Guia de ejercicios Propiedad y resistencia de materiales

Pontificia Universidad Catlica de Chile Escuela de Ingeniera

ING 1024 Propiedades y resistencia de materiales

Practice Guide II

Version 1.0 1st semester, 2015

This guide is a compilation of former tutorials, tests and assignments; the answers are indicated at the end of each problem. Problems are not arranged according to the course schedule rather than difficulty. Should you find any mistake, please contact the teaching team (professors or teaching assistants).

INDICE

Problem 01 Energy of interatomic bond .................................................................................................................... 2

Problem 02 Dissolution of an ionic compound .......................................................................................................... 2

Problem 03 Interatomic bond I .................................................................................................................................. 3

Problem 04 Interatomic bond II ................................................................................................................................. 3

Problem 05 Unit cell I ................................................................................................................................................ 4

Problem 06 Unit cell II ............................................................................................................................................... 4

Problem 07 Unit cell III .............................................................................................................................................. 5

Problem 08 Unit cell IV .............................................................................................................................................. 6

Problem 09 Unitary cell V .......................................................................................................................................... 7

Problem 010 Unitary cells VI ....................................................................................................................................... 7

Problem 011 Unitary cells VII ...................................................................................................................................... 7

Problem 012 Unitary cell VIII ....................................................................................................................................... 7

Problem 013 Unitary cells IX ....................................................................................................................................... 8

Problem 014 Millers indices I ...................................................................................................................................... 8

Problem 015 Millers indices II ................................................................................................................................... 10

Problem 016 Millers indices III .................................................................................................................................. 11

Problem 017 Millers indices IV ................................................................................................................................. 12

Problem 018 Millers indices V .................................................................................................................................. 12

Problem 019 Anisotropy on crystalline structures ..................................................................................................... 13

Problem 020 Burgers vector ..................................................................................................................................... 13

Problem 021 Dislocations .......................................................................................................................................... 14

Problem 022 Sliding system ...................................................................................................................................... 14

ING 1024 Propiedades y Resistencia de Materiales Gua de ejercicios II versin 1,0 1 semestre, 2015

2

PROBLEMS

Problem 01 Energy of interatomic bond

For the ionic bond between a Na cation and a

Cl anion, find the parameters n, b of the energy

of repulsion, for which the equilibrium distance 0r is 283 pm and the bond energy is equal to -4.567

eV.

Considering a situation in which 1 eV of energy is absorbed by the compound, determine the new equilibrium distances. Provide a graphical representation.

Data:

F/m108.85C,101.602e J,101.602 eV1 , m10 pm 1 120-19--19-12

Answer:

731.9n

-112101.014b

r'0 362pm > r0

Problem 02 Dissolution of an ionic compound

Kitchen salt (NaCl) is easily dissolved in water. The dissolution process involves a change in the configurational entropy of water which provides energy for breaking the bonds. Assuming there is enough water to dissolve all the salt (solubility limit not exceeded), determine the total energy used for dissolving 1 mol (60.44 g) of NaCl.

Hint: consider temperature of 0K

Data:

Bond: ionic

Parameters of repulsion force: b = 2.5910-115 Nm11, n = 10

Parameter of attraction force (Coulombs force): Z1=Z2 = 1

Bond length at equilibrium (at 0K): ro = 283 pm

Answer:

kJ 443.76J10376.44E 4


3

Problem 03 Interatomic bond I

Aluminum trifluoride (AlF3) is a salt with a tridimensional structure and a high melting point. However, the aluminum trihalides: aluminum bromide (AlBr3) and aluminum iodide (AlI3), with structures formed by molecular dimers; and aluminum chloride (AlCl3), with a sheet-like structure; have low melting points and high vapor pressure. Based on the given information:

a) What type of interatomic bonds are expected in the compounds? Al-F: Al-Cl, Al-Br, Al-I:

b) Considering that silver fluoride (AgF) and silver chloride (AgCl) have the same characteristics of bonds as AlF3 and AlCl3, respectively, how does it affect the solubility of silver halides in water? Al-F: Al-Cl:

Answer:

a)

Al-F: ionic or ionic covalent tendency

Al-Cl, Al-Br, Al-I: other halides: covalent tendency

b)

Ag-F: soluble or more soluble on water.

Ag-Cl: insoluble o less soluble on water.

Problem 04 Interatomic bond II

For a bar of sodium chloride (monocrystal) under pure tensile stress (see figure), and for the given

parameters of repulsion force between Cl- and Na+ (b = 2.5910-115 Nm11, n = 10, ro = 283 pm), determine the energy necessary for rupture to occur at the indicated crystallographic plane.

Hint: the energy needed to break a crystalline plane can be approximated by the sum of the contributions of each interatomic bond on the plane.

Answer:


4

Energy to break one bond: J107.348-19

0 U

Energy to break 100 cm2 along (100) plane: J092.0

Problem 05 Unit cell I

Determine lattice parameter for the HCP structure i.e. calculate a (edge length) and c (cell height) as a function of r (atomic radius).

Answer:

ra 2

rc 63

4

Problem 06 Unit cell II

Consider the unit cells of diamond and graphite as shown below:

a) Determine theoretical density of both structures (in g/cm3).

b) Compare the result with reference data (indicate source) and discuss possible differences of the results.

Hint1: assume the central atom in the tetrahedron to be in touch with the 4 atoms around it.

Hint2: look for missing data in literature, quote source.


5

Unit cell (left) and tetrahedral structure (right) of diamond. Atoms in the interstitial sites of the FCC cell are located at (1/4, 1/4, 1/4), (3/4, 3/4, 1/4), (3/4, 1/4, 3/4), (1/4, 3/4, 3/4).

Hexagonal structure (left) and unit cell (right) of graphite.

Answer:

3cm

g544.3diamond

3cm

g930.1graphite

Problem 07 Unit cell III

a) Determine cross section area, area occupied by atoms and the planar packing fraction for the planes {100}, {110} and {111} for unit cell of FCC as well as BCC.

b) Determine the length of direction, length occupied by atoms and linear packing factor the directions , y for unit cell of FCC as well as BCC.

Tabulate the answers.

Answer:


6

Problem 08 Unit cell IV

Knowing that chlorine ions in the NaCl structure are located at the FCC sites of the lattice and sodium ions occupy the octahedral interstitials. Determine the density of sodium chloride (NaCl) assuming:

ionic radius of Na+ : 95 pm; ionic radius of Cl- : 181 pm; molar mass chloride: 35,45 g/mol; monalr mass of sodium: 22,99 g/mol.

Hint: the mass of the electron is negligible with respect to that of the nucleus.

Answer:

.g/cm31.2V

m 3

{100} {110} {111}

FCC BCC FCC BCC FCC BCC

rea seccin 28r 316 2r 228 r 3

216 2r 234 r 338 2r

rea tomos 22 rp 2rp 22 rp 22 rp 22 rp 2

2rp

Fraccin de empaquetamiento

0,79 0,59 0,56 0,83 0,91 0,34

FCC BCC FCC BCC FCC BCC

Largo seccin r22 34r r4

324 r r62 r4

Largo tomos r2 r2 r4 r2 r2 r4

Fraccin lineal 0,71 0,87 1,00 0,61 0,41 1,00

Cross section area

Atoms area

Packing fraction

Section length

Atoms length

Linear fraction


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Problem 09 Unit cell V

Determine the value of atomic packing factor of (110) plane in an FCC cell.

Answer:

APF(110)= 55,53%

Problem 010 Unit cell VI

Considering the ratio of atomic radii in a compound structure, determine the likely crystalline structure for MgO (magnesium oxide) and its lattice parameter. rMg2+: 0.075 nm; rO2-: 0.135 nm.

Answer:

FCC, a = 0.42 nm

Problem 011 Unit cell VII

Calculate titaniums theoretical density (titanium atomic radius 0.1445 nm, molar mass 47.9 g/mol).

Also, draw and determine planar density and planar packing factor for the following planes: 0001 , 0002 , 0011 y 0111 .

Answer:

3g/cm66.4

Plane (0001): planar density -2nm8.13 and packing factor of 907.0




Problem 012 Unit cell VIII

Uranium dioxide (UO2) is mostly used for fabricating rods of nuclear fuel. Determine the type of bond likely to be established between the atoms that form the compound as well as the type of the most probable crystalline structure. Also, determine the lattice parameter, the atomic packing factor and the theoretical density for the compound.


8

Answer:

Ionic bond, fluorite-structure like, pm9.528a , 624.0 y 3g/cm12.12

Problem 013 Unit cell IX

Determine the atomic packing factor (APF) of a rhombohedral cell with an edge of 0.268 nm and

angles = = = 60, composed with Zn atoms (metallic atomic radius = 0.134 nm). Compare the obtained value with the atomic packing factor of the cells HCP y FCC (APF: 0.74).

Hint: the height of the described rhombohedral cell is equal to a3

2

Answer:

74.0APF

Problem 014 Millers indices I

Gold (Au) forms a FCC crystalline structure with a density of 19.3 gcm-3 at a room temperature (20C).

a) Draw the unit cell; specifying the value of all angles (, , ) and the proportion among the edges (a ,b, c).

b) Calculate atomic radius of gold in nanometers.

c) In the unit cell, draw the family of directions 111 and list all the directions that belong to this

family.

d) On a different drawing of the unit cell, indicate the point (3

3,

3

2,

2

1 ) and the plane ( 202 ).

e) On a different drawing of the unit cell, indicate the Burgers vector(s) associated with the low-energy sliding system, this is, direction and plane for maximum compression. Name the sliding system.


9

Answer:

a)

b) nm144.04

2ar

c)

Family of directions 111 :

111 , 111 , 111 , 111 , 111 , 111 , 111 , 111

d)


10

e)

Problem 015 Millers indices II

Using the given sketch, draw the following directions and planes inside the unit cell.

a) 211

b) 324

c) 230

d) 421

Answer:

y

x

z


11

Problem 016 Millers indices III

Identify the planes on the following HCP unit cell using Miller-Bravais indices.

a) b)

c) d)


12

Answer:

A: 0211

B: 1101

C: 0121

Problem 017 Millers indices IV

Show how a rotation of the Millers indices may originate the family of planes {100} in a cubic unit cell. Indicate the number of planes associated with this family.

Answer:

{100}: (100), (010), (001), 001 , 010 , 100

Problem 018 Millers indices V

Calculate the linear density on the direction [111] for the zincblende structure. RZn2+ :0.083nm; RS2-: 0.174nm. Consider the existence of a single adjacent ion from the origin at the direction [111], and that the ion is located at (1/4,1/4,1/4)a.

y

x

z


13

Answer:

nm

OZn 22 973.0973.0

Problem 019 Anisotropy of crystalline structures

By using the tonsorial expression shown, calculate the value for Youngs module and Poisson coefficient on the direction for a cubic structure of Cu (FCC).

For this, apply uniaxial stress T and consider values c11 y c12 to be: 2

11

cm

dinas1016.84 and

2

11

cm

dinas1012,14 .

3

2

1

111212

121112

121211

3

2

1

ccc

ccc

ccc

Answer:

GPa7.66E y 42.0 .

Problem 020 Burgers vector

Determine the lattice parameter a, and the magnitude of the Burgers vector for ferrite (iron BCC) and austenite (iron FCC). Iron atomic radius: 0.126 nm.

Answer:

Ferrite:

nma 291.0

nmb 252.0


14

Austenite:

nma 3564.0 nmb 252.0

Problem 021 Dislocations

The following figure shows a scheme of mixed dislocation. Indicate, which part of the dislocation can be considered purely of screw and edge type, respectively. Finally, draw the deformed body after the dislocation has slid through the entire plane (the direction and sense of the progression is the normal to the black dashed line, or the displacement line).

Problem 022 Sliding system

Calculate the number of sliding systems associated to a face centered cubic structure (FCC).

Answer:

There are 12 sliding systems (4 planes x 3 directions)

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