Issue 38
Pubblicazione animata
Frattura ed Integrità Strutturale, 38 (2016); International Journal of the Italian Group of Fracture
Table of Contents
N. Vaysfeld, O. Kryvyi, Z. Zhuravlova On the stress investigation at the edges of the fixed elastic semi-strip ……………………...………. 1 S. Averbeck, E. Kerscher A study of white etching crack formation by compression-torsion experiments …………………...… 12 R. Shravan Kumar, I.S. Nijin, M. Vivek Bharadwaj, G. Rajkumar, Anuradha Banerjee Stress-state dependent cohesive model for fatigue crack growth …………………………………… 19 L. Cui, P. Wang Crack initiation behavior of notched specimens on heat resistant steel under service type loading at high temperature …………….……...................................................................................................... 26 D. Marhabi, N. Benseddiq, G. Mesmacque, Z. Azari, J.M. Nianga Prediction of the critical stress to crack initiation associated to the investigation of fatigue small crack … 36 M. Leitner, F. Grün, Z. Tuncali, R. Steiner, W. Chen Multiaxial fatigue assessment of crankshafts by local stress and critical plane approach ……………... 47 T. Lassen, Z. Mikulski Fatigue methodology for life predictions for the wheel-rail contact area in large offshore turret bearings ... 54 H. Weil, S. Jégou, L. Barrallier, A. Courleux, G. Beck Fatigue modelling for gas nitriding ...…………………………………………………...…… 61 M. A. Meggiolaro, J. T. P. de Castro, H. Wu Incorporation of Mean/Maximum Stress Effects in the Multiaxial Racetrack Filter ……………... 67 S. Hörrmann, A. Adumitroaie, M. Schagerl The effect of ply folds as manufacturing defect on the fatigue life of CFRP materials ……….................. 76 M. Kepka, M. Kepka Jr., J. Chvojan, J. Václavík Structure service life assessment under combined loading using probability approach ……...................... 82 T. Sawada, H. Aoyama Effect of molding processes on multiaxial fatigue strength in short fibre reinforced polymer …………… 92 H. Wu, M. A. Meggiolaro, J. T. P. de Castro Application of the Moment Of Inertia method to the Critical-Plane Approach ………....................... 99
I
Fracture and Structural Integrity, 38 (2016); ISSN 1971-9883
J. Papuga, S. Parma, M. Růžička Systematic validation of experimental data usable for verifying the multiaxial fatigue prediction methods 106 E. Shams, M. Vormwald Fatigue of weld ends under combined in- and out-of-phase multiaxial loading ........................................ 114 M. de Freitas, L. Reis, M. A. Meggiolaro, J. T. P. de Castro Comparison between SSF and Critical-Plane models to predict fatigue lives under multiaxial proportional load histories ................................................................................................................... 121 M. A. Meggiolaro, J. T. P. de Castro, L. F. Martha, L. F. N. Marques A unified rule to estimate multiaxial elastoplastic notch stresses and strains under in-phase proportional loadings………………………………………………………………………………..... 128 A. Znaidi, O. Daghfas, S. Guellouz, R. Nasri Theorical study on mechanical properties of AZ31B Magnesium alloy sheets under multiaxial loading 135 S. Fu, D. Yu, G. Chen, X. Chen Ratcheting of 316L stainless steel thin wire under tension-torsion loading ………….........………… 141 X. Yu, L. Lo, G. Proust Fatigue crack growth of aluminium alloy 7075-T651 under non-proportional mixed mode I and II loads ………......……………………………………………………………………….. 148 M. Springer, M. Nelhiebel, H. E. Pettermann Combined simulation of fatigue crack nucleation and propagation based on a damage indicator ………. 155 A. Bolchoun, C. M. Sonsino, H. Kaufmann, T. Melz Fatigue life assessment of thin-walled welded joints under non-proportional load-time histories by the shear stress rate integral approach …………………………………………….……..……… 162 C. Gourdin, S. Bradaï, S. Courtin, J.C. Le Roux, C. Gardin Equi-biaxial loading effect on austenitic stainless steel fatigue life .......................................................... 170 A. Niesłony A critical analysis of the Mises stress criterion used in frequency domain fatigue life prediction ……… 177 D. Carrella-Payan, B. Magneville, M. Hack, C. Lequesne, T. Naito, Y. Urushiyama, W. Yamazaki, T. Yokozeki, W. Van Paepegem Implementation of fatigue model for unidirectional laminate based on finite element analysis: theory and practice ………………………………………………………………………..………... 184 R. Pezer, I. Trapić Atomistic modeling of different loading paths in single crystal copper and aluminum …........………… 191 M.V. Karuskevich, S.R. Ignatovich, Т.P. Maslak, A. Menou, P.О. Maruschak, S.V. Panin, F. Berto Multi-purpose fatigue sensor. Part 1. Uniaxial and multiaxial fatigue ….........................………… 198
II
Frattura ed Integrità Strutturale, 38 (2016); International Journal of the Italian Group of Fracture
M.V. Karuskevich, S.R. Ignatovich, Т.P. Maslak, A. Menou, P.О. Maruschak, S.V. Panin, F. Berto Multi-purpose fatigue sensor. Part 2. Physical backgrounds for damages accumulation and parameters of their assessment ………………………………………………………………………......………… 205 F. Berto, A. Campagnolo, G. Meneghetti, K. Tanaka Averaged strain energy density-based synthesis of crack initiation life in notched steel bars under torsional fatigue ............................................................................................................................................... 215 S. Suman, A. Kallmeyer, J. Smith Development of a multiaxial fatigue damage parameter and life prediction methodology for non proportional loading …........………………….............................................................................. 224 R. Fincato, S. Tsutsumi Numerical modelling of ductile damage mechanics coupled with an unconventional plasticity model ......... 231 M. Lutovinov, J. Černý, J. Papuga A comparison of methods for calculating notch tip strains and stresses under multiaxial loading ……… 237 S. Tsutsumi, K. Morita, R. Fincato, H. Momii Fatigue life assessment of a non-load carrying fillet joint considering the effects of a cyclic plasticity and weld bead shape …...............................................................................................................……… 244 A. Gryguc, S.K. Shaha, H. Jahed, M. Wells, B. Williams, J. McKinley Tensile and fatigue behaviour of as-forged AZ31B extrusion ….........................................………… 251 T. Inoue, R. Nagao, N. Takeda Random non-proportional fatigue tests with planar tri-axial fatigue testing machine …........………… 259 N.O. Larrosa, R.A. Ainsworth Ductile fracture modelling and J-Q fracture mechanics: a constraint based fracture assessment approach 266 F. Majid, J. Nattaj, M. Elghorba Pressure vessels design methods using the codes, fracture mechanics and multiaxial fatigue …..………... 273 T. Morishita, Y. Murakami, T. Itoh, H. Tanigawa Creep-fatigue life evaluation of high chromium ferritic steel under non-proportional loading …..……… 281 T. Morishita, T. Takaoka, T. Itoh Fatigue strength of SS400 steel under non-proportional loading …..……………………………... 289 M. Sokovikov, D. Bilalov, V. Oborin, V. Chudinov, S. Uvarov, Y. Bayandin, O. Naimark Structural mechanisms of formation of adiabatic shear bands …........……………………………. 296 U. Haider, Z. Bittnar, L. Kopecky, P. Bittnar, J. Němeček, A. Ali, J. Pokorny Mechanical behaviour and durability of high volume fly ash cementitious composites ……………..….. 305 C. Xianmin, S. Qin, D. Hongna, Fan Junling A statistically self-consistent fatigue damage accumulation model including load sequence effects under spectrum loading ………………………………………………………........……….…… 319
III
Fracture and Structural Integrity, 38 (2016); ISSN 1971-9883
I. N. Shardakov, A.A. Bykov, A. P. Shestakov Delamination of carbon-fiber strengthening layer from concrete beam during deformation (infrared thermography) …........……………….………………………………………………….... 331 I. N. Shardakov, A. P. Shestakov, I.O. Glot, A.A. Bykov Process of cracking in reinforced concrete beams (simulation and experiment) …........………………. 339 A. Eberlein, H.A. Richard The effect of varying loading directions and loading levels on crack growth at 2D- and 3D-mixed-mode loadings …........…………………….………………………………………………….... 351 P. Lonetti, A. Pascuzzo A numerical study on the structural integrity of self-anchored cable-stayed suspension bridges ………… 359 S. Bennati, D. Colonna, P.S. Valvo Evaluation of the increased load bearing capacity of steel beams strengthened with pre-stressed FRP laminates ……………………………………………………………………………….. 377 G. S. Serovaev, V. P. Matveenko Numerical study of the response of dynamic parameters to defects in composite structures …………….. 392 A. Shanyavskiy, A. Toushentsov Multiaxial fatigue of in-service aluminium longerons for helicopter rotor-blades …………....……….. 399
IV
Frattura ed Integrità Strutturale, 38 (2016); International Journal of the Italian Group of Fracture
Editor-in-Chief Francesco Iacoviello
(Università di Cassino e del Lazio Meridionale, Italy)
Associate Editors Alfredo Navarro
(Escuela Superior de Ingenieros, Universidad de Sevilla, Spain) (Ecole Nationale Supérieure d'Arts et Métiers, Paris, France)
Thierry Palin-Luc
Luca Susmel John Yates
(University of Sheffield, UK) (University of Manchester, UK)
Guest Editors Andrea Carpinteri
(Università di Parma, Italy; Multiaxial Fatigue and Fracture ) (University of Toledo, USA; Multiaxial Fatigue and Fracture ) (University of Seville, USA; Multiaxial Fatigue and Fracture )
Ali Fatemi
Carlos Navarro Pintado
Advisory Editorial Board Harm Askes
(University of Sheffield, Italy) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy) (University of Plymouth, UK)
Alberto Carpinteri Andrea Carpinteri Donato Firrao M. Neil James Gary Marquis Ashok Saxena Darrell F. Socie Shouwen Yu Ramesh Talreja David Taylor Robert O. Ritchie Cetin Morris Sonsino
(Helsinki University of Technology, Finland)
(University of California, USA)
(Galgotias University, Greater Noida, UP, India; University of Arkansas, USA)
(University of Illinois at Urbana-Champaign, USA)
(Tsinghua University, China) (Fraunhofer LBF, Germany) (Texas A&M University, USA) (University of Dublin, Ireland)
Editorial Board Stefano Beretta
(Politecnico di Milano, Italy)
Nicola Bonora
(Università di Cassino e del Lazio Meridionale, Italy)
Elisabeth Bowman Claudio Dalle Donne Manuel de Freitas Vittorio Di Cocco Giuseppe Ferro Eugenio Giner Tommaso Ghidini Daniele Dini
(University of Sheffield) (EADS, Munich, Germany) (EDAM MIT, Portugal)
(Università di Cassino e del Lazio Meridionale, Italy)
(Imperial College, UK)
(Politecnico di Torino, Italy)
(Universitat Politecnica de Valencia, Spain) (European Space Agency - ESA-ESRIN)
Paolo Lonetti Carmine Maletta Liviu Marsavina
(Università della Calabria, Italy) (Università della Calabria, Italy) (University of Timisoara, Romania) (University of Porto, Portugal)
Lucas Filipe Martins da Silva
Hisao Matsunaga Mahmoud Mostafavi
(Kyushu University, Japan) (University of Sheffield, UK)
Marco Paggi Oleg Plekhov
(IMT Institute for Advanced Studies Lucca, Italy)
(Russian Academy of Sciences, Ural Section, Moscow Russian Federation)
Alessandro Pirondi
(Università di Parma, Italy)
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Fracture and Structural Integrity, 38 (2016); ISSN 1971-9883
Luis Reis
(Instituto Superior Técnico, Portugal)
Giacomo Risitano Roberto Roberti
(Università di Messina, Italy) (Università di Brescia, Italy) (Università di Bologna, Italy) (University of Belgrade, Serbia) (Università di Parma, Italy) (Università di Parma, Italy)
Marco Savoia
Aleksandar Sedmak Andrea Spagnoli Sabrina Vantadori Charles V. White
(Kettering University, Michigan,USA)
VI
Frattura ed Integrità Strutturale, 38 (2016); International Journal of the Italian Group of Fracture
Journal description and aims Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is the official Journal of the Italian Group of Fracture. It is an open-access Journal published on-line every three months (July, October, January, April). Frattura ed Integrità Strutturale encompasses the broad topic of structural integrity, which is based on the mechanics of fatigue and fracture, and is concerned with the reliability and effectiveness of structural components. The aim of the Journal is to promote works and researches on fracture phenomena, as well as the development of new materials and new standards for structural integrity assessment. The Journal is interdisciplinary and accepts contributions from engineers, metallurgists, materials scientists, physicists, chemists, and mathematicians. Contributions Frattura ed Integrità Strutturale is a medium for rapid dissemination of original analytical, numerical and experimental contributions on fracture mechanics and structural integrity. Research works which provide improved understanding of the fracture behaviour of conventional and innovative engineering material systems are welcome. Technical notes, letters and review papers may also be accepted depending on their quality. Special issues containing full-length papers presented during selected conferences or symposia are also solicited by the Editorial Board. Manuscript submission Manuscripts have to be written using a standard word file without any specific format and submitted via e-mail to iacoviello@unicas.it. Papers should be written in English. A confirmation of reception will be sent within 48 hours. The review and the on-line publication process will be concluded within three months from the date of submission. Peer review process Frattura ed Integrità Strutturale adopts a single blind reviewing procedure. The Editor in Chief receives the manuscript and, considering the paper’s main topics, the paper is remitted to a panel of referees involved in those research areas. They can be either external or members of the Editorial Board. Each paper is reviewed by two referees. After evaluation, the referees produce reports about the paper, by which the paper can be: a) accepted without modifications; the Editor in Chief forwards to the corresponding author the result of the reviewing process and the paper is directly submitted to the publishing procedure; b) accepted with minor modifications or corrections (a second review process of the modified paper is not mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. c) accepted with major modifications or corrections (a second review process of the modified paper is mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. d) rejected. The final decision concerning the papers publication belongs to the Editor in Chief and to the Associate Editors. The reviewing process is usually completed within three months. The paper is published in the first issue that is available after the end of the reviewing process.
Publisher Gruppo Italiano Frattura (IGF) http://www.gruppofrattura.it ISSN 1971-8993 Reg. Trib. di Cassino n. 729/07, 30/07/2007
Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0)
VII
Fracture and Structural Integrity, 38 (2016); ISSN 1971-9883
Great news for our journal!
D
ear friend, in this issue you will find the second part of the papers focused on the Multiaxial Fatigue and Fracture . We wish to warmly thanks the three guest editors (Andrea Carpinteri, Ali Fatemi, Carlos Navarro Pintado) for their efforts and all the authors for the really high quality of their papers. I am pleased to inform you that we received great the 2015 Index Copernicus evaluation: the ICV 2015 is equal to 129.04 (ICV2014: 120.66). In 5 years, we always increased our ICVs and this is mainly due to the quality of the published papers … thank you!!! Finally, just a few words about the ECF21 that was held in june in Catania. It was a great success! A great team (first of all, the IGF Ex-Co), enthusiast participants and the wonderful venue were the main ingredients that allowed to organize a successful event, with more than 650 participants and about 700 presentations. For the first time in the ESIS history, almost all the presentations were videorecorded and they are now available in the new ESIS YouTube channel. If you were not able to join us in Catania, you are still in time to join us on web (… don’t miss the social event!!).
Francesco Iacoviello F&IS Chief Editor
VIII
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
Focussed on Multiaxial Fatigue and Fracture
On the stress investigation at the edges of the fixed elastic semi-strip
N. Vaysfeld Odessa Mechnikov University, Institute of Mathematics, Economics and Mechanics vaysfeld@onu.edu.ua O. Kryvyi National University «Odessa Maritime Academy» (NU «OMA») krivoy-odessa@ukr.net, kryvyi-od@math.onma.edu.ua Z. Zhuravlova Odessa Mechnikov University, Institute of Mathematics, Economics and Mechanics zhuravleva@te.net.ua A BSTRACT . The stress state of the elastic fixed semi-strip with the regarding of the singularities at its edge is investigated in the article. The initial boundary problem is reduced to a vector boundary problem in the transformation’s domain by the use of integral Fourier transformation. The one-dimensional vector boundary problem is solved exactly with the help of matrix differential calculations and Green’s matrix apparatus. The problem’s solving was focused at the solving of the singular integral equation (SIE) with the two fixed singularities at the ends of the integration’s interval. The symbol of SIE was constructed and the generalized method of the SIE solving was applied. The stress’ distributions of the semi-strip are investigated. K EYWORDS . Semi-strip; Vector boundary problem; Singular integral equation; Fixed singularity.
Citation: Vaysfeld, N., Kryvyi, O., Zhuravlova, Z., On the stress investigation at the edges of the fixed elastic semi-strip, Frattura ed Integrità Strutturale, 38 (2016) 1 11.
Received: 14.04.2016 Accepted: 09.06.2016 Published: 01.10.2016
Copyright: © 2016 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
I NTRODUCTION
he plain elasticity problems for a semi-strip are important as the model examples for the different engineering applications. So many authors have examined these problems in their works. A short review of the different approaches to the solving of the plane elasticity problems for an elastic semi-strip is given below. The reducing of the problem to the Fredholm's integral equation of the first kind was used in [1] for estimation of the symmetrically loaded semi-strip fixed by the short edge. Another approach based on the construction of the stress function as the combination of the Fourier’s integrals and the series was used in [2, 3]. The problems for a semi-strip considered in [4, 5] were reduced to the singular integral equations which were solved numerically, it leaded to the solving
1
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
of an infinity system of the algebraic equations. The variation method was used for the analogical problem’s solving in [6 8]. The energetic method was applied to the problem of the semi-strip with the free lateral edges and loaded short edge in [9]. In [10] authors constructed a special system of byorthogonal functions, with the help of which they solved the problem on a semi-strip loading at it’s the short edge. The problem for the semi-strip with the free longitudinal sides was solved with the help of the stress function in [11, 12]. The Laplace’s integral transformation was used for the problem’s solving in [13]. The approach based on the use of Fadle-Papkovich functions was applied in [14-16]. In this paper the method, which was worked out by G. Ya. Popov, was used [17]. Accordingly to it the integral transformations were applied directly to the equilibrium equations and boundary conditions of a problem. It leaded the initial problem to one-dimensional boundary problem in the transformation’s domain. The last one was formulated as the vector boundary valued problem and solved exactly with the apparatuses of the matrix differential calculations and Green’s matrix function [18]. The problem was reduced to the singular integral equation’s solving. Investigation of the signature’s nature of the singular integral equation’s solving was under consideration of many famous scientists. Today the new theories are appeared, which describe the solution’s behavior at the particular points [19]. The investigations of the singularities’ nature for the complex medium are continued [20]. But in most studies the authors did not pay attention to the fixed singularities at the angular points of the semi-strip usually, although these singularities play a main role in the estimation of the stress state. One approach that allows to find and to take such singularities into account was proposed in widely known work [21]. It was used in this paper for the fixed singularities’ consideration. The special generalized method, which was proposed in [22, 23], was applied to obtain the solution of the (SIE) with regarding of the solution’s two fixed singularities at the end of the integration’s interval.
Figure 1 : Geometry of the problem.
T HE STATEMENT OF A PROBLEM
T
he elastic ( G is a share module, is a Poison’s coefficient) semi-strip,
x a 0 , 0 is considered. At the y
x a 0, 0 the semi-strip is loaded
edge y
x ( , 0) 0,
x ( , 0)
p x
x a
(1)
,
0
y
xy
where p x ( ) is the known function. At the lateral sides x
y 0, 0 and x a
y , 0 the boundary conditions of the fixed are given
0,
,
,
y
u y
v y
u a y
v a y
(2)
0, 0,
0,
0,
0,
0
2
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
y v x y u x y ( , ) , are the displacements which satisfy the Lame’s equations. The Lame’s
x u x y u x y ( , ) , ,
here
equations are written in the following form [24]
2
2
2
u x y
u x y
v x y x y u x y x y
( , )
( , )
( , ) 0
*
0
2
2
x
y
(3)
2
2
2
v x y
v x y
( , )
( , )
( , ) 0
*
0
2
2
x
y
1
0 * , through the Muskchelishvili constant
0
*
0
where
. After the expression of the constants
,
1
1 2
3 4
, one obtains the system (3) in the another form
2
2
2
u x y
u x y
v x y x y u x y x y
( , )
( , )
( , )
1 1 1 1
2
0
2
2
1
x
y
(4)
2
2
2
v x y
v x y
( , )
( , )
2
( , )
0
2
2
1
x
y
The boundary conditions on the semi-strip’s edge are reformulated with the terms of the displacements
, 0
v x , 0
u x
1
0
G
p x
x a
(5)
2
, 0
x
y
, 0
, 0
u x
v x
0, 0
x a
(6)
y
x
One needs to solve the boundary value problem (2), (4)-(6) to estimate the stress state of the semi-strip.
T HE GENERAL SOLVING SCHEME OF THE PROBLEMS ON THE SEMI - STRIP STRESS STATE ESTIMATION
T
he Fourier’s transformation is applied to the system of Lame’s equation and to the boundary conditions by the scheme u x y u x y dy v x y v x y 0 ( ) cos , ( ) sin , (7)
with the inverse formula
u x y v x y , ,
0
u x v x
( ) cos ( ) sin
y
2
d
(8)
y
The initial problem has the form after this
3
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
2
( -1)
2
3
u x
u x
v x
x '( )
" ( )
( )
' ( )
1
1
1
2 (
1
1)
2
1 1
v x
v x
u x
x ( )
" ( )
( )
' ( )
(9)
1
u a v a (0) 0, ( ) 0 (0) 0, ( ) 0
u
v
Here the new unknown function is inputted x v x
x v x , 0 , ' ' , 0
. As it is seen from the boundary condition
u x
x
(6), , so the condition (6) is satisfied automatically. With the aim to reduce the problem to the vector boundary problem one must input the vectors and the matrixes y , 0 '
3
1 0 1
1
x
'
0
u x v x
x 1 1 1
1
, f x
, P
, Q
y x
.
1 1
1
0
0
1
L y x f x 2
, where L 2
is a
Then the equations in the vector form will be written as the vector equation
differential operator of the second order , I is an identity matrix. The integral transformations also should be applied to the boundary conditions, with the aim to formulate the boundary functionals in the transformations’ domain. As a result the vector boundary problem is constructed L y x f x y y a 2 0 0, 0 (10) L y x Iy x " Qy x ' Py x 2 2 2
T HE SOLVING OF THE VECTOR BOUNDARY VALUE PROBLEM
T
he solution of the vector boundary problem (10) will be searched as the superposition of a homogenous vector equation’s general solution y x 0 and a particular solution of the inhomogeneous one y x 1
y x y x y x 0 1
These solutions were constructed with the help of the matrix differential calculation apparatus earlier [18].
c c 1 2
c c 3 4
y x Y x
y x 1
Y x 2
1
i Y x i , 0,1 are the matrix system of the fundamental matrix solutions [18]:
where
x
x
x
x
x
x
1
x 1
1
1
e
e
Y x 1
Y x 2
,
x
x
x
2
2
1
1
1
1
c i , 1, 4 are founded from the boundary conditions [18].
where constants i
4
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
y x 1
, was constructed the Green’s matrix
For the obtaining of the vector boundary problem’s particular solution function [18]. Elements of matrix are shown in the Appendix A. The inhomogeneous boundary problem’s final form of the solution is constructed
a
c c 1 2
c c 3
y x Y x
G x f ,
d
Y x 2
(11)
( )
1
4 0
The components of (11) can be written in the next form
a 0
a 0
3
1 1
u x Y x c Y x c 11 12 1 1 ( ) 1
Y x c 11
Y x c 12
G x 11
'
G x 12
d
d
,
,
2 2
3 2
4
1
a 0
a 0
3
1 1
v x Y x c Y x c 21 22 1 1 ( ) 1
Y x c 21
Y x c 22
G x 21
'
G x 22
d
d
,
,
2 2
3 2
4
1
where i j G x , , is the Green’s matrix function element in a i row and j column. The integrals with the function are calculated by the parts and the inverse integral transformations’ formulae were applied to the displacements’ transformations.
0
a 0
cos
'
f x 1
u x y ,
y d d
, ,
(12)
0
a 0
cos
'
f x 2
v x y ,
y d d
, ,
where are known functions. The formulae (12) would be the final ones if the unknown function ' is known. For its finding one must satisfy the boundary conditions (5) which are unsatisfied yet. It should be taken into consideration that integrals in these correspondences are conditionally convergent integrals. So, before the differentiating of the displacements’ expressions, at first one must extract the weakly convergence parts at these integrals. The substitution of (12) in the boundary conditions (5) leads to the singular integral equation a f x d r x x a * * * * * * 0 ' , , 0 here the function f x , contains Cauchy’s type singularities and fixed singularities on the both ends of the integration interval. , i i f x g x i , , , , , 1, 2
T HE SOLVING OF THE SINGULAR INTEGRAL EQUATION
T
*
*
x a
a
2
2 ,
is done for the passing to the integration interval
1;1 . As
x
he changing of the variables
a
a
a result the integral equation is transformed to the form
5
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
1
1
2
c
c
x c
c
x
1
1
1
1
c
с
3,1
3,2
3,1
3,2
2
c
d
1
2
2
2
2
2 2
x x
x
2
x
x
x
x
2
2
2
1
1
(13)
1
1 1
x c
1 1
c
x
K x d r x ,
4
4
x 1
d
,
1
3
3
x 2
x
2
1
1
a
here
, K x r x , , are the known regular functions, i
c i , 1, 4 are shown in the Application B.
2
The Eq. (13) is the partial case of the equation with two fixed singularities considered [21]
m k
m k
y
y
1
1
n
c
x y ,
x
x
1
1
c
1
k 2
A x c
x
1
dy
dy
0
i
i
y x
m k
m k k
k
1
y
y
xy
1
1
1
k
k
0
1
1
1
y dy f x
, 0 Re
m k
K x y ,
k
1
which can be rewritten as
1
A x c
x c S x N x N x K x K x y ,
y dy f x
0
1 1
1
1
1
1
where
m
y
1
n
x
1
k
c
k 2
c x 1, 1 lim ,
N x 1
dy c ,
c
,
k k
k
m kk
i
x
1 1
k
1
y
x y
1
2
k
0
1
The symbol of the singular integral Eq. (13) was constructed, which has the following form, where all designations correspond to the designations in [21]
m k
2
c n
c S 1
R
,
k k
2 ,
A A
,
k
0
,
A
(14)
m k
2
c n
,
c S 1
R
,
k k
2 ,
k
0
z , , ,
p S z cth i
1 / ,
z ,
,
m k
k
k 1
1 ,
iz
n
m
,
k
k
k
k
,
k
sh i
z
6
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
1
1 2
G zi
i
4
2 2 G
p
1
i
A z
1
2 1 1
2 1 sinh
z
i
z
i
sinh
p
p
1
1
1
1
1 3
2 1
G zi
zi
i
G zi
i
G z
i
i
16
1
4
2 cosh
p
p
p
p
1
1
1
2 1 1 sinh
2 1 1 sinh
i i
i i
z
z
z
i
1
sinh
p
p
p
here p 2 ,
is found from the known solution of the analogical problem for an edge with the angle of
0.31
openness pi/2 [25]. According to [21] one needs to find the roots of the equation
A z 0
. The found roots of the kernel’s symbol (14)
0.5562 0.3690,
3,4
1.2792 0.2380,
5,6
3.2089 0.7127,
7,8
5.2170 1.0251,...
have the next form: 1,2
,
where k k because of the problem’s statement. The generalized method of SIE solving [22, 23] was applied for the solving of the Eq. (13). According to it the unknown function is expanded by the series in each interval
N
1
k k
s
,
1; 0
k 0 2 1 N k N
(15)
0;1
k k
s
,
where
Re
k
2
1
cos Im ln 1 , sin Im ln 1 , k
N k
k
0,
1
,
Re
2
k
1
k
k 2 1
k N
Re
2
1
k N cos Im ln 1 , sin Im ln 1 , k N
N k N
k
.
,
1
k N
Re
2
1
k 2 1
1 . The Eq. (13) is considered when
The segment
1;1 is divided on N 2 equal segments with the length h N
h
i 2 . After the substitution of the unknown function (15) into the singular integral Eq. (13) one obtains system of the linear algebraic equations relatively to the unknown constants k s k N , 0, 2 1 of the expansion (15). x ih i , N 0, 2 1 1
N 2 1
N 0, 2 1
k ki s d f i , i
(16)
k
0
where ki i d f i k N , , , 0, 2 1 are shown in the Application C. The expression (16) presents the system of N 2 equations with regard of N 2 unknown constants k s . The substitution of the founded constants in the formula (15) and following using of the formulae (12) completes the construction of the problem’s solution.
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N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
T HE RESULTS OF THE NUMERICAL ANALYSES
T
9 61.2781955 10 Pa,
0.33 ). At Fig.2 one can admit
he calculations were done for the elastic semi-strip ( G
that the values of the normal stress at the lateral side x 0 decrease to zero with the increasing of the distance from the semi-strip’s edge. When the semi-strip’s side is a 10 . A similar situation is observed during the analyses of the stress y x , when the semi-strip’s side is a 50 (Fig.4) and a 100 (Fig.6). At Fig.3 one can admit that the absolute values of the normal stress y at the line x a y / 2, 0;10 are higher by its absolute value then normal stress x when the semi-strip’s side is a 10 . A similar situation is observed during the analyses of the stress y x , when the semi-strip’s side is a 50 (Fig.5) and a 100 (Fig.7). As it is seen the stabilization of the stresses y x , is observed when the semi-strip’s side is a 50 (Fig.5) or a 100 (Fig.7). y y x x P / , P /
y
Figure 2 : Normal stresses
y a
a y
a y a / 2, ,
.
Figure 3 : Normal stresses
.
0, ,
0, ,
10
/ 2, ,
10
y
x
y
x
y
.
Figure 5 : Normal stresses
.
Figure 4 : Normal stresses
y a
a y
a y a / 2, ,
0, ,
0, ,
50
/ 2, ,
50
y
x
y
x
8
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
y
y a
a y
a y a / 2, ,
Figure 6 : Normal stresses
0, ,
0, ,
100
.
Figure 7 : Normal stresses
/ 2, ,
100
.
y
x
y
x
C ONCLUSIONS
1. The proposed solving method reduced the initial problem to the singular integral equation, which has the two fixed singularities at the end of the integration’s interval. The special generalized scheme of SIE solving was applied with the aim to take these singularities into consideration. 2. The proposed approach may be applied to the solving of the elasticity mixed problem for the semi-strip with a crack. 3. The analyses of the numerical results show that the taking into consideration the existence of the two fixed singularities of the solution gives the possibility to obtain the numerical result on the distance less than a/1000 to the angular point of the semi-strip in comparison with the usual approach to the solving, allowing to get the stable results only on the distance to the angular points not less than a/10.
A PPENDIX A
a x ch a
ch
x
G x 11
,
sh a
2
1
1
sh a
a x
sh
a x xsh
sh a 2
2
1
a x a
x ach a ch
x
x ash a
a x ch a
ash
ach a ch a 2 1 1
1
G x 12
a x
sh
,
sh a
2
1
x a
sgn
x
a x
x ch a
x
x sh a
ch
a x
sgn
9
N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01
ach a ch a 2 1 1
1
G x 21
a x
sh
,
sh a
2
1
x a
sgn
x
a x
x ch a
x
x sh a
ch
a x
sgn
a x ch a
ch
x
G x 22
,
sh a sh a ch 2
1
x
a x ch a
sh a a x sh 2
2
1
a x a
x a ch a ch
x
x sh a
a x ch a
A PPENDIX B
3 2 1 1 1
4 2 G
2
2
G
G
G
2
2
2
2
2
c 0,
c
c
c
c
,
,
,
,
0
1
3,1
2 1 1
2 1
2 1
4 2 G
4 2 G
4 2 G
1
1
G
G
16
16
c
c
c
c
c
,
,
,
,
3,2
3,1
3,2
4
4
2 1 1
2 1 1
2 1 1
1
1
A PPENDIX C
0
c
c
x
c
c
x
1
1
1
1
c
c
с
i
i
3,1
3,2
3,1
3,2
1
2
2
k
d
ki
2
2
2
2
x
x
x
2
2 2
x
x
x
x
2
2
2
i
i
i
i
i
i
i
1
1 1
1 1
c
x c
x
i
i
4
4
K x d k N i , , 0, 1,
N 0, 2 1
i
3
3
x
x
2
2
i
i
1 0
2
2
c
c
x
c
c
x
1
1
1
1
c
c
с
i
i
3,1
3,2
3,1
3,2
1
k
d
ki
2
2
2
2
x
x
x
2
2 2
x
x
x
x
2
2
2
i
i
i
i
i
i
i
1 1
1 1
c
x c
x
i
i
4
4
K x d k N i , , 0, 1,
N 0, 2 1
i
3
3
x
x
2
2
i
i
r x i ,
f
N 0, 2 1
i
i
R EFERENCES
[1] Vorovich, I. I., Kopasenko, V. V., Some problems of elasticity theory for the semi-strip. (in Russian), Prikladnaya matematica i mekchanica, 30(1) (1966) 128-136. [2] Pickett, G., Jyengar, K. T. S., Stress concentrations in post-tensioned prestressed concrete beams, J. Technol., India, 1(2) (1956). [3] Yamasida, Research of the tensions in semi-infinite strip under acting forces applied to its edge, Trans. Japan. Soc. Mech. Engrs, 20(95) (1954) 466.
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