Stress-strain curves elastic moduli

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 stiffness of a plastic is expressed in terms of a modulus of elasticity. Most values of elastic modulus quoted in technical literature represent the slope of a tangent to the stress-strain curve at the origin (see Fig. 1.6). This is often referred to as Youngs modulus, E, but it should be remembered that for a plastic this will not be a constant and, as mentioned earlier, is only useful for quality... 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

The mechanical properties can be studied by stretching a polymer specimen at constant rate and monitoring the stress produced. The Young (elastic) modulus is determined from the initial linear portion of the stress-strain curve, and other mechanical parameters of interest include the yield and break stresses and the corresponding strain (draw ratio) values. Some of these parameters will be reported in the following paragraphs, referred to as results on thermotropic polybibenzoates with different spacers. The stress-strain plots were obtained at various drawing temperatures and rates. 

Fig. 2-2 (a) Example of the modulus of elasticity determined on the initial straight portion of the stress-strain curve and secant modulus and (b) secant modulus for two different plastics that are 85% of the initial tangent modulus. 

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. 

The generalized stress-strain curve for plastic shown in Fig. 2-7 serves to define several useful qualities that include the tensile strength, modulus (modulus of elasticity) or stiffness (initial straight line slope of... 

The constant is called the modulus of elasticity (E) or Young s modulus (defined by Thomas Young in 1807 although the concept was used by others that included the Roman Empire and Chinese-BC), the elastic modulus, or just the modulus. This modulus is the straight line slope of the initial portion of the stress-strain curve, normally expressed in terms such as MPa or GPa (106 psi or Msi). A... 

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). 

For materials that deviate from the proportionality law even well below the elastic limit, the slope of the tangent to the stress-strain curve at a low stress level is taken as the tensile modulus. When the stress-strain curve displays no proportionality at any stress level, the secant modulus is employed instead of the tensile modulus (Fig. 2-2). The secant modulus is the ratio of stress to corresponding strain, usually at 1% strain or 85% from the initial tangent modulus. 

The test can provide compressive stress, compressive yield, and modulus. Many plastics do not show a true compressive modulus of elasticity. When loaded in compression, they display a deformation, but show almost no elastic portion on a stress-strain curve those types of materials should be compressed with light loads. The data are derived in the same manner as in the tensile test. Compression test specimen usually requires careful edge loading of the test specimens otherwise the edges tend to flour/spread out resulting in inacturate test result readings (2-19). 

Flexural modulus is the force required to deform a material in the elastic bending region. It is essentially a way to characterize stiffness. Urethane elastomers and rigid foams are usually tested in flexural mode via three-point bending and tite flexural (or flex ) modulus is obtained from the initial, linear portion of the resultant stress-strain curve. 

The modulus of elasticity of a material it is the ratio of the stress to the strain produced by the stress in the material. Hooke s law is obeyed by metals but mbber obeys Hooke s law only at small strains in shear. At low strains up to about 15% the stress-strain curve is almost linear, but above 15% the stress and strain are no longer proportional. See Modulus. 

For steel, the modulus of elasticity is the same in the elastic region and yield plateau for static and dynamic response. In the strain hardening region the slope of the stress-strain curve is different for static and dynamic response, although this difference is not important for most structural design applications. 

The tensile modulus can be determined from the slope of the linear portion of this stress-strain curve. If the relationship between stress and strain is linear to the yield point, where deformation continues without an increased load, the modulus of elasticity can be calculated by dividing the yield strength (pascals) by the elongation to yield ... 

The elastic modulus is the slope of the tangent at the origin of the stress/strain curve. The tensile or compression modulus is often called Young s modulus whereas the torsion modulus is often called shear modulus or Coulomb s modulus. 

The effect of gas compression on the uniaxial compression stress-strain curve of closed-cell polymer foams was analysed. The elastic contribution of cell faces to the compressive stress-strain curve is predicted quantitatively, and the effect on the initial Young s modulus is said to be large. The polymer contribution was analysed using a tetrakaidecahedral cell model. It is demonstrated that the cell faces contribute linearly to the Young s modulus, but compressive yielding involves non-linear viscoelastic deformation. 3 refs. 

It is convenient to use a simple weightless Hookean, or ideal, elastic spring with a modulus G and a simple Newtonian (fluid) dashpot or shock absorber having a liquid with a viscosity of 17 as models to demonstrate the deformation of an elastic solid and an ideal liquid, respectively. The stress-strain curves for these models are shown in Figure 14.1. 

A measure of the stiffness of a polymer is the modulus of elasticity (Young s modulus) E. It can be calculated fi om the stress-strain curve as the slope in the linear region of Hooke s law. It should be considered that due to the definition E = o/e for rubberlike materials which show a rather large extension e at quite... 

The stress-strain curves for cortical bones at various strain rates are shown in Figure 5.130. The mechanical behavior is as expected from a composite of linear elastic ceramic reinforcement (HA) and a compliant, ductile polymer matrix (collagen). In fact, the tensile modulus values for bone can be modeled to within a factor of two by a rule-of-mixtures calculation on the basis of a 0.5 volume fraction HA-reinforced... 

A spring with a modulus of G, and a dashpot containing a liquid with a viscosity of rj, have been used as models for Hookean elastic solids and Newtonian liquids, respectively. In these models, the spring stores energy in a reversible process, and the dashpot dissipates energy as heat in an irreversible process. Figure 5.3 is a stress-strain curve for a typical elastomer the straight... 

Typical stress-strain curves for the pure CR gum and the composites containing irradiated and nonirradiated PTFE powder are shown in Fig. 46. The addition of PTFE particles increases the elastic modulus of the CR matrix. In the presence of irradiated PTFE particles, the modulus of the CR matrix increases relative to that of a matrix containing nonirradiated PTFE particles. 

Replicate tests were conducted at 3, 25 and 75% RH and good repeatability was observed. The elastic modulus as function of RH calculated from these stress-strain curves is also shown. The shape of the stress-strain curve can be approximated by two linear segments. It is clear that RH affects the elastic modulus and the yield stress of these MEAs with Nation-type membrane. Note that the elastic modulus more than doubled when the MEA was dried from 75 to 3% RH. However, the yield strain and the slope of the second linear segment are affected to a lesser degree it is notable that the 3% RH condition exhibited the lowest strain-to-failure. Despite some variations, the MEAs tested at all four RH levels were found to be fairly ductile, with strain-to-failure exceeding 100%. The yield stress varies from approximately 12 MPa to 17.5 MPa and the strain-to-failure varies from 86.4 to 152.7%. This is indicative of the initial non-uniformity of the MEA and the presence of initial random defects in the as-fabricated membrane or MEAs. 

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