Electroelasticity of polymer networks

A multiscale analysis of the electromechanical coupling in elastic dielectrics is conducted, starting from the discrete monomer level through the polymer chain and up to the macroscopic level. Three models for the local relations between the molecular dipoles and the electric field that can fit a variety of dipolar monomers are considered. The entropy of the network is accounted for within the framework of statistical mechanics with appropriate kinematic and energetic constraints. At the macroscopic level closed-form explicit expressions for the behaviors of amorphous dielectrics and isotropic polymer networks are determined, none of which admits the commonly assumed linear relation between the polarization and the electric field. The analysis reveals the dependence of the macroscopic coupled behavior on three primary microscopic parameters: the model assumed for the local behavior, the intensity of the local dipole, and the length of the chain. We show how these parameters influence the directional distributions of the monomers and the hence the resulting overall response of the network. In particular, the dependences of the polarization and the polarization induced stress on the deformation of the dielectric are illustrated. More surprisingly, we also reveal a dependence of the stress on the electric field which stems from the kinematic constraint imposed on the chains.     

A nonlocal contact formulation for confined granular systems

We present a nonlocal formulation of contact mechanics that accounts for the interplay of deformations due to multiple contact forces acting on a single particle. The analytical formulation considers the effects of nonlocal mesoscopic deformations characteristic of confined granular systems and, therefore, removes the classical restriction of independent contacts. This is in sharp contrast to traditional contact mechanics theories, which are strictly local and assume that contacts are independent regardless the confinement of the particles. For definiteness, we restrict attention to elastic spheres in the absence of gravitational forces, adhesion or friction. Hence, a notable feature of the nonlocal formulation is that, when nonlocal effects are neglected, it reduces to Hertz theory. Furthermore, we show that, under the preceding assumptions and up to moderate macroscopic deformations, the predictions of the nonlocal contact formulation are in remarkable agreement with detailed finite-element simulations and experimental observations, and in large disagreement with Hertz theory predictions—supporting that the assumption of independent contacts only holds for small deformations. The discrepancy between the extended theory presented in this work and Hertz theory is borne out by studying periodic homogeneous systems and disordered heterogeneous systems.     

Homogeneous rheological behavior of nanoparticle-based melt

The present work explores nonlinear rheological behavior of a strongly viscoelastic paste made of nano-sized polybutadiene particles. Apart from conventional rheometric measurements, particle-tracking velocimetric observations are carried out to determine the macroscopic state of deformation during startup shear and after step strain. Despite its highly nonlinear rheological characteristics, the system shows no sign of inhomogeneous response to large shear deformations in sharp contrast to well-entangled polymeric liquids made of linear chains. Apparently strongly nonlinear rheological behavior can occur in absence of inhomogeneous macroscopic deformation.     

Enhancing the Electro-Mechanical Response of Stacked Dielectric Actuators

Dielectric polymer films subjected to an electric field reduce in thickness and expand in area. A pile up configuration of such films, also known as a stacked dielectric actuator, is capable of exhibiting contractive deformations while subjected to external tensile forces. This work analyzes the capabilities of the stacked actuator according to a new microscopically motivated approach which suggests that the macroscopic response is determined by four microscopic factors—the length of the polymer chains, the local behavior of the monomers, the intensity of the local dipole and the chain-density. With the aim of enhancing the actuators performance, a specific local behavior is assumed and the influence of the remaining three quantities is studied. It is shown that the actuation can be significantly improved with appropriate micro-structural changes. Interestingly, this work demonstrates that these micro-structural alterations depend on the envisaged application.     

分析磁流变流体屈服应力微观力学模型

剪切屈服应力是磁流变流体固化强度的重要指标之一,具有重要的工程意义。基于磁性物理学理论,从磁流变流体在磁场作用下铁磁性固体颗粒极化成链的微观结构出发,探讨磁流变流体中固体颗粒间的相互作用机理,分析研究颗粒间的相互作用力,建立了一种微观力学模型,可用于分析磁流变流体在外加磁场作用下屈服应力及其影响因素的作用效果,揭示磁流变效应的微结构机理,为磁流变流体的性能优化、工程开发及应用磁流变流体提供理论依据。     

Free-energy density functions for nematic elastomers

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.     

Strain gradient plasticity by internal-variable approach with normality structure

This paper presents a constitutive formulation for materials with strain gradient effects by internal-variable approach with normality structure. Specific micro-structural rearrangements are assumed to account for the inelasticity deformations for this class of materials, and enter the constitutive formulations in form of internal variables. It is further assumed that the kinetic evolution of any specific micro-structural rearrangement may be fully determined by the thermodynamic forces associated with that micro-structural rearrangement, by normality relations via a flow potential. Macroscopic gradient-enhanced inelastic behaviours may then be predicted in terms of the microscopic internal variables and their conjugate forces, and thus a micro–macro bridging formulation is available for strain-gradient-characterised materials. The obtained formulations are first applied to crystallographic materials, and a crystal gradient plasticity model is developed to account for the influence of microscopic slip rearrangements on the macroscopic gradient-dependent mechanical behaviour for this class of materials. Micro-cracked geomaterials are also treated with these formulations and a gradient-enhanced damage constitutive model is developed to address the impacts of the evolutions of micro-cracks on the macroscopic inelastic deformations with strain gradient effects for these materials. The available formulations are further compared with other thermodynamic approaches of constitutive developing.     

Statistics of particle interactions in dense granular material under uniaxial compression

Stress evolution in a dense granular material is closely related to interactions of contacting particles. We investigate statistics related to particle interactions and the relationship between the averaged local relative motion and the macroscopic motion. The validity of the Voigt and Reuss assumptions is examined, and extensions to these assumptions are proposed. Effects of history in the dense granular material are investigated. Statistical samples used in this paper are obtained using three-dimensional numerical simulations of dense granular media under uniaxial cyclical compression. The results show that stresses arise mostly from normal forces between particles, and direct contributions from frictional tangential forces between particles are small. Tangential friction, however, significantly increases the particle contact time, and thus reduces the rate of contact breakage. The contact breakage rate is demonstrated to be a stress relaxation rate. Therefore, stress increases significantly with friction between particles as a result of prolonged relaxation time.     

Computer simulation of mechanical properties of carbon nanostructures

The aim of the present paper is the theoretical investigation of the mechanical properties of carbon nanostructures of graphene and single-wall carbon nanotubes by using nanoscopic and macroscopic approaches. The nanoobject structures in free and deformed states were considered and the corresponding energies were computed in the framework of quantum mechanics methods by using the original software package of semi-empirical programsNDDO/sp-spd (developed in the Institute of Applied Mechanics, Russian Academy of Sciences) in parallel computations. The nanostructural deformations were prescribed in the approximation of the mechano-chemical deformation coordinate. The deformation forces were described by the energy gradients in selected coordinates of microscopic deformations. The mechanical characteristics of nanoobjects such as Young’s modulus, rigidity coefficients, works done in deformation, critical stresses, and relative elongations in fracture were calculated in the framework of the macroscopic linear theory of elasticity; the deformation forces determined by quantum mechanical calculations were used in the corresponding relations. It was found that the mechanical characteristics of single-wall carbon nanotubes (CNT) depend on their diameter and chirality, and the deformation properties of a graphene sheet are asymmetric with respect to two normal extension modes directed along the “zigzag” and “armchair” on the sheet edges. The calculated mechanical characteristics are in good agreement with the experimental data known fromthe literature, in both the values and the deformation asymmetrywith respect to different deformation modes.     

Crushing of particles in idealised granular assemblies

Four idealised assemblies of equally sized spherical particles are subjected to a range of macroscopic compressive principal stresses and the contact forces on individual particles are determined. For each set of contact forces the stress fields within individual particles are studied. A failure criterion for brittle materials is imposed and indicates that crushing (or rupture) occurs when the maximum contact force reaches a threshold particle strength value, irrespective of the presence and magnitude of other lesser contact forces acting on the particle and the material properties of the particle. Combining the crushing mechanism with an assembly instability mechanism enables failure surfaces to be drawn in the three-dimensional stress space. A simple spatial averaging technique has been applied to the failure surfaces to remove the effects of assembly anisotropies. Sections of the failure surfaces on π planes have similarities to those commonly used in sand modelling.     

Rheology and flow-birefringence from viscoelastic polymer-clay solutions

 The shear orientation of viscoelastic clay-polymer solutions was investigated by means of rheology and flow birefringence (Δn). The polymer chains are in dynamic adsorption/desorption equilibrium with the clay particles to form a “network”. The elastic behavior of the network was characterized by constant stress, oscillatory shear, and stress relaxation experiments. Constant stress experiments indicated a yield stress upon which shear flow started and no strain recovery could be observed. Oscillatory shear experiments showed a broad elastic region followed by flow when a critical strain was reached. Stress relaxation experiments showed several relaxation times when the same critical strain was reached. Experiments under steady flow characterized the transient behavior of the network. With increasing steady shear rate a pronounced minimum in birefringence was observed at a critical shear rate. The shear rate dependent viscosity showed near power law behavior and no corresponding critical feature. While birefringence detects orientational effects on a microscopic length scale, rheology averages over macroscopic changes in the sample. The same degree of orientation could be achieved under constant shear rate or constant stress conditions. Received: 25 January 2001 Accepted: 22 May 2001     

The role of particle interactions in the rheology of dispersed systems

Summary Certain aspects of the rheological behaviour of dispersions cannot be understood unless attractive forces between the particles are assumed, resulting in the building-up of a network structure.A network model is postulated in which the particles are arranged in chains. During deformation these chains are stretched resulting in breakage of bonds between the particles. In this process and especially in what happens after that, an important question is whether the relative motion of a single particle with respect to the surrounding network is noticeable to only a few particles in the immediate vicinity or to a large number of particles over relatively large distances. The necessary information about the changing network structure during a large deformation at constant shear rate was derived from dielectric measurements in the case of water in oil emulsions and from super-imposed oscillatory shear experiments in the case of fat crystal dispersions.It is shown that the rheologieal behaviour of the water in oil emulsions may be characterized by motion of single particles, whereas in the fat-dispersions in oil collective displacements of large numbers of particles (aggregates) have to be taken into account.In the fat dispersions the magnitude of the forces acting in the network chains is explained in terms ofvan der Waals interaction. In the emulsions such forces determine the behaviour only at sufficiently low rates of shear. At higher shear rates the hydrodynamic interaction between single particles has to be taken into account.Among the quantities, which emerge from these network considerations is the characteristic time related with the motions of the particles or agglomerates. Therefore, time effects in dispersed systems are discussed more extensively.

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Zusammenfassung Bestimmte Aspekte des rheologischen Verhaltens von Dispersionen lassen sich nur unter der Voraussetzung von Anziehungskräften zwischen den Teilchen, die zur Bildung einer Netzwerkstruktur führen, erklären.Es wird ein Netzwerkmodell postuliert, in welchem die Teilchen kettenförmig angeordnet sind. Während der Deformation werden diese Ketten gedehnt, was zum Bruch der Bindungen zwischen den Teilchen führt. In bezug auf diesen Vorgang und besonders auf das, was nachher geschieht, erhebt sich die wichtige Frage, ob die relative Bewegung eines einzelnen Teilchens sich hinsichtlich des umgebenden Netzwerks nur auf einige benachbarte Teilchen auswirkt oder sich auf eine größere Anzahl weiter entfernter Teilchen erstreckt. Die benötigte Information über die Änderungen der Netzwerkstruktur während größerer Deformationen bei konstanter Schergeschwindigkeit wurde an Hand von dielektrischen Messungen für Wasser-in-Öl-Emulsionen und im Falle von Fettkristalldispersionen mit Hilfe von Versuchen mit überlagerter oszillierender Scherung ermittelt. Es zeigt sich, daß das rheologische Verhalten von Wasser-in-Öl-Emulsionen sich durch die Bewegung einzelner Teilchen charakterisieren läßt, während bei Dispersionen in Öl kollektive Bewegungen einer größeren Zahl von Teilchen (Aggregaten) berücksichtigt werden müssen. In den Fettdispersionen wird die Größe der in den Ketten des Netzwerkes ausgeübten Kräfte durchvan derWaalssche Wechselwirkungen erklärt. In den Emulsionen bestimmen derartige Kräfte das Verhalten nur bei genügend niedrigen Schergeschwindigkeiten. Bei höheren Schergeschwindigkeiten ist mit einer hydrodynamischen Wechselwirkung zwischen den einzelnen Teilchen zu rechnen.Unter den Größen, die sich aus diesen Betrachtungen der Netzwerkstruktur ableiten lassen, ist die mit den Bewegungen der Teilchen oder Agglomerate verbundene charakteristische Zeit. Aus diesem Grunde werden die Zeiteffekte in dispersen Systemen ausführlich besprochen.

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Über Spannungsrelaxation und zeitabhängige elastische Bruchvorgänge in orientierten Fasern

Zusammenfassung In der vorliegenden Arbeit wird der Zusammenhang zwischen der Spaltung von Molekülketten in belasteten Nylon-6-Fasern und dem makroskopischen, viskoelastischen Verhalten untersucht. Dazu wird die Morphologie orientierter Fasern diskutiert und auf die Spaltungswahrscheinlichkeit belasteter Einzelketten und Kettenbündel eingegangen. Aus experimentellen (ESR-)Ergebnissen wird hergeleitet, daß die Spaltung von Molekülen unter Last einen verschwindenden Einfluß auf die makroskopische Spannungsrelaxation, und einen beträchtlichen auf den Kriechvorgang hat. Die Bildungsrate freier Radikale (als Folge von Kettenspaltung) scheint unabhängig von der Spannungsrelaxation zu sein. Die zeitabhängige Bildung freier Radikale kann zutreffend durch eine aus den Modellvorstellungen gewonnene Funktion beschrieben werden. Dabei ergibt sich die Verteilung der relativen Längen derjenigen Kettensegmente, die bis zum makroskopischen Bruch der Probe gespalten werden.

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Summary The present paper deals with the correlation between chain scission in loaded Nylon 6 fibers and the macroscopic viscoelastic behavior. Therefore, the morphology of oriented fibers and the probability for scission of stressed single chains and bundles of chains are discussed. It is derived from experimental (ESR-) results that chain scission does hardly contribute to macroscopic stress relaxation, whereas creep is influenced significantly. The rate of formation of free radicals (as a consequence of chain scission) seems to be independent of the amount of stress relaxation. The time-dependent formation of free radicals may well be described by a function derived from the model representation. By optimal curve fitting the distribution of relative lengths of those chains is obtained which are broken until macroscopic failure occurs.

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Vorgetragen auf der Rheologentagung, Bad-Münster am Stein, 28.–30. Mai 1969.     

Photo-induced deformation of active polymer films: Single spot irradiation

Light-activated polymers can undergo complex deformation in response to the combination of mechanical and optical stimuli. These materials are attractive for remote actuation and sensing applications. However, the behavior of such materials subjected to photomechanical patterning is not well understood. In this paper we consider a polymer that operates by photoactivated stress relaxation; at the molecular level, photoinitiation of residual initiator molecules generate free radicals that break and then reform in-chain functionalities of stretched chains in an elastomeric network, which results in macroscopic stress relaxation. We carry out experiments and finite element calculations that demonstrate the sequence of deformation events culminating in the formation of a buckled spot as a result of biaxially stretching the elastomeric film then irradiating a circular region followed by releasing the mechanical constraint. In order to better understand the photomechanics, we analyze a simpler model problem wherein a linear elastic, stress relaxing disk is subjected to (i) radial extension, (ii) irradiation of a concentric circular region, and (iii) release of the applied displacements in (i), which results in deformation and stress redistribution. In the final step, the deformation may transition from planar to buckling out of the plane depending on system parameters. Companion finite element calculations are performed against which our analytical results are in good agreement. Although not directly comparable, the analytic model qualitatively agrees with the experiments. The results of this work provide a useful foundation from which to explore more interesting behavior of periodically photo-mechanically patterned films and other more challenging actuation problems.     

Constitutive equations in finite elasticity of rubbers

A constitutive model is derived for the elastic behavior of rubbers at arbitrary three-dimensional deformations with finite strains. An elastomer is thought of as an incompressible network of flexible chains bridged by permanent junctions that move affinely with the bulk material. With reference to the concept of constrained junctions, the chain ends are assumed to be located at some distances from appropriate junctions. These distances are not fixed, but are altered under deformation. An explicit expression is developed for the distribution function of vectors between junctions (an analog of the end-to-end distribution function for a flexible chain with fixed ends). An analytical formula is obtained for the strain energy density of a polymer network, when the ratio of the mean-square distance between the ends of a chain and appropriate junctions is small compared with the mean-square end-to-end distance of chains. Stress–strain relations are derived by using the laws of thermodynamics. The governing equations involve three adjustable parameters with transparent physical meaning. These parameters are found by fitting experimental data on plain and particle-reinforced elastomers. The model ensures good agreement between the observations at uniaxial tension and the results of numerical simulation, as well as an acceptable prediction of stresses at uniaxial compression, simple shear and pure shear, when its parameters are found by matching observations at uniaxial tensile tests.     

What is behind the plastic strain rate?

The plastic strain rate plays a central role in macroscopic models on elasto-viscoplasticity. In order to discuss the concept behind this quantity, we propose, first, a kinetic toy model to describe the dynamics of sliding layers representative of plastic deformation of single crystalline metals. The dynamic variable is given by the distribution function of relative strains between adjacent layers, and the plastic strain rate emerges as the average hopping rate between energy wells. We demonstrate the behavior of this model under different deformations and how it captures the elastic-to-plastic transition. Second, the kinetic toy model is reduced to a closed evolution equation for the average of the relative strain, allowing us to make a direct link to macroscopic theories. It is shown that the constitutive relation for the plastic strain rate does not only depend on the stress, but also on the macroscopic applied deformation rate, contrary to common practice.     

Structural response of high solidity net cage models in uniform flow

Hydrodynamic loads acting on a fish farm may be affected by the growth of different biofouling organisms, mainly due to increased solidity of the nets. In this paper, the hydrodynamic loads acting on high solidity net cage models subjected to high uniform flow velocities and the corresponding deformation of the net cages are studied. Model tests of net cylinders with various solidities were performed in a flume tank with a simulated current. Standard Morison-type numerical analyses were performed based on the model tests, and their capability of simulating the occurring loads and the observed net cage deformations for different flow velocities was evaluated.Large deformations of the net cage models were observed, and at high velocities the deformations were close to what is physically possible. Net cage deformation appeared to be less dependent on solidity than on flow velocity and weights. Drag forces increased with increasing flow velocity and were dependent on both bottom weights and netting solidity. For the lowest solidity net, drag forces were close to proportional to flow velocity. For the three high solidity nets, the measured drag forces were of similar magnitude, and drag increased less with increasing flow velocity above approximately 0.5 m/s than at lower velocities.This study shows that a basic reduced velocity model is not sufficient to model the interaction between the fluid flow and net (hydroelasticity) for high solidity net cages subjected to high flow velocities.The standard numerical analysis was in general able to make good predictions of the net shape, and was capable of making an acceptable estimate of hydrodynamic loads acting on the lowest solidity net model (Sn=0.19). For high solidities and large deformations, numerical tools should account for changes in water flow and the global drag coefficient of the net.     

Multi-scale constitutive modeling of the small deformations of semi-crystalline polymers

A multi-scale constitutive model for the small deformations of semi-crystalline polymers such as high density Polyethylene is presented. Each macroscopic material point is supposed to be the center of a representative volume element which is an aggregate of randomly oriented composite inclusions. Each inclusion consists of a stack of parallel crystalline lamellae with their adjacent amorphous layers.Micro-mechanically based constitutive equations are developed for each phase. A viscoplastic model is used for the crystalline lamellae. A new nonlinear viscoelastic model for the amorphous phase behavior is proposed. The model takes into account the fact that the presence of crystallites confines the amorphous phase in extremely thin layers where the concentration of chain entanglements is very high. This gives rise to a stress contribution due to elastic distortion of the chains. It is shown that the introduction of chains’ elastic distortion can explain the viscoelastic behavior of crystalline polymers. The stress contribution from elastic stretching of the tie molecules linking the neighboring lamellae is also taken into account.Next, a constitutive model for a single inclusion considered as a laminated composite is proposed. The macroscopic stress-strain behavior for the whole RVE is found via a Sachs homogenization scheme (uniform stress throughout the material is assumed).Computational algorithms are developed based on fully implicit time-discretization schemes.