reference state, monotonically increases with time at active loading, and reaches its ultimate value U0 at large deformations. Evolution of / at unloading is governed by the differential equation
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d/ ¼ að1 /Þ2 ðJ e1 3Þ þ b/; dt
ð8Þ
where a and b are positive parameters. The first term in Eq. (8) characterizes an increase in / driven by strain energy of chains, while the other reflects self-acceleration of plastic flow at retraction. The coefficients a and b remain constant along each unloading path, but they are affected by elastic deformations at the instants when unloading starts. Appropriate dependencies are described by
log a ¼ a0 a1 logðJ e1 3Þ;
log b ¼ b0 b1 logðJ e1 3Þ;
ð9Þ
where am and bm (m ¼ 0; 1) are constants. Changes in / under reloading are determined by the differential equation
d/ ~ n ~ ð1 /Þ2 ðJ e1 3Þ b/ ¼ a dt
ð10Þ
~ and n. The first term in Eq. (10) describe a decrease in / induced by strain energy of chains, ~ , b, with positive parameters a while the other reflects self-deceleration of the sliding process at reloading. The exponent n in Eq. (10) is treated as a con~ remain constant along each reloading path, but depend on elastic ~ and b stant (n ¼ 4 is set in simulation). The coefficients a deformations at the instants when reloading starts. By analogy with Eq. (9), we set
~¼a ~0 a ~ 1 logðJ log a
e1
3Þ;
~¼b ~0 b ~1 logðJ 3Þ; log b e1
ð11Þ
~m (m ¼ 0; 1) are constants. ~ m and b where a Under uniaxial tension with small strains, Eqs. (7), (8), and (10) are transformed into the kinetic equations
d/ ~ð1 /ÞðU0 /Þ; ¼a dt
d/ ¼ A0 ð1 /Þ2 þ b/; dt
d/ ~ 0 ð1 /Þ2 b/ ~ n ¼ A dt
ð12Þ
pffiffiffi ~ 0 ¼ 3a ~ ¼ a 3d=dt, A0 ¼ 3a2e , A ~ 2e , where stands for tensile strain, and e denotes the elastic strain. All differential with a equations (12) have a similar structure that is typical of evolution equations in chemical kinetics. The quadratic terms in these relations reflect interactions between different species (the matrix and filler particles for a reinforced TPE), whereas the terms proportional to / and /n characterize acceleration (slowing down) of the sliding process induced by inter-chain interactions in the matrix. Cyclic loading of a particle-reinforced TPE induces its damage observed as initiation and growth of micro-voids and micro-cracks. Unlike conventional models that associate damage with a decrease in elastic modulus, we suppose that damage accelerates sliding of junctions, while l remains constant (at low-cycle loading). Under deformation with a strain exceeding the maximum strain in the previous cycle max , the function / is governed by Eq. (7), where coefficient U0 (that corresponds to a virgin material) is replaced with U > U0 . The coefficient of proportionality g in the formula U ¼ gU0 is treated as a measure of damage. Its dependence on elastic deformation at the instant when reloading starts is described by the equation
log g ¼ g0 þ g1 logðJ e1 3Þ;
ð13Þ
where gm (m ¼ 0; 1) are constants. Transition from the regime of reloading governed by Eq. (10) to that characterized by modified Eq. (7) is determined by continuity condition for the function /.
~ versus J . (A) Symbols: treatment of observations at first () and second () unloading. (B) Circles: treatment of observations ~, b Fig. 2. Parameters a, b and a e1 at reloading. Solid lines: their approximation by Eqs. (9) and (11).
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A.D. Drozdov / Mechanics Research Communications 36 (2009) 437–443
Fig. 3. Parameter g versus Je1 . Circles: treatment of observations at reloading. Solid line: their approximation by Eq. (13).
Table 1 Adjustable parameters in the constitutive equations. Parameter
Dimension
Value
l
MPa
1.11 0.18 0.6 10.6 0.11 0.92 2:45 1.22 0:47 1.03 2:20 1.12 3:56 102 2:06 102
A
U0 a
a0 a1 b0 b1 a~ 0 a~ 1 ~0 b ~1 b
g0 g1
s1 s1 s1 s1 s1 s1 s1 s1
Fig. 4. Engineering stress r versus tensile strain . (A) Solid lines: results of numerical simulation for cyclic tensile tests with max 1 ¼ 0:4, max 2 ¼ 0:4, max 3 ¼ 1:0 (A1), max 1 ¼ 0:6, max 2 ¼ 0:6, max 3 ¼ 1:0 (A2), and rmin ¼ 0:04 MPa. (B) Solid lines: results of numerical simulation for cyclic tensile tests with max 1 ¼ 0:3, max 2 ¼ 0:5, max 3 ¼ 0:7, max 4 ¼ 1:0 (B1), max 1 ¼ 0:4, max 2 ¼ 0:6, max 3 ¼ 0:8, max 4 ¼ 1:0 (B2), and rmin ¼ 0:06 MPa.
4. Fitting of observations ~0 ; b ~1 ; g ; g that are found by fitting ~0 ; a ~ 1 ; b0 ; b1 ; b Constitutive equations (6)–(13) involve 14 parameters l; A; U0 ; a; a0 ; a1 ; a 0 1 the experimental data depicted in Fig. 1. No thermodynamic restrictions are imposed on these quantities except for (i) the
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A.D. Drozdov / Mechanics Research Communications 36 (2009) 437–443
Fig. 5. Engineering stress r versus tensile strain . Solid line: results of numerical simulation for a cyclic tensile test with max 3 ¼ 0:5, max 4 ¼ 0:6, max 5 ¼ 0:7, max 6 ¼ 0:8, max 7 ¼ 1:0, and rmin ¼ 0:08 MPa.
max 1 ¼ 0:3, max 2 ¼ 0:4,
~ g, and (ii) the condition U < 1 that ensures that sliding of junctions ~ , b, b, natural requirement of positivity of l, A, a, U0 , a, a occurs in the directions of macro-deformation. First, the coefficients l, A, U0 , and a are determined by matching the stress–strain curve at loading of a virgin sample. Afterwards, first unloading paths are approximated, and the parameters a and b are found for each max 1 separately. These quantities are reported in Fig. 2A together with their fits by Eq. (9). Then the first reloading paths are matched, and the ~ are found for each set of observations separately. These coefficients are depicted in Fig. 2B together with ~ and b parameters a their approximation by Eq. (11). The coefficient g is determined by matching the experimental data at reloading in the region where > max 1 . This quantity is presented in Fig. 3, where the results of fitting observations are plotted together with their approximation by Eq. (13). The coefficients a and b found by matching the retration paths of stress–strain for second unloading are depicted in Fig. 2A, which shows no substantial effect of cyclic preloading on these quantities. Material constants are listed in Table 1. 5. Numerical simulation Fig. 1B demonstrates good agreement between the experimental data and the results of numerical simulation based on the stress–strain relations (6)–(13) (the same quality of fitting is reached for observations in tests with other max 1 ). To reveal that the model adequately describes the response of TPE composites in tensile tests with several cycles of loading–unloading and various maximum strains max and minimum stresses rmin , numerical simulation is performed. The results of analysis are presented in Figs. 4 (3 and 4 cycles) and 5 (7 cycles). These figures show that the constitutive equations (i) are stable with respect to excitations of parameters max and rmin and (ii) they correctly reproduce characteristic features of the Mullins effect. 6. Conclusions Experimental data are reported in cyclic tensile tests with a constant strain rate and various maximum strains at room temperature. A constitutive model is derived for the viscoplastic behavior and damage of polymer composites at threedimensional deformations with finite strains. Adjustable parameters in the stress–strain relations are found by fitting the observations. It is demonstrated that the model correctly describes characteristic features of the Mullins effect and it can be employed to predict the viscoplastic behavior of polymer composites in low-cycle tests. This study focuses on the viscoplastic response of TPE composites under tension with a fixed strain rate. The account for strain rate is straightforward when this rate remains relatively large to disregard viscoelastic phenomena: it suffices to treat ~ as functions of _ . Under slow loading, when relaxation of stresses becomes substantial, an ~ , b, and b the coefficients l, a, a, a appropriate modification of stress–strain relation (6) is necessary (Drozdov and Christiansen, 2007b). As the objective of this work is to describe Mullins’ effects, we do not dwell on simulation of cyclic deformations with complicated deformation programs. However, only minor changes in the governing equations are necessary to describe the viscoplastic behavior under time-dependent loading. For example, to predict the response in cyclic tests with stress-con~ to be linear functions of minimum stress ~ and b trolled programs (ratcheting), the model should be modified by assuming a per cycle rmin . Material constants in the stress–strain relations have been found by matching experimental data on a carbon black-filled TPE that demonstrates relatively large plastic strains after the first cycle of deformation. It appears that the constitutive model can be applied to describe observations on particle-reinforced natural and synthetic rubbers as well, as the residual strain
A.D. Drozdov / Mechanics Research Communications 36 (2009) 437–443
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~ in Eq. (10), while the loss of stiffness at reloading is characterized by the coefficients a ~ and b is determined by parameters a and b in Eq. (8) that are chosen separately. Acknowledgement This work was supported by Danish Energy Authority through Project ENS–33033–0096. References Bergstrom, J.S., Kurtz, S.M., Rimnac, C.M., Edidin, A.A., 2002. Constitutive modeling of ultra-high molecular weight polyethylene under large-deformation and cyclic loading conditions. Biomaterials 23, 2329–2343. Bueche, F., 1961. Mullins effect and rubber–filler interaction. J. Appl. Polym. Sci. 5, 271–281. Chagnon, G., Verron, E., Gornet, L., Marckmann, G., Charrier, P., 2004. On the relevance of continuum damage mechanics as applied to the Mullins effect in elastomers. J. Mech. Phys. Solids 52, 1627–1650. Chagnon, G., Verron, E., Marckmann, G., Gornet, L., 2006. Development of new constitutive equations for the Mullins effect in rubber using the network alteration theory. Int. J. Solids Struct. 43, 6817–6831. De Tommasi, D., Puglisi, G., Saccomandi, G., 2006. A micromechanics-based model for the Mullins effect. J. Rheol. 50, 495–512. Diani, J., Brieu, M., Vacherand, J.M., 2006. A damage directional constitutive model for Mullins effect with permanent set and induced anisotropy. Eur. J. Mech. A/Solids 25, 483–496. Drozdov, A.D., 1997. A model of adaptive links in nonlinear viscoelasticity. J. Rheol. 41, 1223–1245. Drozdov, A.D., Christiansen, J.deC., 2007a. Cyclic viscoplasticity of thermoplastic elastomers. Acta Mech. 194, 47–65. Drozdov, A.D., Christiansen, J.deC., 2007b. Viscoelasticity and viscoplasticity of semicrystalline polymers: structure–property relations for high-density polyethylene. Comput. Mater. Sci. 39, 729–751. Drozdov, A.D., Dorfmann, A., 2001. Stress–strain relations in finite viscoelastoplasticity of rigid-rod networks: applications to the Mullins effect. Continuum Mech. Thermodyn. 13, 183–205. Erman, B., Flory, P.J., 1978. Theory of elasticity of polymer networks. II. The effect of geometric constraints on junctions. J. Chem. Phys. 68, 5363–5369. Erman, B., Monnerie, L., 1989. Theory of elasticity of amorphous networks: effects of constraints along chains. Macromolecules 22, 3342–3348. Goktepe, S., Miehe, C., 2005. A micro-macro approach to rubber-like materials. Part III: the micro-sphere model of anisotropic Mullins-type damage. J. Mech. Phys. Solids 53, 2259–2283. Govindjee, S., Simo, J., 1992. Transition from micro-mechanics to computationally efficient phenomenology: carbon black filled rubbers incorporating Mullins’ effect. J. Mech. Phys. Solids 40, 213–233. Horgan, C.O., Ogden, R.W., Saccomandi, G., 2004. A theory of stress softening of elastomers based on finite chain extensibility. Proc. Roy. Soc. A 460, 1737– 1754. Johnson, M.A., Beatty, M.F., 1993. A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments. Continuum Mech. Thermodyn. 5, 301–318. Mullins, L., 1947. Effect of stretching on the properties of rubber. J. Rubber Res. 16, 275–289. Ogden, R.W., Roxburgh, D.G., 1999. A pseudo-elastic model for the Mullins effect in filled rubber. Proc. Roy. Soc. A 455, 2861–2877. Tanaka, F., Edwards, S.F., 1992. Viscoelastic properties of physically cross-linked networks. Transient network theory. Macromolecules 25, 1516–1523.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Mullins’ effect in thermoplastic elastomers: Experiments and modeling A.D. Drozdov * Danish Technological Institute, Plastics Technology, Gregersensvej 1, DK–2630 Taastrup, Denmark
a r t i c l e
i n f o
Article history: Received 25 September 2008 Received in revised form 2 December 2008 Available online 25 December 2008
Keywords: Thermoplastic elastomer Viscoplasticity Damage Mullins’ effect
a b s t r a c t Observations are reported on carbon black-reinforced thermoplastic elastomer in uniaxial cyclic tensile tests with various maximum strains at room temperature. A constitutive model is derived for a polymer composite that accounts for its viscoplastic response and damage. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. It is shown that the model correctly reproduce characteristic features of the Mullins effect. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction This paper deals with experimental investigation and numerical simulation of Mullins’ effects for a carbon black-filled thermoplastic elastomer (TPE). Observations in cyclic tensile tests on particle-reinforced rubbers reveal a number of phenomena that are conventionally referred to as the Mullins effect (Mullins, 1947): (i) unloading and reloading paths of stress–strain diagrams differ substantially (hysteresis of energy), whereas the residual strain (measured when the stress vanishes) remains rather small, (ii) under deformation with a strain-controlled program (the maximum strain per cycle is fixed), the stress monotonically decreases with number of cycles (strain-softening), and (iii) when stretching of a sample proceeds after several cycles of loading– unloading, the stress–strain curve rapidly reaches that for a virgin specimen under the same loading conditions (strainhardening). Despite an agreement that interactions between a rubbery matrix and aggregates of filler provide a physical ground for the Mullins effect (Bueche, 1961), there is no constitutive model that adequately describes the entire set of phenomena. This may be explained by the fact that (i) some effects contradict one another (softening versus hardening), (ii) the account for rubber–filler interactions requires modeling of deformation in the matrix and filler network separately which leads to a substantial increase in the number of material constants, and (iii) Mullins’ phenomena are accompanied by (a) stress relaxation, (b) material anisotropy, and (c) formation, growth, and coalescence of micro-voids and micro-cracks, whose role remains a subject of debate. To develop constitutive equations to be employed in numerical simulation, the attention is conventionally focused on one aspect of observations only. Hysteresis of energy under cyclic loading is adequately described by taking into account alteration (breakage and restoration) of links between polymer chains and between the matrix and aggregates of filler (Chagnon et al., 2006; De Tommasi et al., 2006). This approach is similar to the concept of transient networks in viscoelasticity of polymers (Drozdov, 1997;
* Tel.: +45 72 20 31 42; fax: +45 72 20 31 12. E-mail address: [email protected] 0093-6413/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.12.007
438
A.D. Drozdov / Mechanics Research Communications 36 (2009) 437–443
Tanaka and Edwards, 1992). A variant of this theory was proposed in Drozdov and Dorfmann (2001), where detachment– attachment of links was treated as transition of chains from flexed to extended conformation and vice versa. The latter is closely connected to the two-phase theory of composites (Johnson and Beatty, 1993). Strain-softening of specimens under cyclic deformation is conventionally predicted by accounting for damage (Govindjee and Simo, 1992). In the simplest (pseudo-elastic) model, a filled rubber is treated as an elastic continuum subjected to damage, whose evolution is described by the only internal variable (Ogden and Roxburgh, 1999; Horgan et al., 2004). More sophisticated models accounting for combined effects of damage, viscoelasticity, and viscoplasticity of particle-reinforced rubbers were developed by Chagnon et al. (2004), Diani et al. (2006), and Goktepe and Miehe (2005), to mention a few. In some sense, these constitutive equations may be thought of as variants of the concept of breakage–restoration of links, where breakage (fracture) of chains is characterized by one or several damage variables at the phenomenological level. The objective of this work is three-fold: (i) to study experimentally cyclic deformation of a TPE composite, (ii) to develop a constitutive model and to find adjustable parameters in the stress–strain relations by fitting the observations, and (iii) to perform numerical simulation of the mechanical response and to demonstrate that the model adequately predicts characteristic features of the Mullins effect. 2. Experimental procedure Thermoplastic elastomer Thermoplast K TV5LVZ (a hydrogenated styrene block copolymer–based TPE with addition of polypropylene segments reinforced with 60 to 70 wt.% of carbon black) was purchased from Kraiburg TPE GmgH (Germany). Dumbbell specimens for uniaxial tensile tests (ASTM standard D638) with the cross-sectional area 10.2 3.4 mm were molded by using injection-molding machine Arburg 320C. Mechanical tests were performed at ambient temperature by using a universal testing machine Instron–5569 equipped with 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 r was determined as the ratio of axial force to cross-sectional area of specimens in the stress-free state. The experimental program involved 7 uniaxial tensile tests conducted with a constant strain rate _ ¼ 6:67 102 s1. In the first test, a specimen was loaded up to the maximum strain max ¼ 1:0. In other tests, specimens were stretched up to strains max 1 ¼ 0:3, 0.4, 0.5, 0.6, 0.7, 0.8 with the strain rate _ , unloaded down to the zero stress with the strain rate _ (to avoid buckling of samples, retraction was performed down to the engineering stress rmin ¼ 0:01 MPa), reloaded up to the maximum strain max 2 ¼ 0:9 with the strain rate _ , and unloaded down to the zero stress with the strain rate _ . Each test was conducted on a new specimen. Observations are reported in Fig. 1, where the engineering stress r is plotted versus engineering strain . To avoid overlapping of data, observations for the last unloading are omitted in Fig. 1A (as an example, they are presented in Fig. 1B). The following conclusions are drawn: (i) the stress–strain diagrams at loading and retraction are strongly nonlinear, (ii) the stress–strain curves at first loading and reloading differ substantially, (iii) the residual strain (at which the stress vanishes at unloading) monotonically grows with max 1 , and (iv) the hysteresis energy (area between unloading and subsequent reloading paths) strongly increases with max 1 . 3. Constitutive model With reference to the homogenization concept (Bergstrom et al., 2002), a particle-reinforced TPE is thought of as an equivalent continuum whose response coincides with that of the composite. An incompressible, permanent, non-affine
Fig. 1. Engineering stress r versus tensile strain . (A) Symbols: experimental data in tensile tests with various maximum strains max 1 (solid line – loading, – max 1 ¼ 0:3, – max 1 ¼ 0:4, I – max 1 ¼ 0:5, } – max 1 ¼ 0:6, – max 1 ¼ 0:7, – max 1 ¼ 0:8). (B) Circles: experimental data in a cyclic tensile test with max 1 ¼ 0:3, max 2 ¼ 1:0. Solid line: results of numerical simulation.
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network of chains is chosen as the equivalent medium. Sliding (plastic flow) of junctions between chains under deformation of the non-affine network reflects its viscoplastic behavior. Unlike the standard approach, where damage characterizes a decrease in the elastic moduli (due to breakage of chains), damage serves in the present model as a measure of acceleration of plastic flow of junctions (induced by growth of free volume between chains). Elastic deformation of a chain in an equivalent network is determined by means of the multiplicative decomposition formula
Fe ¼ F F1 p ;
ð1Þ
where F, Fp , Fe denote deformation gradients for macro-deformation, sliding of junctions, and elastic deformation, respece F1 tively, and the dot stands for inner product. The velocity gradient for elastic deformation L e ¼ dF e reads dt
Le ¼ L Fe lp F1 e ;
L¼
dF 1 F ; dt
lp ¼
dFp 1 F : dt p
The rate-of-strain and vorticity tensors for macro-deformation and elastic deformation, respectively, are introduced by the formulas
D¼
1 ðL þ L> Þ; 2
W¼
1 ðL L> Þ; 2
De ¼
1 ðLe þ L>e Þ; 2
W
e
¼
1 ðLe L>e Þ; 2
where > stands for transpose. To describe sliding of junctions, it is postulated that (i) the rate-of-strain tensor for elastic deformation is proportional to the rate-of-strain tensor for macro-deformation
De ¼ ð1 /ÞD;
ð2Þ
where / is a scalar function, and (ii) the vorticity tensor for elastic deformation vanishes, We ¼ 0. The left and right Cauchy– Green tensors for elastic deformation are given by
Be ¼ Fe F>e ;
Ce ¼ F>e Fe :
ð3Þ
The incompressibility condition for elastic deformation implies that the third principal invariants of these tensors equal unity. Their first and second principal invariants read J e1 ¼ Ce : I and J e2 ¼ C1 e : I, where I is the unit tensor, and the colon stands for convolution. Differentiating these relations with respect to time and using Eqs. (2) and (3), we find that
dJ e1 ¼ 2ð1 /ÞBe : D; dt
dJ e2 ¼ 2ð1 /ÞB1 e : D: dt
ð4Þ
Each chain in the equivalent network is modeled as a flexible chain with constrained junctions. Two types of constraints are introduced (Erman and Flory, 1978; Erman and Monnerie, 1989): (i) positions of a chain ends (determined by vectors X1 and X2 ) do not coincide with positions of the junctions (characterized by vectors Q 1 and Q 2 ); the vectors xm ¼ Xm Q m (m ¼ 1; 2) are independent Gaussian random vectors with the zero mean and a given standard deviation, and (ii) the distance between the center of mass of a chain and the middle point of the vector between junctions Q ¼ Q 2 Q 1 is fixed. Under stands for an analog ½ðJ e1 3Þ þ Að3J 2e1 4J e2 15Þ, where l these conditions, the strain energy of a chain reads w ¼ 12 l of the chain rigidity, and A is a dimensionless positive constant (Drozdov and Christiansen, 2007a). Neglecting the energy of inter-chain interaction (this energy is accounted for by means of the incompressibility condition), we calculate the strain energy density per unit volume of the equivalent network as the sum of strain energies of individual chains
W¼
l 2
½ðJ e1 3Þ þ Að3J 2e1 4J
e2
15Þ;
ð5Þ
N, and N stands for the number of chains per unit volume. Inserting Eq. (5) into the Clausius–Duhem inequality where l ¼ l þ R0 : D P 0, where Q stands for internal dissipation per for isothermal deformation of an incompressible medium Q ¼ dW dt 0 unit volume and unit time, and R denotes the deviatoric component of the Cauchy stress tensor R, using Eq. (4), and assuming internal dissipation to vanish, we arrive at the stress–strain relation
R ¼ pI þ lð1 /ÞðW 1 Be W 2 B1 e Þ;
W 1 ¼ 1 þ 6AJ e1 ;
W 2 ¼ 4A;
ð6Þ
where p stands for an unknown pressure. To describe evolution of the coefficient / at cyclic deformation, we distinguish (i) the first loading of a virgin material, (ii) its unloadings, and (iii) subsequent reloadings. Under deformation of a virgin material, / obeys the equation
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / ¼ U0 ½1 expða Je1 3Þ;
ð7Þ
where a and U0 are adjustable parameters. Eq. (7) means that / vanishes in the reference state, monotonically increases with time at active loading, and reaches its ultimate value U0 at large deformations. Evolution of / at unloading is governed by the differential equation
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d/ ¼ að1 /Þ2 ðJ e1 3Þ þ b/; dt
ð8Þ
where a and b are positive parameters. The first term in Eq. (8) characterizes an increase in / driven by strain energy of chains, while the other reflects self-acceleration of plastic flow at retraction. The coefficients a and b remain constant along each unloading path, but they are affected by elastic deformations at the instants when unloading starts. Appropriate dependencies are described by
log a ¼ a0 a1 logðJ e1 3Þ;
log b ¼ b0 b1 logðJ e1 3Þ;
ð9Þ
where am and bm (m ¼ 0; 1) are constants. Changes in / under reloading are determined by the differential equation
d/ ~ n ~ ð1 /Þ2 ðJ e1 3Þ b/ ¼ a dt
ð10Þ
~ and n. The first term in Eq. (10) describe a decrease in / induced by strain energy of chains, ~ , b, with positive parameters a while the other reflects self-deceleration of the sliding process at reloading. The exponent n in Eq. (10) is treated as a con~ remain constant along each reloading path, but depend on elastic ~ and b stant (n ¼ 4 is set in simulation). The coefficients a deformations at the instants when reloading starts. By analogy with Eq. (9), we set
~¼a ~0 a ~ 1 logðJ log a
e1
3Þ;
~¼b ~0 b ~1 logðJ 3Þ; log b e1
ð11Þ
~m (m ¼ 0; 1) are constants. ~ m and b where a Under uniaxial tension with small strains, Eqs. (7), (8), and (10) are transformed into the kinetic equations
d/ ~ð1 /ÞðU0 /Þ; ¼a dt
d/ ¼ A0 ð1 /Þ2 þ b/; dt
d/ ~ 0 ð1 /Þ2 b/ ~ n ¼ A dt
ð12Þ
pffiffiffi ~ 0 ¼ 3a ~ ¼ a 3d=dt, A0 ¼ 3a2e , A ~ 2e , where stands for tensile strain, and e denotes the elastic strain. All differential with a equations (12) have a similar structure that is typical of evolution equations in chemical kinetics. The quadratic terms in these relations reflect interactions between different species (the matrix and filler particles for a reinforced TPE), whereas the terms proportional to / and /n characterize acceleration (slowing down) of the sliding process induced by inter-chain interactions in the matrix. Cyclic loading of a particle-reinforced TPE induces its damage observed as initiation and growth of micro-voids and micro-cracks. Unlike conventional models that associate damage with a decrease in elastic modulus, we suppose that damage accelerates sliding of junctions, while l remains constant (at low-cycle loading). Under deformation with a strain exceeding the maximum strain in the previous cycle max , the function / is governed by Eq. (7), where coefficient U0 (that corresponds to a virgin material) is replaced with U > U0 . The coefficient of proportionality g in the formula U ¼ gU0 is treated as a measure of damage. Its dependence on elastic deformation at the instant when reloading starts is described by the equation
log g ¼ g0 þ g1 logðJ e1 3Þ;
ð13Þ
where gm (m ¼ 0; 1) are constants. Transition from the regime of reloading governed by Eq. (10) to that characterized by modified Eq. (7) is determined by continuity condition for the function /.
~ versus J . (A) Symbols: treatment of observations at first () and second () unloading. (B) Circles: treatment of observations ~, b Fig. 2. Parameters a, b and a e1 at reloading. Solid lines: their approximation by Eqs. (9) and (11).
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Fig. 3. Parameter g versus Je1 . Circles: treatment of observations at reloading. Solid line: their approximation by Eq. (13).
Table 1 Adjustable parameters in the constitutive equations. Parameter
Dimension
Value
l
MPa
1.11 0.18 0.6 10.6 0.11 0.92 2:45 1.22 0:47 1.03 2:20 1.12 3:56 102 2:06 102
A
U0 a
a0 a1 b0 b1 a~ 0 a~ 1 ~0 b ~1 b
g0 g1
s1 s1 s1 s1 s1 s1 s1 s1
Fig. 4. Engineering stress r versus tensile strain . (A) Solid lines: results of numerical simulation for cyclic tensile tests with max 1 ¼ 0:4, max 2 ¼ 0:4, max 3 ¼ 1:0 (A1), max 1 ¼ 0:6, max 2 ¼ 0:6, max 3 ¼ 1:0 (A2), and rmin ¼ 0:04 MPa. (B) Solid lines: results of numerical simulation for cyclic tensile tests with max 1 ¼ 0:3, max 2 ¼ 0:5, max 3 ¼ 0:7, max 4 ¼ 1:0 (B1), max 1 ¼ 0:4, max 2 ¼ 0:6, max 3 ¼ 0:8, max 4 ¼ 1:0 (B2), and rmin ¼ 0:06 MPa.
4. Fitting of observations ~0 ; b ~1 ; g ; g that are found by fitting ~0 ; a ~ 1 ; b0 ; b1 ; b Constitutive equations (6)–(13) involve 14 parameters l; A; U0 ; a; a0 ; a1 ; a 0 1 the experimental data depicted in Fig. 1. No thermodynamic restrictions are imposed on these quantities except for (i) the
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Fig. 5. Engineering stress r versus tensile strain . Solid line: results of numerical simulation for a cyclic tensile test with max 3 ¼ 0:5, max 4 ¼ 0:6, max 5 ¼ 0:7, max 6 ¼ 0:8, max 7 ¼ 1:0, and rmin ¼ 0:08 MPa.
max 1 ¼ 0:3, max 2 ¼ 0:4,
~ g, and (ii) the condition U < 1 that ensures that sliding of junctions ~ , b, b, natural requirement of positivity of l, A, a, U0 , a, a occurs in the directions of macro-deformation. First, the coefficients l, A, U0 , and a are determined by matching the stress–strain curve at loading of a virgin sample. Afterwards, first unloading paths are approximated, and the parameters a and b are found for each max 1 separately. These quantities are reported in Fig. 2A together with their fits by Eq. (9). Then the first reloading paths are matched, and the ~ are found for each set of observations separately. These coefficients are depicted in Fig. 2B together with ~ and b parameters a their approximation by Eq. (11). The coefficient g is determined by matching the experimental data at reloading in the region where > max 1 . This quantity is presented in Fig. 3, where the results of fitting observations are plotted together with their approximation by Eq. (13). The coefficients a and b found by matching the retration paths of stress–strain for second unloading are depicted in Fig. 2A, which shows no substantial effect of cyclic preloading on these quantities. Material constants are listed in Table 1. 5. Numerical simulation Fig. 1B demonstrates good agreement between the experimental data and the results of numerical simulation based on the stress–strain relations (6)–(13) (the same quality of fitting is reached for observations in tests with other max 1 ). To reveal that the model adequately describes the response of TPE composites in tensile tests with several cycles of loading–unloading and various maximum strains max and minimum stresses rmin , numerical simulation is performed. The results of analysis are presented in Figs. 4 (3 and 4 cycles) and 5 (7 cycles). These figures show that the constitutive equations (i) are stable with respect to excitations of parameters max and rmin and (ii) they correctly reproduce characteristic features of the Mullins effect. 6. Conclusions Experimental data are reported in cyclic tensile tests with a constant strain rate and various maximum strains at room temperature. A constitutive model is derived for the viscoplastic behavior and damage of polymer composites at threedimensional deformations with finite strains. Adjustable parameters in the stress–strain relations are found by fitting the observations. It is demonstrated that the model correctly describes characteristic features of the Mullins effect and it can be employed to predict the viscoplastic behavior of polymer composites in low-cycle tests. This study focuses on the viscoplastic response of TPE composites under tension with a fixed strain rate. The account for strain rate is straightforward when this rate remains relatively large to disregard viscoelastic phenomena: it suffices to treat ~ as functions of _ . Under slow loading, when relaxation of stresses becomes substantial, an ~ , b, and b the coefficients l, a, a, a appropriate modification of stress–strain relation (6) is necessary (Drozdov and Christiansen, 2007b). As the objective of this work is to describe Mullins’ effects, we do not dwell on simulation of cyclic deformations with complicated deformation programs. However, only minor changes in the governing equations are necessary to describe the viscoplastic behavior under time-dependent loading. For example, to predict the response in cyclic tests with stress-con~ to be linear functions of minimum stress ~ and b trolled programs (ratcheting), the model should be modified by assuming a per cycle rmin . Material constants in the stress–strain relations have been found by matching experimental data on a carbon black-filled TPE that demonstrates relatively large plastic strains after the first cycle of deformation. It appears that the constitutive model can be applied to describe observations on particle-reinforced natural and synthetic rubbers as well, as the residual strain
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~ in Eq. (10), while the loss of stiffness at reloading is characterized by the coefficients a ~ and b is determined by parameters a and b in Eq. (8) that are chosen separately. Acknowledgement This work was supported by Danish Energy Authority through Project ENS–33033–0096. References Bergstrom, J.S., Kurtz, S.M., Rimnac, C.M., Edidin, A.A., 2002. Constitutive modeling of ultra-high molecular weight polyethylene under large-deformation and cyclic loading conditions. Biomaterials 23, 2329–2343. Bueche, F., 1961. Mullins effect and rubber–filler interaction. J. Appl. Polym. Sci. 5, 271–281. Chagnon, G., Verron, E., Gornet, L., Marckmann, G., Charrier, P., 2004. On the relevance of continuum damage mechanics as applied to the Mullins effect in elastomers. J. Mech. Phys. Solids 52, 1627–1650. Chagnon, G., Verron, E., Marckmann, G., Gornet, L., 2006. Development of new constitutive equations for the Mullins effect in rubber using the network alteration theory. Int. J. Solids Struct. 43, 6817–6831. De Tommasi, D., Puglisi, G., Saccomandi, G., 2006. A micromechanics-based model for the Mullins effect. J. Rheol. 50, 495–512. Diani, J., Brieu, M., Vacherand, J.M., 2006. A damage directional constitutive model for Mullins effect with permanent set and induced anisotropy. Eur. J. Mech. A/Solids 25, 483–496. Drozdov, A.D., 1997. A model of adaptive links in nonlinear viscoelasticity. J. Rheol. 41, 1223–1245. Drozdov, A.D., Christiansen, J.deC., 2007a. Cyclic viscoplasticity of thermoplastic elastomers. Acta Mech. 194, 47–65. Drozdov, A.D., Christiansen, J.deC., 2007b. Viscoelasticity and viscoplasticity of semicrystalline polymers: structure–property relations for high-density polyethylene. Comput. Mater. Sci. 39, 729–751. Drozdov, A.D., Dorfmann, A., 2001. Stress–strain relations in finite viscoelastoplasticity of rigid-rod networks: applications to the Mullins effect. Continuum Mech. Thermodyn. 13, 183–205. Erman, B., Flory, P.J., 1978. Theory of elasticity of polymer networks. II. The effect of geometric constraints on junctions. J. Chem. Phys. 68, 5363–5369. Erman, B., Monnerie, L., 1989. Theory of elasticity of amorphous networks: effects of constraints along chains. Macromolecules 22, 3342–3348. Goktepe, S., Miehe, C., 2005. A micro-macro approach to rubber-like materials. Part III: the micro-sphere model of anisotropic Mullins-type damage. J. Mech. Phys. Solids 53, 2259–2283. Govindjee, S., Simo, J., 1992. Transition from micro-mechanics to computationally efficient phenomenology: carbon black filled rubbers incorporating Mullins’ effect. J. Mech. Phys. Solids 40, 213–233. Horgan, C.O., Ogden, R.W., Saccomandi, G., 2004. A theory of stress softening of elastomers based on finite chain extensibility. Proc. Roy. Soc. A 460, 1737– 1754. Johnson, M.A., Beatty, M.F., 1993. A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments. Continuum Mech. Thermodyn. 5, 301–318. Mullins, L., 1947. Effect of stretching on the properties of rubber. J. Rubber Res. 16, 275–289. Ogden, R.W., Roxburgh, D.G., 1999. A pseudo-elastic model for the Mullins effect in filled rubber. Proc. Roy. Soc. A 455, 2861–2877. Tanaka, F., Edwards, S.F., 1992. Viscoelastic properties of physically cross-linked networks. Transient network theory. Macromolecules 25, 1516–1523.