Transverse shear and pressurization effects in the bending of plates with reinforced cracks

Int J Fract (2006) 142:81–102DOI 10.1007/s10704-006-9027-4

ORIGINAL PAPER

Transverse shear and pressurization effects in the bendingof plates with reinforced cracks

Randal J. Clark · Douglas P. Romilly

Received: 14 June 2006 / Accepted: 23 October 2006 / Published online: 14 December 2006© Springer Science+Business Media B.V. 2006

Abstract The authors develop an eigth-ordermodel for bending of transversally isotropic platesand use integral transforms and a collocationmethod to form a line-spring model for a crackedplate. The eigth-order model allows satisfaction ofthe three standard plate bending boundary condi-tions; the normal moment, twisting moment, andtransverse shear force, and an additional shearstress resultant that allows analysis of transversenormal stresses near the crack tip. The line-springmodel is used to develop geometry correctionfactors for bending of finite-thickness plates,accounting for transverse shear deformation andpressurization of the plate near the crack tip. Theline-spring model is then applied to the problemof a plate with a reinforced crack, and the resultsare used to validate an interpolation solution basedon an energy method. While not explicitly analy-sed, the models are applicable to many problems,including bending of bonded repairs, fracture andfatigue of composite and layered materials, sur-face cracks, crack tip plasticity and crack closureor crack face interaction.

R. J. Clark (B) · D. P. RomillyUniversity of British Columbia, Department ofMechanical Engineering, 2054-6250 Applied ScienceLane, Vancouver, BC V6T 1Z4, Canadae-mail: [email protected]

D. P. Romillye-mail: [email protected]

Keywords Plate bending · Fracture mechanics ·Generalized plane strain · Crack-bridging

1 Introduction

Plates are important structural elements in appli-cations ranging from aircraft and naval structuresto building construction and infrastructure. Whencompared to the analysis of in-plane loading, platebending is complicated by transverse shear defor-mations and transverse loads, leading to higher-order models that are more difficult to solve. Forthis reason, the construction and analysis of platebending models has been an ongoing topic, withresearchers seeking models that are easy to applyand yet capture the essential characteristics of thethree-dimensional behaviour of the plate. Accord-ingly, a range of models exists for engineering anal-ysis of plates. Selection of the correct model foran application depends upon many factors includ-ing the geometry and composition of the plate,whether large stress gradients are present and onthe accuracy required of the analysis.

In the simplest models, transverse shear defor-mations are neglected and the Kirchoff boundarycondition is used to combine the twisting momentand shear force boundary conditions. The resultis a fourth-order governing equation expressed interms of the transverse deflection of the plate. Thisbiharmonic problem is relatively easy to solve, and

82 R. J. Clark, D. P. Romilly

exact solutions exist for many geometries. Moresophisticated models exist that explicitly includeall three standard plate bending boundary condi-tions and account for shear deflections, the mostaccepted being the sixth-order Reissner (1945) andMindlin (1951) models. The Reissner modelfollows from the assumption that the transversestresses in a plate will assume the through-thickness distribution predicted by simple beambending, and is an example of a stress-based solu-tion, whereas the Mindlin model follows from theassumption of a plane strain form for the displace-ment field. More sophisticated models may bedeveloped using higher-order functions to approxi-mate the through-thickness variation of the stressesor strains, and require adoption of extra boundaryconditions to solve the arising differential equa-tions. These models usually employ an infiniteseries expansion of the stresses or strains, and areof interest in the field of fracture mechanics asthey allow an accurate assessment of the stressesnear a crack tip (e.g. Sih 1971). For most engineer-ing problems, stress gradients are small, and theReissner and Mindlin models are considered ade-quate. Other examples of applications requiringmore advanced models are those involving largestress gradients (e.g. higher frequency vibrations),when failure is governed by through-thickness stre-sses, or if there is otherwise the need for veryaccurate solutions.

For plate extension, Kotousov and Wang (2002,2003) developed a dislocation-density solution fora crack in a generalized plane strain plate andfound that, in terms of the stress resultant or aver-age of the through-thickness stresses, the crack tipin plate extension is in a state of plane strain.This is contrary to the common belief that forvery thin plates, the crack tip is effectively in astate of plane stress. This observation has conse-quences for the use of energy methods in fracturemechanics. The pertinent example is the analysis ofreinforced cracks, where the strain energy releaserate may be calculated analytically, but the con-version to stress intensity requires the assumptionof either plane stress or plane strain conditions.Clark and Romilly (2006a) have used the gener-alized plane strain plate model to develop a line-spring model and geometry correction factors forextension of cracked plates. The line-spring model

was then used to investigate the effects of rein-forcing springs bridging the crack faces, and it wasfound that the geometry correction factors couldbe used to include the effects of plate thickness andtransverse material properties on the stress inten-sity and crack-face deflection, leading to improvedaccuracy in the analysis of a reinforced crack. Theproblem of transverse stresses building up near thecrack tip also exists for plate bending, but is notoften recognized due to the already difficultproblem of modelling the shear forces and sheardeflections and assessing the effects of crack facecontact. Accordingly, the fracture mechanics ofbending plates is less well understood than forextension.

In this article, the authors develop an eigth-order plate bending model, which is effectively ageneralized plane strain model for bending. Thisis the simplest model possible that includes theeffects of both transverse normal stresses and sheardeflections. The motivation for this work is todevelop a practical line-spring model and corre-sponding engineering analysis methods that includethese two plate thickness effects and allow the cal-culation of the thickness-averaged bending stressintensity factor. The goal is to expedite the designof bonded composite repairs where the repairedstructure consists of plates containing reinforcedcracks subjected to bending. The authors achievethis by using the integral transform method ofJoseph and Erdogan (1987, 1989) whereby the newplate bending model is reduced to a hyper-singularintegral expressed in terms of the bending stressesalong the plane of the crack and the displacementof the crack faces. The line spring model is thenused to develop equations suitable for the assess-ment of plates containing through-cracks and rein-forced cracks subjected to bending. The resultingequations are accurate, easy to apply, and includeboth transverse shear deflection and transversenormal stress effects.

2 Basic plate bending models and fracture

In this section, the authors describe four basic engi-neering plate bending models. These models aresimilar to the classical plate bending modelsavailable in the literature, and form the basis for the

Transverse shear and pressurization effects 83

eigth-order model developed in subsequent sec-tions. The first two employ the Kirchoff or‘effective shear stress’ boundary condition, and willbe referred to as the plane stress and plane strainplate bending models. In their development, it isassumed that either the transverse stresses or thetransverse strains vanish. The Kirchoff boundarycondition allows the twisting moment and shearforce acting along a cut to be combined, i.e. anartificial condition that satisfies equilibrium butdoes not allow for the explicit inclusion of thetransverse shear forces. For crack problems, thesemodels cannot be trusted. Errors in the range of60–70% may exist, depending on the crack lengthand the thickness and elastic properties of the plate.Another problem is that while the governing equa-tions are equivalent to those arising in the planestress and plane strain plate extension models, theKirchoff boundary condition causes inconsisten-cies between the stress and displacement-baseddefinitions of the stress intensity factor, as observedby Joseph and Erdogan (1989). This occurs becausethe crack tip is a region of large stress gradients,and hence it is important to properly account forthe zero shear force boundary condition on theedge of the crack. For plane stress bending, thedeflection of the crack face and the bending stressat the outer fibre are expressed as follows (Clark2006).

u(y) = 2σ

E

(1 + ν

3 + v

)√a2 − y2, |y| < a, (1)

σ(y) = σy −√y2 − a2√

y2 − a2, |y| > a. (2)

For plane strain plate bending of transversally iso-tropic plates, where η = νz

√E/Ez is an effective

Poisson’s ratio, the stresses are given by (2), andthe crack face deflection is given by:

u(y) = 2σ(

1 + ν

3 + ν − 2η2

)(1 − η2

E

)√a2 − y2,

|y| < a. (3)

The two remaining basic models will be referredto as the plane stress shear deformable plate andthe plane strain shear deformable plate. These areformed by including the transverse shear deflectionin the analysis. These sixth order bending modelssatisfy all three engineering stress resultants; the

normal moment, twisting moment and transverseshear force. While the Reissner model assumesthat the transverse stresses take the form givenby simple beam bending, the plane stress sheardeformable plate is somewhat simplified, as thetransverse normal stresses are assumed to vanish.The plane strain shear deformable plate is identi-cal to the Mindlin plate, where it is assumed thatthe transverse strains vanish. The Reissner plate isusually considered to be more accurate than theMindlin plate, as the assumed transverse stressesmore closely model those found in an actual plate.In the presence of large stress gradients, both theReissner and Mindlin models are far more accu-rate than the Kirchoff model, but their use in frac-ture mechanics is limited by assumptions regardingplane stress or strain, similar to the problem thatexists for extension of cracked plates. For the lim-iting case of a very short crack in the plane stressshear deformable plate, the deflection of the crackface at the outer fibre has the same form as maybe found for plane stress plate extension (Clark2006).

u(y) = 2σE

√a2 − y2, |y| < a. (4)

Similarly, for a very short crack in the plane strainshear deformable plate, the crack face deflectiontakes the form for plane strain plate extension.

u(y) = 2σ(1 − η2)

E

√a2 − y2, |y| < a. (5)

For both types of shear deformable plate, thestresses at the base of the crack assume the formgiven by Eq. 2. It will be shown in subsequent sec-tions that these basic models describe limits on thebehaviour of the eigth-order, or generalized planestrain bending model.

3 An eigth-order plate bending model

Here, we develop an eigth-order plate bendingmodel for a transversally isotropic plate, allow-ing the evaluation of both the transverse sheardeflections and transverse normal stresses. For aplate of thickness 2h, the displacement field may beapproximated by Eq. 6a–c below. The series expan-sion of the transverse displacement is truncatedto produce an engineering model that requires


four boundary conditions and allows the evalua-tion of the effects of transverse pressurization atthe crack tip.

ux = zh

ux(x, y),

uy = zh

uy(x, y),

uz = w(x, y)+ z2

h2 uz(x, y). (6a, b, c)

The displacement field and the stresses in the plateare shown in Fig. 1. The depicted stresses are aschematic representation intended to illustrate oddor even behaviour and the zero-stress boundaryconditions at the edges of the plate.

For displacement-based theories, stresses areexpressed as stress resultants or line-forces. Inte-grating through the thickness, we find expressionsfor the moments and shear forces.

Mxx =∫ h

−hσxxz dz, Myy =

∫ h

−hσyyz dz,

Mxy =∫ h

−hτxyz dz, (7a, b, c)

Vx =∫ h

−hτxzdz, Vy =

∫ h

−hτyzdz. (7d)

These are the loads evaluated in the basic platemodels. Pressurization of the plate may be mod-elled by integration of the transverse normal stressas shown below.

Pzz =∫ h

−hσzzz dz. (7e)

A final set of ‘pinching shear’ stress resultants arisefrom the transverse shear stresses (resulting fromcurvatures in the in-plane stress fields) and act topressurize the plate.

Rx =∫ h

−hτxzz2 dz,

Ry =∫ h

−hτyzz2 dz. (7f, g)

The equilibrium equations may similarly be formu-lated in terms of stress resultants.

Mxx, x + Mxy, y = Vx, Myy,y + Mxy, x = Vy,

Vx, x + Vy, y = −q. (8a, b, c)

A final equilibrium condition arises from thesecond moment of the transverse equilibrium

equations, relating the pressurisation of the plate totransverse loads and ‘pinching shear stressresultants’.

2Pzz = h2q + Rx, x + Ry, y. (8d)

The constitutive laws for a transversally isotropicmaterial may also be integrated over the thickness,and the stress resultants may be expressed in termsof the plate displacements.

Mxx = D[

ux,x + ν + η2

1 − η2 uy,y + νz (1 + ν)

1 − η2

2h

uz

],

(9a)

Myy = D[

uy,y + ν + η2

1 − η2 ux,x + νz (1 + ν)

1 − η2

2h

uz

],

(9b)

Pzz = D

[1 − ν2

1 − η2

(ν2

z

η2

)2h

uz

+νz(1 + ν)

1 − η2

(ux, x + uy, y

)], (9c)

Mxy = 2h2G3

[ux, y + uy, x

], (9d)

Vx = 2Gz

[ux + hw, x + h

3uz, x

], (9e)

Vy = 2Gz

[uy + hw, y + h

3uz, y

]. (9f)

The second moment of the transverse shear stressconstitutive equations results in expressions for thepinching shears.

Rx = 2Gzh2

3

[ux + hw, x + 3h

5uz, x

], (9g)

Ry = 2Gzh2

3

[uy + hw, y + 3h

5uz, y

]. (9h)

The plate bending constant, D, takes the form forplane strain.

D = 4h2

3

(1 − η2

1 − ν − 2η2

)G. (10)

Substitution of the constitutive equations (9) intothe equilibrium equations (8) will yield the gov-erning equations for the plate. It proves useful todefine the functions φ and ψ .

φ = ux, x + uy, y, ψ = ux, y − uy, x. (11)

Eliminating q from the transverse equilibriumequations (8c,d) yields the following.

φ + h∇2w + h3∇2uz = 0, (12)


Fig. 1 Displacements andstresses for theeigth-order plate bendingmodel

2h

uz+ η2

(1 − ν) νzφ=(

1 − ν − 2η2

1 − ν2

)η2Gz

ν2z G

h15

∇2uz.

(13)

Adding the derivatives of Eq. 8a and b, equilibriumin the plane of the plate, and using Eq. 8c to cancelthe shear stress resultants leads to the following:

∇2φ + (1 + ν) νz

1 − η2

2h

∇2uz = 0. (14)

Subtracting the derivatives of Eq. 8a and b leadsto the following additional condition:

∇2ψ − k2ψψ = 0, k2

ψ = 3Gz

h2G. (15)

Decoupling Eqs. 12–14, we find a fourth order gov-erning equation for φ along with the second orderequation for ψ given by Eq. 15.

∇4φ − k2φ∇2φ = 0, k2

φ =(

1 + ν

1 − η2

)30ν2

z G

h2η2Gz.

(16)

The governing equation will be expressed in termsof the average deflection of the plate.

w = 12h

∫ h

−h

[w + z2

h2 uz

]dz = w + 1

3uz. (17)

By Eq. 12, we note that φ = −h∇2w, and we findthe following expression for w.

∇6w − k2w∇4w = 0, where k2

w = k2φ . (18)

Equations (18) and (15) form an eigth-order sys-tem, requiring four boundary conditions. The firstthree are the normal and twisting moments andthe shear force. The fourth is the ‘pinching shear’.For ξ acting along a cut and ς along the outwardnormal, these are:

Mςς = Mςς , Mξς = Mξς , Vζ = Vζ ,

Rς = Rς . (19)

The functions φ and ψ describe the normal andshear deformations of the plate, and lead to asimple derivation of the governing equations, butare not convenient for calculating the moments orshear forces. For this purpose, one may define astress function. From equilibrium, we note that afunction χ must exist such that:

Vx = χ, y, Vy = −χ, x. (20)

From the definition of ψ , and constitutive equa-tions (9e,f), we find the following.

∇2χ = 2Gzψ . (21)

A differential equation for χ may now be foundfrom Eq. 16.

∇4χ − k2ψ∇2χ = 0. (22)

Equations (18), (15) and (22) allow the analysisof transverse shear stresses and pressurization inplate bending.

4 Hyper-singular integral equations

In this section, the governing equations arereduced to a hyper-singular integral relating thecrack face deflection to the bending stresses alongthe crack plane. The authors apply the integraltransform method used by Joseph and Erdogan(1987, 1989) to analyse a Reissner plate. TheFourier transform of a function and its inverse aredefined as follows.

f = f (x,α) =∫ ∞

−∞f (x, y)eiαy dy,

f = f (x, y) = 12π

∫ ∞

−∞f (x,α)e−iαy dα. (23)


Applying this transform, the governing equationsmay be written as ordinary differential equations.(

w, xxxxxx − 3α2w, xxxx + 3α4w, xx − α6w)

−k2w

(w, xxxx − 2α2w, xx + α4w

)= 0, (24)(

ψ , xx − α2ψ)

− k2ψψ = 0. (25)

Keeping only the roots that vanish as x → ∞, onefinds the following expressions for w andψ .

w = 12π

∫ ∞

−∞

(A1(α)e

−|α|x + A2(α)xe−|α|x

+ A3(α)e−Rwx)

e−iαy dα, R2w = α2 + k2

w,

(26)

ψ = 12π

∫ ∞

−∞

(B1(α)e

−Rψx)

e−iαydα,

R2ψ = α2 + k2

ψ . (27)

Symmetry requires vanishing shear stresses alongthe line of the crack. Accordingly:

Mxy(0, y) = 0, Vx(0, y) = 0,

Rx(0, y) = 0. (28a,b,c)

From the constitutive equations, the deflectionsalong the line of the crack are limited as follows:

ux, y + uy,x = 0, ux + hw, x = 0,

uz, x = 0. (29a,b,c)

One must express these in terms of w and ψ . Thefollowing expressions arise from (29a):

ux,y = 12ψ , uy,x = −1

2ψ . (30a,b)

The first symmetry condition arises directly from(29a) by substitution from (29b) and (30b).

ψ + 2hw, xy = 0. (31)

To find the second symmetry condition, we requirean expression for Mxx, x(0, y) under the restrictions(29). The following may be obtained from constit-utive equation (9a).

Mxx, x(0, y) = D[φ, x + 1

2

(1 − ν − 2η2

1 − η2

)ψ, y

].

(32)

The second symmetry condition will then arisefrom equilibrium equation (8a).

φ, x + 12

(1 − ν − 2η2

1 − η2

)ψ, y = 0. (33)

A general equation for the transverse displacementarises from Eqs. 13 and 14.

νz2h

uz + η2

1 − νφ + 1

k2w

(1 − ν − 2η2

1 − ν2

)∇2φ = 0.

(34)

After taking the derivative, the final symmetry con-dition may be found from (29c).

∇2φ, x + k2wη

2(

1 + ν

1 − ν − 2η2

)φ, x = 0. (35)

Substituting (24) and (25) into the symmetry con-ditions eliminates all but one integration constant,and the crack face rotation, β = −w, x(0, y), maybe written in terms of A2(α).

β = −12π

∫ ∞

−∞2

1 − νA2e−iαy dα. (36)

The last integration constant arises from theboundary condition Mxx(0, y) = −M(y), but onemust first find an expression for either ux, x(0, y) oruy, y(0, y). Transforming equation (22) and apply-ing the symmetry conditions, the stress functionχis given by:

χ= 12π

∫ ∞

−∞−8ihGz

k2ψ(1 − ν)

(e−|α|x − e−Rψx

)e−iαy dα.

(37)

From constitutive equations (9e,f) we find expres-sions for ux, x(0, y) and uy, y(0, y).

ux, x = χ, xy

2Gz− hw, xx, uy, y = −χ, xy

2Gz− hw, yy.

(38)

From (9a), we now find an expression for theboundary condition M(y) = −M(0, y).

M(y)=−4h2G3

[ux, x+ ν

1 − νφ− 1

k2w(1 − ν)

∇2φ

].

(39)

An integral equation arises by substituting for ux,xand w.

M(y) = −2h3G3π

(1 + ν

1 − ν

)∫ ∞

−∞f (α)A2e−iαy dα.

(40)

Here, f (α) is given as follows.

f (α) =(

3 + ν

1 + ν

)α + 4α2(α − Rψ)

k2ψ(1 + ν)

+ 1k2

w

(2η2

1 − η2

)(α3 − α4

Rw

). (41)


We may now form a hyper-singular equation forthe crack face rotation. Inverting equation (36) wefind an expression for integration constant A2.

− 2A2

1 − ν=∫ ∞

−∞β(t) eiαt dα. (42)

By substitution, an integral arises relating themoment to the crack face rotation.

M(y) = h3E6π

∫Lβ(t)

∫ ∞

−∞f (α)eiα(t−y)dα dt. (43)

Note the following integral identities, where K0and K2 are modified Bessel functions of the sec-ond kind.∫ ∞

0αeiα(t−y) dα = −1

(t − y)2, (44a)

∫ ∞

0α3eiα(t−y) dα = 6

(t − y)4, (44b)

1k4

∫ ∞

0Rα2eiα(t−y) dα = 1

2

(K2(k |t − y|)

−K0(k |t − y|))+ 3

(k |t − y|)2 K2(k |t − y| ), (44c)

1k4

∫ ∞

0

α4

Reiα(t−y) dα =

(1 + 3

k2 (t − y)2

)

× K2 (k |t − y|) . (44d)

Applying these identities, Eq. (43) may be writ-ten as a hyper-singular integral. The regular andweakly singular parts have been separated into thefunctions Lψ andLφ .

M (y) = −h3E3π(1−η2)

∫Lβ(t)

[1

(t−y)2+k2

ψ

21 − η2

1 + ν

× Lψ(kψ |t−y| )+k2

wη2L(kw |t−y| )

]dt.

(45)

The function Lψ describes the transverse shear def-lections, and function Lφ is the same expressionfound by the authors (Clark and Romilly 2006a)for generalized plane strain plate extension, andaccounts for the effects of pressurization near thecrack tip.

Lψ(z) = −48z4 + 4

z2 + 4(K2(z)− K0(z)

)

+24z2 K2(z), (46)

L(z) = 2[

1 + 3z2

]K2 (z)− 12

z4 − 1z2 . (47)

Equation (45) may be written in terms of the bend-ing stress in the outer fibre as follows.

σ (y) = −hE2π(1 − η2)

∫Lβ(t)

[1

(t − y)2+ k2

ψ

21 − η2

1 + ν

× Lψ(kψ |t − y| )+ k2

wη2L(kw |t − y| )

]dt.

(48)

The equivalent equations for the four basic bend-ing models are provided by Clark (2006).

5 Bending of a cracked plate

Here, the shear deformable plate models are solvedby expanding the crack face deflection into a seriesof Chebychev polynomials. With this expansion, aset of collocation points may be chosen to forma linear system, solvable for the expansion coeffi-cients. First one must normalize the bounds andparameters in the integral equations. We definethe normalized quantities:

u(r) = haβ(t), r = t/a, s = y/a,

κψ = kψa, κw = kwa. (49)

Equations (50) through (52) show the normalizedform of the hyper-singular integrals for the planestress shear deformable plate, the plane strain sheardeformable plate, and the eighth order plate,respectively,

σb (s) = −E2π

∫ 1

−1u(r)

(1

(r − s)2

+ κ2ψ

2(1 + ν)Lψ(κψ |r − s| )

)dr, (50)

σb (s) = −E2π(1 − η2)

∫ 1

−1u(r)

(1

(r − s)2

+κ2ψ

2

(1 − η2

1 + ν

)Lψ(κψ |r − s|)

)dr, (51)


σb (s) = −E2π(1 − η2)

∫ 1

−1u(r)

×(

1(r−s)2

+ κ2ψ

21−η2

1+ν Lψ(κψ |r − s|)+κ2

wη2L(κw |r − s|)

)dr.

(52)

The crack face deflection may be expandedinto Chebyshev polynomials of the second kind,Ui(r) = sin((i + 1) cos−1(r))/

√1 − r2, with the

weight function W(r) = √1 − r2.

u(r) = W(r)N∑

i=0

giUi(r). (53)

The hyper-singular part may be evaluated analyti-cally.

1π

∫ 1

−1

√1 − r2Un(r)

(s − r)2dr

=

⎧⎪⎨⎪⎩

−(n + 1)Un(s), |s| < 1,

(n + 1)

(s−

√s2−1

)n+1

√s2−1

, s > 1.(54)

Equations (50)–(52) may now be written in termsof the expansion.

σb(s) = EN∑

i=0

gi

(i + 1

2Ui(s)− κ2

ψ

4π(1 + ν)

×∫ 1

−1Lψ(κψ |r − s| )W(r)Ui(r)dr

), (55)

σb(s) = E1 − η2

N∑i=0

gi

(i + 1

2Ui(s)− κ2

ψ

4π

(1 − η2

1 + ν

)

×∫ 1

−1Lψ(κψ |r − s|)W(r)Ui(r)dr

), (56)

σb(s) = E1 − η2

N∑i=0

gi

⎛⎝ i+1

2 Ui(s)− κ2ψ

4π

(1−η2

1+ν) ∫ 1

−1 Lψ(κψ |r − s|)W(r)Ui(r)dr

− κ2wη

2

2π

∫ 1−1 Lw(κw |r − s|)W(r)Ui(r)dr

⎞⎠ . (57)

Analytical solutions exist for the logarithmic partsof Lψ(z) and Lw(z), but the regular parts requirenumerical integration. The term Vi,j includes thelogarithmic singularity and may be solved ana-lytically, as demonstrated by Joseph and Erdogan(1987) and by Wang and Rose (1999). To

evaluate the regular part, the authors used Gauss–Chebychev quadrature, as suggested by Joseph andErdogan.

Lψi,j =∫ 1

−1

[Lψ(κψ |ri − s|)− ln(|ri − s|)]

× W(s)Uj(s)ds + Vi,j, (58)

Lwi,j =

∫ 1

−1

[Lw(κw |ri − s|)+ 3

4ln(|ri − s|)

]

× W(s)Uj(s)ds − 34

Vi,j, (59)

Vi,j =∫ 1

−1ln |ri − s| W(s)Uj(s)ds

=⎧⎨⎩

−π2

[−r2

i + 12 + ln 2

], j = 0,

−π2

[Tj(ri)

j − Tj+2(ri)

j+2

], j > 0.

(60)

Selecting collocation points si, a linear system arises,solvable for the coefficients, gj.

σb = Ag. (61)

For the three bending models, the coefficientmatrix may be expressed as follows.

Ai,j = E

(i + 1

2Uj(si)− κ2

ψ

4π(1 + ν)Lψi,j

), (62)

Ai,j = E1 − η2

(i + 1

2Uj(si)− κ2

ψ

4π

(1 − η2

1 + ν

)Lψi,j

),

(63)

Ai,j = E1 − η2

(i + 1

2Uj(si)− κ2

ψ

4π

(1 − η2

1 + ν

)Lψi,j

−κ2wη

2

2πLw

i,j

). (64)

Equations 55–57 are valid along the entire linex = 0, and allow the evaluation of the stresses nearthe crack tip. As for plate extension, the only termscontaining a

√r singularity arise from the hyper-

singular part of the equation. These terms may befound by examining integral identity equation (54)


for the region outside of the crack domain. Theresults for plane stress and plane strain bending(Eq. 65 and 66, respectively) follow from the stress-based definition of the stress intensity. The resultreported is for the stress intensity in the outer fibreof the plate.

K = limr→1+

√2πa(r − 1)σxx(0, r) = E

√πa

2

N∑i=0

giUi(1)

= E√πa

2

N∑i=0

(i + 1)gi, (65)

K = limr→1+

√2πa(r − 1)σxx(0, r) = E

√πa

2(η2 − 1)

N∑i=0

giUi(1)

= E√πa

2(η2 − 1)

N∑i=0

(i + 1)gi. (66)

For the eigth-order model, in the evaluation of thepart of the stress field containing the

√r singular-

ity, the stresses and crack face displacements arerelated by the equations that arise in the analy-sis of plane strain plate extension, meaning thatthe crack tip singular stress field exists in a stateof plane strain. This is analogous to the resultof Kotousov and Wang (2003) for extension of

cracked plates. Equations 65 and 66 allow the eval-uation of the stress intensity for the various platebending models. Table 1 shows properties for se-lected transversely isotropic materials that will beused to evaluate them.

Figure 2 shows stress intensity results for theplane stress shear deformable bending model,which are normalized to the nominal resultK0 = σb

√πa and plotted against the crack length.

For a very long crack, the stress intensity reachesa limit that can be calculated according to the fol-lowing argument. Large stress gradients and hencelarge shear forces only exist near the crack tip.Thus, for a long crack, shear deflections have avanishing contribution to the deflection well awayfrom the crack tip, and the plane stress plate modelgives the correct result. The strain energy releaserate for an increment in crack growth is propor-tional to the work done by the applied stress onthe crack face, which may be calculated consider-ing the deflection given by Eq. 1. For the planestress shear-deformable plate, the crack tip is inplane stress, and the limit stress intensity follows.

K∞ = √g∞E =√

1 + ν

3 + νK0. (67)

Table 1 Materialproperties used to test theGPS model

Material E (GPa) ν Ez (GPa) νz

Isotropic 71.0 0.5 – 0.5Isotropic 71.0 0.33 – 0.33Isotropic 71.0 0.1 – 0.1Quasi-isotropic boron/epoxy 106 0.12 25 0.028Transversely loaded unidirectional boron/epoxy 25 0.019 210 0.21

Fig. 2 Stress intensitiesfor shear deformableplane stress bending

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10 100

Crack Length (a/h)

Str

ess

Inte

nsi

ty(K

/K0)

isotropic 0.1

isotropic 0.3

isotropic 0.5

quasi-iso boron/epoxy

transverse boron/epoxy


Fig. 3 Stress intensitydifference plotted againstnormalized crack length

0

0.25

0.5

0.75

1

0.01 0.1 1 10 100

Normalized Crack Length (c)

Str

ess

Inte

nsi

tyD

iffe

ren

ce[K

d]

isotropic 0.1

isotropic 0.3

isotropic 0.5


trans boron/epoxy

model

The stress intensity result may be plotted againsta normalised crack-length, defined as:

cψ = κψ

√3 + ν

1 + ν. (68)

Applying the normalized crack length and the longcrack limit stress intensity, the difference betweenthe plane stress and the plane stress shear deform-able plate solutions for stress intensity (i.e. denotedKd) collapses onto one curve for all of the testedmaterials.

Kd = (K/K0)− 1√1+ν3+ν − 1

. (69)

Using a statistical analysis package, Eq. 70 was fit toKd. Figure 3 shows that this approximate model fitsthe data reasonably well, with an error not exceed-ing 2.5%.

Kd = c1.2ψ

c1.2ψ + 3.5

. (70)

This relationship can be developed into an expres-sion for a geometry correction factor.

Yk = KK0

= 1 −(

c1.2ψ

c1.2ψ + 3.5

)⎛⎝1 −

√1 + ν

3 + ν

⎞⎠ .

(71)

Similar results may be found for the crack facedisplacement at the outer fibre of the plate andthe centre of the crack. Figure 4 shows this dis-placement normalized to the result for plane stressbending, u0 = 2σba/Eh.

The following normalization collapses theseresults onto one curve.

ud = (u/u0)− 11+ν3+ν − 1

. (72)

Once again, one may develop a simple model thatfits the data remarkably well (Fig. 5).

ud = c1.3ψ

c1.3ψ + 5

. (73)

This relationship can be developed into a geometrycorrection factor, Yr.

Yd = uu0

= 1 −(

c1.3ψ

c1.3ψ + 5

)(1 − 1 + ν

3 + ν

). (74)

Very similar results are found for the plane strainshear deformable plate. The limit stress intensityfactor, crack length normalization, and geometrycorrection factors follow.

K0 = σb√πa, u0 = 2

σba(1 − η2)E

, (75a, b)

K∞ =√

g∞E1 − η2 =

√1 + ν

3 + ν − 2η2 K0,

u∞ =(

1 + ν

3 + ν − 2η2

)u0, (76a, b)

cψ = κψ

√3 + ν − 2η2

1 + ν, (77)

Yk = KK0

=1−(

c1.2

c1.2 + 3.5

)(1−√

1 + ν

3 + ν − 2η2

),

(78)


Fig. 4 Crack facedisplacements forshear-deformable planestress bending

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10 100

Crack length (a/h)

Cra

ckfa

ced

isp

lace

men

t(u

/u0)

isotropic 0.1

isotropic 0.3

isotropic 0.5



Fig. 5 Difference incrack face displacementsplotted against cracklength

0

0.25

0.5

0.75

1

0.01 0.1 1 10 100

Normalized crack length (c)

Dis

pla

cem

ent

dif

fere

nce

( ud)

isotropic 0.1

isotropic 0.3

isotropic 0.5

quasi-isotropic boron/epoxy


model

Yd = uu0

= 1 −(

c1.3

c1.3 + 5

)(1 − 1 + ν

3 + ν − 2η2

).

(79)

It was found above that the crack tip is effec-tively in a state of plane strain, and that the crackface displacement away from the crack tip will ap-proach the displacement given by the plane stressmodel as the crack length gets very large. Thedeveloped eigth-order bending formulation allowsone to investigate this varying effect of shear andtransverse deformations on bending of a crackedplate. Figure 6 shows the stress intensity resultsobtained for the five trial materials analysed. Thesolid lines are the values obtained from the approx-imate model that will be developed over the nextfew pages. The points are data generated by solu-tion of the hyper-singular equation for generalizedplane strain plate bending. The nominal solutions

follow:

K0 = σb√πa, u0 = 2

σba(1 − η2)E

. (80)

Transverse deformations (the difference betweenplane stress and plane strain) result in larger stressintensities, whereas shear deflections decrease thestress intensity and ultimately have a larger effectas crack length increases. The true stress intensityarises as a result of the combination of these ef-fects. For an isotropic plate with a large Poisson’sratio, the increase in stress intensity is large enoughand occurs at a short enough crack length that itresults in a net increase in stress intensity for crackssomewhat smaller than the thickness of the plate.Figure 7 shows similar results for the crack facedisplacement.

No convenient normalization exists to collapsethese results, but they do approach a limit as crack


Fig. 6 Stress intensity foreigth-order plate bendingformulation

0.5

0.6

0.7

0.8

0.9

1

1.1

0.01 0.1 1 10 100

Crack length, a/h

Str

ess

inte

nsi

ty,K

/K0

isotropic 0.1isotropic 0.3isotropic 0.5quasi-iso b/epoxytrans b/epoxyisotropic 0.1 modelisotropic 0.3 modelisotropic 0.5 modelquasi-iso b/epoxy modeltrans b/epoxy model

Fig. 7 Crack facedisplacement foreigth-order plate bendingformulation

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100

Crack length, a/h

Cra

ckfa

ced

isp

lace

men

t,u

/u0

isotropic 0.1isotropic 0.3isotropic 0.5quasi-iso boron/epoxytransverse boron/epoxyisotropic 0.1 modelisotropic 0.3 modelisotropic 0.5 modelquasi-iso boron/epoxytransverse boron/epoxy

length increases. The stress intensity and crack facedisplacement for a long crack follow.

K∞ =√

g∞E1 − η2 =

√1 + ν

3 + ν

K0√1 − η2

,

u∞ =(

1 + ν

3 + ν

)u0

1 − η2 . (81a, b)

Figures 8 and 9 show the generalized plane strainmodel results normalized to these limit values. Theresults are plotted against a normalized cracklength.

cψ = κψ

√3 + ν

1 + ν

√1 − η2. (82)

Correction factors for the generalized planestrain plate arise by multiplication of the correc-tion factors for shear and transverse deformation.

This is indicated by the form of Eqs. 81a and b. Forstress intensity, the correction factor is given by thefollowing:

Yψ

k = 1 −(

c1.2ψ

c1.2ψ + 3.5

)⎛⎝1 −

√1 + ν

3 + ν

⎞⎠ , (83)

Ywk = 1 +

(c1.5

w

c1.5w + 1.5

)(1 − 1√

1 − η2

), (84)

K = Yψ

k Ywk K0. (85)

A similar result may be found for the crack facedisplacement as shown below.

Yψ

d = 1 −(

c1.3ψ

c1.3ψ + 5

)(1 − 1 + ν

3 + ν

), (86)


Fig. 8 Stress intensitydifference plotted againstnormalized crack length

-0.25

0

0.25

0.5

0.75

1

0.01 0.1 1 10 100

Normalized crack length, c

Str

ess

inte

nsi

tyd

iffe

ren

ce,K

d

isotropic 0.1isotropic 0.3isotropic 0.5quasi-iso boron/epoxytrans boron/epoxy

Fig. 9 Crack facedisplacement differenceplotted againstnormalized crack length

-0.25

0

0.25

0.5

0.75

1

0.01 0.1 1 10 100

Normalized crack length, c

Dis

pla

cem

ent

dif

fere

nce

,ud

isotropic 0.1isotropic 0.3isotropic 0.5quasi-isotropic boron/epoxytransverse boron/epoxy

Ywd = 1 +

(c1.5

w

c1.5w + 2.5

)(1 − 1

1 − η2

), (87)

u = Yψ

d Ywd u0. (88)

Equations 83 and 86 only account for transverseshear deformations, and hence one should use the

normalized crack length given by cψ = κψ

√3+ν1+ν .

Note that this should not be confused with the nor-malization given by Eq. 82 which includes a termto account for pressurization.

Figures 8 and 9 show the results obtained by thismethod. Equations 85 and 88 allow for a simpleengineering analysis for bending of cracked finitethickness plates. An additional result is that thecrack tip is observed to be in a state of plane strain,regardless of plate thickness. In the next section,we will develop a model for a crack bridged by

springs, and use these newly developed correctionfactors for stress intensity, strain energy releaserate and crack face displacement to develop designequations for bridged cracks in plates of arbitrarythickness.

6 Comparison to three dimensional finiteelement results

In this section, the thickness-averaged stress inten-sity and the crack face displacement are comparedto the results of a three-dimensional finite elementmodel. The ANSYS finite element software pack-age was used to analyse bending of a 20 mm crackin an isotropic plate with a Poisson’s ratio of 0.495.The plate thickness was varied from 1 to 64 mm,and both the width and height of a quarter-section


Fig. 10 Finite element map showing boundary conditions

of the plate were set at 600 mm, sufficient to min-imize the effects finite plate dimensions on thethrough-thickness averaged bending stress inten-sity. A coarse model of the plate employs fourelements through the thickness of the plate, witha finer sub-model employing ten elements beingused in the region of the crack tip. Figure 10 showsthe planar element distribution for both the coarsemodel and fine sub-model.

The crack tip elements are 20-noded solid brickelements collapsed into quarter-point singularwedges. These elements have an edge length of1.0 mm, which by numerical experimentation wasdetermined to be the smallest element size thatcould be reasonably used for this crack length.In the numerical experiments, the finite elementstress intensity factor results with different sizedcrack tip elements were compared to the exactplane strain results. The 1 mm crack tip elementslead to an underestimate of the stress intensity of2.8%. The results presented below were correctedto account for this error. During this verificationprocess, the nodes of the three-dimensional finiteelement model were constrained in the thicknessdirection, and a membrane stress was applied tothe edge of the model.

The stress intensity factor was calculated by thedisplacement-interpolation method using the dis-placement of the crack face nodes nearest the cracktip (He et al. 1997, Eq. 2). A stress-based methodwas not used because of problems with oscillations

occurring through the thickness of the plate, aproblem that has been observed by others. Notethat in the displacement-interpolation method, asis common in the three-dimensional analysis ofcracks, the crack tip is assumed to be in a stateof plane strain and the stress intensity is calculatedbased upon the theoretical plane-strain solutionof the displacement of the crack faces. The plate-averaged bending stress intensity factor was cal-culated by least-squares regression to results foreach plane of nodes through the thickness of theplate at the crack tip. In this calculation, the nodesat the surface of the plate were given one half theweighting of interior nodes. Figure 11 shows thethrough-thickness variation of the stress-intensityfactor for a selected set of the finite element results.

Note that the effect of the free edge is apparentat the edge of the plate, resulting in a somewhatreduced stress intensity factor at this location. Thiseffect has been noted by many others (e.g. Sih1971), and its characterization requires a higher or-der analysis than that presented in this article. It isalso well known that at the intersection of the crackand the free edge of the plate, the singular stressfield does not observe the classical 1/

√r singularity

except for the special case of ν = 0. Figure 12 showsthe bending stress intensity factor at the edge ofthe plate as determined by a least-squared fit tothe slope of the through-thickness finite elementresults. The result is compared to the bending stressintensity factor determined by the geometry


Fig. 11 Through-thick-ness variation of the stressintensity

Fig. 12 Comparison oftheoretical and finiteelement bending stressintensity factors

50

55

60

65

70

75

80

85

0.01 0.1 1 10 100

a/h

k b[M

Pa

mm

]

8-th order

6-th order

fea, plane strain

fea, plane stress

classical solution

correction factors for the sixth- and eighth-ordermodels described above. For comparison, addi-tional displacement-interpolation method finiteelement results are shown in which the crack tipis assumed to exist in a state of plane stress.

It is readily apparent that the new eighth-ordermodel provides a significant improvement for shortcracks in thick plates. For long cracks in thin plates,the results are more difficult to explain and theaccuracy of the proposed model is difficult to verify.When comparing the plane stress-based displace-ment–interpolation finite element method to thesixth-order model results, the finite element modelpredicts a higher stress intensity. This demonstratesthat some degree of local stiffening and load attrac-tion is occurring that is not predicted by the sixth-order models. Comparing the plane strain based

finite element results to the eigth-order model, thefinite element results are also seen to be higher.The reason for this is that the displacement fieldpredicted using a plane strain model is not identi-cal to that predicted using the eigth-order modelexcept for very short cracks in thick plates, partic-ularly as regards the regular and weakly-singularterms in the generalized plane strain formulation.Here, the finite element results must be seen tobe in error, as the eigth-order results represent anupper-bound on the actual stress intensity. The truesolution should lie between the plane-stress basedfinite element results and the results predicted bythe eigth-order model.

It is possible to show that, as regards the general-ized plane strain plate bending singular stress field,the crack tip is in a state of plane strain regardless


Fig. 13 Comparison oftheoretical and finiteelement model crack facedeflections

0E+00

1E-06

2E-06

3E-06

4E-06

5E-06

6E-06

0.01 0.1 1 10 100

a/h

u[m

]

8-th order

6-th order plane strain

6-th order plane stress

finite element analysis

4-th order plane strain

of the thickness of the plate. Beginning with eitherof Eq. (8d) or (9c), it is possible to show that:

Pzz = 4h3Gνz

3k2wη

2 ∇4w. (89)

By the method used to determine M(y) in Sect. 4,it can also be shown that:

Pzz(y) = 4h3Gνz

3π(1 − η2)

∫L

(β(t)

(t − y)2+ weakly singular

+ reg.terms)

dt. (90)

Comparing with (45) and noting that Mxx(y) =−M(y), the singular stresses at the crack tip arefound to obey the following relationship:

Pzz

Mxx= 2νz. (91)

At the crack tip the singular parts of σxx and σyy

are equal in magnitude, and accordingly the gen-eralized plane strain model predicts that, in termsof the plate-averaged quantities Pzz and Mxx, thesingular stresses at the crack tip exist in a state ofplane strain. This is only true as regards the sin-gular stresses in a small region close to the cracktip, and is in accord with the exact eigenfunctionexpansion results of Hartranft and Sih (1969).

Significant differences between the finite ele-ment and the generalized plane strain plate bend-ing model results were noted above which wouldnot be expected if the crack tip were in a state ofpure plane strain. It is possible that the finite ele-ment method used is sensitive to the lower-orderterms. It is also possible that for very thin plates,

the boundary layer at the plate edges begins tobecome large when compared to the dimensionsof the singular region at the crack tip and theregion dominated by purely plane strain conditionsdecreases in size. For the purposes of this article,it is unfortunate that the standard finite elementapproaches to stress intensity factor determinationare built upon the assumption of the validity ofthe plane stress or plane strain states (i.e. classicalmodels) and that a significant amount of engineer-ing judgement is required in order to interpret theresults.

Figure 13 shows a similar improvement in theprediction of the crack face displacement at thecentre of the crack. Note that the results are verygood for both short cracks in thick plates and longcracks in thin plates. Here, the results for the deflec-tion of the crack face at the outer fibre of the plateare also fitted to the slope of the finite elementresults. Unlike the determination of the stressintensity factor, these results are not subject tointerpretation.

7 Bending of reinforced cracked plates

Here we consider the action of springs bridging thecrack, developing a line-spring model for a gener-alized plane strain plate under bending. Figure 14shows a cracked plate in bending reinforced bysprings. The goal here is to determine the bendingstress intensity for the reinforced crack and to cal-culate the moment carried by the springs bridging


Fig. 14 A bending crack reinforced by springs

the crack, such that the strength of the plate can bedetermined.

To account for crack bridging, Eq. 57 may bemodified to include a stress due to the reinforc-ing springs, which acts to restrict the motion of thecrack faces. Here, ks is the line-spring stiffness, orstress generated by the springs on the outer fibreof the plate per unit deflection of the crack sur-faces. This definition of the bending spring stiffnessleads to line-spring equations with a similar form tothat found for plate extension (Clark and Romilly2006a).

σb (s)− aksu(s) = −E2π(1 − η2)

∫ 1

−1u(r)

(1

(r−s)2+ κ2

ψ

21−η2

1+ν Lψ(κψ |r − s|)+κ2

wη2L(κw |r − s|)

)dr. (92)

Expanding u(r) and choosing a set of colloca-tion points, Eq. 92 will reduce to the linear sys-tem σb = Ag. Here, A is expressed as follows, andincludes a term to account for the springs.

Ai,j = E1 − η2

(i + 1

2Uj(si)− κ2

ψ

4π

(1 − η2

1 + ν

)Lψi,j

−κ2wη

2

2πLw

i,j

)+ aksW(si)Uj(si). (93)

The generalized plane strain plate bending problemmay now be solved for the case of a cracked plate

bridged by springs. The stress intensity may becalculated from the expansion coefficients usingEq. 66. Figure 15 shows results for a cracked iso-tropic plate with σb = 10 MPa, υ = 0.5 and E =72.4 GPa. The results are plotted for a wide rangeof spring stiffnesses, as shown in the legend.

As was the case for plate extension, it is appar-ent that stiffer springs lead to lower stress inten-sities, reaching a limit for long cracks. Applyingthe method of Rose (1982, 1988), one may ana-lytically determine the limit strain energy releaserate for a very long crack. In this case, we examinethe strain energy release rate at the outer fibre ofthe plate. In the limit of a very long crack rein-forced by springs, the energy available for crackgrowth is given solely by considering the energybalance in the springs bridging the crack underthe applied (nominal) bending stresses. The strainenergy release rate may be expressed as follows:

G∞ = 2∫ u

0σb dx = 2

∫ u

0ksx dx = ksu2 = σ 2

b

ks.

(94)

The equations governing bending are identical tothose found for extension. As the crack tip hasbeen shown to exist in a state of plane strain, thelimit stress intensity may now be found.

K∞ =√

G∞E1 − η2 = σb

√E

ks(1 − η2). (95)

An effective crack length may also be defined,including factors to correct for plate geometry.

aeff = E

(Ywk Yψ

k )2π ks(1 − η2)

. (96)

Figure 16 shows the generalized plane strain resultsnormalized with respect to these two theoreticallimits. Note that as the crack becomes longer, thestress intensity limit is obeyed. Based on equation95, the use of the sixth-order Riessner formula-tion would lead to an error in the determinationof the long-crack limit stress intensity analogousto the difference between plane stress and plane


Fig. 15 Stress intensityfrom the line springmodel

0

0.5

1

1.5

2

0.01 0.1 1 10 100Normalized Crack Length (a/h)

Stre

ssIn

tens

ity[M

Pam

]

no springs0.1 GPa/mm1 GPa/mm10 GPa/mm100 GPa/mm1000 GPa/mm

Fig. 16 Normalized stressintensity plotted againstcrack length

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1 10 100 1000

Normalized Crack Length (a/aeff)

No

rmal

ized

Str

ess

Inte

nsi

ty(K

/K)

0.1 GPa/mm

1 GPa/mm

10 GPa/mm

100 GPa/mm

1000 GPa/mm

strain, i.e. on the order of a 5–10% increase formost metals.

It is also possible to define an interpolation forstress intensity between the limit states of a veryshort and a very long crack. For a short crack onemay use Eq. 85, developed for an un-reinforcedcrack. In the long-crack limit, the stress intensitywill approach that determined by Eq. 95. The stressintensity for an arbitrary crack length follows.

Kr = Ywk Yψ

k σb

√πaaeff

a + aeff. (97)

From Eqs. 96 and 97, the transverse shear cor-rection factor (based on the sixth-order models)will have a large role for moderately short cracks

reinforced by relatively weak springs. In this case,based upon the previous analysis of an unreinforcedcrack, the inclusion of the correction factor mayimprove the accuracy of the interpolation modelby as much as 60%.

The effectiveness of the interpolation model forbending may be tested against the line spring model.Figure 17 compares the stress intensity predictedby Eq. 97 normalized with respect to values deter-mined by numerical analysis. The interpolationmodel is observed to lead- to results that are lowby up to 5%, a difference that has also been foundfor extension of reinforced plates. This over-rid-ing trend in the error is principally due to theform of the interpolation model, but other trendsmay be noted that result from the interaction of


Fig. 17 Stress intensitypredicted by line-springand interpolation models

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

0.0001 0.001 0.01 0.1 1 10 100 1000

Normalized Crack Length (a/aeff)

Str

ess

Inte

nsi

tyR

atio

(Kr/K

)

0.1 GPa/mm

1 GPa/mm

10 GPa/mm

100 GPa/mm

1000 GPa/mm

Fig. 18 Crack facedisplacement for variousspring stiffnesses

0.0

0.5

1.0

1.5

0.01 0.1 1 10 100

Normalized crack length (a/h)

Cra

ckfa

ced

isp

lace

men

t,u

[µm

]

no springs

0.1 GPa/mm

1 GPa/mm

10 GPa/mm

100 GPa/mm

1000 GPa/mm

the geometry correction factors and the numericalline-spring model. For long cracks, the numericalmodel cannot accurately model the crack tip andtends to over-predict the stress intensity. At thesame time, shear effects (modelled by the shearcorrection factor) act to reduce the stress intensityat the crack tip. The observed oscillations are dueto the interaction between these effects.

One may apply the same methods to the crackface displacements. Figure 18 shows crack-face dis-placements from the line-spring model. For weaksprings, they approach the nominal solution for anun-reinforced crack. The long crack limit is evidentfor cracks bridged by stiff springs.

The long-crack limit crack face deflection is u∞ =σb/ks. The displacement for short cracks is given byEq. 88. A simple interpolation between these limitsis given by:

ur = u∞u0

u0 + u∞= σb

(2Yw

d Yψ

d (1 − η2)a/E)(1/ks)(

2Ywd Yψ

d (1 − η2)a/E)

+ (1/ks)

= σb

ks + E/(

2Ywd Yψ

d (1 − η2)a) . (98)

Figure 19 shows the ratio of the interpolation modeland line-spring model results for crack displace-ment. This simple interpolation results in an errornot exceeding 7%.

The maximum moment carried by the springs isgiven by Eq. 99.

Mxx = 23

h3ksur. (99)

Knowledge of this moment can be used to pre-dict rupture or fatigue degradation of the spring


Fig. 19 Crack facedisplacement fromline-spring andinterpolation models

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0.001 0.01 0.1 1 10 100 1000

Normalized crack length (a/aeff)

Dis

pla

cem

ent

rati

o(u

r/u)

0.1 GPa/mm

1 GPa/mm

10 GPa/mm

100 GPa/mm

1000 GPa/mm

materials (e.g. patch repair materials) bridging thecrack faces.

8 Discussion

Similar to the case for extension of cracked plates,for plates in bending the crack tip is subject tovery large stress gradients that induce pressuriza-tion of the plate near the crack. As for extension,this pressurization stiffens the plate locally in theregion of the crack tip, effectively increasing thelocal stresses and reducing the overall crack facedeflections from what would be predicted by mod-els that neglect the build-up of transverse stresses.Different from the case of extension of crackedplates, in the solution of plate bending problems,the transverse shear stresses enter into the equi-librium equations and are accompanied by sheardeflections. This additional set of loads and deflec-tions requires the adoption of a higher-order platemodel to obtain an accurate solution, particularlyfor thick plates and plates experiencing large stressgradients. The eigth-order plate model developedin this article is the simplest possible model that caninclude both of these effects, and has been usedto develop a line-spring model which may nowbe used anywhere that line-spring models basedon classical plate models have been applied, forexample, in the analysis of bonded composite re-pairs, surface cracks, crack tip plasticity and crackclosure.

The stresses at the base of a crack in bending,expressed in terms of the moment stress resul-tant, have been shown to exist in generalized planestrain conditions. This study has developed a set ofstandard engineering geometry correction factorsto describe the effects of plate thickness and trans-verse material properties on the stress intensityand crack face deflection, allowing for a simpleand accurate analysis of through-cracks in finitethickness plates. The correction factors accountfor transverse shear deflections and the effects oftransverse stresses caused by curvatures in thestress field in the plane of the plate. This is animportant phenomenon in the analysis of cracks,which contain very large stress gradients. Theseresults are of particular significance for the analy-sis of bridged cracks, where strain energy releaserate arguments are used to determine the stressintensity at the reinforced crack tip, and an appro-priate conversion factor is required to determinethe stress intensity. These results are in accord withthe exact eigenfunction expansion results of Har-tranft and Sih (1969) and Hartranft (1977), whofound that the through-thickness stresses are sin-gular and that plane strain conditions should existat the crack tip, a property which is not possessedby many other approximate theories. Lower-orderplate bending theories are also not capable of mod-elling the local ‘stiffening’ effect caused by thisregion of generalized plane strain, and the au-thors see the eigth-order model as an excellentengineering approach due to its relative ease of


application and the ability to capture the relevantphenomena in a conservative but reasonably accu-rate manner.

As for extension, the long-crack limit stressintensity factor and the interpolation model devel-oped by Rose offer a very effective method for theanalysis of bending of bridged cracks. The inter-polation model for the stress intensity in arepaired crack has been shown to be accurate within5%, a similar result to that found for extensionof reinforced cracks (Clark and Romilly 2006a).This error is principally attributed to the approx-imate nature of the interpolation model, and notto the correction factors introduced in this arti-cle. At its worst, the interpolation model result forthe crack face deflection is low by approximately7%, a difference which should be accounted forwhen using the model to predict rupture or fatiguedegradation of the spring materials bridging thecrack.

It is important to note that this study has notaddressed the subject of crack closure and crackface interaction, a limitation that is general to anybasic cracked-plate plate bending theory. It iswidely known that such an interaction will act to re-duce the bending stress intensity factor at the ten-sion side of the plate (e.g. Liu et al. 2006). The studyof this effect requires the superposition of exten-sional and bending models for cracked plates, andnonlinear analysis is required to determine the areaof contact of the crack surfaces. Several approachesexist to address this problem, including the useof line-spring models (Joseph and Erdogan 1989).The authors have applied the eigth-order platebending model to the analysis of linear coupledbending and extension of reinforced cracks in arecently accepted paper (Clark and Romilly 2006b)that could easily be extended to such a nonlinearanalysis.

9 Concluding remarks

The authors have developed a high-order platemodel and by the method of integral transformshave developed a hyper-singular integral equationuseful for the bending analysis of plates contain-ing cracks. The model allows for the inclusion of

both shear deformations of the plate and pressur-ization of the crack tip under the extreme stressgradients induced by the presence of a crack. Theresulting line-spring model and engineering anal-ysis equations for through-cracks and reinforcedcracks are applicable to a wide range of plate frac-ture problems and provide a significant improve-ment in accuracy over models based on theclassical plate models, and thus should be widelyadopted.

References

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Transverse shear and pressurization effects in the bending of plates with reinforced cracks

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