EMA 4223 Example Problems for Final
Mechanical Behavior
1. A large thick plate of steel is examined by X-ray methods and found to contain no detectable cracks. The equipment can detect a single edge-crack of depth a = 1 mm or greater. Assuming that the plate contains cracks on the limit of detection, determine whether the plate will undergo general yield or will fail by fast fracture before general yielding occurs. What is the stress at which fast fracture will occur? Sketch a graph of the stresses versus crack size and indicate the location of the detectable crack.
[ Kc = 54 MPa Öm ; yield strength = 950 MPa.]
2. The same steel composition in problem 1 is used in an application in which the component is subjected to cyclic loading. Laboratory tests show that the crack growth rate under cyclic loading is given by:
da/dN = A (DK)4,
where A = 4 x 10-13(MPa)-4 m-1. The component is subjected to an alternating stress of range Ds = 180 MPa about a mean tensile stress of Ds / 2. Given that DK = Ds Ö(pa), calculate the number of cycles to failure.
3. (a) Indicate briefly how and why the following would affect the fatigue life of components:
(i) a good surface finish
(ii) the presence of a rivet hole
(iii) a high value for the mean tensile stress
(iv) a corrosive atmosphere
4 The velocity of a crack in a polymer, metal and ceramic subjected to an aggressive environment such as humid air can be represented by:
v = da/dt = A KI n
where a is the crack size and t is the time. A and n are material constants. [Typical values for these constants are : A = 0.03 and n = 2 for polymers; A = 0.5 and n = 20 for ceramics; and A = 0.005 and n =2 for metals.]
(a) Sketch the velocity as a function of stress intensity for a typical ceramic, polymer and metal on the same graph relative to one another.
(b) Sketch the time to failure as a function of the crack size for the three types of materials on the same graph.
5. (a) A plate of steel with a central through-thickness crack of length 16 mm is subjected to a stress of 350 MPa normal to the crack plane. If the yield strength of the material is 1400 MPa, what is the plastic-zone size and the effective stress intensity level at the crack tip.
KIC = 55.5 MPa.
(b) If a second plate of steel with the same crack size and applied stress level is heat treated to provide a yield strength of 385 MPa, would a correction have to be made for the plastic zone size? What is the effective stress intensity factor? State any reasonable assumptions you make.
6. You are a materials engineer who is called upon to design the cantilever beam for an atomic force microscope to be operated above 500C, i.e., polymer based materials cannot be used.
( Attached to this beam will be a silicon or silicon nitride tip during surface measurements.) The design requirements are that the beam must sustain a force, F, of 0.1 N without deflecting more than d (d =1 mm) and that the weight of the structure is minimized. Assume that the beam has a square cross section. The deflection of the beam (ignoring self weight) is:
d = 4 l3 F / E t4
where l is the length of the unsupported beam, E is the elastic modulus and t is the thickness.
[The moment of inertia of a beam with square cross section is t4/12.]
Select a material for
this beam based on principles learned in class. Show all work!
[Select no more than four potential materials and no less than two. Select judiciously!]
7. Modulus of rupture tests were performed on ceramic bars with dimensions l = 100 mm and b = d = 10 mm. The median value of sr was 300 MPa. The ceramic is to be used for components with dimensions l = 50 mm, b = d = 5 mm loaded in simple tension along their length. Calculate the tensile strength, sts, that will give a probability of failure, Pf, of 10-6. Assume m = 10.
[ Hint: for m = 10, sts = sr / 1.73.] l is span length, b is the width and d is the depth of a rectangular beam.
8. A cylindrical steel pressure vessel of 7.5 m diameter and 40 mm wall thickness is to operate at a working pressure of 12 MPa. To prevent fast fracture, the total number of loading cycles from zero to full load and back to zero again must not exceed 3000. A choice of properties are available for several steels: Material A - KIC = 200 MPa m1/2 and sy = 1000MPa; Material B - KIC = 50 MPam1/2 and sy = 500 MPa; and Material C - KIC = 25 MPa m1/2 and sy = 2000 MPa. The growth of the crack by fatigue may be represented approximately by the following equation:
da/dN = A (DK)4,
where A = 4 x 10-14 (MPa)-4 m-1. da/dN is the extent of crack growth per load cycle and DK is the (cyclic) stress intensity factor.
(a) Sketch the loci of general yielding and fast fracture as a function of the crack size for the materials.
(b) Determine which, if any, of the materials can be used for a safe design. State what you consider a safe design.
(c) Find the minimum pressure to which the vessel must be tested before use to guarantee against failure in under 3000 cycles. (Assume the cracks of importance extend parallel to the length of the cylinder.)
9. Show that the onset of necking is given by dse/dee = 0 assuming tensile loading of a rod of cross-sectional area, A, by a force, F. Assume that s = A en is applicable in the range of interest where s and e are the true stress and true strain, respectively.
10. Single crystals of magnesium are prepared in the form of rods that are tested in tension. The six test results are listed below.
f l sA (MPa)
82 11 2.74
63 A 1.06
52 41 0.82
33 50 0.69
29 B 0.96
13 77 C
(a) Calculate the resolved shear stress, trss and normal stresses acting on the slip plane when yielding begins.
(b) Based on your calculations in (a) estimate the values for A, B and C.
(c) From your calculations, does trss or sn control yielding.
(d) Sketch on simulated graph paper the Schmid factor versus the normal stress P/A acting on the rod.
11. The pressure hull of a deep submersible can be modeled as a thin walled spherical pressure vessel. The stress in the pressure vessel wall of thickness, t, is:
s = p r / (2t)
where r is the radius and is fixed at 1m by the design, and p is the external pressure. Determine which material should be selected for the lightest weight vessel that can withstand the water pressure of 7 MPa without yielding. [Use approach taught in class].
12. How do you determine if a material follows the Holloman relationship?
13. Name four strengthening techniques for crystals that yield. Explain the basic mechanism(s) behind the strengthening.
14. A composite material consists of parallel fibers of Young’s modulus Ef in a matrix of Young’s modulus, Em. The volume fraction of fibers is Vf. Derive an expression of Ec, Young’s modulus of the composite which is stressed along the direction of the fibers, in terms of Ef,Em and Vf. Obtain an analogous expression for the density of the composite, rc. Using material parameters given below, find rc and Ec for the following composites: (a) carbon fiber-epoxy resin (Vf = 0.5), (b) steel-concrete (Vf = 0.02).
15. A rectangular bar which has dimensions 20 mm by 10 mm in cross-section of single crystal of magnesium oxide is pulled in tension with a force of 100N. What are the strains that result in the bar in the direction of pulling and perpendicular to the direction of pulling? [Assume for the purpose of the compliance and stiffness constants that the orientation is such that the pulling direction is the principal direction, i.e., the x or 1 (one) direction and the perpendicular directions are the y and z or 2 and 3 directions.]
16. The curve of true stress against true strain for a metal wire approximates to:
s = 350 e 0.4 MPa.
Estimate the work required to take 1 m3 of the wire to necking.
[Note: At the onset of necking, ds/de = s. Also note that the equation is in the form:
s = A en .]
17. An infinitely large sheet is subjected to a gross stress of 300 MPa. There is a central crack 20/p cm long , the material has a yield strength of 750 MPa and a fracture toughness of 75 MPam1/2.
Calculate the stress-intensity factor at the tip of the crack.
(a) Calculate the plastic zone size at the crack tip.
(b) Comment on the validity of the plastic zone correction factor by calculating the effective stress intensity factor and comparing to that in (a).
18. An alloy tie bar has been designed to withstand a stress, s, of 20 MPa at 620°C. Creep tests carried out on specimens of the alloy under these conditions indicated a steady-state creep rate, e, of 3.1 x 10-12 s-1. In service conditions it was found that, for 40% of the running time, the stress and temperature increased to 30 MPa and 650°C. Calculate the average creep-rate under service conditions. Assume n = 5 in the appropriate creep rate equation. R = 8.314 J mol-1 K -1 and Q = 100 kJ mol-1.
19. a. ) Sketch a typical creep rupture diagram. Include at least two temperatures and indicate which is the greater temperature.
(b) Sketch a typical creep curve.
(c) Sketch a relaxation curve. Indicate the creep strain and elastic strain portions of the curve.
20. The secondary stage creep behavior of a high temperature steel at 538°C is given by
de/dt = 1.16 x 10-24 s8
where de/dt = strain rate in h-1 and s = stress in MPa. Predict the creep rate of this steel at a temperature of 500°C at a stress level of 150MPa. The activation energy for creep is 418 kJ/mole and R = 8.314 J mol-1 K -1
21. If the true stress-strain relationship for a material is given by s = Aen, how would you determine the tensile strength of the material? You do not have to perform all of the operations, but demonstrate that you know how to determine the solution including the mechanics of the operations.
22. As a materials engineer you are required to design a glass window for a vacuum chamber. The opening can be adjusted for a circular disc of radius R and thickness t. It is freely supported in a rubber seal around its periphery and subjected to a uniform pressure difference Dp = 0.1 MPa. The material properties of the window glass to be used are summarized in the table. The window is a critical component and requires a failure probability of 10-6. The design life of the component is 1000 hours. The modulus of rupture tests of the glass discs to be used resulted in a mean strength of 100 MPa in a short term (10 minutes) bending test. What are the permissible dimensions for this window? [Assume Poisson’s ratio is 0.25].
23. Calculate the softening temperature for a soda-lime silica glass at which the viscosity is equal to 1013.4 P if the activation energy for viscous flow is 2.5 MJ/ mole and the viscosity at 1000°C is 1014 P. [1 P = 0.1 Pa s}.
24 (a). Sketch the shear strength of a crystal as a function of dislocation density.
24 (b). ty is the dislocation yield strength; sy is the yield strength; and k is the shear yield strength. What is the difference between these and what is their relationship?
25. Single crystal Tungsten is subjected to elastic stresses represented by the matrix
2 -3 1
sij = -3 4 5 MPa
1 5 -1
calculate the corresponding strains. Tungsten is a cubic crystal. The values of stiffness and compliance constants are:
S11 = 0.257 (10-2 GPa –1) C11 = 501 GPa
S44 = 0.66 (10-2 GPa –1) C44 = 151.4 GPa
S12 = -0.073 (10-2 GPa –1) C12 = 198 GPa
26. A line pipe with overall diameter (2 r) of 1 m and 25 mm thickness (t) is constructed from a micro-alloyed steel (KIC = 60 MPam1/2; sy = 600 MPa). Calculate the maximum pressure for which the leak-before-break criterion will be observed. Assume that the semi-elliptical crack (Y = 0.7 ) is along the longitudinal axis of the cylindrical pipe. The longitudinal stress is sL = p r / 2t and the hoop (circumferential) stress is sH = p r / t. NDE can detect cracks that are 3 mm in size and greater. State your reasons for using the values you select.
27. Sketch the general yield and fast fracture loci as a function of flaw size for a pressure-vessel steel ( KIC = 170 MPa m1/2, sy = 1000 MPa ) and an aluminum alloy ( KIC = 25 MPa m1/2 , sy = 400 MPa) on the same graph. (a) Determine the minimum crack size for fracture before yield. (b) Ultrasonic detection can determine cracks that are 3 mm or larger. Are both materials acceptable for use in service? State why or why not.
28. The size of the plastic zone at the crack tip in the general plane-strain case is given by:
ry ={KI2 / (2 p sy2 )} cos2 Q/2 { 4[1-n(1-n)] – 3 cos2 Q/2}
(a) Determine the radius of the plastic zone in the direction of the crack (assume n = 1/3).
(b) Determine the angle Q at which the plastic zone is largest (assume n = 1/3).
29. Explain the difference between the use of the Hall-Petch equation for ductile and brittle failure, i.e., between
sy = s0 + ky D-1/2 and sc = s0c + kc D-1/2 . Make sure you discuss the effect of temperature, strain rate and
grain size.
30. (a) Describe the difference between the appearance of a surface ruptured in a ductile manner if the loading is i.) tension, ii.) flexure and iii.) shear. You may sketch the surfaces and provide a brief explanation.
(b) Describe the typical characterization, i.e., appearance, of a brittle fracture surface for a i.) ceramic, ii.) polymer and iii.) metal. Provide a brief written description of your sketch.