Two-dimensional elastic-plastic cracked finite element...

Indian Journal of Engineering & Materials Sciences Vol. 19, April 2012, pp 95-106

Two-dimensional elastic-plastic cracked finite element for fracture applications

Aysha Kalanad & B N Rao*

Structural Engineering Division, Department of Civil Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

Received August 2011; accepted March 2012

In this paper a novel two-dimensional finite element with a single, non-propagating, open embedded edge crack is developed based on elastic-plastic fracture mechanics. The influence of crack on the local flexibility of the structure is accounted for by the reduction of the element stiffness as a function of the crack length, instead of physically modeling the crack within the element. The proposed element accounts the influence of crack tip plasticity, at the cracked cross-section. The components of the stiffness matrix for the proposed cracked element are derived based on the Castigliano’s first principle. The element is implemented as a user element (UEL) subroutine in ABAQUS. The proposed element can be applied in a wide range of fatigue and fracture applications in order to determine changes in static, dynamic, and stability characteristics of cracked structures.

Keywords: Cracked finite element, Elastic-plastic fracture mechanics, User element, ABAQUS, Natural frequency

The presence of a crack in a structural member reduces the stiffness and increases the damping of the structure. As a consequence, there is a decrease in natural frequencies and modification of the modes of vibration. Several approaches have been used to model the problem of a cracked beam using the finite element method. One-dimensional cracked beam finite elements for vibration studies have been developed by many researchers1-6. With an aim to simulate the crack presence without actually modeling the crack, more recently a two-dimensional cracked finite element was developed by Potirniche et al.7 for fatigue and fracture applications. In this approach the influence of the additional flexibility of the element due to the crack presence was derived from the Castigliano’s first theorem by using fracture mechanics concepts. However, the accuracy of the predicted natural frequency using the cracked finite element developed by Potirniche et al.7 for higher values of crack depth ratios is low. While deriving the components of the stiffness matrix for the cracked element, Potirniche et al.7 assumed that the applied shear forces results only in mode II stress intensity factors (SIF), and accordingly adopted the geometrical factor (for the effect of boundary conditions at free edge) corresponding to pure shear for infinite boundary conditions given in Tada et al.8

It should be noted that the pure shear condition can be reproduced only when the shear force acting along an edge of the cracked element, is accompanied by shear forces acting on three other faces. However when the applied shear force acts along an edge of the cracked element, as considered in Potirniche et al.7, mixed−mode conditions prevail (not the pure shear condition), which results in both mode I and mode II SIFs. In addition the adopted geometrical factor should take into account both the effect of finite size of the cracked element and the effect of boundary conditions at free edges.

In all the papers cited above, it was assumed that the material around the crack tip behaved in a purely elastic manner. In practice, in many materials, the plastic zone appears around the crack tip, and flexibility of the structure increases more than it is observed for a purely elastic material. Krawczuk et al.9,10 developed cracked beam and plate finite elements by taking into account the effect of plasticity ahead of the crack tip. These efforts used Castigliano’s second theorem, which states that the displacements can be obtained by taking the partial derivatives of the strain energy with respect to the correspondingly applied forces. In a standard finite element implementation, the use of Castigliano’s second theorem is not practical because the resulting singular compliance matrix must be inverted in order to obtain the element stiffness matrix.

———————— *Corresponding author (E-mail: [email protected])

INDIAN J ENG. MATER. SCI., APRIL 2012

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This paper presents a novel two-dimensional finite element with a single, non-propagating, open embedded edge crack, which takes into account the influence of the plastic zone ahead of the crack tip on flexibility of the element. The compliance matrix singularity is avoided by employing Castigliano’s first theorem, which states that the applied forces can be obtained by taking the partial derivatives of the strain energy with respect to the corresponding displacements. The element is implemented in the commercial finite element code ABAQUS as user element (UEL) subroutine. The proposed element has potential application in a wide range of analyses in order to determine changes in static, dynamic, and stability characteristics of cracked structures. Elastic-Plastic Cracked Finite Element

Definition of cracked finite element Consider the cracked finite element with the node

numbering and the degrees of freedom per node as shown in Fig. 1. According to the element definition in Fig. 1, the nodal forces and displacements can be written in vector form as:

{ } { } { } { }1 2 8 1 2 8andT TF F F F u u u u= =… …… (1)

The stiffness matrix of the uncracked element is denoted as [S°]. In the case of a cracked element, assuming that a crack of length α is present as illustrated in Fig. 1, the stiffness matrix is denoted as [S] = [S(α)]. The force-displacement equation for an uncracked element is then

{ } { }0 0F S u⎡ ⎤= ⎣ ⎦ … (2)

It is the purpose of this paper to present a formulation for the modified stiffness matrix [S] of a finite element as a function of the crack length α due to a single, non-propagating, open edge crack, based on elastic-plastic fracture mechanics. Due to the crack of length α present at the top edge of the element and the action of the nodal forces, mode I and II SIFs can be defined as

2 3 1 4I IF IF IF IFK K K K K= + = + … (3)

for mode I due to the nodal tensile forces, and

6 7 5 8II IIF IIF IIF IIFK K K K K= + = + … (4) for mode II due to the action of the nodal shear forces. The expressions for the above SIFs will be derived for the nodal forces acting at the right-hand-side of the element on the edge 2-3, i.e., the forces F2, F3, F4 and F7. On similar lines the expressions for the nodal forces acting on the edge 1-4 side of the element can be obtained.

In Fig. 2, the tensile force at node 3 gives a force and a moment, both of which contribute to an opening of the crack. Hence the contribution KIF3 of the nodal force F3 at node 3 is summation of the SIFs given by the force and the resulting bending moment F3h/2 (where h is the element depth), which can be written as

3 3 3

f mIF IF IFK K K= + … (5)

where

3 3

3 33andf mIF f IF m

F FK G K Ght ht

= πα = πα …(6)

( )3

3i.e., 3IF f mFK G Ght

= + πα … (7)

with Gf and Gm being the geometrical factors, and t being the element thickness.

It was proposed by Irwin11 to model the effect of flexibility increasing due to the crack tip yielding by a hypothetical extension of the crack tip rp, as shown in Fig. 3. The radius of the plastic zone around the crack

Fig. 1– Details of two-dimensional cracked finite element

Fig. 2– Force F3 acting on two-dimensional cracked finite element

KALANAD & B N RAO : TWO-DIMENSIONAL ELASTIC-PLASTIC CRACKED FINITE ELEMENT

97

tip rp, for mode I fracture of the crack evaluation (opening mode) can be calculated from the following relationship:

3

2

p pY

12 FIKd r

⎛ ⎞= = ⎜ ⎟⎜ ⎟π σ⎝ ⎠

… (8)

where σY the material yield strength, and dp denotes the diameter of the plastic zone around the crack tip. It was also assumed by Irwin that for a crack with the plastic zone around its tip the SIF can be expressed as follows:

( )3

3p 3IF f m eff

FK G Ght

= + πα … (9)

where

eff prα = α + … (10)

with 22

pY

( 3 )2

f mG Gr

+ ⎛ ⎞σ= α⎜ ⎟σ⎝ ⎠

. As can be observed

from Eq. (9), an iterative solution technique is usually required to solve for the SIF. In the first-order estimate, the geometrical factors are computed in the absence of a plasticity correction; then αeff is obtained from Eq. (10), which in turn is used to compute the geometrical factors Gf and Gm, and subsequently the SIF.

The modified formulation allows calculation of the stress intensity factor for an elastic-plastic crack as a function of the area of the plastic zone around the crack tip described by the ratio σ/σY, with σ being the applied nominal stress equal to F3/ht.

The FRANC2DL finite element code12-15 is used with the J-integral option to extract the SIFs from stress strain fields around the crack tip location. Two-dimensional, finite element models having w/h=2 with

degrees of freedom ranging from 3510 (for the case α/h=0.1) to 4258 (for the case α/h=0.9), along with a ring of six-noded quarter-point elements around the crack tip and eight-noded elements elsewhere are used under plane stress conditions. Figures 4(a)−4(i) show respectively FEM discretization for the cases α/h = 0.1-0.9. The minimum element size at the crack tip location is 0.0025w. Crack length to depth ratios (α/h) are varied from 0.1 to 0.9 with nodal forces applied at various locations on the cracked element. Using the SIFs values obtained from FRANC2DL for α/h ranging from 0.1 to 0.9, and Eq. (6), the geometrical factors Gf and Gm, for the cracked element under tensile and bending loading respectively, are obtained by curve fitting techniques as a function of α/h as follows:

2 3

2.6233 51.173 551.45 2563.7fGh h h hα α α α⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

4 5 6

5883.6 6472.2 2750.4h h hα α α⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠… (11)

and,

2 3

1.6426 18.687 192.08 883.57mGh h h hα α α α⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠4 5 6

2018.3 2213.7 939h h hα α α⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ... (12)

The above given geometrical factors Gf and Gm are validated for other cases with w/h>2.0 by comparing the SIFs values obtained from FRANC2DL with those values obtained using Eq. (6) in conjunction with Eqs (11) and (12). The effect of w/h is found to be practically negligible for w/h≥2.0.

Contrary to the tensile force acting at node 3 (in Fig. 2) as discussed above, in Fig. 5 the nodal force F2 acting at node 2 results in a force that leads to an opening of the crack and a resolved bending moment that leads to the closing of the crack. Hence the contribution KIF2of the nodal force F2 at node 2 can be written as

2 2 2

f mIF IF IFK K K= − … (13)

where

2 2

2 23andf mIF f IF m

F FK G K Ght ht

= πα = πα … (14)

Fig. 3– Plastic zone around crack tip


98

Fig. 4– FEM discretization (a) α/h=0.1, (b) α/h=0.2, (c) α/h=0.3, (d) α/h=0.4, (e) α/h=0.5, (f) α/h=0.6, (g) α/h=0.7, (h) α/h=0.8and (i) α/h=0.98

Fig. 5– Force F2 acting on two-dimensional cracked finite element

( )2

23IF f mFK G Ght

= − πα … (15)

with the geometrical factors Gf and Gm defined in Eqs (11) and (12).

Similar to that discussed above for the case of the tensile force acting at node 3, for the case of the nodal force F2 acting at node 2 with a crack having the plastic zone around its tip the SIF can be expressed as follows:

( )2

2p 3IF f m eff

FK G Ght

= − πα … (16)

which allows calculation of the stress intensity factor for an elastic-plastic crack as a function of the area of


99

Fig. 6– Force F6 acting on two-dimensional cracked finite element the plastic zone around the crack tip described by the ratio σ/σY, with σ being the applied nominal stress equal to F2/ht. In Eq. (16), αeffis same as that defined

in Eq. (10), with ( )2 2

pY

32

f mG Gr

− ⎛ ⎞σ= α⎜ ⎟σ⎝ ⎠

.

In Fig. 6 the nodal force F6 acting at node 2 gives a shear force and a moment (F6w), both of which contribute to mode I and II SIFs, which can be written as

6 6

6 62

6 andIF I IIF IIF w FK G K Gh t ht

= πα = πα … (17)

Using the SIFs values obtained from FRANC2DL for α/h ranging from 0.1 to 0.9, and Eq. (17), the geometrical factors for the cracked element GI and GII respectively, are obtained by curve fitting techniques as a function of α/h as follows:

2 3

0.821 9.344 96.04 441.78IGh h h hα α α α⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠4 5 6

1009.15 1106.85 469.5h h hα α α⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠… (18)

and

2 3

1.018 17.794 162.7 596.45IIGh h h hα α α α⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠4 5 6

1098.3 994.94 353.26h h hα α α⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠… (19)

Similar to that discussed above for the case of the tensile forces acting at node 2 and 3, for the case of mode I fracture (opening mode) of a crack having the plastic zone around its tip the SIF can be expressed as follows:

6pIF I effK G= σ πα … (20)

with σ being the applied nominal stress equal to 6F6w/h2t. In Eq. (20), αeff is same as that defined

in Eq. (10), with 22

pY2

IGr⎛ ⎞σ

= α⎜ ⎟σ⎝ ⎠.

For mode II fracture (sliding mode) the radius of the plastic zone around the crack tip rp, can be calculated from the following relationship:

6

2

p pY

32 IIFKd r

⎛ ⎞= = ⎜ ⎟π σ⎝ ⎠

… (21)

and for a crack with the plastic zone around its tip the SIF can be expressed as follows:

6

6pIIF II eff

FK Ght

= πα … (22)

In Eq. (22), effα is same as that defined in Eq. (10),

with 22

pY

32

IIGr⎛ ⎞σ

= α⎜ ⎟σ⎝ ⎠.

Element stiffness matrix derivation According to the Castigliano’s first theorem the nodal forces Fi can be obtained by taking the partial derivatives of the elastic strain energy (U) with respect to corresponding displacements (Ui)

00andi i

i i

U UF Fu u∂ ∂

= =∂ ∂

… (23)

where the index 0 denotes the undamaged structure. Taking the difference of the above expressions

( )00

i ii

U UF F

u∂ −

− =∂

… (24)

The difference between the elastic strain energies

in the cracked and undamaged cases is related to the SIFs for a fixed displacement case by the relation8

( )0 2 2

0

I IItU U K K daE

α

− = − +′⌠⎮⌡

… (25)

where E’ = E for plane stress, E’ = E /(1-v2) for plane strain, E and v are the modulus of elasticity and Poisson’s ratio.


100

Using Eqs (24) and (25) for the case of force F3 acting at node 3 and for a crack with the plastic zone around its tip

3

3 3

p0 23 3 p p

3 30 0

2 IFIF IF

Kt tF F K da K dau E E u

αα⎛ ⎞ ∂∂ ⎜ ⎟− = =⎜ ⎟′ ′∂ ∂⎝ ⎠

⌠⌠⎮ ⎮⌡ ⌡

… (26)

Substituting Eq. (9) into the above equation and after some simplifications, one obtains

( )20 33 3 32

30

2 3f m effFF F G G a da F

E h t u

α⎡ ⎤ ∂π− = +⎢ ⎥′ ∂⎣ ⎦

∫ … (27)

for a given area of the plastic zone around the crack tip described by the ratio σ/σY. In Eq. (27), αeff is same as that defined in Eq. (10), with

( )2 2

pY

32

f mG Gr a

+ ⎛ ⎞σ= ⎜ ⎟σ⎝ ⎠

.

In the above equation, the nodal forces for the undamaged and for the cracked elements can be separated in the two sides of the previous equation; noting from Eq. (2) that

333

3

F Su∂

=∂

… (28)

the relation between the two forces becomes

( )03 33 33 31F A S F= + … (29)

where A33 is a constant defined as

( )2

33 20

2 3f m effA G G a daE h t

α⎡ ⎤π= +⎢ ⎥′ ⎣ ⎦

∫ … (30)

In Eq. (30), αeff is same as that defined in Eq. (10),

with ( )2 2

pY

32

f mG Gr a

+ ⎛ ⎞σ= ⎜ ⎟σ⎝ ⎠

.

Using Eq. (2), Eq. (29) can be written as

( )8 8

03 33 33 3

1 1

1j j j jj j

S u A S S u= =

= +∑ ∑ … (31)

which is valid only if the coefficients multiplying the independent variables uj on both sides of the above equation, are equal to

( )03 33 33 31j jS A S S= + … (32)

Solving Eq. (32) for S3j the solution is found to be

033 33

3333

1 1 42

A SS

A− + +

= … (33)

and

03

3 033 33

2 for 1,2, ,8 and 3

1 1 4j

j

SS j j

A S= = ≠

+ +…

… (34)

Similarly, for the S2j components, the same procedure indicated above for S3j can be applied, leading to

2

0 22 2 p

2 0

IFtF F K da

u E

α⎛ ⎞∂ ⎜ ⎟− = =⎜ ⎟′∂ ⎝ ⎠

⌠⎮⌡

( )2

22 220

2 3f m effS G G a da FE h t

α⎡ ⎤π−⎢ ⎥′ ⎣ ⎦

∫ … (35)

Thus,

( )202 22 22

0

21 3f m effF S G G a da FE h t

α⎛ ⎞⎡ ⎤π= + −⎜ ⎟⎢ ⎥⎜ ⎟′ ⎣ ⎦⎝ ⎠

∫ … (36)

and the stiffness coefficients are obtained as 0

22 2222

22

1 1 42

A SS

A− + +

= … (37)

and

02

2 022 22

2 for 1,3, ,8 and 2

1 1 4j

j

SS j j

A S= = ≠

+ +…

… (38)

where A22 is defined as

( )2

22 20

2 3f m effA G G a daE h t

α⎡ ⎤π= −⎢ ⎥′ ⎣ ⎦

∫ … (39)

In Eqs (35), (36) and (39), αeff is same as that defined

in Eq. (10), with ( )2 2

pY

32

f mG Gr a

− ⎛ ⎞σ= ⎜ ⎟σ⎝ ⎠

.

For the case of nodal force F6 acting at node 2, the stiffness components S6j can be obtained by adopting the following procedure. Using Castigliano’s first


101

theorem, the difference between the nodal forces in the cracked (Fi) and undamaged (Fi

0) cases can be obtained by taking the partial derivatives of the SIFs with respect to the corresponding displacements (ui) by the relation8

6 6

6 6

p p06 6 p p

6 60 0

2 IF IIFIF IIF

K KtF F K da K daE u u

α α⎡ ⎤∂ ∂⎢ ⎥− = +⎢ ⎥′ ∂ ∂⎢ ⎥⎣ ⎦

⌠ ⌠⎮ ⎮⌡ ⌡

… (40)

Substituting Eqs (20) and (22) into the above equation and after some simplifications, one obtains

20 2 2 6

6 6 62 260 0

2 36I eff II eff

FwF F G a da G a da FE h t h u

α α⎡ ⎤ ∂π− = +⎢ ⎥′ ∂⎣ ⎦

∫ ∫

… (41)

Defining A66 as,

22 2

66 2 20 0

2 36I eff II eff

wA G a da G a daE h t h

α α⎡ ⎤π= +⎢ ⎥′ ⎣ ⎦

∫ ∫ … (42)

and noting that

666

6

F Su∂

=∂

… (43)

the relation between the two nodal forces for the undamaged and cracked elements becomes

( )06 66 66 61F A S F= + … (44)

In Eqs (41) and (42), αeff in the first and second terms (inside the brackets) is same as that defined in Eq.

(10), with 22

pY

12

IGr a⎛ ⎞σ

= + ⎜ ⎟σ⎝ ⎠ and

22

pY

32

IIGr a⎛ ⎞σ

= ⎜ ⎟σ⎝ ⎠

respectively. Using { } { }0 0F S u⎡ ⎤= ⎣ ⎦ , Eq. (44) can be written as

( )8 8

06 66 66 6

1 1

1j j j jj j

S u A S S u= =

= +∑ ∑ … (45)

which is valid only if the coefficients multiplying the independent variables uj on both sides of the above equation are equal, i.e.,

( )06 66 66 61 for 1,2, ,8j jS A S S j= + = … … (46)

Solving Eq. (46) for S6j the solution is found to be

066 66

6666

1 1 42

A SS

A− + +

= … (47)

and

06

6 066 66

2 for 1,2, ,8 and 6

1 1 4j

j

SS j j

A S= = ≠

+ +…

… (48)

Similar formulas can be obtained for all the components S5j, S7j and S8j.

Validation of Cracked Finite Element The proposed improved two-dimensional finite

element with an embedded edge crack is implemented in the commercial finite element code ABAQUS16 as a User Element Fortran subroutine (UEL.f). The performance of the proposed improved finite element is demonstrated by comparing the frequency ratio (ωc/ω) (ratio of the natural frequency of the cracked beam to that of the uncracked beam) versus the crack depth ratio (α/H) (the ratio of the crack depth (α) to the beam height (H)) results obtained using UEL, with the reported results in the literature, for the bending mode, for various crack location ratios (c/L) (ratio of the crack location to the beam length). Example 1

In this numerical example, the following beam cases are considered: (i) simply supported beam with a double edge surface crack at mid-span; (ii) cantilever beam with a surface crack at 20% of the beam span from fixed end; and (iii) simply supported beam with a surface crack at mid-span. In the numerical study it is assumed that the plastic zone around the crack tip does not exist and the crack depth ratio (α/H) is varied from 0 to 0.5. Simply supported beam with a double edge surface crack

In this numerical example, a steel beam4 having length L = 0.575 m, height H = 0.03175 m, thickness t = 0.00952 m with E = 2.06 × 1011 N/m2 and ρ = 7800 kg/m3 is considered. Figure 7 shows typical FEM discretization with 36 standard four node ABAQUS16 elements and one UEL each at the top and the bottom at the top of the beam for c/L = 0.5. Figure 8 shows the first natural frequency ratio (ωc/ω) versus the crack depth ratio (α/H) for simply supported beam with two surface cracks, at the top and bottom edges of the beam at mid-span. Compared with the predictions obtained using the damaged finite


102

Fig. 7– Discretization of simply supported beam with double edge surface crack using 36 standard four node ABAQUS elements and two UELs

Fig. 8– First natural frequency ratio versus the crack depth ratio for simply supported beam with double edge surface crack at mid-span element by Potirniche et al.7, the first natural frequency reduction predicted by the proposed improved two-dimensional finite element matches very well with the experiments results by Chondros et al.4

Cantilever beam with a surface crack

In this numerical example, a steel cantilever beam17

with all the geometric and material properties same as that of simply supported beam with a double edge surface crack except having the height H = 0.0242 m, is considered. Typical FEM discretization with 22 standard four node ABAQUS16 elements and one UEL for c/L = 0.2 measured from fixed end is shown in Fig. 9. Figure 10 shows the first natural frequency ratio (ωc/ω) versus the crack depth ratio (α/H) for steel cantilever beam with crack located at a distance of 20% of the beam length from the fixed end. Compared with the predictions obtained using the damaged finite element by Potirniche et al.7, the first natural frequency reduction predicted by the proposed improved two-dimensional finite element matches very well with the experiments results by Wendtland17. Contrary to the reported results7, the current study showed much deviation in the

Fig. 9– Discretization of cantilever beam with surface crack using 22 standard four node ABAQUS elements and one UEL

Fig. 10– First natural frequency ratio versus the crack depth ratio for cantilever beam with surface crack at 20% of beam length from fixed end predictions obtained using the damaged finite element by Potirniche et al.7, when compared with the experiments results by Wendtland17. Simply supported beam with a surface crack

An aluminum beam4 having the length L = 0.235 m, height H = 0.0254 m, and thickness t = 0.006 m with elastic modulus E = 7.2 × 1010 N/m2 and the density ρ = 2800 kg/m3, is considered. Typical FEM discretization with 8 standard four node ABAQUS16 elements and one UEL for c/L = 0.5 is shown in Fig. 11. Figure 12a shows the first natural frequency ratio (ωc/ω) versus the crack depth ratio (α/H) for simply supported beam with a surface crack at mid-span. Compared with the predictions obtained using the damaged finite element by Potirniche et al.7, the first natural frequency reduction predicted by the


103

Fig. 11– Discretization of simply supported beam with surface crack using 8 standard four node ABAQUS elements and one UEL

Fig. 12– Variation in natural frequency reduction of simply supported beam with surface crack at mid-span (a) first mode with surface crack at mid-span, (b) second mode with surface crack at mid-span and (c) third mode with surface crack at c/L=1/3

proposed improved two-dimensional finite element matches very well with the experiments results by Chondros et al.4

For the simply supported beam case considered above, FEM discretization with the height of UEL equal to the beam height is adopted. In the following, the effect of number of elements along the direction of the beam height on the accuracy in predicting the reduction in the natural frequencies of higher modes is studied.

Noting that if the crack location coincides with the vibration node of one of the modes, the frequency for that mode remains almost unchanged; Fig. 12b shows variation in the natural frequency reduction of second mode of simply supported beam with a surface crack at mid-span with respect to the number of elements along the direction of the beam height, predicted by the proposed improved two-dimensional finite element for the various values of the crack depth ratios (α/H). Similarly, Fig. 12c shows the variation in the natural frequencies reduction of third mode of simply supported beam with a surface crack located at c/L = 1/3 measured from the support, with respect to the number of elements along the direction of the beam height. It can be observed from Figs 12b and 12c that, by adopting two elements along the direction of the beam height, better accuracy can be obtained in the prediction of the reduction in the natural frequencies of higher modes, which are essential for the predicting the crack location and size. Hence, in the crack identification technique presented in the subsequent sections, two elements along the direction of the beam height are adopted and when the crack depth is equal to the height of UEL, the element stiffness is assumed to be zero. Example 2

Owolabi et al.18 tested seven simply supported beam models with cracks. Each beam model was made of an aluminum bar of cross-sectional area 0.0254 m × 0.0254 m with length of 0.650 m. It had the following material properties: Young’s modulus E = 7 × 1010 N/m2, density ρ = 2696 kg/m3, Poisson ratio v = 0.35. In the numerical examples presented below, the same beam is modeled with 101 standard four node elements ABAQUS16 elements and one UEL at the top of the beam. Typical discretization of simply supported beam is


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shown in Fig. 13. In this example the influence of the plastic zone around the crack tip on the intensity of changes in natural frequencies is studied.

First, assuming that the plastic zone around the crack tip does not exist and the crack depth ratio (α/H) is varied from 0 to 0.5. Changes in natural frequencies obtained by applying the elaborated element are compared to results from experimental investigation of Owolabi et al.18 Figures 14a-14c present the frequency response functions of the first three natural frequencies obtained by using the proposed two-dimensional finite element based on elastic-plastic fracture mechanics, plotted in the form of frequency ratio (ωc/ω) versus the crack depth ratio (α/H) for crack located at a distance equal to 50% of the beam length starting from one of the simply supported ends. From Figs 14a-14c it can be noticed that the compatibility obtained between the results of the experimental investigation and the results of FEM analysis using the proposed two-dimensional finite element is excellent.

Figures 15a-15c present the frequency response functions of the first three natural frequencies, obtained using the proposed two-dimensional finite element based on elastic-plastic fracture mechanics, plotted in the form of frequency ratio (ωc/ω) versus the crack location ratio (c/L) for various values of σ/σY ratio, which describes the size of the plastic zone around the crack tip. For these cases the crack depth equal to 0.1 of the beam height is taken.

Figures 16a-16c present the frequency response functions of the first three natural frequencies, obtained using the proposed two-dimensional finite element, plotted in the form of frequency ratio (ωc/ω) versus the crack depth ratio (α/H) for various values of σ/σY ratio. Noting that if the crack location coincides with the vibration node of one of modes, the frequency for that mode remains almost unchanged; for these cases the crack located at a distance of 25% of the beam length starting from one of the simply supported is considered.

Fig. 14– Natural frequency ratio versus crack size for simply supported beam with crack at mid-span (a) mode 1, (b) mode 2 and (c) mode 3

Fig. 13– Discretization of simply supported beam with surface crack using 101 standard four node ABAQUS elements and one UEL


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Fig. 15– Natural frequency ratio versus crack location for simply supported beam with crack depth ratio α/H=0.1 for various values of σ/σY ratio (a) mode 1, (b) mode 2 and (c) mode 3

It can be observed from Figures 15a-15c and 16a−16c that the fall in natural frequencies depends not only on the crack location and depth but also on the size of the plastic zone around the crack tip. Similar observations were reported by Krawczuk et al.9,10

Fig. 16– Natural frequency ratio versus crack depth ratio for simply supported beam with crack at quarter-span for various values of σ/σY ratio (a) mode 1, (b) mode 2 and (c) mode 3 Conclusions

A novel two-dimensional finite element with a single, non-propagating, open embedded edge crack is developed on based on elastic-plastic fracture mechanics. The FRANC2DL finite element code is used with the J-integral option to extract the stress intensity factors from stress strain fields around the crack tip location. The geometric factors for various loading cases of the cracked element for crack depth ratios ranging up to 0.9 are obtained by means of curve fitting techniques; and they are subsequently used to


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obtain the components of the stiffness matrix for the cracked element from the Castigliano’s first theorem, using based on elastic-plastic fracture mechanics concepts. The model takes into account the influence of the plastic zone ahead of the crack tip on flexibility of the element. The compliance matrix singularity was avoided by employing Castigliano’s first theorem, which states that the applied forces can be obtained by taking the partial derivatives of the strain energy with respect to the corresponding displacements. The element is implemented in the commercial finite element code ABAQUS as UEL subroutine. Numerical examples presented shows that the crack tip plasticity increases the predictive capabilities of the model. The proposed element is applicable only for non-hardening materials (i.e. pure elastic-plastic materials).

Acknowledgement The first author would like to acknowledge the

fellowship of the Tata Consultancy Services during course of this research.

Nomenclature c = distanc of crack from end of beam E = modulus of elasticity Fi = nodal force

Gf = geometric factor for single edge notch strip under tension loading

Gm = geometric factor for single edge notch strip under bending loading

GI = mode I geometric factor for vertical nodal force GII = mode II geometric factor for vertical nodal force h = height of element H = height of beam KI , KII = stress intensity factor for mode I and mode II Sij = components of stiffness matrix L = length of beam t = thickness ui = nodal displacement w = width of element α= crack depth v= Poisson’s ratio ρ= density ω = natural frequency of uncracked beam ωc= natural frequency of cracked beam

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