Strips of localization of plastic deformation

Labibov, R. R.; Chernyakov, Yu. A.; Sheveleva, A.; Shevchenko, A. G.

doi:10.1007/s00419-018-1445-z

Cited by 3 publications

(2 citation statements)

References 7 publications

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“…Problems solving of dynamics and strength of machines, elements of technological equipment implies the study of mathematical models described by dynamic boundary-value problems of the theory of elasticity [9,10]. At present, numerical [11,12], qualitative [13,14] and analytical [15,16] approaches to constructing solutions of boundary-value problems of dynamic and static theory of elasticity have been developed.…”

Section: Introductionmentioning

confidence: 99%

The study of mathematical models of the linear theory of elasticity by presenting the fundamental solution in harmonic potentials

Pabyrivskyi¹,

Pabyrivska²,

Пукач³

2020

Math. Model. Comput.

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In this paper, the approaches to the study of mathematical models of the theory of linear elasticity are developed. The general formulation of the 3-dimensional problem based on the representation of the fundamental solution in the form of V. P. Revenko in terms of spatial harmonic functions is considered. The formulation in the harmonic potentials of the 3-dimensional problem of elasticity in a cylindrical coordinate system for bodies bounded by the canonical surface is done. The boundary-value problems of pure rotation and circular symmetry in harmonic potentials are formulated. The mentioned approaches make it possible to obtain analytical solutions to these problems and are the theoretical basis for calculating the strength parameters of technical systems in a way of analysis of their mathematical models.

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Section: Introductionmentioning

confidence: 99%

The study of mathematical models of the linear theory of elasticity by presenting the fundamental solution in harmonic potentials

Pabyrivskyi¹,

Pabyrivska²,

Пукач³

2020

Math. Model. Comput.

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“…A mathematical body of the theory of functions of a com plex variable is widely used for constructing solutions of boundary value problems of theory of plastic deformation [12], fracture mechanics for piezoelectric solids [13,14], pi ezoceramic solids [15], and bimaterials [16] with interfacial cracks under the action of mechanical loads, electromagnetic and thermal fields [17,18].…”

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confidence: 99%

Two-dimensional elastic theory methods for describing the stress state and the modes of elastic boring

Pabyrivskyi¹,

Kuzio²,

Pabyrivska³

et al. 2020

Nauk. visn. nat. hirn. univ.

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Purpose. To find the stressed state and describe the operating regimes of an important element of mining industrial equip ment -an elastic drilling bit -based on the method of finding the solutions of problems of solid mechanics using holomorphic functions of two complex variables.Methodology. Methods for constructing the basic solutions of threedimensional boundary value problems of the theory of elasticity are based on the representation of the fundamental solution of the Lame equations in the PapkovichNeuber form using a scalar spatial harmonic function and a vectoral spatial harmonic function. This made it possible to formulate the boundary value problem in terms of holomorphic functions of complex variables. Based on the representation of the above holomorphic functions in the form polynomial of the n th order in accordance with degrees of the complex variables, corresponding boundary conditions are formulated for basic solutions and integral conditions for the principle moment of the stress vector on the lateral surface being equal to zero are additionally specified.findings. The paper formulates complexconjugate boundary value problems of spatial theory of elasticity in terms of holomor phic functions of two complex variables. We have considered the case when there is one of the spatial coordinates on which the stress tensor does not depend on, for a body whose surface is described by standard curves. For the given finite elastic cylindrical solid (a drilling bit) of canonical crosssection, a set of the basic complex solutions of the n th order and of corresponding vectors of external loads has been constructed. originality. For the first time, a scheme and a method for constructing the basic solutions of static boundary value problems of spatial theory of elasticity have been suggested and the corresponding boundary conditions have been imposed; the real and imaginary parts of the solutions of the basic boundary value problems for a cylindrical drilling bit have been constructed and an analysis of these solutions has been carried out.Practical value. An example of using the developed methods for constructing solutions of boundary value problems of spatial theory of elasticity has been considered for finding an exact analytical solution of a twodimensional boundary value problem that describes the distribution of stresses and corresponding external loads on the lateral surface of a cylindrical drilling bit of a ca nonical crosssection. Such mathematical models and analysis of the external load structure can be effectively used to describe the safe operating modes of mining boring equipment.

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Development of slip line of plasticity localization in construction materials

Labibov,

Khodanen

2024

Visnyk TNTU

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Uniaxial dynamic problem for a rod made with a material with yielding peak is formulated. The model used for formulating the problem takes into account an experimental observation that during the process of yielding a slow wave exists which divides a specimen into domains of elastic and plastic behavior. Process of yielding is described assuming the yielding peak and softening behavior is connected to the process of dislocation release. Discrepancies created during this process in the form of Lüders strips are described as shear lines between structural layers (e.g. ferrite-pearlite boundary, interatomic lattice) under an applied loading in a state of plastic yielding.

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Strips of localization of plastic deformation

Cited by 3 publications

References 7 publications

The study of mathematical models of the linear theory of elasticity by presenting the fundamental solution in harmonic potentials

The study of mathematical models of the linear theory of elasticity by presenting the fundamental solution in harmonic potentials

Two-dimensional elastic theory methods for describing the stress state and the modes of elastic boring

Development of slip line of plasticity localization in construction materials

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