Stresses strain curves

A great deal can be learned about the mechanical properties of materials by stressing them until they fracture or break. The most common mechanical test involving metals or polymers is the tensile test, in which a sample of the solid is stretched. The test uses a standard test piece with a shape dependent on the material to be tested. Metals usually have a central cylindrical section, of known gauge length. 

Knife edge supports Knife-edge supports 

The cross-sectional area of the ceramic specimen is not greatly altered during the test, so that the true stress is measured. In testing a ceramic, the force or load is slowly increased until fracture. 

The Unear part of the stress-strain curve is the elastic region. Here, removal of the load will allow the solid to return to its original dimensions, quite reversibly. In the case of elastomers, this reversibility is maintained over the whole of the stress-strain curve. 

For aU other solids, once the elastic region is passed, the deformation of the solid is not reversed when the stress is removed, and some degree of permanent deformation remains. This is called plastic deformation. For metals, the point at which elastic behaviour changes to plastic behaviour is 

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Figure C2.1.17. Stress-strain curve measured from plane-strain compression of bisphenol-A polycarbonate at 25 ° C. The sample was loaded to a maximum strain and then rapidly unloaded. After unloading, most of the defonnation remains.

A schematic stress-strain curve of an uncrimped, ideal textile fiber is shown in Figure 4. It is from curves such as these that the basic factors that define fiber mechanical properties are obtained. 

Fig. 4. Idealized stress-strain curves of an uncrimped textile fiber point 1 is the proportional limit, point 2 is the yield point, and point 3 is the break or...

The elasticity of a fiber describes its abiUty to return to original dimensions upon release of a deforming stress, and is quantitatively described by the stress or tenacity at the yield point. The final fiber quaUty factor is its toughness, which describes its abiUty to absorb work. Toughness may be quantitatively designated by the work required to mpture the fiber, which may be evaluated from the area under the total stress-strain curve. The usual textile unit for this property is mass pet unit linear density. The toughness index, defined as one-half the product of the stress and strain at break also in units of mass pet unit linear density, is frequentiy used as an approximation of the work required to mpture a fiber. The stress-strain curves of some typical textile fibers ate shown in Figure 5. 

Fig. 5. Stress—strain curves of some textile fibers (17). To convert N/mm to psi, multiply by 145.

Another aspect of plasticity is the time dependent progressive deformation under constant load, known as creep. This process occurs when a fiber is loaded above the yield value and continues over several logarithmic decades of time. The extension under fixed load, or creep, is analogous to the relaxation of stress under fixed extension. Stress relaxation is the process whereby the stress that is generated as a result of a deformation is dissipated as a function of time. Both of these time dependent processes are reflections of plastic flow resulting from various molecular motions in the fiber. As a direct consequence of creep and stress relaxation, the shape of a stress—strain curve is in many cases strongly dependent on the rate of deformation, as is illustrated in Figure 6. 

Fig. 6. The effect of rate of extension on the stress—strain curves of rayon fibers at 65% rh and 20°C. The numbers on the curves give the constant rates of...

The mechanical properties of acryUc and modacryUc fibers are retained very well under wet conditions. This makes these fibers well suited to the stresses of textile processing. Shape retention and maintenance of original bulk in home laundering cycles are also good. Typical stress—strain curves for acryhc and modacryUc fibers are compared with wool, cotton, and the other synthetic fibers in Figure 2. 

The ratio of stress to strain in the initial linear portion of the stress—strain curve indicates the abiUty of a material to resist deformation and return to its original form. This modulus of elasticity, or Young s modulus, is related to many of the mechanical performance characteristics of textile products. The modulus of elasticity can be affected by drawing, ie, elongating the fiber environment, ie, wet or dry, temperature or other procedures. Values for commercial acetate and triacetate fibers are generally in the 2.2—4.0 N/tex (25—45 gf/den) range. 

The abihty of a fiber to absorb energy during straining is measured by the area under the stress—strain curve. Within the proportional limit, ie, the linear region, this property is defined as toughness or work of mpture. For acetate and triacetate the work of mpture is essentially the same at 0.022 N/tex (0.25 gf/den). This is higher than for cotton (0.010 N/tex = 0.113 gf/den), similar to rayon and wool, but less than for nylon (0.076 N/tex = 0.86 gf/den) and silk (0.072 N/tex = 0.81 gf/den) (3). 

Fig. 1. Stress—strain curves A, hard fiber, eg, nylon B, biconstituent nylon—spandex fiber C, mechanical stretch nylon D, spandex fiber E, extruded latex...

Most extmded latex fibers are double covered with hard yams in order to overcome deficiencies of the bare threads such as abrasiveness, color, low power, and lack of dyeabiUty. During covering, the elastic thread is wrapped under stretch which prevents its return to original length when the stretch force is removed thus the fiber operates farther on the stress—strain curve to take advantage of its higher elastic power. Covered mbber fibers are commonly found in narrow fabrics, braids, surgical hosiery, and strip lace. 

Eig. 1. Typical stress—strain curves for cotton and PET fibers. A, industrial B, high tenacity, staple C, regular tenacity, filament D, regular tenacity, staple ... 

Fig. 4. Representative stress—strain curves of spun and drawn PET A, low speed spun-mechanically drawn yam B, 6405 m /min C, 5490 m /min D, 4575...

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

The mechanical piopeities of stmctuial foams and thek variation with polymer composition and density has been reviewed (103). The variation of stmctural foam mechanical properties with density as a function of polymer properties is extracted from stress—strain curves and, owkig to possible anisotropy of the foam, must be considered apparent data. These relations can provide valuable guidance toward arriving at an optimum stmctural foam, however. 

Little error is introduced using the idealized stress—strain diagram (Eig. 4a) to estimate the stresses and strains in partiady plastic cylinders since many steels used in the constmction of pressure vessels have a flat top to their stress—strain curve in the region where the plastic strain is relatively smad. However, this is not tme for large deformations, particularly if the material work hardens, when the pressure can usuady be increased above that corresponding to the codapse pressure before the cylinder bursts. 

Fig. 1. Stress—strain curves for ionomer and polyethylene resins. Test speed is 5 cm/min. The reference matedal is high molecular weight conventional...

Fig. 2. Typical stress—strain curves of nonwoven fabrics, where (—) is woven (-), thermally bonded nonwoven and (-), needle-punched...

This concept is explained by Figure 12 which shows the uniaxial stress— strain curve for a ductile material such as carbon steel. If the stress level is at the yield stress B or above, the problem is no longer a linear one. 

Fig. 12. Uniaxial stress—strain curve for an elastic plastic material. See text.

As a pipeline is heated, strains of such a magnitude are iaduced iato it as to accommodate the thermal expansion of the pipe caused by temperature. In the elastic range, these strains are proportional to the stresses. Above the yield stress, the internal strains stiU absorb the thermal expansions, but the stress, g computed from strain 2 by elastic theory, is a fictitious stress. The actual stress is and it depends on the shape of the stress-strain curve. Failure, however, does not occur until is reached which corresponds to a fictitious stress of many times the yield stress. 

Fig. 4. Types of stress—strain curves (a) soft and weak (b) hard and brittle (c) soft and tough (d) hard and strong and (e) hard and tough.

Fig. 2. Stress—strain curve for standard polycarbonate resin at 23°C where the points A, B, and C correspond to the proportional limit (27.6 MPa), the yield point (62 MPa), and the ultimate strength (65.5 MPa), respectively. To convert MPa to psi, multiply by 145.

Typical stress—strain curves are shown in Figure 3 (181). The stress— strain curve has three regions. At low strains, below about 10%, these materials are considered to be essentially elastic. At strains up to 300%, orientation occurs which degrades the crystalline regions causing substantial permanent set. 

Fig. 3. Stress—strain curve of typical polyesterether elastomer showing the three main regions (I, II, and III) (181), where A is the slope (Young s modulus)...

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