APPLICATION OF TENSOR CALCULUS OPERATIONS FOR ELASTIC BODIES

ABSTRACT

In this article, the history of scientific research on elastic bodies and the current work, their stability evaluation, tensor calculations for elastic bodies are explained through examples.

АННОТАЦИЯ

В данной статье на примерах поясняется история научных исследований упругих тел и современные работы, оценка их устойчивости, тензорные расчеты упругих тел.

Keywords: External and internal forces, hypothesis, elastic body, stress, stress tensor, principal stresses, stress tensor invariants, stability, theory of elasticity.

Ключевые слова: Внешние и внутренние силы, гипотеза, упругое тело, напряжение, тензор напряжений, главные напряжения, инварианты тензора напряжений, устойчивость, теория упругости.

We can find the first information about determining the strength of buildings in the manuscripts of the great medieval scientist Leonardo da Vinci (1452-1519). However, the great Italian scientist Galileo Galilei (1564-1642) is rightfully recognized as the founder of the theory of stability.

In the XVII-XVIII centuries, the science of resistance of materials mainly developed. Hooke, Mariotte, Jacob, Bernoulli, Leonard Euler, Coulomb, Jung and others contributed greatly to the development of this science. Thus, by the second half of the 19th century, the basis of the science of material resistance was created. By this time, it became necessary to create a general theory of a deformable spatial body. French scientist Navye was the first to deal with this issue . In 1821, Navye founded the theory of elasticity with his lecture on the equilibrium equation of a deformable rigid spatial body at the Paris Academy of Sciences.

Scientists such as Cauchy, Lyame also contributed greatly to the formation of the theory of elasticity as a science.

The next (XIX century) development of the theory of elasticity is associated with the names of many scientists, such as Russian scientist D.I. Juravsky, French scientist Saint-Venant, Russian academician A.V.Gadolin, Professor X.S.Golovin, French scientist Bussineska, Gers, F.K. Yasinsky.

In the theory of elasticity, a real body with a complex structure is replaced by a homogeneous body, not taking into account the fact that bodies are made up of a complex structure of atoms. The Saint-Venant principle is widely used in the theory of elasticity. According to Saint-Venant's principle, the stress at a point a certain distance from the external load does not depend on how the load is applied or the nature of the load.

Another requirement of the theory of elasticity is the assumption that an elastic body is cohesive. In this case, the elastic body is assumed to be continuous throughout its volume. In the theory of elasticity, a deformable body is considered to be purely elastic, that is, it is assumed that the body returns to its original state after an external load is removed.

All operations of the theory of elasticity are based on the correct proportionality between stress and strain, that is, Hooke's law. It should be noted that, in general, the relationship between stress and strain is not strictly proportional. But in the case of small stress and small deformation, if the relationship between them is taken as linear, that is, correctly proportional, then a big mistake is not allowed.

An elastic body with uniform characteristics is said to be a body in which the elasticity characteristics of any point of this body are the same in all directions.

An elastic body with different properties is a body whose elasticity properties are the same at all points and are different in directions. The classical theory of elasticity requires a body to have the six listed properties.

The first of these properties is the contiguity hypothesis, which means that a body adjacent to a deformer must remain contiguous after deformation. In this, any part of the body, including a very small particle, does not have gaps and discontinuities. The assumption of continuity allows us to consider displacements and deformations as a continuous function of coordinates, and to check them it is possible to use the apparatus of continuous functions of mathematics. If the body is ideally elastic, the relationships between strains and stresses are assumed to be linear. In doing so, a bridge of mutual value is established between stresses and strains for each value of temperature and independent of time.

The elastic body is sufficiently uniform is assumed to have. In other words, displacements of body points are small compared to its linear dimensions, relative elongations (shortenings) and displacement angles are required to be relatively small. The same requirement and the linearity of the relationship between stresses and strains make it possible to apply the principle of independence of force effects. It is known that this principle allows to calculate the effect of the system of forces acting on the body as the sum of the individual effects of each of the forces entering the system.

An ideal elastic body should be homogeneous. This means that under the influence of the same stresses, the same deformations occur at all points of the body. In other words, homogeneity requires considering the quantities characterizing the elastic properties of the body as constant over its entire volume.

lying on an arbitrary field defined by the state vector of a deformed solid body . For this, we separate from the body an elementary volume in the form of a tetrahedron, whose dimensions are infinitesimally small and whose three edges are normal to the fourth edge consisting of coordinate planes (Fig. 1.1).

Figure 1. Scheme

the basis vectors of the Cartesian coordinate system are denoted by , and the direction cosines of the vector

of the tetrahedron coordinate plains with top - top descender legs in it perpendicular was coordinate Read it number with define : with  to the perpendicular was turn on surface , and  s with  and  to the arrows perpendicular was of yqs surfaces is  determined with while  to normal perpendicular was the fourth turn on surface is determined. Then, as we know from the geometry course,

                                                       (1.2)

equality is appropriate. In addition to surface forces, the edges of the element are affected by volume. But since we are interested in the stress state of the point, we will infinitely reduce the size of the tetrahedron. In this case, the volume forces also tend to zero, and eventually these forces become zero when the tetrahedron tends to the point. That's why we don't consider volume forces.

Under the influence of external surface forces, the considered element is in a state of equilibrium. Then the principal vector of external forces should be equal to zero, i.e

                                              (1.3)

equality holds. If we put the expressions (1.1) and (1.2) into the equation (1.3) and remember that the sum is calculated by repeated indices (property of the "gung" index), from the equation (1.3)

                                                      (1.4)

we get an expression.

This equality shows that the stress vector in an arbitrary field with a normal to the body through the point is completely determined by the stress vectors in the coordinate planes and three coordinate planes each passing through the point. Now, using the fact that any vector can be expanded in terms of basis vectors, and we write the voltage vectors in terms of basis vectors as follows:

                                          (1.5)

here are the projections of the magnitude vector on the coordinate axes or the components of the stress vector on the coordinate plane; -components of the stress vector on an arbitrary field . In general , the nine components of the three stress vectors in the three coordinate planes will play an important role in our further practice.

Now substituting the first expression of (1.5) into (1.4).

we get the formula. We put the resulting expression (1.5) on the left side of the second of the expressions: and from the condition of equality of vectors

                                                         (1.6)

we form the expression.

the nine components of the three vectors are the components of the second color tensor. This tensor is called the stress tensor and is denoted by It is accepted to write this tensor in matrix form as follows:

or                                        (1.7)

the first index of the components of the stress tensor corresponds to the index of the coordinate axis perpendicular to the field of the stress vector , that is, the first index indicates that this component of the tensor acts in the plane perpendicular to the axis. The second index indicates the effect of the component in the direction of the coordinate axis. For example: the component acts in the direction of the axis in the plane parallel to the plane perpendicular to the component, in the direction perpendicular to the component . Therefore, (1.7) elements of the matrix directed along the coordinate axes ( do not sum along) or the normal stresses in the coordinate plane or normal components of the stress tensor is called The remaining elements of the matrix are the experimental stresses in the coordinate planes or stress components of the stress tensor is called

Thus, the components of the stress tensor are normal stresses at a given point and . Therefore, the components of the stress tensor completely determine the state of stress at a given point of the body. In other words, if the tensor is known, determining the projections of the stress vector on any field passing through the considered point is easily solved using the formula (1.6):

                                            (1.8)

If the field under consideration overlaps with the surface surrounding the body or is a part of it, the components of the stress tensor will differ from the components of the external surface forces acting on the surface of the body. Then equations (1.8) are conditions on the surface of the body they call it. They connect external forces with internal ones.

It should be noted here that in order to describe the state of stress at a point of the body with planes perpendicular to the coordinate axes around this point, they consider separately the elementary volume consisting of a parallelepiped whose sides are parallel to the coordinate axes and whose edges tend to zero. In this case, the effects of corresponding voltages on each side are depicted by arrows (Fig. 1.2).

Figure 2. Scheme

In the picture, in the Cartesian coordinate system, 1.6 - a picture and typical designations 1.6. - b The picture depicts a parallelepiped.

In conclusion, when solving problems of the theory of elasticity, coordinate systems other than the Cartesian coordinate system are used, in particular, cylindrical and spherical coordinate systems.

References:

1. Rikholmurodov, XXXudoynazarov "Theory of elasticity" Part I-II. Tashkent , 2003 .
2. Mamatkulov Sh. Lectures on " Theory of elasticity " . T.: University, 1995 .
3. Tymoshenko S.P., Guder J. The field of theory. M., Mir, 1975.
4. Alexandrov A.V. Potapov V.D. "Osnovy teorii uprugosti i plastichnosti" M.Vys.shk. 1990 g. 400 St.