AlllDb 0 5 3 b 4 5

NBSIR 85-3038

DUCTILE TEARING STABILITY ANALYSIS

OF A SHIP STRUCTURE CONTAINING A

CRACK ARRESTER STRAKE

Reference

PUBLICATIONS

A. V. Clark, Jr. D. T. Read

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

January 1986

-QC

100

- U 5 6

85-3038

1986

NBS RESEARCH INFORMATION CENTER NBSIR 85-3038

DUCTILE TEARING STABILITY ANALYSIS

OF A SHIP STRUCTURE CONTAINING A

CRACK ARRESTER STRAKE

A. V. Clark, Jr. D. T. Read

Fracture and Deformation Division

Institute for Materials Science and Engineering

National Bureau of Standards

Boulder, Colorado 80303

January 1986

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

INTRODUCTION 1

Overview of the Interaction Between Crack Arrester Strakes

and Ship Structure 1

NBS Tests and Analyses 4

Problem Areas in Crack Reinitiation/Ductile Tearing 5

CRACK ARRESTER DESIGN METHODOLOGY 6

DUCTILE MATERIALS CHARACTERIZATION 7

Ductile Tearing Instability 7

Generalization of Stability Criterion 8

Relation Between dP/d6 , da/d 6 , and Tmat 11

STRUCTURAL MODEL (STATIC) 1 4

Modeling the Ship Structure With Springs 15

Simplified Model for Quasistatic Stability Criterion 16

DYNAMIC RELOADING FOLLOWING ARREST 18

STRUCTURAL MODEL (DYNAMIC) 19

Special Cases 21

General Case 24

DIRECTIONS FOR FUTURE RESEARCH 26

CONCLUSIONS 29

ACKNOWLEDGMENTS 30

APPENDIX 30

REFERENCES 32

iii

DUCTILE TEARING STABILITY ANALYSIS OF A SHIP STRUCTURE

CONTAINING A CRACK ARRESTER STRAKE

A. V. Clark, Jr., and D. T. Read

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

An analysis is presented for a structure made up of a crack arrester

plate embedded in a ship structure. The crack arrester material is

specified by its crack arrest temperature, its strength, and its tearing

modulus Tmat . The remainder of the structure is characterized as a set of

springs and lumped masses.

A stability condition is derived which states that the loaddisplacement curve of the structure as a whole must increase monotonically. An approximate quasistatic stability criterion sets a minimum

material tearing modulus value that depends on the structural stiffness.

Higher stiffness promotes effective crack arrest. A calculation including

dynamic effects requires forward integration of a set of differential

equations describing the fracture process and the motion of the structure.

Repeating such a calculation for different values of crack arrester widths

could indicate the minimum width needed for a successful (stable) crack

arrest.

Key words: arrester strakes, crack arresters, ductile fracture, ductile

tearing instability, structural instability.

INTRODUCTION

Overview of the Interaction Between Crack Arrester Strakes and Ship

Structure

Crack arrester strakes are the last line of defense in preventing

catastrophic failure of surface ships due to crack propagation. For

reasons of economy, current practice for many types of ships is to

fabricate the bulk of the ship structure (hull, decks, etc.) from materials

1
which can be thought of as brittle (low fracture toughness). To prevent

total failure of the ship structure due to crack propagation, arrester

strakes are welded into the hull and deck at critical locations. Current

practice is to install arrester strakes at the gunwale and at the turn of

the bilge [1], thus dividing the ship into four sections. It has been

recommended that at least two additional arresters be placed in ships with

a beam of 37 m (120 ft) or less and three additional arresters for ships

having a greater beam [2],

For simplicity, the ship structure can be idealized as a hull girder

in the shape of a box beam with varying section modulus (varying stiffness

and area moment of inertia). The hull girder is subjected to a variety of

in-service loads. However, the primary loads are those due to wave action

[1]. The primary hull girder loads result from the differences between

the dead load of the ship and buoyancy forces. The ship is assumed to be

balanced on a standard wave with the crest centered amidships (hogging) or the trough centered amidships (sagging). Either of those conditions

results in a bending moment applied to the ship. The bending moment

is used to calculate the longitudinal bending, or primary, stresses, once

the hull girder section modulus is known.

The primary stresses typically have a frequency on the order of a few

cycles/second, since they are associated with sea states. These stresses

may, under unfavorable conditions, cause crack propagation. Even if some other dynamic loading (accident, explosion, etc.) initiates crack propagation, the primary stresses still act as crack driving forces, since they

are applied quasistatically during the crack propagation event.

As the crack propagates through a structural element (deck, hull, etc.)

the compliance of that element rapidly increases. Because the primary

stresses are quasistatically applied, the mass of the element begins to

accelerate as the compliance increases. When the crack is arrested, the

inertia of the mass will cause a dynamic load to be applied to the

arrester strake. The mass of the cracked structural element has been

accelerating as the compliance increases; to decelerate the mass will

require a force to be exerted by the uncracked portion of the structure.

From the principle of action and reaction (Newton's third law), an inertial force will therefore be exerted by the mass on the uncracked

2
structure, and in particular, on the uncracked ligament of the arrester

strake.

The problem of crack arrester design is twofold. First, a suitable

arrester material must be chosen. Second, the physical dimensions

(length, width, thickness) of the arrester must be determined.

In order to certify a material as suitable for use in crack arresters,

it is necessary to show that the material has enough fracture toughness to

arrest a crack for values of stress intensity corresponding to the crackdriving force in an actual vessel. Because the crack arrest toughness

changes rapidly over a small temperature interval, this requirement

implies a requirement on the ductile-brittle transition temperature. The

material must also have a high tearing modulus so that it will prevent

failure of the arrester by ductile tearing instability in an arrestreinitiation event.

We will show later that the required value of tearing modulus for

stability depends upon the stiffness of certain elements of the ship

structure.

Current practice is to place crack arresters along the amidships 3/5

of the vessel [1]; this criterion determines the arrester length. For an arrester in the form of a plate, the thickness should be chosen to match

the thickness of the deck (or hull) in which the arrester is embedded;

otherwise, stress concentrations will occur due to discontinuity in deck

thickness. (It is assumed that the elastic properties of the crack

arrester and the deck and hull plating are the same). Consequently, the

only remaining design variable is the arrester width.

Obviously, the width must be such that the arrester can stop a running

crack; i.e., w > a

Q where w = arrester width, a

Q

= crack length at arrest.

This inequality is merely a lower bound an arrester width, however.

Because of dynamic reloading of the remaining ligament, the possibility of

reinitiation and consequent failure by ductile tearing exists; the

arrester width must be large enough to prevent failure by ductile tearing

instability.

3
NBS Tests and Analyses

A previous report documented a test methodology developed at NBS in

support of efforts for certifying new materials for use in crack arresters

[33. A novel specimen design was developed which allowed a crack to run through a brittle material and then encounter a step in fracture toughness. This simulates (locally) the situation in an actual vessel where a crack runs through brittle plating in the deck (or hull) and is arrested

by a tough ductile arrester strake.

Furthermore, the NBS tests were designed to simulate the dynamic

reloading which would be expected to occur in a vessel upon crack arrest.

This was accomplished by using a spring-loaded double-cantilever beam

(DCB) as the specimen design [ 3 ]; see figure 1. This configuration allows

large crack lengths to be achieved using relatively low loads. The spring

in the load train prevents large load drop as the crack runs (and specimen

compliance increases). It also provides a means for dynamic reloading of

the remaining ligament after arrest has occurred. This simulates, in a qualitative sense, dynamic reloading that would be experienced by an arrester strake in service.

In [ 3 ], the response of the DCB specimen was calculated during the

crack run event on the assumption of negligible ductility. After arrest,

and upon reloading, extensive plastic deformation occurs; the strain rates

and crack velocity are now much lower. It was assumed that fracture

parameters such as the J-integral and the tearing modulus are still valid

during the reinitiation/ductile tearing event.

Two methods were used in [ 3 ] to calculate J. In the first method,

a quasistatic analysis of the DCB specimen was performed assuming a plastic hinge had developed upon crack reinitiation. The J-integral was split

into an elastic and a plastic part:

where J

e

= K

2 /E, K is the stress intensity factor and E is Young's

modulus. Both J

Q and J

p

depend upon the crack length, a(t), in a complicated fashion; it is necessary to know a(t) from test records to calculate

J with this approach.

4
The second method uses a lumped-mass viscoelastic-plastic model to

account for the mass and stiffness of the specimen plus load train; see

figure 2. It also accounts for elevation of the yield stress due to strainrate effects. The effect of crack propagation is modelled by a sliding

frictional device (shown in fig. 2 as P

s ) which slides freely when the

load in the specimen exceeds the limit load, P^, and is rigid when the

load is less than P

L

. The limit load is also a function of the crack

length.

In [ 3 ], the limit load was calculated on the assumption that a(t) =

a

Q ; i.e., the crack length was assumed to be the same value during tearing

as at arrest. (Obviously, the measured value of a(t) could be input to

the model of [ 3 ] if desired.) Note that the model represents (qualitatively) the situation in a ship. The mass M and stiffness k represent the

effective mass and stiffness of the structure external to the arrester,

which has its own mass M

s and stiffness ks

. The motion of the lumped-mass model was calculated and then used to

determine a dynamic J-integral and applied tearing modulus. The resulting

values of J were believed to be too large [ 3 ], leading to questions of the

validity of this model.

Problem Areas in Crack Reinitiation/Ductile Tearing

From the discussion, it is seen that several problems exist with the

approaches taken to model the crack reinitiation/ductile tearing event:

(1) The quasistatic approach neglects dynamic effects as a possible

crack-driving force.

(2) The lumped-mass model qualitatively models the effect of the ship

structure by a single mass and spring in series with the arrester.

A method is needed to determine the actual values of the effective

mass and stiffness of the ship structure.

(3) Both models require the crack length to be input as a function of

time. There is no way of calculating a(t).

(*0 Both models have J(t) as output. Dynamic J has not yet been shown

to be a valid fracture parameter [ 3 ].

5
This report summarizes recent research which attempts to overcome the

four problems listed above. To deal with (1) and (2), the interaction

between the arrester and the ship structure is considered. There must be

continuity of displacements and tractions at the interface between

arrester and structure. This fact is used to show how the effect of the

inertia and stiffness of the structural elements of the ship (e.g. hull,

deck) should be incorporated into a simple model consisting of masses and

springs.

To solve problems (3) and (4), a tearing modulus approach is used. It

is assumed that the material's tearing modulus, T

t

( a ), is known. Under

certain conditions (explored later in this report), knowledge of the

tearing modulus, coupled with a calculation of the response of the system

of masses and springs representing arrester and ship, allows a calculation

of a(t) to be made. Obviously, if a(t) > w at any time, the arrester has

failed. What is desired in the arrester design is that a(t) < w when the

crack driving force vanishes (or becomes constant).

CRACK ARRESTER DESIGN METHODOLOGY

We assume that a material has been chosen which will arrest a running

crack. The material must have sufficient fracture toughness to force a brittle-to-ductile fracture mode transition in order to arrest the crack

at the minimum service temperature. It must also be sufficiently tough to

arrest a crack which may be reinitiated upon dynamic reloading of the

uncracked arrester ligament.

As mentioned previously, the design task is to size the arrester.

In particular, the width of the arrester must be chosen so that any

reinitiated crack will be arrested inside the arrester. Failure to arrest

a reinitiated crack will result in the crack propagating from the arrester

into brittle regions of the ship structure with potentially catastrophic

consequences.

A procedure for crack arrester design is shown in figure 3. The

designer chooses an initial crack arrester width, w, for the arrester

material. The material should be characterized by its tearing modulus,

and its load-displacement curve during tearing.

6
A simple dynamic model for the crack run-arrest event has been

previously developed by NBS [ 3 ]. The outputs of the model are crack

opening displacement, 6, and its time derivative, 6 (dots over a quantity

will denote time derivatives in this paper). This model works reasonably

well for fracture specimens which are double cantilever beams. It is

assumed in this report that similar calculations have been done for the

ship structure to estimate 6 at arrest (6=0 at arrest by definition). The

arrest value of 6 becomes an initial value for the dynamic ductile tearing

calculations.

The structural model referred to in figure 3 pertains to the actual

ship structure in which the arrester is embedded. It should include the

effects of the inertia and stiffness of the hull and deck, since these

structural members interact with the arrester. The internal friction

(damping) of these structural elements must also be included.

Using all the above inputs, the dynamic tearing model calculates crack

length, a(t) and COD, 6, upon reinitiation. If 6 and a are bounded (in

particular, if w > a), then the system is stable; i.e., the arrester has

stopped the crack. In this case, the design is adequate. If the system is

unstable (w < a), then the arrester width w must be increased (or a new

arrester material chosen). The process described above is then repeated

until a stable design is found.

In this report, we will develop a structural model for the shipstructure-crack arrester interaction. We will develop the dynamic tearing

model, and show how tearing instability is governed by dynamic crack

driving forces, material tearing modulus, and structural stiffness.

DUCTILE MATERIALS CHARACTERIZATION

Ductile Tearing Instability

The simplest case of ductile tearing instability, that of a fracture

specimen in series or parallel with a spring, has been considered by Ernst

[4], and Kaiser and Carlsson [5]. By requiring that the total load

carried be montonically increasing (specimen and spring in parallel),

Ernst derived the stability criterion

7
-dP/d 6 < k ( 1 )

where P = load carried by specimen and k = spring stiffness. For the

specimen and spring in series Ernst found that the stability criterion (1)

could be derived by requiring that the total displacement be monotonically

increasing under load.

In this section, we generalize these results to any combination of

springs in series and/or parallel with the specimen. We do this by

showing that the stability criterion can be derived from a fundamental

principle, conservation of energy.

We note that the instability criterion (1) only holds for the case of

quasistatic loading. In fact, Ernst [4] points out that eq (1) can be used

to predict only the onset of instability, and not the end of instability;

to predict the latter, inertial effects must be accounted for. In fact,

it is the inclusion of these inertial terms that lead us to the formulation of our dynamic model.

Generalization of Stability Criterion

For simplicity, we begin with the case of a fracture specimen in

series with a linear spring. This system is subjected to an applied load,

P 0

; see figure 4.

For a quasistatically applied load, the increment of external work

applied (W

e

) to the system is equal to the increment of work (W^nt ) represented by the area under the specimen's P-6 record. The latter work

actually involves three terms: (1) elastic strain energy, (2) plastic

work, (3) energy to create new surface as the crack propagates in the

specimen. Obviously, the last two terms are irreversible. Energy conser- vation requires that

MWe- Wlnt ) - 0 (2)

where A denotes a small increment.

For simplicity, we assume that the effect of bending in the specimen

can be neglected due to constraint by the surrounding structure. Also,

we assume the arrester material is sufficiently tough so that a 0 <<w.

8
Then the crack opening displacement 6 is approximately the same as the

crack mouth opening displacement; i.e., there is no rotation of the

specimen.

For a monotonic increase in 6 (no elastic unloading/reloading), the area under the P -6 curve is path independent; consequently, the work done

on the specimen (W

g ) is

6 W

s

= P( 6 ) d 6 (3) o

The total work Wint done on specimen and spring is just Ws plus the strain

energy k(U- 6)^/2 stored in the spring where U is the displacement of one terminal of the spring. The external work We is just P eU. Consequently,

( 2 ) gives

A < W

e

' wint) " 3 < we" Wint )AU + 3(We

- Wint ) A 6 = 0 (4)

3U 36

Since Au and A 6 can be varied independently, we require that

3(w

e

- wint) = 0 . 3( we

“ wint) = 0 (5)

3U 36

Taking the derivatives gives the two equations of equilibrium

P e

= k (U- 6 )

P = k (U- 6 )

for the spring and specimen, respectively.

Furthermore, we note that We does not depend upon 6 . This allows us

to consider whether the energy stored in the specimen-spring system is at

a maximum (unstable) or a minimum (unstable). For stability, we require

that

( 6 a)

( 6 b)

^ ^int ~ 0 (7a)

36

9
(7b)

We note that the requirement

( 8 )

leads to the condition k £ 0, which requires a positive spring constant.

which is the same as the criterion (1) derived by Ernst.

A simple physical interpretation can be given to the stability condition. Suppose that the fracture specimen in figure 4 is an isolated crack

arrester strake containing a crack. Suppose also that a point mass M is

placed at the terminal which connects the spring to the arrester; see figure 5. As the crack grows in the arrester, the load P carried by the

arrester drops (dP<0). The drop in load, dP, would accelerate the mass to

the right in figure 5. However, as the load drops in the arrester, the COD

5 increases by d6, thus compressing the spring. Consequently, the spring

exerts a reaction force kdS on to the left in figure 5. If the spring is

stiff enough, it will restrain any acceleration of M; the condition for

this restraint is just the stability condition, eq (9).

This analysis can obviously be extended to more complex situations.

An example is given in the Appendix, where a fracture specimen has both a spring in series and in parallel. The example models (approximately) the

quasistatic tearing of an arrester placed in the hull of a ship. It is

interesting to note that for the example in the Appendix, the spring in

series with the arrester controls the stability of the system, rather than

the spring in parallel.

Since W

in^

= / Pd6 + k (U-S)^/^, the stability condition (7b) requires

that

dP/dS + k > 0 (9)

10
A typical set of P-6 curves is shown in figure 6, with initial crack

length aQ as a parameter. Obviously, as a0 increases the maximum load

carried by the specimen decreases. We are concerned with the falling

portion of the curve, for which dP/d6 < 0. If the slope becomes too large

(in the negative sense) as the crack advances (6 increasing), instability

may occur. Obviously, for the quasistatic case, it is necessary to choose

an initial aQ small enough that when the load is applied, -dP/d6< k.

There are (at least) two possible ways of generating P-6 curves. One

is to make specimens of the appropriate geometry with different initial

crack lengths, load them to failure, and measure load vs. COD during the

test. Another method makes use of the material's tearing modulus, Tmat>

as outlined in the next section.

Relation Between dP/d6, da/d6 and Tmat

The material's tearing modulus, Tmat , is a measure of resistance to

ductile tearing, and is defined by [6]

T

mat ( d,J mat) . da ( 10 )

where o Q is the flow stress.

Since J is a function of both crack length a and COD, 6, we have

T

mat [(— ) + (— ) (—)]

3a 6 36 a da (ID

showing that the rate of crack growth, da, can be related to the increment

of COD, d6 , if the functional form of J is given, and if Tmat is known.

In like fashion the load carried by the specimen depends on a and 6, so that

<lv ( f>6 <at>

a

( 12 )

Solving for da/d6 from equation (11) and substituting into (12) gives
( 3P/ 3a ) ( <3 J / 3 6

)

6 a (13) d 6 36 a o mat

E

Ernst [4] has pointed out that the partial derivatives appearing in

eq ( 13 ) are "calibration functions", that is, they can be obtained from

finite-element calculations or from specimen tests. These functions do

not depend upon the material's resistance to crack growth; all the information about crack growth resistance is contained in T

In the most general case, it may not be possible to obtain closed form

solutions for the "calibration functions." However for some simple

geometries, it is possible to do so. A simple case is that of a crack

which has propagated across the deck of a ship and has been arrested by an arrester strake. For simplicity, it is assumed that the crack propagates

in a symmetrical fashion, so that the deck can (locally) be approximated

as a center-cracked panel (CCP) with inhomogeneous fracture resistance

properties; see figure 7.

It is assumed that the arrester strakes have arrested the crack, and

that the remaining ligament is fully plastic when the load is reapplied.

In this case, the load carried by the panel (deck plus embedded arrester

strake) is just the limit load, P^. For a CCP in plane strain we have [6]:

where w = width of deck, o Q

= flow stress, B = thickness. For this simple

case, we have, for the slope of the load displacement curve (or stiffness)

dP/d6

,

P L

= ao (w - a) B (14)

dP/d6 = -c0B ( da/d6

)

(15)

Note that this stiffness has a negative value as the crack length

increases

.

12
As pointed out by Paris et al. [6],

d6 = a dJ/oQ

= (a/o0 ) (dJ/da) da ( 1 6

)

where a is a constant. Combining eqs (15) and (16) gives

dP/d6 = - (oQ

2B/a) (dJ/da)' 1 (17a)

or, using the definition Tmat = (E/o0

2

) (dJ/da),

dP/d6 = - BE/aTmat (a) (17b)

We note that eq (17b) is equal to the value of dP/d6 predicted by

eq (13) for the case P = P L , a=1 and (3J/3a) (5 <<a0

2 Tmat (a)/E. We can show

that the inequality holds by the following argument'.

We use the well known relation [7]

6 J = -J

(3P/3a)

fi d6 (18)

o

The interval of integration can be divided into two regions: from 6=0 to

6 = 6nsy> and from 6= 6 NS y

to <5 = 6. Here 6 NSy

= value of 6 at net section

yield.

For 0 £ 6 £ ^ms Y

we are P er'f

> orming the integration, eq (18), in the

elastic region; in this region we have J

e

= K /E where J

e

= value of J in

the elastic region. For a center-cracked panel [8],

K

2 = a 2 ua sec (ira/2w) (19)

where a= remote stress. The above formula is a valid approximation for

a/w £0.9; for larger values, it over-predicts the value of K. Assuming

eq (19) is in its range of validity,

3J

( 3a

: ) =

6 (—

)

^ E

; sec t—

)

^2w ; tan ( g)] (20)

13
For example, if a/w = 0.9, the right-hand side of eq (20) becomes approximately 60tto 2 /E. For this quantity to be much less than (a 2/g) 7 ^ ma u requires Tmat >> ioo (o/oG ) 2

. As an example, we consider the case of HY-80 steel. The initial

tearing modulus (before tearing occurs) is about 100, dropping to about 40

after several millimeters of tearing [9]. Using the (conservative) value

of Tmat = 40 and setting o 0

= 550 MPa (80 ksi) for HY-80, the inequality

T mat >>100 (o/o0 ) 2 re Quires that the remote stress 0 be much less than

about 345 MPa (50 ksi). This requirement is almost certainly met for the case of quasistatic loading of the deck due to hogging and sagging as the

ship is loaded in bending by buoyancy forces due to wave action. Dynamic

reloading may cause an increase in the remote stresses, as will be discussed subsequently.

STRUCTURAL MODEL (STATIC)

Since the arrester strake is embedded in the ship, structural interaction between the arrester and the ship structure must be accounted for.

From the simplest design standpoint, the ship structure is a hull girder

of varying section modulus. Regarding loads applied to the ship, "...the

most important loading imposed upon a ship (is) the longitudinal bending

resulting from local weight-buoyancy imbalances over the length of the

ship" [1]. In addition, there are loads imposed due to ship motion in

various sea states, wave slap loads, etc. For the purpose of this report,

all applied loads will be treated as quasistatic. That is, they are assumed to be (approximately) constant during the crack run-arrest event,

and during subsequent reinitiation and ductile tearing. The only loads

assumed to act dynamically are inertial loads caused when the crack

arrests. More will be said about this later in the "Dynamic Model"

section.

As a first approximation to a structural model, we consider the hull

girder to be a box beam, as shown in figure 8. Current practice is to place

arrester strakes along the amidships 3/5 length of the ship [1]. The

strakes are placed at the gunwale and at the turn of the bilge, thus

dividing the ship into four segments [1]. For simplicity, only the

arresters at the gunwales are shown in figure 8.
It should be noted that even this simple approximation to the ship

structure is more complex than the configuration tested in [ 3]

It

should also be noted that this model is intended as a starting point for

incorporation of more realistic interaction between strake and ship

structure. It is anticipated that further refinements will be made to the

model based on inputs from the NAVSEA ship structures community.

The box beam in figure 8 is subjected to longitudinal bending stresses,

o, due to the weight-buoyancy imbalances which create hogging and sagging.

These are the primary stresses acting on the hull [1], and for simplicity

only these will be considered in this report.

Obviously, if one is concerned with modeling all structural details,

even using the box beam model would involve a fairly large computer code,

with elastic-plastic elements necessary to model the arrester. Such a complicated model would be accurate, but unattractive from a design

standpoint.

Modeling the Ship Structure With Springs

A simpler model can be developed by considering the arrester as a free

body, with constraint forces exerted upon it by the adjacent ship

structure; see figure 9. The axial stress o is due to the longitudinal

bending stress applied at the end of the box beam (shown in fig. 8) and

transmitted through the deck to the arrester. The shear stress t applied

along the interface between the deck and arrester is due to the tearing of

the deck. A free body diagram of the deck shows that a shear stress x must

arise to balance the remote stress o. Furthermore, there will be another shear stress x acting between the

arrester and the hull. As the arrester material flows plastically along

slip lines at ±45° to the crack axis, it tries to shear the adjacent hull

material. For simplicity, we show only the component of shear stress

acting along the axis of the ship. Since the slip lines are at ±45° to

this axis, there will also be a component of shear normal to the ship's

axis.

To generate the simple ship structure-crack arrester interaction model

we replace the effect of the distributed stresses with concentrated forces

15
applied through springs connected in series and in parallel with the

arrester, as shown in figure 10.

The springs k dp are (conceptually) connected to the arrester on either

side of the crack through rigid, massless bars welded to the arrester.

These springs represent the (concentrated) effect of the shear stress x acting between the arrester and the deck. One terminal of the spring has

displacement 5 (COD), the other has displacement U The latter is the

displacement of the deck along its centerline, and is due (in part) to

application of force Pp. This force is the (concentrated) effect of the

longitudinal bending stress o applied to the centerline of the deck. If

the deck has width L and the arrester width is w, then P p

= a(L-2w)B.

The displacement of the deck centerline, U

p

, will be resisted by the

hull girder in bending. This resistance is represented by spring kHp ; one terminal of k^ p is connected to "ground" or a position of zero displacement.

This position is the neutral axis of the hull girder as in shown in

figure 9.

In like fashion, there is a force P

g

= awB applied to the arrester

through spring k dS . This spring represents the stiffness of the deck

material in the region between the end of the arrester and the end of the

hull girder (recall that the arrester is placed in the amidships 3/5 of

the ship [1]). The deck material in question is shown in the region

marked "A" in figure 8. The displacement of the material in region "A" is

denoted by U_. This displacement is also resisted by bending of the hull

O girder, represented by the spring kHS in figure 9.

The effect of the shear stress x acting between the arrester and the

hull (shown in fig. 9) is represented by spring kH

in figure 10. Note that

kH and kHS are both connected to "ground", which is the neutral axis of

the hull.

It should also be noted that since we model the deck as a CCP, there

will be symmetry about the crack axis.

Simplified Model for Quasistatic Stability Criterion

The model in figure 10 can be simplified even further by recalling

that we are only interested in deriving a stability condition analogous to

eq (9). Consequently, the only generalized coordinate of interest is the

16
COD, 6. We also can make use of the fact that, for two springs k

1 and k 2

in series, the combined spring constant kj is given by

k T = ( 1 /k i + 1/k 2 )

1

( 22 )

which is analogous to the case of two impedances in parallel in electrical

circuit theory.

Since we are concerned with stability (as opposed to equilibrium), we can ignore applied loads P

s and Pp. The reasoning behind this statement

is as follows. Equilibrium at points having displacements Up, us , and 6 can be derived by setting 3W/3u

p

, 3W/3u s , and 3W/36 to zero, respectively;

here W = Wg

- W^ nt . These conditions give three equilibrium equations

For stability, we require that W^

n p assume a minimum, so we require

to zero. In calculating these derivatives, terms involving P

s and P will

obviously vanish since applied forces are not functions of the displacements.

Consequently, we can ignore the applied forces and combine the springs

k H p and k dp into spring k

p

:

The spring k^ is still in parallel with kH

, so the effective spring k

s in

series with the arrester is

p p

= kHP u

p

+ k dP ( U

p

-(S )

P s= kHS US+ k ds (Us~ 6)

P = _kHS <5 " k ds^ 6_us^

_ k dP ( 5_up)

(23a)

(23b)

(23c)

that 9 2w

i nt/3u

p

2

» s2w int /3US

2 and ^ 2wint / ^ <s2 a11 be Sreater than or equal

kp - [(1/k^jp) + (1/k^p)] ^ (24)

Likewise, springs k ds and kHS can be combined into k^:

(25)

(26)

17
Using the above results gives a greatly simplified model for stability

calculations, as shown in figure 11. Note that the coordinates Ug and Up are absent from the figure, having been essentially absorbed when the effective spring constants k

g and kp were calculated. Also, the stability

conditions 9^W^

n p/3Up 2

^ 0, S^W^^/SUg 2

£ 0 give the trivial requirements

to kpjp + k,jp£0, k,j S

+ k^g £ 0. Therefore, no loss of information results

in ignoring the coordinates U U

s in stability calculations.

For the model shown in figure 11,

W

int = < k

p

+ k s> (<$/2) 2 + /Pd6 (27)

so that the stability criterion 9 2 W

int /36 2 S0 gives

- dP/d6 < (k

s

+ k

p

). (28)

Consequently, the structural stiffness k s

+ k

p

must be at least as large

as the absolute value of the effective tearing stiffness dP/d6 to ensure stability in the quasistatic case. The quasistatic case will be reached

after all inertial effects associated with the crack arrest/reinitiation

event have died out. To see what additional requirements are necessary we must consider the dynamic reloading of the arrester.

DYNAMIC RELOADING FOLLOWING ARREST

The model depicted in figure 10 accounts (approximately) for the static

effect of the ship structure on the arrester strake. It does not account

for dynamic effects.

During the crack run event, the compliance of the deck is rapidly

increasing. Since the remote bending stresses due to hogging/ sagging are quasistatically applied during crack propagation, the center of mass of

the deck experiences an acceleration. When the arrester stops the crack,

the compliance quickly reaches a final value; the stiffnesses of the deck

and uncracked ligament of the arrester try to restrain the motion of the mass center. The result is an inertial force which is rapidly applied to

the remaining ligament of the arrester.

18
A simple model which accounts for these inertial effects is shown in

figure 12. This model is the same as the static model of figure 10, except

that spring k dp is split in two and masses, M, inserted. For simplicity,

the total mass of the deck is divided into quarters; the inertial force

associated with each quarter is applied to one of the crack faces of the

arrester strakes through springs k

dl . Consequently, each mass, M, shown

in figure 12 is 1/4 the total deck mass. This division of mass assumes that only the lowest mode of vibration

of the deck is excited by the crack run/arrest event. Laboratory tests on SEN and DCB specimens [3,10] showed that the total measured loads and

displacements during crack arrest-reinitiation events were approximated

reasonably well by considering only the lowest modes. If further analysis

of ship structures proves this assumption wrong, it may be necessary to

perform a modal analysis and consider the effect on the arrester of the

inertia associated with each normal mode on the arrester.

STRUCTURAL MODEL (DYNAMIC)

Having modeled the statics and dynamics of the hull girder and having

characterized the arrester strake's ductile fracture resistance properties, it is now possible to generate a mathematical model to calculate

ductile tearing. Since dynamic reloading occurs upon crack arrest/

reinitiation, the static form of energy conservation eq (5) can no longer be used. Rather, the dynamic form

d/dt (W

e

- T - Wint ) = 0 (29)

must be used, where T = kinetic energy.

We note that there are now four coordinates U

s

, U

p

, U, and 6, where U

is the displacement of mass, M; see figure 12. Consequently, there will be

four equations of motion. However, we will show that for stability calculations, we can reduce the set to just two equations involving U and 6,

just as we were able to reduce the static model of figure 10 to the simpler

model of figure 1 1

.

The dynamic energy conservation equation generates four equations of

motion since

19
3W

6) At (30) AW = ,dW, , 3W Ts 3W • 3W 3W

( dt} At ( 3U S U

s

+ au

p

U

P

+ 3U

U + 36

6) At (30)

where W =

• W

e

" w int " T * Since Us At = - AUS , and AUS , AU

p

, etc. can be

varied independently, we have

aw

3u

p

= 0 “ p p

" [kH p U

p

+ k d2 (U

p

- U)] (31a)

3W/3U S

= 0 = P s

- Ck ds (U s -6) + kHS Us ] (31b)

3W/3U = 0 = M U + k d2 (U-U

p

) + k dl (U-6) (31c)

3W/36 = 0 = P + k

d1 (5-U) + k ds (6-Ug) + kH

6 (31 d)

Considerable simplification of the static model occurred because (a)

Pp and P g were applied (prescribed) forces so that dPs

= dP

p

=0; (b) springs

in series having common terminals at nodes U

s and U

p

could be combined.

We use a similar approach to simplify the dynamic stability calculations.

Since dP

p

= 0, we have from eq (31a)

dU

P

-

k d2 dU (32a)

kHP + k d2

likewise, (31b) gives

dUg = k ds d6 ( 32 b)

k ds + kHS

The differential forms of eqs (31c) and (31 d) are just

M dU + (k d2 + k

d1 ) dU - k

dl d6 - k d2 dU

p

= 0 (32c)

dP + < k

d1 + k ds + kH> ds - k

d1 dU - k ds dus

0 <32d>

Using (24a) and (24b) gives the simpler results

20
(33a)

MdU + [k dl +

( k d2 kHP — } dU = k

dl do

k d2 + kHP

dP +[kH

+ k d1 + ( k HS k ds )] d6 = kd1 dU . (33b)

kHS + k ds

The effect of the springs kpjp and k d2 (connected in parallel at Up) Up

simplifies to a single spring with stiffness k d2 kH p/(k d2 + kH p). The same is true of springs kHS and k ds connected to node U s

. Consequently,

we have a much simpler stability model, as shown in figure 13* The new

spring constant k d ^ is defined by

k d3 -

kHP k d2 (3Ha)

kHP + k d2

and from (25) and (26),

k s

= Kh

+ ( kHS k ds ). (34b)

k HS + k ds

Special Cases

Before considering the general solution of eqs (33)

>

it is worthwhile

to consider some special cases. For the static case, dU= 0, which corresponds to the state of the system after oscillations of mass M have been

damped out. For this case eq (33a) gives

dU = (k d -| / (k d -| + k d^) )do . (35)

Considering a free body diagram of node 6 shows that stability results if

-dP < k

s d6 + k

dl (d5-dU) (36)

or, using eq (35),

21
( 37 ) -dP/dd <[k s

+ ( K d1 k d3 )].

k d1 + k d3

We can give a physical interpretation to eq (37). For dU = 0, the springs

k

d1 and can be combined into spring constant k d -| k d3/( k d1 + k d3^ = k 'p*

This spring is in parallel with spring k s

. Hence, the combined spring

constant resisting the load drop in the arrester is k' p

+ k s , so the drop

in load -dP is resisted by the force (k'

p

+ k s )d6. Equation (37b) states

that if the spring force exceeds the load drop -dP, stability will occur.

An equivalent interpretation is that the spring stiffness k_ + kA must a p

be greater than the tearing stiffness -dP/d6. This is merely a restatement of the stability condition (28) derived for the static case. In

fact, a necessary condition for the validity of the dynamic model of

figure 13 is that it gives the same quasistatic stability criterion as the

model of figure 1 1

.

Another special case occurs when U ^ 0 , but dP/d6 = constant. For

example, [5] presents data for a center-cracked panel showing a region

of the P-6 plot where dP/dS = constant. We note that eq (17b) gives

dP/d6 = constant when Tmat (a) = constant.

Combining (33a) and (33b) gives

MdU + (k^ + k

d1 ) dU =

^ k

d1 2

j dU * (38)

dP/d6 + k

s

+ k

dl

which describes the free vibration of the system shown in figure 13 when

dP/d6 = constant. For stability, the natural frequency of the system must

be real, rather than imaginary; that is, if U = then co must be

real. Substitution of the above form of U into eq (38) gives

-Mo)2

+ [k d3 + k

d1 , dP/d6 i _+__k

v a )] = 0 (39) dP/d6 + k s

+ k

d1

The condition for real w is just that the effective spring constant in

braces in (39) be positive:

k d3 + k

d1 ( dP/d6 + k s

) > 0 (40a)

dP/dS + k s

+ k d i

22
Rearranging the above leads to the requirement

-1 (40b)

Requiring that the first term in parenthesis be >0 is equivalent to

requiring that motion of mass M be stable. An instability results when

-dP/d6 >(k

s

+ k d -| ) ; the total spring force resisting motion of mass M

drops. This causes a change in the resonant frequency of the system and

also an increase in the amplitude of oscillation of mass M.

However, it is obvious that this instability will end. The motion of mass M will still be resisted by spring k^ and (combined) spring

k s

k d1 / ( k

s

+ k

d1 ) • The amplitude of oscillations will increase temporarily

as tearing occurs, but will reach a final value (end of instability of mass M)

.

Furthermore, we note that there is a fundamental difference between

motion of mass M and the node having displacement 6. Only if 6 increases

without bound will a tearing unstability of the arrester occur. This

tearing unstability will not occur if the second term in eq (40b) is

positive. Requiring the second term to be positive is exactly the same criterion expressed by eq (37).

Since dP/d6 and the tearing modulus are related through eq (17b), the

stability criterion for the special cases considered here can be restated

as

where kp = k^ k^Xk^ + k^. Clearly, the stability criterion is

violated when Tmat 0. Since Tmat is proportional to dJ/da, T mar will

vanish when the slope of the material J-R curve is zero.

BE

<(K

S k’) (36c)

23
General Case

We now consider the most general case, for which dP/dc ^ constant. In

this case we must solve the differential equations of motion:

M dU + (k d1 + k d3 )dU = k

d1 do (41a)

d p + (k

s

+ k

d1 )d6 = k d1 dU

.

(41b)

We have three unknowns: U, 6, and P. We can obtain a well posed problem

(number of unknowns equals number of equations) if we add eqs (15)

and (17b):

dP = - [BE/aTmat (a)] d6 (17b)

da = - dP/[o0

B] = E d6/[o0aTmat (a) ] (15)

We see that dP/d6 -*•-«, and da -> + « when Tmat = 0; i.e., failure of the

arrester occurs when the tearing modulus vanishes.

We can integrate eqs (41), (17b), and (15) to calculate all

unknowns, provided that T ma ^(a) is given, and the initial conditions

U(t=0), U(t=0), a(t=0) are known. The initial conditions presumably can

be obtained from crack arrest calculation such as those performed in

[3,10]. We assume for simplicity that 5(t=0) is negligible, since the

COD at arrest will be much smaller than the COD which occurs during

ductile tearing, for a material with a high toughness.

Denoting the initial conditions by U

Q , 0

Q

, a

Q

, we have, for t=At

(first time increment)

M[U( At ) - U

Q

] + (k dl + k d3 ) [U(At)-U0

] (42a)

= k

d1 6 ( At

)

k

s

+ k

d1

" BE 6(At) = k

d1

" U

o

] (42b)

aTmat (a

o>

where we have combined ( 4 1 b ) and (17b). Equations (42) can be solved for

U( At ) , 6(At) (we use U(At) = [U(At) - U

Q ]/At); the new crack length a(At)

24
is computed from

a( At) = a

Q

+ E 6 (At)

°o a^mat ( ao ) (42c)

The process is then repeated for t = 2At using the updated value of T mat :

M [U( 2 At ) - U( At ) ] + [k d1 + k d3 ] [U( 2At ) - U(At)]

= k d1 [6(2At) - 6 ( At ) ] (43a)

[k

3

+ k dl - BE ] [6 ( 2At ) - 6 ( At ) ] aTmat( a ( At ))

= k

dl [U( 2At ) - U( At ) ] (43b)

a(2At) = a( At ) + E[6(2At)-6(At)] (43c)

°o a Tmat

(

a ( At )

)

Since we are considering ductile tearing, we must require da^O in our calculations. From eq (15), we see that da^O if d6^0, since Tmat (a) is

always greater than (or equal to) zero. However, during unloading, d6 can

be negative so eqs (15) (42c), and (43c) are only valid during elasticplastic loading.

Unloading of the arrester occurs when dU and dU are negative; the

forces associated with motion of the center of mass of the deck try to

close up the crack. During unloading, eq (17b) is to be replaced by

dP = k e (a) d6 (44)

where k

g (a) is the elastic stiffness of the arrester.:

k e (a) = BE(w-a) . (45)

25
Consequently, during unloading we have the set of differential

equations

MdU + (k d1 + k d3 )dU = k

dl d6

(k

e (a) + k

s

) d6 = k

dl dU

(46a)

(46b)

da = 0 (46c)

Proceeding in this fashion, we can calculate 6 and a and see if they

increase without bound, or whether they reach asymtotic values. The

former case indicates instability (6, a -> «>) whereas the latter case corresponds to stability (provided that the final value of a(t) is less

than the arrester width, w)

.

DIRECTIONS FOR FUTURE RESEARCH

The model proposed here for dynamic ductile tearing instability is a preliminary one. There are several areas where future research is

necessary to prove its validity.

First, a method must be devised to calculate the stiffnesses of the

various springs in the model. A necessary first step in this direction is

to obtain inputs from members of the NAVSEA structures community, especially

since these will be the ultimate users of the crack arrest/reinitiation model.

If simple but reasonably accurate approximations to the ship structure

can be made (such as the box beam approach used here) then methods such as finite element analysis could be used to calculate the effective spring

constants. To use the box beam model (hull girder) approach requires

knowledge of the elastic and geometric properties of structural elements

such as hull and deck plating, plus the effect of stiffeners, bulkheads,

etc

.

A second area of research concerns the modelling of structural

damping/ internal friction, and also modelling of strain-rate effects in

the arrester. These effects will cause the oscillations of the masses in

the model to decay. As the model is currently configured, no damping is

26
included. It may therefore be overly conservative, because damping could

reduce the intertial forces that tend to reinitiate cracking.

It should be noted that the effect of damping was included in the

viscoelastic lumped-mass model of [ 3 ]. This was done by incorporating

a viscious damping element (dashpot) to account for strain-rate elevation

effects. The damping coefficient was calculated by assuming an effective

(tensile) strain rate, e, which was set equal to the ratio of the rate of

change of CTOD to specimen thickness. The CTOD was calculated on the

assumption of formation of a plastic hinge form a limit-load analysis. The

elevation of the flow stress was assumed to be linearly related to i, and

hence to the rate of change of CTOD.

This approach can be used in the current model with some modification.

The current model assumes net section yield on reinitiation, but without

rotation, i.e., the COD and CTOD are equal and there is no plastic hinge.

Slip lines are assumed to develop, emanating from the crack tip. (Recall

that the deck was modeled as a center-cracked panel with the crack tips

partway through the arresters; see fig. 9). Material flows plastically

along these slip bands, at ±45° to the crack axis. Assuming the displacement is uniform along the slip lines, the corresponding strain e is 6/b,

where b is the remaining ligament.

Assume that flow stress elevation is given by a Q

= aQ

° + C

1 e and

if 6 /6»b/b, then

We note that for a material with a high fracture toughness, the unequality

Since the load P carried by the arrester equals the limit load, P^,

and since P L depends on the flow stress (see equation (14)), we have

(47)

• • 6/6>>b/b should be satisfied; a large amount of plastic deformation must

• • occur (large 5/6), with relatively small crack advance (b/b small).

P = P(6,5, a). The relation between dP and d6 now becomes

( 48 )

Setting P = B (

o

0

° + c-j 6/b) (w-a) and using equation (48) gives

27
dP/d6 - B[C^d6/do - 08 da/do] (49)

where we assume C^6/b<<o0

°. Comparing eq (49) with eq (15) we see that

the effect of strain-rate elevation is to make dP dependent upon the

incremental strain rate d<5/b.

When eq (49) is substituted into the equations of motion, eqs (33),

the term C^Bd6 will cause oscillations of of mass M to decay. Likewise,

the inclusion of structural damping should also add a term C

2 dU to

eqs (33). which also causes decay of oscillations.

A third area of further research is the integration of the eqs (43)

(for da>o) and eqs (46) (da =0). To perform the integration for an actual structure, the values of M and of the spring stiffnesses must be

known.

Finally, a test of the dynamic ductile tearing model could be done by

comparing predicted values of 6 and a with those obtained from laboratory

tests, such as those reported in [ 3 ]. We note, however, that the specimen

geometry used in [ 3 ] was that of a DCB in deep bending with external

spring and mass. This is not the same structural model as that treated

here; namely the case of a CCP which is an element of a box beam. What is

required is a reformulation of the basic ideas developed here to fit the

DCB test geometry. For example, the load carried by the DCB specimen

could be idealized as the limit load, P L (a) (assuming NSY). Then eqs (13)

could be used to calculate dP/d6 for this geometry, using the material J-R curve and the relation (3J/36)

a

= -B

-1 (3P/3a)

(S . The various spring constants appropriate to the DCB test configuration

have already been calculated in [ 3 ]. Inequalities corresponding to the

quasistatic case, eq (37), could be derived and used to predict tearing

instability. These predictions could then be compared to the test results

of [31.

28
CONCLUSIONS

We have proposed a methodology for design of crack arrester strakes.

Because of current practices, the only design variable (for a given

arrester material) is the arrester width, w. To perform the design, it is necessary to:

(a) Determine the mass and stiffnesses of structural elements (deck,

hull, etc.),

(b) Calculate the displacement and velocities of masses at arrest, and

(c) Know the crack arrester material's J-R curve. Steps (a) - (c) are to be used as inputs to the dynamic ductile tearing model; the model consists of a set of differential equations which

allow calculation of 6 , a, and displacement of masses. In the most

general case, stability occurs when 6 is bounded and the final value of a

is < w. The dynamic model is based in part on a generalization of the tearing

instability model of [4 and 5]. The generalization allows us to treat

the cases of (a) multiple springs in series and parallel with the crack

arrester, (b) dynamic reloading of the arrester ligament.

We treat the ship structure as a box beam with a CCP as one of the

structural elements. A free body diagram of the box beam shows how the

various structural elements interact with the arresters, which are part of

the CCP.

The arrester becomes a nonlinear spring element with stiffness dP/d6.

Catastrophic failure of the arrester always occurs when dP/d6 -0, or when Tmat * °- In both the quasistatic case, and the case dP/d6 = constant, stability

of the arrester will occur when

-dP/dS < (k

s

+ k£) . (37)

The spring constant k

g represents the combined stiffness of the deck in

series with the arrester and also a portion of the hull; k^ is the

combined stiffness of the deck in parallel with the arrester and the

remaining portion of the hull.

29
Satisfaction of the inequality, eq (37), is a necessary condition, but not

a sufficient condition for stability in the general dynamic case. Dynamic

reloading of the arrester by acceleration of e.g. the mass of the deck

must be accounted for.

The model proposed here is preliminary. Actual values of stiffnesses

for realistic ship structures are needed. Once these are obtained, a computer code could be written to integrate the equations governing

dynamic ductile tearing in the most general case. In the meantime, it may

be worthwhile to adapt the model to test configurations (such as that of

[3] and compare its stability predictions with the results of actual

experiments. The resulting equations of motion would be somewhat

different since the test configuration of [ 3 ] is a DCB , and the structural

model developed here is for a CCP in a box beam. However, the basic ides

should transfer from one type of geometry or loading system to another.

ACKNOWLEDGMENTS

This work was supported as part of the Fracture Control Technology

Program under the sponsorship of Mr. Charles Zanis, Naval Sea Systems

Command. The effort was directed by Mr. John P. Gudas, David Taylor Naval

Ship R & D Center, under Program Element 62761N, Task Area Sf-6l -544-504

.

APPENDIX

Here we consider the case of a crack which has propagated through a portion of the hull, has been arrested by an arrester strake, and has

reinitiated. Treating the ship as a hull girder (or box beam), we have

the situation shown in figure A-1. We assume for simplicity that the

cracked side of the hull acts as a plate with negligible interaction with

the remainder of the box beam. Because of the primary (bending) stresses

applied to the hull, this plate will be subjected to a bending moment, M,

applied at its ends.

Recall that the arrester is assumed placed in the midship 3/5 of the

vessel [1]. Consequently, there will be a portion of the plate between

the ends of the arrester and the ends of the plate labelled "A" in

30
figure A-1 . We model the stiffness of this portion of the hull by springs

of stiffness k

sl as shown in figure A-2. The uncracked portion of the hull

(its remaining ligament) is assumed to have stiffness kp ; this region is

labelled "c". Finally, there will be an effective stiffness k s (associated with region "B" in figure A-2), connected between the arrester and

the neutral axis of the plate in bending.

We assume that the springs are connected to a rigid member, which

idealizes the effect of bulkheads in the ship. Those members rotate

(rigidly) through angle 6, so the external work done is 2Re. Assuming the

deformation of the arrester (fracture specimen) is adequately represented

by the COD, 6, the internal work done is

W

int = -fP d<s + ksl (6/2 - eJ^) 2 + k s2 ( 6/2) 2 + 2 k

p

(e£ 2 ) 2 (A-1 )

The springs k

s1 and k

p are assumed connected at distances , S. 2 , respectively, from the neutral axis.

The equations of equilibrium can be derived from

9(W

e -Wint) 3< we 90 36

" W = 0

(A-2)

The first of these gives

M = k s (6/2 - 0 S,-,) H

1 -2(k p

eS, 2 )£ 2 (A-3a)

which is moment equililbrium for the rigid members in figure A-2. The

second of eq (A-2) gives

P + k

sl (6/2-0S,

1 ) + k s2 (6/2) = 0 (A-3b)

which is equilibrium of forces applied to the terminal connecting the

arrester with springs k

s1 , k s2 . For stability, we require 3 2Wint/30

2

£ 0, 3 2 W

int /36 2 > 0. The first

inequality gives

k

s1 + k s2 ^2 = 0 (A-JJa)

31
which is automatically satisfied. The second inequality gives

dP/do + 1/2 (k s1 + k s2 ) > 0. (A-4b)

Consequently, the combined stiffness (k s1 + k s2 ) of the springs k s1 and

k s2 in parallel must be larger than the tearing stiffness -2dP/d6 of the

specimen during ductile tearing ( reinitiation in tearing mode).

Note that the uncracked portion of the hull does not contribute to the

stability of the arrester in ductile tearing for the case considered here;

the stiffness kp does not appear in stability criterion, eq (A-4b).

However, the crack length will decrease the value of k s2 , so clearly the

problem of ductile tearing instability becomes worse for longer crack

jumps in this case.

REFERENCES

[1] Structual Design Manual for Naval Surface Ships , NAVSEA

0900-LP-097-401 0 , Naval Ship Engineering Center, Dept, of Navy,

Washington, DC 20362, Dec. 1976.

[2] Fracture-Control Guidelines for Welded Steel Ship Hulls , Ship

Structures Committee, SSC-244, U.S. Coast Guard HQ, Washington, DC,

197-4.

[3] R. B. King, T. Teramoto, D. T. Read, "Second Interim Progress

Report: Materials Selection Criteria for Crack Arrester Strakes in

Naval Vessels," NBSIR 84-3012, Oct. 1984.

[4] H. A. Ernst, "Some Salient Features of the Tearing Instability

Theory," Elastic-Plastic Fracture: Second Symposium, Volume

II-Fracture Resistance Curves and Engineering Applications , ASTM STP

803, C. F. Shih and J. P. Gudas, eds., ASTM, Philadelphia, PA, 1981,

pp. 133-155.

[5] S. Kaiser, A. J. Carlsson, "Studies of Different Criteria for Crack

Growth Instability in Ductile Materials," Elastic-Plastic Fracture:

Second Symposium, Volume II-Fracture Resistance Cruves and

Engineering Applications , ASTM STP 803» C. F. Shih and J. P. Gudas,

eds., ASTM, Philadelphia, PA, 1981, pp. 58-79.

[6] P. C. Paris, H. Tada, A. Zahoor, H. Ernst, "The Theory of

Instability of the Tearing Mode of Elastic-Plastic Crack Growth,"

32
Elastic-Plastic Fracture ASTM STP 668, J. D. Landes, J. A. Begley,

G. A. Clarke, eds., ASTM, Philadelphia, PA, pp. 5~36.

[7] J. R. Rice, P. C. Paris, J. G. Merkle, "Some Further Results of

J-Integral Analysis and Estimates," Progress in Flaw Growth and

Fracture Toughness Testing , ASTM STP 536, ASTM, Philadelphia, PA.

[8] D. Broek, Elementary Engineering Fracture Mechanics , Martinus

Nijhoff, The Hague, Boston, 1982.

[9] M. G. Vasilaros, J. P. Gudas, J. A. Joyce, "Experimental Evaluation

of Tearing Instability Phenomena for Structural Materials," Rep. No.

NUREG/CR-2570, U.S. Nuc. Reg. Comm., Washington, DC 1982.

[10] T. Teramoto, D. T. Read, R. B. King, "Fracture Mechanics Parameters

in Crack Arrest Specimens," National Bureau of Standards, Boulder,

CO (in review process).

33
Load

Jack

Modified double-cantilever beam (DCB) specimen used in NBS tests.

Figure 1

34
«L

Lumped-mass viscoelastic-plastic dynamic model of DCB specimen.

Figure 2
Flow chart for crack arrester design procedure.

Figure 3

36
eo

•H3 r\ 0) oo

4-1 3

•H •H

V4 3 u a

CO

>•>

4-J

X •H •u t—I

•H

•HX

cd CO 4J cu • CO •H (U 3 Pj

<U CU 4J CO

A

cd X) u c cd X •H o

CO3 3

T—1

cu rH

t-H £ cd •H •H c <r O 3 O (U cu cu

4-1 a, 4H 3

CO

X 3 x <u 00

cu <u •H

CO 3 o Ch 3 3 4J

4J e a

X

41 cd CU

4-J 3 4J

CO >4-4 CJ

CU CO >4-1 *x O X 1—1 3

cd CO CO u 4-1 •H CO c •H

cd CO .c 3 o O

0) o B E

cu cu D- CO B >,

•h cn

cn

37
COo

r—1 • 3 1 rQ 00 3 o 3

•H 3 03 •H e o- a> (-4

3-1 03 4-1

a

0) cd 03 •u tn 00 43 3 u o 43 3 X rH

a) a)

X

x •H 3

<uU 3 4-1

uU 03 l-i 3 cd S 3

i—iCJ *H !*!

a CJ 03 XI 03 n cd o. 03 S3 O C/3 X3 n O

cn • 4-1 to • 00

23 c c <-o S 03 -rl Xi e ^

4J •H cd in 3 CJ 03 03

•rl 03 4-1

O 3 O U 3-1 a 03 O O 3

•U M-4 00

0) •H

03 3-i 03 3 Pm 3 3 3 O O 4-1 x) •H

•H CJ 4-1 CJ cd •> O

•H S-l p4 3

4-4 <4-4 "3 3

CJ 3

•H o •> 4-1 4-1 4-1 00 '

» 3 3

00 3 •H 3 3 O 03 o •h g 3

S-i 3 3

00

a CJ 3 03 •H 40 A

4-1 00X

a 3 03

X

cd •H &,

4-4

O *i 03 CJ 3-i 4-4 <D 03 rO 3 o 3 3 3 3 O o O 03 6

CM CJ

g 3

38
&CC

CL

39

Load-displacement

curves

for

hypothetical

fracture

specimen,

with

ini

crack

length,

a

,

as

parameter.

The

slope

indicated

is

the

effective

tearing

stiffness,

dP/d6,

which

is

negative

due

to

load

drop

on

teari
o CU

4-1 Pi3

XJ •3 3 CJ g cO 3 Pi 3 4-1 CO3 toC 3 • CO •H CO •H >. 3

Pi 44

44 CO 3 a > 3 3 4-> o 3: 3 u

•H 3 » & 3

44 4-1 O /—V 3 3 p-l 3 3 CJ 3

CJ CJ 3 '—

3 r-^ 3=

•u iH 44 3

•H 3 3 3 & 3 3 3 3 3 &0

PP CPa •H

•H C3 43 X) 0

CO 3 u

44

4-1

O 3 O 3 3 3 X) 44 3 o 1 3 3 3 3

X) 3 3

4J 3

4-1 3 43 o 3 60 O 3

I—1 O

Cl) 3 •u X)o CD g 3

cu 3

i-H >

cp 3 g 43

•H 3

CO 43

40
41

Section

modulus

(bending

stiffness)

and

mass

can

vary

along

length

of

box

beam.
Exploded view of ship structure showing

interaction stresses between structural components.

Figure 9

42
43

Ship

structure

-

crack

arrester

interaction

model

for

quasistatic

loading.
m LU

3

o

•HP

cd

i—

I

3u

•H

4=

cd u

cno

•H

4J

Cd 4J

CO •H

CO

cd3

crM

o

CH

O)3

3

bO

•rH

fl-l

d

QJCO3

3d

OBd

3

•H

14-4 H

O-S

C/0

44
QP45

Dynamic

model

for

ship

structure

-

crack

arrester

interaction.

Model

includes

masses,

M,

to

account

for

inertia

forces

due

to

acceleration

of

deck.
UJ

46

Simplified

dynamic

model

for

stability

calculations.
Arrester

47

Crack

has

been

arrested

by

crack

arrester

strake.
©A6

12

•CO T3

a)

cu 4344

rH3

43

M-lO

CO3O

•H3 • 3

44

N o •H 3 f—1 3 3 O CU "d 'd •H

cu 3 (U

CO 3

0) CO 3u 43 3 O

•rl 4=

4-J &

•H CM

01 1 CO <

i—i 3

T—iCU 3 3 rg 3

43 0 3 3 oo

14-1 3 •H O

-a

Pm rH •H

CU 00

T3 •H O 3 0 O

T3 4-1 CU •H 33

M-l 3

•H rH rH

o- CM 0 CM H 3 cn

co

•rl IS

•us

0)6o

00c

•Hs

0) PQ

48
MBS-1I4A »REV. 2 -6C > U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See in structions)

1. PUBLICATION OR

REPORT NO.

NBSIR 85-3038

2. Performing Organ. Report No 3. Publ i cation Date

January 1986

4. TITLE and SUBTITLE

Ductile Tearing Stability Analysis of a Ship Structure Containing a Crack

Arrester Strake

5. AUTHOR(S)

A. V. Clark, Jr., and D. T. Read

6. PERFORMING ORGANIZATION (If joint or other than N BS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final Report

Sept. '84 to Sept. '85

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

David Taylor Naval Ship R & D Center

Annapolis, Maryland

10. SUPPLEMENTARY NOTES

J Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) An analysis is presented for a structure made up of a crack arrester

plate embedded in a ship structure. The crack arrester material is specified

by its crack arrest temperature, its strength, and its tearing modulus T^ The remainder of the structure is characterized as a set of springs and

lumped masses.

mat

A stability condition is derived which states that the load-displacement

curve of the structure as a whole must increase monotonically. An approximate

quasistatic stability criterion sets a minimum material tearing modulus value

that depends on the structural stiffness. Higher stiffness promotes effective

crack arrest. A calculation including dynamic effects requires forward integration of a set of differential equations describing the fracture process

and the motion of the structure. Repeating such a calculation for different

values of crack arrester widths could indicate the minimum width needed for a successful (stable) crack arrest.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s, arrester strakes; crack arresters; ductile fracture; ductile tearing instability;

structural instability

13. AVAILABILITY

I | Unlimited

I I For Official Distribution. Do Not Release to NTIS

I I Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

I"*! Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

54

15. Price