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Journal of Applied Mechanics | Volume 74 | Issue 2 | TECHNICAL PAPERS
TECHNICAL PAPERS

Modeling Heterogeneous Media With Microstructures of Different Scales

[+] Author and Article Information
C. T. Sun

School of Aeronautics and Astronautics,  Purdue University, West Lafayette, IN 47907sun@ecn.purdue.edu

G. L. Huang

School of Aeronautics and Astronautics,  Purdue University, West Lafayette, IN 47907

J. Appl. Mech 74(2), 203-209 (Jan 24, 2006) (7 pages) doi:10.1115/1.2188536 History: Received July 02, 2005; Revised January 24, 2006
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Citing Articles

The objective of this paper is to extend the framework of the continuum theory so that it can capture the properties that are embedded in the microstructure or nanostructure and still keep its simplicity and efficiency. The model thus developed is capable of accounting for local deformation of microstructures in solids especially their micro- (local) inertia effect. The essence underlying this approach is the introduction of a set of bridging functions that relate the local deformation of microstructures to the macrokinematic variables. Once the solution of the macroscopically homogeneous continuum is obtained, the solutions in the microstructures are obtained through the use of these bridging functions. Propagation of waves of different wavelengths is considered and the dispersion curve is used to evaluate the accuracy of the model. The model is also employed to study wave reflection and transmission at the boundary of two media with microstructures of very different scales.
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Copyright © 2007 by American Society of Mechanical Engineers
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References

1
Cosserat, E., and Cosserat, F., 1909, "Theorie des Corps Deformables", A. Hermann & Fils, Paris.
2
Toupin, R. A., 1962, ”Elastic Materials With Couple-Stresses,” Arch. Ration. Mech. Anal.  [CrossRef], 11 , pp. 385–414.
3
Mindlin, R. D., 1964, “Micro-Structure in Linear Elasticity,” Arch. Ration. Mech. Anal.  [CrossRef], 16 , pp. 51–78.
4
Eringen, A. C., and Suhubi, E. S., 1964, “Nonlinear Theory of Micro-Elastic Solids,” Int. J. Eng. Sci.  [CrossRef], 2 , pp. 189–203.
5
Sun, C. T., Achenbach, J. D., and Herrmann, G., 1968, “Continuum Theory for a Laminated Medium,” ASME J. Appl. Mech., 35 , pp. 467–475.
6
Achenbach, J. D., Sun, C. T., and Herrmann, G., 1968, “On the Vibrations of a Laminated Body,” ASME J. Appl. Mech., 35 , pp. 689–696.
7
Chen, Y. P., and Lee, J. D., 2003, “Connecting Molecular Dynamics to Micromorphic Theory. (I). Instantaneous and Averaged Mechanical Variables,” Physica A  [CrossRef], 322 , pp. 359–376.
8
Kunin, I. A., 1982, "Elastic Media with Microstructure I, One Dimensional Models", Springer-Verlag, Berlin.
9
Pasternak, E., and Muhlhaus, H. B., 2005, “Generalised Homogenisation Procedures for Granular Materials,” J. Eng. Math.  [CrossRef], 52 , pp. 199–229.
10
Nesterenko, V. F., 2001, "Dynamics of Heterogeneous Materials", Springer-Verlag, New York.
11
Wang, Z. P., and Sun, C. T., 2002, “Modeling Micro-Inertia in Heterogeneous Materials under Dynamic Loading,” Wave Motion  [CrossRef], 36 , pp. 473–485.
12
Sinclair, J., 1971, “Improved Atomistic Model of a bcc Dislocation Core,” J. Appl. Phys.  [CrossRef], 42 , pp. 5321–5329.
13
Mullins, M., and Dokainish, M. A., 1982, “Simulation of the (0 0 1) Plane Crack in a-Iron Employing a New Boundary Scheme,” Philos. Mag. A, 46 , pp. 771–787.
14
Tadmor, E. B., Ortiz, M., and Phillips, R., 1996, “Quasicontinuum Analysis of Defects in Solids,” Philos. Mag. A, 73 , pp. 1529–1563.
15
Broughton, J. Q., Abraham, F. F., Bernstein, N., and Kaxiras, E., 1999, “Concurrent Coupling of Length Scales: Methodology and Application,” Phys. Rev. B  [CrossRef], 60 , pp. 2391–2403.
16
Rudd, R. E., and Broughton, J. Q., 2000, “Concurrent Coupling of Length Scales in Solid State Systems,” Phys. Status Solidi B  [CrossRef], 217 , pp. 251–291.
17
E, W. N., and Huang, Z. Y., 2001, “Matching Conditions in Atomistic-Continuum Modeling of Materials,” Phys. Rev. Lett.  [CrossRef], 87 , p. 135501.
18
Wagner, G. J., and Liu, W. K., 2003, “Coupling of Atomic and Continuum Simulations using a Bridging Scale Decomposition,” J. Comput. Phys.  [CrossRef], 190 , pp. 249–274.
19
Xiao, S. P., and Belytschko, T., 2004, “Coupling Methods for Continuum Model With Molecular Model,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 193 , pp. 1645–1669.
20
Irving, J. H., and Kirkwood, J. G., 1950, “The Statistical Mechanical Theory of Transport Processes,” J. Chem. Phys.  [CrossRef], 18 , pp. 817–829.
21
Lutsko, J. F., 1988, “Stress and Elastic Constants in Anisotropic Solids: Molecular Dynamic Techniques,” J. Appl. Phys.  [CrossRef], 64 , pp. 1152–1164.
22
Cormier, J., Rickman, J. M., and Delph, T. J., 2001, “Stress Calculation in Atomistic Simulations of Perfect and Imperfect Solids,” J. Appl. Phys.  [CrossRef], 89 , pp. 99–104.
23
Zhou, M., 2003, “A New Look at the Atomic Level Virial Stress: on Continuum-Molecular System Equivalence,” Proc. R. Soc. London, Ser. A  [CrossRef], 459 , pp. 2347–2392.

Figures

Grahic Jump Location
+A layered medium attached to an elastic medium
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A layered medium attached to an elastic medium
Figure 8

A layered medium attached to an elastic medium

Grahic Jump Location
+Ratio of stresses of the incident and transmitted waves at the interface of the layer medium∕homogeneous solid system
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Ratio of stresses of the incident and transmitted waves at the interface of the layer medium∕homogeneous solid system
Figure 9

Ratio of stresses of the incident and transmitted waves at the interface of the layer medium∕homogeneous solid system

Grahic Jump Location
+Effectiveness of wave transmission in layered medium∕representative continuum system
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Effectiveness of wave transmission in layered medium∕representative continuum system
Figure 10

Effectiveness of wave transmission in layered medium∕representative continuum system

Grahic Jump Location
+Representative unit cell of the lattice system
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Representative unit cell of the lattice system
Figure 3

Representative unit cell of the lattice system

Grahic Jump Location
+Comparison of the dispersive curves obtained with the lattice model and the microinertia continuum model
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Comparison of the dispersive curves obtained with the lattice model and the microinertia continuum model
Figure 4

Comparison of the dispersive curves obtained with the lattice model and the microinertia continuum model

Grahic Jump Location
+A lattice system connected to a homogeneous elastic solid
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A lattice system connected to a homogeneous elastic solid
Figure 5

A lattice system connected to a homogeneous elastic solid

Grahic Jump Location
+Ratio of stresses of the incident and transmitted waves at the interface in the lattice∕homogeneous solid system
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Ratio of stresses of the incident and transmitted waves at the interface in the lattice∕homogeneous solid system
Figure 6

Ratio of stresses of the incident and transmitted waves at the interface in the lattice∕homogeneous solid system

Grahic Jump Location
+Effectiveness of wave transmission in lattice∕representative continuum system
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Effectiveness of wave transmission in lattice∕representative continuum system
Figure 7

Effectiveness of wave transmission in lattice∕representative continuum system

Grahic Jump Location
+Representation of a representative volume element of a heterogeneous solid by a homogeneous solid
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Representation of a representative volume element of a heterogeneous solid by a homogeneous solid
Figure 1

Representation of a representative volume element of a heterogeneous solid by a homogeneous solid

Grahic Jump Location
+The lattice system
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The lattice system
Figure 2

The lattice system

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