**Annette D. Karstensen**

^{(1)}, Anthony Horn^{(2)}and Martin Goldthorpe^{(1)}^{(1)}TWI, Cambridge, UK

^{(2)}Corus, Swinden Laboratory, Rotherham, UK

Paper presented at 2nd International Symposium on High Strength Steel (PRESS), Stiklestad, Verdal, Norway, 23-24 April, 2002

**Abstract:** This paper uses finite element analysis to study how different structural geometries and loading modes, combined with different levels of weld yield strength mismatch, affect the constraint indexing parameter *Q* for a crack tip located in a weld heat affected zone. A normal 'V' weld and a straight-sided weld in a single edge notch plate geometry are modelled. These are subjected to either uniaxial tension or three-point bending. The tip of the crack is located in a stronger, narrower weld grain coarsened heat affected zone. The study confirms that tensile loaded geometries lose the unique, single-parameter crack tip fields at much smaller levels of CTOD or J than bending geometries. The range of single-parameter characterisation in under-matched welds for these geometries is more limited than suggested previously in the literature as plasticity spreads away from the crack tip location. Single-parameter control of the crack tip is more limited in such circumstances.

## 1. Introduction

Geometry independence of fracture toughness is seen in test specimens and structures that impose a high level of plastic constraint at the crack tip. These specimens include the single edge notch bend specimen and the compact test specimen. Materials that have low toughness and/or high yield strength are more likely to give geometry-independent or transferable measurements of toughness. On the other hand, cracked geometries that impose a predominantly uniform tensile stress on the cracked section generally experience a loss of plastic constraint at the crack tip. This is associated with a breakdown of single-parameter controlled fields at relatively low values of J/d_{n}σ

_{o}or δ/d

_{n}where the parameter

*d*is a function of material properties (primarily the degree of strain hardening) but also depends weakly on the geometry. For engineering steels

_{n}*d*typically has values between 0.4 and unity. δ is the crack tip opening displacement, and

_{n}*J*is the J-integral, σ

*is the yield stress of the material where the crack is located. The breakdown of single-parameter characterisation causes an increase in the*

_{o}**effective**fracture toughness for that cracked geometry. Materials with high toughness combined with low strength are more likely to exhibit such geometry-dependent toughness. A two-parameter approach to describe the loss of constraint due to geometry dependence was developed by Bilby et al (1986)

^{[1]}and Hancock and co-workers.

^{[2,3,4] }

### 1.1 The constraint parameters

O'Dowd and Shih^{[5,6]}showed that elastic-plastic crack tip fields could be characterised by using a second constraint indexing parameter,

*Q*, in addition to

*J*. Further development of two-parameter characterisation using

*Q*was carried out by Karstensen et al.

^{[7,8]}

*Q*is determined for a particular cracked structure subjected to a given load by taking the difference between the full elastic-plastic crack tip solution for a particular structural loading and a reference field. This latter is usually the solution for conditions of plane strain, small-scale yielding at the tip of a semi-infinite crack located in the material of interest.

*Q*takes into account the reduction (or increase) of crack tip constraint due to global effects of geometry and loading. A modification of this approach that takes into account local effects of weld strength mismatch and weld geometry is known as the

*J-Q-M*model.

^{[9]}

In the spirit of O'Dowd and Shih, an operational definition of *Q* was originally taken to be as follows in this work:

*Equation.1.*is the hoop stress for the actual geometry in a radial co-ordinate system (

*r, θ*) centred at the crack tip, and

*Equation.2.*is the hoop stress for conditions of small-scale yielding in the same material. The radial distance chosen,

*r*=2

*J*/ σ

*, is the approximate position of the peak stress that occurs ahead of a blunting crack in an elastic-plastic material.*

_{o}

**Equation.1.**

**Equation.2.**

During the calculation some inconsistencies were discovered when comparing *J* determined by methods used in BS 7448 ^{[10]} with large strain finite element analyses. An alternative definition of *Q* in terms of δ was therefore adopted as follows:

*d*=0.5, Eq.[1] and [2] would give the same result for

_{n}*Q*.

## 2. Finite Element Analyses

### 2.1. Welded plate geometries

**Geometries studied**

*Figure 1*shows the four different cases investigated. These are a normal 'V' weld (left hand side of

*Fig.1*) and a straight-sided 'V' weld (right hand side) in a single edge notch plate geometry subjected to either uniaxial tension or three-point bending.

**Finite element models**

^{[11]}The three-dimensional finite element meshes of the two plate geometries were constructed using eight-noded, first order elements of type C3D8H.

*Figure 2*shows general views of the meshes used for both the normal 'V' weld and the straight-sided weld. By taking advantage of symmetry, only half of the thickness is modelled.

**Fig.1. The four different geometries considered. These are either a normal 'V' weld or a straight-sided 'V' weld in a plate containing a surface breaking half thickness crack. The plate is subjected to either three-point bend or uniform tensile loading.**

*Table 1*shows the relative dimensions of the two different geometries. The specimen width is

*W*=50mm.

**Table 1 Relative dimensions of the weld specimens analysed**

Ratio of specimen thickness to width of cracked section: | B/W=1.00 |

Ratio of specimen length to width of cracked section: | 2H/W=2.00 |

Ratio of crack depth to width of cracked section: | a/W=0.50 |

Ratio of weld cap width to width of cracked section: | w =1.36_{c}/W |

Ratio of weld root width to width of cracked section: | w =0.20_{r}/W |

Ratio of GCHAZ width to width of cracked section: | h/W=0.10 |

**Fig.2 General views of the meshes used for**

*Figure 3*shows the finite element mesh closer to the crack tip for the normal 'V' mesh.

Large strain, incremental, elastic-plastic analyses are conducted by updating the geometry of the model at each load increment. The incremental analyses are controlled to provide output of results at frequent intervals of loading.

**Fig.3. The mesh surrounding the crack tip for the normal 'V' weld, showing the innermost mesh surrounding the crack tip**

**Loading and boundary conditions**

The meshes that model three-point bending are loaded by means of prescribed displacements of lines of nodes through the thickness of the model, as indicated in *Fig.1*. The meshes that model tensile loading are loaded by imposing prescribed uniform and parallel opening displacements to the left and right-hand faces of the plates, see *Fig.1*.

The displacements of the nodes on the back faces of each model are restrained from through-thickness displacements to model the plane of symmetry at half thickness. Appropriate boundary conditions are applied to prevent rigid body movement of each model.

### 2.2. Boundary layer analysis of small-scale yielding

To obtain the various reference solutions for small-scale yielding,*Equation.2.*in [1] and [2] above) a so-called boundary layer model is set up to simulate contained yielding at a crack tip in homogeneous GCHAZ material. This is a semi-circular mesh modelling a crack in a semi-infinite half space, with prescribed displacements applied on the boundary nodes. The displacement field is controlled by the stress intensity factor,

*K*. This boundary layer model is the limiting case of a very long crack in a very large structure, subjected to a low level of applied load.

### 2.3. Material properties

The parent steel modelled is a BS 7919 Grade 450EMZ plate produced by Corus via the quenched and tempered process. For each of the four different plate geometries studied, analyses are undertaken to investigate weld yield strength over-match (OM), even-match (EM) and under-match (UM). In all cases the Young's modulus is set to 207000MPa and Poisson's ratio to 0.3. Experimentally measured true stress versus true strain behaviour is used for the parent plate, simulated GCHAZ microstructure and an over-matched weld metal at -30°C. No experimental data existed for an under-matched weld metal, so the entire stress strain curve was scaled from the over matched weld data, to give the same degree of mismatch. The resulting stress strain curves used in the analyses are shown in*Fig.4*. In this case, the GCHAZ exhibits higher stress/strain behaviour than parent steel or overmatching weld metal. Each of the four curves has almost the same limit of linear stress-strain behaviour at about 500MPa, with exception of the under-matched weld which has limit of 341MPa. In the elastic-plastic range the materials exhibit different strain hardening behaviour.

**Fig.4. True stress versus true strain behaviour of the materials modelled**

## 3. Results

The results for the bend loading are plotted in*Fig.5*in which the effect of weld strength mismatch is illustrated for the normal 'V'-weld and the straight-sided 'V'-weld. Similar illustrations of the effect of weld strength mismatch for the tensile geometry are shown in

*Fig.6*.

*Table 2*gives levels of

*Q*for two different levels of crack tip opening displacement ( δ=0.167mm and δ=0.068mm). The results are discussed in more detail in the following section.

**Fig.5. Comparison of results for constraint parameter Q for welds subject to bend loading for different levels of weld strength mismatch: normal 'V' weld (left), straight-sided 'V' weld (right).**

*Table 2* summarises the result for a low level of applied low (high level of *b*/δ = 400), and for a higher level of applied load *b*/δ = 150, where *b* is the remaining ligament (W-a). Note value marked * has been forward extrapolated)

Loading | Weld mismatching | Q | |||
---|---|---|---|---|---|

δ = 0.068mm( b/δ = 400) | δ= 0.167mm( b/δ = 150) | ||||

Straight sided'V' | Normal sided'V' | Straight sided'V' | Normal sided'V' | ||

Tension | Even | -0.406 | -0.515* | -0.796 | -0.810 |

Undermatch | -0.579 | -0.625* | -1.013 | -1.11 | |

Overmatch | -0.380 | -0.399 | -0.754 | -0.752 | |

Bending | Even | 0.028 | -0.066 | -0.258 | -0.244 |

Undermatch | -0.118 | -0.138 | -0.543 | -0.410 | |

Overmatch | 0.040 | -0.055 | -0.227 | -0.206 |

## 4. Discussion of results

### 4.1. Difference between bending and tensile loading

*Table 2*shows that the bending-loaded geometries exhibit

*Q*values of close to zero or slightly negative that tend to become more negative with decreasing

*b/ δ*. In contrast, the tensile-loaded cases display more negative

*Q*values for all load levels studied. This illustrates the well-known phenomenon that tensile loaded geometries lose the single-parameter crack tip fields (where

*Q*≈0) at much smaller levels of

*δ*or

*J*than do those subject to bending.

^{[12,13]}

**Fig.6 Comparison of results for constraint parameter Q for welds subject to tensile load for different levels of weld strength mismatch: normal 'V' weld (a), straight-sided 'V' weld (b).**

### 4.2. Loss of constraint in the bending loaded weld geometries

The constraint behaviour in bending is a reflection of the rapid increase in the size of crack tip plastic zone compared with the dimensions of the welded specimen.McMeeking and Parks ^{[12]} and Shih and German ^{[13]} have previously studied this loss of the *J*-controlled crack tip fields using finite element analysis. They concluded that, in bending, a minimum value of *b*σ_{o}/J equal to 25 is necessary to maintain J-characterisation. Shih and German used, as the criterion for loss of J-control, a 10% change from the HRR fields at a distance ahead of the crack tip equal to r = 2*J*/ σ_{o},or about *4* δ. This is the normalised distance at which *Q* is determined. Using a similar criteria as Shih and German, but taking account of the strain hardening and assuming a peak stress ahead of the crack tip equal to three times the yield stress, single-parameter characterisation can be assumed to break down at values of *Q* below -0.3. The breakdown of single-parameter characterisation using a criteria of *Q*=-0.3 is given in *Table 3* below.

*Table 3* Limits of single-parameter characterisation for weld geometries loaded in bending.

Weld strength mismatch | b/δ( Q=-0.3) |
---|---|

Straight sided 'V' | |

Even | 135 |

Undermatch | 250 |

Overmatch | 128 |

From *Table 3* it can be seen that breakdown of under-matched welds occurs at a much lower level of deformation (higher *b/ δ* values) than for even- and overmatched. However, it must be noted that the results from the welded plate geometry with a narrow GCHAZ are compared with a small-scale yielding solution for a crack completely surrounded by GCHAZ material. Single-parameter control is likely to be maintained at higher relative loads if the materials were completely homogeneous in the weld.

The effect of the loss of single-parameter controlled fields depends on the mode of failure of the material. It is known that cleavage failure is very sensitive to the principal stress ahead of the crack tip ^{[14]} and a large increase in effective toughness is possible when unique, single-parameter control of the crack tip fields breaks down.

### 4.3. Loss of constraint in the tensile-loaded weld geometries

Even at relatively large values of*b/ δ*, the tensile geometries show negative values of

*Q*, indicating a loss of unique, single-parameter, crack tip fields (

*Fig.6*). McMeeking and Parks

^{[12]}and Shih and German

^{[13]}suggested that, in tension, a minimum value of

*b*σ

_{o}/

*J*equal to 200 is necessary to maintain

*J*-control and thus ensure geometry-independent measurements of fracture toughness. The present results suggest that that

*J*-dominance is lost for values of

*b/ δ*as high as 500 for these tensile loaded welded plate geometries. Again, it must be noted that the welded plate results are compared with a small-scale yielding solution for all GCHAZ material.

### 4.4. Effect of weld preparation

The results show a tendency for the straight-sided bending geometry to have slightly more negative values of*Q*than the normal 'V' weld preparation (see

*Table 2*). It is not clear whether this is due to a real difference in crack tip constraint or another effect; such as a change in the asymmetry of loading of the crack tip. This latter effect may result in a change in the effective crack driving force, δ, and so the value of

*Q*via Eq.[2].

Conversely, the normal 'V' weld in tension exhibits slightly more negative values of *Q* than the straight-sided case, particularly for the under-matched weld. As noted above, these effects of weld preparation may be the result of a change in crack driving force, due to asymmetric loading of the crack, rather than a change in constraint.

### 4.5. Effect of weld mis-match

*Figure 5*illustrates the effect of mismatch for both normal and straight-sided 'V' welds subject to bending loading. At all values of

*b/ δ*the under-matched weld gives more negative

*Q*than the over-matched and even-matched cases. This reduced

*Q*is probably due to the lower strength weld metal adjacent to the cracked GCHAZ reducing the maximum stress generated ahead of the crack tip. This effect is also seen in tensile loading (

*Fig.6*). Pisarski and Harrison

^{[15]}concluded that when the crack is located in a softer under-matched GCHAZ, strains will be concentrated into this narrow, lower strength region, resulting in high triaxiality stresses which will speedup the onset of cleavage. On the other hand, when the strength of the GCHAZ is higher, as in this case, the strain is shed into the weaker surrounding material and will slow down the onset of cleavage.

## 5. Conclusions

This study uses finite element analyses to investigate how different structural geometries and loading modes, combined with different levels of weld yield strength mismatch, affect the constraint indexing parameter*Q*for a crack tip located in a weld heat affected zone. A normal 'V' weld and a straight-sided 'V' weld in a single edge notch plate geometry are modelled. These are subjected to either uniaxial tension or three-point bending. The tip of the crack is located in weld grain coarsened heat affected zone which is stronger than the parent steel or weld metal in this case. The following conclusions are reached:

- The study confirms that tension-loaded geometries lose the unique, single-parameter controlled, crack tip fields at much smaller levels of CTOD or
*J*than bending-loaded geometries. - In three-point bending, single-parameter crack tip fields are lost for values of
*b/ δ*between about 100 and 250, or*b*σ_{o}/*J*between about 50 and 125. This loss of control is most important for the under-matched weld. - In the under-matched tensile geometry, single-parameter fields are lost for values of
*b/ δ*as high as 500. - These results from welded plate geometries with a stronger, narrow GCHAZ are compared with a small-scale yielding solution for a crack completely surrounded by GCHAZ material. Single-parameter dominance would be expected to be maintained at higher relative loads in homogeneous materials.
- The straight-sided bending geometry has slightly more negative values of
*Q*than the normal 'V' weld preparation. Conversely, the normal tensile geometry has slightly more negative values of*Q*than the straight-sided case, particularly for the under-matched weld. Though it is not clear whether this is due to a real difference in crack tip constraint or another effect, such as a change in the asymmetry of loading of the crack tip. - For both normal and straight-sided 'V' welds subject to either bending and tensile loading, the under-matched weld tends to give more negative
*Q*than the over-matched and even-matched cases.

## 6. References

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*Size Effect in Fracture*, Inst. Mech. Eng., London, UK, pp.37-46. - Betegón C and Hancock J W (1991): 'Two-parameter characterisation of elastic-plastic crack-tip fields',
*Journal of Applied Mechanics***58**, pp.104-110. - Du Z Z and Hancock J W (1991): 'The effect of non-singular stresses on crack tip constraint'.
*Journal of Mechanics and Physics of Solids***39**, pp.555-567. - Al-Ani M and Hancock J (1991): 'J-dominance of short cracks in tension and bending'.
*Journal of Mechanics and Physics of Solids***39**, No.1, pp.23-43. - O'Dowd N P and Shih C F (1991): 'Family of crack-tip fields characterised by a triaxiality parameter: Part I - Structure of applications'.
*Journal of Mechanics and Physics of Solids*,**39**, pp.989-1015. - O'Dowd N P and Shih C F (1992): 'Family of crack-tip fields characterised by a triaxiality parameter: Part II - Fracture applications'.
*Journal of Mechanics and Physics of Solids*,**40**, pp.939-963. - Karstensen A (1996): 'Constraint estimation schemes in fracture mechanics'. Department of Mechanical Engineering, University of Glasgow. PhD Thesis.
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*ICF9 Vol 4 Theoretical and Computational Fracture mechanics and New Directions*p.2007 Editors B.L. Karihaloo, Y.W. Mai, M.I. Ripley and R. O. Ritchie. - Thaulow C and Toyoda M (1997): 'Strength mis-match effect on fracture behaviour of HAZ', mis-matching of interfaces and welds, Edited by K.H. Schwalbe and M.Kocak,
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