SOLUTIONS

Chap. I. Sec. 4

1. G(t) = Gge-t/TH(t). Gg = I/Jg• T = Jl'. Fluid since J'(CXl»O.

2. J(t) = Je[l - exp(-t/T')]H(t). J e = I/Ge• T' = TI/Ge• Solidsince Je < OIl.

3. If the relaxation modulus of the spring is G1H(t) and thatof the Maxwell element is G2exp(-t/nH(t). then

The compliance is

J(t) = [Je - (Je - Jg)exp(-t/T')]H(t).

where J e = I/G1• J g = 1/(G1+G2). T l = T + (G2T/G1).

Chap. I. Sec. 7

2. (J,.)n+l = (l/T')e-t/T ' (t/T,)n/n!

OIl / ' OIlr (-G J,.)n = -(G /T')e-t T r (l/n!)(-G t/T,)n .n=l g g n=O g

Solutions

With Gg = I, the series converges to e-t/T '. Then

G(t) = (1/2)[1 + exp(-2t/T')]H{t).

5. Upper bound gives I/l(t) 4; average of G on (O,t).

Chap. II, Sec. 3

175

5. Re s > 0; Re s = O. Singular at s = O. Pole if p is aninteger, branch point otherwise.

Chapter II, Sec. 4

2(e). arctan(c/s).

Chap. II, Sec. 5

2. Yes. Same solution.

4. (1/2)! = ifi/2 (so (-1/2)! = iii).

Chap. III, Sec. 2

1. 170 = (Gg - Ge)T. a = Je = I/Ge. b = Je - J g, J g = l/Gg.

T' = T + 170/Ge = GgT/Ge.

2. Let 17' and 17" be approximations from modulus andcompliance, respectively. Then

17 I = Got~-Pp/{1 - p), 17" = l(/t~-Pp/{1 + p),

T = to/2(2 - p){l - p), T' = to{l + p)/2(2 + p).

With loGo = I/p!(-p)!, then 17' /17" = 1 + O(p2) and

(T + 17' /Ge)/T' = 1 + 4p + O(p2), not so good.

176

Chap. III, Sec. 3

(a) Solid. J(t) = (2/1I)"T.

Solutions

(b) Fluid. J(t) - (2/1I)"T for small t. Since 7} = iii and T =1/2, J(t) - (t + 1/2)/ifi for large t.

(c) Solid. G(t) = 5(t) + g(t) where g(O) = I and get) ­I /log(1+ I ) for large t.

(d) Fluid. G(O) = I. For t large, G - 1/4t3/ 2•

Chap. III, Sec. 6

2. p(t) = t/(et - I). p(O) = I, p(<<» = O. Approximation shouldbe good for large t, bad for small t. G(t) = I + 5(t) exact.

G1(t) = 1/(1 - e-t),

G2(t) = (et/nt)sin[nt/(et - I)].

p(2) = 0.313. G(2) = I. G1(2) = 1.16. G2(2) = 0.98.

3. q(t) = 9t/(et + 9). q(O) = q(<<» = 0 but q(2.1) = 1.1(maximum). With q actually greater than 1, secondapproximation is worse than the first near t = 2.1.

J2 < J 1 < J so second approximation is always worse thanfirst.

Chap. III, Sec. 9

Let JI = J~.J~. Then J~+ (the restriction to t > 0) iscompletely monotone. Also

sJ - (sJ)1/2 and L[J' J- sJ - J1/22- 2+ - 2 g'

Then after some manipulation,

Solutions 177

Since -tJ2+ is non-negative, its transform is completelymonotone.

Chap. IV, Sec. la

1. The term F90G(t) is zero for t > 0 and the equation is notvalid at t =O.

Chap. IV, Sec. Ie

log sO = log G*(w) + p log(s/iw) + (p' /2)[log(s/iw)]2 + ...,p' = dp/d(1og w). Terms of order a:p' neglected, i.e.,p2p '. Power-like behavior used in supposing that p' issmall.

Chap. IV, Sec. 2

Chap. IV, Sec. 2f

1. The exponent is st - y(p/1'I )1/2/(S), /(s) = s(sJ1'Io)1/2.

Saddle point at /'(s) = (1'Io/p) 192t / y. For t/y large this issatisfied at small s, where

/(s) = {s[l + sT/2 + 0(s2)].

The saddle point is at So = (p/1'Io)(y/2t)2. Cubic term inexponent is negligible if {t/y)2 » 3pT/81'10' and in thatcase T drops out too.

Chap. IV, Additional Problems

1. See E.H. Lee and I. Kanter, J. Appl. Phys. 24, 1115-22(1953).

178

2. (a) u = U(y)exp(iwt),

U(y) = A sin(ky)/sin(kh), k = (pw2/G·)1/2.

U = Ay/h if hk « n/2, i.e. hw « c(w)n/2.

Solutions

(d) U ~ (A/2)exp[-(h - y)(W5/2c + iw/c»).Need frequency high enough that h5 » 2c/w.

3. See C.R. Norman and A.C. Pipkin, Trans. Soc. Rheo. ll,335-45 (1967).

5. See Pipkin, A.C., Phys. Fl. 7, 1143-6 (1964).

Chap. V, Sec. 3

Chap. V, Sec. 4

2. Thinner, more like an incompressible material.

3. M = K + 4G/3. N;; I/K if G « K. Assumingincompressibility would neglect everything of interest.

Chap. V, Sec. 7

1. On unloading after equilibrium, Gee = PLe - PLg'

Chap. VI, Sec. 2

2. With e- eo = a~, y = hy·, t = bt·, get a = I/Bp, b = Ch2/K,and

g2 = K-lh2Bp~wJ2[wa(eO»)'

3. gy = 2 exp(~0/2)argtanh[1 - exp(~ - ~0»)1/2.

Solutions

Chap. VI, Sec. 3

Chap. VI, Sec. 4

(hIR)2 ~ 12tlte for hlR small.

Chap. VII, Sec. 9

179

See B. Maxwell and R.P. Chartoff, Trans. Soc. Rheo. 9,41·52 (1965); T.N.C. Abbot and K. Walters, J. Fl. Mech. 40,205-13 (1970); and A.C. Pipkin and R.I. Tanner, Ann.Rev. Fl. Mech. 9, 13-32 (1977).

Chap. VIII, Sec. 2

wT = co: Finite elasticity with instantaneous modulus.wT = 0: Finite elasticity with equilibrium modulus.

Chap. VIII, Sec. 3

1. a = -yTGi 1 - e-t /T ).

Chap. VIII, Sec. 4

I; 1J.(t)(t212)dt.

Chap. VIII, Sec. 5

Viscometric flow.

Chap. VIII, Sec. 6

3. See Chap. IX, Sec. 6.

180

4. .Q. = -pI + TlA 1 - TlOTA2 + TlO(T + T·)A~ ,

n = no + 2(C2 + CS)12•

s. Need aT« 1 and bT « 1.

Al = a(L!. + {D + b({li + 10)-

42 = 2a2{ { + 2b2kk + ab(ik + liD.

4s = 3a2b({li + ~), ~ = 6a2b2kk, .1n = Q (n~5).

Solutions

Chap. VIII, Sec. 7

2. u = U(y)!(t), U(y) = sin(nny/H). Let F = p-l(nn/H)2. Then

/(s) = s/(s2 + FsG). See Chap. IV, Sec. 1.

Chap. VIII, Sec. 11

2. See Pipkin, A.C., Proc. 4th Int. Congo Rheo. 1, 213-22(1965).

4. See S. I. Chou and A. C. Pipkin, Acta Mech. 4, 164-9 (1967).

Chap. IX, Sec. 2

fGR/ 2

QGs =8n o2r(o)do. See M. Mooney, J. Rheo., 2, 210-22o

(1931).

Chap. IX, Sec. 6

INDEX

Abbot. T. N. C.• 179Anisotropic fluids. 128-130Anisotropic materials. 79.

117-119.128-130Arithmetic and geometric

mean. 109Asymptotic approximations

asymptotic scale invariance.44. 60

behavior of transforms atsmall arguments. 34-39.41. 44. 60

behavior of transforms atlarge arguments. 41. 44.72

saddle point integration.59. 65

series. 63. 136-138stationary phase. 69. 71Watson's lemma. 63wide saddle points. 65-66

Attenuation ofoscillations. 52-56pulses. 62. 70sinusoidal waves. 67-68

Beta function. 30Blasius' problem. 172Boltzmann principle (see

Superposition)Boundary layers

in high-frequencyoscillations. 74

in steady flows. 171-173Boundary value problems.

88-90for incompressible

materials. 95-96with synchronous

moduli. 96-97for slow flow. 145-156

Bulk compliance. 82. 112­114

relation to bulk modulus.82.84

Bulk longitudinal compli­ance and modulus. 87. 93

Bulk mod ul us. 81approximations. 85-87.

113

182

[Bulk modulus]comparison to shear

modulus, 85relation to bulk com­

pliance, 82, 84

Cantey, D. E., 106Carson transforms, 41Centripetal effects, 168-

171Channel flow

starting and stopping, 75oscillatory, 75spurious instability, 146­

147viscometric, 159-160

Chou, S. I., 180Circular flow, 168-171

kinematics, 161stress, 168

Compatibili ty conditions,96

Completely monotonefunctions, 47-50

Complex compliance,modulus, viscosity(see Dynamic compli­ance, modulus,viscosity)

Compliance, (see Bulk, Bulklongitudinal, Complex,Compressional, Dynamic,Equilibrium, Extensional,Glass, Instantaneous, Loss,Oscillatory, Shear, Storage,Tensile, Torsional,Volume)

Compressibility, 81-82, 113Compressional compliance and

modulus (see Bulk com­pliance and Bulkmodulus)

Index

Conductivity, 103Cone-and-plate viscometer,

170Conservation of

angular momentum, 80energy, 103momentum, 78

Consistency relations bet­ween constitutiveequations, 134, 135

elastic and linear visco­elastic, 134

linear and slow visco­elastic, 135, 141

slow viscoelastic andviscometric, 135, 155,168

Constitutive equations (seeStress-strain relations,Heat flux, Internalenergy

Convolutiondefinition, 15integral equations, 15, 29Laplace transform of,

28,29input-output relations,

57-58Cooling rod, 110-112Correspondence principle,

86,88Couette flow (see Circular

flow)Creep compliance (see

Compliance)Creep function, 8 (see also

Compliance)Creep integral, 13, 82Creep recovery, 14, 39, 75,

91Creeping flow (see Quasi­

static approximation,Stokes' flow)

Index

Dashpot,9Deformation gradient, 118

relative, 119Deviatoric stress and

strain, 81Diffusion

of heat, 102-106, 110of momentum, 60

Dimensional analysis, 53, 82,91, 104, 112, 153

Dirac delta, 5Dispersion, 70Dissipation of energy, 19, 102­

106, 128Dynamic compliance and

modulus, 16-21approximate relation to

modulus, 45low-frequency behavior,

34, 37temperature dependence, 101

Dynamic viscosity, 18, 21, 127

Eirich, F. R., 1, 102Elasticity theory, 77, 112Elastic-viscoelastic corres-

pondence principle, 88-90Energy

conservation, 103, 110internal, 103

Energy storage and dissipa­tion, 19, 102

Equilibriumapproach to, 13, 78equation, 78, 95, 148

Equilibrium modulus andcompliance, 8-9

Ericksen, J. L., 130, 139, 143Euler's constant, 41Expansion, thermal, 112-114Extension, 87Extensional compliance and

modulus (see Bulk

183

longitudinal compli­ance and modulus)

Factorials, 25reflection formula, 30

Ferry, J. D., 1, 100, 102,113

Flow diagnosis, 132-136,139, 157

diagram, 133Fluids

definition, 38Fourier's law, 103Fourier series, 22Fourier transforms, 22, 23

application to wavepropagation, 68

Free surface, 170Frequency, characteristic,

133, 138, 157

Glass modulus and compli-ance, 8, 100

Glassy state, 99Glass transition, 99-101Graphical methods, 46, 56,

101-102Gurtin, ME., 1

Heat conductionflux, 103

Heaviside unit step func-tion, 5

Helical flow, 161Helicoidal flow, 162Huang, N. C., 106

Incompressibility, 85, 95-97,121, 129, 140-141

Indentation, 90-91Inequalities

for modulus and compli­ance, 15, 16

184

[Inequalities]for relaxation and retarda­

tion times, 37for time and material

time, 109Instantaneous modulus and

compliance (see Glassmodulus and compli­ance)

Integral equationsapproximate solution, 42-44convolution, 14-15Fourier, 23Volterra, 14-15, 109, 110

Internal energy, 103Inversion integral, 23, 24

applications, 53-56, 58-59,63-65, 68-71

Isotropic materials, 8, 80,112-113, 121, 140-141

Heration, 15, 16, 56

Kanter, 1, 177Kearsley, E., 156Kelvin model, 10Kolsky, H., 61, 65, 66Kovacs, A., 113Kreis. A., 66

Lame constants. 80Landel, R. F.• 100Langlois. W. E., 156Laplace transforms. 24-26

as response in exponentialshearing. 19

two-sided, 23formulas, 26-28

Leaderman, H., 102Lee, E. H., 86. 106. 114, 177Limit theorems

for fluids. 33-35, 38-39for solids. 36-37, 38-39for transforms, 34. 41. 44. 72

Index

Liquid crystals, 128-130Lodge, A. S.. 1Logarithmic decrement, 55Longitudinal compliance

and modulus (see Bulklongitudinal compli­ance and modulus)

Loss angle, 18relation to log-log slope,

45-46relation to logarithmic

decrement, 55-56Loss compliance, modulus.

and tangent, 18

Matrix exponentials. 125­127

polynomials, 140-141Maxwell-Chartoff rheo­

meter. 28. 179Maxwell model. 9

response functions, 21Models, 9-11. 21Modulus. 8 (see Bulk. Bulk

longitudinal, Complex.Compressional,Dynamic. Equilibrium.Extensional. Glass.Instantaneous. Lame,Loss. Oscillatory.Shear. Storage. Tensile.Torsional, Volume,Young's)

Momentsof compliance. 37of relaxation modulus,

34. 36, 137theorems. 34-39

Mooney, M., 160, 180Morland. L. W.• 106. 114Muki. R., 114

Navier-Stokes equation.135, 140-141

Index

Normal stresseseffects in boundary

layers, 171-173effects in circular flows,

168-170effects in tube flow,

153-156, 163-166pressure-hole error, 152­

155response functions, 158, 167

Norman, C. R., 178

Oldroyd, J. G., 145Orowan's rule, 39Orthogonal transformation,

117Oscillatory modulus and com­

pliance (see Dynamicmodulus and compliance)

Perturbationsordinary, 145-151singular, 65

Phase velocity, 67-68maximum, 70

Plane flowsslow viscoelastic flow,

151-153uniform velocity gradient,

125Poiseuille flow (see Tube

flow and Channel flow)Poisson's ratio, 83-85Power-law approximations,

54-56, 60, 67-72Precursor wave, 72Pressure-hole errors, 152-155Pressurization of tubes, 91-94Pulse propagation, 56-73

equation of motion, 58response function, 57, 60, 62longitudinal, 61velocity, 61, 66, 71

185

[Pulse propagation]universal pulse shape,

61, 67in slabs, 73-74

Punch problems, 90-91

Quasi-elastic approxima­tion, 89-90

in tension, 84Quasi-static deformations,

77-78

Rabinowitsch, B., 160Reciprocal relations bet­

weencomplex modulus and

compliance, 17equilibrium modulus and

compliance, 9glass modulus and com­

pliance, 9Laplace transforms, 20modulus and compli­

ance, 15power-law approxima­

tions, 32, 43time and frequency, 46

Recovery after deforma­tion, 13, 39, 75, 91

Reduced time (see Time,Material)

Relaxation of stress, 4-6(see Modulus, Stressrelaxation)

Relaxation spectrum, 47Relaxation time

mean, 6, 34, 35, 36, 141effect of temperature,

99-100, 106-109Resid ual stress, 108Response functions, 8, 57

(see also Modulus,Compliance,

186

[Response functions]Viscosity, Normalstress)

Retardation spectrum, 50Retardation time, mean, 37Rivlin, R. S., 139, 141, 142,

143, 156Rivlin-Ericksen tensors, 139,

143-145, 166-167Rogers, T. G., 114Rotation, 116, 117-119, 129­

130Rubbery state, 99-100Runaway, 102-106

Saddle point integration,59, 65

wide saddle points, 65-67Scale invariance, 39-41

approximate, 41-44, 55-56,60, 67-72

Scaling formula for trans­forms, 26

Schapery, R. A., 42, 106Schapery's rule, 42, 86, 90

Secondary flowcentripetal, 168-171in tubes, 156, 163-168

Second-order effects, 123,124, 143, 146

Shearamount of, 5characteristic amplitude,

133, 138, 153Shear axes, 158-162Shear compliance and modulus,

8, 81-82approximate relations bet­

ween, 33-46effect of temperature on,

99-102inequalities satisfied by,

15-16

Index

[Shear compliance]relation between, 14-15relation to dynamic

compliance andmodulus, 16-21

use in superpositionintegrals, 12-13

Shear rate, 122, 124, 149,158-162, 166

Shearing stress, 5Shifting theorems, 27Sinusoidal oscillation

in shear, 16-19with increasing ampli-

tude, 19-21in torsion, 52-56waves, 67-68of slab, 74of fluid in channel, 75,

146-147thermal effects, 102-106in steady flow, 126, 128

Skew rectilinear flow, 161Slip surfaces, 160-162Slow viscoelastic flow, 131­

156asymptotic approxima­

tions, 136-138boundary layers, 171-173constitutive equations,

138-142Solids, definition, 38Specific heat, 103Spectrum

relaxation, 47retardation, 50

Spring, 9Standard linear solid, 11Stationary phase, 69, 71Steady shearing

viscosity, 8, 34-35, 132,158

Sternberg, E., 1, 114

Index

Stokes' flow, 96, 149Storage compliance, 18Storage of energy, 19Storage modulus, 18Strain

deviatoric, 81finite, 118-Il9infini tesimal, 80relative, 119, 144

Stress analysis, 77-97Stress convection, 150-151Stress relaxation, 4-6

(see Modulus)function, 8integral, 12-13, 79moduli, 79-82

Stress-strain relationsanisotropic materials, 79,

119, 130isotropic materials, 80large rotations, 117-Il9multiple integral, 135slow flow, 138-142thermal stress, 112viscometric flow, 159, 167,

172Stress tensor, 78

deviator, 81Summation convention, 78Superposition, 11-13, 57, 79,

81, 88-89Synchronous moduli, 84

boundary value problems,96-97

Tanner, R I., 151, 173, 179Temperature, effect on

creep, 98equilibrium modulus, 98observation of modulus and

compliance, 98-102relaxa tion, 99-100volume, 112-114

187

Tensile compliance andmodulus, 13, 82-87

Tensile response in cooling,110-112

Thermal effects, 98-Il4Thermal expansion, 112-114Thermorheologically simple

behavior (see Time­temperature shift)

Tilted plates, 166Time

effect on observations,4-7, 61

material, 106relaxation, 6, 34, 35, 36,

141retardation, 37translation invariance, II

Time-temperature shift,98-102

with variable tempera-ture, 106-109

Tobolsky, A. V., 102Torsion, f ini te, Il6Torsional flow (see Heli-

coidal flow)Torsional modulus, 52, 82Torsional oscillations, 52-56Transform inversion

approximate, 41-42integrals, 23-24

Translation invariance, IITransversely isotropic

rnaterials, 128-130Tube flow

slow flow, 153-156viscometric, 160, 163-166,

168Tube under internal pres­

sure, 91-94Two-sided Laplace trans­

form, 23

188

Uniform velocity gradient,124-128

Universal pulse shape, 61, 67

Viscometric flow,B3, 157-173Viscosity, 7

apparent, 131, 158dynamic, 18, 21, 127extensional, 83integral of modulus, 34kinematic, 58limit of dynamic modulus,

34steady-shearing, 8, 34, 131,

158temperature-dependence,

99-101Voigt model, 10Volterra integral equations,

14-16, 111Volume relaxation

under pressure, 81after temperature change,

109, 112-114Volumetric response

after change in tempera­ture, 112-114

under pressure (see Bulkcompliance)

Walters, K., 128, 179Wave propagation (see Pulse

propagation, Sinusoidaloscillation)

Weissenberg, K., 160, 165Williams, M L., 100WLF equation, 100Woo, T. C., 114

Young's modulus (seeTensile modulus)

Index

Applied Mathematical Sciences

I. John: Partial Differential Equations, 4th ed.2. Sirovich: Techniques of Asymptotic Analysis.3. Hale: Theory of Functional Differential Equations, 2nd ed.4. Percus: Combinatorial Methods.5. von Mises/Friedrichs: Fluid Dynamics.6. Freiberger/Grenander: A Short Course in Computational Probability

and Statistics.7. Pipkin: Lectures on Viscoelasticity Theory.9. Friedrichs: Spectral Theory of Operators in Hilbert Space.

II. Wolovich: Linear Multivariable Systems.12. Berkovitz: Optimal Control Theory.13. Bluman/Cole: Similarity Methods for Differential Equations.14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions

and Almost Periodic Solutions.15. Braun: Differential Equations and Their Applications, 3rd ed.16. Lefschetz: Applications of Algebraic Topology.17. CollatzlWetterling: Optimization Problems.18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I.20. Driver: Ordinary and Delay Differential Equations.21. Courant/Friedrichs: Supersonic Flow and Shock Waves.22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method.23. Lamperti: Stochastic Processes: A Survey of the Mathematical

Theory.24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II.25. Davies: Integral Transforms and Their Applications, 2nd ed.26. Kushner/Clark: Stochastic Approximation Methods for Constrained

and Unconstrained Systems.27. de Boor: A Practical Guide to Splines.28. Keilson: Markov Chain Models-Rarity and Exponentiality.29. de Veubeke: A Course in Elasticity.30. Sniatycki: Geometric Quantization and Quantum Mechanics.31. Reid: Sturmian Theory for Ordinary Differential Equations.32. Meis/Markowitz: Numerical Solution of Partial Differential Equations.33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III.34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics.35. Carr: Applications of Centre Manifold Theory.36. Bengtsson/GhiIlKiillen: Dynamic Meterology: Data Assimilation

Methods.37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces.38. Lichtenberg/Lieberman: Regular and Stochastic Motion.

Applied Mathematical Sciences

39. Piccini/Stampacchia/Vidossich: Ordinary Differential Equations in R".40. Naylor/Sell: Linear Operator Theory in Engineering and Science.41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange

Attractors.42. Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems and

Bifurcations of Vector Fields.43. Ockendon/Tayler: Inviscid Fluid Flows.44. Pazy: Semigroups of Linear Operators and Applications to Partial

Differential Equations.45. Glashoff/Gustafson: Linear Optimization and Approximation: An

Introduction to the Theoretical Analysis and Numerical Treatment ofSemi-Infinite Programs.

46. Wilcox: Scattering Theory for Diffraction Gratings.47. Hale et al.: An Introduction to Infinite Dimensional Dynamical

Systems--Geometric Theory.48. Murray: Asymptotic Analysis.49. Ladyzhenskaya: The Boundary-Value Problems of Mathematical

Physics.50. Wilcox: Sound Propagation in Stratified Fluids.51. Golubitsky/Schaeffer: Bifurcation and Groups in Bifurcation Theory,

Vol. I.52. Chipot: Variational Inequalities and Flow in Porous Media.53. Majda: Compressible Fluid Flow and Systems of Conservation Laws

in Several Space Variables.54. Wasow: Linear Turning Point Theory.55. Yosida: Operational Calculus: A Theory of Hyperfunctions.56. Chang/Howes: Nonlinear Singular Perturbation Phenomena: Theory

and Applications.57. Reinhardt: Analysis of Approximation Methods for Differential and

Integral Equations.58. Dwoyer/HussainilVoigt (eds.): Theoretical Approaches to Turbulence.59. SanderslVerhulst: Averaging Methods in Nonlinear Dynamical

Systems.60. Ghil/Childress: Topics in Geophysical Fluid Dynamics: Atmospheric

Dynamics, Dynamo Theory and Climate Dynamics.61. Sattinger/Weaver: Lie Groups and Algebras with Applications to

Physics, Geometry, and Mechanics.