Uniaxial elastomers

Uniaxial deformations give prolate (needle-shaped) ellipsoids, and biaxial deformations give oblate (disc-shaped) ellipsoids [220,221], Prolate particles can be thought of as a conceptual bridge between the roughly spherical particles used to reinforce elastomers and the long fibers frequently used for this purpose in thermoplastics and thermosets. Similarly, oblate particles can be considered as analogues of the much-studied clay platelets used to reinforce a variety of materials [70-73], but with dimensions that are controllable. In the case of non-spherical particles, their orientations are also of considerable importance. One interest here is the anisotropic reinforcements such particles provide, and there have been simulations to better understand the mechanical properties of such composites [86,222],... 

Figure 2. Scattering intensity versus azimuthal angle for a uniaxially oriented elastomer, X is 3, x is 0.2. Phantom network where , f is 3 A, f is 4 V, f is 10. Crosslink junctions fixed, X.

The first SANS experiments on end-linked elastomers with a well-defined functionality were carried out by Hinkley et al, (22). Hydroxy-terminated polybutadiene was crosslinked by a trifunctional isocyanate, and the resultant polymer was uniaxially stretched. 

A theoretical investigation of the use of NMR lineshape second moments in determining elastomer chain configurations has been undertaken. Monte Carlo chains have been generated by computer using a modified rotational isomeric state (RIS) theory in which parameters have been included which simulate bulk uniaxial deformation. The behavior of the model for a hypothetical poly(methylene) system and for a real poly(p-fluorostyrene) system has been examined. Excluded volume effects are described. Initial experimental approaches are discussed. 

For elastomers, factorizability holds out to large strains (57,58). For glassy and crystalline polymers the data confirm what would be expected from stress relaxation—beyond the linear range the creep depends on the stress level. In some cases, factorizability holds over only limited ranges of stress or time scale. One way of describing this nonlinear behavior in uniaxial tensile creep, especially for high modulus/low creep polymers, is by a power... 

Note 3 For elastomers, which are assumed incompressible, the modulus is often evaluated in uniaxial tensile or compressive deformation using X - as the strain function (where X is the uniaxial deformation ratio). In the limit of zero deformation the shear modulus is evaluated as... 

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

The theoretical concept explained above is rather universal. It has few restrictions in terms of materials used for wrinkle formation. The main requirement is that the Young s modulus of the film is large as compared to the substrate elastomer. In addition, if micron- and sub-micron wavelengths are desired, the film thickness should be controllable on the nanoscale. In most of the work, polydimethylsiloxane (PDMS) was used as elastomeric substrate. Experimentally, both transient wrinkles, which only exist when macroscopic strains are applied, and permanent wrinkles, which remain in the absence of macroscopic strains, can be produced. Both cases are illustrated in Figs. 3 and 4 for the simplest situation of a uniaxial deformation. 

A powerful technique for the study of orientation and dynamics in viscoelastic media is line shape analysis in deuteron NMR spectroscopy [1]. For example, the average orientation of chain segments in elastomer networks upon macroscopic strain can be determined by this technique [22-31]. For a non-deformed rubber, a single resonance line in the deuterium NMR spectrum is observed [26] while the spectrum splits into a well-defined doublet structure under uniaxial deformation. It was shown that the usual network constraint on the end-to-end vector determines the deuterium line shape under deformation, while the interchain (excluded volume) interactions lead to splitting [26-31]. Deuterium NMR is thus able to monitor the average segmental orientation due to the crosslinks and mean field separately [31]. 

Deuterated PB networks filled with carbon black have been investigated recently [74]. The 2H NMR lineshape is different from that in unfilled elastomers an asymmetric doublet is observed as the sample is uniaxially stretched (A,=1.8)). This asymmetry is related to the presence of carbon black fillers, which induce magnetic inhomogeneities. 

The success of the developed model in predicting uniaxial and equi-biaxi-al stress strain curves correctly emphasizes the role of filler networking in deriving a constitutive material law of reinforced rubbers that covers the deformation behavior up to large strains. Since different deformation modes can be described with a single set of material parameters, the model appears well suited for being implemented into a finite element (FE) code for simulations of three-dimensional, complex deformations of elastomer materials in the quasi-static Emit. 

The non-linear response of elastomers to stress can also be handled by abandoning molecular theories and using continuum mechanics. In this approach, the restrictions imposed by Hooke s law are eliminated and the derivation proceeds through the strain energy using something called strain invariants (you don t want to know ). The result, called the Mooney-Rivlin equation, can be written (for uniaxial extension)—Equation 13-60 ... 

Equation (4-46) is valid for small extensions only. The actual behavior of real cross-linked elastomers in uniaxial extension is described by the Mooney-Rivlin equation which is similar in form to Eq. (4-46) ... 

This opens the possibility to tailor block copolymers with a wide variety of LC phases and phase transition temperatures. A interesting possibility is the preparation of thermoplastic LC elastomers of the ABA-type with amorphous A-blocks having a high Tg and an elastomeric LC B-block with low Tg. An uniform director orientation can be achieved in these systems by stress as shown recently for chemically crosslinked elastomers (12). Various applications of these systems in which optical uniaxiality and transparency are induced by strain can be envisaged. 

Cross-linked liquid crystalline polymers with the optical axis being macroscopically and uniformly aligned are called liquid single crystalline elastomers (LSCE). Without an external field cross-linking of linear liquid crystalline polymers result in macroscopically non-ordered polydomain samples with an isotropic director orientation. The networks behave like crystal powder with respect to their optical properties. Applying a uniaxial strain to the polydomain network causes a reorientation process and the director of liquid crystalline elastomers becomes macroscopically aligned by the mechanical deformation. The samples become optically transparent (Figure 9.7). This process, however, does not lead to a permanent orientation of the director. 

Equation 11.41 describes the engineering (nominal) stress a on an elastomer as a function of its draw ratio X. The measure of deformation that is most commonly used by engineers, the engineering (nominal) strain , equals (X-l). Under uniaxial tension, it describes the fraction by which a specimen has been extended relative to its initial (undeformed) dimensions. 

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

Figure 6-3. A unit cube of elastomer (a) in the unstrained state (b) in the homogeneous strained state (c) under uniaxial extension.

According to the statistical theory of rubber elasticity, the elastic stress of an elastomer under uniaxial extension is directly proportional to the concentration... 

The equilibrium small-strain elastic behavior of an "incompressible" rubbery network polymer can be specified by a single number—either the shear modulus or the Young s modulus (which for an incompressible elastomer is equal to 3. This modulus being known, the stress-strain behavior in uniaxial tension, biaxial tension, shear, or compression can be calculated in a simple manner. (If compressibility is taken into account, two moduli are required and the bulk modulus. ) The relation between elastic properties and molecular architecture becomes a simple relation between two numbers the shear modulus and the cross-link density (or the... 

Figure 22.2 General description of the uniaxial load/deformation behavior for (a) flexible plastics and (b) elastomers. Source Adapted with permission of John Wiley Sons, Inc., from Odian G. Principles of Polymerization. 4th ed. New York Wiley-Interscience 2004 [1].

Fig. 1.10 Actuation of dielectric elastomer devices with biaxial and uniaxial westraliL Uniaxial prestrain results in preferential in-plane strain in the direction perpendicular to the applied prestrain direction [1]. SPIE Press 2004, reprinted with permission...

Fig. 3.13 Effect of electrode type, humidity, maximum operating field and strain on the lifetime of dielectric elastomer transducers a Electrodes distribution of circular high strain actuators operated with different electrodes formulations (3M VHB 4910 film, 50% RH, 300% X 300% prestrain, actuation real strain 30-40% at 5 Hz, Max field 140 MV/min). b Humidity difference in high-fleld lifetimes for six circular actuators, three in open air and three in a dry environment (VHB 4910, 300% x 300% prestrain, IHz, Max field 140 MV/ min), c Electric field average life time versus electric field of high-humidity actuators (VHB 4910, 100% RH, 300% x 300% prestrain, 5% uniaxial strain at 5 Hz), d Strain lifetime of ten actuators with differing strain operated at high humidity (VB 4910, 100% RH, 300% X 300% prestrain, uniaxial strain at 5 Hz)...

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