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Article history:Available online 26 November 2011

Keywords:Crack initiationResidual stressCrack compliance methodModied SEN specimen

and costly. Nevertheless, with the passing of time, the improvingof technology and skills has allowed a better application of numer-ous and diverse materials. In particular, the extended increase onthe use of metal technology has provided the chance of a fasterand better development. However, the use of metals in miscella-neous applications has caused the number of accidents and casual-ties to reach unknown levels. In these sense there has been

development of failure could be divided in two basic parts, initiationand spread [2]. Additionally, there is a great number of external andinternal factors that contribute to the nucleation and propagation ofa crack [3]. Slip bands or dislocations and surface scratches can beconsidered as internal effects, while as external factors areconsidered the effect of forces and deformations. Nevertheless,when the development, performance and effect of a crack is ana-lyzed, prior loadhistory in thematerial is not considered extensivelyor in a sufcient manner. To consider prior load history in thecomponent raises the difculty of the problem in a substantial

Corresponding author.

Theoretical and Applied Fracture Mechanics 56 (2011) 188199

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.e lE-mail address: guiurri@hotmail.com (G. Urriolagoitia-Sosa). 2011 Elsevier Ltd. All rights reserved.

1. Introduction

It iswell knownworldwide and through themankind technologydevelopment that the application of materials in engineering de-signs has posed diverse problems [1]. At the beginning of technolog-ical development, mankind set its goal to solve the problem ofshaping thematerials. Latter, the necessity was both the productionand shaping ofmaterials. Even so, formany centuries the productionand manufacture of diverse components was extremely laborious

abundant fatalities produced by; cars, trains, boats, vessels and air-plane failures, construction and structure breakdowns, componentspoor design, etc. In fact, themain cause of all these accidents has notbeen entirely due to a poor design, but to a lack of understanding ofmaterial deciencies in a form of pre-existing aws that tend tonucleate cracks and propagate fractures.

This condition has been gradually corrected by a developmentand implementation of a new (at that time) science that is calledFracture Mechanics. In this sense, it has been well documented that0167-8442/$ - see front matter 2011 Elsevier Ltd. Adoi:10.1016/j.tafmec.2011.11.007a b s t r a c t

The understanding of how materials fail is still today a fundamental research problem for scientist andengineers. The main concern is the assessment of the necessary conditions to propagate a crack that willeventually lead to failure. Nevertheless, this kind of analysis tends to be more complicated, when a priorloading history in the material is taken into consideration and it will be extremely important to recognizeall the factors involved in this process. In this work, a numerical simulation and experimental evaluationof the induction of residual stresses, which change the crack initiation conditions, in a modied compacttensile specimen is presented. Several analyses were carried out; an initial evaluation (numerical andexperimental) was performed in a specimen without a crack and this was used for the estimation of aresidual stress eld produced by an overload; three more cases were simulated and a crack was intro-duced in each specimen (1 mm, 5 mm and 10 mm long, respectively). The overload was then appliedto set up a residual stress eld into the component; furthermore, in each case the Crack ComplianceMethod (CCM) was applied to measure the induced residual stress eld. By performing this numericalsimulation, the accuracy of the CCM can be evaluated and later corroborated by experimental procedure.On the other hand, elasticplastic nite element analysis was utilized for the residual stress estimation.The analyses were based on the mechanical properties of a biocompatible material (AISI 316L). Theobtained results provided signicant data about diverse factors, like; the manner in which a residualstress eld could modify the crack initiation conditions, the convenient set up for the induction of a ben-ecial residual stresses eld, as well as useful information that can be applied for the experimental imple-mentation in this research. Finally, some benecial aspects of residual stresses are discussed.Edicio 5, 2do Piso, Col. Lindavista, CP 07738, Mxico DF, MexicoCrack-compliance method for assessing rloading/unloading history: Numerical an

G. Urriolagoitia-Sosa , B. Romero-ngeles, L.H. HernG. Urriolagoitia-CaldernInstituto Politcnico Nacional, Seccin de Estudios de Posgrado e Investigacin, Escuela S

Theoretical and Appli

journal homepage: wwwll rights reserved.idual stress due toexperimental analysis

dez-Gmez, C. Torres-Torres,

rior de Ingeniera Mecnica y Elctrica, Unidad Profesional Adolfo Lpez Mateos,

SciVerse ScienceDirect

Fracture Mechanics

sevier .com/locate / tafmec

Applway. This is why the simplest way to analyze failure and its conse-quences is to consider the specimen free of previous load history.But on the other hand, the manufacture of components will alwaysleave inside the material an induced stress or strain eld and thiseld will interact with the development of all sorts of defects [4].

The induction of previous load history into the component isbased on the effect leaved by the application of an external agentabove the yielding strength of the material when the load is re-moved. The introduction of previous history can be divided intwo great groups; homogenous loading or non-homogenous load-ing. The consequence of a homogenous loading derives into strainhardening and Bauschinger effect; meanwhile the consequences ofnon homogeneous loading are the introduction of residual stresses.In both cases, the consequences of the application and removal ofthe external agent could contribute into the material either in abenecial and/or a detrimental manner. Strain hardening andBauschinger effect can be found in the material at the same time,if the component has been strengthen by tensile strain hardening,Bauschinger effect (that is a change of the yield strength value ofthe material) will be found in the compressive behavior and viceversa. In relation to residual stresses, they are also detrimentaland benecial, as tensile and compressive stresses are applied to-gether and tend to auto-equilibrate them self [5]. So, in the processof manufacturing pieces and components it is very important toidentify the outcome that a particular fabrication process couldadd to the material.

On the other hand, it is very difcult to measure the grade ofstrain hardening and Bauschinger effect that a material has. It isproposed that the best manner to do it, is to apply a four pointbending test in a produced beam specimen of the material in ques-tion and compare it to a specimen partially or fully annealed madefrom the same material [6]. This procedure will provide either anincrease in the yield stress produced by the strain hardening pro-cedure or the decrease of the mechanical resistance themselvesoriginated by the Bauschinger effect into the component. In con-trast, the quantication of the introduction of a residual stress eldin a component can be performed by a great number of methods ortechniques. These techniques are divided into three groups;destructive, semi-destructive and non destructive methods. Themain difference between each group is related to the structuraldamage caused to obtain the residual stress eld in the component,which in the non-destructive methods the specimens residualstress eld can be acquired and brought back to service, in thesemi-destructive techniques the component can be evaluateddepending on the technique used to control the damage could becontrolled, whilst the destructive procedure completely harmsthe component making it un-useful for service.

In the group of destructive techniques, there is one procedurethat has called the attention of several researchers; this is theCrack Compliance Method (CCM). The name came from the simi-larity of this technique to the compliance method for measuringcrack length in a fatigue or fractured specimen [7]; a known loadis applied to a cracked specimen, and the resulting strain is usedto determine the crack length (in the CCM the crack length isknown and the measured strain is used to calculate the residualstress eld acting into the component). In the engineering environ-ment the CCM is also known as; Fracture Mechanics Approach,Successive Cracking Method, Slotting Method, Rectilinear GrooveMethod, etc. The CCM adds unique new capabilities to the currentdetermination of residual stress measurement procedures. Com-pared to other destructive methods, this technique offers increasedspatial resolution of residual stresses and greater than before sen-sitivity to low stresses. Additionally, the sub-millimeter spatial res-

G. Urriolagoitia-Sosa et al. / Theoretical andolution provided by the CCM cannot currently be matched by themost common non-destructive techniques (X-ray or neutrondiffraction). Other CCM advantages include a simple analyticaltechnique to determine the stress intensity factor caused by acrack in a residual stress eld and the ability to measure crack clo-sure stresses. Furthermore, the CCM can be applied fairly easilywith commonly available equipment (strain gauges and electricdischarge or conventional machining) and it is extremely cheap,when it is compared to other methods [7].

In this paper, it is presented the numerical simulation andexperimental evaluation of the introduction of a residual stresseld with the objective to modify the strength of the material.Which could improve the mechanical resistance of the componentby setting a tensile overload, which at the beginning of its actioncan propitiate the nucleation or propagation of a crack, but whenthe application of the external agent is ended it would leave a ben-ecial residual stress eld. Also, in this research paper it is pre-sented a numerical evaluation of the CCM and the determinationof the possible residual stress acting on the component. Addition-ally, it will be corroborated the exactitude of the application of theCCM by an experimental procedure.

2. Theoretical basis of the crack compliance method [5]

The analytical solution using the CCM can be carried out onlywhen the relaxed strain readings have been obtained from cuttinga component with inherent residual stresses. In general, the anal-ysis for the determination of the residual stress eld from thestrain data collected is performed in two stages; the forward solu-tion stage, followed by the inverse solution stage. These solutionsare based on linear isotropic material considerations.

In this section a brief summary of the theory relative to the CCMused in this research is presented. Let the unknown residual stressdistribution in the beam be represented by the summation of annth order polynomial series as:

ryx Xni0

AiPix 1

where Ai are the coefcient that have to be obtained and Pi are apower series, x0,x1,x2, . . .xn, etc. Legendre polynomials are also used.However, the CCM includes a step which assumes that the stressdistribution, ry(x) = Pi(x), interacting with the crack is known. Thisknown stress eld is used to obtain the crack compliance functionC by using Castiglianos approach. Therefore, it is required the eval-uation of the change in the strain energy due to the presence of thecrack and the virtual force. One alternative is by means of the StrainEnergy Density. Its main factor, S, is direction sensitive. It estab-lishes the direction of least resistance for crack initiation. The sta-tionary value of Smin can be used as an intrinsic materialparameter, whose value at the point of crack instability is indepen-dent of crack geometry and loading [8]. In the case of an elasticmaterial, the expression of the intensity of the strain energy densityeld is:

S a11K2I 2a12KIKII a22K2II a33K2III 2

This criterion is based on the local density of the energy eld at thecrack tip and it is not required any assumption on the direction inwhich the energy is released. This is suitable for mixed mode load-ing. For the problem at hand, KI = ra1/2; KII = KIII = 0, because thespecimen is under mode I. In this way, S can be combined withthe theorem of Castigliano. The displacement u(a,s) can be deter-mined by taking a derivative with respect to the virtual force, as [9]:

ua; s 12

@u@F

F0

1E0

Z a0

KI@KIFa; s

@FdaF0

3

ied Fracture Mechanics 56 (2011) 188199 189Differentiating now with respect to the distance s, the strain in thex-direction is given by [9]:

3. Material and test specimen

The material used in this work is stainless steel AISI 316L, whichis one of the most utilized steel in the area of biomechanics. Med-ical grade stainless steel AISI 316L is presently used extensively inmedicine for implants. This grade of stainless steel has been usedto avoid and/or minimize the danger of pitting corrosion by mak-ing the high potentials at which pitting occurs highly improbable,although this generally depends on the individual human beinginvolved.

To characterize the stainless steel AISI 316L material, four beamspecimens; 10 mm high by 6.35 mm thick by 250 mm length wereprepared [14]. The beams were stress relieved by annealing toeliminate prior loading history [15]. The stressstrain curve for thismaterial was obtained by four point bending tests (Fig. 1) [16].

The main advantages on this procedure are the simultaneousevaluation of tensile and compress behavior; also it is possible todetermine if the material has prior loading history. The Youngsmodulus and Poisson ratio obtained from this tests and used forthe application of the CCM was 190 MPa and 0.28 respectively.

Badr, has designed a special keyhole specimen to measure themaintainability of residual stresses induced by an overload [17].The specimen is basically a modied compact-tension specimenper ASTM standard E 647-91 with special requirements as

Applied Fracture Mechanics 56 (2011) 188199eaj; s 1E0Z a0

KIa@2KIFa; s@F@s

da

F0

4

The strain e(a,s) (where a = crack length and s is the distance be-tween the location of the strain gauge and the crack plane) due tothe stress elds Pi(x) is known as the compliance function Ci(aj,s)and is given by:

Ciaj; s 1E0Z aj0

KIa @2KIFa; s@F@s

da 5

Due to the linearity of KIF with F, the second term under the integralsign in Eq. (5) is the same as Z(a) in:

Za BF

@KIF@s

s0

6

With B = 1, therefore it can be written:

Ciaj; s 1E0Z aj0

KIaZada 7

KI(a) is the stress intensity factor due to the residual stress eldwhen the crack depth in the beam is equal to a and KIF(a) is thestress intensity factor corresponding to the same depth due to a pairof virtual forces F applied tangentially at a position on the beamwhere strain measurements will be taken during the CCM cuttingof the slot (where Z(a) is a geometry dependant function (Eq. (3)):

Za @2KIFa; s@F@s

8

By following the weight function approach, KI(a) and KIF(a) can beexpressed as [10]:

KIa Z a0

hx; aryxdx 9

Za 4:283Z a0

hx; a1 2xdx 10

where ry(x) = Pi(x) and h(x,a) is known as the weight function [11].So, the ryF(x) is the stress eld due to the virtual force F. Once theCi(a,s) solutions are determined the expected strain due to thestress components in Eq. (1) can be obtained as:

eaj; s Xni0

AiCiaj; s 11

The unknown terms Ai are determined so that the strains givenby Eq. (11) match those strains measured in the experiment duringcutting, this is e(aj,s)actual. In order to minimize the average errorover all data points for an nth order approximation, the methodof least squares is used to obtain the values of Ai. Therefore thenumber of cutting increments m is chosen to be greater than theorder of the polynomial, i.e.m > n. This work used n = 7 with 8 con-stants Ai andm = 9, this being the number of experimental slot cut-ting depths at which strain readings were collected. The leastsquare solution is obtained by minimizing the square of the errorrelative to the unknown constant Ai [12]:

@

@ai

Xmjl

ea; sacyual XnK0

AkCkaj; s" #2

0 i 0; . . . . . .n 12

This gives [H]{A} = {J} where [H] = [C]T[C] and {J} = [C]T{ej}actual[13] gives a linear set of simultaneous solutions from which Ai val-

190 G. Urriolagoitia-Sosa et al. / Theoretical andues are determined and Eq. (1) is then used to determine the resid-ual stress distribution. The numerical procedure was implementedin a FORTRAN program. Fig. 1. Mechanical characterization of stainless steel AISI 316L.

Stress (MPa) Strain

370 0.001947425 0.005104471 0.008171491.7 0.009828

ApplG. Urriolagoitia-Sosa et al. / Theoretical andindicated in ASTM E 399. The main feature of this new experimen-tal sample is a circular hole introduced at the notch tip (Fig. 2). Itcan be clearly observed from Fig. 2, that the addition of the holewill produce a stress concentration into some section of the spec-imen when the tensile load is applied. This stress concentrationwill not only produce a non homogeneous loading (inducing aresidual stress eld when the load is removed), but it will also pro-duce a high stress concentration action that could promote veryeasily crack nucleation and propagation.

Fig. 2. Modied compact tensile specimen SEN.

Fig. 3. Mechanical properties stainless steel AISI 316L [18,19].Table 1Stressstrain data used for the numerical simulation [19].

ied Fracture Mechanics 56 (2011) 188199 191Apparently, the main reason for the modication of the compactSEN specimen is to ensure (up to certain degree) that the manufac-turing process will not produce additional cracks or structural de-fects into the material, while is possible in the standard compact

513.5 0.011726525 0.012756546 0.015000558 0.016500567 0.018000575.8 0.019500584.5 0.021000590.5 0.022500596.85 0.024000

Fig. 4. Modied compact tensile specimen SEN modeled.

Appl192 G. Urriolagoitia-Sosa et al. / Theoretical andtensile specimen. The main characteristic on the modication ofthe specimen is a hole inducted at the notch tip, which when a suf-ciently large tensile overload is applied to the specimen, makesthe material at the concentration zone to behave plastically anddeform while the surrounding material remains elastic. Upon theremoval of the overload, the elastic energy surrounding the mate-rial compresses it at the stress concentration section inducing acompressive residual stress eld. As was shown in the numericalanalysis presented in this research, the induced residual stress eldby an overload can be simulated using the nite element method.For this work only one half of the specimen was modeled and ana-lyzed due to the symmetry of the geometry, this saves computa-tional resources [18].

4. Statement of the problem and numerical simulation

The part of the work presented in this paper is aimed to estab-lish the effect of prior history loading in the development of a crackand how a benecial residual stress eld could increase themechanical resistance of the material. In this paper, four studycases based in numerical simulations are presented. The rst caseof study was a tensile specimen set to endure a load above itsyielding stress considering the introduction of a compressive resid-ual stress eld at the stress concentration zone (Fig. 2, Point A). Thenext three cases of study, consider the presence of a crack beforethe application of a tensile overload. The length of the crack wasset to be 1 mm, 5 mm and 10 mm respectively for each case. Themain objective of introducing the crack was to establish the effectthat such a defect would have in the development of the residual

Fig. 5. Numerical evaluatied Fracture Mechanics 56 (2011) 188199stress eld after removing the applied load. The mechanical prop-erties employed for the numerical analysis were the ones obtainedby four point bending tests performed on beams manufacturedfrom a stainless steel AISI 316L (Fig. 3a) [16].

The elastic properties of the material were set up as follows;Youngs modulus of 190000 N/mm2 (E) and Poisson ratio of 0.28(t). For the elasto-plastic condition a kinematic hardening rulewas applied and the mechanical properties were introduced in atabular manner (Fig. 3b and Table 1). The yield stress was set upat 370 MPa and the yield strain was considered at 1947 le. Themaximum stress and the corresponding strain was set to be596.85 MPa and 24,000 le [19].

A general model consisting of a non-linear kinematichardening and isotropic hardening components was used for theanalysis [5]:

da C 1r0

r adple cadple 13

r0 rj0 Q1 1 eeplb

14

where e is the equivalent plastic strain, a is the back-stress, C is theinitial kinematic hardening modulus, c determines the rate at whichkinematic modulus decreases with plastic deformation, r0 is thecurrent yield stress, r|o is the initial yield stress, Q1 is the maxi-mum change in the size of the yield surface and b denes the rateat which the size of the yield surface changes as plastic strainingdevelops. Eq. (13) describes the translation of the yield surface inthe stress space due to the back-stress a, while Eq. (14) describes

ion of residual stress.

ApplG. Urriolagoitia-Sosa et al. / Theoretical andthe change of the equivalent stress dening the size of the yield sur-face r0, as a function of plastic deformation.

To save computational resources all the numerical simulationsperformed in this work were carried out in a symmetrical mannerand all the specimens were modeled in 2D (Fig. 4). Quadratic orderelements (Plane 183) with 8 nodes were used and plane stress anal-ysis was performed. The numerical model was developed by nodes,lines and areas. The specimenwas loaded (in all four cases) in tensileformand the forcewasuniformlydistributedamong17nodeson theloading keyholewith amagnitudeof 100 Neach (Fig. 4a). Thebaseofthe specimen (from Point A to the strain relaxation measurementpoint (Fig. 2)) was specially prepared with a rectangular zone(elements dimensions 0.5 mmwidth by 1 mm large), this zone willbe employed later in the analysis to simulate the introduction of aslot and to evaluate the performance of the CCM, and strain relaxa-tion data will be obtained at the rear node of the base line (Fig. 3).Boundary conditionswere applied at the bottom line in a symmetri-cal manner [18]. When the cases of study required considering theintroduction of a crack, the lack of application of boundary condi-tions at the near end of the bottom line is used to simulate the

the maximum value of the compressive stress is in this region

that will be represented as the cut or slot. The creation of the sim-

Fig. 6. Specimen preparation for the evaluation of the CCM by the FEM.ulated slot produces the rearrangement of the residual stress eldinside the component over the material that has not been affectedby the cut. The reshufe of the residual stress eld will produce astrain or deformation relaxation, which can be measured by astrain gauge ahead in the slot growth. The strain data used forthe CCM in the numerical analysis is obtained at a node locatedahead of the direction of the cutting procedure (Fig. 2) [25]. Thestrain data used in the CCM is only the elastic strain value, notthe complete (elastic and plastic) strain measured. The elasticstrain measured is then introduced into the CCM developed pro-gram and the original stress eld acting inside the component isobtained.and will decrease with the length until becomes a tensile stress.This is an obvious observation, because residual stresses areauto-equilibrant, which means that compressive and tensile stres-ses have to interact together to exist.

6. CCM numerical evaluation

After the numerical simulation for the induction of the residualstress eld was nished, a numerical reproduction of the behaviorof the CCM was carried out. The main objective to perform thisevaluation was to assess the exactitude of the CCM against numer-ical data obtained in the analysis. From this evaluation it is possi-ble, later on, to establish a better experimental practice so as tond the percentage error on the application of the CCM.

The theory and the experimental tests to apply the CCM havebeen widely explained by several authors [2023]. Additionally,there has been development in a new way to apply the CCM inan experimental form and also in a numerical manner [24]. Forthe development of the numerical procedure a simulation of thecut is created by the deletion of nite elements in the area or zoneintroduction of a crack (Fig. 4a) and this lack on the application ofsymmetry will depend on the length of the crack to be simulated.

The numerical analysis for each case of study was performed bya two step procedure. The rst step was done by applying the loadup an established value of 100 N per active node and sends it to besolved. In the second step of the analysis, the load was removedand once again the simulation was sent to be solved. It is importantto mention that the numerical analysis has to be performed unin-terrupted, otherwise the result data from the loading stage to theunloading stage will not be added (Fig. 4b). The same considerationhas to be taken later, when it is performed the elimination ofelements to produce the evaluation of the CCM.

5. Results for the cases of residual stress eld induction

The rst case of study deals with the introduction of a residualstress eld into a specimen free of a crack (Fig. 5a). For the nextthree cases of study, a numerical analysis was performed on the ef-fect of the introduction of a crack with different lengths (1 mm,5 mm and 10 mm, respectively) for each case of study. The residualstress elds obtained by numerical simulation can be observed inFig. 5.

From Fig. 5ad, it can be seen that after a loading process hasnished and the external agent has been removed, a residual stresseld has been inducted into the component. In all four cases it canbe seen a compressive stress value at the left part of the graphics,which corresponds to the stress concentration point at the cracktip depending on the case of study. It can also be observed, that

ied Fracture Mechanics 56 (2011) 188199 193For all cases presented in this paper, the numerical evaluation ofthe CCM was performed by simulating the introduction of a slot,which would cause a modication, by auto-equilibrium, of the

Appl194 G. Urriolagoitia-Sosa et al. / Theoretical andresidual stress acting on the base line, producing a relaxation in thematerial. The relaxation will produce strain data (elastic and plas-tic), which can be used by the CCM to determine the acting residualstress eld. In Fig. 6 are presents in detail the general specimenmodel with the prepared area for the simulation of the introduc-tion of the slot and the zone where the inducted crack was posed(for the second, third and fourth cases of study).

In Fig. 6a it is presented the general design of the specimen bylines. Therefore, in Fig. 6b it can be observed the rectangular zone,where, by deleting elements of 0.5 mm high and 1 mm long the

Fig. 7. Residual stress comparis

Fig. 8. Cracked modied compied Fracture Mechanics 56 (2011) 188199creation of a cut or slot can be simulated. On the other hand, inFig. 6c it is shown the meshed specimen the lack of boundary con-ditions at the bottom of the component to simulate the effect of thecrack and the relaxation consequence caused by the introduction ofthe slot by deleting the elements. Meanwhile in Fig. 6d, presents azoom of the same condition for the last gure. It is important tostate, that the numerical simulation has to be carried out by asequence of several different steps, from the consideration of theapplication of the load and unload process (acting stresses andintroduction of residual stresses), the total or partial application

on between FEM and CCM.

act tensile specimens SEN.

r co

ApplFig. 9. Slot width evaluation fo

G. Urriolagoitia-Sosa et al. / Theoretical andof the boundary conditions (specimen without or with crack) andthe deletion of the elements one by one to obtain the strain datato evaluate the CCM.

After the simulation of the slot was performed, the elastic straindata that will be used for the evaluation of the CCM was collectedand Fig. 6 shows the obtained results. This information, which isused by the CCM program, has to be fed into the commercial pro-gram as a polynomial equation of the 7th order. In fact, this equa-tion depicts the curve that best ts the results obtained by thenumerical simulation; this paper presents only the graphic results.The residual stress results obtained by the CCM for each one ofthese cases are presented in Fig. 7. These gures show as well com-parisons against results obtained from numerical simulation byFEM. From Fig. 7, it is possible to see that the rst value foundby the use of the CCM is not at the surface of the specimen, butat 0.1 mm from the total width of the specimen, this is becausethe program is set to produce only nine points along the evaluatedwidth of the beam and the reason to chose only nine data pointswas based in the precision of the programming. If a procedure isset to divide the results in more points, the precision will be lessaccurate because it will tend to produce more decimal errors andby dividing the results in fewer points, the obtained data wouldbe insufcient to produce an accurate result.

Fig. 10. Representation of the way tmpact tensile specimens SEN.

ied Fracture Mechanics 56 (2011) 188199 1957. Experimental introduction of the residual stress eld

For the development of the experimental analysis a set of six-teen specimens were manufactured and there were divided intofour groups of 4 specimens each. The entire set of specimens wasstress released by a heat treatment process at 600 C for half anhour and slowly cooled down inside a furnace [15]. It is importantto highlight, that a procedure to avoid possible oxidation duringthe heat treatment process was performed, which includes theintroduction of the specimens into a semi-vacuum metallic bagmade from thin sheets of medium Carbon steel. Additionally, themain reason for applying a heat treatment process was to elimi-nate any kind of prior loading history acting into the material,which can cause the indetermination of the actual original residualstress eld induced into the material by the application of the overloading process.

Four cases of study were proposed for the development of theexperimental analysis. For the rst case of study, it was includedthe introduction of a residual stress eld into a specimen free ofa crack. For the next three cases, an analysis was performed toevaluate the effect of an existing crack (with different lengths1 mm, 5 mm and 10 mm, respectively) into the specimen and thestate of a residual stress eld, which would modify the conditions

he specimen is support and cut.

Appl196 G. Urriolagoitia-Sosa et al. / Theoretical andfor crack propagation. In this sense, the generation of the crack intothe specimen was performed by a wire electro discharge machine(EDM) in a manner of a slot (Fig. 8).

The decision to use the electric discharge machine was to pro-duce a slot that will function as a crack, but according the introduc-tion of an additional stress eld into the material by cold workprocedures. Additionally, great care was taken to induce a crackthat was no thicker than 1 mm for all cases. In Fig. 10 are presentedall the specimens showing the introduction of the crack and themeasured data obtained for each one of the pieces. In these guresit was relevant to ensure that the slot was as thin as possible, for allcases the thickness of the slot was between 0.315 mm to 0.36 mm,some examples are shown in Fig. 9 [18].

All the specimens were prepared with a strain gauge at the rearsurface (with respect to the stress concentration hole (Fig. 2)). Thisgauge will be used mainly to measure the strain relaxation causedby the introduction of the slot and is applied by CCM for the eval-uation of the residual stress eld induced into the material. Also,the strain gauge can be used to determine the magnitude of theoverload applied to introduce the residual stress eld, keepingidentical magnitudes of the overload for all study cases. Addition-ally, for the specimens free of an initial crack, a strain gauge wasapplied at the holes border (Fig. 2) to measure the strain effectdue to the application of the tensile load and to correlate theexperimental analysis with the numerical evaluation.

In a previous numerical simulation research a 100 N load wasdistributed on 17 nodes at the loading keyhole (numerical part ofthis study). The resulting numerical strain obtained at the tip ofthe concentration hole was of 5675 le and was taken as the base

Fig. 11. Residual stress comparison between experimental evaluations againstnumerical analysis.ied Fracture Mechanics 56 (2011) 188199for the determination of the experimental load, which was foundto be 975 N. The load was applied in a tensile way by a servo-hydraulic device, with a capacity of 100 kN and was the same forall the experimental study cases. This overload magnitude is largeenough to produce a localized plasticity effect near the stress con-centration zone while elasticity will remain around the rest of thespecimen, so by unloading the system a residual stress eld isinduced.

8. Specimen preparation for the application of the crackcompliance method

After the induction of the residual stress eld was performed,the results obtained by the numerical analysis were used to assessthe manner in which the specimen is stress released by the intro-duction of the cut. With this information, it was possible to deter-mine the best way to hold the specimen to perform the cutting ofthe slot and obtain the strain relaxation for the application of theCCM. So, it was decided to clamp the specimen at one and supportit at the other side as shown in Fig. 10.

The cut was done by a wire EDM (CHARMILLES, model ROBOFIL)in sequential steps 1 mm depth for all cases, and the strain relaxa-tion data was obtained using a Wheatstone bridge conguration.Also, the strain gauges were protected with M-Coat-A Air DryingPolyurethane coating as rst stage and nishing with an M-Coat-B Nitride Rubber coating. This procedure is to ensure the encapsu-lation of the strain gauge when it is submerged into the dielectricliquid used by the EDM machine.

Fig. 12. Residual stress comparison between experimental evaluations againstnumerical analysis.

ApplG. Urriolagoitia-Sosa et al. / Theoretical and9. Experimental cases of study and results

The theory, application and performance of the CCM have beenextensively explained by several authors [4,7,9]. The rst experi-mental case of study in this work is the introduction of a residualstress eld into a specimen free of crack, Fig. 11a. Also this rstexperimental case was used to set up and evaluate the accuracyof the CCM. For this rst case, three specimens were prepared,overloaded and induced with a residual stress eld. In Fig. 11bare presented the strain relaxation behavior against the length ofthe cut, which is used to determine, by the CCM, the original stateof the residual stress eld.

For the next three experimental study cases, an analysis wasperformed on the effect of the introduction of a residual stress eldin a specimen with a crack of different lengths (1 mm, 5 mm and10 mm, respectively). For all three cases, three specimens wereprepared respectively. The results obtained by the application ofthe CCM and also a comparison against the numerical evaluationfor the case of a specimen with a crack length of 1 mm can beobserved in Fig. 12a. The obtained strain relaxation used for theCCM to determine the introduced original residual stress eldcan also be observed in Fig. 12b.

In Figs. 13 and 14 it can be observed the residual stress resultsobtained from CCM evaluation for specimens with a 5 mm and10 mm crack lengths. It is important to mention that three speci-mens were prepared with strain gauges for each particular caseof study. In Figs. 13 and 14 it is also shown the strain relaxationcaused by the induction of the slot into the material, which isproduced by the re-accommodation of the residual stress eld into

Fig. 13. Residual stress comparison between experimental evaluations againstnumerical analysis.ied Fracture Mechanics 56 (2011) 188199 197the uncut material. The results obtained by the application of theCCM and also a comparison against the numerical evaluation foreach specic case of study with a 5 mm and 10 mm crack lengthare shown in these two gures.

10. Conclusions

This research was performed to validate the use of FEM in theintroduction of residual stresses and to validate its application tothe CCM. The numerical data obtained facilitates the experimentalprocedure for the induction of residual stress elds and the appli-cation to the CCM. The modication of the compact tensile speci-men SEN, will provide a more controllable set up for theintroduction of residual stresses, this is because, if the specimenis kept with a notch tip, it cannot ensure the nucleation of thecrack. The main idea of this research was to evaluate the effect ofa crack with the introduction of residual stress elds, which hasto be done by reasonably controlling that no defects are present.From Fig. 5 it can be concluded, that after the effect of the load isremoved from the specimens with a crack, a benecial residualstress eld has been induced at the zone or surface of the stressconcentration. So, to be able to propagate the crack will requiresufcient energy to overcome the compressive residual stress eldintroduced into the component by the loading process. Apparently,the combination of loading the crack and removing the load effect,allows the material to raise its mechanical resistance. Neverthe-less, much care has to be taken into consideration, as a crack thereis in the specimen and this could accumulate plastic energy due tosubsequent loading cycles prior to propagation.

Fig. 14. Residual stress comparison between experimental evaluations againstnumerical analysis.

Additionally, the exactitude on the application of the CCM has the introduction of a residual stress eld) can extend the working

198 G. Urriolagoitia-Sosa et al. / Theoretical and Applied Fracture Mechanics 56 (2011) 188199been evaluated by the use of FEM. Signicant data has been ob-tained for the experimental procedure of the CCM and a propermanner to establish its application. On the other hand it can be ob-served in Fig. 7, that similar residual stress elds between FEM andCCM have been achieved. Nonetheless, it can be observed in Fig. 7,that both ends of the curve for the calculated residual stress eldare not as accurate as those in the middle part of the specimen.It is thought that the mismatch observed at the opposite end ofthe rear location is due to the remote position in which the relax-ation effect is evaluated, as the strain relaxation location cannot befully determined. With respect to the discrepancy in residual stressresults at the end near to the strain relaxation location it has beenconcluded that this originates by the fact that the structural integ-rity of the material has been compromised by the introduction ofthe cut, and there is only a small part of material left.

The validation for the experimental application of the CCM onthe determination of induced residual stress elds has been pre-sented in this paper. This research has proved that the CCM canbe applied to components other than beams, pipes and regularplates. It has also been established that the CCM can be applied tospecimens wider than 10 mm, this investigation was applied tomaterial thickness of 23 mm, 28 mm, 32 mm and 33 mm (whichrepresents specimens with crack lengths of 10 mm, 5 mm, and1 mm, and a specimen without a crack).

The development of a previous numerical analysis, which sim-ulated the problem applied to this experimental procedure, pro-vided important and signicant data. The numerical studyproduced results that were directly applied on the experimentaltesting and have been useful to simplify the CCM procedure. Also,it was very helpful to dene the correct manner to hold and sup-port the piece at the moment of the introduction of the slot. Fur-thermore, the numerical investigation is a powerful tool thatleads to the expected results, which later were corroborated bythe experimental procedure and have proved that the introductionof a residual stress eld could enhance the mechanical resistanceof the material.

It has been corroborated the importance in the use of the mod-ied compact tensile specimen SEN, which has provided a morecontrollable set up for the introduction of residual stresses andthe development of a methodology that effectively arrests crackpropagation. Nevertheless, the main objective of this researchwas to evaluate the effect of a crack with the introduction of theresidual stress eld. From Figs. 1114 it can be concluded that afterthe effect of the load is removed from specimens with a crack, abenecial residual stress eld has been induced. On the other handit can be concluded from Figs. 1114, that there are similar resid-uals stress elds between both the numerical simulation and theexperimental procedure. Nonetheless, it can be observed in Figs.1114 that at the ends of the specimen the calculation of the resid-ual stress elds are not as accurate as in the middle part of thespecimen. The mismatch at the end opposite to the rear locationcould be caused by the effect of the remote position of the effectand the relaxation cannot be totally read at the strain relaxationlocation. With respect to the discrepancy in residual stress resultsat the end near to the point of collection of relaxation data, a pos-sible explanation could be based on the fact that the structuralintegrity has been compromised by the introduction of the cutand only 1 mm of material has been left. It has been corroboratedthat the combination of loading the crack and removing the loadeffect has allowed the material to gain mechanical resistance.Additionally, it could be said, that a mechanical procedure (likelife of a component after a crack has been successively loaded andunloaded.

Acknowledgements

The authors gratefully acknowledge the nancial support fromthe Mexican government by the Consejo Nacional de Ciencia y Tec-nologa and the Instituto Politcnico Nacional.

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G. Urriolagoitia-Sosa et al. / Theoretical and Applied Fracture Mechanics 56 (2011) 188199 199

Crack-compliance method for assessing residual stress due to loading/unloading history: Numerical and experimental analysis1 Introduction2 Theoretical basis of the crack compliance method [5]3 Material and test specimen4 Statement of the problem and numerical simulation5 Results for the cases of residual stress field induction6 CCM numerical evaluation7 Experimental introduction of the residual stress field8 Specimen preparation for the application of the crack compliance method9 Experimental cases of study and results10 ConclusionsAcknowledgementsReferences