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14PHYSICS AND MATHEMATICS
MATHEMATICAL MODEL SHEAR LINES FOR PLASTIC DEFORMATION
Domichev K.
Kiev International University, professor Chair of Computer Science Candidate of technical sciences, Petrov A.
Dnieper National University named after Oles Honchar,
Senior researcher, Chair of Theoretical and Computer Mechanics, PhD,
Steblyanko P.
University of Customs and Finance, professor, Chair of Cybersecurity and Information Technology Doctor of Physical and Mathematical Sciences
Abstract
The paper considers issue of mathematical modeling of shear line for the strip during plastic deformations.
The problem is solved in a nonlinear formulation. The described model of shear band formation resembles the model of dislocation motion, when the "quantum" of plastic shear is defined as the displacement of the dislocation by a distance equal to the length of the Burgers vector. The model assumes that plastic deformation for polycrystalline material develops due to plastic displacements of individual grains (crystallites), and the "quantum" of plastic deformation is defined as the slip within a pair of grains.
The proposed campaign allows us to describe flow site and subsequent hardening, as well as the effect of the Bauschinger effect on the flow site and on the site of hardening under cyclic loading.
Keywords: mathematical modeling, nonlinear problem statement, Luders' band, plasticity, deformation.
— /
A shear line moving at a constant speed.
Consider a shear line moving a < x - Vt < a with a constant velocity V in the positive direction xy along y = 0. The flat stress state and the flat deformed state are considered. Loads at infinity consist of tangential stresses T = T , as well
x>
as in the general case of normal stresses, 7 = 7y
but in this case we will assume that normal and tangential stresses at infinity are absent. It is also believed that there is no friction between the banks of the landslide line.
Due to the fact that the offset occurs between the start and end edges, the finite amount of slip will be
distributed along y =
0 for X < Vt — a . Thus, two adjacent particles located at each edge of the plane and in contact before the arrival of the initial y = 0 and
final edges will be reported at a distance
2A from
each other after passing the shear band. Assume that is a constant on X < Vt — a , y = 0.
In the future it will be accepted that V < Cs.
Thus, the speed of movement of the crack will be pre-sound. The validity of this assumption will become apparent when cracks with velocities between c S-wave
velocity and C P-minute velocity are considered.
Considering a semi-infinite body y > 0, we obtain a mixed boundary value problem. The boundary conditions for will be as follows
7y = 0 to all X
^ = 0 for |x - Vt\ <
a
du
— = 0 for
dx
x
- Vt| > a
Localization of plastic deformation shear
It is known that in materials with a clearly defined yield point, in the presence of a non-uniform stress field, isolated flux bands may appear. These bands occupy a small volume of the body compared to the elastic part. The corresponding discontinuous problems of the linear theory of elasticity are considered in [1]. In [2], an original concept was proposed that considers cracks-cuts in elastic bodies (surfaces of discontinuities of normal displacements) as nontrivial states of equilibrium of a physically nonlinear elastic medium. This concept was used to study the bands of localization of plastic deformation for a homogeneous stress field in a homogeneous material, provided that the deformation diagram of the material has a peak tooth under conditions of hard loading (Fig. 1).
A similar pattern of deformation can occur in the case of piecewise homogeneous materials. However, the localization bands of plastic deformation in such cases will first appear in the intermediate layers, ensuring the integrity of the components of the material. This can be explained by the fact that these intermediate layers obtained by welding or gluing dissimilar materials are usually the weakest components of composites. Consider further the bands of localization of plastic de-
formation in the region of separation of the two materials, provided that the deformation diagram of the layer is marked as a peak tooth.
Shear model for the Luders band To model the development of Luders bands in mild steel consisting of ferrite with inclusions of harder
perlite, consider the slip^ = yd where ^ - shear
deformation, d - the characteristic size of the considered area) of two adjacent grains of perlite and ferrite
T = Txy / yc exp(-y / yc) + T2 (1 + ay / yc)[l - exp(-y / yc)],
under shear forces T (T = rd where T - tangent tension). Consideration of the elastic-plastic model for ferrite grain and brittle fracture for the dependence of shear stresses of perlite grain between two grains T on shear deformation can y be expressed as a ratio that generalizes Novozhilov's model [3] for type I crack in an elastic solid:
where, T , T2, a - material constants, yc is the deformation for the shear strength Tc .
tc =t exp(-l) + T2(l - a)[ 1 - exp(-1)]
Additions in correspond to disturbance of perlite grain and accordingly elastic-plastic deformation of ferrite.
The system of two grains can be kept in three equilibrium states, marked as 1, 2 and 3 on the stress-strain curve T ~ y (Fig. 1). The first state means the upward slope of the stress-strain curve T ~ y ; the second state corresponds to the descending segment of the slope strengthen. Points 2, 3 are states of stable equilibrium, and 1 is unstable. A pair of grains interacting along a descending segment of the stress-strain curve T ~ y inevitably passes into the state of hardening at point 3. If all pairs of grains of two adjacent layers intersecting the body are transformed into such a state, then the whole body passes into a state of ideal plasticity. Thus, in an elastic body in a state of stable elastic deformation, the interaction determined by the law of the descending stress-strain curve T ~ y can exist
only locally. In this regard, such areas can be described as displacement rupture lines in a solid or shear bands.
Stress, t
All grains are in a state of stable interaction, described by the law of ascending stress-strain T ~ y around these lines, with no rupture of displacements.
In the theoretical study of equilibrium deformations of elastic-plastic bodies, it is always possible to interpret the body as a continuous medium using the methods of the theory of plasticity. However, we can take into account not only the forms of equilibrium, when all grains interact according to the law of the ascending (stable) branch of the curve, T ~ y but also
such forms when the body breaks gaps between the banks of which interact according to the law of the descending branch of the curve T ~ y (Fig. 1). The shape and size of these gaps are unknown in advance. They can be determined from the equations of the theory of elasticity when the problem on the banks of each line of rupture of the corresponding boundary conditions arising from the law at y > y .
Strain, 7
O 7. 73
Fig. 1 The stress-strain T ~ y diagram is based on the ratio
We consider the solution of this nonlinear problem under the following assumptions:
1) the relationship between stresses and strains on
the ascending (stable) part of the curve ( Y < Yc ), ie in
the area of the body where its continuity is preserved, corresponds to Hooke's law;
2) the equations of equilibrium and the formulas connecting deformations with displacements are accepted in the linear form, ie the problem is interpreted as geometrically linear;
3) on the segment fc < f <f (strengthening
begins when reached f3) for the T ~ y simplest approximation is used:
T = TCH (y —yc),
where H ( x ) - Heaviside function 1, if x > 0,
(0.1)
H ( x ) =
0, if x < 0.
(0.2)
We introduce the yield strength Ts , which is determined from the ratio
f3
\(t-ts ) dy = 0,
requiring that the area of the approximating curve (Fig. 1) be zero in the interval fc < f <fj . This condition is equivalent to the requirement of approximation of the dependence to give the same value of the surface energy density. The accepted simplifications lead to the linearization of all equations of problems and allow to obtain its approximate solution.
The described model of shear band formation resembles the model of dislocation motion, when the "quantum" of plastic shear is defined as the displacement of the dislocation by a distance equal to the length of the Burgers vector b0 . The model assumes that plastic deformation for polycrystalline material develops due to plastic displacements of individual grains (crystallites), and the "quantum" of plastic deformation is
defined as the slip within a pair of grains 8 .
Only tangential stresses will be considered within the localization band. Given the above simplifications, we can assume that at the edges of the localization band
in the region 0 < X < l are active only tangential
stresses Ts , and in areas l < |x| < b tangential
stresses can vary between them Ts Tc . This model of
the band bending load is similar to the Dugdale model, but with the significant difference that the stresses
within the band are not equal to zero (Ts > 0).
The proposed campaign allows us to describe the flow site and subsequent hardening, as well as the effect of the Bauschinger effect on the flow site and on the site of hardening under cyclic loading.
Conclusions. The paper describes a model of shear band formation, which resembles the model of dislocation motion, when the "quantum" of plastic shear is defined as the displacement of the dislocation by a distance equal to the length of the Burgers vector. The model assumes that the plastic deformation for pol-ycrystalline material develops due to plastic displacements of individual grains (crystallites), and the "quantum" of plastic deformation is defined as the slip within a pair of grains. at the site of fluidity and at the site of hardening under cyclic loading. This approach can be widely used on functional materials [4].
References
1. Ю. И. Кадашевич, В. В. Новожилов, Теория пластичности, учитывающая эффект Баушин-гера, Докл. АН СССР, 1957, том 117, номер 4, -с.586-588
2. В. В. Новожилов, Ю. И. Кадашевич, Ю. А. Черняков, "Теория пластичности, учитывающая микродеформации", Докл. АН СССР, 284:4 (1985), - с. 821-823
3. Новожилов В.В. Теория упру гости /. - Л: Судпромгиз, 1958. - 371 с.
4. Petrov A. Development of the method with enhanced accuracy for solving problems from the theory of thermo-psevdoelastic-plasticity / А. Petrov, Yu. Chernyakov, P. Steblyanko, K. Demichev, V. Hay-durov // Eastern-European Journal of Enterprise Technologies. 2018. Vol. 4/7 (94). P. 25-33.
SCIENTIFIC APPROACHES AND PRINCIPLES OF ENTREPRENEURIAL COMPETENCE
FORMATION IN PHYSICS LESSONS
Mukha A.
PhD student,
Sumy State A. S. Makarenko Pedagogical University,
teacher of physics,
Sumy comprehensive school №15 named after D. Turbin,
Sumy region, Ukraine ORCID ID: 0000-0002-2892-7495
У
с
Abstract
The article deals with the formation of entrepreneurial competence of students in physics lessons. It focuses on the structure of entrepreneurial competence. The main attention is focused on highlighting the potential of scientific approaches and principles to the formation of entrepreneurial competence. The expediency of applying axiological, problem-based, system-activity, competence-based, personality-oriented and acmeological approaches as an important methodological basis for the formation of entrepreneurial competence in physics lessons is substantiated.
