olid tate Theory hysics 545
Mechanical properties of materials. Basics. tress and strain. Basic definitions. Normal and hear stresses. Elastic constants. tress tensor. Young modulus. rystal symmetry and elasticity tensor. Relations between the strength of chemical bonds and elasticity. Thermal expansion.
Deformation is change of linear dimensions under external force. ELATI DEFORMATION is reversible and involves bond stretching. Bond stretching Relaxed stress-free solid Material in compression. The springs resist tension. Material in tension. The springs resist elongation.
F Bond springs. train. F F F Irrespective of the number of the springs in the chain, each spring carriers the same load! Each individual spring will elongate by the same amount! Hence, strain is defined as elongation for given unit! > strain is dimensionless! F F L L elongation train original length L L
Bond springs. tress. F train Force cross sectional area a b ross sectional area F
Z Y tress. General case. X zz zy normal stresses: xx ; yy ; zz zx yz yy xz xy yx xz shear stresses: xz ; yz ; xy z x z y z z onditions of equilibrium : ) zz - z z, xx - x x, yy - y y ) xy yx ; xz zx ; zy yx xx yx zx xy yy zy xz yz zz 6 5 6 4 5 4 3 ik ki
train. General case. Z Y X u uu x +u y +u z + x u x u x x xx + x u y u y x xy + i k k i ik x u x u ik ki 3 4 5 4 6 5 6 zz zy zx yz yy yx xz xy xx
tress-train relations. In general, stress causes displacement in the directions that are parallel and perpendicular to the stress direction. Therefore: xx xx xxxx + yy xxyy + zz xxzz + xy xxxy + + + 33 + 66 + 55 + 44 3 4 5 6 3 4 5 6 ik are elastic compliances. 3 4 5 6 3 3 33 43 53 63 4 4 34 44 54 64 5 4 34 44 55 65 5 4 34 44 55 66 ik are elastic coefficients (stiffness constants) 3 4 5 6 xz xxxz + i ik k k k ik k i yz xxyz
Number of independent elastic constants. ik and c ik are not independent.. ince the stress is invariant with respect to inversion: ik ki ik ki and matrices contain no more than independent element.. ymmetry of crystals further restricts the number independent elements of and matrices. Triclinic Monoclini c Orthorho mbic Tetragonal Trigonal (rhomboh edral) Hexagonal ubic (Isotropic) ymmetry none 4 ( I) 3 ( I) 6 ( or 6 ) 4 3 Number of ik and ik 3 9 7(6) 6(7) 5 3
Examples. Trigonal (rhombohedral) symmetry (quartz). ubic crystal 3 3 33 4 44 4 44 ( 4 How to deduce the relations between ik from symmetry? ) (see rystallographic tables. Example: in cubic crystals xxyyzz> 33 ) Isotropic case. 44 44 4
Elastic constants (Ga) for different crystals Na B 7. 6. 3 33 44 66 K B 3.7 3.4.88 Be H 9 6.7 4 336 63 Mg H 59.7 6..7 6.7 6.7 Al F 7 6 8 In Tetr. 44.5 39.5 4.5 44.4 6.6. Dmd 76 5 576 i Dmd 66 64 8 u F 68 75 Au F 86 57 4 B 9 9 43 Ta B 67 6 8 W B 5 98 5 Fe B 34 36 8 o H 37 65 3 358 75.3 Lil Nal 49.4.8 4.6 Nal Nal 48.7.4.6 A 4.5 44 is the anisotropy ratio. A 6.7..9.57 3..9.78.55..4.8.88 B F H Dmd Nal
Generalized Hooke s law. For a certain direction under uniaxial loading E is called Young s modulus (uniaxial modulus). For a cubic crystal load in () direction E is i i E E ( )( + ) ( + ) For shear stresses one can write the same relation G ( ) i i G ( ) G is called shear modulus.
ompressibility. If body is subjected by the compressive stress equal in all directions (hydrostatic pressure) it will deform without changing its shape. The change of the volume defines isothermal compressibility. K B/K is called bulk modulus. For a cubic crystal Where does the resistance to compression comes from? B + 3 For metals resistance to compression comes from the electrostatic repulsion of electrons. Homework: try to prove that BnE f /3. Hint: estimate change of the free energy of electrons due to change of volume. In ionic crystals resistance to compression comes from the mutual repulsion of inner shell electrons. Homework: which crystal do you expect to be more compressible LiF or si?
oisson ratio. Let us consider a solid under uniaxial stress xx. Then the deformations are given as xx xx xxxx ( ) yy xx xxyy ( ) tress along one direction causes deformation along other directions! The ratio ν yy xx xxyy xxxxx ν. If a body is isotropic and does not change its volume then ν ½.. If a body does not change its shape then ν. is called oisson ratio 3. ν is a function of direction! In B lattices ν() >, whereas v() <! Example uzn (βbrass): ν().39, v().39. > tension along () causes expansion of the crystal in the transverse direction! Hard-sphere model predicts v(, ) and v(,)+. (The same model does not predict such a large values for F and H). 4. Most of the materials have ν..4. Homework: rove both claims.
Isotropic crystals. For isotropic crystals there are only two independent elastic constants normal modulus E (Young s modulus) and shear modulus G. G ( ) E E ν G E G( ν + ). Most of the polycrystalline materials (metals, ceramics, polymers, glasses) are isotropic.. ast majority of crystals is anisotropic. 3. Elastic (fully recoverable) deformations are usually very small for metals, ceramics and glasses (<.). 4. olids with a very large range of elastic deformations(~.-5) are called rubbers.
Relations between crystal structure and elastic properties. Al r u Fe Nb Ta W E(Ga) 7.3 79 3 5 86 4 G(Ga) 6. 5 48.3 8.6 37.5 69. 6 B(Ga) 76. 94 34 7 67 39 ν.345..343.93.397.34.8
Thermal expansion. Gruneisen constant. Thermal expansion is a result of anharmonic atomic interactions. Linear thermal expansion coefficient as: T α 3 Homework: why is necessary to put /3 in the definition of α? The mechanical energy associated with thermal expansion is 3Bα. ince T U T U U T T B α 3 For most of the materials Gruneisen constant, γ, is ~, positive, independent of temperature and represents the degree of unharmonicity of atomic interactions. Materials with large bulk modulus (hard) have small thermal expansion coefficient. B 3 γ α.8.8 5. 7 Fe. 6.5 4.4 34 u. 3. 4. 76. Al γ α -6 K - J/(molK) B Ga γ α U U B 3 T T T
Thermal expansion. Materials aspects. For most of the materials α polymers >α metals α ionic >α covalent 5.4 α( -6 K - ) Ag 8.9 Au 4. Mg 4.8 Ni 3.4 t 8.8 Ti 8.6 Diamond.8 Graphite 6.7 Graphite -. i 4.68 Ge 5.8 GaAs i 3 N 4.7 Nal 39.6 olystyrene 7 olypropylene 68 γ.4..5.6.3..96.3.3.8.4.7.75 For most of the materials γ> and α >. However, since thermal expansion reflects unharmonicity of atomic interactions it is not obligatory for α and γ to be positive
Negative thermal expansion of ZrW O 8. Lattice parameter a (open circles) Grüneisen parameter, γ, (crosses) honon energy The room-temperature crystal structure of ZrW O 8 is a primitive cubic Bravais lattice with 44 atoms per unit cell arranged according to space group 3. ZrW O 8 consists of corner-sharing WO 4 tetrahedra and ZrO 6 octahedra. Each corner of the ZrO 6 octahedra is shared with one WO 4 tetrahedron, whereas one corner of each WO 4 tetrahedra remains unshared.