The numerical solution of nonlinear problems on deformation and buckling of atomic lattices

Korobeinikov, S. N.

doi:10.1023/b:frac.0000040995.13933.e0

Cited by 9 publications

(5 citation statements)

References 2 publications

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“…Заметим, что уравнения механики деформируемого твердого тела [31], используемые при численном моделировании методом конечных элементов, не содержат характерный линейный размер d структуры материала. При дискретизации расчетной области естественным образом вводится структурный параметр, совпадающий с размером конечного элемента.…”

Section: сравнение численных и аналитических результатовunclassified

Simulation of Elastoplastic Fracture of a Center Cracked Plate

Astapov

Кургузов

2023

PNRPU Mechanics Bulletin

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The strength of a square plate with a central crack at normal separation was studied within the framework of the Neuber–Novozhilov approach using a modified Leonov–Panasyuk–Dugdale model using an additional parameter, the diameter of the plasticity zone (width of the pre-fracture zone). As a model of a deformable solid body, a model of an ideal elastic-plastic material with a limiting relative elongation was chosen. This class of materials includes, for example, low-alloy steels used in structures operating at temperatures below the cold brittleness threshold. In the presence of a singular feature in the stress field in the vicinity of the crack tip, it is proposed to use a two-parameter discrete integral strength criterion. The deformation fracture criterion is formulated at the tip of a real crack, and the force criterion for normal stresses, taking into account averaging, is formulated at the tip of a model crack. The lengths of real and model cracks differ by the length of the pre-fracture zone. The constitutive equations of the analytical model are analyzed in detail depending on the characteristic linear dimension of the material structure. Simple formulas suitable for verification calculations for the critical breaking load and the length of the pre-fracture zone are obtained. Numerical modeling of the propagation of plasticity zones in square plates under quasi-static loading has been performed. In the numerical model, the updated Lagrangian formulation of the equations of mechanics of a deformable solid body is used, which is most preferable for modeling the deformation of bodies made of an elastic-plastic material at large deformation. The plastic zone in the vicinity of the crack tip is obtained by the finite element method. The results of analytical and numerical prediction of plate fracture under plane deformation are compared. It is shown that the results of numerical experiments are in good agreement with the results of calculations using the analytical model of fracture of materials with a structure under normal separation. Diagrams of quasi-brittle and quasi-ductile fracture of a structured plate are constructed.

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Section: сравнение численных и аналитических результатовunclassified

Simulation of Elastoplastic Fracture of a Center Cracked Plate

Astapov

Кургузов

2023

PNRPU Mechanics Bulletin

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“…is satisfied, are referred to as singular points (see e.g. [68, 99]). (Hereinafter, we use the left superscript of a quantity to denote the time at which this quantity is considered.)…”

Section: Quasi-static Motion Equations and Buckling Criteria For Slgssmentioning

confidence: 99%

“…In computer simulations of the deformation of discrete structures, singular points are usually turning or bifurcation points on the integral curve (see e.g. [68, 99]). Let a defect of the matrix τ K be equal to I ¯ , and let W i ( i = 1 to I ¯ , 1 ≤ I ¯ ≤ NEQ ) be the vectors forming the basis of the null-space of the matrix τ K , in other words,…”

Section: Quasi-static Motion Equations and Buckling Criteria For Slgssmentioning

confidence: 99%

Quasi-static buckling simulation of single-layer graphene sheets by the molecular mechanics method

Korobeynikov

Alyokhin

Аннин

et al. 2014

Mathematics and Mechanics of Solids

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This paper presents a quasi-static nonlinear buckling analysis of compressed single-layer graphene sheets (SLGSs) using the molecular mechanics method. Bonded interactions between carbon atoms are simulated using a modified parameter set of the DREIDING force field that leads to better agreement between simulated mechanical properties of graphene and reference literature data than the standard parameter set of this force field (see Mayo et al., J Phys Chem 1990; 94: 8897–8909). Identification of constraints of atoms of the SLGS edges with the boundary conditions of clamped and simply supported thin plates is made. The buckling loads and modes obtained by linear and nonlinear buckling analysis of a compressed quadratic SLGS with a side length of 6 nm are shown to be close to each other. In addition, it has been found by nonlinear buckling analysis that only equilibrium configurations with modes of initial post-buckling deformed configurations correlated with the one-half-wave column-like buckling mode have stable equilibrium configurations for clamped and simply supported SLGSs. As the edges of a simply supported SLGS approach each other, the geometry of this mode of post-buckling deformation with inclusion of the non-bonded van der Waals (vdW) interactions between carbon atoms becomes closer to the geometry of a single-walled carbon nanotube, and without inclusion of the vdW interactions, this mode has the geometry of a cylinder with a drop-shaped cross-section.

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“…Numerical methods for solving general 2/D and 3/D problems of quasistatic deformation of atomic lattices have been developed in Korobeynikov [4], Dluzewski and Traczykowski [5], Korobeynikov [6]. In the typical case, the Cauchy problem is solved in the form (1) where is the displacement vector of atoms in a lattice, is the vector of internal forces acting on lattice atoms, is the symmetric tangential stiffness matrix of a lattice, a superimposed dot denotes derivative of the magnitude with respect to the monotonically increasing deformation parameter .…”

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confidence: 99%

“…Presented procedures of solution continuation through singular points of integral curves require as exact definition of a matrix as possible in order to improve a convergence as well as for prevent a divergence of iteration processes applied for the solution refinement. The expressions refined in comparison with expressions given in [4,6], which account for both tension/compression of segments connecting atomic pairs and their rotations have been proposed by Korobeynikov…”

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confidence: 99%

Determination of Equilibrium Configurations of Atomic Lattices at Quasistatic Deformation

Korobeynikov

Fracture of Nano and Engineering Materials and Structures

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The numerical solution of nonlinear problems on deformation and buckling of atomic lattices

Cited by 9 publications

References 2 publications

Simulation of Elastoplastic Fracture of a Center Cracked Plate

Simulation of Elastoplastic Fracture of a Center Cracked Plate

Quasi-static buckling simulation of single-layer graphene sheets by the molecular mechanics method

Determination of Equilibrium Configurations of Atomic Lattices at Quasistatic Deformation

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