# The effect of residual stress on a centre-cracked plate under uniaxial loading

- 555 Downloads

## Abstract

The behaviour of a crack in the centre of a plate subject to a far-field applied stress perpendicular to the crack surface has been studied. The plate contains an initial, self-equilibrated residual stress, symmetric to the central position of the crack. The component of the residual stress perpendicular to the crack at the centre of the plate can be tensile or compressive. Elastic and elastic–plastic material behaviours have been considered and crack closure effects have been included in the analyses. For elastic behaviour a series of analyses based on stress intensity factor solutions have been developed to calculate the crack opening and the stress intensity factor for cracks of different lengths relative to the size of the residual stress field. Different magnitudes of applied stress relative to the magnitude of the residual stress were applied. Crack behaviour maps have been developed that show the behaviour of the crack for different crack lengths and magnitudes of applied stress. For elastic–plastic behaviour a strip yield model has been used to develop a similar set of analyses to those for the elastic case. The results compare favourably with those produced by finite element analysis. The work provides the basis for a first estimate of the likelihood of fracture for a component containing residual stress and subject to applied load.

## Keywords

Fracture mechanics Residual stress Finite element analysis Strip yield model## 1 Introduction

In an engineering component containing a crack, residual stresses interact with stresses generated by applied loading in a complex manner. Typically, numerical techniques are required to predict the likelihood of fracture, providing results that are particular to the set of conditions considered. The work we describe here represents an attempt to understand the general behaviour of the crack for a simple geometry, loading and residual stress distribution that provides insight into the results of more complex analyses.

In general, a residual stress distribution is characterised by the magnitude and location of the peak residual stress and the spatial extent of its distribution. As an example, consider the as-welded residual stresses generated by a butt-weld between two identical plates. The magnitude of the peak residual stresses depends on the welding parameters and the material properties, including the phase transformation behaviour. Assuming symmetry the in-plane location of the peak residual stress is the centreline of the weld. Finally, the spatial extent of the residual stress distribution depends on the geometry of the weld and the thickness of the plates. We now introduce a through-thickness crack into the plate, symmetric about the weld centreline and subject the plate to a uniaxial applied stress in the direction parallel to the weld centreline. When the crack length is large compared to the extent of the residual stress distribution, the residual stress can be neglected, and a fracture analysis based only on the magnitude of the applied stress. Conversely, when the crack length is small compared to the extent of the residual stress distribution, the fracture analysis must use the sum of the residual stress and applied stress. When the crack length is similar to the extent of the residual stress distribution the analysis is more difficult: the crack is subject to stresses that vary with position and may in general cause parts of the crack surface to be in contact.

Terada (1976) provided an early analysis of the effect of residual stress on a crack. He proposed a residual stress distribution to represent a butt-weld between two plates and then developed a method to evaluate the stress intensity factor assuming elastic behaviour. Tada and Paris (1983) extended Terada’s analysis to different residual stress distributions and provided an alternative method to evaluate the stress intensity factor. Terada and Najajima (1985) considered a crack asymmetrically located within the residual stress field. Chell and Ewing (1977) discussed the effect of plasticity on fracture when residual stresses exist. Wu and Carlsson (1984) calculated stress intensity factors for half-elliptical surface cracks in residual stress fields demonstrating the non-conservatism of two-dimensional analyses. Labeas and Diamantakos (2009) used finite element analysis to explore the stress intensity factor for a crack of varying length embedded in a residual stress field. Bouchard and Withers (2006) and Bouchard et al. (2012) used a Fourier approach to study the influence of the size of the residual stress distribution on the stress intensity factor. In general, previous work has not explored the behaviour of the crack as the magnitude and spacial extent of the residual stress has been varied and therefore has been unable to draw general conclusions about the behaviour of the crack.

## 2 Residual stress field

*R*centred at the origin surrounded by compressive tangential stress that decays to zero away from the origin. Figure 1a plots the stresses \(\sigma _{rr} \) and \(\sigma _{\theta \theta } \) versus

*r*and Fig. 1b shows the geometry. The minimum value of \(\sigma _{\theta \theta } \) is \(-2\sigma _{RS} \hbox {e}^{-3/2}\) at \(x=\sqrt{3}R\). When the residual stress field is introduced into an elastic–plastic material, the magnitude of \(\sigma _{RS} \) must be less than the yield stress of the material to avoid plasticity.

The eigenstrain method is an alternative approach to the use of a stress function for the introduction of a residual stress in a finite element analysis, see for example Ribeiro and Hill (2016). “Appendix A” provides a calculation of the eigenstrain corresponding to the stress function of Eq. (3).

Although the residual stress produced by the stress function of Eq. (3) is of simple form, it matches closely residual stress fields produced by a variety of different methods. For example, Fig. 2a shows the centre hole drilling results of Formby and Griffiths (1977) for a 4 mm thick rectangular steel plate heated locally between two circular copper heaters with the geometry in Fig. 2b. The heating causes the steel to yield which leaves a tensile residual stress in the centre of the plate once it cools. The results of a series of residual stress measurements are shown in Fig. 3a for a 10 mm thick circular stainless steel disc that has been compressed between two circular tool steel indenters as shown in Fig. 3b (Pagliaro et al. 2009). In this case the residual stress in the centre of the plate is compressive. Finally, Fig. 4a shows contour and neutron diffraction measurements of residual stress in a butt-welded steel plate (Balakrishnan et al. 2018). Figure 4b shows the dimensions of the plate and Fig. 4c the detail of the weld. In all sets of experiments the dimensions of the plates were not large enough for the residual stress to decay to zero. Furthermore, the residual stress distributions in Figs. 2 and Fig. 4 will not be axisymmetric.

*x*is equivalent to that of \(\sigma _{\theta \theta } \) versus

*r*shown in Fig. 1a and is identical to that used by Terada (1976). Other stress functions could be used to generate a residual stress field and would give different detailed results to those presented here. Our aim however is to explore the effect of a simple residual stress field on the behaviour of a crack, although our simple residual stress field includes all the general attributes of a residual stress field.

Although the distribution of the residual stress has been determined, the method of its creation has not been defined. Therefore, the effect of the process that caused the residual stress on the subsequent material properties, the state of hardening for example, has not been included. In the work described here the material is taken to be elastic-perfectly plastic, but in general parameters defining the history of the material in addition to the residual stresses would have to be prescribed (Lei et al. 2000).

## 3 Elastic crack behaviour

### 3.1 Tensile residual stress

Figure 6 shows the behaviour of the crack for tensile residual stress and tensile and compressive applied stress. The half-length of the crack *c* is normalised with respect to the size of the tensile region of the residual stress field *R* and the applied stress \(\sigma _{\mathrm {APP}} \) is normalised with respect to \(\sigma _{\mathrm {RS}} \). As an example, the line segment *AC* in Fig. 6 forms the boundary between two regions of crack behaviour. Above the line, for higher magnitudes of applied stress, the crack is fully open. Below the line the crack is partially open: closed at the tip but open in the centre. These two crack states are defined in Fig. 5.

For all sizes of crack, the crack is always closed if the applied stress is compressive and greater in magnitude than the value of the residual stress at \(x=0\), that is \(\sigma _{\mathrm {APP}} <-\sigma _{\mathrm {RS}} \). As the magnitude of the compressive applied stress is reduced, the crack opens first at \(x=0\) when the sum of the applied and residual stresses equals zero. The boundary between the fully closed and partially open regions of crack behaviour, the line *AB*, is therefore given by \({\sigma _{\mathrm {APP}} }/{\sigma _{\mathrm {RS}} }=-1\)

*AC*. This line is given by the condition that the total stress intensity factor \(K_{\mathrm {TOT}} \) given by the sum of stress intensity factors due to the applied stress \(K_{\mathrm {APP}} \) and the residual stress \(K_{\mathrm {RS}} \) is equal to zero:

*CD*. This line is given by the condition that the total opening displacement at the point of first contact \(\delta _{\mathrm {TOT}} \) given by the sum of the displacement due to the applied stress \(\delta _{\mathrm {APP}} \) and to the residual stress \(\delta _{\mathrm {RS}} \) is equal to zero:

For crack half-lengths greater than \(\sqrt{5}R\) a small regime of crack behaviour exists as shown in Fig. 5 where the crack is partially open: open at the tips and in the centre but closed between these points. This regime is bounded by the two lines *CD* and *CE*. To use closed form expressions to find the line *CE* requires a solution for the stress intensity factors at the multiple tips of three collinear cracks. We have not attempted to do this, instead the line *CE* in Fig. 4 has been found using finite element analysis.

*X*the crack begins to open at the centre but is closed at the tip. At the point marked

*Y*, when \({\sigma _{\mathrm {APP}} }/{\sigma _{\mathrm {RS}} }\approx -0.4446\), the crack is open completely. The stress intensity factor is zero for points between

*X*and

*Y*. For points above

*Y*the total stress intensity factor \(K_{\mathrm {TOT}} \) is calculated by

*Z*in Fig. 6, \(K_{\mathrm {APP}} =0\) and \(K_{\mathrm {TOT}} =K_{RS} \approx 0.4446\;\sigma _{\mathrm {RS}} \sqrt{\pi c}\). As shown in Fig. 7, once the crack opens at the tip, the total stress intensity varies linearly with the applied stress.

*a*(see Fig. 5 for the geometry of a partially open crack, closed at the tip and open at the centre) is calculated by the condition that the stress intensity factor at the point where the crack closes (\(x=a)\) must be zero. Therefore

*a*is calculated by

*c*is replaced by

*a*. Figure 7 also shows the half-length of the open portion of the crack

*a*versus the applied stress. The crack begins to open at an applied stress of \({\sigma _{\mathrm {APP}} }/{\sigma _{\mathrm {RS}} }=-1\) and is open completely (\(a/c=1)\) for applied stresses higher than \({\sigma _{\mathrm {APP}} }/{\sigma _{\mathrm {RS}} }\approx -0.4446\).

The finite element results shown in Fig. 7 for comparison were evaluated using the Abaqus 6.11 finite element system using a quarter mesh with symmetry constraints applied on the lower and left hand edges of the mesh as shown in Fig. 8. The residual stress distribution of Eq. (5) was prescribed using the SIGINI user subroutine facility in Abaqus. The width of the fine region of elements around the crack was four times the half-length of the crack and the width of the complete mesh 75 times the half-length of the crack. Plane stress conditions were used with linear quadrilateral elements CPS4. Calculations of stress intensity factors in this work used the JEDI code (Beardsmore 2008). This uses a contour integral evaluation and conveniently defines the contours automatically.

### 3.2 Compressive residual stress

The behaviour of the crack for negative \(\sigma _{\mathrm {RS}} \), that is when the tangential residual stress is compressive at the centre, is shown in Fig. 9. The half-length of the crack *c* is normalised with respect to the size of the compressive region of the residual stress field *R* and the applied stress \(\sigma _{\mathrm {APP}} \) is normalised with respect to \(\left| {\sigma _{\mathrm {RS}} } \right| \).

*AC*in the figure are therefore given by the condition that the sum of the residual stress and the applied stress equals zero at the crack tip:

*CE*is given by

*CD*are given by simultaneous solution of the equations

*a*and \(\sigma _{\mathrm {APP}} \) for prescribed values for

*c*and \(\sigma _{\mathrm {RS}} \). The stress intensity factors in Eq. (13) are calculated using the expressions in “Appendix B” and are given by

*AB*. This line is given by the condition that the total opening displacement \(\delta _{\mathrm {TOT}} \) at the centre of the crack due to the applied stress \(\delta _{\mathrm {APP}} \) and the residual stress \(\delta _{\mathrm {RS}} \) is equal to zero:

*X*in Fig. 9 the crack begins to open at the tip but is closed at the centre. At the point marked

*Y*the crack is open completely. The applied stress corresponding to point

*Y*is calculated to be given by \({\sigma _{\mathrm {APP}} }/{\left| {\sigma _{\mathrm {RS}} } \right| }\approx 0.7910\). The procedure for calculating the stress intensity factor for points between

*X*and

*Y*is first to choose a value for the partially open half-length

*a*(see Fig. 5 for the geometry of a partially open crack, open at the tip and closed at the centre) where \(0<a<c\). The applied stress corresponding to the chosen value of

*a*is determined from the condition that the stress intensity factor at

*a*is zero:

*a*is determined, the stress intensity factor \(K_{\mathrm {TOT}}^c \) at the tip of the crack is calculated by

*Y*in Fig. 9 the crack is open completely and the stress intensity factor is

Figure 10 also shows the half-length of the open portion of the crack *a* normalised by *c* for a crack of half-length \(c=R\) versus the applied stress. At the point marked *X* in Fig. 9 the crack begins to open at the tip and \(a=c\). At the point marked *Y* the crack just opens fully and \(a=0\). The applied stress corresponding to each value of *a* has already been found during the calculation of the stress intensity factor.

## 4 Strip yield model

We will now use the strip yield model of Dugdale (1960) and Barrenblatt (1962) to examine the effect of small-scale yielding on the behaviour of a crack in a residual stress field. We will not present revised crack behaviour maps valid for small-scale yielding since in general these maps are similar to the maps for elastic behaviour of Figs. 6 and 9. Instead we choose one crack length defined by \(c=R\) and study the behaviour of the crack as the applied load is increased from an initial state where the crack is fully closed. The approach described in this section can be used for other crack lengths.

### 4.1 Tensile residual stress

For tensile residual stress we consider an initial state where the applied stress is sufficient to ensure the crack is fully closed. For a crack of half-length \(c=R\) the applied stress \(\sigma _{\mathrm {APP}} \) must be less than (more compressive than) \(-\sigma _{\mathrm {RS}} \). As the applied stress is increased the crack opens first in the centre, as described in the previous section. It is only when the crack has opened fully to the tip that the small-scale yielding behaviour is different to the elastic behaviour. When the applied stress is larger than that to open the crack fully the strip yield model may be used to calculate the effective stress intensity factor.

*J*is the value of the

*J*-integral evaluated using the JEDI procedure (Beardsmore 2008). Very good agreement is obtained, even for applied stresses approaching the yield stress where large plastic zones develop and small-scale yielding conditions certainly do not exist.

### 4.2 Compressive residual stress

Again, we consider an initial state where the applied stress is sufficient to ensure the crack is fully closed. For a crack of half-length \(c=R\) the applied stress \(\sigma _{\mathrm {APP}} \) must be less than zero. For the case of a compressive residual stress the crack opens first at the tips and therefore the small-scale yielding behaviour will be different to the elastic behaviour. The strip yield model may still be used to calculate the effective stress intensity factor but complicated by the need to use a twin crack geometry. When the applied stress is high enough to open the crack completely, the same strip yield model as for tensile residual stress case can be used.

*a*. Again, evaluation of the effective stress intensity factor requires the crack opening displacement to be calculated at \(x=c\). The total crack opening displacement \(\delta _{\mathrm {TOT}} \) is evaluated as the sum of the crack opening due to the applied stress \(\delta _{\mathrm {APP}} \), the residual stress \(\delta _{\mathrm {RS}} \) and the stresses in the yielded zone \(\delta _{\mathrm{Y}} \). That is

*b*replaced by

*c*, and \(K_x^d \) is given by Eq. (34). Similarly, for pairs of unit point forces applied to the crack at \(x=\pm \,d\) where \(c<d<c+\rho \) we calculate the opening at \(x=c\) to be

*AB*slightly.

## 5 Discussion

### 5.1 Effect of residual stress on crack behaviour

Results have been obtained for a residual stress field derived from a closed form expression which allows the magnitude and extent of the field to be varied. Alternatively, the residual stresses may be obtained from a finite element simulation of a process that creates residual stress, welding for example, or indeed from a set of experimental measurements. The techniques described in this paper can then be used to provide exact values for the stress intensity factor for elastic material behaviour or approximate values for elastic–plastic behaviour. Although the detail of the crack behaviour will depend on the precise form of the residual stress distribution, general statements can be made of the crack behaviour based on the results presented here.

For a tensile residual stress field and with no superimposed applied stress, reference to the line *ACE* in Fig. 6 shows that the crack is open at the tip and therefore the stress intensity factor is greater than zero for almost all crack lengths, although the magnitude varies with the length of the crack. In Fig. 7 it can be seen that for elastic behaviour and for a crack of half-length \(c=R\), once the crack is open the stress intensity factor increases linearly with applied stress. For elastic–plastic behaviour, Fig. 12 shows that the effective stress intensity factor is greater than the elastic value as the applied stress approaches the yield stress.

For a compressive residual stress field, reference to the line *ACD* in Fig. 9 shows that with no superimposed applied stress the crack is closed for crack lengths less that the size of the residual stress field and open at the tips for cracks greater than the size of the field. For elastic behaviour, Fig. 10 indicates that for a crack of half-length \(c=R\), partial opening of the crack causes a nonlinear variation of stress intensity factor with applied stress until the crack opens completely whereupon the variation becomes linear. Again, it can be seen in Fig. 14 that elastic–plastic behaviour causes the effective stress intensity factors to be higher than the elastic value as the applied stress increases, but in comparison to the case of tensile residual stress the transition occurs at lower values of applied stress.

For both tensile and compressive residual stress, when the crack is partially open, open at the tips but closed elsewhere, the stress intensity factor is higher than would be calculated if crack closure was neglected.

Comparison of strip yield (SY) and finite element (FE) normalised effective stress intensity factors \({K_{\mathrm {EFF}} }/{\sigma _{\mathrm {APP}} \sqrt{\pi R}}\) for plane stress and plane strain with a yield stress \({\sigma _{\mathrm{Y}} }/{\sigma _{\mathrm {APP}} =1.5}\) for no residual stress, tensile residual stress of magnitude \({\sigma _{\mathrm {RS}} }/{\sigma _{\mathrm {APP}} }=1\) and compressive residual stress of magnitude \({\left| {\sigma _{\mathrm {RS}} } \right| }/{\sigma _{\mathrm {APP}} }=1\)

Crack length | Plane condition | No residual stress | Tensile residual stress | Compressive residual stress |
---|---|---|---|---|

\(c/R=0.6\) | Plane stress (SY) | 0.8709 | 1.415 | 0.1923 |

Plane stress (FE) | 0.8450 | 1.406 | 0.1973 | |

Plane strain (FE) | 0.8106 | 1.141 | 0.4301 | |

\(c/R=1.0\) | Plane stress (SY) | 1.124 | 1.401 | 0.7694 |

Plane stress (FE) | 1.102 | 1.406 | 0.8150 | |

Plane strain (FE) | 1.045 | 1.230 | 0.8609 | |

\(c/R=1.6\) | Plane stress (SY) | 1.422 | 1.371 | 1.481 |

Plane stress (FE) | 1.384 | 1.348 | 1.403 | |

Plane strain (FE) | 1.327 | 1.276 | 1.334 |

Our results can be used to provide a first estimate of the likelihood of fracture for a crack in a residual stress field. They also enable the stress intensity factor range to be calculated from a given cycle of applied stress to allow predictions of fatigue crack growth to be made.

### 5.2 Influence of plane conditions

The results of the analysis of the elastic behaviour of the crack do not depend on plane conditions, that is identical results are obtained for plane strain as for plane stress. Although crack opening displacements are calculated using the effective modulus \({E}'\) which has a different value for plane strain compared to plane stress, these displacements are only used to find the boundary between different regions of crack behaviour. The positions of these boundaries are unaffected by the value of the effective modulus.

The strip yield model used previously does however assume plane stress conditions. The size of the strip yield zone calculated in the model depends only on the stress normal to the crack. If plane strain conditions exist, stresses would be developed perpendicular to the plane and these would influence the calculation of the strip yield zone. Although, Neimitz (2004) has described modifications to the strip yield model to account for variations of in-plane and out-of-plane constraint, no attempt has been made here to extend the analysis to plane strain. Rice (1974) remarked the effect of constraint on the size of the plastic zone was much more marked than the effect on the effective stress intensity factor. Nevertheless, a finite element study has been carried out to explore the sensitivity of the results to plane conditions.

Table 1 shows the results of the strip yield model and the finite element study for the case of no residual stress, tensile residual stress and compressive residual stress. Values of the normalised effective stress intensity factor are given for three different crack sizes \(c/R=0.6,\;1.0,\;1.6\) and one magnitude of the yield stress relative to the applied stress defined by \({\sigma _{\mathrm{Y}} }/{\sigma _{\mathrm {APP}} =}1.5\).

For the case of no residual stress, effective stress intensities are higher for longer cracks but changing the plane condition does not have a significant effect, although the plane stress values are slightly higher than plane strain. For the range of conditions considered in the table, the strip yield results are within 3% of the finite element results for plane stress. For the case of tensile residual stress, the magnitude of the residual stress field is defined by \({\sigma _{\mathrm {RS}} }/{\sigma _{\mathrm {APP}} }=1\). Again, plane conditions do not have a significant effect and the agreement between the strip yield and finite element results is very good.

### 5.3 Effect of crack length

The equations presented in Sections 3 and 4 of this paper may be used to find the stress intensity factors for cracks of different lengths, as presented in Fig. 17. The figure shows normalised stress intensity factor versus normalised crack length for elastic material behaviour for cases of no residual stress, and tensile and compressive residual stress with \(\left| {\sigma _{\mathrm {RS}} } \right| =\sigma _{\mathrm {APP}} \). Figure 17 also shows normalised effective stress intensity factors for elastic–plastic behaviour where \(\sigma _{\mathrm{Y}} =1.5\;\sigma _{\mathrm {APP}} \). These elastic–plastic results strictly are only valid for a static crack, that is where the crack has been introduced into a residual stress field all in one go. Sherry et al. (2006) suggest that the effective stress intensity factor for a crack that has been introduced progressively into a residual stress field is less than that for a static crack of the same length. In the figure, symbols are used to differentiate between LEFM and strip-yield results and do not represent FE calculated values.

### 5.4 Comparison with structural integrity procedures

*V*factor to account for the interaction between the applied and residual stresses due to plasticity. The

*V*factor allows the fracture ratio, \(K_r \) to be calculated which is used in the subsequent assessment of structural integrity:

*V*factor closer to zero in Eq. (25) results in the calculation of a smaller fracture ratio and suggests a safer loading condition. Recommended procedures to determine

*V*factors can be found in R6 or BS 7910. Both codes suggest that

*V*should be taken to be zero if the residual stress is compressive, although this can be seen to be overly conservative by reference to Fig. 17 (at \(c/R=0.5\), the stress intensity factor for combined applied and compressive residual stress is much less than that for applied stress only).

*V*factor to be estimated without detailed finite element analysis using the equation

Figure 18 compares the *V* factors obtained by the simplified procedures in R6 and BS 7910 with those obtained using Eq. (26) for the case of tensile residual stress. The *V* factors predicted using Eq. (26) are much less than those estimated using the simplified procedures, suggesting these procedures result in conservative assessment outcomes. However, *V* factors obtained using Eq. (26) are calculated using an elastic-perfectly plastic material model while the simplified procedures make an account for hardening.

## 6 Conclusions

Analytical and finite element methods have been used to map the elastic behaviour of a crack in a residual stress field. Depending on the length of the crack and the magnitude of the superimposed applied stress the crack may be fully open, partially open and open at the tip, partially open but closed at the tip or fully closed. Of course, a non-zero stress intensity factor only develops when the crack is open at the tip.

Effective stress intensity factors for an elastic-perfectly plastic crack in a residual stress field under plane stress conditions have been calculated using strip yield and finite element methods with good agreement, even for high levels of applied stress approaching the yield stress. The crack opening behaviour is similar to that of an elastic crack.

Plane stress and plane strain conditions for an elastic-perfectly crack in a residual stress field have been compared using finite element analysis for a range of crack lengths and applied stresses. The effective stress intensity factors are similar except that under plane stress conditions and compressive residual stress some yielding occurs away from the crack tip leading to significantly lower stress intensity factors for shorter crack lengths.

## Notes

### Acknowledgements

Numerical evaluation of the expressions in this paper were carried out using Mathematica 9.0. Finite element analysis used Abaqus 6.11. We are grateful to DW Beardsmore of Wood plc, UK for providing access to the JEDI code. MB Prime provided the residual stress measurements used in Fig. 3, while AN Vasileiou provided the measurements used in Fig. 4. G. Wu was a research student at the University of Bristol funded by EURATOM (Grant Number 249648). C.J. Aird was a post-doctoral research associate at the University of Bristol funded by EPSRC (Grant EP/F029748).

## Supplementary material

## References

- Balakrishnan J, Vasileiou AN, Francis JA, Smith MC, Roy MJ, Callaghan MD (2018) Residual stress distributions in arc, laser and electron-beam welds in 30 mm thick SA508 steel: a cross-process comparison. Int J Press Vess Pip 162:59–70CrossRefGoogle Scholar
- Barrenblatt DS (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–125CrossRefGoogle Scholar
- Beardsmore DW (2008) A code for the calculation of J for cracks inserted in initial strain fields for the role of J and Q in the predictions of crack extension and fracture. In: Proceedings of the ASME 2008 pressure vessels and piping conference, Chicago, USA, paper no. PVP2008-61169, pp 955–966Google Scholar
- Becker W (1997) Closed-form modeling of the unloaded mode I Dugdale crack. Eng Fract Mech 57:355–364CrossRefGoogle Scholar
- Bhargava RR, Hasan S (2011) Crack opening displacement for two unequal straight cracks with coalesced plastic zones—a modified Dugdale approach. Appl Math Model 35:3788–3796CrossRefGoogle Scholar
- Bouchard PJ, Withers PJ (2006) Identification of residual stress length scales in welds for fracture assessment. In: Proceedings of the 16th European conference on fracture, pp 163–176Google Scholar
- Bouchard PJ, Budden PJ, Withers PJ (2012) Fourier basis for the engineering assessment of cracks in residual stress fields. Eng Fract Mech 91:37–50CrossRefGoogle Scholar
- BSI (2013) BS 7910:2013, Guide on methods for assessing the susceptibility of flaws in structures, Incorporating Amendment 1 and Corrigenda 1 and 2Google Scholar
- Burdekin FM, Stone DEW (1966) The crack opening displacement approach to fracture mechanics in yielding materials. J Strain Anal Eng Des 1:145–153CrossRefGoogle Scholar
- Chang Dh, Kotousov A (2012) A strip yield model for two collinear cracks. Eng Fract Mech 90:121–128CrossRefGoogle Scholar
- Chell GG (1976) Bilby, Cottrell and Swinden model solutions for centre and edge cracked plates subject to arbitrary mode I loading. Int J Fract 12:135–147Google Scholar
- Chell GG, Ewing DJF (1977) The role of thermal and residual stresses in linear elastic and post yield fracture mechanics. Int J Fract 13:467–479CrossRefGoogle Scholar
- Chen YZ (2014) Numerical solution of a Dugdale-type crack problem for two cracks in series. Eng Fract Mech 117:152–158CrossRefGoogle Scholar
- Collins RA, Cartwright DJ (2001) An analytical solution for two equal-length collinear strip yield cracks. Eng Fract Mech 68:915–924CrossRefGoogle Scholar
- Daniewicz SR (1994) A closed-form small-scale yielding collinear strip yield model for strain hardening materials. Eng Fract Mech 49:95–103CrossRefGoogle Scholar
- DeWald AT, Hill MR (2006) Multi-axial contour method for mapping residual stresses in continuously processed bodies. Exp Mech 46:473–490CrossRefGoogle Scholar
- Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8:100–104CrossRefGoogle Scholar
- EDF Energy (2001) R6 Revision 4. Assessment of the integrity of structures containing defects, 2001, with amendments to 2014Google Scholar
- Erdogan F (1962) On the stress distribution in plates with collinear cuts under arbitrary loads. In: Proceedings of the 4th National US congress on applied mechanics, pp 547–553Google Scholar
- Formby CL, Griffiths JR (1977) The role of residual stress in the fracture of steel. In: Residual stresses in welded construction and their effects. The Welding Institute, London, pp 359–374Google Scholar
- Goudar DM, Smith DJ (2013) Validation of mechanical strain relaxation methods for stress measurement. Exp Mech 53:267–286CrossRefGoogle Scholar
- Kfouri AP (1979) Crack separation energy-rates for the DBCS model under biaxial modes of loading. J Mech Phys Solids 27:135–150CrossRefGoogle Scholar
- Kim YJ, Oh CG, Kim JS, Jin TE (2006) A simplified method to estimate elastic–plastic fracture mechanics parameters under combined primary and secondary stresses. Key Eng Mater 306–308:655–660CrossRefGoogle Scholar
- Labeas G, Diamantakos I (2009) Numerical investigation of through crack behaviour under welding residual stresses. Eng Fract Mech 76:1691–1702CrossRefGoogle Scholar
- Lei Y, O’Dowd NP, Webster GA (2000) Fracture mechanics analysis of a crack in a residual stress field. Int J Fract 106:195–216CrossRefGoogle Scholar
- Liu QC (1998) Plastic strip model of cracked weld joint with residual stresses. Theor Appl Fract Mech 30:51–63CrossRefGoogle Scholar
- Lowengrub M, Srivastava KN (1968) On two coplanar Griffith cracks in an infinite elastic medium. Int J Eng Sci 6:359–362CrossRefGoogle Scholar
- Mahmoudi AH, Truman CE, Smith DJ, Pavier MJ (2011) The effect of plasticity on the ability of the deep hole drilling technique to measure residual stress. Int J Mech Sci 53:978–988CrossRefGoogle Scholar
- Maiti M, De SK, Palit SS (1979) In the opening of two coplanar Griffith cracks. Int J Fract 15:41–46CrossRefGoogle Scholar
- Neimitz A (2004) Modification of Dugdale model to include the work hardening and in- and out-of-plane constraints. Eng Fract Mech 71:1585–1600CrossRefGoogle Scholar
- Nishimura T (2002) Strip yield analysis of two collinear unequal cracks in an infinite sheet. Eng Fract Mech 69:1173–1191CrossRefGoogle Scholar
- Pagliaro P, Prime MB, Clausen B, Lovato ML, Zuccarello B (2009) Known residual stress specimens using opposed indentation. J Eng Mater Technol 131:031002–031002CrossRefGoogle Scholar
- Radayev YN, Stepanova LV (2001) On the effect of the residual stresses on the crack opening displacement in a cracked sheet. Int J Fract 107:329–360CrossRefGoogle Scholar
- Ribeiro RL, Hill MR (2016) A benchmark fracture mechanics solution for a two-dimensional eigenstrain problem considering residual stress, the stress intensity factor and superposition. Eng Fract Mech 163:313–326CrossRefGoogle Scholar
- Rice JR (1974) Limitations to the small-scale yielding approximation for crack tip plasticity. J Mech Phys Solids 22:17–26CrossRefGoogle Scholar
- Sherry AH, Fonseca JQ, Goldthorpe MR, Taylor K (2006) Measurement and modelling of residual stress effect on cracks. Fatigue Fract Eng Mater Struct 30:243–257CrossRefGoogle Scholar
- Smith E (1964) The spread of plasticity between two cracks. Int J Eng Sci 2:379–387CrossRefGoogle Scholar
- Smith MC, Smith AC (2009) NeT bead-on-plate round robin: comparison of residual stress predictions. Int J Press Vess Pip 86:79–95CrossRefGoogle Scholar
- Tada H, Paris PC (1983) The stress intensity factor for a crack perpendicular to the welding-bead. Int J Fract 21:279–284CrossRefGoogle Scholar
- Tada H, Paris PC, Irwin GR (2000) The stress analysis of cracks handbook, 3rd edn. ASME Press, New YorkCrossRefGoogle Scholar
- Terada H (1976) An analysis of the stress intensity factor of a crack perpendicular to the welding bead. Eng Fract Mech 8:441–111CrossRefGoogle Scholar
- Terada H, Najajima T (1985) Analysis of stress intensity factor of a crack approaching welding bead. Int J Fract 27:83–90Google Scholar
- Theocaris PS (1983) Dugdale model for two collinear unequal cracks. Eng Fract Mech 18:545–559CrossRefGoogle Scholar
- Timoshenko PS, Goodier JN (1982) Theory of elasticity, 3rd edn. McGraw-Hill, New York Google Scholar
- Tranter CJ (1961) The opening of a pair of coplanar Griffith cracks under internal pressure. Q J Mech Appl Math 14:283–292CrossRefGoogle Scholar
- Wang GS (1999) A strip yield analysis of fatigue crack growth in the residual stress field. Int J Fract 96:247–277CrossRefGoogle Scholar
- Wu XR, Carlsson J (1984) Welding residual stress intensity factors for half-elliptical surface cracks in thin and thick plates. Eng Fract Mech 19:407–426CrossRefGoogle Scholar
- Xu W, Wu XR, Wang H (2011) Weight functions and strip yield solution for two equal-length collinear cracks in an infinite sheet. Eng Fract Mech 78:2356–2368CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.