# Inkrementalnoe the equation of a bending down of non-uniform physically nonlinear plate of a variable thickness

Let's consider a rectangular plate of a variable thickness the sizes a? b, executed from a nelinejno-deformed non-uniform material. The plate is under the influence of the distributed lateral load q (h,).

We will select system of co-ordinates so that the plane hu coincided with the top plane of a plate. A plate datum origin it is compatible to one of vertexes of a contour of a plate. A co-ordinate axis zнаправим on a vertical downwards, and axes h and at, accordingly along the parties and and b. The thickness of a plate on a contour a constant also is designated as h, 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 of a plate h (h,) we will set in the form of sinusoidal velaroida (14).In this dissertational work plates of an average thickness are considered. Plates concern such type at which thickness h, is in limits (0,1 - 0,2) a_{mm}, [81, 82, 84].

Movings of points of a median plane along an axis z, namely a plate deflection, we will designate as wи we will consider its positive if it coincides with this axis.

To increase efficiency of use of a material of a plate and to raise its load-carrying capacity, we will create in extreme top and bottom fibres on its thickness of heterogeneity. 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 23 and 24 fragments of a plate of a variable thickness with two-sided and single-sided heterogeneity on a thickness, where h - a thickness of a plate in the centre are resulted.

Drawing 23. A fragment of a plate of a variable thickness at two-sided heterogeneity on

To thickness

Drawing 24. A fragment of a plate of a variable thickness at single-sided heterogeneity on

To thickness

Heterogeneity of a material in extreme top and plate base layers are created so that the curved lines describing change prochnostnyh of characteristics of a material in non-uniform 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 inkrementalnogo the equations of a bending down of a plate of a variable thickness executed from a nelinejno-deformed non-uniform material it is necessary to consider three groups inkrementalnyh the equations which have been written down through increments of required functions [81, 82, 84]: static (the balance equations), geometrical, physical.

At a conclusion inkrementalnogo the equations of a bending down of a plate it is believed fair following hypotheses [82]:

1. At a plate bending down its thickness not изменяется:В

Result of that the plate deflection does not depend on co-ordinate z, that is w = w (h,) •

2.

As plates of an average thickness are considered, plate deflections w (h,) are small in comparison with its thickness geometrically linear and physically nonlinear problem Is considered.3. The hypothesis of straight normals of G.Kirhgofa from which is fair follows, that angular deformations are absent, that is

4. The median surface of a plate at its bending down does not change the

The sizes, that is if

Inkrementalnoe the equation of balance of an element of a median plane of the plate, received in works [82], looks as follows:

According to a classical kinematic model of straight normals of G.Kirhgofa of an increment flexural deformatsijv a layer

Plates with co-ordinate on a thickness zбудут to have the following appearance [81, 82, 84]:

At construction inkrementalnoj a mathematical model for physically nonlinear non-uniform plate of a variable thickness, as a basis, we use the theory small uprugoplasticheskih A.A.Ilyushin's deformations [39]. For the description of a mechanical properties of a material in non-uniform layers on a thickness of a plate, applying the phenomenological approach, within front

Heterogeneity (hardening) function is entered funktsijaeta will be

To consider change of time resistance of a material on a thickness of a non-uniform layer.

Function introduction neodnorodnostiv inkrementalnye the physical

Parities it is carried out through communication between an increment tenzora-deviatora naprjazheniji an increment tenzora-deviatora deformations ΔD which is resulted in the formula (20 [39].

As at the decision of problems of a bending down of physically nonlinear non-uniform plates of a variable thickness as a parent material the material with the nonlinear chart of a warping charts of a warping of a material in a non-uniform layer too will be nonlinear is used.

According to the theory small uprugoplasticheskih inkrementalnye physical parities for physically nonlinear non-uniform plate of a variable thickness look like deformations:

Delivering (55) in (56) we will receive inkrementalnye physical parities expressed through an increment of required function of a deflection:

Substituting inkrementalnye physical parities (57) in formulas (25) we will receive expressions for increments bending and twisting the moments expressed through an increment of required function of a deflection:

71

Where D_{k} - a variable along spatial co-ordinates h, at rigidness of a plate at a bending down which is defined under the following formula:

In expression (59) variable rigidness of plate D_{k}справедлива for a case of two-sided heterogeneity on a thickness (fig. 23) to use this formula at single-sided heterogeneity (fig. 24) is necessary to replace in nihtakim with image variable rigidness of plate Dпри

Single-sided heterogeneity will look like:

Position of points of a surface of a centroid of a plate at single-sided heterogeneity on a thickness (fig. 24) is defined from a condition (29). Solving the equation (29) we find co-ordinates of points of a neutral surface of a plate z.

Having a full complex of static, geometrical and physical parities it is possible to receive inkrementalnoe the differential equation of a bending down of physically nonlinear non-uniform plate of a variable thickness.

Substituting expressions for bending and twisting the moments (58) in the equation of balance (54) and having spent corresponding mathematical transformations we will receive the basic inkrementalnoe the equation of a bending down of physically nonlinear non-uniform plate of a variable thickness in a following sort [51, 81, 82, 84]:

72

In the developed sort inkrementalnoe the differential equation (61) looks as follows:

Using parities for dimensionless parametres (41) - (43) and (50) we will result expressions (55), (57) - (63) in a dimensionless sort.

Increments of deformations of a plate (55), taking into account the accepted marking-offs, in the dimensionless form look like:

The increments of pressure (57) expressed through required function of a deflection, in the dimensionless form, taking into account the accepted marking-offs, look like:

Expressions for definition of increments bending and twisting the moments of a plate (58) in a dimensionless sort, taking into account the accepted marking-offs, look as follows:

73

Variable rigidness of a plate (59), taking into account the accepted marking-offs, in a dimensionless sort looks as follows:

Expression (60) in the dimensionless form registers in a following sort:

In formulas (66) - (68) dimensionless parametre D^^{6}имеет a sort:

Linear inkrementalnoe the differential equation of a bending down of non-uniform physically nonlinear plate of a variable thickness (61) in a dimensionless sort, taking into account the accepted marking-offs, looks as follows:

In the developed sort the equation (69) registers as follows:

Where - the biharmonic operator in the dimensionless

To the form.

According to order of the differential equation (69), for the decision of a regional problem, it is necessary to formulate in each point of a contour of a plate two boundary conditions [82]. As the equation (69) is resulted in a dimensionless sort also boundary conditions are formulated through an increment of required dimensionless function of a deflection

Parities (64) - (66) and the equation (69) in the form of a method of final differences are resulted in appendix A.

2.3.

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