Statistical Theory Of Elasticity Research Articles

Identifying the weak spots in\xa0packaging paper: local variations in grammage, fiber orientation and density and the resulting local strain and failure under load

Measured local paper structure—i.e. local basis weight, local thickness, local density and local fiber orientation—has been linked to local strain and local material failure (local temperature increase due to energy dissipation upon fiber–fiber bond failure) measured during tensile testing. The data has been spatially linked through data map registration delivering several thousand 1times 1,hbox{mm}^2 paper regions, each containing all measured properties. The relation between local paper structure and resulting local deformation and failure is studied with regression models. Multiple linear regression modeling was used to identify the paper structure related drivers for local concentrations of strain under load and local concentrations of material failure, which are both starting to occur considerably before rupture of the paper. Analyzing the development of local strain in paper we found that regions with higher basis weight and higher fiber orientation in load direction tend to exhibit considerably lower strain during tensile testing. Furthermore, the relation between local strain and local grammage can be predicted with the statistical theory of elasticity. Also regions with higher density have lower local strain, but not as pronounced. The findings for local fiber–fiber bond failure of paper are similar but not equivalent. The strongest correlation exists with local grammage. Local density and local fiber orientation show in turn weaker correlation with local bond failure. Local variations in paper thickness were not relevant in any case. These findings are highlighting the relevance of local fiber orientation and local density variations as structural mechanisms governing paper failure. In the past the focus has been mostly on paper formation. Together with local grammage (formation) they are responsible for the weak spots in paper, and thus cause local concentrations of paper strain and the initiation of failure under tensile load.

Improvement of mechanical properties of in situ-prepared HTPE binder in propellants.

A new type of hydroxyl-terminal block copolymer (HTPE) binder with excellent mechanical properties was prepared using an in situ preparation method. Compared with traditional HTPE binder preparation, this method involves relatively simple operations, which not only reduces costs, but also does not require a complicated synthesis process to prepare the HTPE prepolymer intermediate. Thus, it is expected to replace the binder for HTPE propellants. The mechanical properties, crosslinking density, hydrogen bonding, and thermal performances of the prepared HTPE binders were investigated through tensile testing, low-field nuclear magnetic resonance (LF-NMR), Fourier-transform infrared spectroscopy (FTIR), and differential scanning calorimetry (DSC) analysis. The ultimate tensile strength (σm) of the in situ-prepared HTPE binder was 1.83 MPa, the fracture elongation (εb) was 371.61%, and the strength increased by 80% compared to the HTPE binders. The crosslink density (Ve) decreased with an increasing content of PEG and/or TDI. The proportion of H-bonds formed by the imino groups increased with the content of PEG and TDI and reached 81.49% at PEG and TDI contents of 50% and 80%, respectively, indicating a positive correlation between the H-bonds and σm. Based on the statistical theory of elasticity, the integrity of the curing networks showed that the contents of PEG and TDI affected the integrity of the curing networks. The DSC data of the in situ-prepared HTPE binder showed a lower glass transition temperature. Finally, compared to HTPE propellant, the strength and elongation of the in situ-prepared HTPE propellant increased by 206% and 135%, respectively. This exciting result greatly enhances the feasibility of the in situ HTPE preparation method.

Struktura fizyczna i topologia sieci przestrzennych elastomerow

Two groups of chosen models of elastomers networks have been discussed on the basis of literature data and own research results. The first group of models encloses: statistical theory of elasticity, Mooney-Rivlin model, model with constraints on chain and junctions, tube model by Heinrich, Straube and Helmis as well as Tschoegl model. They use simple experimental measurements of stress - strain dependence. However the concentration of chemical crosslinking joints of network, N-c, and crosslinking efficiency E-c, calculated for NR and IR cured with dicumyl peroxide (DCP) are much higher than the values resulting from the known mechanism of these elastomers; curing with a peroxide (E-ct = 2) (Table 1, Figs. 3 and 4). In the second groups of models, by Charlesby-Pinner (concerning the rubber showing M-w/M-n approximate to 2) and by Langley and Pearson (any value of M-w/M-n), there is necessary (for proper analysis) to do labour-consuming determinations of sol content as well as molecular weight and polydispersity of a rubber before curing, but this way determinated values of N-c and E-c, in case of curing NR and IR with DCP [(E-c = 1.8-2.0), agree with theoretical values (Table 1)]. These methods can be recommended to complex analysis of physical structure and topology of a network. In networks of cured elastomers there are both the chemical and permanent topological joints. Their part in total number of network joint reaches up to 80% when chemical crosslinking degree is not high. Key words: networks of elastomers, chemical and topological joints, chain entanglement, comparison of network models, crosslinking efficiency.

Double‐step retraction of a polyisoprene vulcanizate: Comparison with molecular models

AbstractLarge stress peaks have been observed by subjecting a weakly crosslinked polyisoprene vulcanizate to double‐step extensions in opposite directions. The resulting retractions are well described by a BKZ constitutive equation with a strain‐dependent term identical to that for deviations from the statistical elasticity theory at equilibrium. This unexpected nonlinear behavior suggests that disengagement of pendant chains and attenuation of junction constraints both contribute to the stress relaxation of random networks.

The non‐linear viscoelastic behavior of a carbon‐black filled elastomer

The stress relaxation behavior of a carbon-black-filled elastomer is shown to be independent of the state of deformation, and the temperature dependence of its static modulus is in good accord with that predicted by the statistical theory of elasticity. The storage and loss moduli are found to be separable functions of frequency and overall strain effects in both simple tension and pure shear. A separability of static deformational and dynamic strain amplitude effects is found for the storage modulus, but could not be determined for the loss modulus. However, the loss tangent is found to be a function of the static state of deformation. From these experimental results, it is shown that an existing viscoelasticity theory for an unfilled system when specialized to uniaxial extensions has a form which can be made suitable for representing the dynamic behavior of filled systems by the introduction of two functions of the dynamic strain amplitude.

The nonlinear viscoelastic behavior of a carbon‐black‐filled elastomer

AbstractThe stress relaxation behavior of a carbon‐black‐filled elastomer is shown to be independent of the state of deformation, and the temperature dependence of its static modulus is in good accord with that predicted by the statistical theory of elasticity. The storage and loss moduli are found to be separable functions of frequency and overall strain effects in both simple tension and pure shear. A separability of static deformational and dynamic strain amplitude effects is found for the storage modulus, but could not be determined for the loss modulus. However, the loss tangent is found to be a function of the static state of deformation. From these experimental results, it is shown that an existing viscoelasticity theory for an unfilled system when specialized to uniaxial extensions has a form which can be made suitable for representing the dynamic behavior of filled systems by the introduction of two functions of the dynamic strain amplitude.

Concerning existence theorems for some problems of elasticity theory

We prove an existence theorem for the statistical elasticity theory equation for a homogeneous incompressible medium and its extension to the second and third boundary value problem case. We demonstrate, in the case of the first, second, and third problems that, as λ→∞ the solution of the elasticity theory equation with Lame constants λ andμ converges to the solutions of the respective equations for incompressible material. An existence theorem in the rectangle is demonstrated for the third boundary value problem inwq2.

Initial stresses in reinforced and filled polymers

The methods of the statistical theory of elasticity are used to calculate the average stresses in the components of uniaxially reinforced and randomly filled polymers arising in the process of fabricating the material as a result of the different linear expansion coefficients of the components and chemical shrinkage of the resin. Expressions are also obtained for the macroscopic characteristics of the media — moduli of elasticity, linear expansion coefficients, and shrinkage coefficients. The results are compared with experimental data and with the results of computations based on the equations proposed by other authors. The relations obtained for the dependence of the stresses in the components on their properties offer a satisfactory explanation for the experimentally established positive correlation between the shrinkage and the compressive strength of a composite based on furan resin.

Statistical theory of elasticity of reinforced plastics with internal stresses of shrinkage origin

The statistical boundary value problem of the theory of elasticity of macrohomogeneous composite media in the natural (unstressed) starting state is extended to media with internal stresses of shrinkage origin. It is established that the moduli of elasticity of the composite do not depend on the magnitude of the shrinkage stresses. The conditions, under which shrinkage of the resin in materials of the glass-reinforced plastic type does not lead to warping, are determined. Applications of the results to the computation of structural reliability characteristics are noted.

Statistical theory of elasticity of reinforced plastics

A method of solving statistical boundary value problems in the stresses is proposed for microinhomogeneous composite media simulating the structure and properties of reinforced plastics. The coefficients of variation of the physical properties of the medium are not assumed small. Numerical examples are presented.

Stress-Strain Behavior of Natural Rubber Vulcanized in the Swollen State

Abstract Samples of natural rubber were vulcanized in the presence of n-decane, decalin and o-chlorobenzene. The diluents were then removed, and the force-extension characteristics of the samples studied over the range 1.1<α<2.0. The elastic behavior of the solution-vulcanized elastomers appear to be in much closer agreement with the statistical theory of elasticity than is the case for vulcanizates prepared in the dry state.

Stress-strain behaviour of natural rubber vulcanized in the swollen state

Samples of natural rubber were vulcanized in the presence of n-decane, decalin and o-chlorobenzene. The diluents were then removed, and the force-extension characteristics of the samples studied over the range 1·1 < α < 2·0. The elastic behaviour of the solution-vulcanized elastomers appear to be in much closer agreement with the statistical theory of elasticity than is the case for vulcanizates prepared in the dry state.

Comparison of Stress-Strain Relations of Polybutadiene Vulcanizates in Compression and in Extension

Abstract According to elementary statistical theory of rubber elasticity, the modulus of rigidity τ/(λ - λ−2) is constant. The phenomenological theory predicts that this modulus is in first approximation equal to the sum of a constant (2C1) and a deformation-dependent term, 2C2. The dependence on elongation of experimental modulus data of many elastomers agrees more or less with the C2 term. Several attempts have been made to explain this dependence on the basis of molecular phenomena or by refinements of the statistical theory of elasticity. At present there is no consensus of opinion on the origin or interpretation of the deformation-dependent term. Almost all of the data on this dependence have been generated in extension experiments.

Transient Changes in Topology and Energy on Extension of Polybutadiene Networks

Abstract The theory of elasticity of polymer networks has been developed along two lines. The phenomenological approach leads to the Mooney-Rivlin relation between stress and extension ratio for uniaxial extension. The statistical theory of elasticity, based on a model for polymer molecules, predicts a similar relation with one of the constants zero. Actual elastic properties of rubbers do not agree fully with either theory. Experimental results are reported obtained with quantitatively cured polybutadiene and polyisoprene vulcanizates. These data are near-equilibrium results through the use of a cyclic stress sequence which largely eliminates the influence of long-time creep. The dependence of the initial modulus and the parameters of the Mooney-Rivlin relation on the chemical nature and the degree of branching of the polymer, the type of cross-links, and temperature has been investigated. A possible relation between the energy component of the elastic force and one of the parameters is discussed. These ...

Stress-Strain Relation of Pure-Gum Rubber Vulcanizates in Compression and Tension

Abstract The empirical function of Martin, Roth, and Stiehler represented by Eq. (1) where A has the value of 0.38 may be regarded as an adequate representation of the available experimental data covering both the compression and tension of pure-gum vulcanizates. The stress and strain are to be measured after a constant time of creep. The approximate validity extends over the range 0.5&amp;lt;L&amp;lt;3.5. The success of the single empirical function in representing data obtained in both compression and tension over a range as great as this is regarded as very significant. In the range of values of L from 0.5 to 1.0 (compression) the empirical function gives results in agreement with the predictions of the statistical theory of rubber elasticity. After representing the stress and strain in a transition region extending from L=1.0 to 1.5, the empirical function gives approximately constant coefficients C1and C2 in the Mooney equation over the range from 1.5 to about 3.5. The behavior of the empirical function in the transition region from L=1.0 to L=1.5 shows that the statistical theory of elasticity fails to represent the experimental data even at the lowest elongations while proving satisfactory in the compression region. The Mooney-Rivlin equation also fails to furnish adequate representation of the experimental data at low elongations. The most satisfactory method of determining Young's modulus from experimental observations of stress and strain in pure-gum vulcanizates involves a plot of log F/(L−1−L−2) against (L−L−1). For observations within the range of L=0.75 to 2.0 the simpler plot of F/(L−1−L−2) against (L - 1) is thoroughly satisfactory. In both cases M is obtained from the intercept, and the constant A in Eq. (1) is determined from the slope.

Stress Relaxation of Peroxide and Sulfur Vulcanizates of Natural Rubber

Abstract Direct chemical investigation of the degradation of polymeric networks is usually impracticable owing to the experimental difficulty of insolubility and to the fact that reaction at only a small proportion of network units is sufficient to cause marked alteration of network structure. Resort has, therefore, to be made to the measurement of physical changes brought about by the chemical reactions. Change of tension at constant extension is a useful measure for this purpose, since statistical elasticity theory predicts a direct proportionality between the tension and the number of chains supporting the stress. The present purpose is: (1) to describe a stress relaxometer possessing certain advantages over previous models; (2) to report on the stress relaxation of peroxide-crosslinked rubber, which can be considered from a chemical viewpoint to be the simplest possible rubber network; (3) to point out some complications in the relaxation behavior of sulfur vulcanizates, and (4) to interpret the shapes of stress relaxation curves.

The Thermodynamics of a Strained Elastomer. III. The Thermal Coefficient of Modulus and the Statistical Theory of Elasticity

Abstract Published thermoelasticity data are critically analyzed by means of thermodynamic equations developed in the first paper of this series. Equations known to be of the proper form are fitted by least squares to the experimental data, which were first corrected to allow for the changes in shape when tested under variable temperature and constant length. If the measurements are accepted as truly reversible, the data show that several different elastomers depart considerably in some respects from the predictions of the statistical theory of elasticity. Some thermoelasticity measurements in torsion are reported which seem to give the most reliable evidence that stress is proportional to absolute temperature.