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ScienceDirect Procedia Engineering 153 (2016) 151 – 156

XXV Polish – Russian – Slovak Seminar “Theoretical Foundation of Civil Engineering”

Application of the methods of the theory similarity and dimensional analysis for research the local stress-strain state in the neighborhood of an irregular point of the boundary L.U. Frishtera, * a

Moscow State University of Civil Engineering (National Research University), 26 Yaroslavskoye Shosse, Moscow, 129337, Russia

Abstract In this paper, the methods of similarity theory and dimensional analysis analyzes the features of solving the problem of elasticity, caused by a form of boundary or "geometric factor" and the finite break of specified internally strains emerging in an irregular point of the boundary. Given the similarity criteria for the self-similar solution of the elasticity problem in a neighborhood of irregular points on a singular line of the elastic body of the border. Due to self-solve the elastic problem, stress, strain, displacement in a neighborhood of an irregular point of the boundary admit of the group similarity and functions possess the property of homogeneity, characterized by the fact that these functions can be represented in the form of power complexes. The properties of similarity and homogeneity must have an experimental solution, resulting in the model as the fringe pattern by photoelasticity. Therefore, sequence stripes in some neighborhood of irregular point of the boundary should have the property of similarity, homogeneity as well as stress and be represented in the form of λ power complexes, m ~ C λ r , which is confirmed by research of experimental data. © 2016 The Authors. Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Peer-review responsibility of the organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Foundation Foundationunder of Civil Engineering”. of Civil Engineering”. Keywords: self-similarity solutions, similarity criteria, stress concentration, homogeneity of the stresses, strains, displacements, sequence stripes;

This article discusses the features of solving the problem of elasticity, caused by a form of boundary or "geometric factor" and the finite break of preassigned forced strains emerging in an irregular point of the boundary.

* Corresponding author. Tel.: 8-916-999-57-24; fax: 8-499-183-28-74. E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Foundation of Civil Engineering”.

doi:10.1016/j.proeng.2016.08.095

152

L.U. Frishter / Procedia Engineering 153 (2016) 151 – 156

Such features of the stress-strain states occur in structures and constructions having different variants of design figuration border: special lines, dots, for example, the incoming angle, etc. The active forced deformation doesn’t satisfy the compatibility conditions, have a finite break (jump) on line (surface) contact areas, emerging at irregular point (line) border, which gives rise to stress concentration. The relevance of the research stress from the action such incompatible strain arises in the study of stress-strain state of structural elements by temperature gradients, temperature changes in the joints of dissimilar materials with different coefficients of thermal expansion, the hopping change distortions have a finite break in joints of areas with different mechanical properties, as well as the mounting and fabrication sequence structures, the interference fit and others. Features of the stress-strain state of buildings and structures, having "structural heterogeneity" and discontinuous forced deformations are determined in the polymer model by methods of photoelasticity [1,2,3] as the stress concentrators, which are the subject of this article. The geometric stress concentrators, causing features of stress-strained state of an elastic body, determined by the irregular points (lines) of boundary of the following types: Ⱥ) the break points (lines) of the first derivatives of the functions defining the boundary lines (surface)of the elastic body, such as the corner points (lines), the polyhedral angles, the conical points; B) points (lines) of the boundary of composite body owned by line (surface) of contact between two areas on which there is a jump the forced deformation or specified volumetric forces; C) points (lines) of the boundary of composite body owned by the line of contact of two homogeneous isotropic media with different constant values of the physico-mechanical properties; D) changes in the character of homogeneous boundary conditions. In the neighborhood of an irregular point (line) homogeneous boundary conditions are given. Displacements of elastic body are continuous at a singular point (line) and their surroundings. The irregular point (line) of boundary of the body and their neighborhoods arises a feature of the stress-strain state due to factors A) - D), which has a local character, and a distancing by irregular point (line) of the boundary damped. Outside the influence irregular point (line) of the boundary of the body, the solution of the elastic problem is smooth. Questions behavior of solutions of the Laplace equation, Poisson and elliptic equations for areas with nonsmooth boundaries were studied by Kondratiev V.A., Fufaev V.V., Williams V.L., Uflyand Y. S., Kalandia A.I., Cherepanov G.P., Bogey D.B., Aksentyan D.C., Alexandrov A.Y. [4-11] and others. Analytical methods of calculation (VA Kondratiev, Williams VL, Uflyand YS [5,7,8], and others. Authors) suggest that in the neighborhood of irregular points of the boundary solution of general elliptic boundary value problem is presented in the form of an asymptotic series and endlessly differentiable function. The terms of this series contain solutions of the homogeneous boundary value problems for model areas: the wedge or cone. In this paper, using methods of the theory of dimensions [12-14] are studied the orders of stresses changes, deformations and displacements depending on the coordinates of the point at the approach to the irregular point of the boundary. Consider the resolving a system of equations of the elasticity problem in a small neighborhood of an irregular point Oδ (0 ) on a singular line of boundary of the elastic body V . Let us write the equations of initial system to dimensionless form, using the following expression:

ξ = ξ0 ξ ,

(1)

where ξ – the considered quantity, ξ0 – the characteristic value of the quantity, ξ – the dimensionless value of this quantity. We will consider:

σ = σ 0 σ ; ε = ε 0 ε ; U = U 0U ; l l x = 0 x; y = 0 y; z = l0 z , t t where t – dimensionless parameter similarity group,

(2) (3)

is introduced for the analysis of "the degree of approximation to the irregular point". We obtain a system of equations of elasticity theory problem [3,12-14] in the dimensionless form:

153


∂σ ij

⎧F l ⎫ + ⎨ 0 0 ⎬F = 0, j ∂j ⎩ tσ 0 ⎭ ⎧ U t ⎫ ⎛ ∂U ∂U j ⎞ ⎟, 2ε ij = ⎨ 0 ⎬ ⎜ i + ∂i ⎟⎠ ⎩ ε 0l0 ⎭ ⎜⎝ ∂ j ∑ σ ij n j = 0; ∑

j

∑ σ ij n j j

(4)

(5)

L

ΓΒ

⎧σ ⎫ Β = ⎨ 01 ⎬σ in ⎩ σ0 ⎭

,

(6)

ΓΒ

⎧σ 0 ⎫ 1 ⎧ ε ⎫ 0 ⎧α T ⎫ [(1 + ν 0ν )σ ij − ν 0ν sδ ij ] + ⎨ 01 ⎬ ε ij + ⎨ 0 0 ⎬α Tσ ij , ⎬ ⎩ ε 0 ⎭ EE0 ⎩ ε0 ⎭ ⎩ ε0 ⎭ l l where i, j ∈ Oδ (0) , i, j = x, y, z; i, j = x, y, z; x = 0 x; y = 0 y; z = l0 z . t t

ε ij = ⎨

(7)

In order to obtain a homogeneous boundary value problem and go to the self-similar solution is necessary and sufficient that performed (in accordance with the system of equations 4-7), the following conditions:

F0l0 = 0 or F0 l 0 ~ tσ 0 tσ 0

or

F0 ~

tσ 0 . l0

(8)

Condition (8) means that the change in stresses, and coordinates with Oδ (0) the pursuit of an irregular point is much faster than change of volumetric forces in a small volume of this neighborhood. Criterion (8) suggests that there are parameters t , l0 under which the neighborhood of an irregular point in the influence of given volumetric forces in the stress can be neglected. According to (Equation 6) conditions must be satisfied:

σ 01 = 0 or σ 01 ~ σ 0 , t → ∞ , (9) σ0 σ 01 = 1 or σ 01 ~ σ 0 , t → 1 , (10) σ0 where σ 01 – the characteristic value for the stress in GV cross section Β ⊃ Oδ (0) of the body V . Condition (9) indicates that when approaching at irregular point, stress σ 0 , resulting a singular solution to the problem, change is much faster than the stress σ 01 , resulting the influence ofbe cut off body parts in the neighborhood Oδ (0) , i.e. the initial load or "general stress field."

As the distance from the irregular points of the boundary change orders of stresses σ 0 , σ 01 equalized and at

some position section or parameter value t stresses σ 0 and σ 01 are equivalent according (10). According to the criteria of the relations (equation 7) for the self-similar solutions should be performed:

⎧σ 0 ⎫ ⎨ ⎬ = 1; ⎩ ε0 ⎭

⎧ ε 01 ⎫ ⎨ ⎬ = 0; ⎩ ε0 ⎭

or when t → ∞ ,

⎧α 0T0 ⎫ ⎨ ⎬ = 0; ⎩ ε0 ⎭

σ 0 ~ ε 0 ε 01 ~ ε 0 ; α 0T0 ~ ε 0

where ε 01 – the characteristic value set the forced deformations ε ij0 .

(11) (12)

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According to the criteria (11,12) defined forced strain

ε 01 , α 0T0 in changes much more slowly when

approaching the irregular point or with the increase of the parameter t , than the strain ε 0 , resulting irregular point of the boundary O . Therefore, when t → ∞ , i.e. the closer to the irregular point O , the less significant defined strain by comparison with "singular" strains. Condition σ 0 ε 0 when t → ∞ means that the stresses and strains are of the same order of change over the coordinates in the neighborhood the singular point O , i.e. in Oδ (0) can talk about the equivalence of the order changes over the coordinates for the stresses and strains. As the distance from the irregular point O , i.e. when t → 1 , defined forced strains ε 01 , α 0T0 have the order changes like strain ε 0 . According to relations (5) criteria must be met:

⎧ U 0t ⎫ ⎨ ⎬ =1 ⎩ ε 0l0 ⎭

or

U 0 t ~ ε 0 l0

or

U0 ~

ε 0 l0 t

.

(13)

According to the (13) order of change of movement functions by the coordinates of point Oδ (0) is one more than order change of deformations, and according to the equivalence of condition (11), (12) and the order of stresses changes by point of coordinates in the Oδ (0) . If displacement in Oδ (0) have the order change r λ , where r – distance to the irregular point O (0,0) in a

section orthogonal to a particular line, then strain and, therefore, stress have the order change r λ −1 . When λ ∈ (0,1) limited movement and stress and strain are feature order λ − 1 , in this case possible evaluation:

U 0 = O(r λ ); σ 0 = O((r − r0 )λ −1 ) , where r ≠ r0 and considered Oδ (0) without the most points O and some of its small neighborhood.

(14)

Conclusion. Given the criteria associated with influences :

F0l0 →0; tσ 0

σ 01 → 0; σ0

ε 01 α 0T0 → 0; →0, ε0 ε0

(15)

and criteria arising from the given equation:

σ0 → 1; ε0

U 0t →1, ε 0l0

(16)

elastic problem in a neighborhood of an irregular point on a special boundary line is reduced to homogeneous boundary (singular) the problem of having self-similar solution. According to the representation of the solution of the elastic problem in a neighborhood point of special line of the boundary, we consider two homogeneous plane problems: plane strain and antiplane strain. Let us give the ratio of change orders U 0 , σ 0 , ε 0 by the coordinates of point in the Oδ (0) , using work [3,1214]. Let displacement U ( x), V ( x), x = x, y ∈ Oδ (0) , satisfy the homogeneous boundary-value elastic problem, then U (tx), V (tx), t − a real parameter, t > 0 , also satisfy these equations. By the linearity of the homogeneous boundary value problem solutions this problem.

U (tx ) t1

;

V (tx ) t1

, t1 – a real parameter, t1 >

0 , also satisfy

155


Therefore, the solution of the homogeneous problem U ( x), V ( x) generates a general solution to this problem

U (tx) ; t1

V (tx) . Many desired functions U ( x), V ( x) admits the similarity group (self- similar solutions), t1

according to the property [4, 12-14] should be performed:

U (tx) = U ( x); t1

V (tx) = V ( x) , t1

(17)

x = x, y ∈ Oδ (0) , t1 > 0 , t > 0 . The parameters t1 > 0 , t > 0 satisfy the equations homogeneous boundary value problem and the relations (17), should therefore be interdependent. Differentiating relation (17) over the variable t , assuming t1 = f (t ) a single of the functions:

−

1 dt1 1 d (U (tx)) 1 dt1 d (U (tx)) U (tx) + x = 0 or U (tx) = x, 2 dt d ( tx ) dt d (tx) t1 t1 t1

(18)

Separating the variables in (18), we obtain:

dU (tx) d (tx) dt1 t = , U (tx) (tx) t1 dt or

d ln U (tx) = d ln tx

(19) λ

d ln t1 . d ln t

(20)

d ln t1 = λ ; t > 0; t1 > 0 , relations (20) rewritten: d ln t

Designating

λ

d ln U (tx) = d ln tx .

(21)

Integrating (21), we obtain the order changes of movement depending on the coordinates of the point x = x, y ∈ Oδ (0) :

U (tx ) ~ C (tx )

λ

or

U ( x1 ) ~ C ( x1 ) ; V ( x1 ) ~ C ( x1 ) , λ

λ

(22)

where C – arbitrary constant, x1 = tx, y1 = ty , x1 , y1 ∈ Oδ (0) . When λ ∈ (0,1) displacement U ( x1 , y1 ), V ( x1 , y1 ) in Oδ (0) continuous, limited according to (22) have the property of homogeneity. According to the criteria (16), the stresses and strains have the order changes in the coordinates of a point on a unit smaller than a displacement:

σ ij ~ C1λx1λ −1 ; ε ij ~ C1λx1λ −1 , x1 = x1 , y1 ∈ Oδ (0); x1 = tx; y1 = ty . When λ ∈ (0,1) stress and strain in Oδ (0) have a singularity of order λ − 1 . The solution of homogeneous plane problems in the neighborhood of a nonregular point on a special line should be sought in the polar coordinate system as:

U = r λ f (θ );

V = r λ g (θ );

W = r λ1 p (θ ),

which is consistent with the decisions given in the paper [2, 15-20]. Due to self-decision elastic stress problem, strain, displacement in a neighborhood of an irregular point of

156


the boundary allow similarity group and have the property of homogeneity functions characterized the fact that these functions can be represented in the form of power complexes. The same properties (similarity homogeneity) should have an experimental solution, resulting in the model as the fringe pattern by photoelasticity. Therefore, in some stripes orders neighborhood irregular point of the boundary as well as stress and should have the property of similarity, homogeneity and be represented in the form of power λ complexes, m ~ Cλr as evidenced by studies of experimental data [15-20]. References [1].Dyurelli, D. Riley, Introduction to photomechanics, trans. with Eng. ed. NI Prigorovsky, Mir, Moscow, 1970. [2].S.P. Tymoshenko, J. Goodier, Elasticity Theory, Nauka, Moscow, 1975, 576 p. [3].G.S. Hesin, G.S. Vardanyan and etc., Photoelasticity method. In 3 Vols., Vol.3., Modelling of creep. The study of thermal stresses. Stroyizdat, M., 1975, p. 310. [4].G.P. Cherepanov, Brittle fracture mechanics, Nauka, Moscow, 1974, 640 p. [5].V.A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, (1967) Proceedings of the Moscow Mathematical Society, 16, p. 209-292. [6].V.V. Fufaev, the Dirichlet problem for domains with angles (1960) Reports of the Academy of Sciences, 1 (131), p. 37. [7].M.L. Williams, Stress singularities resulting from various boundary conditions in angular corners of plates in extension(1952) J. Appl. Mech., 4 (19), p. 526. [8].Ya.S. Uflyand, Integral transforms in problems of elasticity theory, Nauka, Moscow-Leningrad, 1963. [9].A.I. Kalandia, Notes on features elastic solutions near the corners (1969). Applied Mathematics and Mechanics, 1 (33), p. 132-134. [10].O.K. Aksentyan, Features stress-strain state of the plate in the neighborhood of the edge (1967) Applied Mathematics and Mechanics, 1 (31), p. 178-186. [11].K.S. Chobanyan, S.H. Gevorgyan, Behavior field strength around the corner in terms of section lines in the problem of plane deformation of the composite elastic body (1971), ProcHHGLQJVRIWKH$FDGHP\RI6FLHQFHVRIWKH$UPHQLDQ665ʋ;;,9 S-23. [12].L.I. Sedov, Similarity and dimensional methods in mechanics, Nauka, Moscow, 1972, 440 p. [13].G.S. Vardanyan, Fundamentals of similarity theory and dimensional analysis. IISS, M., 1977, 121c. [14].G.S. Vardanyan, Applied mechanics: the use of methods of similarity theory and dimensional analysis to modeling the mechanics of a deformable solid, INFRA-M, Moscow, 2016, 174 p. [15].L.U. Frishter, The study the stress state structures in a geometric stress concentration area under the influence of internally discontinuous deformations (2011), Journal of Research Center Building, 3-4, S. 135-145. [16].L.U. Frishter, V.N.Savostyanov, About the possibilities of analysis of experimental solution elastic problem in the area of stress concentration (2011), Journal of MSUCE, 1, p. 50-55. [17].L.U. Frishter. Estimates of solutions of the homogeneous plane problem of elasticity theory in the neighborhood of a singular point of the boundary (2012) Journal of MSUCE, 2, pp 20-24. [18].L.U. Frishter, P.S. Ivanov, A.S. Isajkin, G.E. Shablinskij. Analysis Of Complex Impacts On Stress-Strain State Of The Wall Chamber Lock. XXIV R-S-P Seminar, Theoretical Foundation Of Civil Engineering (24RSP) (Tfoce 2015), p. 215-219. [19].L.U. Frishter, V.A .Vatansky. The solution of the homogeneous problem of the theory of elasticity zone notch plane of the boundary (2013) Journal of MSUCE, 8, 51-58. [20].L.U. Frishter. Analysis of stress-strain state at the top of the rectangular wedge on the example of the experimental solutions M.Frohta (2014), Journal of MSUCE, 5, p. 57-62.