Stress-Strain Behaviors

Characterization of stress-strain behavior of poly(ether ester)s is of a great interest from both practical and fundamental point of view. Poly(ether ester)s exhibit a high tensile strains comparable to chemically crosslinked rubbers ranging from 500 to 800 %, while their tensile stress is higher than that of vulcanized rubbers, i.e. 20 to 50 MPa 

Property Test method G 4774W G5544 H5556 HRT 5555HS 

Source Hytrel , Product Information, Publication H-81098 Design Guide - Module V, (2000) E. I. du Pont de Nemours and Company. 

Hooke s law-relationship between engineering stress and engineering strain for elastic deformation (tension and compression) 

The degree to which a structure deforms or strains depends on the magnitude of an imposed stress. For most metals that are stressed in tension and at relatively low levels, stress and strain are proportional to each other through the relationship 

Room-T emperature Elastic and Shear Moduli and Poisson s Ratio for Various Metal Alloys 

This is known as Hooke s law, and the constant of proportionahty E (GPa or psi) is the modulus of elasticity, or Young s modulus. For most typical metals, the magnitude of this modulus ranges between 45 GPa (6.5 X 10 psi), for magnesium, and 407 GPa (59 X 10 psi), for tungsten. Modulus of elasticity values for several metals at room temperature are presented in Table 6.1. 

Metal Alloy Modulus of Elasticity Shear Modulus Poisson s Ratio 

Calcium silicate microparticles increase the modulus more efficiently than the nanoparticles. In this case, a silane-treatment of the CaSiOj surface ensures good filler-matrix bonding and favors stress transfer via the interface. The flexural strength decreases slightly, but remains above that of the neat matrix. The fracture surfaces of these composites (EP/AljOj/CaSiOj) occur extensively rugged and 

For nanocomposites it is surprisingly observed that the strain at maximum force and also the strain to break behave not exactly as it would be expected. They 

This assumption makes it possible to caleulate the change in entropy on deformation of a single chain for a specified macroseopie strain. A summation over all ehains gives the macroscopic change in entropy of the rubber block, and the subsequent application of Eq. (10.3.3) yields the desired force or stress corresponding to the imposed strain. Let us illustrate this process for some idealized situations. The more general case will be eonsidered later. 

Consider, for example, a normal force F acting perpendicular to one face of an initially unstrained cube of rubber of edge Iq. Under the influence of this force. 

Because the chain is randomly oriented before it is stretched, 

The change in entropy AS of all the chains in the cube of rubber is Mg times the change in entropy of a single chain. In view of Eqs. (10.4.1) and (10.4.2), this quantity is 

Because rubber is incompressible, its volume does not change on deformation. Therefore, it must be true that 

Designers of most structures specify material stresses and strains well within the proportional/elastic limit. Where required (with no or limited experience on a particular type product materialwise and/or processwise) this practice builds in a margin of safety to accommodate the effects of improper material processing conditions and/or unforeseen loads and environmental factors. This practice also allows the designer to use design equations based on the assumptions of small deformation and purely elastic material behavior. Other important properties derived from stress-strain data that are used include modulus of elasticity and tensile strength. 

Not all plastics can be converted into practical fibers, however, because the intermolecular forces or crystallization tendency may be too weak to achieve useful stable fibers by axial orientation. Synthetic fibers are 

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The effect of temperature on PSF tensile stress—strain behavior is depicted in Figure 4. The resin continues to exhibit useful mechanical properties at temperatures up to 160°C under prolonged or repeated thermal exposure. PES and PPSF extend this temperature limit to about 180°C. The dependence of flexural moduli on temperature for polysulfones is shown in Figure 5 with comparison to other engineering thermoplastics. 

Stress—Strain Curve. Other than the necessity for adequate tensile strength to allow processibiUty and adequate finished fabric strength, the performance characteristics of many textile items are governed by properties of fibers measured at relatively low strains (up to 5% extension) and by the change ia these properties as a function of varyiag environmental conditions (48). Thus, the whole stress—strain behavior of fibers from 2ero to ultimate extension should be studied, and various parameters should be selected to identify characteristics that can be related to performance. 

Typical patterns of stress—strain behavior and the relationship of molecular motion on stress—strain behavior have been discussed (10,18,19,21,49—51). At times, it becomes desirable to characterize stress—strain behavior numerically so that a large amount of information can be condensed and many fibers exhibiting different behaviors can be compared. Procedures for measurement of stress—strain parameters are described ia ASTMD3822 andD2101 (10). 

Table 10-56 gives values for the modulus of elasticity for nonmetals however, no specific stress-limiting criteria or methods of stress analysis are presented. Stress-strain behavior of most nonmetals differs considerably from that of metals and is less well-defined for mathematic analysis. The piping system should be designed and laid out so that flexural stresses resulting from displacement due to expansion, contraction, and other movement are minimized. This concept requires special attention to supports, terminals, and other restraints. 

Figure 6.1. Stress-strain behavior of shock-loaded copper compared to the annealed starting condition illustrating an enhanced flow stress following shock-wave deformation compared to quasi-static deformation (based on an equivalent strain basis).

Figure 6.12. Stress-strain behavior of shoek-loaded NijAl as a funetion of peak pressure.

P.S. Follansbee, The Rate Dependence of Structure Evolution in Copper and its Influence on the Stress-Strain Behavior at Very High Strain Rates, in Impact Loading and Dynamic Behavior of Materials (edited by C.Y. Chiem, H.-D. Kunze, and L.W. Meyer), Springer-Verlag, New York, 1988, pp. 315-322, Vol. 1. 

The properties of the lamina constituents, the fibers and the matrix, have been only briefly discussed so far. Their stress-strain behavior is typified as one of the four classes depicted in Figure 1-8. Fibers generally exhibit linear elastic behavior, although reinforcing steel bars in concrete are more nearly elastic-pertectly plastic. Aluminum, as well as... 

Several experiments will now be described from which the foregoing basic stiffness and strength information can be obtained. For many, but not all, composite materials, the stress-strain behavior is linear from zero load to the ultimate or fracture load. Such linear behavior is typical for glass-epoxy composite materials and is quite reasonable for boron-epoxy and graphite-epoxy composite materials except for the shear behavior that is very nonlinear to fracture. 

Robert M. Jones ar Harold S. Morgan, Analysis of Nonlinear Stress-Strain Behavior of Fiber-Reinforced Composite Materials, AIAA Journal, December 1977, pp. 1669-1676. 

First, the stress-strain behavior of an individual lamina is reviewed in Section 4.2.1, and expressed in equation form for the k " lamina of a laminate. Then, the variations of stress and strain through the thicyiess of the laminate are determined in Section 4.2.2. Finally, the relation of the laminate forces and moments to the strains and curvatures is found in Section 4.2.3 where the laminate stiffnesses are the link from the... 

Harold S. Morgan and Robert M. Jones, Analysis of Nonlinear Stress-Strain Behavior of Laminated Fiber-Reinforced Composite Materials, Proceedings of the 1978 International Conference on Composite Materials, Bryan R. Noton, Robert A. Signorelli, Kenneth N. Street, and Leslie N. Phillips (Editors), Toronto, Canada, 16-20 April 1978, American Institute of Mining, Metallurgical a Petroleum Engineers, New York, 1978, pp. 337-352. 

Harold S. Morgan and Robert M. Jones, Buckling of Rectangular Cross-Ply Laminated Plates with Nonlinear Stress-Strain Behavior, Journal of Applied Mechanics, September 1979, pp. 637-643. 

Let s address the issue of nonlinear material behavior, i.e., nonlinear stress-strain behavior. Where does this nonlinear material behavior come from Generally, any of the matrix-dominated properties will exhibit some degree of material nonlinearity because a matrix material is generally a plastic material, such as a resin or even a metal in a metal-matrix composite. For example, in a boron-aluminum composite material, recognize that the aluminum matrix is a metal with an inherently nonlinear stress-strain curve. Thus, the matrix-dominated properties, 3 and Gj2i generally have some level of nonlinear stress-strain curve. 

These differences on the stress-strain behavior of P7MB and PDTMB show the marked influence of the mesomorphic state on the mechanical properties of a polymer. When increasing the drawing temperatures and simultaneously decreasing the strain rate, PDTMB exhibits a behavior nearly elastomeric with relatively low modulus and high draw ratios. On the contrary, P7MB displays the mechanical behavior typical of a semicrystalline polymer. 

Hydrogen effect on the mechanical properties discussed below was studied on several a and a+fi alloys with the following nominal composition of metallic components (Russian trade marks given in parentheses) commercial titanium of nominal purity 99.3% (VTl-0), Ti-6Al-2Zr-1.5V-lMo (VT20), Ti-6A1-4.5V (VT6), Ti-6Al-2.5Mo-2Cr (VT3-1), Ti-4Al-1.5Mn (OT4), Ti-6.5Al-4Mo-2Sn-0.6W-0.2Si (VT25u) and others. The main features of their stress-strain behavior due to hydrogenation were much similar, but some individuality was characteristic of each alloy. 

This stress-strain behavior is consistent with the optic metallographic data which evidenced partial redistribution of hydrogen over the powder particles when the compacting temperature was increased to 400°C and uniform hydrogen distribution on additional annealing or during plastic deformation at T > 500°C. 

Hydrogen drastically modifies the strain-rate dependence of the stress-strain behavior. 

Avoiding structural failure can depend in part on the ability to predict performance of materials. When required designers have developed sophisticated computer methods for calculating stresses in complex structures using different materials. These computational methods have replaced the oversimplified models of materials behavior relied upon previously. The result is early comprehensive analysis of the effects of temperature, loading rate, environment, and material defects on structural reliability. This information is supported by stress-strain behavior data collected in actual materials evaluations. 

Consequently, changing the temperature or the strain rate of a TP may have a considerable effect on its observed stress-strain behavior. At lower temperatures or higher strain rates, the stress-strain curve of a TP may exhibit a steeper initial slope and a higher yield stress. In the extreme, the stress-strain curve may show the minor deviation from initial linearity and the lower failure strain characteristic of a brittle material. 

Brittleness Brittle materials exhibit tensile stress-strain behavior different from that illustrated in Fig. 2-13. Specimens of such materials fracture without appreciable material yielding. Thus, the tensile stress-strain curves of brittle materials often show relatively little deviation from the initial linearity, relatively low strain at failure, and no point of zero slope. Different materials may exhibit significantly different tensile stress-strain behavior when exposed to different factors such as the same temperature and strain rate or at different temperatures. Tensile stress-strain data obtained per ASTM for several plastics at room temperature are shown in Table 2-3. 

Test rate and property The test rate or cross-head rate is the speed at which the movable cross-member of a testing machine moves in relation to the fixed cross-member. The speed of such tests is typically reported in cm/min. (in./min.). An increase in strain rate typically results in an increase yield point and ultimate strength. Figure 2-14 provides examples of the different test rates and temperatures on basic tensile stress-strain behaviors of plastics where (a) is at different testing rates per ASTM D 638 for a polycarbonate, (b) is the effects of tensile test-... 

The stress-strain behavior of plastics in flexure generally follows from the behavior observed in tension and compression for either unreinforced or reinforced plastics. The flexural modulus of elasticity is nominally the average between the tension and compression moduli. The flexural yield point is generally that which is observed in tension, but this is not easily discerned, because the strain gradient in the flexural RP sample essentially eliminates any abrupt change in the flexural stress-strain relationship when the extreme fibers start to yield. 

The majority of tests to evaluate the characteristics of plastics are performed in tension or flexure hence, the compressive stress-strain behavior of many plastics is not well described. Generally, the behavior in compression is different from that in tension, but the stress-strain response in compression is usually close enough to that of tension so that possible differences can be neglected (Fig. 2-19). The compression modulus is not always reported, since defining a stress at... 

Fig. 2-19 Comparison of tensile and compression stress-strain behavior of TPs.

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