6. MANUAL DEL SISTEMA DE GESTION DE SEGURIDAD Y SALUD OCUPACIONAL EN EL
6.2 POLITICA DE SEGURIDAD Y SALUD OCUPACIONAL
We conclude this section with a discussion on the macroscopic response and texture evolution in HDPE under uniaxial tension. The loading conditions considered in this subsection differ from the uniaxial compression case of subsection 4.5.2only in that the applied deformation rate along the axis of the cylinder is 𝐷33>0. The associated von Misses equivalent stress𝜎𝑒and the equivalent strain measure𝜀𝑒 are defined by (4.112)1 and (4.112)2, respectively.
Figure4.20compares the theoretical predictions of the LCC estimate (4.80), the Lee et al. [80] estimate and the Nikolov et al. [110] estimate for uniaxial tension of HDPE at a constant defor- mation rate 𝐷33 = 10−3𝑠−1. For further comparison, this figure includes also the corresponding experimental results from the works of G’Sell and Jonas [46] and Hiss et al. [54]. As already mentioned, these experiments have been performed for HDPE materials with a different molecular weight than the HDPE for which the material parameters of the models were chosen. Therefore, a quantitative comparison of the estimates with the experimental data is not entirely fair. It is observed that the estimate of Lee et al. [80] is softer and closer to the experimental results than the other two estimates up a strain near 𝜀𝑒 = 1. However, shortly after this value the model of Lee et al. [80] predicts a strong macroscopic hardening, leading to a significant deviation from the experimental data. On the other hand, the LCC estimate and the estimate of Nikolov et al. [110] are in a good qualitative agreement with each other as well as with the experimental results in the entire range of deformations considered. The estimate of Nikolov et al. [110] is softer and in slightly better agreement with the experimental data than the LCC estimate. Comparing the results of Fig. 4.20with the corresponding results of Fig. 4.7, we observe that the response of HDPE under tension is qualitatively similar to that under compression, but the material is significantly stiffer under tension than under compression.
In analogy with the results of Fig. 4.8for the morphological texture evolution under uniaxial compression, Figure 4.21 shows the evolution of the aspect ratios of the distributional ellipsoid
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Exp. (G’Sell and Jonas, 1979)Exp. (Hiss et al., 1999) Lee et al., 1993 Nikolov et al., 2006 LCC model
(M
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σ
eε
eFigure 4.20: Macroscopic response of HDPE under uniaxial tension at a constant deformation rate𝐷33= 10−3𝑠−1. Comparison of the LCC estimate (4.80), the estimate of Lee et al. [80] and the estimate of
Nikolov et al. [110] with the experimental results of G’Sell and Jonas [46] and Hiss et al. [54]. The applied equivalent stress measure𝜎𝑒 is plotted as a function of the applied equivalent strain measure𝜀𝑒, with𝜎𝑒
and𝜀𝑒defined by (4.112)1 and (4.112)2, respectively.
(4.8) for uniaxial tension, as predicted by the LCC model. Once again, results are presented both for the 500-grain system of Fig. 4.3and for a refined 1000-grain system. The aspect ratios𝑤max and𝑤min shown in Fig. 4.21are in this case defined by
𝑤max= 𝜆3
𝜆max
, 𝑤min= 𝜆3
𝜆min
, (4.116)
where we recall that 𝜆max and 𝜆min are respectively the maximum and minimum macroscopic stretches in the plane normal to the loading axis e3 and 𝜆3 is the corresponding stretch along
e3. It is also recalled that the difference between 𝑤max and 𝑤min provides a measure for the deviation of the material response from perfectly transversely isotropic behavior corresponding to
𝑤max=𝑤min= Exp[3𝜀e/2] and represented in Fig. 4.21with the continuous, thin line. We observe that the difference between 𝑤max and 𝑤min is negligible up to a strain value 𝜀𝑒 = 1, after which it becomes somewhat more evident and increases with increasing strain. Similar to the case of uniaxial compression, the difference between 𝑤max and 𝑤min for the 1000-grain system (dotted, thin curves) is smaller than the corresponding difference for the 500-grain system (continuous, thick curves), suggesting once again that the deviation of the material response from transverse isotropy
0 0.5 1 1.5 2 0 5 10 15 20 A spe ct r at ios 3 e/ 2
e
ε 500 grains 1000 grains eε
maxw
minw
Figure 4.21: Morphological texture evolution in HDPE under uniaxial tension as predicted by the LCC model. The maximum𝑤maxand minimum𝑤minaspect ratios of the distributional ellipsoid (4.8), defined respectively by (4.116)1 and (4.116)2, are plotted as functions of the applied equivalent strain𝜀𝑒. Results
are shown both for the 500-grain system of Fig. 4.3and also for a refined 1000-grain system. For com- parison, the aspect ratio 𝑤max =𝑤min = Exp[3𝜀e/2] corresponding to a perfectly transversely isotropic material is also shown.
at large strains is probably due to the corresponding deviation of the undeformed specimen from the isotropic behavior.
Figure4.22shows the stereographic projection representations of the lamellar normalsn(𝑟)at the applied strain values𝜀𝑒= 0.4,𝜀𝑒= 0.8,𝜀𝑒= 1.3 and𝜀𝑒= 2.1, as predicted by the LCC model. In contrast with the uniaxial compression case, here we observe that the lamellar normals n(𝑟) rotate monotonically towards the radial direction with increasing strain 𝜀𝑒. This evolution of the normalsn(𝑟)leads progressively to configurations corresponding to stiffer deformation modes for the laminates. Note, in particular, that at𝜀𝑒= 2.1 the lamination orientationsn(𝑟)are practically perpendicular to the tensile direction. In this configuration, the laminates are expected to exhibit their stiffest possible response (Taylor-type). Hence, similar to uniaxial compression, the lamellar texture evolution is expected to have a stiffening effect on the overall response of HDPE under uniaxial tension at large strains. Taking into account that the rotation of the laminates takes place through interlamellar shear, the predicted evolution of the normalsn(𝑟)is consistent with physical experience.
Next, we consider the predictions of the LCC model for the crystallographic texture evolution. Figure4.23shows the stereographic projection representations of the (001), (100) and (010) plane poles—i.e., the c(𝑟), a(𝑟) andb(𝑟) crystallographic axes, respectively—as well as the {011} plane
1 2 Lam. 1 2 Lam. 1 2 Lam. 1 2 Lam. 𝜀𝑒= 0.4 𝜀𝑒= 0.8 𝜀𝑒= 1.3 𝜀𝑒= 2.1 Figure 4.22: Lamellar texture evolution in HDPE under uniaxial tension as predicted by the LCC model. Stereographic projection figures of the lamination orientations n(𝑟) are shown at various values of the
applied equivalent strain𝜀𝑒. The direction of tension is along the 3−axis.
poles at the applied strain values𝜀𝑒= 0.4,𝜀𝑒= 0.8,𝜀𝑒= 1.3 and𝜀𝑒= 2.1. We observe the gradual development of a very strong texture component formed through an intense rotation of the c(𝑟) axes towards the loading directione3combined with a corresponding rotation of thea(𝑟)andb(𝑟) axes towards the radial direction. Note, in particular, that at 𝜀𝑒 = 2.1 thec(𝑟) axes are almost perfectly aligned with the tensile axise3. The formation of this texture suggests that the dominant deformation mechanism in the associated crystals is chain slip, since the rotation of the chain axes
c(𝑟)—in addition to any possible contribution from the rotation of the lamination orientationsn(𝑟)
(Fig. 4.22)—takes place through crystallographic chain slip. Another interesting observation from Fig. 4.23is that the maximum of the strong texture component of the{011}plane poles is aligned with the tensile axise3 at𝜀𝑒= 0.4 and𝜀𝑒= 0.8, but it is tilted slightly away from it at𝜀𝑒= 1.3 and at the strain level𝜀𝑒= 2.1 it forms an angle approximately 20𝑜withe3.
From Fig. 4.23we also observe the development of a second texture component, which is much weaker than the one discussed above. This texture consists of a small fraction of crystals whose chain axesc(𝑟) remain practically fixed throughout the deformation at an orientation nearly per- pendicular to the loading axise3, presumably because the associated resolved shear stresses are not sufficient to activate chain slip. Theb(𝑟)anda(𝑟)axes of these crystals rotate respectively towards and away the loading axis, which in turn suggests that the dominant deformation mechanism in these crystals is (100)[010] transverse slip. In particular, at the strain level 𝜀𝑒 = 2.1 we observe that the b(𝑟) anda(𝑟) axes are oriented at 25𝑜 and 65𝑜, respectively, with respect to the loading direction e3.
Figure4.24reproduces the experimentally determined (Li et al. [83]) crystallographic texture in a specimen of HDPE subjected to uniaxial tension up to a strain value𝜀𝑒= 2.1. The relevant stereographic projections of the (001), (100) and (010) plane poles in both the undeformed and deformed, after relaxation, sample are shown. It is emphasized that the undeformed specimen of
1 2 (001) 1 2 (001) 1 2 (001) 1 2 (001) 𝜀𝑒= 0.4 𝜀𝑒= 0.8 𝜀𝑒= 1.3 𝜀𝑒= 2.1 1 2 (100) 1 2 (100) 1 2 (100) 1 2 (100) 𝜀𝑒= 0.4 𝜀𝑒= 0.8 𝜀𝑒= 1.3 𝜀𝑒= 2.1 1 2 (010) 1 2 (010) 1 2 (010) 1 2 (010) 𝜀𝑒= 0.4 𝜀𝑒= 0.8 𝜀𝑒= 1.3 𝜀𝑒= 2.1 1 2 {011} 1 2 {011} 1 2 {011} 1 2 {011} 𝜀𝑒= 0.4 𝜀𝑒= 0.8 𝜀𝑒= 1.3 𝜀𝑒= 2.1 Figure 4.23: Crystallographic texture evolution in HDPE under uniaxial tension as predicted by the LCC model. Stereographic projection figures of the (001), (100), (010) and {011} plane poles are shown at various values of the applied equivalent strain𝜀𝑒. The direction of tension is along the 3−axis.
Fig. 4.24is strongly textured and, therefore, a quantitative comparison of these results with the corresponding LCC predictions of Fig. 4.23 is not entirely appropriate. The dominant feature of
(001) (100) (010)
𝜀𝑒= 0 𝜀𝑒= 0 𝜀𝑒= 0
(001) (100) (010)
𝜀𝑒= 2.1 𝜀𝑒= 2.1 𝜀𝑒= 2.1
Figure 4.24: Crystallographic texture in the HDPE specimen used by Li et al. [83] in their uniaxial tension test. The corresponding stereographic projection figures of the (001), (100) and (010) plane poles are shown in the undeformed and deformed to𝜀𝑒= 2.1 sample, after relaxation. The direction of tension
is normal to the plane shown.
the results of Fig. 4.24is the formation of a very strong texture component in which thec(𝑟)-axes are almost perfectly aligned with the tensile axis (normal to the pole figures) and the a(𝑟)- and
b(𝑟)-axes are oriented towards the radial direction. From Fig. 4.23, it is recalled that this texture component is also the dominant feature of the corresponding crystalographic texture evolution predicted be the LCC model. As reported by Li et al. [83], their results show, in addition, the formation of a weaker texture component—in which the axes c(𝑟), a(𝑟) and b(𝑟) are respective oriented at about 25𝑜, 90𝑜 and 65𝑜 with respect to the loading axis—as well as an even weaker component—in which the axes b(𝑟) are aligned andc(𝑟),a(𝑟) are normal to the tensile axis. The weaker texture components may be due to the initial texture of the material. These later textures are not predicted by the LCC estimate.
The evolution of the strong crystallographic texture component discussed above is expected to have a significant hardening effect on the overall response of the composite. This becomes
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e 2 1 3310
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−Figure 4.25: The effect of the deformation rate𝐷33 on the LCC estimate (4.80) for the macroscopic response of HDPE under uniaxial tension. The applied equivalent stress𝜎𝑒is plotted as a function of the
applied equivalent strain𝜀𝑒 for𝐷33= 10−4𝑠−1,𝐷33= 10−3𝑠−1 and𝐷33= 10−2𝑠−1.
immediately obvious by taking into account the rotation of the chain axesc(𝑟)towards the tensile direction with increasing strain and recalling that the c(𝑟) axes correspond to an inextensible direction for the crystals. This observation is probably the main reason for which the macroscopic response of a HDPE material under tension is much stiffer than its response under compression (compare Figs. 4.20and4.7).
Finally, we consider the effect of the applied applied deformation rate𝐷33on the macroscopic response of HDPE under uniaxial tenstion, as predicted by the LCC model (4.80). Fig. 4.25
shows results for the cases 𝐷33 = 10−2𝑠−1, 𝐷33 = 10−3𝑠−1 and 𝐷33 = 10−4𝑠−1. Once again, we observe that increasing the deformation rate 𝐷33 results in a stiffer macroscopic response for the composite. This observation in consistent with corresponding experimental findings (see, e.g., G’Sell and Jonas [46], Hiss et al. [54]).
4.6
Concluding remarks
In this chapter we have developed a constitutive model for the macroscopic response and texture evolution of semi-crystalline polymers under general finite strain loading histories. In the context of this model, the semi-crystalline polymer is idealized as a two-scale composite with a granular
meso-structure and a lamellar micro-structure, incorporating fine sub-structural information at both length-scales. The constitutive behavior of both the crystalline and the amorphous phase is characterized by means of viscoplastic models. It is important to emphasize, however, that the effect elastic strains in the amorphous material is also accounted for by means of a back-stress model. The instantaneous effective behavior of this material has been determined by means of the “linear comparison composite” (LCC) methods, introduced in the context of more general two-scale viscoplastic systems in chapter2. Specifically, the LCC estimate for the semi-crystalline polymer has been derived by employing a “secant” type of linearization for the amorphous phase (subsection 2.6.1) and a “generalized-secant” type of linearization (subsection2.8.1) for the crys- talline phase. The effective behavior of the associated two-scale LCC is determined sequentially by making combined use of the well-suited self-consistent estimate for the granular meso-structure and the well-known exact solution for the lamellar (grain) micro-structure. As a byproduct of these homogenization estimates for the LCC, we generated corresponding estimates for the associated grain- and phase-average deformation rate and spin tensors, which were in turn used to establish appropriate evolution laws for the internal variables of the semi-crystalline polymer.
Applications on “high-density polyethylene” (HDPE) materials were considered in great detail. Based on the experimental results of Bartczak et al. [10] for the uniaxial compression test of HDPE, we argued that the micro-mechanical mechanism responsible for the intense macroscopic hardening of the material at large strains is the hardening of the crystalline phase due to the Coulomb effect (Bartczak et al. [8]) on the critical resolve shear stresses (CRSSs) of the crystals. To the best of our knowledge, this interpretation is novel and of crucial importance in modeling the constitutive behavior of the crystalline phase. In passing, it is remarked, in the context of earlier theoretical works such as the models proposed by Lee et al. [80] and Nikolov et al. [110], that it has been implicitly assumed that the underlying mechanism for the macroscopic hardening of HDPE under compressive loadings is the hardening of the amorphous material. The use of the Coulomb yield criterion (4.109) for the CRSSs requires consideration of a non-associative constitutive relation for the crystalline phase. Unfortunately, the LCC model proposed here—and, more generally, the LCC methods of chapter2—are restricted to associative constitutive models for the phases. These observations motivated our prescription of the strain-hardening relation (4.77) for the CRSSs in the context of the LCC model. This prescription is by no means equivalent to the Coulomb yield criterion (4.109), but it has the advantage of attributing the aforementioned hardening behavior at large strains to the corresponding hardening of the crystalline phase, which is an important difference between our model and the models of Lee et al. [80] and Nikolov et al. [110].
The predictions of the LCC model for the macroscopic response and texture evolution in HDPE were confronted with corresponding experimental results and compared with the associated predic- tions of the models of Lee et al. [80] and Nikolov et al. [110] for uniaxial compression, simple shear
and uniaxial tension loading conditions. The LCC estimates were found to be in a good agreement with the experimental results for all loadings and to improve, in some cases significantly, over the predictions of the earlier models.