ISSN 1068-798X, Russian Engineering Research, 2009, Vol. 29, No. 8, pp. 761–768. © Allerton Press, Inc., 2009.Original Russian Text © A.A. Ostsemin, 2009, published in Vestnik Mashinostroeniya, 2009, No. 8, pp. 21–28.

761

It is important to determine the safety of structureswith crack-like defects.

In contrast to cracks, we know that stress concentra-tors (scratches, tears, cuts), welding defects (gaps, slaginclusions, pores), and corrosion defects (pitting), how-ever serious, have a finite radius of curvature

ρ

. Theshape of the defect has great influence on the nucleationand development of fatigue-corrosion cracks in weldseams of angled, T-shaped, and butt type and in thebasic metal in structures, pipelines, and casings [1].

The use of linear failure mechanics in predictingbrittle failure entails determining the stress-intensitycoefficients for structures and plates with crack-likedefects. Metal structures often contain arbitrarily ori-ented defects whose tip radius

ρ

differs significantlyfrom the radius of the fatigue crack.

Currently, we lack effective means of estimating thestrength of parts with defects. On account of therounded tips of such defects, standard crack theory can-not be applied [1–7]. On the other hand, the large stressand strain gradients in the region of the concentratorsprevent the application of classical stress-concentrationtheory [3, 6].

CALCULATING THE STRESS-TENSOR COMPONENTS, THE PRIMARY STRESS,

AND THE STRESS INTENSITY FOR CRACK-LIKE DEFECTS

We assume that the stress distribution at the tip of acut (radius

ρ

) is analogous to that at the tip of a crack ina body loaded by tensile stress far from the crack butshifted horizontally from its tip by

ρ

/2. For an infiniteplate with an elliptical or hyperbolic cut (with smallrounding radius

ρ

), simple approximate formulas (of

satisfactory accuracy) for the stress distribution in nor-mal rupture were presented in [1]

(1)

where

K

I

is the stress-intensity coefficient;

r

,

θ

are thepolar coordinates of the given point;

x

,

y

are rectangularcoordinates.

When

θ

= 0 and

r

=

ρ

/2, the stress at the tip of the

concentrator with elastic material is

σ

y

= 2

K

I

/

according to Eq. (1). The strain is

ε

= 2

K

I

/

E

( ) [7];this corresponds to classical strength theory fordeformable bodies with structural stress concentrators.The effectiveness of Eqs. (1), taking account of theradius

ρ

, is confirmed by photoelastic data [8]. Takingaccount of Eqs. (1), the primary stresses

σ

1

and

σ

2

inthe vivacity of a cut (radius

ρ

) take the form

(2)

When

ρ

= 0, we obtain the primary stresses

σ

1, 2

forcracks from Eq. (2) [2–7].

σx

K I

2πr------------- θ

2--- 1

θ2--- 3

2---θsinsin–cos=

–K I

2πr------------- ρ

2r----- 3

2---θ;cos

σy

K I

2πr------------- θ

2--- 1

θ2--- 3

2---θsinsin+cos=

+K I

2πr------------- ρ

2r----- 3

2---θ;cos

τxy

K I

2πr------------- θ

2--- θ

2--- 3

2---θ

K I

2πr------------- ρ

2r----- 3

2---θ,sin–cossincos=

⎭⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎫

πρ,

πρ

σ1 2,K I

2πr------------- θ

2---cos

θ2--- θ

2---2cos2sin

ρ2r-----⎝ ⎠

⎛ ⎞2

+±⎝ ⎠⎛ ⎞ .=

Stress State and Stress-Intensity Coefficients in Structures with Crack-Like Defects by Holographic Interferometry

A. A. Ostsemin

South Ural Scientific Production Center, Chelyabinsk

Abstract

—The stress–strain state and stress-intensity coefficients in a plate at the tip of a crack-like defect areinvestigated on the basis of photoelasticity and holographic interferometry. The components of the stress tensor,the primary stress, the stress intensity, and the elastic strain at the tip of a cut are determined. Various methodsof determining the stress-intensity coefficient from maps of isolines of the total primary stress, maximum tan-gential stress, and absolute path differences at the tip of an elliptical cut are analyzed. New analytical formulasare obtained for determining the stress-intensity coefficient on the basis of linear failure mechanics.

DOI:

10.3103/S1068798X09080036

762

RUSSIAN ENGINEERING RESEARCH

Vol. 29

No. 8

2009

OSTSEMIN

Taking account of Eq. (2), the difference of the pri-mary stresses takes the form

(3)

When

ρ

= 0, we obtain a formula for the cracks fromEq. (3) [2–4, 9].

Holographic interferometry permits measurementof the sum of primary stresses, which does not dependon the cut radius

ρ

[10]

(4)

This reduces the error in determining the result fromthe map of isolines of the total primary stress as a resultof the finite rounding radius at the tip of the cut. In addi-tion, holographic interferometry offers high sensitivityin determining the stress-intensity coefficient using thetotal-stress isolines and the isolines of the absolute pathdifferences. Another benefit is the separate determina-tion of the primary stresses for a model of highly elasticmaterial with a finite radius of the cut’s tip if we requireincreased accuracy and reliability of the results and theanalysis of the sample’s stress state. The sensitivity ofholographic interferometry significantly exceeds thatof the photoelastic method, which is characterized by aset of total-stress isolines of higher order than the set ofpath-difference isolines for the same stress-intensitycoefficients at equal distances from the cut tip in mod-els of equal thickness [10].

The photoelastic method permits measurement ofthe maximum tangential stress

τ

max

, which may bederived from Eqs. (1) [8, 10]

From the Tresk condition

τ

max

=

σ

T

/2, the radius ofthe plastic zone is

When

ρ

= 0, we obtain a formula for cracks

σ1 σ2–2K I

2πr------------- θ

2--- θ

2---2cos2sin

ρ2r-----⎝ ⎠

⎛ ⎞2

+ .=

σ1 σ2+K I

2πr-------------2

θ2---.cos=

τmax

σx

2-----⎝ ⎠

⎛ ⎞ σy

2-----⎝ ⎠

⎛ ⎞–2

τxy2+=

= K I

2 2πr---------------- θ2sin

ρr---⎝ ⎠

⎛ ⎞2

+ .

rK I

2

2πσT2

----------------- θ2sinρr---⎝ ⎠

⎛ ⎞2

+⎝ ⎠⎛ ⎞ .=

τmax

K I

2 2πr---------------- θ.sin=

For a plane stress state (σz = 0), taking account ofEqs. (1), we may write the stress intensity in the vicin-ity of the cut in the form

(5)

From Eq. (5), the coordinate r of the isolines of thestress intensity σi is related to the angle θ and to σi

(6)

Substituting ρ = 0 into Eq. (6), we obtain a formula forthe plastic-zone radius at the crack tip [2–4]

(7)

We may calculate the plastic-zone radius r (ρ = 0)from Eqs. (7) and (6). From Eq. (7), r(ρ) will be largerfor cracks.

The elastic strain in the region of the cut is deter-mined from the stress on the basis of Hooke’s law for aplane stress state (σz = 0), taking account of Eqs. (1)

(8)

where µ is Poisson’s ratio.

When ρ = 0, Eqs. (8) reduces to the formula forcracks [3].

In investigating the stress in pipes, rings, and disks,it is expedient to use polar coordinates. In solving two-dimensional problems in polar coordinates, we may

σi

K I

2πr------------- θ

2--- 1 3 θ

2---2sin 3

ρ2r-----⎝ ⎠

⎛ ⎞2 1

θ2---2cos

--------------+ + .cos=

r1

2π------

K I

σi

-----⎝ ⎠⎛ ⎞

2 θ2---2cos=

× 1 3 θ2---2sin 3

ρ2r-----⎝ ⎠

⎛ ⎞2 1

θ2---2cos

--------------+ + .

r1

2π------

K I

σi

-----⎝ ⎠⎛ ⎞

2 θ2---2 1 3 θ

2---2sin+⎝ ⎠

⎛ ⎞ .cos=

εx1E---

K I

2πr------------- θ

2--- 1 µ–( )c---cos

⎩⎨⎧

=

–θ2--- 3

2---θ 1 µ+( )sinsin

K I

2πr------------- ρ

2r----- 3

2---θ 1 µ+( )cos–

⎭⎬⎫

;

εy1E---

K I

2πr------------- θ

2--- 1 µ–( )c---cos

⎩⎨⎧

=

+θ2--- 3

2---θ 1 µ+( )sinsin

K I

2πr------------- ρ

2r----- 3

2---θ 1 µ+( )cos+

⎭⎬⎫

,

εz2µE

------K I

2πr------------- θ

2---,cos–=

⎭⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎫

RUSSIAN ENGINEERING RESEARCH Vol. 29 No. 8 2009

STRESS STATE AND STRESS-INTENSITY COEFFICIENTS IN STRUCTURES 763

switch to the stress components σr, σθ, τrθ from Eqs. (1),for normal-fracture defects

(9)

When ρ = 0, expressions for cracks are obtained fromEqs. (9) [4].

For transverse-shear defects, we write formulasobtained on the basis of [1]

(10)

From Eqs. (9) and (10), by algebraic transformation,we obtain expressions for the stress in the immediatevicinity of the tip of an inclined crack-like defect(radius ρ)

(11)

where λ = KII/KI.

Equations (11) is a generalization of the expressionsfor a crack with ρ = 0 in [6].

Determination of the stress-intensity coefficient ismathematically challenging [2–7]. Because the calcula-tion scheme must be idealized, the accuracy is signifi-cantly reduced. Therefore, experimental determinationof the stress-intensity coefficient is preferable for com-plex structures containing stress concentrators, cuts,scratches, welding defects, and pitting. The mainrequirement here is that it must be possible to investi-gate stress fields with large gradients in the vicinity ofthe cuts [2–7]. This requirement is satisfied by the pho-

σr

K I

2πr------------- 1

2--- θ

2--- 3 θcos–( )cos

ρ2r----- θ

2---cos– ;=

σθK I

2πr------------- 1

2--- θ

2--- 1 θcos+( )cos

ρ2r----- θ

2---cos+ ;=

τrθK I

2πr------------- 1

2--- θ θ

2---cossin

ρ2r----- θ

2---sin+ .=

⎭⎪⎪⎪⎪⎬⎪⎪⎪⎪⎫

σθK II

2πr------------- 3

2--- θ

2--- θ

2---sin

ρ2r----- θ

2---sin+cos– ;=

τrθK II

2πr------------- 3

2--- θ

2--- θcoscos

12--- θ

2---cos–

ρ2r----- θ

2---cos– .=

⎭⎪⎪⎬⎪⎪⎫

σθK I

2πr------------- θ

2--- 1 θcos+

2---------------------

3λ2

------ θsin–⎝ ⎠⎛ ⎞cos

⎩⎨⎧

=

+ρ2r----- θ

2---cos λ θ

2---sin+⎝ ⎠

⎛ ⎞⎭⎬⎫

;

τrθK I

2πr------------- 1

2--- θ

2--- θsin λ 3 θcos 1–( )+[ ]cos

⎩⎨⎧

=

+ρ2r----- θ

2---sin λ θ

2---cos–⎝ ⎠

⎛ ⎞⎭⎬⎫

,⎭⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎫

toelastic method [8–11] and by holographic interferom-etry [12–16]. These methods are widely used in deter-mining the stress-intensity coefficient. In solving planeproblems, holographic interferometry based on maps ofthe absolute path differences is significantly preferableto other methods [17–19].

DETERMINING THE STRESS-INTENSITY COEFFICIENTS FOR A STRIP WITH A CENTRAL

DEFECT AND STRESS DISTRIBUTION

The goal here is to develop methods of determiningthe stress-intensity coefficient for structures with crack-like defects (radius ρ) by holographic interferometry.According to the Neumann method, on the basis ofHooke’s law, assuming small deformation, we obtainthe Favre equations, relating the numbers of the inter-ference band in the map of absolute path differencesand the primary stresses σ1, σ2, for a plate with planestress [17–19]

(12)

where a, b are optical constants of the material, deter-mined by calibration experiments [7]; N1, N2 are thenumbers of bands in the maps of absolute path differ-ences with, respectively, vertical (α = 0) and horizontal(α = 90°) polarizations of the reference beam.

The map of absolute path differences is most expe-diently analyzed along the axis of the cut. In that case,with θ = 0, Eq. (2) takes the form

(13)

Taking into account that σ1 = σx, σ2 = σx at a dis-tance r > rmin, for a plate with a crack-like defect, weobtain the following result from Eqs. (12) and (13) forthe first map of absolute path differences

(14)

where N1i is the number of the band at the axis of thecut (θ = 0), at a distance ri from the cut tip. FromEq. (14), we obtain an expression for the stress-inten-sity coefficient in the case of a plate with a central cut

(15)

As we know, in the extension of a finite plate with acrack, we may write [2–5]

(16)

N1 aσ1 bσ2;+=

N2 aσ2 bσ1,+= ⎭⎬⎫

σ1 2,K I

2πri

-------------- 1ρ

2ri

------±⎝ ⎠⎛ ⎞ .=

N1i

K I

2πri

-------------- a bρ

2ri

------ a b–( )+ + ,=

K Ii

N1i 2πri

a bρ

2ri

------ a b–( )+ +------------------------------------------------.=

K I σr πl f 1,=

764

RUSSIAN ENGINEERING RESEARCH Vol. 29 No. 8 2009

OSTSEMIN

where σr is the rated stress far from the cut; l is the half-length of the cut; f1 is a correction function dependingon the sample geometry and the type of load.

As ρ 0, we may use the precise formula fordetermining the stress-intensity coefficient from thenormal-stress concentration. We may use Eq. (15) todetermine the stress-intensity coefficient for bodieswith a crack as ρ 0.

From Eqs. (15) and (16), we obtain

(17)

An analogous expression is obtained for the secondmap of absolute path differences (N2i).

Thus, to determine f1i at the axis of the cut, we deter-mine the coordinates ri of the bands of absolute path dif-ferences (N1i, N2i), and calculate f1i from Eq. (17). For therelative cut length l/B = 0.3 and 0.5 (Fig. 1), f1i deter-mined from map N1 best agrees with the Feddersen the-oretical solution [5]. This may be explained in that theprimary stress σ1 may be more precisely determinedfrom Eq. (15) on moving away from the crack-likedefect, and a in Eq. (12) is larger. For the same reason,f1 determined from map N2 always has a smaller theo-retical value. Thus, to determine f1, it is better to usemap N1. For epoxy, the linearity is disrupted at stressesof 20–25 MPa. When the rated stress σr0 = 5 MPa and thehalf-length of the cut is 6.5 mm, we find that rmin =0.12 mm. We know that the stress states due to a crackand a cut with finite tip radius (ρ = 0.15 mm) will be the

f 1i

N1i 2πri/l

σr a bρ

2ri

------ a b–( )+ +-----------------------------------------------------.=

same when r > ρ/4,. i.e., more than 0.04 mm [3, 4]. Theminimal zone at the tip of the cut in which the stress stateis not determined is around 0.2 mm. The measurementerror is no more than 2.5%.

The distributions of the experimental primarystresses σ1 (curve 1) and σ2 (curve 2) and the theoreticalstress σ1 (curve 3) along the axis of the cut in a platewith a single lateral cut are shown in Fig. 1, when l/B =0.39; ρ = 0.15 mm; σr = 18 MPa. The experimental val-ues of σ1 and σ2 are obtained from maps of the absolutepath differences; the theoretical values are determinedfrom Eq. (13), when θ = 0.

Photoelastic determination of the stress-intensitycoefficient is based on the isolines of maximum tangen-tial stress τmax in plane models of the photoelastic mate-rial. Using Eq. (3) for a plane with a cut (radius ρ) thatfails by normal fracture, we may write the correctionfunction from Eq. (16) in the form

(18)

where

is the optical constant of the model material; ni isthe order of the band of tangential-stress isolines at thepoint with coordinates ri, θi; σr is the rated stress; t is thethickness of the model. From Eq. (18), when ρ = 0, wefind that [9]

(19)

From photographs of the tangential-stress isolinesalong the line perpendicular to the cut at its tip, whenθ = π/2, we measure the distance r from the center ofeach band ni to the cut tip. From the results, we plot

curves in the coordinates and theslope of the linear section corresponds to f1.

To determine f1 by this method, plates with a centralcut (radius ρ = 0.15 mm) undergo tensile testing. Themeasurements are made on a universal holographic unitconsisting of a light source, an LG-38 laser, optical ele-ments for directing and shaping the light beams, and adevice for fixing and loading the plate.

The experiment was described in [15]; the maps ofisolines of the absolute path differences may be foundin [14, 16].

The maps of the absolute path differences arerecorded by two exposures using focused-image holo-grams. To isolate the map of tangential-stress isolines,the test beam is depolarized using a diffuser, while thereference beam is plane-polarized. A plate made of

f 1σ1 σ2–( ) 2ri/l

σr2M-------------------------------------

σ01.0( )ni 2ri/ltσr2M

-------------------------------,= =

Mθ2--- θ

2---2cos2sin

ρ2ri

------⎝ ⎠⎛ ⎞ 2

+⎝ ⎠⎛ ⎞

1/2

;=

σ01.0( )

fσ0

1.0( )ni 2ri/ltσr θsin

-------------------------------.=

σ01.0( )ni/ σrt( ) l/2ri;

10

8

6

4

2

01 2 3 4 5

1

2

3

2lA

x

y

r, mm

σ, MPa

σr

σy

σx

2b

σ1

σ2

θ

Fig. 1. Experimental σ1 (1) and σ2 (2) and theoretical σ1 (3)distributions of the primary stress along the cut axis in aplate with a single lateral cut when l/B = 0.39, ρ = 0.15 mm,σr = 18 MPa.

0.6

RUSSIAN ENGINEERING RESEARCH Vol. 29 No. 8 2009

STRESS STATE AND STRESS-INTENSITY COEFFICIENTS IN STRUCTURES 765

rigid optically sensitive material based on ED-20 epoxyis considered.

Difficulties arise when determining f1 from the slope

of the linear section in the coordinates

and by the procedure of [9]. Specifically, theslip of the curve does not remain constant on switchingfrom one set of tangential-stress isolines to another overthe whole range of ri. There, the method of determiningf1 is adapted as follows: values of f1i calculated fromEq. (18) are used to plot the dependence of f1i on ri.

From the map of tangential-stress isolines, f1i isdetermined using Eq. (18). Table 1 presents the experi-mental values determined from the tangential-stressisolines and from Eq. (19) and the theoretical valuesobtained from the Feddersen formula [5]. Analysis ofthe results shows that, especially when l/B = 0.1–0.5Eq. (18) is in good agreement with the Irwin approxi-mate solution

the relative error is no greater than 2%. With decreasein ri, f1 from Eq. (19) is in better agreement with theFeddersen solution [4, 5]. The coefficients f1 calculatedby the Irwin, Isida, and Feddersen methods for finiteplate width were compared in [4]. In the range l/B =0.3–0.5, the Feddersen value is 0.02–0.03 greater thanthe Irvin result. The difference of all the experimentaland theoretical values of f1 is no more than 2%.

DETERMINING THE STRESS INTENSITY COEFFICIENTS AND PRIMARY STRESSES

FOR AN INCLINED CUT IN A PLATE

Currently, methods of determining the stress-inten-sity coefficient when normal rupture stress acts at thecrack tip have been well developed and tested. The bestof these methods take account of not only the singularterm but also high-order terms in the Taylor-seriesexpansion of the Westergaard function [20].

To verify the proposed method of determining KIand KII, we conduct an experiment on a plate with aninclined elliptical cut, under a mixed load (l = 13.4 mm,ρ = 0.1 mm; β = 122.5°; 2B = 96 mm; σr = 2.1 MPa).For the sake of convenience in analyzing the interfer-ence patterns and to increase the accuracy of the formu-las from the map of tangential-stress isolines, we useexpressions for the difference σ1 – σ2 based on the for-mulas in [20]. These expressions take account of thesecond term in the Williams eigenfunction-series for-mula for the stress components in the plane case. In therectangular coordinate system xy, the contribution ofthis term does not depend on the distance from the

σ01.0( )ni/ σrt( )

l/2ri

f 12Bπl------- π

2--- l

B---tan=

crack tip, along the axis of the cut (θ = 0) and perpen-dicular to the cut (θ = π/2), respectively

(20)

(21)

where m = σ(1 – η)cos2β is the biaxial-loading param-eter (β is measured from the OY axis); η is the biaxial-load parameter, equal to the ratio of the horizontal andvertical loads [20].

In the photoelastic method, the difference in pri-mary stresses in the photoelastic method is given by theformula [17]

(22)

Thus, in the map of tangential-stress isolinesobtained on loading a model with an inclined crack-likedefect, we may determine KII and hence KI usingEqs. (20) and (22). To this end, the distance ri from thecenter of each tangential-stress isoline to the tip of thecut is measured on the photographs of the tangential-stress isolines along a cut passing through the tip vertexat an angle θ = 0. Substituting these values intoEq. (20), we determine KII. Then, from Eq. (21), whenθ = π/2, we calculate KI from the photographs of thetangential-stress isolines, taking account of the biaxial-load parameter m = σ(1 – η)cos2β.

Neglecting the influence of the load Px and settingm = 0, we may obtain a simpler expression for the tan-

σ1 σ2– 2K II

2

2πri

---------- m2

4------+ , θ 0;= =

σ1 σ2–

= 2K I

2 K II2+

8πri

--------------------K IK II

22πri

------------------2K II K I–

2 2πri

-------------------------- m4----– m–+ ,

θ π2---,=

σ1 σ2– σ01.0( )ni/t.=

Table 1. Values of f1

l/B

Theoretical value from the Feddersen

formula f1 =

Experimental value

from the Irwin formula

from Eq. (19)

0.3 1.059 1.04 1.08

0.4 1.11 1.08 1.13

0.5 1.18 1.15 1.21

0.6 1.30 1.21 1.33

0.7 1.48 1.34 1.51

0.8 1.8 1.57 1.83

π2--- l

B---sec 2B

πl------- π

2--- l

B---tan

766

RUSSIAN ENGINEERING RESEARCH Vol. 29 No. 8 2009

OSTSEMIN

gential-stress isolines from Eq. (21) [16]

(23)

(24)

For plates with inclined defects [5, 21]

(25)

where fI and fII are correction functions taking accountof the finite plate width.

Taking account of Eq. (25), we determine fII fromthe map of tangential-stress isolines corresponding toEq. (23), by dividing the experimental KII values bysinβcosβ. We determine KI from Eq. (24) when θ = π/2.To calculate fI, we divide KI by sin2β. Analogous resultsfor plates with crack inclination β = 22.5°, 45°, and 75°may be obtained by boundary collocation [21] and bythe photoelastic method [22, 23]. The experiment givescloser values of fI and fII, which are higher than for thecollocation method [21].

We may derive formulas for the primary stresses andmaximum tangential stresses from the expressions forthe stress tensor in the vicinity of the inclined-crack tipin the plate [20]

(26)

σ1 σ2–2K II

2πri

-------------- when θ 0;= =

σ1 σ2–K I

2 K II2+

2πri

------------------------ when θ π2---.= =

K I σ πl β2 f I l/B( );sin=

K II σ πl β β f II l/B( ),cossin=

σ1 2,K I

2πri

-------------- m2----

K II2

2πri

---------- m2

4------+±+=

when θ 0;=

σ1 2,K I K II–

4πri

------------------- m2----+=

±K I

2 K II2+

8πri

--------------------K IK II

22πri

------------------2K II K I–

2 2πri

-------------------------- m4----– m–+

when θ π2---;=

τmax2 K I

2

8πr--------- θ2sin

K II2

8πr--------- 4 3 θ2sin–( )+=

+K IK II

2πr------------- θ θ

2---cossin

K I

2 2πr---------------- θ 3θ

2------sinsin–

+K II

2 2πr---------------- θ 3θ

2------cossin 2 θ

2---sin+⎝ ⎠

⎛ ⎞

–σ 1 η–( ) 2αcos

4------------------------------------- σ 1 η–( ) 2α.cos

⎭⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎫

The maps N1i and N2i of absolute path differencesmay expediently be analyzed along the longitudinalaxis of the cut (θ = 0). In this case, taking account of theformulas in [20] and Eqs. (12) and (26) for KI and KII inthe direction of the cut (θ = 0), we find that

(27)

(28)

Note that Eq. (28) is a generalization of the result in[16] when m = 0 and the result in [15] when KII = 0.

Determination of the stress-intensity coefficientcalls for plotting graphs of N1i and N2i from the maps ofabsolute path differences along the axis of the cut;determining N1i and N2i at points ri by extrapolation;and plotting graphs of KI(ri) and KII(ri) in the rangermin–rmax, with subsequent extrapolation by the least-squares method to the cut vertex ri = 0 by the least-squares method [4, 21]. In the last step, we must takeaccount of the influence of the cut radius ρ, as outlinedin [8, 13]. The stress-intensity coefficient is determinedon a universal holographic unit [13]. There is no biaxialload. From the map of absolute path differences, KI andKII are determined from Eqs. (27) and (28). The resultsfor KI and KII are, respectively, 8.9% and 2–3.3% higherthan the values in [15]. On moving away from the tip of

the cut—i.e., with increase in the differenceincreases. Setting m = 0, we obtain KI = 0.199 MPa m1/2

and KII = 0.137 MPa m1/2. From Eqs. (27) and (28), wefind that KI = 0.216 MPa m1/2 and KII = 0.138 MPa m1/2,if we take account of the biaxial-loading parameterm/4 = –0.221 MPa according to [15]. The experimentalvalues of the stress-intensity coefficients when m = 0agree with the theoretical values to within 4.7% [20]

(29)

As an illustration, we consider a plate with radial cutsbeginning at the edge of a load-free circular hole. Experi-ments are conducted with a plate (thickness t = 3.87 mm,width 2B = 100 mm), containing a hole (diameter D =10 mm, cut radius ρ = 0.15 mm) under rated tensilestresses σr = 4.6, 3.05, and 2.41 MPa, when the relativelength of the radial cuts W = 2l/D = 0.34, 3.08, and 4.92,respectively. The optical constants of the model are a =0.616 band/MPa; b = 0.473 band/MPa. To increase themeasurement accuracy, each map of absolute path differ-ences is recorded three times and the results are averagedover the coordinates of the corresponding bands. From

K Ii

N1i N2i+2 a b+( )---------------------- m

2----– 2πri;=

K IIi

N1i N2i–2 a b–( )---------------------

2 m2

4------–

⎩ ⎭⎨ ⎬⎧ ⎫

2πri.=

ri/ 2l( ),

K Iσ πl

2------------- 1 η+( ) 1 η–( ) 2βcos–[ ];=

K IIσ πl

2------------- 1 η–( ) 2β.sin=

⎭⎪⎪⎬⎪⎪⎫

RUSSIAN ENGINEERING RESEARCH Vol. 29 No. 8 2009

STRESS STATE AND STRESS-INTENSITY COEFFICIENTS IN STRUCTURES 767

N1 and N2 for the maps of absolute path differences(Fig. 2), we determine the correction function f1i of Kcalibration and plot a graph of f1i against ri (Fig. 3). Wesee that f1i may be approximated by a straight line. Thevalue of f1e is closer to the Bow theoretical solution [4].For W = 0.34, f1i is in better agreement with the Bowsolution obtained by conformal mapping [4]. Table 2gives the experimental f2e values for three values of W,together with the Bow theoretical values f1t [4]. Analy-sis of the experimental data indicates good agreementwith the theoretical results; the relative error is no morethan 1.9%.

If the crack is modeled by a cut with a tip radius ρ,the stress distribution when r < rmin differs from thestress for a real crack: σ2 begins to decrease in zone A(shaded zone in Fig. 1) and vanishes at the free contourat the tip of the cut; σ1 is determined by the stress con-centration. Analysis of the error of the photoelasticmethod in a plate with a cut [1] may be found in [8].

When rmin > 3ρ/sinθ, the error is 1%. For ρ = 0.05 mm,the error appears at a distance rmin > 0.2 mm [8]. Fromthe stress curves (Fig. 1), we obtain the experimentalvalue rmin = (3.5–4)ρ. This is consistent with the region

D

–2 –2

2B

l

12

3

45

2 1

Fig. 2. Map of absolute path differences for a plate with radial cuts beginning at the edge of a load-free circular hole.

2.4

2.2

2.00 1 2 3 4

f1i

ri

1

2 α = 90°

α = 0

2.1

Fig. 3. Determining the experimental value of f1i from N1 (1)and N2 (2).

768

RUSSIAN ENGINEERING RESEARCH Vol. 29 No. 8 2009

OSTSEMIN

of investigation rmin > 3ρ and the results in [8]. Withvariation in cut radius from 0.1 to 0.25 mm and in cutlength from 3 to 30 mm, the same expression applies.The value of rmax depends on the agreement of the the-oretical and experimental distributions of the stress σ1.When these distributions are the same, KIi dependsalmost linearly on ri. When ri > rmax, the linearity is dis-rupted, and extrapolation on the basis of points beyondrmax increases the error in determining the stress-inten-sity coefficient [8, 13].

CONCLUSIONS

1. Analysis permits generalization of the methods ofdetermining KI and KII for crack-like defects with aradius of curvature by the photoelastic method and byholographic interferometry, for plates with an inclineddefect. The proposed formulas for the stress-intensitycoefficients are in good agreement with theoretical andexperimental data. Their use is simple and convenient.

2. Holographic interferometry permits the experi-mental determination of the stress and the stress-inten-sity coefficient for plates with radial cracks starting atthe edge of a load-free crack. The correction coefficientagrees with the Bow theoretical solution obtained byconformal mapping with 1.9% error.

3. The method of determining the experimental KIvalues by holographic interferometry, in tests of plateswith inclined and central elliptical cuts, has beenimproved. The error is no more than 4.7%.

REFERENCES1. Creager, M. and Paris, P.C., Elastic Field Equation for

Blunt Cracks with Reference to Stress Corrosion Cracking,Int. J. Fract. Mechan., 1967, vol. 3, no. 3, pp. 247–252.

2. Krasovskii, A.Ya., Khrupkost’ metallov pri nizkikhtemperaturakh (Brittleness of Metals at Low Tempera-tures), Kiev: Naukova Dumka, 1980.

3. Makhutov, N.A., Deformatsionnye kriterii razrusheniyai raschet elementov konstruktsii na prochnost’ (Defor-mational Failure Criteria and Strength Calculations ofStructural Elements), Moscow: Mashinostroenie, 1981.

4. Broek, D., Elementary Engineering Failure Mechanics,Dordrecht: Kluwer, 1978.

5. Cherepanov, G.P., Mekhanika khrupkogo razruysheniya(Mechanics of Brittle Failure), Moscow: Nauka, 1974.

6. Sih, G.C. and Liebowitz, H., Mathematical Theories of Brit-tle Fracture, Vol. 2, Fracture, New York: Academic, 1968.

7. Panasyuk, V.V., Mekhanika kvazikhrupkogo razrush-eniya materialov (Mechanics of Quasi-Brittle Failure),Kiev: Naukova Dumka, 1991.

8. Doyle, J.E. and Kamle, S., Error Analysis of Photoelas-ticity in Fracture Mechanics, Experim. Mechan., 1981,vol. 21, no. 11, pp. 429–435.

9. Bakshi, O. A., Zaitsev, N.L., Googe, S.Yu., et al., Deter-mining the Stress-Intensity Coefficients KI by Photoelas-tic Methods, Zavod. Lab., 1980, no. 3, pp. 280–282.

10. Ovchinnikov, A.V., Safarov, Yu.S., and Garlinskii, R.N.,Determining the Stress-Intensity Coefficients by Holo-graphic Interferometry, Fiz.-Khim. Mekhan. Mater.,1983, no. 2, pp. 59–63.

11. Dolgopolov, V.V. and Shilov, S.E., Determining theStress-Intensity Coefficients in Cracked Structures bythe Photoelastic Method, Probl. Prochn, 1975, no. 2,pp. 108–110.

12. Ostsemin, A.A., Deniskin, S.A., Sitnikov, L.L., et al.,Determining the Stress State of Bodies with Defects byHolographic Photoelasticity, Probl. Prochn., 1982,no. 10, pp. 77–81.

13. Ostsemin, A.A., Deniskin, S.A., and Sitnikov, L.L.,Determining the Stress-Intensity Coefficient by Photoelas-tic Modeling, Probl. Prochn., 1990, no. 1, pp. 33–37.

14. Ostsemin, A.A., Two-Parameter Determination of theStress-Intensity Coefficients for an Inclined Crack byHolographic Interferometry, Zavod. Lab., 1991, no. 12,pp. 43–48.

15. Ostsemin, A.A., Two-Parameter Determination of theStress-Intensity Coefficient KI by Holographic Interfer-ometry, Zavod. Lab., 2008, no. 3, pp. 47–50.

16. Ostsemin, A.A., Deniskin, S.A., Sitnikov, L.L., et al.,Determination of the Stress-Intensity Coefficients for anInclined Crack by Holographic Interferometry, Zavod.Lab., 1987, no. 12, pp. 66–68.

17. Metod fotouprugosti (Photoelastic Method),Strel’chuk, N.A. and Khesin, Kh.L., Eds., Moscow:Stroiizdat, 1975, vol. 2.

18. Aleksandrov, A.Ya. and Akhmetzyanov, M.Kh., UsingLasers for Separate Determination of the Stress inPolarization-Optical Measurements, Prikl. Mekh. Tekh.Fiz., 1967, no. 5, pp. 120–122.

19. Zhavoronok, I.V., Polarizational–Holographic Investiga-tion of the Stress Fields from Maps of the Bands ofAbsolute Path Difference, Eksperimental’nye metodyissledovanii deformatsii i napryazhenii (ExperimentalMethods of Investigating Stress and Strain), Kiev: IESim. E.O. Patona, 1983, pp. 89–99.

20. Eftis, J. and Subramonian, N., The Inclined Crack underBiaxial Load, Eng. Fract. Mech., 1978, vol. 10, no. 1,pp. 43–63.

21. Wilson, W.K., Numerical Method of Determining theStress-Intensity Coefficients for an Internal Crack in aFinite Plate, Teor. Rasch., 1971, no. 4, pp. 217–222.

22. Bakshin, O.A., Zaitsev, N.L., and Googe, S.Yu., Determiningthe Stress-Intensity Coefficients KI and KII by the Photo-elastic Method, Zavod. Lab., 1981, no. 4, pp. 73–76.

23. Razumovskii, I.A. and Koksharov, I.I., Determining theStress-Intensity Coefficients in Mixed Loading on theBasis of Polarization-Optical Data, Mashinovedenie,1987, no. 2, pp. 44–50.

Table 2. Values of f1 determined by various methods

W

Bow method [4],

f1 = Proposed

method (f1e)Error, %

0.34 2.074 2.078 0.23.08 1.150 1.140 0.94.92 1.073 1.094 1.9

D2l----- 1+