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Journal articles on the topic 'Continuum Elasticity'

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Author: Grafiati

Published: 11 March 2023

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1

Charlotte, M., and L. Truskinovsky. "Towards multi-scale continuum elasticity theory." Continuum Mechanics and Thermodynamics 20, no. 3 (June 17, 2008): 133–61. http://dx.doi.org/10.1007/s00161-008-0075-z.

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2

Tarasov, Vasily E. "Fractional Gradient Elasticity from Spatial Dispersion Law." ISRN Condensed Matter Physics 2014 (April 3, 2014): 1–13. http://dx.doi.org/10.1155/2014/794097.

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Nonlocal elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak nonlocality) and the integral nonlocal models (strong nonlocality). This paper focuses on the fractional generalization of gradient elasticity that allows us to describe a weak nonlocality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We demonstrate how the continuum limit transforms the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in Riesz's form. A weak nonlocality of power-law type in the nonlocal elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The continuum equations with derivatives of noninteger orders, which are obtained from the lattice model, can be considered as a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.

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3

Haq, Omer, and Sergei Shabanov. "Bound States in the Continuum in Elasticity." Wave Motion 103 (June 2021): 102718. http://dx.doi.org/10.1016/j.wavemoti.2021.102718.

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4

OKAMOTO, Hirosuke. "Dispersion and continuum models of powder elasticity." Journal of the Society of Powder Technology, Japan 27, no. 3 (1990): 146–52. http://dx.doi.org/10.4164/sptj.27.146.

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5

Lerner, Edan, Eric DeGiuli, Gustavo Düring, and Matthieu Wyart. "Breakdown of continuum elasticity in amorphous solids." Soft Matter 10, no. 28 (2014): 5085. http://dx.doi.org/10.1039/c4sm00311j.

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6

Okamoto, Hirosuke. "Dispersion and Continuum Models of powder Elasticity [Translated]†." KONA Powder and Particle Journal 9 (1991): 28–35. http://dx.doi.org/10.14356/kona.1991008.

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7

Wang, Jing. "Effect of Temperature on Elasticity of Silicon Nanowires." Key Engineering Materials 483 (June 2011): 526–31. http://dx.doi.org/10.4028/www.scientific.net/kem.483.526.

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A semi-continuum approach is developed for mechanical analysis of a silicon nanowire, which captures the atomistic physics and retains the efficiency of continuum models. By using the Keating model, the strain energy of the nanowire required in the semi-continuum approach is obtained. Young’s modulus of the silicon (001) nanowire along [100] direction is obtained by the developed semi-continuum approach. Young’s modulus decreases dramatically as the size of a silicon nanowire width and thickness scaling down, especially at several nanometers, which is different from its bulk counterpart. The semi-continuum approach is extended to perform a mechanical analysis of the silicon nanowire at finite temperature. Taking into account the variations of the lattice parameter and the bond length with the temperature, the strain energy of the system is computed by using Keating anharmonic model. The dependence of young’s modulus of the nanowire on temperature is predicted, and it exhibits a negative temperature coefficient.

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8

Lakes, R. "Experimental Micro Mechanics Methods for Conventional and Negative Poisson’s Ratio Cellular Solids as Cosserat Continua." Journal of Engineering Materials and Technology 113, no. 1 (January 1, 1991): 148–55. http://dx.doi.org/10.1115/1.2903371.

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Continuum representations of micromechanical phenomena in structured materials are described, with emphasis on cellular solids. These phenomena are interpreted in light of Cosserat elasticity, a generalized continuum theory which admits degrees of freedom not present in classical elasticity. These are the rotation of points in the material, and a couple per unit area or couple stress. Experimental work in this area is reviewed, and other interpretation schemes are discussed. The applicability of Cosserat elasticity to cellular solids and fibrous composite materials is considered as is the application of related generalized continuum theories. New experimental results are presented for foam materials with negative Poisson’s ratios.

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9

Gao, Bin, Yu Zhou Sun, and Shen Li. "Higher-Order Elasticity Constants of Carbon Nanotubes." Advanced Materials Research 815 (October 2013): 516–19. http://dx.doi.org/10.4028/www.scientific.net/amr.815.516.

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In this paper, the higher-order elasticity constants are evaluated in the theoretical scheme of higher-order continuum. A single-walled carbon nanotube is treated as a higher-order continuum cylindrical tube with a thin wall, and the representative cell is chosen as a triangle unit that contains four carbon atoms. The Brenner potential is employed to describe the C-C atomic interaction, and the higher-order constitutive relationship is derived by virtue of the higher-order Cauchy-Born rule. The higher-order elasticity constants of carbon nanotubes are evaluated based on the derived higher-order constitutive model, which can provide a foundation for the further analysis of the mechanical properties of carbon nanotubes in the theoretical scheme of higher-order continuum.

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10

Wang, Jing. "Size-Dependence of Elasticity of Phosphorus-Doped Silicon Nano-Plates." Advanced Materials Research 486 (March 2012): 80–83. http://dx.doi.org/10.4028/www.scientific.net/amr.486.80.

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Elasticity of phosphorus (P)-doped silicon nanoplates has been investigated by a semi-continuum approach which captures the atomistic physics and retains the efficiency of continuum models. Youngs modulus of silicon (001) nanoplates along [10 direction is obtained by the developed semi-continuum approach. The results show that P-doping has an effect on the elasticity of silicon nanoplates, especially with the variation of doping concentration. The model predicts that Youngs moduli of P-doped silicon nanoplates are size-dependence.

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11

Mesarovic, Sinisa Dj. "Lattice continuum and diffusional creep." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2188 (April 2016): 20160039. http://dx.doi.org/10.1098/rspa.2016.0039.

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Diffusional creep is characterized by growth/disappearance of lattice planes at the crystal boundaries that serve as sources/sinks of vacancies, and by diffusion of vacancies. The lattice continuum theory developed here represents a natural and intuitive framework for the analysis of diffusion in crystals and lattice growth/loss at the boundaries. The formulation includes the definition of the Lagrangian reference configuration for the newly created lattice, the transport theorem and the definition of the creep rate tensor for a polycrystal as a piecewise uniform, discontinuous field. The values associated with each crystalline grain are related to the normal diffusional flux at grain boundaries. The governing equations for Nabarro–Herring creep are derived with coupled diffusion and elasticity with compositional eigenstrain. Both, bulk diffusional dissipation and boundary dissipation accompanying vacancy nucleation and absorption, are considered, but the latter is found to be negligible. For periodic arrangements of grains, diffusion formally decouples from elasticity but at the cost of a complicated boundary condition. The equilibrium of deviatorically stressed polycrystals is impossible without inclusion of interface energies. The secondary creep rate estimates correspond to the standard Nabarro–Herring model, and the volumetric creep is small. The initial (primary) creep rate is estimated to be much larger than the secondary creep rate.

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12

Lee, Sunmi, Russel E. Caflisch, and Young‐Ju Lee. "Exact Artificial Boundary Conditions for Continuum and Discrete Elasticity." SIAM Journal on Applied Mathematics 66, no. 5 (January 2006): 1749–75. http://dx.doi.org/10.1137/050644252.

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13

Barraza-Lopez, Salvador, Alejandro A. Pacheco Sanjuan, Zhengfei Wang, and Mihajlo Vanević. "Strain-engineering of graphene's electronic structure beyond continuum elasticity." Solid State Communications 166 (July 2013): 70–75. http://dx.doi.org/10.1016/j.ssc.2013.05.002.

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14

Stark, Sebastian, and Peter Neumeister. "A continuum model for sintering processes incorporating elasticity effects." Mechanics of Materials 122 (July 2018): 26–41. http://dx.doi.org/10.1016/j.mechmat.2018.04.001.

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15

Sept, David, and Fred MacKintosh. "Microtubule elasticity: Connecting all-atom simulations with continuum mechanics." Biophysical Journal 96, no. 3 (February 2009): 131a—132a. http://dx.doi.org/10.1016/j.bpj.2008.12.596.

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16

Chow, C. L., and June Wang. "An anisotropic theory of elasticity for continuum damage mechanics." International Journal of Fracture 33, no. 1 (January 1987): 3–16. http://dx.doi.org/10.1007/bf00034895.

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17

KHALDJIGITOV, ABDUVALI, AZIZ QALANDAROV, NIK MOHD ASRI NIK LONG, and ZAINIDIN ESHQUVATOV. "NUMERICAL SOLUTION OF 1D AND 2D THERMOELASTIC COUPLED PROBLEMS." International Journal of Modern Physics: Conference Series 09 (January 2012): 503–10. http://dx.doi.org/10.1142/s2010194512005594.

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Study the heats propagation in a solid, liquid continuums is an actual problems. The liquid continuum may be considered as a biomaterial. The present investigation is devoted to the study of 1D and 2D dynamic coupled thermo elasticity problems. In case of coupled problems the motion and heat conduction equations are considered together. For numerical solution of thermo elasticity problems an explicit and implicit schemes are constructed. The explicit and implicit schemes by using recurrent formulas and the "consecutive" methods are solved. Comparison of two results shows a good coincidence.

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18

Luskin, Mitchell, and Christoph Ortner. "Atomistic-to-continuum coupling." Acta Numerica 22 (April 2, 2013): 397–508. http://dx.doi.org/10.1017/s0962492913000068.

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Atomistic-to-continuum (a/c) coupling methods are a class of computational multiscale schemes that combine the accuracy of atomistic models with the efficiency of continuum elasticity. They are increasingly being utilized in materials science to study the fundamental mechanisms of material failure such as crack propagation and plasticity, which are governed by the interaction between crystal defects and long-range elastic fields.In the construction of a/c coupling methods, various approximation errors are committed. A rigorous numerical analysis approach that classifies and quantifies these errors can give confidence in the simulation results, as well as enable optimization of the numerical methods for accuracy and computational cost. In this article, we present such a numerical analysis framework, which is inspired by recent research activity.

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19

Ostoja-Starzewski, Martinos, X. Du, Z. F. Khisaeva, and W. Li. "On the Size of Representative Volume Element in Elastic, Plastic, Thermoelastic and Permeable Random Microstructures." Materials Science Forum 539-543 (March 2007): 201–6. http://dx.doi.org/10.4028/www.scientific.net/msf.539-543.201.

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The Representative Volume Element (so-called RVE) is the corner stone of continuum mechanics. In this paper we examine the scaling to RVE in linear elasticity, finite elasticity, elasto-plasticity, thermoelasticity, and permeability of random composite materials.

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20

Bottani, C. E. "Equations of Continuum Elasticity Including Both Dislocation Motion and Production." Europhysics Letters (EPL) 9, no. 8 (August 15, 1989): 785–90. http://dx.doi.org/10.1209/0295-5075/9/8/008.

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21

Liu, Liping. "An energy formulation of continuum magneto-electro-elasticity with applications." Journal of the Mechanics and Physics of Solids 63 (February 2014): 451–80. http://dx.doi.org/10.1016/j.jmps.2013.08.001.

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22

Gao, Jian. "An asymmetric theory of nonlocal elasticity—Part 2. Continuum field." International Journal of Solids and Structures 36, no. 20 (July 1999): 2959–71. http://dx.doi.org/10.1016/s0020-7683(97)00322-3.

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23

Miri, MirFaez, and Nicolas Rivier. "Continuum elasticity with topological defects, including dislocations and extra-matter." Journal of Physics A: Mathematical and General 35, no. 7 (February 11, 2002): 1727–39. http://dx.doi.org/10.1088/0305-4470/35/7/317.

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24

Conti, Sergio, and Ekhard K. H. Salje. "Surface structure of ferroelastic domain walls: a continuum elasticity approach." Journal of Physics: Condensed Matter 13, no. 39 (September 13, 2001): L847—L854. http://dx.doi.org/10.1088/0953-8984/13/39/103.

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25

Nan, Ce-Wen, and D. M. Smith. "Non-universal elasticity exponent for three-dimensional continuum percolation systems." Materials Science and Engineering: B 10, no. 1 (September 1991): L1—L3. http://dx.doi.org/10.1016/0921-5107(91)90099-h.

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26

Sumelka, Wojciech, Krzysztof Szajek, and Tomasz Łodygowski. "Plane strain and plane stress elasticity under fractional continuum mechanics." Archive of Applied Mechanics 85, no. 9-10 (November 27, 2014): 1527–44. http://dx.doi.org/10.1007/s00419-014-0949-4.

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27

Davison, Lee. "Continuum Modeling." MRS Bulletin 13, no. 2 (February 1988): 16–21. http://dx.doi.org/10.1557/s0883769400066318.

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Common experience with the thermomechanical properties and behavior of matter is usually at the continuum level, i.e., observations concern such phenomena as elastic and inelastic deformation of solids, flow of fluids, and conduction of heat. Continuum descriptions of these phenomena are expressed in terms of partial differential equations representing the principles of balance of mass, momentum, and energy. Since the basic principles apply to all materials, it is apparent that they alone will not suffice for solving specific problems. The peculiarities of individual materials are expressed in terms of constitutive equations, of which Hooke's law of elasticity and Newton's law of viscosity are examples. For the most part, “continuum modeling” refers to the process of devising constitutive equations. This work requires exercise of physical insight at both the macroscopic and microscopic levels, consideration of experimental observations, and application of formal mathematical principles.Much of materials science is devoted to development and application of materials that have been selected for, or designed to have, useful properties different from those of existing materials. Many materials are of interest precisely because they have unusual properties. For example, paints and other coatings are often designed to flow easily when spread, but resist running afterward. Much work is devoted to analyzing manufacturing processes, whether they be processing of foodstuffs, forging metals, drawing films and fibers, or curing polymers. Properties and physical states of materials that are important during manufacture are often very different from those desired under service conditions.

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28

Ciambella, J., and G. Saccomandi. "A continuum hyperelastic model for auxetic materials." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2163 (March 8, 2014): 20130691. http://dx.doi.org/10.1098/rspa.2013.0691.

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We propose a simple mathematical model to describe isotropic auxetic materials in the framework of the classical theory of nonlinear elasticity. The model is derived from the Blatz–Ko constitutive equation for compressible foams and makes use of a non-monotonic Poisson function. An application to the modelling of auxetic foams is considered and it is shown that the material behaviour is adequately described with only three constitutive parameters.

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29

Tarasov, Vasily E., and Elias C. Aifantis. "Toward fractional gradient elasticity." Journal of the Mechanical Behavior of Materials 23, no. 1-2 (May 1, 2014): 41–46. http://dx.doi.org/10.1515/jmbm-2014-0006.

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AbstractThe use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The second involves the Riesz fractional derivative in three dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, in a Riesz-type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.

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30

Neff, Patrizio, Angela Madeo, Gabriele Barbagallo, Marco Valerio d'Agostino, Rafael Abreu, and Ionel-Dumitrel Ghiba. "Real wave propagation in the isotropic-relaxed micromorphic model." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2197 (January 2017): 20160790. http://dx.doi.org/10.1098/rspa.2016.0790.

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For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.

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31

MARUYAMA, K., K. OKUMURA, H. YAMAUCHI, and S. MIYAZIMA. "CRITICAL EXPONENTS OF ELASTICITY IN A CONTINUUM PERCOLATION SYSTEM (INVERSE SWISS-CHEESE MODEL)." Fractals 01, no. 04 (December 1993): 904–7. http://dx.doi.org/10.1142/s0218348x93000940.

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Critical exponents of elastic constant of Inverse Swiss-cheese model (Continuum percolation problem) is experimentally investigated following our previous experimental studies on electric conductivity and permeability in a continuum percolation system.

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32

Julien, Scott E., Ann Lii-Rosales, Kai-Tak Wan, Yong Han, Michael C. Tringides, James W. Evans, and Patricia A. Thiel. "Squeezed nanocrystals: equilibrium configuration of metal clusters embedded beneath the surface of a layered material." Nanoscale 11, no. 13 (2019): 6445–52. http://dx.doi.org/10.1039/c8nr10549a.

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33

Fruchart, Michel, Colin Scheibner, and Vincenzo Vitelli. "Odd Viscosity and Odd Elasticity." Annual Review of Condensed Matter Physics 14, no. 1 (March 10, 2023): 471–510. http://dx.doi.org/10.1146/annurev-conmatphys-040821-125506.

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Elasticity typically refers to a material's ability to store energy, whereas viscosity refers to a material's tendency to dissipate it. In this review, we discuss fluids and solids for which this is not the case. These materials display additional linear response coefficients known as odd viscosity and odd elasticity. We first introduce odd viscosity and odd elasticity from a continuum perspective, with an emphasis on their rich phenomenology, including transverse responses, modified dislocation dynamics, and topological waves. We then provide an overview of systems that display odd viscosity and odd elasticity. These systems range from quantum fluids and astrophysical gases to active and driven matter. Finally, we comment on microscopic mechanisms by which odd viscosity and odd elasticity arise.

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34

E. Tarasov, Vasily. "Three-Dimensional Lattice Approach to Fractional Generalization of Continuum Gradient Elasticity." Progress in Fractional Differentiation and Applications 1, no. 4 (October 1, 2015): 243–58. http://dx.doi.org/10.18576/pfda/010402.

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35

Terentjev, E. M., and M. Warner. "Continuum theory of elasticity and piezoelectric effects in smectic A elastomers." Journal de Physique II 4, no. 1 (January 1994): 111–26. http://dx.doi.org/10.1051/jp2:1994118.

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36

Brandt, O., K. Ploog, R. Bierwolf, and M. Hohenstein. "Breakdown of continuum elasticity theory in the limit of monatomic films." Physical Review Letters 68, no. 9 (March 2, 1992): 1339–42. http://dx.doi.org/10.1103/physrevlett.68.1339.

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37

Ciambella, Jacopo, Abderrezak Bezazi, Giuseppe Saccomandi, and Fabrizio Scarpa. "Nonlinear elasticity of auxetic open cell foams modeled as continuum solids." Journal of Applied Physics 117, no. 18 (May 14, 2015): 184902. http://dx.doi.org/10.1063/1.4921101.

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38

Skatulla, S., A. Arockiarajan, and C. Sansour. "A nonlinear generalized continuum approach for electro-elasticity including scale effects." Journal of the Mechanics and Physics of Solids 57, no. 1 (January 2009): 137–60. http://dx.doi.org/10.1016/j.jmps.2008.09.014.

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39

Puthanpurayil, Arun M., Athol J. Carr, and Rajesh P. Dhakal. "Application of nonlocal elasticity continuum damping models in nonlinear dynamic analysis." Bulletin of Earthquake Engineering 16, no. 12 (June 29, 2018): 6269–97. http://dx.doi.org/10.1007/s10518-018-0412-y.

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40

Puthanpurayil, Arun M., Oren Lavan, Athol J. Carr, and Rajesh P. Dhakal. "Application of local elasticity continuum damping models in nonlinear dynamic analysis." Bulletin of Earthquake Engineering 16, no. 12 (July 18, 2018): 6365–91. http://dx.doi.org/10.1007/s10518-018-0424-7.

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41

Yakobson, B. I., C. J. Brabec, and J. Bernholc. "Structural mechanics of carbon nanotubes: From continuum elasticity to atomistic fracture." Journal of Computer-Aided Materials Design 3, no. 1-3 (August 1996): 173–82. http://dx.doi.org/10.1007/bf01185652.

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42

Kupferman, Raz, Elihu Olami, and Reuven Segev. "Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies." Journal of Elasticity 128, no. 1 (January 9, 2017): 61–84. http://dx.doi.org/10.1007/s10659-016-9617-y.

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43

Mielke, Alexander, and Lev Truskinovsky. "From Discrete Visco-Elasticity to Continuum Rate-Independent Plasticity: Rigorous Results." Archive for Rational Mechanics and Analysis 203, no. 2 (September 16, 2011): 577–619. http://dx.doi.org/10.1007/s00205-011-0460-9.

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44

Terzi, M. Mert, Kaushik Dayal, Luca Deseri, and Markus Deserno. "Revisiting the Link between Lipid Membrane Elasticity and Microscopic Continuum Models." Biophysical Journal 108, no. 2 (January 2015): 87a—88a. http://dx.doi.org/10.1016/j.bpj.2014.11.510.

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45

Tarasov, Vasily E. "General lattice model of gradient elasticity." Modern Physics Letters B 28, no. 07 (March 13, 2014): 1450054. http://dx.doi.org/10.1142/s0217984914500547.

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In this paper, new lattice model for the gradient elasticity is suggested. This lattice model gives a microstructural basis for second-order strain-gradient elasticity of continuum that is described by the linear elastic constitutive relation with the negative sign in front of the gradient. Moreover, the suggested lattice model allows us to have a unified description of gradient models with positive and negative signs of the strain gradient terms. Possible generalizations of this model for the high-order gradient elasticity and three-dimensional case are also suggested.

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46

Asgari, Meisam. "Micro-mechanical, continuum-mechanical, and AFM-based descriptions of elasticity in open cylindrical micellar filaments." Soft Matter 13, no. 39 (2017): 7112–28. http://dx.doi.org/10.1039/c7sm00911a.

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47

Chan, Youn-Sha, Glaucio H. Paulino, and Albert C. Fannjiang. "Change of Constitutive Relations due to Interaction Between Strain-Gradient Effect and Material Gradation." Journal of Applied Mechanics 73, no. 5 (July 14, 2005): 871–75. http://dx.doi.org/10.1115/1.2041658.

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For classical elasticity, the constitutive equations (Hooke’s law) have the same functional form for both homogeneous and nonhomogeneous materials. However, for strain-gradient elasticity, such is not the case. This paper shows that for strain-gradient elasticity with volumetric and surface energy (Casal’s continuum), extra terms appear in the constitutive equations which are associated with the interaction between the material gradation and the nonlocal effect of strain gradient. The corresponding governing partial differential equations are derived and their solutions are discussed.

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48

Kalampakas, Antonios, and Elias C. Aifantis. "A note on the discrete approach for generalized continuum models." Journal of the Mechanical Behavior of Materials 23, no. 5-6 (December 1, 2014): 181–83. http://dx.doi.org/10.1515/jmbm-2014-0020.

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AbstractGeneralized continuum theories for materials and processes have been introduced in order to account in a phenomenological manner for microstructural effects. Their drawback mainly rests in the determination of the extra phenomenological coefficients through experiments and simulations. It is shown here that a graphical representation of the local topology describing deformation models can be used to deduce restrictions on the phenomenological coefficients of the gradient elasticity continuum theories.

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49

Ospina Holguín, Javier Humberto. "The Cobb-Douglas function for a continuum model." Cuadernos de Economía 36, no. 70 (January 1, 2017): 1–18. http://dx.doi.org/10.15446/cuad.econ.v36n70.49052.

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This paper introduces two formal equivalent definitions of the Cobb-Douglas function for a continuum model based on a generalization of the Constant Elasticity of Substitution (CES) function for a continuum under not necessarily constant returns to scale and based on principles of product calculus. New properties are developed, and to illustrate the potential of using the product integral and its functional derivative, it is shown how the profit maximization problem of a single competitive firm using a continuum of factors of production can be solved in a manner that is completely analogous to the one used in the discrete case.

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50

Banks, Harvey Thomas, Shuhua Hu, and Zackary R. Kenz. "A Brief Review of Elasticity and Viscoelasticity for Solids." Advances in Applied Mathematics and Mechanics 3, no. 1 (February 2011): 1–51. http://dx.doi.org/10.4208/aamm.10-m1030.

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AbstractThere are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.

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