# the decision of a problem of a bending down of physically nonlinear flat envelopment at two-sided heterogeneity on a thickness

In this research it is necessary to estimate, on concrete examples, influence of two-sided heterogeneity on a thickness of a flat envelopment on its is intense-deformed. We will consider square in the plan a flat envelopment under the influence of a uniformly distributed lateral load.

In the calculations resulted more low we believe, that the considered flat envelopment on all contour leans on absolutely rigid in the plane and absolutely flexible diaphragms from the plane. These boundary conditions correspond to conditions of a sort (94). A material of a flat envelopment - penoaljuminy, made of alloy АК7. Pairs experimental values of the nonlinear chart σ_{i}- ε

_{i}приведенные in table 1, we approximate cubic splines. The task in view decision we will consider in a dimensionless sort.

At the decision of a problem of a bending down of a flat envelopment we believe, that non-uniform layers of an identical thickness settle down in extreme top and bottom fibres of a flat envelopment, that is the variant of two-sided heterogeneity on a thickness (hardening of extreme top and bottom fibres) is considered. A thickness of a layer of heterogeneity in a dimensionless sort it is accepted equal S ζ = 0,2.

For the description prochnostnyh material characteristics in non-uniform layers we use function of heterogeneity of a sort (105).

For an estimation of character of change of parametres tensely - a state of strain of a flat envelopment at two-sided hardening it is necessary to solve system inkrementalnyh the differential equations of a sort (52) with variables zhestkostnymi in parametres of a sort (45) and boundary conditions (94).

For increase of accuracy of received results and error decrease linearizatsii for the decision of the put nonlinear regional problem it was used DMPVP by V.V. Petrov thus loading broke into 10 parts. For decision specification at each stage DMPVP it was used the Ministry of Taxes and Tax Collection before achievement of an accuracy requirement of the decision (in the examples of calculation considered more low there were enough 5 iterations). For numerical realisation considered inkrementalnoj systems of the equations (52) the method of final differences with a grid 32 was used? 32. The used size of a grid is selected on the basis of comparison of results of calculation with application of grids of other sizes and the analysis of results received in works [51, 52]. Numerical realisation of particular integrals for variables zhestkostnyh parametres of a sort (45) was carried out with use of a method of Simpson with splitting of area of integration into 387 points.

In the calculations resulted more low we believe, that considered flat envelopments have identical parametres dimensionless krivizn k ξ = k _{η}. Sizes intensivnostej, operating dimensionless design loads, are selected from a condition of affinity of the maximum intensity of deformations to a limit value for a considered material in an initial state.

Results of calculation of non-uniform physically nonlinear flat envelopments with kriviznami 40, 60 and 80 are more low resulted. In drawings 55 and 56, 57 and 58, 59 and 60 are resulted epjury intensity of pressure on a thickness of a flat envelopment with skewness parametres, accordingly, k ξ = k _{η} = 40,60,80, in points with σ_{i}, _{max}по

To the bottom fibres (fig.

55, 57, 59) and on the top fibres (fig. 56, 58, 60). The marking-offs resulted on epjurah correspond to size of factor of hardening K.

By results of the analysis epjur - 60, it is visible to intensity of the pressure resulted in drawings 55, that with growth of factor of hardening K, that is at K> 1, intensity of pressure in an envelopment parent material decreases. For a detailed estimation of character of change of pressure on border of front of heterogeneity more low in table 2 it is resulted, in a percentage parity, a difference showing how much pressure on front border decrease at K = 1 (an envelopment in an initial state).

Table 2 - Results of comparison of pressure on front border

Drawing 55 (k ξ = k _{η} = 40) | |||||

- | K | 1 | 2 | 3 | 4 |

ζ =-0,8 | % | 0 | -17,46 | -28,08 | -35,57 |

ζ =-0,2 | % | 0 | -15,8 | -25,79 | -33,02 |

Drawing 56 (k ξ = k _{η} = 40) | |||||

ζ =-0,8 | % | 0 | -17,17 | -27,68 | -35,14 |

ζ =-0,2 | % | 0 | -16,61 | -26,96 | -34,37 |

Drawing 57 (k ξ = k _{η} = 60) | |||||

ζ =-0,8 | % | 0 | -16,80 | -27,18 | -34,58 |

ζ =-0,2 | % | 0 | -15,29 | -25,07 | -32,19 |

Drawing 58 (k ξ = k _{η} = 60) | |||||

ζ =-0,8 | % | 0 | -17,06 | -27,5 | -34,93 |

ζ =-0,2 | % | 0 | -16,47 | -26,75 | -34,13 |

Drawing 59 (k ξ = k _{η} = 80) | |||||

ζ =-0,8 | % | 0 | -16,28 | -26,48 | -33,80 |

ζ =-0,2 | % | 0 | -14,96 | -24,59 | -31,64 |

Drawing 60 (k ξ = k _{η} = 80) | |||||

ζ =-0,8 | % | 0 | -16,97 | -27,36 | -34,77 |

ζ =-0,2 | % | 0 | -16,36 | -26,60 | -33,94 |

Also from the analysis epjur, resulted in drawings 55 - 60, presence of layers of heterogeneity, in the form of strengthening layers is visible, that, leads to essential redistribution of pressure on a thickness. And redistribution occurs in such a manner that in the field of a pressure parent material decrease, but thus in working areas of heterogeneity of pressure on the contrary essentially increase. This circumstance gives the chance, by hardening variation in factor, to regulate the is intense-deformed condition of a flat envelopment. As well as in the previous case for more detailed estimation of character of change of pressure on

Surfaces of a flat envelopment we will result in table 3, in percentage, a difference showing how much pressure increase by envelopment surfaces at K = 1.

Table 3 - Results of comparison of pressure on a surface of a flat envelopment

Drawing 55 (k_{2} = k _{η} = 40) | |||||

- | K | 1 | 2 | 3 | 4 |

ζ =-1,0 | % | 0 | 62,73 | 110,91 | 150,49 |

ζ =0 | % | 0 | 67,79 | 121,57 | 166,42 |

Drawing 56 (k ξ = k _{η} = 40) | |||||

ζ =-1,0 | % | 0 | 65,43 | 116,75 | 159,54 |

ζ =0 | % | 0 | 67,53 | 120,58 | 164,65 |

Drawing 57 (k ξ = k _{η} = 60) | |||||

ζ =-1,0 | % | 0 | 64,72 | 114,92 | 156,33 |

ζ =0 | % | 0 | 68,90 | 123,78 | 169,83 |

Drawing 58 (k ξ = k _{η} = 60) | |||||

ζ =-1,0 | % | 0 | 65,42 | 116,61 | 159,25 |

ζ =0 | % | 0 | 67,66 | 120,91 | 165,21 |

Drawing 59 (k ξ = k _{η} = 80) | |||||

ζ =-1,0 | % | 0 | 66,23 | 117,94 | 160,79 |

ζ =0 | % | 0 | 69,77 | 125,64 | 172,63 |

Drawing 60 (k ξ = k _{η} = 80) | |||||

ζ =-1,0 | % | 0 | 65,45 | 116,69 | 159,27 |

ζ =0 | % | 0 | 67,78 | 121,19 | 165,68 |

Let's consider influence of two-sided heterogeneity on a thickness of a flat envelopment on its deflections and moments of flexion. For this purpose more low in drawings 61 and 62, 63 and 64, 65 and 66 we will result epjury deflections and moments of flexion in flat envelopments along a line η = 0,5. Marking-offs epjur, correspond to size of factor of hardening K.

Drawing 61. Epjury deflections in the flat

To envelopment (k ξ = k _{η} = 40)

Drawing 62. Epjury moments of flexion in

To envelopment (k ξ = k _{η} = 40)

Drawing 64. Epjury moments of flexion in an envelopment (k ξ = k _{η} = 60)

Drawing 63. Epjury deflections in a flat envelopment (k ξ = k _{η} = 60)

From the analysis epjur the deflections resulted in drawings 61, 63, 65 it is visible, that with growth of factor of hardening deflections in a flat envelopment decrease, however from the analysis of results of rationing of these epjur to unit in the envelopment centre, follows, that qualitative changes in epjurah deflections thus do not occur. Also from the analysis epjur deflections it is visible, that with growth of skewness influence of non-uniform layers remains former, peak values change only.

The greatest interest for the analysis cause epjury the moments of flexion, resulted on drawings 62, 64 and 66 from which it is visible, that with growth of factor of hardening moments of flexion in quarters increase, and their peak values thus are slightly displaced towards the centre of a flat envelopment. From the analysis epjur also it is visible, that with increase of parametre of skewness of an envelopment and factor of hardening moments of flexion in the centre at first decrease, and then in process of skewness increase increase. For a detailed estimation of degree of change of deflections and moments of flexion in the centre of an envelopment more low in table 4 it is resulted, in a percentage parity, a difference, at K = 1.

From the analysis of numerical files for function of efforts follows, that presence of two-sided heterogeneity on a thickness of a flat envelopment does not influence distribution of function of efforts. More low in table 4 results of comparison of peak values for function of efforts are resulted at various values of factor of hardening and parametre of skewness of an envelopment.

However in spite of the fact that function of efforts does not undergo any changes, axial thrusts thus receive both quantitative, and qualitative changes. In drawings 67, 69 and 71 are resulted epjury axial thrusts in a flat envelopment, and for an estimation of qualitative changes in drawings 68, 70 and 72 are resulted, accordingly, same epjury, but normirovannye in unit in the centre of a flat envelopment.

Drawing 67. Epjury axial thrusts in an envelopment (k ^ = k _{η} = 40)

Drawing 68. Epjury axial thrusts, normirovannye to unit (k ⅛ = k _{η} = 40)

Drawing 69. Epjury axial thrusts in an envelopment (k ξ = k _{η} = 60)

Drawing 70. Epjury axial thrusts, normirovannye to unit (k ^ = k _{η} = 60)

Drawing 71. Epjury axial thrusts in an envelopment (k ^ = k _{η} = 80)

Drawing 72. Epjury axial thrusts, normirovannye to unit (k ⅛ = k _{η} = 80)

From the analysis epjur in drawings 67, 69 and 71 it is visible, that with growth of factor of hardening axial thrusts in quarters decrease, and in the centre of a flat envelopment on the contrary increase. However from results resulted in drawings 67 - 72 it is visible, that with increase in skewness of a flat envelopment influence of non-uniform layers on axial thrusts decreases. More low in table 4 results of comparison of values of axial thrusts in the envelopment centre are resulted at various values of factor of hardening and skewness parametres.

Table 4 - Results of comparison of deflections, moments of flexion, functions of efforts and axial thrusts in the centre of a flat envelopment

Drawing 61 and 62, 67 (k ξ = k _{η} = 40) | ||||

K | 1 | 2 | 3 | 4 |

u, % | 0 | -15,48 | -25,26 | -32,35 |

M, % | 0 | -6,10 | -15,66 | -25,38 |

ψ, % | 0 | -0,05 | -0,12 | -0,18 |

N, % | 0 | 2,03 | 3,24 | 4,08 |

Drawing 63 and 64, 69 (k ^ = k _{η} = 60) | ||||

u, % | 0 | -16,23 | -26,24 | -33,40 |

M, % | 0 | 28,67 | 45,47 | 56,63 |

ψ, % | 0 | 0,07 | 0,11 | 0,14 |

N, % | 0 | 0,99 | 1,71 | 2,26 |

Drawing 65 and 66, 71 (k ξ = k _{η} = 80) | ||||

u, % | 0 | -16,82 | -27,09 | -34,38 |

M, % | 0 | 60,31 | 103,13 | 135,78 |

ψ, % | 0 | 0,04 | 0,07 | 0,09 |

N, % | 0 | 0,16 | 0,35 | 0,53 |

## More on topic the decision of a problem of a bending down of physically nonlinear flat envelopment at two-sided heterogeneity on a thickness:

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