Departures of the Elastic Behavior of Rubbers in Simple Extension from the Kinetic Theory

1955 ◽  
Vol 28 (1) ◽  
pp. 24-35 ◽  
Author(s):  
S. M. Gumbrell ◽  
L. Mullins ◽  
R. S. Rivlin
Keyword(s):  
Kinetic Theory ◽  
Volume Fraction ◽  
Single Parameter ◽  
Elastic Behavior ◽  
Simple Extension ◽  
Stress Strain ◽  
Equilibrium Stress ◽  
Strain Behavior

Abstract It is shown that the equilibrium stress-strain behavior of highly swollen rubber vulcanizates in simple extension agrees with the predictions of the kinetic theory. The departures from these predictions which are found in dry or lightly swollen rubbers have been investigated, and it is shown that they can be described in terms of a single parameter C2. The magnitude of this parameter is large in dry rubbers, and decreases to zero at high degrees of swelling ; this decrease occurs linearly with decrease in the volume fraction of rubber. The value of C2 is found to be independent of the nature of the rubber polymer, of the degree of vulcanization, and of the nature of the swelling liquid. The possible significance of this parameter is discussed in light of these observations.

Theoretical Model for the Elastic Behavior of Filler-Reinforced Vulcanized Rubbers

1957 ◽  
Vol 30 (2) ◽  
pp. 555-571 ◽  
Author(s):  
L. Mullins ◽  
N. R. Tobin
Keyword(s):  
Elastic Properties ◽  
Tensile Tests ◽  
Elastic Behavior ◽  
Simple Extension ◽  
Stress Strain ◽  
Empirical Relationships ◽  
Equilibrium Stress ◽  
Strain Behavior ◽  
Strain Data ◽  
Limited Applicability

Abstract One of the more important advances in rubber science during the past twenty years has been the development of quantitative theories describing the elastic properties of pure-gum vulcanized rubbers. As a result it is now possible to account for their equilibrium stress-strain behavior with considerable success. There is, however, no adequate theory to describe the elastic properties of filler-reinforced rubber vulcanizates and the work described herein is an attempt to provide a basis for such a theory. When a reinforcing filler is added to rubber it produces a large increase in the stiffness of the vulcanizate. This increase is reduced and may be substantially destroyed by deformation. Numerous attempts have been made to describe the increase of stiffness resulting from the introduction of fillers and relationships describing the dependence of the modulus on the concentration and particle shape of the filler have been developed. However, these do not take into account the softening which results from previous deformation and thus have limited applicability. Recently Blanchard and Parkinson have attempted to develop empirical relationships to describe the elastic properties in simple extension of reinforced rubber vulcanizates after they have been previously deformed. They started with the appropriate stress-strain relationships from the classical kinetic theory and introduced two curve-fitting parameters to describe stress-strain data obtained in conventional tensile tests on a Goodbrand machine. In this way they were able to fit the course of the stress-strain data obtained after previous extension at extensions less than those previously applied and to describe the dependence of the parameters on previous deformation. Unfortunately, the significance of the parameters is obscure.

Stress-Strain Equations for Some Near-Eutectic Tin-Lead Solders

1991 ◽  
Vol 113 (4) ◽  
pp. 475-484 ◽  
Author(s):  
K. P. Jen ◽  
J. N. Majerus
Keyword(s):  
Strain Rate ◽  
Printed Circuit Boards ◽  
Volume Fraction ◽  
Total Error ◽  
Tensile Tests ◽  
Elastic Behavior ◽  
Plastic Behavior ◽  
Strain Rates ◽  
Stress Strain ◽  
Engineering Strain

This paper presents the evaluation of the stress-strain behavior, as a function of strain-rate, for three tin-lead solders at room temperature. This behavior is critically needed for reliability analysis of printed circuit boards (PCB) since handbooks list minimal mechanical properties for the eutectic solder used in PCBs. Furthermore, most handbook data are for stable eutectic microstructure whereas PCB solder has a metastable microstructure. All three materials were purchased as “eutectics.” However, chemical analysis, volume fraction determination, and microhardness tests show some major variations between the three materials. Two of the materials have a eutectic composition, and one does not. The true stress-strain equations of one eutectic and the one noneutectic material are determined from compressive tests at engineering strain-rates between 0.0002/s and 0.2/s. The second eutectic material is evaluated using tensile tests with strain-rates between 0.00017/s and 0.042/s. The materials appear to exhibit linear elastic behavior only at extremely small strains, i.e., less than 0.0005. However, this “elastic” behavior showed considerable variation, and depended upon the strain rate. In both tension and compression the eutectic alloy exhibits nonlinear plastic behavior, i.e., strain-softening followed by strain-hardening, which depends upon the strain rate. A quadratic equation σy = σy(ε˚/ε˚0) + A(ε˚/ε˚0)ε + B(ε˚/ε˚0)ε2 fit to the data gives correlation coefficients R2 > 0.91. The coefficients σy(ε˚/ε˚0), A(ε˚/ε˚0), B(ε˚/ε˚0) are fitted functions of the normalized engineering strain rate ε˚/ε˚0. Replicated experiments are used at each strain-rate so that a measure of the statistical variation could be estimated. Measures of error associated with the regression analysis are also obtained so that an estimate of the total error in the stress-strain relations can be made.

Correlation of Large Longitudinal Deformations with Different Strain Histories

1967 ◽  
Vol 40 (2) ◽  
pp. 506-516 ◽  
Author(s):  
L. J. Zapas ◽  
T. Craft
Keyword(s):  
Stress Relaxation ◽  
Single Step ◽  
Strain History ◽  
Simple Extension ◽  
Stress Strain ◽  
Fluid Equation ◽  
Elastic Fluid ◽  
Strain Behavior ◽  
Elastomeric Materials ◽  
Single Integral

Abstract In 1963 Bernstein, Kearsley, and Zapas1 presented a theory of an elastic fluid which gave the correct stress-relaxation response for a large variety of elastomeric materials, including vulcanized rubbers. A principle attractiveness of this theory is its relative simplicity; with a single integral in time, it describes the stress-strain behavior for all types of deformation histories. In the case of simple extension, it predicts the behavior in any uniaxial strain history from the results of single step stress-relaxation experiments which cover the same range of extension and time. We designed a series of experiments to check the validity of this theory and found, as is shown in this paper, excellent agreement with experiment in all cases. We are aware that experiments cannot prove a theory. From our results, however, we feel strongly that a single integral expression with a nonlinear integrand such as the BKZ elastic fluid equation is sufficient to describe the stress-strain behavior of elastomeric materials.

Inelastic Strain Accumulation in Cortical Bone During Rapid Transient Tensile Loading

1999 ◽  
Vol 121 (6) ◽  
pp. 616-621 ◽  
Author(s):  
M. T. Fondrk ◽  
E. H. Bahniuk ◽  
D. T. Davy
Keyword(s):  
Cortical Bone ◽  
Axial Strain ◽  
Human Bone ◽  
Elastic Behavior ◽  
Transverse Strain ◽  
Bovine Bone ◽  
Stress Strain ◽  
Nonlinear Strain ◽  
Strain Behavior ◽  
Nonlinear Stress

An experimental study examined the tensile stress-strain behavior of cortical bone during rapid load cycles to high strain amplitudes. Machined bovine and human cortical bone samples were subjected to loading cycles at a nominal load/unload rate of ±420 MPa/s. Loads were reversed at pre-selected strain levels such that load cycles were typically completed in 0.5-0.7 seconds. Axial strain behavior demonstrated considerable nonlinearity in the first load cycle, while transverse strain behavior was essentially linear. For the human bone 29.1 percent (S.D. = 4.7 percent), and for the bovine bone 35.1 percent (S.D. = 10.8 percent) of the maximum nonlinear strain accumulated after load reversal, where nonlinear strain was defined as the difference between total strain and strain corresponding to linear elastic behavior. Average residual axial strain on unloading was 35.4 percent (S.D. = 1.2 percent) for human bone and 35.1 percent (S.D. = 2.9 percent) of maximum nonlinear strain. Corresponding significant volumetric strains and residual volumetric strains were found. The results support the conclusions that the nonlinear stress-strain behavior observed during creep loading also occurs during transient loading at physiological rates. The volume increases suggest that damage accumulation, i.e., new internal surfaces and voids, plays a major role in this behavior. The residual volume increases and associated disruptions in the internal structure of bone provide a potential stimulus for a biological repair response.

scholarly journals Experimentation of the Bilinear Elastic Behavior of Plain-Woven GFRP Composite with Embedded SMA Wires

Polymers ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 405 ◽  
Author(s):  
Zhiwei Sun ◽  
Yingjie Xu ◽  
Wenzhi Wang
Keyword(s):  
Tensile Strain ◽  
Tensile Tests ◽  
Elastic Behavior ◽  
Uniaxial Tensile ◽  
Stress Strain ◽  
Strain Behavior ◽  
Reinforced Polymer ◽  
Resin Injection ◽  
Stress Threshold ◽  
Gfrp Composite

In this paper, a plain-woven glass-fabric-reinforced polymer (GFRP) composite with embedded shape memory alloy (SMA) wires is investigated by means of experiments. The vacuum-assisted resin injection (VARI) method is utilized to fabricate the composite specimens. Quasi-static uniaxial tensile tests are then carried out to evaluate the influence of SMA reinforcement on the stress–strain behavior of the composite. Only the elastic behavior of the composite is considered in the present study. The tensile strain in all the experiments is kept below 2.5% to avoid debonding of the SMA-resin interface, which would lead to failure of the composite. Stress–strain curves are obtained and shown to present a bilinear behavior due to phase transformation taking place in the SMA wires beyond a certain stress threshold.

Biodegradable polyhydroxybutyrate as a polyol for elastomeric polyurethanes

2012 ◽  
Vol 66 (9) ◽  
Author(s):  
Lucy Vojtová ◽  
Vojtěch Kupka ◽  
Jan Žídek ◽  
Jaromír Wasserbauer ◽  
Petr Sedláček ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Mechanical Properties ◽  
Volume Fraction ◽  
Stress Strain ◽  
Strain Behavior ◽  
Feed Stock ◽  
Measured Stress ◽  
Strain Data ◽  
Polyether Polyol ◽  
Filled Composite

AbstractIn the proposed work, new elastomeric bio-polyol based polyurethanes (bio-PUs) with specific mechanical properties were prepared by a one-shot process without the presence of a solvent. Commercial non-degradable polyether polyol derived from petrochemical feed stock was partly (in the amount of 1 mass %, 5 mass %, and 10 mass %) substituted by the biodegradable polyhydroxybutyrate (PHB). Morphology of elastomeric PU composites was evaluated by scanning electron microscopy and mechanical properties of the prepared samples were obtained by both tensile measurements and prediction via the Mooney-Rivlin equation. Electron microscopy proved that the prepared materials have the character of a particle filled composite material, where PHB particles are regular with their size of about 1–2 μm in diameter. Tensile measurements demonstrated that the Young’s modulus, tensile stress at break, and tensile strain at break of each sample increase with the increase of the volume fraction of the filler. From the measured stress-strain data, the first and the second term of the Mooney-Rivlin equation were calculated. The obtained constants were applied to recalculate the stress-strain curves. It was found that the Mooney-Rivlin equation corresponds well with the stress-strain behavior of the prepared specimens.

Effects of Water on the Mechanical Properties of Paper and their Relationship to the Treatment of Paper

1992 ◽  
Vol 267 ◽  
Author(s):  
Timothy Vitale
Keyword(s):  
Mechanical Properties ◽  
Tensile Strength ◽  
Tap Water ◽  
Water Immersion ◽  
Exchange Process ◽  
Elastic Behavior ◽  
Elastic Fibers ◽  
Plastic Behavior ◽  
Stress Strain ◽  
Strain Behavior

ABSTRACTThrough a series of experiments the mechanical properties of paper are explored. Hydrogen bonding is fundamental to the performance of paper and its disruption results in distinctive stress-strain behavior. Stress-strain curves were generated from which tensile strength, Young's modulus, percent stretch, and work (tensile energy absorption) were obtained.It was found that the contribution of the fiber to the mechanical properties of paper is primarily elastic. Fibers are many times stronger than paper. Only fibers which have been severely deteriorated show measurable changes in stress-strain behavior. Fiber deterioration results in characteristically different stress-strain behavior than that which results from disruption of interfiber bonding.Water immersion results in the disruption of interfiber bonds in paper, leaving only 2-3% of dry tensile strength. Interfiber bonds make a profound contribution to the mechanical properties of the paper. Aqueous treatment is shown to be a radical treatment, altering the original dried-in properties of the sheet. The release of structural bonds and dried-in strains during wetting and the subsequent reformation of interfiber bonds during drying are shown to be independent of water purity, be it ultrapure water, tap water, or water containing washing aids such as Ca(OH)2, NaOH, CaCO3 or Na2CO3.The effects of immersion in organic solvents was explored. Solvents have effects on mechanical properties which are approximately proportional to the degree of swelling caused by the solvent. Water, the liquid which caused the greatest swelling of the liquids evaluated, is shown to be the most disruptive liquid followed by methanol and acetone; toluene caused virtually no change.To explore the behavior of interfiber bonds paper was taken through a solvent exchange process. A sample was immersed in water and then taken through separate ethanol and acetone immersions to toluene, and dried. The result was a sheet with little bonding and decreases in all mechanical properties. To explore the surface tension and capillary action effects of water, the solvent-exchanged sheet was re-immersed in water. Upon drying, interfiber bonding was reintroduced which resulted in the regain of mechanical properties lost.A paradigm for the mechanical behavior of paper is developed. Fibers contribute elastic behavior and interfiber bonds are a principal source of plastic behavior.

scholarly journals Mechanical Properties of Fiber Reinforced Lightweight Concrete Containing Surfactant

2010 ◽  
Vol 2010 ◽  
pp. 1-8 ◽  
Author(s):  
Yoo-Jae Kim ◽  
Jiong Hu ◽  
Soon-Jae Lee ◽  
Byung-Hee You
Keyword(s):  
Mechanical Properties ◽  
Lightweight Concrete ◽  
Volume Fraction ◽  
Lightweight Aggregate ◽  
Unit Weight ◽  
Fiber Reinforced ◽  
Compression Tests ◽  
Stress Strain ◽  
Strain Behavior ◽  
Toughness Index

Fiber reinforced aerated lightweight concrete (FALC) was developed to reduce concrete's density and to improve its fire resistance, thermal conductivity, and energy absorption. Compression tests were performed to determine basic properties of FALC. The primary independent variables were the types and volume fraction of fibers, and the amount of air in the concrete. Polypropylene and carbon fibers were investigated at 0, 1, 2, 3, and 4% volume ratios. The lightweight aggregate used was made of expanded clay. A self-compaction agent was used to reduce the water-cement ratio and keep good workability. A surfactant was also added to introduce air into the concrete. This study provides basic information regarding the mechanical properties of FALC and compares FALC with fiber reinforced lightweight concrete. The properties investigated include the unit weight, uniaxial compressive strength, modulus of elasticity, and toughness index. Based on the properties, a stress-strain prediction model was proposed. It was demonstrated that the proposed model accurately predicts the stress-strain behavior of FALC.

Stress Softening in Rubber Vulcanizates. Part I. Use of a Strain Amplification Factor to Describe Elastic Behavior of Filler-Reinforced Vulcanized Rubber

1966 ◽  
Vol 39 (4) ◽  
pp. 799-813 ◽  
Author(s):  
L. Mullins ◽  
N. R. Tobin
Keyword(s):  
Volume Concentration ◽  
Spherical Particles ◽  
Elastic Behavior ◽  
Simple Extension ◽  
Factor X ◽  
Stress Strain ◽  
Vulcanized Rubber ◽  
Stress Strain Relation ◽  
Strain Data ◽  
Volume Swelling

Abstract Measurements of Young's modulus of vulcanized rubbers containing thermal carbon black show the predicted dependence on the volume concentration given by relationships derived for a suspension of spherical particles, for example, that due to Guth, Simha, and Gold which gives E=E0 (1+2.5c+14.1c2). A simple interpretation of the results is that the strain in the rubber is increased by the presence of filler so that the ratio of the average strain to the measured overall strain is given by the factor X=1+2.5c+14.1c2. This factor was used to analyze simple extension stress-strain data obtained at larger extensions. For this purpose the Mooney—Rivlin relation was used to describe the behavior of the rubber phase. Values of C1 independent of the volume concentration and in close accord with measurements of the equilibrium volume swelling of the rubbers were obtained. Values of λ* were also consistent with those of #5#. Analysis of stress—strain data obtained on rubbers containing smaller particle-sized carbon blacks is more complex. For these materials the relation due to Guth, viz., E=E0 (1+2.5c+14.1c2), was chosen. By the choice of suitable values of f, good agreement with the Mooney—Rivlin stress—strain relation was achieved at volume concentrations less than about 0.15.

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