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Solutions Manual for Manufacturing Engineering and Technology

Edition by Kalpakjian, Schmid

ISBN: 9780138309466

Manufacturing Engineering and Technology

NINTH

EDITION

Serope Kalpakjian

Illinois Institute of Technology

Steven R. Schmid

University of North Carolina at Charlotte

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Preface

This solutions manual is intended to assist instructors in the organization of assignments and discussions associated with courses in manufacturing engineering, and using the textbook Manufacturing Engineering and Technology, 9th ed. In addition to these solutions, instructors can ﬁnd other education resources at the Prentice Hall maintained website, as discussed and connected with links in the e-book.

Manufacturing presents a number of challenges and opportunities to instructors. As a topic of study it is exciting because of its breadth and unending ability to provide fascinating opportunities for research, analysis, and creativity. Literally every discipline and sub-discipline in engineering has strong ties to man ufacturing, and a number of universities have used design and manufacturing as the basis of a capstone course that culminates a mechanical engineering bachelor’s degree. To students of manufacturing, it is, at ﬁrst, a ﬁeld so enormous that any semester or academic year sequence in manufacturing can do nothing but scratch the surface of the subject. This perception is absolutely true: Manufacturing, like so many other areas of specialization within engineering, truly is an area where lifelong learning is necessary. -

As educators, we have a responsibility to prepare our students as best we can for a life of continued education. Lifelong learning need not be restricted to formal classroom training, but it should be impressed upon students that they need to continually and systematically examine the physical world in order to achieve continued levels of improvement.

A challenging question, and perhaps one without any one good answer is: how should one teach an effective manufacturing course? We have seen examples of many successful strategies, some based solely on analytical methods, others involving surveys of manufacturing, and again others emphasizing the impact of manufacturing on engineering design. Most instructors develop hybrid approaches that are not restricted to any one area. We have attempted to include problems at the end of every chapter to accommodate each of these approaches, and it is our hope that instructors will ﬁnd good homework assignments in the book.

Manufacturing is a challenge to instructors. There are a number of courses, such as statics, dynam ics, solid and ﬂuid mechanics, etc., where topics for study are broken down into small enough portions and where closed-form, quantitative problems are routinely solved by students and by faculty during lec tures. Such problems are important for learning concepts, and they give students a sense of security in that absolute answers can be determined.

In manufacturing practice, such closed-form solutions do exist, but they are relatively rare. Usually, multiple disciplines are blended, and the information available is insufficient to truly optimize a desired outcome. In practice, manufacturing engineers need to apply good judgment after they have researched a problem as best they can given budgetary and time restrictions. These difficult open-ended problems are much more demanding than closed-form solutions, and require a different mindset. Instead of considering anumberasvalidorinvalid(usuallybycheckingagainsttheanswerprovidedinthebookorbytheinstruc tor), an open-ended problem can be evaluated only with respect to whether or not the result is reasonable and if good scientific methods were used to obtain the result.

This textbook has been intentionally designed with a large number of open-ended problems. Such problems are, in our experience, extremely valuable when teaching manufacturing engineering. However, a solutions manual is well-suited for closed-form solutions, not open-ended problems. We have attempted to describe the pitfalls and methods that will be valuable in solving the open-ended problems, but by their nature, it is difﬁcult to give a correct answer to these problems. Often, such problems have as the ﬁrst sentence in the solution “By the student”. We will describe acceptable solutions or approaches to obtaining those solutions, but it should be recognized that many potential answers exist for these problems.

Bloom’s taxonomy of learning objectives, illustrated in Fig. 1, suggest that there is a hierarchy of skills that students can acquire. The chapter-ending problems have been designed with Bloom’s taxonomy in mind, with Review Questions and Qualitative Problems emphasizing lower-order thinking skills, and

Synthesis, Design & Projects

Figure 1: Bloom’s taxonomy of learning objectives. Source: Anderson, L.W., and Krathwold, D.R., A tax onomy for learning, teaching and assessing: A revision of Bloom’s Taxonomy of educational objectives: Complete Edition, New York, Longman, 2001.

Quantitative Problems and Synthesis, Design and Projects to exercise higher-order thinking skills. It is recognized that truly effective courses will address all Bloom levels; the chapter-ending problems have been designed to assist instructors in this task.

A recent challenge has involved the treatment of artiﬁcial intelligence language models that are able to respond to queries by collecting language on the Internet and formulating solutions. The language models are impressive, and promise to become more sophisticated with time. They do represent a disruption to the established methods of teaching engineering topics such as manufacturing. The authors recognize that this solutions manual, and in fact the entire book, is available on the Internet, and therefore an AI language model can formulate answers to queries based in part on the book contents. As educators, the challenge is to teach, and avoid students obtaining AI-generated homework solutions. There are potential approaches, but one that has been included often in the chapter-ending problems involves asking a student to conduct a literature search and write a one-page summary of the topic; they then are asked to obtain an AI-generated solution; they then must compare the two and explain why they are different. We have heard of other approaches as well. Some professors (with smaller class sizes) will assign students a 5-10 minute presentation every week, on a topic corresponding to course or book content. Students are encouraged to use AI language models, as they still must learn the material by teaching it.

We encourage faculty to communicate with us and to give us feedback in any of the areas of the book.

Steven Schmid schmid.2@nd.edu

Chapter 1 The Structure of Metals

Qualitative Problems

1.26. Explain your understanding of why the study of the crystal structure of metals is important.

The study of crystal structure is important for a number of reasons. Basically, the crystal structure inﬂuences a material’s performance from both a design and manufacturing standpoint. For example, the number of slip systems in a crystal has a direct bearing on the ability of a metal to undergo plastic deformation without fracture. Similarly, the crystal structure has a bearing on strength, ductility and corrosion resistance. Metals with face-centered cubic structure, for example, tend to be ductile whereas hexagonal close-packed metals tend to be brittle. The crystal structure and size of atom determines the largest interstitial sites, which has a bearing on the ability of that material to form alloys, and with which materials, as interstitials or substitutionals.

1.27. What is the signiﬁcance of the fact that some metals undergo allotropism?

Allotropism, also called polymorphism, is discussed in Section 1.3, and refers to a change in crystal structure for a metal. Since properties vary with crystal structures, allotropism is useful and essential in heat treating of metals to achieve desired properties (Chapter 4). A major application is hardening of steel, which involves the change in iron from the fcc structure to the bcc structure (see Fig. 1.3). By heating the steel to the fcc structure and quenching, it develops into martensite, which is a very hard, hence strong, structure.

1.28. Is it possible for two pieces of the same metal to have different recrystallization temperatures? Is it possible for recrystallization to take place in some regions of a part before it does in other regions of the same part? Explain.

Two pieces of the same metal can have different recrystallization temperatures if the pieces have been cold worked to different amounts. The piece that was cold worked to a greater extent (higher strains), will have more internal energy (stored energy) to drive the recrystallization process, hence its recrystallization temperature will be lower. Recrystallization may also occur in some regions of the part before others if it has been unevenly strained (since varying amounts of cold work have different recrystallization temperatures), or if the part has different thicknesses in various sections. The thinner sections will heat up to the recrystallization temperature faster.

1.29. Describe your understanding of why different crystal structures exhibit different strengths and ductili ties. -

Different crystal structures have different slip systems, which consist of a slip plane (the closest packed plane) and a slip direction (the close-packed direction). The fcc structure has 12 slip sys tems, bcc has 48, and hcp has 3. The ductility of a metal depends on how many of the slip systems can be operative. In general, fcc and bcc structures possess higher ductility than hcp structures, because -

they have more slip systems. The shear strength of a metal decreases for decreasing b/a ratio (b is inversely proportional to atomic density in the slip plane and a is the plane spacing), and the b/a ratio depends on the slip system of the chemical structure (see Section 1.4).

1.30. A cold-worked piece of metal has been recrystallized. When tested, it is found to be anisotropic. Explain the probable reason.

The anisotropy of the workpiece is likely due to preferred orientation remaining from the recrystal lization process. Copper is an example of a metal that has a very strong preferred orientation after annealing. Also, it has been shown that below a critical amount of plastic deformation, typically 5%, no recrystallization occurs.

1.31. What materials and structures can you think of (other than metals) that exhibit anisotropic behavior?

This is an open-ended problem and the students should be encouraged to develop their own answers. However, some examples of anisotropic materials are wood, polymers that have been cold worked, bone, any woven material (such as cloth) and composite materials.

1.32. Two parts have been made of the same material, but one was formed by cold working and the other by hot working. Explain the differences you might observe between the two.

There are a large number of differences that will be seen between the two materials, including:

1. The cold worked material will have a higher strength than the hot worked material, and this will be more pronounced for materials with high strain hardening exponents.

2. Since hardness (see Section 2.6.2) is related to strength, the cold worked material will also have a higher hardness.

3. The cold worked material will have smaller grains and the grains will be elongated.

4. The hot worked material will probably have fewer dislocations, and they will be more evenly distributed.

5. The cold worked material can have a superior surface ﬁnish when in an as-formed condition. Also, it can have better tolerances.

6. A cold worked material will have a lower recrystallization temperature than a hot worked ma terial. -

1.33. Explain the importance of homologous temperature.

The homologous temperature is deﬁned as the ratio of a metal’s current temperature to its melting temperature on an absolute scale (Kelvin or Rankine, not Celsius or Fahrenheit). This is important for determining whether or not the metal will encounter recrystallization or grain growth. The homolo gous temperature is more important than actual temperature, because recrystallization occurs at very different temperatures for different metals, but the homologous temperature effects are fairly consis tent. The homologous temperature therefore allows one to distinguish between cold, warm, and hot working, as discussed in Table 1.2.

1.34. Do you think it might be important to know whether a raw material to be used in a manufacturing process has anisotropic properties? What about anisotropy in the ﬁnished product? Explain.

Anisotropy is important in cold-working processes, especially sheet-metal forming where the mate rial’s properties should preferably be uniform in the plane of the sheet and stronger in the thickness direction. As shown in Section 16.7, these characteristics allow for deep drawing of parts (like bev erage cans) without earing, tearing, or cracking in the forming operations involved. In a finished part, anisotropy is important so that the strongest direction of the part can be designed to support the largestloadinservice. Also, theefficiencyoftransformerscanbeimprovedbyusingasheetsteelwith anisotropy that can reduce magnetic hysteresis losses. Hysteresis is well known in ferromagnetic ma terials. When an external magnetic field is applied to a ferromagnet, the ferromagnet absorbs some

of the external ﬁeld. When sheet steel is highly anisotropic, it contains small grains and a crystal lographic orientation that is far more uniform than for isotropic materials, and this orientation will reduce magnetic hysteresis losses.

1.35. What is the difference between an interstitial atom and a substitutional atom?

The difference can be seen in Fig. 1.8. A substitutional atom replaces an atom in the repeating lattice without distortion. An interstitial does not ﬁt in the normal lattice; it can ﬁt in the gap between atoms, or else it distorts the lattice. Examples of substitutionally are copper-nickel, gold-silver, and molybdenum-tungsten. Interstitials can be self-interstitials, but common other examples are carbon, lithium, sodium, and nitrogen.

1.36. Explain why the strength of a polycrystalline metal at room temperature decreases as its grain size increases.

Strength increases as more entanglements of dislocations occur with grain boundaries (Section 1.4.2). Metals with larger grains have less grain-boundary area per unit volume, and hence will not be as able to generate as many entanglements at grain boundaries, thus the strength will be lower.

1.37. Describe the technique you would use to reduce the orange-peel effect on the surface of workpieces.

Orange peel is surface roughening induced by plastic strain. There are a number of ways of reducing the orange peel effect, including:

• Performing all forming operations without a lubricant, or else a very thin lubricant ﬁlm (smaller than the desired roughness) and very smooth tooling. The goal is to have the surface roughness of the tooling imparted onto the workpiece.

• Large grains exacerbate orange peel, so the use of small grained materials would reduce orange peel.

• If deformation processes can be designed so that the surfaces see no deformation, then there would be no orange peel. For example, upsetting beneath ﬂat dies can lead to a reduction in thickness with very little surface strains beneath the platen (see Fig. 14.3).

• Finishing operations can remove orange peel effects.

1.38. Whatisthesigniﬁcanceofthefactthatsuchmetalsasleadandtinhavearecrystallizationtemperature that is about room temperature?

Recrystallization around room temperature prevents these metals from work hardening when cold worked. This characteristic prevents their strengthening and hardening, thus requiring a recrystal lization cycle to restore their ductility. This behavior is also useful in experimental veriﬁcation of analytical results concerning force and energy requirements in metalworking processes (see Part III of the text).

1.39. It was stated in this chapter that twinning usually occurs in hcp materials, but Fig. 1.6b shows twinning in a rectangular array of atoms. Can you explain the discrepancy?

The hcp unit cell shown in Fig. 1.6a has a hexagon on the top and bottom surfaces. However, an intersectingplanethatisverticalinthisﬁgurewouldintersectatomsinarectangulararrayasdepicted in Fig. 1.6b. Thus, twinning occurs in hcp materials, but not in the hexagonal (close packed) plane such as in the top of the unit cell.

1.40. It has been noted that the more a metal has been cold worked, the less it strain hardens. Explain why this is the case.

This phenomenon can be observed in stress-strain curves, such as those shown in Figs. 2.2 and 2.5. Recall that the main effects of cold working are that grains become elongated and that the average grain size becomes smaller (as grains break down) with strain. Strain hardening occurs when dislo cations interfere with each other and with grain boundaries. When a metal is annealed, the grains -

are large, and a small strain results in grains moving relatively easily at ﬁrst, but they increasingly interfere with each other as strain increases. This explains that there is strain hardening for annealed materials at low strain. To understand why there is less strain hardening at higher levels of cold work, consider the extreme case of a very highly cold-worked material, with very small grains and very many dislocations that already interfere with each other. For this highly cold-worked material, the stress cannot be increased much more with strain, because the dislocations have nowhere else to go-they already interfere with each other and are pinned at grain boundaries.

1.41. Is it possible to cold work a metal at temperatures above the boiling point of water? Explain.

The metallurgical distinction between cold and hot working is associated with the homologous temper ature Cold working is associated with plastic deformation of a metal when it is below one-third of its melting temperature on an absolute scale. At the boiling point of water, the temperature is 100°C, or 373 Kelvin. If this value is one-third the melting temperature, then a metal would have to have a melting temperature of 1119K, or 846°C. As can be seen in Table 3.1, there are many such metals. -

1.42. Comment on your observations regarding Fig. 1.14.

This is an open-ended problem with many potential answers. Students may choose to address this problem by focusing on the shape of individual curves or their relation to each other. The instructor may wish to focus the students on a curve or two, or ask if the ﬁgure would give the same trends for a material that is quickly heated, held at that temperature for a few seconds, and then quenched, or alternatively for one that is maintained at the temperatures for very long times.

1.43. Is it possible for a metal to be completely isotropic? Explain.

This answer can be answered only if isotropy is deﬁned within limits. For example:

• A single crystal of a metal has an inherent an unavoidable anisotropy. Thus, at a length scale that is on the order of a material’s grain size, a metal will always be anisotropic.

• A metal with elongated grains will have a lower strength and hardness in one direction than in others, and this is unavoidable.

• However, a metal that contains a large number of small and equiaxed grains will have the ﬁrst two effects essentially made very small; the metal may be isotropic within measurement limits.

• Annealing can lead to equiaxed grains, and depending on the measurement limits, this can essentially result in an isotropic metal.

• A metal with a very small grain size (i.e., a metal glass) can have no apparent crystal structure or slip systems, and can be essentially isotropic.

1.44. Referring to Fig. 1.1, assume you can make a ball bearing from a single crystal. What advantages and disadvantages would such a bearing have?

Thisisachallengingproblem, andisonethatcanbereintroducedattheendof Chapters1-4. Students shouldbeencouragedtoprovideanswersthatarebasedontheirunderstandingandexperience; these are intended to start a discussion.

The advantages of a single crystal bearing include:

• Corrosion generally starts at a grain boundary, where loosely packed atoms are more susceptible to chemical attack; therefore, the bearing could be more corrosion resistant.

• Fatigue cracks may be more difﬁcult to form, but with some materials are more difﬁcult to propogate through a grain than along a grain boundary.

• The strength could be higher than a polycrystalline metal.

• Distortion of the bearing at elevated temperatures may not be an issue.

Disadvantages include:

• Theductilityofthebearingmaybesuspect, sinceplasticdeformationassociatedwithdislocation motion are not present.

• Because of the anisotropy of the material, producing a perfect sphere may be very difﬁcult.

• Cost may be a serious concern.

1.45. Referring to Fig. 1.10, explain why edge dislocations cannot cross grain boundaries using appropriate sketches.

Figure 1.10 shows that dislocations can move inside a lattice in close-packed directions. At grain boundaries, it should be recognized that the direction of the lattice changes on each side; there is then no natural direction for a dislocation to travel. This is shown below for a simple cubic arrangement. Note that the grain on the left (blue) and the grain on the right (red) are rotated 15° from each other. The grain boundary in between has only occasional direct contact between atoms, and has a large gap where dislocations would have a difﬁcult time moving and translating across. Note that grain boundaries often have impurities, which are not shown.

1.46. Refer to Eq. (1.1), and describe the experiments you would conduct to evaluate the constants Syi and k

Eq. (1.1) is the Hall-Petch equation, and relates the strength of a material to its grain size. It is possible to evaluate the two constants with only two experiments, but much more accuracy is achieved with more experiments. Selecting how many experiments are needed is the ﬁrst decision that must be made, and this is often discussed in detail in references on design of experiments. To obtain data that can evaluate Syi and k, one needs to obtain a material with different grain sizes, d, and then measure the strength, Sy. Once those experiments are analyzed, any mathematics package can be used to obtain Syi and k. However, varying d is challenging, as is measuring grain size without ruining a test specimen. The following tests can provide estimates of Syi:

1. A tension test, as described in Section 2.2, can be conducted.

2. A compression test can be performed, although care must be taken to avoid errors associated with friction.

3. Three or four-point bending can be conducted, but the strengths measured will be somewhat different, as described in Section 2.5.

4. Torsiontestscanbeconducted,withtheshearyieldstrengthconvertedtoauniaxialyieldstrength using relations from solid mechanics (stress transformation/Mohr’s circle).

5. Strength can also be developed through less common approaches, such as biaxial tension for sheets,measurementofextrusionforceforsolidsections,measurementofprocessenergyneeded in machining, etc.

The grain size needs to be varied. This can be accomplished by:

1. Annealing workpieces to varying degree. Workpieces with identical thermal histories should have similar grain sizes, which can be conﬁrmed on a suitable microscope.

2. Workpieces can be cold worked to reduce grain size. For example, equal channel angular extrusion (ECAE), as discussed in Section 15.4.2, can produce a more reﬁned grain structure.

Quantitative Problems

1.47. How many atoms are in a single repeating cell of a fcc crystal structure? How many in a repeating cell of an hcp structure?

For an fcc structure, refer to Fig. 1.4a. The atoms at each corner are shared by eight unit cells, and there are eight of these atoms. Therefore, the corners contribute one total atom. The atoms on the faces are each shared by two cells, and there are six of these atoms. Therefore, the atoms on the faces contribute a total of three atoms to the unit cell. Therefore, the total number of atoms in an fcc unit cell is four atoms.

For the hcp, refer to Fig. 1.5. The atoms on the periphery of the top and bottom are each shared by six cells, and there are 12 of these atoms (on top and bottom), for a contribution of two atoms. The atoms in the center of the hexagon are shared by two cells, and there are two of these atoms, for a net contribution of one atom. There are also three atoms fully contained in the unit cell. Therefore, there are six atoms in an hcp unit cell.

1.48. The atomic weight of aluminum is 26.98, meaning that 6 023 x 1023 atoms weigh 26.98 grams. The density of aluminum is 2700 kg/m3 , and pure aluminum forms fcc crystals. Estimate the diameter of an aluminum atom.

Consider the face of the fcc unit cell, which consists of a right triangle with side length a √ and hy potenuse of 4r. From the Pythagorean theorem, a = 4r/ 2. Therefore, the volume of the unit cell is -

Each fcc unit cell has four atoms (see Prob. 1.45), and each atom has a mass of

The density inside a fcc unit cell is ρ = Mass V = 2700 kg/m3 = 4(4.48 x 10-26 kg) 22.63r3

Solving for r yields r = 1.43 x 10-10 m, or 0.143 nm Note that the accepted value is 0.14 nm; this is very close, with the very slight difference attributable to a number of factors, including impurities in crystal structure and a concentration of mass in the nucleus of the atom.

The following Matlab script conﬁrms the calculations.

AW=26.98; m=AW/6.023e23/1000; rho=2700; r=(4*m/22.63/rho)ˆ(1/3);

1.49. Plot the data given in Table 1.1 in terms of grains/mm2 vs. grains/mm3 , and discuss your observations.

The plot is shown below. It can be seen that the grains per cubic millimeter increases faster than the grainspersquaremillimeter. Thisrelationshipistobeexpectedsincethevolumeofanequiaxedgrain depends on the diameter cubed, whereas its area depends on the diameter squared.

Grains/mm3

1.50. A strip of metal is reduced from 60 mm in thickness to 30 mm by cold working; a similar strip is reduced from 50 mm to 30 mm. Which of these cold-worked strips will recrystallize at a lower temperature? Why?

The metal that is reduced to 30 mm by cold working will recrystallize at a lower temperature. The more the cold work the lower the temperature required for recrystallization. This is because the number of dislocations and energy stored in the material increases with cold work. Thus, when recrystallizing a more highly cold worked material, this energy can be recovered and less energy needs to be imparted to the material.

1.51. The ball of a ballpoint pen is 0.5 mm in diameter and has an ASTM grain size of 10. How many grains are there in the ball?

From Table 1.1, a metal with an ASTM grain size of 10 has about 520,000 grains/mm3 . The volume of the ball is

Multiplying the volume by the grains per cubic millimeter gives the number of grains in the paper clip as about 2.18 million.

1.52. How many grains are on the surface of the head of a pin? Assume that the head of a pin is spherical with a 1-mm diameter and has an ASTM grain size of 11.

Note that the surface area of a sphere is given by A = 4πr2 . Therefore, a 1 mm diameter head has a surface area of

From Eq. (1.2), the number of grains per area is

= 2n-1 = 211-1 = 1024

This is the number of grains per 0.0645 mm2 of actual area (see Section 1.5.1); therefore the number of grains on the surface is

g = 1024 0.0645(π) = 207 grains

1.53. The unit cells shown in Figs. 1.3 through 1.5 can be represented by tennis balls arranged in various conﬁgurations in a box. In such an arrangement, the atomic packing factor (APF) is deﬁned as the ratio of the sum of the volumes of the atoms to the volume of the unit cell. Show that the APF is 0.68 for the bcc structure and 0.74 for the fcc structure.

1. bcc unit cell Note that the bcc unit cell in Fig. 1.3a has 2 atoms inside of it; one inside the unit cell and eight atoms that each have one-eighth of their volume inside the unit cell. Therefore the volume of atoms inside the cell is 8πr3/3, since thevolume of a sphere is √ 4πr3/3 Note that the diagonal of a face of a unit cell has a length of a 2, which can be easily determined from the Pythagorean theorem. Using √ that diagonal and the height of a results in the determination of the diagonal of the cube as a 3. Since there are four radii across that diagonal, it can be deduced that

The volume of the unit cell is a3 , so

Therefore, the atomic packing factor is

2. fcc unit cell. For the fcc cell, there are four atoms in the cell, so the volume of atoms inside the fcc unit cell is 16πr3/3. On a face of the fcc cell, it can be shown from the Pythagorean theorem that the hypotenuse is √a 2. Also, there are four radii across the diameter, so that

Therefore, the volume of the unit cell is

so that the atomic packing factor is

1.54. Show that the lattice constant a in Fig. 1.4a is related to the atomic radius by the formula , where R is the radius of the atom as depicted by the tennis-ball model.

For a face centered cubic unit cell as shown in Fig. 1.4a, the Pythagorean theorem yields a 2 + a 2 = (4r)2 Therefore,

The largest hole is shown in the sketch below. Note that this hole occurs in other locations, in fact in three other locations of this sketch.

1.55. Show that, for the fcc unit cell, the radius r of the largest hole is given by r = 0 414R Determine the size of the largest hole for the iron atoms in the fcc structure. Largest hole diameter = 2r

For a face centered cubic unit cell as shown in Fig. 1.4a, the Pythagorean theorem yields

Therefore,

Also, the side dimension a is

Therefore, substituting for a,

Since the atomic radius of iron is 0.125 nm, the largest hole in an iron lattice is 0.0517 nm.

1.56. A technician determines that the grain size of a certain etched specimen is 8. Upon further checking, it is found that the magniﬁcation used was 125x, instead of the 100x that is required by the ASTM standards. Determine the correct grain size.

If the grain size is 8, then there are 2048 grains per square millimeter (see Table 1.1). However, the magniﬁcation was too large, meaning that too small of an area was examined. Instead of 2048 grains persquaremillimeter, itis2048grainsper1/1.25mmsquare, whichisanarea64%smaller. Therefore, there really are 2048/0.64=3200 grains per mm2 , which corresponds to a grain size between 8 and 9.

1.57. If the diameter of the aluminum atom is 0.28 nm, how many atoms are there in a grain of ASTM grain size 6?

If the grain size is 6, there are 8200 grains per cubic millimeter of aluminum-see Table 1.1. Each grain has a volume of 1/8200 = 1 220 x 10-4 mm3 Note that for an fcc material there are four atoms per unit cell (see solution to Prob. 1.47), with a total volume of 16πR3/3 √ , and that the diagonal, a, of the ( ) unit cell is given by a = 2 2 R Hence,

APFfcc = 16πR3/3 (2R√2)3 = 0 74 ()

Note that as long as all the atoms in the unit cell have the same size, the atomic packing factor does not depend on the atomic radius. Therefore, the volume of the grain which is taken up by atoms is (1 220 x 10-4)(0 74) = 9 02 x 10-5 mm3 Recall that 1 mm=106 nm. If the diameter of an aluminum atom is 0.28 nm, then its radius is 0.14 nm or 0 14 x 10-6 mm. The volume of an aluminum atom is then V = 4πR3/3 = 4π(0 25 x 10-6)3/3 = 6 54 x 10-20 mm3

Dividing the volume of aluminum in the grain by the volume of an aluminum atom yields the total number of atoms in the grain as (9.02 x 10-5)/(6.54 x 10-20) = 1.38 x 1015 .

1.58. Does water have a homologous temperature? What is the highest temperature where water (ice) can get cold worked?

Allmaterialswithameltingpointhaveahomologoustemperature,deﬁnedastheratiooftheworking temperature to the melting temperature on an absolute scale. If water had strengthening mechanisms like metals, it could be cold worked up to one-third of its melting temperature. Since water melts at 0°C or 273 K, it could be cold worked up to 273/3=91 K =-182 °C. However, water is a complicated material with many possible crystal structures, and does not in general display cold working like metals.

1.59. The atomic radius of iron is 0.125 nm, while that of a carbon atom is 0.070 nm. Can a carbon atom ﬁt inside a steel bcc structure without distorting the neighboring atoms?

Consider the bcc cell shown in Fig. 1.3a. The diagonal is 4R. The length of a side of the bcc unit cell is a, so that

or a = 2.31R. Consider a diagonal on a face. Since there are two radii on an edge, the size of the opening is 0 31R. For iron, this would be (0.31)(0.125)=0.0387 nm, which is too small for a carbon atom to ﬁt without distortion. The iron lattice does get distorted, which is one of the strengthening mechanisms of carbon in iron.

1.60. The following data are obtained in tension tests of brass:

Does the material follow the Hall-Petch effect? If so, what is the value of k?

First, itis obviousfromthistablethatthematerial becomes strongerasthegrainsizedecreases, which is the expected result. However, it is not clear whether Eq. (1.1) is applicable. It is possible to plot the yield stress as a function of grain diameter, but it is better to plot it as a function of d-1/2 , as follows:

The least-squares curve ﬁt for a straight line is Y = 35 22 + 458d-1/2

with an R factor of 0.990. This suggests that a linear curve ﬁt is proper, and it can be concluded that the material does follow the Hall-Petch effect, with a value of k = 458 MPa-√μm.

QuantitativeProblems

1.61. Estimate the atomic radius for the following materials and data: (a) Gold (atomic weight = 196.97, density = 19,320 kg/m3; (b) silver (atomic weight=107.87 grams/mole, density = 10,500 kg/m3); (c) titanium (atomic weight = 47.87 grams per mole, density = 4506 kg/m3).

Refer to Problem 1.46. Aluminum and silver are fcc materials; titanium can be fcc or hcp. The radii for aluminum is found as r = 1 44A ˚ ; 1.45 Afor ˚ silver, and 1.46 for Titanium. The following Matlab code can be used to evaluate other materials:

M=47.87/(6.023e23)/1000; rho=4506; r=(4*M/rho/22.63)ˆ0.3333;

1.62. A simple cubic structure involves atoms located at the cube corners that are in contact with each other along the cube edges. Make a sketch of a simple cubic structure, and calculate its atomic packing factor.

The sketch is as follows:

The edge has a length of two times the atom radius, so that the volume of the cube is 8a3 There is one atom in the structure, which would take a volume of πa3 . Therefore, the atomic packing factor is

1.63. Estimate the ASTM grain size number for a 300 mm (12 in.) silicon wafer used to produce computer chips.

The notion of a grain is not useful for silicon wafers, but it provides an interesting demonstration of the ASTM grain size number. If a cross section of a 300-mm diameter is considered, it would have an area of A = πd2/4 = π(300)2/4 = 70, 680 mm2 . 0.0645 mm would represent 9.12 x 10-7 grains. Therefore,

9.12 x 10-7 = 2n-1 log 9 12 x 10-7 = (n - 1)log 2

or n = -19.

1.64. Pure copper and pure titanium follow the Hall-Petch equation. For copper, Syi = 24 MPa and k = 0 12 MPa-m1/2 For titanium, Syi = 80 MPa and k = 0 40 MPa-m1/2 (a) Plot the yield strength of these metals as a function of grain size for ASTM grain sizes of 0 to 10. (b) Explain which material would see greater strengthening from a reduction in grain size, as in cold working.

Note from Eq. (1.2) that for an ASTM grain size of-3, there would be N = 2-4 = 0 0625 grains in 0.0645 mm2 . Hence, each grain would have an area of around 1.03 mm2 . Using A = πd2/4 would

give an estimate of a grain of diameter 1.15 mm. For a grain size of 11, the grain diameter is similarly estimated as d = 0.008955 mm = 8.955 x 10-6 m. The Hall-Petch equation is given by Sy = Syi + kd-1/2

The desired plot is as follows:

As can be seen, titanium strengthens more with a decrease in grain size (higher ASTM grain size number).

1.65. Derive an equation for yield strength as a function of grain size number assuming the material of interest follows the Hall-Petch effect.

There are a number of approaches that can be used to obtain a useful relation. Note from Eq. (1.2) and the text preceeding this equation, that the number of grains visible in 0.0645 mm2 is N , and it is given by N = 2n-1 From this deﬁnition, the area per grain is (0.0645 mm2)/N. If the grains are approximated as circles, then the diameter of an average grain is obtained from:

πd2 4 = 0.0645 mm2 N

so that d = / (4)(0 0645) πN

Simplifying and substituting for N, d = 0.0821 2n-1 / where n is the grain size number and d is in millimeters. Substituting this value into Eq. (1.1) yields Sy = Syi + kd-1/2 = Syi + k (/ 0 0821 2n-1 )-1/2 = Syi + k ( 2n-1 0 0821 )1/4

This could be simpliﬁed further if desired, but it is an acceptable form of the desired relationship. Note that depending on the units used, d may need to be converted to meters.

Synthesis, Design, and Projects

1.66. By stretching a thin strip of polished metal, as in a tension-testing machine, demonstrate and comment on what happens to its reﬂectivity as the strip is being stretched.

The polished surface is initially smooth, which allows light to be reflected uniformly across the surface. As the metal is stretched, the reflective surface of the polished sheet metal will begin to become dull. The slip and twin bands developed at the surface cause roughening (see Fig. 1.7), which tends to scatter the reflected light.

1.67. Draw some analogies to mechanical ﬁbering—for example, layers of thin dough sprinkled with ﬂour or melted butter between each layer.

A wide variety of acceptable answers are possible based on the student’s experience and creativity. Some examples of mechanical ﬁbering include: (a) food products such as lasagna, where layers of noodles bound sauce, or pastries with many thin layers, such as baklava; (b) log cabins, where tree trunks are oriented to construct walls and then sealed with a matrix; and (c) straw-reinforced mud.

1.68. Draw some analogies to the phenomenon of hot shortness.

Some analogies to hot shortness include: (a) a brick wall with deteriorating mortar between the bricks, (b) time-released medicine, where a slowly soluble matrix surrounds doses of quickly soluble medicine, and (c) an Oreo cookie at room temperature compared to a frozen cookie.

1.69. Obtain a number of small balls made of plastic, wood, marble, or metal, and arrange them with your hands or glue them together to represent the crystal structures shown in Figs. 1.3–1.5. Comment on your observations.

Bythestudent. Therearemanypossiblecomments, includingtherelativedensitiesofthethreecrystal structures (hcp is clearly densest). Also, the ingenious and simple solid-ball models are striking when performing such demonstrations.

1.70. Take a deck of playing cards, place a rubber band around it, and then slip the cards against each other to represent Figs. 1.6a and 1.7. If you repeat the same experiment with more and more rubber bands around the same deck, what are you accomplishing as far as the behavior of the deck is concerned? By the student. With an increased number of rubber bands, you are physically increasing the friction force between each card. This is analogous to increasing the magnitude of the shear stress required to cause slip. Furthermore, the greater the number of rubber bands, the higher the shear or elastic modulus of the material (see Section 2.4). This problem can be taken to a very effective extreme by using small C-clamps to highly compress the cards; the result is an object that acts like one solid, with much higher stiffness than the loose cards.

1.71. Give examples in which anisotropy is scale dependent. For example, a wire rope can contain annealed wires that are isotropic on a microscopic scale, but the rope as a whole is anisotropic.

All materials may behave in an anisotropic manner when considered at atomic scales, but when taken as a continuum, many materials are isotropic. Other examples include:

• Clothing, which overall appears to be isotropic, but clearly has anisotropy deﬁned by the directionofthethreadsinthecloth. Thisanisotropicbehaviorcanbeveriﬁedbypullingsmallpatches of the cloth in different directions.

• Wood has directionality (orthotropic) but it can be ignored for many applications.

• Human skin: it appears isotropic at large length scales, but microscopically it consists of cells with varying strengths within the cell.

1.72. ThemovementofanedgedislocationwasdescribedinSection1.4.1bymeansofananalogyinvolving a hump in a carpet on the floor and how the whole carpet can eventually be moved by moving the hump forward. Recall that the entanglement of dislocations was described in terms of two humps at different angles. Use a piece of cloth placed on a flat table to demonstrate these phenomena. By the student. This can be clearly demonstrated, especially with a cloth that is compliant (flexible) buthas highfriction witha flatsurface. Two methodsofensuring thisis thecase are(a)to usea cotton material (as found in T-shirts) and wetting it before conducting the experiments, or (b) spraying the

bottom side of the fabric with temporary adhesives, as found in most arts and ofﬁce supply stores. The experiments (single and two lumps) can then be conducted and observations made.

1.73. If you wanted to strengthen a material, would you wish to have it consist of one grain, or would you want it to have grains that contain the minimum number of atoms? Explain. Most likely, the desire would be to have grains with the minimum number of atoms, as suggested by Eq. (1.1) (The Hall-Petch equation). This is the rationale for the use of metallic glasses, with no identiﬁable grain structure.