# Stress-Strain Behavior

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1 Stress-Strain Behavior 6.3 A specimen of aluminum having a rectangular cross section 10 mm 1.7 mm (0.4 in. 0.5 in.) is pulled in tension with 35,500 N (8000 lb f ) force, producing only elastic deformation. Calculate the resulting strain. This problem calls for us to calculate the elastic strain that results for an aluminum specimen stressed in tension. The cross-sectional area is just (10 mm) (1.7 mm) 17 mm ( m 0.0 in. ); also, the elastic modulus for Al is given in Table 6.1 as 69 GPa (or N/m ). Combining Equations 6.1 and 6.5 and solving for the strain yields ε σ E F A 0 E 35,500 N ( m )( N/m ) 6.5 A steel bar 100 mm (4.0 in.) long and having a square cross section 0 mm (0.8 in.) on an edge is pulled in tension with a load of 89,000 N (0,000 lb f ), and experiences an elongation of 0.10 mm ( in.). Assuming that the deformation is entirely elastic, calculate the elastic modulus of the steel. This problem asks us to compute the elastic modulus of steel. For a square cross-section, A 0 b 0, where b 0 is the edge length. Combining Equations 6.1, 6., and 6.5 and solving for E, leads to F E σ ε A 0 l F b 0 l (89,000 N)( m) ( m) ( m) N/m 3 GPa ( psi)

2 6.7 For a bronze alloy, the stress at which plastic deformation begins is 75 MPa (40,000 psi), and the modulus of elasticity is 115 GPa ( psi). (a) What is the maximum load that may be applied to a specimen with a cross-sectional area of 35 mm (0.5 in. ) without plastic deformation? (b) If the original specimen length is 115 mm (4.5 in.), what is the maximum length to which it may be stretched without causing plastic deformation? (a) This portion of the problem calls for a determination of the maximum load that can be applied without plastic deformation (F y ). Taking the yield strength to be 75 MPa, and employment of Equation 6.1 leads to F y σ y A 0 ( N/m )( m ) 89,375 N (0,000 lb f ) (b) The maximum length to which the sample may be deformed without plastic deformation is determined from Equations 6. and 6.5 as l i 1 + σ E 75 MPa (115 mm) mm (4.51 in.) MPa Elastic Properties of Materials 6.15 A cylindrical specimen of aluminum having a diameter of 19 mm (0.75 in.) and length of 00 mm (8.0 in.) is deformed elastically in tension with a force of 48,800 N (11,000 lb f ). Using the data contained in Table 6.1, determine the following: (a) The amount by which this specimen will elongate in the direction of the applied stress. (b) The change in diameter of the specimen. Will the diameter increase or decrease?

3 (a) We are asked, in this portion of the problem, to determine the elongation of a cylindrical specimen of aluminum. Combining Equations 6.1, 6., and 6.5, leads to F π d 0 4 σ Eε E l Or, solving for l (and realizing that E 69 GPa, Table 6.1), yields l 4F π d 0 E (4)(48,800 N)( m) (π)( m) ( N /m ) m 0.50 mm (0.0 in.) (b) We are now called upon to determine the change in diameter, d. Using Equation 6.8 ν ε x ε z d / d 0 l / From Table 6.1, for aluminum, ν Now, solving the above expression for d yields d ν l d 0 (0.33)(0.50 mm)(19 mm) 00 mm mm ( in.) The diameter will decrease A cylindrical specimen of a hypothetical metal alloy is stressed in compression. If its original and final diameters are and 0.05 mm, respectively, and its final length is mm, compute its original length if the deformation is totally elastic. The elastic and shear moduli for this alloy are 105 GPa and 39.7 GPa, respectively.

4 This problem asks that we compute the original length of a cylindrical specimen that is stressed in compression. It is first convenient to compute the lateral strain ε x as ε x d 0.05 mm mm d mm In order to determine the longitudinal strain ε z we need Poisson's ratio, which may be computed using Equation 6.9; solving for ν yields ν E G MPa ()( MPa) Now ε z may be computed from Equation 6.8 as ε z ε x ν Now solving for using Equation 6. l i 1 + ε z mm 75.5 mm A brass alloy is known to have a yield strength of 75 MPa (40,000 psi), a tensile strength of 380 MPa (55,000 psi), and an elastic modulus of 103 GPa ( psi). A cylindrical specimen of this alloy 1.7 mm (0.50 in.) in diameter and 50 mm (10.0 in.) long is stressed in tension and found to elongate 7.6 mm (0.30 in.). On the basis of the information given, is it possible to compute the magnitude of the load that is necessary to produce this change in length? If so, calculate the load. If not, explain why. We are asked to ascertain whether or not it is possible to compute, for brass, the magnitude of the load necessary to produce an elongation of 7.6 mm (0.30 in.). It is first necessary to compute the strain at yielding from the yield strength and the elastic modulus, and then the strain experienced by the test specimen. Then, if

5 ε(test) < ε(yield) deformation is elastic, and the load may be computed using Equations 6.1 and 6.5. However, if ε(test) > ε(yield) computation of the load is not possible inasmuch as deformation is plastic and we have neither a stress-strain plot nor a mathematical expression relating plastic stress and strain. We compute these two strain values as ε(test) l 7.6 mm 50 mm 0.03 and ε(yield) σ y E 75 MPa MPa Therefore, computation of the load is not possible since ε(test) > ε(yield). Tensile Properties 6.5 Figure 6.1 shows the tensile engineering stress strain behavior for a steel alloy. (a) What is the modulus of elasticity? (b) What is the proportional limit? (c) What is the yield strength at a strain offset of 0.00? (d) What is the tensile strength? Using the stress-strain plot for a steel alloy (Figure 6.1), we are asked to determine several of its mechanical characteristics. (a) The elastic modulus is just the slope of the initial linear portion of the curve; or, from the inset and using Equation 6.10 E σ σ 1 ε ε 1 (00 0) MPa ( ) MPa 00 GPa ( psi) The value given in Table 6.1 is 07 GPa. (b) The proportional limit is the stress level at which linearity of the stress-strain curve ends, which is approximately 300 MPa (43,500 psi). (c) The 0.00 strain offset line intersects the stress-strain curve at approximately 400 MPa (58,000 psi).

6 (d) The tensile strength (the maximum on the curve) is approximately 515 MPa (74,700 psi). 6.7 A load of 85,000 N (19,100 lb f ) is applied to a cylindrical specimen of a steel alloy (displaying the stress strain behavior shown in Figure 6.1) that has a cross-sectional diameter of 15 mm (0.59 in.). (a) Will the specimen experience elastic and/or plastic deformation? Why? (b) If the original specimen length is 50 mm (10 in.), how much will it increase in length when this load is applied? This problem asks us to determine the deformation characteristics of a steel specimen, the stress-strain behavior for which is shown in Figure 6.1. (a) In order to ascertain whether the deformation is elastic or plastic, we must first compute the stress, then locate it on the stress-strain curve, and, finally, note whether this point is on the elastic or plastic region. Thus, from Equation 6.1 σ F A 0 85, 000 N π m N/m 481 MPa (69,900 psi) The 481 MPa point is beyond the linear portion of the curve, and, therefore, the deformation will be both elastic and plastic. (b) This portion of the problem asks us to compute the increase in specimen length. From the stress-strain curve, the strain at 481 MPa is approximately Thus, from Equation 6. l ε (0.0135)(50 mm) 3.4 mm (0.135 in.) 6.35 A cylindrical metal specimen having an original diameter of 1.8 mm (0.505 in.) and gauge length of mm (.000 in.) is pulled in tension until fracture occurs. The diameter at the point of fracture is 6.60 mm (0.60 in.), and the fractured gauge length is 7.14 mm (.840 in.). Calculate the ductility in terms of percent reduction in area and percent elongation.

7 This problem calls for the computation of ductility in both percent reduction in area and percent elongation. Percent reduction in area is computed using Equation 6.1 as %RA π d 0 π d f π d in which d 0 and d f are, respectively, the original and fracture cross-sectional areas. Thus, %RA 1.8 mm π 6.60 mm π 1.8 mm π % While, for percent elongation, we use Equation 6.11 as %EL l f mm mm mm 100 4% Elastic Recovery After Plastic Deformation 6.49 A cylindrical specimen of a brass alloy 7.5 mm (0.30 in.) in diameter and 90.0 mm (3.54 in.) long is pulled in tension with a force of 6000 N (1350 lb f ); the force is subsequently released. (a) Compute the final length of the specimen at this time. The tensile stress strain behavior for this alloy is shown in Figure 6.1. (b) Compute the final specimen length when the load is increased to 16,500 N (3700 lb f ) and then released. (a) In order to determine the final length of the brass specimen when the load is released, it first becomes necessary to compute the applied stress using Equation 6.1; thus

8 σ F F A 0 π d N π m 136 MPa (19,000 psi) Upon locating this point on the stress-strain curve (Figure 6.1), we note that it is in the linear, elastic region; therefore, when the load is released the specimen will return to its original length of 90 mm (3.54 in.). (b) In this portion of the problem we are asked to calculate the final length, after load release, when the load is increased to 16,500 N (3700 lb f ). Again, computing the stress σ 16,500 N π m 373 MPa (5,300 psi) The point on the stress-strain curve corresponding to this stress is in the plastic region. We are able to estimate the amount of permanent strain by drawing a straight line parallel to the linear elastic region; this line intersects the strain axis at a strain of about 0.08 which is the amount of plastic strain. The final specimen length l i may be determined from a rearranged form of Equation 6. as l i (1 + ε) (90 mm)( ) 97.0 mm (3.8 in.)

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