1

THE PHILOSOPHY OF CONSTRAINT CORRECTION

by

Christian Thaulowa, Erling Østby

b, Bård Nyhus

b, Vigdis Olden

b and Zhiliang Zhang

b

Abstract

The Failure Assessment Diagrams (FAD) used in BS 7910:1999 (Guide on methods

for assessing the acceptability of flaws in fusion welded structures) represent high

structural constraint applications. The standard gives literature references for

constraint correction methods, based on T and Q, but none of these are included in

the standard. There is evidently a need to present a framework for a practical

application of constraint corrections. The paper presents constraint correction

parameters and demonstrates the JQM Approach with reference to a 690-steel. With

the increased use of FE calculations in the industry, a method for direct calculations,

with high accuracy and low costs, is presented.

presented at

2nd International Symposium on High

Strength Steel

23-24 April 2002, Stiklestad, Norway

a The Norwegian University of Science and Technology

b SINTEF Materials technology

2

Cracks

All materials will contain cracks or defects. The question is: When will cracks be of

practical interest? Under which conditions will cracks influence upon the behaviour of

structures and components? When can we ignore the existence of cracks?

Structural engineers normally judges the capacity or ultimate strength of a structure

on the basis of a load-deflection diagram, where the maximum load or plastic collapse

load is considered as the limit. The next step then is to impose a partial safety factor

on this limit load combined with minimum tensile eleongation requirements.

If we now introduce cracks in the structure, this can influence the load bearing

capacity, Figure 1, either by brittle fracture, ductile tearing, plastic collapse or

combinations of these failure modes.

Traditional structural design compares the design stress with the flow properties of the

material, which is normally taken to some fraction of the yield stress. A material is

assumed to be adequate if its strength is greater than the expected applied stress. In

fracture mechanics there are two structural variables, design stress and flaw size, and

fracture toughness replaces strength as the material resistance property. Fracture

mechanics quantifies the critical combination of these three variables.

In fabrication with steel and aluminium, the welded joints represents the most critical

region. This is where cracks normally appears and regions of the weld metal or the

heat affected zone can have low toughness. The weld metal, heat affected zone and

the base material will have different material properties, and this mismatch in strength

will influence the failure conditions. The effect of material mismatch on fracture

depends upon the crack size, the location of the crack, the strength mismatch and the

fracture toughness.

For cracks located at the fusion line in steel weldments, weld metal overmatch is

recommended if ductile behaviour can be guaranteed. If, however, brittle fracture can

occur, evenmatch seems most favourable in order to avoid brittle fracture initiation

from the heat affected zone Thaulow et al (7, 8).

In the welding of high strength steels, the probability of even- or undermatch

increases, Figure 2, and it will be of importance to quantify the effects of mismatch.

In this paper we first shortly introduce the principle of constraint and transferability.

We the presents the JQM Approach and show how the approach can be applied for

the 690-steel investigated in the PRESS project. At the end we introduce effective

ways of applying constraint corrections, and the new company LINKftr.

Constraint

The starting point in fracture mechanics analysis is to consider a crack of a certain

size located in a component or specimen. An external load is applied and the

component is loaded until it fails. During loading a plastic zone develops from the

3

crack tip, and at a certain load net section yielding occurs as the plastic zone reaches

the through thickness surface.

As long as the plastic zone at the crack tip is limited compared with the geometry of

the component or specimen, socalled small scale yielding, a single parameter fracture

mechanics approach can be applied. K, J or CTOD characterizes the crack tip

conditions and can be used as geometry independent fracture criterion.

The geometry dependence under linear elastic conditions for five standard fracture

mechanics geometries are plotted in Figure 3. The pure tensile specimens, DENT and

CCT, have the lowest constraint , while specimens dominated by bending have the

highest constraint. Standard fracture mechanics testing procedures are based on the

specimens with high constraintin order to reproduce the worst case conditions.

However, the single parameter fracture mechanics breaks down in the presence of

excessive plasticity, and fracture toughness will now depend on the size, geometry

and mode of loading.

McClintock (1) was one of the first to examine the near crack tip stress field under

fully plastic conditions for various specimen geometries and non-hardening materials,

Figure 4. For small scale yielding, the maximum stress is approximately three times

the yield stress, while a centre cracked panel under tension is incapable of maintaining

significant triaxiality. These effects are, however, less severe when strain hardening is

taken into account. We notice that the DENT specimen, with low constraint under

linear elastic conditions, Figure 3, now reach high stresses because of the interference

between the two fields of deformation.

The history of constraint is how to deal with crack tip stresses under fully plastic

conditons. The aim is to find a parameter that characterize the stress-strain fields, so

that results from one test geometry can be transferred to another geometry.

One approach has been to restrict the application of fracture mechanics to high

constraint since single-parameter fracture mechanics may be approximately valid in

the presence of significantly plasticity, provided the specimens maintains a relatively

high level of triaxiality. Most laboratory fracture mechanics specimens, as three-point

bending and compact tension, represent this high triaxiality conditions.

A more basic approach has been to define the crack tip triaxialtity as the ratio between

the hydrostatic stress, or first invariant of the stress tensor, which does not cause any

plastic deformation, and the Mises effective stress, which is the square root of the

second invariant of the deviatoric stress being responsible for plastic flow. This

parameter has been extensively applied to describe ductile crack initiation and growth.

There are a number of mathematical models for void growth and coalescence, where

the two most widely referenced models were published by Rice and Tracy (2) and

Gurson (3). They found an exponential dependence of the void growth rate on the

stress triaxiality, 0/ h . Here h is the hydrostatic stress, and 0 is the yield stress.

The yield stress has later, Needleman and Tvergaard (4), been substituted with e ,

the Mises effective stress.

4

Brocks and Schmitt (5) has intoduced the parameter h for the ratio, and proposed this

ehh /

as the second parameter needed, in addition to J, to quantify the geometry dependence

of ductile crack growth. They also argue that the obvious disadvantage, that it is a

field quantity and requires 3D elastic-plastic FE-solutions, can be overcome by

extrapolation schemes and inexpencive computer power.

Another constraint parameter is the T-stress, Larsson and Carlsson (9), Du and

Hancock (10). This is a non-singular linear elastic stress component parallell to the

crack. The T-stress charcterizes the local crack tip stress field for linear elastic

material, and the global in-plane constraint of a specimen with respect to

predominantly local small scale yielding conditions. It has however been argued that

the T-stress also can be applied under plastic conditions, Betegon and hancock (11).

T increases or lower the hydrostatic stress by

Tfr

K Ih

3

1

2

The idea of adding a second term has been taken over in elastic-plastic fracture

mechanics by defining the so-called Q parameter, O`Dowd and Shih (12, 13)

0

0

Tyyyy

Q

The solution for yy is obtained by FE calculations.

The Q parameter, like the T stress, is supposed to characterize the geometry

dependent constraint. Both quantities affect the hydrostatic stress in the same way, i.e.

negative values lower, positive values raise the hydrostatic stress.

The Failure Assessment Diagrams (FAD) used in BS 7910:1999 (Guide on methods

for assessing the acceptability of flaws in fusion welded structures) represent high

structural constraint applications. The standard gives litterature references for

constraint correction methods, based on T and Q, but none of these are included in

the standard: "The FADs represent high structural constraint applications. When

toughness is measured using standard procedures, it is possible to modify the FAD to

account for lower constraint. Alternatively, it is possible to maintain the use of a high

constraint FAD and account for lower structural constraints using appropriate test

geometries." There is evidently a need to present a framework for a practical

application of constraint corrections.

5

JQM-Approach

The JQM Approach quantifies the crack tip stress fields in dependence of geometry

(size, crack depth, global geometry and mode of loading), the Q parameter, and

material (yield strength and hardening exponent), called M, Figure 5, Zhang et al

(14), Thaulow et al (15). The Approach is based on the exsisting J-Q theory and the

RKR brittle failure criterion , but is further developed to take material mismatch into

account.

The stress field is expressed with three terms

)12()( 1_01_0

0,0 M

ij

Q

ij

TM

ijij fMfQ

where =0 for mismatch ratio m= 0 2 0 1_ _ 1 (weld metal overmatch), and =1

for m<1, 0_1 is the yield stress of the reference material and fij

M represent the

angular functions of the difference fields, which depend only on the properties of the

reference material.

The first term sets the size scale of the local deformation, with reference to the

validity range of one-parameter fracture mechanics. The effect of geometry and

mismatch is scaled by the other two terms, Figure 6.

The constraint of the different fracture mechanics specimens can now be presented as

J vs Q+M, Figure 7. The fracture toughness obtained under standarised high

constraint conditions ( 0Q ) can be transferred to more structural relevant lower

constraint conditions.

The methodology will now be presented with reference to the 690-steel investigated

in the PRESS-project.

JQM constraint correction for 690 steel

The material data for the 690-steel are presented in Figure 8. Notched tensile testing

was applied to derive the stress-strain input data, Olden et al (16).

Three specimen geometries were selected, Figure 9. The idea was to cover a large

variation in constraint with as simple test specimens as possible. The shallow notched

SENT specimen has been extensively used the last year, and the testing methodology

is now well established, Nyhus et al (17).

We have to distinguish between the crack driving force (expressed as J) induced at the

crack tip because of loading and mismatch (applied J), and the material resistance

(expressed as fracture toughness J).

6

The evolution of constraint for the three test specimens as function of applied J, is

presented in Figure 10. When we add the mismatch effect, for cracks located at the

fusion line, the constraint increases with weld metal overmatch, Figure 11. At a

certain load ductile crack growth can be experienced. The effect of limited ductile

crack growth has been examined, Østby et al (18), and an increase in constraint is

observed, Figure 12. The constraint effect on ductile crack growth, J-R curves, has

been further evaluated, Zhang et al (19) and Nyhus et al (20).

In order to establish material resistance curves, a large testing program has been

performed, Nyhus et al (21). The lower bound toughness results from the heat

affected zone shows that the toughness increases significantly as the constraint is

reduced, Figure 13. The M parameter is not included in this calculation because it is

close to evenmatch conditions.

By comparing the applied- and resistance curves, Figure 14, we can now determine

the critical conditions for brittle fracture.

We can now select a structural component of interest, introduce a crack, and calculate

the constraint and check if brittle fracture will take place.

Discussion

At present stage FE calculations are needed in order to calculate the constraint. But

two approaches have been suggested to make the calculations simpler, more effective

and less time consuming.

The first is an engineering or simplified approach where the need for calculations is

reduced to a minimum. Polynoms for a range of typical stress-strain curves are

calculated in beforehand and presented for practical use.

The other approach is the so-called direct calculations. The 3D crack geometry is

represented by a so-called linespring FE element. This element is introduced in a shell

FE analysis of a structure at critical locations.

A new company, named LINKftr as, has been established with the aim to develop

software for direct calculations. The LINKftr concept is to link detailed crack tip

calculations with the structural response, with the linespring as the transfer-element,

Figure 15. The introduction of linespring elements will not influence on the

calculation capacity for the shell element geometry; hence, high accuracy can be

obtained at low costs, Figure 16. And not to forget: the routines will be easy to use.

Acknowledgement

This work is a part of the research project PRESS (Prediction of Structural Behavior

on the Basis of Small Scale Testing), with financial support from the Norwegian

Research Council and EU. The authors wish to thank colleagues from industry and

research institutes for the close cooperation.

7

Conclusions

Single-parameter fracture mechanics breaks down in the presence of excessive

plasticity, and a second term has to be added to the standard K, J or CTOD in

order to quantify the constraint.

The J-Q-M Approach quantifies the crack tip stress fields in dependence of

geometry (size, crack depth, global geometry and mode of loading), the Q

parameter, and material (yield strength and hardening exponent), called M.

Three simple specimen geometries have been selected to cover a wide range of

constraint: SENB (a/W=0.5), SENB (a/W=0.2) and SENT (a/W=0.2). Both

calculations and test results reveal that the specimens are good candidates for

future standardization with respect to constraint corrections.

The constraint correction procedures must be easy to perform, have high accuracy

and low costs. A new company, LINKftr, has been established with the aim to

develop software for direct calculations.

References

(1) McClintock, F.A. "Plasticity Aspects of Fracture." Fracture: An Advanced

Treatise, Vol. 3, Academic Press, New York, 1971, pp.47-225

(2) Rice, J.R. and Tracy, D.M. "On the ductile enlargement of voids in triaxial stress

fields." Journal of the Mechanics and Physics of Solids, Vol. 17, 1969, pp.201-217

(3) Gurson, A.L. "Continuum Theory of Ductile Rupture by Void Nucleation and

Growth: Part 1-Yield Criteria and Flow Rules for Porous Ductile Media." Journal of

Engineering Materials and Technology, Vol. 99, 1977, pp.2-15

(4) Needleman, A. and Tvergaard, V. "An Analysis of Ductile Rupture in Notched

Bars." Journal of Mechanics and Physics of Solids, Vol. 32, 1984, pp.461-490.

(5) Brocks, W. and Schmitt, W. "The Second Parameter in J-R Curves: Constraint or

Triaxiality?" Second Symposium on Constraint Effects, ASTM STP 1244, 1994

(6) Anderson, t.L. "Fracture mechanics. Fundamentals and applications" 2nd edition,

1995, CRC Press, Florida, USA.

(7) Thaulow,C. ”Effect of weld metal over- and undermatch on fracture resistance of

pipeline girth welds”. Open seminar on Deep Water Pipelines and Flowlines, , 21

October 1999, Trondheim, Norway

8

(8) Thaulow,C., Hauge, M, Zhang,Z.L.,Ranestad,Ø. and Fattorini,F.:” On the

interrelationship between fracture toughness and material mismatch for cracks located

at the fusion line of weldments”. Engineering Fracture Mechanics, 64 (1999) pp.367-

382.

(9) S.G. Larsson, A.J. Carlsson, Influence of non-singular stress terms and specimen

geometry on small-scale yielding, Journal of the Mechanics and Physics of Solids 21

(1973) 263-277

(10) Z.Z. Du, J. W. Hancock, The effect of non-singular stresses on crack-tip

constraint, Journal of the Mechanics and Physics of Solids 39 (1991) 555-567

(11) C. Betegón, J.W. Hancock, Two-parameter charaterisation of elastic-plastic

crack-tip fields, Journal of Applied Mechanics 58 (1991) 23-43

(12) O’Dowd N. P. and Shih C. F., Family of crack-tip fields characterised by a

triaxility parameter: Part I - Structure of fields. J. Mech. Phys. Solids, 39, 989-

1015, (1991).

(13) O’Dowd N. P. & Shih C. F., Family of crack tip fields characterised by a

triaxility parameter-Part II. Fracture applications, J. Mech. Phys. Solids, 40, 939-963

(1992).

(14) Zhang, Z.L., Hauge, M. and Thaulow, C.: "Two Parameter Characterisation of

the Near Tip Stress Fields for a Bi-Material Elastic-Plastic Interface Crack", Int.

Journal of Fracture, 79:65-83, 1996.

(15) Thaulow, C., Zhang, Z.L., Ranestad, Ø. and Hauge,M., “J-Q-M approach for

failure assessment of fusion line cracks: two material and three material models”.

ASTM STP 1360, Fatigue and Fracture Mechanics: 30th

Volume. St.Louis. June 1998.

(16) Olden, V. "Notch tensile testing og high strength steel weldments." 2nd

International Symposium on High Strength Steel, 23-24 April, 2002, Verdal, Norway

(17) Nyhus, B., Polanco, M., Knagenhjelm, H.O. and hauge, M. "A more efficient

engineering critical assessment for pipes based on testing of single edge notch tension

specimens." 6th International Pipeline Conference&Exhibition, Merida, Mexico,

November, 2001

(18) Østby, E , Nyhus, B, Thaulow, C, Olden, V. and Zhang,Z.L. "The effect of

geometry and ductile crack growth on the near-tip constraint level."

2nd International Symposium on High Strength Steel, 23-24 April, 2002, Verdal,

Norway

(19) Zhang Z.L.,Thaulow,C. and Ødegård J.”A Complete Gurson Model Approach

for Ductile Fracture.” Engineering Fracture Mechanics. 67, 155-168, 2000

9

(20) Nyhus,B., Zhang, Z.L. and Thaulow, C. "Normalisation of Material Crack

Resistance Curves by the T Stress." 2nd International Symposium on High Strength

Steel, 23-24 April, 2002, Verdal, Norway

(21) Nyhus, B and Østby, E. "SENT Testing of High Strength Steel." 2nd

International Symposium on High Strength Steel, 23-24 April, 2002, Verdal, Norway

(22) Chiesa, M., Nyhus, B., Skallerud, B. and Thaulow, C "Efficient Fracture

Assessment of Pipelines. A Constraint Corrected SENT Specimen Approach",

Engineering Fracture Mechanics, 68, 527-547, 2001

(23) Chiesa, M., Skallerud, B. and Thaulow, C. "Line spring elements in a yield

strength mismatch situation with application to welded wide plates", Engineering

Fracture Mechanics, 68, No 8, 2001

10

Figure 1 Load vs displacement of a structural component. The normal structural

design (no-crack assumption) is schematically compared when cracks are included

(brittle fracture, ductile tearing, plastic collapse).

Figure 2 Distribution of yield strength for base material (BM) and weld metal (WM)

for two classes of steel.

Brittlefracture ?

No-crack assumption

Global displacement

Glo

bal

load With possible cracks

450

MPa

690MP

a

BM BM WM WM

11

Figure 3 Linear elastic solutions for standard fracture mechanics test specimens. The

figure is based upon Anderson (6)

Low constraint

High constraint

12

Figure 4 Plastic deformation pattern in small scale yielding (a) and slip line patterns

under fully plastic conditions in three fracture mechanics test geometries. The

estimated local stresses are based on the slip line analyses of McClintock (1), and

apply only to non-hardening materials. From Anderson (6).

13

Geometryconstraint specimen

geometry cracksize loadingmode

Mismatchconstraint strength

mismatch hardeningmismatch generalmismatch

J-Q theory J-Mtheory

J-Q-M theory

WM

BM or HAZ

Figure 5 The JQM theory

Figure 6 The JQM theory. The actual geometry and mismatch is always compared

with a reference solution, representing small scale yielding and homogeneous

material.

Same failure

condition

RKR

criterion

Jref, Q=0, M=0Japplied, Q, M

14

Figure 7 Fracture toughness as a function of the constraint

Fracture

Toughness

Constraint Q+M

15

Figure 8 Chemical composition and mechanical data for the 690-steel investigated in

the PRESS project.

Figure 9 The three standard test specimen geometries examined in the PRESS

project

FR

AC

TU

RE

TO

UG

HN

ES

S

[J, K

, C

TO

D]

GEOMETRY / CONSTRAINT [Q]

SENB (a/W = 0.5)

SENT clamped

(a/W=0.2)

SENB (a/W = 0.2)

690 MPa steel

C Si Mn P S Al N Cu Mo Ni Cr V Nb Ti B

.16(.17)

.430(.36)

1.20(1.35)

.013(.015)

.001(.003)

.039 (-)

.008(.008)

.035(.037)

.31(.43)

.34(1.00)

.425(1.00)

.043(-)

.026(-)

.003(-)

.0002(-)

1.0 mm

FL

Notched tensile testing Stress-strain curves

Chemical composition

16

Figure 10 Evolution of constraint in the three test specimens

Figure 11 The effect of mismatch on the constraint. m=1.3 represents 30% weld

metal overmatch when HAZ is considered as the critical material (or reference

material, Figure 6).

17

Figure 12 Effect of ductile crack growth on the crack tip stress and the constraint.

Figure 13 Lower bound fracture toughness for the 690-steel with the crack located at

the fusion line.

Ductile crackgrowth initiation

Increase in local crack tipconstraint due to ductilecrack growth

18

Figure 14 Comparison between the crack driving force and the fracture toughness

test results for the three specimen test geometries

19

Figure 15 The LINKftr concept. Detailed crack tip calculations are linked to the

structural response with linespring as the transfer-element.

Figure 16 Application of direct fracture calculations based on linespring elements.

Procedures are developed by LINKftr.

3D FE

calculations

Shell elements FE

calculations

with line spring

Analytical

equations

(CrackWise)

Acc

ura

cy

Co

sts

Costs

Accuracy

LINK ftr

THE LINK BETWEEN LOCAL FAILURE AND STRUCTURAL RESPONSE THE LINK BETWEEN LOCAL FAILURE AND STRUCTURAL RESPONSE

LINK failure LINK transfer LINK respons

B B A A

B B A A

Q Q Q Q

q q q q

D D D D D D D D D D D D D D D D

2 1 2 1

2 1 2 1

44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11

) ( ) ( h a

D ep ij Line Spring Tangent stiffness

matrix B A

i q , Generalized displacements at nodes A and B

i Q Generalized Force at nodes A and B, in tension and in bending ( N,M),

transfer LINK failure LINK B B A A B B A A Q Q Q Q q q q q D D D D D D D D D D D D D D D D 2 1 2 1 2 1 2 1 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 ) ( ) ( h a D ep ij Line Spring Tangent stiffness matrix B A i q , Generalized displacements at nodes A and B i Q Generalized Force at nodes A and B, in tension and in bending ( N,M),