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Journal articles on the topic 'Effective elastic constants'

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Author: Grafiati

Published: 30 May 2022

Last updated: 31 May 2022

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1

Duquennoy, Marc, Mohammadi Ouaftouh, Dany Devos, Frédéric Jenot, and Mohamed Ourak. "Effective elastic constants in acoustoelasticity." Applied Physics Letters 92, no. 24 (June 16, 2008): 244105. http://dx.doi.org/10.1063/1.2945882.

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2

Grimsditch, M. "Effective elastic constants of superlattices." Physical Review B 31, no. 10 (May 15, 1985): 6818–19. http://dx.doi.org/10.1103/physrevb.31.6818.

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3

Kim, Jin-Yeon. "Effective elastic constants of anisotropic multilayers." Mechanics Research Communications 28, no. 1 (January 2001): 97–101. http://dx.doi.org/10.1016/s0093-6413(01)00149-5.

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4

Bosher, S. H. B., and D. J. Dunstan. "Effective elastic constants in nonlinear elasticity." Journal of Applied Physics 97, no. 10 (May 15, 2005): 103505. http://dx.doi.org/10.1063/1.1894586.

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5

Bonilla, Luis L. "Effective elastic constants of polycrystalline aggregates." Journal of the Mechanics and Physics of Solids 33, no. 3 (January 1985): 227–40. http://dx.doi.org/10.1016/0022-5096(85)90013-4.

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6

Dunstan, D. J., S. H. B. Bosher, and J. R. Downes. "Effective thermodynamic elastic constants under finite deformation." Applied Physics Letters 80, no. 15 (April 15, 2002): 2672–74. http://dx.doi.org/10.1063/1.1469658.

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7

Akcakaya, E., and G. W. Farnell. "Effective elastic and piezoelectric constants of superlattices." Journal of Applied Physics 64, no. 9 (November 1988): 4469–73. http://dx.doi.org/10.1063/1.341270.

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8

Bulut, Osman, Necla Kadioglu, and Senol Ataoglu. "Absolute effective elastic constants of composite materials." Structural Engineering and Mechanics 57, no. 5 (March 10, 2016): 897–920. http://dx.doi.org/10.12989/sem.2016.57.5.897.

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9

Sun, Yi, Gao Ying Kang, Ding Cui, and Jing Ran Ge. "Study on Effective Elastic Constants of Homogenization Tube Sheet." Advanced Materials Research 430-432 (January 2012): 158–63. http://dx.doi.org/10.4028/www.scientific.net/amr.430-432.158.

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The expressions of the effective elastic constants of composite material with cylindrical inclusions are derived based on M-T method, and it can be used in discussing the approximate range of effective elastic constant of air. Moreover, it is possible to homogenize tube-sheet by making use of the expression. The numerical result obtained is in good agreement with effective elastic constant adopted by the ASME code. It demonstrates that the approach is effective and accurate. At the last, the relationship between effective elastic and thickness of the tube-sheet is discussed.

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10

Li, Bao Feng, Jian Zheng, Xin Hua Ni, Ying Chen Ma, and Jing Zhang. "Effective Elastic Constants of Fiber-Eutectics and Transformation Particles Composite Ceramic." Advanced Materials Research 177 (December 2010): 182–85. http://dx.doi.org/10.4028/www.scientific.net/amr.177.182.

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The composite ceramics is composed of fiber-eutectics, transformation particles and matrix particles. First, the recessive expression between the effective stress in fiber-eutectic and the flexibility increment tensor is obtained according to the four-phase model. Second, the analytical formula which contains elastic constant of the fiber-eutectic is obtained applying Taylor’s formula. The eutectic is transverse isotropy, so there are five elastic constants. Third, the effective elastic constants of composite ceramics are predicted. The result shows that the elastic modulus of composite ceramic is reduced with the increase of fibers fraction and fibers diameter.

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11

Sun, C. T., and Sijian Li. "Three-Dimensional Effective Elastic Constants for Thick Laminates." Journal of Composite Materials 22, no. 7 (July 1988): 629–39. http://dx.doi.org/10.1177/002199838802200703.

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12

Olson, Tamara, and Marco Avellaneda. "Effective dielectric and elastic constants of piezoelectric polycrystals." Journal of Applied Physics 71, no. 9 (May 1992): 4455–64. http://dx.doi.org/10.1063/1.350788.

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13

Reinert, Th, H. Kiewel, Hans Joachim Bunge, and L. Fritsche. "Calculation of Effective Elastic Constants for Polycrystalline Materials." Materials Science Forum 273-275 (February 1998): 617–24. http://dx.doi.org/10.4028/www.scientific.net/msf.273-275.617.

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14

Grimsditch, M., and F. Nizzoli. "Effective elastic constants of superlattices of any symmetry." Physical Review B 33, no. 8 (April 15, 1986): 5891–92. http://dx.doi.org/10.1103/physrevb.33.5891.

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15

Golovchan, V. T., and V. I. Kushch. "Elastic equilibrium and effective elastic constants of a cross-reinforced fiber composite." International Applied Mechanics 28, no. 1 (January 1992): 40–48. http://dx.doi.org/10.1007/bf00847328.

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16

Kim, Jin O., Jan D. Achenbach, Meenam Shinn, and Scott A. Barnett. "Effective Elastic Constants of Superlattice Films Measured by Line-Focus Acoustic Microscopy." Journal of Engineering Materials and Technology 117, no. 4 (October 1, 1995): 395–401. http://dx.doi.org/10.1115/1.2804732.

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The effective elastic constants of single-crystal nitride superlattice films have been determined by calculation and by measurement methods. The calculation method uses formulas to calculate the effective elastic constants of superlattices from the measured elastic constants of the constituent layers. The calculated effective elastic constants are tested by comparing the corresponding surface acoustic wave (SAW) velocities calculated for thin-film/substrate systems with the corresponding SAW velocities measured by line-focus acoustic microscopy (LFAM). The measurement method determines the effective elastic constants of the superlattices directly from the SAW velocity dispersion data measured by LFAM. Two kinds of superlattice films are considered: one has relatively flat and sharp interfaces between layers, and the other has rough interfaces with interdiffusion. The calculation method has yielded very good results for the superlattices with flat and sharp interfaces but not for the superlattices with rough interfaces. The measurement method yields results for both kinds, with the restriction that the constituent layers have similar crystal symmetries.

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17

O'Rourke, John P., Marc S. Ingber, and Martin W. Weiser. "The Effective Elastic Constants of Solids Containing Spherical Exclusions." Journal of Composite Materials 31, no. 9 (May 1997): 910–34. http://dx.doi.org/10.1177/002199839703100905.

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18

Wang, Wei, and Iwona Jasiuk. "Effective Elastic Constants of Particulate Composites with Inhomogeneous Interphases." Journal of Composite Materials 32, no. 15 (August 1998): 1391–424. http://dx.doi.org/10.1177/002199839803201503.

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19

Kiewel, H., L. Fritsche, and T. Reinert. "Calculation of nonlinear effective elastic constants of polycrystalline materials." Journal of Applied Physics 79, no. 8 (1996): 3963. http://dx.doi.org/10.1063/1.361823.

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20

Drygaś, Piotr, Simon Gluzman, Vladimir Mityushev, and Wojciech Nawalaniec. "Effective elastic constants of hexagonal array of soft fibers." Computational Materials Science 139 (November 2017): 395–405. http://dx.doi.org/10.1016/j.commatsci.2017.08.009.

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21

Fu, Yun Wei, Xie Quan Liu, Xin Hua Ni, and Hong Na Cao. "Elastic Constants of Eutectic Composite Ceramic Containing Parallel Lamellar Inclusion." Applied Mechanics and Materials 251 (December 2012): 285–88. http://dx.doi.org/10.4028/www.scientific.net/amm.251.285.

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The eutectic composite ceramic is composed of parallel lamellar inclusions distributed in the matrix. First, the recessive expression for the effective stress and the flexibility increment tensor of eutectic composite ceramic are obtained according to the four-phase model. Second, the analytical formula which contains elastic constant is given by applying Taylor’s formula. The eutectic composite ceramic is transverse isotropy, so there are five elastic constants. Third, the effective elastic constants of composite ceramics are predicted.

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22

Turik, A. V., L. A. Reznitchenko, A. I. Chernobabov, G. S. Radchenko, S. A. Turik, and M. G. Radchenko. "Elastic Constants Relaxation in Disordered Heterogeneous Systems." Solid State Phenomena 115 (August 2006): 215–20. http://dx.doi.org/10.4028/www.scientific.net/ssp.115.215.

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Using self-consistent effective-medium theory, we studied the complex elastic compliances of conducting disordered heterogeneous piezoelectric-polymer systems. The considered system is a random mixture of piezoelectric spheroids and polymer ones with the same orientation. The proximate cause of the effective elastic constants frequency dependencies was considered. The nature of the obtained spectra was analyzed.

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23

Liu, Lili, Cai Chen, Dingxing Liu, Zhengquan Hu, Gang Xu, and Rui Wang. "Nonlinear Elasticity of Borocarbide Superconductor YNi2B2C: A First-Principles Study." Advances in Materials Science and Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/9038151.

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First-principles calculations combined with homogeneous deformation methods are used to investigate the second- and third-order elastic constants of YNi2B2C with tetragonal structure. The predicted lattice constants and second-order elastic constants of YNi2B2C agree well with the available data. The effective second-order elastic constants are obtained from the second- and third-order elastic constants for YNi2B2C. Based on the effective second-order elastic constants, Pugh’s modulus ratio, Poisson’s ratio, and Vickers hardness of YNi2B2C under high pressure are further investigated. It is shown that the ductility of YNi2B2C increases with increasing pressure.

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24

Chwał, Małgorzata. "Numerical Evaluation of Effective Material Constants for CNT-Based Polymeric Nanocomposites." Advanced Materials Research 849 (November 2013): 88–93. http://dx.doi.org/10.4028/www.scientific.net/amr.849.88.

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The effective material constants for CNT-based polymeric composites are studied. The analysis is based on the elasticity theory involving a spatial square representative volume element and the finite element method. The transversally isotropic body having aligned and uniformly distributed long carbon nanotubes is assumed. The perfect bonding between the carbon nanotubes and the matrix are considered. For such a material the five elastic material constants is needed to completely describe the elastic behavior. Related to the calculated material constants, the results are given and compared with the other models presented in the literature. Generally, the increase of the effective material constants normalized by the matrix modulus is observed.

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25

Zhonghai, Xu, He Xiaodong, Li Zhihe, Wang Di, and Wang Rongguo. "Simulating Effective Engineering Elastic Constants of the Stator Core Lamination." Polymers and Polymer Composites 19, no. 4-5 (June 2011): 421–26. http://dx.doi.org/10.1177/0967391111019004-526.

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26

Lee, Usik. "Effective Elastic Compliances and Engineering Constants for Damaged Isotropic Solids." International Journal of Damage Mechanics 8, no. 2 (April 1999): 138–52. http://dx.doi.org/10.1177/105678959900800203.

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27

Whitney, James M. "Effective Elastic Constants of Bidirectional Laminates Containing Transverse Ply Cracks." Journal of Composite Materials 34, no. 11 (June 2000): 954–78. http://dx.doi.org/10.1177/002199830003401103.

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28

Yu, Wei, Meijuan Xin, Xi Liang, and Huijian Li. "Numerical Investigation into Effective Elastic Constants of MHS/EP Composite." Journal of Materials Engineering and Performance 21, no. 10 (January 31, 2012): 2038–43. http://dx.doi.org/10.1007/s11665-012-0137-z.

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29

Guan, Z. P., X. W. Fan, H. Xia, S. S. Jiang, and X. K. Zhang. "Effective elastic constants in EnZeZnSe1 − S strained-layer superlattices." Journal of Crystal Growth 159, no. 1-4 (February 1996): 485–88. http://dx.doi.org/10.1016/0022-0248(95)00728-8.

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30

Kim, Jin O., Jan D. Achenbach, Meenam Shinn, and Scott A. Barnett. "Effective elastic constants and acoustic properties of single-crystal TiN/NbN superlattices." Journal of Materials Research 7, no. 8 (August 1992): 2248–56. http://dx.doi.org/10.1557/jmr.1992.2248.

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Using the measured elastic constants of TiN and NbN single crystals with cubic symmetry, the effective elastic constants of single-crystal TiN/NbN superlattice films with tetragonal symmetry, namely c11, c12, c13, c33, c44, and c66 have been calculated for various thickness ratios of the layers. Using a line-focus acoustic microscope, measurements of surface acoustic waves (SAWs) have been carried out on single-crystal TiN/NbN superlattice films grown on the (001) plane of cubic crystal MgO substrates. The phase velocities measured as functions of the angle of propagation display the expected anisotropic nature of cubic crystals. Dispersion curves of SAWs propagating along the symmetry axes have been obtained by measuring wave velocities for various film thicknesses and frequencies. The SAW dispersion curves calculated from the effectiveelastic constants and the effective mass density of the superlattice films show very good agreement with experimental results. The results of this paper exhibit no anomalous dependence of the elastic constants on the superlattice period of TiN/NbN superlattices.

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31

Dundurs, John, and Iwona Jasiuk. "Effective Elastic Moduli of Composite Materials: Reduced Parameter Dependence." Applied Mechanics Reviews 50, no. 11S (November 1, 1997): S39—S43. http://dx.doi.org/10.1115/1.3101847.

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In this paper, we focus on the effective elastic constants of composite materials and pay attention to the possibility of reducing the number of independent variables. Surprisingly, this important issue has hardly been explored before. In our analysis, we rely on a new result in plane elasticity due to Cherkaev, Lurie, and Milton (1992), and use Dundurs constants (Dundurs, 1967, 1969). As an example, we consider a result for the effective elastic moduli of a composite containing a dilute concentration of perfectly-bonded circular inclusions.

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32

Yoon, Y. J., G. Yang, and S. C. Cowin. "Estimation of the effective transversely isotropic elastic constants of a material from known values of the material's orthotropic elastic constants." Biomechanics and Modeling in Mechanobiology 1, no. 1 (June 1, 2002): 83–93. http://dx.doi.org/10.1007/s10237-002-0008-x.

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33

Lee, Usik, and Deokki Youn. "Effective Material Properties of Damaged Elastic Solids." Key Engineering Materials 324-325 (November 2006): 1185–88. http://dx.doi.org/10.4028/www.scientific.net/kem.324-325.1185.

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By using a continuum modeling approach based on the equivalent elliptical crack representation of a local damage and the strain energy equivalence principle, the effective elastic compliances and the effective engineering constants are derived in closed forms in terms of the virgin (undamaged) elastic properties and a scalar damage variable for damaged two- and threedimensional isotropic solids. It is shown that the effective Young’s modulus in the direction normal to the crack surfaces is always smaller than its intact value.

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34

Mullen, R. "Monte Carlo simulation of effective elastic constants of polycrystalline thin films." Acta Materialia 45, no. 6 (June 1997): 2247–55. http://dx.doi.org/10.1016/s1359-6454(96)00366-7.

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35

Dickinson, L. C., G. L. Farley, and M. K. Hinders. "Prediction of Effective Three-Dimensional Elastic Constants of Translaminar Reinforced Composites." Journal of Composite Materials 33, no. 11 (June 1999): 1002–29. http://dx.doi.org/10.1177/002199839903301104.

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36

Chao, Chen-Ping, and Shive K. Chaturvedi. "A Thermo-Micromechanical Theory for Effective Elastic Constants of Composite Materials." Journal of Reinforced Plastics and Composites 16, no. 5 (March 1997): 392–413. http://dx.doi.org/10.1177/073168449701600501.

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37

Dutta, T., T. K. Ballabh, and T. R. Middya. "Green function calculation of effective elastic constants of textured polycrystalline materials." Journal of Physics D: Applied Physics 26, no. 4 (April 14, 1993): 667–75. http://dx.doi.org/10.1088/0022-3727/26/4/020.

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38

Akçakaya, E., G. W. Farnell, and E. L. Adler. "Dynamic approach for finding effective elastic and piezoelectric constants of superlattices." Journal of Applied Physics 68, no. 3 (August 1990): 1009–12. http://dx.doi.org/10.1063/1.346736.

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39

Sarkar, Sudeshna, T. K. Ballabh, T. R. Middya, and A. N. Basu. "T-matrix approach to effective nonlinear elastic constants of heterogeneous materials." Physical Review B 54, no. 6 (August 1, 1996): 3926–31. http://dx.doi.org/10.1103/physrevb.54.3926.

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40

Whitney, James M. "Effective Thermo-Elastic Constants for Monoclinic Laminates Containing Transverse Ply Cracks." Journal of Reinforced Plastics and Composites 22, no. 3 (February 2003): 203–27. http://dx.doi.org/10.1177/0731684403022003849.

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41

Ledbetter, Hassel, Subhendu Datta, and Martin Dunn. "Elastic Properties of Particle-Occlusion Composites: Measurements and Modeling." Journal of Engineering Materials and Technology 117, no. 4 (October 1, 1995): 402–7. http://dx.doi.org/10.1115/1.2804733.

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We review some of our recent studies on the effective elastic constants of composites where the occluded phase is spherical or spheroidal (oblate or prolate). From constituent properties and phase geometry, we estimated the composites’ effective macroscopic elastic constants using two models: the Ledbetter-Datta scattered-plane-wave ensemble-average model and the Mori-Tanaka effective-field model. We measured elastic constants by three principal methods: resonance, pulse-echo, and acoustic-resonance spectroscopy. We show microstructures, measurements, and model calculations for five representative composites: SiCp/Al, Al2O3-mullitep/Al, Al2O3p/ mullite, graphitep/ferrite (cast iron), voids/Ti.

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42

Daley, T. M., M. A. Schoenberg, J. Rutqvist, and K. T. Nihei. "Fractured reservoirs: An analysis of coupled elastodynamic and permeability changes from pore-pressure variation." GEOPHYSICS 71, no. 5 (September 2006): O33—O41. http://dx.doi.org/10.1190/1.2231108.

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Equivalent-medium theories can describe the elastic compliance and fluid-permeability tensors of a layer containing closely spaced parallel fractures embedded in an isotropic background. We propose a relationship between effective stress (background or lithostatic stress minus pore pressure) and both permeability and elastic constants. This relationship uses an exponential-decay function that captures the expected asymptotic behavior, i.e., low effective stress gives high elastic compliance and high fluid permeability, while high effective stress gives low elastic compliance and low fluid permeability. The exponential-decay constants are estimated for physically realistic conditions. With relationships coupling pore pressure to permeability and elastic constants, we are able to couple hydromechanical and elastodynamic modeling codes. A specific coupled simulation is demonstrated where fluid injection in a fractured reservoir causes spatially and temporally varying changes in pore pressure, permeability, and elastic constants. These elastic constants are used in a 3D finite-difference code to demonstrate time-lapse seismic monitoring with different acquisition geometries. Changes in amplitude and traveltime are seen in surface seismic P-to-S reflections as a function of offset and azimuth, as well as in vertical seismic profile P-to-S reflections and in crosswell converted S-waves. These observed changes in the seismic response demonstrate seismic monitoring of fluid injection in the fractured reservoir.

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43

Rao, R. Ramji, and A. Padmaja. "Fourth-order elastic constants and second pressure derivatives of white tin." Canadian Journal of Physics 69, no. 7 (July 1, 1991): 801–7. http://dx.doi.org/10.1139/p91-131.

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The expressions for the 25 fourth-order elastic constants of a body-centered tetragonal crystal are derived with interactions extending to fifth neighbours. The expressions for its effective second-order elastic constants are obtained in terms of its natural second-, third-, and fourth-order elastic constants using the finite strain elasticity theory. The fourth-order elastic constants and the second pressure derivatives of white tin are evaluated using these formulae. All the second pressure derivatives of white tin except that of C11 are positive. The second pressure derivatives C12, C13, and C33 are large suggesting that phase transformation will occur in white tin when subjected to hydrostatic pressure.

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44

Hwang, K. C., M. D. Xue, X. F. Wen, and G. Chen. "Stresses of Thick Perforated Plates With Reinforcement of Tubes and Their Effective Elastic Constants." Journal of Pressure Vessel Technology 114, no. 3 (August 1, 1992): 271–79. http://dx.doi.org/10.1115/1.2929041.

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Based on the concept of equivalent solid plate, this paper deals with thick perforated plates with triangular or square patterns of holes reinforced by tubes. The results obtained show that the tubes connected (by welding or expanding) to the perforated plates lead to a noticeable stiffening effect which is neglected or considerably underestimated by current design codes. The stresses of tubesheets calculated based on the effective elastic constants given by this method are in better agreement with the experimental results than those based on the effective elastic constants given by current codes.

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45

Ma, Tian Fei, Hong Xia Li, Xin Fu Wang, and Guo Qi Liu. "Micromechanics Analysis of Elastic Modulus of Alumina-Carbon Refractories." Advanced Materials Research 1095 (March 2015): 175–79. http://dx.doi.org/10.4028/www.scientific.net/amr.1095.175.

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Based on the micromechanics theory, Alumina-carbon refractories were regarded as resin-carbon bonded composites, including alumina, graphite and pores derived from particles packing gaps and phenolic resin pyrolysis. Graphite was regarded as isotropic spherical inclusions; particles packing gaps and phenolic resin pyrolysis pores were regarded as pore phase all together. Applying Mori-Tanaka multi-phase spherical inclusion method, firstly, elastic constants of resin-carbon phase were computed reversely by the elastic constants known alumina-carbon refractories, alumina and graphite, and then the effective elastic modulus of alumina-carbon refractories were estimated by the calculated elastic constants of resin-carbon and other raw materials. The results show that: the predicted elastic modulus by Mori-Tanaka model are higher than the experimental measurement values; resin carbon residue and pores have a great influence on effective elastic modulus of alumina-carbon refractories.

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46

Jasiuk, I., J. Chen, and M. F. Thorpe. "Elastic Moduli of Two Dimensional Materials With Polygonal and Elliptical Holes." Applied Mechanics Reviews 47, no. 1S (January 1, 1994): S18—S28. http://dx.doi.org/10.1115/1.3122813.

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We study the effective elastic moduli of two-dimensional composite materials containing polygonal holes. In the analysis we use a complex variable method of elasticity involving a conformal transformation. Then we take a far field result and derive the effective elastic constants of composites with a dilute concentration of polygonal holes. In the discussion we use the recently-stated Cherkaev-Lurie-Milton theorem, which gives general relations between the effective elastic constants of two-dimensional composites. We also discuss known results for elliptical holes in the context of the present work.

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47

Hasheminejad, S. M., and M. Maleki. "Effect of Interface Anisotropy on Elastic Wave Propagation in Particulate Composites." Journal of Mechanics 24, no. 1 (March 2008): 79–93. http://dx.doi.org/10.1017/s1727719100001581.

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ABSTRACTThe scattering of time harmonic plane longitudinal and transverse elastic waves in a composite consisting of randomly distributed identical isotropic spherical inclusions embedded in an isotropic matrix with anisotropic interface layers is examined. The interface region is modeled as a spherically isotropic shell of finite thickness with five independent elastic constants. The Frobenius power series solution method is utilized to deal with the interface anisotropy and the effect of random distribution of particulates in the composite medium is taken into account via a recently developed generalized self-consistent multiple scattering model. Numerical values of phase velocities and attenuations of coherent plane waves as well as the effective elastic constants are obtained for a moderately wide range of frequencies, particle concentrations, and interface anisotropies. The numerical results reveal the significant dependence of phase velocities and effective elastic constants on the interface properties. They show that interface anisotropy can moderately depress the effective phase velocities and the elastic moduli, but leave effective attenuation nearly unaffected, especially at low and intermediate frequencies. Limiting cases are considered and good agreements with recent solutions have been obtained.

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48

Quiroga Mendez, Jabid E., Octavio Andrés González-Estrada, and Diego F. Villegas. "Stressed Cylinder Dispersion Curves Based on Effective Elastic Constants and SAFE Method." Key Engineering Materials 774 (August 2018): 295–302. http://dx.doi.org/10.4028/www.scientific.net/kem.774.295.

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A Semi-Analytical Finite Element (SAFE) formulation is applied to determinethe dispersion curves in homogeneous and isotropic cylindrical waveguides subject touniaxial stress. Bulk waves are required for estimating the guided wave dispersion curvesand acoustoelasticity states a stress dependence of the ultrasound bulk velocities. Therefore,acoustoelasticity influences the wave field of the guided waves. Effective Elastic Constants(EEC) has emerged as a less complex alternative to deal with the acoustoelasticity; allowinga stressed material to be assumed as an unstressed material with EEC which considers thedisturbance linked to the presence of stress. In this approach the isotropic specimen subjectto load is studied by proposing an equivalent stress-free with a modified elasticity matrixwhich terms are the EEC. EEC provides an approximate stress-strain relation facilitating thedetermination of the dispersion curves using the well-studied numerical solution for the stressfreecases reducing the complexity of the numerical implementation. Therefore, a numericalmethod combining the SAFE and EEC is presented as a tool for the dispersion curve generationin stressed cylindrical specimens. The results of this methodology are verified by comparingthem with an approach previously reported in the literature based on SAFE including the fullstrain-displacement relation

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49

SU, Shu-lan, Qiu-hua RAO, and Yue-hui HE. "Theoretical prediction of effective elastic constants for new intermetallic compound porous material." Transactions of Nonferrous Metals Society of China 23, no. 4 (April 2013): 1090–97. http://dx.doi.org/10.1016/s1003-6326(13)62570-4.

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50

Xia, Hua, X. K. Zhang, An Hu, S. S. Jiang, R. W. Peng, Wei Zhang, Duan Feng, et al. "Effective elastic constants and phonon spectrum in metallic Ta/Al quasiperiodic superlattices." Physical Review B 47, no. 7 (February 15, 1993): 3890–95. http://dx.doi.org/10.1103/physrevb.47.3890.

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