Reflection and refraction of harmonic SH-waves from the interface of two dissimilar media with microheterogeneity is studied. The effect of the microheterogeneity on the overall behavior of the media is taken into account by adding higher order displacement gradients in the stress-strain relationship. It is found that a harmonic wave reflects back with the same angle of the incident wave, like in a classical case. However, it is found that the direction of propagation of the refracted wave is dependent on the wave number. It is also shown that the critical angle for which the incident wave cannot be transmitted to the other half plane is dependent on the wave number.

References

1.
Ament
,
W. S.
, 1953, “
Sound Propagation in Gross Mixtures
,”
J. Acoust. Soc. Am.
,
25
, pp.
638
641
.
2.
Postma
,
G. W.
, 1955, “
Wave Propagation in a Stratified Medium
,”
Geophysics
,
20
(
4
), pp.
780
806
.
3.
Nemat-Nasser
,
A.
, and
Hori
,
M.
, 1999,
Micromechanics: Overall Properties of Heterogeneous Materials
,
Elsevier
,
New York.
4.
Ament
,
W. S.
, 1959, “
Wave Propagation in Suspensions
,” U.S. Naval Research Lab, Report No. 5307.
5.
Mal
,
A. K.
, and
Knopoff
,
L.
, 1967, “
Elastic Wave Velocities in Two-Component Systems
,”
J. Inst. Math. Appl.
,
3
, pp.
376
387
.
6.
Kuster
,
G. T.
, and
Toksöz
,
M. N.
, 1974, “
Velocity and Attenuation of Seismic Waves in Two-Phase Media: Part 1. Theoretical Formulations
,”
Geophysics
,
39
(
5
), pp.
587
606
.
7.
Biot
,
M. A.
, 1956, “
Theory of Propagation of Elastic Waves in Fluid Saturated Porous Solid I: Low Frequency Range
,”
J. Acoust. Soc. Am.
,
28
, pp.
168
178.
8.
Biot
,
M. A.
, 1956, “
Theory of Propagation of Elastic Waves in Fluid Saturated Porous Solid II: Higher Frequency Range
,”
J. Acoust. Soc. Am.
,
28
, pp
179
191
.
9.
Hill
,
R.
, 1952, “
The Elastic Behaviour of a Crystalline Aggregate
,”
Proc. Phys. Soc., London, Sect. A
,
65
, pp.
349
354
.
10.
Kröner
,
E.
, 1953, “
Das Fundamentalintegral Der Anisotropen Elastischen Differentialgleic-Hungen
,”
Z. Phys.
,
136
, pp.
402
410
.
11.
Hershey
,
A. V.
, 1954, “
The Elasticity of an Isotropic Aggregate of Anisotropic Cubic Crystals
,”
J. Appl. Mech.
,
21
, pp.
236
240
.
12.
Hashin
,
Z.
, 1964, “
Theory of Mechanical Behaviour of Heterogeneous Media
,”
Appl. Mech. Rev.
,
17
, pp.
1
9
.
13.
Budiansky
,
B.
, 1965, “
On the Elastic Moduli of Some Heterogeneous Materials
,”
J. Mech. Phys. Solids
,
13
, pp.
223
227
.
14.
Walpole
,
L. J.
, 1966, “
On Bounds for the Overall Elastic Moduli of Inhomogeneous Systems—I
,”
J. Mech. Phys. Solids
,
14
, pp.
151
162
.
15.
Willis
,
J. R.
, 1977, “
Bounds and Self-Consistent Estimates for the Overall Properties of Anisotropic Composites
,”
J. Mech. Phys. Solids
,
25
, pp.
185
202
.
16.
Christensen
,
R. M.
, 1979,
Mechanics of Composite Materials
,
Wiley-Interscience
,
New York.
17.
Nabarro
,
F. R. N.
, 1979,
Dislocations in Solids, Vol. 1: The Elasticity Theory
,
North-Holland
,
Amsterdam.
18.
Walpole
,
L. J.
, 1981, “
Elastic Behavior of Composite Materials: Theoretical Foundations
,”
Adv. Appl. Mech.
,
21
, pp.
169
242
.
19.
Willis
,
J. R.
, 1981, “
Variational and Related Methods for the Overall Properties of Composites
,”
Adv. Appl. Mech.
,
21
, pp.
1
78
.
20.
Bilby
,
B. A.
,
Miller
,
K. J.
, and
Willis
,
J. R.
, eds., 1985,
Fundamentals of Deformation and Fracture—Eshelby Memorial Symposium
,
Cambridge University Press
,
Cambridge
.
21.
Mura
,
T.
, 1987,
Micromechanics of Defects in Solids
, 2nd ed.,
Martinus Nijhoff Publishers
,
Dordrecht, The Netherlands
.
22.
Weng
,
G. J.
,
Taya
,
M.
, and
Abé
,
H.
, eds., 1990,
Micromechanics and Inhomogeneity—The T. Mura 65th Anniversary Volume
,
Springer-Verlag
,
New York.
23.
Beran
,
M. J.
, 1968,
Statistical Continuum Theories
,
Wiley-Interscience
,
New York.
24.
Beran
,
M. J.
, 1971, “
Application of Statistical Theories of Heterogeneous Materials
,”
Phys. Status Solidi A
,
6
, pp.
365
384
.
25.
Kröner
,
E.
, 1971,
Statistical Continuum Mechanics
,
Springer–Verlag
,
Berlin.
26.
Batchelor
,
G. K.
, 1974, “
Transport Properties of Two-Phase Materials With Random Structure
,”
Annu. Rev. Fluid Mech.
,
6
, pp.
227
255
.
27.
McCoy
,
J. J.
, 1981, “
Macroscopic Response of Continua With Random Microstructure
,”
Mechanics Today
, Vol.
6
,
S.
Nemat-Nasser
, ed.,
Pergamon
,
Oxford
, pp.
1
40
.
28.
Beran
,
M. J.
, 1974, “
Application of Statistical Theories for the Determination of Thermal, Electrical and Magnetic Properties of Heterogeneous Media
,”
Composite Materials
, Vol.
2
,
G. P.
Sedneckyj
, ed.,
Academic Press
,
New York
, pp.
209
249
.
29.
Beran
,
M. J.
, and
McCoy
,
J. J.
, 1970, “
Mean Field Variations in a Statistical Sample of Heterogeneous Linearly Elastic Solids
,”
Int. J. Solids Struct.
,
6
,
pp
1035–
1054
.
30.
Levin
,
V. M.
, 1971, “
The Relation Between Mathematical Expectations of Stress and Strain Tensors in Elastic Microheterogeneous Media
,”
Prikladnaya Matematikay Mekanika
,
35
, pp.
694
701
(English translation from Russian).
31.
Torquato
,
S.
, 1991, “
Random Heterogeneous Media: Microstructure and Improved Bounds on Effective Properties
,”
Appl. Mech. Rev.
,
42
, pp.
37
76
.
32.
Coleman
,
B.
, and
Gurtin
,
M.
, 1967, “
Thermodynamics With Internal State Variables
,”
J. Chem. Phys.
,
47
, pp.
597
613
.
33.
Rice
,
J. R.
, 1971, “
Inelastic Constitutive Relations for Solids: An Internal–Variable Theory and its Applications to Metal Plasticity
,”
J. Mech. Phys. Solids
,
19
, pp.
433
455
.
34.
Talreja
,
R.
, 1985, “
A Continuum Mechanics Characterization of Damage in Composite Materials
,”
Proc. R. Soc. London, Ser. A
,
399
, pp.
195
216
.
35.
Ju
,
J. W.
, ed., 1992,
Recent Advances in Damage Mechanics and Plasticity
, Vol.
132
,
ASME
,
New York
.
36.
Krajcinovic
,
D.
, 1989, “
Damage Mechanics
,”
Mech. Mater.
,
8
, pp.
117
197
.
37.
Krajcinovic
,
D.
, 1996,
Damage Mechanics
,
North-Holland
,
New York.
38.
Truesdell
,
C.
, and
Toupin
,
R. A.
, 1960, “
The Classical Field Theories
,”
Handbuch der Physik
,
Springer-Verlag
,
Berlin
.
39.
Bowen
,
R. W.
, 1976, “
Theory of Mixtures
,”
Continuum Physics
, Vol.
4
,
A. C.
Eringen
, ed.,
Academic Press
,
New York
.
40.
Bedford
,
A.
, and
Stern
,
M.
, 1972, “
Multi-Continuum Theory for Composite Elastic Materials
,”
Acta Mech.
,
14
, pp.
85
102
.
41.
Stern
,
M.
, and
Bedford
,
A.
, 1972, “
Wave Propagation in Elastic Laminates Using a Multi-Continuum Theory
,”
Acta Mech.
,
15
, pp.
22
38
.
42.
Bedford
,
A.
,
Sutherland
,
H. J.
, and
Linge
,
R.
, 1972, “
On Theoretical and Experimental Wave Propagation in a Fiber-Reinforced Elastic Material
,”
J. Appl. Mech.
,
39
(
2
), pp.
597
598
.
43.
Hegemier
,
G. A.
,
Gurtman
,
G. A.
, and
Nayfeh
,
A. H.
, 1973, “
A Continuum Mixture Theory of Wave Propagation in Laminated and Fiber-Reinforced Composites
,”
Int. J. Solids Struct.
,
9
, pp.
395
414
.
44.
Nayfeh
,
A. H.
, and
Gurtman
,
G. A.
, 1974, “
A Continuum Approach to the Propagation of Shear Waves in Laminated Wave Guides
,”
J. Appl. Mech.
,
41
(
1
), pp.
106
110
.
45.
McNiven
,
H. D.
, and
Mengi
,
Y., A.
, 1979, “
Mathematical Model for the Linear Dynamic Behavior of two Phase Periodic Materials
,”
Int. J. Solids Struct.
,
15
, pp.
271
280
.
46.
McNiven
,
H. D.
, and
Mengi
,
Y.
, 1979, “
A Mixture Theory for Elastic Laminated Composites
,”
Int. J. Solids Struct.
,
15
, pp.
281
302
.
47.
McNiven
,
H. D.
, and
Mengi
,
Y.
, 1979, “
Propagation of Transient Waves in Elastic Laminated Composites
,”
Int. J. Solids Struct.
,
15
, pp.
303
318
.
48.
Altan
,
B. S.
, and
Subhash
,
G.
, 2002, “
A Nonlocal Formulation Based on a Novel Averaging Scheme Applicable to Nanostructured Materials
,”
Mech. Mater.
,
35
, pp.
281
294
.
49.
Voigt
,
W.
, 1894,
Theoretische Studien ueber die Elastizitaetverhaeltnisse der Krystalle. Abh. Ges. Wiss. Gottingen
, Vol.
34
(
1887
); pp.
72
79
.
50.
Cosserat
,
E.
, and
Cosserat
,
F.
, 1909,
Theorie de Corps Deformables
,
A.
Hermann
, ed.,
Scientific Library A. Hermann and Sons
,
Paris
.
51.
Mindlin
,
R. D.
, and
Eshel
,
N. N.
, 1968, “
On First Strain-Gradient Theories in Linear Elasticity
,”
Int. J. Solids Struct.
,
1
, pp.
109
124
.
52.
Ru
,
C. Q.
, and
Aifantis
,
E. C.
, 1993, “
A Simple Approach to Solve Boundary-Value Problems in Gradient Elasticity
,”
Acta Mech.
,
10
, pp.
59
68
.
53.
Ting
,
T. C. T.
, 2005, “
The Polarization Vectors at the Interface and the Secular Equation for Stoneley Waves in Monoclinic Biomaterials
,”
Proc. R. Soc., London, Ser. A
,
461
, pp.
711
731
.
54.
Jerzak
,
W.
,
Siegmann
,
W. L.
, and
Collins
,
M. D.
, 2005, “
Modeling Rayleigh and Stoneley Waves and Other Interface and Boundary Effects With the Parabolic Equation
,”
J. Acoust. Soc. Am.
,
117
(
6
), pp.
3497
3503
.
You do not currently have access to this content.