Mechanics of Fatigue and Fracture Assignment Help
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Problems Problem 1. The S-N curve for an elastic material is characterized by the Basquin relationship, b σ a = ⋅ C Nf , where C is a material constant, σa is the stress amplitude, Nf is the number of fully reversed stress cycles to failure, and b is the Basquin exponent approximately equal to –0.09. When the stress amplitude is equal to the ultimate tensile strength of the material in this alloy, the fatigue life is ¼ cycle. If a specimen spends 70% of its life subject to alternating stress levels equal to its fatigue endurance limit σe, 20% at 1.1 σe, and 10% at 1.2 σe, estimate its fatigue life using the Palmgren-Miner linear damage rule. Problem 2. Explain why the modified Goodman diagram can be rewritten in terms of the endurance limit, σe, as:
Problem 3. A circular rod with a uniform cross-sectional area of 20 cm2 is subjected to a mean axial force of 120 kN. The fatigue strength of the material, σa = σfs, is 250 MPa after 106 cycles of fully reversed loading and σTS = 500 MPa. Using the different procedures
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discussed in class, estimate the allowable amplitude of force for which the shaft would be designed to withstand at least one million fatigue cycles. State all of your assumptions clearly. Problem 4. A metallic material is shot-peened in an attempt to improve its fatigue life. The shotpeening process results in a compressive residual stress of 250 MPa at the surface of the material. The material has the following monotonic and fatigue characteristics in the as-fabricated condition (before shot-peening): E = 210 GPa, A’ = 1000 MPa, σf’ = 1100 MPa, εf’ = 1.0, nf = 0.13, b = -0.08, and c = -0.63. (See the discussions related to Equations 3.5, 7.1, and 8.5 for the definition of these variables). On a log-log plot, show the variation of the total strain amplitude ∆ ε/2 with the number of reversals to failure, 2Nf, for the material in both the as-fabricated and shot-peened conditions. Problem 5. Assume a large plate with an initially non-closing notch is subjected to cyclic compression loading. How do the stress cycles below affect the crack length vs. number of cycles (a vs. N) curve?
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Problem 6. Discuss a set of conditions under which the application of a tensile overload to a component can result in (i) an improvement, (ii) a reduction, or (iii) no change in fatigue life. Provide at least one practical example for each situation. Problem 7. Problem 14.2 from S. Suresh, Fatigue of Materials. solidworksassignmenthelp.com
Solutions Solution 1.We are told that the S-N curve for an elastic material follows the Basquin relationship, i.e.
Where b is approximately equal to —0.09. Say that the total lifetime of the component is n cycles. We are told that it spends 70% of its life at the endurance limit ao 20 % at 1.1 cre, and 10 % at 1.2 a, By definition, the lifetime at the endurance limit is Nf = 107 cycles so that:
Say that the lifetime at a. = 1.1a. is N1 cycles. N1 is given by:
Using the definition of a. we find that:
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Similarly, if the lifetime at a. = 1.2a. is N2, N2 is given by: We use the Palmgren-Miner law so that
Solving for ra, n = 4.912 x 106. We can use the information given (failure occurs after 1/4 cycle) to determine the relationship between at and o'n (not required for this problem but still interesting . . . )
You should not necessarily assume that ae 0.35au — 0.50aTs. That only applies for some materials (some steels and copper alloys). Solution 2. Explain why the modified Goodman diagram can be re-written in terms of the endurance limit, as solidworksassignmenthelp.com
where aelo„,.0 is the endurance limit for zero mean stress cyclic loading. Sodium The modified Goodman equation states that:
What does this equation mean? Say we apply a certain stress with no mean stress (call that stress adom=0) and the component has a certain lifetime. Say we now have a situation with a mean stress am. This equation tells us the stress a. we can apply and have the lifetime be the same as in the case with no mean stress This applies to any stress aa, in particular we may let it be the endurance limit a, and then we obtain the desired relationship, i.e
Since this a constant life relationship, the lifetime will be 107 cycles for both a, (with no mean stress applied) and aelem.0 (with mean stress applied), so both stresses (a. and acle„,.0) do indeed represent the endurance limits. solidworksassignmenthelp.com
Solution 3. A circular cylindrical rod with a uniform cross-sectional area of 20 cm2 is subjected to a mean axial force of 120 kN. The fatigue strength of the material, era = ars is 250 MPa after 106 cycles of fully reversed loading and aim = 500 MPa. Using the different procedures discussed in class, estimate the allowable amplitude of force for which the shaft should be designed to withstand at least one million fatigue cycles. State all your assumptions clearly. Solution The different expressions we have to assess the influence of mean stresses are:
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Applying the Soderberg criterion requires a bit more thought since it includes the yield strength (which is not given) rather than the tensile strength. Depending on the details of the material behavior, the yield stress an could be the same as the tensile strength aTs (for very brittle materials) or as low as 0.5aTs (for very ductile materials). I will assume that ays = 0.7aTs = 350 MPa. Thus the Soderberg criterion gives:
As you can see, you get significantly different answers depending on the model used. The Soderberg gives the most conservative result, while Gerber is the least conservative.
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Solution 4.
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Solution 5.
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Solution 6.
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Solution 7.
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