appendixa - springer978-0-387-22643-9/1.pdf · appendixa here we present a brief account of the...
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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.
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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