Appendix A

Here we present a brief account of the derivation of the Riemann–Christoffelcurvature tensor, and the Laplacian of the stress tensor in curvilinear coordinates.For the background necessary we refer the readers, who are not familiar with tensoranalysis, to Eringen [1980, Appendix C], Eringen [1971], or any books on tensorcalculus.

In the curvilinear coordinates xk , the metric tensor is denoted gkl(x) so that thesquare of the element of arclength is given by

ds2 = gkl dxk dxl. (A.1)

The inverse gkl of gkl is the solution of

gkigil = δkl . (A.2)

By means of gkl and gkl we can raise and lower the indices of any tensor to obtainthe covariant, mixed, and contravariant components, e.g.,

Akl = gkiAil, Akl = gkjAl

j ,

Akl = gjlA

kj , Akl = gikAil . (A.3)

In orthogonal curvilinear coordinates, gkl and gkl possess only three componentseach, and they are related to each other by

gk k = 1

gkk, gkl = 0, k �= l, (A.4)

352 Appendix A

where the underbars denote the suspension of the summation on repeated indices.If zk denotes the Cartesian coordinates with base vectors ik , and xk the curvilinearcoordinates with base vectors gk(x), the partial derivative of a vector u(z) inrectangular coordinates read

∂u∂zl

= ∂(ukik)∂zl

= ∂uk

∂zlik.

However, in the curvilinear coordinates xk , we have

∂u∂xl

= ∂(ukgk)∂xl

= ∂uk

∂xlgk + uk

∂gk∂xl

,

but∂gk∂xl

= ∂

∂xl

(∂zn

∂xkin

)= ∂2zn

∂xk ∂xlin.

Upon replacing in = (∂xk/∂zn)gk , we obtain

∂gk∂xl

={i

k l

}gi , (A.5)

where

{i

k l

}is the Christoffel symbol of the second kind defined by

{i

k l

}= ∂2zn

∂xk ∂xl

∂xi

∂zn. (A.6)

The Christoffel symbol of the second kind is related to the first kind [ i j, k ] by

[ i j, k ] = gkr

{r

i j

},

{k

i j

}= gkr [ i j, r ]. (A.7)

Using (A.6) we can express both symbols in terms of a metric tensor

[ i j, k ] = 12

(∂gik

∂xj+ ∂gjk

∂xi− ∂gij

∂xk

). (A.8)

Using (A.5) we can express the partial derivative of a vector in curvilinear coor-dinates by

∂u∂xi

= ∂

∂xi(ukgk) =

(∂uk

∂xi+{k

i j

}uj)

gk ≡ uk;igk,

where uk;i , defined by

uk;i = uk,i +{k

i j

}uj , (A.9)

Appendix A 353

is called the covariant derivative of a contravariant vector. The covariant derivativeof a covariant vector is similarly obtained

uk;i = uk,i −{j

k i

}uj . (A.10)

Covariant derivatives are tensors and can be covariant differentiated further.Thus, for example, covariant derivatives of a second-order tensor are given by

treating each index as a vector index, e.g.,

Akl;i ≡ Akl

,i +{k

i j

}Ajl +

{l

i j

}Akj ,

Akl;i ≡ Ak

l,i −{j

l i

}Ak

j +{k

i j

}Ajl,

Akl;i ≡ Akl,i −{j

k i

}Ajl −

{j

l i

}Akj . (A.11)

It is simple to verify that the covariant derivatives of metric tensors vanish, i.e.,

gkl;i = gkl;i = 0. (A.12)

Consequently, we raise and lower the indices of (A.11) by using metric tensors,e.g.,

Akl;i = A

kj

;igjl, Akj

;i = Akl;ig

jl . (A.13)

Covariant derivatives of higher-order tensors follow the same rules displayed inthe compositions of (A.11).

I. Riemann–Christoffel Curvature Tensor

From calculus we know that the order of mixed partial derivatives is not important,i.e.,

∂2φ

∂xi ∂xj= ∂2φ

∂xj ∂xi.

The question then arises: Under what conditions does the second-order covariantpartial derivative commute? For example: When can we write

Ak;lm = Ak;ml?

The answer is found by forming both sides of this equation and subtracting onefrom the other.

We have

Ak;l = Ak,l −{r

k l

}Ar,

and

Ak;lm = (Ak;l ),m −{r

k m

}Ar;l −

{r

l m

}Ak;r ,

354 Appendix A

or

Ak;lm = Akl,m −{r

k l

},m

Ar −{r

k l

}Ar,m −

{r

k m

}Ar,l

+{r

k m

}{s

r l

}As −

{r

l m

}Ak,r +

{r

l m

}{s

k r

}As. (A.14)

Interchanging the indices l and m and subtracting we obtain

Ak;lm − Ak;ml = RrklmAr, (A.15)

where

Rrklm =

{r

k m

},l

−{r

k l

},m

+{s

k m

}{r

s l

}−{s

k l

}{r

s m

}. (A.16)

This fourth-order tensor is called the Riemann–Christoffel curvature tensor. Clearly,Rr

klm is independent of the vector Ar . It is formed in terms of the metric tensoronly. Hence, we have proved:

Theorem. Cross covariant derivatives of any vector commute if and only if theRiemann–Christoffel tensor vanishes identically.

By lowering the index r , we obtain a fourth-order tensor

Rklmn = gkrRrlmn, (A.17)

which is known as the curvature tensor. In three dimensions the nonvanishingcomponents of Rklmn are six: R1212, R1313, R2323, R1213, R2123, and R3132. Intwo dimensions the only nonvanishing component is R1212. These constitute thecompatibility conditions when the metric tensor is ckl or CKL.

The Riemann–Christoffel curvature tensor was used by Einstein to develop histheory of general relativity.

II. Laplacian of a Tensor

We wish to calculate the Laplacian of a second-order symmetric tensor. To thisend, we need a second covariant derivative, e.g.,

Akl;ij =

[Akl

,i +{k

i r

}Arl +

{l

i r

}Akr

],j

+{k

j n

}[Anl

,i +{n

i r

}Arl +

{l

i r

}Anr

]+{l

j n

}[Akn

,i +{k

i r

}Arn +

{n

i r

}Akr

]−{n

i j

}[Akl

,n +{k

n r

}Arl +

{l

n r

}Akr

]. (A.18)

Appendix A 355

The Laplacian of a contravariant tensor Akl is given by

∇2Akl = Akl;ij g

ij . (A.19)

Consequently, in orthogonal curvilinear coordinates, we have

∇2Akl =∑i

1

gii

{[Akl

,i +{k

i r

}Arl +

{l

i r

}Akr

],i

+{k

i n

}[Anl

,i +{n

i r

}Arl +

{l

i r

}Anr

]+{l

i n

}[Akn

,i +{k

i r

}Arn +

{n

i r

}Akr

]−{n

i i

}[Akl

,n +{k

n r

}Arl +

{l

n r

}Akr

]}. (A.20)

We must now replace Akl by its physical components, given by A(k)(l):

Akl = A(k)(l)/

√gk kgl l . (A.21)

With this then, (A.20) gives the physical components of the Laplacian (∇2A)(k)(l)

in the orthogonal curvilinear coordinates

(∇2A)(k)(l) =√

gk kgl l∑i

∑n

∑r

{1

gii

[(A(k)(l)√

gk kgl,l

),i

+{k

i r

}A(r)(l)√

grrgl l+{l

i r

}A(k)(r)√

gk kgrr

],i

+ 1

gii

{k

i n

}

×[(

A(n)(l)√

gnngl l

),i

+{n

i r

}A(r)(l)√

grrgl l+{l

i r

}A(n)(r)√

gnngrr

]

+ 1

gii

{l

i n

}[(A(k)(n)√

gk kgnn

),i

+{k

i r

}A(r)(n)√

grrgnn

+{n

i r

}A(k)(r)√

gk kgrr

]

− 1

gii

{n

i i

}[A(k)(l)√

gk kgl l+{k

n r

}A(r)(l)√

grrgl l

+{l

n r

}A(k)(r)√

gk kgrr

]}, (A.22)

where the summation is suspended on the indices k and l. For the indices i, n, andr the conventional summation applies.

356 Appendix A

In orthogonal curvilinear coordinates we have

ds2 = g11(dx1)2 + g22(dx

2)2 + g33(dx3)2,

gk k = 1

gk k,{

l

k k

}= 1

2gl l

∂gk k

∂xl,

{k

k l

}= ∂

∂xl(ln

√gk k),{

k

k k

}= ∂

∂xk(ln

√gk k),

{k

l m

}= 0, k �= l �= m. (A.23)

For example, in the cylindrical coordinates (r, θ, z), we have

g11 = g11 = g33 = g33 = 1, g22 = 1

g22 = r2,{21 2

}={

22 1

}= 1

r,

{12 2

}= −r all other

{k

l m

}= 0. (A.24)

Using these in (A.22), we calculate the Laplacian of the stress tensor tkl in cylin-drical coordinates

(∇2t)rr = ∇2trr − 4

r2

∂trθ

∂θ− 2

r2 (trr − tθθ ),

(∇2t)θθ = ∇2tθθ + 4

r2

∂trθ

∂θ+ 2

r2 (trr − tθθ ),

(∇2t)rθ = ∇2trθ − 4

r2 trθ + 2

r2

∂

∂θ(trr − tθθ ),

(∇2t)rz = ∇2trz − 1

r2 trz − 2

r2

∂tθz

∂θ,

(∇2t)θz = ∇2tθz − 1

r2 tθz + 2

r2

∂trz

∂θ

(∇2t)zz = ∇2tzz, (A.25)

where

∇2f = ∂2f

∂r2 + 1

r

∂f

∂r+ 1

r2

∂2f

∂θ2 + ∂2f

∂z2 . (A.26)

These equations were given by Povstenko [1995] without their derivations.

References

Agranovich, V.M. and Ginsburg, V.L. [1984]: Crystal Optics with Spatial Dis-persion and Excitons, 2nd ed., Springer-Verlag, Berlin.

Altan, B.S. [1984]: Uniqueness in linear theory of nonlocal elasticity, Bull. Tech.Univ. Istanbul, 37, 373–385.

Altan, B.S. [1990]: Some theorems in nonlocal thermoelasticity, J. ThermalStresses, 13, 207–221.

Ari, N. and Eringen, A.C. [1983]: Nonlocal stress field at Griffith crack, Cryst.Lattice Def. Amorph. Mat., 10, 33–38.

Artan, R. [1996]: Nonlocal elastic half-plane loaded by a concentrated force, Int.J. Engng. Sci., 34, 3, 943–950.

Brillouin, L. [1946]: Wave Propagation in Periodic Structures, Dover, NewYork.Chambers, R.G. [1952]: Proc. Roy. Soc. London A215, 481.Chan, D.Y.C. and Horn, R.G. [1985]: The drainage of thin liquid films between

solid surfaces, J. Chem. Phys., 83, 10, 5311–5324.Chandrasekhar, S. [1950]: Radiative Transfer, Oxford University Press, Lon-

don.Chandrasekhar, S. [1992]: Liquid Crystals, 2nd ed., Cambridge University

Press, Cambridge.Chirita, S. [1976]: On some boundary value problems in nonlocal elasticity,

Amale stiinfice ale Universitatii “AL. I. CUZA” din Iasi Tomul, xxii, s.Ia, f.2.Chirita, S. [1976a]: Nonlocal effects on the stress distribution in an elastic half-

space, Accad. Naz. Lincei, serie viii, LX fasc. 6, 802–807.Coleman, B.D. and Noll, W. [1961]: Foundations of linear viscoelasticity, Rev.

Mod. Phys., 33, 239.

358 References

Collins, P.J. [1990]: Liquid Crystals, Princeton University Press, Princeton, NJ.Cottrell, A.H. and Bilby, B.A. [1949]: Dislocation theory of yielding and strain

ageing of iron, Proc. Phys. Soc. A, 62, 49.Cracium, B. [1996]: On nonlocal thermoelasticity, Am. St. Univ., Ovidus Con-

stanta, 5 (1), 29–36.De Gennes, P.G. and Prost, J. [1993]: Physics of Liquid Crystals, 2nd ed., Claren-

don Press, Oxford.De Groot, S.R. and Suttorp, L.G. [1972]: Foundations of Electrodynamics,

North-Holland, Amsterdam.De Jeu, W.H. [1980]: Physical Properties of Liquid Crystalline Materials, Gordon

and Breach, New York.Demiray H. and Eringen, A.C. [1978]: On nonlocal diffusion of gases, Arch.

Mech., 30, 65–77.De Wit, R. [1973]: Theory of disclinations: IV. straight disclinations, Res. Nat.

Bur. Standards, 77A, 607.Driessen, A., Griessen, R., Koeman, N., Salomons, E., Brown, R., de Groot,

D.G., Heeck, K., Hemmes, H., and Rector, J. [1987]: Pressure dependenceof the Tc of Y Ba2Cu3O7 up to 170 kbar, Phys. Rev. B. 36 (16), 8329.

Edelen, D.G.B. [1993]: The College Station Lectures on Thermodynamics, TexasA & M University, College Station, TX.

Elliott, H.A. [1947]: An analysis of conditions for rupture due to Griffith cracks,Proc. Phys. Soc., A59, 208–223.

Emslie, A.G., Bonner, F.T., and Peck, L.G. [1958]: J. Appl. Phys., 29, 858.Eringen, A.C. [1964]: Simple microfluids, Int. J. Engng. Sci., 2, 2, 189–217.Eringen, A.C. [1965]: Theory of micropolar continua, in Developments in Me-

chanics and Materials, Proc. 9th Midwestern Conference, University of Wis-consin, Aug. 16–18, 1965 (Huang, T.C. and Johnson, M.W., Jr., eds.), Wiley,New York, 1967, pp. 23–40.

Eringen, A.C. [1966]: A unified theory of thermomechanical materials, Int. J.Engng. Sci., 4, 179–202.

Eringen, A.C. [1966a]: Linear theory of micropolar elasticity, J. Math. Mech.,15, 909–923.

Eringen, A.C. [1966b]: Mechanics of Micromorphic Materials, Proc. 11th Int.Congr. Appl. Mech., Munich, Germany, 1964 (Gortler, H. ed.), Springer-Verlag, Berlin, pp. 131–138.

Eringen, A.C. [1967]: Mechanics of Continua, Wiley, New York.Eringen, A.C. [1969]: Micropolar fluids with stretch, Int. J. Engng. Sci., 7, 115–

128.Eringen, A.C. [1970]: Balance laws of micromorphic mechanics, Int. J. Engrg.

Sci., 8, 10, 819–828.Eringen, A.C. [1971]: Continuum Physics, Vol. I (Eringen, A.C., ed.), Academic

Press, New York.Eringen, A.C. [1971a]: Micropolar elastic soids with stretch, Prof. Mustafa Inan

Anisma, Ari Kitapevi Matbaasi, pp. 1–18, Istanbul, Turkey.

References 359

Eringen, A.C. [1972a]: Linear theory of nonlocal elasticity and dispersion ofplane waves, Int. J. Engng. Sci., 10, 5, 425–435.

Eringen, A.C. [1972b]: On nonlocal fluid mechanics, Int. J. Engrg. Sci., 10, 6,561–575.

Eringen, A.C. [1972c]: Nonlocal polar elastic continua, Int. J. Engrg. Sci., 10,1–16.

Eringen, A.C. [1973]: Linear theory of nonlocal microelasticity and dispersionof plane waves, Lett. Appl. Engng. Sci., 1, 2, 129–146.

Eringen, A.C. [1973a]: Theory of nonlocal electromagnetic elastic solids, J. Math.Phys., 14,6, 733–740.

Eringen, A.C. [1973b]: On nonlocal microfluid mechanics, Int. J. Engrg. Sci., 11,2, 291–306.

Eringen, A.C. [1974]: Theory of nonlocal thermoelasticity, Int. J. Engrg. Sci., 12,1063–1077.

Eringen, A.C. [1974a]: Memory dependent nonlocal elastic solids, Lett. Appl.Engng. Sci., 2, 3, 145–159.

Eringen, A.C. [1974b]: Memory-dependent nonlocal thermoelastic solids, Lett.Appl. Engng. Sci., 2, 3, 145–149.

Eringen, A.C. [1974c]: Nonlocal elasticity and waves, in Continuum MechanicsAspect of Geodynamics and Rock Fracture Mechanics (Thoft-Christensen,ed.), pp. 81–105.

Eringen, A.C. [1975]: Continuum Physics, Vol. II, Sect. 1.3, Academic Press,New York.

Eringen, A.C. [1975a]: Nonlocal micropolar elastic moduli, Lett. Appl. Engng.Sci., 3, 5, 385–393.

Eringen, A.C. [1976]: Continuum Physics,Vol. IV (Eringen,A.C., ed.),AcademicPress, New York.

Eringen, A.C. [1977a]: Edge dislocation on nonlocal elasticity, Int. J. Engng. Sci.,15, 177–183.

Eringen, A.C. [1977b]: Screw dislocation in nonlocal elasticity, J. Phys. D. Appl.Phys., 10, 671–678.

Eringen, A.C. [1978]: Line crack subject to shear, Int. J. Fracture, 14, 4, 367–379.Eringen, A.C. [1978a]: Micropolar theory of liquid crystals, in Liquid Crystals

and Ordered Fluids, Vol. 3, pp. 443–473 (Johnson, J.F. and Porter, R.S., eds.),Plenum, New York.

Eringen, A.C. [1979]: Line crack subject to antiplane shear, Engng. FractureMech., 12, 211–219.

Eringen, A.C. [1979a]: Electrodynamics of cholesteric liquid crystals, Mol. Cryst.Liq. Cryst., 54, 21–44.

Eringen, A.C. [1979b]: Continuum theory of nematic liquid crystals subject toelectromagnetic fields, J. Math. Phys., 20, 12, 2671–2681.

Eringen, A.C. [1980]: Mechanics of Continua, 2nd ed., Krieger, Melbourne, FL.Eringen, A.C. [1981]: Nonlocal continuum theory of liquid crystals, Mol. Cryst.

Liq. Cryst., 75, 321–343.

360 References

Eringen, A.C. [1983]: Interaction of a dislocation with a crack, J. Appl. Phys.,54, 12, 6811–6817.

Eringen, A.C. [1983a]: On differential equations of nonlocal elasticity and solu-tions of screw dislocation and surface waves, J. Appl. Phys., 54, 9, 4703–4710.

Eringen, A.C. [1984]: On continuous distributions of dislocations in nonlocalelasticity, J. Math. Phys., 25, 11, 3235–3249.

Eringen, A.C. [1984a]: Theory of nonlocal piezoelectricity, J. Math. Phys., 25,3, 717–727.

Eringen, A.C. [1984b]: Point charge, infra-red dispersion and conduction in non-local piezoelectricity, in The Mechanical Behavior of Electromagnetic SolidContinua (Maugin, G.A., ed.), North-Holland, Elsevier Science, pp. 187–196.

Eringen, A.C. [1984c]: Electrodynamics of memory-dependent nonlocal elasticcontinua, J. Math. Phys., 25, 11, 3235–3249.

Eringen, A.C. [1986]: Nonlocal Inviscid magneto-hydrodynamics and dispersionof Alfven waves, Bull. Tech. Univ. Istanbul, 39, 393–408.

Eringen, A.C. [1987a]: Theory of nonlocal elasticity and some applications, Res.Mechanica, 21, 313–342.

Eringen, A.C. [1987b]: Nonlocal dispersion of lattice dynamics and applications,in Constitutive Models of Deformation (Chandra, J. and Srivastare, R.P., eds.),SIAM, Philadelphia, 1987, pp. 58–80.

Eringen, A.C. [1990]: Memory-dependent nonlocal electrodynamics, in Mechan-ical Modelling of New Electromagnetic Materials, Proceedings of IUTAMSymposium (Hsieh, R.K.T., ed.), pp. 45–49.

Eringen, A.C. [1990a]: On screw dislocation and yield, in Elasticity (Eason, G.and Ogden, R.W., eds.), Ellis Horwood, Chichester, UK, pp. 87–89.

Eringen, A.C. [1991]: Memory-dependent nonlocal electromagnetic elastic solidsand superconductivity, J. Math. Phys., 32, 3, 787–796.

Eringen, A.C. [1991a]: Memory-dependent orientable nonlocal micropolar fluids,Int. J. Engng. Sci., 29, 12, 1515–1529.

Eringen, A.C. [1992]: Vistas of nonlocal continuum physics, Int. J. Engng. Sci.,30, 1551–1565.

Eringen, A.C. [1992a]: Balance laws of micromorphic continua revisited, Int. J.Engng. Sci., 30, 1551–1565.

Eringen, A.C. [1993]: Vistas of nonlocal electrodynamics, AMD, Vol. 161/MD,Vol. 42, Mechanics of Electromagnetic Materials and Structures,ASME, NewYork, NY, pp. 1–20.

Eringen, A.C. [1993a]: An assessment of director and micropolar theories ofliquid crystals, Int. J. Engrg. Sci., 31, 605–615.

Eringen, A.C. [1994]: Disclinations in liquid crystals, Bull. Tech. Univ. Istanbul,47, 1–21.

Eringen, A.C. [1997a]: A unified continuum theory of liquid crystals, ARI, 50,73–84.

Eringen, A.C. [1997b]: A unified continuum theory of electrodynamics of liquidcrystals, Int. J. Engng. Sci., 35, 12/13, 1137–1157.

References 361

Eringen, A.C. [1998a]: A mixture theory of electromagnetism and superconduc-tivity, Int. J. Engng. Sci., 36, 5/6, 525–543.

Eringen, A.C. [1999]: Microcontinuum Field Theories, Vol. I: Foundations andSolids, Springer-Verlag, New York.

Eringen, A.C. [2000]: A unified continuum theory for electrodynamics of poly-meric liquid crystals, Int. J. Engrg. Sci., 38, 959–987.

Eringen, A.C. [2001]: Microcontinuum Field Theories, Vol. II. Fluent Media,Springer-Verlag, New York.

Eringen, A.C. and Altan, B. [1989]: Plastic yielding at crack tip, J. Appl. Phys.,65, 15, 4587–4594.

Eringen, A.C. and Balta, F. [1978]: Screw dislocation in nonlocal hexagonalelastic crystals, Crystal Lattice Defects, 7, 183–189.

Eringen, A.C. and Balta, F. [1979]: Penetration of a half-space by a rectangularcylinder, J. Appl. Mech., 46, 3, 587–591.

Eringen, A.C. and Balta, F. [1979a]: Edge dislocation in nonlocal hexagonalelastic crystals. Crystal Lattice Defects, 8, 73–80.

Eringen, A.C. and Edelen, D.G.B. [1972]: On nonlocal elasticity, Int. J. Engrg.Sci., 10, 3, 233–248.

Eringen, A.C. and Kim, B.S. [1974a]: Stress concentration at the tip of crack,Mech. Res. Comm., 1, 4, 233–237.

Eringen, A.C. and Kim, B.S. [1974b]: On the problem of crack tip in nonlocalelasticity, in Continuum Mechanics Aspects of Geodynamics and Rock Frac-ture Mechanics (Thoft-Christensen, ed.), Reidel, Dordrecht, pp. 107–113.

Eringen, A.C. and Maugin, G. [1989]: Electrodynamics of Continua, Springer-Verlag, New York.

Eringen, A.C. and Okada, K. [1995]: A lubrication theory for fluids with mi-crostructures, Int. J. Engrg. Sci., 33, 15, 2297–2308.

Eringen, A.C., Speziale, C.A., and Kim, B.S. [1977]: Crack tip problem in non-local elasticity, J. Mech. Phys. Solids, 25, 339–355.

Eringen, A.C. and Suhubi, E.S. [1964]: Nonlinear theory of simple microelasticsolids, I and II, Int. J. Engng. Sci., 2, 189–205 and 389–404.

Eshelby, J.D. [1961]: Elastic inclusions and inhomogeneities, in Prog. SolidMech., North-Holland, Amsterdam, pp. 89–140.

Forcada, M.L. and Mate, C.M. [1993]: J. Colloid Interface Sci., 160, 218.Friedman, N. and Katz, M. [1966]: Representation theorem for additive func-

tionals, Arch. Rat. Mech. Anal., 21, 49–59.Gairola, B.K.D. [1982]: The Nonlocal Continuum Theory of Lattice Defects

(Rogula, D., ed.), Springer-Verlag, Berlin, p. 59.Gao, F. [1990]: Screw dislocation in a bimedium in nonlocal elasticity, J. Phys.

D. Appl. Phys., 23, 3, 328–333.Ghatak, A.K. and Kothari, L.S. [1972]: An Introduction to Lattice Dynamics,

Addison-Wesley, Reading, MA.Gohar, A.Y. [1979]: A microscopic fracture study in the two-dimensional trian-

gular lattice, Ph.D. Thesis, City University of New York.

362 References

Gradshteyn, I.S. and Ryzhik, I.W. [1965]: Tables of Integrals, Series and Prod-ucts, Academic Press, New York.

Gray, G.W. and Winsor, P.A. [1974]: Liquid Crystals and Plastic Crystals, Vols.1 and 2, Ellis Horwood, Chichester, UK.

Griessen, R. [1987]: Pressure dependence of high Tc superconductors, Phys. Rev.B, 36, 5284.

Hearmon, R.F.S. [1966, 1969]: The elastic constants of non-piezoelectric crys-tals, in Landolt–Bornstein, Numerical Data and Functional Relationships inScience (Hellwage, K.H., ed.), Vols. III/1 and III/2, Springer-Verlag, Berlin.

Hirth, J.P. and Lothe, J. [1968]: Theory of Dislocations, McGraw-Hill, NewYork.

Hopfield, J.J. and Thomas, D.G. [1961]: Phys. Rev., 122–135.Hopfield J.J. [1969]: Free and bound exitons, in Elementary Exitations in Solids

(Maradudin, A.A. and Nardelli, G.F., eds.), Phenum, NewYork, pp. 413–436.Iesan, D. [1977]: Reciprocal theorems and variational theorems in nonlocal elas-

todynamics, Int. J. Engrg. Sci., 15, 693–699.Israelashvili, J.N. [1986a]: Measurement of the viscosity of thin fluid films

between two surfaces with and without adsorbed polymers, Colloid PolymerSci., 264, 1060–1065.

Israelashvili, J.N. [1986b]: Measurement of the viscosity of liquids in very thinfilms, J. Colloid Interface Sci., 110, 263.

Joynson, R.E. [1954]: Elastic spectrum of zinc from the temperature scatteringof X-rays, Phys. Rev., 94, 851–855.

Kelly, A. [1966]: Strong Solids, Clarendon Press, Oxford.Kelly, A., Tyson, W.R., and Cottrell, A.H. [1967]: Ductile and brittle crystals,

Phil. Mag., 15, 567–586.Ketterson, J.B. and Song, S. [1999]: Superconductivity, Cambridge University

Press.King, H.W. [1966]: Quantitative size-factors for metallic solid solutions, J. Mater.

Sci., 1, 79–90.Kiral, E. and Eringen, A.C. [1990]: Constitutive Equations of Nonlinear Elec-

tromagnetic Elastic Solids, Springer-Verlag, New York.Kittel, C. [1974]: Introduction to Solid State Physics, 4th ed., Wiley, New York,

p. 38.Kröner, E. [1954]: Die isotropen Spannungsfunktionen der dreidimensionalen

Elastizitatstheorie, Z. Phys., 139, 175–188.Kröner, E. [1981]: Continuum theory of defects, in Physics of Defects (Balian,

R. et al., eds.), North-Holland, Amsterdam.Kröner, E. and Datta, B.K. [1966]: Nichtlokale elastostatik: Ableitung aus der

gittertheorie, Z. Phys., 196, 203–211.Krumhansi, J.A. [1965]: Generalized continuum field representations for lattice

vibrations, in Lattice Dynamics (Wallis, R.F., ed.), Pergamon Press, Oxford.

References 363

Krumhansi, J.A. [1968]: Some considerations on the relations between solid statephysics and generalized continuum mechanics, in Mechanics of GeneralizedContinua (Kroner, E., ed.), Springer-Verlag, Berlin, pp. 298–311.

Kunin, I.A. [1966]: Model of elastic medium with simple structure and spacedispersion (in Russian), Prikl. Mat. Mekh., 30, 542–550.

Kunin, I.A. [1983]: Elastic Media with Microstructure II, Springer-Verlag, Berlin.Lardner, R.W. [1974]: Mathematical Theory of Dislocations and Fracture, Uni-

versity of Toronto Press, Toronto.London F. and London, H. [1935]: Proc. Roy. Soc. (London), A149, 71.London, F. [1950]: Superfluids, Vol. 1, Wiley, New York.Louat, N.P. [1965]: Proc. Int. Conf. on Fracture, Sendai, Japan, 117.Lawn, B.R. and Wilshaw, T.R. [1975]: Fracture of Brittle Solids, Cambridge

University Press, London.Maradudin, A.A., Montroll, E.W., Weiss, G.H., and Ipatova, I.P. [1971]: The-

ory of Lattice Dynamics in the Harmonic Approximations, 2nd ed., AcademicPress, New York.

McCay, B.M. and Narasimhan, M.N.L. [1981]: Theory of nonlocal electromag-netic fluids, Arch. Mech., 33, 3, 365–384.

Meyer, R.B. [1969]: Piezoelectric effects in liquid crystals, Phys. Rev. Lett., 22,18, 914–921.

Montfort, J.P. and Haziioannou, G. [1988]: Equilibruim and dynamic behaviorof thin-films in a perfluorinated polyether, J. Chem. Phys., 88, 11, 7187–7196.

Nabarro, F.R.N. [1967]: Theory of Crystal Dislocations, Clarendon Press, Ox-ford.

Narasimhan, M.N.L. and McCay, B.M. [1981]: Dispersion of surface waves innonlocal dielectric fluids, Arch. Mech., 33, 385–400.

Nowinski, J.L. [1990]: On a three-dimensional Kelvin problem for an elasticnonlocal medium, Acta Mechanica, 84, 77–87.

Oevel, P.J. and Schröter, J. [1981]: Balance equations for micromorphic mate-rials, J. Stat. Phys., 25, 4, 645–662.

Ohr, S.M. and Chang, S.J. [1982]: Direct observations of crack tip dislocationbehavior during tensile and cyclic deformation, Proc. Second Int. Symposiumon Defects, Fracture and Fatigue, Martinus Nijhoff, Amsterdam.

Ohr, S.M. and Chang, S.J. [1982]: Dislocation-free zone model of fracture com-parison with experiments, J. Appl. Phys., 53, 8, 5645–5651.

Pan, K.L. [1995a]: Interaction of a circular second-phase particle and a screwdislocation in nonlocal elasticity, Acta Mechanica, 111, 74–84.

Pan, K.L. [1995b]: Interaction of a dislocation with a surface crack in nonlocalelasticity, Int. J. of Fracture, 69, 307–318.

Pan, K.L. [1996a]: Interaction energy of dislocation and point defect in BCC iron,Radiation Effects and Defects in Solids, 139, 147–154.

Pan, K.L. [1996b]: Interaction of a dislocation and an inclusion in nonlocal elas-ticity, Int. J. Engng. Sci., 34, 14, 1675–1688.

Pippard, A.B. [1953]: Proc. Roy. Soc. London, A 216, 547.

364 References

Pippard, A.B. [1965]: The Dynamics of Conduction Electrons, Gordon and Breach,New York.

Povstenko, Yu.Z. [1995]: Circular dislocation loops in nonlocal elasticity, J.Phys. D. Appl. Phys., 28, 105–111.

Povstenko, Yu.Z. [1995a]: Straight disclinations in nonlocal elasticity, Int. J.Engng. Sci., 33, 4, 575–582.

Rice, J. and Thomson, R. [1974]: Ductile versus brittle behaviour in crystals, Phil.Mag., 29, 73–98.

Shannon, C.E. [1949]: Communication in the presence of noise, Proc. I.R.E., 37,1, 10–21.

Sneddon, I.N. [1951]: Fourier Transforms, McGraw-Hill, New York.Spencer, A.J.M. [1971]: Theory of invariants, in Continuum Physics, Vol. I, Part

III (Eringen, A.C., ed.), Academic Press, New York.Speziale, C. and Eringen, A.C. [1981]: Nonlocal fluid mechanics description of

wall turbulence, Comput. Math. Appl., 7, 27–42.Teodosiu, C. [1982]: Elastic Models of Crystal Defects, Springer-Verlag, Berlin.Tricomi, F.G. [1951]: On the finite Hilbert transformation, Quart. J. Math. Oxford,

2, 199.van Bentum, P.J.M., van Kempen, H., van de Leemput, Perenboom, J.A.A.J.,

Schreurs, L.W.M. and Teunissen [1987]: High field measurements on thehigh-Tc superconductors, Phys. Rev. B., 36 (16), 8329.

Vörös, G. and Kovács, I. [1995]: Elastic interaction between point defects anddislocations in quasi-continuum, Phil. Mag. A, 72, 4, 949–961.

Wright, J.D. [1995]: Molecular Crystals, 2nd ed., Cambridge University Press,New York.

Zhou, Shu-Ang [1999]: Electrodynamics of Solids and Microwave Superconduc-tivity, Wiley, New York.

Index

Acceleration field, 262Admissibility, Axiom of, 37Admissible thermoelastic state, 91Airy’s stress function, 126Alfven velocity, 221Alfven waves, 220–221Ampère’s law, 50Amphiphilic molecules, 338, 339Angular momentum, balance of, 56Anomalous skin effect, 209Antiplane strain, 127–128Approximate models in nonlocal lin-

ear elasticity, 98–105Atomic models, matching dispersion

curves with, 98–100Attenuating neighborhood hypothe-

sis, 34–35, 178, 197, 233Axial vector, 6

Balanceof angular momentum, 56of energy, 15–24, 56of moment of momentum, 22,

274of momentum, 22, 56, 274

of momentum moments, 22, 274Balance laws

electromagnetic, 49–53of liquid crystals, 340local, 22mechanical, 56–58, 347of microcontinuum mechanics,

272of micromorphic continua,

271–274of micropolar continua,

275–276of microstretch continua,

274–275nonlocal, see Nonlocal balance

lawsof nonlocal linear electromag-

netic theory, 194for thermomechanics, 18–21

Body loads, 15–16Bold brackets, 17Born–Kármán lattice model, 98–99Bravais lattice, 79Burger’s vector, 106

366 Index

Cartesian base vectors, 2Cartesian coordinates, 1Cauchy data, 84Cauchy deformation tensor, 5Cauchy–Green deformation tensors,

right and left, 6Causality, Axiom of, 32–33Channel flow in nonlocal fluid dy-

namics, 181–184Characteristic determinant, 7Characteristic lengths, viii

internal, 77Chiral nematics, 338Cholesterics, 338Christoffel symbol, 352Classical field theories, vClassical stress field, 144Cohesive distance, 22Cohesive shear stress, 141Cohesive zone, 34Compatibility conditions, 12–13

in nonlocal microcontinua,260–261

Concentrated force, nonlocal elastichalf-plane under, 166–171

Conservationof magnetic flux, 50of mass, 22, 56, 265, 271of microinertia, 265, 271of microstretch inertia, 274

Constitutive axioms, 31–38Constitutive-dependent variables, 33Constitutive equations, vii, 31–47

for electromagnetic solid media,199–200

of isotropic solids, 85–86linear, 73–77of memory-dependent nonlocal

electromagnetic elastic solids,62–66

of memory-dependent nonlocalelectromagnetic thermovis-cous fluids, 66–70

of memory-dependent nonlocalmicromorphic elastic solids,278–285

of memory-dependent nonlocalmicropolar elastic solids,293–296

of memory-dependent nonlocalmicropolar electromagneticelastic solids, 325–334

of memory-dependent nonlocalmicrostretch elastic solids,286–293

of memory-dependent nonlocalthermoelastic solids, 38–43

of memory-dependent nonlocalthermofluids, 43–47

of nematic liquid crystals, 343–347

of nonlocal electromagnetic liq-uid crystals, 340–343

of nonlocal fluid dynamics,178–179

of nonlocal linear thermo-micro-polar elasticity, 302

of nonlocal micropolar elastic-ity, 296–297

of rigid media, 242–243Constitutive residual, 27Continuity equation for incompress-

ible fluids, 185–186Continuity requirements, 88Continuous distribution of disloca-

tions, 123–132Continuous media, theory of, 2Continuum field theories, nonlocal,

see Nonlocal continuumfield theories

Continuum mechanics, vConvolution, 86

nonlocal micropolar elasticityformulation by means of,301

Convolution product, 93Cosserat elasticity, 257n

Index 367

Couple, 53–57at discontinuity surface, 55

Couple stress tensor, 269, 270Covariant derivatives, 353Cracks, interaction of dislocations with,

144–153

Debye continuum, 81Debye frequency, 99Debye screening, 204Defects

interaction between dislocationsand, 153–161

interaction energy between, 156–160

kinematical tensor of, 154Deformation gradients, 2Deformation-rate tensors, 9, 25, 263–

264Deformation tensor, 258Determinism, Axiom of, 33Dielectric tensor, 200

models for, 200–204Difference histories, 41, 282–283Dilatation center, 159Dipoles, force, 153Director theories, viiDisclination, 161

straight wedge, 161–163Discontinuity surface

couple at, 55electromagnetic force at, 55power at, 55small, 155

Dislocation density, true, 123Dislocation free zone, 108, 111Dislocations, 153

continuous distribution of, 123–132

interaction of with cracks, 144–153

with defects, 153–161stress fields for special distribu-

tions of, 128–132

Dispersion curves, matching,with atomic models, 98–100

Displacement potentials in nonlocalmicrocontinua, 320–321

Displacement vector, 73Dissipation, postulate of, 42Dissipation function density, 195Dissipation inequality, 40Dissipation potential, 26–29, 41, 46,

280

E-M, see Electromagnetic entriesEddy currents, 209Edge dislocation, 112–116, see also

Dislocations along line seg-ment, 128

Eigenvalues, 7Elastic distortion, 123Elastic poles, 155Elastic-solid dielectric tensor model,

201–203Elasticity, nonlocal, ixElectromagnetic (E-M) balance laws,

49–53Electromagnetic fields, absence of,

242Electromagnetic force, 53–57

at discontinuity surface, 55Electromagnetic solid media,

constitutive equations for,199–200

Electromagnetic theory, vnonlocal, 49–58

Energy, 23Energy balance, 15–24, 56

principle of, 267Energy balance law, v, 15–24Equipresence, Axiom of, 33Eulerian strain tensor, 6Exponential-integral function, 162Extrinsic body loads, 15–16

Fading Memory, Axiom of, 36–37Fading-memory hypothesis, 36, 233

368 Index

Faraday’s law, 50Field equations

of memory-dependent nonlocalelectromagnetic elastic solids,243–245, 252

of memory-dependent nonlocalelectromagnetic thermovis-cous fluids, 234–235

of memory-dependent nonlocalmicropolar electromagneticelastic solids, 335–336

of nematic liquid crystals, 347–349

of nonlocal fluid dynamics, 179–181

of nonlocal linear elasticity, 82–87

of nonlocal micropolar elastic-ity, 297–300

Finger deformation tensor, 5Finite microrotation tensor, 257Force dipoles, 153Fracture criterion, 129Free charge density, 54Function space, 38Fundamental solution, 165–166Fundamental solutions in nonlocal mi-

crocontinua, 323–324

Galerkin representation, 171generalization of, 322

Gauss’ law, 50Gradient theories, viiGreen deformation tensor, 5Green function

of linear differential operator, 103–105

nonlocal, 126Green–Gauss theorem, 17Griffith crack, 132

hoop stress near tip of, 136, 137nonlocal stress field at, 132–137

Gyrotropic media, 210–212

Hall current, 28

Heat input, 23Helmholtz free energy, 25Hookean stress, 100Hookean stress components, 135Hooke’s law, 81Hoop stress near tip of Griffith crack,

136, 137Hydrodynamic lubrication problem,

184

Ideal fluids, 47Incompatibility tensor, 124Incompressible fluids, 181

continuity equation for, 185–186Inertia in nonlocal microcontinua,

264–266Influence function, 34Interaction energy between defects,

156–160Interatomic attractions, viiiInternal characteristic length, 77Internal energy density, 16Inverse microdeformation tensor, 256Inverse motion, 2Isotropic media, material moduli for,

227Isotropic micropolar media, 295–296Isotropic microstretch media, 291–

293Isotropic solids, 75–77

constitutive equations of, 85–86material moduli for, 285

Jump conditions, 23, 194Jump discontinuities, 55

Kelvin problem, 165Kelvin–Voigt model, 227–228Kernel function, 76–77Kinematical tensor of defects, 154Kinematically admissible states, 91,

311Kinematics, 254–261Kinetic energy, in nonlocal micro-

continua, 264–266

Index 369

Kinetic energy density per unit mass,266

LA (longitudinal acoustic branch), 319Lagrange’s equation, 79Lagrangian strain tensor, 6Laplacian of tensors, 354–356Lattice dynamical foundations of lin-

ear elasticity, 78–82Left stretch tensor, 6Limited nonlocality, viiLine crack subject to shear, 138–143Line distribution, 127Linear chains, 100–102Linear constitutive equations, 73–77

of memory-dependent nonlocalelectromagnetic elastic solids,237–243

of memory-dependent nonlocalelectromagnetic thermovis-cous fluids, 231–234

of memory-dependent nonlocalmicrostretch elastic solids,287–291

of memory-dependent nonlocalthermoelastic solids, 223–228

of micromorphic elastic solids,281–285

of nonlocal linear electromag-netic theory, 195–198

Linear differential operator, Green func-tion of, 103–105

Linear function space, 91Liquid crystals

balance laws of, 340description of, 337–340nematic, see Nematic liquid crys-

talsnonlocal continuum theory of,

337–349polymeric, 338

LO (longitudinal optic branch), 319Local balance laws, 22Local media, 242

with memory, 242Local theory of superconductivity,

218–219Localization, 21London depth, 212London equation

first, 212second, 213

London gauge, 213Longitudinal acoustic branch (LA),

319Longitudinal optic branch (LO), 319Lubricant film flow on rotating disk,

189–192Lubrication, in microscopic channels,

184–189Lubrication problem, hydrodynamic,

184

Macromotion, 254Magnetic flux, conservation of, 50Magnetic vector potential, 219Magnetization vector, 54Magnetoelectric effect, 239

nonlocal, 197Magnetohydrodynamic (MHD)

waves, 220–221Mass

conservation of, 22, 56, 265, 271in nonlocal microcontinua,

264–266Mass density, 43Mass density residuals, 18Material derivative, 8, 262Material frame-indifferent quantities,

10Material Invariance, Axiom of,

33–34Material moduli, 75

for isotropic media, 227for isotropic solids, 285

Material particles, 254–255Material points of body, viiMaterial stability, 82Material tensors, 38

370 Index

Maxwell equations, 50, 336Mechanical balance laws, 56–58, 347Mechanical variables, independent,

32Media

with absorption dielectric ten-sor model, 203

gyrotropic, 210–212isotropic, see Isotropic media isotropic

micropolar, 295–296isotropic microstretch, 291–293local, see Local media

Meissner experiments, 212Memory

Axiom of, 35–36local media with, 242nonlocal elastic solids without,

42–43nonlocal electromagnetic fluids

without, 251–252nonlocal electromagnetic solids

without, 65–66nonlocal media without, 242nonlocal thermoviscous fluids with-

out, 69–70thermoviscous fluids without, 47

Memory-dependence, viiMemory-dependent nonlocal electro-

magnetic elastic solids, 237–245, 247–252

constitutive equations of, 62–66,247–252

field equations of, 243–245, 252linear constitutive equations of,

237–243Memory-dependent nonlocal electro-

magnetic thermoviscous flu-ids, 231–235

constitutive equations of, 66–70field equations of, 234–235linear constitutive equations of,

231–234Memory-dependent nonlocal micro-

morphic elastic solids, con-

stitutive equations of, 278–285

Memory-dependent nonlocal microp-olar elastic solids, consti-tutive equations of, 293–296

Memory-dependent nonlocal microp-olar electromagnetic elas-tic solids, 325–336

constitutive equations of, 325–334

field equations of, 335–336Memory-dependent nonlocal

microstretch elastic solidsconstitutiveequations of, 286–293

linear constitutive equations of,287–291

Memory-dependent nonlocal Peltiereffect, 251

Memory-dependent nonlocal Seebeckeffect, 251

Memory-dependent nonlocal thermoe-lastic solids, 223–229

boundary-initial value problemsof, 228–229

constitutive equations of, 38–43linear constitutive equations of,

223–228Memory-dependent nonlocal thermo-

fluids, constitutive equationsof, 43–47

Memory functionals, 29MHD (magnetohydrodynamic) waves,

220–221Microcontinuum mechanics, balance

law of, 272Microdeformation tensor, 256, 258Microelements, 201Microgyration tensor, 262Microinertia, conservation of, 265,

271Microinertia tensors, 265Micromorphic continua, 255–256

Index 371

balance laws of, 271–274strain measures of, 258–259

Micromorphic elastic solids, linearconstitutive equations of, 281–285

Micromotion, 254Micropolar continua, 257

balance laws of, 275–276strain measures of, 260

Micropolar media, isotropic, 295–296Micropolar moduli, nonlocal, 311–

316Microscopic channels, lubrication in,

184–189Microstress moments, 267Microstress tensor, 268Microstretch continua, 257

balance laws of, 274–275strain measures of, 259–260

Microstretch inertia, conservation of,274

Microstretch media, isotropic, 291–293

Microstretch rotary inertia, 266Microstretch scaler inertia, 266Mixed boundary-initial value prob-

lem, general, 235Moment of momentum, balance of,

22, 274Momentum, balance of, 22, 56, 274Momentum moments, balance of, 22,

274Motion, 1–5

inverse, 2

Navier equations, 100Neighborhood, Axiom of, 34–35Nematic liquid crystals, 337–338

constitutive equations of, 343–347

field equations of, 347–349Nondimensional shear stress, 121–

122Nonlocal balance laws, reduction of,

23–24

Nonlocal continuum field theories, vdefined, viilattice dynamical foundations of,

78–82literature on, ix–x

Nonlocal continuum theory of liquidcrystals, 337–349

Nonlocal elastic half-plane under con-centrated force, 166–171

Nonlocal elastic half-space, rigidstamp on, 171–175

Nonlocal elastic solids, 38without memory, 42–43

Nonlocal elasticity, ixNonlocal electromagnetic fluids with-

out memory, 251–252Nonlocal electromagnetic liquid crys-

tals, constitutive equationsof, 340–343

Nonlocal electromagnetic solids, with-out memory, 65–66

Nonlocal electromagnetic theory, 49–58

Nonlocal fluid dynamics, 177–192channel flow in, 181–184constitutive equations of, 178–

179field equations of, 179–181

Nonlocal Green function, 126Nonlocal hexagonal elastic solids,

screw dislocation in, 116–123

Nonlocal linear elasticity, 71–175approximate models in, 98–105field equations of, 82–87uniqueness theorem of, 87–91

Nonlocal linear electromagnetic the-ory, 193–221

balance laws of, 194linear constitutive equations of,

195–198point charge in, 204

Nonlocal linear thermo-micropolarelasticity, 301–305

372 Index

constitutive equations of, 302uniqueness theorem for, 305

Nonlocal magnetoelectric effect, 197Nonlocal media with no memory, 242Nonlocal microcontinua, 253–324

compatibility conditions in, 260–261

displacement potentials in, 320–321

fundamental solutions in, 323–324

inertia in, 264–266kinetic energy in, 264–266mass in, 264–266propagation of plane waves in,

316–320reciprocal theorem for,

306–308variational principles for, 308–

311Nonlocal micropolar elasticity, 296–

301boundary conditions, 299constitutive equations of, 296–

297field equations of, 297–300formulation by means of convo-

lution, 301initial conditions, 299–300

Nonlocal micropolar moduli, 311–316

Nonlocal Peltier effect, 197memory-dependent, 251

Nonlocal pyroelectricity, 197Nonlocal residuals, 271

nature of, 21–22Nonlocal Seebeck effect, 197

memory-dependent, 251Nonlocal solid media with absorp-

tion dielectric tensor model,204

Nonlocal stress field at Griffith crack,132–137

Nonlocal theory of superconductiv-ity, 214–220

Nonlocal thermoviscous fluids with-out memory, 69–70

Nonlocality, vconcept of, viiilimited, vii

Nonsimple materials of gradient type,34

Normed distance, 36

Objective tensors, 10–12Objectivity, 11

Axiom of, 33Onsager postulate, 28, 224Onsager reciprocity relations, 27Optical activity, 211Optical waves, 205–206Oscillator model, 201

Peach–Koehler formula, 126Peltier effect, nonlocal, see Nonlocal

Peltier effectPerfect body, 24Perfect continuum, 24Permutation symbols, 3Phospholipids, 338Piezoelectric effect, 239Piezoelectricity, Voigt’s, 238Piezomagnetic effect, 239Piola deformation tensor, 5Pippard’s theory of superconductiv-

ity, 213, 219–220Plane strain, 126–127Plane waves, propagation of, in non-

local microcontinua, 316–320

Point charge in nonlocal linear elec-tromagnetic theory, 204

Point defects, 153–156, see also De-fects

Poiseuille flow profile, 182Polaritons, 206–209Polarization vector, 54Poles, elastic, 155

Index 373

Polymeric liquid crystals, 338Power, 53–57

at discontinuity surface, 55Power and energy theorem, 92–93Poynting vectors, 54Pyroelectricity, nonlocal, 197Pyromagnetic effect, 197

Quasi-continuum, defined, 155–156Quasi-continuum representation, 79

Reciprocal theorem, 93–94for nonlocal microcontinua, 306–

308References, 357–364Response functionals, viiResponse objects, viiRiemann–Christoffel curvature ten-

sor, derivation of, 351–354Right stretch tensor, 6Rigid body motions, restrictions for,

37–38Rigid body susceptibility, 206Rigid media, constitutive equations

of, 242–243Rigid stamp on nonlocal elastic half-

space, 171–175Rotating disk, lubricant film flow on,

189–192

Screw dislocation(s), 106–112, seealso Dislocations

along line segments, 129distribution of, 109–110in half-plane, 110–111in nonlocal hexagonal elastic solids,

116–123uniform distribution of, along

circle, 130–132Second law of thermodynamics, 24–

26, 57–58, 276–278Second sound, 28Seebeck effect, nonlocal, see Nonlo-

cal Seebeck effectShannon’s theorem, 80

Shear, line crack subject to, 138–143Shear-plane model, 188Shear stress

cohesive, 141nondimensional, 121–122

Shifters, 257Simple lattice, 79Slowly varying fields, 98Smectic liquid crystals, 337–338Smooth Memory, Axiom of, 36Smooth neighborhood hypothesis, 34Solution of the mixed problems, 91Somigliana-type representation,

164–165Spin-inertia per unit mass, 266Spin tensor, 10Spring-dashpot dielectric tensor

model, 200–201Stokes’ theorem, 17Straight-edge dislocation, 112–116,

see also DislocationsStraight wedge disclination, 161–163Strain energy density, 74, 281Strain invariants, 7Strain measures

of micromorphic continua, 258–259

of micropolar continua, 260of microstretch continua, 259–

260Stress, 15–29, 267Stress field(s)

classical, 144for special distributions of dis-

locations, 128–132Stress intensity factors, critical, 150Stress moment tensor, 268Stress-strain relations, 32Stress tensor, 23Stretch, 7Superconductivity, 212–213

local theory of, 218–219nonlocal theory of, 214–220

374 Index

Pippard’s theory of, 213, 219–220

Surface heat, 180Surface loads, 15–16Surface traction residual, 180Symmetric function, 39

TA (transverse acoustic branch), 319Temperature change, 73Tensors

Laplacian of, 354–356objective, 10–12time-rate of, 8–10, 261–264

Thermodynamic equilibrium, defined,25–26

Thermodynamic flux, 26, 41, 62, 277Thermodynamic force, 26, 41, 62,

277Thermodynamic pressure, 342Thermodynamics, second law of, 24–

26, 57–58, 276–278Thermomechanics, balance laws for,

18–21Thermostatic equilibrium, 277Thermostatic flux, 62Thermostatic force, 62Thermoviscous fluids without mem-

ory, 47

Time-rate of tensors, 8–10, 261–264Time-symmetric terms, 196Titchmarch’s theorem, 86TO (transverse optic branch), 319Transport theorems, 17Transverse acoustic branch (TA), 319Transverse optic branch (TO), 319True dislocation density, 123Twist elasticity, 340

Uniqueness theoremfor nonlocal linear elasticity,

87–91for nonlocal linear thermo-

micropolar elasticity, 305

Variational principles, 95–98for nonlocal microcontinua,

308–311Voigt’s piezoelectricity, 238Volterra dislocation, 115Volume defects, 153, see also

DefectsVolume loads, 15–16Vorticity vector, 10

Wryness tensor, 258

Errata 375

Errata for “Microcontinuum Field Theories I:Foundations and Solids"

Page Location Misprint Correction

xiii line 17 Axisymmetric Antisymmetric

5 Eq. (1.2.4) XK = Xk(x, t) XK = XK(x, t)

10 last sentence In order , in order

16 Eq. (1.5.23) ckl = jckl ckl = jckl

21 Eq. (1.7.6) · · · + εlmncpmγnq · · · + εlmncpmγnq

22 Definition 1. f (x, ", t) f (X, ", t)

26 Eq. (1.8.29) 3ν ≡ ν ≡27 Eq. (1.8.33) CKL = 6j2νδKL CKL = 6j2νδKL

27 Eq. (1.8.33)4 CKL = 6j2νδKL CKL = 2j2νδKL

27 Eq. (1.8.34)2 ckl = νδkl ckl = 13 δkl

28 Eq. (1.8.43) φlk = φl =29 Eq. (1.9.3) Qlm(t) Qln(t)

43 line 16 tklm mklm

49 Eq. (2.3.1)∫∂V−σ

ρhθ

∫∂V−σ

ρhθdv

53 Line after Eq. (2.4.9) U(Y, 0) U(Y, ω)

55 Eq. (2.4.18-a) �[Y (t − x); · · · ] �[Y (t − s); · · · ]55 Eq. (2.4.18-b) P [Y (t − x); · · · ] P [Y (t − s); · · · ]67 Eqs. (3.3.14) & (3.3.15) 3ρj ρj

86 Eq. (4.1.13-a) (∇E) · B (∇E) · P

86 Eq. (4.1.13-b) WE = FE · v + ρE · · · WE = ρE · · ·94 Eq. (4.4.3) {· · · ,EK(t), BK(t)} {· · · , EK(t), BK(t)}94 Eq. (4.4.5) of RPk Ek EK94 Eq. (4.4.5) of RMk Bk BK

94–96 Eqs. (4.4.6) & (4.4.13) of DPk EK EK94–96 Eqs. (4.4.6) & (4.4.13) of DMk BK BK

94–96 In Section 4.4) � has a new definition namely:

� = ε − θη + ρ−1MkBk − ρ−10 MkBk − ρ−1

0 �KEK97, 98 Eqs. (4.4.5) & (4.5.11) of DPk Ek Ek97, 98 Eqs. (4.4.5) & (4.5.11) of DMk Bk Bk

376 Errata

Page Location Misprint Correction

108 last line Mαβ �= Mαβ Mαβ �= Mβα

136 Theorem 2 {U, φ, T } {u, φ, T }209 Eq. (5.21.27) (O

√c − r) O(

√c − r)

235 Eq. (5.27.2) un(x1, x1, t) un(x1, x2, t)

250 Eq. (6.1.1) j = 1 + 3φ + · j = 1 + φ + ·250 Eq. (6.1.2) �K � 3γkδkK �K = γkδkK

250 Eq. (6.1.3) γk = 3φ,k γk = φ,k

250 Eq. (6.1.3) e = 3φ e = φ

253 Eq. (6.1.21) mkl = αφr,r + · · · mkl = αφr,r δkl + · · ·273 Eq. (7.1.15) �T = · · · − ρC0T

2

2T0�T = · · · − ρC0

T0T 2

289 item (e) λTEkl

λT Ek

289 item (e) λTBkl

λT Bk

Errata for “Microcontinuum Field Theories II:Foundations and Solids"

Page Location Misprint Correction

19 Line 6 Lukaszewicz Łukaszewicz

95 Line 7 Megneto- Magneto-

289 Eqs. (17.7.18) ...− p − 12 (a1 − a2)ν

′′1 = 0 . . .− p + 1

2 (a1 − a2)ν′′1 = 0

290 Eq. (17.7.23) . . . (λ2 − a23h

2), . . . (λ2 − a23h

2) = 0,

291 Eq. (17.7.26) f (y) = 1µν+κν [(σν − κν)Q1 + . . . . . . f (y) = 1

λν+µν [(−µν + κν + (σν

− κν)Q1 + σνQ2)(y2 − 1)+ . . .

291 Eq. (17.7.30) i22 = . . . i12 = . . .

319 [6] Lukaszewicz Łukaszewicz

326 [95] Lukaszewicz Łukaszewicz