Total stress tensor can be divided into two components: hydrostatic or mean stress tensor (om) involving only pure tension or compression and deviatoric stress tensor

(o’j) representing pure shear with no normal components:

It is to be noted that pure hydrostatic stress does not lead to plastic deformation and finite deviatoric stresses are needed for any plastic deformation to take place.

A.4

Normal and Shear Stresses on a Given Plane (Cut-Surface Method)

Given Oj in reference system 12 3, n is the unit vector normal to the plane = njn2n3, m is the unit vector in the plane = m1m2m3, oN is the normal stress along n, and t is the shear stress along m (see Figure A.2).

Note: П ■ m — 0, n2 + n2 + n| = 1, and m1 + m2 + m2 = 1. If П — 1, 2, 5 ) П — 1/^30, 2Д/30, 5Д/30 , n is a unit vector, where /12 + 22 + 52 = /30 so that n1 + n2 + n2 = 1.

First, we find the stress vector (S) {the stress vector is the vector force per unit area acting on the cut}:

S1	011	012	013
S2	— 021	022	023
S3	031	032	033

n1 3^ n2 ) Si = Oikm; n3 k=1

that is, S1 = o11n1 + o12n2 + o13n3, and so on. oN and t are given as follows: oN = S ■ n = S1n1 + S2n2 + S3n3 and

t — S ■ m = S1m1 + S2m2 + S3m3,

Figure A.2 Designations for normal and shear stress calculations.

and tmax occurs when n, S, and m are in the same plane:

Thus, given the stress tensor, we need only two elastic constants E and n, since G is related to E and n:

G = E/2(1 + n). (A. 14)

We now note that the volume change or dilatation is given by

Д =(1- + e1)(1 + e2)(1 + e3) — 1 — e1 + e2 + e3

since e’s << 1. Note that Д is the first invariant of the strain tensor and mean strain em = Д/3.

Thus, bulk modulus

(A.17)

Note that the derivative of U0 with respect to any strain component equals the cor­responding stress component:

— 1Д T 2Gex — Sx

and similarly

A.7

Generalized Hooke’s Law

eij Sijklakl:

Here Sjki is the elastic compliance tensor (fourth rank) and

aij Cijklekh

where Cijkl is elastic stiffness (or elastic constants).

Crystal symmetry reduces the number of independent terms: cubic — 3, hexago­nal — 5, and so on, and these are related to E and G.

Details on these compliance and stiffness coefficients are beyond the scope of this book and may be found elsewhere.

Appendix B

B.1

SI Units

Quantity	Unit	Symbol
Length	Meter	m
Mass	Kilogram	kg
Time	Second	s
Electric current	Ampere	A
Temperature	Kelvin	K
Amount of substance	Mole	mol
Luminous intensity	Candela	cd
Plane angle	Radian	rad
Solid angle	Steradian	sr

B.2

Some Derived Units

Quantity	Special name	Symbol	Equivalence in Other Base derived units units
Force, load, weight	Newton	N	—	kgm s-2
Stress, strength, pressure	Pascal	Pa	N m-2	kgm-1 s-2
Frequency	Hertz	Hz	—	s-1
Energy, work, heat	Joule	J	Nm	kg m2 s-2
Power	Watt	W	Js-1	kg m2 s-3
Electric charge	Coulomb	C	—	A s
Electric potential/voltage	Volt	V	WA-1	kg m2 s-3 A-1 (continued)

An Introduction to Nuclear Materials: Fundamentals and Applications, First Edition. K. Linga Murty and Indrajit Charit © 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

Quantity	Special name	Symbol	Equivalence in Other Base derived units units
Resistance	Ohm	V	VA-1		kgm2s-3 A-2
Capacitance	Farad	F	C V-1		kg-1 m-2 s4A2
Magnetic flux	Weber	Wb	Vs		kgm2 s-2 A-1
Magnetic flux density	Tesla	T	Wb m	-2	kgs-2 A-1
Inductance	Henry	H	Wb A-	1	kg m2 s-2 A-2

B.3 Standard Unit Prefixes and Their Multiples and Submultiples
Name	Multiplication factor	Symbol
Atto	10-18	a
Femto	10-15	f
Pico	10-12	p
Nano	10-9	n
Micro	10-6	m
Milli	10-3	m
Kilo	103	k
Mega	106	M
Giga	109	G
Tera	1012	T

B.4

Some Unit Conversion Factors

B.4.1

Length

1 inch = 25.4 mm 1 nm= 10-9m 1 mm = 10-6 m 1A= 10-10m = 0.1 nm

B.4.2

Temperature

T (K) = T (°C) + 273.15 T (°C) = [T (°F) — 32]/1.8

B.4.3

Mass

1 Mg= 103kg 1kg = 10~3 Mg

1kg = 103g 1kg = 2.205 lbm

1g = 10~3kg

B.4.4

Force

1kgf = 9.81 N 1lb = 4.448 N 1 dyne = 10~5 N

B.4.5

Stress

1 ksi (i. e., 103 psi) = 6.89 MPa 1 MPa= 1N mm-2 = 145 psi 1Pa = 10 dyn cm-2 1 ton in.-2 = 15.46 MPa 1 atm = 0.101325 MPa 1bar = 0.1 MPa 1 Torr (mmHg) = 133.3 MPa

B.4.6

Energy, Work, and Heat

1 eV atom 1 = 96.49 kJ mol 1 1 cal = 4.184 J 1 Btu = 252.0 cal 1 erg = 10-7 J

B.4.7

Miscellaneous

1° = rad

57.3

1 g cm-3 = 1000 kg m~3 1 poise = 0.1 Pas 1 ksi in.1/2 = 1.10 MN m~3/2

B.5 Selected Physical Properties ofMetals (Including Metalloids) Symbol Atomic number Atomic weight Density at 20 °C (gcm-3) Melting point (°C) Al 13 26.98 2.7 660 Sb 51 121.76 6.68 630.5 As 33 74.91 5.727 814 Ba 56 137.36 3.5 710 Be 4 9.013 1.845 1284 Bi 83 209.0 9.8 271.2 B 5 10.82 2.34 2300 Cd 48 112.41 8.65 320.9 Ca 20 40.08 1.54 851 Ce 58 140.13 6.66 795 Cs 55 132.91 1.873 28.5 Cr 24 52.01 7.19 1875 Co 27 58.94 8.90 1493 Nb 41 92.91 9.57 2468 Cu 29 63.54 8.94 1083 Dy 66 162.51 8.536 1407 Er 68 167.21 9.051 1497 Eu 63 152.0 5.259 826 Gd 64 157.26 7.895 1312 Ga 31 69.72 5.907 29.75 Ge 32 72.60 5.32 936 Au 79 197.2 19.32 1063 Hf 72 178.50 13.29 2150 Ho 67 164.94 8.803 1461 In 49 114.82 7.31 156.6 Ir 77 192.2 22.42 2410 Fe 26 55.85 7.87 1535 La 57 138.92 6.174 920 Pb 82 207.21 11.34 327.4 Li 3 6.940 0.534 179 Lu 71 174.99 9.842 1652 Mg 12 24.32 1.74 651 Mn 25 54.94 7.44 1244 Hg 80 200.61 13.55 -38.87 Mo 42 95.95 10.22 2610 Nd 60 144.27 7.004 1024 Ni 28 58.69 8.9 1452 Os 76 190.2 22.5 3000 Pd 46 106.4 12.02 1552 (continued)

Symbol	Atomic number	Atomic weight	Density at 20 °C (gcm~3)	Melting point (°C)
Pt	78	195.09	21.40	1769
Pu	94	239.11	19.84	639.5
K	19	39.10	0.87	63.7
Pr	59	140.92	6.782	935
Pm	61	145	7.264	1035
Re	75	186.22	21.02	3180
Rh	45	102.91	12.44	1960
Ru	44	101.1	12.4	2250
Sm	62	150.35	7.536	1072
Sc	21	44.96	2.99	1539
Se	34	78.96	4.79	217
Si	14	28.09	2.33	1410
Ag	47	107.873	10.49	960.5
Na	11	22.991	0.97	97.9
Sr	38	87.63	2.6	770
Ta	73	180.95	16.6	2996
Te	52	127.61	6.25	449.5
Tb	65	158.93	8.272	1356
Tl	81	204.39	11.85	303
Th	90	232.05	11.66	1750
Tm	69	168.94	9.332	1545
Sn	50	118.7	7.3	232
Ti	22	47.90	4.54	1668
W	74	183.92	19.3	3410
U	92	238.07	19.07	1132
Yb	70	173.04	6.977	824
Y	39	88.92	4.472	1509
Zn	30	65.38	7.133	419.5
Zr	40	91.22	6.45	1852

Adapted from Ref. [1].

B.6 Thermal Neutron (0.025 eV) Absorption Cross Sections of Some Elements Element Absorption cross section (b) Element Absorption cross section (b) Element Absorption cross section (b) C 0.0035 Ni 4.49 Ho 64.7 Be 0.0076 Sb 4.91 Lu 74 Bi 0.034 V 5.08 Am 75.3 Mg 0.063 Ti 6.09 Re 89.7 Si 0.171 Pd 6.90 Au 98.7 Pb 0.171 Th232 7.56 Tm 100 Zr 0.185 U 7.57 Hf 104.1 Al 0.231 La 8.97 Rh 144.6 H 0.332 Pt 10.3 Np 175.9 Sn 0.626 Pr 11.5 Er 159 Ce 0.630 Se 11.7 In 193.8 Zn 0.110 Mn 13.3 Pu240 289.6 Nb 1.15 Os 16.0 Ir 425 Y 1.28 W 18.3 B 767 Ge 2.20 Ta 20.6 Dy 994 Fe 2.56 Tb 23.4 Pu239 1017.3 Ru 2.56 Sc 27.5 Pu241 1400 (*) Mo 2.48 Co 37.2 Cd 2520 Cr 3.05 Yb 34.8 Eu 4530 Tl 3.43 Nd 50.5 Sm 5922 Cu 3.78 Ag 63.3 Gd 49 700 Te 4.7 [Source: Special feature section of neutron scattering lengths and cross sections of the elements and isotopes, Neutron News, 3 (1992) 29-37]

B.7 Mechanical Properties of Some Important Metals and Alloys Alloy Young’s modulus (GPa) Poisson’s ratio UTS (MPa) YS (MPa) Elongation to fracture (%) Fracture toughness (MPa m1/2) Al 2024 T851 72.4 0.33 455 400 5 26.4 Al 7075 T651 72 0.33 570 505 11 24.2 Al 7178T651 73 0.33 605 540 10 23.1 Ti-6Al-4V 113.8 0.342 1860 148 14 55 (grade 5) Alpha annealed 100 0.3 620 500 15 100 Ti-3Al-2.5V 702 Zirconium 99.3 0.35 379 207 16 — Stainless steel 205 — 745 470 22 60.4 4340 Stainless steel 193 0.29 505 215 70 — 304 Tool steel H11 210 — 1990 1650 9 — (hot worked) Maraging steel 200 — 1864 1737 17.4 — Superalloy — — 860 310 10 — CoCrWNi Superalloy H-X — — 690 276 40 — nickel

B.8

Mechanical Properties of Some Important Ceramics

Ceramic	Young’s modulus (GPa)	Poisson’s ratio	Hardness (HV)	Tensile strength (MPa)	Compressive strength (MPa)	Flexural strength (MPa)	Fracture toughness (MPa m1/2)
Silicon nitride	320	0.28	1800	350-415	2100-3500	930	6
Silicon carbide	450	0.17	2300	390-450	1035-1725	634	4.3
Tungsten carbide	627	0.21	1600	344	1400-2100	1930	—
MgO-stabilized ZrO2	200	0.3	1200	352	1750	620	11
Boron carbide	450	0.27	2700	—	470	450	3.0
Titanium diboride	556	0.11	2700	—	470	277	6.9

Note: The above values are indicative only. Adapted from Ref. [2].

References

[1] For example, B10 + n1 ! Li7 + He4.

[2] Using these data, evaluate the parameters n and Q in the power-law creep equation.

ii) Identify the underlying creep mechanism.

iii) If in another example, a thin polycrystalline sample exhibited essentially the same n and Q, what would be the controlling creep mechanism?

[3] Relaxation (or excess) volume is generally asso­ciated with a defect leading to some form of internal stress and is expressed in terms of atomic volume (V). Relaxation volume basi­cally measures the distortion volume induced in the lattice due to the insertion of a vacancy or interstitial. Most calculations of relaxation