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Derivation of Born/von Karman difference equations through consistent lattice angular interactions

CHAOS SOLITONS & FRACTALS 2025年 195卷

作者： Challamel, Noel Nguyen, H. P. Wang, C. M. Ruta, Giuseppe Univ Bretagne Sud Ctr Rech CNRS UMR 6027 IRDL Rue St MaudeBP92116 F-56100 Lorient France Van Lang Univ Inst Computat Sci & Artificial Intelligence Lab Computat Civil Engn Ho Chi Minh City Vietnam Van Lang Univ Fac Civil Engn Sch Technol Ho Chi Minh City Vietnam Univ Queensland Sch Civil Engn St Lucia Qld 4072 Australia Univ Roma La Sapienza Dipartimento Ingn Strutturale & Geotecn Rome Italy

This paper investigates statics and dynamics of two-dimensional (2D) linear elastic lattices and their continuum approximations. Focus is on the mixed differential-difference equations proposed by Born and von Karman in 1912 for cubic lattices with both central and non-central interactions, applied here to 2D lattices. The non-central interaction introduced by Born and von Karman, classified as a shear interaction, corresponds to a framedependent (non-objective) angular interaction. A consistent connection between discrete and continuum elasticity can be achieved by incorporating the proper non-central (angular) interactions introduced by Gazis et al. in 1960. Inspired by Hrennikoff's truss scheme, we constructed an alternative lattice model, based on Gazis et al. formulation, that is augmented with additional objective internal angular interactions. The difference operators associated with both consistent lattices, i.e. Gazis et al. lattice and the new Hrennikoff inspired lattice, are shown to differ. The conditions of positive definiteness of the associated potential energy of each lattice, are discussed. The new lattice successfully bridges the gap between discrete and continuum elasticity, while being governed by the same mixed differential-difference equations proposed by Born and von Karman. It can reproduce macroscopic auxetic behaviour, while preserving the positive definiteness of its discrete potential energy. This finding resolves the long-standing inconsistency of Born and von Karman's equations, which, originally derived from flawed physical assumptions, can now be justified through correct mathematical reasoning.

关键词： Lattice/discrete elasticity continuum elasticity Elastodynamics Difference equations Partial differential equations In-plane vibrations Eigenvalue problem Discrete plate

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Analytical solutions for free vibrations of rectangular cuboid elastic lattices and their continuous approximations

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JOURNAL OF SOUND AND VIBRATION 2025年 600卷

作者： Nguyen, H. P. Challamel, Noel Wang, C. M. Van Lang Univ Inst Computat Sci & Artificial Intelligence Lab Computat Civil Engn Ho Chi Minh City Vietnam Van Lang Univ Fac Civil Engn Sch Technol Ho Chi Minh City Vietnam Univ Bretagne Sud Ctr Rech IRDL CNRS UMR 6027 Rue St MaudeBP92116 F-56321 Lorient France Univ Queensland Sch Civil Engn St Lucia Qld 4072 Australia

This paper presents analytical solutions for the free vibration of an elastic cuboid lattice (rectangular parallelepiped) with sliding supports along its planar boundaries. The lattice model includes both central and angular interactions. The free vibration problem is solved by formulating a 3D difference eigenvalue problem, and exact solutions for the eigenfrequencies and eigenmodes are derived analytically using a trigonometric discrete displacement field. A cubic equation for the eigenfrequency squares is obtained, with solutions determined using Cardano's formula. The derived exact solutions for the finite cuboid lattice are validated by comparison with solutions obtained from a discrete algebraic method, calibrated for accurate stiffness and mass properties both within the lattice and at its boundaries. Furthermore, these exact solutions for the 3D lattice are benchmarked against analytical solutions for the corresponding 3D continuum, based on classical elasticity, lattice-based gradient elasticity, and lattice-based nonlocal elasticity theories, demonstrating their accuracy and reliability for free vibration analysis.

关键词： Three-dimensional lattice continuum elasticity Partial differential equations Eigenvalue problem Strain gradient elasticity Nonlocal elasticity Discrete elasticity

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Membrane Tension Inhibits Lipid Mixing by Increasing the Hemifusion Stalk Energy

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ACS NANO 2023年第19期17卷 18942-18951页

作者： Shendrik, Petr Golani, Gonen Dharan, Raviv Schwarz, Ulrich S. Sorkin, Raya Tel Aviv Univ Raymond & Beverly Sackler Fac Exact Sci Sch Chem IL-6997801 Tel Aviv Israel Tel Aviv Univ Ctr Phys & Chem Living Syst IL-6997801 Tel Aviv Israel Heidelberg Univ Inst Theoret Phys D-69120 Heidelberg Germany Heidelberg Univ BioQuant Ctr Quantitat Biol D-69120 Heidelberg Germany

Fusion of biological membranes is fundamental in various physiological events. The fusion process involves several intermediate stages with energy barriers that are tightly dependent on the mechanical and physical properties of the system, one of which is membrane tension. As previously established, the late stages of fusion, including hemifusion diaphragm and pore expansions, are favored by membrane tension. However, a current understanding of how the energy barrier of earlier fusion stages is affected by membrane tension is lacking. Here, we apply a newly developed experimental approach combining micropipette-aspirated giant unilamellar vesicles and optically trapped membrane-coated beads, revealing that membrane tension inhibits lipid mixing. We show that lipid mixing is 6 times slower under a tension of 0.12 mN/m compared with tension-free membranes. Furthermore, using continuum elastic theory, we calculate the dependence of the hemifusion stalk formation energy on membrane tension and intermembrane distance and find the increase in the corresponding energy barrier to be 1.6 k(B)T in our setting, which can explain the increase in lipid mixing time delay. Finally, we show that tension can be a significant factor in the stalk energy if the pre-fusion intermembrane distance is on the order of several nanometers, while for membranes that are tightly docked, tension has a negligible effect.

关键词： membrane fusion tension micropipette aspiration optical tweezers continuum elasticity

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Universal Behavior of Atomistic Strain in Self-Assembled Quantum Dots

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IEEE JOURNAL OF QUANTUM ELECTRONICS 2016年第7期52卷 1-8页

作者： Ilatikhameneh, Hesameddin Ameen, Tarek A. Klimeck, Gerhard Rahman, Rajib Purdue Univ Dept Elect & Comp Engn W Lafayette IN 47907 USA Purdue Univ Network Computat Nanotechnol Ctr W Lafayette IN 47907 USA

Self-assembled quantum dots (QDs) are highly strained heterostructures. The lattice strain significantly modifies the electronic and optical properties of these devices. A universal behavior is observed in atomistic strain simulations (in terms of both strain magnitude and profile) of QDs with different shapes and materials. In this paper, this universal behavior is investigated by atomistic as well as analytic continuum models. Atomistic strain simulations are very accurate but computationally expensive. On the other hand, analytic continuum solutions are based on assumptions that significantly reduce the accuracy of the strain calculations, but are very fast. Both techniques indicate that the strain depends on the aspect ratio (AR) of the QDs, and not on the individual dimensions. Thus, simple closed-form equations are introduced which directly provide the atomistic strain values inside the QD as a function of the AR and the material parameters. Moreover, the conduction and valence band edges E-C/V and their effective masses m*(C/V) of the QDs are dictated by the strain and AR consequently. The universal dependence of atomistic strain on the AR is useful in many ways. Not only does it reduce the computational cost of atomistic simulations significantly, but it also provides information about the optical transitions of QDs given the knowledge of E-C/V and m*(C/V) from AR. Finally, these expressions are used to calculate optical transition wavelengths in InAs/GaAs QDs, and the results agree well with the experimental measurements and atomistic simulations.

关键词： Self-assembled quantum dots Stranski-Krastanov atomistic strain analytical continuum strain continuum elasticity optical transition

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Maxwell Lattices and Topological Mechanics

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ANNUAL REVIEW OF CONDENSED MATTER PHYSICS, VOL 9 2018年第1期9卷 413-433页

作者： Mao, Xiaoming Lubensky, Tom C. Univ Michigan Dept Phys Ann Arbor MI 48109 USA Univ Michigan Dept Phys Ann Arbor MI 48109 USA Univ Penn Dept Phys & Astron Philadelphia PA 19104 USA

This is a review on the emergent field of topological mechanics, where concepts from electronic topological states of matter are applied to mechanics. We focus on the subcategory of topological mechanics of Maxwell lattices, which are mechanical frames having average coordination numbers equal to twice their spatial dimension, < z > = 2d, leaving them on the verge of mechanical instability. We start by introducing examples of Maxwell lattices, describing their elastic properties, and discussing their general classification. We then focus on topological phonon modes of these lattices by reviewing recent theoretical progress on one-dimensional chains and two-dimensional lattices that exhibit topologically protected zero-frequency phonon modes on edges and domainwalls. We also proposemetamaterials based on Maxwell lattices with unusual topologically protected mechanical properties.

关键词： mechanics of frames continuum elasticity zero modes surface mode mechanical metamaterials

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Discrete and continuous models of linear elasticity: history and connections

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continuum MECHANICS AND THERMODYNAMICS 2023年第2期35卷 347-391页

作者： Challamel, Noel Zhang, Y. P. Wang, C. M. Ruta, Giuseppe dell'Isola, Francesco Univ Bretagne Sud IRDL CNRS UMR 6027 Ctr Rech Rue St MaudeBP92116 F-56321 Lorient France Univ Queensland Sch Civil Engn St Lucia Qld 4072 Australia Univ Roma La Sapienza Dipartimento Ingn Strut & Geotecn Rome Italy Grp Nazl Fis Matemat Rome Italy Univ Aquila Laquila Italy Int Res Ctr Math & Mech Complex Syst M&MOCS Laquila Italy

This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed differential-difference equation of a one-dimensional (1D) lattice and the partial differential equation of the associated continuum. A consistent three-dimensional (3D) generalization of this model was given much later: Poincare and Voigt reconciled the molecular and the continuum approaches at the end of the nineteenth century, but only in 1912 Born and von Karman presented the mixed differential-difference equations of discrete isotropic elasticity. Their model is a 3D generalization of Lagrange's 1D lattice and considers longitudinal, diagonal and shear elastic springs among particles, so the associated continuum is characterized by three elastic constants. Born and von Karman proved that the lattice equations converge to Navier's partial differential ones asymptotically, thus being a formulation of continuous elasticity in terms of spatial finite differences, as for Lagrange's 1D lattice. Neglecting shear springs in Born-Karman's lattice equals to Navier's assumption of pure central forces among molecules: in the limit, the lattice behaves as a one-parameter isotropic solid ("rari-constant" theory: equal Lame parameters, or, equivalently, Poisson's ratio upsilon = 1/4). Hrennikoff and McHenry revisited the lattice approach with pure central interactions using a plane truss;the equivalent Born-Karman's lattice in plane stress in the limit tends to a continuum with Poisson's ratio upsilon = 1/3. Contrary to McHenry-Hrennikoff's truss, Born-Karman's lattice leads to a "free" Poisson's ratio bounded by its "limit' bound (upsilon = 1/4 for plane strain or 3D elasticity;upsilon = 1/3 for plane stress elasticity). Unfortunately, Born

关键词： Lattice/discrete elasticity continuum elasticity Elastodynamics Discrete elasticity Difference equations Partial differential equations Microstructured solids Constrained kinematics Metamaterials

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Quantification of the effects of misfit strain on the energy states of zinc-blende spherical core/shell quantum dots

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JOURNAL OF THE KOREAN PHYSICAL SOCIETY 2021年第1期79卷 87-94页

作者： Nguyen, Anh Tay Seo, Youjung Park, Seoung-Hwan Pak, Y. Eugene SUNY Korea Dept Mech Engn Incheon South Korea Seoul Natl Univ Grad Sch Convergence Sci & Technol Nano Sci & Technol Suwon 443270 South Korea Catholic Univ Daegu Dept Elect Engn Hayang 712702 Kyeongbuk South Korea

The effects of lattice-mismatched strain on the optical properties of type-I core/shell quantum dot nanostructures are investigated theoretically using the simple continuum elasticity and the effective mass quantum mechanics models. The misfit strains are calculated based on the spherically anisotropic linear elasticity solution for a misfitted inclusion embedded in a finite matrix. The effective mass quantum mechanics model for the case of a CdSe/CdS core/shell quantum dot (QD) shows that the subband energy is blue shifted by the compressive strain in the QD core resulting from the lattice-mismatched heterojunction. Using a very simple spherically anisotropic strain model, we calculated the subband, interband transition, and binding energies for a zinc-blende (ZB) CdSe/CdS QD. These results compare favorably with those obtained from the finite element analysis (FEA) considering the rectilinear cubic crystal anisotropy.

关键词： Quantum dot Quantum mechanics continuum elasticity Lattice misfit Spherically symmetric model Ground-state energy shift

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Physically based deformable models in computer graphics

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COMPUTER GRAPHICS FORUM 2006年第4期25卷 809-836页

作者： Nealen, Andrew Mueller, Matthias Keiser, Richard Boxerman, Eddy Carlson, Mark Tech Univ Berlin Berlin Germany ETH Comp Graph Lab Zurich Switzerland Ubisoft Montreal Montreal PQ Canada

physically based deformable models have been widely embraced by the Computer Graphics community. Many problems outlined in a previous survey by Gibson and Mirtich have been addressed, thereby making these models interesting and useful for both offline and real-time applications, such as motion pictures and video games. In this paper, we present the most significant contributions of the past decade, which produce such impressive and perceivably realistic animations and simulations: finite element/difference/volume methods, mass-spring systems, mesh-free methods, coupled particle systems and reduced deformable models-based on modal analysis. For completeness, we also make a connection to the simulation of other continua, such as fluids, gases and melting objects. Since time integration is inherent to all simulated phenomena, the general notion of time discretization is treated separately, while specifics are left to the respective models. Finally, we discuss areas of application, such as elastoplastic deformation and fracture, cloth and hair animation, virtual surgery simulation, interactive entertainment and fluid/smoke animation, and also suggest areas for future research.

关键词： physically based animation deformation continuum elasticity time integration FEM mass-spring mesh-free methods modal analysis fluid animation

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Simulation of inhomogeneous strain in Ge-Si core-shell nanowires

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Frontiers of Electrical and Electronic Engineering in China 2009年第3期4卷 342-347页

作者： Yuhui HE Yuning ZHAO Chun FAN Xiaoyan LIU Ruqi HAN Institute of Microelectronics Peking UniversityBeijing 100871China Computer Center of Peking University Beijing 100871China

This paper studies the elastic deformation field in lattice-mismatched Ge-Si core-shell nanowires(NWs).Infinite wires with a cylindrical cross section under the assumption of translational symmetry are *** strain distributions are found by minimizing the elastic energy per unit cell using finite element *** paper finds that the trace of the strain is discontinuous with a simple,almost piecewise variation between core and shell,whereas the individual components of the strain can exhibit complex *** simulation results are prerequisite of strained band structure calculation,and pave a way for further investigation of strain effect on the related transport property simulation.

关键词： core-shell nanowire strain continuum elasticity

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Physically based deformable models in computer graphics

Physically based deformable models in computer graphics

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18th Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI 2005)

作者： Nealen, Andrew Mueller, Matthias Keiser, Richard Boxerman, Eddy Carlson, Mark Tech Univ Berlin Berlin Germany ETH Comp Graph Lab Zurich Switzerland Ubisoft Montreal Montreal PQ Canada

关键词： physically based animation deformation continuum elasticity time integration FEM mass-spring mesh-free methods modal analysis fluid animation

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