Towards a macroscopic model for the finite-strain mechanical response of semi-crystalline polymers

Transcription

1 Towards a macroscopic model for the finite-strain mechanical response of semi-crystalline polymers D.J.A. Senden MT 9.5 Eindhoven University of Technology Department of Mechanical Engineering Section Polymer Technology prof. dr. ir. H.E.H. Meijer dr. ir. L.E. Govaert dr. ir. J.A.W. van Dommelen Eindhoven, February 27, 29

2 Contents Contents Abstract i ii 1 Introduction 1 2 Eindhoven Glassy Polymer Model Framework Kinematics Stress calculation Materials and Methods Experimental setup Numerical methods Results and Discussion Poly(methyl-methacrylate) Isotactic polypropylene Towards Modeling of Semi-Crystalline Polymers Constitutive modeling Model behavior: framework A vs. framework B Viscoelastic strain hardening: a gedankenexperiment Viscoelastic strain hardening in 3-D Conclusion 37 Bibliography 39 A Compression Molding Protocol 42 B Relaxation Spectrum of PMMA 43 C Relaxation Spectrum of ipp 44 D Evolution of Plastic Deformation 45 E Integration Scheme: Framework A 46 F Integration Scheme: Framework B 47 i

3 Abstract This study aims to initiate the development of a phenomenological constitutive model that offers the possibility to describe and predict the macroscopic mechanical behavior of semi-crystalline polymers. The investigation is commenced by comparing the experimentally observed intrinsic mechanical behavior of isotactic polypropylene with the results from simulations using the Eindhoven Glassy Polymer model in order to exactly pinpoint the aspects of the behavior that primarily require attention. Based on this comparison, a list of requirements is formulated which the new constitutive model must be able to comply with in order to accurately capture the mechanical response of semi-crystalline polymers. The ability of two different constitutive frameworks to meet these requirements is evaluated. Special attention is granted to the type of modeling that can be used to describe the strain hardening behavior of polymers in general. In this connection, the concept of a deformation-dependent activation volume is introduced and its implications for the mechanical behavior are discussed. ii

4 Chapter 1 Introduction Products and components that are manufactured from polymeric materials are ubiquitous, from mass-produced commodities to high-tech biomedical devices. Particularly for load bearing applications it is important to understand the mechanical behavior of such components, as well as the mechanisms that undermine their structural integrity and may eventually lead to catastrophic failure. The ability to accurately predict the mechanical performance of a product has great industrial significance as it allows structural engineers to better tailor their designs to the required properties. For amorphous polymers, such a tool was recently developed, employing numerical simulations of the injection molding process combined with finite element simulations of the product with its loading conditions during use [1, 2]. Although the industrial application is still in its infancy, the method has proven to possess great potential for amorphous polymers [1]. However, from the two main classes of polymers, amorphous and semi-crystalline, the latter is most substantial in terms of its commercial application. Consequently, the question arises whether it is feasible to devise a similar numerical procedure for products made from semi-crystalline polymers. Prediction of the mechanical performance of a polymeric product requires a constitutive model, relating stresses to strains, to enable the determination of the mechanical response of the material for a certain loading geometry. Semi-crystalline polymers consist of both amorphous and crystalline domains and the overall mechanical behavior is strongly dependent on the microstructure (crystallinity, crystallographic and morphological textures), which is formed during the manufacturing process in which the molten polymer solidifies [3, 4]. Hitherto, a majority of the modeling efforts concerning these materials have therefore concentrated on micromechanical approaches that recognize the influence of the microstructural composition [5 7]. Unfortunately, these models are inherently computationally complex and therefore expensive, making them unsuitable for industrial applications where real-life, complex geometries need to be evaluated. A constitutive model that describes the macroscopic mechanical behavior of semi-crystalline polymers and which is based on a continuum mechanics approach circumvents these practical problems, which is beneficial for its applicability in industry. Nevertheless, as the mechanical behavior of these materials depends heavily on their microstructure, so will the model parameters that shape the mechanical response of a phenomenological macroscopic model. Micromechanical models play an important role in understanding the relation between microscopic and macroscopic deformation behavior. At present, the field of macroscopic constitutive modeling of the mechanical behavior of semicrystalline polymers is largely unexplored. One of the few exceptions is a model for the anisotropic yielding of these type of polymers, which was recently presented by Van Erp et al. [8]. However, a substantial number of experimental observations concerning the macroscopic mechanical response of semi-crystalline polymers have been published. The macroscopic deformation behavior of isotactic polypropylene (ipp), which is the reference material on which this study focusses, is qualitatively quite similar to the behavior that is observed for amorphous polymers as it exhibits non-linear viscoelastic behavior up to the yield point, followed by intrinsic strain softening, 1

5 which is countered by strain hardening [9 12]. It was noted that, in general, semi-crystalline polymers exhibit less pronounced strain softening behavior and weaker strain hardening behavior than amorphous polymers [11, 12]. The work of Schrauwen et al. [9] is particularly interesting since it investigates the correlations between the microstructure (crystallinity and lamellar thickness) and macroscopic deformation phenomena (yielding and strain hardening). The yield kinetics of ipp were observed to involve two separate relaxation mechanisms (α and β), depending on the experimental conditions at which they were measured. The α-process is generally associated with the relaxation of the amorphous fraction in the vicinity of the crystalline domains, caused by mobility within these crystals, whereas the β-process is associated with main chain segmental mobility (the glass transition) of the amorphous domains [13, 14]. The contribution of the β-process is revealed when the yield stress is measured at high strain rates [13, 15, 16], low temperatures [13, 15, 16] and/or high pressures [17]. It was shown that the flow theory of Ree and Eyring is well capable of describing this so-called thermorheologically complex deformation behavior [13, 15, 17]. It was already noted that the microstructure of semi-crystalline polymers develops during processing, generally yielding high levels of anisotropy in the final product since the molten polymer is stretched and sheared to a large extent in many manufacturing processes [18]. Moreover, the microstructure evolves during plastic deformation, thereby creating an additional contribution to the anisotropy in the material [4, 19]. This last phenomenon leads to a strong Bauschinger effect in both semi-crystalline and amorphous polymers [2]. Over the past two decades, the Eindhoven Glassy Polymer (EGP) model was developed for a phenomenological description of the macroscopic mechanical behavior of amorphous polymers [2, 21, 22]. At present, the model has become well-established as it has proven to be capable of describing and predicting the mechanical response of amorphous polymers with great accuracy. Therefore, it would be a sensible point of departure for the development of a constitutive model for semi-crystalline polymers to evaluate the abilities of the EGP model to describe their mechanical behavior and thus exactly pinpoint the deficiencies of the model with regard to this type of polymers. The theoretical aspects of the EGP model are dealt with in Chapter 2 and the experimental and numerical methods that were used in this part of the study are described in Chapter 3. In Chapter 4, the abilities of the EGP model to describe the material behavior of poly(methylmethacrylate) (PMMA) is treated as a case study and subsequently the performance of the model with respect to the mechanical response of ipp is evaluated, yielding a number of requirements for the model in order to describe the response of semi-crystalline polymers. Finally, the road towards the development of a macroscopic constitutive model specifically for semi-crystalline polymers is explored in Chapter 5. For this purpose, the characteristics of two different constitutive frameworks are compared, with a focus on the modeling of the strain hardening behavior of these materials. 2

6 Chapter 2 Eindhoven Glassy Polymer Model 2.1 Framework The Eindhoven Glassy Polymer (EGP) model was developed over the past two decades and it has proven to be capable of excellently describing and predicting the mechanical behavior of amorphous polymers [2, 21, 22]. Depending on the complexity of the material behavior and the desired level of accuracy in the prediction, different versions of the model can be utilized. The most basic form of the model, as described by Klompen et al. [21], features a single non-linear viscoelastic Maxwell element in parallel with a neo-hookean spring, representing the flow contribution and the hardening contribution to the total stress, respectively. Its 1-D mechanical analogue is depicted in Figure 2.1a. The single, stress-dependent relaxation time that governs the non-linear viscoelastic response of this model prior to the yield point, fails to accurately describe the actual mechanical response of many polymers in this pre-yield regime. This problem can be overcome by using a spectrum of multiple relaxation times and moduli, as proposed by Tervoort et al. [23], which implies that the single Maxwell element is replaced by a series of parallel Maxwell elements, or viscoelastic modes [24]. The number of modes that is required depends on the material behavior and the desired accuracy, in most cases it is a number in the order of 1 to 2. The 1-D mechanical analogue for the general case of n viscoelastic modes is depicted in Figure 2.1b. i = n α i = 2 β i = 1 (a) (b) (c) Figure 2.1: Mechanical analogues (1D) of the various forms of the EGP model: (a) single mode, (b) multi mode and (c) multi process. Both of the previously described models are well able to account for the dependence on deformation rate and temperature of the mechanical behavior of glassy polymers, but only when the material displays thermorheologically simple behavior, i.e. when the mechanical response is dominated by only one molecular relaxation process. This is a valid assumption, for instance, for the deformation behavior of polycarbonate at room temperature and at moderate strain rates. Most polymers, 3

7 however, exhibit thermorheologically complex behavior, indicating that at least two molecular relaxation processes (significantly) contribute to the deformation behavior. An extension of the EGP model that allows for incorporation of multiple relaxation processes was introduced by Van Breemen et al. [25], based on the ideas developed earlier by Klompen et al. [15]. This extension comprises two parallel Maxwell modes, as depicted in Figure 2.1c, one for each relaxation process (α and β) that is usually encountered in glassy polymer behavior. The crucial difference between the use of multiple modes and multiple processes is that for the former, the relaxation kinetics are the same for all modes, whereas for the latter, these kinetics are essentially different for each process. It is also possible to combine the concepts of multi mode and multi process modeling, yielding a constitutive model in which each of the relaxation processes has its own spectrum of relaxation times and moduli associated with it. 2.2 Kinematics The basic kinematics for all representations of the EGP model are the same and build on the concept of a virtual, stress-free intermediate state, as postulated by Lee [26]. This implies a multiplicative decomposition of the total deformation gradient tensor F into an elastic and a plastic contribution (subscripts e and p, respectively): F = F e F p (2.1) Using the above definition, equations can be derived for the elastic and plastic velocity gradient tensors, L e and L p, respectively: L = F F 1 = F e Fe 1 + F e F p Fp 1 Fe 1 = L e + L p (2.2) In itself, the decomposition of Equation (2.1) is not unique; it remains undetermined what portion of the total rotation is associated with the elastic and the plastic part. It was shown by Boyce et al. [27] that several methods can be used to obtain uniqueness of the decomposition, without loss of generality. Here, this is achieved by taking the plastic spin tensor equal to the null tensor, i.e. Ω p =, implying that the plastic deformation rate tensor D p is identical to the plastic velocity gradient tensor: L p = D p (2.3) The volume ratio J, capturing the volumetric strains in the material, is governed solely by the elastic part of the deformation, since the plastic deformation is assumed to be incompressible: J = det (F e ) (2.4) Non-volumetric deformations are described using the isochoric left Cauchy-Green deformation tensor B, which is defined as: B = F F T (2.5) 4

8 where the superscript T denotes the transpose of a tensor and the isochoric part of the deformation gradient tensor is defined as: F = J 1/3 F. Non-volumetric elastic strains are described using the isochoric part of the elastic left Cauchy-Green deformation tensor B e, which is calculated completely analogous to Equation (2.5), using the isochoric part of the elastic deformation gradient tensor F e. 2.3 Stress calculation Obviously, the stress calculation according to the EGP model is different for each representation of the model, i.e. it depends on the number of modes and processes that are used. In this section, a generalized description of the model is presented which is applicable in all of the aforementioned cases. In the following, x denotes the process (e.g. x = α, β for many polymer glasses) and i denotes the mode number. The total Cauchy stress tensor is additively decomposed in two contributions: driving stress σ s and hardening stress σ r. The driving stress, associated with segmental motions of the polymer chains, is split in a hydrostatic and a deviatoric part. The strain hardening stress contribution is generally attributed to stretching of the entangled network of polymer chains. Essentially, this is a 3-D equivalent of the stress-split that was originally proposed by Haward and Thackray [28]: σ = σ s + σ r = σ h s + σ d s + σ r (2.6) where the superscripts d and h denote the deviatoric and the hydrostatic part of a tensor, respectively. The deviatoric driving stress and the hardening stress together shape the characteristic intrinsic behavior of amorphous polymers. This is illustrated in Figure 2.2 where their respective contributions to the total (deviatoric) stress (1-D) are qualitatively depicted. It is clear that the deviatoric driving stress accounts for the rate-dependent yielding which is typical for glassy polymer behavior; the phenomenon of intrinsic strain softening is also incorporated herein. With increasing deformation it becomes constant, allowing the rubber-elastic strain hardening response to dominate the large strain mechanical behavior. The hydrostatic part of the driving stress is straightforwardly defined as: σ h s = κ (J 1) I (2.7) It is assumed that the bulk modulus is a single constant for the whole system, implying that the hydrostatic stress is only a function of the elastic deformation, regardless of the number of modes and processes. The contribution of the entangled network is modeled with a neo-hookean rubber-elastic model, governed by the hardening modulus G r [29, 3]: σ r = G r Bd (2.8) The total deviatoric driving stress is calculated by, for each process, summing the contributions from each Maxwell element and subsequently summing the total deviatoric driving stresses associated with each of the processes: m n x σs d = σs d x,i (2.9) x=1 i=1 5

9 total true stress driving hardening true strain Figure 2.2: Qualitative illustration of the additive decomposition (1-D) of the total (deviatoric) stress into driving and hardening stress contributions. where the total number of processes and modes (unique for each process) involved are denoted with m and n x, respectively. Following Baaijens [31], the deviatoric driving stress of a mode is defined as the stress in the elastic spring in the Maxwell element: σ d s x,i = G x,i Bd ex,i (2.1) in which G x,i denotes the shear modulus that corresponds to a process x and mode i. The nonlinear viscoelastic nature of the EGP model is reflected in the non-newtonian flow rule, which couples the driving stress to the plastic deformation rate: D px,i = σ d s x,i 2 η x,i ( τ x, p, T, S x ) = G x,i 2 η x,i ( τ x, p, T, S x ) B d e x,i (2.11) Herein, the modal viscosities η x,i depend on equivalent stress τ x, pressure p, temperature T and a parameter S x that describes the thermomechanical state of the material, including the effects of intrinsic strain softening and physical ageing. It is important to note that the viscosities of all modes corresponding with a particular relaxation process are activated by the total stress associated with that process. The stress and temperature dependence of the viscosities is based on Eyring s flow theory [32] and the incorporation of pressure dependency and intrinsic strain softening is defined in accordance with Govaert et al. [33]: [ Ux η x,i = η x,i exp RT + µp ] ( τ x / τ x ) + S x ( γ p ) τ x sinh ( τ x / τ x ) (2.12) where the pressure dependence parameter µ is a model parameter and the initial modal viscosities are denoted by η x,i. The universal gas constant R and the activation energy U x govern the dependence on absolute temperature T. The temperature dependence of the characteristic stress τ x is governed by the activation volume V x [15]: 6

10 τ x = k T Vx (2.13) where k is Boltzmann s constant. When the temperature dependence of the material behavior is not of interest, the characteristic stress τ x can be regarded as a model parameter and a simplified version of Equation (2.12) can be used: [ ] µp η x,i = η ( τ x / τ x ) x,i exp + S x ( γ p ) τ x sinh ( τ x / τ x ) (2.14) Note that the parameter τ x is different for each process, while it is the same for each mode associated to a particular process. This characteristic embodies the aforementioned difference between processes and relaxation modes, namely that all modes associated to a process have the same relaxation kinetics, whereas these differ between separate processes. The equivalent stress τ x and pressure p are defined as: τ x = 1 2 σd s x : σ d s x (2.15) p = 1 tr (σ) (2.16) 3 In Equation (2.12) it is stated that the state parameter S x only depends on the equivalent plastic strain that has accumulated in the material, which is true when physical ageing is not taken into account. In that case, the state parameter only evolves as a result of intrinsic strain softening. The kinetics of this evolution may vary between different molecular relaxation processes, despite the fact that for all processes the onset of softening is triggered at the same level of equivalent plastic strain. Following Govaert et al. [2], the evolution of thermomechanical state is defined as: (1 + (r,x exp ( γ p )) r r 2,x 1 1,x r ) 1,x S x = S,x r (2.17) (1 + r r,x 1,x) 2,x 1 r 1,x in which the three parameters r, r 1 and r 2 govern the kinetics of intrinsic strain softening and, as mentioned, can be different for each process involved. The equivalent plastic strain can be calculated by integration of the equivalent plastic strain rate, which is defined as: γ p = τ 1 η 1 (2.18) where the subscript 1 indicates that the equivalent plastic strain rate is coupled to the mode that determines the macroscopic yield point, since that is the mode that marks the onset of plastic flow. Basically, this is the mode with the highest initial viscosity, regardless of the process to which the mode is associated. 7

11 Chapter 3 Materials and Methods 3.1 Experimental setup Materials In this study, two polymeric materials are considered: poly(methyl-methacrylate) (PMMA) and isotactic polypropylene (ipp). The former is an amorphous polymer which is used as a case study to illustrate the capabilities of the EGP model to describe glassy polymer behavior, using a combined multi mode and multi process approach. The latter is a semi-crystalline polymer which is used to obtain a detailed impression of what deficiencies the EGP model displays when it is applied to this class of polymers. The experimental results for PMMA were adopted from Klompen [34], the details of the experimental procedure are therefore not discussed here. The ipp grade that was used in this study is commercially known as Borealis HD61CF. Prior to sample processing, the granules were dried in a vacuum oven at 6 C for 24 hours. Flat plates were compression molded using a 6 mm thick mold, which, due to substantial shrinkage, resulted in a minimum final plate thickness just above 5 mm. In order to obtain a good, isotropic and homogeneous plate, a specific protocol was developed, which was strictly followed during the preparation of all samples used in this study. A detailed description hereof is presented in Appendix A. Next, the cylindrically shaped compression samples (Ø4 mm x 4 mm) were machined from these plates, erasing all surface effects that possibly resulted from the compression molding. Uniaxial compression tests On the ipp samples, uniaxial compression experiments were conducted at constant true strain rates ranging from 1 5 to 1 2 s 1. These tests were performed on a servo-hydraulic MTS Elastomer Testing System 81 using two parallel, flat steel plates. In order to prevent any bulging of the sample and thereby ensuring homogeneous deformation, friction was reduced by using polished flat plates on which a lubricating PTFE spray (Griffon TF89) was applied. Moreover, a layer of PTFE tape (3M 548) was placed between the sample and the lubricated steel plates. The measurements were performed at three different ambient temperatures (5, 2 and 4 C), using a thermostatically controlled oven. The finite stiffness of the testing equipment was measured and corrected for in a real-time feedback loop to ensure a correct prescription of the true strain rate. True stresses were computed from the force-displacement output by assuming isochoric deformation. 8

12 3.2 Numerical methods Material characterization The characterization of the material parameters is done using the intrinsic deformation behavior of the polymer as it is obtained from uniaxial compression experiments. The procedure with which the material parameters are determined from the measurement data was outlined by Klompen et al. [21]. In this section, special attention is given to the determination of the spectrum of initial viscosities and moduli that constitute the non-linear relaxation modulus of the material. Traditionally, this is done by performing a large number of creep experiments with different loads and constructing a master curve of the creep compliance by time-stress superposition. Subsequently, the relaxation modulus of the material can be calculated from the creep compliance by invoking the correspondence principle [23]. With the method used here, the relaxation spectrum can be determined directly from a uniaxial compression experiment when only one relaxation process is active. Details of this method are described by Van Breemen et al. [24]. Here, it is roughly outlined. The non-linear viscoelastic behavior of a series of n parallel Maxwell elements can be described by a Boltzmann integral to calculate the uniaxial stress at a certain time t [35]: σ (t) = t i=1 n [ E i exp ( ψ )] ψ ε (t ) dt (3.1) τ i with ψ = t dt a σ (σ (t )) and ψ = t dt a σ (σ (t )) (3.2) It is emphasized that τ i denotes the relaxation time of mode i, not to be confused with the equivalent stress of that mode as defined in Chapter 2. The time variable t spans the entire history of deformation. The time-stress superposition principle is invoked to calculate the nonlinear (i.e. stress-accelerated) time variables ψ and ψ, which account for the non-linearity of the viscoelastic response. For uniaxial stress conditions, the stress-dependent shift function a σ (σ) is defined as [15, 23]: a σ (σ) = σ/σ sinh (σ/σ ) with σ = 3 3 µ τ (3.3) When a discrete spectrum of relaxation times τ i is now selected, the only remaining unknowns in Equation (3.1) are the corresponding moduli E i. All other variables can be calculated from the experimentally obtained intrinsic pre-yield response. To determine these moduli, Equation (3.1) is transformed into the following system of equations: σ = MẼ i (3.4) where the column contains the measured values of the true stress at every experimental time point. The matrix σ M contains the contributions of each relaxation time τ i for every data point in σ. This system of equations can be solved by using a non-negative least squares method, automatically imposing a non-negativity constraint on the moduli E i, which is required for obvious physical reasons. The constitutive model (Chapter 2) requires the relaxation spectrum to consist of shear moduli G i and initial viscosities η i, making a conversion of the above determined moduli E i necessary. Since 9

13 these are not simply elastic moduli, but rather a part of a relaxation spectrum, the conversion is performed by making use of a procedure to interconvert viscoelastic response functions based on the correspondence principle [36]. Finally, the modal initial viscosities are calculated according to: η i = τ i G i (3.5) where it should be noted that the initial viscosities η i are associated with a specific reference temperature and thermomechanical state. These contributions to the initial viscosities can be isolated, yielding a spectrum of initial viscosities as it is defined in Equation (2.12): ( η i = η i exp U ) ( = η RT i exp ( S ) exp U ) RT (3.6) Non-linear finite element method The finite element software package MSC.Marc/Mentat (version 25r3) was used to perform the uniaxial compression simulations with the EGP model. The axisymmetric nature of the experiment is exploited in the simulations, reducing the complexity and calculation time, by making use of a 2- D mesh and imposing rotational symmetry constraints. Since the deformation in the experiments is (within experimental error) homogeneous, a mesh consisting of a single four-noded quadrilateral element suffices in this case. In accordance with the experiments, the deformation is applied under a constant true strain rate. To be able to calculate the mechanical response as it is predicted by the EGP model, an implementation of this model in the Marc/Mentat subroutine HYPELA2 (Van Breemen [37, 38]) was used. 1

14 Chapter 4 Results and Discussion 4.1 Poly(methyl-methacrylate) In this section, the material behavior of PMMA is used as a case study to illustrate the ability of the EGP model to describe and predict the mechanical behavior of an amorphous polymer by employing a combined multi mode and multi process approach. The uniaxial compression data that are used for this purpose are adopted from Klompen [34] and are depicted in Figure 4.1a as true stress - true strain curves for a broad range of deformation rates. Two distinct points in these curves are of particular interest: the yield stress (maximum in true stress prior to strain softening) and the lower yield stress (minimum in true stress subsequent to strain softening). In Figure 4.1b, these stresses are plotted versus the logarithm of true strain rate. In both figures, a small portion of the data is depicted in grey, indicating that this data is unreliable because the specimens heat up during deformation at this stain rate. compressive true stress [MPa] compressive true strain [ ] (a) compressive true yield stress [MPa] compressive true strain rate [s 1 ] (b) Figure 4.1: (a) Experimental uniaxial compression curves for PMMA at true strain rates ranging from to 1 1 s 1. (b) Experimental values for the yield stress ( ) and the lower yield stress ( ) of PMMA in uniaxial compression. Data are adopted from Klompen [34]. The data depicted in grey is unreliable, due to deformation-induced heating of the specimens. 11

15 Across the entire range of deformation rates, the large strain behavior of the material shifts to higher stresses with a constant factor for increasing strain rates, i.e. the lower yield stress linearly depends on the logarithm of the strain rate. This indicates that that portion of the mechanical behavior is, in these conditions, dominated by a single molecular relaxation process α, which is associated with the glass transition. The increase in yield stress with increasing strain rate, however, is constant for low deformation rates, but shifts to a larger value for the highest rates applied. This can be seen in Figure 4.1b, where the straight line formed by the data points shows a clear slope change. This indicates that the material exhibits thermorheologically complex deformation behavior in this regime, meaning that in this case there are two molecular relaxation processes (α and β) actively contributing to the material response, which was also noted by other researchers [39, 4]. The β-process is related to secondary glass transitions, which is the reason for the observed difference in relaxation kinetics between these two processes. In the systematic procedure that was followed to acquire the model parameters, the characteristic stresses τ α and τ β are determined first, since they are directly coupled to the dependence of the yield stress on the logarithm of the strain rate, as depicted in Figure 4.1b. The value of the pressure dependence parameter µ was adopted from Klompen [34]. Next, the strain hardening modulus G r can be fitted on the large strain regime of the mechanical response [21]. The bulk modulus κ of the material is determined using the initial Young s modulus (E = 18 MPa) and Poisson s ratio (ν =.33) determined by Klompen [34]. Subsequently, the discrete spectrum of relaxation modes, associated with the α-process, is computed according to the procedure that was outlined in Section 3.2. The spectrum is determined from the measurement at 1 4 s 1, because the contribution of the β-process is negligible at that strain rate. Moreover, it is emphasized that the relaxation spectrum is associated with the driving stress, i.e. the hardening stress contribution must be subtracted from the measurement data before fitting the spectrum. For PMMA, the spectrum associated with the α-process consists of 12 modes, the numerical values of which are tabulated in Appendix B. A list of all model parameters is given in Table 4.1. compressive true stress [MPa] compressive true strain [ ] (a) shear relaxation modulus [MPa] exp( S ) time [s] (b) Figure 4.2: (a) Experimental uniaxial compression curve ( ) for PMMA at a true strain rate of 1 4 s 1, and simulation results using the directly fitted α-relaxation spectrum (dashed line), the S -shifted relaxation spectrum (dash-dotted line) and including the evolution of thermomechanical state (solid line). (b) Shear relaxation moduli associated with the directly fitted α-relaxation spectrum (dashed line) and the S -shifted relaxation spectrum (solid line). 12

16 In Figure 4.2a, the experimentally obtained mechanical response for a true strain rate of 1 4 s 1 is depicted, along with the results from a simulation (dashed line) using the α-relaxation spectrum as it was obtained with the aforementioned fitting procedure and neglecting the evolution of thermomechanical state (i.e. S = ). It is clear that the pre-yield regime, including the yield point, are described well with this spectrum, but the large strain response is grossly overpredicted. The reason for this is that the spectrum is fitted on experimental data from a sample with a certain thermomechanical state, implying that the initial viscosities in the spectrum are a factor exp (S ) too high, cf. Equation (3.6) [24]. The value of S can be obtained by shifting the spectrum until the model response coincides with the experimental response at large strains (dash-dotted line in Figure 4.2a), since there is no influence of thermomechanical state in that regime, due to mechanical rejuvenation. This shift is also depicted in Figure 4.2b, where the shear relaxation moduli associated with the directly fitted and the S -shifted α-relaxation spectra are given. compressive true stress [MPa] compressive true strain [ ] (a) compressive true yield stress [MPa] compressive true strain rate [s 1 ] (b) Figure 4.3: (a) Experimental uniaxial compression curves for PMMA at true strain rates of 1 4 ( ), 1 3 ( ) and 1 2 ( ) s 1, and simulation results using multi process (solid lines) and single process (dashed lines) versions of the EGP model. (b) Experimental values for the compressive yield stress ( ) and the lower yield stress ( ) for a wide range of true strain rates, and simulation results using the multi process (solid lines) and single process (dashed lines) versions of the EGP model. In Figure 4.3a, the results of finite element simulations based on the single process (dashed lines) and multi process (solid lines) versions of the EGP model are depicted together with the experimental data for true strain rates of 1 4, 1 3 and 1 2 s 1. The α-process is described with the aforementioned relaxation spectrum, whereas the β-process is described with only a single mode. The softening kinetics and the initial state of both processes are in this case fitted with the same set of parameters: S, r, r 1 and r 2 (cf. Table 4.1). For the lowest strain rate, the single process and multi process calculations coincide, which is a logical consequence of the fact that, as mentioned, the influence of the β-process on the mechanical behavior of PMMA is negligible for this deformation rate. At high strain rates, the single process model fails to describe the increased dependence of the yield stress on the applied strain rate, but the incorporation of a second relaxation process (i.e. the β-process) in the model proves to be an adequate solution for this problem. This is also illustrated in Figure 4.3b, where the single process approach is clearly unable to capture the slope change in the dependence of the yield stress on the logarithm of the strain rate. In the large strain regime of the material behavior, it appears that the strain hardening modulus that is observed in the measurements, is structurally higher than that predicted by 13

17 the simulations. This mismatch, although small, seems to be related to the influence of the strain rate on the strain hardening behavior of the material. Whether or not this effect is also present in the material behavior of semi-crystalline polymers, is investigated in the next section, where the abilities of the EGP model to describe the mechanical response of this type of polymers are evaluated. Table 4.1: Model parameters for PMMA. κ [MPa] G r [MPa] G α,tot [MPa] η α,tot [MPa s] G β,tot [MPa] η β,tot [MPa s] τ α [MPa] τ β [MPa] µ [-] S [-] r [-] r 1 [-] r 2 [-] Isotactic polypropylene The ability of the EGP model to describe the mechanical behavior of semi-crystalline polymers is evaluated by making use of the intrinsic material behavior of ipp. This behavior, as measured in uniaxial compression tests at 2 C, is depicted in Figure 4.4 through true stress - true strain curves for different strain rates. The general characteristics of the material behavior are very similar to what was found for PMMA, with a local maximum in stress at the yield point, subsequent strain softening and a final upswing in stress due to strain hardening. 6 compressive true stress [MPa] compressive true strain [ ] Figure 4.4: Experimental uniaxial compression curves for ipp at true strain rates of 1 5, 1 4, 1 3 and 1 2 s 1. The measurements were conducted at an ambient temperature of 2 C. To investigate if the mechanical response is influenced by a single or multiple relaxation processes, the yield stresses and lower yield stresses are plotted versus the logarithm of the true strain rate in Figure 4.5. Note that the stress values are scaled with the absolute temperature at which they were measured, producing (within experimental error) parallel lines, in accordance with Eyring s 14

18 flow theory. For the lowest measurement temperature (5 C), the yield stress values form a linear curve, indicating thermorheologically simple behavior. The corresponding lower yield stresses also form a straight line, but with a different slope. This is only possible if the lower yield behavior is dominated by a particular relaxation process (e.g. α), while the yield behavior is governed by two relaxation processes (e.g. α and β). At lower deformation rates and/or higher ambient temperatures, the contribution of the β-process to the yield stress should diminish and, ultimately, become negligible. For the measurements at 4 C, the slopes of the linear curves that are formed by the yield stresses and lower yield stresses, are equal in the lower strain rate regime. This supports the notion that, in the range of experimental conditions considered here, two separate relaxation processes actively contribute to the mechanical response of ipp. The presence of two active relaxation mechanisms in the mechanical behavior of ipp was also noted by Roetling [13]. The α-process is in this case associated with the relaxation of the amorphous fraction that is a result of translational mobility of the crystal stem, whereas the β-process is associated with main chain mobility (the glass transition) of the amorphous fraction [13, 14]. compressive true yield stress / absolute temperature [MPa K 1 ] T = 5 C T = 2 C T = 4 C compressive true strain rate [s 1 ] Figure 4.5: Experimental values for the yield stress ( ) and the lower yield stress ( ) of ipp in uniaxial compression, scaled with the absolute temperature at which they were measured. The measurements were conducted at ambient temperatures of 5, 2 and 4 C, as indicated in the graph. Also depicted are the yield stresses (black lines) and lower yield stresses (red lines), calculated with a combined multi process and multi mode version of the EGP model. The procedure with which the model parameters for ipp were determined, is similar to the one described in Section 4.1 for PMMA. However, in contrast to PMMA, where the initial state (S,x ) of the material and the softening parameters (r,x, r 1,x and r 2,x ) are equal for both processes x, the material behavior of ipp requires these parameters to be different for each of the two processes involved. This can be explained with the fact that, as mentioned before, the physical nature of the molecular relaxation processes differs for amorphous and semi-crystalline polymers. Moreover, the pressure dependence of the viscosity is not taken into account, i.e. µ =, which has no influence on the results since the material behavior is evaluated only in uniaxial compression. For ipp, the α-relaxation spectrum is determined from the measurements that were conducted at a strain rate of 1 3 s 1 and an ambient temperature of 4 C, because the contribution of the β-process is negligible in that case. The relaxation spectrum, consisting of 12 modes, was shifted with a factor exp (S,α ) in order to acquire the relaxation spectrum associated with the rejuvenated (i.e. S α = ) state, for a reference temperature of 4 C. The relaxation spectrum at other temperatures can be obtained by shifting it with a factor exp ( ) U α RT, cf. Equation (3.6). The numerical values of the spectrum are tabulated in Appendix C. In Figure 4.6b, the resulting shear relaxation moduli are depicted for all three measurement temperatures. The corresponding mechanical responses, 15

19 as they are predicted by a multi mode, single process version of the EGP model, are presented in Figure 4.6a. From these figures, it can be concluded that the temperature dependence of the α-relaxation spectrum is captured well, since the lower yield stress values are quite accurately predicted. As expected, it is obvious that a single process approach fails to describe the yield stresses at lower temperatures. compressive true stress [MPa] compressive true strain [ ] (a) shear relaxation modulus [MPa] time [s] (b) Figure 4.6: (a) Experimental uniaxial compression curves for ipp at a true strain rate of 1 3 s 1 and at ambient temperatures of 5 ( ), 2 ( ) and 4 C ( ). Simulation results using a multi mode, single process version of the EGP model are also depicted (lines). (b) Shear relaxation moduli associated with the α-relaxation spectrum for three reference temperatures: 5 ( ), 2 ( ) and 4 C ( ). In Figure 4.7a-c, the results from finite element simulations of uniaxial compression tests are depicted, along with the experimentally obtained mechanical responses. A single mode approach is used for the β-process. On the whole, the EGP model describes the experimental data quite well, especially if one bears in mind that this experimental data encompasses a wide range of deformation rates and three different temperatures, which is fitted with a single, consistent parameter set. A list of all model parameters is given in Table 4.2. In the EGP model, the strain hardening modulus is assumed to be a constant, which is independent of strain rate and temperature. In the simulations of the material behavior of ipp, see Figure 4.7a-c, this modulus was fitted on the measurements at 4 C, where it indeed describes the experimental data well. At low temperatures and high rates, however, the experimentally obtained strain hardening modulus appears to be progressively higher than it is predicted by the simulations. This phenomenon was also observed for PMMA in the previous section, but for ipp the effects are much more pronounced. At large deformations, friction between the steel plates and the sample might come into play, causing an upswing in stress in this regime, especially at low temperatures, while the simulations assume a frictionless contact. It is, however, too simplistic to attribute the entire phenomenon to friction effects because of the care that was taken to reduce friction in the experiments and the systematic nature of the increase in strain hardening modulus with both strain rate and temperature. Furthermore, it is an effect that was also observed by Wendlandt et al. [41] for various glassy polymers, who reduced friction in a similar way and verified this by video-controlling their experiment. This observation suggests that strain hardening is not a fully elastic, but rather a viscoelastic process. Based on these findings, it can be concluded that the EGP model is capable of describing the mechanical behavior of semi-crystalline polymers with a reasonable level of accuracy, although some aspects of the behavior are not accounted for. 16

20 compressive true stress [MPa] T = 5 C compressive true strain [ ] (a) compressive true stress [MPa] T = 2 C compressive true strain [ ] (b) 8 compressive true stress [MPa] T = 4 C compressive true strain [ ] (c) Figure 4.7: Experimental uniaxial compression curves for ipp at true strain rates of 1 5 ( ), 1 4 ( ), 1 3 ( ) and 1 2 ( ) s 1, and simulation results using a combined multi mode, multi process version of the EGP model (solid lines) for three different ambient temperatures: (a) 5, (b) 2 and (c) 4 C. Table 4.2: Model parameters for ipp. κ [MPa] G r [MPa] Vα [nm 3 ] Vβ [nm3 ] µ [-] U α [kj mol 1 ] η α,tot [MPa s] G α,tot [MPa] U β [kj mol 1 ] η β,tot [MPa s] G β,tot [MPa] S,α [-] r,α [-] r 1,α [-] r 2,α [-] S,β [-] r,β [-] r 1,β [-] r 2,β [-]

21 Chapter 5 Towards Modeling of Semi-Crystalline Polymers In this chapter, the road towards a constitutive model that is capable of describing and predicting the macroscopic mechanical behavior of semi-crystalline polymers is explored. In the previous section, characteristic phenomena that must be taken into account in such a model in order to enable a quantitative description of the mechanical behavior for a broad range of experimental conditions, were discussed. Here, these are summarized: 1. Uniaxial deformation responses exhibit a rate-dependent upswing in stress at large deformations. 2. A spectrum of relaxation modes is required to accurately capture the pre-yield behavior. 3. Multiple relaxation mechanisms (processes) contribute to the mechanical response. 4. Strain softening kinetics are different for each relaxation process that is involved. In the previous sections, the attention was focussed solely on isotropic material behavior. The vast majority of products made from semi-crystalline polymers, however, will possess some sort of anisotropy due to microstructural orientation. Therefore, it is essential that the proposed model enables the incorporation of anisotropic material behavior. A state of anisotropy can also be obtained in an initially isotropic specimen (e.g. a tensile bar) when it is uniaxially extended; the polymeric chains will then preferentially orient themselves along the deformation axis, yielding a state of transverse isotropy. The effect of this orientation on the material behavior of both glassy and semi-crystalline polymers is considerable [19]. If a specimen is pre-oriented in uniaxial extension and subsequently subjected to tensile or compressive uniaxial deformation along the same loading axis, a large difference is observed between the mechanical responses in tension and compression, a phenomenon which is termed the Bauschinger effect [2]. This is visualized in Figure 5.1b, where the results from uniaxial tension and compression experiments on rejuvenated polycarbonate, adopted from Weltevreden [2], are depicted for specimens with a pre-orientation in uniaxial tension of.6 true strain. It is obvious that the Bauschinger effect is indeed dramatic, when these results are compared with Figure 5.1a, where similar experimental results for isotropic, rejuvenated polycarbonate are depicted, adopted from Tervoort et al. [29]. Note that the absolute value of the true stress is plotted versus the absolute value of the strain measure λ 2 λ 1, where λ denotes the draw ratio, yielding straight lines for a neo-hookean response. 18

22 15 15 absolute true stress [MPa] 1 5 absolute true stress [MPa] λ 2 λ 1 [ ] λ 2 λ 1 [ ] (a) (b) Figure 5.1: Results from uniaxial tension ( ) and compression ( ) experiments on rejuvenated polycarbonate at a constant true strain rate of 1 2 s 1. (a) Measurements on isotropic specimens, data adopted from Tervoort et al. [29]. (b) Measurements on pre-oriented specimens (.6 true strain in uniaxial tension), data adopted from Weltevreden [2]. The difference between the mechanical response in tension and compression is quite small for the isotropic specimens. There is a constant offset between them, with the yield stress in compression being slightly higher than in tension, due to the influence of the hydrostatic pressure [33]. Furthermore, the strain hardening response is neo-hookean, with the same hardening modulus in compression and tension [29]. The mechanical response of the pre-oriented samples is entirely different. In compression, the strain hardening is almost absent directly after the yield point and is mildly visible again at large strains. The tensile behavior, on the contrary, displays an increased elastic modulus, a dramatic increase in yield stress and much more pronounced strain hardening behavior. From these observations, two additions to the aforementioned list of requirements arise: 5. The Bauschinger effect in (semi-crystalline) polymers is substantial. 6. It should be possible to define the initial state of anisotropy of a material and, ultimately, also the evolution of anisotropy during plastic deformation. 5.1 Constitutive modeling Two possibilities for the constitutive framework The development of a constitutive model starts with the choice of a framework. In the constitutive modeling of the large strain mechanical behavior of polymers, there basically exist two possibilities for the constitutive framework: the one used in the Eindhoven Glassy Polymer (EGP) model [32] and the one used in the Boyce-Parks-Argon (BPA) model [42]. The former (referred to as framework A) was already discussed in Chapter 2 and the difference with the latter (framework B) is that the strain hardening spring is placed in parallel with only the non-linear dashpot in that case, cf. Figure 5.2a-b. This results in a distinct discrepancy in the responses of these frameworks, because the interplay between the viscoelastic part of the deformation and the elastic strain hardening contribution is different. In framework A, the strain hardening behavior is governed by the entire deformation, whereas it is governed solely by the plastic deformation in framework B. 19