PAMM · Proc. Appl. Math. Mech. 15, 221 – 222 (2015) / DOI 10.1002/pamm.201510101
Stress concentration control in the problem of plane elasticity theory Nana Odishelidze1,∗ , Francisco Criado Aldeanueva2,∗∗ , and Francisco Criado2,∗∗∗ 1 2
Iv. Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences, Georgia Malaga University, Spain
In engineering practice, one of the important problems is the problem of finding full-strength contours which permits to control stress concentration at the hole boundary. The article addresses the mixed problem of plane elasticity theory for doubly-connected domain with partially unknown boundary conditions. In the presented work the stress state of the given body and full-strength contours were defined. c 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Introduction
Boundary-value problems of the plane elasticity theory for infinite plates weakened with full strength holes are reflected in [1,2]. It’s proven that for infinite domains tangential normal stresses (tangential normal moments) the minimum of maximal value will be obtained on such contours, where this value maintains the constant value. These contours are named full-strength contours. The solvability of these problems provides controlling stress optimal distribution selecting the appropriate hole boundary. For finite domains the axis-symmetric and cycle-symmetric problems of the plane theory of elasticity and plate bending with partially unknown boundaries are studied in [3, 4, 6, 7] . The article addresses the mixed problem of plane elasticity theory for doubly-connected domain which outer boundary presents the rhombus boundary, whose diagonals lie at the coordinate axes OX and OY , the internal boundary is required full-strength hole, whose symmetric axes are the rhombus diagonals.
2
Statement of the Problem
Let to every link of the broken line ( outer boundary of the given body) be applied absolutely smooth rigid stamps with rectilinear bases which displace to the normal under the action of concentrated normally compressive forces P applied to the stamp midpoints. There is no friction between the given elastic body and stamps. The unknown full-strength contour is free from outer actions. Under the above assumptions, the tangential stresses τns = 0 are zero along the entire boundary of the domain S and the normal displacements of every link of external boundary vn = v = const. Consider the following problem: Find the shape of the unknown hole and the stress state of the given body such that the tangential normal stress arising at it would take constant value: σs = K = const. Since the problem is axially symmetric, then to investigate the state problem, it is sufficient to consider the curvilinear quadrangle A1 A2 A3 A4 which be denoted by D. The normal displacements and the tangential stresses are equal to zero νn = τns = 0 at each segment [A1 A notations Γ1 = A1 A2 , Γ2 = A2 A3 ,RΓ3 = A3 A4 , γ = R 2 ], [A3 A4 ] . Let Rintroduce the following R S3 A4 A1 , Γ = j=1 Γj .P1 = Γ1 σn ds, P2 = Γ2 σn ds, P3 = Γ3 σn ds.σn is the normal stress. P2 = Γ2 σn ds = −P . Since the D is in the equilibrium state, then we have: P1 = P2 cos β = −P cos β, P3 = P2 sin β = −P sin β, where β = 6 A1 A2 A3 . The most effective methods for studying this problem are the methods of the theory of analytical functions of a complex variable. On the basis of the well-known Kolosov-Muskelishvili’s formulas [5], the problem reduces to finding the functions ψ, ϕ which are holomophic in the domain D with the following conditions Ree−iα(t) χϕ (t) − tϕ´(t) − ψ (t) = 2µνn (t) ,
Ree−iα(t) ϕ (t) + tϕ´(t) + ψ (t) = c (t) ,
t ∈ Γ,
t ∈ Γ,
(1)
(2)
∗
Corresponding author: e-mail
[email protected], phone +995 593 103 999 Campus El Ejido, Department of Applied Physics II. ∗∗∗ Campus Teatinos, Department of Statistics and Operational Research. ∗∗
c 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
222
Section 4: Structural mechanics
ϕ (t) + tϕ´(t) + ψ (t) = 0,
t ∈ γ,
(3)
σn + σs K = , 4 4
t ∈ γ,
(4)
Reϕ´(t) =
where - χ, µ are elasticity constants, c(t) is a piecewise – constant function, α(t) - is the angle formed between the external normal n to contour and the abscissa axis Ox.
3
Results
The considered problem with partially unknown boundaries is reduced to the known boundary value problem of the theory of analytic functions by means of the developed method. The solutions are presented in quadratures. Full-strength contours are constructed. The Mathcad is used for construction and numerical calculations. Here as an illustration: some graphics of full-strength contours are presented for various parameters.
Fig. 1: The full-strength contours: K=-9.196
Fig. 2: The full-strength contours : K=–18.41
Fig. 3: The full-strength contours: K=-11.8
References [1] N. B. Banichuk, Optimization of elastic solids. (Russian) Nauka, Moscow, 1980 [2] G. Cherepanov, Inverse problem of plane theory of elastisity. (Russian) Prikl. Mat. Mekh. No. 6, 38(1974),pp. 963–980; [3] R. Bantsuri., On one mixed problem of the plane theory of elasticity with a partially unknown boundary Proc. A. Razmadze Math. Inst.,Vol. 140, 2006, pp.9–16. [4] R. Bantsuri, Solution of the mixed problem of plate bending for a multi-connected domain with partially unknown boundary in the presence of cyclic symmetry, Proc. A. Razmadze Math. Inst., Vol. 145, 2007, pp.9–22. [5] N. I. Muskhelishvili., Some Basic problems of mathematical theory of elasticity, (Russian), Nauka, Moscow 1966. [6] N. Odishelidze, F. Criado-Aldeanueva, F. Criado, J. M. Sanchez. On one contact problem of plane elasticity for a doubly connected domain: application to a hexagon. Zeitschrift fur Angewandte Mathematik und Physik, 2013, Vol. 64, No.1, pp. 193–200 [7] N. Odishelidze, F. Criado-Aldeanueva, J.M. Sanchez . A mixed problem of plate bending for a regular octagon weakened with a required full-strength hole. Acta Mechanica, 2013, vol. 224, No. 1, pp. 183-192, impact 1.247 c 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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