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A viscoelastic-viscoplastic constitutive model for glassy polymers informed by molecular dynamics simulations
Abstract
This contribution presents a phenomenological viscoelastic-viscoplastic constitutive model informed by coarse-grained (CG) molecular dynamics (MD) simulations of pure glassy polystyrene (PS). In contrast to experiments, where viscoplasticity is caused by various effects simultaneously, these effects can be decomposed in MD simulations by adjusting the MD system. In the MD simulations considered here, neither bond breakage nor cross-links are introduced; instead, we focus on the intermolecular interaction of polymer chains. We employ a thermo-dynamically consistent generalized Maxwell framework parallelly comprising an elastic, a viscoelastic, and several elasto-viscoplastic modules with different yield stress to capture the viscoelastic and the viscoplastic mechanical behavior simultaneously. The yield stresses decrease with the maximum deformation the MD system has experienced. The constitutive model presented here is based on 10 material parameters, which can be identified by a few data sets and fits well the CG MD simulations of PS under uniaxial and biaxial deformation with time-proportional and cyclic loading conditions in a wide range of strain rates (0.1%/ns-20%/ns).
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... Hence, cyclic loadings for small strain ranges will be considered in the subsequent section. with focus on small amplitudes; all data have been generated withε max xx = 1 % ns −1 ; data from [9] and [11] A quantification of inelastic effects can be obtained from Figure 5.4: There, stress-strain curves are depicted from time-periodic uniaxial tension tests with different maximum amplitudes, cf. [9]. ...
... Influence of strain rate at small strains: a) Cauchy stress in tensile direction σ xx over Green-Lagrange strain in tensile direction ε xx for time-proportional uniaxial tension tests; b) Effective Young's modulus E at small strains derived as the slope of the secant between 0 and 1 % strain from time-proportional uniaxial tension; data from[9] and[11] a . ...
... In[11], the Cauchy stress σ is used rather than the Kirchhoff stress τ . However, since τ = Jσ and thus τ F = J σ F as well as tr (τ ) = J tr (σ), σ can be substituted by τ . ...
The present thesis subsumes recent research activities of the "Capriccio" group, which has been established by the author at the Institute of Applied Mechanics of the Friedrich-Alexander-Universität Erlangen-Nürnberg in 2018. It focuses on multiscale modelling and simulation of polymers and is interested, among others, in particle-based material descriptions and molecular dynamics pseudo experiments, constitutive modelling of polymers, material characterization and model calibration, polymer nanocomposites and fracture of polymers. To this end, the group further advances the Capriccio method, which is a domain-decomposition multiscale technique specifically designed for amorphous materials and which has proven to be a suitable simulation tool to couple a continuum with a particle domain treated by molecular dynamics (MD). This approach subdivides the entire domain of interest into sub-domains of different resolution and thus confines the computationally expensive particle-based treatment to regions of specific interest: Typically, these are located in the vicinity of interfaces, crack tips, or other types of discontinuities. In contrast, the remaining regions are modelled as a continuum requiring significantly less computational effort. Such an approach, however, requires a careful consideration of the transition region between the different domains involved, but is a powerful tool for material modelling and simulation. With such a technique at hand, the system sizes which molecular dynamics is usually capable of can be significantly enlarged. Hence, representative volume elements of, e.g., heterogeneous materials like polymer nanocomposites, come within reach and form the basis for further upscaling techniques. The first part of this theses concentrates on an enhanced version of the Capriccio method in a purely elastic setting. The second part provides a profound data basis by MD pseudo experiments of a polymer model system at coarse-grained resolution. It turns out that the material behaviour is by far too complex to be captured by simple elastic descriptions and even available inelastic models are not able to adequately reproduce the MD simulation data. Hence, another part of this thesis focuses on the development of a viscoelastic-viscoplastic constitutive law which is able to appropriately model the material behaviour. With this novel constitutive description at hand, the Capriccio method is extended to inelasticity and yields promising results. Finally, the present thesis applies the multiscale framework introduced here to polymer nancomposites and detects material parameter profiles in the interphase between the filler particles and the bulk polymer. The approach presented here is rather unconventional and uses goal data obtained from coupled simulations using the Capriccio method. There, the interphase regions are described at particle resolution and base exclusively on the interactions between the particles. By means of an appropriate, purely continuum surrogate model, the material property profiles in the interphase are eventually detected. This procedure enables insights into regions which are not accessible by experiments and thus is a promising, novel methodology to identify the material behaviour informed by molecular dynamics simulations. Based on these findings, this thesis closes with an outlook to future research activities of the Capriccio group and embeds them into a larger scientific context. There, envisioned studies in the fields of multiscale treatment of nanocomposites and fracture, integration of modelling, simulation, and experiments as well as of advanced design of polymer materials are addressed.
... However, under larger deformation, an inelastic constitutive model is required to capture the inelastic mechanical behavior of polymers. To this end, a VE-VP constitutive model is proposed by Zhao et al. 29 based on CG MD simulations of glassy PS implemented in Ries et al. 30 In general, the constitutive model 29 has the form Ψ = Ψ (F, ) and (7) ...
... with a set of internal variables (t) and their given initial values (t = 0), which is subjected to the evolution function f. The specific formulation of the VE-VP constitutive model is presented in Appendix A, compare the study by Zhao et al. 29 for more details. The Capriccio method can be extended for simulations of inelastic polymers by substituting the strain energy density in (2) by the VE-VP model (7) and taking into account the evolution of the internal variables in (8). ...
... It has an external edge length of 30 nm and an internal edge length of about 17.3 nm and comprises 936 nodes and 604 tri-linear hexahedral elements with 2 × 2 × 2 Gauss points each. The isothermal constitutive model introduced in our preceding study 29 is employed in the continuum domain, which is based on the MD simulation of PS under PBC at 100 K. 30 A brief summary of the VE-VP constitutive model together with its parameters is given in Appendix A. In the FE domain, which is calculated implicitly, an explicit Euler time integration is used for solving the evolution equations of the constitutive model at each Gauss point since the incremental strain per load step Δ ls has to be chosen small to couple with the MD domain, which is sufficient for the stability of the solution in our case. With larger time step sizes, the unconditionally stable explicit algorithm 42,43 for such constitutive models can be used without additional computational effort. ...
In this contribution, we present a partitioned‐domain method coupling a particle domain and a continuum domain for multiscale simulations of inelastic amorphous polymers under isothermal conditions. In the continuum domain, a viscoelastic‐viscoplastic constitutive model calibrated from previous molecular dynamics (MD) simulations is employed to capture the inelastic properties of the polymer. Due to the material's rate‐dependence, a temporal coupling scheme is introduced. The influence of the time‐related parameters on the computational cost and accuracy is discussed. With appropriate parameters, multiscale simulations of glassy polystyrene under various loading conditions are implemented to showcase the method's capabilities to capture the mechanical behavior of polymers with different strain rates and with non‐affine deformations of the MD domain.
... Related works exploring the role of individual chains in polymer deformation are due to Kausch et al. [46], and more recently to Grasinger [47] and Mulderrig et al. [48]. Eventually, some publications address viscous polymers [49,50]. 40 By controlling the composition and microstructural distribution of polymers, it is possible to tailor their properties for specific applications [51,52,53]. ...
... The authors indicated that viscous relaxation entails a remarkable reduction of the flexoelectric response, for example, with an increase in temperature. From a chemical perspective, Boschetto et al. provided an integrative theoretical and experimental vision of effect that 50 the different lengths of functional blocks had in the effective performance of organic photovoltaic block copolymers [56]. A related work owes to Liedel et al. and their review on the reorientation mechanisms that experience the macromolecules in block copolymers when they are subjected to external electric fields [57]. ...
Flexoelectric sensors respond with an electric polarization when they are subjected to inhomogeneous deformation. When these are based on polymeric materials, the electro-mechanical coupling relates to the loss of microstructural symmetries of the polar polymer chains. The object of this work is to explore the mechanisms that induce flexoelectricity in soft (co)polymers via a micromechanical model that explicitly considers the spatial composition distribution of polymer chains. To this end, we analyze the deformation of the Arruda-Boyce 8-chain cell under inhomogeneous deformation and its influence on the overall polarization resulting from the polar monomers. Cells with random-walk chains and tailored degree of entanglement feed Monte Carlo simulations to inform an effective chain dipole that drives the flexoelectric polarization. Homogeneous stretch/compression of the cell accompanying bending deformation reinforce the flexoelectric response. In addition, we focus on stochastic chain-growth processes and the insertion of different types of polar monomers within a same chain. Here, we introduce a model where reactivity ratios of real copolymerization reactions control the resulting flexoelectric coupling. Since the framework allows to relate the effective flexoelectric polarization to any structure isomer distribution, it is especially convenient to model copolymers. Overall, the model is proposed as a versatile framework to explore the intricacies of flexoelectric (co)polymers and to foster the design of flexoelectric devices from real manufacturing parameters.
... A comparison with the momentary stress profiles from molecular-dynamic-reference simulation runs revealed some shortcomings in the predictive modelling at its current stage. From a mathematical viewpoint, the use of more mechanical elements has enabled more accurate fitting [23][24][25][26][27][28]. However, the larger the number of parameters, the more challenging the physical interpretation [27]. ...
... From a mathematical viewpoint, the use of more mechanical elements has enabled more accurate fitting [23][24][25][26][27][28]. However, the larger the number of parameters, the more challenging the physical interpretation [27]. In the present contribution, we therefore did not further expand the constitutive model, but instead analyzed the physical foundations of the observed discrepancies. ...
Epoxy resin plasticity and damage was studied from molecular dynamic simulations and interpreted by the help of constitutive modelling. For the latter, we suggested a physically motivated approach that aims at interpolating two well-defined limiting cases; namely, pulling at the vanishing strain rate and very rapid deformation; here, taken as 50% of the speed of sound of the material. In turn, to consider 0.1–10-m/s-scale deformation rates, we employed a simple relaxation model featuring exponential stress decay with a relaxation time of 1.5 ns. As benchmarks, deformation and strain reversal runs were performed by molecular dynamic simulations using two different strain rates. Our analyses show the importance of molecular rearrangements within the epoxy network loops for rationalizing the strain-rate dependence of plasticity and residual stress upon strain reversal. To this end, our constitutive model reasonably reproduced experimental data of elastic and visco-elastic epoxy deformation, along with the maximum stress experienced before fracturing. Moreover, we show the importance of introducing damage elements for mimicking the mechanical behavior of epoxy resins.
... This study is complementary to the characterization [33,34], and complex continuum modeling of polystyrene using a novel viscoelastic-viscoplastic constitutive law [35]. ...
... This study demonstrates that a constitutive law on the nanoscale, as it is necessary for the modeling of an RVE in micromechanics, can be derived on the basis of MD simulations. This contribution is complementary to an earlier characterization and modeling of polystyrene as an exemplary matrix material [33][34][35]. Together with our extensive efforts to characterize the interphase between polymer and filler particles [25,26], we can now provide continuum mechanical descriptions of all constituents of a nanocomposite. This enables us to set up RVEs to apply the molecular-level findings on the microscale. ...
The addition of fillers can significantly improve the mechanical behavior of polymers. The responsible mechanisms at the molecular level can be well assessed by particle-based simulation techniques, such as molecular dynamics. However, the high computational cost of these simulations prevents the study of macroscopic samples. Continuum-based approaches, particularly micromechanics, offer a more efficient alternative but require precise constitutive models for all constituents, which are usually unavailable at these small length scales. In this contribution, we derive a molecular-dynamics-informed constitutive law by employing a characterization strategy introduced in a previous publication. We choose silicon dioxide (silica) as an exemplary filler material used in polymer composites and perform uniaxial and shear deformation tests with molecular dynamics. The material exhibits elastoplastic behavior with a pronounced anisotropy. Based on the pseudo-experimental data, we calibrate an anisotropic elastic constitutive law and reproduce the material response for small strains accurately. The study validates the characterization strategy that facilitates the calibration of constitutive laws from molecular dynamics simulations. Furthermore, the obtained material model for coarse-grained silica forms the basis for future continuum-based investigations of polymer nanocomposites. In general, the presented transition from a fine-scale particle model to a coarse and computationally efficient continuum description adds to the body of knowledge of molecular science as well as the engineering community.
... In this work, we perform mode I and mode III fracture simulations using the concurrent Capriccio method [34]- [37], which employs elements of the Arlequin method and couples a particle-based model to a continuum model. The former is solved using MD, while the latter is discretized and solved using the finite element method (FEM). ...
The fundamental understanding of the root causes of failure requires information on atomic-scale processes. Molecular dynamics (MD) simulations are widely used to provide these insights while maintaining chemical specificity. However, the boundary conditions that can be applied in MD fracture simulations are limited to the linear-elastic fracture mechanics (LEFM) far-field solution, posing restrictions in simulating bending or shearing deformations commonly applied in classical experimental setups. This has to date prevented the accurate and precise prediction of key fracture quantities such as critical stress intensity factors. To overcome these limitations, we here apply the domain-decomposition Capriccio method to couple atomistic MD samples representing silica glass synthesized at various quenching rates with the finite element (FE) method and perform both mode I (tension, three- and four-point bending) and mode III simulations at different loading rates. We investigate multiple criteria to best identify the onset of crack propagation based on the virial stress, the number of pair interactions, the kinetic energy/temperature, the crack velocity, and the crack opening displacement. Based on this, we propose a novel protocol that allows us to determine the fracture toughness of silica glass under mode I and III conditions from atomistic data with increased fidelity, with our results being in good agreement with experimental data and predictions from LEFM. Overall, our contribution demonstrates how coupled FE-MD simulations enable chemically specific quantitative predictions of the fracture behavior of amorphous materials under arbitrary mechanical loading conditions.
... However, as we focus on general properties of glassy polymers, an arbitrary traditional CG bead-spring model is sufficient for the purpose of this study. Compared to other widely-used generic bead-spring models such as the the Kremer-Grest model [35] and the morse model [36], the primary reason for using such a CG model is that various previous work [37,38,39,15,40] has provided substantial useful information for choosing its parameters. ...
We perform molecular dynamics simulations under uniaxial tension to investigate the micromechanisms underlying strain hardening in glassy polymers. By decomposing the stress into virial components associated with pair, bond, and angle interactions, we identify the primary contributors to strain hardening as the stretching of polymer bonds. Interestingly, rather than the average bond stretch, we find that the key contributions to stress response come from a subset of bonds at the upper tail of the stretch distribution. Our results demonstrate that the stress in the hardening region can be correlated with the average stretch of the most extended bonds in each polymer chain, independent of temperatures and strain rates. These bonds, which we denote as load-bearing bonds, allow us to define a local load-bearing deformation gradient in continuum mechanics that captures their contribution to the hardening stress tensor. Building on this insight, we incorporate the load-bearing mechanism into a constitutive framework with orientation-induced back stress, developing a model that accurately reproduces the stress response of the molecular systems over a wide range of temperatures and strain rates in their glassy state.
... The viscoelastic and viscoplastic characteristics of polymer composites have been extensively explored in the literature [10][11][12][13][14]. Whenever the strain is recovered and ...
The generation of micro-nano structures on a polymer is a growing industrial-sector application due to its utility in optics, microfluidic chips, photovoltaic, and medical sectors. The filling behavior of polymers in micro-nano dies differs significantly from that of bulk polymer deformation due to the scaling law effect. Deformation at various scales affects the mechanical, electrical, and other properties of polymers. As the demand for micro and submicron components grows, it becomes increasingly important to develop a process for producing small components at a lower cost and in less time. As a result, understanding the dynamics of polymer behavior/deformation in micro-nano cavities (or dies) and various forming methods capable of mass-producing such micro components becomes critical. As a result, the goal of this chapter is to give the reader a basic understanding of the hot embossing micro-nano forming process in comparison to other micro-forming polymer techniques. The process parameters associated with the hot embossing process are thoroughly discussed, as is their impact on the process. In addition, the mathematical description of viscoelastic behavior and polymer viscosity will pave the way for a better understanding of the process simulation later in this chapter.
... Furthermore, certain special engineering materials such as surrounding rocks from underground, asphalt, and concrete exhibit time-independent stress-strain relationships under normal pressure or temperature conditions. [2][3][4] However, as pressure or temperature increases to a certain extent, noticeable viscoelastic characteristics may emerge. Therefore, researching viscoelastic materials' stress performance and analysis methods 5 holds significant scientific and engineering importance. ...
This paper introduces a conceptual framework for finite element analysis of generalized Maxwell viscoelastic materials based on the initial stress approach. The proposed method facilitates the explicit and convenient determination of mechanical parameters for viscoelastic materials by directly utilizing relaxation test data. Each time step’s viscoelastic relaxation stress is treated as the initial stress, and a recursive calculation formula for the material’s initial stress is established, relying solely on the relaxation stress and strain from the previous time step. Based on this, the study outlines the computational steps for initial-stress-based finite element analysis of elastic materials. The proposed algorithm’s accuracy and efficiency are validated through a classical one-dimensional axial rod example. Results demonstrate that the introduced initial stress-type finite element analysis maintains the stiffness matrix equal to the initial stiffness matrix throughout the calculation process, significantly enhancing the efficiency of finite element analysis for viscoelastic materials while reducing the computational resource required. The conceptual framework improves the efficiency and accuracy of analyzing the mechanical parameters of viscoelastic materials using relaxation test data.
... At lower temperatures, the absence of explicitly modeled side groups in this potential results in reduced friction and confinement between the polymer segments. Consequently, the glass transition temperature is significantly lower, with an approximate value of g ≈ 170 K, which is considerably below the values observed in atomistic simulations (Lyulin and Michels, 2002) and experiments (Kaliappan and Cappella, 2005), where g is around 370 K. Additionally, this CG model renders lower yield stresses (Rahimi et al., 2012;Ries et al., 2019;Zhao et al., 2021b) compared to full atomistic simulations (Lyulin et al., 2004), likely due to the same underlying reason. Moreover, the temperature-sensitive secondary relaxation (Bauwens-Crowet et al., 1969) originating from the side groups is deactivated in this CG model. ...
In this paper, we investigate the time-temperature correlation of amorphous thermoplastics at large strains based on coarse-grained molecular dynamics simulations. This correlation behavior is characterized by the strain hardening modulus in uniaxial tension simulations at different strain rates across the glass transition region. The temperature regime is divided into a melt zone, a glassy zone, and a transition zone between them, according to the storage modulus calculated from dynamic mechanical analysis (DMA) at small strains. In the melt zone, the existence of time-temperature superposition (TTS) at large strains is verified by constructing a master curve of the hardening modulus. The obtained shift factors are then compared to those from DMA at small strains, showing that the TTS behavior is transferable between small and large strains. In the glassy zone, the effects of time and temperature are not superposable at large strains but still can be correlated. To demonstrate this correlation behavior, we introduce a level set of the hardening modulus with a variable pair of strain rate and temperature. Pairs lying in the same level result in coincident stress-strain curves at large strains. The transferability of the correlation behavior between large and small strains is validated by comparing these stress-strain curves at small strains in the pre-yield region. In the transition zone, the correlation behavior is studied with both aforementioned methods, showing that TTS is applicable to large strains but not transferable to small strains. Finally, we propose a phenomenological constitutive model for uniaxial tension to demonstrate the time-temperature correlation at large strains, considering different constant strain rates and temperatures.
... For example, current wind turbines feature rotor blades of 100 m dimensions, thanks to tailor-made developments of high-performance epoxy composites. 2 The wealth of engineering and experimental characterization studies 3 is flanked by a range of modeling and simulation approaches encompassing atomistic considerations to continuum models. 4 Among these, continuum-type models based on empirical materials parameters clearly represent the largest body of simulation studies-indeed the layout of machine parts critically depends on finite-elements modeling. 2 In turn, to gain in-depth understanding of material-specific properties, molecular simulation studies offer fundamental insights at atomic resolution. ...
We outline a coarse-grained model of epoxy resins (bisphenol-F-diglycidyl-ether/3,5-diethyltoluene-2,4-diamine) to describe elastic and plastic deformation, cavitation, and fracture at the μm scale. For this, molecular scale simulation data collected from quantum and molecular mechanics studies are coarsened into an effective interaction potential featuring a single type of beads that mimic 100 nm scale building blocks of the material. Our model allows bridging the time–length scale problem toward experimental tensile testing, thus effectively reproducing the deformation and fracture characteristics observed for strain rates of 10⁻¹ to 10⁻⁵ s⁻¹. This paves the way to analyzing viscoelastic deformation, plastic behavior, and yielding characteristics by means of “post-atomistic” simulation models that retain the molecular mechanics of the underlying epoxy resin at length scales of 0.1–10 µm.
... This model was verified by comparison of the prediction with the tests of PC. Other researchers [13,[91][92][93][94][95][96][97][98] have also made significant contributions in this regard. ...
Most of the stress–strain relationships of thermoplastic polymers for aeronautical composites tend to be nonlinear and sensitive to strain rate and temperature, so accurate constitutive models are urgently required. Classical and machine learning-based constitutive models for thermoplastic polymers are compared and discussed. In addition, some typical models have been recovered and compared by authors to evaluate the performance of classical and machine learning-based constitutive models, so that the advantages and shortcomings of these models can be demonstrated. By reviewing constitutive models, it was found that the equations of physical constitutive models are derived according to thermodynamical principles, so the physical constitutive models can describe the deformation mechanism at the microscopic level. The phenomenological constitutive models may combine the macroscopic phenomena and theories of physical models, and good performance and wide range of applications can be realized. In addition, phenomenological constitutive models combined with machine learning algorithms have attracted attentions of investigators, and these models perform well in predicting the stress–strain relationships. In the future, the constitutive models combining the theories of physical constitutive models, phenomenological constitutive models, and machine learning algorithms will be increasingly attractive as some challenging issues are effectively addressed.
... The viscoelastic and viscoplastic characteristics of polymer composites have been extensively explored in the literature [93][94][95][96][97]. Whenever the strain is recovered and is thermodynamically reproducible after deformation, the term elasticity is employed [98]. ...
Micro-hot embossing (micro-HE) of polymeric materials creates exact micro/nanoscaled designs. Micro-HE processes include plate-to-plate (P2P), roll-to-roll (R2R), and roll-to-plate (R2P). Micro-HE is preferred for large-scale production of micro-patterns on polymer substrates. However, the lack of simulation models for optimization and component design prevents the broad use of this technology. As the size of the micro patterns decreases from micron to sub-micron, it improves performance features. Micro-HE cannot be analyzed using software tools like injection molding since there is no macroscopic equivalent. Commercial simulation software covers injection molding and associated processes. No commercial tool covers all micro-HE process steps, variations, and boundary conditions. According to the authors, such review articles aren't in the literature. This article summarizes the simulation work in the micro-HE process field related to replication accuracy, and mold filling behaviour. In addition to this, various models were discussed based on properties of material, based on various forces participating in the HE process, and gives a detailed idea about mold-filling behavior and demolding analysis. Finally challenges and future scope related to modelling and simulation work in field of hot embossing has been presented.
... The present study forms the basis for future investigations. In the next step, we intend to calibrate continuum mechanical constitutive laws for polymer and filler analogous to Zhao et al. [80] and Ries et al. [26] based on the present results. These continuum models are a prerequisite for the identification of the property gradients within the interphase, as described, for example, in our previous studies [27,35]. ...
The addition of nano-sized filler particles enhances the mechanical performance of polymers. The resulting properties of the polymer nanocomposite depend on a complex interplay of influence factors such as material pairing, filler size, and content, as well as filler-matrix adhesion. As a complement to experimental studies, numerical methods, such as molecular dynamics (MD), facilitate an isolated examination of the individual factors in order to understand their interaction better. However, particle-based simulations are, in general, computationally very expensive, rendering a thorough investigation of nanocomposites’ mechanical behavior both expensive and time-consuming. Therefore, this paper presents a fast coarsegrained MD model for a generic nanoparticle-reinforced thermoplastic. First, we examine the matrix and filler phase individually, which exhibit isotropic elasto-viscoplastic and anisotropic elastic behavior, respectively. Based on this, we demonstrate that the effect of filler size, filler content, and filler-matrix adhesion on the stiffness and strength of the nanocomposite corresponds very well with experimental findings in the literature. Consequently, the presented computationally efficient MD model enables the analysis of a generic polymer nanocomposite. In addition to the obtained insights into mechanical behavior, the material characterization provides the basis for a future continuum mechanical description, which bridges the gap to the engineering scale.
... The outcome was an elastoplastic interphase with saturation profiles for Young's modulus E and yield stress σ y , as shown in Figure 4 (4 layers). In order to limit the complexity of the investigations to a feasible level, we neglect the viscous, i.e., time-dependent, effects observed in [43], [45], [47], [61], which will be investigated in more detail in future studies. ...
The enhanced mechanical behavior of polymer nanocomposites with spherical filler particles is attributed to the formation of matrix-filler interphases. The nano-scale leads to particularly high interphase volume fractions while rendering experimental investigations extremely difficult. Previously, we introduced a molecular dynamics-based interphase model capturing the crucial spatial profiles of elastic and inelastic properties inside the interphase. This contribution demonstrates that our model captures polymer nanocomposites’ essential characteristics reported from experiments. To this end, we thoroughly verify and validate the model before discussing the resulting local plastic strain distribution. Furthermore, we obtain a reinforcement in terms of the overall stiffness for smaller particles and higher filler contents, while the influence of particle spacing seems negligible, matching experimental observations in the literature. This paper proposes a methodology to unravel the underlying complex mechanical behavior of polymer nanocomposites and to translate the findings into engineering quantities accessible to a broader audience and technical applications.
... A more recent approach is the thorough mechanical characterization of polymers by applying a set of different load cases (time-proportional, time-periodic, relaxation, creep) to MD samples. 5,6 The results allow the calibration of sophisticated inelastic material models, e.g., the viscoelastic-viscoplastic model proposed by Zhao et al. 7 Such a precise MD-informed continuum mechanical constitutive model, which is the result of a sequential multiscale strategy, is a prerequisite for many concurrent multiscale methods. 8 One of the most prominent concurrent multiscale methods is the quasicontinuum method, 9,10 which introduces the concept of representative atoms (repatoms). ...
Atomistic-to-continuum coupling methods are used to unravel molecular mechanisms of polymers and polymer composites. These multiscale techniques advantageously combine the computational efficiency of continuum approaches while keeping the accuracy of particle-based methods. The Capriccio method [Pfaller et al. Comput. Methods Appl. Mech. Eng. 2013, 260, 109−129.] is a well-proven multiscale technique, which connects finite elements (FE) with molecular dynamics (MD) in a partitioned-domain approach. A vital aspect of these multiscale methods is to provide physically sound boundary conditions to the particle domain suppressing any interface effects at the domain boundary occurring due to the coupling. These interfacial coupling artifacts still pose a significant problem, especially for amorphous polymers due to their highly irregular microstructure. We solve this problem by extending the particle-continuum interface by a layer of passive atoms which move with the outer continuum, thereby providing the missing interactions with a surrounding polymer bulk to the inner particle region. This solution allows us to successfully reproduce structural and mechanical properties obtained under conventional periodic boundary conditions, like density, stress, Young’s modulus, and Poisson’s ratio. Furthermore, we demonstrate the application of a nonaffine deformation by means of a simple bending test. In general, our revised method provides a framework to apply complex deformations for molecular scientists, while it allows the engineering community to examine challenging phenomena such as fracture behavior at a molecular level.
... The former determines quantities on the fine scale that are applied in a subsequent analysis on the coarse scale. As an example, a material characterization based on molecular dynamics (MD) pseudo-experiments [11,12] is used to subsequently calibrate a continuum mechanical constitutive law [13] enabling the analysis of significantly larger samples, e.g., with the finite element method (FE). In contrast, concurrent multiscale approaches apply both resolutions simultaneously and are thus suited for the analysis of complex problems. ...
... To this end, a thorough characterization of the particle domain's mechanical properties is required to enable choice and proper calibration of the constitutive law for the continuum. Based on MD pseudo-experiments as detailed 155 in [77,78,79], appropriate constitutive descriptions have been chosen, and an inverse parameter identification has been carried out [80]. A crucial aspect of this procedure is to carefully choose the load range in which the continuum model is valid: In general, larger deformations require more sophisticated models but enable to reduce the size of regions to be treated entirely on the particle scale. ...
This contribution introduces an unconventional procedure to characterize spatial profiles of elastic and inelastic properties inside polymer interphases around nanoparticles. Interphases denote those regions in the polymer matrix whose mechanical properties are influenced by the filler surfaces and thus deviate from the bulk properties. They are of particular relevance in case of nano-sized filler particles with a comparatively large surface-to-volume ratio and hence can explain the frequent observation that the overall properties of polymer nanocomposites cannot be determined by classical mixing rules, which only consider the behavior of the individual constituents.Interphase characterization for nanocomposites poses hardly solvable challenges to the experimenter and is still an unsolved problem in many cases. Instead of real experiments, we perform pseudo experiments using our recently developed Capriccio method, which is an MD-FE domain-decomposition tool specifically designed for amorphous polymers. These pseudo-experimental data then serve as input for a typical inverse parameter identification. With this procedure, spatially varying mechanical properties inside the polymer are, for the first time, translated into intuitively understandable profiles of continuum mechanical parameters.As a model material, we employ silica-enforced polystyrene, for which our procedure reveals exponential saturation profiles for Young's modulus and the yield stress inside the interphase, where the former takes about seven times the bulk value at the particle surface and the latter roughly triples. Interestingly, hardening coefficient and Poisson's ratio of the polymer remain nearly constant inside the interphase. Besides gaining insight into the constitutive influence of filler particles, these unexpected and intriguing results also offer interesting explanatory options for the failure behavior of polymer nanocomposites.
This work introduces a method for generating generalized structures of amorphous polymers using simulated polymerization and molecular dynamics equilibration, with a particular focus on amorphous polymers. The techniques and algorithms used in this method are described in the main text, and example input scripts are provided for the GMXPolymer code, which is based on the GROMACS molecular dynamics package. To demonstrate the efficacy of our method, we apply it to different glassy polymers exhibiting varying degrees of functionality, polarity, and rigidity. The reliability of the method is validated by comparing simulation results with experimental data in various structural and thermal properties, both of which show excellent agreement.
This work implements the GMXPolymer simulated polymerization algorithm on the GROMACS program. GMXPolymer code controls the main polymerization loop. The energy minimizations and molecular dynamics simulations use the GROMACS program called by the GMXPolymer code. A new ITP file is generated when a new bond is formed, and the necessary additions to the ITP file are made to include new bonds, angles, and dihedrals. In preparing the ITP file of the monomer, the charge of the reactive atom must be modified before the code runs so that it is a correct value after bonding.
In this study, a micromechanical cyclic constitutive model for accurately describing the ratchetting deformation of polymeric composites is proposed. The model is based on the mean‐field homogenization method and incorporates the nonlinear viscoelastic‐viscoplastic (VE‐VP) cyclic constitutive model of the polymeric matrix and the imperfect interface. First, the nonlinear VE‐VP model was linearized using the incremental affine method and then was introduced into the homogenization process. The imperfect interface was represented using a linear spring model. A modified Mori‐Tanaka model was developed to incorporate the nonlinear VE‐VP model and imperfect interface into the micromechanical constitutive model. To implement the proposed micromechanical model, a numerical algorithm was developed. The model is capable of capturing the influence of the imperfect interface on the ratchetting deformation of polymeric composites. Furthermore, it is applicable to simulate various loading histories, including monotonic tension, stress relaxation, creep, and uniaxial ratchetting loads. The proposed model was validated by comparing its simulations with those obtained from the full‐field finite element method (FEM) model. The proposed model can predict the ratchetting deformation of polymeric composites and promote the practical application of these material in structures.
Highlights
A micromechanical cyclic constitutive model is developed to describe the ratchetting deformation of polymeric composites with imperfect interface.
Nonlinear viscoelastic‐viscoplastic deformation of polymeric matrix is considered.
A modified Mori‐Tanaka method is employed to incorporate the nonlinear constitutive model and imperfect interface into the process of homogenization.
Good agreement between simulations of the proposed model and the full‐field FEM model under various loading histories.
This contribution presents a partitioned‐domain particle‐continuum coupling method for amorphous polymers with multiple particle‐based domains. The coupling method treats the particle‐based domains with molecular dynamics (MD) simulations and the continuum domain discretized by the Finite Element (FE) method. In the continuum domain, a viscoelastic‐viscoplastic (VE‐VP) constitutive model derived from MD simulation results of the polymer at molecular resolution is employed. The effects of the minimum distances between the domains, the distribution and the number of the MD domains as well as the strain rates are studied under uniaxial tension. This method is a precursor for multiscale simulations of polymer‐based nanocomposites (PNC).
The complex nonlinear constitutive behavior of amorphous polymers has been extensively investigated in numerous experiments, displaying a strong dependence on the thermal history and loading path. Theoretical models have been proposed to conjoin the underlying mechanisms to certain aspects of mechanical features, such as yielding, strain hardening, physical aging, and the Bauschinger effect. However, an urgent need for theoretical understanding capable of fulfilling the comprehensive predictions of the featured constitutive behaviors of amorphous polymers is to be addressed. In this paper, we develop a micromechanical mean-field shear transformation zone (STZ) model to systematically describe the stress responses of amorphous polymers. The dynamics of STZ is affected by the nonequilibrium structural state described by effective temperature and also related to the anisotropic internal structure induced by the microscopical deformation. Our model exhibits the ability to fully capture the behaviors of amorphous polymers in constant rate/switch rate loading, creep, relaxation, and loading-unloading-reloading cycles. The model is further verified by reproducing the Bauschinger effect in experiments, which remains a challenge to other widely used models. Our results show that the deformation-induced material polarization, in the form of an anisotropic distribution of STZs, is the governing mechanism of the Bauschinger effect. This work establishes a comprehensive relationship between the microscopic mechanisms and the constitutive behaviors of deforming amorphous polymers, which advances the fundamental understanding of complex mechanical behaviors in amorphous glassy polymers.
In the context of fracture simulations of polymers, the molecular mechanisms in the vicinity of the crack tip are of particular interest. Nevertheless, to keep the computational cost to a minimum, a coarser resolution must be used in the remaining regions of the numerical sample. For the specific case of amorphous polymers, the Capriccio method bridges the gap between the length and time scales involved at the different levels of resolution by concurrently coupling molecular dynamics (MD) with the finite element method (FEM). Within the scope of the Capriccio approach, the coupling to the molecular MD region introduces non-periodic, so-called stochastic boundary conditions (SBC). In similarity to typical simulations under periodic boundary conditions (PBC), the SBC MD simulations must reach an equilibrium state before mechanical loads are exerted on the coupled systems. In this contribution, we hence extensively study the equilibration properties of non-periodic MD samples using the Capriccio method. We demonstrate that the relaxation behavior of an MD-FE coupled MD domain utilizing non-periodic boundary conditions is rather insensitive to the specific coupling parameters of the method chosen to implement the boundary conditions. The behavior of an exemplary system equilibrated with the parameter set considered as optimal is further studied under uniaxial tension and we observe some peculiarities in view of creep and relaxation phenomena. This raises important questions to be addressed in the further development of the Capriccio method.
In this work, a fully thermomechanical coupling constitutive theory is developed for amorphous glassy polymers to explain and predict their complex temperature- and rate-related nonlinear behavior during finite deformations. The foundation for the theory is the original kinematic and thermodynamic framework derived by Bouvard et al. (2013), in which internal state variables with clear physical significance are applied to describe the deformation mechanism of polymers. In contrast to the viewpoint of Bouvard et al. (2013), in the proposed model, we further divide the entanglements into two kinds of molecular microstructures, “permanent entanglements” and “dynamic entanglements”, and describe how their evolution during deformation is regarded as the internal mechanism for the material's macroscopic mechanical properties. Further, a series of new constitutive equations related to “dynamic entanglements” are proposed, which account for the strain softening caused by the dissociation of “dynamic entanglements”, reflect the influence of temperature on the dissociation rate of “dynamic entanglements” and include the effect of rate-dependent glass transition temperature (θg) on the microstructure. The equations related to “permanent entanglements” are also illustrated, which are improved based on the the work of Ames et al. (2009). The predictive capacity of the proposed thermomechanical coupling constitutive theory has been numerically realized by writing a user subroutine for finite element program. Taking polycarbonate (PC) as an example, the simulations are compared with experimental results for compression, shear, tension, and creep at different temperatures and strain rates. It is demonstrated that our proposed constitutive model with a more detailed physical meaning can well reproduce nonlinearity before yield, yield peak, strain softening, hardening and unloading behavior for amorphous glassy polymers under different stress states.
Pre-deformed glassy polymers exhibit distinct stress responses with opposite loading directions, referred to as the Bauschinger effect. Although this phenomenon has been known for decades, the underlying microscopic origin remains largely elusive. In this work, we perform coarse-grained molecular dynamics (CGMD) tension and compression simulations on a typical glassy polymer polycarbonate. The intermedia variables of self-entanglement and network orientation are extracted to describe the internal microstructure change during deformation. The results show that the competition between intra-chain deformation and inter-chain friction leads to the occurrence of yielding, while strain hardening is governed by the increase of inter-chain friction. Motived by the physical mechanisms revealed by the CGMD simulations, we further develop a mean-field shear transformation zone (STZ) model which contains the crucial internal variable of self-entanglement. The theoretical model well captures the yielding, strain hardening and the Bauschinger effect observed in MD simulations. By comparing the mechanical responses of the polycarbonates under tension and compression, we contribute the substantial Bauschinger effect to the distinct deformation mechanisms in these loading processes. The increase in yield strength during tensile-reloading is governed by the decrease of self-entanglement, which leads to enhanced inter-chain friction, while the decreased yield strength during compressive-reloading is associated with the increase of self-entanglement, causing reduced inter-chain friction. Overall, this work promotes the fundamental understanding of the complex mechanical responses of glassy polymers and also provides a new continuum-level theoretical framework for amorphous solids.
The expansion of the application of polymers instead of metals has warranted the evaluation and modeling of their mechanical behaviors under various conditions. In the present study, the strain-rate- and temperature-dependent nonlinear mechanical behavior of a glassy polymer from small to large strain ranges is investigated experimentally and numerically. The effects of strain rate and temperature on the mechanical responses and development of the strain field under monotonic and cyclic uniaxial tensile tests of epoxy are evaluated experimentally. In addition, the stress relaxation test is performed to evaluate the time-dependent mechanical behavior of epoxy. In the previous work (Uchida et al. 2019), the concept of the transient network (TN) theory was applied to demonstrate the nonlinear behavior of the thermosetting glassy polymer. In this model, the time-dependent mechanical behavior of the thermosetting glassy polymer was assumed to be characterized by the debonding and rebonding of intermolecular secondary bonds (SBs). In the present study, the TN model is extended to represent the temperature- and strain rate-dependent mechanical behavior of the thermosetting glassy polymer below the glass transition temperature Tg. The decrease in the deformation resistance of a polymer at a higher temperature is represented by a decrease in the SB density. Furthermore, the fixation parameter is introduced to accurately predict the residual strain in the cyclic tensile test. The mechanical responses in the monotonic and cyclic tensile tests and the stress relaxation test predicted by the extended TN model are consistent with the experimental results. The proposed model is further verified using additional experimental results provided in references. The obtained results demonstrate the nonlinear deformation behaviors of glassy polymers under wide ranges of temperature and strain rate.
This contribution introduces an unconventional procedure to characterize spatial profiles of elastic and inelastic properties inside polymer interphases around nanoparticles. Interphases denote those regions in the polymer matrix whose mechanical properties are influenced by the filler surfaces and thus deviate from the bulk properties. They are of particular relevance in case of nano-sized filler particles with a comparatively large surface-to-volume ratio and hence can explain the frequent observation that the overall properties of polymer nanocomposites cannot be determined by classical mixing rules, which only consider the behavior of the individual constituents.
Interphase characterization for nanocomposites poses hardly solvable challenges to the experimenter and is still an unsolved problem in many cases. Instead of real experiments, we perform pseudo experiments using our recently developed Capriccio method, which is an MD-FE domain-decomposition tool specifically designed for amorphous polymers. These pseudo-experimental data then serve as input for a typical inverse parameter identification. With this procedure, spatially varying mechanical properties inside the polymer are, for the first time, translated into intuitively understandable profiles of continuum mechanical parameters.
As a model material, we employ silica-enforced polystyrene, for which our procedure reveals exponential saturation profiles for Young’s modulus and the yield stress inside the interphase, where the former takes about seven times the bulk value at the particle surface and the latter roughly triples. Interestingly, hardening coefficient and Poisson’s ratio of the polymer remain nearly constant inside the interphase. Besides gaining insight into the constitutive influence of filler particles, these unexpected and intriguing results also offer interesting explanatory options for the failure behavior of polymer nanocomposites.
In modern engineering applications, plastics play an important role for instance in the field of lightweight constructions or as substitutes for classical materials like wood, metal, or glass. Typically, they consist of organic polymers, which are long-chained molecules comprising numerous monomers as repeat units. In addition, polymers frequently contain fillers, plasticisers, or colourants to achieve and to adjust specific properties. In recent years, new techniques have been established to produce and to disperse filler particles in the range of nanometres, which corresponds to the typical dimensions of the monomers. Experiments reveal that these so-called “nanofillers” may significantly toughen polymers, improve their fatigue lifetime, and enhance control of their thermodynamical properties, even for low filler contents in terms of mass or volume. This cannot be explained by a simple rule of mixture, but is traced back to the very large ratio of surface to volume in case of nanofillers and to the associated processes at the molecular level. The effective design of such “nanocomposites” is demanding and often requires timeconsuming mechanical testing. For a better understanding of the relevant parameters and in order to improve the process of material development, it is beneficial to substitute “real” experiments by numerical simulations. To this end, sophisticated computation techniques are required that account for the specific processes taking place at the level of atoms and molecules. Particle-based strategies, as for instance employed in physical chemistry, are able to consider the atomistic structure in detail and thus permit to simulate material behaviour at atomistic length scales. However, it is still not possible to apply these techniques to large-scale systems relevant in engineering. There, the material behaviour of structures is typically described by continuum approaches, which, on the other hand, cannot account explicitly for the processes at the atomistic level. To overcome this, the present thesis proposes a novel coupling scheme to incorporate particle-based simulations into continuum-based methods. In particular, it links molecular dynamics as a standard tool in physical chemistry with the finite element method, which is nowadays widely used in engineering applications. This multiscale simulation approach has been developed jointly by the Theoretical Physical Chemistry Group at the Technische Universität Darmstadt and the Chair of Applied Mechanics at the Friedrich-Alexander-Universität Erlangen-Nürnberg. Thus, it bases upon expertise in atomistic simulation as well as continuum mechanics, whereby crucial modifications of established techniques in both fields had to be developed. Two sample systems, modelling pure polystyrene and a polystyrene-silica nanocomposite, are studied numerically and prove the suitability of the new approach. In this context, various parameters of the proposed method and its implementation are investigated. Based on this, a number of options to improve this multiscale technique are discussed and relevant issues for future research are summarised.
This contribution introduces an unconventional procedure to characterize spatial profiles of elastic and inelastic properties inside polymer interphases around nanoparticles. Interphases denote those regions in the polymer matrix whose mechanical properties are influenced by the filler surfaces and thus deviate from the bulk properties. They are of particular relevance in case of nano-sized filler particles with a comparatively large surface-to-volume ratio and hence can explain the frequent observation that the overall properties of polymer nanocomposites cannot be determined by classical mixing rules, which only consider the behavior of the individual constituents.Interphase characterization for nanocomposites poses hardly solvable challenges to the experimenter and is still an unsolved problem in many cases. Instead of real experiments, we perform pseudo experiments using our recently developed Capriccio method, which is an MD-FE domain-decomposition tool specifically designed for amorphous polymers. These pseudo-experimental data then serve as input for a typical inverse parameter identification. With this procedure, spatially varying mechanical properties inside the polymer are, for the first time, translated into intuitively understandable profiles of continuum mechanical parameters.As a model material, we employ silica-enforced polystyrene, for which our procedure reveals exponential saturation profiles for Young's modulus and the yield stress inside the interphase, where the former takes about seven times the bulk value at the particle surface and the latter roughly triples. Interestingly, hardening coefficient and Poisson's ratio of the polymer remain nearly constant inside the interphase. Besides gaining insight into the constitutive influence of filler particles, these unexpected and intriguing results also offer interesting explanatory options for the failure behavior of polymer nanocomposites.
In this contribution we present an extension of the multiscale Capriccio method towards inelasticity. This enables coupled simulations of a particle domain embedded into a continuum with particular focus on polymer systems. Starting from the method's initial implementation of pure elasticity, we substitute the nonlinear elastic continuum constitutive law by a recently developed viscoelastic‐viscoplastic one which is able to capture the mechanical behaviour of the particle system in a much larger strain range. Furthermore, we discuss numerical aspects like the choice of time steps and iteration numbers.
In this contribution, we present a characterization methodology to obtain pseudo experimental deformation data from CG MD simulations of polymers as an inevitable prerequisite to choose and calibrate continuum mechanical constitutive laws. Without restriction of generality, we employ a well established CG model of atactic polystyrene as exemplary model system and simulate its mechanical behavior under various uniaxial tension and compression load cases. To demonstrate the applicability of the obtained data, we exemplarily calibrate a viscoelastic continuum mechanical constitutive law. We conclude our contribution by a thorough discussion of the findings obtained in the numerical pseudo experiments and give an outline of subsequent research activities. Thus, this work contributes to the field of multiscale simulation methods and adds a specific application to the body of knowledge of CG MD simulations.
In this work, we have developed a multiscale computational algorithm to couple finite element method with an open source molecular dynamics code – the large scale atomic/molecular massively parallel simulator (LAMMPS) – to perform hierarchical multiscale simulations in highly scalable parallel computations. The algorithm was firstly verified by performing simulations of single crystal copper deformation, and a good agreement with the well-established method was confirmed. Then, we applied the multiscale method to simulate mechanical responses of a polymeric material composed of multi-million fine scale atoms inside the representative unit cells (r-cell) against uniaxial loading. It was observed that the method can successfully capture plastic deformation in the polymer at macroscale, and reproduces the double yield points typical in polymeric materials, strain localization and necking deformation after the second yield point. In addition, parallel scalability of the multiscale algorithm was examined up to around 100 thousand processors with 10 million particles, and an almost ideal strong scaling was achieved thanks to LAMMPS parallel architecture.
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In this paper, a new constitutive model is proposed for the behavior of thermoplastic polymers under non-isothermal conditions. The model couples linear viscoelasticity, viscoplasticity and thermal effects. It is formulated within the framework of irreversible thermodynamics. The total strain is the sum of viscoelastic, viscoplastic and thermal strains. General hereditary integrals describe the thermo-viscoelastic response. The viscoplastic part accounts for both isotropic and kinematic hardenings. The stress-strain response and the material self-heating are predicted and compared to experimental data on Polyamide 66 (PA66) and Polypropylene (PP). Good agreement between the numerical simulations and experimental data was obtained for the two materials.
We perform a systematic application of the hybrid particle-field molecular dynamics technique [Milano et al, J. Chem. Phys. 2009, 130, 214106] to study interfacial properties and potential of mean force (PMF) for separating nanoparticles (NPs) in a melt. Specifically, we consider Silica NPs bare or with Polystyrene chains, aiming to shed light on the interactions among free and grafted chains affecting the dispersion of NPs in the nanocomposite. The proposed hybrid models show good performances in catching the local structure of the chains, and in particular their density profiles, documenting the existence of the “wet-brush-to-dry-brush” transition. By using these models, the PMF between pairs of ungrafted and grafted NPs in Polystyrene matrix are calculated. Moreover, we estimate the three-particle contribution to the total PMF and its role in regulating the phase separation on the nanometer scale. In particular, the multi-particle contribution to the PMF is able to give an explanation of the complex experimental morphologies observed at low grafting densities. More in general, we propose this approach and the models utilized here for a molecular understanding of specific systems and the impact of the chemical nature of the systems on the composite final properties.
Reptation theory has been highly successful in explaining the unusual material properties of entangled polymer solutions. It reduces the complex many-body dynamics to a single-polymer description, where each polymer is envisaged to be confined to a tube through which it moves in a snake-like fashion. For flexible polymers, reptation theory has been amply confirmed by both experiments and simulations. In contrast, for semiflexible polymers, experimental and numerical tests are either limited to the onset of reptation, or were performed for tracer polymers in a fixed, static matrix. Here, we report Brownian dynamics simulations of entangled solutions of semiflexible polymers, which show that curvilinear motion along a tube (reptation) is no longer the dominant mode of dynamics. Instead, we find that polymers disentangle due to correlated constraint release, which leads to equilibration of internal bending modes before polymers diffuse the full tube length. The physical mechanism underlying terminal stress relaxation is rotational diffusion mediated by disentanglement rather than curvilinear motion along a tube.
Highly-crosslinked epoxy resins are used as matrix material for high-performance composites, typically developed for weight-reduction purposes in structural aerospace applications. If the clever assembling of different materials into large integrated composite panels can lead to outstanding mechanical properties with a guaranteed weight reduction, it also involves complex deformation mechanisms and failure scenarios which are not yet fully understood today. The development of more predictive modeling capabilities based on multi-scale modeling strategies therefore requires a better modeling of the epoxy matrix behavior.
In this thesis, a thorough study of the mechanical and fracture behavior of the HexFlow RTM6 epoxy resin, certified for aeronautics, is performed. A wide mechanical testing campaign provides the basis for the identification of a phenomenological elastic-viscoelastic constitutive model, exhibiting most of the mechanical features of usual amorphous thermoplastic polymer systems. This model provides good prediction of the RTM6 deformation response under monotonic loading, accounting for rate, temperature and pressure dependence. The failure behavior of the resin is also accounted for through a macroscopic fracture criterion.
Nevertheless, the unloading behavior and complex time-dependent mechanisms (e.g. creep, recovery and rate-reversal phenomena) are rarely well accounted for by phenomenological models, stressing a lack of physical insights in the modeling approach. A new physics-based mesoscopic modeling approach, inspired from the concept of shear transformation zones as the vehicle for elementary plastic deformation, is developed by coupling a FE framework with a time-dependent Monte-Carlo kinetic model. The proposed model, which requires only 6 parameters, captures all the observed experimental trends of RTM6. This strongly supports the theoretical foundations of this relatively simple micro-mechanical approach and opens the route to promising applications and extensions in the field of glassy polymers in general.
A large strain hyperelastic phenomenological constitutive model is proposed to model the highly nonlinear, rate-dependent mechanical behavior of amorphous glassy polymers under isothermal conditions. A corotational formulation is used through the total Lagrange formalism. At small strains, the viscoelastic behavior is captured using the generalized Maxwell model. At large strains beyond a viscoelastic limit characterized by a pressure-sensitive yield function, which is extended from the Drucker-Prager one, a viscoplastic region follows. The viscoplastic flow is governed by a non-associated Perzyna-type flow rule incorporating this pressure-sensitive yield function and a quadratic flow potential in order to capture the volumetric deformation during the plastic process. The stress reduction phenomena arising from the post-peak plateau and during the failure stage are considered in the context of a continuum damage mechanics approach. The post-peak softening is modeled by an internal scalar, so-called softening variable, whose evolution is governed by a saturation law. When the softening variable is saturated, the rehardening stage is naturally obtained since the isotropic and kinematic hardening phenomena are still developing. Beyond the onset of failure characterized by a pressure-sensitive failure criterion, the damage process leading to the total failure is controlled by a second internal scalar, so-called failure variable. The final failure occurs when the failure variable reaches its critical value. To avoid the loss of solution uniqueness when dealing with the continuum damage mechanics formalism, a non-local implicit gradient formulation is used for both the softening and failure variables, leading to a multi-mechanism non-local damage continuum. The pressure sensitivity considered in both the yield and failure conditions allows for the distinction under compression and tension loading conditions. It is shown through experimental comparisons that the proposed constitutive model has the ability to capture the complex behavior of amorphous glassy polymers, including their failure.
Although traditional constitutive models for rubbery elastic materials are incompressible, many materials that demonstrate nonlinear elastic behavior are somewhat compressible. Clearly important in hydrostatic deformations, compressibility can also significantly affect the response of elastomers in applications for which several boundaries are rigidly fixed, such as bushings, or triaxial states of stress are realized. Compressibility is also important for convergence of finite element simulations in which a rubbery elastic constitutive law is in use. Volume changes that reflect compressibility have been observed historically in both uniaxial tension and hydrostatic compression tests; however, there appear to be no data obtained from both types of tests on the same material by which to validate a compressible hyperelastic law. In this paper, we propose a new compressible hyperelastic constitutive law for elastomers and other rubbery materials in which entropy and internal energy changes contribute to the volume change. Using data from the literature, we show that this law is capable of reproducing both the pressure-volume response of elastomers in hydrostatic compression, as well as the stress-stretch and volume change-stretch data of elastomers in uniaxial tension.
Molecular dynamics simulations are performed to study compressive yielding behavior of epoxy-amine cross-linked polymer networks in the low temperature glassy state. The simulations show a sharp drop in stresses after the elastic regime which was identified to occur due to the activation of wedge disclinations. For the first time in literature, both the chemistry and geometry (critical segment length, angles, bond torsions) involved in the molecular mechanism of compressive yielding have been measured. We analyze these results in the context of the Argon theory which is based on a linear elastic model of wedge disclination. The kink distance calculated using this simple theory gives a surprisingly good match to the results seen from the complex molecular simulation. The yield stress versus temperature predictions of Argon theory are directly compared with molecular simulation results. Finally, the use of Argon theory for extracting yield stresses at quasi-static strain rates from high rate molecular simulations is investigated. (c) 2013 Elsevier Ltd. All rights reserved.
In this work, we construct a new physics-based model of rubber
elasticity to capture the strain softening, strain hardening, and
deformation-state dependent response of rubber materials undergoing
finite deformations. This model is unique in its ability to capture
large-stretch mechanical behavior with parameters that are connected to
the polymer chemistry and can also be easily identified with the
important characteristics of the macroscopic stress-stretch
response. The microscopic picture consists of two components: a
crosslinked network of Langevin chains and an entangled network with
chains confined to a nonaffine tube. These represent, respectively,
changes in entropy due to thermally averaged chain conformations and
changes in entropy due to the magnitude of these conformational
fluctuations. A simple analytical form for the strain energy density is
obtained using Rubinstein and Panyukov's single-chain description of
network behavior. The model only depends on three parameters that
together define the initial modulus, extent of strain softening, and the
onset of strain hardening. Fits to large stretch data for natural
rubber, silicone rubber, VHB 4905 (polyacrylate rubber), and b186 rubber
(a carbon black-filled rubber) are presented, and a comparison is made
with other similar constitutive models of large-stretch rubber
elasticity. We demonstrate that the proposed model provides a complete
description of elastomers undergoing large deformations for different
applied loading configurations. Moreover, since the strain energy is
obtained using a clear set of physical assumptions, this model may be
tested and used to interpret the results of computer simulation and
experiments on polymers of known microscopic structure.
A new simulation technique is introduced to couple a flexible particle domain as encountered in soft-matter systems and a continuum which is solved by the Finite Element (FE) method. The particle domain is simulated by a molecular dynamics (MD) method in coarse grained (CG) representation. On the basis of computational experiences from a previous study, a staggered coupling procedure has been chosen. The proposed MD–FE coupling approximates the continuum as a static region while the MD particle space is treated as a dynamical ensemble. The information transfer between MD and FE domains is realized by a coupling region which contains, in particular, additional auxiliary particles, so-called anchor points. Each anchor point is harmonically bonded to a standard MD particle in the coupling region. This type of interaction offers a straightforward access to force gradients at the anchor points that are required in the developed hybrid approach. Time-averaged forces and force gradients from the MD domain are transmitted to the continuum. A static coupling procedure, based on the Arlequin framework, between the FE domain and the anchor points provides new anchor point positions in the MD–FE coupling region. The capability of the new simulation procedure has been quantified for an atactic polystyrene (PS) sample and for a PS-silica nanocomposite, both simulated in CG representation. Numerical data are given in the linear elastic regime which is conserved up to 3% strain. The convergence of the MD–FE coupling procedure has been demonstrated for quantities such as reaction forces or the Cauchy stress which have been determined both in the bare FE domain and in the coupled system. Possible applications of the hybrid method are shortly mentioned.
In this paper we present a theory of finite deformation viscoelasticity. The presentation is not restricted to small perturbations from the elastic equilibrium in contrast to many viscoelasticity theories. The fundamental hypothesis of our model is the multiplicative viscoelastic decomposition of Sidoroff (1974). This hypothesis is combined with the assumption of a viscoelastic potential to give a model that is formally similar to finite associative elasto-plasticity. Examples are given to compare the present proposal to an alternative formulation in the literature for the cases of uniaxial plane strain relaxation and creep.
We demonstrate how an iterative method for potential inversion from distribution functions developed for simple liquid systems can be generalized to polymer systems. It uses the differences in the potentials of mean force between the distribution functions generated from a guessed potential and the true distribution functions to improve the effective potential successively. The optimization algorithm is very powerful: convergence is reached for every trial function in few iterations. As an extensive test case we coarse-grained an atomistic all-atom model of polyisoprene (PI) using a 13:1 reduction of the degrees of freedom. This procedure was performed for PI solutions as well as for a PI melt. Comparisons of the obtained force fields are drawn. They prove that it is not possible to use a single force field for different concentration regimes.
This contribution introduces an unconventional procedure to characterize spatial profiles of elastic and inelastic properties inside polymer interphases around nanoparticles. Interphases denote those regions in the polymer matrix whose mechanical properties are influenced by the filler surfaces and thus deviate from the bulk properties. They are of particular relevance in case of nano-sized filler particles with a comparatively large surface-to-volume ratio and hence can explain the frequent observation that the overall properties of polymer nanocomposites cannot be determined by classical mixing rules, which only consider the behavior of the individual constituents.
Interphase characterization for nanocomposites poses hardly solvable challenges to the experimenter and is still an unsolved problem in many cases. Instead of real experiments, we perform pseudo experiments using our recently developed Capriccio method, which is an MD-FE domain-decomposition tool specifically designed for amorphous polymers. These pseudo-experimental data then serve as input for a typical inverse parameter identification. With this procedure, spatially varying mechanical properties inside the polymer are, for the first time, translated into intuitively understandable profiles of continuum mechanical parameters.
As a model material, we employ silica-enforced polystyrene, for which our procedure reveals exponential saturation profiles for Young’s modulus and the yield stress inside the interphase, where the former takes about seven times the bulk value at the particle surface and the latter roughly triples. Interestingly, hardening coefficient and Poisson’s ratio of the polymer remain nearly constant inside the interphase. Besides gaining insight into the constitutive influence of filler particles, these unexpected and intriguing results also offer interesting explanatory options for the failure behavior of polymer nanocomposites.
Nano-filled polymers are becoming more and more important to meet the continuously growing requirements of modern engineering problems. The investigation of these composite materials at the molecular level, however, is either prohibitively expensive or just impossible. Multiscale approaches offer an elegant way to analyze such nanocomposites by significantly reducing computational costs compared to fully molecular simulations. When coupling different time and length scales, however, it is in particular important to ensure that the same material description is applied at each level of resolution. The Capriccio method (Pfaller et al, 2012, 2013), for instance, couples a particle domain modeled with molecular dynamics (MD) with a finite element based continuum description and has been used i.a. to investigate the effects of nano-sized silica additives embedded in atactic polystyrene (PS), cf. Pfaller et al (2016); Liu et al (2017). However, a simple hyperelastic constitutive law is used so far for the continuum description which is not capable to fully match the behavior of the particle domain. To overcome this issue and to enable further optimization of the coupling scheme, the material model used for the continuum should be derived directly from pure MD simulations under thermodynamic conditions identical to those used by the Capriccio method. To this end, we analyze the material response of pure PS under uniaxial deformation using strain-controlled MD simulations (Ries et al, 2019). Analogously, we perform simulations under pure shear deformation to obtain a comprehensive understanding of the material behavior. As a result, the present PS shows viscoelastic characteristics for small strains, whereas viscoplasticity is observed for larger deformations. The insights gained and data generated are used to select a suitable material model whose parameters have to be identified in a subsequent parameter optimization.
In this paper, we thoroughly develop strategies to improve the Capriccio method, which is a very promising numerical tool for multiscale simulations of amorphous polymers and polymer composites. It is a coupling technique that uses a partitioned-domain approach to link regions of high resolution, i.e. at the atomistic or molecular level, with a surrounding continuum description. We discuss in detail the differences between the Capriccio method in its original implementation (Pfaller et al. in Comput Methods Appl Mech Eng 260:109–129, 2013) and the ideal case. Based on reference systems, we investigate the sources of these differences and provide strategies for optimisation. By means of numerical examples, we prove the suitability of our considerations and demonstrate significant reductions of the original differences. We conclude the paper with an overview of our current work in progress and prospective future activities.
In elastic and plastic deformation of glassy polymers and with the increase of deformation, the material would exhibit elastic, viscoelastic, and viscoelastic-viscoplastic (VE-VP) responses in sequence. These deformation behaviors should be constitutively modeled in such a way to accurately and efficiently simulate many important physical behaviors and phenomena involved in polymer deformation and processing. The elastic and viscoelastic deformation behavior in the pre-yield region can be well represented by the existing constitutive models. However, the VE-VP response in the post-yield region, such as stress relaxation and strain recovery behaviors, cannot be well modeled yet. In this research, a series of uniaxial compression tests of the stress relaxation and the loading-unloading-recovery behaviors of glassy polymer were firstly carried out. The experimental phenomena show strain-rate-dependent characteristics that cannot be explained by the thermally-activated viscosity/viscoplasticity theory. Therefore, it is proposed that a glassy polymer can be treated as a glassy network linked by the secondary bonds and entanglements; while the glassy network is locally yielded and elastically distorted in the pre-yield region, giving rise to a glassy network resistance (GNR) which accounts for the strain-rate-dependent characteristics. A finite strain thermodynamically-based constitutive model was then proposed to incorporate the effects of the GNR. Finally, the newly proposed model was calibrated and its predictive capability was elaborated with the comparison of experiments as well as two widely reported models. The predicted results demonstrate that the GNR is critical in controlling the post-yield stress relaxation and strain recovery. In addition, the nonlinear evolution of the GNR plays a key role in controlling the nonlinear pre-peak hardening and the nonlinear unloading curve. This research thus not only provides a new constitutive theory for modeling of the VE-VP deformation behavior of glassy polymers, but also advances the understanding of the mechanical responses of glassy polymers in deformation process.
Semi-crystalline polymeric composites are increasingly used as bearing material in the biomedical sector, mainly because of their specific mechanical properties and the new advances in 3D printing technologies that allows for customised devices. Among these applications, total or partial prostheses for surgical purposes must consider the influence of temperature and loading rate. This paper proposes a new constitutive model for semi-crystalline polymers, commonly used as matrix material in a wide variety of biomedical composites, that enables reliable predictions under a wide range of loading conditions. Most of the recent models present limitations to predict the non-linear behaviour of the polymer when it is exposed to large deformations at high strain rates. The proposed model takes into account characteristic behaviours of injected and 3D printed thermoplastic polymers such as material hardening due to strain rate sensitivity, thermal softening, thermal expansion and combines viscoelastic and viscoplastic responses. These viscous-behaviours are relevant for biomedical applications where temperature evolution is expected during the deformation process due to heat generation induced by inelastic dissipation, being essential the thermo-mechanical coupling consideration. The constitutive model is formulated for finite deformations within a thermodynamically consistent framework. Additionally, the model is implemented in a finite element code and its parameters are identified for two biomedical polymers: ultra-high-molecular-weight-polyethylene (UHMWPE) and high density polyethylene (HDPE). Finally, the influence of viscous behaviours on dynamic deformation of semi-crystalline polymeric matrices is analysed. This constitutive model predicts the mechanical behaviour of semi-crystalline polymeric matrices for a wide range of strain rate and temperature conditions, allowing for the optimisation of new composite materials potentially used as effective joint replacement prostheses.
A large deformation viscoelastic-viscoplastic (VE-VP) constitutive framework is proposed for polymers. It is generic assuming isotropy and isothermal conditions and is developed with respect to the reference configuration by satisfying the Clausius-Duhem non-negative dissipation inequality. The Helmholtz free energy is the sum of four contributions: VE, VP, softening and hyperelastic re-hardening. The VE part has an integral form which takes into account the history of a rather non-classical VE strain measure. Various stress measures are computed. The flow rule and the VP response have general expressions in terms of the deviatoric and hydrostatic parts of the Mandel stress tensor. General Prony series are considered for the time-dependent VE moduli. Fully implicit time integration algorithms are presented. Application of the model is made to a glassy polymer -RTM6 Epoxy – where a strongly nonlinear finite strain response is observed in compression-dominated deformation mode. The results of numerical simulations are presented and compared to experimental data.
In this work, a phenomenological model for thermoplastic polymers involving several mechanisms is proposed. The constitutive equations lie within the framework of thermodynamics and account for both viscoelasticty, viscoplasticity and ductile damage. An implicit numerical scheme utilizing the ”return mapping algorithm” is provided along with the formulation of the tangent operator. The parameters of the developed model are experimentally identified through a gradient-based inverse method using three strain-controlled configurations. The model validation is achieved by comparing numerical results with experimental data obtained on a cyclic loading configuration test. Finally, the capabilities of the proposed model are demonstrated with a series of numerical simulations where complex cyclic and non-proportional loading conditions are applied as well as with a structural FE application.
In this work, we proposed a Cauchy-Born rule (CBR) based multiscale model to study mechanical properties of amorphous materials. In this work, we combine a coarse-grained Parrinello-Rahman (CG-PR) method and the Multiscale Cohesive Zone Model (MCZM) method to model the Lennard-Jones (L-J) binary glass and amorphous silicon (a-Si) solid. The proposed CG-PR method applies the CBR to a representative volume element of an amorphous material with representative microstructure pattern, whose side dimension is about twice of the cutoff distance of interatomic interaction. Numerical simulations were carried out, and it is found that CG-RP method can reproduce the stress-strain relations extrapolated from large scale MD simulations for both L-J binary glass as well as amorphous silicon (a-Si).
A recently developed hybrid method is employed to study the mechanical behavior of silica-polystyrene nanocomposites (NCs) under uniaxial elongation. The hybrid method couples a particle domain to a continuum domain. The region of physical interest, i.e., the interphase around a nanoparticle (NP), is treated at molecular resolution, while the surrounding elastic continuum is handled with a finite-element approach. In the present paper we analyze the polymer behavior in the neighborhood of one or two nanoparticle(s) at molecular resolution. The coarse-grained hybrid method allows us to simulate a large polymer matrix region surrounding the nanoparticles. We consider NCs with dilute concentration of NPs embedded in an atactic polystyrene matrix formed by 300 chains with 200 monomer beads. The overall orientation of polymer segments relative to the deformation direction is determined in the neighborhood of the nanoparticle to investigate the polymer response to this perturbation. Calculations of strainlike quantities give insight into the deformation behavior of a system with two NPs and show that the applied strain and the nanoparticle distance have significant influence on the deformation behavior. Finally, we investigate to what extent a continuum-based description may account for the specific effects occurring in the interphase between the polymer matrix and the NPs.
The field of viscoplastic constitutive modeling is evolving, and new models are developed every year by different research groups. This chapter attempts to present some of the most useful models that are available for different polymers, and the trends of where this field is going in the future.
In this Part I, of a two-part paper, we present a detailed continuum-mechanical development of a thermo-mechanically coupled elas-to-viscoplasticity theory to model the strain rate and temperature dependent large-deformation response of amorphous polymeric materials. Such a theory, when further specialized (Part II) should be useful for modeling and simulation of the thermo-mechanical response of components and structures made from such materials, as well as for modeling a variety of polymer processing operations.
A series of uniaxial tests (including multilevel loading-unloading recovery, creep-recovery, and cyclic tension-compression/tension ones) were performed to investigate the monotonic and cyclic viscoelastic-viscoplastic deformations of polycarbonate (PC) polymer at room temperature. The results show that the PC exhibits strong nonlinearity and rate-dependence, and obvious ratchetting occurs during the stress-controlled cyclic tension-compression/tension tests with nonzero mean stress, which comes from both the viscoelasticity and viscoplasticity of the PC. Based on the experimental observation, a nonlinear viscoelastic-viscoplastic cyclic constitutive model is then constructed. The viscoelastic part of the proposed model is constructed by extending the Schapery's nonlinear viscoelastic model, and the viscoplastic one is established by adopting the Ohno-Abdel-Karim's nonlinear kinematic hardening rule to describe the accumulation of irrecoverable viscoplastic strain produced during cyclic loading. Furthermore, the dependence of elastic compliance of the PC on the accumulated viscoplastic strain is considered. Finally, the capability of the proposed model is verified by comparing the predicted results with the corresponding experimental ones of the PC. It is shown that the proposed model provides reasonable predictions to the various deformation characteristics of the PC presented in the multilevel loading-unloading recovery, creep-recovery, and cyclic tension-compression/tension tests.
Very few polymer mechanics problems are solved with only pen and paper today, and virtually all academic research and industrial work relies heavily on finite element simulations and specialized computer software. Introducing and demonstrating the utility of computational tools and simulations, Mechanics of Solid Polymers provides a modern view of how solid polymers behave, how they can be experimentally characterized, and how to predict their behavior in different load environments. Reflecting the significant progress made in the understanding of polymer behaviour over the last two decades, this book will discuss recent developments and compare them to classical theories. The book shows how best to make use of commercially available finite element software to solve polymer mechanics problems, introducing readers to the current state of the art in predicting failure using a combination of experiment and computational techniques. Case studies and example Matlab code are also included. As industry and academia are increasingly reliant on advanced computational mechanics software to implement sophisticated constitutive models - and authoritative information is hard to find in one place - this book provides engineers with what they need to know to make best use of the technology available.
A review of constitutive models for the finite deformation response of rubbery materials is given. Several recent and classic statistical mechanics and continuum mechanics models of incompressible rubber elasticity are discussed and compared to experimental data. A hybrid of the Flory-Erman model for low stretch deformation and the Arruda-Boyce model for large stretch deformation is shown to give an accurate, predictive description of Treloar's classical data over the entire stretch range for all deformation states. The modeling of compressibility is also address.
Performing coarse-grained molecular dynamics simulations, the local dynamics of free and grafted polystyrene chains surrounding a spherical silica nanoparticle has been investigated, where the silica nanoparticle was either bare or grafted with 80-monomer polystyrene chains. The effect of the free (matrix) chain molecular weight and grafting density on the relaxation time of both the free and grafted polystyrene chains has been investigated. Furthermore, we have analyzed the local mobility of the grafted chains at different separations from the nanoparticle surface, as well as on the mean square displacement of the nanoparticles. Proximity to the surface, confinement by the surface, increased grafting density and increased matrix chain length were found to slow down the dynamics of the chain monomers and hence to increase the corresponding relaxation times. “Drying” of the grafted network of the nanoparticle via increasing the free chain lengths, which is known to shrink the brush-height, was found to slow down the relaxation of the brushes, too. The thickness of the interphase, beyond which the polymers showed bulklike behavior, was 2 nm for a bare nanoparticle, corresponding to four monomer layers, for all matrix chain lengths investigated. It increased to 3 nm for grafted nanoparticles depending on the grafting density and the matrix chain molecular weight.
Micromechanical analyses of unidirectional continuous-fibre reinforced composite materials were performed to study the mechanisms of deformation and fracture of the constituents, and their influence on the mechanical properties of the composite. Special focus was given to the matrix material behaviour as well as to the interface between constituents. The matrix was modelled using a pressure dependent, elasto-plastic thermodynamically consistent damage model. Cohesive elements were used to model the interface between matrix and fibres. Part I of this paper details the continuum model developed for a typical epoxy matrix. Part II will focus on micromechanical analyses of composite materials and the estimation of its elastic and strength properties.
The aim of this paper is to develop a selfconsistent theory of rubber-like materials consisting of networks of non-Gaussian chain molecules. Three kinds of series developments are derived for the distribution function of perfectly flexible single chains from the Fourier integral solution of Rayleigh; namely, (1) long chains with actual extension much less than the maximum extension, (2) long chains with actual extension comparable to the maximum extension, and (3) short chains. In the non-Gaussian network theory, the leading term of the series (2) is used as the starting point for the individual chains of the network. Calculations are made for the case where the free junctions are moving with no restriction, and for the case where the free junctions are assumed to be at their most probable positions. The final expressions of the elastic energy for the two cases are compared, and it is shown that the percentage difference of the two expressions is of the order 1/n (n being the average number of links per chain), which is negligible for sufficiently large n. Finally an expression of the elastic energy is obtained with the assumption that all junctions are fixed and is shown to be, in general, a function of three strain invariants. The interdependence of the coefficients of the invariants is shown. Comparison of theory and experiment is given. Because of the interdependence of the coefficients only part of the observed deviations from Gaussian theory can be explained by our molecular theory. The remaining discrepancies must be ascribed to van der Waals forces. This should show up in the (not yet investigated) temperature dependence of these discrepancies.
A viscoelastic-viscoplastic constitutive model for isotropic materials undergoing isothermal infinitesimal deformation is proposed. The model is based on the assumption that the total strain rate is decomposable into a viscoelastic and a viscoplastic portion. Consequently, the model consists of a linear viscoelastic model in series with a modified plasticity model. This modified plasticity model adopts the classical Drucker-Prager yield surface with isotropic hardening and the associative flow rule of the invicid theory of plasticity. However, hardening is assumed to be a function of both the viscoplastic strain as well as the total strain rate. In this manner, the proposed model acquires the advantage of having both the initial and the subsequent yield surfaces to be a function of the strain rate, a property which has its experimental supportive evidence for viscoplastic materials such as polymers and some metals at highly elevated temperatures.A finite-element algorithm is developed to implement the constitutive equation derived in Part I. This algorithm adopts a combination of the tangent stiffness matrix and the initial load approach. The method of treating the transitional region between viscoelastic and viscoelastic-viscoplastic behavior is given. The details of implementation is described. Convergence of the computation scheme is discussed.Two examples are calculated numerically to demonstrate the strain rate and the pressure effects on the mechanical behavior of some viscoelastic-viscoplastic material. Results show that essential features in the stress-strain diagram obtained experimentally are exhibited by the model.
1. General Physical Properties of Rubber 2. Internal Energy and Entropy Changes on Deformation 3. The Elasticity of Long-Chain Molecules 4. The Elasticity of a Molecular Network 5Ex5 Experimental Examination of the Statistical Theory 6. Non-Gaussian Chain Statistics and Network Theory 7. Swelling Phenomena 8. Cross-linking and Modulus 9. Photoelastic Properties of Rubbers 10. The General Strain: Phenomenological Theory 11. Alternative Forms of Strain-Energy Function 12. Large-Deformation Theory: Shear and Torsion 13. Thermodynamic Analysis of Gaussian Network
The volume changes accompanying extension of peroxide vulcanizates of natural gum rubber were measured using a dilatometer technique. Measurements of the force‐extension behavior and compressibilities were made on the same samples for the range of extension and volume change covered in the volume experiments. A constant compressibility was found; however, the volume changes accompanying extension were not proportional to the isotropic part of the stress. Thus, the strain energy cannot be separated into a sum of two parts, one due to the shear and one due to the dilatation.
In some circumstances, elastic-plastic deformation occurs in which both components of strain are finite. Such situations fall outside the scope of classical plasticity theory which assumes either infinitesimal strains or plastic-rigid theory for large strains. The present theory modifies the kinematics to include finite elastic and plastic strain components. For situations requiring this generalization, dilatational influences are usually significant including thermo-mechanical coupling. This is introduced through the consideration of two coupled thermodynamic systems: one comprising thermo-elasticity at finite strain and the other the irreversible process of dissipation and absorption of plastic work. The present paper generalizes a previous theory to permit arbitrary deformation histories. (Author)
Molecular dynamics computer simulations have been carried out of a chemically realistic many-chain nonentangled model of glassy atactic polystyrene under the influence of uniaxial mechanical deformation. Both the initial elastic and the postyield (up to 100% of the deformation) behavior have been simulated. The Poisson ratio, the Young modulus, and the temperature dependence of the yield peak are well reproduced. The simulated strain-hardening modulus is in quantitative agreement with existing experiments. The deformationally induced anisotropy in the global and local segmental orientation is accompanied by an anisotropy of the local translational mobility: the mean-square translational displacement of the individual segments in the direction of the deformation is drastically increased just beyond the yield point as compared to the isotropic sample. The mechanical deformation of a quenched sample leads to an almost complete erasure of the aging history.
Silica nanoparticles (NPs) embedded in atactic polystyrene (PS) are simulated using coarse-grained (CG) potentials obtained via iterative Boltzmann inversion (IBI). The potentials are parametrized and validated on polystyrene of 2 kDa (i.e., chains containing 20 monomers). It is shown that the CG potentials are transferable between different systems. The structure of the polymer chains is strongly influenced by the NP. Layering, chain expansion, and preferential orientation of segments as well as of entire chains are found. The extent of the structural perturbation depends on the details of the system: bare NPs vs NPs grafted with PS chains, grafting density (0, 0.5, and 1 chains/nm2), length of the grafted chains (2 and 8 kDa), and the matrix chains (2–20 kDa). For example, there is a change in the swelling state for the grafted corona (8 kDa, 1 chains/nm2), when the matrix polymer is changed from 2 to > 8 kDa. This phenomenon, sometimes called “wet brush to dry brush transition”, is in good agreement with neutron scattering investigations. Another example is the behavior of the radius of gyration of free polymer chains close to the NP. Short chains expand compared to the bulk, whereas chains whose unperturbed radius of gyration is larger than that of the NP contract.
In this article, we present coarse-grained potentials of ethylbenzene developed at 298 K and of amorphous polystyrene developed at 500 K by the pressure-corrected iterative Boltzmann inversion method. The potentials are optimized against the fully atomistic simulations until the radial distribution functions generated from coarse-grained simulations are consistent with atomistic simulations. In the coarse-grained polystyrene melts of different chain lengths, the Flory exponent of 0.58 is obtained for chain statistics. Both potentials of polystyrene and ethylbenzene are transferable over a broad range of temperature. The thermal expansion coefficients of the fully atomistic simulations are well reproduced in the coarse-grained models for both systems. However, for the case of ethylbenzene, the coarse-grained potential is temperature-dependent. The potential needs to be modified by a temperature factor of √ T/T 0 when it is transferred to other temperatures; T 0) 298 K is the temperature at which the coarse-grained potential has been developed. For the case of polystyrene, the coarse-grained potential is temperature-independent. An optimum geometrical combination rule is proposed with the combination constant x) 0.4 for mutual interactions between the polystyrene monomer and ethylbenzene molecules in their mixtures at different composition and different temperature.
The mechanical behavior of polystyrene and a silica-polystyrene nanocomposite under uniaxial elongation has been studied using a coarse-grained molecular dynamics technique. The Young's modulus, the Poisson ratio and the stress-strain curve of polystyrene have been computed for a range of temperatures, below and above the glass transition temperature. The predicted temperature dependence of the Young's modulus of polystyrene is compared to experimental data and predictions from atomistic simulations. The observed mechanical behavior of the nanocomposite is related to the local structure of the polymer matrix around the nanoparticles. Local segmental orientational and structural parameters of the deforming matrix have been calculated as a function of distance from nanoparticle's surface. A thorough analysis of these parameters reveals that the segments close to the silica nanoparticle's surface are stiffer than those in the bulk. The thickness of the nanoparticle-matrix interphase layer is estimated. The Young's modulus of the nanocomposite has been obtained for several nanoparticle volume fractions. The addition of nanoparticles results in an enhanced Young's modulus. A linear relation describes adequately the dependence of Young's modulus on the nanoparticle volume fraction.
In this paper, different formulations of finite isotropic hyperelastic material laws for compressible solids are considered. Material laws with an additive split of the hyperelastic strain energy function into isochoric parts and volumetric parts are often used in the numerical treatment of nearly incompressible solids. It will be shown that this formulation leads to unphysical results in the simple tension problem when we do not restrict ourselves to nearly incompressible materials.
The relaxation process of viscous flow may be visualized as the sudden shifting of some small patch on one side of a shear surface with respect to the neighboring material on the other side of the shear surface. Any shear surface will divide a mosaic of such patches lying on the two sides of the surface. Except for the simplest systems, this mosaic of patches will be heterogeneous and can be described by groups each characterized by its mean relaxation time βn, by xn the fractional area of the shear surface which the group occupies and by αn, a characteristic shear volume divided by kT. The resulting generalized expression for viscosity is η= n ∑ n=1 xnβn αn sinh -1βns˙ βns˙ , where s˙ is the rate of shear. This equation is applied to masticated natural rubber, polystyrene, X-672 GR-S, X-518 GR-S rubber, and Vistanex LM-S polyisobutylene. All applications give good agreement with experiment. The known criticisms of Eyring's simple relaxation theory for viscous flow are reviewed, and are apparently taken care of in this general treatment.
Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944, Trans. Faraday Soc. 40, 59) illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.
The material behaviour of rubber at the micro level is usually described by means of statistical mechanics. In particular, the Neo-Hooke model has been derived in this fashion. The micromechanical modelling can be extended to include also the breaking and reforming of chains. One possible approach at this level is the so-called transient network theory. Using certain assumptions for the chain distributions, one arrives at a continuum mechanical model of finite viscoelasticity which is based on the multiplicative decomposition of the deformation gradient. This means that the inelastic part of the deformation is regarded as an elastic isomorphism. Further, the considerations at the micro level give information about the temperature dependence of the mechanical material parameters. For instance, it can be shown easily that the shear modulus depends approximately linearly on the temperature. This fact has important consequences for thermo-mechanical coupling which have not yet been discussed in detail in the literature.
In this Part I, of a two-part paper, we present a detailed continuum-mechanical development of a thermomechanically
coupled elasto-viscoplasticity theory to model the strain rate and temperature dependent largedeformation
response of amorphous polymeric materials. Such a theory, when further specialized (Part II)
should be useful for modeling and simulation of the thermo-mechanical response of components and structures
made from such materials, as well as for modeling a variety of polymer processing operations.