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An elastic molecular model for rubber inelasticity
... At present, there are many research studies on the static stiffness calculation of rubber components under the condition of small deformation at room temperature [1][2][3][4][5][6][7]. ...
... e following formula can be obtained by combining formulas (2) and (24): ...
Rubber spring plays an important role in improving train performance, so the study of rubber spring is one of the focuses of train dynamics. The vertical characteristic parameters of rubber spring are affected by prepressure significantly, as a result of varying parameters of static stiffness, dynamic stiffness, periodic energy consumption, damping coefficient, and so on. In order to use the theoretical method to calculate the precise static stiffness and predict the dynamic characteristics and to reduce the workload of the rubber spring performance test, this paper takes the annular rubber pad as an example to study with different prepressures. In this paper, the convexity coefficient correction formula (simply called the CCCF) for static stiffness calculation and the dynamic fiducial conversion coefficient (simply called the DFCC) method based on different prepressures are proposed. Through further analysis, the accuracy of CCCF and DFCC is proved both theoretically and experimentally. The results have shown precise prediction of the variation of prepressure on rubber spring parameters by using CCCF and DFCC and can be used as the reference of accurate vertical dynamic-static characteristics of the rubber spring.
... Other proposed mechanisms include, but are not limited to, the breakage [13,14] or slippage of network chains connecting filler particles [15,16], debonding of chains adsorbed on the surface of particles [17,18], movement of free chains superimposed on the relaxed network [19], and macromolecular disentanglement [20], and so on. The Mullins effect has been modeled based on microstructural alternations of filler-rubber interface [21,22], filler network [23][24][25] hyperelasticity [19,[26][27][28] for accounting for the "permanent set" [29][30][31], hysteresis loss [32], and the change of maximum stress during cyclic deformation [33]. However, the extreme assertion of softening as side effects of the filler meets insurmountable difficulties [34] in explaining the same event occurred in non-crystallizing gums [35,36] and their vulcanizates [37]. ...
... The normalized plot of a s E s /(1-φ) against A f ε pre in Fig. 6(b) characterizes the energy loss associating with softening of the rubber phase at a given velocity while the parameter 1a s shown in inset in Fig. 6(b) represents the percentage of energy loss associating with the filler-induced softening. While energy loss of the recovery hysteresis at given microscopic strain is similar for the rubber phase of different crosslinking degrees [27], the energy loss associating with softening of this phase at A f ε pre > 2 becomes increasingly large at high crosslinking degrees. It is of significance that the strain amplification factor A f is nearly independent of sulfur content and approximately follows the Guth-Gold equation A f = 1 + 2.5kφ + 14.1(kφ) 2 with a filler aspect ratio of k = 1.4. ...
The Mullins effect is not involved in the "filler network"
... Other proposed mechanisms include, but are not limited to, the breakage [13,14] or slippage of network chains connecting filler particles [15,16], debonding of chains adsorbed on the surface of particles [17,18], movement of free chains superimposed on the relaxed network [19], and macromolecular disentanglement [20], and so on. The Mullins effect has been modeled based on microstructural alternations of filler-rubber interface [21,22], filler network [23][24][25] hyperelasticity [19,[26][27][28] for accounting for the "permanent set" [29][30][31], hysteresis loss [32], and the change of maximum stress during cyclic deformation [33]. However, the extreme assertion of softening as side effects of the filler meets insurmountable difficulties [34] in explaining the same event occurred in non-crystallizing gums [35,36] and their vulcanizates [37]. ...
... The normalized plot of a s E s /(1-φ) against A f ε pre in Fig. 6(b) characterizes the energy loss associating with softening of the rubber phase at a given velocity while the parameter 1a s shown in inset in Fig. 6(b) represents the percentage of energy loss associating with the filler-induced softening. While energy loss of the recovery hysteresis at given microscopic strain is similar for the rubber phase of different crosslinking degrees [27], the energy loss associating with softening of this phase at A f ε pre > 2 becomes increasingly large at high crosslinking degrees. It is of significance that the strain amplification factor A f is nearly independent of sulfur content and approximately follows the Guth-Gold equation A f = 1 + 2.5kφ + 14.1(kφ) 2 with a filler aspect ratio of k = 1.4. ...
Mullins effect of carbon black filled nitrile butadiene rubber compounds and vulcanizates during cyclic tensile deformation is investigated and the influences of thermal annealing of the stretched samples, the deformation velocity and history and the solvent swelling are discussed. The energy losses associated with recovery hysteresis and softening accompanying the Mullins effect at a given crosshead velocity are quantitatively analyzed. The results show that the recovery hysteresis involves in microscopic deformation of the rubber phase in the nanocomposites but it disappears in the swollen samples. On the other hand, the softening is associated with both the rubber and filler phases; its energy loss increases with increasing filler content and decreases significantly by swelling. Furthermore, the softening of samples stretched at room temperatures is recoverable after conditioning at 80 °C but the recovery is retarded significantly by filler. The investigation allows gaining recognition of the important role played by the viscoelastic rubber matrix and the influence of the filler.
... Due to softening, the picture of the deformation distribution in the tyre varies. Contact between tyre rubber and road (specifically in considering Mullins effect) can lead to local material instability problems, due to negative tangential bond stiffnesses [58]. The algorithm was developed for performing computational experiments. ...
... Disadvantageous is the comparatively high effort to generate simulation models. Sample applications of the MDFEM include carbon nanotubes [9] as well as elastomers [17]. ...
A combined analytical and numerical study of oscillating elastic waves propagating in a 1D hyperelastic material modeled by the Lennard–Jones hyperelastic potential reveals that, due to different velocities of oscillating waves, the faster parts of the initially harmonic wave overtake the slower moving parts with formation and propagation of multiple shock wave fronts. These shock fronts cause the mechanical energy to decay with the release of heat. Thus, it is shown that in the considered purely mechanical system without viscous or dry friction, the mechanical energy can dissipate. The observed phenomenon opens up the possibility for creating a new type of vibration isolators without viscous or dry friction dampers. The use of the Lennard–Jones hyperelastic potential together with the Yeoh polynomial potential has recently been proposed for modeling different rubber-based cross-linked polymers.
The performance of fiber reinforced plastics (FRP), in particular their outstanding mechanical properties, motivate research into possibilities of further enhancement due to their high lightweight potential. Nanotechnology opens a new horizon for further improvement. The research presented in this book shows a significant augmentation of the matrix dominated properties by means of nanoparticle reinforced matrices. The results clearly demonstrate the capabilities of these continuous fiber reinforced nanocomposites as well as the need to gain a comprehensive understanding of the acting principles and mechanisms depending on the material, the surface functionalisation of the particles and the dispersive properties of the matrix. The chapter gives an overview on the state of research as the starting point of the research work and the theses obtained, which contribute to a comprehensive understanding of the acting mechanisms of nano-scaled additives to polymer matrices of continuous fiber reinforced polymer composites with respect to improved matrix dominated properties, damage tolerance combined with unchanged processability.
In this contribution, the transition in the thermo-mechanical properties of unvulcanized rubber during the vulcanization (curing) process is modeled by a material model for unvulcanized and vulcanized rubber (two-phase material) in the context of finite strain thermo-mechanics within the finite element method (FEM). For the vulcanization rate, a constitutive approach with few parameters is proposed representing the monotonic increase of the degree of vulcanization as a function of current temperature and the curing history. The material model taking into account the two phases of rubber uses different constitutive approaches for the isochoric part and the volumetric part for which a multiplicative split of the deformation gradient is considered. The isochoric part of the model consists of a hyperelastic spring and a dashpot device in series forming a Maxwell element with properties depending on the degree of vulcanization and temperature. Inelastic isochoric deformations during the forming and vulcanization process of rubber parts can be captured. The volumetric part consists of a hyperelastic spring and takes into account volume changes due to mechanical loads and temperature variations (thermal expansion). The proposed material model is formulated based on the displacement field, the temperature field, the degree of vulcanization and its kinetics. It is implemented into the thermo-mechanically coupled framework of the FEM in 3D for finite strains.
Gummiwerkstoffe zeigen unter Belastung ein hochkomplexes Verhalten. Besonders markant sind dabei die von der Belastungsgeschichte und Belastungsrichtung abhängigen Entfestigungseffekte. Für die Entwicklung zuverlässiger und optimierter Produkte aus Gummi ist ein vertieftes Verständnis dieser Eigenschaften essentiell. Sie lassen sich aber nur schwer mit den Eigenschaften der bei der Herstellung des Werkstoffs verwendeten Grundkomponenten in Zusammenhang bringen. Die genauen Vorgänge auf molekularer Ebene, die zu dem typischen Materialverhalten führen, sind unbekannt.
In der Arbeit wird als Erklärungsansatz die von Ihlemann entwickelte Theorie selbstorganisierender Bindungsmuster untersucht. Die zentrale These der Theorie besagt, dass sich unter Deformation infolge eines Selbstorganisationsprozesses eine inhomogene Verteilung schwacher physikalischer Bindungen im Material einstellt. Dieses Bindungsmuster verändert sich mit dem Deformationsvorgang und stellt damit das Gedächtnis des Materials dar. In der Arbeit werden zunächst allgemeine Aspekte selbstorganisierender Systeme untersucht und die Theorie anschließend unter diesen Aspekten analysiert. Außerdem wird der Erklärungsansatz geeignet erweitert und demonstriert, dass sich auch neue messtechnische Befunde damit problemlos erklären lassen. Insgesamt wird gezeigt, dass die Theorie selbstorganisierender Bindungsmuster als überzeugende Begründung für eine Vielzahl beobachteter Eigenschaften von Gummiwerkstoff geeignet ist.
Um die Theorie zu testen, wurde ein Simulationsprogramm erstellt. Es verwendet eine starke Abstraktion der Molekularstruktur von gefülltem Gummiwerkstoff. Die einzelnen Modellelemente wurden dabei so einfach wie möglich gehalten. Sie sind alle elastisch, es existiert aber eine Regel zum dynamischen Einfügen und Entfernen der physikalischen Bindungen. Daher können alle inelastischen Eigenschaften des Modells direkt mit einer Veränderung der Bindungsstruktur assoziiert werden. Es werden Verfahren vorgestellt, mit denen die zeitliche und deformationsabhängige Evolution des Modellzustands verfolgt werden kann. Damit werden die Abläufe von Messungen nachgebildet. Die Resultate der Simulation werden mit den Messergebnissen verglichen. Dabei zeigt sich, dass mit dem Programm eine Vielzahl typischer inelastischer Eigenschaften von gefülltem Gummiwerkstoff reproduziert werden kann. Dieses Resultat ist bemerkenswert, weil keines der Grundelemente des Modells diese Eigenschaften aufweist. Eine Analyse des Modells zeigt eindeutig, dass das Modellverhalten auf einem Selbstorganisationsvorgang des Modellelements, welches die physikalischen Bindungen repräsentiert, basiert. Der Selbstorganisationsvorgang verläuft genau so, wie von der Theorie vorhergesagt. Die erstaunliche Ähnlichkeit zum Materialverhalten und die geringe Anzahl von getroffenen Annahmen ist ein starkes Indiz dafür, dass in Gummiwerkstoff ebenfalls ein solcher Selbstorganisationsprozess abläuft und das Materialverhalten maßgeblich beeinflusst.
Two amorphous polymer blocks, one is made of polyethylene the other cis-1,4 polybutadiene, are subjected to cyclic deformation by means of molecular dynamics simulation. Each role of the bond stretch, bending, torsion and van der Waals is investigated in detail, revealing that the bond stretch and van der Waals dominate the hysteresis of both of the polyethylene and polybutadiene. Then the different behavior of both polymers is discussed from the viewpoint of change in the torsional nodes such as gauche⇔trans transition.
The molecular mechanics (MM) method is used to determine the frequencies and natural vibration shapes and to determine the buckling critical parameters and the postcritical deformation shapes of single-walled carbon nanotubes with twisted ends. The following two variants of the MM method are used: the standard MM method and the mixed method of molecular mechanics/molecular structure mechanics method (MM/MSM). Computer simulation shows that the MM/MSM method allows one to obtain acceptable values of frequencies and natural vibration shapes as well as of critical angles of twist, appropriate buckling modes, and postcritical deformation configurations of nanotubes compared with the same characteristics of nanotube free vibrations and buckling obtained by the standard MM method.
In molecular mechanics, the formalism of the finite element method can be exploited in order to analyze the behavior of atomic structures in a computationally efficient way. Based on the atom-related consideration of the atomic interactions, a direct correlation between the type of the underlying interatomic potential and the design of the related finite element is established. Each type of potential is represented by a specific finite element. A general formulation that unifies the various finite elements is proposed. Arbitrary diagonal- and cross-terms dependent on bond length, valence angle, dihedral angle, improper dihedral angle and inversion angle can also be considered. The finite elements are formulated in a geometrically exact setting; the related formulas are stated in detail. The mesh generation can be performed using well-known procedures typically used in molecular dynamics. Although adjacent elements overlap, a double counting of the element contributions (as a result of the assembly process) cannot occur a priori. As a consequence, the assembly process can be performed efficiently line by line. The presented formulation can easily be implemented in standard finite element codes; thus, already existing features (e.g. equation solver, visualization of the numerical results) can be employed. The formulation is applied to various interatomic potentials that are frequently used to describe the mechanical behavior of carbon nanotubes. The effectiveness and robustness of this method are demonstrated by means of several numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.
Kinetics of conformational change of a semiflexible polymer under mechanical external field were investigated with Langevin dynamics simulations. It is found that a semiflexible polymer exhibits large hysteresis in mechanical folding/unfolding cycle even with a slow operation, whereas in a flexible polymer, the hysteresis almost disappears at a sufficiently slow operation. This suggests that the essential features of the structural transition of a semiflexible polymer should be interpreted at least on a two-dimensional phase space. The appearance of such large hysteresis is discussed in relation to different pathways in the loading and unloading processes. By using a minimal two-variable model, the hysteresis loop is described in terms of different pathways on the transition between two stable states.
A method of approach to the correlation of theory and experiment for incompressible isotropic elastic solids under finite strain was developed in a previous paper (Ogden 1972). Here, the results of that work are extended to incorporate the effects of compressibility (under isothermal conditions). The strain-energy function constructed for incompressible materials is augmented by a function of the density ratio with the result that experimental data on the compressibility of rubberlike materials are adequately accounted for. At the same time the good fit of the strain-energy function arising in the incompressibility theory to the data in simple tension, pure shear and equibiaxial tension is maintained in the compressible theory without any change in the values of the material constants. A full discussion of inequalities which may reasonably be imposed upon the material parameters occurring in the compressible theory is included.
In order to understand the underlying mechanisms of inelastic material behavior and nonlinear surface interactions, which can be observed on macro-scale as damping, softening, fracture, delamination, frictional contact etc., it is necessary to examine the molecular scale. Force fields can be applied to simulate the rearrangement of chemical and physical bonds. However, a simulation of the atomic interactions is very costly so that classical molecular dynamics (MD) is restricted to structures containing a low number of atoms such as carbon nanotubes. The objective of this paper is to show how MD simulations can be integrated into the finite element method (FEM) which is used to simulate engineering structures such as an aircraft panel or a vehicle chassis. A new type of finite element is required for force fields that include multi-body potentials. These elements take into account not only bond stretch but also bending, torsion and inversion without using rotational degrees of freedom. Since natural lengths and angles are implemented as intrinsic material parameters, the developed molecular dynamic finite element method (MDFEM) starts with a conformational analysis. By means of carbon nan-otubes and elastomeric material it is demonstrated that this pre-step is needed to find an equilibrium configuration before the structure can be deformed in a succeeding loading step.
A new Molecular Dynamics Finite Element Method (MDFEM) with a coupled mechanical‐charge/dipole formulation is proposed. The equilibrium equations of Molecular Dynamics (MD) are embedded exactly within the computationally more favourable Finite Element Method (FEM). This MDFEM can readily implement any force field because the constitutive relations are explicitly uncoupled from the corresponding geometric element topologies. This formal uncoupling allows to differentiate between chemical‐constitutive, geometric and mixed‐mode instabilities. Different force fields, including bond‐order reactive and polarisable fluctuating charge–dipole potentials, are implemented exactly in both explicit and implicit dynamic commercial finite element code. The implicit formulation allows for larger length and time scales and more varied eigenvalue‐based solution strategies.
The proposed multi‐physics and multi‐scale compatible MDFEM is shown to be equivalent to MD, as demonstrated by examples of fracture in carbon nanotubes (CNT), and electric charge distribution in graphene, but at a considerably reduced computational cost. The proposed MDFEM is shown to scale linearly, with concurrent continuum FEM multi‐scale couplings allowing for further computational savings. Moreover, novel conformational analyses of pillared graphene structures (PGS) are produced. The proposed model finds potential applications in the parametric topology and numerical design studies of nano‐structures for desired electro‐mechanical properties (e.g. stiffness, toughness and electric field induced vibrational/electron‐emission properties). Copyright © 2014 John Wiley & Sons, Ltd.
The mechanical response of single-walled helically coiled carbon nanotubes (CCNTs) under large axial deformations is examined using a molecular dynamics finite element method. The 3D reference configuration of CCNTs is determined based on a 2D graphene layer using conformal mapping. Three sets of analyses are performed to fully describe the mechanical response of (n, n) CCNTs under elongation up to the bond breaking point and compression down to the solid length or the onset of buckling instability. First, the strain dependency of the mechanical properties of individual CCNTs during deformation is investigated by calculating the stress-strain curve and the spring constant of the CCNTs for the entire load range. Significant responses including brittle fracture under tension and buckling instability under compression are observed. Second, to examine the size dependence of the mechanical properties, several CCNTs with different geometric parameters are constructed, and their spring constant, fracture strain, fracture load, and energy storage density are determined. All CCNTs exhibit a superelasticity of 50-66%. A comparison between the mechanical properties of CCNTs and those of carbon nanotubes (CNTs) reveals that the fracture load and energy storage per atom of CCNTs is lower than that of the corresponding armchair CNTs.
The authors report the parameters for a new generic force field, DREIDING, that they find useful for predicting structures and dynamics of organic, biological, and main-group inorganic molecules. The philosophy in DREIDING is to use general force constants and geometry parameters based on simple hybridization considerations rather than individual force constants and geometric parameters that depend on the particular combination of atoms involved in the bond, angle, or torsion terms. Thus all bond distances are derived from atomic radii, and there is only one force constant each for bonds, angles, and inversions and only six different values for torsional barriers. Parameters are defined for all possible combinations of atoms and new atoms can be added to the force field rather simply. This paper reports the parameters for the nonmetallic main-group elements (B, C, N, O, F columns for the C, Si, Ge, and Sn rows) plus H and a few metals (Na, Ca, Zn, Fe). The accuracy of the DREIDING force field is tested by comparing with (i) 76 accurately determined crystal structures of organic compounds involving H, C, N, O, F, P, S, Cl, and Br, (ii) rotational barriers of a number of molecules, and (iii) relative conformational energies and barriers of a number of molecules. The authors find excellent results for these systems.
Nanoscale engineering has been developing rapidly. However, experimental investigations at the nanoscale level are very difficult to conduct. This research seeks to employ the same model to investigate an atomic-scale structure for tensile and modal analyses, based on atomistic–continuum mechanics (ACM) and a finite element method (FEM). The ACM transfers an originally discrete atomic structure into an equilibrium continuum model using atomistic–continuum transfer elements. All interatomic forces, described by the empirical potential functions, can be transferred into springs to form the atomic structure. The spring network models were also widely utilized in FEM based nano-structure studies. Thus, this paper attempts to explore ACM using three examples including silicon, carbon nanotube, and copper. All of the results are validated by bulk properties or literature.
A novel strain-energy function which is a simple cubic equation in the invariant (I1−3) is proposed for the characterization of the elastic properties of carbon-black-filled rubber vulcanizates. Conceptually, the proposed function is a material model with a shear modulus which varies with deformation. This contrasts with the neo-Hookean and Mooney-Rivlin models which have a constant shear modulus. The variation of shear modulus with deformation is commonly observed with filled rubbers. Initially, the modulus falls with increasing deformation, leading to a flattening of the shear stress/strain curve. At large deformations, the modulus rises again due to finite extensibility of the network, accentuated by the strain amplication effect of the filler. This characteristic behavior of filled rubbers may be described approximately by the proposed strain-energy function by requiring the coefficient C20 to be negative, while the coefficients C10 and C30 are positive. The use of the proposed strain-energy ...
An exact solution is obtained for the Schroedinger equation representing the motions of the nuclei in a diatomic molecule, when the potential energy function is assumed to be of a form similar to those required by Heitler and London and others. The allowed vibrational energy levels are found to be given by the formula E(n)=Ee+hω0(n+12)−hω0x(n+12)2, which is known to express the experimental values quite accurately. The empirical law relating the normal molecular separation r0 and the classical vibration frequency ω0 is shown to be r03ω0=K to within a probable error of 4 percent, where K is the same constant for all diatomic molecules and for all electronic levels. By means of this law, and by means of the solution above, the experimental data for many of the electronic levels of various molecules are analyzed and a table of constants is obtained from which the potential energy curves can be plotted. The changes in the above mentioned vibrational levels due to molecular rotation are found to agree with the Kratzer formula to the first approximation.
Two nano-blocks of polyethylene (PE) are made and subjected to cyclic deformation with various loading conditions, i.e., strain vs. stress control, zero lateral strain vs. zero lateral stress, and different load amplitude by using the coarse grained molecular dynamics simulation. The one block is filled with 1000 random coil chains of (long chain), while the other 10 000 chains of (short chain). The random coil chains are freely grown and relaxed in a simulation box of , then compressed to and relaxed to obtain a stress-free equilibrium. Under the zero lateral stress condition, , the long-chain block shows a leaf-like hysteresis curve both in the stress- and strain-controlled cyclic loading. The area of the hysteresis loop increases as the maximum load is changed to , and , respectively. The “Mullin's effect” is also observed, i.e., the stress–strain curve depicts lower path in the 2nd or later loading, although the target is never a rubber with filler. Under the zero lateral strain condition, , the long-chain block shows little hysteresis with the stress amplitude of 100 MPa, while it shows rapid or unstable elongation around at in the simulation of and . The short-chain block also shows unstable elongation under the condition even with the stress amplitude of 100 MPa, noting that it has an upper yield point of at and lower one around at . On the other hand, the short-chain block is stretched without remarkable stress increase up to the strain around 1.0, under the lateral condition of . Then the block shows “strain hardening” and comes up to the external stress of 100 MPa. It is worth noting that the block shows a leaf-like hysteresis in the 2nd or later cycle; the stress goes back to zero around in the unloading process and rises up immediately when the load is reversed, as same as the long-chain block.
The purpose of this discussion, then, is to show how the nature of
the strain energy function can be deduced from experiments on rubbery materials.
Many attempts have been made to reproduce theoretically the
stress-strain curves obtained from experiments on the isothermal
deformation of highly elastic 'rubberlike' materials. The existence of a
strain-energy function has usually been postulated, and the
simplifications appropriate to the assumptions of isotropy and
incompressibility have been exploited. However, the usual practice of
writing the strain energy as a function of two independent strain
invariants has, in general, the effect of complicating the associated
mathematical analysis (this is particularly evident in relation to the
calculation of instantaneous moduli of elasticity) and, consequently,
the basic elegance and simplicity of isotropic elasticity is sacrificed.
Furthermore, recently proposed special forms of the strain-energy
function are rather complicated functions of two invariants. The purpose
of this paper is, while making full use of the inherent simplicity of
isotropic elasticity, to construct a strain-energy function which: (i)
provides an adequate representation of the mechanical response of
rubberlike solids, and (ii) is simple enough to be amenable to
mathematical analysis. A strain-energy function which is a linear
combination of strain invariants defined by φ
(α)=(a1α+a2α+a3α-3)/α
is proposed; and the principal stretches a1,a2 and
a3 are used as independent variables subject to the
incompressibility constraint a1a2a3=1.
Principal axes techniques are used where appropriate. An excellent
agreement between this theory and the experimental data from simple
tension, pure shear and equibiaxial tension tests is demonstrated. It is
also shown that the present theory has certain repercussions in respect
of the constitutive inequality proposed by Hill (1968a, 1970b).
Geometrical and different physical nonlinearities have to be taken into account in order to model elastomeric material. In the present contribution, a novel approach of rubber elasticity is introduced. The approach considers the topological constraints as well as the limited extensibility of network chains in filled rubber. Subsequently, a formulation of finite viscoelastic damage is derived. With this description at hand, softening effects as well as time and frequency dependency of the material can be characterized. The goal of the present paper is twofold. Firstly, the constitutive formulations are introduced. Moreover, computational aspects are stressed. The shown approaches are presented in a format ready for a finite element implementation.
Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944, Trans. Faraday Soc. 40, 59) illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.
The theory of self-organizing linkage patterns is an effort to explain the physical reasons for the complicated behavior of industrial rubber materials, in particular under large deformations. To check the consequences of the theory, calculated data of a Trial Program, especially developed to this purpose, are presented.
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Influence of cross-linking density
- Fig
Fig. 19. Influence of cross-linking density.
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