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Computational Materials Science 82 (2014) 415–426
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Computational Materials Science
journal homepage: www.elsevier .com/locate /commatsci
A micromechanical study on the deformation kinetics of orientedsemicrystalline polymers
0927-0256/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2013.09.068
⇑ Corresponding author. Tel.: +31 40 2474521; fax: +31 40 2447355.E-mail address: [email protected] (J.A.W. van Dommelen).
A. Sedighiamiri a, D.J.A. Senden a, D. Tranchida b, L.E. Govaert a, J.A.W. van Dommelen a,⇑a Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsb Borealis Polyolefine GmbH, 4021 Linz, Austria
a r t i c l e i n f o a b s t r a c t
Article history:Received 11 July 2013Received in revised form 19 August 2013Accepted 14 September 2013Available online 9 November 2013
Keywords:Composite inclusion modelEyring flow ruleMicromechanical modelingOriented semicrystalline polymersSlip kineticsStructure–property relation
The mechanical response of extruded and injection molded semicrystalline materials, in which an ori-ented microstructure is commonly observed, depends on the direction of loading. Plasticdeformation and failure are, therefore, both anisotropic. The ability of a micromechanical model,including the characterization of the kinetics of crystallographic slip and amorphous yield, to describethe yield kinetics of oriented high-density polyethylene tapes with different draw ratios isevaluated here. The initial morphology of the material is characterized with wide-angle X-ray diffrac-tion experiments, which evidence a strong alignment of molecular chains with the drawing directionfor specimens produced with a large draw ratio. Anisotropic crystal plasticity alone in a two-phaseframework proves not able to quantitatively describe the macroscopic mechanical response in thesolid state hot drawn samples. Most likely, these deviations are related to process-inducedorientation within the amorphous domains. Therefore, the influence of loading angle dependent yieldkinetics for the amorphous phase is evaluated, and indeed the description improves considerably.Finally, the possibilities for characterizing the properties of different crystallographic slip systemsare discussed.
� 2013 Elsevier B.V. All rights reserved.
1. Introduction
Due to their low costs, low weight and ease of processing, semi-crystalline polymeric materials are increasingly being used to re-place many structural and non-structural metallic parts.However, currently the inability to accurately predict and controllifetime of these materials, and the strong dependence of mechan-ical properties on processing conditions, are limiting factors for theuse of polymers in load-bearing applications. To overcome theselimiting factors it is required to thoroughly understand the under-lying mechanisms of deformation and failure in relation to themicrostructure and processing conditions. The mode and time offailure of polymeric products depends strongly on the molecularweight distribution of the polymer, the molecular orientation andthe thermal history [1–4]. For semicrystalline polymers, structuralfeatures, such as degree of crystallinity, crystal size and orienta-tion, that strongly influence their mechanical properties [5–12],depend on the way by which the polymer is shaped into the finalproduct, and may vary drastically with any subtle change in theprocessing conditions, such as flow rate, pressure, thermal history.For example, an important process for the production of plastic
parts is injection molding, during which the material is subjectedto shear and elongational flow that induces an oriented micro-structure prior to crystallization, leading to anisotropic mechanicalbehavior, which plays a crucial role in the performance and failureof polymer products. Therefore, the ability to use polymer productsin high-end applications is uniquely linked to the capability to pre-dict the elasto-viscoplastic behavior resulting from the underlyingmicrostructure.
Several micromechanically-based models of increasing com-plexity have been developed with the goal of establishing a struc-ture–property relation for semicrystalline polymers [13–22]. Thebasic idea of these models has been to generically relate themechanical properties at the macroscopic level to both micro-scopic constitutive and morphological properties, and they havebeen successfully applied to semicrystalline polymers with an ini-tially isotropic structure to predict their texture evolution andyield kinetics.
While most of the micromechanically based models describedthe mechanical response qualitatively, Sedighiamiri et al. [20–22]focused on improving the quantitative predictive abilities of themicromechanical model of van Dommelen et al. [18,19]. A key is-sue in achieving that goal is the fine-tuning of kinetics of slip alongcrystallographic planes, which have been re-evaluated and charac-terized for an initially isotropic semicrystalline polyethylene. Theysuccessfully predicted the strain rate dependence of the yield
0 0.2 0.4 0.6 0.80
5
10
15
20
25
engineering strain [-]
engi
neer
ing
stre
ss [M
Pa]
10-3 s-125 oC
10-4 s-1
10-3 s-150 oC
10-4 s-1
10-4 s-180 oC
10-5 s-1
a
10-5 10-4 10-3 10-2 10-10
10
20
30
25 oC50 oC
65 oC80 oC
strain rate [s-1]
yiel
d st
ress
[MPa
]
bFig. 1. (a) Tensile response of isotropic HDPE, at various temperatures and strain rates. (b) Temperature dependence of the tensile yield kinetics of isotropic HDPE. Symbolsindicate experimental data and lines show the micromechanical model prediction [22].
416 A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426
stress, as shown in Fig. 1, and the stress dependence of time-to-failure under various temperatures [22]. However, because of theisotropy of the material considered, a full characterization of thekinetics of the individual slip systems proved impossible.
Having quantitatively predicted the short and long-term failurekinetics of initially isotropic semicrystalline HDPE, this study aimsat extending the approach to initially oriented semicrystallinepolymers and their deformation kinetics. The stacked lamellarmorphology, commonly observed in extruded semicrystallinematerials [23,24], gives rise to a strong influence of the loadingdirection relative to the original extrusion direction on the defor-mation kinetics, causing plastic deformation and failure to beanisotropic [25,26]. This was confirmed by van Dommelen et al.[27], who performed tensile tests on melt-extruded HDPE samplesat different angles with respect to the extrusion direction. Their re-sults illustrate that when loaded in the extrusion direction, ahomogeneous deformation was observed throughout the entiresample, whereas a neck was formed when the material was loadedperpendicular to the extrusion direction. The composite inclusionmodel was used to describe these effects in a qualitative sense.
The present study addresses two main objectives. First, it isinvestigated whether the previously characterized slip systemsusing isotropic material [21,22] can also be used to quantitativelydescribe the response of anisotropic systems with a micromechan-ical model. Second, the possibility of using oriented systems forcharacterizing the properties of different crystallographic slip sys-tems is explored. For this purpose, the mechanical behavior at themacroscopic scale is modeled by an aggregate of layered two-phase composite inclusions. The initial orientation distribution isgenerated based on wide angle X-ray scattering measurements ofcrystalline orientation distributions for hot-drawn tapes. Model re-sults based on these orientation distributions are compared to re-sults of mechanical tests on these samples. Finally, the necessity ofmodeling an anisotropic amorphous domain for an accurate quan-titative description is further discussed.
2. Experimental
2.1. Material
The semicrystalline material used in this study is a high densitypolyethylene, supplied by SABIC (HDPE M80063CO), withMw = 70,000 [g/mol] and Mn = 11,000 [g/mol]. HDPE tapes weremono-extruded using a Collin E20-T single screw extruder fittedwith a 100 mm die. The extrudate was quenched on Collin CR-72
rolls at a temperature of 15 �C and then collected on a spool. Solidstate drawing was performed at a temperature of 120 �C, and dif-ferent tapes with draw ratios k of 1, 4 and 6 were made at a drawrate of approximately 1 s�1. The thickness of the undrawn tape(k = 1) was 130 lm.
2.2. Crystallinity
Wide angle X-ray scattering (WAXS) experiments were per-formed on the tapes and the weight fraction of crystals of the ori-ented samples was determined to be 60%, 66% and 63% for tapeswith draw ratios of 1, 4 and 6, respectively. Fig. 2 gives the re-corded WAXS pattern of HDPE tapes with different draw ratios.
The tape with a draw ratio of k = 1 is found to be isotropic, as itsWAXS pattern shows full Debye rings of the reflection of the crys-tallographic planes. The volumetric degree of crystallinity is ob-tained to be approximately 57% from the weight fraction ofcrystals for this sample, assuming qc = 1000 kg/m3, for the crystaldensity and qa = 855 kg/m3, for the isotropic amorphous density[28]. WAXS patterns of the other two samples (Figs. 2b and c),however, indicate a rather strong orientation. Bartczak et al. [29]estimated the density of the amorphous component in uniaxiallyoriented HDPE samples. A density of about qa = 960 kg/m3 was ob-served for a HDPE, compressed to a compression ratio of 6.4, whichis very close to the density of the crystals, suggesting a very highdegree of ordering of the molecular chains in the amorphous frac-tion. Due to lack of information about the density of the orientedamorphous fraction, which makes the estimation of the volumetricdegree of crystallinity uncertain, the volumetric degree of crystal-linity of all samples is set to be 60% in this study. A concise discus-sion on the effect of crystallinity on the deformation kinetics willbe given in Section 4.
2.3. Orientation measurement
The textures of oriented HDPE tapes were analyzed by addi-tional wide angle X-ray scattering experiments, and pole figureswere constructed to characterize the three dimensional crystallo-graphic orientation distribution with a Bruker D8 Discoverequipped with General Area Detector Diffraction System and oper-ating at 40 kV and 40 mA. The (200) and (110) diffraction reflec-tions were analyzed.
The resulting pole figures of orientation distributions of twocrystallographic planes are shown in Fig. 3 for oriented sampleswith draw ratios of 4 and 6. It is observed that the (200) plane
a b cFig. 2. WAXS patterns of HDPE tapes with a draw ratio of k = 1 (a), k = 4 (b) and k = 6 (c).
MD
TD
MD
TD
(110)(200)
a
MD
TD
MD
TD
(110)(200)
bFig. 3. Pole figures of orientation distribution for HDPE tape with draw ratios of 4 (a) and 6 (b).
Table 1Hermans’ orientation distribution factors for crystallographic directions.
Draw ratio fa fb fc
4 �0.40 �0.29 0.706 �0.45 �0.48 0.94
A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426 417
normals, corresponding to the crystallographic a-axes, and (110)plane normals are oriented in the ND-TD plane, being perpendicu-lar to the draw direction, with an intensity maximum in the center.HDPE possesses an orthorhombic unit cell, i.e. the crystallographicdirections are orthogonal. Therefore, the crystallographic b-axes,corresponding to the lamellar growth directions, should also beoriented perpendicular to the draw direction, and the crystallo-graphic c-axes, denoting the molecular chain axes, are orientedpreferentially in the draw direction, corresponding to a stackedlamellar morphology.
For uniaxially oriented systems, a quantitative measure of ori-entation can be given by the Hermans’ orientation factor (fH), de-fined as [30]:
fH ¼3hcos2 /i � 1
2ð1Þ
with / the angle between the unit whose orientation state is ofinterest (e.g. crystallographic axes) and a reference axis (e.g. fiberor machine direction). fH = 1 indicates perfect orientation, fH = 0
indicates random orientation, and fH = �0.5 indicates an orthogonalorientation. Table 1 lists the obtained Hermans’ orientation distri-bution factors for the principal crystallographic axes, calculatedusing Wilchinsky’s method [31] and the orthogonality relationship.
2.4. Mechanical testing
For uniaxial tensile tests, dogbone-shaped samples were cutfrom the tapes with different angles with respect to the machinedirection (drawing direction), namely 0�, 20� and 50�, as shownschematically in Fig. 4. Tensile tests were performed at constant,
5= 0° 2= 0° = 0°
3
1
2 MD
Fig. 4. Schematic illustration of the orientation of samples cut from the tapes withdifferent angles with respect to the machine direction.
418 A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426
linear strain rates varying from 10�4 s�1 up to 10�2 s�1 at roomtemperature.
3. Micromechanical modeling
The constitutive behavior of semicrystalline material is mod-eled by an aggregate of a discrete number of layered two-phasecomposite inclusions [14,15,18]. Each composite inclusion is com-prised of a crystalline lamella and an amorphous layer. A micro-structural elasto-viscoplastic constitutive model is used for bothconstituent phases. In this section, the material models for theelastic and viscoplastic behavior of each phase together with theaggregate model are briefly summarized. The constitutive materialmodels of each domain and the set of equations that compose thecomposite inclusion model are given in A. A more detailed discus-sion of the models is given by van Dommelen et al. [18] and Sedig-hiamiri et al. [22].
3.1. Crystalline phase
The crystalline phase of semicrystalline polymers consists ofregularly ordered molecular chains. The crystal structure causesthe crystalline phase to possess highly anisotropic elastic proper-ties with a high elastic modulus in the chain direction, and to haveplastic deformation primarily governed by crystallographic slip upto moderate strains [6,32–34].
The elastic component of the deformation in the crystalline do-main is modeled linearly and is characterized by an anisotropicfourth order elasticity tensor. The viscoplastic behavior of the crys-talline phase is described by a rate-dependent crystal plasticitymodel. In this model, the plastic flow rate of the crystalline lamella,consisting of a single crystal, is composed of the contributions of allphysically distinct slip systems, being 8 for high density polyethyl-ene. The shear rate of each slip system is assumed to be related tothe corresponding resolved shear stress via a viscoplastic Eyringflow rule [35]:
_c ¼ _c�0ðTÞ sinhssc
0
� �; ð2Þ
where
_c�0ðTÞ ¼ _c0 exp�DURT
� �ð3Þ
captures the temperature dependence of the slip kinetics, with DUthe activation energy of the slip system, R the universal gas constantand T the absolute temperature.
3.2. Amorphous phase
The amorphous phase of semicrystalline polymers consists ofan assembly of randomly coiled macromolecules, strongly con-nected to their adjacent crystalline lamellae through many chainscrossing the crystalline/amorphous interface, such as tight or loosechain ends, cilia, and intercrystalline links [36]. In the case of poly-
ethylene, the glass transition temperature of the amorphous phaseis below room temperature. The amorphous phase should, there-fore, be in the rubbery state at room temperature. However, someliterature [37] suggests that the interlamellar layers in a semicrys-talline polymeric material may be stiffer than purely bulk amor-phous material due to the confinement of amorphous layersbetween thick crystalline lamellae. Boyd [38,39] showed that forthe amorphous fraction in semicrystalline polymers, the relaxedmodulus associated with the glass-rubber relaxation is very highdue to the constraints by the neighboring crystals.
Furthermore, due to random thermal fluctuations, chain seg-ments in the crystalline phase can migrate into the amorphous do-main. Similarly, chain ends or loose chain folds can escape from theamorphous phase into the crystalline lamellae. This process ofchain diffusion [40], generally referred to as a-relaxation, resultsin redistribution of the chain segments that constitute the interla-mellar layer. Therefore, the relaxation of the interlamellar materialalso occurs in a rate-dependent process at a rate controlled bytranslational mobility of chain segments within the crystals [39].In line with that, also the amorphous phase is modeled as elasto-viscoplastic.
The elastic deformation of the amorphous phase is modeled bya generalized neo-Hookean relationship, characterized by shearand bulk moduli. An Eyring flow relation is used to relate the effec-tive shear strain rate and the effective shear stress:
_c ¼ _c�0ðTÞ sinhs
asa0
� �; ð4Þ
with a = 1.15 a constant accounting for the difference in the defini-tion of the effective shear stress of the amorphous phase and the re-solved shear stress of the crystalline phase, making themcomparable for a uniaxial stress state. The plastic rate of stretchingis defined by an associated flow rule [18]. The Arruda–Boyce eight-chain network rubber elasticity model [41] is used to account fororientation induced hardening [42].
3.3. Composite inclusion model
The mechanical behavior of the material at the macroscopicscale is modeled by an aggregate of layered two-phase compositeinclusions, as proposed by Lee et al. [14,15] for rigid-viscoplasticmaterial behavior. The stress and deformation fields in each sepa-rate phase of the composite inclusion are assumed to be piecewisehomogeneous; but, they can be different between the two coupledphases. The inclusion-averaged Cauchy stress and the inclusion-averaged deformation gradient are given by the volume-averageof the fields of the constituent phases. Furthermore, the crystallinelamella and the amorphous layer are assumed to be kinematicallycompatible and in equilibrium across the interface.
To relate the volume-averaged mechanical response of eachlayered composite inclusion to the imposed boundary conditionsfor an aggregate of inclusions, a hybrid local–global interactionlaw [14,15,18] is formulated. The consistency conditions forequilibrium and compatibility of the aggregate are maintained byenforcing the macroscopic stress and deformation to equal thevolume-average of the stress and deformation fields of thecomposite inclusions.
In the hybrid interaction model, local–global compatibility con-ditions are formulated for the projections of the inclusion-averagedfields for which intra-inclusion equilibrium conditions wereformulated. Inversely, local–global equilibrium conditions areassumed for the components of the inclusion-averaged fields thatare subjected to intra-inclusion compatibility.
(100)3
2
(010)3
2
(001)3
2
3
2
nI
a
(100)3
2
(010)3
2
(001)3
2
3
2
nI
bFig. 5. Equal area projection pole figures of the principal crystallographic and lamellar orientation distributions used to represent HDPE tape with draw ratios of 4 (a) and 6(b). The draw direction is vertical.
Table 2Hermans’ orientation distribution factors of crystallographic directions, used torepresent drawn HDPE tape.
Sample Draw ratio fa fb fc
1 4 �0.38 �0.32 0.732 6 �0.47 �0.47 0.94
Table 3Elastic and hardening parameters of the amorphous phase.
Ga (MPa) Ka (MPa) lR (MPa) N
65 3000 1 49
Table 4Elastic constants (GPa) for the crystalline phase of PE [50].
C11 C22 C33 C12 C13 C23 C44 C55 C66
7.99 9.92 315.92 3.28 1.13 2.14 3.62 1.62 3.19
10-5 10-4 10-3 10-2 10-10
5
10
15
20
shear rate [s-1]
]aPM[
sserts raehs(100)[010], {110}<110>, {110}[001](010)[001], (010)[100]
amorphous
(100)[001]
_
25oC
80oC
Fig. 6. Crystallographic slip and amorphous yield kinetics, used to describe theyield kinetics of the isotropic material at two different temperatures.
Table 5Yield mechanisms in the constituent phases, and their reference shear rates andactivation energies, at Tref = 23 �C.
Process I Process II
Mechanism _c�0;ref ðs�1Þ DUI (kJ mol�1) _c�0;ref ðs�1Þ DUII (kJ mol�1)
(100)[001] 8 � 10�35 512 2 � 10�5 105(010)[001] 2 � 10�49 762 2 � 10�5 105{110}[001] 2 � 10�49 762 2 � 10�5 105(100)[010] 2 � 10�49 762 2 � 10�5 105(010)[1 00] 2 � 10�49 762 2 � 10�5 105{110}h1 �10i 2 � 10�49 762 2 � 10�5 105Amorphous 6 � 10�39 516 2 � 10�5 105
A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426 419
4. Modeling oriented material
The initial crystalline orientations of the oriented HDPE tapeswith different draw ratios are generated according to the experi-mentally obtained pole figures of orientation distributions andHermans’ orientation distribution factors. For initially randomlyoriented materials, possessing a spherulitic morphology, the anglebetween the chain direction~c and the lamellar normal direction~nI
is known to be 35� [43,44], corresponding to the {201} planes.However, for oriented systems, the molecular chains are assumedto be more aligned with the lamellar normals. Shinozaki andGroves [45,46] studied the structure of polyethylene and polypro-pylene sheets, oriented by hot drawing at a temperature range of105–120 �C, after an elongation of approximately 600%. The finalmaterial was observed to be oriented with molecular chains paral-lel to the initial draw direction, and the lamellae roughly perpen-dicular to the draw direction and, therefore, normal to themolecular chain axis. Their finding is also supported by experimen-tal WAXS and SAXS data [47–49] on structural changes during hot
drawing of semicrystalline polymers, which show a process oftransformation of crystalline lamellae into much smaller folded-chain crystalline blocks, with both their crystalline lamellar normaland chain orientation almost parallel to the draw direction, whichare connected by somewhat extended amorphous chains. In linewith that, a reduced chain tilt angle of 5�, with lamellar surfacesagain of the {h0 l} type, is assumed.
Fig. 5 shows equal area projection pole figures of the principalcrystallographic lattice directions together with the initial lamellar
10-5 10-4 10-3 10-2 10-10
20
40
60
80
100
120
strain rate [s -1 ]
yiel
d st
ress
[MPa
]
θ= 2θ= 5θ= 10θ= 19θ= 35
a
10-5 10-4 10-3 10-2 10-10
20
40
60
80
100
120
strain rate [s -1]
yiel
d st
ress
[MPa
]
= 70%
= 65%
= 60%
= 55%f c
f c
f c
f c
bFig. 7. (a) The effect of varying chain tilt h on the tensile yield kinetics for a constant crystallinity of fc = 60%. (b) The effect of varying crystallinity on the tensile yield kineticsfor a constant chain tilt of h = 5�. The loading direction is aligned with the draw direction of the sample with draw ratio of 6.
(100)[001]
b
c(100)[010]
b
c{110}<110>
b
c{110}[001]
b
c
_
0
5
10
15[MPa]
(a)
(100)[001]
b
c(100)[010]
b
c{110}<110>
b
c{110}[001]
b
c
_
0
5
10
15[MPa]
(b)
(100)[001]
b
c(100)[010]
b
c{110}<110>
b
c{110}[001]
b
c
_
0
5
10
15[MPa]
(c)
(100)[001]
b
c(100)[010]
b
c{110}<110>
b
c{110}[001]
b
c
_
0
5
10
15[MPa]
(d)
Fig. 8. Inverse pole figures showing the resolved shear stress of a number of slip systems at the engineering yield stress under a constant strain rate of 10�3 s�1 in tension at23 �C, with various load angles of (a) 0�, (b) 20�, (c) 50� and (d) 90�, for k = 6. The kinetics are the same as in Fig. 6. Colors represent the magnitude of the resolved shear stressof the specified slip system for each inclusion, and the location of the markers indicates the loading direction relative to the crystallographic axes of each lamella. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
420 A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426
orientations. The oriented structure of the material is representedby an aggregate of 500 composite inclusions. The resulting Her-mans’ orientation distribution factors listed in Table 2 show goodagreement with experimentally obtained values in Section 2.3.
The elasto-viscoplastic composite inclusion model is used toinvestigate the effect of the morphological factors, namely the tiltangle and the degree of crystallinity, on the mechanical behavior ofthe material. In the model, the elastic deformation of the interla-
10-5 10-4 10-3 10-2 10-10
5
10
15
20
25
30
shear rate [s -1]
shea
r stre
ss [M
Pa]
(010)[001], {110}[001],(100)[001](100)[010], {110}<110>, (010)[100]amorphous
_
a
10-5 10-4 10-3 10-2 10-10
20
40
60
80
100
120
strain rate [s -1]
yiel
d st
ress
[MPa
]
λ= 6,T= 23 C
φ= 0
φ= 90
b
10-5 10-4 10-3 10-2 10-10
5
10
15
20
25
30
shear rate [s -1]
shea
r stre
ss [M
Pa]
(100)[010], {110}<110>, (010)[100](010)[001], {110}[001]amorphous(100)[001]
_
c
10-5 10-4 10-3 10-2 10-10
20
40
60
80
100
120
strain rate [s -1]
yiel
d st
ress
[MPa
]
λ= 6,T= 23 C
φ= 0
φ= 90
dFig. 9. Crystallographic slip and amorphous yield kinetics with (a) increased chain slip kinetics and (c) increased transverse slip kinetics, and (b) and (d) the effect on the yieldstress for different loading angles. Dashed lines show the model prediction corresponding to the kinetics of (a) and (c), and solid lines give the prediction corresponding to thekinetics of Fig. 6.
A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426 421
mellar layer is characterized by the shear and bulk moduli, Ga andKa, respectively. The elastic parameters as well as the hardeningproperties of the amorphous phase, used in this study, are summa-rized in Table 3, with lR and N being the constant proportional tothe initial stiffness and the number of rigid links between entan-glements in the Arruda–Boyce eight-chain model [41]. For ortho-rhombic PE crystals, the anisotropic fourth order elasticity tensorinvolves nine independent elastic constant. The constants used inthis study are given in Table 4.
Fig. 6 shows the crystallographic slip kinetics and the amor-phous yield kinetics at 23 �C and 85�C. The evaluation of the kinet-ics at the microscopic level was performed, taking into account thetensile kinetics of the HDPE tape with draw ratio k = 1 (isotropic).For an isotropic system, at the macroscopic level, the effective con-tribution of all slip systems is observed, accompanied by the yieldkinetics of the amorphous domains. As a result, it is not possible touniquely characterize and distinguish between the effect of indi-vidual slip systems for an isotropic system [22] and, therefore,the properties of some slip systems are assumed to be identicalin the micromechanical model. Table 5 summarizes the yieldkinetics of all slip systems1 and the amorphous phase, in terms oftheir activation energies and the reference shear rates _c�0;ref , charac-terized at a reference temperature Tref, for two relaxation processes.A more elaborated discussion on the contribution of two relaxationprocesses to the yield kinetics is given in [22]. The temperature
1 Note that the 6 types of crystallographically different slip systems of Table 5represent 8 physically different slip systems.
dependence of the reference shear rates can then be calculatedusing:
_c�0ðTÞ ¼ _c�0;ref exp�DU
R1T� 1
Tref
� �� �; ð5Þ
with Tref being 23 �C in this study. The values of the characteristicshear stresses for the relaxation processes within the crystallineand amorphous phases in the Eyring model are set tosc;I
0 ¼ sa;I0 ¼ 0:06 MPa and sc;II
0 ¼ sa;II0 ¼ 0:85 MPa.
A systematic study on the effect of the tilt angle as well as thedegree of crystallinity on the model prediction of the tensile yieldkinetics of the sample with draw ratio of 6, loaded in drawingdirection, is illustrated in Fig. 7. As can be seen, the deformationkinetics are predominantly influenced by the chain tilt angle,whereas the crystallinity has a much smaller effect on the kinet-ics. Therefore the assumption of a volumetric degree of crystallin-ity of 60% for all samples would not drastically affect theirdeformation kinetics. Furthermore, the results are observed tonot be very sensitive to making the chain tilt angle smaller than5�.
Unlike for isotropic systems, for initially oriented semicrystal-line polymers, it is, indeed, possible to distinguish between dif-ferent slip systems. Fig. 8 illustrates the resolved shear stressof a selected number of slip systems at the engineering yieldstress (maximum in local stress) under a constant applied strainrate of 10�3 s�1 for the sample with k = 6 in tension at 23 �C, forvarious loading angles. In these pole figures, the location ofmarkers shows the loading direction relative to the crystallo-graphic axes of each lamella and colors represent the magnitude
10-5 10-4 10-3 10-2 10-10
50
100
150
200
250λ= 1,T= 23 C
φ= 0φ= 20φ= 50
strain rate [s -1]
yiel
d st
ress
[MPa
]
a
10-5 10-4 10-3 10-2 10-10
50
100
150
200
250λ= 4,T= 23 C
φ= 0φ= 20φ= 50
strain rate [s -1 ]
yiel
d st
ress
[MPa
]
b
10-5 10-4 10-3 10-2 10-10
50
100
150
200
250λ= 6,T= 23 C
φ= 0φ= 20φ= 50
strain rate [s -1]
yiel
d st
ress
[MPa
]
cFig. 10. The model prediction of the tensile yield kinetics of the material with (a) k = 1, (b) k = 4 and (c) k = 6, at various loading angles. Lines show the micromechanical modelprediction and symbols indicate experimental data.
422 A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426
of the resolved shear stress of the specified slip system for eachinclusion. As can be seen, for a 0� load angle, the stress is mainlyresolved on the chain slip systems, namely (100)[001] and{110}[001], whereas the amount of resolved shear stress onthe transverse slip systems, namely (100)[010] andf110gh1 �10i, is less significant. As the load angle increases to90�, a change in the shear stress on the slip systems is observed.At a 90� load angle, the stress is mostly resolved on the(100)[010] and f110gh1 �10i transverse slip systems and the(100)[001] and {110}[001] chain slip systems show a negligibleresolved shear stress. The effect of the properties of differentgroups of slip systems on the macroscopic deformation kineticsis shown in Fig. 9. The results are compared to those corre-sponding to the slip and amorphous yield kinetics of Fig. 6, asa reference to indicate the effect of different slip systems. Firstthe level of the chain slip kinetics is increased, while the levelof the transverse slip kinetics is the same as in Fig. 6, seeFig. 9a. The effect on the macroscopic yield kinetics is depictedin Fig. 9b, which shows an increase in the 0� yield kinetics,whereas the 90� yield kinetics do not change. In the sameway, by increasing the level of the transverse slip systems, whilekeeping the chain slip systems the same (Fig. 9c), the 90� yieldkinetics increase, whereas the 0� yield kinetics remain almostthe same, as shown in Fig. 9d. These results are supported bythe inverse pole figures showing the resolved shear stress of slipsystems in Fig. 8, and provides a promising method for charac-terization of the kinetics of the individual slip systems, whichis a major step towards modeling anisotropic properties of semi-crystalline polymers.
5. Results and discussion
The mechanical response of HDPE tapes with various micro-structures subjected to uniaxial tension is investigated here for arange of strain rates and loading angles, using the composite inclu-sion model. The starting point is formed by characterization of thekinetics of the crystallographic slip systems and the deformationkinetics of the amorphous domain at the microscopic level toquantitatively describe the macroscopic plastic flow kinetics forthe material with draw ratio k = 1, which are both structurallyand mechanically found to be isotropic. The model prediction ofthe tensile yield kinetics of the HDPE tape with draw ratio k = 1can be seen in Fig. 10a, which shows a similar strain rate depen-dence of the yield stress for various loading angles, as observedin experimental data. The application of the kinetics used for theisotropic sample (Fig. 6), to the oriented samples with draw ratiosk = 4 and k = 6 is depicted in Fig. 10b and c, respectively. As can beseen, the model result for the tensile yield kinetics of the orientedsamples at a loading angle of 50� is reasonably good, whereas atlower loading angles, 20� and 0�, the kinetics are not describedwell. It is worth pointing out that despite several attempts, a un-ique set of crystallographic slip kinetics, which can describe thedeformation kinetics at all three draw ratios, was not found. Theresults show that the anisotropy of the crystalline phase alone isapparently not sufficient to capture the anisotropic macroscopicyield kinetics. This remarkable observation could be rationalizedby the fact that in these drawn tapes the anisotropy is deforma-tion-induced, causing not only the crystalline phase but also theamorphous domains to stretch and orient in the drawing direction.
10-6 10-5 10-4 10-3 10-2 10-1 1000
10
20
30
40
50
60
strain rate [s -1]
]aPM[
ssertsdlei y
eurttension,φ= 0tension,φ= 90
compression, φ= 0compression, φ= 90
a
10-5 10-4 10-3 10-2 10-10
20
40
60
80
100tension
compressionφ= 0
φ= 90
strain rate [s -1]
true
yiel
d st
ress
[MPa
]
bFig. 11. (a) Tensile and compressive yield kinetics of a rolled PE to a draw ratio of 2, with different loading angles. Symbols show experimental data and lines are a guide tothe eye. (b) Micromechanical model prediction of the tensile and compressive yield kinetics of the sample with draw ratio k = 6, with different loading angles.
10-5 10-4 10-3 10-2 10-10
5
10
15
20
25
30
shear rate [s -1]
shea
r stre
ss [M
Pa]
am .φ= 0
am .φ= 20
am .φ= 50
a
10-5 10-4 10-3 10-2 10-10
50
100
150
200
250
strain rate [s-1]
yiel
d st
ress
[MPa
]
λ = 4, T= 23 C
φ= 0φ= 20φ= 50
b
10-5 10-4 10-3 10-2 10-10
5
10
15
20
25
30
shear rate [s-1]
]aPM[
sserts raeh s
am .φ= 0
am .φ= 20
am .φ= 50
c
10-5 10-4 10-3 10-2 10-10
50
100
150
200
250
strain rate [s -1 ]
yiel
d st
ress
[MPa
]
λ = 6, T= 23 C
φ= 0φ= 20φ= 50
dFig. 12. Crystallographic slip and varying amorphous yield kinetics, used to describe the yield kinetics of the material with (a) k = 4 and (c) k = 6 at room temperature.Micromechanical model prediction of the anisotropic yield kinetics of the material with (b) k = 4 and (d) k = 6. Lines show the micromechanical model prediction and symbolsindicate experimental data. Slip kinetics are the same as in Fig. 6, with the solid line showing the kinetics of (100)[001] chain slips, and the dash-dot line denoting theremaining kinetics.
A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426 423
One important ‘‘tell-tale’’ indicator of the presence of an ori-ented amorphous phase is the so-called Bauschinger effect in ori-ented polymers [51–54], i.e. a difference between thecompressive and tensile yield stress. The larger the difference,the higher the level of orientation. In isotropic polymers, the yieldstress in compression is slightly higher than in tension due to theeffect of hydrostatic pressure [55,56,22]. However, for orientedpolymers, the Bauschinger effect completely overwhelms the rela-tively small hydrostatic pressure effect, and the yield stress in ten-
sion becomes much higher than in compression. Fig. 11a shows thetensile and compressive true yield stresses of a pre-oriented poly-ethylene, rolled to a draw ratio of 2. It is observed that the tensileand compressive yield kinetics for a loading angle of 90� relative tothe rolling direction are relatively similar, whereas, when the load-ing direction is aligned with the sample’s orientation direction, thetensile yield stress is almost 3 times higher than the compressiveone. It should be noted that the rolled PE has a lower draw ratiothan the HDPE tapes in this study, and therefore, much lower
Table 6The reference shear rates and characteristic shear stresses of the amorphous phase forvarying loading angles at Tref = 23 �C.
Sample k = 4 Sample k = 6
Loading angle _c�0;ref ðs�1Þ sa0 ðMPaÞ _c�0;ref ðs�1Þ sa
0 ðMPaÞ0� 1.6 � 10�4 6.3 1.9 � 10�4 7.720� 5.1 � 10�5 1.4 1.2 � 10�4 3.450� 2.0 � 10�5 0.84 2 � 10�5 0.84
424 A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426
tensile yield stresses. Thus, one should expect a more pronouncedBauschinger effect for the oriented HDPE tapes with draw ratios of4 and 6.
The micromechanical model prediction of the tensile and com-pressive yield kinetics, with the crystallographic slip and amor-phous yield kinetics of Fig. 6, is illustrated in Fig. 11b. For ease ofinterpretation, the hydrostatic pressure effect is left out of consid-eration in this exercise. As also observed in experimental data, thepredicted tensile and compressive yield kinetics for a loading angleof 90� are similar. However, for the 0� loading direction, a small dif-ference between the compressive and tensile yield stress is ob-served, which is due to the difference in the evolution of theinitial microstructure in tension and compression. It is worthpointing out that the hydrostatic pressure effect would even fur-ther decrease this small difference. One possible explanation forthe significant difference in the mechanical response of the ori-ented semicrystalline polymers in uniaxial tension and compres-sion is that the molecular chains in the amorphous domainbecome oriented, and thereby, cause the oriented amorphousphase to possess anisotropic elastic and viscoplastic properties.The presence of oriented amorphous domains in plastically de-formed semicrystalline polymers has been demonstrated usingX-ray diffraction and infrared dichroism studies on the molecularorientation [29,57].
The effect of introducing an oriented amorphous phase com-bined with the contribution of the anisotropic crystalline phaseon the yield kinetics of the oriented samples is mimicked by usingdifferent kinetics for the amorphous phase for different loading an-gles and is given in Fig. 12. The crystallographic slip kinetics are ta-ken the same as the ones given in Fig. 6. The reference shear rates_c�0;ref and the characteristic shear stresses s0 used for process II inevery load angle are summarized in Table 6. The properties of pro-cess I are set to be the same as in Table 5. A higher modulus is as-sumed for the 0� loading angle, since the elastic properties alsobecome anisotropic for oriented amorphous domains [58,59]. As
10-5 10-4 10-3 10-2 10-10
10
20
30
40
50
60
70
strain rate [s-1]
yiel
d st
ress
[MPa
]
λ = 4, T= 80 C
φ= 0φ= 20φ= 50
aFig. 13. Micromechanical model prediction of the deformation kinetics at 80 �C with (a)indicate experimental data.
can be seen, by assuming a loading angle dependent yield kineticsfor the amorphous phase of the oriented HDPE tapes, it is possibleto describe the macroscopic deformation kinetics of two tapes withdifferent degrees of orientation at various load angles. However, itshould be noted that for the oriented material with k = 4, the 0�yield kinetics are not predicted well at high strain rates. This couldpartly be due to the effect of the initial orientation distribution. Thegenerated initial orientations were quantified by the experimen-tally obtained Hermans’ orientation factor. There is not a uniquesolution for a Hermans’ orientation factor of approximately 0.7and different distributions can result in the same number.
The model prediction of the deformation kinetics of two ori-ented samples at 80 �C for various load angles is shown inFig. 13. The kinetics of the amorphous phase are again assumedindependent of the loading angle. It is observed that althoughthe 20� and 50� deformation kinetics are described reasonablywell, the 0� kinetics for both oriented samples are underestimated.This could be an indication of the influence of the oriented amor-phous phase on the macroscopic flow kinetics, even at elevatedtemperatures.
The results show that the prediction of anisotropic yield kinet-ics with high values of yield stress for a 0� load angle, requires thedescription of anisotropic yield kinetics for the amorphous domain,due to the presence of oriented extended chains. For modeling pur-pose, the amorphous phase in this study is modeled both elasti-cally and viscoplastically isotropic. However, in order toquantitatively predict the deformation kinetics of oriented semi-crystalline polymers, expressions should be used which are ableto describe the plastic flow properly, taking into account an aniso-tropic behavior of the amorphous phase.
6. Conclusions
The deformation of semicrystalline polymers is the result of theinterplay of various mechanisms at different levels. A quantitativeand accurate prediction of the mechanical behavior of these mate-rials requires a coupled and detailed modeling of the various defor-mation mechanisms in the heterogeneous microstructure. In thisstudy, a multi-scale micromechanical model is used to describethe deformation kinetics of hot-drawn oriented semicrystallinepolyethylene tapes with different initial microstructures. The mod-el represents the microstructure as an aggregate of layered com-posite inclusions, each consisting of a crystalline lamella, whichis mechanically coupled to its adjacent amorphous layer. The crys-talline phase is modeled as anisotropic elastic with plastic flow
10-5 10-4 10-3 10-2 10-10
10
20
30
40
50
60
70
strain rate [s-1 ]
yiel
d st
ress
[MPa
]
λ = 6, T= 80 C
φ= 0
bk = 4 and (b) k = 6. Lines show the micromechanical model prediction and symbols
A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426 425
governed by crystallographic slip. The amorphous phase is as-sumed to be isotropic elastic with a rate dependent plastic flowand strain hardening resulting from molecular orientation. To re-late the volume-averaged mechanical behavior of each layeredcomposite inclusion to the aggregate of composite inclusions, a hy-brid local–global interaction law is used.
The initial orientation distribution for the micromechanicalmodel was generated based on pole figures from wide-angleX-ray diffraction experiments. A key issue in the initial morphol-ogy of oriented HDPE was found to be the angle between the chaindirection and the lamellar normal direction, also referred to aschain tilt, which showed a strong influence on the yield kinetics.
Unlike initially isotropic semicrystalline polymers, orientedpolymers look promising for characterizing the properties of indi-vidual crystallographic slip systems. The resolved shear stress of anumber of slip systems for various loading angles revealed the fea-sibility of distinguishing between the contribution of different slipsystems. However, the results of the micromechanical model withisotropic yield kinetics for the amorphous phase showed that theorientation of the crystalline phase alone is not sufficient to ac-count for the macroscopically anisotropic yield kinetics. The addi-tional contribution of the anisotropic yield kinetics of theinterlamellar material is needed to quantitatively describe theanisotropic mechanical response of the tapes. This is attributedto the orientation of the solid state drawn tapes being deforma-tion-induced, which means that not only the crystalline phase,but also the chains in the amorphous domains are oriented. Toinvestigate the effect of an oriented amorphous phase on the mac-roscopic plastic flow kinetics, an isotropic but loading angle depen-dent yield kinetics for the amorphous phase was taken, whichenabled to describe the deformation kinetics of the oriented tapesat various loading angles. Therefore, a critical factor for predictionof the anisotropic mechanical response of oriented HDPE tapes ap-pears to be the dominant influence of the yield kinetics of theamorphous phase on the macroscopic yield kinetics, and an accu-rate description of the constitutive behavior of the oriented amor-phous phase is required.
Furthermore, characterizing different crystallographic slip sys-tems by a combination of the micromechanical model and experi-mental data of oriented samples is a challenging task due to thepresence of oriented amorphous phase, and requires a gooddescription of the orientation and the mechanical anisotropy ofthe amorphous domains.
Appendix A. Multi-scale constitutive model
In this appendix, the material models used for the constituentphases, and the composite inclusion model for semicrystallinepolymers are presented.
A.1. Crystalline phase
The elastic behavior of the crystalline phase is characterized bya fourth order elasticity tensor 4Cc which linearly relates the sec-ond Piola–Kirchhoff stress tensor sc and the Green–Lagrange straintensor Ec
e:
sc¼4Cc : Ece; ðA:1Þ
with
sc ¼ JceFc�1
e � rc � Fc�T
e and Ece ¼
12
FcT
e � Fce � I
� �; ðA:2Þ
with Jce ¼ detðFc
eÞ the volume ratio, rc the Cauchy stress tensor and Ithe second order identity tensor. The viscoplastic component of the
deformation in the crystalline phase, composed of the contributionsof all Ns physically distinct slip systems, is given by:
Lcp ¼ _Fc
p � Fc�1
p ¼XNs
a¼1
_caPa0; Pa
0 ¼~sa0 �~na
0; ðA:3Þ
with Pa0 the nonsymmetric Schmid tensor, given by the dyadic prod-
uct of the unit slip direction~sa0 and the unit slip plane normal ~na
0 ofthe ath slip system. The resolved shear rate _ca of each slip system isassumed to be related to the corresponding resolved shear stress sa,via a modified Eyring flow relation:
_ca ¼ _ca0 exp
�DUa
RT
� �sinh
sa
sc0
� �; ðA:4Þ
with
sa ¼ Jrc : Pa ¼ sc � Cce : Pa
0; ðA:5Þ
where Cce denotes the elastic right Cauchy-Green deformation
tensor.
A.2. Amorphous phase
The elastic behavior of the amorphous phase is modeled by ageneralized neo-Hookean relationship:
rae ¼
Ga
Jae
~Bad
e þ KaðJae � 1ÞI; ðA:6Þ
where the superscript ‘‘d’’ denotes the deviatoric part, ~Fae ¼ Ja�
13
e Fae is
the isochoric elastic deformation gradient tensor and ~Bae ¼ ~Fa
e � ~FaT
e isthe isochoric elastic left Cauchy-Green deformation tensor. Ga and Ja
e
are the shear modulus and bulk modulus, respectively.A viscoplastic Eyring flow rule is employed to relate the effec-
tive shear strain rate _cap to the effective shear stress of the amor-
phous phase sa, defined as
sa ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12rad� : rad
�
rwith ra
� ¼ RaT
e � ra � Rae � Ha; ðA:7Þ
with Rae the rotation tensor, obtained from the polar decomposition
of Fae and Ha a back stress tensor, which accounts for orientation-in-
duced hardening and is given by:
Ha ¼ lR
ffiffiffiffiNp
kchL�1 kchffiffiffiffi
Np� �
Bap � k2
chI� �
; ðA:8Þ
where kch ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 trðBa
pÞq
represents the stretch of each chain in theeight-chain network model and L�1 is the inverse of the Langevinfunction. The plastic rate of deformation Da
p is then defined as:
Dap ¼
_cap
sa rad
� ; ðA:9Þ
where the effective shear rate _cap, is given by:
_cap ¼ _ca
0 exp�DUa
RT
� �sinh
sa
asa0
� �: ðA:10Þ
A.3. Composite inclusion model
The inclusion-averaged deformation gradient FI and Cauchystress rI of each individual composite are given by:
F I ¼ f c0 Fc þ ð1� f c
0 ÞFa; ðA:11Þ
rI ¼ f crc þ ð1� f cÞra: ðA:12Þ
The two phases are assumed to be kinematically compatible and inequilibrium across the interface. The compatibility and equilibriumconditions on the interface can be written as:
426 A. Sedighiamiri et al. / Computational Materials Science 82 (2014) 415–426
Fa �~xI0 ¼ Fc �~xI
0 ¼ F I �~xI0; ðA:13Þ
ra �~nI ¼ rc �~nI ¼ rI �~nI; ðA:14Þ
where~nI ¼~eI3 denotes the unit normal vector of the interface, and~xI
0
denotes an arbitrary vector in the plane of interface, spanned by~eI1
and~eI2 such that~eI
1 �~eI2 ¼~eI
3.
A.4. Hybrid interaction law
In the hybrid interaction model, six auxiliary deformation-likeunknowns U are introduced. The macroscopic Cauchy stress tensor,the macroscopic right stretch tensor an the macroscopic rotationtensor are denoted by �r; �U and �R, respectively. Then, for anaggregate of NI composite inclusions, the following local–globalinteraction conditions are used:
U Ii
�~nIi
0 ¼ U �~nIi
0 ; i ¼ 1; . . . ;NI; ðA:15Þ~xIi � rIi �~xIi ¼~xIi � �r �~xIi
; i ¼ 1; . . . ;NI; ðA:16Þ
RIi
¼ �R; i ¼ 1; . . . ;NI; ðA:17Þ
in combination with the consistency conditions:
�r ¼XNI
i¼1
f IirIi; ðA:18Þ
�U ¼ JJR
!13XNI
i¼1
f Ii
0 UIi
; ðA:19Þ
with J ¼PNI
i¼1f Ii
0 JIi and JR ¼ detPNI
i¼1f Ii
0 FIi� �
.
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