The effects of temperature and molecular weight on the mechanical response and strength of elastomers

Summary Constitutive equations are derived for the mechanical behavior of rubbery polymers at finite strains. The model is based on the concept of rigid-rod networks, where breakage of chains is treated as bond scission. Adjustable parameters in the stress-strain relations are found by fitting observations in tensile tests for ethylene-octene copolymers. It is revealed that the constitutive equations correctly describe stress-strain curves up to the break points. Young's modulus and the critical strength per bond monotonically decrease with temperature and increase with molecular weight. Received: 17 January 2001/Accepted: 27 February 2001     

Disorder effects on the strain response of model polymer networks

Molecular dynamics simulations are used to investigate polymer networks made by either end-linking or randomly crosslinking a melt of linear precursor chains. The resulting network structures are very different, since end-linking leads to nearly ideal monodisperse networks, while random crosslinking leads to polydisperse networks, characterized by an exponential strand length distribution. Networks with average strand length 20 and 100 were generated. These networks were used to study the effects of disorder in the network connectivity on observables averaged either over the entire network or selected sub-structures. Heterogeneities in the randomly crosslinked networks cause significant differences in the localization of monomers, however, neither the localization of crosslinks nor the microscopic strain response are significantly affected. Compared to end-linked networks, randomly crosslinked networks have a slightly increased tube diameter, and as a result a slightly decreased shear modulus, but otherwise identical stress-strain behavior. For the investigated systems, we conclude that the microscopic strain response, tube diameter, and stress-strain relation are all insensitive to the heterogeneities due to the linking process by which the network were made.     

Constitutive modeling for mechanical behavior of PMMA microcellular foams

Constitutive equations for nonlinear tensile behavior of PMMA foams were studied. Five viscoelastic models composed of elastic and viscous components were accounted for the modeling of the constitutive equations. The developed constitutive equations are expressed in terms of material properties and foam properties such as strain, strain rate, elastic modulus, relative density of foam, and relaxation time constant. It was found that the stress-strain behaviors by Generalized Maxwell model, Three Element model and Burgers model could be described by the constitutive equation obtained from the Maxwell model. For the verification of the constitutive model, poly(methyl methacrylate) (PMMA) microcellular foams were manufactured using batch process method, and then uniaxial tensile tests were performed. The stress-strain curves by experiment were compared with the theoretical results by the constitutive equation. It was demonstrated that nonlinear tensile stress-strain behaviors of PMMA foams were well described by the constitutive equation.     

Elasticity of real networks: Comparison of molecular theory with experiments

Predictions of the theory of elasticity of real networks are compared with results of experiments. The shape of the stress-strain curve for networks in the dry and swollen states and over wide ranges of strain, both in tension and compression, agrees with results of calculations based on the theory. The theory is also in close agreement with results of multiaxial stress-strain experiments and with the predictions of the degree of crosslinking obtained from measurements of the modulus. The theory may additionally be applied to the analysis of birefringence. The assumption of the linear additivity of the elastic and mixing free energies in a swollen network leads to results which are in agreement with experimental findings on different crosslinked systems.     

Analysis of nonlinearly elastic cantilever snap beams for assembly of plastic parts

Accounting for the nonlinear stress-strain behavior of plastics is important for good design of a snap joint. A constitutive equation for approximating the nonlinear stress-strain behavior of plastics is introduced in this paper. The equations for the stress, strain, and deflection of a cantilever beam, based upon the newly introduced constitutive equation are presented. In comparison with the experimental results, the cantilever deflection predicted from the nonlinear analysis is found to be about 50% more accurate than the corresponding predictions from the linear analysis.     

A predictive model for the mechanical behavior of particulate composites

A method for predicting the stress-strain and volumetric behavior of particulate composites from constituent properties has been developed for large values of strain. This approach allows a simple model for systems in which damage occurs without resorting to complicated constitutive equations. An energy balance derived from the first law of thermodynamics and the equations of linear elasticity calculates critical strain values at which filler particles will dewet when subjected to uniaxial tension and superimposed pressure. Calculations of critical strains over the entire strain history using reevaluated material properties accounting for the damage yield highly nonlinear stress-strain and volumetric curves. Experimentally observed dependences on particle size, filler concentration, matrix and filler properties, and superimposed pressure are correctly predicted. The method has no adjustable parameters, and allows several idealized models of the dewetting process to be examined. Comparisons of model predictions to experimental data show good agreement.     

Characterization of nonideal networks by stress-strain measurements at large extensions

It is shown how a characterization of unfilled, amorphous rubber networks can be evaluated from uniaxial stress-strain measurement data. Beside the cross-linking density, the relative scission probability during the curing procedure is evaluated, which determines the amount of dangling free chain ends and the number of trapped entanglements. These values are found from the C₂ term of the Mooney-Rivlin equation by using the predictions of a tube model. A necessary requirement for applying stress-strain measurements at large extensions is the consideration of the finite extensibility component of the reduced stress. It is taken into account by using a numerical procedure, which derives from a series expansion of the inverse Langevin approximation. The experimental results found at natural rubber networks cross-linked with thiuram (TMTD) and peroxid (DCP) show that network defects cannot be neglected in the DCP networks. They are assumed to be connected to the worse tensil strength properties compared to the TMTD networks. © 1993 John Wiley & Sons, Inc.     

An experimental investigation of the large-strain tensile behavior of neat and rubber-toughened polycarbonate

The large-strain tensile behavior of polycarbonate and polycarbonate filled with several volume fractions (f) of rubber particles is studied via an optical technique. Digital image correlation is used to determine, in two dimensions, the local displacement gradients and full-field displacements during a uniaxial tension test. Full-field strain contours, macroscopic true stress-strain behavior, and local volumetric strain are reduced from the raw test data. Full-field strain contours exhibit a decreasing degree of localization with increasing f. The true stress-strain results show a decrease in modulus, yield stress, post-yield strain softening, and subsequent strain hardening with increasing f. The volumetric strain decreases with increasing f as well. In the case of the neat polymer, comparisons are made to a three-dimensional finite element simulation.     

Prediction of stress–relaxation and creep of linear aromatic polyesters related to poly(ethylene terephthalate) in the nonlinear range

A study of the nonlinear viscoelastic properties of polymers was made using the Halsey-Eyring model with one- and three-dimensional mathematical analyses. Various parameters to calculate stress–relaxation and creep could be calculated from a single stress–strain curve of the same polymer. The parameters so calculated reconstituted the stress-strain curves, the one-dimensional equations yielding the better fit. The same constants were applied to predict stress-relaxation and creep. The fit using the three-dimensional equations is much better for stress-relaxation and creep than the one-dimensional equations.     

Constitutive relations for compressible glassy polymers

Constitutive equations are derived for the isothermal viscoelastic response of glassy polymers at small strains. The model employs the concept of temporary polymeric networks for shear relaxation and the diffusion mechanism for volume recovery of compressible materials. Adjustable parameters are found by fitting experimental data for poly(methyl methacrylate) and polycarbonate in uniaxial tensile tests. It is demonstrated that the stress-strain relations correctly predict nonmonotonic changes in the specific volume observed in creep tests.     

Comparison of recent rubber-elasticity theories with biaxial stress-strain data: the slip-link theory of Edwards and Vilgis

The Edwards-Vilgis (EV) slip-link theory (1986) derives the elastic free energy of a rubber-like network model containing stable and sliding network junctions (crosslinks and slip-links) and predicts both low-strain softening and high-strain hardening. The four-parameter stress-strain relations calculated by the theory for geometrically different deformation modes up to high strains were tested experimentally using published biaxial stress-strain data on simple covalently crosslinked networks. For networks with low degrees of strain softening and low extensibilities, the experimental dependencies could be described rather well but, generally, a simultaneous satisfactory fit to uniaxial, pure shear and equibiaxial data was not obtained. Systematic experiment-theory deviations exceeding 10% were observed and some of the parameters had a tendency to assume values lying outside the reasonably expected range. The prediction of a pronounced maximum in the strain dependence of stress supported by slip-links seems to be a reason for the discrepancy. Also, modeling of the high-strain singularity in entropy is done in the EV theory using a rather simple approximation. As a result, the finite extensibility contribution to the stress of a slip-link-free network model becomes improbably high and significantly exceeds that following, at a given modulus and locking stretch, from the rigorously derived Langevin-statistics-based eight-chain-network elasticity theory of Arruda and Boyce.     

Effect of strain rate and temperature on stress-strain properties of EVA-LDPE blends and the mechanism of strain hardening

Stress-strain behaviour of different blends of poly(ethylene-co-vinyl acetate) (EVA) (28 wt.-% VA content) and low density polyethylene (LDPE) is studied under various strain rates and temperatures. It is found that stress-strain plots of such semicrystalline polymer blends consist of three parts, namely, elastic or Hookeian region, region of chain slippage and region of strain hardening. Decrease in strain rate has an increasing effect on the strain hardening region. Increase in measurement temperature adversely affects the whole stress-strain plot. It is apparent from the study that at an elevated temperature the process of strain hardening is dependent on the crystalline melting point of the major component in the blend. The X-ray and DSC studies reveal that the process of strain hardening is mainly due to a change in internal order of crystallites in LDPE and LDPE-rich blends, whereas in EVA and EVA-rich blends it is due to induced crystallization in the amorphous phase.     

Measuring Hydro-Static Dependent Constitutive Behaviour of Adhesives Using a Bend Specimen

A simple bend test and associated equations that determine, simultaneously, both the tensile and compressive uniaxial stress-strain behaviour of a bulk adhesive are presented. From this, the ratio of the flow stress in compression to tension (S) can be found. Such data are required if a meaningful elastoplastic stress analysis of an adhesive joint is to be carried out. Experimental results for both the tensile and compressive stress-strain data are obtained for an epoxy specimen using this technique. The tensile data are compared with the results from uniaxial tensile tests on flat dumbell specimens of the same material and the good correlation that is found indicates that this is a reliable technique. Values of the ratio (S) for this adhesive are calculated as a function of both strain hardening and work hardening parameters. These results indicate that this technique complements standard test techniques and provides an elegant method not only of obtaining the ratio (S) but also of deriving uniaxial stress-strain curves.     

Latex interpenetrating polymer networks based on acrylic polymers. III. Synthesis variations

Two latex interpenetrating polymer networks, one from a supposedly compatible pair and the other from an incompatible pair of polymers, were prepared by two-stage emulsion polymerization. The synthesis conditions in each case were varied by altering the ratio of the glassy polymer to the rubbery polymer and also by reversing the order of synthesis. Fabricated samples of these latex interpenetrating polymer networks were subjected to hardness, stress-strain, and dynamic mechanical measurements. Hardness and stress-strain measurements showed that, although the second formed polymer dominated the final mechanical properties, the compatibilities of the polymer pairs played an important role. Dynamic mechanical analysis for the inverse systems showed evidence of enhanced mixing.     

Molecular modelling of the elastic behaviour of poly(ethylene terephthalate) network chains

The Monte-Carlo (MC) method developed to model the elastomeric stress-strain behaviour of polyethylene (PE) and poly(dimethyl siloxane) (PDMS) networks and the stress-optical behaviour of PE networks is now applied to the stress-strain behaviour of poly(ethylene terephthalate) (PET) networks. In keeping with the previous results for PE and PDMS networks, increases in the proportions of fully extended chains with macroscopic deformation are found to give rise to steady decreases in the rates of Helmholtz energy changes, causing reductions in moduli at moderate macroscopic deformations. There is no need to invoke a transition from affine to phantom chain behaviour as deformation increases.By using rotational-isomeric-state (RIS) models of the network chains and the MC method, stress-strain behaviour can be related to chemical structure. In this respect, the greater conformational flexibility of the PET chain leads to lower network moduli and smaller deviations from Gaussian network behaviour than for PE networks. In addition, the stiff, aromatic section of the PET repeat unit structure is seen to endow particular characteristics on the end-to-end distribution functions of PET chains. These characteristics are taken fully into account in evaluating the elastomeric properties of the PET networks. Subsequent publications will apply the present results to interpreting the measured stress-strain and the stress-optical properties of entangled PET melts.     

受冻融混凝土动态抗压特性试验研究

通过对受冻融混凝土试件进行单轴动态抗压试验,分析了冻融次数(0、25、50和75次)及应变速率(10-5 s-1、10-4 s-1和10-3 s-1)对混凝土抗压强度、峰值应变和应力-应变曲线的影响.结果表明:不同应变速率下受冻融混凝土的破坏特点有所差异,随着应变率的提高,试件断面更加平整,有较多粗骨料发生破坏;相同冻融次数下,随着应变率的增大,抗压强度明显增加,而峰值应变减小,应力-应变全曲线的峰值点呈前移和上升的趋势,曲线上升段斜率逐渐增大且长度也随之增大,下降段越为陡峭,表现明显的脆性特点;相同应变率下,随着冻融次数的增加,抗压强度均有所降低,而峰值应变逐渐增大,应力-应变全曲线的峰值点表现为明显下降和后移的趋势,其曲线下包面积减小,吸收能量的能力变差;通过对试验结果分析,建立了考虑应变率影响的受冻混凝土受压应力-应变曲线方程.     

复合材料开口补强计算的有限元分析

对开口翻边补强工艺在大开口、中厚板情况下的静力学性能进行分析。分别用二维八节点面单元和三维八节点体单元,对复合材料的应力—应变响应进行模拟。对两种计算结果进行对比,两种算法的位移和应变结果非常接近,但应力的结果差异较大。     

Elasticity of urethane networks

Various polymer networks have been prepared based on poly(propylene oxide) which has been linked either by urethane groups or ester groups. These networks have been examined by stress-strain measurements to find the degree of deviation from ideal rubber elasticity (as specified by the statistical theory) and by neutron scattering to find the mobility of the poly(propylene oxide) chain. The stress-strain results showed the urethane-linked networks to be identical to the ester-linked networks as long as there were no short chain polyols in the formulation. Networks containing short chain polyols have a greater deviation from ideal rubber elasticity. The neutron scattering results showed the ester-linked poly(propylene oxide) to have the same mobility as an uncrosslinked poly(propylene oxide), whereas urethane-linked networks have a lower mobility, similar to an analogous urethane-linked polymer. A small effect of crosslinking was observed. The results are explained in terms of some phase separation occurring in the polyurethanes only when short chain polyols are present. In the simple urethane-linked networks the poly(propylene oxide) chain mobility is reduced by hydrogen bonding to the urethane groups, but this has no effect on the equilibrium stress-strain properties.     

The effect of strain rate on the stress–strain curve of oriented polymers. I. Presentation of experimental results

The stress-strain curves of viscose, nylon 6.6, poly(ethylene terephthalate), polyacrylonitrile, and polypropylene have been determined at a large number of different strain rates between 10^?4 and 330 sec.^?1. The shape of these stress-strain curves and its change with strain rate is shown to depend upon whether the material is tested above or below its glass temperature. The stress-strain curves of materials tested below their glass temperature consists of an initial straight portion followed by a yield point at a few per cent strain. The breaking strain is only slightly affected by strain rate, and the energy to rupture increases with increasing rate. For materials tested above their glass temperature the initial portion of the stress-strain curves in nonlinear, and the yield strain is much higher than for the other materials. There is a small range of strain rate, in which the breaking strain falls sharply to the yield strain with increasing rate, and the energy to rupture also decreases. Outside this range the energy to rupture increases with increasing rate.     

Dynamic mechanical properties of polyurethane elastomers using a nonmetallic Hopkinson bar

A modified split Hopkinson-bar apparatus, in which the striker and input/output bars are made of polycarbonate instead of metal, was used to study three typical examples of a high-density flexible polyurethane elastomer (PORON) in sheet form. This variation of the device reduces a mismatch in impedance between the input/output bars and the specimen, thus allowing the stress in the specimen to reach a uniform state before significant engineering strain is induced. Dynamic compressive stress-strain curves were obtained from the measured incident, transmitted, and reflected waves. This article presents the behavior of these foams as a function of strain rate; for PORON 4701-05-20125-1637 under strain rates of 2.67 × 10⁻³ s⁻¹ to 4500 s⁻¹, the stress-strain response can be described by a function comprising a rate-dependent modulus and a strain-dependent factor, while for PORON 4701-01-30125-1604 and 4701-12-30062-1604, only loading at high strain rates yields similar characteristics. Empirical equations were derived to characterize these mechanical properties; in addition, characteristics relating to energy-absorption capability as well as deformation under approximately constant stress were also studied. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 66: 619–631, 1997