Figure 1.7(a) demonstrates three concepts of the stress-strain behavior of elasto- plastic materials. The lowest point shown on the diagram is the proportional limit, defined as the transition point between linear and nonlinear behavior. The second point is the elastic limit, which is the transition between elastic and plastic behavior. However, most materials do not display an abrupt change in behavior from elas- tic to plastic. Rather, there is a gradual, almost imperceptible transition between the behaviors, making it difficult to locate an exact transition point (Polowski and Ripling, 2005). For this reason, arbitrary methods such as the offset and the exten- sion methods, are used to identify the elastic limit, thereby defining the yield stress (yield strength). In the offset method, a specified offset is measured on the abscissa, and a line with a slope equal to the initial tangent modulus is drawn through this Elastic limit 0.2% offset 0.5% extension yield strength yield strength Proportional limit Strain, % Strain, % 0.2% 0.5% (a) (b) FIGURE 1.7 Methods for estimating yield stress: (a) offset method and (b) extension method. point. The point where this line intersects the stress-strain curve is the offset yield stress of the material, as seen in Figure 1.7(a). Different offsets are used for different materials (Table 1.2). The extension yield stress is located where a vertical projec- tion, at a specified strain level, intersects the stress-strain curve. Figure 1.7(b) shows the yield stress corresponding to 0.5% extension. Stress Stress

The representative curves of the applied stress versus the number of cycles for different metals are shown in Figure Q2. Analyze both metals in terms of fatigue properties. 500 Stress amplitude (MN/m²) 400 300 200 100 0 1045 steel 10³ 104 2014-T6 aluminium 10⁹ 106 107 Number of cycles, N₂ Figure Q2: Representative curves of stress amplitude vs number of cycles 108 10⁹

A cylindrical specimen of a metal alloy 47.5 mm long and 9.72 mm in diameter is stressed in tension. A true stress of 397 MPa causes the specimen to plastically elongate to a length of 54.3 mm. If it is known that the strain-hardening exponent for this alloy is 0.2, calculate the true stress (in MPa) necessary to plastically elongate a specimen of this same material from a length of 47.5 mm to a length of 55.8 mm. i MPa

A brass alloy rod having a cross sectional area of 0.24 in.2and a modulus of 16 * 106 psi is subjected to a tensile load. Plastic deformation was observed to begin at a load of 8,944 lb.a. Determine the maximum stress that can be applied without plastic deformation.b. If the maximum length to which a specimen may be stretched without causing plastic deformation is 3.28 in., what is the original specimen length?