σ m using Equation 8.1 given that σ

Transcription

1 8. Etimate the theoretical fracture trength of a brittle material if it i known that fracture occur by the propagation of an elliptically haped urface crack of length 0.8 mm and having a tip radiu of curvature of mm when a tre of 100 MPa i applied. In order to etimate the theoretical fracture trength of thi material it i neceary to calculate m uing Equation 8.1 given that 0 = 100 MPa, a = 0.8 mm, and ρ t = mm. Thu, m a = 0 ρ t 1/ 4 = ()(100 MPa) = MPa = 37 GPa 1/ 0.8 mm mm

2 8.3 If the pecific urface energy for oda-lime gla i 0.30 J/m, uing data contained in Table 1.5, compute the critical tre required for the propagation of a urface crack of length 0.05 mm. We may determine the critical tre required for the propagation of an urface crack in odalime gla uing Equation 8.3; taking the value of 69 GPa (Table 1.5) a the modulu of elaticity, we get E γ c = π a 1/ 1/ 9 ()( N/m )(0.30 N/m) 6 = = N/m = 16. MPa ( ) ( π) m

3 8.5 A pecimen of a 4340 teel alloy having a plane train fracture toughne of 45 MPa m i expoed to a tre of 1000 MPa. Will thi pecimen experience fracture if it i known that the larget urface crack i 0.76 mm long? Why or why not? Aume that the parameter Y ha a value of 1.0. Thi problem ak u to determine whether or not the 4340 teel alloy pecimen will fracture when expoed to a tre of 1000 MPa, given the value of K Ic, Y, and the larget value of a in the material. Thi require that we olve for c from Equation 8.6. Thu KIc 45 MPa m c = = = 91 MPa Y π a (1.0) ( π) m ( ) Therefore, fracture will mot likely occur becaue thi pecimen will tolerate a tre of 97 MPa before fracture, which i le than the applied tre of 1000 MPa.

4 8.8 A large plate i fabricated from a teel alloy that ha a plane train fracture toughne of 55 MPa m. If, during ervice ue, the plate i expoed to a tenile tre of 00 MPa, determine the minimum length of a urface crack that will lead to fracture. Aume a value of 1.0 for Y. For thi problem, we are given value of K Ic ( 55 MPa m), (00 MPa), and Y (1.0) for a large plate and are aked to determine the minimum length of a urface crack that will lead to fracture. All we need do i to olve for a c uing Equation 8.7; therefore a c 1 KIc 1 55 MPa m = 0.04 m 4 mm π = = = Y π (1.0)(00 MPa)

5 8.13 Following i tabulated data that were gathered from a erie of Charpy impact tet on a tempered 4140 teel alloy. Temperature ( C) Impact Energy (J) (a) Plot the data a impact energy veru temperature. (b) Determine a ductile-to-brittle tranition temperature a that temperature correponding to the average of the maximum and minimum impact energie. (c) Determine a ductile-to-brittle tranition temperature a that temperature at which the impact energy i 70 J. The plot of impact energy veru temperature i hown below.

6 (b) The average of the maximum and minimum impact energie from the data i 89.3 J + 5 J Average = = 57. J A indicated on the plot by the one et of dahed line, the ductile-to-brittle tranition temperature according to thi criterion i about 75 C (198 K). (c) Alo, a noted on the plot by the other et of dahed line, the ductile-to-brittle tranition temperature for an impact energy of 70 J i about 55 C (18Κ).

7 8.16 An 8.0 mm diameter cylindrical rod fabricated from a red bra alloy (Figure 8.34) i ubjected to revered tenion compreion load cycling along it axi. If the maximum tenile and compreive load are N and 7500 N, repectively, determine it fatigue life. Aume that the tre plotted in Figure 8.34 i tre amplitude. We are aked to determine the fatigue life for a cylindrical red bra rod given it diameter (8.0 mm) and the maximum tenile and compreive load (+7500 N and 7500 N, repectively). The firt thing that i neceary i to calculate value of max and min uing Equation 6.1. Thu F = = F max max max A0 d 0 π 7500 N = = = m ( π) N/m 150 MPa = F min min d0 π 7500 N = = = m ( π) N/m 150 MPa (, 500 pi) Now it become neceary to compute the tre amplitude uing Equation 8.16 a a max min 150 MPa ( 150 MPa) = = = 150 MPa From Figure 8.34, f for the red bra, the number of cycle to failure at thi tre amplitude i about cycle.

8 8.18 The fatigue data for a bra alloy are given a follow: Stre Amplitude (MPa) Cycle to Failure (a) (b) (c) Make an S N plot (tre amplitude veru logarithm cycle to failure) uing thee data. Determine the fatigue trength at cycle. Determine the fatigue life for 00 MPa. (a) The fatigue data for thi alloy are plotted below. (b) A indicated by the A et of dahed line on the plot, the fatigue trength at cycle [log ( ) = 5.7] i about 50 MPa. (c) A noted by the B et of dahed line, the fatigue life for 00 MPa i about 10 6 cycle (i.e., the log of the lifetime i about 6.3).

9 8.5 Lit four meaure that may be taken to increae the reitance to fatigue of a metal alloy. Four meaure that may be taken to increae the fatigue reitance of a metal alloy are: (1) Polih the urface to remove tre amplification ite. () Reduce the number of internal defect (pore, etc.) by mean of altering proceing and fabrication technique. (3) Modify the deign to eliminate notche and udden contour change. (4) Harden the outer urface of the tructure by cae hardening (carburizing, nitriding) or hot peening.

10 8.9 For a cylindrical S-590 alloy pecimen (Figure 8.31) originally 10 mm in diameter and 505 mm long, what tenile load i neceary to produce a total elongation of 145 mm after,000 h at 730 C (1003 K)? Aume that the um of intantaneou and primary creep elongation i 8.6 mm. It i firt neceary to calculate the teady tate creep rate o that we may utilize Figure 8.31 in order to determine the tenile tre. The teady tate elongation, l, i jut the difference between the total elongation and the um of the intantaneou and primary creep elongation; that i, l = 145 mm 8.6 mm = mm Now the teady tate creep rate, i jut t l l t 0 = = = mm 505 mm,000 h = h 1 Employing the 730 C line in Figure 8.31, a teady tate creep rate of h 1 correpond to a tre of about 00 MPa [ince log ( ) =.87]. From thi we may compute the tenile load uing Equation 6.1 a F d0 A0 π = = m = ( N/m )( π) = 15, 700 N

11 8.33 (a) Etimate the activation energy for creep (i.e., Q c in Equation 8.0) for the S-590 alloy having the teady-tate creep behavior hown in Figure Ue data taken at a tre level of 300 MPa and temperature of 650 C and 730 C. Aume that the tre exponent n i independent of temperature. (b) Etimate at 600 C and 300 MPa. (a) We are aked to etimate the activation energy for creep for the S-590 alloy having the teadytate creep behavior hown in Figure 8.31, uing data taken at = 300 MPa and temperature of 650 C (93 K) and 730 C (1003 K). Since i a contant, Equation 8.0 take the form n Q Q = exp c exp c K = K RT RT where K i now a contant. (Note: the exponent n ha about the ame value at thee two temperature per Problem 8.3.) Taking natural logarithm of the above expreion ln = ln K Qc RT For the cae in which we have creep data at two temperature (denoted a T 1 and T ) and their correponding teady-tate creep rate ( and 1 ), it i poible to et up two imultaneou equation of the form a above, with two unknown, namely K and Q c. Solving for Q c yield Qc = R ( ln ln ) T1 T Let u chooe T 1 a 650 C (93 K) and T a 730 C (1003 K); then from Figure 8.31, at = 300 MPa, = h 1 and 1 = h 1. Subtitution of thee value into the above equation lead to Q c = 5 ( ) ( ) (8.31 J/mol K) ln ln K 1003 K = 480,000 J/mol