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Combined implementing the hole-drilling method and reflection hologram interferometry for residual stresses determination in cylindrical shells and tubes

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This article was srcinally published in a journal published byElsevier, and the attached copy is provided by Elsevier for theauthor’s beneﬁt and for the beneﬁt of the author’s institution, fornon-commercial research and educational use including withoutlimitation use in instruction at your institution, sending it to speciﬁccolleagues that you know, and providing a copy to your institution’sadministrator.All other uses, reproduction and distribution, including withoutlimitation commercial reprints, selling or licensing copies or access,or posting on open internet sites, your personal or institution’swebsite or repository, are prohibited. For exceptions, permissionmay be sought for such use through Elsevier’s permissions site at:http://www.elsevier.com/locate/permissionusematerial
A u t h o r ' s p e r s o n a l c o p y
Optics and Lasers in Engineering 45 (2007) 661–676
Combined implementing the hole-drilling method and reﬂectionhologram interferometry for residual stresses determination incylindrical shells and tubes
V.V. Balalov, V.S. Pisarev
, V.G. Moshensky
Moscow State University of Service Industries, Cherkizovo Moscow reg., 141200 Russia
Available online 30 October 2006
Abstract
Main features inherent in simpliﬁed approach to residual stresses determination in cylindrical shells and tubes, external diameter of which is not less than 60mm, by combing the hole-drilling method and reﬂection hologram interferometry are discussed in detail. Initialexperimental information in a form of hole diameter increments in principal stress directions is derived from high-quality reﬂectionholograms recorded near cylindrical objects of intermediate curvature value. Converting measured parameters into required stress valuesis based on the transition model that corresponds to plane stress conditions of pure membrane type. The technique developed is capableof determining residual stress component values within 5% accuracy in an absence of stress gradients over the probe hole diameter whena type of residual stress ﬁeld corresponds to the transition model adopted. The accuracy analysis involved is based on matrix formulationof conventionally direct problem and an assumption on a pure membrane character of residual stress ﬁeld under study for thin-walledshell. Required error estimations in a case of inspecting thick-walled cylindrical tube are obtained by combining the above-mentionedapproach and an analogy of reconstructed fringe patterns with actual and artiﬁcial interferograms, which follow from drilling blind holeof the same geometrical parameters in thick-walled plates. Experimental veriﬁcation of the developed approach is founded upon adetermination of actual stresses in thin-walled cylindrical shell and obtaining residual stress distributions at the proximity of welded jointin thick-walled cylindrical tube.
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2006 Elsevier Ltd. All rights reserved.
Keywords:
Reﬂection hologram interferometry; The hole drilling method; Residual stresses; Cylindrical shells and tubes; Interference fringe patterns;Veriﬁcation of the transition model
1. Introduction
Residual stresses are usually deﬁned as those stresses,which remain in a structural component after manufactur-ing and processing in the absence of external loading orthermal gradients. In other words, a body subjected toresidual stresses inﬂuence is stationary and at equilibriumwith its surroundings. This circumstance means thatresidual stresses are more difﬁcult for theoretical ornumerical prediction than in-service stresses with whichthey are superimposed. Modern analytical and especiallycomputational techniques supported with advancedmathematical and numerical procedures such as ﬁnite-element- and boundary-element-method are capable of reliable estimating the stresses to which a component issubjected in service. But a power of these techniquesmainly depends on a degree of the accuracy within thatexternal inﬂuences, which are responsible for stressesarising, are prescribed in the course of numericalsimulation. Reasonable accuracy of the determination of material mechanical properties in both elastic andelasto-plastic deformation ranges is the second factoressential for a correctness of the computational process.Available approaches are quite reliable for a componentloading by forces of any nature especially in elasticdeformation range. But a determination of elasto-plasticand especially thermal stresses is often accompanied withserious difﬁculties, which are inﬂuenced by both above-mentioned factors.
ARTICLE IN PRESS
0143-8166/$-see front matter
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2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.optlaseng.2006.08.011
Corresponding author. Fax: +74957776332.
E-mail address:
pisarev@tsagi.ru (V.S. Pisarev).
A u t h o r ' s p e r s o n a l c o p y
Residual stress evaluation represents by itself the mostsophisticated problem in both a mechanical formulationand modelling prediction. The point is that residual stressesare inherent in most manufacturing processes involvingmaterial deformation, heat treatment, machining orprocessing operations that transform the shape or changethe properties of a material. In many situations both asource and a history of residual stress arising cannot beestablished within the accuracy that is large enough forreliable numerical simulation. Such a simulation has to bebased on prescribing a wide set of unknown parameters,which cannot be characterised without doubts. Thus aproblem of incorporating predictive residual stress analysisinto design, which is of the great importance in aerospace,nuclear and other critical engineering industries, rigorouslydemands involving experimental information obtainedwith a high degree of reliability [1,2].That is why a wide set of experimental techniques basedon interferometric measurements of the material responseon small hole drilling was developed for residual stresscharacterisation during more than last two decades. Theﬁrst steps in this direction, which are mainly founded upondifferent holographic interferometric methods, have de-monstrated some evident advantages of whole-ﬁeld ap-proach comparing with traditional point-wise techniquesbased on using strain gage rosette [3–11]. Implementingholographic interferometric techniques for a measurementof the required deformation parameters gives the maximumpossible sensitivity of the hole drilling method with respectto residual stresses determination. At ﬁrst, this is attributedto a high sensitivity of the measurement procedure to allthree components of strain-induced displacements. Second,a high quality of holographic interferograms allows usobtaining reliable quantitative results at the edge of smallhole immediately. Deriving reliable quantitative results ispossible starting from a hole of 1mm diameter. Thus acombination of the hole-drilling method with furtherholographic interferometric measurements represents byitself the least destructive method from all other destructivemethods of residual stresses determination.Some further developments are related to looking forways of semi-automated or fully automated measurementsof the required deformation parameters with simultaneousincreasing an accuracy of their determination. In most of cases, these parameters can be reliably expressed proceed-ing from in-plane displacement components ﬁelds at thehole proximity. Created approaches of the ﬁrst type aremainly based on single-axis holographic interferometry[12–16]. The second class of developed research toolsincludes Moire’ interferometry and analogous opticaltechniques [17–22]. Fully automated procedures of initialdata acquiring and further processing are founded uponelectronic and digital speckle pattern interferometry[23–33]. The third development direction assumes con-structing a measurement procedure that is capable of obtaining maximum possible volume of information forfurther highly accurate deriving residual stress compo-nents. Such an approach is based on reﬂection holograminterferometry [34,35]. Essential features of reﬂectionhologram interferometry with respect to residual stressesdetermination in plane structures by through or blind holedrilling have been previously established and comprehen-sively demonstrated [36–38]. Among them unique cap-ability of recognising principal strain directions at a stageof fringe patterns reconstruction is of great metrologicalimportance.All above-mentioned techniques can be implemented forresidual stresses characterisation in objects bounded withplane or quasi-plane surfaces. They are reliably substan-tiated in the metrological sense with respect to adetermination of initial experimental information that ispresented in a form of displacement component ﬁelds atthe hole proximity. But the main problem of anymeasurement procedure resides in the fact that derivingresidual stress components from initial experimental data isa non-direct process. This means that measured parametershave to be converted into required residual stress values.Such a transition always demands of introducing somemechanical relations. In many cases, these relations cannotbe experimentally or analytically established withoutdoubts. A set of the above-mentioned dependencies isusually deﬁned as a transition model [36–41]. A form of thetransition model directly depends on a type of residualstresses ﬁeld under study that is unknown in general case[42,43]. In most of practical cases, a choice of the transitionmodel could be only based on measurement data obtainedover a single face of the structure (one-side measurements).This is especially true with respect to cylindrical shells andtubes of small external diameter.Thus we can say that most of approaches to residualstresses deriving from data of interferometric measure-ments are founded upon prescribing a deﬁnite type of theresidual stress state under study if a single both blind andthrough hole is drilled. Moreover, in most of cases this typerepresents a pure membrane stress ﬁeld that corresponds tothe plane stress conditions on the object face of interest.Such an approach demands a posteriori analysis of accuracy of the results obtained especially if thin-walledor curved structures have to be inspected. Three mainways, which are suitable for this purpose, can berecognised. The ﬁrst of them is initially proposed andrealised in work [35] and then is implemented in works[24,25,33]. Its essence resides in holding uniform actualstress ﬁeld with known parameters over a large area of theobject surface. The second approach consists of investiga-tion of residual stress ﬁeld, parameters of which can bereliably established proceeding from the known solution[13–14]. Drilling holes in stress ﬁeld of any from two above-mentioned types allows us a direct comparison of obtainedand given actual or residual stress values. But we shouldkeep in mind that a positive result of such comparing mightbe of limited metrological value because a real inverseproblem is substituted with a conventionally directproblem.
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662
A u t h o r ' s p e r s o n a l c o p y
The third and the most sophisticated way for a choiceand veriﬁcation of the transition model is based onconstructing so-called reference fringe patterns of thesecond kind [40,41]. A ﬁne coincidence between actualand reference interferograms takes place when the transi-tion model involved corresponds to the type of residualstress ﬁeld of interest [13,14,28–31,36–39]. If such acorrelation is not achieved a question on the accuracy of determination of residual stress components remains open.Unfortunately, the latter situation is typical for welded joints of both plane and curved structures of differentthickness [36–42]. In mechanical sense this means thatresidual stress ﬁeld of interest consists of both membraneand bending components for thin-walled structures and ischaracterised by a presence of strain gradients for thick-walled structures. In such cases there is no uniquecorrespondence between a form of each speciﬁc fringepattern and a type of the residual stress state.In other words, an identiﬁcation of the transition modelin rigorous sense represents by itself an inverse problem[44]. This means that the results of residual stressesdetermination might be often obtained within considerableerrors, which follow from both experimental procedureand, mainly, incorrect formulation of each speciﬁc problem[39,42]. Thus increasing an accuracy of residual stressesderiving from initial experimental data demands, asminimum, recognising the type of residual stress state atthe point of hole drilling. Reﬂection hologram interfero-metry is the most reliable tool for this purpose. We can saythat reconstructed holographic fringe patterns carry themost possible information volume in a sense of residualstresses determination comparing with any other opticalinterferometric techniques of whole-ﬁeld nature. This isattributed to a high sensitivity of any holographicinterferometric technique to out-of-plane displacementcomponent. This fact in turn gives us a remarkablecapability of recognising a type of residual stress ﬁeld atthe point of interest [38,39].A problem of correct residual stresses deriving can besolved for plane thin-walled structures in general form[42,43]. Developed approach includes a correct formulationof the inverse problem on a base of two-side initial datawith a measurement procedure on each side has to beperformed in two principal strain directions. But this way isof great technical complexity when curved cylindrical shellsand tubes are of interest. That is why a determination of residual stress components in cylindrical objects of inter-mediate curvature by using one-side initial data is the ﬁrstmain subject of presented work. To do this, the techniquesdeveloped and the results obtained in the course of residualstresses determination in plane structures by means of one-side measurements [36–39] have to be expanded forresidual stresses determination in structures bounded bycurved surfaces with minimal changing and reﬁning.Emphasis is made in overcoming both technical andmetrological difﬁculties, which are inherent in inspection of bodies bounded with curved surfaces. First a procedure of reconstructing high-quality fringe patterns in the caseconsidered is more difﬁcult comparing with the case of plane surfaces. This resides in distinctions in diffuseproperties between plane and curved object made fromthe same material and having the same micro relief. Such adifference negatively inﬂuences on a fringe quality when acurvature of non-plane surface is increased. Thus the ﬁrstproblem to be solved is creating experimental technologythat is capable of obtaining fringe patterns related tocurved surfaces, a quality of which is high enough forreliable deriving initial information. The second problem tobe overcome for an accurate determination of residualstress components in cylindrical shells and tubes is a correctchoice of the transition model for a wide variety of parameters to be prescribed. Common list of theseparameters for plane and curved structures includes a typeof probe hole (through or blind), a type of residual stressesﬁeld, and an explicit form of numerical model used for acalculation of residual stress components. But in a case of non-plane object a form of the above-mentioned modelmight considerably depend on a value of the surfacecurvature. This means that each speciﬁc version of transition model, a choice of which is based on one-sidedata, should be very carefully veriﬁed proceeding fromdifferent criteria. An analysis of some from possible waysof transition model veriﬁcation is the second main subjectof presented work.
2. Formulation of the problem and main relations
There is the basic assumption, which is mainly used forderiving residual stress values in plane structural elementsby means of the hole drilling method. It consists of the factthat a small hole is made in two-dimensional stress ﬁeld,which is uniform independently of each from two principalstress directions on the object surface [35,36]. This meansthat a possible inﬂuence of stress gradients at the holeproximity is not taken into account. The superpositionprincipal for local displacement and strain ﬁelds corre-sponding to each in-plane stress tensor component is alsopostulated. A small area of the surface to be investigated,which is under the plane stress conditions, is shown inFig. 1. Components
s
1
and
s
2
on the object surface shalldenote the stress ﬁeld that satisﬁes the above-mentionedconditions. A through or blind hole of diameter 2
r
0
is madeat some point of the surface area under consideration. Thecentre of this hole is a conventional point where theresidual or actual stresses must be determined. LocalCartesian co-ordinate system (
x
1
,
x
2
), the axes of whichcoincide with the directions of the principal strains
e
1
and
e
2
, respectively, is also shown in Fig. 1. These axes aredirected along the axes of symmetry of all interferencefringe patterns, which are presented in this work asillustrations.A decomposition of components of tensor of principalstresses (
s
1
,
s
2
) and strains (
e
1
,
e
2
), which are related to thehole-drilling method, are conventionally shown in Fig. 1.
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A u t h o r ' s p e r s o n a l c o p y
Roman numerals denote mechanical state of the objectsurface. State I represents a two-axes loading of theelement of the material volume containing the probe hole.State II is an initial deformed state of the object surfacearea of interest caused by two-dimensional ﬁeld of residualstress before drilling a hole. The components of this stressﬁeld are the principal residual stresses
s
1
and
s
2
to bedetermined. State III is related to residual stress energyrelease after hole drilling and can be represented as adifference between State I and II:I
II
¼
III
:
(1)Relation (1) in general form describes a residual stressesenergy release after hole drilling. Corresponding para-meters of State III have to be derived from interferencefringe patterns and those of States I and II should bedescribed analytically or numerically proceeding from theelasticity theory relations.The most convenient and powerful way for derivingparameters, which are essential for further calculation of required residual stress values, resides in representation of Relation (1) in terms of local deformation of the hole edge[36–43]. This approach is based on considering a distribu-tion of circumferential strain component
e
j
. along a real orconventional edge of small circular hole. Requireddistribution, when the plane stress conditions are valid,can be expressed as [35,36]
j
¼
1
r
0
q
v
q
j
cos
j
q
u
q
j
sin
j
, (2)where
r
0
is a hole radius;
j
is the polar angle in the co-ordinate system referred to the hole centre;
u
and
v
are thein-plane displacement components of points belonging tothe hole edge in the directions of co-ordinate axes
x
1
and
x
2
, respectively (see Fig. 2). Accordingly to the developedapproach the values of residual strain components arederived from Eq. (2) by
1
¼
j
¼
90
;
2
¼
j
¼
0. (3)Notation (3) allows us obtaining Relation (1) in requiredform:
I1
II1
¼
III1
;
I2
II2
¼
III2
. (4)The principal strains
I1
and
I2
related to State I can bealways expressed in a form of solution of the strain/stressconcentration problem in elastic deformation range:
I1
¼
a
1
s
1
E
a
2
s
2
E
;
I2
¼
a
1
s
2
E
a
2
s
1
E
, (5)where
a
1
and
a
2
are generalised strain concentrationfactors;
E
is the elasticity modulus of the material. Withinthe approach developed the values of strain concentrationfactors have quite evident mechanical interpretation thatfollows from Relations (2) and (3):
a
i
¼ ð
1
Þ
i
1
I
i
II
i
, (6)where
i
¼
1, 2 indicates principal strain direction. Notethat the values of deformation parameters in Formulation(6) are referred to one-axis stress state.
ARTICLE IN PRESS
ϕ
1234I
X
1
II
X
1
X
2
X
2
X
2
III
X
1
(a) (b) (c)
Fig. 1. Conventional scheme related to implementing the hole-drilling method for residual stresses determination; (a) State I, (b) State II, (c) State III.
σ
2
2
r
0
σ
1
σ
1
σ
2
ε
1
ε
2
φ
x
1
, u x
2
,
Fig. 2. Notation used in the course of residual stresses determination.
V.V. Balalov et al. / Optics and Lasers in Engineering 45 (2007) 661–676
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