STRESS CONCENTRATION AT SHARP AND ROUNDED V … · 2015-05-31 · STRESS CONCENTRATION AT SHARP AND...

Proceedings of the 6th International Conference on Mechanics and Materials in Design,

Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015

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PAPER REF: 5545

STRESS CONCENTRATION AT SHARP AND ROUNDED V-NOTCHES

IN ORTHOTROPIC PLATES

Mykhaylo Savruk1, Andrzej Kazberuk

1, Marta Kosior-Kazberuk

2(*)

1Faculty of Mechanical Engineering, Bialystok University of Technology, Bialystok, Poland

2Faculty of Civil and Environmental Engineering, Bialystok University of Technology, Bialystok, Poland

(*)Email: [email protected]

ABSTRACT

The solution of elastostatics problem for orthotropic plate with an infinite rounded V-shaped

notch was obtained by singular integral equation method. This solution was used to find the

dependence between the stress intensity factor at the tip of sharp V-notch, the stress

concentration factor for rounded V-notch, and the radius of curvature in notch vertex. This

relationship could be treated as asymptotic for problems of finite bodies with notches rounded

with small radius of curvature.

Keywords: stress intensity factor, stress concentration factor, sharp and rounded V-shaped

notches, singular integral equation.

INTRODUCTION

Nowadays fibre reinforced composites are widely used in engineering structures. These

structures are commonly considered as orthotropic materials, thus the solutions of notch

problems in this type of materials are attractive. The problem of the distribution of singular

stress fields near the sharp or rounded wedge vertex is important in fracture mechanics of

bodies with angular corners. The solutions of this type of eigenvalue problems of the theory

of elasticity are utilized for wedge-shaped regions where asymptotic stress field at infinity is

expressed in terms of stress intensity factors at the wedge vertex (Cherepanov, 1979,

Benthem, 1987, Lazzarin, 1996, Savruk, 2010, Zappalorto, 2015).

The aim of this work is to extend already established method of unified approach to problems

of stress concentration in the vicinity of sharp and rounded vertices of V-notches in

anisotropic materials.

The essence of the method of unified approach (Savruk, 2007) is based on the calculation of

stresses at the vertex of V-notch rounded with the small radius of curvature. Then the limit

transition to sharp notch is performed. The knowledge of the relationship between the stress

concentration factors at the vertex of rounded notch and the stress intensity factors at the

vertex of a corresponding sharp notch makes this transition possible. These relationships were

established for the symmetric deformation state (mode I) (Savruk, 2006), antisymmetric

(mode II) (Savruk, 2011) and antiplane deformations (mode III) (Savruk, 2012). These

relationships can be treated as asymptotic for a wide class of problems of finite two- and

three-dimensional notched bodies. The precision and effectiveness of the method has been

repeatedly demonstrated (Savruk, Kazberuk, 2008-2013). It was also demonstrated that the

maximum stress at the vertex of the rounded notch depends not only on the radius of

curvature at the notch vertex and the opening angle of the notch, but the shape of the notch in

the certain vicinity of the tip as well (i.e. applying the formula addressed to parabolic notch to

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U-notch problem leads to imprecise stress estimation). Thus our approach adopts the simplest

and unambiguous definition of the rounded V-notch: the notch edges are straight lines

connected with circular arc.

The relationships between stress concentration and stress intensity factors allow to figure out

the stress intensity factors for sharp notch performing the calculations of the stress

concentration factors for rounded notch. Calculations of this type can be performed by any of

the commonly used tools like, for example, finite element method.

Knowing the stress intensity factors, the stress concentration factors at the rounded notch

vertex can be precisely estimated (for small radius of curvature). This feature can be used to

verify the calculation of stress concentration at the rounded notch vertex, as well as to design

the approximate formulas used in the research and engineering practice.

STRESS FIELD NEAR SHARP V-NOTCH IN ORTHOTROPIC PLATE

Hook’s law for plane stress state in orthotropic material has the form (Lekhnitskii, 1968)

2 2 2 2 2 2 2

2 21 2 3 1 2 3 31 2

2 21 1, , ,

2 2

+ − + −= + = + =

xx xx yy yy yy xx xy xy

x x xE E E

γ γ γ γ γ γ γε σ σ ε γ γ σ σ ε σ

where γ1, γ2, γ3 are dimensionless constants. In the terms of the engineering constants they can

be expressed as:

2 2 2 2 2

1 2 1 2 3, 2 ,2 2

= + = − =

x x xxy

y

E E E

E G Gγ γ γ γ ν γ ,

where E1, E2 are Young’s modulus alongside principal material axes, G is shear modulus and

v is Poisson’s ratio.

The stress function F(x,y) fulfils the following elliptic differential equation of the forth order

4 4 4

2 2 2 2

1 2 1 24 2 2 4( ) 0

∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂F F F

y x y xγ γ γ γ . (1)

The characteristic equation corresponding to (1)

4 2 2 2 2 2

1 2 1 2( ) 0+ + + =µ γ γ µ γ γ (2)

has two complex roots ( 0)= >k k kiµ γ γ and = −k kiµ γ (k=1,2).

For the V-notch with the symmetry axis Ox parallel to one of the principal material axes of

the orthotropic material, as shown in the Fig. 1, the stress field components are expressed as

follows (Lekhnitskii, 1968):

0 2 0 2 0

1 1 1 2 2 2

0 0 0

1 1 2 2

0 0 0

1 1 1 2 2 2

2 ( ) ( ) ,

2 ( ) ( ) ,

2 ( ) ( ) .

= − ℜ Φ + Φ

= ℜ Φ + Φ

= ℑ Φ + Φ

x

y

xy

z z

z z

z z

σ γ γ

σ

τ γ γ

(3)



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Fig. 1 - Semi-infinite sharp V-notch in orthotropic plate

Complex potentials 0

1 1( )Φ z and 0

2 2( )Φ z (where = +k kz x i yγ ) for symmetric deformation

mode (mode I) have the form (Savruk, 2014):

[ ]

I

I

I

I

I

V0 2 I1 1 1

2 1

V0 1 I2 2 2

2 1

1 /22 2 2

I

( )( ) ,

2[ ( ) ( )] (2 )

( )( ) ,

2[ ( ) ( )] (2 )

cos ( ) sin ( ) cos (2 ) ( ) ,

( ) arctan( tan ), / 2 / 2, 1, 2.

−

−

−

Φ =−

Φ = −−

= + − = + ≤ ≤ =

k k k

k k

R Kz z

R R

R Kz z

R R

R

k

λλ

λλ

λ

αα α π

αα α π

α γ α λ β α

β α π γ α π α π

(4)

Notch stress intensity factor V

IK we define as

( ) IV

I0

lim 2 ( ,0) .→

= rK r r

λθθπ σ (5)

Stress singularity order Iλ is the smallest real root of the following characteristic

equation (Savruk, 2014):

( ) ( ) [ ]

( ) [ ]

2 2 2 2

2 1 2 1 2

2 2

1 2 1

tan 1 tan tan (2 ) ( )

1 tan tan (2 ) ( ) 0.

− − + − +

+ + − =

γ γ α γ γ α λ β α

γ γ α λ β α (6)

The form of complex potentials (3) and characteristic equation (5) are valid only for 1 2≠γ γ .

The relationships for degenerate material ( 1 2=γ γ ) was obtained by limit transition and were

presented by Savruk, Kazberuk (2014) .

STRESS CONCENTRATION AT THE VERTEX OF ROUNDED V-NOTCH

Let us consider an elastic orthotropic plane with a rounded semi-infinite V-notch (Fig. 2a).

The notch opening angle is 2β (0<β<π/2). The radius of curvature at notch vertex is ρ . The

axis Ox is the axis of symmetry of the notch contour. Principal material axes are parallel to the

axes of the coordinate system Oxy. The edges are free of load and the asymptotic stress field

at the infinity is defined by complex potentials (3).

To find the stress distribution alongside the notch edge we use approach described in

papers (Savruk, 2006, 2011). Using the superposition method we look for the solution of the

problem of smooth curvilinear infinite crack in orthotropic plane subjected to asymptotic load

(Fig. 2b).

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Fig. 2 - a) Semi-infinite rounded V-notch b) infinite crack cutting contour L in an orthotropic plate

Thus, the boundary condition on the crack edges is as follows:

( ), ,+ + − −+ = + = ∈n ns n nsi i p t t Lσ τ σ τ (7)

where σn is normal and τn is tangential stress component alongside left (+) or right (−) crack

edge.

The expression on the right-hand side of the equation (6) has the form (Ioakimidis, 1977):

2 0 2 0 2 0 2 0

1 1 2 2 1 1 2 2

0 0

1 1 2 2

( ) (1 ) ( ) (1 ) ( ) (1 ) ( ) (1 ) ( )

2 ( ) ( ) ,

= −ℜ − Φ + − Φ − ℜ + Φ + + Φ +

+ ℑ Φ + Φ

dtp t t t t t

dt

dti t t

dt

γ γ γ γ

γ γ (8)

The boundary equation is solved using singular integral equation method (Savruk, 1981). The

formulation of the singular integral equations for two-dimensional anisotropic plates with

curvilinear cracks is based on (Ioakimidis, 1977) and was discussed in details in the

work (Savruk, 2014). Below, we present only necessary relationships leading to integral

equations.

We take the representation of the solution in the form of Cauchy-type

integrals (Muskhelishvili, 1977):

( )1

( ) , , 1, 2.2

′Φ = ∈ =

−∫

k

k kk k L k k k

k k

t d t L ki t

φ ττ

π τ (9)

where the unknown function ( )′k kφ τ is a derivative of the displacement jump function on the

crack contour. Function 2 2( )′φ τ one can express via 1 1( )′φ τ using the relation (Ioakimidis,

1977):

2 2 2 2 1 2 1 1 1 1 2 1 1 12 ( ) ( ) ( ) ( ) ( )′ ′ ′= − + + −i t dt i t dt i t dtγ φ γ γ φ γ γ φ (10

which results from the condition of continuity of stresses in passing through the crack contour

L. The singular integral equation, which satisfies equation (7) has the form (Savruk, 2014)



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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1( , ) ( ) ( , ) ( ) ( ), , ′ ′+ = ∈ ∫L K t d L t d P t t Lτ φ τ τ τ φ τ τ

π (11)

where

1 2 1 21 1 1

1 1 2 2

1 2 1 21 1 1

1 1 2 2

1 1 2 2

( ) 1 1( , ) ,

2

( ) 1 1( , ) ,

2

1( ) ( ) (1 ) ( ) (1 ) ( ) .

2

−= +

− −

+= −

− −

= = + − −

i dt dtK t

t dt dtt

i dt dtL t

dt dtt t

dtP t P t p t p t

dt

γ γτ

τ τ

γ γτ

τ τ

γ γ

(12)

The integral equation (11) has the unique solution (within the class of functions with an

integrable singularity at the ends of the contour L1) under the supplementary

condition (Savruk, 1981)

1 1 1 1( ) 0.′ =∫L dφ τ τ (13

For further computations it is convenient to express unknown function ( )′k kφ τ in another form

1 2 1 11 1 1 2 1 1 2 2 1 2 2 2 1 1 1 1

2 1 2 2 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,2

+′ ′ ′ ′ ′ ′= − = − = + − % % % %

dt dtii i t t

dt dt

γ γφ τ γ γ φ τ φ τ γ γ φ τ φ φ

γ γ γ (14)

so the system of integral equations (11), (13) is as follows:

1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1

1( , ) ( ) ( , ) ( ) ( ),

( ) 0

, ′ ′+ = ∈ ′ =

∫

∫

% %% %

%

L

L

K t d L t d P t t L

d

τ φ τ τ τ φ τ τπ

φ τ τ

(15)

where

1 21 1 1 1 1 1

1 2 1 1 2 2

1 2 1 21 1 1 1 1 1

1 2 1 2 1 1 2 2

1 1 1 1( , ) ( , ) ,

( ) 2

( )1 1 1( , ) ( , ) .

( ) 2( )

= = +

− − −

+= = − −

− − − − −

%

%

dt dtK t K t

i t dt dtt

dt dtL t L t

i dt dtt t

τ τγ γ τ τ

γ γτ τ

γ γ γ γ τ τ

(16

We define the contour L in the form of parametric equation ( )=t ω ξ for 11− < <ξ .

Substituting tk in the following manner

1

( ) (1 ) ( ) (1 ) ( ) , 1,22

k k k kt kω ξ γ ω ξ γ ω ξ = = + + − = (17)

we obtain the system (15) in the canonical form:

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1

1 1 1 1 11

1

11

1( , ) ( ) ( , ) ( ) ( ), 1 1,

( ) 0,

−

−

′ ′+ = − ≤ ≤ ′ =

∫

∫

% %% %K L d P

d

ξ η φ ξ ξ η φ ξ ξ η ηπ

φ ξ ξ (18)

where

( ) ( )

( ) ( )1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

( , ) ( ), ( ) , ( , ) ( ), ( ) ,

( ) ( ) ( ), ( ) ( ) .

= =

′ ′ ′= =

% % % %

% %

K K L L

P P

ξ η ω ξ ω η ξ η ω ξ ω η

φ ξ φ ω ξ ω ξ η ω η

We assume that unknown function 1( )′φ ξ is unbounded at the ends of interval of integration

11

2

( )( )

1′ =

−%

u ξφ ξ

ξ (19)

and solve the system of integral equations (18) using the method of mechanical

quadratures (see e.g. Savruk, 1981, Linkov 2002) and Gauss-Chebyshev formulas. The result

is the complex system of 2n linear equations with 2n unknowns 1( )ku ξ (k=1,…,2n):

2

1 1 1 1 1

1

2

1

1

1( , ) ( ) ( , ) ( ) ( ),

2

( ) 0,

=

=

+ = =

∑

∑

% %n

k m k k m k m

k

n

k

k

K u L u Pn

u

ξ η ξ ξ η ξ η

ξ (20)

where coordinates of quadrature nodes and points of collocation are calculated according to

formulas:

(2 1)

cos , 1, , 2 , cos , 1, , 2 1.4 2

−= = … = = … −k m

k mk n m n

n n

π πξ η (21)

The problem is symmetrical with respect to Ox axis, thus using the relation

1 1( ) ( ),− =u uξ ξ (22)

we reduce order of the system (20) by two. Resulting system has the form:

1 1 1

1

1 1

1

1( , ) ( ) ( , ) ( ) ( ), 1, , ,

2

1( ) ( ) 0

2

=

=

+ = = … + =

∑

∑

n

k m k k m k m

k

n

k k

k

M u N u P m nn

u un

ξ η ξ ξ η ξ η

ξ ξ (23

where

( , ) ( , ) ( , ), ( , ) ( , ) ( , )= + − = + −k m k m k m k m k m k mM M N N N Mξ η ξ η ξ η ξ η ξ η ξ η (24



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The stress components on the crack contour L are calculated according to formulas:

0 2 2

1 1 1 2 2 2

0

1 1 2 2

0

1 1 1 2 2 2

2 ( ) ( ) ,

2 ( ) ( ) ,

2 ( ) ( ) ,

± ± ±

± ± ±

± ± ±

= − ℜ Φ + Φ

= + ℜ Φ + Φ

= + ℑ Φ + Φ

x x

y y

xy xy

z z

z z

z z

σ σ γ γ

σ σ

τ τ γ γ

(25)

where boundary values of the Cauchy-type integral ( )Φk kz are obtained using Sokhotski-

Plemelj theorem

2

211 2

( )1 1( ) ( ) , , 1,2.

2 2

( )( )1 1( ( )) ,

2( ) 2 ( ) ( )( ) 1

(2 1)cos , 1, , 2

4

±

±

=

′′Φ = ± + ∈ =

−

Φ = − ± +

− −′ − −

= = …

∫

∑

k

k kk k k k k k k

Lk k

nk jk

k k

j k j kk

j

t t d t L ki t

uiu

n

jj n

n

φ τφ τ

π τ

ξηω η

γ γ ω ξ ω ηω η η

πξ

(26)

The values of 1( )ku ξ are the solutions of the linear equation system (23) with respect to

symmetry condition (22). The values of 2 ( )ku ξ are calculated from transformed equation (14)

taking into account that 2

2 2( ) ( ) / 1′ = −% uφ ξ ξ ξ . Thus

2 1 2 1 1 2 1

2

1( ) ( ) ( ) ( ) ( )

2 = − + + − u u uξ γ γ ξ γ γ ξ

γ (27)

The values of ( )ku η ( )≠ jη ξ are calculated using Langrange interpolation formulas over

Chebyshev nodes jξ (see Savruk, 1981).

The edges of the notch are free of load ( 0= =n nsσ τ ) and the sum of normal stresses is

invariant, i.e. + = +n s x yσ σ σ σ , so the normal traction alongside notch edge is equal:

I

V

II( ) ( ) ( ) ( ) ( ),

(2 )s s x y

KRλσ η σ η σ η σ η η

πρ− − −= = + = (28

where I ( )R η is dimensionless normal stress at point η.

At the vertex of rounded notch max( 0) ( )= =s sσ η σ and I I( 0)= =R Rη is the stress rounding

factor (Benthem, 1981, Savruk, 2006), which value (for isotropic material) depends on the

notch opening angle and notch shape in a certain vicinity of the notch vertex (Savruk, 2006).

The asymptotic relationship

I

I

V

Imax I( )

(2 )s

KR

λλσ ρ

π−= (29)

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between the stress concentration factor for rounded notch and the stress intensity factor for

sharp notch has two main purposes (Savruk, 2007, Savruk, 2010): if we know the stress

intensity factor we could estimate the maximum stress at the vertex of V-notch rounded with

small radius of curvature, or we can obtain the stress intensity factor

I

IV

I max0

I

(2 )lim( )sK

R

λλπ

σ ρ→

=ñ

(30)

performing the calculations for rounded notch with diminishing curvature radius.

NUMERICAL RESULTS

The calculations were performed for three arbitrary chosen orthotropic materials: hard wood,

graphite/epoxy and glass/epoxy unidirectional composites. The material principal axes of

orthotropy are parallel to the notch Oxy coordinate system. Two material orientations were

considered: fibres parallel to Ox (Ex=E1) and fibres perpendicular to Ox (Ex=E2). The elastic

properties of the materials are shown in Tab. 1. The notch singularity order Iλ , calculated

according to the equation (6), as the function of notch angle is shown in Fig. 3.

Table 1 - Elastic properties of the exemplary orthotropic materials, M1 — wood (pine, white (Kretschmann,

2010)), M2 — glass/epoxy unidirectional composite Scotchply 1002 (Tan, 1994), M3 — graphite/epoxy

unidirectional composite IM6/F584 (Tan, 1994)

Material E2/E1 G/E1 ν Ex=E1 Ex=E2

γ1 γ2 γ1 γ2

M1 0.038 0.048 0.34 4.3291 1.1850 0.8439 0.2310

M2 0.215 0.107 0.26 2.8704 0.7513 1.3310 0.3484

M3 0.056 0.030 0.33 5.6225 0.7508 1.3319 0.1779

Fig. 3 - The sharp V-notch singularity order λI as a function of the notch opening angle 2β for sample orthotropic

materials; solid lines Ex=E1, dashed lines Ex=E2



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The rounded V-notch smooth contour L consists the straight semi-infinite edges joined with

circular arc (Savruk, 2006, 2011, 2012). The radius of the arc is ρ. Parametric equation which

describe contour L has the following form:

[ ]

[ ]( )

sin ( )cos cos ( )sin , ,

cos sin , ,

sin ( )cos cos ( )sin , ,

B B B

B B

B B B

i

i

i

ξ

β ζ ζ β β ζ ζ β ζ ζω ρ ζ ζ ζ ζ ζ

β ζ ζ β β ζ ζ β ζ ζ

+ + − − + −∞ < < −

= + − ≤ ≤ − − + + − < < ∞

(31)

where 2/ (1 )= − γζ ξ ξ , / 2= −Bζ π β is an angular coordinate of the circular arc end point, γ

is an arbitrary positive number (here γ=3/2), chosen experimentally to obtain the best

convergence of the solution.

We also consider contour LH as hyperbolic notch (Benthem, 1987)

2cos 1

( ) cot , , 1 1.cos cos 2 2

ie ξα α αω ξ ρ α π β ξ

α ξα

= + = − − < < − (32)

The comparison of the notch contours in the vicinity of notch apex is shown in Fig. 4.

Fig. 4 - Comparison of the shape of rounded V-notch L and hyperbolic notch LH of the

same radius of curvature at the notch vertex

The stresses at the notch vertex were calculated for the notch opening angle varying in the

range 5 2 175° °≤ ≤β . Dimensionless stress rounding factors are shown in Fig. 5.

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Fig. 5 - Dimensionless stress concentration factors RI at the rounded notch vertex as a function of notch opening

angle 2β for sample materials a) Ex=E1, b) Ex=E2; solid lines — rounded V-notch (31), dashed lines —

hyperbolic notch (32)

For hyperbolic notch and opening angle 2 2.5°=β , calculated values agree well (relative

difference is less then 0.1%) with results obtained analytically for the problem of elliptical

hole in orthotropic plane subjected to uniform tension (Lekhnitskii, 1981, Chiang, 1994):

1 2I

1 2

( 0) 2+

= =HRγ γ

βγ γ

(33)

The relative differences in the stress concentration values between hyperbolic notch and the

rounded V-notch (straight edges and circular arc apex) exceed 10% and depend on the

material orientation and the notch opening angle.

CONCLUSIONS

The asymptotic problem of semi-infinite rounded V-notch in the orthotropic plane subjected

to symmetrical deformation (mode I) was solved. For three different orthotropic materials the

relationship between stress intensity factor for sharp V-notch and stress concentration factor

at the vertex of rounded V-notch (with the same opening angle) was established.

The results shown that stress concentration factor at the vertex of rounded V-notch strongly

depends on the notch shape in the vicinity of notch apex. For orthotropic materials the relative

difference in maximum stress at the notch vertex resulting from notch geometry exceeds 10%

(i.e. is significantly greater than for isotropic materials) and depends on the elastic properties

of material.

ACKNOWLEDGMENTS

This work was financially supported by National Science Centre (Poland) under the project

DEC-2011/03/B/ST8/06456.

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STRESS CONCENTRATION AT SHARP AND ROUNDED V … · 2015-05-31 · STRESS CONCENTRATION AT SHARP AND...

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