- Email: [email protected]
Stress- and strain-controlled cyclic deformation of polypropylene
Computational Materials Science 64 (2012) 198–202
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Stress- and strain-controlled cyclic deformation of polypropylene A.D. Drozdov ⇑ Department of Plastics Technology, Danish Technological Institute Gregersensvej 1, Taastrup 2630, Denmark
a r t i c l e
i n f o
Article history: Received 15 October 2011 Accepted 24 February 2012 Available online 19 March 2012 Keywords: Polypropylene Viscoelasticity Viscoplasticity Constitutive modeling
a b s t r a c t Experimental data are reported on isotactic polypropylene in uniaxial tensile cyclic tests with stress- and strain-controlled programs. A constitutive model is developed that adequately describes stress–strain diagrams, and its parameters are found by fitting the observations. Numerical simulation demonstrates that fatigue failure in ratcheting tests is induced by pronounced growth of plastic strain in crystallites. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction This paper deals with experimental investigation, constitutive modeling, and numerical simulation of the viscoelastic and viscoplastic responses of semicrystalline polymers in cyclic tests with stress- and strain-controlled programs. We begin with the analysis of stress-controlled cyclic tensile tests (ratcheting between some maximum rmax and minimum rmin stresses). Observations show that maximum max and minimum min strains per cycle grow monotonically with number of cycles n until an instant when the sample breaks. The question of interest is why fatigue failure occurs under the stress-controlled deformation program? The conventional answer is that breakage is induced by damage accumulation: some non-decreasing measure of damage exists that equals zero for an undeformed specimen and which increases monotonically with n until it reaches some ultimate value corresponding to failure of the specimen [1]. Growth of damage variable in semicrystalline polymers is attributed to (i) lamellar fragmentation [2] and (ii) nucleation and growth of micro-voids [3]. As damage accumulation cannot be measured directly in mechanical tests, it is traditionally associated [4] with hysteresis energy (area between subsequent loading and unloading paths of a stress–strain diagram). Failure in ratcheting tests with relatively large rmax occurs within the experimental time-scale (50–100 cycles) when maximum strain per cycle reaches its critical value c. This value is employed as a simple criterion for failure: breakage occurs when ratcheting strain exceeds c. This condition, however, is not unique: a number of different criteria for fracture are available in the literature [5,6]. ⇑ Tel.: +45 72 20 31 42; fax: +45 72 20 31 12. E-mail address: [email protected] 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2012.02.035
Let us now turn to strain-controlled tensile cyclic tests (oscillations between a maximum strain max and the minimum stress rmin). As c is located below necking strain n for uniaxial tension, the maximum strain per cycle max in these tests can be chosen equal or even exceeding the critical strain. Observations in these tests show that (i) maximum stress per cycle rmax decreases only slightly with n, (ii) minimum strain per cycle min strongly exceeds that in corresponding stress-controlled tests, (iii) at each cycle, hysteresis energy is substantially higher than in experiments with the stress-controlled program, but no breakage of specimens is observed within the experimental time-scale (which means that traditional fracture criteria are inapplicable). The objective of this study is threefold: (i) to report experimental data on isotactic polypropylene in stress- and strain-controlled tests, (ii) to develop constitutive equations in cyclic viscoelasticity and viscoplasticity of semicrystalline polymers that correctly describe the observations, and (iii) to provide an explanation for the difference in the mechanical response in these two types of tests by comparison of results of numerical simulation. 2. Experimental results Isotactic polypropylene Moplen HP 400R (density 0.90 g/cm3, melt flow rate 25 g/10 min, melting temperature Tm = 161 °C, degree of crystallinity 47%) was purchased from Basell Polyolefins (Switzerland). Dumbbell specimens for mechanical tests (ASTM standard D-638) with cross-sectional area 10.1 mm 4.2 mm were molded by using injection-molding machine Arburg 320C. Mechanical tests were conducted by means of universal testing machine Instron-5569 equipped with an electro-mechanical sensor for control of longitudinal strains. Tensile force was measured by 5 kN load cell. The engineering stress r was determined as
A.D. Drozdov / Computational Materials Science 64 (2012) 198–202
the ratio of axial force to cross-sectional area of undeformed specimens. Uniaxial tensile cyclic tests were conducted at room temperature. In the first test (ratcheting), a specimen was stretched with cross-head speed 100 mm/min (that corresponded to strain rate _ ¼ 1:7 102 s1 Þ up to the maximum stress rmax = 32 MPa, unloaded down to the minimum stress rmin = 1 MPa with the same cross-head speed, reloaded up to rmax, retracted down to rmin, etc. Breakage under ratcheting occurs at the 48th cycle. The experimental stress–strain diagram for the stress-controlled test is depicted in Fig. 1, where stress r is plotted versus strain . The following conclusions are drawn: (i) loading and unloading paths of stress–strain diagrams are strongly nonlinear, (ii) hysteresis energy per cycle increases with n, (iii) the stress– strain diagram slowly rotates clockwise when number of cycles increases.
A
B
Fig. 1. Stress r versus strain . Symbols: experimental data in the stress-controlled test. Solid lines: results of numerical simulation. (A) 46 cycles and (B) n = 1, 2(); n = 20 (); n = 40 (q).
Fig. 2. Maximum max and minimum min strains versus number of cycles n. Unfilled circles: observations in the stress-controlled test. Filled circles: results of numerical simulation.
199
Evolution of maximum max and minimum min strains per cycle with n is illustrated in Fig. 2. The diagrams max(n) and min(n) may be split into three intervals. Along the first (primary fatigue), the curves max(n) and min(n) are convex. Along the other interval (secondary fatigue), max and min grow with n linearly. At the last interval (tertiary fatigue), the dependencies max(n) and min(n) become nonlinear and concave. In the other test with strain-controlled loading, a specimen was stretched with the same cross-head speed 100 mm/min up to the maximum strain max = 0.16, unloaded down to the minimum stress rmin = 1 MPa, reloaded up to max, retracted down to rmin, etc. The program involved 150 cycles of loading–retraction. No signs of damage were observed in the sample except for slight whitening. The stress–strain diagram for the first 50 cycles of oscillations is depicted in Fig. 3. This figure reveals that (i) hysteresis energy
A
B
Fig. 3. Stress r versus strain . Symbols: experimental data in the strain-controlled test. Solid lines: results of numerical simulation. (A) 50 cycles and (B) n = 1, 2 (); n = 20 (); n = 40 (q).
Fig. 4. Maximum stress rmax and minimum strain min versus number of cycles n. Circles: observations in the strain-controlled test. Solid lines: predictions of the model.
200
A.D. Drozdov / Computational Materials Science 64 (2012) 198–202
strongly decreases with number of cycles, and (ii) the stress–strain curve slowly rotates counter-clockwise with growth of n. Evolution of maximum stress per cycle rmax and minimum strain per cycle min is reported in Fig. 4. This figure shows that (i) rmax decreases monotonically with n and becomes practically independent of n after a dozen of cycles, and (ii) min noticeably increases with n. The minimum stress rmin = 1 MPa was chosen instead of min r = 0 to avoid buckling of specimens. The cross-head speed 100 mm/min was selected to ensure that duration of cyclic tests does not exceed that of conventional short-term creep and relaxation tests. Each test was carried out on a new specimen and repeated by twice. Observations reveal good reproducibility of measurements: deviations between maximum stresses and minimum strains measured on different specimens did not exceed 3%. The terms ‘‘stress-controlled’’ and ‘‘strain-controlled’’ are used in a broad sense. Strictly speaking, the loading programs under investigation are not stress- and strain-controlled. The ratcheting test was conducted with a constant strain (not stress) rate, while unloading in the cyclic test with fixed max was performed down to the minimum stress (not strain). 3. Constitutive model A two-step approach is employed to describe the response of semicrystalline polymers [7]. At the first step, constitutive equations are derived by using the Clausius–Duhem inequality for the viscoelastic and viscoplastic behavior of polypropylene along an individual cycle of loading–retraction. At the other step, some parameters in the stress–strain relations are allowed to change with number of cycles. Their evolution induced by damage accumulation is described by kinetic equations which are validated by comparison of the model predictions with observations. Stress–strain relations in viscoelastoplasticity of semicrystalline polymers under an arbitrary three-dimensional deformation with small strains have been developed in [8,9]. In what follows, these equations are simplified for uniaxial tension. Strain is presented as the sum of elastic e and plastic p strains
¼ e þ p : The plastic strain
ð1Þ
p is split into the sum of two components
p ¼ p1 þ p2 ;
ð2Þ
that reflect plastic deformation in the crystalline and amorphous phases, respectively. The strain rate for plastic deformation in the crystalline phase is proportional to that for macro-deformation
dp1 d ¼/ : dt dt
ð3Þ
The coefficient / obeys the differential equation
d/ d ¼ að1 /Þ2 ; dt dt
ð4Þ
where a adopts different values a1 and a2 under loading and retraction. The strain rate for plastic deformation in the amorphous matrix is governed by the equation
dp2 ¼S dt
e Rp2
Z 0
1
d f ðv ÞZðt; v Þdv ; dt
ð5Þ
where R and S adopts different positive values R1 = 1, R2, and S1, S2, under loading and unloading. The integral term in Eq. (5) accounts for the effect of viscoelasticity on plastic flow in the amorphous phase. The function f(v) describes inhomogeneity of the amorphous phase induced by the presence of spherulites. A quasi-Gaussian expression is adopted for this function
1 v 2 ðv P 0Þ; f ðv Þ ¼ f0 exp 2 R
f ðv Þ ¼ 0 ðv < 0Þ;
ð6Þ
where R > 0 characterizes distribution of relaxation times, and f0 is R1 determined from normalization condition 0 f ðv Þdv ¼ 1. The function Z(t,v) obeys the differential equation
@Z ðt; v Þ ¼ Cðv Þ½e ðtÞ Zðt; v Þ; @t
Zð0; v Þ ¼ 0;
ð7Þ
where C(v) = cexp (v), and c denotes relaxation rate. The stress r reads
rðtÞ ¼ Eð1 /ðtÞÞ e ðtÞ
Z
1
f ðv ÞZðt; v Þdv ;
ð8Þ
0
where E stands for Young’s modulus. Eqs. (1)–(8) with eight adjustable parameters, E, a1, a2, R2, S1, S2, c, R, describe uniaxial tension with an arbitrary deformation program (t). The following scenario is proposed for evolution of these quantities: 1. Parameters E, c, and R are independent of n. The assumption that E remains constant distinguishes this approach from conventional concepts in damage mechanics that postulate a decay in elastic modulus under deformation. 2. After an initial transition period (about 10 cycles) coefficients a1 and a2 coincide (a1 = a2 = a1) and become independent of n. The quantity a1 equals zero for the strain-controlled program and remains positive for the stress-controlled program. 3. Coefficient R2 (this parameter characterizes energy of interchain interactions) monotonically decreases and vanishes at large n. 4. After the initial transition period, coefficients S1 and S2 (that characterize rates of plastic strain in the amorphous matrix under loading and retraction) linearly evolve with n
S1 ¼ S10 þ S11 n;
S2 ¼ S20 þ S21 n;
ð9Þ
where S10, S11, S20, S21 are constants. After an initial transition period, mechanical behavior of a semicrystalline polymer under the strain-controlled program is determined by five adjustable parameters: (i) E stands for elastic modulus, (ii) c determines relaxation rate, (iii) R characterizes distribution of relaxation times, (iv) S1 and S2 denote rates of plastic flow in the amorphous matrix. The viscoelastoplastic response under the stress-controlled program is characterized by the same quantities and coefficient a1. 4. Adjustable parameters Coefficients c and R are determined by fitting observations in relaxation tests; they read c = 0.11 s1, R = 13.1 [9]. The modulus E = 2.15 GPa is found by approximation of the first loading path of the stress–strain curve depicted in Fig. 3. Adjustable parameters a1, a2, R2, S1, and S2 are determined by matching observations depicted in Figs. 1 and 3 (50 cycles of loading–retraction) following the procedure described in [7]. Each loading path is fitted by two parameters, a1 and S1, and each unloading path is approximated by three parameters, a2, R2, and S2. The number of parameters is reduced when some of these quantities vanish. For example, when a1 = R2 = 0, each loading and unloading path is matched by means of one parameter only: S1 for loading and S2 for retraction. Coefficients a1 and a2 are plotted versus number of cycles in Fig. 5. For the stress-controlled program, these coefficients coincide and become independent of n (a1 = 9.0) after 5 cycles, whereas for
A.D. Drozdov / Computational Materials Science 64 (2012) 198–202
A
B
201
5. Numerical simulation To assess accuracy of fitting, results of numerical simulation are presented together with experimental stress–strain curves in Figs. 1 and 3. These figures reveal that quality of approximation is not reduced with number of cycles. To demonstrate that the model accurately predicts the viscoelastoplastic response in strain-controlled tests with large (noticeably exceeding that used in the fitting procedure) number of cycles, simulation is conducted of the stress–strain relations for 150 cycles of oscillations. Results of the numerical analysis are
A
Fig. 5. Parameters a1() and a2() versus number of cycles n. Symbols: treatment of observations in cyclic tests. (A) stress-controlled program and (B) strain-controlled program.
B A
B Fig. 7. Parameter S1 versus number of cycles n. Circles: treatment of observations in cyclic tests. Solid lines: approximation of the data by Eq. (9). (A) stress-controlled program and (B) strain-controlled program.
A
Fig. 6. Parameter R2 versus number of cycles n. Circles: treatment of observations in cyclic tests. (A) stress-controlled program and (B) strain-controlled program.
the strain-controlled program, a1 and a2 vanish (a1 = 0) after 10 cycles of oscillations. Parameter R2 is plotted versus number of cycles in Fig. 6. Under the strain-controlled deformation with a relatively large max, this quantity strongly decays and vanishes after 10 cycles of loading– retraction (severe damage accumulation). Under the stress-controlled deformation with mild conditions, R2 decreases with n, but remains positive up to the breakage point. Evolution of S1 with number of cycles is illustrated in Fig. 7 which demonstrates that S1(n) strongly decreases in the stresscontrolled test and grows in the strain-controlled experiment. The experimental dependencies S2(n) are depicted in Fig. 8 which shows similar responses in the stress- and strain-controlled tests.
B
Fig. 8. Parameter S2 versus number of cycles n. Circles: treatment of observations in cyclic tests. Solid lines: approximation of the data by Eq. (9). (A) stress-controlled program; (B) strain-controlled program.
202
A.D. Drozdov / Computational Materials Science 64 (2012) 198–202
A
To understand why fatigue failure occurs in stress-controlled tests and no failure is observed in strain-controlled experiments, maximum max and minimum min plastic strains per cycle are p p calculated and plotted versus logarithm (log = log10) of n in Fig. 9. Results of simulation for the strain-controlled test are approximated by
max ¼ cmax þ cmax log n; min ¼ cmin þ cmin log n; p 0 1 p 0 1
B
Fig. 9. Maximum max ðÞ and minimum min p p ðÞ plastic strains versus number of cycles n. Symbols: treatment of observations in cyclic tests. Solid lines: approximation of the data by Eq. (10). (A) stress-controlled program; (B) strain-controlled program.
ð10Þ
min where cmax ; cmax ; cmin are constants. Fig. 9 reveals a pronounced 0 1 0 ; c1 difference between evolution of plastic strains under stress- and strain-controlled loading programs: in the former case, both plastic strains increase strongly with n, especially, at the stage of tertiary fatigue, whereas in the latter case, max remains independent of n, p while min grows linearly with log n. p To comprehend which component of the plastic strain increases substantially under ratcheting, maximum and minimum plastic strains in the crystalline p1 and amorphous p2 phases are reported in Fig. 10. This figure shows that fatigue failure is driven by a pronounced growth of maximum plastic strain in the crystalline phase max p1 caused by lamellar fragmentation under loading.
6. Concluding remarks Experimental data are reported on isotactic polypropylene in uniaxial tensile cyclic tests with stress- and strain-controlled loading programs. It is demonstrated that breakage of a specimen occurs in the ratcheting test after 48 cycles, whereas no sign of failure is observed after 150 cycles of oscillations with the strain-controlled program and maximum strain max coinciding with strain at breakage under the stress-controlled program. To rationalize these observations, a constitutive model is suggested in cyclic viscoelastoplasticity of semicrystalline polymers. Adjustable parameters are found by fitting experimental stress– strain curves (50 cycles of loading–retraction). A pronounced difference is revealed between evolution of plastic strains under stress- and strain-controlled deformations. In the former case, maximum plastic strain per cycle increases strongly with number of cycles, while in the latter case, it remains practically constant. The growth of plastic strain under ratcheting is induced by severe plastic flow in the crystalline phase that is associated with lamellar fragmentation. References
Fig. 10. Maximum max and minimum min plastic strains in the crystalline (p1 ) p p and amorphous (p2 °) phases versus number of cycles n. Symbols: predictions of the model for the stress-controlled test.
depicted in Fig. 4. Good agreement between predictions of the model and experimental data confirms that Eq. (9) adequately describes functions S1(n) and S2(n).
[1] L.M. Kachanov, Introduction to continuum damage mechanics, Martinus Nijhoff Publishers, Dordrecht, 1986. [2] N. Stribeck, U. Nochel, S. Funari, T. Schubert, A. Timmann, Macromol. Chem. Phys. 209 (2008) 1992–2002. [3] A. Pawlak, A. Galeski, J. Polym. Sci. B: Polym. Phys. 48 (2010) 1271–1280. [4] R.M. Guedes (Ed.), Creep and fatigue in polymer matrix composites, Woodhead Publ., Philadelphia, PA, 2011. [5] N.L. Post, S.W. Case, J.J. Lesko, Int. J. Fatigue 30 (2008) 2064–2086. [6] A.C. Orifici, I. Herszberg, R.S. Thomson, Compos. Struct. 86 (2008) 165–176. [7] A.D. Drozdov, Comput. Mater. Sci. 50 (2011) 1991–2000. [8] A.D. Drozdov, Int. J. Solids Struct. 48 (2011) 2026–2040. [9] A.D. Drozdov, J.deC. Christiansen, C.-G. Potarniche, Acta Mech. 221 (2011) 201– 222.
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Stress- and strain-controlled cyclic deformation of polypropylene A.D. Drozdov ⇑ Department of Plastics Technology, Danish Technological Institute Gregersensvej 1, Taastrup 2630, Denmark
a r t i c l e
i n f o
Article history: Received 15 October 2011 Accepted 24 February 2012 Available online 19 March 2012 Keywords: Polypropylene Viscoelasticity Viscoplasticity Constitutive modeling
a b s t r a c t Experimental data are reported on isotactic polypropylene in uniaxial tensile cyclic tests with stress- and strain-controlled programs. A constitutive model is developed that adequately describes stress–strain diagrams, and its parameters are found by fitting the observations. Numerical simulation demonstrates that fatigue failure in ratcheting tests is induced by pronounced growth of plastic strain in crystallites. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction This paper deals with experimental investigation, constitutive modeling, and numerical simulation of the viscoelastic and viscoplastic responses of semicrystalline polymers in cyclic tests with stress- and strain-controlled programs. We begin with the analysis of stress-controlled cyclic tensile tests (ratcheting between some maximum rmax and minimum rmin stresses). Observations show that maximum max and minimum min strains per cycle grow monotonically with number of cycles n until an instant when the sample breaks. The question of interest is why fatigue failure occurs under the stress-controlled deformation program? The conventional answer is that breakage is induced by damage accumulation: some non-decreasing measure of damage exists that equals zero for an undeformed specimen and which increases monotonically with n until it reaches some ultimate value corresponding to failure of the specimen [1]. Growth of damage variable in semicrystalline polymers is attributed to (i) lamellar fragmentation [2] and (ii) nucleation and growth of micro-voids [3]. As damage accumulation cannot be measured directly in mechanical tests, it is traditionally associated [4] with hysteresis energy (area between subsequent loading and unloading paths of a stress–strain diagram). Failure in ratcheting tests with relatively large rmax occurs within the experimental time-scale (50–100 cycles) when maximum strain per cycle reaches its critical value c. This value is employed as a simple criterion for failure: breakage occurs when ratcheting strain exceeds c. This condition, however, is not unique: a number of different criteria for fracture are available in the literature [5,6]. ⇑ Tel.: +45 72 20 31 42; fax: +45 72 20 31 12. E-mail address: [email protected] 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2012.02.035
Let us now turn to strain-controlled tensile cyclic tests (oscillations between a maximum strain max and the minimum stress rmin). As c is located below necking strain n for uniaxial tension, the maximum strain per cycle max in these tests can be chosen equal or even exceeding the critical strain. Observations in these tests show that (i) maximum stress per cycle rmax decreases only slightly with n, (ii) minimum strain per cycle min strongly exceeds that in corresponding stress-controlled tests, (iii) at each cycle, hysteresis energy is substantially higher than in experiments with the stress-controlled program, but no breakage of specimens is observed within the experimental time-scale (which means that traditional fracture criteria are inapplicable). The objective of this study is threefold: (i) to report experimental data on isotactic polypropylene in stress- and strain-controlled tests, (ii) to develop constitutive equations in cyclic viscoelasticity and viscoplasticity of semicrystalline polymers that correctly describe the observations, and (iii) to provide an explanation for the difference in the mechanical response in these two types of tests by comparison of results of numerical simulation. 2. Experimental results Isotactic polypropylene Moplen HP 400R (density 0.90 g/cm3, melt flow rate 25 g/10 min, melting temperature Tm = 161 °C, degree of crystallinity 47%) was purchased from Basell Polyolefins (Switzerland). Dumbbell specimens for mechanical tests (ASTM standard D-638) with cross-sectional area 10.1 mm 4.2 mm were molded by using injection-molding machine Arburg 320C. Mechanical tests were conducted by means of universal testing machine Instron-5569 equipped with an electro-mechanical sensor for control of longitudinal strains. Tensile force was measured by 5 kN load cell. The engineering stress r was determined as
A.D. Drozdov / Computational Materials Science 64 (2012) 198–202
the ratio of axial force to cross-sectional area of undeformed specimens. Uniaxial tensile cyclic tests were conducted at room temperature. In the first test (ratcheting), a specimen was stretched with cross-head speed 100 mm/min (that corresponded to strain rate _ ¼ 1:7 102 s1 Þ up to the maximum stress rmax = 32 MPa, unloaded down to the minimum stress rmin = 1 MPa with the same cross-head speed, reloaded up to rmax, retracted down to rmin, etc. Breakage under ratcheting occurs at the 48th cycle. The experimental stress–strain diagram for the stress-controlled test is depicted in Fig. 1, where stress r is plotted versus strain . The following conclusions are drawn: (i) loading and unloading paths of stress–strain diagrams are strongly nonlinear, (ii) hysteresis energy per cycle increases with n, (iii) the stress– strain diagram slowly rotates clockwise when number of cycles increases.
A
B
Fig. 1. Stress r versus strain . Symbols: experimental data in the stress-controlled test. Solid lines: results of numerical simulation. (A) 46 cycles and (B) n = 1, 2(); n = 20 (); n = 40 (q).
Fig. 2. Maximum max and minimum min strains versus number of cycles n. Unfilled circles: observations in the stress-controlled test. Filled circles: results of numerical simulation.
199
Evolution of maximum max and minimum min strains per cycle with n is illustrated in Fig. 2. The diagrams max(n) and min(n) may be split into three intervals. Along the first (primary fatigue), the curves max(n) and min(n) are convex. Along the other interval (secondary fatigue), max and min grow with n linearly. At the last interval (tertiary fatigue), the dependencies max(n) and min(n) become nonlinear and concave. In the other test with strain-controlled loading, a specimen was stretched with the same cross-head speed 100 mm/min up to the maximum strain max = 0.16, unloaded down to the minimum stress rmin = 1 MPa, reloaded up to max, retracted down to rmin, etc. The program involved 150 cycles of loading–retraction. No signs of damage were observed in the sample except for slight whitening. The stress–strain diagram for the first 50 cycles of oscillations is depicted in Fig. 3. This figure reveals that (i) hysteresis energy
A
B
Fig. 3. Stress r versus strain . Symbols: experimental data in the strain-controlled test. Solid lines: results of numerical simulation. (A) 50 cycles and (B) n = 1, 2 (); n = 20 (); n = 40 (q).
Fig. 4. Maximum stress rmax and minimum strain min versus number of cycles n. Circles: observations in the strain-controlled test. Solid lines: predictions of the model.
200
A.D. Drozdov / Computational Materials Science 64 (2012) 198–202
strongly decreases with number of cycles, and (ii) the stress–strain curve slowly rotates counter-clockwise with growth of n. Evolution of maximum stress per cycle rmax and minimum strain per cycle min is reported in Fig. 4. This figure shows that (i) rmax decreases monotonically with n and becomes practically independent of n after a dozen of cycles, and (ii) min noticeably increases with n. The minimum stress rmin = 1 MPa was chosen instead of min r = 0 to avoid buckling of specimens. The cross-head speed 100 mm/min was selected to ensure that duration of cyclic tests does not exceed that of conventional short-term creep and relaxation tests. Each test was carried out on a new specimen and repeated by twice. Observations reveal good reproducibility of measurements: deviations between maximum stresses and minimum strains measured on different specimens did not exceed 3%. The terms ‘‘stress-controlled’’ and ‘‘strain-controlled’’ are used in a broad sense. Strictly speaking, the loading programs under investigation are not stress- and strain-controlled. The ratcheting test was conducted with a constant strain (not stress) rate, while unloading in the cyclic test with fixed max was performed down to the minimum stress (not strain). 3. Constitutive model A two-step approach is employed to describe the response of semicrystalline polymers [7]. At the first step, constitutive equations are derived by using the Clausius–Duhem inequality for the viscoelastic and viscoplastic behavior of polypropylene along an individual cycle of loading–retraction. At the other step, some parameters in the stress–strain relations are allowed to change with number of cycles. Their evolution induced by damage accumulation is described by kinetic equations which are validated by comparison of the model predictions with observations. Stress–strain relations in viscoelastoplasticity of semicrystalline polymers under an arbitrary three-dimensional deformation with small strains have been developed in [8,9]. In what follows, these equations are simplified for uniaxial tension. Strain is presented as the sum of elastic e and plastic p strains
¼ e þ p : The plastic strain
ð1Þ
p is split into the sum of two components
p ¼ p1 þ p2 ;
ð2Þ
that reflect plastic deformation in the crystalline and amorphous phases, respectively. The strain rate for plastic deformation in the crystalline phase is proportional to that for macro-deformation
dp1 d ¼/ : dt dt
ð3Þ
The coefficient / obeys the differential equation
d/ d ¼ að1 /Þ2 ; dt dt
ð4Þ
where a adopts different values a1 and a2 under loading and retraction. The strain rate for plastic deformation in the amorphous matrix is governed by the equation
dp2 ¼S dt
e Rp2
Z 0
1
d f ðv ÞZðt; v Þdv ; dt
ð5Þ
where R and S adopts different positive values R1 = 1, R2, and S1, S2, under loading and unloading. The integral term in Eq. (5) accounts for the effect of viscoelasticity on plastic flow in the amorphous phase. The function f(v) describes inhomogeneity of the amorphous phase induced by the presence of spherulites. A quasi-Gaussian expression is adopted for this function
1 v 2 ðv P 0Þ; f ðv Þ ¼ f0 exp 2 R
f ðv Þ ¼ 0 ðv < 0Þ;
ð6Þ
where R > 0 characterizes distribution of relaxation times, and f0 is R1 determined from normalization condition 0 f ðv Þdv ¼ 1. The function Z(t,v) obeys the differential equation
@Z ðt; v Þ ¼ Cðv Þ½e ðtÞ Zðt; v Þ; @t
Zð0; v Þ ¼ 0;
ð7Þ
where C(v) = cexp (v), and c denotes relaxation rate. The stress r reads
rðtÞ ¼ Eð1 /ðtÞÞ e ðtÞ
Z
1
f ðv ÞZðt; v Þdv ;
ð8Þ
0
where E stands for Young’s modulus. Eqs. (1)–(8) with eight adjustable parameters, E, a1, a2, R2, S1, S2, c, R, describe uniaxial tension with an arbitrary deformation program (t). The following scenario is proposed for evolution of these quantities: 1. Parameters E, c, and R are independent of n. The assumption that E remains constant distinguishes this approach from conventional concepts in damage mechanics that postulate a decay in elastic modulus under deformation. 2. After an initial transition period (about 10 cycles) coefficients a1 and a2 coincide (a1 = a2 = a1) and become independent of n. The quantity a1 equals zero for the strain-controlled program and remains positive for the stress-controlled program. 3. Coefficient R2 (this parameter characterizes energy of interchain interactions) monotonically decreases and vanishes at large n. 4. After the initial transition period, coefficients S1 and S2 (that characterize rates of plastic strain in the amorphous matrix under loading and retraction) linearly evolve with n
S1 ¼ S10 þ S11 n;
S2 ¼ S20 þ S21 n;
ð9Þ
where S10, S11, S20, S21 are constants. After an initial transition period, mechanical behavior of a semicrystalline polymer under the strain-controlled program is determined by five adjustable parameters: (i) E stands for elastic modulus, (ii) c determines relaxation rate, (iii) R characterizes distribution of relaxation times, (iv) S1 and S2 denote rates of plastic flow in the amorphous matrix. The viscoelastoplastic response under the stress-controlled program is characterized by the same quantities and coefficient a1. 4. Adjustable parameters Coefficients c and R are determined by fitting observations in relaxation tests; they read c = 0.11 s1, R = 13.1 [9]. The modulus E = 2.15 GPa is found by approximation of the first loading path of the stress–strain curve depicted in Fig. 3. Adjustable parameters a1, a2, R2, S1, and S2 are determined by matching observations depicted in Figs. 1 and 3 (50 cycles of loading–retraction) following the procedure described in [7]. Each loading path is fitted by two parameters, a1 and S1, and each unloading path is approximated by three parameters, a2, R2, and S2. The number of parameters is reduced when some of these quantities vanish. For example, when a1 = R2 = 0, each loading and unloading path is matched by means of one parameter only: S1 for loading and S2 for retraction. Coefficients a1 and a2 are plotted versus number of cycles in Fig. 5. For the stress-controlled program, these coefficients coincide and become independent of n (a1 = 9.0) after 5 cycles, whereas for
A.D. Drozdov / Computational Materials Science 64 (2012) 198–202
A
B
201
5. Numerical simulation To assess accuracy of fitting, results of numerical simulation are presented together with experimental stress–strain curves in Figs. 1 and 3. These figures reveal that quality of approximation is not reduced with number of cycles. To demonstrate that the model accurately predicts the viscoelastoplastic response in strain-controlled tests with large (noticeably exceeding that used in the fitting procedure) number of cycles, simulation is conducted of the stress–strain relations for 150 cycles of oscillations. Results of the numerical analysis are
A
Fig. 5. Parameters a1() and a2() versus number of cycles n. Symbols: treatment of observations in cyclic tests. (A) stress-controlled program and (B) strain-controlled program.
B A
B Fig. 7. Parameter S1 versus number of cycles n. Circles: treatment of observations in cyclic tests. Solid lines: approximation of the data by Eq. (9). (A) stress-controlled program and (B) strain-controlled program.
A
Fig. 6. Parameter R2 versus number of cycles n. Circles: treatment of observations in cyclic tests. (A) stress-controlled program and (B) strain-controlled program.
the strain-controlled program, a1 and a2 vanish (a1 = 0) after 10 cycles of oscillations. Parameter R2 is plotted versus number of cycles in Fig. 6. Under the strain-controlled deformation with a relatively large max, this quantity strongly decays and vanishes after 10 cycles of loading– retraction (severe damage accumulation). Under the stress-controlled deformation with mild conditions, R2 decreases with n, but remains positive up to the breakage point. Evolution of S1 with number of cycles is illustrated in Fig. 7 which demonstrates that S1(n) strongly decreases in the stresscontrolled test and grows in the strain-controlled experiment. The experimental dependencies S2(n) are depicted in Fig. 8 which shows similar responses in the stress- and strain-controlled tests.
B
Fig. 8. Parameter S2 versus number of cycles n. Circles: treatment of observations in cyclic tests. Solid lines: approximation of the data by Eq. (9). (A) stress-controlled program; (B) strain-controlled program.
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A
To understand why fatigue failure occurs in stress-controlled tests and no failure is observed in strain-controlled experiments, maximum max and minimum min plastic strains per cycle are p p calculated and plotted versus logarithm (log = log10) of n in Fig. 9. Results of simulation for the strain-controlled test are approximated by
max ¼ cmax þ cmax log n; min ¼ cmin þ cmin log n; p 0 1 p 0 1
B
Fig. 9. Maximum max ðÞ and minimum min p p ðÞ plastic strains versus number of cycles n. Symbols: treatment of observations in cyclic tests. Solid lines: approximation of the data by Eq. (10). (A) stress-controlled program; (B) strain-controlled program.
ð10Þ
min where cmax ; cmax ; cmin are constants. Fig. 9 reveals a pronounced 0 1 0 ; c1 difference between evolution of plastic strains under stress- and strain-controlled loading programs: in the former case, both plastic strains increase strongly with n, especially, at the stage of tertiary fatigue, whereas in the latter case, max remains independent of n, p while min grows linearly with log n. p To comprehend which component of the plastic strain increases substantially under ratcheting, maximum and minimum plastic strains in the crystalline p1 and amorphous p2 phases are reported in Fig. 10. This figure shows that fatigue failure is driven by a pronounced growth of maximum plastic strain in the crystalline phase max p1 caused by lamellar fragmentation under loading.
6. Concluding remarks Experimental data are reported on isotactic polypropylene in uniaxial tensile cyclic tests with stress- and strain-controlled loading programs. It is demonstrated that breakage of a specimen occurs in the ratcheting test after 48 cycles, whereas no sign of failure is observed after 150 cycles of oscillations with the strain-controlled program and maximum strain max coinciding with strain at breakage under the stress-controlled program. To rationalize these observations, a constitutive model is suggested in cyclic viscoelastoplasticity of semicrystalline polymers. Adjustable parameters are found by fitting experimental stress– strain curves (50 cycles of loading–retraction). A pronounced difference is revealed between evolution of plastic strains under stress- and strain-controlled deformations. In the former case, maximum plastic strain per cycle increases strongly with number of cycles, while in the latter case, it remains practically constant. The growth of plastic strain under ratcheting is induced by severe plastic flow in the crystalline phase that is associated with lamellar fragmentation. References
Fig. 10. Maximum max and minimum min plastic strains in the crystalline (p1 ) p p and amorphous (p2 °) phases versus number of cycles n. Symbols: predictions of the model for the stress-controlled test.
depicted in Fig. 4. Good agreement between predictions of the model and experimental data confirms that Eq. (9) adequately describes functions S1(n) and S2(n).
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