Finite deformation constitutive model for macro-yield behavior of amorphous glassy polymers with a molecular entanglement-based internal-state variable

International Journal of Mechanical Sciences 161–162 (2019) 105064

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Finite deformation constitutive model for macro-yield behavior of amorphous glassy polymers with a molecular entanglement-based internal-state variable Han Jiang∗, Chengkai Jiang Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China

a r t i c l e

i n f o

Keywords: Glassy polymer Constitutive modeling Viscoplastic material Finite deformation Molecular entanglement

a b s t r a c t To describe the macro-yield behavior of amorphous glassy polymers which is significantly affected by thermomechanical history and deformation state, an elasto-viscoplastic constitutive model incorporating a molecular entanglement-based internal-state variable is proposed. While the classical topological entanglement governs the benchmark mechanical performances such as the pre-yield linearity and the post-yield softening plateau, the secondary short-range microstructure, named as the sub-entanglement owing to the local interaction between neighboring molecular chains, is taken as the intrinsic source of yield peak. The microstructure related deformation resistance can be decomposed into the topological entanglement related one, 𝑠̃te , and the sub-entanglement related one, 𝑠̃se , respectively. During the macro-yield procedure, 𝑠̃te is assumed as constant because the topological entanglement rarely disentangles, while 𝑠̃se evolves with the gradually dissociating sub-entanglement. The evolution equation of 𝑠̃se is presented and its physical meaning is clarified. Validated with the experimental data of amorphous glassy polymers in literature, the proposed constitutive model can not only well describe the macroyield behavior, but also reasonably explain the effects of thermo-mechanical history and deformation state on the macro-yield behavior.

1. Introduction Amorphous glassy polymers have been extensively utilized as structural components in various engineering fields such as aerospace, automobile, medical and optical devices which are often subjected to various complex loading conditions. The finite deformation behavior of amorphous glassy polymers has been widely concerned. As schematically shown by the solid line in Fig. 1(a), its stress-strain curve owns a pronounced macro-yield peak consisting of three parts, i.e., a preyield nonlinearity, a yield peak point and a post-yield softening region. The macro-yield peak of amorphous glassy polymers has been experimentally found to be strongly correlated to thermo-mechanical history [1–6]. For instance, a more pronounce macro-yield peak is found with a longer annealing/aging time. On the contrary, the macro-yield peak shrinks, even completely diminishes, if the specimen has previously undergone a severe plastic deformation. Meanwhile, the deformation states were observed to affect the macro-yield response of amorphous polymers. For example, Arruda and Boyce [7] found that, at the same loading rate, the macro-yield peak under the plane strain compression was higher than that under the uniaxial compression. Therefore, a constitu-

∗

tive model to describe the complex macro-yield behavior of amorphous glassy polymers is necessary. In recent decades, researchers have made great achievement in constitutive modeling of the finite deformation behavior of amorphous glassy polymers. Duan et al. [8], Jeridi et al. [9], Zaïri et al. [10], Hu et al. [11], and Wang et al. [12], etc., proposed the phenomenological models mainly focusing on the macroscopic deformation responses. The understanding of microscopic deformation mechanism is important for modeling the complex macro-yield behavior of amorphous glassy polymers. Haward and Thackray [13] adopted the physical-based Eyring model [14] to establish a one-dimensional constitutive model for amorphous glassy polymers. They considered the yield behavior as a thermally activated process and the deformation resistance as a result of the state/configuration transition of a single molecular chain. Their pioneering work has been extended into three-dimensional version by Boyce et al. [15]; Anand and Gurtin [16]; Mirkhalaf et al. [17]; Zhang and To [18]; Li and Buckley [19]; van Breemen et al. [20]; Tervoort et al. [21], among others. Govaert and co-workers [21,22] proposed the single-mode EGP (Eindhoven glassy polymer) model to describe the macro-yield process of amorphous glassy polymers. Mirkhalaf et al.

Corresponding author. E-mail address: [email protected] (H. Jiang).

https://doi.org/10.1016/j.ijmecsci.2019.105064 Received 3 April 2019; Received in revised form 2 August 2019; Accepted 5 August 2019 Available online 5 August 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

H. Jiang and C. Jiang

International Journal of Mechanical Sciences 161–162 (2019) 105064

Fig. 1. Schematics of (a) the typical finite deformation stress-strain curve of amorphous glassy polymers and (b) the microstructures in amorphous glassy polymers (circular box: topological entanglement (TE) and square boxes: sub-entanglement (SE)).

[17] reimplemented the single-mode EGP model and introduced a Lode angle parameter to improve its capability to predict the post-yield responses of amorphous polymers under various stress states. Furthermore, van Breemen et al. [20] extended the single-mode EGP model by introducing a spectrum of linear relaxation times, which shifted to shorter time scales under the stress effect, to represent the pre-yield nonlinearity response. To describe the plastic flow during the macroyield process, Boyce et al. [15] assumed that the deformation resistance comes from the ‘double-kink’ transition of molecular chain with the constraint of surrounding media. They extended Argon model [23] to effectively address the hydrostatic pressure effect and the post-yield softening by introducing a phenomenological evolution equation to describe the intermolecular deformation resistance. The extended Argon model was widely utilized to describe the macro-yield behavior [7,24– 29]. Recently, Poulain et al. [30] modified the original Argon model by setting the exponential factor 5/6 as a free parameter to account the effect of temperature-dependent elastic modulus on the plastic flow of amorphous glassy epoxy. Bernard et al. [31] demonstrated that the Argon’s model can be extended to a wide range of temperatures and strain rates by introducing an appropriate model of Young’s modulus with the effects of temperature and strain rate. On the basis of the cooperative model [32], Richetion et al. [33] proposed a thermally activated model with the consideration of time-temperature superposition principle, which can describe the dependence of yield peak point on a wide range of temperatures and loading rates. This model was also utilized to capture the macro-yield behavior of various types of amorphous polymers [34–37]. It should be pointed out that most of those models focus on the yield peak point alone, the effect of microstructure and its evolution on the macro-yield behavior of amorphous glassy polymers has not thoroughly considered yet. The internal-state variable theory in finite deformation framework has been widely utilized to describe the macroscopic deformation behavior of polymeric materials [16,36,38,39]. The internal-state variables are introduced to represent the underlying microscopic mechanism(s) of the macroscopic deformation procedure of polymer materials. Hasan et al. [40] suggested that the local free volume may dominate the macroyield behavior and the increase of free volume content results in the post-yield softening of amorphous glassy polymers. Adopting the free volume content as a scalar internal-state variable, Anand and Gurtin [16] proposed a thermodynamical framework of amorphous materials based on the principle of virtual power. They established the coupling equations between the free volume content and the intermolecular deformation resistance to describe the macro-yield peak of amorphous glassy polymers. This model was successfully extended to account the nonlinear viscoelasticity effect [41], as well as the thermo-mechanical coupling effect below and span the glass-transition temperature [37,42– 44] of amorphous polymers. The free volume terminology has also been

adopted by Nada et al. [45], Voyiadjis and Samadi-Dooki [46] and Hasan et al. [40]. Hasan and Boyce [47] and Miehe et al. [48] used the evolution of free volume distribution rather than its content to describe the viscoplastic flow during the macro-yield process. The free volume theory assumes that the increase of free volume content results in the post-yield softening. Meanwhile, van Melick et al. [49], Wang and Shen [50], Broutman and Patil [51], Xie et al. [52] have experimentally found that the free volume content of polymeric materials may decrease during the macro-yield process. Considering the microstructures evolution, Bouvard et al. [36] developed a noteworthy constitutive model with the internal-state variables and accurately captured the high strain rate responses of the polycarbonate (PC) under the SHPB compression. Fleischhauer et al. [53] and Bowden and Raha [54] employed the dislocation theory from the metallic materials to study the yield and plastic flow of amorphous glassy polymers. Nada et al. [45] proposed a molecular chain plasticity model which is somewhat similar to the crystal plasticity theory of metals. Voyiadjis and Samadi-Dooki [46] suggested that the gradual reduction of the nucleation energy of shear transition zone, which is initially proposed for the metallic glasses [55], was the main cause for the formation of post-yield softening behavior of amorphous glassy polymers. While these models can phenomenologically reproduce the macro-yield peak, the borrowed concepts from metals is not really suitable for mimicking the microstructures evolution of amorphous glassy polymers. To investigate the influence of thermo-mechanical history on the macro-yield behavior of amorphous glassy polymers, Hasan et al. [40] introduced the density of free volume sites as a scalar internal variable and phenomenologically capture the influence of thermal history on the yield peak point and post-yield softening. Based on the aggregation-fragmentation theory, Drozdov [56] reasonably explained the effect of physical aging on the mechanical performances of amorphous glassy polymers. Voyiadjis and Samadi-Dooki [46] discussed the effect of thermal history on the macro-yield behavior of amorphous glassy polymers from the viewpoint of nucleation energy of shear transition zone. Introducing a scalar state variable to represent the thermal history dependent characteristic of macro-yield behavior, Klompen et al. [57] discussed the effect of aging time on the yield peak point of the PC. Xiao and Nguyen [4], Semkiv and Hutter [3] and Das et al. [58] introduced the effective temperature [59] as a thermodynamic state variable to describe the configurational/structural relaxation of molecular chain and explained the effects of deformation and temperature history on the macro-yield behavior of amorphous polymers. However, the physical background of the effective temperature theory is far from settled as discussed by Leuzzi [60]. To this end, the influence of thermo-mechanical history on the molecular microstructures’ status and evolution, which is necessary to understand the macro-yield behavior of amorphous glassy polymers, needs to be explored further.

H. Jiang and C. Jiang

In this work, to understand the underlying micro-mechanism of the macro-yield behavior of amorphous glassy polymers, two categories of molecular entanglement microstructures, i.e. the topological entanglement and the sub-entanglement, are introduced. These two types of molecular entanglements collaboratively determine the intrinsic plastic flow resistance of amorphous polymers during yielding. By introducing the sub-entanglement microstructure and its evolution as the internal physical reasons, this paper provides a new alternative perspective for the micro-mechanism of macro-yielding. Then, following the distinguished constitutive framework of the yield behavior of amorphous glassy polymers in literature [15,36,42,61], a constitutive model with an entanglement-based scalar internal-state variable is proposed. The capability of the proposed model to predict the macro-yield behavior of amorphous glassy polymers is validated. The effects of thermomechanical history and deformation state are also discussed. 2. Microstructures and their effects on the macro-yield behavior of amorphous glassy polymers Since the microstructures’ characteristics and evolution play a crucial role in the macroscopic behavior of polymeric materials, the understanding of polymers’ microscopic deformation mechanism(s) is fundamental for establishing a physical-based constitutive model. In this section, the molecular microstructures of amorphous glassy polymers are clarified according to the corresponding experimental results in literature. Then the correlation between these microstructures, as well as their evolution, and the macro-yield behavior are discussed. 2.1. Molecular entanglement of amorphous glassy polymers It is generally acknowledged that the molecular chains of amorphous glassy polymers are randomly coiled in the three-dimensional space and cannot form obvious long-range ordered microstructure such as the crystals in semi-crystalline polymers [15]. The topological entanglement (TE), sometimes called as permanent entanglement [62], is schematically shown in the circular boxes in Fig. 1(b) which roughly contains several hundred (∼102 ) monomer units [63]. Below the glass transition temperature (Tg ), the molecular chain mobility is rather low as in the ‘frozen’ state. The spatial structure of TE is quite stable and cannot be easily disentangled. Using MD simulation method, Mahajan et al. [64] demonstrated that even above Tg , the entanglement density almost keeps unchanged. Therefore, TE in amorphous glassy polymers, which can be comparable with the cross-link in rubbery materials, can serve as anchor points to construct the molecular structure network. TE density, depending on the molecular weight (Mw) of molecular chain [65], significantly affects the macroscopic mechanical properties of polymers. Generally speaking, above a critical value of Mw which varies by polymer chain type, the polymers with larger Mw should own a higher TE density and then superior mechanical performances, i.e., higher yield peak point and better deformation resistance [66]. However, Xi et al. [67] experimentally found that the PC with a larger Mw showed a lower yield strength than that with a smaller one. This finding cannot be explained using the concept of TE alone. Meanwhile, it was also found that the polymers with a longer aging time or a higher aging temperature exhibited a higher and more pronounced yield peak [68,69], Since the molecular chain cannot form additional topological entanglement during the physical aging, the TE density, as well as the deformation resistance related to TE, should keep almost unchanged. The effect of physical aging on the macro-yield behavior cannot be explained by the perspective of TE. Certain microstructure should exist and affect the macro-yield behavior of amorphous polymers. Actually, the concept that polymers may consist of two groups of microstructure, i.e., one remains intact during deformation while another one evolves with the inelastic deformation, has been adopted for rubberlike materials [70,71], amorphous polymers at rubbery state [72], semicrystalline polymers [73] and double-network hydrogels [74,75], and

International Journal of Mechanical Sciences 161–162 (2019) 105064

so on. Schoon and Kretschmer [76] observed the short-range ordered nodular structures, whose size is about 3∼10nm, in amorphous glassy polymers. More experimental evidence of the existence of short-range ordered microstructures in amorphous glassy polymers has been found by Yeh [77], Wendorff [78], Neki and Geil [79] and Červinka et al. [80]. Recently, Likhtman and Ponmurugan [81] adopted the molecular dynamics (MD) method to analyze the local interaction of the adjacent molecular chains. They proposed a concept of ‘microscopic entanglement’ for the space configurations of neighboring molecular chains. When polymers were cooled from the molten state into the glassy state, the ‘microscopic entanglement’ survive and form a new type of molecular microstructure totally different from topological entanglement. MD simulation performed by Mahajan et al. [64] and Hossain et al. [82] also found that the macro-yield behavior of amorphous polymers is correlated with the intermolecular interaction related to Van der Waals force. The temporary mechanical entanglement, as a secondary bond, are found to affect material properties [83]. The secondary microstructures and their evolution in amorphous glassy polymers associated with macroscopic deformation procedure have also been explored. Raha and Bowden [84] studied the evolution of molecular network structure of amorphous poly (methyl methacrylate (PMMA) under the plane strain compression with photoelasticity method. The cohesive point, which was taken as a kind of weak bonding interaction between the segment units of neighboring chains, was proposed. The number of cohesive points was found to decrease gradually with the increase of plastic deformation. The so called cohensional entanglement [63,85], has been proposed by Qian [63] as a type of joint owing to the local parallel alignment of a few segments to interpret the effect of physical aging on the macro-yield behavior of amorphous glassy polymers, and by Lv et al. [85] as the mechanism for a higher glass transition temperature due to the increased hydrogen bonding in PMMA/SMA blend films. Thus, it can be safely assumed that, in addition to the classically acknowledged topological entanglement, the secondary short-range microstructure does exist in the amorphous glassy polymers. Those secondary microstructures (schematically shown in the square boxes in Fig. 1(b)), owing to the relatively weak local interaction (such as Van der Waals force and hydrogen bond etc.) between neighboring molecular segments, are named as the sub-entanglement (SE) to hint its relatively weaker physical nature comparing to the topological one. The relatively weak sub-entanglement which can dissociate or regenerate by loading and thermal treatment, rather than topological entanglement, is assumed to be the main contribution factor of postyield strain softening. It plays a vital role in constructing the complex molecular structure of amorphous polymers, and can considerably affect polymers’ internal deformation resistance and macroscopic mechanical properties as well. 2.2. Effects of molecular entanglement microstructures on macro-yield behavior In this subsection, we discuss the effect of molecular entanglement on the macro-yield behavior of amorphous polymers under the moderate deformation with temperature range below Tg . As discussed above, the topological entanglement of amorphous polymers will remain intact under arbitrary thermo-mechanical history. The corresponding deformation resistance related to TE does not change either. The sub-entanglement will vary with the deformation or thermal treatment process, therefore resulting in the change of the intrinsic deformation resistance and the macroscopic deformation responses of polymers. Fig. 2(a) shows the effect of thermal treatment on the macro-yield behavior of amorphous Polystyrene (PS). It can be found that the quenched specimen does show a lower macro-yield peak than the annealed one. This is because during the quenching process, the molten polymeric material is rapidly cooled to the glass state, resulting in a rapid reduction

H. Jiang and C. Jiang

International Journal of Mechanical Sciences 161–162 (2019) 105064

Fig. 2. Effect of thermal treatment process on macro-yield behavior and DSC endothermic peak of PS material (Experimental data form Hasan and Boyce [86]): (a) the stress-strain curves; (b) the thermogram curves.

of molecular chain mobility. Due to this quick cooling down, little SE has opportunity to form. In contrast, the temperature drop of annealing is much slower than that of quenching. The molecular chains have enough time to mobilize and generate new SE, which can accumulate gradually. Therefore, SE density in the annealed specimen is higher than that in the quenched one, leading that the annealed specimen shows a more pronounced yield peak. Fig. 3(a) shows the effect of mechanical deformation, as well as the succeeding thermal treatment after pre-deformation, on the macroyield behavior of amorphous glassy polymers. For an annealed specimen (Case 1 in Fig. 3(a)) been loaded beyond its macro-yield peak, if it is loaded again (Case 2 in Fig. 3(a)), the macro-yield peak vanishes, while the height of post-yield softening plateau remains. This deformation process, also called mechanical rejuvenation [1,49,64], seems to erase the thermal-mechanical history. That is to say, the pre-strain dissociates the SE formed by previous annealing. The disappearance of SE related secondary deformation resistance results in the vanishment of yield peak in the subsequent loading process. If the pre-deformed specimen, which owns little SE or yield peak, is reheated for a certain time below Tg , the disappeared macro-yield revives (Case 3 in Fig. 3(a)). A longer thermal treatment time brings back a higher macro-yield peak (Case 4 in Fig. 3(a)). It is due to the fact that SE has been regenerated and accumulated during the thermal treatment after pre-deformation. The longer the heat treatment is, the larger the SE density is, a more pronounced macro-yield peak will exist. There is no doubt that the variation of SE density of amorphous polymers is an important intrinsic reason for the formation or disappearance of the thermo-mechanical history-dependent macro-yield peak. Figs. 2(a) and 3(a) also show that as long as the specimens are loaded to the softening plateau, their post-yield flow stresses tend to be similar at the same strain regardless of the previous thermo-mechanical history.

It implied that the softening plateau is generally independent on the thermo-mechanical history as well as SE. In the view of microstructures, the benchmark mechanical properties, including the pre-yield linearity and the post-yield softening plateau, are dominated by the topological entanglement which is independent on the thermo-mechanical history. In the following section, keeping those benchmark mechanical properties unchanged, the SE density (N) is taken as the internal-state variable controlling the macro-yield peak to establish the constitutive model of amorphous glassy polymers. 2.3. Characterization of SE and its evolution during the macro-yield procedure The change of polymers’ microstructure state is always accompanied by the storage or release of internal energy. DSC (Differential Scanning Calorimeter) measurement, as a simple and effective technique, has been utilized to characterize the material’s internal thermodynamic status [86]. For various types of amorphous glassy polymers, such as PS [6], Polyethylene terephthalate(PET) [50], PC and PMMA [86], the effect of thermo-mechanical history on the macro-yield peak and the additional DSC endothermic peak near Tg has been experimentally investigated. Experimental results demonstrated that the influence of thermomechanical history on the DSC endothermic peak was consistent with that on the yield peak. As shown in Figs. 2 and 3, PS was chosen as an example to analyze the correlation of DSC endothermic peak and macro-yield peak. The macro-yield peak of the quenched specimen is rather small, if existed, as in Fig. 2(a). At the same time, the phase transition stage of the quenched specimen shows no/little additional endothermic peak as shown in Fig. 2(b). However, the annealed specimen does show a pronounced DSC endothermic peak and macro-yield peak. As shown in Fig. 3. Effects of mechanical deformation and post-deformation thermal treatment on macro-yield behavior and DSC endothermic peak of 𝛼-PS material (Experimental data from Song et al. [6]): (a) the stress-strain curves; (b) the thermogram curves.

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International Journal of Mechanical Sciences 161–162 (2019) 105064

Fig. 4. Effect of tensile deformation on the additional DSC endothermic peak during the macro-yield process on 𝛼-PS (data from Song et al. [6]): (a) the tensile stress-strain curve (b) the DSC thermogram curves at different tensile strain levels (c) the evolution of endothermic peak values as a function of applied strain level (the dash line showing the tendency of the value of endothermic peaks).

Fig. 3, once the specimen is loaded beyond the post-yield softening region, the additional endothermic peak and the macro-yield peak vanish simultaneously. Then, the DSC endothermic peak and the macro-yield peak restore simultaneously as long as the pre-deformed specimen is aged. And both of them will arise simultaneously with the increase of aging time. Adam et al. [69], Haward et al. [87] and Hasan and Boyce [86] found that the energy associated with the additional DSC endothermic peak roughly equals to the applied external work needed to eliminate the macro-yield peak. Therefore, for the macro-yield process, the evolution of SE density can be qualitatively characterized as the evolution of DSC endothermic peak with the external pre-deformation. The PS specimens were stretched to different strain levels and then unloaded to zero stress, as shown in Fig. 4(a) and(b). It can be found that the additional endothermic peak of PS does not change until the tensile strain becomes larger than 3%. This suggests that at the elastic region, little sub-entanglement is dissociated and SE density keeps unchanged. Raha and Bowden [84] also experimentally proved that the change of molecular network of amorphous polymers was associated with the inelastic deformation. As the tensile strain continues to increase, the DSC endothermic peak gradually decreases. Finally, the endothermic peak diminishes when the tensile strain reaches beyond 7% which is close to the plateau after the post-yield softening. The integration of additional DSC endothermic peaks at various levels of applied strain, taken as an indirect characterization of the magnitude of SE density, are plotted in Fig. 4(c). It is clear that the SE density decreases with the inelastic deformation during the macro-yield process. An explicit exponential-form equation is adopted to phenomenologically describe the evolution of SE density during the macro-yield process ( ) 𝑁 = 𝑁0 exp −𝑘𝜀p (1) Here, N0 is the initial SE density reflecting the effect of thermomechanical history on the microstructural status of polymers. k is the dissociation rate of SE during the macro-yield process, which is an intrinsic material property related to the chain tacticity etc., indicating

Fig. 5. One-dimensional rheological model of the proposed constitutive model.

SE’s capability to resist the external deformation. 𝜀p is the equivalent plastic strain. Hasan et al. [40] found that the index representing the microscopic status of glassy polymers by PLAS was consistent under different loading rates. Therefore, in this paper, the dissociation of SE is assumed to be independent on strain rate.

3. Constitutive model of the macro-yield behavior of amorphous glassy polymers with an entanglement-based internal variable As discussed in Section 2, the topological entanglement and the sub-entanglement collaboratively determine the macro-yield behavior of amorphous glassy polymers. The magnitude of the macro-yield peak is correlated with SE and its evolution. In this section, under the finite deformation thermodynamics framework, considering SE as an internal-state variable, a three-dimensional elasto-viscoplastic constitutive model is proposed. Fig. 5 shows its one-dimensional rheological model which is a combination of a linear spring and a viscoplastic dashpot in series.

H. Jiang and C. Jiang

International Journal of Mechanical Sciences 161–162 (2019) 105064

3.1. Finite deformation constitutive framework

According to Eq. (10), the plastic part of velocity gradient Lp can be rewritten as,

Considering an arbitrary material point X in a homogeneous deformable body ℜR occupied in a fixed reference configuration, and x = y(X, t) is the corresponding arbitrary material point in the current configuration via a smooth one-to-one mapping of X. The deformation gradient F, velocity 𝜐 and velocity gradient L, respectively, are given as

𝐋 p =𝐃 p

𝜕𝐱 𝐅 = ∇𝑦 = , 𝜐 = 𝑦̇ , 𝐋 = grad𝜐 = 𝐅̇ 𝐅−1 𝜕𝐗

(2)

Here, the symbol ( • ) represents the material time derivative of scalar or tensor fields. The deformation gradient F is multiplicative decomposed as [88]1

(11)

Dp

with is symmetric and deviatoric. The elastic strain Ee is defined as the logarithmic Hencky strain tensor 𝐄e = ln 𝐔e

(12)

Based on Anand and Gurtin [16], for an isotropic elasto-viscoplastic solid material, the Cauchy stress 𝝈 in the current configuration is defined as T 1 𝝈 = 𝐑e 𝐌e 𝐑e (13) 𝐽

where, Ue and Ve are the right and left elastic symmetric stretch tensors respectively. Up and Vp are the right and left plastic symmetric stretch tensors respectively. Re and Rp are the elastic and plastic rotation tensors respectively. Substituting Eq. (3) into Eq. (2) 3 , the velocity gradient L is written as

where, Me is the Mandel stress in the intermediate stress-free configuration, which is work-conjugated with Ee . Anand [90,91] generalized the classical isotropic elastic constitutive model into the finite deformation framework based on the logarithmic strain measure, which has been utilized to describe the elastic response of amorphous polymers during finite deformation process [36,41,42]. Following this idea, in this work, the Mandel stress is defined as Me = 2G(Ee )′ + K(trEe )1. Here, G and K are the elastic shear and elastic bulk modulus, respectively. trEe is the trace of tensor Ee . (𝐄e )′ = 𝐄e − 13 (tr 𝐄e )𝟏 is the deviatoric part of tensor Ee . 1 is the second-order unit tensor. To describe the underlying plastic flow during the macro-yield process, the evolution equation of Fp obtained from Eq. (5) can be written as

−1 𝐋 = 𝐅̇ 𝐅−1 = 𝐋e + 𝐅e 𝐋p 𝐅𝑒

𝐅̇ p = 𝐋p 𝐅p

𝐅 = 𝐅e 𝐅p

(3)

where, Fe and Fp are the elastic and plastic deformation gradients, respectively. The polar decompositions of Fe and Fp are respectively 𝐅e = 𝐑e 𝐔e = 𝐕e 𝐑e and 𝐅p = 𝐑p 𝐔p = 𝐕p 𝐑p

(4)

(5)

−1 −1 with, 𝐋e =𝐅̇ e 𝐅𝑒 and 𝐋p =𝐅̇ p 𝐅p are the elastic and plastic parts of L respectively. Further, the elastic stretching tensor De and elastic spin tensor We are respectively defined as ( ) ( ) T T 1 e 1 e 𝐋 + 𝐋e , 𝐖e = 𝐋 − 𝐋e (6) 𝐃e = 2 2

Similarly, the plastic stretching tensor Dp and plastic spin tensor Wp are respectively defined as ( ) ( ) T T 1 p 1 p 𝐃p = 𝐋 + 𝐋p , 𝐖p = 𝐋 − 𝐋p (7) 2 2 So that, Le =De + We and Lp =Dp + Wp . Referring to the thermodynamical framework proposed by Anand and Gurtin [16], Gurtin and Anand [38], for an initially isotropic elastoviscoplastic amorphous glassy polymer, two basic assumptions related to plastic flow are given. Firstly, we assume that the plastic flow is incompressible, det 𝐅p = 1 and 𝑡𝑟 𝐋p = 0

where, 𝛾̇ p is the shear strain rate. (𝐌e )′ = 𝐌e − 13 (tr 𝐌e )𝟏 is the deviatoric part of Mandel stress tensor Me . 𝜏̄ is the effective shear stress and defined as √ 1 |( )′ | 1 𝜏̄ = √ | 𝐌e | = √ (16) (𝐌 e )′ ∶ (𝐌 e )′ | | 2 2 and a simple power-law model for shear strain rate 𝛾̇ p is utilized 𝛾̇ p = 𝛾̇ 0

( )1 𝜏̄ 𝑚 𝑠

(17)

where, 𝛾̇ 0 is the reference shear strain rate. s is the plastic flow resistance correlated to the microstructures’ state and their evolution during the macro-yield process of amorphous glassy polymers. m is the strain rate sensitive parameter. 3.2. Evolution equation of the plastic flow resistance during macro-yield behavior

(9)

Secondly, according the well-accepted assumption as proved in [15,16,38,89], the plastic flow is taken to be irrotational, 𝐖p = 𝟎

According to Eq. (11), Eq. (14) can be rewritten as 𝐅̇ p = 𝐃p 𝐅p . Here, Dp is defined as the following form [16] ( e ′) (𝐌 ) 𝐃p = 𝛾̇ p (15) 2𝜏̄

(8)

Then, 𝐽 = det 𝐅 = det 𝐅e

(14)

(10)

1 There was an interesting private discussion between Prof. R. Hill and Prof. Zhuping Huang back to 1992. Both of them agreed that the multiplicative decomposition of F was suitable for single-crystalline materials but not necessary for polycrystalline materials. For a Maxwell element consisting of a spring and a dashpot in series, after the suddenly unloading, the spring immediately return to its original stress-free state, while the dashpot would keep its current state which can be taken as the intermediate configuration and the multiplicative decomposition of F is suitable. However, for a system with two (or more) Maxwell elements, the intermediate configuration no longer explicitly existed.

The macro-yield behavior of amorphous glassy polymers also depends on the current deformation state. Arruda and Boyce [7] investigated the effect of uniaxial compression and plane strain compression on the deformation behavior of amorphous glassy polymers. Experimental results showed that the macro-yield peak and the post-yield softening plateau of the plane strain compression are higher than those of the uniaxial compression [7]. This phenomenon is due to the additional restriction from plane strain compression as well as the polymers’ intrinsic pressure sensitivity. To describe the macro-yield behavior of amorphous glassy polymers under different deformation states with one set of material parameters, the hydrostatic pressure effect needs to be considered. Similar to Boyce et al. [15], the material’s plastic flow deformation resistance s considering the hydrostatic pressure effect can be written as, 𝑠 = 𝑠̂ + 𝛼p p

(18)

H. Jiang and C. Jiang

International Journal of Mechanical Sciences 161–162 (2019) 105064

where, 𝑠̂ is the plastic flow resistance associated with the molecular entanglements, 𝛼 p is the pressure sensitivity coefficient, p = − 31 tr (𝝈) is hydrostatic pressure. Referring to Boyce et al. [15], the evolution equation of plastic flow resistance 𝑠̂, which is related to microstructure evolution during the macro-yield process, is written as ( ) 𝑠̂ p 𝑠̂̇ = ℎ0 1 − 𝛾̇ (19) 𝑠̃ and the initial condition 𝑠̂ = 𝑠̂0 when 𝛾 p = 0

(20)

where, 𝑠̂0 is the initial resistance related to microstructure evolution. h0 is a material parameter related to the onset of pre-yield nonlinearity. 𝑠̃ is the intrinsic deformation resistance depending on the current molecular entanglement density. Fig. 6. The stress-strain curve for material parameters determination.

3.3. Evolution equation of entanglement-based intrinsic resistance during the macro-yield process Two types of molecular entanglements, i.e., topological entanglement and sub-entanglement, exist in amorphous polymers. While TE determines the benchmark mechanical behavior of amorphous glassy polymers, the dissociable SE controls the macro-yield peak. The intrinsic deformation resistance 𝑠̃ of amorphous glassy polymers can be decomposed into two individual parts, i.e., the TE related resistance 𝑠̃te and the SE related one 𝑠̃se , respectively 𝑠̃ = 𝑠̃te + 𝑠̃se

(21)

For topological entanglement, MD analysis demonstrated that the TE density kept unchanged under moderate deformation process [82,92,93]. Thus, 𝑠̃te can be assumed as a constant here. Meanwhile, 𝑠̃se evolves as a function of SE density N 𝑠̃se = 𝑠̃se (𝑁 )

(22)

The internal variable 𝑠̃se characterizes the deformation resistance related to the evolution of sub-entanglement during yielding. During the deformation process, the relative slippage of molecular chains leads to the gradual dissociation of sub-entanglement, which in due course reduces 𝑠̃se . Without loss of generality, Eq. (22) can be rewritten with the help of Eq. (1) ( ( )) ( ) ( ) 𝑠̃se = 𝑠̃se 𝑁0 exp −𝑘𝜀p = 𝑠̃0se 𝑁0 exp −𝑘𝜀p (23) where, 𝑠̃0se is material’s initial intrinsic deformation resistance depending on its original thermodynamic status and previous thermo-mechanical history. 𝜀p is the local accumulated inelastic strain and defined as 𝜀p =

∫

𝜀̇ p d𝑡

(24)

√ with 𝜀̇ p = 32 𝐃p ∶ 𝐃p is the effective plastic strain rate. It should be noted that if not considering the SE micro-structure, i.e., ignoring the evolution of 𝑠̃se in Eq. 19, the proposed model can be reduced to the one of Boyce et al. [15]. 3.4. Identification of material parameters In this section, the approach to identify the material parameters of the proposed constitutive model is summarized. The material parameters need to be determined are: { } 𝐺, 𝐾, 𝑚, 𝛼p , 𝛾̇ 0 , 𝑠̂0 , 𝑠̃te , ℎ0 , 𝑘, 𝑠̃0se where, G and K are the elastic shear modulus and bulk modulus respectively. m is the strain rate sensitive coefficient. 𝛼 p is the pressure sensitivity coefficient. 𝛾̇ 0 is the reference shear strain rate. 𝑠̂0 is the initial resistance related to microstructure evolution. 𝑠̃te is the intrinsic resistance related to topological entanglement. h0 is the parameter related to the initial slope of pre-yield nonlinearity. k is the dissociation

rate of sub-entanglement. 𝑠̃0se is the initial intrinsic resistance related to sub-entanglement. All above material parameters can be estimated from uniaxial tension/compression stress-strain curve. Based on von Mises equivalence theory, the correlation between the equivalent shear stress 𝜏 and uniaxial stress 𝜎, as well as the equivalent shear strain 𝛾 and uniaxial strain 𝜀 are, respectively √ √ 𝜏 = 𝜎∕ 3 and 𝛾 = 3𝜀 (25) With the help of Eq. (25), the material parameters {𝛼p , 𝛾̇ 0 , 𝑠̂0 , 𝑠̃te , ℎ0 , 𝑠̃0se } for the three-dimensional constitutive model can be converted from the corresponding values in the one-dimensional form √ √ 1 1 1 1D 𝛼p = 3𝛼p1D , 𝛾̇ 0 = 3𝜀̇ 0 , 𝑠̂0 = √ 𝑠̂1D , 𝑠̃te = √ 𝑠̃1D te , ℎ0 = 3 ℎ0 , 0 3 3 1 𝑠̃0se = √ 𝑠̃0se1D 3 the remaining material parameters are the same in both cases. 𝐸 𝐸 1) G and K are calculated from 𝐺 = 2(1+ and 𝐾 = 3(1−2 respectively. 𝜈) 𝜈) Here, E and 𝜈 are the elastic modulus and Poisson’s ratio, respectively, obtained from the uniaxial tension/compression test. The values of Poisson’s ratio of PC, 𝛼-PS and PMMA are defined as 0.42 and 0.34 and 0.34, respectively from literature [94].

2) m can be calculated approximately from 𝑚 ≈

ln(𝜎𝑦 ∕𝜎𝑦 ) 1

2

ln(𝜀̇ 1 ∕𝜀̇ 2 )

. Here, 𝜎𝑦1

and 𝜎𝑦2 are the yield strengths at the strain rates of 𝜀̇ 1 and 𝜀̇ 2 , respectively. Since the rate-dependent macro-yield behavior is not the main interest in this work, m is set to 0.001. 3) 𝛼 p is the pressure sensitivity coefficient which reflects the effect of mean stress on the deformation response. The experimental studies by Rabinowitz et al. [95] and Spitzig and Richmond [96] found that the yield stress 𝜎 y of polymers is linearly correlated with the mean pressure p, i.e., 𝜎 y = 𝜎 y0 + 𝛼 p p. Here, 𝜎 y0 is the yield stress at zero pressure. Thus, 𝛼 p can be estimated from the plot of 𝜎 y as a linear function of pressure p. For uniaxial tension or compression, neglecting the pressure sensitivity coefficient shows little effect on the estimation of the macro-yield behavior. For simplicity, this parameter is taken as zero in these situations. 4) 𝜀̇ 0 is defined as the prescribed loading rate of the uniaxial stressstrain curve for material parameters determination. 5) 𝑠̂1D is the initial resistance related to microstructure evolution. At the 0 beginning point of microstructure evolution, the material’s plastic 𝛼 flow resistance 𝑠 = 𝑠̂0 − 3p tr (𝝈). The microstructure evolution only correlates to plastic deformation. For the uniaxial loading case, the flow stress at the initial point of plastic deformation is 𝜎 i , as shown in Fig. 6. Let s1D = |𝜎 i |, then 𝑠̂1D = |𝜎𝑖 | + 13 𝛼p1D ⋅ sign(𝜎𝑖 ). Here, |𝜎 i | is 0 the absolute value, sign(𝜎 i ) = 𝜎 i for tension, sign(𝜎 i ) = −𝜎 i for compression.

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International Journal of Mechanical Sciences 161–162 (2019) 105064

Fig. 7. Effect of the initial intrinsic resistance related to SE 𝑠̃0se on the macro-yield behavior: (a) the stress-strain curves and (b) evolution of normalized 𝑠̃se .

6) 𝑠̃1D te is the intrinsic resistance related to TE. At the post-yield softening plateau, SE is almost completely dissociated by the inelastic strain, while TE density keeps unchanged. The plastic flow re𝛼 sistance s reaches its saturation value, i.e., 𝑠 = 𝑠̃te − 3p tr (𝝈). Thus, 1 1D 𝑠̃1D te = |𝜎plateau | + 3 𝛼p ⋅ sign(𝜎plateau ). Here, |𝜎 plateau | is the absolute value, sign(𝜎 plateau ) = 𝜎 plateau for tension, sign(𝜎 plateau ) = −𝜎 plateau for compression. 7) h0 defines the initial slope of pre-yield nonlinearity. To identify h0 , only the pre-yield portion of stress-strain curve is needed. In this stage, the plastic flow resistance s monotonically increases from its initial value 𝑠̂0 to its maximum value 𝑠̂m . Eq. (19) can be written in the following form

d𝑠̂

(

1−

𝑠̂ 𝑠̃

) = ℎ0 d𝛾 p

(26)

To integrate Eq. (26) on both sides, we obtain ( ( ) ( )) 𝑠̂ 𝑠̂ −𝑠̃ ln 1 − m − ln 1 − 0 = ℎ0 𝛾 p 𝑠̃ 𝑠̃

(27)

Thus, under the uniaxial condition, h0 can be expressed as ( ) ( ) 𝑠1D 𝑠1D 0 ln 1 − peak − ln 1 − 𝑠̃ 𝑠̃ ℎ0 = −𝑠̃ 𝜀peak − 𝜀𝑖

(28)

8) From Eq. (23), the dissociation rate of SE, k, can be expressed as ( ) 𝑠̃se 1 𝑘 = − p ln (29) 𝜀 𝑠̃0se

as ln(

𝑠̃se ) 𝑠̃0se

=(

𝑠̃se 𝑠̃0se

− 1) − 12 (

𝑠̃se 𝑠̃0se

− 1)2 + 13 (

𝑠̃se ) 𝑠̃0se

𝑠̃se 𝑠̃0se

in Eq. (29) can be expressed

− 1)3 + 𝑜((

𝑠̃se 𝑠̃0se

3

− 1) ). And only

the first three terms are considered in this work. During the macro-yield process, SE almost dissociates completely at the post-yield softening 𝑠̃ p 11 plateau. This implies that when 𝜀p → 𝜀plateau , 𝑠̃se → 0. Then ln( se 0 ) ≈ − 6 . Thereby, k can be estimated from Eq. (30), 𝑘≈

𝑠̃se

11 p 6𝜀plateau

𝜀p

𝐴peak =

∫ 0

𝑠̃se d𝜀 = − p

𝑠̃0se

𝜀p

𝑘 ∫ 0

𝑠̃0 ( ( ( ) ( ) ) ) exp −𝑘𝜀p d −𝑘𝜀p = − se exp −𝑘𝜀p − 1 𝑘 (31)

For the uniaxial loading condition, Eq. (31) can be rewritten as 𝑠̃0se1D =

𝑘𝐴peak ( ) p 1 − exp −𝑘𝜀plateau

(32)

Here, Apeak is the area integration of the stress-strain curve associated with macro-yield peak. For the purpose of simplification, Apeak may be calculated from the triangle area composing by the initial plastic flow point 𝜎 i , the macro-yield peak point 𝜎 peak and the post-yield softening point 𝜎 plateau , as shown in Fig. 6. 4. Effect of the SE-related material parameters on the macro-yield behavior

Here, 𝑠̃=𝑠max . 𝜀peak and 𝜀i are the strain levels of macro-yield peak point and initial plastic flow point, respectively. 𝑠1D and 𝑠1D 𝑖 are the peak stress levels of macro-yield peak point and initial plastic flow point, respectively.

Based on Taylor series expansion, ln(

magnitude of macro-yield peak. The required energy to dissociate the existed SE can be assumed to be equal to the external work associated with macro-yield peak. Thus,

(30)

p

Here, 𝜀plateau is the plastic strain of post-yield softening plateau. 9) As the initial intrinsic deformation resistance depending on material’s original thermodynamic status, i.e. previous thermomechanical history, 𝑠̃0se is closely correlated to the corresponding

This section will discuss the effects of the two material parameters related to SE, i.e., initial intrinsic resistance related to SE (𝑠̃0se ) and the dissociation rate of SE (k), on the macro-yield behavior of amorphous glassy polymers. They are the respectively. 𝑠̃0se represents the contribution of SE to the intrinsic resistance of amorphous polymers. Its value depends on the magnitude of SE density and the bonding strength of SE. Fig. 7(a) shows the simulated macroyield stress-strain curves with different 𝑠̃0se . A larger 𝑠̃0se corresponds to a more pronounced macro-yield peak. For the same loading condition, the larger the 𝑠̃0se , the larger the 𝑠̃se is (as shown in Fig. 7(b)). Thus, the larger the corresponding plastic flow resistance is, the higher the deformation response should be. Although the value of 𝑠̃0se significantly affects the slope of post-yield softening stage, it has little effect on the strain level of post-yield softening plateau. Actually, it can be found from Eq. (1) that 𝑠̃se will eventually converge to zero no matter what 𝑠̃0se is. As long as 𝑠̃se approaches zero, the stress-strain response reaches the post-yield softening plateau. It is also clear that 𝑠̃0se has no influence on the pre-yield linearity and the height of post-yield softening plateau which are the intrinsic benchmark behavior related to TE. It should be noted that the initial intrinsic resistance related to SE, 𝑠̃0se (𝑁0 ), can be altered by the thermo-mechanical procedure. Reflecting the decreasing rate of materials’ intrinsic resistance against the inelastic deformation, the dissociation rate of SE represents the inherent characteristic of sub-entanglement against external stimuli. Fig. 8(a) shows its effects on the macro-yield behavior. A lower macroyield peak is observed for a larger k. This is due to the fact that the

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International Journal of Mechanical Sciences 161–162 (2019) 105064

Fig. 8. Effect of the dissociation rate of SE k on the macro-yield behavior: (a) the stress-strain curves and (b) evolution of normalized 𝑠̃se .

very early dissociation of molecular network of amorphous polymers occurs before the macro-yield peak point [6,84,87]. For a given amorphous glassy polymer under certain thermodynamic status, a larger k means an easier and faster dissociation of sub-entanglement, suggesting a lower capability against the external deformation. Under the same loading case, a quick reduction of the magnitude of SE density means a small current 𝑠̃se , thus a lower macro-yield peak is given as shown in Fig. (8). Different from 𝑠̃0se , k affects not only the slope of post-yield softening, but also the strain level of post-yield softening plateau, as shown in Fig. 8(a). The larger the k, the more rapid the reduction of 𝑠̃se is. Thus a sharper slope of post-yield softening and a smaller strain level of postyield softening plateau can be observed. As one material parameter only involving SE, k shows no influence on the pre-yield linearity, as well as the height of post-yield softening plateau. Furthermore, it can be seen from Figs. 7(b) and 8(b) that 𝑠̃se keeps unchanged in the elastic deformation region and is only correlated to the plastic portion of the applied deformation. Parameters 𝑠̃0se and k have a good potential to depict the experimental observed macro-yield behavior of amorphous glassy polymers. 5. Results and discussion In this section, the capability of the proposed constitutive model to simulate and predict the effects of thermal history, mechanical history and loading mode on the macro-yield behavior of amorphous glassy polymers is verified. 5.1. Effect of thermal history on the macro-yield behavior Klompen et al. [57] investigated the uniaxial compressive deformation behavior of PC under two thermal treatment conditions, i.e., annealing (Case A) and quenching (Case B), respectively. The corresponding experimental stress-strain curves were shown in Fig. 9. The material parameters of PC under two thermal treatment conditions were identified using the approach in Section 3.4. Since only the uniaxial loading case is involved, the effect of hydrostatic pressure is neglected here, i.e., 𝛼 p = 0. The material parameters of PCs are listed in Table 1. As shown in Fig. 9, the simulation results of macro-yield behavior agrees well with the experimental finding for both thermal treatment conditions. The thermal treatment processes mainly affects the magnitude of macro-yield peak and shows no significant influence on other deformation responses. This phenomenon is also being reflected by the material parameters, as given in Table 1. Except the parameter 𝑠̃0se , other material parameters are the same for the two cases. The annealed PC specimen (Case A) has a higher initial SE density than the quenched one (Case B), which means a larger initial SE related intrinsic deformation resistance than the quenched one. It is confirmed that the intro-

Fig. 9. Simulated and experimental results of the effect of thermal history on the macro-yield behavior of PC. (Case A: the annealed state; Case B: the quenched state). Table 1 Material parameters (with experimental data from Klompen et al. [57]). Basic physical parameters Elastic shear modulus G 0.59 GPa Elastic bulk modulus K 3.5 GPa 1.73 × 10− 4 s−1 Reference shear strain rate 𝛾̇ 0 Strain rate sensitive parameter m 0.001 The yield and post-yielding softening parameters 18.82 MPa Initial resistance related to microstructures’ evolution 𝑠̂0 Intrinsic deformation resistance related to 29.73 MPa TE 𝑠̃te Case A 17.32 MPa Initial intrinsic deformation resistance Case B 13.27 MPa related to SE 𝑠̃0se Dissociation rate of sub-entanglement k 16 h0 1.43 GPa

duced secondary microstructure, SE, and its evolution can successfully interpret the underlying mechanism of the macro-yield peak, as well as the effect of thermal history on the thermodynamic state/microstructure state of amorphous glassy polymer. Glassy polymers, which is usually in a thermodynamic nonequilibrium state, will slowly change its mechanical properties by physics aging below Tg . Both high temperature and long aging time will give the molecular chains a larger probability to form more SE. Experimental observation found that a higher aging temperature and/or a longer aging time does produce a more pronounced macro-yield peak of amorphous polymers [68]. This pronounced yield peak by aging means an increasing of SE density, as well as the initial intrinsic resistance.

H. Jiang and C. Jiang

Fig. 10. Simulated and experimental results of the effect of mechanical history on the macro-yield peak of 𝛼-PS. (Case C: the specimen aged at 85 °C for 30 h without pre-stretching; Case D: the specimen aged at 85 °C for 30 h followed by a pre-stretching beyond macro-yield peak). Table 2 Material parameters (Experimental data of 𝛼-PS from Song et al. [6]). Basic physical parameters Elastic shear modulus G 0.18 GPa Elastic bulk modulus K 0.51 GPa 7.22 × 10− 4 s−1 Reference shear strain rate 𝛾̇ 0 Strain rate sensitive parameter m 0.001 The yield and post-yielding softening parameters 4.33 MPa Initial resistance related to microstructures’ evolution 𝑠̂0 Intrinsic deformation resistance related to 6.66 MPa TE 𝑠̃te Case C 7.91 MPa Initial intrinsic deformation resistance Case D 1.04 MPa related to SE 𝑠̃0se Dissociation rate of sub-entanglement k 68 h0 1.2 GPa

5.2. Effect of mechanical history on the macro-yield behavior Song et al. [6] experimentally studied the tensile behavior of 𝛼-PS material with different pre-deformations. The 𝛼-PS specimens are aged at 85 °C for 30 h without pre-stretching (Case C) and aged at 85 °C for 30 h followed by a pre-stretching beyond macro-yield peak (Case D),. Fig. 10 shows the obtained stress-strain curves of 𝛼-PS. The material parameters of 𝛼-PS material with two different pre-deformations were obtained by the approach given in Section 3.4. Since only the uniaxial deformation is discussed here, the effect of hydrostatic pressure is not considered, i.e., 𝛼 p = 0. The material parameters of 𝛼-PSs are listed in Table 2. Except the initial intrinsic deformation resistance 𝑠̃0se , the rest of material parameters are the same for the two cases. As shown in Fig. 10, for the two pre-deformed cases, the stress-strain curves obtained by the proposed model are in a good agreement with the experimental ones. Similarly, the pre-deformation only affects the magnitude of the macro-yield peak. The pre-deformation dissociates the originally existed sub-entanglement due to aging process The 𝑠̃0se without pre-stretching (Case C) is larger than that with a pre-stretching (Case D), resulting in a lower initial intrinsic deformation resistance. This furtherly confirms that the sub-entanglement microstructure and its evolution is the underlying source of macro-yield peak of amorphous glassy polymers. The proposed constitutive model with SE as an internal-state variable can well describe the influence of mechanical history on the macro-yield behavior of amorphous glassy polymer. To investigate the effect of pre-deformation on the macro-yield behavior of amorphous glassy polymers, we numerically simulate the macro-yield behavior of 𝛼-PS with different level of pre-strains via the proposed model. As illustrated in Fig. 11(a), the macro-yield peak of 2% pre-stretched sample is almost same as that of 0% pre-strain, which

International Journal of Mechanical Sciences 161–162 (2019) 105064

Fig. 11. Simulated stress-strain curves of the effect of pre-strain on the macroyield behavior of 𝛼-PS.

suggests that little rejuvenation occurs. A rapid reduction of macro-yield peak occurs when the level of pre-strain is near to yield peak (∼5%). As the pre-strain keeps increasing (>= 7%), the macro-yield peak approximately vanishes. The effect of pre-strain on macro-yield peak saturates since most sub-entanglement has been dissociated. The correlation between the area of macro-yield peak and SE density is similar to the evolution procedure of the additional DSC endothermic peak as shown in Fig. 4.

5.3. Effect of loading mode on the macro-yield behavior To verify the reasonability and capability of the proposed constitutive model, the macro-yield deformation behavior of amorphous glassy polymers under two different loading modes, i.e., the uniaxial compression and the plane strain compression, are discussed in this subsection. The loading mode could have significant effect on the mechanical performance of amorphous glassy polymers [7,97]. Arruda and Boyce [7] performed the uniaxial compression and plane strain compression to investigate the finite deformation behavior of PMMA and PC, respectively. For both loading modes, the strain rates for PMMA and PC are 0.001 s−1 and 0.01 s−1 , respectively. According to the experimental stress-strain curves shown in Fig. 12, it can be found that the macroyield responses for the plane strain compression are much higher than that for uniaxial compression. The uniaxial compression stress-strain curves of PMMA and PC were chosen as the benchmark to determine the corresponding material parameters for macro-yield behavior, respectively. The material parameters are identified by the approach in Section 3.4 and listed in Table 3. The pressure sensitivity coefficients 𝛼 p are 0.353 for PMMA [95] and 0.08 for PC [96], respectively. Fig. 12(a) and (b) show the comparison of the simulation results by the proposed model and experimental ones under two different loading modes for PMMA and PC, respectively. It can be found that the proposed model not only reasonably mimics the uniaxial compression macro-yield behavior, but also gives a quite good prediction on the plane strain compression macro-yield behavior. In this paper, a secondary microstructure with a clear physical background, i.e., sub-entanglement of amorphous glassy polymers, is introduced. The SE microstructure, formed by the local interaction of segment units of adjacent molecular chains, and its evolution are the underlying deformation mechanism and resistance of macro-yield peak. The initial SE density correlated to the current thermodynamic status can be utilized to represent the influence of thermo-mechanical history on the molecular network of amorphous glassy polymers. The proposed finite elasto-viscoplastic constitutive model using SE as the internal-state variable not only well reproduces the macro-yield peak, but also shows a good capability to describe the effects of thermo-mechanical history and

H. Jiang and C. Jiang

International Journal of Mechanical Sciences 161–162 (2019) 105064

Fig. 12. Effect of loading mode on the macro-yield behavior of amorphous glassy polymers: (a) PMMA and (b) PC. Table 3 Material parameters (Experimental data of PMMA and PC from Arruda and Boyce [7]) PMMA Basic physical parameters Elastic shear modulus G 0.605 GPa Elastic bulk modulus K 1.58 GPa 1.73 × 10− 3 s−1 Reference shear strain rate 𝛾̇ 0 Strain rate sensitive parameter m 0.001 The yield and post-yielding softening parameters 34.64 MPa Initial resistance related to microstructures’ evolution 𝑠̂0 Intrinsic deformation resistance related to 44.28 MPa TE 𝑠̃te Initial intrinsic deformation resistance 10.68 MPa related to SE 𝑠̃0se Dissociation rate of sub-entanglement k 7.4 Pressure sensitive parameter 𝛼 p 0.204 2.13 GPa h0

loading mode on the macro-yield behavior of amorphous glassy polymers. It should be noted that while current work mainly focuses on the discussion of the novel alternative of the micro-mechanism of macroyielding, the viscoelastic properties such as creep and relaxation of amorphous glassy polymers have not been considered. Undoubtfully, the rate/temperature dependence of yield behavior of glassy polymers is very important and its underlying microscopic mechanisms should be furtherly explored. The consideration of viscoelastic characteristic can make the proposed model a more powerful tool to describe the deformation behavior of glassy polymers. 6. Summary In this paper, a new perspective on the macro-yield behavior of amorphous glassy polymers is introduced considering the evolution of secondary sub-entanglement. Then, an elasto-viscoplastic finite deformation constitutive model with an entanglement-based internal-state variable is proposed. The proposed model can well describe the effects of thermo-mechanical history on the macro-yield behavior of amorphous glassy polymers. 1) While the classical topological entanglement is responsible for the benchmark mechanical behavior of amorphous glassy polymers, a secondary short-range microstructure, i.e., the sub-entanglement formed by the local interactions between neighboring molecular segments, is the intrinsic cause of macro-yield peak of amorphous glassy polymers. 2) An evolution equation of SE density to describe the intrinsic deformation resistance associated with the macro-yield process is pro-

PC 0.535 GPa 1.39 GPa 1.73 × 10− 2 s−1 0.001 27.14 MPa 32.22 MPa 12.07 MPa 19.0 0.08 2.07 GPa

posed. It is verified that SE and its evolution can reasonably depict the underlying mechanism(s) of macro-yield peak of amorphous glassy polymers. 3) The proposed constitutive model considering SE as an internal-state variable can not only depict the macro-yield behavior, but also well describe the effects of thermo-mechanical history and deformation state on amorphous glassy polymers. Acknowledgments This work was supported by the National Natural Science Foundation of China (11872322, 11472231) and the 2016 Doctoral Innovation Funds of Southwest Jiaotong University is also greatly appreciated. The authors also acknowledge the fruitful discussions with Professor Zhuping Huang of Perking University. References [1] Govaert LE, van Melick HGH, Meijer HEH. Temporary toughening of polystyrene through mechanical pre-conditioning. Polymer 2001;42:1271–4. [2] van Melick HGH, Govaert LE, Meijer HEH. Localisation phenomena in glassy polymers: Influence of thermal and mechanical history. Polymer 2003;44:3579–91. [3] Semkiv M, Hutter M. Modeling aging and mechanical rejuvenation of amorphous solids. J Non-Equilib Thermodyn 2016;41:79–88. [4] Xiao R, Nguyen TD. An effective temperature theory for the nonequilibrium behavior of amorphous polymers. J Mech Phys Solids 2015;82:62–81. [5] Zhang R, Bai P, Lei D, Xiao R. Aging-dependent strain localization in amorphous glassy polymers: From necking to shear banding. Int J Solids Struct 2018;146:203–13. [6] Song R, Zhao J, Yang J, Xin LS, Fan QR. The study of stress-yielding of aged atactic polystyrene (a-PS) by differential scanning calorimetry. Macromol Chem Phys 2001;202:512–15. [7] Arruda EM, Boyce MC. Evolution of plastic anisotropy in amorphous polymers during finite straining. Int J Plast 1993;9:697–720.

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