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Modeling steady state rate- and temperature-dependent strain hardening behavior of glassy polymers
... Anand et al. [13,14] generalized this formulation by incorporating a unimodular orientation tensor A as an internal variable of the free energy density for orientation and proposed that the evolution of A is subjected to a dynamic recovery term driven by the back stress. Additional approaches have also been proposed, including models with deformation-dependent viscosity [1,3], a viscous model that uses strain rate as an external variable to capture steady-state mechanical behavior [15], and models that represent orientation through shear transformation zones (STZs) [16,17,18]. While these models successfully replicate the stress response of glassy polymers in the hardening region under various conditions, they remain largely phenomenological, each grounded in different assumptions about the origin of hardening stress. ...
... Finally, we develop a constitutive model within the framework established by [14], incorporating orientation-induced back stress. Compared to the model proposed by [14] and various other hardening models [26,1,3,16,17,18,15], the improvement presented in this paper lies in providing a clear physical basis for the stress response in the hardening region, as well as a well-defined interpretation of the deformation gradient decomposition F = F l F r , where the contribution of F l to the stress can be directly extracted from MD simulations. ...
... 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 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.
Adhesively bonded joints using epoxy resin are nowadays often replacing welding in offshore applications for safety reasons. During its lifetime, the bonded joint epoxy is submitted to severe environmental and loading conditions, such as humidity, water uptake, thermal aging, and complex loading conditions affecting its mechanical performance. We investigate here the dynamic mechanical behavior of an epoxy resin at large strain over a wide range of temperatures. Analysis of the elastic modulus, yield strain, yield stress and plastic flow as a function of the temperature and strain rate is carried out. Unlike the elastic modulus and the yield stress showing strong sensitivities to the temperature and the strain rate, the plastic flow appears to have a limited sensitivity to the temperature and the strain rate. Numerical modeling is used to determine the yield stress and elastic modulus variations over the glass transition temperature and good agreement is observed between numerical predictions and experimental results.
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.
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.
Atomistic simulation is a useful method for studying material science phenomena. Examination of the state of a simulated material and the determination of its mechanical properties is accomplished by inspecting the stress field within the material. However, stress is inherently a continuum concept and has been proven difficult to define in a physically reasonable manner at the atomic scale. In this paper, an expression for continuum mechanical stress in atomistic systems derived by Hardy is compared with the expression for atomic stress taken from the virial theorem. Hardy's stress expression is evaluated at a fixed spatial point and uses a localization function to dictate how nearby atoms contribute to the stress at that point; thereby performing a local spatial averaging. For systems subjected to deformation, finite temperature, or both, the Hardy description of stress as a function of increasing characteristic volume displays a quicker convergence to values expected from continuum theory than volume averages of the local virial stress. Results are presented on extending Hardy's spatial averaging technique to include temporal averaging for finite temperature systems. Finally, the behaviour of Hardy's expression near a free surface is examined, and is found to be consistent with the mechanical definition for stress.
Molecular simulations, when they are used to understand properties characterizing the mechanical strength of solid materials, such as stress-strain relation or Born stability criterion, by using elastic constants, are sometimes seriously time consuming. In order to resolve this problem, we propose an efficient simulation approach under constant external stress and temperature, modifying Parrinello-Rahman (PR) method using useful sampling techniques developed recently-massive Nose-Hoover chain method and hybrid Monte Carlo method. Test calculations on the Ni crystal employing the embedded atom method have shown that our method greatly improved the efficiency in sampling the elastic properties compared with the conventional PR method.
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.
The amorphous polymers present remarkable temperature- and rate-dependent deformation behaviors. Based on a combination of the multiple relaxation viscoelastic-viscoplastic model and the three-element viscoelastic model, a constitutive model was constructed to describe the changes in mechanical properties of amorphous polymers from below to above glass transition temperature (θg). In this model, an exponential evolution equation of volume fraction was constructed to reflect the changes in glassy and rubbery phases at different temperatures. The proposed model was implemented into ABAQUS using the user-defined material subroutine (VUMAT). The strain-softening after yield and rate-dependent behaviors above and below θg were reasonably captured by the present model. Meanwhile, the creep and relaxation behaviors of the material were described. Finally, the processes of the tensile deformation of a dumbbell plate with a circular hole and the rate-dependent pressed film molding were simulated by the VUMAT. The results show that the implemented model can reasonably predict the structural responses of amorphous polymers.
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.
Glassy polymers exhibit a strong thermomechanical coupling when subjected to mechanical loading. A homogenous strain distribution can be achieved in uniaxial compression conditions. However, a clear temperature increase is observed when the loading rate is relatively high, which further results in a decrease in stress due to thermal softening. In tensile tests, necking instability can easily occur. A temperature change is accompanied by the nucleation and propagation of necked regions. In this work, we apply the effective temperature theory, which can capture the nonequilibrium structure evolution of amorphous polymers, to investigate the thermomechanical behavior of glassy polymers. We demonstrate that a finite element model based on the effective temperature theory can well capture the stress response and the temperature increase of polycarbonate (PC) compressed at different loading rates. We further combine experiments and simulations to investigate the effects of the loading rate and aging treatment on the necking behavior in amorphous glassy polymer poly(ethylene terephthalate)-glycol (PETG) in uniaxial tension loading tests. Through employing digital image correlation and infrared thermometry, the strain distribution and the temperature field can be fully characterized during the formation and propagation of necked regions. The model captures all important features of localization behavior observed in experiments, including the force–displacement relationship, and the strain and temperature distribution with necking propagation. However, it underestimates the maximum strain and temperature when the displacement is large. Both experimental and simulative results indicate that increasing loading rate and aging time can induce an increase of the intrinsic strain softening due to a more active structural state caused by mechanical rejuvenation. This further leads to more pronounced localized behavior and a ductile–brittle transition. Our work proves that the effective temperature model is a powerful theoretical framework to predict the complex thermomechanical response of glassy polymers.
Understanding the mechanical responses of glassy polymers remains a grand scientific challenge for many decades, which plays a central role in the application of these materials In this work, we have performed stress relaxation tests on a series of glassy polymers. An abnormal rate-dependent stress relaxation response is revealed, featuring as a larger stress drop and smaller quasi-steady-state stress with increasing the initial loading rate. To explore the underlying deformation mechanism, we further carry out a coarse-grained molecular dynamics simulation on glassy polymers, which qualitatively demonstrates the experimental observations. The underlying physical mechanism is revealed to be that a larger initial loading rate can lead to a more-activated structure state of polymers. Based on these findings, we develop a physically-based model by introducing the effective temperature model into the shear transformation zone (STZ) theory. The effective temperature, as an internal variable, can describe the deformation-induced structural evolution of amorphous polymers. The Adam-Gibbs model is then adopted to relate the effective temperature with the transformation rates of STZ sites. Comparison with experimental results shows that the micro-mechanical model can capture the rate-dependent stress–strain relationship in the loading process as well as the abnormal stress relaxation responses. In comparison, it is challenging to rationalize this abnormal behavior based on classic viscoplastic models due to a lack of physical mechanisms. Thus, this work promotes our understanding of the origin of complex mechanical responses of glassy polymers that are strongly related to the evolution of microstructure upon deformation.
Since the classical molecular dynamics simulator LAMMPS was released as an open source code in 2004, it has become a widely-used tool for particle-based modeling of materials at length scales ranging from atomic to mesoscale to continuum. Reasons for its popularity are that it provides a wide variety of particle interaction models for different materials, that it runs on any platform from a single CPU core to the largest supercomputers with accelerators, and that it gives users control over simulation details, either via the input script or by adding code for new interatomic potentials, constraints, diagnostics, or other features needed for their models. As a result, hundreds of people have contributed new capabilities to LAMMPS and it has grown from fifty thousand lines of code in 2004 to a million lines today. In this paper several of the fundamental algorithms used in LAMMPS are described along with the design strategies which have made it flexible for both users and developers. We also highlight some capabilities recently added to the code which were enabled by this flexibility, including dynamic load balancing, on-the-fly visualization, magnetic spin dynamics models, and quantum-accuracy machine learning interatomic potentials.
Program Summary
Program Title: Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)
CPC Library link to program files: https://doi.org/10.17632/cxbxs9btsv.1
Developer's repository link: https://github.com/lammps/lammps
Licensing provisions: GPLv2
Programming language: C++, Python, C, Fortran
Supplementary material: https://www.lammps.org
Nature of problem: Many science applications in physics, chemistry, materials science, and related fields require parallel, scalable, and efficient generation of long, stable classical particle dynamics trajectories. Within this common problem definition, there lies a great diversity of use cases, distinguished by different particle interaction models, external constraints, as well as timescales and lengthscales ranging from atomic to mesoscale to macroscopic.
Solution method: The LAMMPS code uses neighbor lists, parallel spatial decomposition, and parallel FFTs for long-range Coulombic interactions [1]. The time integration algorithm is based on the Størmer-Verlet symplectic integrator [2], which provides better stability than higher-order non-symplectic methods. In addition, LAMMPS supports a wide range of interatomic potentials, constraints, diagnostics, software interfaces, and pre- and post-processing features.
Additional comments including restrictions and unusual features: This paper serves as the definitive reference for the LAMMPS code.
References
[1]S. Plimpton. Fast parallel algorithms for short-range molecular dynamics. J. Comp. Phys., 117:1–19, 1995.
[2]L. Verlet. Computer experiments on classical fluids: I. Thermodynamical properties of Lennard–Jones molecules. Phys. Rev., 159:98–103, 1967.
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).
(Free access link until July 16, 2021:
https://authors.elsevier.com/a/1d8MX_UECrkz1)
The potential and the interest of the development of mechanical approach based on concepts issued from the physics of polymers, as well as of the use of the time temperature equivalence principle, are illustrated. To achieve that point, a revisited constitutive model [1], [2] was used to model the mechanical behaviour of amorphous PMMA with different molecular weights. The model accounts for the elastic contribution of an equivalent network which experiences inelastic mechanisms coming from the evolution of internal state variables when the polymer is deformed. The experimental database included non-monotonic tensile tests at targeted “equivalent strain rate at reference temperature” coupled with DIC for obtaining local boundary conditions. Model exhibited good capabilities to capture the mechanical response of the material at different temperatures and strain rates corresponding to material state ranging from the end of the glassy state to near-liquid state going through viscoelastic and rubbery regime. Analysis of the parameters allowed introducing empirical equations to consider the time/temperature dependence into the model. It was possible to pretty well reproduce the behaviour of PMMAs from rubbery like domains to their glassy state with one unique formalism and one unique and of reduced number set of parameters. Effect of molar mass and crosslinking are discussed.
The rate- and temperature-dependence of the macro-yield behavior, as one of the important concerns for the investigation and application of amorphous glassy polymers, is very significant below and near the glass transition temperature (Tg). Two molecular chain microstructures, i.e., sub-entanglement (SE) and topological entanglement (TE) and their evolution, are introduced as the inherent micro-mechanisms for the rate- and temperature-dependent macro-yield performance. With the introduced entanglement microstructures-based internal variables, an elasto-viscoplastic finite deformation constitutive model of amorphous glassy polymers is proposed to describe its rate- and temperature-dependent macro-yield response. The rate-dependence of Tg and its effect on the microstructures’ thermo-mechanical state and evolution are also taken into account. Comparing with the experimental results of PMMA in literature, the proposed model shows a good capability to simulate and predict the macro-yield behavior at various strain rates (from 0.0003s⁻¹ to 0.1s⁻¹) and temperatures (from room temperature to Tg). This work illustrated that the introduction of SE and TE as the inherent micro-mechanisms is a reasonable interpretation for the macro-yield behavior of amorphous glassy polymers.
The Kremer−Grest (KG) polymer model is a
standard model for studying generic polymer properties in
molecular dynamics simulations. It owes its popularity to its
simplicity and computational efficiency, rather than its ability to
represent specific polymers species and conditions. Here we show
that by tuning the chain stiffness it is possible to adapt the KG
model to model melts of real polymers. In particular, we provide
mapping relations from KG to SI units for a wide range of
commodity polymers. The connection between the experimental
and the KG melts is made at the Kuhn scale, i.e., at the crossover
from the chemistry-specific small scale to the universal large scale
behavior. We expect Kuhn scale-mapped KG models to faithfully represent universal properties dominated by the large scale
conformational statistics and dynamics of flexible polymers. In particular, we observe very good agreement between entanglement
moduli of our KG models and the experimental moduli of the target polymers.
The low recovery stress has limited utilizing shape-memory polymers (SMPs) as actuators. In this work, we demonstrate that the resultant recovery stress of amorphous SMPs can be significantly increased through cold programming. A three-dimensional model is formulated to describe the thermo-mechanical behavior and shape-memory performance of amorphous polymers in large deformation. The constitutive relationship is derived based on the two-temperature thermodynamic framework employing an effective temperature as a thermodynamic state variable to describe the nonequilibrium structure of amorphous polymers. The model also incorporates the molecular orientation with a relaxation mechanism to describe strain hardening. The model is applied to simulate the stress-strain relationship and stress relaxation behaviors of an amorphous SMP in the programming process and the stress response in the constrained recovery process. The model can accurately reproduce the measured stress response at different temperatures and loading rates. Both experimental and simulation results show that polymers deformed at a larger loading rate exhibit a lower stress in the holding period, though simulation pronouncedly underestimates the magnitude of stress drop in the stress relaxation process. The model also captures the magnitude and location of the peak stress during the constrained recovery process for SMPs programmed at different temperatures and with different applied strains.
A novel viscous dissipation potential is proposed for the visco-hyperelastic constitutive modeling of short-time memory responses of soft materials, which can capture both linear and nonlinear large deformation behaviors over a wide range of strain rates. The proposed potential is compatible with objectivity and continuum thermodynamics principles, consists of physically motivated model parameters, and adds the capability of modeling strain rate sensitivity in the small strain regime, which is currently not possible with available external state variable driven viscous dissipation potentials. By combining the proposed viscous dissipation potential with the Mooney-Rivlin strain energy density function, a visco-hyperelastic relation is formulated and fit to the rate-dependent tensile stress-strain data of human patellar tendon, which was previously modeled using an existing viscous dissipation potential. It is demonstrated that the proposed model offers improvements in fitting accuracy and prevents possible thermodynamic instabilities in quasi-static hyperelastic models from corrupting the dynamic response. In addition, the uncomplicated mathematical form of the model and the accompanying multi-step multi-start optimization procedure helps prevent numerical instabilities. Multi-deformation mode fitting of human brain gray matter under all three primary deformation modes (compression, tension and shear) is also considered using a visco-hyperelastic model based on the proposed potential and the semi-empirical Gent-Gent strain energy density function. It is shown that visco-hyperelastic models based on the proposed viscous dissipation potential capture all the essential features of the stress-strain data with unique optimal model parameters, giving reasonable accuracy in both single and multiple deformation mode cases. Further, it is demonstrated that the proposed model is stable and robust with respect to both the choice of the hyperelastic strain energy density and the availability of data from multiple deformation modes.
In this paper, we present an experimental study on strain hardening of amorphous thermosets. A series of amorphous polymers is synthesized with similar glass transition regions and different network densities. Uniaxial compression tests are then performed at two different strain rates spanning the glass transition region. The results show that a more pronounced hardening response can be observed as decreasing temperature and increasing strain rate and network density. We also use the Neo-Hookean model and Arruda-Boyce model to fit strain hardening responses. The Neo-Hookean model can only describe strain hardening of the lightly cross-linked polymers, while the Arruda-Boyce model can well describe hardening behaviors of all amorphous networks. The locking stretch of the Arruda-Boyce model decreases significantly with increasing network density. However, for each amorphous network, the locking stretch is the same regardless of the deformation temperature and rate. The hardening modulus exhibits a sharp transition with temperature. The transition behaviors of hardening modulus also vary with the network density. For lightly crosslinked networks, the hardening modulus changes 60 times with temperature. In contrast, for heavily crosslinked polymers, the hardening modulus in the glassy state is only 2 times of that in the rubbery state. Different from the results from molecular dynamic simulation in literatures, the hardening modulus of polymers in the glassy state does not necessarily increase with network density. Rather, the more significant hardening behaviors in more heavily crosslinked polymers are attributed to a lower value of the stretch limit.
The relevance of equivalent strain rate at reference temperature derived from time/temperature superposition principle is validated as a constitutive parameter at large strain for PMMAs of different chain architecture. Shift factors were obtained from DMTA at infinitesimal strain, then identified according to Williams-Landel-Ferry or Arrhenius equations and finally extended to large deformations. Mechanical behaviour was characterized under cyclic tensile loading. So-called 3D digital image correlation was used to measure local strain. It is demonstrated that for different experimental conditions having same equivalent strain rate, the macroscopic behaviour will be the same. This was validated for elastoplastic, viscoelastic and rubbery behaviours. Such experimental observations indicate that time/temperature superposition at low strain can be extended for large deformation for PMMA. Additionally, the study opens a new way of addressing the temperature and strain rate dependencies in constitutive model by using the equivalent strain rate at reference temperature as a unique parameter.
Due to the lack of the long-range order in their molecular structure, amorphous polymers possess a considerable free volume content in their inter-molecular space. During finite deformation, these free volume holes serve as the potential sites for localized permanent plastic deformation inclusions which are called shear transformation zones (STZs). While the free volume content has been experimentally shown to increase during the course of plastic straining in glassy polymers, thermal analysis of stored energy due to the deformation shows that the STZ nucleation energy decreases at large plastic strains. The evolution of the free volume, and the STZs number density and nucleation energy during the finite straining are formulated in this paper in order to investigate the uniaxial post-yield softening-hardening behavior of the glassy polymers. This study shows that the reduction of the STZ nucleation energy, which is correlated with the free volume increase, brings about the post-yield primary softening of the amorphous polymers up to the steady-state strain value; and the secondary hardening is a result of the increased number density of the STZs, which is required for large plastic strains, while their nucleation energy is stabilized beyond the steady-state strain. The evolutions of the free volume content and STZ nucleation energy are also used to demonstrate the effect of the strain rate, temperature, and thermal history of the sample on its post-yield behavior. The obtained results from the model are compared with the experimental observations on poly(methyl methacrylate) which show a satisfactory consonance.
The elasto-viscoplasticity of amorphous solids is modeled, with a focus on the effects of physical aging and mechanical rejuvenation. Using nonequilibrium thermodynamics, the concept of kinetic and configurational subsystems has been employed. The Hamiltonian structure of reversible dynamics is exploited to derive a constitutive relation for the stress tensor. Furthermore, it is demonstrated that accounting for mechanical rejuvenation results in a modification of the driving force for viscoplastic flow.
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.
Amorphous polymers lack an organized microstructure, yet they exhibit structural evolution, where physical properties change with time, temperature, and inelastic deformation. To describe the influence of structural evolution on the mechanical behavior of amorphous polymers, we developed a thermomechanical theory that introduces the effective temperature as a thermodynamic state variable representing the nonequilibrium configurational structure. The theory couples the evolution of the effective temperature and internal state variables to describe the temperature-dependent and rate-dependent inelastic response through the glass transition. We applied the theory to model the effect of temperature, strain rate, aging time, and plastic pre-deformation on the uniaxial compression response and enthalpy change with temperature of an acrylate network. The results showed excellent agreement with experiments and demonstrate the ability of the effective temperature theory to explain the complex thermomechanical behavior of amorphous polymers.
The time-temperature superposition property of an amorphous polymer acrylate network is characterized at infinitesimal strain by standard dynamic mechanical analysis tests. Comparison of the shift factors determined in uniaxial tension and in torsion shows that both tests provide equivalent time-temperature superposition properties. More interestingly, finite strain uniaxial tension tests run until break at constant strain rate show that the acrylate network exhibits the same time-temperature superposition property at finite strain as at infinitesimal strain. Such original experimental evidence provides new insight for finite strain constitutive modelling of polymer amorphous networks.
Human spine ligaments show a highly non-linear, strain rate dependent biomechanical behavior under tensile tests. A visco-hyperelastic fiber-reinforced constitutive model was accordingly developed for human ligaments, in which the energy density function is decomposed into two parts. The first part represents the elastic strain energy stored in the soft tissue, and the second part denotes the energy dissipated due to its inherent viscous characteristics. The model is applied to various human spinal ligaments including the anterior and posterior longitudinal ligaments, ligamentum flavum, capsular ligament, and interspinous ligament. Material parameters for each type of ligament were obtained by curve-fitting with corresponding experimental data available in the literature. The results indicate that the model presented here can properly characterize the visco-hyperelastic biomechanical behavior of human spine ligaments.
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.
There is an ever-growing need for predictive models for the
elasto-viscoplastic deformation of solids. Our goal in this paper is to
incorporate recently developed out-of-equilibrium statistical concepts into a
thermodynamically consistent, finite-deformation, continuum framework for
deforming amorphous solids. The basic premise is that the configurational
degrees of freedom of the material --- the part of the internal energy/entropy
that corresponds to mechanically stable microscopic configurations --- are
characterized by a configurational temperature that might differ from that of
the vibrational degrees of freedom, which equilibrate rapidly with an external
heat bath. This results in an approximate internal energy decomposition into
weakly interacting configurational and vibrational subsystems, which exchange
energy following a Fourier-like law, leading to a thermomechanical framework
permitting two well-defined temperatures. In this framework, internal variables
that carry information about the state of the material equilibrate with the
configurational subsystem, are explicitly associated with energy and entropy of
their own, and couple to a viscoplastic flow rule. The coefficients that
determine the rate of flow of entropy and heat between different internal
systems are proposed to explicitly depend on the rate of irreversible
deformation. As an application of this framework, we discuss two constitutive
models for the response of glassy materials, a simple phenomenological model
and a model related to the concept of Shear-Transformation-Zones as the basis
for internal variables. The models account for several salient features of
glassy deformation phenomenology. Directions for future investigation are
briefly discussed.
The modeling and quantification of a viscous contribution to strain hardening is discussed. Traditional strain hardening models, based on rubber-elasticity, show serious deviations from the experimentally observed large-strain response of glassy polymers. Therefore, to capture both the strain rate and temperature dependence of strain hardening correctly, a part of the rubber-elastic strain hardening is replaced with a viscous contribution. This is realized by introducing a deformation-dependent viscosity in the Eindhoven Glassy Polymer model. Verification of the accuracy of the model proposed is done using a set of uniaxial compression tests on polycarbonate at different strain rates and temperatures. An accurate description of all experimental data results and, moreover, the model quantitatively captures the Bauschinger effect observed in oriented polycarbonate. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012
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 micro-macro approach to the non-affine micro-sphere model of rubber elasticity was discussed. The motivating key evidence for the need of non-affine network models are shortcomings of the affinity assumption with respect to locking-stretch characteristics. The computational effort is absolutely competitive with purely macroscopic models of rubber elasticity formulated in terms of spectral decomposition of macroscopic strain measures. The overall model contains five effective material parameters, obtained from the single chain statistics and properties of the network.
Modularly invariant equations of motion are derived that generate the isothermal–isobaric ensemble as their phase space averages. Isotropic volume fluctuations and fully flexible simulation cells as well as a hybrid scheme that naturally combines the two motions are considered. The resulting methods are tested on two problems, a particle in a one-dimensional periodic potential and a spherical model of C60 in the solid/fluid phase.
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)
Since to form a hole the size of a molecule in a liquid requires almost the same increase in free energy as to vaporize a molecule, the concentration of vapor above the liquid is a measure of such ``molecular'' holes in the liquid. This provides an explanation of the law of rectilinear diameters of Cailletet and Mathias. The theory of reaction rates yields an equation for absolute viscosity applicable to cases involving activation energies where the usual theory of energy transfer does not apply. This equation reduces to a number of the successful empirical equations under the appropriate limiting conditions. The increase of viscosity with shearing stress is explained. The same theory yields an equation for the diffusion coefficient which when combined with the viscosity and applied to the results of Orr and Butler for the diffusion of heavy into light water gives a satisfactory and suggestive interpretation. The usual theories for diffusion coefficients and absolute electrical conductance should be replaced by those developed here when ion and solvent molecule are of about the same size.
This article introduces a simple and fast method to reinsert atomistic details into mesoscale models of polymers with rigid side groups. We describe our backmapping scheme from a coarse-grained (CG) resolution to an atomistic picture in the framework of molecular dynamics (MD) simulations of a silica–atactic polystyrene (PS) composite. The CG model of Qian et al. [ Macromolecules 2008, 41, 9919] has been used in the coarse-graining; it combines the atoms of one repeat unit of PS to a CG bead. In the reverse mapping only the centers of mass of these units and their chiralities are known. We show that this information is sufficient for the reverse mapping which requires simple geometrical and mechanical considerations. The capability of the suggested method is demonstrated by comparing MD results from the original atomistic model with those emerging from the reverse mapping. Because of its simplicity, the suggested technique offers the opportunity to study relaxed structures of melt chains with large molecular weights.
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.
Molecular dynamics (MD) simulations of bulk atactic polystyrene have been performed for chains up to 320 monomer units in a temperature range from 100 to 600 K and in a broad pressure range from 0.1 to 1000 MPa. The MD-determined specific volume vs temperature curves are in a good agreement with experimental PVT data at different values of applied pressure, but the measured glass-transition temperature, Tg, is displaced to somewhat higher temperature than the longer time experimental value. Local translational mobility has been investigated by measuring the mean-square translational displacements of monomers as a function of time. The long-time asymptotic slope of these dependencies is close to 0.6 at T > Tg, showing diffusive behavior. The cage effect, when local translational motions are essentially frozen in the glassy state, has been studied. The characteristic time of cage release does not depend on molecular weight, but the duration of the crossover to the diffusive regime increases almost linearly with increasing chain molecular weight, for both the backbone monomers and phenyl side groups.
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.
The strain hardening behavior of model polymer glasses is studied with simulations over a wide range of entanglement densities, temperatures, strain rates, and chain lengths. Entangled polymers deform affinely at scales larger than the entanglement length as assumed in entropic network models of strain hardening. The dependence of strain hardening on strain and entanglement density is also consistent with these models, but the temperature dependence has the opposite trend. The dependence on temperature, rate, and interaction strength can instead be understood as reflecting changes in the flow stress. Microscopic analysis of local rearrangements and the primitive paths between entanglements is used to test models of strain hardening. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3487–3500, 2006
The nature of strain hardening in glassy polymers is investigated by studying the mechanical response of oriented polycarbonate in uniaxial extension and compression. The yield stress in extension is observed to increase strongly with predeformation, whereas it slightly decreases in compression (the so-called Bauschinger effect). Moreover, oriented specimens tend to display increased strain hardening in extension, whereas this nearly vanishes in compression. It is shown that these observations can be captured by the introduction of a viscous contribution to strain hardening in terms of a deformation dependence of the flow stress. This can originate either from a deformation-induced change in activation volume, as observed for isotactic polypropylene, or from a deformation-induced change of the rate constant, as observed for polycarbonate, which causes the room temperature yield kinetics of this material to shift from the into the (+β) regime. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1483–1491, 2010
A new Lagrangian formulation is introduced. It can be used to make molecular dynamics (MD) calculations on systems under the most general, externally applied, conditions of stress. In this formulation the MD cell shape and size can change according to dynamical equations given by this Lagrangian. This new MD technique is well suited to the study of structural transformations in solids under external stress and at finite temperature. As an example of the use of this technique we show how a single crystal of Ni behaves under uniform uniaxial compressive and tensile loads. This work confirms some of the results of static (i.e., zero temperature) calculations reported in the literature. We also show that some results regarding the stress‐strain relation obtained by static calculations are invalid at finite temperature. We find that, under compressive loading, our model of Ni shows a bifurcation in its stress‐strain relation; this bifurcation provides a link in configuration space between cubic and hexagonal close packing. It is suggested that such a transformation could perhaps be observed experimentally under extreme conditions of shock.
A polymeric product generally possesses a residual state of orientation incurred due to processing. The pre-oriented polymer exhibits a highly anisotropic mechanical behavior as characterised by a yield stress and post yield deformation which are both direction dependent. In this paper, polycarbonate (PC) has been subjected to various magnitudes of pre-orientation via uniaxial compression. The subsequent anisotropic yield and post yield behavior is then experimentally determined, also by compression. A constitutive model of glassy polymer deformation proposed by two of the authors is found to be successfully predictive of the prominent features of the observed experimental behavior.