On a molecular statistical basis for Ogden's model of rubber elasticity

In this paper, a link is established between the statistical theory of long chain molecules and Ogden's phenomenological model of rubber elasticity. It has been shown by several authors in the past that many invariant-based phenomenological models for rubber-like materials are related to the classical statistical theories. The essential means to reach this reconciliation were methods to account for a non-affine deformation of polymer chains in the network, appropriate techniques to calculate their averaged response, and an approximation of the inverse Langevin function appearing in the non-Gaussian statistical theory. It is shown in this paper that the very same approach, if appropriately implemented, allows to express the strain-energy function of Ogden's material in terms of physical constants characterising the polymer chain and network, together with few additional parameters that account for the non-affine deformation of the polymer chains. Particularly, it is shown that Ogden's model can be represented as a non-affine non-Gaussian 3-chain model with topological constraints.     

A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage

A micromechanically based non-affine network model for finite rubber elasticity and viscoelasticity was discussed in Parts I and II [Miehe, C., Göktepe, S., Lulei, F., 2004. A micro-macro approach to rubber-like materials. Part I: The non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617-2660; Miehe, C., Göktepe, S., 2005. A micro-macro approach to rubber-like materials. Part II: Viscoelasticity model for polymer networks. J. Mech. Phys. Solids, published on-line, doi:10.1016/j.jmps.2005.04.006.] of this work. In this follow-up contribution, we further extend the micro-sphere network model such that it incorporates a deformation-induced softening commonly referred to as the Mullins effect. To this end, a continuum formulation is constructed by a superimposed modeling of a crosslink-to-crosslink (CC) and a particle-to-particle (PP) network. The former is described by the non-affine elastic network model proposed in Part I. The Mullins-type damage phenomenon is embedded into the PP network and micromechanically motivated by a breakdown of bonds between chains and filler particles. Key idea of the constitutive approach is a two-step procedure that includes (i) the set up of micromechanically based constitutive models for a single chain orientation and (ii) the definition of the macroscopic stress response by a directly evaluated homogenization of state variables defined on a micro-sphere of space orientations. In contrast to previous works on the Mullins effect, our formulation inherently describes a deformation-induced anisotropy of the damage as observed in experiments. We show that the experimentally observed permanent set in stress-strain diagrams is achieved by our model in a natural way as an anisotropy effect. The performance of the model is demonstrated by means of several numerical experiments including the solution of boundary-value problems.     

Finite element modelling of rubber-like polymers based on chain statistics

In this work finite element simulations are conducted based on the micro structure of polymers in order to transfer the information of the micro level to the macro level. The micro structure of polymers is characterized by chain-like macromolecules linked together at certain points. In this way an irregular three-dimensional network is formed. Many authors use the tool of statistical mechanics to describe the deformation behaviour of the entire network. Most of these concepts can be reformulated as traditional continuum mechanical formulations. They are, however, restricted to affine deformation, regular chain arrangements and purely elastic material behaviour. For this reason, in the present contribution, we propose a new finite element-based simulation method for polymer networks which enables us to include non-affinity and arbitrary chain configurations. It can be easily extended to include chain breakage and reconnection.The polymer structure to be investigated, e.g. a rubber boot or a seal, is discretized by means of tetrahedral elements. To each edge of a tetrahedral element one truss element is attached which models the force–stretch behaviour of a bundle of polymer chains. Each of these tetrahedral unit cells represents the micro mechanical material behaviour in a certain point of the network. The proposed method provides the possibility to observe how changes at the microscopic level influence the macroscopic material behaviour. Such information is especially valuable for the polymer industry.     

A constitutive model of polyacrylate interpenetrating polymer networks for dielectric elastomers

A physically based method is proposed to represent interpenetrating polymer networks and their electromechanical behavior. The mechanical behavior of the material is nonlinear elastic and the electromechanical coupling arises from electrostatic effects often called the Maxwell stress effect. Ha et al. have synthesized interpenetrating polymer networks (IPNs) that invalidate the need for an external pre-stretch mechanism in dielectric elastomers. IPNs of acrylic elastomer and 1, 6-hexanediol diacrylate were successfully synthesized to create free-standing films with preserved prestretch. This results in a dual polymer network, with one polymer network in tension and the other in compression. The prestretch is preserved chemically in the dominant network. The internal prestretch is accompanied by an overall stiffening of the dual polymer network leading to compromised actuation strains. A mechanistically simple representation of the networks is proposed by means of a model of two springs in parallel, replaced by an equivalent single spring. A material parameter is introduced to account for the effect of the weight percent of the secondary network. The effect of the additive on the preserved prestretch in the primary network and hence the overall stress strain response is determined. Specifically, a modified Ogden strain energy function is proposed that describes the mechanical behavior of the new interpenetrating polymer network. The electromechanical response of the material is described using a previously presented constitutive formulation that works well for single network polymers. The model results indicate that ideally an interpenetrating polymer network DE should not stiffen when the secondary network is formed to avoid reduced actuation strains.     

A nonlinear constitutive model for magnetostrictive materials

A general nonlinear constitutive model is proposed for magnetostrictive materials, based on the important physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to the magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress. To avoid the complicity of determining the tensor function describing the nonlinear elastic strain part, this paper proposes a simplified model by means of linearizing the nonlinear function. For the convenience of engineering applications, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are, respectively, given for an isotropic material and their applicable ranges are also discussed. By comparison with the experimental data of a Terfenol-D rod, it is found that the proposed model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation further illustrates that, for either magnetostrictive rods or thin films, the proposed model can effectively describe the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves, while none of the known models can capture all of them. Therefore, the proposed model enjoys higher precision and wider applicability than the previous models, especially in the region of the high field.The project supported by the National Natural Science Foundation of China (10132010 and 90405005)     

Suspension-based rheological modeling of crystallizing polymer melts

The applicability of suspension models to polymer crystallization is discussed. Although direct numerical simulations of flowing particle-filled melts are useful for gaining understanding about the rheological phenomena involved, they are computationally expensive. A more coarse-grained suspension model, which can relate the parameters in a constitutive equation for the two-phase material to morphological features, such as the volume fractions of differently shaped crystallites and the rheological properties of both phases, will be more practical in numerical polymer processing simulations. General issues, concerning the modeling of linear and nonlinear viscoelastic phenomena induced by rigid and deformable particles, are discussed. A phenomenological extension of linear viscoelastic suspension models into the nonlinear regime is proposed. A number of linear viscoelastic models for deformable particles are discussed, focusing on their possibilities in the context of polymer crystallization. The predictions of the most suitable model are compared to direct numerical simulation results and experimental data.     

Global and local large-deformation response of sub-micron,soft- and hard-particle filled polycarbonate

Since polymers play an increasingly important role in both structural and tribological applications, understanding their intrinsic mechanical response is key. Therefore in the last few decades much effort has been devoted into the development of constitutive models that capture the polymers' intrinsic mechanical response quantitatively. An example is the Eindhoven Glassy Polymer model. In practice most polymers are filled, e.g. with hard particles or fibers, with colorants, or with soft particles that serve as impact modifiers. To characterize the influence of type and amount of filler particles on the intrinsic mechanical response, we designed model systems of polycarbonate with different volume fractions of small, order 100 nm sized, either hard or soft particles, and tested them in lubricated uniaxial compression experiments. To reveal the local effects on interparticle level, three-dimensional representative volume elements (RVEs) were constructed. The matrix material is modeled with the EGP model and the fillers with their individual mechanical properties. It is first shown that (only) 32 particles are sufficient to capture the statistical variations in these systems. Comparing the simulated response of the RVEs with the experiments demonstrates that in the small strain regime the stress is under-predicted since the polymer matrix is modeled by using only one single relaxation time. The yield- and the large strain response is captured well for the soft-particle filled systems while, for the hard-particles at increased filler loadings, the predictions are less accurate. This is likely caused by polymer–filler interactions that result in accelerated physical aging of the polymer matrix close to the surfaces. Modifying the S_a-parameter, that captures the thermodynamic state of the polymer matrix, allows us to correctly predict the macroscopic response after yield. The simulations reveal that all rate-dependencies of the different filled systems originate from that of the polymer matrix. Finally, an onset is presented to predict local and global failure based on critical events on the microlevel, that are likely to cause the over-prediction in the large-strain response of the hard-particle filled systems.     

Role of droplet bridging on the stability of particle-containing immiscible polymer blends

The effect of micron-sized hydrophobic calcium carbonate particles on the stabilization of polydimethylsiloxane (PDMS)/polyisobutylene (PIB) immiscible model blends is investigated in this study. The analytical splitting of bulk and liquid–liquid interface contributions from the droplet bridging one is successfully performed due to the negligible contribution of hydrophobic microparticles to the bulk rheology of phases. The presence of particles at the fluid–fluid interface is supported by wetting parameter calculation and verified by optical microscopy observations. Moreover, direct visualizations shows that particles are able to form clusters of droplets by simultaneously adsorbing at two fluid–fluid interfaces and glue-dispersed droplets together, probably due to the patchy interactions induced by heterogeneous distribution of particles along the interface. Rheological studies show that the flow-induced coalescence is slowed down upon addition of particles and almost suppressed with the addition of 4 wt% particles. The linear viscoelastic response is modeled to estimate interfacial tension by considering the contribution of particle-induced droplet aggregation in addition to bulk and droplet deformation ones. From linear and nonlinear viscoelastic responses, the improved stability of filled polymer blends is attributed to the interfacial rheology and/or the bridged structure of droplets, even though the interfacial area is not fully covered by particles. Furthermore, Doi–Ohta scaling relations are investigated by employing stress growth response upon step-up of shear flow.     

A model for compression-weakening materials and the elastic fields due to contractile cells

We construct a homogeneous, nonlinear elastic constitutive law that models aspects of the mechanical behavior of inhomogeneous fibrin networks. Fibers in such networks buckle when in compression. We model this as a loss of stiffness in compression in the stress–strain relations of the homogeneous constitutive model. Problems that model a contracting biological cell in a finite matrix are solved. It is found that matrix displacements and stresses induced by cell contraction decay slower (with distance from the cell) in a compression weakening material than linear elasticity would predict. This points toward a mechanism for long-range cell mechanosensing. In contrast, an expanding cell would induce displacements that decay faster than in a linear elastic matrix.     

先张预应力CFL加固RC梁预应力损失的实验研究

由于预应力碳纤维增强复合材料(CFRP)加固钢筋混凝土(RC)结构技术能够充分发挥其优异的力学性能而备受关注。本文以预应力碳纤维薄板(CFL)加固RC桥梁结构为研究对象,研制了采用先张法对CFL施加预应力的实验装置,提出了先张预应力CFL加固RC梁的实验方法及其预应力损失的监测方法,并在3个预应力水平下对RC梁实施了预应力CFL加固,测试和分析了CFL的预应力损失演化规律。研究结果表明,本文提出的先张预应力CFL加固RC梁以及预应力损失测试的方法是可行和有效的。     

Molecular simulation guided constitutive modeling on finite strain viscoelasticity of elastomers

Viscoelasticity characterizes the most important mechanical behavior of elastomers. Understanding the viscoelasticity, especially finite strain viscoelasticity, of elastomers is the key for continuation of their dedicated use in industrial applications. In this work, we present a mechanistic and physics-based constitutive model to describe and design the finite strain viscoelastic behavior of elastomers. Mathematically, the viscoelasticity of elastomers has been decomposed into hyperelastic and viscous parts, which are attributed to the nonlinear deformation of the cross-linked polymer network and the diffusion of free chains, respectively. The hyperelastic deformation of a cross-linked polymer network is governed by the cross-linking density, the molecular weight of the polymer strands between cross-linkages, and the amount of entanglements between different chains, which we observe through large scale molecular dynamics (MD) simulations. Moreover, a recently developed non-affine network model (Davidson and Goulbourne, 2013) is confirmed in the current work to be able to capture these key physical mechanisms using MD simulation. The energy dissipation during a loading and unloading process of elastomers is governed by the diffusion of free chains, which can be understood through their reptation dynamics. The viscous stress can be formulated using the classical tube model (Doi and Edwards, 1986); however, it cannot be used to capture the energy dissipation during finite deformation. By considering the tube deformation during this process, as observed from the MD simulations, we propose a modified tube model to account for the finite deformation behavior of free chains. Combing the non-affine network model for hyperelasticity and modified tube model for viscosity, both understood by molecular simulations, we develop a mechanism-based constitutive model for finite strain viscoelasticity of elastomers. All the parameters in the proposed constitutive model have physical meanings, which are signatures of polymer chemistry, physics or dynamics. Therefore, parametric materials design concepts can be easily gleaned from the model, which is also demonstrated in this study. The finite strain viscoelasticity obtained from our simulations agrees qualitatively with experimental data on both un-vulcanized and vulcanized rubbers, which captures the effects of cross-linking density, the molecular weight of the polymer chain and the strain rate.     

Relaxation effects of slip in shear flow of linear molten polymers

The transient shear response of a linear molten polymer (linear low-density polyethylene) in the nonlinear domain was studied using a true shear (sliding plate) rheometer with different gap spacings to detect slip effects. It was found that nonlinear viscoelasticity is further complicated by wall slip phenomena. Experimental evidence suggested that static slip models coupled with Wagner’s constitutive equation cannot adequately describe the experimental data at large and fast shear deformations. A new dynamic slip model involving multiple slip relaxation times is proposed in this paper, together with a method to assess the model parameters. Significant improvement in predicting the stress response is demonstrated by several examples of start-up of steady shear and large-amplitude oscillatory tests of a linear low-density polyethylene.     

Three-dimensional modeling of the mechanical property of linearly elastic open cell foams

Three-dimensional Voronoi models are developed to investigate the mechanical behavior of linearly elastic open cell foams. Dependence of the Young’s modulus, Poisson’s ratio and bulk modulus of the foams on the relative density is evaluated through finite element analysis. Obtained results show that in the low density regime the Young’s modulus and bulk modulus of random Voronoi foams can be well represented by those of Kelvin foams, and are sensitive to the geometric imperfections inherent in the microstructure of foams. In contrast, the compressive plateau stress of the foams is less sensitive to the imperfections. Failure surface of the foams subject to multi-axial compression is determined and is found to comply with the maximum compressive principal stress criterion, consistent with available experimental observations on polymer foams. Numerical results also show that elastic buckling of cell edges at microscopic level is the dominant mechanism responsible for the compressive failure of elastic open cell foams.     

A finite deformation nonlinear thermo-elastic model that mimics plasticity during monotonic loading

We present and study a nonlinear thermo-elastic constitutive model that under monotonic loading closely reproduces the response seen in plasticity, showing the initial stiff elastic response, kneeing as if yielding, and then showing response resembling post-yield hardening. The proposed large deformation thermo-elastic response model is constructed based on four physically identifiable mechanical parameters, that are closely related to the parameters used to construct plasticity models, thermal expansion parameters and two thermodynamic parameters. The four mechanical parameters are the initial elastic shear and bulk moduli, the yield point in shear, the hardening slope in shear. The thermodynamic parameters are the heat capacity at a reference temperature and its rate of change with changes of temperature. The model can be considered an alternate to deformation plasticity models currently used and, as such, can be used as a lightweight substitute for plasticity modeling in certain analysis. Since the proposed model is thermodynamically based, not only thermal effects are integrated into the model, but also the stress is calculated in terms of the applied deformation, allowing the model to be integrated with other models when conducting numerical analysis. We study the response of the proposed model under simple shear, uniaxial extension, confined compression, partially-confined compression, and biaxial extension. We incorporate the elastic model into ABAQUS using its UMAT subroutine for solid elements and using UHYPER for shell elements. We compare the large deformation response from the proposed elastic model with J2-plasticity, and with plasticity and deformation plasticity models implemented in ABAQUS. The model in most cases compares very favorably to all such models. This comparison is done for both homogeneous and non-homogeneous problems including the case of a cantilever beam under tip loading. We show that for the problems that it applies to, the models run in approximately one tenth the computational time and with one tenth the number of iterations needed to conduct the analysis using the plasticity model in ABAQUS.     

聚合物分子链微结构-力学性能关系的数据驱动模型

聚合物一般由随机分布的大分子链组成,分子链的分布、缠绕、交联等微结构状态显著影响聚合物的力学和物理性能。本文通过数据驱动方法,建立了聚合物分子链微结构-力学性能关系。使用有限元方法建立了两种分子链的随机微结构模型并得到了其力学性能。基于微结构-力学性能关系建立数据集,以聚合物的随机分子链微结构为输入,以聚合物的弹性刚度为响应输出,进行数据驱动模型的训练和验证。得到了精度满意的微结构-力学性能关系的分析结果。结果表明,通过数据驱动方法研究聚合物的弹性刚度问题是可靠的。     

Influence of dispersion procedure on rheological properties of aqueous solutions of high molecular weight PEO

The linear and nonlinear viscoelastic behaviors of poly(ethylene oxide) (PEO) in aqueous media have been investigated as a function of concentration and molecular weight. A particular interest has been paid to study the effect of turbulent flow under stirring, inducing both shear and elongational stresses, on the rheological behavior of the polymer solutions. The comparison of intrinsic viscosity and viscoelastic properties between shaken and stirred PEO solutions is discussed at the molecular scale in terms of chain scission and aggregation. Results point out that the effect of the mechanical history on the rheological response of PEO solutions depends also on the concentration regime and molecular weight. Indeed, the influence of the dispersion procedure vanishes by decreasing both the concentration and the molecular weight.