# Inkrementalnye the equations of a bending down of non-uniform physically nonlinear flat envelopment of a variable thickness

Let's consider rectangular in the plan a flat envelopment the sizes a? b, executed from physically nonlinear non-uniform material and Rectangular system of co-ordinates being under the influence of a distributed lateral load x, y, zсовместим with an upper surface of a flat envelopment. An axis zнаправим on a normal to an upper surface of a flat envelopment towards a centre of curvature. Co-ordinate axes xи yсовместим with directions of lines of the main things krivizn envelopments, accordingly k_{x}и k • the Median not deformed surface of a flat envelopment on the rectangular

The thickness of a flat envelopment on a contour a constant also is designated as h_{0}, and in other points of the plan it depends on co-ordinates h, at and is designated in the form of function h (h,). A variable thickness h (h,) we will set in the form of sinusoidal velaroida [50] therefore this function will assume the following air: gdebezrazmernyj parametre of a relative thickness, h_{c} - a thickness

Flat envelopment in the centre.

In this dissertational work thin flat envelopments at which relation - the least are considered only

Radius of curvature of a median surface of an envelopment, - accordingly, the greatest skewness of a median surface of a flat envelopment. The deflection w is considered positive if it is directed to a centre of curvature of a flat envelopment.

To increase efficiency of use of a material of a flat envelopment and to raise its load-carrying capacity, we will create in extreme top and base layers on a thickness of a design of heterogeneity by means of the certain processing methods which are not considered in given dissertational work. These layers we will name the non-uniform (strengthened) layers, and a line limiting the basic (initial) material from a non-uniform layer we will name heterogeneity front. More low in drawings 21 and 22 fragments of a flat envelopment of a variable thickness with two-sided and single-sided heterogeneity with following marking-offs are resulted: 5 thickness of a non-uniform layer, - front co-ordinate

Heterogeneity, - co-ordinate of a centroid of sectional view.

Drawing 21. A fragment of a flat envelopment of a variable thickness at the two-sided

Heterogeneity on a thickness

Drawing 22.

A fragment of a flat envelopment of a variable thickness at the single-sidedHeterogeneity on a thickness

Heterogeneity is created so that the curved lines describing change prochnostnyh of characteristics of a material in strengthened and in the basic layer, had a general tangent line in points of front of heterogeneity. In this case concentrators of pressure in places of contact of the basic and non-uniform materials are not formed. In this connection in each point of a layer of heterogeneity of the chart of a warping will be various.

For a conclusion inkrementalnyh the equations of a bending down of physically nonlinear non-uniform flat envelopment of a variable thickness it is necessary to consider three groups inkrementalnyh the equations which have been written down through increments of required functions: static (the balance equations), geometrical, physical.

The system of the equations defining balance of an element of a flat envelopment has been received by V.Z.Vlasovym in work [24] and led inkrementalnoj

To the form, with use of differential Gato, V.V. Petrov in work [82] also looks as follows:

Lateral load.

The geometrical parities connecting increments of deformations with increments of movings, for flat envelopments register in the following form:

Double derivation on variables xи y, we will exclude increments of tangential movings ΔU, ΔVи we will receive as a result a continuity equation of deformations of a median surface of a flat envelopment:

With co-ordinate on a thickness zпринимают the following sort:

58

Flat envelopment, and expressions defining them written down through an increment of a deflection look like:

At construction of physical parities for a flat envelopment executed from a nelinejno-deformed non-uniform material we take for a basis the theory small uprugoplasticheskih A.A.Ilyushin's deformations [39]. For the account of heterogeneity of a material properties of a flat envelopment, applying the phenomenological approach, we enter within heterogeneity front funktsijukotoraja will consider change of time resistance

Material on a thickness of a non-uniform layer.

Function neodnorodnostivvodim in inkrementalnye the physical

The parities received on the basis of the theory small uprugoplasticheskih of deformations. According to this theory for an incompressible material with a Poisson's ratio equal 0,5, communication between an increment tenzora - deviatora pressure ΔD_{σ}и an increment tenzora-deviatora deformations Δ∕.). Will look like [82]:

Pressure, intensity of deformations. As at the decision of problems as a parent material the material with the nonlinear chart is used

59 warpings, that, accordingly, charts of a warping of a material in a non-uniform layer too are nonlinear.

Let's write down inkrementalnye physical parities connecting increments of pressure with increments of deformations for a flat envelopment of a variable thickness executed of a nelinejno-deformed non-uniform material:

Where - increments normal and tangent lines of pressure.

Expressions defining increments axial and shifting efforts in a flat envelopment look as follows:

Substituting inkrementalnye physical parities (21) in expressions for increments axial and shifting efforts (22) we will receive:

Where - variable zhestkostnye parametres of a flat envelopment of a variable

Thickness which are defined under following formulas:

Expressions for increments of efforts momentnoj groups in a flat envelopment have the following appearance:

60

Substituting inkrementalnye physical parities (21) in expressions for increments bending and twisting the moments (25) we will receive following expressions:

Where, I_{3k} - variable zhestkostnoj parametre of a flat envelopment which is defined under the formula:

In formulas (24) and (27) variable zhestkostnye parametres I_{2k}и I_{3k}записаны for a case of two-sided heterogeneity on a thickness (fig. 21) to use these formulas at single-sided heterogeneity (fig. 22) is necessary to replace in them 0.5h (x,) on z_{c}. Therefore variable zhestkostnye parametres Iи Iпри of single-sided heterogeneity will assume the following air:

Position of points of a surface of a centroid of a flat envelopment at single-sided heterogeneity on a thickness (fig. 22) is found from a condition, that the resulted static moment of sectional view concerning a neutral axis is equal to zero, whence we have a following condition:

61

Solving the equation (29) we find co-ordinates of points of a neutral surface of a flat envelopment z_{c}.

In the linear theory of flat envelopments in efforts membrannoj groups neglect changes krivizn envelopments, and in efforts momentnoj groups - deformations of a median surface. In works [78, 82] it is shown, that this assumption is fair and in case of small nelinejno-elastic deformations. According to this assumption it is accepted zhestkostnoj parametre ∕_{2fc}равным to zero. Therefore expressions for definition of increments of efforts membrannoj and momentnoj groups will assume the following air:

In the formula (30) from first three equations defining increments axial (membrannyh) efforts we will receive expressions for definition of an increment of deformations of a flat envelopment:

Where - variable zhestkostnoj parametre of a flat envelopment.

For a conclusion of system of the resolving equations in the mixed form, we enter into consideration an increment of function of efforts Δ ^ (h,) and we will write down an increment of efforts membrannoj groups in a following sort:

Substituting formulas (32) in the right part of expressions (31) we will receive formulas for definition of an increment of deformations of a flat envelopment the functions of efforts expressed through an increment:

Substituting formulas (19) in expressions for increments bending and twisting the moments in the formula (30), we will receive increments of efforts momentnoj groups expressed through an increment of a deflection of a flat envelopment:

Substituting expressions (19) and (33) in parities (21) we will receive inkrementalnye physical parities for a flat envelopment expressed through increments of required functions:

Having a full complex of static, geometrical and physical parities it is possible to receive system of the equations of a bending down of flat envelopments of a variable thickness executed of a nelinejno-deformed non-uniform material.

Substituting in first two equations of balance in the formula (15) expressions (32) we are convinced, that they are completely satisfied, and the third equation of balance takes the following form:

63

Substituting expressions for bending and twisting the moments (34) in the equation (36) and having spent corresponding mathematical transformations we will receive the basic inkrementalnoe the differential equation of balance in a following sort:

Substituting increments of deformations (33) in the equation (17) we will receive the basic inkrementalnoe a differential continuity equation of deformations of a flat envelopment in the following form:

Consolidating the received equations (37) and (38) it is received, linear inkrementalnuju system of the equations for calculation of flat envelopments of a variable thickness executed from a nelinejno-deformed non-uniform material:

Let's result expressions (14), (24), (27) - (29), (33), (34), (35) and (40) in a dimensionless sort. Increments of required dimensionless functions we will define following expressions:

64

Where E - the initial module (elastic modulus).

Expressions for dimensionless variables we will write down in a sort:

For transition to a dimensionless sort we will enter following dimensionless parametres:

Function of a variable thickness (14) in the dimensionless form looks like:

Expressions for variables zhestkostnyh parametres (24) and (27), taking into account the accepted marking-offs, in a dimensionless sort look as follows:

Expression for rigidness I^в to the formula (28) in the dimensionless form will look like:

Where - dimensionless kasatelnyj modul. - dimensionless co-ordinate

Points of a neutral surface (axis), - dimensionless function

Heterogeneity.

The equation (29) in the dimensionless form looks as follows:

65

Where - a dimensionless thickness of a non-uniform layer.

Expressions for definition of an increment of deformations of a flat envelopment (33), taking into account the accepted marking-offs, in the dimensionless form register as follows:

Expressions for definition of increments bending and twisting the moments of a flat envelopment (34), taking into account the accepted marking-offs, in a dimensionless sort look as follows:

The increments of pressure (35) expressed through required functions, in the dimensionless form, taking into account the accepted marking-offs, look like:

In formulas (49) and (50) for a translation of expressions in a dimensionless sort following dimensionless sizes are accepted:

66

Considering the resulted parities (41) - (43) we will write down linear inkrementalnuju system of the equations for calculation of flat envelopments of a variable thickness executed of a nelinejno-deformed non-uniform material in a dimensionless sort:

According to order of system of the differential equations (52) it is necessary to formulate on four boundary conditions in each point of a contour of a flat envelopment [51, 52, 82, 83, 84]. As the system of the equations (52) is resulted in a dimensionless sort also boundary conditions are formulated in increments of required dimensionless functions

Parities (48) - (50) and system of the equations (52) in the form of a method of final differences are resulted in appendix A.

2.2.

## More on topic Inkrementalnye the equations of a bending down of non-uniform physically nonlinear flat envelopment of a variable thickness:

- the decision of a problem of a bending down of physically nonlinear flat envelopment of a variable thickness
- the decision of a problem of a bending down of physically nonlinear flat envelopment at two-sided heterogeneity on a thickness
- Inkrementalnoe the equation of a bending down of non-uniform physically nonlinear girder of a variable thickness
- Inkrementalnoe the equation of a bending down of non-uniform physically nonlinear plate of a variable thickness
- 4.1. The decision of a problem of a bending down of physically nonlinear girder of a variable thickness
- the decision of a problem of a bending down of physically nonlinear plate of a variable thickness
- MISHCHENKO Novel Viktorovich. CALCULATION of NON-UNIFORM PHYSICALLY NONLINEAR THIN-WALL SPATIAL DESIGNS of the VARIABLE THICKNESS. The DISSERTATION on competition of a scientific degree of a Cand.Tech.Sci. Saratov - 2018, 2018
- the decision of a problem of a bending down of physically nonlinear plate at two-sided and single-sided heterogeneity on a thickness
- 3.1. The decision of a problem of a bending down of physically nonlinear girder at two-sided and single-sided heterogeneity on a thickness
**CHAPTER 4. NUMERICAL REALIZATION OF THE MATHEMATICAL MODEL AND THE ANALYSIS OF RESULTS OF CALCULATION OF PHYSICALLY NONLINEAR THIN-WALL SPATIAL DESIGNS OF THE VARIABLE THICKNESS****CHAPTER 3. NUMERICAL REALIZATION OF THE MATHEMATICAL MODEL AND THE ANALYSIS OF RESULTS OF CALCULATION OF NON-UNIFORM PHYSICALLY NONLINEAR THIN-WALL SPATIAL DESIGNS****CHAPTER 2. THE EQUATIONS DEFINING TENSELY - DEFORMED THIN-WALL SPATIAL DESIGNS WITH CHANGING ON THICKNESS prochnostnymi BY CHARACTERISTICS****CHAPTER 1. NON-UNIFORM DESIGNS WITH CHANGING ON THEIR THICKNESS prochnostnymi HARAKTRISTIKAMI. MATHEMATICAL MODELS OF CALCULATION (THE REVIEW OF THE CONDITION OF THE QUESTION)**- boundary conditions for flat envelopments
- the Basic parities in the form of a method of final differences for flat envelopments, plates and girders