Статті в журналах: "Elastic Lattice Model"

1

Zabulionis, Darius, and Vytautas Rimša. "A Lattice Model for Elastic Particulate Composites." Materials 11, no. 9 (September 1, 2018): 1584. http://dx.doi.org/10.3390/ma11091584.

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In the present article, a version of the lattice or spring network method is proposed to model the mechanical response of elastic particulate composites with a high volume fraction of spherical particles and with a much weaker matrix compared to the stiffness of the particles. The main subject of the article is the determination of the axial stiffnesses of the springs of the cell. A comparison of the mechanical response of a three-dimensional particulate composite cube obtained using the finite element method and the proposed methodology showed that the efficiency of the proposed methodology increases with an increasing volume fraction of the particles.

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2

Saito, Yukio. "Three-dimensional elastic lattice model of heteroepitaxy." Surface Science 586, no. 1-3 (July 2005): 83–95. http://dx.doi.org/10.1016/j.susc.2005.05.004.

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3

Pal, Raj Kumar, Federico Bonetto, Luca Dieci, and Massimo Ruzzene. "A study of deformation localization in nonlinear elastic square lattices under compression." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2127 (July 23, 2018): 20170140. http://dx.doi.org/10.1098/rsta.2017.0140.

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The paper investigates localized deformation patterns resulting from the onset of instabilities in lattice structures. The study is motivated by previous observations on discrete hexagonal lattices, where a variety of localized deformations were found depending on loading configuration, lattice parameters and boundary conditions. These studies are conducted on other lattice structures, with the objective of identifying and investigating minimal models that exhibit localization, hysteresis and path-dependent behaviour. To this end, we first consider a two-dimensional square lattice consisting of point masses connected by in-plane axial springs and vertical ground springs, which may be considered as a discrete description of an elastic membrane supported by an elastic substrate. Results illustrate that, depending on the relative values of the spring constants, the lattice exhibits in-plane or out-of-plane instabilities leading to localized deformations. This model is further simplified by considering the one-dimensional case of a spring–mass chain sitting on an elastic foundation. A bifurcation analysis of this lattice identifies the stable and unstable branches and sheds light on the mechanism of transition from affine deformation to global or diffuse deformation to localized deformation. Finally, the lattice is further reduced to a minimal four-mass model, which exhibits a deformation qualitatively similar to that in the central part of a longer chain. In contrast to the widespread assumption that localization is induced by defects or imperfections in a structure, this work illustrates that such phenomena can arise in perfect lattices as a consequence of the mode shapes at the bifurcation points. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

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4

Colquitt, D. J., I. S. Jones, N. V. Movchan, and A. B. Movchan. "Dispersion and localization of elastic waves in materials with microstructure." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2134 (May 11, 2011): 2874–95. http://dx.doi.org/10.1098/rspa.2011.0126.

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This paper considers the interaction of elastic waves with materials with microstructure. The paper presents a mathematical model of elastic waves within a lattice system incorporating rotational motions and interaction between different lattice elements through elastic links. The waves are dispersive and the lattice system itself is heterogeneous, i.e. the elastic stiffness and/or mass are non-uniformly distributed. For such systems, one can identify stop bands, representing the intervals of frequencies of waves, which become evanescent and cannot propagate through the structure. Filtering properties of such lattices are studied in this paper. Defect modes are created by removing a periodic array of elastic links, which leads to localization within a macro-cell. Special attention is given to the evaluation of the effective group velocities and to the study of standing waves within the system. Analytical estimates are accompanied by numerical simulations and analysis of dispersion surfaces. We also consider an example showing the focusing and the creation of an image point by a flat elastic ‘lens’ formed from a finite micropolar lattice system.

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5

Johnson, R. A., and D. J. Oh. "Analytic embedded atom method model for bcc metals." Journal of Materials Research 4, no. 5 (October 1989): 1195–201. http://dx.doi.org/10.1557/jmr.1989.1195.

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The requirements for fitting bcc metals within the EAM format are discussed and, for comparative purposes, the EAM format is cast in a normalized form. A general embedding function is defined and an analytic first- and second-neighbor model is presented. The parameters in the model are determined from the cohesive energy, the equilibrium lattice constant, the three elastic constants, and the unrelaxed vacancy formation energy. Increasing the elastic constants, increasing the elastic anisotropy ratio, and decreasing the unrelaxed vacancy formation energy favor stability of a close-packed lattice over bcc. A stable bcc lattice relative to close packing is found for nine bcc metals, but this scheme cannot generate a model for Cr because the elastic constants of Cr require a negative curvature of the embedding function.

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6

Giraud, Laurent, Dominique d'HumièRes, and Pierre Lallemand. "A Lattice-Boltzmann Model for Visco-Elasticity." International Journal of Modern Physics C 08, no. 04 (August 1997): 805–15. http://dx.doi.org/10.1142/s0129183197000692.

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The classical lattice-Boltzmann scheme is extended in an attempt to represent visco-elastic fluids in two dimensions. At each lattice site, two new quantities are added. A suitable coupling of these quantities with the viscous stress tensor leads to a nonzero shear modulus and visco-elastic effects. A Chapman–Enskog expansion gives us the equilibrium populations and conditions for isotropy of the model. A finite wave vector analysis is needed to study the relaxation of sound waves and to determine the dependence of the transport coefficients upon the frequency.

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7

Pouget, J., A. Aşkar, and G. A. Maugin. "Lattice model for elastic ferroelectric crystals: Microscopic approach." Physical Review B 33, no. 9 (May 1, 1986): 6304–19. http://dx.doi.org/10.1103/physrevb.33.6304.

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8

Pouget, J., A. Aşkar, and G. A. Maugin. "Lattice model for elastic ferroelectric crystals: Continuum approximation." Physical Review B 33, no. 9 (May 1, 1986): 6320–25. http://dx.doi.org/10.1103/physrevb.33.6320.

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9

Karamoozian, Aminreza, Chin An Tan, and Liangmo Wang. "Homogenized modeling and micromechanics analysis of thin-walled lattice plate structures for brake discs." Journal of Sandwich Structures & Materials 22, no. 2 (February 22, 2018): 423–60. http://dx.doi.org/10.1177/1099636218757670.

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Periodic cellular structures, especially lattice designs, have potential to improve the cooling performance of brake disk system. In this paper, the method of two scales asymptotic homogenization was used to indicate the effective elastic stiffnesses of lattice plates structures. The arbitrary topology of lattice core cells connected to the back and front plates which are made of generally orthotropic materials, due to use in brake disc design. This starts with the derivation of general shell model with consideration of the set of unit cell problems and then making use of the model to determine the analytical equations and calculate the effective elastic properties of lattice plate concerning the connected back and front plates. The analytical and numerical method allows determining the stiffness properties and the internal forces in the trusses and plates of the lattice. Three types of core-based lattice plates, which are pyramidal, X-type and I-type lattices, have been studied. The I-type lattice is characterized here for the first time with particular attention on the role that the cell trusses and plates plays on the stiffness and strength properties. The lattice designs are finite element characterized and compared with the numerical and experimentally validated pyramidal and X-type lattices under identical conditions. The I-type lattice provides 4% higher strength more than the other lattice types with 9% higher truss fraction coefficient. Results show that the stiffness and yield strength of the lattices depend upon the stress–strain response of the parent alloy of trusses and plates, the truss mass fraction coefficient, the face carriers thickness and the core elements parameters. The study described here is limited to a linear analysis of lattice properties. Geometric nonlinearities, however, have a considerable impact on the effective behavior of a lattice and will be the subject of future studies.

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10

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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11

Li, Yuchen, Noël Challamel, and Isaac Elishakoff. "Stochastic analysis of lattice, nonlocal continuous beams in vibration." Vietnam Journal of Mechanics 43, no. 2 (June 28, 2021): 139–70. http://dx.doi.org/10.15625/0866-7136/15671.

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In this paper, we study the stochastic behavior of some lattice beams, called Hencky bar-chain model and their non-local continuous beam approximations. Hencky bar-chain model is a beam lattice composed of rigid segments, connected by some homogeneous rotational elastic links. In the present stochastic analysis, the stiffness of these elastic links is treated as a continuous random variable, with given probability density function. The fundamental eigenfrequency of the linear difference eigenvalue problem is also a random variable in this context. The reliability is defined as the probability that this fundamental frequency is less than an excitation frequency. This reliability function is exactly calculated for the lattice beam in conjunction with various boundary conditions. An exponential distribution is considered for the random stiffness of the elastic links. The stochastic lattice model is then compared to a stochastic nonlocal beam model, based on the continualization of the difference equation of the lattice model. The efficiency of the nonlocal beam model to approximate the lattice beam model is shown in presence of rotational elastic link randomness. We also compare such stochastic function with the one of a continuous local Euler-Bernoulli beam, with a special emphasis on scale effect in presence of randomness. Scale effect is captured both in deterministic and non-deterministic frameworks.

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12

Cai, Zhi-Xiong, David O. Welch, and Girija S. Dubey. "Isothermal Elastic Constants of Flux-Line Lattice in Layered Superconductors." International Journal of Modern Physics B 12, no. 29n31 (December 20, 1998): 2974–81. http://dx.doi.org/10.1142/s0217979298001897.

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A model of the effective interaction between the magnetic flux-lines in a layered superconductor is derived from the Lawrence–Doniach model. We show analytically that the intralayer interaction energy can be evaluated using the Ewald summation technique. The melting of flux line lattices is studied using Langevin dynamics simulation of the model with various values of interlayer coupling strength and pinning intensities. The thermal fluctuation terms of the isothermal shear modulus are found to increase sharply at the melting transition temperature for systems with or without pinning, while the structural order parameters were close to zero at all temperatures for systems with strong pinning. The melting of the pinned flux-line lattice is discussed in the context of its elastic properties.

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13

LU, CHUNSHENG, DAVID VERE-JONES, HIDEKI TAKAYASU, ALEX YU TRETYAKOV, and MISAKO TAKAYASU. "SPATIO-TEMPORAL SEISMICITY IN AN ELASTIC BLOCK LATTICE MODEL." Fractals 07, no. 03 (September 1999): 301–11. http://dx.doi.org/10.1142/s0218348x9900030x.

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An elastic block lattice model is proposed to simulate the spatio-temporal seismicity and stress patterns in the Earth's brittle crust. The famous Gutenberg-Richter magnitude-frequency law in seismology is reproduced. The synthetic catalogs generated by this model are analyzed by using a linked stress release model, which incorporates the stress transfer and spatial interactions. The results highlight the triggering mechanism of earthquake occurrence and the evidence that the crust may lie in a near-critical or self-organized critical state due to the long-range spatial interaction of elastic stress. The spatio-temporal complexity of seismicity is closely related to both nonlinear dynamics of faults and heterogeneities in a seismic region.

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14

Uemura, Hideaki, Yukio Saito, and Makio Uwaha. "Elastic Interaction in a Two-Dimensional Discrete Lattice Model." Journal of the Physical Society of Japan 70, no. 3 (March 15, 2001): 743–52. http://dx.doi.org/10.1143/jpsj.70.743.

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15

Cheng, Cheng, Ning Dai, Weiping Gu, Bai Xu, and Jing Xu. "Hierarchical Lattice Modeling Method with Gradient Functions." Mobile Information Systems 2022 (June 1, 2022): 1–14. http://dx.doi.org/10.1155/2022/1359472.

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Lattice structure materials have great application potential in biomedical, aerospace, and other fields. In fact, designing multilevel structure parametrically to achieve excellent mechanical performance is a challenging task. In this paper, a hierarchical lattice modeling method with gradient functions is proposed on the basis of the multilevel structural characteristics of organisms to solve the problem of the coexistence of high hardness, high strength, and high toughness. The multilevel body space is filled with matching lattices in accordance with the given dynamic parameters, and the design is optimized in accordance with the simulation results. Thus, the desired composite functional structure is generated. In fact, the proposed method is divided into two stages, that is, preprocessing and design. In the former stage, the interpolation method is used to establish a lattice unit model family retrieved by parameters, such as elastic modulus and impact toughness. In the latter stage, the multilevel 3D reconstruction model is initially filled with the gradient multilevel lattices through the global optimization technique with an isosurface modeling algorithm. The porosity, rod diameter, and other parameters of the lattice structure are regulated in accordance with the results of finite element analysis through image mapping, meeting the requirements of biomechanical characteristics. Our empirical evaluation and experimental results demonstrate that the proposed method can control the relationship between porous structure porosity, elastic modulus, and impact toughness and design bionic multispace microstructures close to biodental displacement and deformation.

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16

Hu, Xiaolin, and Xiaofeng Jia. "A dynamic lattice method for elastic seismic modeling in anisotropic media." GEOPHYSICS 81, no. 3 (May 2016): T131—T143. http://dx.doi.org/10.1190/geo2015-0511.1.

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Basing on a particle network model, we have developed a dynamic lattice approach to simulate seismic wave propagation in transversely isotropic (TI) media with tilted symmetry axis (TTI). Different from other wave-equation-based numerical methods, the dynamic lattice approach calculates the micromechanical interactions between particles in the lattice instead of solving the wave equation. We implement Lagrange’s equations to transform these interactions into elastic forces acting upon each particle. By solving the equations of motion, we obtain the disturbances of particles. Therefore, seismic wave motions in continua are approximated by displacements of these particles. Elastic features of the continuum are represented by properties of the particle lattice, including the physical properties of particles and the micromechanical interactions between particles. In the case of TI media, it is a challenge to find the correct particle lattice model that can reflect the anisotropic nature of TI media. We have determined the theoretical connection between the TI medium and the particle lattice model, allowing us to model elastic seismic waves in heterogeneous TI media. Furthermore, we have linearized the propagator to improve the CPU efficiency of our method for seismic wave simulation. We have applied the method to reverse time migration on a TI model to test its usefulness in complex media imaging.

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17

Regueiro, Richard A., Douglas J. Bammann, Esteban B. Marin, and Krishna Garikipati. "A Nonlocal Phenomenological Anisotropic Finite Deformation Plasticity Model Accounting for Dislocation Defects." Journal of Engineering Materials and Technology 124, no. 3 (June 10, 2002): 380–87. http://dx.doi.org/10.1115/1.1480410.

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A phenomenological, polycrystalline version of a nonlocal crystal plasticity model is formulated. The presence of geometrically necessary dislocations (GNDs) at, or near, grain boundaries is modeled as elastic lattice curvature through a curl of the elastic part of the deformation gradient. This spatial gradient of an internal state variable introduces a length scale, turning the local form of the model, an ordinary differential equation (ODE), into a nonlocal form, a partial differential equation (PDE) requiring boundary conditions. Small lattice elastic stretching results from the presence of dislocations and from macroscopic external loading. Finite deformation results from large plastic slip and large rotations. The thermodynamics and constitutive assumptions are written in the intermediate configuration in order to place the plasticity equations in the proper configuration for finite deformation analysis.

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18

Srivastava, Umesh Chandra, and Shyamendra Pratap Singh. "Structural and Vibrational Properties of Solid Naphthalene (C10H8) by Use of VTBFS Model." Oriental Journal Of Chemistry 38, no. 3 (June 30, 2022): 762–65. http://dx.doi.org/10.13005/ojc/380329.

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In this article author’s are reporting lattice dynamical properties of naphthalene (C10H8) by use of VTBFS model. we also report the combined density of states (CDS), dispersion relation and elastic properties of naphthalene. So, use of the present model the lattice property of naphthalene is reported successfully. Moreover, within our reported result the lattice modes exhibit more drastic changes in the structural and vibrational properties of naphthalene.

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19

Gallyamov, I. I., and L. F. Yusupova. "Magnetization of an elastic ferromagnet." Journal of Physics: Conference Series 2061, no. 1 (October 1, 2021): 012026. http://dx.doi.org/10.1088/1742-6596/2061/1/012026.

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Abstract At the macroscopic level, ferromagnetism is a quantum mechanical phenomenon. To describe magnetic materials, it is necessary to create a heuristic model that in terms of continuum mechanics describes the interaction between the lattice continuum, which is a carrier of deformations, and the magnetization field, which is associated with the spin continuum through the gyromagnetic effect. According to the laws of quantum mechanics, each individual particle is associated with a magnetic moment and an internal angular momentum – spin. Electrons mainly contribute to the magnetic moment of the atom. Therefore, the continuum is continuously associated with the discrete distribution of individual spins in a real ferromagnetic body known as the electron spin continuum. In addition, it is necessary to formulate field equations that, together with Maxwell’s equations, describe the electron spin continuum. After that, it is necessary to consider the interaction between the lattice continuum and the electron spin continuum. Elastic ferromagnets should be described with due regard to the spin density and couple stresses. The spin system is a carrier of the magnetic properties, and the mechanical properties are associated with the lattice. Thus, spin–lattice interactions indicate the relationship between magnetic and mechanical properties.

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20

Romano, S. "Elastic Constants and Pair Potentials for Nematogenic Lattice Models." International Journal of Modern Physics B 12, no. 22 (September 10, 1998): 2305–23. http://dx.doi.org/10.1142/s0217979298001344.

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Director configurations in nematic Liquid Crystals can be determined by minimizing their elastic free-energy density, on the basis of elastic constants and of specific boundary conditions; in some published cases, this has been obtained by numerical procedures where the elastic free-energy density plays the same role as the overall potential energy in a standard Monte Carlo simulation. The "potentials" used in these papers are short-ranged but, in general, not pairwise additive, unless the three elastic constants are set to a common value, thus reducing the potential to the well-known Lebwohl–Lasher lattice model.On the other hand, one can construct, possibly in different ways, a lattice model with pairwise additive interactions, approximately reproducing the elastic free-energy density, where parameters defining the pair potential are expressed as linear combinations of elastic constants; a nematogenic pair interaction of this kind, originally proposed by Gruhn and Hess (T. Gruhn and S. Hess, Z. Naturforsch.A51, 1 (1996)), has been investigated here by Monte Carlo simulation with periodic boundary conditions, i.e. aimed at the resulting bulk behavior.

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21

Xia, Muming, Hui Zhou, Qingqing Li, Hanming Chen, Yufeng Wang, and Shucheng Wang. "A General 3D Lattice Spring Model for Modeling Elastic Waves." Bulletin of the Seismological Society of America 107, no. 5 (September 25, 2017): 2194–212. http://dx.doi.org/10.1785/0120170024.

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22

Yan, Guangwu, Tingting Li, and Xianli Yin. "Lattice Boltzmann model for elastic thin plate with small deflection." Computers & Mathematics with Applications 63, no. 8 (April 2012): 1305–18. http://dx.doi.org/10.1016/j.camwa.2012.01.015.

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23

Sengsri, Pasakorn, Hao Fu, and Sakdirat Kaewunruen. "Mechanical Properties and Energy-Absorption Capability of a 3D-Printed TPMS Sandwich Lattice Model for Meta-Functional Composite Bridge Bearing Applications." Journal of Composites Science 6, no. 3 (February 24, 2022): 71. http://dx.doi.org/10.3390/jcs6030071.

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This paper reports on a proposed novel 3D-printed sandwich lattice model using a triply periodic minimal surface (TPMS) structure for meta-functional composite bridge bearings (MFCBBs). It could be implemented in bridge systems, including buildings and railway bridges. A TMPS structure offers a high performance to density ratio under different loading. Compared to typical elastomeric bridge bearings with any reinforcements, the use of 3D-printed TPMS sandwich lattices could potentially lead to a substantial reduction in both manufacturing cost and weight, but also to a significant increase in recyclability with their better mechanical properties (compressive, crushing, energy absorption, vibration, and sound attenuation). This paper shows predictions from a numerical study performed to examine the behaviour of a TPMS sandwich lattice model under two different loading conditions for bridge bearing applications. The validation of the modelling is compared with experimental results to ensure the possibility of designing and fabricating a 3D-printed TPMS sandwich lattice for practical use. In general, the compressive experimental and numerical load–displacement behaviour of the TPMS unit cell are in excellent agreement within the elastic limit region. Moreover, its failure mode for bridge bearing applications has been identified as an elastic–plastic and hysteretic failure behaviour under uniaxial compression and combined compression–shear loading, respectively.

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24

Shi, Jiale, Hythem Sidky, and Jonathan K. Whitmer. "Novel elastic response in twist-bend nematic models." Soft Matter 15, no. 41 (2019): 8219–26. http://dx.doi.org/10.1039/c9sm01395d.

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25

Mahbod, Mahshid, Masoud Asgari, and Christian Mittelstedt. "Architected functionally graded porous lattice structures for optimized elastic-plastic behavior." Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications 234, no. 8 (May 28, 2020): 1099–116. http://dx.doi.org/10.1177/1464420720923004.

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In this paper, the elastic–plastic mechanical properties of regular and functionally graded additively manufactured porous structures made by a double pyramid dodecahedron unit cell are investigated. The elastic moduli and also energy absorption are evaluated via finite element analysis. Experimental compression tests are performed which demonstrated the accuracy of numerical simulations. Next, single and multi-objective optimizations are performed in order to propose optimized structural designs. Surrogated models are developed for both elastic and plastic mechanical properties. The results show that elastic moduli and the plastic behavior of the lattice structures are considerably affected by the cell geometry and relative density of layers. Consequently, the optimization leads to a significantly better performance of both regular and functionally graded porous structures. The optimization of regular lattice structures leads to great improvement in both elastic and plastic properties. Specific energy absorption, maximum stress, and the elastic moduli in x- and y-directions are improved by 24%, 79%, 56%, and 9%, respectively, compared to the base model. In addition, in the functionally graded optimized models, specific energy absorption and normalized maximum stress are improved by 64% and 56%, respectively, in comparison with the base models.

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26

Xia, Muming, Hui Zhou, Hanming Chen, Qingchen Zhang, and Qingqing Li. "A rectangular-grid lattice spring model for modeling elastic waves in Poisson’s solids." GEOPHYSICS 83, no. 2 (March 1, 2018): T69—T86. http://dx.doi.org/10.1190/geo2016-0414.1.

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The lattice spring model (LSM) combined with the velocity Verlet algorithm is a newly developed scheme for modeling elastic wave propagation in solid media. Unlike conventional wave equations, LSM is established on the basis of micromechanics of the subsurface media, which enjoys better dynamic characteristics of elastic systems. We develop a rectangular-grid LSM scheme for elastic waves simulation in Poisson’s solids, and the direction-dependent elastic constants are deduced to keep the isotropy of the discrete grid. The stability condition and numerical dispersion properties of LSM are discussed and compared with other numerical methods. The 2D and 3D numerical simulations are carried out using the rectangular-grid LSM, as well as the second- and fourth-order accuracy staggered finite-difference method (FDM). Wavefields obtained by LSM are fairly similar with those by analytical solution and FDM, which demonstrates the correctness of the proposed scheme and its capability of modeling elastic wave propagation in heterogeneous media. Moreover, we perform plane P-wave simulation through a semi-infinite cavity model via LSM and FDM, the recorded wavefield snapshots indicate that our proposed rectangular-grid LSM obtains more reasonable wavefield details compared with those of FDM, especially in media with high compliance and structure complexity. Our main contribution lies in offering an alternative simulation scheme for modeling elastic wave propagation in media with some kinds of complexities, which conventional FDM may fail to simulate.

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27

Mahbod, Mahshid, and Masoud Asgari. "Multiobjective optimization of a newly developed additively manufactured functionally graded anisotropic porous lattice structure." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 234, no. 11 (February 11, 2020): 2233–55. http://dx.doi.org/10.1177/0954406220903743.

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In this paper, the elastic behavior of uniform and functionally graded porous lattice structures made by a double pyramid dodecahedron unit cell is investigated. Analytical solutions are derived in order to estimate the elastic moduli of the proposed structures in two directions. The analytical solution is validated by finite element simulations and experimental tests while the results show good agreement in general. The average difference between the numerical and analytical values of elastic modulus is under 14.44%, while the average error of experimental test and analytical solution is 15.69%. A comprehensive optimization is performed by considering elastic moduli in two different directions as objective functions. Various uniform lattice structures with different relative densities are optimized using NSGA-II algorithm as well as lattice structures with graded distribution of porosity. A variety of optimal designs are achieved by multiobjective optimization algorithm and the best point of the Pareto front is selected by the TOPSIS method. Furthermore, the functionally graded lattice structures are optimized by considering desirable relative densities in each layer and applying constructive constraints. Different distribution patterns of relative density are considered in layers in order to present the flexible design capability of the developed structure. The obtained results show that the elastic modulus is significantly dependent on the relative density of each layer as well as cell configuration. Also, different lattice structures could be achieved by applying desirable prescribed distribution of properties. A comparison between optimized and base model indicated that elastic moduli was considerably improved in optimized models. In optimization of uniform models, [Formula: see text] was increased by 115%, 89%, and 69% in optimized structures for relative densities of 10%, 30%, and 50%, respectively. Moreover, [Formula: see text] was improved in optimized models by 27%, 24%, and 18% for relative densities of 10%, 30%, and 50%, respectively.

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28

BEDI, S. S., MAJOR SINGH, and JASPAL SINGH. "THERMOELASTIC BEHAVIOUR OF FLUORITE SOLIDS USING THREE-BODY FORCE SHELL MODEL." International Journal of Modern Physics B 06, no. 19 (October 10, 1992): 3179–88. http://dx.doi.org/10.1142/s0217979292002437.

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The expressions for the second and third order elastic constants of fluorite lattice are derived using Lundqvist three-body potential incorporating thermal phonon pressure and inter-sublattice displacement through the shell model. Theoretically calculated values of the third order elastic constants are compared with the theoretical results of other workers and experiments. First order pressure derivatives of the second order elastic constants calculated using Thurston and Brugger relations for Ca 1-x Sr x F 2 and Sr 1-x Ba x F 2 are found to be in agreement with the experiment.

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29

Andrianov, Igor V., Vladyslav V. Danishevskyy, and Graham Rogerson. "Vibrations of nonlinear elastic lattices: low- and high-frequency dynamic models, internal resonances and modes coupling." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2236 (April 2020): 20190532. http://dx.doi.org/10.1098/rspa.2019.0532.

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We aim to study how the interplay between the effects of nonlinearity and heterogeneity can influence on the distribution and localization of energy in discrete lattice-type structures. As the classical example, vibrations of a cubically nonlinear elastic lattice are considered. In contrast with many other authors, who dealt with infinite and periodic lattices, we examine a finite-size model. Supposing the length of the lattice to be much larger than the distance between the particles, continuous macroscopic equations suitable to describe both low- and high-frequency motions are derived. Acoustic and optical vibrations are studied asymptotically by the method of multiple time scales. For numerical simulations, the Runge–Kutta fourth-order method is employed. Internal resonances and energy exchange between the vibrating modes are predicted and analysed. It is shown that the decrease in the number of particles restricts energy transfers to higher-order modes and prevents the equipartition of energy between all degrees of freedom. The conditions for a possible reduction in the original nonlinear system are also discussed.

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30

Ryzhkin, M. I., A. A. Levchenko, and I. A. Ryzhkin. "Peierls Instability of the Lieb Lattice." JETP Letters 116, no. 5 (September 2022): 307–12. http://dx.doi.org/10.1134/s002136402260152x.

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It is shown that the energy of the electron system in the two-dimensional Lieb lattice decreases owing to displacements of the edge atoms from the lattice sites along the edges. This decrease in the electron energy gives rise to soft phonon modes, anharmonic phonons, and to a lattice instability. Under certain conditions, the decrease in the electron energy can exceed the increase in the elastic energy of the ion lattice, and the total energy as a function of the displacements of edge atoms takes the form of a double-well potential. As a result, in the case of a pronounced instability, a partially ordered sublattice of edge atoms arises with the number of equilibrium positions twice as large as the number of atoms. The quantum tunneling of edge atoms between equilibrium positions results in the formation of quantum tunneling modes. The possible experimental manifestations of such instability and the extension of the model under study to the three-dimensional lattices are discussed.

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31

Leitner, D., S. Wassertheurer, M. Hessinger, and A. Holzinger. "A Lattice Boltzmann Model for pulsative blood flow in elastic vessels." e & i Elektrotechnik und Informationstechnik 123, no. 4 (April 2006): 152–55. http://dx.doi.org/10.1007/s00502-006-0332.

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32

Singh, Shyamendra Pratap, and U. C. Srivastava. "Lattice Dynamical Study and Elastic Property of Europium Telluride (Eute) Crystal." Oriental Journal Of Chemistry 37, no. 5 (October 30, 2021): 1091–95. http://dx.doi.org/10.13005/ojc/370511.

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In the present work authors are reporting complete lattice dynamical properties of Europium telluride (EuTe). The present model works on three body rigid ion model & three body rigid shell model (TRIM & TRSM). The short-range overlap repulsion is operative up to the second neighboring ions. An excellent agreement has been obtained between theory and experiment for their all-phonon properties of (EuTe) like phonon dispersion curves, Debye temperature variations, two-phonon IR/Raman spectra, third-fourth order lattice constant, pressure derivative and anharmonic elastic properties.

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33

Hou, Mengjie, Jinxing Liu, and Ai Kah Soh. "Modeling lattice metamaterials with deformable joints as an elastic micropolar continuum." AIP Advances 12, no. 6 (June 1, 2022): 065116. http://dx.doi.org/10.1063/5.0093094.

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Planar lattice metamaterials, such as periodic beam networks, are often considered as the micropolar continuum, where each material point has two translational degrees of freedom and one rotational degree of freedom. The joints through which bars are linked to one another are generally approximated as rigid. This study focuses on lattices with complex-structured deformable joints. The deformation field in each joint is obtained by conducting structural analyses. Once the “stiffness matrix” of the joint-centered unit cell is obtained by the finite element method, it can be used as the input for the standard procedure of calculating micropolar elastic moduli that are based on the equivalence of strain energy. As a result, effective moduli can be expressed in a semi-analytical form, meaning that only the cell structural stiffness is given numerically. The present model is validated by comparison to the FEM simulations. Particularly, the auxetic and anisotropic properties are discussed for various lattice metamaterials with deformable joints. We then take the obtained effective moduli as inputs to the in-house micropolar FEM code and obtain results agreeing well with the FEM structural simulations.

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34

Verma, A., M. L. Verma, and R. P. S. Rathore. "Elastic Behavior and Lattice Vibronics of bcc Phase Metals." Modern Physics Letters B 11, no. 05 (February 20, 1997): 209–18. http://dx.doi.org/10.1142/s0217984997000281.

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A model in real space has been developed by extending the generalized form of the exponential potential known as extended generalized exponential potential (EGEP) to account for (a) the correct nature of repulsive and attractive components of forces for all the separations in general and that of small separations in particular (b) the dielectric screening functions in an alternative and simpler form through the parameter m (c) the three-body forces such as volume forces in an indirect way in the framework of EGEP through the parameter n. The model is employed to compute the cohesive energy, phonon spectra and second-order elastic constants in group IV bcc metals. The predictions show good agreement with experimental findings.

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35

Bursac´, Predrag, C. Victoria McGrath, Solomon R. Eisenberg, and Dimitrije Stamenovic´. "A Microstructural Model of Elastostatic Properties of Articular Cartilage in Confined Compression." Journal of Biomechanical Engineering 122, no. 4 (March 30, 2000): 347–53. http://dx.doi.org/10.1115/1.1286561.

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A microstructural model of cartilage was developed to investigate the relative contribution of tissue matrix components to its elastostatic properties. Cartilage was depicted as a tensed collagen lattice pressurized by the Donnan osmotic swelling pressure of proteoglycans. As a first step in modeling the collagen lattice, two-dimensional networks of tensed, elastic, interconnected cables were studied as conceptual models. The models were subjected to the boundary conditions of confined compression and stress–strain curves and elastic moduli were obtained as a function of a two-dimensional equivalent of swelling pressure. Model predictions were compared to equilibrium confined compression moduli of calf cartilage obtained at different bath concentrations ranging from 0.01 to 0.50 M NaCl. It was found that a triangular cable network provided the most consistent correspondence to the experimental data. The model showed that the cartilage collagen network remained tensed under large confined compression strains and could therefore support shear stress. The model also predicted that the elastic moduli increased with increasing swelling pressure in a manner qualitatively similar to experimental observations. Although the model did not preclude potential contributions of other tissue components and mechanisms, the consistency of model predictions with experimental observations suggests that the cartilage collagen network, prestressed by proteoglycan swelling pressure, plays an important role in supporting compression. [S0148-0731(00)00704-4]

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36

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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37

Phlip, Thresiamma, C. S. Menon, and K. Indulekha. "Higher Order Elastic Constants, Gruneisen Parameters and Lattice Thermal Expansion of Trigonal Calcite." E-Journal of Chemistry 2, no. 4 (2005): 207–17. http://dx.doi.org/10.1155/2005/913794.

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The second- and third-order elastic constants of trigonal calcite have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second- and third-order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The seven second-order elastic constants and the fourteen non-vanishing third-order elastic constants of trigonal calcite are obtained. The second-order elastic constants C11, which corresponds to the elastic stiffness along the basal plane of the crystal is greater than C33, which corresponds to the elastic stiffness tensor component along the c-axis of the crystal. First order pressure derivatives of the second-order elastic constants of calcite are evaluated. The higher order elastic constants are used to find the generalized Gruneisen parameters of the elastic waves propagating in different directions in calcite. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with available reported values.

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38

Alwattar, Tahseen, and Ahsan Mian. "Development of an Elastic Material Model for BCC Lattice Cell Structures Using Finite Element Analysis and Neural Networks Approaches." Journal of Composites Science 3, no. 2 (April 1, 2019): 33. http://dx.doi.org/10.3390/jcs3020033.

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Lattice cell structures (LCS) are being investigated for applications in sandwich composites. To obtain an optimized design, finite element analysis (FEA) -based computational approach can be used for detailed analyses of such structures, sometime at full scale. However, developing a large-scale model for a lattice-based structure is computationally expensive. If an equivalent solid FEA model can be developed using the equivalent solid mechanical properties of a lattice structure, the computational time will be greatly reduced. The main idea of this research is to develop a material model which is equivalent to the mechanical response of a lattice structure. In this study, the mechanical behavior of a body centered cubic (BCC) configuration under compression and within elastic limit is considered. First, the FEA approach and theoretical calculations are used on a single unit cell BCC for several cases (different strut diameters and cell sizes) to predict equivalent solid properties. The results are then used to develop a neural network (NN) model so that the equivalent solid properties of a BCC lattice of any configuration can be predicted. The input data of NN are bulk material properties and output data are equivalent solid mechanical properties. Two separate FEA models are then developed for samples under compression: one with 5 × 5 × 4 cell BCC and one completely solid with equivalent solid properties obtained from NN. In addition, 5 × 5 × 4 cell BCC LCS specimens are fabricated on a Fused Deposition Modeling uPrint SEplus 3D printer using Acrylonitrile Butadiene Styrene (ABS) and tested under compression. Experimental load-displacement behavior and the results obtained from both the FEA models are in good agreement within the elastic limit.

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39

Li, Ming Tian, Xia Ting Feng, and Hui Zhou. "2D Vector Cellular Automata Model for Simulating Fracture of Rock under Tensile Condition." Key Engineering Materials 261-263 (April 2004): 705–10. http://dx.doi.org/10.4028/www.scientific.net/kem.261-263.705.

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Based on the cellular automata of the plane truss structure, a 2D cellular automata model is presented to simulate the fracture of rock at meso-level. Cellular automata are made up of cell, states, lattice, neighbor and rule. Rock is divided into lattice in which each lattice point presents a cell. Each cell is assumed to connect with several cells, which are called as its neighbors, in virtue of truss elements. The truss elements can adopt some different simple local laws, i.e. constitutive law, which may be elastic or elastic-plastic and the simple fracture rule. It also can adopt different mechanical properties, which present their heterogeneity and anisotropy. This model can make full use of the advantages of cellular automata such as its intrinsic parallelism, localization and so on. In the meantime, as a powerful tool to analyze the nonlinear, complex system, cellular automata can be used to study the nonlinear, complex fracture process. The model is used to simulate the direct tensile of the rock plates, the complete fracture process and the stress-strain curves are attained which are accordance with the experiment.

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40

Jaszczak, J. A., and D. Wolf. "On the elastic behavior of composition-modulated superlattices." Journal of Materials Research 6, no. 6 (June 1991): 1207–18. http://dx.doi.org/10.1557/jmr.1991.1207.

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Atomistic computer simulations are used to systematically investigate the role of interfacial disorder on the elastic behavior of composition-modulated superlattices of fcc metals, represented by simple Lennard–Jones potentials. The structures, energies, and average elastic properties of four types of superlattices with various degrees of interfacial disorder are computed as a function of the modulation wavelength along [001]. The four superlattice types studied include perfectly coherent, incoherent, and two types derived from these by introducing relative twists about [001] between alternating layers. A 20% lattice-parameter mismatch between the two modulating materials is assumed. Results are compared with our earlier work on unsupported thin films, grain-boundary superlattices, and incoherent superlattices with a 10% lattice-parameter mismatch. The degree of structural disorder at the interfaces is found to correlate well with the magnitude of the elastic anomalies, which cannot be accounted for by anisotropic lattice-parameter changes alone. The grain-boundary superlattices studied earlier are found to provide a good model limit for the elastic behavior of interfacially disordered dissimilar-material superlattices.

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41

Price, Geoffrey D., Stephen C. Parker, and Maurice Leslie. "The lattice dynamics of forsterite." Mineralogical Magazine 51, no. 359 (March 1987): 157–70. http://dx.doi.org/10.1180/minmag.1987.051.359.18.

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AbstractWe use an approach based upon the atomistic or Born model of solids, in which potential functions represent the interactions between atoms in a structure, to calculate the infrared and Raman vibrational frequencies of forsterite. We investigate a variety of interatomic potentials, and find that although all the potentials used reproduce the structural and elastic behaviour of forsterite, only one potential (THB1) accurately predicts its lattice dynamics. This potential includes ‘bond-bending’ terms, that model the directionality of the Si-O bond, which we suggest plays a major role in determining the structural and physical properties of silicates. The potential was derived empirically from the structural and physical data of simple oxides, and its ability to model the lattice dynamics of forsterite is a significant advance over previous, force-constant models, which have been simply derived by fitting to the spectroscopic data that they aim to model. The success that we have had in predicting the lattice dynamics of forsterite indicates that the potential provides the previously elusive yet fundamental, quantitative link between the microscopic or atomistic behaviour of a mineral and its macroscopic or bulk thermodynamic properties.

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42

Booske, John H., Reid F. Cooper, and Ian Dobson. "Mechanisms for nonthermal effects on ionic mobility during microwave processing of crystalline solids." Journal of Materials Research 7, no. 2 (February 1992): 495–501. http://dx.doi.org/10.1557/jmr.1992.0495.

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Models for nonthermal effects on ionic motion during microwave heating of crystalline solids are considered to explain the anomolous reductions of activation energy for diffusion and the overall faster kinetics noted in microwave sintering experiments and other microwave processing studies. We propose that radiation energy couples into low (microwave) frequency elastic lattice oscillations, generating a nonthermal phonon distribution that enhances ion mobility and thus diffusion rates. Viewed in this manner, it is argued that the effect of the microwaves would not be to reduce the activation energy, but rather to make the use of a Boltzmann thermal model inappropriate for the inference of activation energy from sintering-rate or tracer-diffusion data. A highly simplified linear oscillator lattice model is used to qualitatively explore coupling from microwave photons to lattice oscillations. The linear mechanism possibilities include resonant coupling to weak-bond surface and point defect modes, and nonresonant coupling to zero-frequency displacement modes. Nonlinear mechanisms such as inverse Brillouin scattering are suggested for resonant coupling of electromagnetic and elastic traveling waves in crystalline solids. The models suggest that nonthermal effects should be more pronounced in polycrystalline (rather than single crystal) forms, and at elevated bulk temperatures.

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43

Baksamawi, Hosam Alden, Mostapha Ariane, Alexander Brill, Daniele Vigolo, and Alessio Alexiadis. "Modelling Particle Agglomeration on through Elastic Valves under Flow." ChemEngineering 5, no. 3 (July 26, 2021): 40. http://dx.doi.org/10.3390/chemengineering5030040.

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This work proposes a model of particle agglomeration in elastic valves replicating the geometry and the fluid dynamics of a venous valve. The fluid dynamics is simulated with Smooth Particle Hydrodynamics, the elastic leaflets of the valve with the Lattice Spring Model, while agglomeration is modelled with a 4-2 Lennard-Jones potential. All the models are combined together within a single Discrete Multiphysics framework. The results show that particle agglomeration occurs near the leaflets, supporting the hypothesis, proposed in previous experimental work, that clot formation in deep venous thrombosis is driven by the fluid dynamics in the valve.

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44

Philip, Thresiamma, C. S. Menon, and K. Indulekha. "Higher Order Elastic Constants, Gruneisen Parameters and Lattice Thermal Expansion of Lithium Niobate." E-Journal of Chemistry 3, no. 3 (2006): 122–33. http://dx.doi.org/10.1155/2006/842320.

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The second and third-order elastic constants and pressure derivatives of second- order elastic constants of trigonal LiNbO3(lithium niobate) have been obtained using the deformation theory. The strain energy density estimated using finite strain elasticity is compared with the strain dependent lattice energy density obtained from the elastic continuum model approximation. The second-order elastic constants and the non-vanishing third-order elastic constants along with the pressure derivatives of trigonal LiNbO3are obtained in the present work. The second and third-order elastic constants are compared with available experimental values. The second-order elastic constant C11which corresponds to the elastic stiffness along the basal plane of the crystal is less than C33which corresponds to the elastic stiffness tensor component along thec-axis of the crystal. The pressure derivatives, dC'ij/dp obtained in the present work, indicate that trigonal LiNbO3is compressible. The higher order elastic constants are used to find the generalized Gruneisen parameters of the elastic waves propagating in different directions in LiNbO3. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with available reported values.

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45

Upadhyaya, S. C., J. C. Upadhyaya, and R. Shyam. "Model-potential study of the lattice dynamics and elastic constants of theNi0.55Pd0.45alloy." Physical Review B 44, no. 1 (July 1, 1991): 122–29. http://dx.doi.org/10.1103/physrevb.44.122.

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46

Stelitano, Davide, and Daniel H. Rothman. "Fluctuations of elastic interfaces in fluids: Theory, lattice-Boltzmann model, and simulation." Physical Review E 62, no. 5 (November 1, 2000): 6667–80. http://dx.doi.org/10.1103/physreve.62.6667.

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47

Katsuno, Hiroyasu, Makio Uwaha, and Yukio Saito. "Heteroepitaxial growth modes with dislocations in a two-dimensional elastic lattice model." Surface Science 602, no. 22 (November 2008): 3461–66. http://dx.doi.org/10.1016/j.susc.2008.03.049.

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48

Saleem, Mohammad, Muhammad Rafique, Haris Rashid, and Fazal-E-Aleem. "The Chou-Yang model, lattice quantum chromodynamics and hyperon-proton elastic scattering." Pramana 29, no. 5 (November 1987): 469–83. http://dx.doi.org/10.1007/bf02845787.

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49

Hyunjune Yim and Younghoon Sohn. "Numerical simulation and visualization of elastic waves using mass-spring lattice model." IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 47, no. 3 (May 2000): 549–58. http://dx.doi.org/10.1109/58.842041.

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50

Thapa, SRB. "Study of lattice dynamics of silicon." BIBECHANA 9 (December 5, 2012): 13–17. http://dx.doi.org/10.3126/bibechana.v9i0.7145.

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Lattice dynamics of Si has been investigated by using a Urey-Bradley Valence Force Field (UVFF) model which is a phenomenological model. In this model following interactions are taken into account: (i) the central force due to bond-stretching (ii) the angular force due to bond bending (iii) central force between non-bonded atoms (iv) the force due to interaction of one internal co-ordinate to adjacent internal coordinate. Calculated results of phonon dispersion curves, Debye Characteristic temperature, microscopic elastic constants and Bulk modulus of Si are compared with experimental results giving fairly good agreement. DOI: http://dx.doi.org/10.3126/bibechana.v9i0.7145BIBECHANA 9 (2013) 13-17

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