Analysis of the tensile stress-strain behavior of elastomers at constant strain rates. I. Criteria for separability of the time and strain effects

Considering the case where the relaxation time spectrum is preserved at finite deformations, a theoretical analysis of the tensile stress-strain relation of elastomers at constant strain rates has been carried out. The finite strain effect is taken into account by replacing the Cauchy strain by a general strain function, ?(?), in the Boltzmann superposition integral. The analysis shows that there are two cases where the time and strain effects are separable when: (1) the segment of the stress relaxation modulus which coincides with the experimental time of stretching can be represented by a single power law; and (2) the general strain function, ?(?), is linearly proportional to the Cauchy strain. Separability of the time and strain effects, therefore, can be achieved by adjusting the stretching time (or strain) and temperature, if the relaxation time spectrum remains unchanged by the deformation. The tensile stress-strain relations derived from the theoretical analysis were applied to analyze data on a crosslinked styrene butadiene rubber obtained in the temperature range ?40 to 60°C. Γ(?), which describes the strain dependence of tensile stress, B_?, the ratio of isochronal stresses at different strains, and a_i, slope of a segment of the relaxation modulus E_i(t) on log t plot, were obtained directly from the experiment. Values of Γ(?), B_? and a_i obtained at ?40°C are quite different from those obtained at ?30°C or higher. Results obtained from our analysis are generally in agreement with those obtained by an empirical method for analyzing the experimental data.     

Stress–strain behavior of SAN/glass bead composites above the glass transition temperature

Relaxation and stress–strain behavior of SAN–glass bead composites are studied above the glass transition temperature. The strain imposed on the polymeric matrix of the composite is defined as ?_p = ?_c/(1 ? ?^?). Stress relaxation data for the filled polymer which is independent of strain can be calculated by multiplying the relaxation modulus (at a certain strain) by (1 + ?_p). Stress–strain curves at constant strain rate and for different concentrations of the filler can be shifted to form a master curve independent of filler content if the tensile stress is plotted versus ?^p. The relaxation modulus increases with increasing the filler concentration and can be predicted by a modified Kerner equation at 110°C.     

Calculation of stress–strain curves from relaxation data in the rubbery flow region

Stress relaxation curves for polysulfone and Lexan polycarbonate are only time dependent at a constant temperature if strain is defined as ?_H = In (l/l₀) and the “true” cross-sectional area A = A₀/(1 + ?) is used. The strain-independent stress relaxation curves can be used to calculate stress–strain curves at different rates of strain according to the linear viscoelastic theory. The agreement between experimental and calculated stress–strain curves is good at least up to about 60% strain in the range of 0.01 to 0.2 in./min rate of extension if an average rate of strain defined by ?_H = 1/t ln(l/l₀) is used.     

Effect of sparse long‐chain branching on the step‐strain behavior of a series of well‐defined polyethylenes

The effect of sparse long chain branching, LCB, on the shear step‐strain relaxation modulus is analyzed using a series of eight high‐density polyethylene (HDPE) resins. Strains of 1 to 1250% are imposed on materials with LCB content ranging from zero to 3.33 LCB per 10,000 carbon atoms. All materials are observed to obey time–strain separation beyond some characteristic time, τ_k. The presence of LCB is observed to increase the value of τ_k relative to the linear resin. The behavior of the relaxation modulus at times shorter than τ_k is investigated by an analysis of the enhancement seen in the linear relaxation modulus, G⁰(t), as a function of strain and LCB content. This enhancement is seen to (1) increase with increasing strain in all resins, (2) be significantly larger in the sparsely branched HDPE resins relative to the linear HDPE resin, and (3) increase in magnitude with increasing LCB content. The shape and smoothness of the damping function is also investigated. The finite rise time to impose the desired strain is compared to the Rouse relaxation time of linear HDPE resins studied. Sparse LCB is found to increase the magnitude of the relaxation modulus at short times relative to the linear resin. POLYM. ENG. SCI., 2010. © 2010 Society of Plastics Engineers     

Stress relaxation of elongated strip of poly(vinylidene fluoride) in ethyl acetate vapor

Poly(vinylidene fluoride) films in ethyl acetate vapor were studied at 30°C for vapor pressures of p = 0, 12, 30, 40, 66, 85 torr and elongations ε = 4.5%, 7%, 9.5%, 19%, 29%, and 44%. A cyclic experiment was also performed at ε = 7% and p = 40 torr for three sorption/desorption cycles. Assuming, as a first approximation, that the stress relaxation of the “dry” and “wet” polymers is proportional to the elastic strain, ε_el, empirical calculations were performed and compared to experimental results. In general, the presence of a vapor or gas in a polymer matrix enhances the stress relaxation by softening or plasticizing the polymer and transforming a portion of the elastic strain, ε_el, into the plastic strain, ε_pl. As the transformation continues, the sorption and stress of the wetted elongated sample change simultaneously with time until an optical “overshoot” and a mechanical or stress “undershoot” is observed. This result seems to be the consequence of the differential change of the stress of the “dry” and “wetted” samples with sorption time, τ = t ? t_o(t = time; t_o = initial sorption time), which depends on the differential time dependencies of the transformations of the elastic and plastic strains.     

Thermodynamic interactions of solvents with styrene–butadiene–styrene triblock copolymers

Fundamental thermodynamic interaction data for various solvents with two styrene–butadiene–styrene triblock copolymers (Kraton D-1101 and Kraton D-1300X) have been collected by the use of inverse gas chromatography at infinite dilution. Experimental results are presented for nine D-1101/solvent systems and nine D-1300X/solvent systems at 308, 328. and 348 K. Weight-fraction activity coefficients and Flory–Huggins χ interaction parameters have been calculated from the retention volumes. The χ parameter is used as a measure of the strength of interaction and therefore as a guide in the prediction of polymer–solvent compatibility. In addition, partial molar heats of mixing, ΔH_m^∞, and heats of solution, ΔH_s, were determined. The Hildebrand–Scatchard solubility theory was combined with the Flory theory in order to estimate the solubility parameter of the thermoplastic rubbers at the three different temperatures.     

Comparison of the viscous and elastic components of two ABS materials with creep,stress relaxation and constant strain rate measurements using the universal viscoelastic model

In general, the universal viscoelastic model evaluated in this study was found to adequately predict constant strain rate, creep, and/or stress relaxation measurements from the constants determined from constant strain rate measurements. The elastic and viscous components for two acrylonitrile–butadiene–styrene (ABS) viscoelastic materials were also easily isolated using this new universal viscoelastic model. The creep measurements for ABS‐A (25383‐A) and ABS‐N (LL‐4102‐N) at three different stresses allowed elucidation of the common creep intercept strain of the calculated creep slopes that was designated as the “projected elastic limit.” Once the values for n and β were evaluated from creep measurements, then the creep variation of the universal viscoelastic model yielded a reasonably good fit of the measured creep data for both ABS‐A and ABS‐N. The extensional viscosity constant λ_E was found to be 7.2% greater for ABS‐A than for ABS‐N. Consequently, ABS‐N was found to have a lower extensional viscosity in secondary creep than that of ABS‐A at any specific strain rate. The value of the efficiency of yield energy dissipation n for ABS‐N as determined from creep measurements was also 37.6% larger than the value of n for ABS‐A. In addition, the projected elastic limit ?_I for ABS‐A was 2% greater than the projected elastic limit for ABS‐N. These observations indicated that ABS‐A should be slightly more solidlike than ABS‐N. However, both ABS‐A and ABS‐N were significantly more solidlike than liquidlike because both of their values for the efficiency of yield energy dissipation n were very close to zero. In general, values of n range from 0 < n < 1 with a material characterized as being essentially pure elastic having a value of n = 0. Using the yield strain as the failure condition for constant strain rate and stress relaxation measurements and the strain at critical creep, the failure condition for creep, it was found that the universal viscoelastic model allowed these failure criteria to yield remarkably good agreement on a projected time scale. This agreement resulted even though separate and independent data were used to evaluate these three different techniques for both ABS‐A and ABS‐N. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 90: 1298–1318, 2003     

Elastomeric and mechanical properties of poly-m-carboranylenesiloxanes

A new high-temperature elastomer, SiB-2, has been investigated by stress relaxation, modulus-temperature, and volume--temperature techniques. SiB-2 was found to be more stable than a related elastomer, radiation-cured silicone rubber, having about twice as long as a chemical relaxation time at 250°C. Possible mechanisms to account for this increased stability are discussed. At low temperatures, T_g for SiB-2 was estimated at –34°C., which compares well with T_i = ?30°C. for this polymer. By comparison, SiB-3 has T_i = ?60°C., while phenyl-modified SiB-4 was found to have T_i = ?25°C. T_m for SiB-2 was estimated to be + 56°C.     

Nonlinear stress relaxation of silica filled solution‐polymerized styrene–butadiene rubber compounds

The stress relaxation of silica (SiO₂) filled solution‐polymerized styrene–butadiene rubber (SSBR) has been investigated at shear strains located in the nonlinear viscoelastic regions. When the characteristic separability times are exceeded, the nonlinear shear relaxation modulus can be factorized into separate strain‐ and time‐dependent functions. Moreover, the shear strain dependence of the damping function becomes strong with an increase in the SiO₂ volume fraction. On the other hand, a strain amplification factor related to nondeformable SiO₂ particles can be applied to account for the local strain of the rubbery matrix. Furthermore, it is believed that the damping function is a function of the localized deformation of the rubbery matrix independent of the SiO₂ content. The fact that the time–strain separability holds for both the unfilled SSBR and the filled compound indicates that the nonlinear relaxation is dominated by the rubbery matrix, and this implies that the presence of the particles can hardly qualitatively modify the dynamics of the polymer. It is thought that the filler–rubber interaction induces a coexistence of the filler network with the entanglement network of the rubbery phase, both being responsible for the nonlinear relaxation. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Maxwell Fluid Model for Generation of Stress–Strain Curves of Viscoelastic Solid Rocket Propellants

Solid rocket propellants are modeled as Maxwell Fluid with single spring and single dashpot in series. Complete stress–strain curve is generated for case‐bonded composite propellant formulations by taking suitable values of spring constant and damping coefficient. Propellants from same lot are tested at different strain rate. It is observed that change in spring constant, representing elastic part is very small with strain rate but damping constant varies significantly with variation in strain rate. For a typical propellant formulation, when strain rate is varied from 0.00037 to 0.185 per second, spring constant (K) changed from 5.5 to 7.9 MPa, but damping coefficient (D) varied from 1400 to 4 MPas. For all strain rates, stress–strain curve is generated using developed Maxwell model and close matching with actual test curve is observed. This indicates validity of Maxwell fluid model for case‐bonded solid propellant formulations. It is observed that with increases in strain rate, spring constant increases but damping coefficient decreases representing solid rocket propellant as a true viscoelastic material. It is also established that at higher strain rate, damping coefficient becomes negligible as compared to spring constant. It is also observed that variation of spring constant is logarithmic with strain rate and that of damping coefficient follows a power law. The correlation coefficients are introduced to ascertain spring constants and damping coefficients at any strain rate from that at a reference strain rate. Correlation for spring constant needs a coefficient “H,” which is function of propellant formulation alone and not of test conditions and the equation developed is K₂=(K₁‐H)×{ln(dε₂/dt)/ln(dε₁/dt)}+H. Similarly for damping coefficient (D) also another constant “S” is introduced and prediction formula is given by D₂=D₁×{(dε₂/dt)/(dε₁/dt)}^S. Evaluating constants “H” and “S” at different strain rates validate this mathematical formulation for different propellant formulations. Close matching of test and predicted stress–strain curve indicates propellant behavior as viscoelastic Maxwell Fluid. Uniqueness of approach is to predict complete stress–strain curves, which are not attempted by any other researchers.     

Phenomenology of plastic recovery in high polymer glasses

Deformations in isotropic, strain-free polymer glasses are usually completely recoverable (at the test temperature or after warming to T_g), in sharp contrast with the behavior of low molecular weight glasses and crystals. The apparent ‘plastic strain’ which remains at the end of a creep or stress relaxation experiment does not recover at the test temperature, but only after the sample is heated. It is proposed that the long time scales needed for entanglement reorganization in the glass are responsible for this delayed recovery. A phenomenological network model for thermally activated strain recovery in polymer glasses is analyzed. A superposition relation between the stress and the strain history using a KWW (stretched exponential) memory kernel is employed. The recovery of plastic (i.e. residual) strain in non-crosslinked amorphous thermoplastics is a two-step process that may be interpreted in terms of the network model. In particular, recovery at sub-T_g temperatures is associated with entanglement slippage, while recovery near-T_g is believed to involve reorganization at or near chain ends.     

EQUILIBRIUM OF PALLADIUM(II) EXTRACTION WITH 1-OCTYLTHEOBROMINE

ABSTRACT

1-Octyltheobromine was synthesized to study extraction equiliburium of palladium(II). To evaluate its efficiency as an extractant, the extraction of palladium(II) from acidic chloride media was studied at 303K using toluene. The extraction of palladium(II) from hydrochloric acid media by 1-octyltheobromine exhibited high selectivity for palladium(II) over the platinum group metals. The stoichiometrics of the extraction of palladium(II) with 1-octyltheobromine was elucidated by examining the effects of hydrochloric acid, chloride ion, hydrogen ion, extractant and metal ion concentrations on its extractability. Palladium(II) was found to be extracted as two molecules of 1-octyltheobromine reacted with PdCl₂ as follows: PdCl₂ +{2}¯RN ? ¯PdCl₂(RN)₂. The extraction equilibrium constant was K=2.72?×?10⁸(mol?dm^?3)^?2. The complex of 1-octyltheobromine with PdCl₂ was confirmed by mass spectrometric analysis. The stripping of palladium(II) was performed over 60% by a single batchwise treatment with an aqueous solution of thiourea or ammonia.     

Thermal stresses and birefringence in quenched tubes and rods: Simulation and experiment

Measurements and simulations of the radial distribution of the thermal birefringence components, Δn and n_θθ ? n_rr, and the average birefringence, <n_zz ? n_θθ>, in free quenched tubes and rods of polystyrene (PS) and polycarbonate (PC) at different initial temperatures were carried out. The thermal stress and birefringence components were simulated using the linear viscoelastic and photoviscoelastic constitutive equations combined with the first‐order rate equation for volume relaxation and the master curves for the Young's relaxation modulus and strain‐optical coefficient functions of polymers. The numerical procedures used to discretize the governing equations using finite difference method were described. The obtained numerical results provided the evolution of stress and birefringence components with time during and after quenching and an explanation of the measured residual birefringence distribution in quenched tubes and rods. It was also found that the thickness of the slices removed from the samples to measure the thermal birefringence components, Δn and n_θθ ? n_rr, was critical, in particular, when the initial temperatures were close to the glass transition temperature of polymers. With an increase of the initial temperature during quenching, a better agreement between the simulated and measured birefringence components was obtained. POLYM. ENG. SCI., 2011. © 2010 Society of Plastics Engineers     

The coupling of deformations in a class of viscoelastic solids to temperature changes

A class of viscoelastic materials is defined by the parametric constitutive equations σ = σ(ε,t) and T = T(ε,t). It is shown that the creep rates and relaxation rates of such materials are determined by their thermal properties and the ambient temperature, and that the assumptions of constant stress of strain respectively and constant temperature are incompatible.     

Dielectric relaxation spectra of chemically treated woods

The changes in the dielectric properties of absolutely dried Sitka spruce (Picea sitchensis Carr.) wood caused by chemical treatments were investigated. Eight kinds of chemical treatments with various levels of weight percentage gain (WPG) were performed. Through the application of the Cole–Cole circular arc law to the results of the dielectric measurements in the frequency range of 1 kHz to 1 MHz at ?60°C, the relaxation spectrum was calculated. The relaxation magnitude (ε₀ ? ε_∞) was reduced by formalization, acetylation, propylene oxide, and phenol–formaldehyde resin treatments, in which chemical reactions occurred between the OH groups in the cell wall and the added reagents. On the other hand, the generalized relaxation time (τ_m) decreased with increasing WPG, except for acetylation, for which τ_m decreased up to a WPG level of 20% and then increased. In poly(ethylene glycol) impregnation, (ε₀ ? ε_∞) decreased with increasing WPG up to about 50% and then increased, whereas τ_m linearly decreased with increasing WPG. No significant changes in these parameters were recognized for the wood methyl methacrylate composite and heat treatment. With the steam treatment, τ_m increased. The distribution of the relaxation times was broad in acetylation and narrow in propylene oxide treatment and poly(ethylene glycol) impregnation. However, it remained almost unchanged in the other treatments. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 96: 37–43, 2005     

Experimental and numerical investigation of fracture of ABS polymeric material for different sample's thickness using a new loading device

The fracture behavior of ABS (acrylonitrile butadiene styrene) polymeric material has been investigated under the full range of in‐plane loading conditions using a new loading device to obtain more reliable results. Loading conditions from pure mode‐I through various mixed‐mode I/II ratios up to pure mode‐II have been generated using the proposed new loading device for the same specimen geometry. From the experimentally measured critical loads, the mode‐I, mode‐II, and the various mixed‐mode I/II critical energy release rates have been determined at different loading angles from 0° to 90°. Using the FE results, nondimensional stress intensity factors were applied to the specimen. The primary objectives of this study were to develop a new loading device to determine the mixed‐mode fracture toughness K_IC and K_IIC of ABS polymeric material. Another goal was to obtain stress intensity and strain energy release rates solutions associated with the crack, and to examine effects of thickness and geometric variables, particularly under mixed‐mode loading conditions. It was found that the thickness of the 10 mm specimen satisfied the plane strain condition with average fracture toughness ≈4.32 MPa·m^1/2 under pure mode‐I loading and ≈1.42 MPa·m^1/2 for pure mode‐II loading. POLYM. ENG. SCI., 54:2086–2096, 2014. © 2013 Society of Plastics Engineers     

Residual stress craze and crack formation in poly(methyl methacrylate)

Samples of poly(methyl methacrylate) with a central circular hole are compressed, and crazes form on or after unloading, provided that the strain attains or exceeds a threshold value ?_t. Crazes induced in air are transformed rapidly to cracks, but environmental crazes are more stable. These residual stress crazes form at the diameter of the hole on a plane perpendicular to the applied stress direction. In contrast, during loading, crazes form on the vertical plane containing the hole axis. Unloading crazes are relatively insensitive to changes in strain rate, whereas loading erazes have a pronounced rate dependence. Environmental residual stress crazing exhibits an apparent rate sensitivity at constant time, but the critical applied strain ?_t is essentially constant, irrespective of rate, if the sample is in contact with the environment for a sufficiently long time to ensure that the minimum ?_t is obtained. Residual stress crazes appear to initiate at the equator of the hole, and the maximum tensile residual strain, indicated by a strain gauge, occurs in this position.     

Some viscoelastic properties of ABS polymer

The discrete relaxation spectrum of an ABS (acrylonitrile–butadiene–styrene) polymer at 190°C. was calculated by using results from tensile relaxation moduli and the principle of reduced variables. The shift factor was found to conform well to the WLF equation, and the free volume fraction at T_g was calculated to be 0.026 in good agreement with the universal value. The values of the thermal expansion coefficient of free volume were calculated to be 9.8 X 10^-4 deg.^?1 and 7.0 × 10^?4 deg.^?1, respectively, from the WLF coefficients and from dilatometric results. The width of the entanglement plateau of the relaxation spectrum was observed to be a factor of approximately 2 larger than that calculated from molecular weights between entanglement couplings determined either from rubber elasticity theory or from an assumed molecular model which discounts the presence of the butadiene in the ABS system. By using Pao's theory, flow curves at 190°C. were calculated both from the discrete relaxation spectrum and from the dynamic modulus. These curves were essentially identical. However, the stress values of these curves were found to be about a decade higher than those experimentally determined from capillary flow measurements. Nevertheless, the shapes of the curves are in good agreement, and an explanation is suggested for existing discrepancies. Flow instability, processing variables, and residual strains are discussed in light of the flow curves and the calculated recoverable shear strains.     

Multiaxial ratcheting‐fatigue interaction on acrylonitrile‐butadiene‐styrene terpolymer

A series of multiaxial tests on an industrial acrylonitrile‐butadiene‐styrene (ABS) tube were conducted at room temperature. The ratcheting strain accumulation was found to be closely related to the shear strain amplitude and axial mean stress. The ratcheting strain and its rate increased with increasing shear strain amplitude as well as axial mean stress. At the same time, the ratcheting strain accumulation was retarded with a negative mean stress, and the ratcheting shakedown appeared when the negative mean stress reached ?10 MPa. To understand the influences of loading history on axial ratcheting strain, a series of multistep loading tests under constant symmetrical torsion at different axial mean stresses of 5, 0, and ?5 MPa were performed. The results showed that both the compressive viscous strain and strain relaxation of ABS could lead to the decrease of axial ratcheting strain. Furthermore, the compressive viscous strain and strain relaxation are independent of the loading history. Considering the influence of mean stress, a modified fatigue life model based on the Basquin law was presented to predict the pure torsional fatigue and multiaxial fatigue of ABS. The results of the modified fatigue life model show it can give relatively accurate fatigue life predictions. POLYM. ENG. SCI., 55:664–671, 2015. © 2014 Society of Plastics Engineers     

抗湿滑轮胎胎面胶配方的研究

研究羧基改性溶聚丁苯橡胶(SSBR)、丁腈橡胶(NBR)和高性能橡胶助剂Nanoprene在抗湿滑轮胎胎面胶中的应用。结果表明:在胎面胶配方中,以SSBR等量替代ESBR,胶料的门尼焦烧时间和t90延长,硫化胶的300%定伸应力和撕裂强度增大,湿抓着性提高,压缩生热降低,耐磨性能变化不大;加入20份Nanoprene,胶料的门尼焦烧时间和t90延长,硫化胶的300%定伸应力增大,湿抓着性提高,压缩生热降低,耐磨性能下降;加入5份NBR,胶料的硫化特性、硫化胶的物理性能和压缩生热变化不大,湿抓着性提高,耐磨性能下降。