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Micromechanical modeling of isotropic elastic behavior of semicrystalline polymer
Abstract
Considering semicrystalline polymers as heterogeneous materials consisting of an amorphous phase and crystallites, several micromechanical models have been tested to predict their elastic behavior. Two representations have been considered: crystallites embedded in a matrix and a layered-composite aggregate. Several homogenization schemes have been used in these representations. Firstly, comparisons between the models and experiments show that the micromechanics approach applies at this scale. Secondly, the results differ according to the rubbery or glassy state of the amorphous phase. Finally, the results suggest that the spherulitic mesostructure does not affect the material behavior while infinitesimal elastic strains are considered.
... In two-phase and three-phase models, two material representations have been used for mean-field modeling of semi-crystalline polymers: ( ) the classical Eshelby's inclusion-matrix representation (Eshelby, 1957), and ( ) a layered composite inclusion representation (Lee et al., 1993b), specifically developed for semi-crystalline polymers. The first representation is often applied in the context of elasticity (Guan and Pitchumani, 2004;Bédoui et al., 2004Bédoui et al., , 2006, while the second one is used to predict micro-structure evolution in the context of elasticity , viscoelasticity (Diani et al., 2008), plasticity (Lee et al., 1993b) andelasto-viscoplasticity (van Dommelen et al., 2003c). figure 8 schematically shows the concepts of inclusion and aggregate in these two representations. ...
... In addition, six more equations are obtained from the macroscopic equilibrium condition (̄= ⟨ ⟩) (Sedighiamiri et al., 2010). The two hybrid models (̂-inclusion and̂-inclusion models) are similar, with the difference that in the hybrid̂-inclusion model (Bahloul et al., 2021), in Eq. (8),̂is replaced by (macroscopic right stretch tensor), and̄is replaced bŷas an unknown auxiliary stress field (Bédoui et al., 2006;Bahloul et al., 2021). ...
... Single-crystal-like constitutive equations were used in each lamella, where the shear of the amorphous layer is adopted as an additional slip system, and obtained better results than the bi-crystal approach proposed by Lee et al. (1993b,a). Bédoui et al. (2006) considered the two above mentioned micromechanical representations, proposed in Bédoui et al. (2004), to investigate the isotropic elastic properties of three different semi-crystalline polymers, namely PP, HDPE and PET. Since the materials were considered to be isotropic, randomly distributed inclusions were modeled in both representations. ...
Highly heterogeneous and complex micro-structure of semi-crystalline polymers challenges accurate prediction of their macroscopic behavior. Micro-mechanical models establish a relationship between the micro-structure and macroscopic properties (structure–property relationship), and are able of predicting not only the macro-scale behavior, but also the evolution of the micro-structure. Therefore, micro-mechanical modeling can be used as a virtual experiment to predict the overall behavior of semi-crystalline polymers, where the effect of any single micro-structural parameter can be investigated. These parameters include morphological information about distribution of amorphous and crystalline phases, and constitutive properties of both phases. In this review paper, two main categories of micro-mechanical models, including mean-field and full-field models, are reviewed in detail. Three different groups of mean-field models, namely single-phase, two-phase, and three-phase models are discussed. Besides, the morphology of semi-crystalline polymers together with different deformation mechanisms, involved in different deformation regimes, are illustrated.
... [20][21][22][23]). The other approaches, based on micromechanics, are divided into two families: the mean-field models, such as self-consistent model [24][25][26][27], ̅ -inclusion [28,29] and the full field models, where the effective properties are computed by numerical homogenization [30][31][32][33] on REV of the spherulitic microstructure. It is important to note than Teixeira-Pinto et al. have ...
... Here, the properties of the amorphous phase come from the tensile tests on the amorphous injected plates ( 11 = 22 = /(1 − 2 ) , 12 = /(1 − 2 ) and 33 = /2(1 + ) ) and the properties of the crystal comes from the literature [26]. ...
... In Mori-Tanaka estimation and the first self-consistent estimation (SC1), we consider phases (amorphous phase and the crystalline orientation). For the second self-consistent estimation (SC2), we consider phases with the elastic properties from the homogenization of crystal amorphous multi-layer stack for each given crystalline orientation like Bédoui et al. [26]. ...
A 2D multiscale numerical strategy is presented in this paper. It allows to generate a representative elementary volume (REV) with a spherulitic microstructure used to predict the elastic properties of PET using a 2-scale numerical homogenization scheme. Because of the rapid crystallization kinetics of PET, DSC and optical microscopy were combined with empirical laws to estimate the crystallization kinetic parameters used to generate the REVs. Our framework allows estimating the elastic properties identified by tensile tests for several specimens crystallized at different temperatures. In addition, the comparison with mean-field models from the literature confirms that the Young's modulus of PET does not only depend on the crystallinity volume ratio but also on the crystal organization in the spherulites. The main advantage of this study is to provide a strategy for estimating the elastic properties that can be transposed to many semi-crystalline polymers with a spherulitic microstructure. Nevertheless, the numerical framework presented in the article is limited to semi-crystalline polymers with a spherulitic microstructure, which crystallize under isothermal condition.
... Micromechanical modelling based on Eshelby theory [14] has also been developed within microstructures context. Applied to polymer materials, this modelling allowed conclusive results on prediction of elastic properties for Polypropylene (PP) or Polyethylene (PE) [15,16]. In this approach, the crystalline lamellae are ellipsoidal inclusions whereas the amorphous phase is the matrix. ...
... In this approach, the crystalline lamellae are ellipsoidal inclusions whereas the amorphous phase is the matrix. The same authors have also developed two-phase models where the inclusions are dispersed randomly in the amorphous phase [16]. A three-phase model has also been developed by Gueguen et al. [17] through a generalization of the double-inclusion model. ...
... Elastic stiffness of additional structures of type B and C are assessed with different degrees of crystallinity. The obtained mechanical properties are compared with experimental results [16,32] and also to a micromechanical approach proposed by Bédoui [16] as shown in Fig. 8. ...
Nowadays, computational resources allow carrying out mechanical calculations on complex multi-scale mate-rials. Finite Element (FE) calculations can especially be directly performed on microstructures of materials. Thiswork is a first attempt to analyse the impact of the crystalline architecture at a mesoscopic scale on the mac-roscopic elastic properties of Semi-Crystalline Polymers (SCP). Such polymers can be considered biphasic ma-terials, which are composed of an amorphous phase embedded in a crystalline network. The material studiedhere is Polyethylene (PE).Molecular Dynamics (MD) calculations are carried out on a 100% crystallized Polyethylene model to de-termine the elastic properties of the crystalline regions of the material. 3D mesostructures of the typical layout ofthe spherulitic crystalline network of Semi-Crystalline Polymers are then constructed from experimental ob-servations. These material data and this geometrical description are then integrated in computations with theFinite Element method on elementary volumes to finally determine the macroscopic elastic properties of thematerial. In this work, which is a first attempt to test such a multi-scale workflow, no amorphous phase isconsidered. Different 3D architectures are compared demonstrating the role of the crystalline arrangement onthe stiffness of the material. Three main types of mesostructures have been analysed: crystalline lamellae dis-posed in a complete random arrangement, crystalline lamellae disposed in a spherulite arrangement, crystallinelamellae with branches disposed in a spherulite arrangement. It appears that the 3D configuration of the la-mellae, as well as the presence of branches, have an influence on the macroscopic elastic properties of thematerial. Then, comparisons with experimental data suggest that the macroscopic elastic properties can berepresented with a purely cohesive crystalline network for crystalline degree up to about 50%. This resultquestions the role of the amorphous phase on the elastic properties of such systems.
... 8). The same results but with more experimental reconciliation are reported in [6]. As with most anisotropic materials in matrix-inclusion schemes, simplifying assumptions need to be made, which may decouple the material anisotropy from the microstructure's morphological anisotropy, as discussed in Sect. ...
... According to this observation, [38] employ isotropic phase materials with the decisive Young modulus that is encountered in-plane. They find, contrary to Bédoui et al. [5,6] the best model to be an amorphous-inclusion-in-crystalline-matrixscheme. However, at high volume fractions, the experimental result for PP160 (annealed at 160°C), which has the highest crystallinity, lies outside the Hashin-Shtrikman and even the Voigt/Reuss bounds. ...
... Further, very long ellipsoids may basically break the scale separation. Bédoui et al. [5] worked with largest-to-smallest half axes ratios of up to 140, but experimental results indicate thicknesses of PP lamellae of 10 nm at length values up to 4 µm [6]. One can see that there is a considerable scale separation between these two dimensions, i.e. a violation of the Micro-Mini-Macro principle [22], which may render the oblate ellipsoid model with extreme aspect ratios unfit. ...
Calculating the effective elastic properties of semi-crystalline polymers is challenging due to a pronounced phase stiffness contrast, complex micro-structures and extreme anisotropies of the crystalline phases. To estimate the effective Young modulus of isotactic polypropylene depending on the crystallinity, we construct complex and simple homogenization schemes that unify the micro-scale interaction mechanisms that have been identified by Bartenev and Valishin [4], Bédoui et al. [5], Parenteau et al. [38]. This is achieved by a two-step homogenization approach, namely a local laminate stiffness that is then subjected to orientation averaging. The resulting estimates show good agreement to experimental results. They further suggest that the relatively high effective stiffness of isotactic polypropylene when compared to polyethylene may in part be a result of the stronger lateral confinement effect on the laminate level.
... Some models, based on Eshelby's inclusion theory and micromechanical homogenisation framework, approximate a matrix-inclusion microstructure where crystalline lamellae or amorphous domains are dispersed in a rubbery or crystalline matrix, depending on the ratio of crystallinity. These frameworks are typically limited to describing the material behaviour within the elastic region (see, e.g., [34][35][36]), and only extending into the small-strain post-yield regime (as, e.g., in [37,38]). ...
... By numerically calculating the value of where = ln( ); i.e., the gradient of the true stress-true strain curve, at the beginning of the deformation, we obtained the predicted Young's modulus for the = 50%, 60% and 70% specimens. These values have been enumerated in Table 3. Happily, Bédoui et al. [34] also report on the values of the Young's modulus for their PE specimens at various crystallinity ratios. For comparison, those values have also been given in Table 3. ...
... The inner morphology of the spherulite might be modelled and accounted for. Some attempts already exist within the frame of micromechanics that are based on simplified microstructure and that are most often restricted to an elastic behavior [31,32]. Recently, such kind of approach was coupled with a numerical simulation of spherulite growth to model elastic components as a function of crystallization [33]. ...
... Schematically, the crystallizing medium can appear as composed of individual spherulites whose number and radius vary for low crystallization temperatures (high cooling rates) or coalescing spherulites for higher temperatures (low cooling rates). Based on what is known in the field of micromechanics [32,33], this should have a significant impact and make a simple correlation with the crystallinity rate irrelevant. This may also explain the apparent inconsistency of results in the literature regarding this correlation. ...
This paper deals with the viscoelastic behavior during crystallization and melting of semicrystalline polymers with the aim of later modeling the residual stresses after processing in case where crystallization occurs in quasi static conditions (in additive manufacturing for example). Despite of an abundant literature on polymer crystallization, the current state of scientific knowledge does not yet allow ab initio modeling. Therefore, an alternative and pragmatic way has been explored to propose a first approximation of the impact of crystallization and melting on the storage and the loss moduli during crystallization-melting-crystallization cycle. An experimental approach, combining DSC, optical microscopy and oscillatory shear rheology was used to define macroscopic parameters related to the microstructure. These parameters have been integrated into a phenomenological model. Isothermal measurements were used to describe the general framework, and crystallization at a constant cooling rate was used to evaluate the feasibility of a general approach. It can be concluded that relying solely on the crystalline fraction is inadequate to model the rheology. Instead accounting for the microstructure at the spherulitic level could be more useful. Additionally, the results obtained from the experiments help to enhance our understanding of the correlations between crystallization kinetics and its mechanical effects.
... The inner morphology of the spherulite might be modeled and accounted for. Some attempts already exist within the frame of micromechanics that are based on simplified microstructure and that are most often restricted to an elastic behavior [31,32]. Recently, this kind of approach was coupled with numerical simulation of spherulite growth to model elastic components as a function of crystallization [33]. ...
... Schematically, the crystallizing medium can appear as composed of individual spherulites whose number and radius vary for low crystallization temperatures (high cooling rates) or coalescing spherulites for higher temperatures (low cooling rates). Based on what is known in the field of micromechanics [32,33], this should have a significant impact and make a simple correlation with the crystallinity rate irrelevant. This may also explain the apparent inconsistency of results in the literature regarding this correlation. ...
This paper deals with the viscoelastic behavior during crystallization and melting of semicrystalline polymers, with the aim of later modeling the residual stresses after processing in cases where crystallization occurs in quasi-static conditions (in additive manufacturing for example). Despite an abundant literature on polymer crystallization, the current state of scientific knowledge does not yet allow ab initio modeling. Therefore, an alternative and pragmatic way has been explored to propose a first approximation of the impact of crystallization and melting on the storage and loss moduli during crystallization–melting–crystallization cycles. An experimental approach, combining DSC, optical microscopy and oscillatory shear rheology, was used to define macroscopic parameters related to the microstructure. These parameters have been integrated into a phenomenological model. Isothermal measurements were used to describe the general framework, and crystallization at a constant cooling rate was used to evaluate the feasibility of a general approach. It can be concluded that relying solely on the crystalline fraction is inadequate to model the rheology. Instead, accounting for the microstructure at the spherulitic level could be more useful. Additionally, the results obtained from the experiments help to enhance our understanding of the correlations between crystallization kinetics and its mechanical effects.
... [220] to model the viscoelastic behavior of semicrystalline polymers at small deformations and by van Dommelen et al. [183] to model the elasto-viscoplastic behavior of semicrystalline polymers under large deformations. More recently, Bedoui et al. [221,222]. show that a classical inclusion / matrix model ( Figure 17) associated with a differential scheme gives satisfactory results for predicting the elastic behavior of isotropic semicrystalline polymers. Amorphous phase Crystalline lamella ...
... Some very sophisticated models have already been performed to simulate the mechanical behavior of semi-crystalline polymers at large deformations, but could these models simulate at their limit the simple elastic behavior of isotropic semicrystalline polymers? Bedoui et al. [221,222] have considered fully anisotropic crystallites embedded in a matrix and a layeredcomposite aggregate; several homogenization schemes have been used in those representations and the amorphous phase modulus varied. Finally only one model gives satisfactory trends and results for both modulus ( Figure 18) and Poisson ratio ( Figure 19): the crystalline lamellae embedded in a matrix associated with the differential scheme. ...
... These microstructural variations significantly influence its physical and mechanical properties [4][5][6]. Consequently, understanding the crystallization behavior of PET and its associated microstructure has been a focal point of research in recent years [7][8][9][10]. ...
Poly(ethylene terephthalate) (PET) is widely used in packaging due to its mechanical properties, which are closely linked to its microstructure. To better understand how PET crystallizes under stretching, we studied its behavior during biaxial tensile test near the glass transition temperature (Tg). A biaxial testing machine with heating system was developed to perform in-situ tests under a synchrotron beam. Small-angle X-ray scattering (SAXS) analysis was conducted under constant width (CW) and equal-biaxial (EB) loading conditions to examine macromolecular orientation and crystallization. A two-step homogenization approach was developed to estimate the elastic properties of stretched PET based on the observed microstructural changes. This study highlights the relationship between the microstructure and mechanical properties of PET, particularly the effects of strain-induced crystallization and molecular orientation. Additionally, a parametric uncertainty study on the elastic properties of the oriented amorphous phase was carried out to test the sensitivity of the elastic moduli. Monte Carlo simulations confirmed the convergence of the model, supporting the robustness of the predictions in terms of both means and standard deviation statistical data.
... Most attempts, however, are more phenomenologically oriented. Some of the attempts that can be mentioned are Duan et al. (2001), Bedoui et al. (2006), Hartmann (2006), Ayoub et al. (2010), Zeng et al. (2010), Polanco-Loria et al. (2010), Balieu et al. (2013), Krairi and Doghri (2014), Abdul-Hameed et al. (2014), Garcia-Gonzalez et al. (2017) and Kroon and Rubin (2023). ...
... Their results showed the importance of capturing the semicrystalline microstructure for predicting accurate elastic properties. Bédoui and coworkers used several classical micromechanics homogenization methods to predict the effective elastic properties of semicrystalline isotactic polypropylene, polyethylene, and polyethylene terephthalate (PET) [9]. They then extended this investigation to consider the viscoelastic behavior of PET [10]. ...
A multiscale repeating unit cell model of a single spherulite containing four disparate length scales was developed to predict the thermoelastic behavior of semicrystalline thermoplastic materials for composite aerospace applications. The continuum level scales were fully coupled and modeled using the generalized method of cells and the high-fidelity generalized method of cells micromechanics theories. Data from molecular dynamics simulations were used as inputs for the amorphous and crystalline constituents in the multiscale continuum models. Effective Young’s modulus, shear modulus, Poisson’s ratio, coefficient of thermal expansion, and thermal conductivity were predicted for polyether ether ketone and polyether ketone ketone, showing good agreement with the available experimental data from the open literature. Moreover, it is shown that predicted properties are fairly insensitive to the fidelity of the micromechanics model used at the highest continuum scale or the assumed shape of the spherulite.
... Most attempts, however, are more phenomenologically oriented. Some of the attempts that can be mentioned are Duan, Saigal, Greif and Zimmerman (2001), Bedoui, Diani, Regnier and Seiler (2006), Hartmann (2006), Ayoub, Zairi, Nait-Abdelaziz and Gloaguen (2010), Zeng, Grognec, Lacrampe and Krawczak (2010), Polanco-Loria, Clausen, Berstad and Hopperstad (2010), Balieu, Lauro, Bennani, Delille, Matsumoto and Mottola (2013), Krairi and Doghri (2014), Abdul-Hameed, Messager, Zairi and Nait-Abdelaziz (2014),Garcia-Gonzalez, Zaera and Arias (2017) and Kroon and Rubin (2023). ...
... These microstructures have a strong influence on the physical and the mechanical properties (Mandelkern, 2004;Reiter and Sommer, 2003;Piorkowska and Rutledge, 2013). Therefore, the study of crystallization of polymer and the related microstructure has been widely studied in the past few years (Jabarin, 1984;Groeninckx et al., 1974;van Drongelen et al., 2016;Chevalier and Marco, 2007;Bédoui et al., 2006). ...
Mechanical properties induced by the thermoforming processes of polymeric materials are strongly related to the changes of morphology during the process. In particular, during the stretch blow molding process, molecular chain are submitted to large biaxial elongations and the material Poly Ethylene Terephthalate (PET for short) is reinforced: its Young modulus for example can be multiplied by 3 or 4 from the preform to the final bottle. In this study, thanks to SAXS measurements on PET bottles and on biaxially stretched samples under the X-ray beam, we propose a simple model to represent the induced microstructure. With a two steps homogenization managed quasi-analytically, we can predict the induced Young modulus and compare them to the measured modulus from the PET bottle. The main result is that the amorphous phase does not remain isotropic and the orientation of the amorphous phase needs to be taken into account.
... They are treated as an Eshelby-type inclusion problem in which the reinforced semi-crystalline material system is regarded as a continuous amorphous matrix (phase am) in which discrete ellipsoidal inclusions are randomly dispersed and oriented, namely, the (α and β) crystalline phases and the particles assumed initially to be perfectly bonded at interfaces. Such a composite-type representation of the semi-crystalline structure was previously employed to predict different behaviors in the context of elasticity [30][31][32], plasticity [33] and plasticity with strain-induced phase transformation [34]. The progressive interfacial debonding and α → β phase transition are the two prevalent deformation mechanisms considered in the present work and introduced into the micromechanical framework. ...
Polyvinylidene fluoride or polyvinylidene difluoride (PVDF) is a piezoelectric semi-crystalline polymer whose electro-mechanical properties may be modulated via strain-induced α → β phase transition and the incorporation of polarized inorganic particles. The present work focuses on the constitutive representation of PVDF-based piezo-composites developed within the continuum-based micromechanical framework and considering the combined effects of particle reinforcement, α → β phase transition, and debonding along the interface between the PVDF matrix and the particles under increasing deformation. The micromechanics-based model is applied to available experimental data of PVDF filled with various concentrations of barium titanate (BaTiO3) particles. After its identification and predictability verification, the model is used to provide a better understanding of the separate and synergistic effects of BaTiO3 particle reinforcement and the micromechanical deformation processes on the electro-mechanical properties of PVDF-based piezo-composites.
... For instance, Parks and Ahzi (1990), Lee et al. (1993), van Dommelen et al. (2003), and Nikolov et al. (2006) proposed micromechanically motivated constitutive models for semi-crystalline polymers, where the contributions from the crystalline and amorphous phases were represented using mixture theory. Other attempts include the works by Duan et al. (2001), Bedoui et al. (2006), Ayoub et al. (2010), Zeng et al. (2010), Polanco-Loria et al. (2010), Balieu et al. (2013), Krairi and Doghri (2014), Abdul-Hameed et al. (2014), and Garcia-Gonzalez et al. (2017). Most models assume isothermal conditions, but a few models account for temperature changes as well (e.g. ...
... Their results showed the importance of capturing the semicrystalline microstructure for predicting accurate elastic properties. Bédoui and co-workers used several classical micromechanics homogenization methods to predict the effective elastic properties of semicrystalline isotactic polypropylene, polyethylene, and polyethylene terephthalate (PET) [9]. They then extended this investigation to consider the viscoelastic behavior of PET in ref. [10]. ...
... Nonetheless, the effect of environmental factors, such as humidity and temperature, would not consider the amorphous and crystalline domains individually, setting a limitation in this approach. In the second category, a micromechanical approach is considered that treats the domains as a two-phase composite composed of an amorphous matrix and crystalline inclusions [49][50][51][52][53][54][55][56][57][58][59][60][61][62]. A shortcoming of this approach is the excessive number of material parameters it renders and the consequent decrease in fitting accuracy. ...
High density polyethylene (HDPE) can show viscoelastic-viscoplastic behaviors under monotonic loads and a stress softening after reloading under cyclic ones. This sets a challenge in simultaneously representing such response in material constitutive models. In addition, due to the adoption of novel accelerated tests at higher temperatures, e.g., 95 °C, the need for a higher temperature calibration is motivated. Therefore, the objective of this study is threefold: (i) to investigate the capability of the three network viscoplastic (TNV) model in capturing HDPE thermo-viscoplasticity under monotonic and cyclic loads, (ii) to report observations on HDPE at various strain-rates and temperatures from 23 °C to 95 °C including the α-relaxation region (iii) to explore the ratcheting behavior of HDPE, i.e., cyclic creep. The FEA analysis based on the calibrated TNV model was successfully able to predict the HDPE behavior under static, quasi-static and dynamic loads. The predicted strain range Δε and mid-range strain εs of the cyclic creep showed good agreements. This implies that the TNV model can be a reliable candidate for HDPE engineering assessments. Findings of this work will have many industrial applications, e.g., products manufacturers or resin producers, in which HDPE is used under complex loads. Similar procedures can be followed for other thermoplastics which lays the basis for establishing a standard calibration guideline.
... Bedoui et al. proposed a micromechanical model to predict the elastic behavior of semi-crystalline polymers. It was found that the microstructure does not affect the mechanical behavior of these polymers when dealing with infinitesimal elastic deformation [11]. Zhang et al. put forward an orthotropic finite element calculation scheme based on the multiplicative decomposition, and carried out finite element numerical simulation analysis based on the tensile necking process of polymer samples [12]. ...
A series of uniaxial ratcheting experiments were carried out for HDPE specimens under cyclic compressive loads. The effects of stress range, temperature and stress rate were studied in detail. Results show that the compressive stress–strain curves are nonlinear at different stress and temperature levels. Moreover, the factors affecting the cyclic hardening effect are decreased in an order of stress range, temperature and stress rate. The viscoplastic effect of HDPE only takes place in high-temperature and high stress range. Furthermore, a Universal Ratcheting Model (URM) was developed to predict ratcheting strains under different cases. The ratcheting stress threshold value and related ratcheting evolution parameters of HPDE were deduced. A good agreement was obtained between those predicted and tested data under various temperature and stress levels.
... In a second approach, the material system is treated as an Eshelby inclusion problem by seeing the crystals as reinforcing ellipsoidal inclusions embedded into a continuous amorphous matrix. The Eshelby-type inclusion approach was mainly used for the initial elastic behavior [26,[28][29][30][31][32][33], the linear viscoelastic behavior [34], the initial yield behavior [35] and the postyield behavior [36]. ...
In this paper, a micromechanics-based constitutive representation of the deformation-induced phase transformation in polyethylene terephthalate is proposed and verified under biaxial loading paths. The model, formulated within the Eshelby inclusion theory and the micromechanics framework, considers the material system as a two-phase medium, in which the active interactions between the continuous amorphous phase and the discrete newly formed crystalline domains are explicitly considered. The Duvaut–Lions viscoplastic approach is employed in order to introduce the rate-dependency of the yielding behavior. The model parameters are identified from uniaxial data in terms of stress–strain curves and crystallization kinetics at two different strain rates and two different temperatures above glass transition temperature. Then, it is shown that the model predictions are in good agreement with available experimental results under equal biaxial and constant width conditions. The role of the crystallization on the intrinsic properties is emphasized thanks to the model considering the different loading parameters in terms of mechanical path, strain rate and temperature.
... This interaction is sensitive 50 to temperature as well as to the rhythm of deformation and it can generate a double yield point remarkably visible in 51 macroscopic stress-strain curves. 52 The double yield (DY) phenomenon has been reported for polyamide (PA) [3,[12][13][14][15][16][17], polytrimethylenetereph-53 thalate (PTT), polybutyleneterephthalate (PBT) [18] and polyethylene (PE) [19][20][21][22]. Fig. 1 presents a typical stress-54 strain curves for nylon 101 with two clear yield points [12]. ...
The double yield (DY) phenomenon observed in a wide variety of semi-crystalline polymers (SCP) adds difficulties in the material characterization. In this paper, a constitutive model, termed as explicit semi-crystalline polymer (ESCP) model, is proposed to study DY phenomenon as well as the rate- and temperature-dependent thermomechanical response below the glass transition temperature. The underlying yield kinetics due to the morphological changes of the spherulite micro-structure is represented by a rheological analogue described by a physically-based amorphous intermolecular resistance and a rate-independent crystalline interlamellar resistance. Independently-identified viscoelastic response and network resistance are also implemented to complete the model description. The activation and disclosure of the crystalline component depend on the saturated state of amorphous phase. The proposed model is validated against ex- perimental data obtained from different authors for three commonly used SCPs: nylon 101, LDPE and PA6. A straightforward parameter identification procedure, requiring a minimum number of calibration tests, is presented to illustrate the model usage. The thermomechanical-coupled analyses provide satisfactory predictions using simulated models of a cylinder compression and dogbone tensile tests at different rates, where the self-heating and thermal softening effects are naturally captured by the model.
... When cooled from the melt, semicrystalline polymers regularly exhibit spherulitic morphologies comprising radial arrangements of broad thin crystalline lamellae infilled with amorphous layers, and the overall chemo-thermomechanical properties depend on the local properties of the constituents and crystallographic evolutions [17][18][19][20][21]. The homogenizationbased method was demonstrated effective on well predicting overall properties of semicrystalline polymers with detailed description of micro-features, as done in the works [18,20,[22][23][24][25][26][27]. ...
This work provide theoretical understandings for the enzyme-degradable PCL, and assist its structural designs and engineering applications. An energy-dependent evolution model is developed to reflect the enzyme-triggered decrystallization of crystals and the further dissolution by applying a chain-broken chemical reaction. Then, the mechanical properties of the enzyme-degradable semicrystalline PCL is modelled through the homogenization-based procedure by the volume-average of a collection of laminated inclusions with crystals and amorphous phase. A dual-phase-lag diffusion model is advanced to solve the enzyme concentrations in the PCL. The model is calibrated by the experiments and then applied for the chemomechanical properties of the PCL when under enzyme conditions. Some numerical examples are conducted to discuss effects of the enzyme concentration and the crystallinity on the crystallographic axe evolution as well as the overall chemomechanical properties of the semicrystalline PCL.
... This analytical function can be applied to composites with isotropic fillers and orthotropic or transversely isotropic matrices. In our study we employed SiO 2 fused quartz (amorphous) that belongs to the group of isotropic materials [32,33]. Therefore, we can compare our results to the effective thermal conductivity of the 3D Mori-Tanaka model, k MT , for isotropic matrices according to the equation, ...
The effective thermal conductivity of composites made up of VO2 (SiO2) spherical particles randomly distributed and embedded in a SiO2 (VO2) matrix is numerically studied in a range of temperatures around the metal-insulator transition of VO2. This is done by means of three-dimensional finite element simulations for different concentrations and sizes of the particles as well as various interface thermal resistances. Our results are validated against the Mori-Tanaka analytical model. In addition, we develop a numerical method to calculate the heat storage capacity for composites with VO2 particles dispersed into a SiO 2 matrix. It is shown that: i) The effective thermal conductivity of VO 2 /SiO 2 composites increases with the VO 2 particles' size, while the one of SiO2 /VO2 composites is pretty much independent of the SiO 2 particles' radius. ii) At the VO 2 transition temperature (342.5 K), the effective thermal conductivity of VO 2 /SiO 2 composites increases significantly at a rate of 2.7 × 10 − 3 Wm − 1 K − 2, such that its value doubles up the SiO 2 matrix thermal conductivity at the particle concentration of 40.2%. By contrast, the effective thermal conductivity of SiO 2 /VO 2 composites decreases at a rate of 8.6 × 10 − 3 Wm − 1 K − 2. iii) The effective thermal conductivity is strongly affected by the thermal resistance in VO 2 /SiO 2 composites, by contrast, the resistance effect does not play an important role for particle volume fractions of SiO2 up to 34.1% in SiO2 /VO2 composites. The Mori-Tanaka model and our simulations predict the same trend of the effective thermal conductivity values of VO2/SiO2 composites. However, the analytical model fails when the matrix is made up of VO2 and the volumetric fraction of SiO2 exceeds 34.1%. The latent heat storage capacity of VO2/SiO2 composites increases with the VO2 particles' concentration, such that at 40.2%, it takes the value of 24553 J kg − 1 (486.7 cal mol − 1), which is about half that of the pure VO2.
... This model was extended in [9] by considering the amorphous and crystalline fractions within the spherulites with a laminar composite model. Bédoui et al. [10] compared two different types of the representation of the lamellar microstructure of the spherulites: the laminar composite model and crystallites embedded in an amorphous matrix. Recently, Glüge et al. [11] estimated the effective Young's modulus depending on the crystallinity. ...
A multiscale simulation method for the determination of mechanical properties of semi-crystalline polymers is presented. First, a four-phase model of crystallization of semi-crystalline polymers is introduced, which is based on the crystallization model of Strobl. From this, a simulation on the nanoscale is derived, which models the formation of lamellae and spherulites during the cooling of the polymer by using a cellular automaton. In the solidified state, mechanical properties are assigned to the formed phases and thus the mechanical behavior of the nanoscale is determined by a finite element (FE) simulation. At this scale, simulations can only be performed up to a simulation range of a few square micrometers. Therefore, the dependence of the mechanical properties on the degree of crystallization is determined by means of homogenization. At the microscale, the cooling of the polymer is simulated by a cellular automaton according to evolution equations. In combination with the mechanical properties determined by homogenization, the mechanical behavior of a macroscopic component can be predicted.
... As an example, biobased poly(butylene succinate) presents a glass transition temperature (T g ) of − 37 • C and a Young's modulus of approximately 368 MPa [8]. These values are far below the most common fossil-based polyester, namely polyethylene terephthalate (PET) with a T g of 76 • C [9] and Young's modulus above 2600 MPa [10]. In order to improve these properties, aromatic and semi-aromatic biobased polyesters are now gaining a lot of attention since the aromaticity brings higher thermal resistance and rigidity linked to higher transition temperatures. ...
Enzymatic polymerization is a promising route for a greener synthesis of biobased polyesters. However, this approach is still often limited to aliphatic polyesters, which present limited properties such as low thermal resistance. In this context and till now, introduction of aromatic monomers into polyesters very often resulted in low molar mass chains. Herein, aliphatic-aromatic copolyesters based on dimethyl-2,5-furandicarboxylate were enzymatically synthesized using immobilized Candida antarctica lipase B with a particular focus on the influence of the solvent used. Two series of poly(hexylene adipate)-co-(hexylene furanoate) and poly(butylene adipate)-co-(butylene furanoate) copolyesters were successfully synthesized in diphenyl ether, displaying high molar masses (Mn up to 19 000 g.mol⁻¹) for aromatic monomer contents up to 70 and 50%, respectively. High aromatic content resulted in reduced molar masses due to a loss of solubility of the growing aromatic chains. Replacing diphenyl ether by acetophenone, which had never been used before as solvent for enzymatic synthesis of polyesters, led to significantly enhanced solubility and thus an increase in the average molar masses of poly(hexylene furanoate) and poly(butylene furanoate). The resulting polyesters showed greater thermal stability and higher Tg, offering new perspectives for expanding the potential applications of such enzymatically-produced biobased aromatic copolyesters.
... Experimental data of Young's modulus of Polyethylene as a function of the crystallinity ratio[22],[23] and[24].The mean strain in all the RVEs is equal to the same imposed macro strain of 345 0.03. The strains in the amorphous phase exceed the mean value, and increase with an increasing crystallinity ratio. ...
An enhanced phase field model recently proposed by the authors enables to model polymer crystallization and to predict spherulite growth and representative semi-crystalline micro-structures in 2D or 3D. After solidification, the value of the phase field variable in each subcell (2D pixel or 3D voxel) indicates whether the material is amorphous or crystalline. In the present paper, full-field micromechanical simulations are conducted on the microstructures generated by the enhanced phase field model in order to predict the effective mechanical properties. A Fast Fourier Transform (FFT) method with periodic boundary conditions is used. Care is taken to obtain representative volume elements (RVEs) by computing the number of subcells in each spherulite and the number of spherulite nucleations in each RVE. Numerical FFT predictions of elastic properties for a range of crystallinity ratios are validated against experimental data and compared to simpler composite inclusion models. Finite element simulations of thermal shrinkage are also shown. The approach was numerically implemented in a research version of the Digimat software.
... PP displays the same trend: an increase in 5% in crystalline ratio leads to an increase of 500 MPa of Young modulus [49] . However, in polymers with their glassy amorphous phase such as PEKK, PLA or PA11 [19,50,51] , the mechanical properties in the elastic domain depend less on crystallinity because the contrast of elastic modulus between crystalline phase and amorphous phase is hundred to thousand times lower [52] . At this point, it seems that PBT behaves more similarly to PA11 than to PE or PP consistently with the glassy nature of its amorphous phase. ...
This paper reports the study of embrittlement of PBT submitted either to thermal or hydrolytic ageing. All changes were followed up by tensile tests, rheometry in molten state and gel permeation chromatography for molar mass changes, SAXS and DSC experiments for crystallinity changes. Both kind of ageing were shown to induce predominant chain scissions with moderate crystallinity increase, in great part due to annealing. The combination of all results were used to establish a Mw - χc embrittlement window helping for a determination of an end of life criterion.
... The dependency of mechanical properties of PP on the DoC and cooling rate was studied in refs. [2,5,[12][13][14][15][16][17][18][19][20][21][22][23]. These data, compared together in ref. [16], focus on elastic properties, demonstrating a trend of monotonically increasing Young's modulus of PP with increase of DoC: for DoC in the range 0.30 to 0.65 Young's modulus is in the range from 0.8 to 2.0 GPa. ...
The article experimentally investigates the interface strength of glass fibers in polypropylene (PP) with two different levels of crystallinity. The different degrees of crystallinity, 46.6% and 52.5%, were achieved using fast (quenching, ~4500°C/min) and slow (~2°C/min) cooling, respectively, during production of the PP film. The degree of crystallinity was measured using the differential scanning calorimetry. The mechanical properties of the films were characterized with the dynamic mechanical analysis and with a tensile test on Deben micro‐tester. Interfacial shear strength (IFSS) of glass fiber/PP was determined using fiber fragmentation test. Interfacial normal strength (IFNS) was determined using inverse identification based on finite‐element modeling of transverse tensile loading of a single fiber (Deben micro‐tester with digital image correlation‐based observation of the debonding). The measurements have confirmed the expected trend in mechanical properties of the film: increase of the storage and Young's moduli (room temperature) with the increase of degree of crystallinity, accompanied by the decrease in the loss modulus and tan δ. Interfacial strength followed the trend of the PP stiffness: both the IFSS and IFNS values for the PP with high crystallinity (slow cooling) are three (IFSS) and four (IFNS) times higher than those for the low crystallinity (fast cooling) case.
... Most of the existing research works [3,7,8,16] about the ejection deformation only consider elastic deformation at room temperature. They all treated the injection-molded product as a homogeneous linear elastic part. ...
Early ejection of injection-molded plastic parts aims to shorten the production cycle. However, the early ejection process may induce unfavorable deformations. No commercial simulation tool is available to predict the transitional mechanical properties of the plastic part and the possible deformations induced by early ejection. The authors propose a method to simulate the early ejection process and the plastic part’s mechanical response by integrating Moldflow™ and Ansys™. A real industrial case study is provided to show the procedure and its validation. Compared to the stand-alone molding simulation, the proposed method can predict the part’s final dimensions more accurately. By considering the ejection-induced deformations at the mold design stage, the mold can be better designed to facilitate early ejection, while the part is only partial solidified.
Shape memory polymers (SMPs) are a class of intelligent materials capable of recovering their original shape in
response to external stimuli. This study employs a modified Mori-Tanaka (MMT) model to predict the effective
thermomechanical behavior of SMPs. By utilizing a homogenization procedure, a constitutive equation describing the
evolution of the effective behavior of SMPs under thermomechanical loading was proposed. The model accounted for the SMP’s dual-phase structure, consisting of active and frozen phases, and determined the effective stiffness by
considering each phase’s shape and volume fraction. Unlike existing phase transition models, the proposed model
incorporates the interaction between phases and the phase transition process throughout the thermomechanical cycle.
The model was implemented in the UMAT user subroutine of the ABAQUS software to simulate the mechanical
behavior of SMPs. Investigations into various inclusion phase shapes revealed that an ellipsoidal shape most accurately
represents the morphology of the inclusion phase. While shape recovery is influenced by inelastic strain, the stress
response of the present model showed improved agreement with experimental results due to the consideration of phase
interactions during transformation. Application of the proposed model to the auxetic behavior of a re-entrant structure
fabricated from PLA demonstrated that varying Poisson’s ratios and cell-opening factors (CoF) can be achieved by
programming different deformation magnitudes. The most negative Poisson’s ratio (-0.64) was obtained at a 70° reentrant
angle induced by a 20 mm pre-displacement. Additionally, the formulation was extended to simulate particle
release, highlighting its potential application in drug delivery. The findings suggested that microstructure and nonuniform
deformation significantly influence the cell-opening factor.
The mechanical behavior of thermoplastics is strongly rate-dependent, and oftentimes it is difficult to find constitutive models that can accurately describe their behavior in the small to moderate strain regime. In this paper, a hyperelastic network model (modified neo-Hookean) and a set of experiments are presented. The testing consists of monotonic tensile loading as well as stress relaxation and zero stress creep. Two materials were tested, polyoxymethylene (POM) and recycled polypropylene (rPP), representing one more rigid and brittle and one softer and more ductile semi-crystalline polymer. The model contains two main novelties. The first novelty is that the stiffness is allowed to vary with the elastic deformation (in contrast to a standard neo-Hookean model). The second novelty is that the exponent governing viscous relaxation is allowed to vary with the viscous deformation. The basic features of the new model are illustrated, and the model was fitted to the experimental data. The model proved to be able to describe the experimental results well.
Spherulite is the main microstructure feature of semicrystalline polymers, exhibiting a non-axisymmetric geometry with a sheaf structure in the center. In the initial stage of growth, spherulite is manifested initiation from the sheaf structure with a certain orientation. The size and orientation of sheaf structure are affected by various processing parameters. Previous research considered spherulite as completely radially symmetric structures, ignoring the effects of anisotropic sheaf on mechanical properties. In this paper, an anisotropic micromechanical model for evaluating the mechanical properties of semicrystalline polymers was proposed. The microstructure of single spherulite was modeled with different sheaf sizes, orientations, and crystallinities. The viscoplastic constitutive model together with the Arruda–Boyce model was used to describe the micromechanical behaviors of the crystalline lamellae and amorphous lamellae, respectively. The evolution of inhomogeneous plastic deformation, interlayer deformation and slip activities in spherulite was observed under tension. The results shown in this work enhance the understanding of the microstructure–property relationship of semicrystalline polymers, which, in turn, guides the development of advanced manufacturing techniques for tailoring mechanical properties.
This article presents a multi-scale plastic-damage model for strain-induced morphological anisotropy in semi-crystalline polyethylene formulated within a continuum-based micromechanical framework. The crystallographic shear in the crystalline lamellae and the molecular alignment/relaxation of the amorphous phase are two underlying inelastic processes integrated in the constitutive representation. The cavitation damage accumulation related to the progressive nucleation and anisotropic growth of nano-sized cavities in the amorphous phase is also integrated and treated separately for the elastic-viscoplastic intermolecular interactions and for the viscohyperelastic network interactions. The mechanical coupling between the deformation modes in the amorphous and crystalline domains is obtained by considering the crystalline-amorphous interfacial interaction in the micro-macro homogenization procedure. The model output is compared to tensile experimental observations from the literature of high-density polyethylene, in terms of stress–strain response and inelastic volumetric strain, during stretching and stretching-retraction-recovery sequences upon large-strain plastic deformation at different strain rates and temperatures. The effect of strain-induced morphological anisotropy on the internal cavitation damage distributions is analyzed by means of pole figures. The key role of the cavitation damage on the macroscopic and internal responses is studied.
The present contribution examines the ability of a constitutive model to capture polyethylene response variation with crystallinity. The model considers the elastic-viscoplastic crystal deformation and anisotropy due to the crystallographic texturing along with the amorphous chains network elastic-viscoplastic-viscohyperelastic deformation. The coupling between the two phases is performed by means of a multi-scale homogenization-based approach in which the interfacial interaction is considered. The model is applied on semi-crystalline polyethylene systems containing a broad range of crystallinities. Both monotonic and oligo-cyclic tensile loading sequences are considered upon large-strain plastic deformation. The model is found able to correctly capture the gradual transition of the rate-dependent monotonic response from a thermoplastic-like behavior at high crystallinity to a nearly elastomeric-like behavior at low crystallinity. The model abilities to capture cyclically loaded polyethylene systems are critically discussed. The influential and decisive role of the amorphous chains network alignment/relaxation on the cyclic stretching is emphasized by analyzing local stresses and stretches.
The aim of this article is to investigate how the stress triaxiality affects the plastic deformation behavior of high-density polyethylene using an approach combining experiments and micromechanics-based modeling. The stress-strain behavior along with the cavitation damage accumulation are experimentally quantified under well-controlled transversal response of hourglass-shaped tensile specimens with different curvature radii in order to set different triaxial stress states in the median cross-section. A constitutive elastic-plastic-damage representation is then presented within a continuum-based micromechanical framework. The model, constrained by the same boundary conditions as the experimental tests, is used to examine the stress triaxiality effects on the separate and synergistic effects of plasticity and cavitation damage micromechanisms that govern the macro-response.
Currently Polyethylene terephthalate (PET) foam is the most promising structural core materials, and the tensile mechanical properties are one of its important application indicators. Herein, environmental-friendly supercritical CO2 (ScCO2) extrusion foaming was adopted to prepare PET foam. Aiming at investigating the influence of crystals on the mechanical properties, isothermal treatment in the post-process was used to improve the crystallization process of PET foams. Due to the crystal perfection proceeds via migration and rejection of the structural defects at the crystallites induced by slow crystallization, the crystallinity increased rapidly with the rise of isothermal temperature, especially above the glass transition temperature (Tg). Qualitatively, it can be concluded that the crystalline phase contents have an intimate positive correlation with the tensile modulus, meanwhile, the shape ratio of the crystal have no significant effects on the tensile modulus. In addition, a coupling scheme of aggregate two-layered composite inclusion model and Simone-Gibson equation was first proposed to quantify the mathematical relationship between crystallization and tensile modulus of PET foam, which realized basic agreement.
Ultra-high molecular weight polyethylene has a low X-ray attenuation, hence, the performance of the polyethylene implants used for joint replacements cannot be directly investigated using X-ray-based imaging techniques. In this study, the X-ray attenuation of polyethylene was increased by diffusing an FDA-approved oil-based contrast agent (Lipiodol ultra fluid) into the surface of the samples, and the suitability of this novel radiopaque ultra-high molecular weight polyethylene for clinical applications was examined. Different levels of radiopacity were created by controlling the diffusion parameters, and the level of radiopacity was quantified from computed tomography scans and reported in Hounsfield units. The physical, chemical and tensile properties of the radiopaque ultra-high molecular weight polyethylene were examined and compared to untreated and thermally treated controls. The results of this study confirmed that for the samples treated at 115°C or less the diffusion of the contrast agent did not significantly alter the crystallinity ( p = 0.7) or melting point ( p = 0.4) of the polyethylene. Concomitantly, the tensile properties were not significantly different from the control samples ( p > 0.05 for all properties). In conclusion, the radiopaque ultra-high molecular weight polyethylene treated for less than 18 h at a temperature of 115°C or below is a promising candidate for joint replacement applications as it can be identified in a standard X-ray while retaining the tensile properties of clinically used radiolucent ultra-high molecular weight polyethylene.
The internal structure of injection molded model components made from a commercial non-nucleated isotactic polypropylene grade is investigated by wide angle X-ray diffraction and polarized-light optical microscopy. Changes in the polymorphic state, degree of crystallinity and spherulitic superstructure are quantified depending on the distance from the outer surface d. Interrelations between structure and local cooling rate as obtained from numerical simulations are considered. The influence of structural features on mechanical properties is studied using thin quasi-homogenous films microtomed at different depth from the injection molded component. A significant variation of the storage part of the tensile modulus E′ and the toughness Wt depending on the local semi-crystalline state is observed. A pronounced maximum in Wt is found about 200μm below the surface where a fine spherulitic superstructure occurs. The changes in E′ are moderate (values ranging from 1.4 to 1.8 MPa) and show a maximum at intermediate depth although the degree of crystallinity is continuously increasing from ≈ 27% at the surface to ≈ 45% in the core region. Relations between structural features and mechanical properties measured at small and large deformation (E′ and Wt) are discussed. General conclusion is that a deeper understanding of relations between structural state of semi-crystalline polymers determined by the processing conditions and mechanical parameters is of major importance for predicting the properties of injection molded components.
In the injection molding process of semi-crystalline polymers, the melt undergoes a complex deformation and cooling history which results in an inhomogeneous distribution of spherulites in the component. To evaluate these inhomogeneities in an isotactic polypropylene (α-iPP) part, an integrated multi-scale simulation approach has been developed in Laschet et al.. (2017a). This approach combines a macroscale mold filling and heat transfer analysis with a crystallization model at the microscale and a two-scale homogenization scheme in order to calculate microstructure dependent elastic properties of the PP component. This approach is extended here by adding some features improving the local Hooke matrix predictions. A link between Molecular Dynamics (MD) simulations and the homogenization scheme is achieved in order to obtain unknown properties of the pure amorphous and crystalline phases. Next, the accuracy of the crystallization model is improved by considering not only isothermal but also athermal nucleation. The spherulite growth is now evaluated by a continuous cellular growth algorithm. Then, several numerical improvements have been integrated in the homogenization tool. The main ones are the implementation of stabilized mixed finite elements to handle accurately the quasi-incompressibility of the amorphous phase and the application of periodic boundary conditions via a master-slave projection algorithm. Eventually, the distribution of effective mechanical properties over the component thickness is compared for different crystallization scenarios with measured elastic moduli.
In this paper, a new constitutive model is proposed for the behavior of thermoplastic polymers under non-isothermal conditions. The model couples linear viscoelasticity, viscoplasticity and thermal effects. It is formulated within the framework of irreversible thermodynamics. The total strain is the sum of viscoelastic, viscoplastic and thermal strains. General hereditary integrals describe the thermo-viscoelastic response. The viscoplastic part accounts for both isotropic and kinematic hardenings. The stress-strain response and the material self-heating are predicted and compared to experimental data on Polyamide 66 (PA66) and Polypropylene (PP). Good agreement between the numerical simulations and experimental data was obtained for the two materials.
Laser scribing can be used to enhance the flexibility of polymer films for flexible device applications. To optimize the bending curvature by controlling the scribing parameters—the depth, number, and interval of the scribed grooves, finite element analysis was conducted on the bending tests of scribed polyethylene terephthalate films. Moreover, the influences of the parameters on the stress/strain near the grooves were investigated. The maximum stress/strain and curvature generally increased with an increase in depth, whereas these values decreased with an increase in number and intervals. However, to maintain the mechanical stability of the films, the parameters were limited. The optimization results revealed that the maximum value of the curvature was 2.6 mm⁻¹ at depth = 40 and intervals = 25 μm, for number = 7. An empirical equation relating the curvature to depth and intervals was also provided. The results of the analysis are useful for the design of laser-scribed grooves on various polymer films, for the enhancement of their bending curvature, while minimizing the mechanical instability.
Calculation of normal mode frequencies and three-dimensional elastic constants is made for the first time for isotactic polypropylene crystal on the basis of lattice dynamical theory. The vibrational frequencies are found in relatively good agreement with the observed values for both the internal and external modes. The calculated Young’s modulus along the chain axis is 40.1 GPa which agrees well with the X-ray observed crystallite modulus ca. 40 GPa. The anisotropic curves of Young’s modulus and linear compressibility in the plane perpendicular to the chain axis are also calculated but agreement with the values observed at room temperature is not good. This discrepancy is erased successfully by considering a large anharmonic effect of methyl torsional modes.
The elastic properties of interlamellar bridges in semicrystalline polyethylene (PE) were estimated from the molecular-mechanics calculations on the assumption that the energy loading of a chain backbone represents the principal deformation mechanism. The calculations result in the force–length functions featuring abrupt discontinuities due to sequential annihilation of the defects by the conformational transitions. The correlation of the chain elastic moduli E with the concentration of defects in the chain and with the chain extension ratio x were established. The distribution functions ζ(E) of Young's moduli of interlamellar bridges in semicrystalline PE were calculated by using the literature data on the chain length distributions of tie molecules. The impact of the distribution function of moduli ζ(E) on the overall elastic response of solid PE materials was examined, particularly in cases of the stacked lamellae morphology involving so-called hard elastic PE.
This study presents an experimental investigation into the strain-induced crystalline microstructure, under biaxial elongation in poly(ethylene terephthalate) (PET), using wide angle X-ray diffraction (WAXD). We examined how the microstructure of a polymer subjected to a complex strain field evolves in terms of its crystalline ratio, its molecular orientation and the size of its crystallite. PET injection-molded specimens have been subjected to biaxial elongation tests, both equibiaxial and sequential, at different drawing speeds, draw ratios and temperatures above and close to Tg. The strain field was determined using a home-developed image correlation technique that has allowed us to determine all the strain components at each point of the specimen, even with a non-homogeneous strain field. To minimize the effect of quiescent crystallization, specimens are quickly heated with infrared and the temperature was regulated during the test. At the end of the deformation process, the specimens were quenched to room temperature. Their microstructures were later investigated, using both differential densimetry and WAXD with a synchrotron beam. Influences of strain rate, temperature and strain path sequence on the size of the crystallites and their orientation are evaluated.
The crystalline lamellae of semi-crystalline polymers are highly anisotropic. For instance, the stiffness in the chain direction of a polyethylene crystalline lamella is more than an order of magnitude higher than the stiffness in the transverse directions. The use of the classical bounds (lower and upper bounds) leads to far-apart estimates of the effective elastics properties. These bounds account for the composite nature of semi-crystalline polymers only through the volume fractions of each of the two phases, and neglect the microstructural morphology. Based on the consideration of a two-phase composite inclusion (crystalline lamella + amorphous layer) as the basic element constituting semi-crystalline polymers only through the volume fractions of each of the two phases, and neglect the microstructural morphology. Based on the consideration of a two-phase composite inclusion (crystalline lamella + amorphous layer) as the basic element constituting semi-crystalline polymers, were propose new estimates for the effective elastic properties of these materials. These new averaging schemes account for the microstructural morphology and the texture. We apply our proposed models to predict the effective elastic constants of isotropic and oriented polyethylene. Our results are compared to those of other models.
Deformation mechanism of polyethylene terephthalate (PET) gel and melt films was investigated in terms of morphological aspects by scanning electron microscopy, small angle X-ray scattering, and wide angle X-ray difiraction. The morphological properties were discussed in relation to the temperature dependence of the dynamic complex modulus. The storage modulus of the melt film with a draw ratio of 20 was 19.5 GPa at 20°C, while that of the gel film with a draw ratio of 7 was 16.2 GPa. The apparent crystal lattice modulus of the (1̅05) plane was measured by X-ray diffraction. Using the apparent value, the real value of the crystal lattice modulus was estimated to be 118 GPa through a somewhat complicated geometrical treatment. The storage moduli for both the specimens are much lower than the crystal lattice modulus. In order to study the large difference, the crystallite orientation was estimated in terms of the second order orientation factor of the c-axis and the theoretical analysis of small angle X-ray scattering patterns. It turns out that under oriented crystallization, the preferential orientation of the c-axis is attributed to the rotation of crystallites with respect to the stretching direction but not to a crystal transformation from a folded to a fibrous type. Such a orientation mode is thought to hamper ultradrawing to produce high modulus PET films.
Part 1 Overall properties of heterogeneous solids: aggregate properties and averaging methods aggregate properties, averaging methods elastic solids with microcavities and microcracks linearly elastic solids, elastic solids with traction-free defects, elastic solids with micrcavities, elastic solids with microcracks elastic solids with micro-inclusions overall elastic modulus and compliance tensors, examples o elastic solids with elastic micro-inclusions, upper and lower bounds for overall elastic moduli, self-consistent differential and related averaging methods, Eshelby's tensor and related topics solids with periodic microstructure general properties and field equations, overall properties of solids with periodic microstructure, mirror-image decomposition of periodic fields. Part 2 Introduction to basic elements of elasticity theory: foundations geometric foundations, kinematic foundations, dynamic foundations, constitutive relations elastostatic problems of linear elasticity boundary-value problems and extremum principles three-dimensional problems solution of singular problems. Appendix: references.
Intimate relationships between the structure and the mechanical properties of polymer crystals are discussed from the molecular theoretical point of view. (1) The Young's modulus along the chain axis is dependent largely on the molecular conformation and the force constants. Some typical polymers including polyethylene, polyoxymethylene, poly(p-phenylene benzobisoxazole) and cellulose are discussed. (2) Three-dimensional anisotropy of the Young's modulus is discussed in relation to the packing mode of the chains. In the case of isotactic polypropylene crystal, the important role of anharmonic torsional vibrational modes of the methyl groups is discussed, which significantly governs the anisotropy of the elastic constants. (3) The molecular deformation mechanism was predicted lattice-dynamically and proved experimentally on the basis of vibrational spectroscopic measurements. The direct experimental evaluation of the theoretically predicted atomic displacements was performed for the first time through the refined X-ray structural analysis of polydiacetylene single crystal under the application of tensile stress. (4) The molecular design of novel polymer materials with three-dimensionally high Young's moduli is made starting from the various types of conventional polymer crystals such as poly(p-phenylene benzobisoxazole), polyacetylene, poly(p-phenylene), orthorhombic polyethylene and cellulose. Some of these crosslinked polymer crystals were found to possess a Young's modulus exceeding that of diamond crystal.
A micromechanically-based composite model is proposed to study large plastic deformation and texture evolution in semi-crystalline polymers. The microstructure of many semi-crystalline polymers consists of co-existing crystalline and amorphous phases locally associated with each other in a fine plate-like morphological structure. An aggregate of two-phase composite inclusions is used to model these materials. Special consideration is given to molecular chain inextensibility within the crystalline phase. The introduction of a back stress tensor in the constitutive model of the amorphous phase accounts for hardening due to deformation-induced molecular alignment. Interface compatibility and traction equilibrium are enforced within each composite inclusion. A Sachs-like model and two newly-developed self-consistent-like hybrid models are proposed to relate volume-average deformation and stress within the two-phase composite inclusion to the remote (macroscopic) fields. Applications of these composite models arc made to predict stress strain behavior and texture evolution in initially isolropic high density polyethylene (HOPE) under different modes of straining.
It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabulated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants different from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.
A micromechanically-based composite model is proposed to study large plastic deformation and texture evolution in semi-crystalline polymers. The microstructure of many semi-crystalline polymers consists of co-existing crystalline and amorphous phases locally associated with each other in a fine plate-like morphological structure. An aggregate of two-phase composite inclusions is used to model these materials. Special consideration is given to molecular chain inextensibility within the crystalline phase. The introduction of a back stress tensor in the constitutive model of the amorphous phase accounts for hardening due to deformation-induced molecular alignment. Interface compatibility and traction equilibrium are enforced within each composite inclusion. A Sachs-like model and two newly-developed self-consistent-like hybrid models are proposed to relate volume-average deformation and stress within the two-phase composite inclusion to the remote (macroscopic) fields. Applications of these composite models arc made to predict stress strain behavior and texture evolution in initially isolropic high density polyethylene (HOPE) under different modes of straining.
New results are derived for the effective elastic properties of composite materials composed of a continuous matrix phase containing a highly concentrated suspension of rigid spherical inclusions. The composite is of the type that admits the limiting case of full packing of the inclusion phase corresponding to a polydisperse suspension. The resulting analytical forms from several different theoretical micro-mechanics models are found to vary widely and this provides a highly discriminating means of comparing them with respect to their physical significance. The various models also are evaluated with critical experimental data. Some extensions of the results are provided for fiber composite materials.
Synchrotron small-angle X-ray scattering (SAXS) studies were carried out to investigate effects of thickness of injection-moulded isotactic polypropylene (iPP) plates on shear-induced morphology and morphological distribution through the depth direction of the plates. Different levels of effective shear flow are imposed on the iPP melt by changing the thickness of the plates although the injection moulding is performed with the same injection ram speed and the same melt and mould temperatures. Shish-kebab-like morphology is found roughly 100 μm from the surface of plates, regardless of the thickness of plates. However, the type of shish-kebab-like morphology is very sensitive to the thickness. The shish-kebab structure at the surface region can be changed into the kebab-structure only or random crystalline lamellae as the thickness of the plate increases. The preferential orientation of crystalline lamellae along the flow direction strongly depends on the thickness of the plate, although the melt-shear does not significantly enhance the degree of linear crystallinity. It is also found that in the core region, the slow relaxation of polymer chains in the thick plate results in a higher degree of linear crystallinity. The results indicate that the shear-induced morphology is strongly dependent on effective shear flow and should be described individually.
The isotacticity dependence of the spherulitic morphology of isotactic polypropylene (iPP) with high isotacticity between 98.9% and 99.8% in isotactic pentad fraction ([mmmm]%) was studied. Conventional iPP with low isotacticity, [mmmm]=93.5%, was used as a reference. Isothermal crystallization was carried out at 125°C–150°C. Morphological observations were carried out by polarizing optical microscopy and transmission electron microscopy. Nearly all the spherulites showed an α-form crystal structure and negative birefringence (Δn). The radial lamellar (R-lamellar) fraction, fR, estimated from Δn increased with increase in isotacticity and crystallization temperature. This result indicates that the degree of cross-hatching composed of R-lamellae and tangential lamellae (T-lamellae) decreased with increase in isotacticity and crystallization temperature. It was shown for the first time that the spherulite fully occupied by R-lamella, i.e. fR≈1, was obtained when the sample with the highest isotacticity was crystallized at 150°C. This means that there are few cross-hatched lamellae. The origin of cross-hatching is discussed, focusing on the role of configurational defects within a molecular chain and the mobility of molecules in the crystallization process.
The technique of permanganic etching reveals lamellar detail in polyethylene and other polyolefines allowing representative melt-crystallized morphologies to be studied with the electron microscope. When etching with the original recipe is prolonged artefacts on a scale of ∼ 10 μm can develop and have probably been misinterpreted as genuine features in some instances. It is emphasized that the conditions of etching should be adjusted to suit the needs of individual specimens. The morphology of artefacts is demonstrated so that they may be recognized as such should they occur. A simple modification of the etchant by incorporating orthophosphoric acid has been found which avoids formation of artefacts on polyethylene.
Bounds on the elastic constants are derived for semicrystalline polymers whose local morphology is lamellar. Local response matrices (stiffness and compliance) are formulated in three dimensions that simultaneously incorporate uniform in-plane strain and additive forces from layer to layer of crystalline and amorphous phases and uniform stress and additive displacements normal to the lamellar surfaces. Spatial averaging of the stiffness and compliance matrices under the assumption of axially symmetric orientation gives the upper and lower bounds on the longitudinal and transverse tensile moduli and the axial and transverse shear moduli as functions of the separate phase elastic constants, the volume percent crystallinity, and the moments of the orientation 〈cos2θ〉 and 〈cos4θ〉. The bounds are much tighter than the Voight upper and Reuss lower bounds that do not recognize phase geometry. Using the known crystal elastic constants of polyethylene, sample calculations on isotropic unoriented materials show that the divergence of bounds at high crystallinity necessitated by the extreme crystal anisotropy shows up only at very high crystallinity. At low temperature the bounds are tight enough to specify G1, the amorphous modulus, from the measured G and the known crystal elastic constants. At higher temperatures and lower G, the bounds are not tight enough for this purpose but the shear modulus versus crystallinity and temperature data are well fitted by the lamellar lower bound using a temperature-dependent, crystallinity-independent G1.
The aim of this article is to try to explain why isotactic polypropylene (PP) is stiffer than high density polyethylene (HDPE) despite the fact that this latter is more crystalline and that its crystallites are stiffer than PP ones. Two micromechanical models were chosen for their ability to represent semi-crystalline polymers. The first one is a differential scheme in which ellipsoidal crystallites are randomly dispersed in an amorphous matrix. The second one is a self-consistent scheme where the material is considered as an aggregate of randomly oriented two layered-phase composite inclusions (crystalline–amorphous). Experiment-model comparisons are clearly in favor of the first model. This latter demonstrates the key importance of the crystalline lamellae aspect ratio on the elastic properties of semi-crystalline polymers.
Theoretical values for the thermomechanical properties of poly(ethylene terephthalate) (PET) are determined self-consistently using the pcff force field to compute the potential energy and quasiharmonic lattice dynamics to determine the vibrational free energy. Complete sets of lattice constants, thermal expansion coefficients, elastic properties, and Grüneisen coefficients are reported between 0 and 400 K for the triclinic PET unit cell. Mean square displacement matrices for the constituent atoms of PET were determined, from which a theoretical B-factor for X-ray scattering of 4.0 Å2 at 300 K is estimated. The 50% probability ellipsoids for thermal vibration of all atoms in the asymmetric unit are computed. Calculated lattice parameters at 300 K agree with experimental data, to within the accuracy of the method. Calculated elastic constants for a transverse isotropic composite agree with data from X-ray and ultrasonic velocity measurements on highly oriented samples. The tensile elastic stiffness constants are temperature-dependent, while the shear stiffnesses are roughly constant in the range 0−400 K. Thermal contraction along the chain direction is observed in PET, consistent with results for other polymer crystals possessing chains in fully extended conformations. The driving force for contraction is entropic in origin, arising from negative γ3 and γ6 Grüneisen coefficients.
This paper aims to provide an overview of observed situations in heterogeneous materials, both in terms of the geometry of the multiphase structure (topology, shapes and scales) and in terms of the mechanical contrast between the constituents. An overview of some tools available in micromechanics will be given in order to answer the following two questions: What is the best model class for a given material? What is the best model material to test a mechanical theory? Some remarks which probably require new conceptual approaches will be outlined in the conclusion.
The five independent elastic moduli C11, C12, C13, C33, and C44 of oriented high-density polyethylene with draw ratio λ from 1 to 27 have been determined from −60 to 100°C by an ultrasonic method at 10 MHz. At low temperature the sharp rise in the axial extensional modulus C33 with increasing λ and the slight changes in the other moduli result from chain alignment and the increase in the number of intercrystalline bridges connecting the crystalline blocks. At high temperature (say, 100°C) the transverse extensional modulus C11, as well as the axial (C44) and transverse (C66) shear moduli, also show substantial increases, reflecting the prominent reinforcing effect of stiff crystalline bridges in this temperature region where the amorphous matrix is rubbery. If the crystalline bridges are regarded as the fiber phase, the mechanical behavior can be understood in terms of the Halpin–Tsai equation for aligned short-fiber composites.
The fabrication techniques now available for the production of highly oriented polymers are reviewed. These techniques include tensile drawing from both melt-spun and gel-spun polymers, extrusion under pressure from the melt, and hydrostatic extrusion, ram extrusion and die drawing in the solid phase. In addition, lyotropic and thermotropic liquid crystalline polymers offer new routes to very stiff and strong polymers.
Following the review of processing methods, an account is given of low strain mechanical behaviour and its relationship to structure, thermal properties (including thermal conductivity and thermal expansion behaviour) and barrier properties (permeability to liquids and gases and solubility).
Samples of poly(etherether ketone) (PEEK) were subjected to large plastic deformations under uniaxial tension and simple shear by means of a new video-controlled testing method at constant true strain rate. The “equivalent” stress-strain curves obtained under the two loading modes are close at the yield point, but diverge drastically at large strains, with a rapidly increasing hardening in tension and a moderate hardening under simple shear. X-ray diffraction goniometry shows that these contrasting behaviors are associated with the different textures developed in the crystallite orientations. Under tension, the PEEK lamellae are progressively tilted in such a way that the chain axis becomes oriented parallel to the tensile axis; in the other mode, the final chain orientation is near to the shear axis. DSC analyses of deformed samples in both modes are carried out. The results show that the tension loading induces a fragmentation of the thin lamellae, while the shear mode generates less fragmentation. A quantitative model is presented that involves a composite approach: (i) the viscoplastic deformation of the crystalline lamellae, which is controlled by chain slip and transverse slip systems on planes parallel to the c axis, and (ii) the hyperelastic deformation of the amorphous phase, which depends on the affine unfolding of statistically distributed subchains. A discussion of the influence of the CRSS values on the stress-strain curves and textures is developed by means of this model.
Mechanical relaxation has been studied at 0.67 cps in linear polyethylene (LPE) and polytetrafluoroethylene (PTFE) between −190 and −20°C. Specimens were prepared by use of various thermal treatments to produce in LPE a range of crystalline fractions from 0.690 to 0.825 and in PTFE from 0.615 to 0.870. An empirical theory is proposed relating the modulus of the crystalline–amorphous composite solid to the moduli and the volume fractions of the two phases. The empirical theory is shown to be in accord with the bounds of Hill and of Hashin and Shtrikman. The theory is used to determine the magnitudes of the crystalline and amorphous components of the low temperature relaxations in LPE and PTFE from measurements of logarithmic decrement and shear modulus. In PTFE the γ relaxation occurs in the amorphous fraction alone. In LPE the γ relaxation is a composite one, formed from the superposition of a small crystal relaxation and a large amorphous relaxation. For the crystal relaxation in LPE the ratio of relaxed to unrelaxed modulus equals 0.78; for the amorphous relaxation, the ratio equals 0.23. In a specimen of LPE with crystal fraction 0.69 the crystal contribution to the relaxation is 25% of the total. The magnitude of the unrelaxed modulus of the crystal fraction of LPE (modulus of polycrystalline LPE at −190°C) is in reasonable agreement with theoretical calculations of Odajima and Maeda.
A large collection of data on Young's modulus and density of unfilled polyethylenes at ambient conditions has been compared with various competing theoretical mixing rules developed for composite micromechanics. The objective was to see if such theories usefully predict the dependence of stiffness on crystalline content in an archetypal isotropic semicrystalline thermoplastic polymer above its glass trnsition temperature. It was found that the self-consistent scheme derived by Hill and Budiansky from continuum micromechanics appears to have valid application to this system. The scheme naturally and coherently incorporates information on bulk and shear moduli and Poisson's ratios, while giving a good account of the main trend in the Young's modulus data. Conversely, other theoretical models frequently invoked in the polymer literature were explicitly found to be unsuitable for representing principal features of modulus-density relationships dectated by the data.
It is shown that the entanglement junction may be modeled as a binary hooking contact of Kuhn nodes between two chains. The entanglement behavior is thus determined by chain tortuosity and given by Nv = (1/β)C, where Nv is the number of real or virtual skeletal bonds in an entanglement strand, C∞ is the characteristic ratio, α = 2 is the number of hooks involved at an entanglement junction, and β = 1/3 is the fraction of binary hooking configurations out of all possible configurations at a binary nodal contact. In other words, we have Nv = 3C, which is verified experimentally for 44 polymers, covering a wide variety of skeletal, pendant, and stereoisomeric (tacticity) structures. Since C∞ may be estimated by group additivity, the present equation may be used to predict the entanglement behavior from chemical structure.
This paper presents a micromechanical analysis of the elastic properties of semicrystalline thermoplastic materials. A lamellar stack aggregate model reported in the literature is used to derive tighter bounds and a self-consistent scheme for the elastic modulus, and it is shown that the existing geometric models of the microstructures are not effective in predicting experimentally measured modulus of semicrystalline materials. Toward addressing this limitation, a model based on Mori-Tanaka's mean field theory is developed by treating the semicrystalline materials as short-fiber reinforced composites, in which the lamella crystalline phase is modeled as randomly embedded anisotropic ellipsoidal inclusions, and the amorphous phase as an isotropic matrix. The lamellae are characterized by two independent aspect ratios from three distinct geometric axes in general. Existing morphological studies on polyethylene (PE) and a syndiotactic polystyrene (sPS) are used to deduce the corresponding lamella aspect ratios, based on which the theoretical model is applied to predict the elastic modulus of the two material systems. The model predictions are shown to compare well with the reported measurements on the elastic moduli of PE and sPS. Polym. Eng. Sci. 44:433–451, 2004. © 2004 Society of Plastics Engineers.
A micromechanically based constitutive model for the elasto-viscoplastic deformation and tex-ture evolution of semi-crystalline polymers is developed. The model idealizes the microstruc-ture to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of ÿnite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic ow occurring via crystallographic slip. The amor-phous phase is modeled as isotropic elastic with plastic ow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic ÿelds by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress–strain behav-ior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the eeect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading–reloading cycles are presented.
The measurement of the elastic constants of several highly oriented thermoplastic polymer fibres is described. The method makes use of the hot-compaction process, developed and patented in this laboratory, which enables a solid section of highly oriented polymer to be produced from an aggregate of highly oriented fibres. As only a small fraction of the original fibre is melted and recrystallized during the process, the compacted materials offer a unique opportunity for measuring fibre properties in the bulk. An ultrasonic immersion technique is used to measure the elastic properties of the compacted materials, from which the properties of the polymer fibres are inferred. The experimentally determined fibre elastic properties have been compared with other oriented polymer materials to assess any similarities in elastic anisotropy between different methods for producing fibre orientation, and compared with theoretical upper limits for the fibre elastic properties based on theoretical estimates for the polymer crystal unit cell appropriately averaged for hexagonal symmetry using the aggregate model.
This study contains a combined application of three different techniques for the study of injection moulded polyethylene (PE), showing an oriented shish-kebab structure: small angle X-ray scattering (SAXS), low frequency Raman spectroscopy (LAM) and transmission electron microscopy (TEM), A series of linear PEs and molecular weights in the range 51000–478000 has been investigated and two injection temperatures have been used (T
m=144 and 210 C). SAXS patterns from the highly oriented regions show the presence of either one axial long period (L
1) or two (L
1 and L
2) depending on molecular weight (M
w) and T
m. It is shown that L
1 and L
2 increase with M
w up to a given critical molecular weight M
c. Above M
c, L
1 and L
2 remain constant. Raman results qualitatively confirm the existence of two separate distributions of straight-length chain segments for those samples having molecular weights above the critical value. Shorter segments are shown to be more abundant than the longer ones. In the lowest molecular weight sample, results from SAXS, TEM and Raman spectroscopy seem to be consistent with each other, although in some cases a tilted molecular arrangement within the lamellae has to be invoked. On the other hand, in case of the highest molecular weight sample, the length of the short straight-chain segments derived from Raman spectroscopy agrees well with the double periodicity obtained from SAXS. On the contrary, long periods measured from TEM only correspond to the shorter SAXS periodicity. This result is discussed by assuming the occurrence of crystalline bridges among adjacent lamellae.
The changes of supramolecular structure of PET fibres upon supercritical fluid (SCF) dyeing, conventional water dyeing, hot
air thermofixation at 130°C, respectively were studied. The following observations were made; the increase in crystallinity
is nearly alike with the SCF and water treated sample, while hot air results in lower crystallinity change. There is only
a very slight change in the periodical structure along the fibre axis, determined by the long period. The diminution of the
long period by the treatment at 130°C is nearly alike with SCF and hot-air treatment, while water medium decreases the long
period less. The strongest influence on dimensions of crystallites is observed with water, while hot air and supercritical
CO2 cause a smaller diminution of the crystallite sizes. After the treatments a slight disorientation is observed irrespective
of the medium used. Perhaps water represents the medium where the relaxation process occurs easier and a slightly greater
fall of orientation function is detected. Compared to SCF treatment of PET fibres both other treatment media cause a more
pronounced change of the micro-void system and the inhomogeneity form factor additionally increases with the latter two treatment
media. From only very slight differences between the samples treated at 130°C in different media a conclusion follows: obviously
the medium used is not the main influence, but temperature used has the greatest influence.
A multi-scale constitutive model for the small deformations of semi-crystalline polymers such as high density Polyethylene is presented. Each macroscopic material point is supposed to be the center of a representative volume element which is an aggregate of randomly oriented composite inclusions. Each inclusion consists of a stack of parallel crystalline lamellae with their adjacent amorphous layers.Micro-mechanically based constitutive equations are developed for each phase. A viscoplastic model is used for the crystalline lamellae. A new nonlinear viscoelastic model for the amorphous phase behavior is proposed. The model takes into account the fact that the presence of crystallites confines the amorphous phase in extremely thin layers where the concentration of chain entanglements is very high. This gives rise to a stress contribution due to elastic distortion of the chains. It is shown that the introduction of chains’ elastic distortion can explain the viscoelastic behavior of crystalline polymers. The stress contribution from elastic stretching of the tie molecules linking the neighboring lamellae is also taken into account.Next, a constitutive model for a single inclusion considered as a laminated composite is proposed. The macroscopic stress–strain behavior for the whole RVE is found via a Sachs homogenization scheme (uniform stress throughout the material is assumed).Computational algorithms are developed based on fully implicit time-discretization schemes.
In this paper, the influence of processing conditions on the spatial distribution of the molecular orientation was determined within the depth of the thickness of injection molded isotactic polypropylene (iPP) plates. Small 35 μm-thick slices were microtomed from the surface to the core of 1 and 3 mm-thick plates. The orientation functions along the three crystallographic axes were determined on the slices from IR dichroism measurements and WAXS pole figures. It was found that the orientation of the amorphous phase was low and the crystalline orientation had a maximum in the shearing layer, which was solidified during the filling stage. The plate thickness seemed to govern the global level of orientation, while the injection speed determined the thickness of the shearing layer without changing the maximum of orientation. Changing the mold temperature from 20 to 40 °C did not modify the molecular orientation. A specific bimodal crystalline orientation was found in the shearing layer. This crystalline structure continued in the post-filling layer, but the local symmetry axes tilted towards the core.
In previous papers (Hashin and Shtrikman 1961a, b) the authors have established some new variational principles in the linear theory of elasticity for isotropic nonhomogeneous bodies with prescribed surface displacements. These have been applied to the theory of the elastic behaviour of multiphase materials. The present work is concerned with the generalization of these principles to anisotropic and nonhomogeneous elasticity‡ and the establishment of similar variational principles for a body with prescribed surface tractions.An accompanying paper deals with the application of these variational principles to the theory of the elastic behaviour of polycrystals.
Having noted an important role of image stress in work hardening of dispersion hardened materials, (1,3) the present paper discusses a method of calculating the average internal stress in the matrix of a material containing inclusions with transformation strain. It is shown that the average stress in the matrix is uniform throughout the material and independent of the position of the domain where the average treatment is carried out. It is also shown that the actual stress in the matrix is the average stress plus the locally fluctuating stress, the average of which vanishes in the matrix. Average elastic energy is also considered by taking into account the effects of the interaction among the inclusions and of the presence of the free boundary.
A micromechanically based composite model which we have recently proposed is employed to study large plastic deformation and texture evolution in initially isotropic high density polyethylene (HDPE) under different modes of straining. Attention is focused on the macroscopic stress-strain response and the evolution of crystallographic, morphological and macromolecular textures in HDPE subject to uniaxial tension and compression, simple shear and plane strain compression. Comparison of the predicted results with experimental observations (e.g. stress-strain measurements, wide-angle X-ray scattering and small-angle X-ray scattering studies of deformed material) shows excellent agreement in nearly all respects.
The overall elastic moduli of some solid composite materials are evaluated, first by bounding them precisely, and secondly by a ‘self-consistent’ estimate. Transversely isotropic inclusions of ‘needle’ and ‘disc’ shapes are particularly considered, at both random and aligned orientations, and at arbitrary volume concentration.
Novel morphology of isotactic polypropylene (iPP) crystal, which is quite different from that of usual spherulite, has been observed by scanning electron microscope (SEM). The crystals show ‘bamboo leaf-like (BL)’ shape with α-monoclinic high crystallinity. The BL crystals are formed by neither melt nor glass crystallization, but by a complicated annealing process that goes through mesomorphic phase of iPP. Substrates are not essential for the formation of BL crystals, since the BL crystals are formed both on glass surface and free surface as well as in bulk. Along with the annealing process, a possible explanation for the mechanism of the formation of the BL crystal is proposed.
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