ME 108 -- Homework 6

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1 University of California, Berkeley Department of Mechanical Engineering ME 108: Mechanical Properties of Engineering Materials Fall 2023 Homework 6: Fracture Due: Monday October 23rd , 11.59pm on Gradescope Please upload your responses as PDF, Word .DOCX, or image-format files (PNG, JPG, etc). Please also tag your response to each part of each question in Gradescope so that the reader can find them easily. If you find you need an extension, please request it using this short Google Form . Completing the form automatically grants the extension, within reason. We ask for advance warning so that we can think of ways to support students as much as possible, and so that we can pause the release of homework solutions for a few days if needed. Please don’t hesitate to ask for an extension if you need one. The Google Form is intended to handle routine extension requests efficiently. However, please do communicate early with your GSI and/or me in person or by e-mail if you find yourself struggling with the material or feel overwhelmed. Total points: 40 1. Fracture concepts and terminology [10 points] [Graded by completion] 1.1. Describe and distinguish between modes I, II and III fracture. [1] 1.2. The stress intensity factor, K , is defined by 𝐾𝐾 = 𝑌𝑌𝜎𝜎√𝜋𝜋𝑎𝑎 , where 𝑎𝑎 is the crack length (full crack length for an edge crack; or the half-length for a crack in the center of a component). Explain how this quantity can be used to predict whether or not a component will fail by fast fracture. [2] 1.3. For cracks that are small compared to the component they are in, what value can we usually approximate Y to? [1] 1.4. If you wanted to get a more accurate value for Y for a particular component and crack geometry, how might you go about doing this? [1] 1.5. Explain what is physically meant by the strain energy release rate, G , in a cracked component. [1] 1.6. What is the difference between stress intensity factor and critical stress intensity factor ? [1] 1.7. Why is fracture toughness , e.g. 𝐾𝐾 𝐼𝐼𝑐𝑐 , more widely used in engineering analysis of fracture than toughness , e.g. 𝐺𝐺 𝐼𝐼𝑐𝑐 ? [2] 1.8. Pair up the following four terms into two pairs of terms that mean the same as each other: (a) toughness, (b) fracture toughness, (c) critical stress intensity factor, (d) critical strain energy release rate. [1] 2. Failing by yield or fast fracture? [4 points] [Graded by completion] 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. The steel has a fracture toughness K C of 53 MN m –3/2 and a yield strength of 950 MPa. 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 failure occurs? [Ashby & Jones vol. 1, ed. 2, Q29] 3. Fracture in a bending beam [10 points] [Graded by completion] Two wooden beams are butt-jointed using an epoxy adhesive (see figure below). The adhesive was stirred before application, entraining air bubbles which, under pressure in forming the joint, deform to flat, penny-shaped disks of diameter 2 a = 2 mm.

2 3.1. If the beam has the dimensions shown, and epoxy has a fracture toughness of 0.5 MN m –3/2 , calculate the maximum load F that the beam can support. Assume 𝐾𝐾 = 𝜎𝜎√𝜋𝜋𝑎𝑎 for the disk-shaped bubbles. [6] 3.2. Below what value of yield stress would the glue fail by yielding instead of fast fracture? [4] [Adapted from Ashby & Jones vol. 1, ed. 2, Q30] 4. Charpy test lab video [8 points] Please watch the video entitled ‘Fracture toughness’ in Media Gallery on bCourses. 4.1. Pause at about the 4-minute mark. Predict, in descending order, the fracture toughnesses of the samples that were (a) quenched but not tempered; (b) heat-treated (i.e. quenched and tempered); (c) not heat-treated. Explain briefly why you predicted this order. [2] 4.2. Finish watching the video and look at the data (the measured values appear on the screen at about 14:20). Were you correct in part 5.1? [1] 4.3. Discuss which mode(s) of fracture are being tested by a Charpy test, and why. [3] 4.4. Could you carry out a fracture test that involved gradually raising the applied stress to a critical load for fast fracture (as opposed to subjecting the sample to an impact, as in the Charpy test)? If so, describe how the test would determine 𝐾𝐾 𝐶𝐶 . [2] 5. Design of a Mode III fracture toughness test [8 points] [Graded by creativity] Sketch and explain an idea for a physical testing configuration or machine that could be used to measure the mode III fracture toughness of a particular steel. Your method should seek to isolate mode III fracture as far as possible – i.e. to measure mode III, rather than modes I or II. You can base your idea on your understanding of the tearing nature of mode III fracture. If you refer to any information sources to inspire or guide your design, please cite them. You can require the steel test sample to be prepared in any geometry you see fit. Optional extra question for practice [ungraded] 6. Designing for a leak-before-break criterion In pressure vessels such as steam boilers, a common design approach is ‘leak before break’, where the critical crack length for fast fracture is designed to be larger than the wall thickness of the vessel, so that it is impossible for a crack ever to reach its critical length. Instead, if a crack does happen to grow so that it spans the wall thickness of the vessel, it will cause the contents to leak out, relieving the pressure. While not completely safe, this failure mode would usually be considered preferable to an explosive fracture.

3 A cylidrical pressure vessel has an inner diameter of 150 mm and a wall thickness of 5 mm, and it contains a pressure of 20 MPa. The safety factor against yielding must be at least two. Also, a leak-before-break criterion must be met, with a safety factor of at least nine on critical crack length, meaning that the critical crack length must be at least nine times the wall thickness. 6.1. Does the vessel meet the design criteria if 300-M steel is used, with 0.2% proof stress of 1670 MPa and fracture toughness 𝐾𝐾 𝐼𝐼𝐶𝐶 of 77 MPa √ m ? Explain briefly. 6.2. What is the minimum fracture toughness needed for the material in this application? 6.3. What is the safety factor on 𝐾𝐾 relative to 𝐾𝐾 𝐼𝐼𝐶𝐶 , due to the safety factor of nine on critical crack length? [Adapted from Dowling Q8.36]

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