Proportional limit

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 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. 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.

Platinum [7440-06-4], Pt, detracts from the gold color, produckig an undeskable grayish-red color kicreased platinum produces a platinum-colored ahoy. Platinum kicreases strength, proportional limit, and solidification temperatures reduces grain size and produces a heat-treatable ahoy with gold. It has a useful range of 0—18 wt %. 

Modulus of elasticity (E) the ratio of the unit stress to the unit strain within the proportional limits of a material in tension or compression. Refer to Figure 30.1. 

Proportional limit the point on the stress-strain curve at which will commence the deviation in the stress-strain relationship from a straight line to a parabolic curve (Figure 30.1). 

Note When the elastic modulus is used, the elastic limit or proportional limit should be used with it in the formula. When the plastic modulus or Secant Modulus is used, it should be used with the corresponding yield strength. 

Stress-strain curves at the conditions of product application. If applicable, this would usually indicate the toughness of material by sizing up the area under the curve (Chapter 2). It would also show the proportional limit, yield point, corresponding elongations, and other relevant data. 

When the magnitude of deformation is not too great, viscoelastic behavior of plastics is often observed to be linear, i.e., the elastic part of the response is Hookean and the viscous part is Newtonian. Hookean response relates to the modulus of elasticity where the ratio of normal stress to corresponding strain occurs below the proportional limit of the material where it follows Hooke s law. Newtonian response is where the stress-strain curve is a straight line. 

Proportional limit A material s proportional limit is the greatest stress at which it is capable of sustaining an applied load without deviating from the proportionality of a stress-strain straight line (Fig. 2-2). 

Modulus of elasticity Most materials, including plastics and metals, have deformation proportional to their loads below the proportional limit. Since stress is proportional to load and strain to deformation, this implies that stress is proportional to strain. Hooke s Law, developed in 1676, follows that this straight line (Fig. 2-2) of proportionality is calculated as ... 

Ductility A typical tensile stress-strain curve of many ductile plastics is shown in Fig. 2-13. As strain increases, stress initially increases approximately proportionately (from point 0 to point A). For this reason, point A is called the proportional limit of the material. From point 0 to point B, the behavior of the material is purely elastic but beyond point B, the material exhibits an... 

The moduli of elasticity, G for shear and E for tension, are ratios of stress to strain as measured within the proportional limits of the material. Thus the modulus is really a measure of the rigidity for shear of a material or its stiffness in tension and compression. For shear or torsion, the modulus analogous to that for tension is called the shear modulus or the modulus of rigidity, or sometimes the transverse modulus. 

In simple beam-bending theory a number of assumptions must be made, namely that (1) the beam is initially straight, unstressed, and symmetrical (2) its proportional limit is not exceeded (3) Young s modulus for the material is the same in both tension and compression and (4) all deflections are small so that planar cross-sections remain planar before and after bending. The maximum stress... 

The important tensile modulus (modulus of elasticity) is another property derived from the stress-strain curve. The speed of testing, unless otherwise indicated is 0.2 in./min, with the exception of molded or laminated TS materials in which the speed is 0.05 in./min. The tensile modulus is the ratio of stress to corresponding strain below the proportional limit of a material and is expressed in psi (pounds per square inch) or MPa (mega-Pascal) (Fig. 2-7). 

The proportional limit is the greatest stress that a material is capable of sustaining without any deviation of the proportionality law. It is located on the stress-strain curve below the elastic limit. The elastic limit is the great-... 

The second major assumption is that the material is elastic, meaning that the strains are directly proportional to the stresses applied and when the load is removed the deformation will disappear. In engineering terms the material is assumed to obey Hooke s Law. This assumption is probably a close approximation of the material s actual behavior in direct stress below its proportional limit, particularly in tension, if the fibers are stiff and elastic in the Hookean sense and carry essentially all the stress. This assumption is probably less valid in shear, where the plastic carries a substantial portion of the stress. The plastic may then undergo plastic flow, leading to creep or relaxation of the stresses, especially when the stresses are high. 

When an engineering plastic is used with the structural foam process, the material produced exhibits behavior that is easily predictable over a large range of temperatures. Its stress-strain curve shows a significantly linearly elastic region like other Hookean materials, up to its proportional limit. However, since thermoplastics are viscoelastic in nature, their properties are dependent on time, temperature, and the strain rate. The ratio of stress and strain is linear at low strain levels of 1 to 2%, and standard elastic design... 

Hooke s Law It is the ratio of normal stress to corresponding strain (straight line) for stresses below the proportional limit of the material. 

The term elastic limit is mainly a definition. It describes a stress which, if exceeded, will influence plastic deformation. Experimentally, the elastic limit is practically unattainable because it is a limit. Either it has not been reached or it is overreached. Ideally, the elastic limit and proportional limit are the same. 

It can be easily shown that equivalent expressions for the strain energy per unit volume are U=o e/2 and U = a2/2 E. The modulus of elastic resilience, Ur, of a material is defined as the strain energy absorbed per unit volume when it is stressed to its proportional limit. Thus,... 

It may be pointed out that the term yield point is sometimes erroneously used as a synonym for elastic limit and proportional limit As it has been described in the paragraphs above it is actually a phenomenon that occurs in only a very small number of cases in tensile testing. As it has also been observed in the description that graphically and experimentally, it is an anomalous behaviour in which there is a strain occurring with no increase in stress. 

Fig. 21 The cross-section and fracture surface of SA-Tyrannohex. This shows relatively large fracture energy (-2000 J/m2), higher proportional limit (about 120 MPa) and high tensile strength (200 MPa)...

When a linear relationship between the stress and strain is no longer present, the proportional limit is reached. On the diagram this is the highest point on the linear portion of the graph or where the curve no longer is a straight line. The material at this point is still elastic. The proportional limit is sometimes called the yield point. 

At some stress level above the proportional limit the material will no longer return to its original shape it will be permanently deformed. This region beyond the yield point is called the plastic range. 

Sugar maple was reacted with propylene and butylene oxide (Rowell etal., 1982). The modulus of elasticity (MOE) and modulus of rupture (MOR), fibre stress at proportional limit, and maximum crushing strength all exhibited a reduction, compared to unmodified samples. Nilsson and Rowell (1983) reacted ponderosa pine with butylene oxide and exposed the wood in an unsterile soil decay test. At low WPGs, severe surface decay due to soft rot and tunnelling bacteria was observed. Such attack was reduced at 15 % WPG,... 

Figure 3.3 shows representative stress-strain curves for a variety of polymeric materials. At normal use temperatures, such as room temperature, rigid polymers such as polystyrene (PS) exhibit a rapid increase in stress with increasing strain until sample failure. This behavior is typical of brittle polymers with weak interchain secondary bonding. As shown in the top curve in Figure 3.3, the initial stress-strain relation in such polymers is approximately linear and can be described in terms of Hooke s law, i.e., S = Ee, where E is Young s modulus, typically defined as the slope of the stress-strain plot. At higher stresses, the plot becomes nonlinear. The point at which this occurs is called the proportional limit. 

Some solids exhibit what is called elastic hysteresis, in which the variables corresponding to II and B in the magnetic case are the stress and the strain or deformation. Elastic bodies such as metals operating at stresses below the proportional limit also undergo hysteresis. 

Autographic load-deformation curves are often drawn during the test. From such a curve, the modulus of elasticity, proportional limit, and yield strength can be determined,... 

The maximum stress that is developed without deviation from proportionality of stress to strain is the proportional limit (the stress corresponding to load A). The maximum stress that can be applied without causing permanent deformation upon release of the load is die elastic limit. Usually, there is little difference between the proportional limit and die elastic limit. Both are dependent on the sensitivity of the measuring devices used and certain details of testing technique. For this reason, the yield strength is generally used as a practical measure of the elastic properties of metals. 

Syntactic foams manufactured from hollow glass or silica microspheres and an epoxide, phenolic or other matrix resin represent a class of lightweight structural materials used for buoyancy purposes, insulation and packaging. The effect of silanes on the mechanical properties of syntactic foams at a nominal density of 0.35 g/cm3 is shown in Tables 14-16. The Proportional Limit is defined as the greatest stress which the foam is capable of sustaining without any deviation from proportionality of stress to strain (Hooke s Law). 

Addition level (w/r) Density (g/cm ) Yield stress (MPa) Yield strain (%) Proportional limit (MPa/%) Compressive modulus (MPa)... 

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