Numerical and experimental investigation of inclined cracks in ship plating

NUMERICAL AND EXPERIMENTAL INVESTIGATION OF INCLINED CRACKS IN SHIP

ABSTRACT

Cracks originate in ships and offshore structures in arbitrary directions. Occurrence of inclined cracks lead to a mixed mode problem of fracture mechanics. The paper deals with the propagation of inclined cracks in fatigue

loading. The analysis of t is attempted with the help of ANSYS software. Numerical treatment of the paper is

limited to the application of the displacement extrapolation

method and the stiffness derivative method. Whereas the former is inhuilt in the ANSYS software, the latter is implemented in it. While so doing considerable effort has been put in by writing macros to reduce drastically the

manual work involved in the calculation. Numerical results

have been presented for plates with inclined cracks with a

number of varying parameters under tension at two opposite edges. Experiments have been conducted on cracked plates

with inclined cracks under inpiane cyclic tensile loading. Comparison for the position of the crack tip with the

number of cycles has been made between theoretical and experimental curves.

1. INTRODUCTiON

Occurrence of cracks is rather common in offshore

structures and ships. Structures made of high strength materials may often have low margin of safety. As such in service stresses may be high enough to induce cracks particularly if preexisting flaws or high stress concen-trations are present. Due to a continuous application of cyclic loading resulting from the waves, these cracks develop a tendency to propagate. In ship structures they may originate in arbitrary directions. In order to study the

integrity of the structure, the analysis of crack initiation and subsequent crack propagation is of paramount ¶mportance.

Deift University of Technology

i

Ship Hydromechanics Laboratory

Library

Mekelweg 2, 2628 CD Deift

The Netherlands

PLATING

Jacobus H. VINK Madhujit MUKHOPADHYAY Bart BOON

Ship Structures Laboratory, Faculty of Mechanical Engineering and Marine Technology Deift University of Technology, Delft, The Netherlands

(on leave from Indian Institute of Technology, Kharagpur)

Traditional methods for dealing with cracks are concentrated on the application of fracture mechanics in

opening or Mode J failure mechanism. But experience has

indicated many failures occurring from mixed mode

loading. Randomly oriented cracks are frequently observed

in variou', types of ships.

The paper deals with numerical and experimental investigation of propagation of inclined cracks in ship plating. The inclination of the crack with respect to the

loading axis results in a mixed mode problem. The work in

this area has been reviewed in the literature [1]. But the

problem is far from being solved. For example, the

maximum tangential stress and the maximum strain energy

density criteria are extensively used in mixed mode crack growth problems for determining the direction of the crack

propagation. However, experimental results do exist which do not pertain to those criteria [21.

The general purpose finite element software ANSYS has been used in carrying out the numerical analysis of fatigue

crack propagation in mixed mode. ANSYS provides the numerical values of stress intensity factors based on the displacement extrapolation method. One of the objectives of the present investigation is to implement the stiffness derivative method in ANSYS. While so doing attempts have been made to re.strict the manual actions for per-forming the calculation to a hare minimum.

Experimental investigation on plates with mixed mode problems are not many [I]. Experiments have now been

conducted on rectangular plates subjected to cyclic tensile

loading. The inclination of the crack in the plate is varied. Comparison has been made between the theoretical and

experimental curves obtained from fatigue testing.

2.. FORMULATION

Zi Mixed Mode Stress Intensty Factors

Two methods have been applied for the determination of

mixed mode stress intensity factors with

the help of

ANSYS software. They are: the displacement extrapolation method and the stiffness derivative method.

The method of extrapolating the displacement along the crack face is one of the earliest approaches for determining the stress intensity factor using the finite element technique

[3,4,51. Mixed mode stress intensity factors can be obtained

from ANSYS by this method by executing the appropriate commands. lt will not be further discussed here. In the

literature it is suggested that

this method is not very

accurate.

The stiffness derivative method is well documented [6]. It has been applied to the mixed mode problem by various investigators [T Il]. The implementation of the method in

ANSYS is presented here. Whereas it originally would have

taken a large number of steps for the application of the stiffness derivative method in the software, the relevant macros can now be activated with one command. The

salient

features of the discussion are centered on the

curtailment of steps of calculations as a computational

strategy. First, the stiffness derivative method is discussed

briefly.

The energy release rate for the extension of the crack is given by:

G= -(a(qaa) = _{F61Ta1s1IaaH81} (1)

where [ô] is the displacement vector, [S] is the stiffness

matrix, a is the crack length and U is the potential energy.

In order to apply the method, [Ô] is determined for a crack length a and the stiffness matrix is evaluated. The

method is also known as the virtual crack extension method

as the crack length is increased by a small amount n the

plane nf the crack and a new stiffness matrix is evaluated. The stiffness derivative is written approximately as

as

(1[1)

₍₂₎

8a ôa

Combining equations (I) and (2) yields the value of G. lt may he noted that it would not he necessary to change

all the elements in the mesh. The crack growth can be

accomodated by moving the crack tip node thus affecting

the elements at the crack tip only and leaving the rest of

the mesh intact (Fig. 1). Instead of moving the crack tip

only, the first row of elements surrounding the crack tip

within A0 in Fig. I is given a rigid body displacement and

the elements outside the region A1 are not moved. This

approach is more accurate as the theoretical crack tip

stresses are infinte. The change in the elastic energy will be (lue to the deformation of hands between A0 and A1. The

possibility of choosing different rings as flexible makes the approach more general.

For a single mode fracture, the desired stress intensity

factor K1 is obtained from the relation

G=aK/

(3)

where ai= H= E/(l-v2) for plane strain and H = E for

plane stress.

For the determination of the individual stress intensity

factor in the case of mixed mode loading, conservation of the energy in the cracked elastic body is considered 181. Total energy release rate in mixed mode crack extension is given by

1= G, .4G,,= G

(4)

where the J-integral has been defined by Rice [121. As JG in equation (4), all computations have been performed with G and not J. G = a (K,2 + (5) Therefore,

G, = a K7 and G,,

a K

2 (6) r'

A combined equilibrium state, designated by (0), is

consi-dered now, as resulting from the superposition of two

independent equilibrium states denoted by superscripts (1) and (2). The superimposed state is written as

Gt0t = G"

G2 .

(7)

Combining equations (6) and (7) leads to

= 2 aI K»t j(2) + K,J'1 K,2) J (8)

Solution of the individual stress intensities K1 and K11 for

state (1) with the help of equation (8) is possible in two stages. First, the theoretical solution of a cracked body

subjected to mode I deformation only is used as the second state denoted by (2)

and J(J(2) _{= 0 (9)}

For this auxiliary second state, displacements can he

calculated from r 9 9

cos - (I - 2î+sin2)

2 (IO) 1,12)

G\ 2ir

2 ¿

r

_{cos (22j-cos2}

9 6 )

where r =v for plane strain and r = v/( I + y) for plane stress.

Equation (X) simplifies with the help of equation (9) into

= 2aK»

(12)

In order to solve K1") it is required to determine the strain

energy release rate with the help of the ring for the following cases

0°' = strain energy release rate for the combined case (1,2) 0°' = strain energy release rate for the original mixed mode

case,

= strain energy release rate for the theoretical

displace-ment field.

A similar procedure is adopted in the evaluation of K,,") with the help of the theoretical solution for a cracked body

under mode 11 deformation only.

3. IMPLEMENTATION IN ANSYS

The above procedure of the stiffness derivative method is

implemented in ANSYS as follows.

A circular ring of 8 noded quadrilateral elements is gene-rated around the triangular crack tip elements (Fig. I). This

ring has been assumed as the deformahie ring. The six

noded triangular cracktip elements have their nodes at the

radial sides shifted to quarterpoint positions, (Fig. I). The

mesh is generated by ANSYS and the number of divisions

is specified for the ring. The plate is loaded in its own plane. For the calculation of K,, five identical deformable rings are copied from the main ring one after another with

the node numbers and the element numbers incremented for

each ring. Two consequtive rings are needed for any

particular displacement for the evaluation of the differential

quantity given by equation (2). The evaluation of K,

involves three displacement cases as per equation (7). The

first one pertaining to the loading of the plate and the

displacements of the flexible ring are as obtained from the finite element solution and the strain energy release rate for this case is given by G" in Fig. 2. The displacements at the

outer and inner radii for case (1 ,b) are kept same as the main ring for case (I,a) with the help of constraint equations. Except the original ring, the modulus of elasticity of all other rings are drastically reduced so that the overall solution is not affected.

The next case indicated in Fig. 2 involves forced

displacements which are equivalent to the theoretical

dis-placements at the nodes of the flexible region as given by

equations (10) and (11). The strain energy release rate for

this case is given by 02). Finally, the combined case indicated by (0) in

Fig. 2 yields G''. Once M"" is

determined, K, can be obtained from equation (12).

In fact for the determination of K, the value of G"

remains the same, but four more rings are needed as

mentioned above, as the equations for the equilibrium state

for the case with K,12 =0 and K,," = I will be different

from equations (10 and Il). As the relevant university

release of ANSYS 5.3 is restricted to a maximum of 16000

nodes, the addition of four more rings to the five as given

in Fig. 2 will exceed the limit. As such the evaluation of

two stress intensity factors for mixed mode crack propagation has to be done in two stages due to the limitation of the memory requirement. Suitable MACROs have been written to implement the calculation scheme in the ANSYS software. Had this scheme of calculation not

been adopted, the whole analysis would have required the

solution of each ring separately, the time and effort of which would have been niuch more than the whole

computation done in two stages.

4. ANALYSIS OF FATIGUE CRACK PROPAGATION

There are two major aspects in mixed mode fatigue crack

growth: crack growth direction and crack growth rate. Various criteria and parameters have been proposed for both [l}.

The maximum tangential stress criterion due to Erdogan

and Sih 1131 has been adopted here for the sake of

simplicity and support from many experiments. As per this

criterion the crack propagation starts from the crack tip along the radial direction O=O on which the tangential

stress 0, becomes maximum (Fig. 3). Mathematically, this

crack growth direction is given by

a'i7i,

<0

(13)

30 302

Substituting Westergaard's expression for On, the following equation for the crack growth results [141

K,sin8 +K,,(3cos0- 1) = 0

(14)

The solution of equation (14) is given by

sinO

c-3 si 8s2

(15)

1-9s

where s = K,,/K1, and with K,, > O the crack will deflect

downwards.

Specimens tested are rectangular plates with a central inclined crack subjected to tensile cyclic loading (Fig. 4).

It may be noted that though the maximum tangential stress criterion has been adopted for the crack growth direction in

this investigation, any of the other crack growth criterion

mentioned in Ref I could also have been attempted as well.

However, some of them require data which are not

available from the present calculation method.

For mixed mode loading the fatigue crack growth rate

equation proposed by Tanaka 121 has been adopted

r C(K,.,)m

(16)

where for the combined mode I and mode 11 loading is expressed as

and C and m are material constants.

Once K1 and K11 have been evaluated for a plate with a predetermined crack, can be calculated. Next, the number of cycles ¿N needed for a crack extension ¿a is

given by

AN-

¿a

(

AN can be summed tip for the successive extensions of the crack, and after calculation of the crack growth direction, the new cracked configuration can he evaluated. The steps

involved in the computation of the fatigue crack

propagation are now as follows:

L For a predetermined crack, K1 and K11 are calculated. ¿Kff is calculated from equation (17).

Using the C and m values of the material, calculate for that crack propagation from equation (16).

Calculate the number of cycles for the crack extension AN from equation (18) for an extension of ¿a.

Add ¿ N to the number of cycles already calculated.

Based on K1 and K11 values, calculate O for the direction

of the crack extension for next iteration from equation (14). Evaluate the new crack tip positioi.

MESH GENERATiON FOR PLATES W[TH CRACK PROPAGATiON

A MACRO has been written

with the objective of

analysing the plate for the crack propagation by iterating on

the extension of the crack. For this, key points are to be

selected ¡uditiously for the model so that for the extended crack the key points are automatically' generated (Fig. 5). The niesh density at the four corner key points numbered

I to 4 in Fig. 5 are specified and were kept constant for the

entire range of the crack propagation. The remaining is left

to the discreti(on of the automatic mesh generator of

ANSYS.

The plate is divided into two equal halves by a vertical

plane at half width and only one half of the plate

is

analysed in view of the economy of the solution. For this

purpose appropriate boundary conditions are ptit in the

model: the horizontal and the vertical displacements at the vertical centre line of the upper half of the model arc equal

in magnitude, but opposite in sign to its lower half. The

horizontal displacement at the vertical centre line at the top and the bottom of the plate are restrained (Fig. 6).

8noded isoparametric elements, denoted PLANE82 by ANSYS. have been used along with 6-noded isoparametric singular elements at the crack tip. A typical mesh is shown in Fig. 7.

TEST SPECIMEN AND EXPERIMENTS

Two specimens given in Fig. 4 have been tested with 0 30 and 45 degrees respectively. They are subjected to

uni-(18)

axial cyclic loading varying within the maximum and

minimum values of tension loading. The plates are made of steel which conforms to the criteria specified by the Lloyds Register for the normal strength steel of Grade D.

The specimen is put in the grips of a 1000 kN MTS

Testing Machire. A software has been used to control and monitor various details

of the testing. They are

the

frequency, amplitude, mean value and their variations.

Length and direction of the crack were measured by visual inspection. In addition the crack length was monitored by

means of direct current potential drop measurement and

printed at intervals. Furthermore, this crack monitoring was used for automatic crack front marking at predetermined crack lengths to enable an accurate correlation of the crack length afterwards. The frequency of the loading was kept

at 6 Hz. A sketch of the experimental details is given in

Fig. 8.

RESULTS

There are two parts

of this

investigation. First, a

rectangular plate, subjected to uniform tensile loading, is analysed for a number of varying parameters. The ratio of half length of the crack over width, a/W, has been varied frani 0.2 o 0.8 and the inclination of the crack was varied

from 0° in steps of 15° to 75° . Other variables which

control the mesh generator of ANSYS are the mesh density at corners of the plate, the radius of the circle at the crack tip and the number of circumferential divisions of the mesh

at the crack tip. K1 and K11 have been obtained both by the

displacement extrapolation method as well as by the

stiffness derivative technique. Results are compared with

the theoretical values for the particular plate [15,161. Values

of stress intensity factors for plates having a/W = 0.2, 0.5 and 0.8 are given in Tables 1-3 respectively. Subscript 'en' stands for the the values obtained by the stiffness derivative method; subscript 'th' refers to the theoretical values and the subscript 'kc' refers to those obtained by the displacement

extrapolation approach.

A careful look into the Tables 1-3 will immediately reveal

that the displacement extrapolation method appears to be as

accurate as the stiffness derivative method through the entire range of variables studied. Some of the variables

have less

influence on the computation of the stress

intensity factors such as the mesh density of the corner key points. The radius of the crack tip elements (delr) and the number of elements at 180° at the crack tip (nthet) have a

considerable bearing on the accurate evaluation of both

stress intensity factors. The numerical values obtained by

both these methods compare well with the theoretical

values for afW ratios = 0.2. 0.5 and 0.8.

Two specimens having dimensions as given in Fig. 4, but with different inclination of the crack, have been tested. As already mentioned, the crack extension was electronically

monitored. A siniiliarity has been observed with both

mens regarding the angle of propagation of the crack. From the experiment it has been found that the crack propagated

almost horizontally from the initial crack tip. In the

theoretical calculation the angle of propagation of the crack

with the extension of the crack tended to he almost

horizontal from the beginning of the crack propagation too. The values of C and m as used for the theoretical

calcula-tion of in Equation (16) are mean values and were

puhlisheïy Dijkstra l7]. Based on this calculation the

curve for the crack extension against the number of cycles

has been plotted and compared with the experimental curve in Fig. 9 for both initial crack inclinations. Fig. 10

gives crack paths for these cases. A careful look into these figures will immediately reveal that the agreement between the theoretical and experimental curves is reasonably well

for the entire range of the number of cycles. 8. CONCLUSIONS

The paper presents a theoretical and experimental

inves-tigation on the influence of inclined cracks on the fatigue

strength of ship plating. Two methods have been applied to

the determination of stress intensity factors for the mixed

mode loading: the displacement extrapolation method and

the stiffness derivative method. Based on a number of varying parameters such as the crack inclination, crack width, mesh density and the number of divisions at the

crack tip the accuracy of both the methods appear to be of

the same order. Numerical analysis has been carried out with the software ANSYS. An innovative approach has

been presented to drastically reduce the calculation steps of the stiffness derivative technique. Experiments of two plates

having inclined cracks with varying angles under cyclic

loading have been conducted. For these specimens tensile

forces were applied along two opposite edges. The propagation of cracks for all the specimens was almost

horizontal which tallied with theoretical prediction.

Maximum tangential stress criterion has been applied for

determining the angle of crack extension. Curves have been

plotted for the crack extension with the number of cycles.

It has been found that the correlation between the

theoretical and experimental values are reasonably good.

ACKNOWLEDGEMENT

Authors express their sincere gratitude

to Mr Ben

Buisman, Managing Director of the Ship Structures Laboratory for preparing the details of the experiniental facilities. They are also grateful to Mr Ruud Vonk, Mr Ton

Vredeveldt, Mr Jan Turkenhurg and Mr Marcel Zoutzeling.

REFERENCES

I. Qui, J. and Fatenii, A., "Mixed Mode Fatigue Crack

Growth: A Literature Survey, Engineering Fracture

Mechanics, Vol. 55, 1996, pp. 969-990.

2. Tanaka, K.. " Fatigue Crack Propagation from a crack

Inclined to the Cyclic Tensile Axis", Engineering Fracture Mechanics, Vol. 6, 1974, pp. 493-507.

Bank-SilI;, L., "Application of the Finite Element

Method o Linear Elastic Fracture Mechanics", Applied Mechanics Review, Vol. 44, 1991, pp. 447-461.

Carpenter, WC., "Extrapolation Techniques for deter-mining Stress rntensity Factors", Engineering Fracture

Mechanics,VoI. 18, 1983, pp. 325-332.

Lim, 1W., Johnston, 1W. and Choi, S.K., " Comparison between Various Displacements Based Stress Intensity Factor Computation Techniques", International Journal of

Fracture, Vol. 58, ¡992, pp. 193-210. ,,/t. (

₍

Parks, DM., "A Stiffness Derivative Technique forth Determination of Crack Tip Stress Intensity Factor",

International Journal of Fracture, Vol. 10, 1974, pp. 487-502.

l-Iarnoush, SA, and Reza Salami, M., "A Stiffness Derivative Technique to Determine Mixed Mode Stress

Intensity Factors for Anisotropic Solids", Engineering

Fracture Mechanics, Vol. 44, 1993, pp. 297-305.

Yau, J.F., Wang, S.S. and Corten, HT., "A Mixed-mode

Crack Analysis of Isotropic Solids using Conservation Laws of Elasticity", ASME Journal of Applied Mechanics,

Vol. 47, 1980, pp. 335-341.

Matos, P.P.L., Meeking, R.M., Charalambides, P.G. and

Drory, M.D., " A Method for Calculating Stress Intensities in Bimaterial Fracture", International Journal of Fracture, Vol. lO, 1989, pp. 235-254.

Chen, F.H.K and Shield, R.T., "Conservation Laws of Elasticity of J-integral Type", ZAMP, Vo!. 28, 1977, pp. 1-22.

II. Bui, H.D.," Associated Path Independent J-integrals for Separating Mixed Modes", Journal of Mechanics Physics and Solids, Vol. 31 1983, pp. 439-448.

¡2. Rice, JR., " A Path-Independent Integral and Appro-ximate Analysis of Strain Concentrations by Notches and

Cracks", Journal of Applied Mechanics, Vol. 35, 1968, pp. 379-381.

Erdogan, F. and Sib, G.C., "On the Crack Extension in Plates under Plane Loading and Transverse Shear", ASME Journal of Basic Engineering, Vol. 85, 1963, pp. 5 19-525.

Gdoutous. E.E., "Fracture Mechanics Criteria and Applications", Kluwer, The Netherlands, 1990.

IS. Rooke, D.P. and Cartwright, D.S.. "Compendium of Stress Intensity factors", HMSO, London, 1976.

Tada, H., Paris, P. and Irwin, G., "The Stress Analysis of Cracks Handbook", Del Research Corporation, Penn,

USA, 1973.

Dijkstra, OD. and Snijder, H.H.. "Fatigue Crack Growth Models and their Constants", IBBC-TNO rep.

Table I: Stress intensity factors for strips with slanted center crack

Table Il: Stress intensity factors for strips with slanted center crack

Table Ill: Stress intensity factors for strips with slanted center crack

Description: a = Half cracklength

(1h) = theoretical value theta r Crack angle (deg

(kc) displ extrapolator nthet = No of eIern at 180 deg

(en) r energy method deltafdelr = Cracktip shi9 ratio

w = Half width of strip delr Radius of crack tip elements

H = Half height of strip MD = Mesh density at corners

Constants:

W 199

Nr 200

Vink, Mukhopadhyay, Boon

a/W = 0.50 theta = 45 degr.

deli MD nthetl K1-th ModelK1-kc Kl-en If K2-fh ModeLK2-kc K2-en

00060 010 12 7669 7671 7719 6841 68-42 6900 00075 010 12 7669 7677 7672 68-41 6834 6893 00100 010 12 7669 7674 7721 6841 5937 5894 00150 010 12 7669 7677 7722 6841 6825 6885 00200 010 12 7669 7679 7723 6841 6821 6882 00250 010 12 7669 7681 7724 6841 6820 6882 00300 010 12 7659 7682 7723 6841 6818 6880 00350 010 12 7669 7682 7723 6841 6816 6880 00400 010 12 7669 7682 7723 6841 6814 6879 01006 010 12 7669 7694 7719 6841 6796 6874 02000 010 12 7669 7726 7712 6841 6771 3868 02 020 8 7660 7726 7712 6841 6771 6911 00050 040 8 7669 7463 7565 6841 6814 6940 00050 040 12 7669 7608 7672 6841 6837 6893 02000 040 4 7669 7726 8056 6841 6639 7151 0 2000 0 40 6 7 669 7 743 7 838 6841 6 730 6975 o 2000 040 8 7 669 7 729 7 760 6841 6 748 6900 02000 040 18 7669 7 718 7685 6841 6782 6853 a/W 0.80 nthet = 12 o.e e

deli MD theta K1-th K1-kc K1en K2 -1h K2-kc K2-en

00100 0 10 O 29 789 28 780 28 964 00100 0 20 O 28 789 28445 28 (552 00100 010 15 26 205 26007 26172 4 867 5 261 5 308 O 0050 020 15 26 205 25 796 25 980 4 867 5 143 5 200 0010G 010 30 19 737 19 712 19838 8 719 8 748 8 828 00100 020 30 19 737 19 653 19787 8719 8 885 8876 00100 010 45 12603 12622 12701 9 998 9 957 10042 00100 020 45 12 603 12587 12673 9 998 10020 lO 105 00100 020 60 6 151 6 166 6 205 8 703 8 715 8 786 00100 020 75 1 517 1 632 1 639 5 057 5 065 5 107 aJW = 0.20 nthet = 12

!Ail!AL1aAc1

00100 015 0 8 122 8114 8 166 01000 030 0 8122 7994 8062 o oioo 0 10 15 7 591 7 573 7637 1 990 1 999 2 007 00100 020 15 7591 7373 7620 199(1 1999 2017 00750 020 15 7591 7538 7589 1990 1993 2009 00100 030 15 7591 7485 7539 1990 2059 2078 00100 010 30 6127 6127 6154 3462 3462 3492 01000 030 30 6127 6067 6118 3462 3554 3576 00100 010 45 4 107 4 107 4 127 4020 4010 4056 o t000 030 45 4107 4042 4071 4020 4)46 4189 00100 020 60 2065 2035 2045 3501 3532 3560

o itj

030 60 2065 1 982 1 994 3501 3591 3621 00050 025 75 0556 0551 0552 2029 2058 2074 00100 030 75 0556 0526 0527 2029 2044 2063 00120 030 75 0556 0543 0517 2029 2048 2100 00100 035 75 0556 0543 0544 2029 2051 2068 00025 040 75 0556 0562 0567 2029 2124 2144 00100 040 75 0556 0531 0531 2029 2601 2077

C r

C k

Y

/\e

Fig

1: Crack Lip zone

X

Fig 3 Stress system near cracktip

11111111

_0000000000

₀₀₀₀₀

00000

(n =

-where: &nc

siraun energy iii ihr rung of ekmcnLs

u .Sone120 -

- -.

-6a (;ili -! 5riwb e&hi i'1 [fluai ion (7): GO r Gr r ri

Fig. 2: mpIementation o the lormulatitin

in ANSYS Ar 2 A.r I 12 loo -line nr Y - keypoint nr. Fig. 5 Keypoints and

Ijres o( cracked model

i FU«b Z, U,tU, VI U'u-U' ScneO01 - Sene' -- --- ---ôa a

Fig. 6. Boundary conditions

Original mcoiel

Copy ol ring

(opy ol ring

Copy rit ring

Copy cl ring

Copy of ong

'rrh loading

mirer r i rig h i

Ieri

Inner ring noi shifted

inner ong shrired

Inner ring noi

hiIied

Inner nng shifted

[Ala) = Emodrilua

Eli h)-E(l .i)c

F(.a)=E(L a)c E(Z.h)=El 3Cc E(Oa)=E(i u)e° E(Ub)=E(l a)e

C1Lu

C4L,bJ C.Ln'LQiU j,rc Loir1,ng cnditiori \auIyiica1r.se [ir K,'°=l K1..(J CQullunc:d i;i I Öfl 1ÛÛ Fig

4: Test specimen

Ii

i i i III Iv V uJ Sena" Sene1 Scrue°"

Fig. 7 A typical mesh 54 52 46 44 32 °20 IR ID r-'

-______,_____

-______,____

- 45 4

---,----

-

- 454 SIn

- -

- DOd

- - -

-

- 304 So"

-

---u

u

--

u

¶6 14 ¶0

t

0*0 .*4 OS'L.0 Lf0

*?Oft*uUCS c0*re0t ,,&*tIluØf L0*I*th. *C0*t

C0.F00t p&«I.

Fig. 8

Test set-up and measurement

1 D54ECT CURRENT _{i 'CEI. DROP} PER SO N COMPUtER

HIERFoCE

or

ÇJNCIION GENERRUON 0'O MEASURE(NIS

300 400 500 20 O ¶00 2130 CsI1O30

FIg 9: Crack growth complson

0

20

C,.d *,drloo X-Op roo)

Fig. IO: Crick path simulation

¶00

40

60