A Mathematical Treatment of a Theory of Rubber Structure

1935 ◽  
Vol 8 (1) ◽  
pp. 23-38
Author(s):  
T. R. Griffith
Keyword(s):  
Elastic Modulus ◽  
Plastic Flow ◽  
Strain Curve ◽  
The Other ◽  
Stress Axis ◽  
Mathematical Treatment ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Joule Effect ◽  
Vulcanized Rubber

Abstract A brief consideration of the work that has been done on the structure of rubber convinces, one that the elasticity is wholly or at least mainly explained by a consideration of the kinetics involved. The fact that when a strip of stretched rubber, one end of which is free, contracts when it is warmed, contrary to the behavior of most bodies, and that it becomes warmed on stretching, commonly known as the Gough-Joule effect, pp. 453–461, would lead one to suspect .that there is a connection between the kinetic energy of the rubber molecule and its elasticity. Lundal, Bouasse, Hyde, Somerville and Cope, Partenheimer and Whitby and Katz have reported observations, principally stress-strain curves, which show that vulcanized rubber has a lower modulus of elasticity at higher temperatures, i. e., it becomes easier to stretch as the temperature is raised. On the other hand, Schmulewitsch, Stevens, and Williams found that the elastic modulus increases with the temperature. Williams shows that the softening of vulcanized rubber with rise of temperature is due to an increase of plasticity. In order to get rid of plastic flow, he first stretches the specimen several times to within about 50 per cent of its breaking elongation, and then obtains an autographic stress-strain curve of the rubber stretched very quickly. He finds that in this case the rubber actually becomes stiffer with rise of temperature, increasing temperatures causing the stress-strain curves to lean progressively more and more toward the stress axis. He concludes that rise of temperature has two effects, one a softening due to increase of plasticity, rendering plastic flow more easy, the other an actual stiffening of the rubber due to rise of temperature. It is not easy to explain the latter effect on any theory which does not take kinetics into account.

A Contribution to the Characterization of the Plastic-Elastic State

1939 ◽  
Vol 12 (4) ◽  
pp. 799-804 ◽  
Author(s):  
E. Rohde
Keyword(s):  
Characteristic Property ◽  
Strain Curve ◽  
The Other ◽  
Elastic State ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Vulcanized Rubber ◽  
Other Hand ◽  
Puzzling Problem

Abstract The manner in which vulcanized rubber can be deformed and yet return almost completely to its original dimensions after the stress is released is a unique and characteristic property. Technically the problem in testing rubber is to evaluate this property and to define it in terms of the factors which are concerned. To define completely this property of rubber whereby it is susceptible to deformation, it is necessary to know the stress, the elongation, the energy expended, the energy lost, the time and the temperature. The stress, elongation and energy expended are closely related and are characterized by the stress-strain curve, which in turn depends on the time and temperature. In addition, it must be borne in mind that rubber can be deformed either by tension or by pressure, but this will not be discussed further here. On the other hand a rather puzzling problem will be considered, the solution of which brings out the fact that the three variables involved in any deformation, viz.: (1) The time or frequency. (2) The temperature. (3) The interrelated factors: stress, elongation and energy expended, must be varied considerably in order to characterize the phenomena of deformation and that when this is done, unexpected results are obtained.

The Behavior of Raw Rubber When Stretched Isothermally

1938 ◽  
Vol 11 (4) ◽  
pp. 647-652 ◽  
Author(s):  
H. Hintenberger ◽  
W. Neumann
Keyword(s):  
Room Temperature ◽  
Strain Curve ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Steep Ascent ◽  
Flow Effect ◽  
Vulcanized Rubber ◽  
Flow Phenomena ◽  
Entropy Effect ◽  
The Subject

Abstract The S-shaped form of the stress-strain curve of rubber is today explained in a quite satisfactory way. In the first part of the curve, i. e., the gradual ascent, work must be expended because of the van der Waals forces of attraction of the molecules; in the second part, i. e., the steep ascent, the elasticity is chiefly an entropy effect, which is finally exceeded by crystallization phenomena. The phenomenon of crystallization itself has been the subject of extensive investigations, but in most cases vulcanized rubber has been employed, and because of the various accelerators and fillers which the rubber has contained, the products have been rather ill-defined. It is evident that the phenomena involved in crystallization would be much more clearly defined if the substance under investigation were to be in a higher state of purity. If experiments are carried out with raw rubber, a flow effect is added to the various other phenomena. As a result of this flow effect, Rosbaud and Schmidt, and Hauser and Rosbaud as well, found that the stress-strain curve depends on the rate of elongation at very low extensions, with a greater stiffness at high rates of elongation. As found recently by Kirsch, there is no evidence of any flow phenomena in vulcanized rubber at room temperature. Most investigations have been so carried out that the stress has been measured at a definite elongation. It was therefore of interest to determine the elongation at constant stress, and the changes in this relation with time and with temperature, of various types of raw rubber.

A MATHEMATICAL TREATMENT OF A THEORY OF RUBBER STRUCTURE

1934 ◽  
Vol 10 (5) ◽  
pp. 486-520 ◽  
Author(s):  
T. R. Griffith
Keyword(s):  
Average Length ◽  
Average Distance ◽  
Strain Curve ◽  
Molecular Chain ◽  
Mathematical Work ◽  
Mathematical Treatment ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Starting Point ◽  
Junction Points

The theory that the elasticity of rubber is due to the heat vibrations of very long chain molecules, bound to one another at occasional points along their length, but able to move freely relatively to one another at all other points, is susceptible of mathematical treatment. In the present treatment it is assumed that the rubber molecule has a restricted rotation about the axis formed by joining two adjacent junction points.A stress-strain curve has been developed mathematically on this assumption, and this curve, which is a reasonably close approximation to the curve obtained experimentally, serves as a standard with which to compare the rubber stress-strain curve and as a starting point for further mathematical work on the structure of rubber. The discrepancy between the mathematical and the experimental curve is explained on the very probable assumption that there is a wave motion or other vibration along the length of the rotating chain, as well as a rotation of the chain as a whole.An explanation of the peculiar S-shape of the beginning of the experimental stress-strain curve develops automatically from the mathematically deduced relation between stress and strain, and it is also shown why the S-shape appears to vanish when the calculation of the stress is based on the actual cross section of the stretched rubber.In addition, the following values, calculated from the above assumption and X-ray data, of certain constants, were obtained:(i) The average distance between junction points.(ii) The average length of molecular chain between junction points and, incidentally, the ratio between the number of freely swinging carbon atoms and those bound at junction points. This gives the number of freely swinging carbon atoms on the molecular chain between junction points and an idea of the length of the rubber molecule.(iii) The quantity of kinetic energy per cubic centimetre causing the elastic effect in rubber.(iv) The percentage of sulphur necessary to form the junction points in vulcanized rubber and, consequently, the minimum quantity of sulphur needed for vulcanization, both for hard and soft rubber. This minimum agrees closely with practical experiment.

Aging of Stretched Rubber

1928 ◽  
Vol 1 (1) ◽  
pp. 106-112
Author(s):  
Arthur Kelly ◽  
Bert S. Taylor ◽  
Webster N. Jones
Keyword(s):  
Direct Relationship ◽  
Strain Curve ◽  
Ultra Violet ◽  
The Other ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Test Strips ◽  
Ultra Violet Light ◽  
Violet Light ◽  
Early Stages

Abstract Sunlight aging under tension of many compounds including the following has been investigated: tire tread shoe upper, tube stocks, golf ball thread, jar rubber, solid tire, bathing cap stock, channel rubber. With some of these stocks the sunlight aging as been compared with unstretched samples by Geer oven, Bierer bomb, and ultra-violet light methods. The stretching of the test strips accelerates deterioration in sunlight, ultra-violet light, and Geer oven. Stretched samples have not yet been tested in the Bierer bomb. The rate of deterioration was not proportional to the degree of stretch in any of the stocks in the early stages of exposure. In sunlight there is a critical elongation for each stock at which the deterioration progresses more rapidly than at any other in the early stages of aging. No direct relationship was found between the results of sunlight aging and the other methods employed. Stretched strips aged in ultra-violet light were found to give softer stress-strain curves than the unaged samples, whereas sunlight aging under the same conditions stiffens the stress-strain curve.

Reinforcement of Butyl with Carbon Black. II. Thermally vs. Chemically Oxidized Blacks

1964 ◽  
Vol 37 (4) ◽  
pp. 1034-1048 ◽  
Author(s):  
A. M. Gessler
Keyword(s):  
Carbon Black ◽  
Chemical Reactivity ◽  
Preceding Paper ◽  
Strain Curve ◽  
Convincing Evidence ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Carbon Particles ◽  
Vulcanized Rubber ◽  
Oxygen Functionality

Abstract The effect of oxidized blacks on the stress-strain properties and bound-rubber content of butyl and SBR was discussed in the preceding paper. Oxidized blacks, when compared with similar untreated blacks, were shown to have a greatly increased reinforcing capacity in butyl. Oxygen functionality on carbon black, it was therefore concluded, is essential in butyl to produce the chemical reactivity which is required between polymer and black if high-order reinforcement is to be obtained. Oxygen functionality on carbon black, it was also demonstrated, is not only not required for enhanced reinforcement in SBR, but it is in fact a deterrent, because it exerts severe restraining effects on the cure of the resulting vulcanizates as well. These interesting results were proposed to provide qualitative but convincing evidence that carbon-polymer bonding, which we believe is requisite to reinforcement, is achieved by different mechanisms in butyl and SBR. In butyl, the unique sensitivity of the stress-strain curve to reinforcing effects was used to speculate on the disposition of carbon blacks in “filled” and reinforced vulcanizates, respectively. With oxidized blacks, reinforcement effects were pictured as stiffening effects which, starting with the gum vulcanizates, caused the stress-strain curve to be shifted without intrinsic changes in its shape. The resulting “reinforced gum,” it was suggested, derived its physical characteristics from the fact that carbon black was included in the vulcanized rubber network. With untreated blacks, in “filled” systems, carbon black was pictured as being enmeshed or entangled in an independently formed vulcanized rubber network. The stiffening effects in this case were attributed to viscous contributions arising from steric restrictions which the occluded carbon particles were thought to impose on both initial movements and the subsequent orientation of network chains when the sample was extended.

Thermodynamics of Crystallization in High Polymers. VI. Incipient Crystallization in Stretched Vulcanized Rubber

1950 ◽  
Vol 23 (3) ◽  
pp. 576-580 ◽  
Author(s):  
Thomas G. Fox ◽  
Paul J. Flory ◽  
Robert E. Marshall
Keyword(s):  
Theoretical Prediction ◽  
Experimental Determination ◽  
Strain Curve ◽  
Steep Slope ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Hydrostatic Method ◽  
Vulcanized Rubber ◽  
High Polymers

Abstract Experimental determination of the elongation at which crystallization commences in vulcanized rubber has been attempted through measurement of density changes by a hydrostatic method. The critical elongation for incipient crystallization appears to depend on the temperature, in approximate accordance with theoretical prediction. Crystallization sets in at an elongation well below that at which the stress-strain curve assumes a steep slope.

The Research of Test on Stress-Strain Constitutive Relations of Straw Concrete

2014 ◽  
Vol 584-586 ◽  
pp. 987-992
Author(s):  
Wei Liu ◽  
Wei Xi ◽  
Yi Lu Zhang
Keyword(s):  
Mechanical Properties ◽  
Compressive Strength ◽  
Elastic Modulus ◽  
Fly Ash ◽  
Constitutive Relations ◽  
Strain Curve ◽  
Green Building ◽  
Corn Straw ◽  
Stress Strain Curve ◽  
Stress Strain

As a new green building material, straw concrete are introduced about its mechanical properties and characteristics. Mechanical properties test such as prism compressive strength, elastic modulus and Poisson's ratios use standard prismatic blocks. Under different rate of corn straw, cement, sand and fly ash, test gets the full stress-strain curve. Results show that with increase of volume of corn straw, the prism compressive strength reduces significantly. Comparing with natural concrete, elastic modulus of straw concrete can reduces greatly. Poisson’s ratio reduces with increase of volume of corn straw. Fly ash could improve property of the material and replace cement, but excessive replacement will reduce the strength of material.

Research on Compressive Deformation Characters of Mortar-Free Grouted Concrete Masonry

2014 ◽  
Vol 1004-1005 ◽  
pp. 1531-1536 ◽  
Author(s):  
Xi Xi He ◽  
Ye Lin
Keyword(s):  
Elastic Modulus ◽  
Compressive Stress ◽  
Test Data ◽  
Strain Curve ◽  
Compressive Deformation ◽  
Filling Rate ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Concrete Masonry

Compressive experiments on mortar-free grouted concrete masonry composed with hollow blocks were studies in this essay. Characteristics of compressive stress-strain curve were analyzed by utilizing test data of 15 specimens with 100% filling rate of grouted concrete. Further more, elastic modulus formula was proposed according to results of previous and present work.

scholarly journals The mechanism of plastic deformation of crystals. Part II.—Comparison with observations

1934 ◽  
Vol 145 (855) ◽  
pp. 388-404 ◽  
Keyword(s):  
Elastic Limit ◽  
Large Range ◽  
Strain Curve ◽  
Pure Metal ◽  
The Other ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Essential Characteristic ◽  
Small Strains ◽  
Plastic Stress

According to the theory give in Part I the strain-hardening or plastic stress strain curve for a pure metal should be parabola. In figs. 1, 2, and 3, Part I, Parabolas are drawn, the parameters being chosen so that they lie as close as possible to the points which represent actual observations. It will be seen that for aluminium and gold the agreement is good. For a single crystal of copper the agreement is not good, but, on the other hand, the plastic stress-strain curve for polycrystalline specimens of copper which is shown in fig. 1 is very nearly parabolic over a large range. The observations for iron seem to show that there is a small finite elastic limit, i. e. , S T may be finite. Parabolas corresponding with the existence of a small elastic limit and with no elastic limit have been drawn. It seems that the observed points lie rather closer to the former curve. In any case, the observed curves have the essential characteristic of the theoretical ones that they are very steep at small strains, but get less and less steep as the strain increases.

scholarly journals Stress–Strain Curve and Carbonation Resistance of Recycled Aggregate Concrete after Using Different RCA Treatment Techniques

2021 ◽  
Vol 11 (9) ◽  
pp. 4283
Author(s):  
Long Li ◽  
Dongxing Xuan ◽  
Chisun Poon
Keyword(s):  
Elastic Modulus ◽  
Strain Curve ◽  
Peak Stress ◽  
Recycled Aggregate Concrete ◽  
Recycled Aggregate ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Aggregate Concrete ◽  
Treatment Techniques ◽  
Carbonation Resistance

Five recycled coarse aggregate (RCA) treatment techniques including flow-through carbonation, pressurized carbonation, wet carbonation, nano silica (NS) pre-spraying and combined pressurized carbonation with NS pre-spraying, were utilized to improve the performance of recycled aggregate concrete (RAC). The characteristics of the stress–strain curves of RACs including peak stress, peak strain, elastic modulus, ultimate strain and toughness were evaluated after using the above RCA treatment techniques. A theoretical model for natural aggregate concrete was used to analyse the stress–strain curve of RAC. Additionally, the carbonation resistance of RAC after using different RCA treatment techniques were investigated. The results showed that the calculated stress–strain curve of RAC based on the theoretical model matched well with the experimental results. Among the three types of carbonation techniques, pressurized carbonation caused the highest improvement in peak stress and elastic modulus of RAC, followed by flow-through carbonation, the last was wet carbonation. The NS pre-spraying method contributed to even higher improvement in peak stress and elastic modulus of RAC than the pressurized carbonation method. The combined pressurized carbonation with NS pre-spraying exhibited the highest enhancement of RAC because both the RCA and the new interface transition zone (ITZ) were improved. The carbonation resistance of RAC was improved after using all the studied RCA treatment techniques.

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