Effect of Residual Stress on Cleavage Fracture Toughness by Using Cohesive Zone Model

Fatigue & Fracture of Engineering Materials & Structures doi: 10.1111/j.1460-2695.2011.01550.x

Effect of residual stress on cleavage fracture toughness by using cohesive zone model

X. B. REN1,Z.L.ZHANG1 andB.NYHUS2 1Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), N-7491, Trondheim, Norway, 2SINTEF Materials and Chemistry, N-7465 Trondheim, Norway

Received in final form 13 Dec 2010

ABSTRACT This study presents the effect of residual stresses on cleavage fracture toughness by using the cohesive zone model under mode I, plane stain conditions. Modified boundary layer simulations were performed with the remote boundary conditions governed by the elastic K-field and T-stress. The eigenstrain method was used to introduce residual stresses into the finite element model. A layer of cohesive elements was deployed ahead of the crack tip to simulate the fracture process zone. A bilinear traction–separation-law was used to characterize the behaviour of the cohesive elements. It was assumed that the initiation of the crack occurs when the opening stress drops to zero at the first integration point of the first cohesive element ahead of the crack tip. Results show that tensile residual stresses can decrease the cleavage fracture toughness significantly. The effect of the weld zone size on cleavage fracture toughness was also investigated, and it has been found that the initiation toughness is the linear function of the size of the geometrically similar weld. Results also show that the effect of the residual stress is stronger for negative T-stress while its effect is relatively smaller for positive T-stress. The influence of damage parameters and material hardening was also studied.

Keywords cleavage toughness; cohesive zone model; eigenstrain method; modified boundary layer model; residual stress.

INTRODUCTION its highly localized character of the failure mechanism.5 Understanding how residual stresses influence the cleav- Residual stresses associated with welding are unavoidable age fracture behaviour becomes more and more impor- and play an important role in structural integrity of engi- tant when high-strength steels are increasingly utilized in neering components. It has been shown that the residual offshore industry. stresses influence both the crack driving force1 and crack- Experimental work undertaken by Mirzaee-Sisan et al.7 tip constraint.2–4 Crack-tip constraint arises from differ- indicated an apparent reduction in mean cleavage fracture ences in the level of opening stress near the crack tip and toughness of an A553-B ferritic steel of 50% from con- affects the transferability of material toughness, which is ventional fracture toughness data due to residual stresses. a key issue in application of fracture mechanics to assess Panontin and Hill6 utilized the Ritchie–Knott–Rice the integrity of structural components.5 Cleavage fracture (RKR)8 model to predict the effect of residual stresses featured with negligible plastic tearing before final failure on brittle fracture initiation and found that the constraint is often the most dangerous failure mode. It occurs by the generated by the residual stress decreases the initiation unstable propagation of microcracks formed within grain toughness of brittle fracture. boundary particles by twinning or slip dislocation pile-ups Micromechanical models using continuum representa- and then grows into the ferrite matrix under the action of tion of stress and strain are generally used to predict lo- tensile stress.6 The cleavage fracture toughness exhibits cal conditions for cleavage fracture. For cleavage fracture sensitivity to the local stress and deformation fields due to to happen, the opening stress should reach the critical σ value c at a certain distance from the crack tip rc or 9 Correspondence: Z. L. Zhang. E-mail: [email protected] within a certain volume in front of the crack tip. This

592 c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 RESBrittle 593 physical scale must be considered in studying the mi- cromechanisms of fracture in order to consider mi- crostructural features necessary for the physical failure mechanism. It was assumed in the RKR model that occur- rence of cleavage fracture requires that the opening stress σ σ ∗ 22 at the crack tip exceeds the fracture stress f over a ∗ critical distance lb which is on the order of the grain size of the steel. Previous studies concerning cleavage fracture indicate that the critical fracture stress ranges from three to four times the yield strength of the material, and that Fig. 1 Schematic plot of the assumption made in this study. is relatively independent of temperature and strain rate. Estimates of the characteristic length or distance in mild steels range from 2 to 5 grain diameters.6 However, in real elastic–plastic materials, large plastic deformations are of- introduced. The cohesive zone model was used to model ten necessary to initiate the cleavage fracture. Therefore, the fracture process zone (FPZ). The material property Neimitz et al.9 proposed an alternative formulation of the mismatch between the weld metal and base metal has not RKR criterion. The cleavage fracture was demonstrated been taken into account. as a synergistic action of the stress and deformation at the critical moment. It has been demonstrated that for fracture to occur it is not sufficient that the opening stress Finite element model reaches the critical value alone, but it is also necessary that The MBL model used for this study consists of a weld the location of this maximum from the crack tip must be region located in the centre of the model and an outer ≥ over the distance l lc,wherelc is considered as a material base metal region, and a sharp crack was embedded in the parameter. centre of weld. The load was applied to the remote edges The current study investigated the effect of the residual of the model through a displacement field (u, v) governed stresses on the cleavage fracture toughness by employing by the elastic asymptotic stress field of a plane strain mode a modified boundary layer (MBL) model under mode I I crack: plane strain conditions with the remote boundary gov- + v erned by elastic K-field and T-stress. The cohesive zone ,θ = 1 r 1θ − v − θ u(r ) K I cos (3 4 cos ) model was utilized to simulate the cleavage fracture, and E 2π 2 the failure was assumed to occur when the opening stress 1 − v2 +T r cos θ at the first integration point of the first cohesive element E (1) ahead of the crack tip drops to zero, as illustrated in Fig. 1. 1 + v r 1 v(r,θ) = K sin θ (3 − 4v − cos θ) The assumption is based on the modified RKR criterion I E 2π 2 proposed by Neimitz et al.,9 and the large strain effect was v(1 + v) −T r sin θ, also considered in the analyses. Based on this assumption, E the effect of the residual stresses on cleavage toughness = / − v2 was investigated by comparing the case including residual where K I EJ 1 under plane strain condition, stress effect with the reference case. E is Young’s modulus, ν is Poisson’s ratio, and r and θ are polar coordinates centred at the crack tip with θ = 0 corresponding to the crack tip. NUMERICAL PROCEDURE The finite element computations were performed using ABAQUS.10 The radius of the MBL model was taken as Problem description 200 mm. A layer of uniform-sized cohesive elements was This study concerns an ideal problem. A large round deployed along the central line ahead of the crack tip to cylinder with a weld in the centre was studied. The cylin- simulate the fracture process. The length of the cohesive der was simulated by a 2D plane strain MBL model with element-layer is 20 mm, and the size of the uniform co- the remote boundary governed by the elastic K-field and hesive element lc is 0.1 mm. The thickness of the cohesive − T-stress. The analysis procedure consists of the following elements is 2.5 × 10 4 mm. The weld metal and base metal steps: (1) enforce a welding procedure, which introduce region of the model was meshed by standard full integra- a residual stress field; (2) introduce a sharp crack; (3) ap- tion four-node 2D plane strain elements. The cohesive ply the external load, as illustrated in Fig. 2. It should elements are standard cohesive element COH2D4. The be noted that the contact between the upper and lower finite element model has 4992 elements and the meshes free surfaces were considered when the residual stress was areshowninFig.3.

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 594 X. B. REN et al.

Fig. 2 Illustration of the problem, (a) the round cylinder; (b) welding at the centre and a sharp crack is introduced and (c) the external load applied.

207,000 MPa, yield stress σ0 is 460 MPa, which corre- sponds to a typical modern pipeline steel. ε0 = σ0/E is the yield strain. n is the plastic strain hardening exponent, and low n represents low hardening. Different thermal expansion coefficients of the base metal and weld metal are assumed to introduce the residual stresses into the model by so-called eigenstrain method. It should be noted that the thermal expansion coefficients here are not physical thermal coefficients, but are used to introduce the residual stress into the computational model. The eigenstrain method was also called ‘inherent strain’ method when first introduced by Ueda et al.11 The concept of the eigenstrain method is that the source of residual stress is an incompatible strain field caused by plastic deformation, thermal strains and phase-transformation, etc.12 Thus, if the distribution of the eigenstrain is known, the distribution of residual stresses can be obtained through linear elastic calcula- tion by using the finite element method. Our approach is to set the eigenstrain values equal to the thermal expansion coefficients for different regions first. The model is then loaded by applying a unit temperature decrease, thereby introducing the residual stress field of interest. Similar approaches were also applied in the literature,13,14 and their results showed that the distributions of residual stresses obtained by such approaches agreed well with the experimental results.

Fig. 3 Finite element meshes for the MBL model, (a) global view Cohesive zone model and (b) crack-tip region and the illustration of the weld region. An important issue when considering failure is the ob- servation that most engineering materials are not per- Material fectly in the Griffith sense, but display some ductility after reaching the strength limit.15 In fact, there exists a small The weld metal and base metal were assumed to have the FPZ in which small-scale-yielding, micro-cracking and same elastic–plastic properties. Rate-independent power void initiation, growth and coalescence may take place. law strain hardening materials were assumed to have the A proper process zone model is then needed to charac- following form: terize the fracture process and describe the local fracture ε p n behaviour. Among the various process models, the cohe- σ = σ + , f 0 1 ε (2) sive zone model seems particularly attractive for practical 0 application because it is applicable to a wide range of σ ε p 16 where f is the flow stress; is the equivalent plastic materials and fracture mechanisms. The cohesive zone strain, the Poisson’s ratio ν is 0.3, Young’s modulus is model was introduced by Barenblatt17 and Dugdale18 for

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 RESBrittle 595

RESULTS AND DISCUSSION Cleavage fracture toughness exhibits a strong sensitivity to the local stress and deformation fields due to its highly localized character. Residual stresses affect both the crack driving forces and crack tip constraint,1,3,4,6 which may further influence the cleavage fracture toughness. Therefore, the effect of residual stresses on cleavage fracture toughness was investigated in this study. The contour J-integral26 was utilized as the measure of the cleavage fracture toughness. Lei et al.1 and Lei27 showed Fig. 4 Schematic plot of the TSL used in the analysis. that the J-integral shows path-dependence with the presence of the residual stresses. In our study, the computed far-field J-integral by ABAQUS shows practically path- elastic–plastic fracture in ductile metals and for quasi- independence beyond the large strain region for the cases brittle materials by Hillerborg et al.19 under the name of investigated. The J-integral in the following means the fictitious crack model. calculated J-integral. The fundamental concept of the cohesive zone model is a so-called traction–separation law (TSL), which is a function described by the cohesive stress (σ ) and separa- Residual stress fields tion (δ).20 The area under the traction–separation relation represents the cohesive energy 0. The basic parameters As described before, we used the eigenstrain method to necessary to describe the TSL are two among the critical introduce residual stresses into the finite element model. σ δ traction max, the critical separation c and the cohesive A rectangular weld region was constructed in the centre 21 energy 0. of the MBL model, as illustrated in Fig. 3. The thermal α One of the key problems in the application of the co- expansion coefficient of the base metal b was assumed to hesive zone model is the choice of the TSL within the be zero, and it was assumed to be orthogonal and repre- cohesive zone. Needleman first proposed a polynomial sented by α11 and α22 for weld metal. The ratio α11/α22 22 law, and later an exponential law was introduced by Xu was fixed to be 2, and by setting α22 = –0.0005, 0.0005, and Needleman.23 Tvergaard and Hutchinson21 proposed 0.001 and 0.002, four residual stress fields were generated a trapezoidal law for ductile fracture. The TSL used in and designated as RsField0, RsField1, RsField2 and Rs- this paper is a bilinear relationship between the traction Field3, respectively, as shown in Fig. 5. It should be noted and the separation, as shown in Fig. 4, which is charac- that the eigenstrain values selected here are taken from teristic of brittle materials.16 The dominant parameters the experimental measurement results in literature.28,29 are cohesive energy 0 and the maximum cohesive stress The residual stress fields generated by these values have σ max. similar distribution to that shown in Ref. [30]. To obtain When a cohesive zone model is employed to simulate accurate distribution of the residual stress fields by eigen- the cracking behaviour of a brittle thin interface, the soft- strain method, one should carry out the experiments to ening part of the TSL may cause some problems to the measure the distribution of the eigenstrain. However, the solution algorithm. A snap-back instability can occur de- main objective of this study is to investigate the effect of pending on interface thickness, stiffness and the length of the residual stresses; the prediction of the real distribution the elements adjacent to the cohesive zone.24 If a discon- of the residual stress field is outside the scope. Note that tinuity of the response occurs, the simulation can stop. the stress components are normalized by the yield stress, A possible solution is the viscous regularization method and the distance from the crack tip x is normalized by the 24 proposed by Chaboche et al. which consists in intro- size of the uniform element size lc. ducing a fictitious viscosity parameter in the constitutive It can be seen that the negative eigenstrain value intro- equation of the cohesive elements. In turn, the conver- duces the compressive residual stress at the weld region gence of the solution can be achieved by dissipating excess while the positive ones generate tensile residual stresses. energy; but the value of the viscosity parameter should Both tensile and compressive residual stresses parallel to be small enough to not affect the results. Pezzotta and the crack front converge to zero far from the crack tip. Zhang25 demonstrated that when the viscosity value v The opening residual stresses are self-balanced ahead of ≤ 1.0E-5 the predicted failure becomes independent of the crack tip. There is a sharp turning point in the dis- the viscosity parameter when other parameters are fixed. tribution of the opening residual stresses, which is the Thus, the value was used for all the calculations in this region where eigenstrain discontinuities have been intro- study. duced into the FE model, namely a weld metal–base metal

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 596 X. B. REN et al.

Fig. 6 Cleavage toughness as the function of the crack growth length when the incremental plasticity model is used for surrounding materials. E/σ 0 = 450, ν = 0.3; n = 0.1; 0 = 100 N/mm, σ max = 3σ 0.

N/mm were selected and the maximum cohesive stress σ max was set to be 3σ 0. The cleavage toughness was plotted as the function of crack growth length in Fig. 6. It can be seen that the crack growth resistances are almost flat for both with and without residual stress cases. Figure 6 also indicates that the cohesive zone model with bilinear TSL is applicable to predict the cleavage fracture toughness. In the following context, the initiation fracture toughness (Jc) predicted according to the assumption made in first section will be investigated. Residual stresses may influence both the FPZ and plasticity of surrounding materials. Therefore, three different constitutive models, Fig. 5 Residual stress distributions in the MBL model, (a) stress i.e. incremental plasticity, deformation plasticity and elas- component parallel to the crack plane; (b) normal to the crack tic, were employed to characterize different behaviour of plane. Four different residual stress cases were considered, where surrounding materials. The relationship between JC and RsField0 is compressive and the remaining three are tensile. the eigenstrain value α22 is shown in Fig. 7. It should be α / α = 2, α =−0.0005 is for RsField0, α = 0.0005, 0.001 11 22 22 22 noted that α = 0 represents the reference case without and 0.002 for RsField1, RsField2 and Rsfield3, respectively. 22 residual stresses. As shown in Fig. 7, the compressive residual stress (α22 boundary. The tensile residual stresses also show similar- < 0) increases the cleavage fracture toughness while the ity, and the level of the tensile residual stress increases tensile residual stresses decrease the cleavage fracture with the increase of α22. Due to the crack-tip singularity, toughness. With the increase of tensile residual stress, σ 11 is about 960 MPa and σ 22 is about 1380 MPa at the the cleavage fracture toughness decreases. Also observe crack tip for RsField3. that the effect of residual stress on fracture toughness is almost the same for elastic and deformation plasticity surrounding materials. However, the cleavage fracture Effect of residual stresses on cleavage fracture toughness for surrounding materials predicted with incre- toughness mental plasticity is significantly larger than in other cases. Cleavage fracture is a crucial failure mode in practice, and When the surrounding material is elastic or character- understanding how residual stresses affect the cleavage ized by deformation plasticity model, the cleavage frac- fracture is very important. The effect of residual stresses, ture toughness without residual stress (α22 = 0) equals as shown in Fig. 5, on cleavage fracture toughness was in- to cohesive energy 0, which represents energy needed vestigated in this section. Cohesive parameters 0 = 100 to advance the crack in the absence of plasticity. For

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 RESBrittle 597

Fig. 9 Plastic zone size for εp = 1% at the crack initiation, and the incremental plasticity model is used for surrounding materials. E/σ 0 = 450, ν = 0.3; n = 0.1; 0 = 100 N/mm, σ max = 3σ 0.

Fig. 7 Cleavage fracture toughness as the function of α . = 0.1; 22 n , which indicates the decreasing of the plastic zone = 100 N/mm, σ = 3σ . 0 0 max 0 size of surrounding materials. When residual stress is present, it may influence the plastic deformation of the surrounding materials. Figure 9 shows the plastic zone size when cleavage fracture occurs for different residual stresses. Figure 9 shows that the compressive residual stress field, i.e. RsField0, both enlarges the maximum radius of the plastic zone and causes plastic zone to swing forward. In contrast, tensile residual stress fields cause the plastic zone to decrease in size and rotate backward. Similar behaviour has been reported by Du and Hancock31 who investigated the effect of T-stress on the crack-tip constraint. Negative T-stress indicates loss of the crack-tip constraint while the positive T-stress has the opposite effect. Thus, we may conclude that the compressive residual stress reduces the crack-tip constraint and enlarges the plastic zone, which in turn enhances the cleavage fracture Fig. 8 Plasticity contributions from the surrounding materials at toughness significantly. Unlike the compressive resid- = = σ = σ the crack initiation. n 0.1; 0 100 N/mm, max 3 0. ual stress, tensile residual stresses increased the crack-tip constraint and reduced the cleavage toughness. incremental plasticity material model, it can be seen that cleavage fracture toughness is larger than 1. For elastic Effect of weld zone size on cleavage fracture and deformation plasticity model, the deformation can toughness return back to the original state when the load is re- moved. However, when an incremental plasticity model The length scale of residual stress field may play an im- is used and unloading occurs, the plastic deformation portant role on the effect of residual stress on cleavage will be retained and the energy will be dissipated, which fracture toughness. To better demonstrate this, three ge- in turn increases the fracture toughness. In the follow- ometrically similar rectangular weld regions were con- ing, the incremental plasticity model has been used for structed, as shown in Fig. 10. The size of the weld is the study. In order to better understand the contribu- designated as c. Eigenstrain values α11 = 0.004 and α22 tion of plasticity of surrounding material, the effect of = 0.002 have been used to generate residual stress field different σ 0 on cleavage fracture toughness in the ab- for all welds. Residual stress fields represented by Size1, sence of residual stresses was investigated. Figure 8 shows Size2 and Size3, respectively, are shown in Fig. 11, in the relationship between cleavage fracture toughness and which residual stresses are normalized by the yield stress σ 0/E. and the distance from the crack tip is normalized by lc. It can be seen that with the increase of yield stress, the Residual stresses are tensile in the weld metal and show cleavage fracture toughness decreases and approaches to similar feature as the previous residual stresses showed in

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 598 X. B. REN et al.

Fig. 10 Schematic plot of different weld zone sizes considered in the study.

Fig. 5. With the increase of the weld zone size, residual stress components parallel to the crack plane increases, and the size of the tensile-dominated region of opening stress also increases. Figure 12 shows that the cleavage fracture toughness decreases with the increase of the weld region size c,which can be expected because both the residual stress level and tensile-dominated residual stress region increases with the increase of the weld zone size. Furthermore, it is interesting to observe that the relationship between the cleavage fracture toughness and the weld zone size can be fitted by = + a linear function, i.e. y –0.05432x 1.409, which can Fig. 11 Residual stress distributions in the MBL model for possibly be used to predict the effect of residual stress on different weld zone sizes, (a) components parallel to the crack the cleavage fracture toughness for geometrically similar plane, and (b) normal to the crack plane. lc is the uniform element σ = ν = = = welds. size close to the crack tip. E/ 0 450, 0.3; n 0.1; 0 100 N/mm, σ max = 3σ 0; α11 = 0.004, α22 = 0.002.

Effect of residual stresses on cleavage fracture for weaker hardening material. Hence, the effect of plastic toughness for different hardening dissipation becomes significant, and a larger reduction of In this study, the effect of the residual stress on the cleav- the cleavage fracture toughness can be expected. age fracture toughness was investigated for three hardening exponents, i.e. n = 0.05, 0.1 and 0.2. Residual stress Effect of damage parameters field with α11 = 0.004 and α22 = 0.002, i.e. RsField3, was introduced for study. The relationship between the The cohesive energy 0 and the maximum cohesive stress cleavage fracture toughness and the hardening exponent σ max are two dominant parameters of the TSL. The effect for both with and without residual stress is showed in of the residual stress on the cleavage fracture toughness Fig. 13, and the difference between two curves represents may vary for different cohesive zone parameters. In this the effect of residual stress. study, we firstly investigated the effect of the residual As shown in Fig. 13, the existence of residual stress re- stress on the cleavage fracture toughness for different 0 duces the cleavage fracture toughness for all the cases with the same σ max, and then for the same 0 with dif- analysed. However, the effect of the residual stress de- ferent σ max. Residual stress field with α11 = 0.004 and creases with the increase of material hardening. As is α22 = 0.002, i.e. RsField3, was introduced for the study. known, fully developed plastic zone is easier to be achieved Figure 14 shows the comparison of the cleavage fracture

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 RESBrittle 599

Fig. 12 Effect of weld zone size on cleavage fracture toughness. E/σ 0 = 450, ν = 0.3; n = 0.1; 0 = 100 N/mm, σ max = 3σ 0; α11 = 0.004, α22 = 0.002.

Fig. 14 Relationship between the fracture toughness and cohesive energy, (a) absolute difference between the case with and without residual stress, and (b) normalized values. E/σ 0 = 450, ν = 0.3; n = Fig. 13 Effect of residual stress on cleavage fracture toughness for 0.1; σ max = 3σ 0; α11 = 0.004, α22 = 0.002. different hardening. E/σ 0 = 450, ν = 0.3; 0 = 100 N/mm, σ max = 3σ 0; α11 = 0.004, α22 = 0.002. As shown in Fig. 15a the residual stress increases the length of FPZ when 0 < 150 N/mm, beyond which the toughness Jc for both with and without residual stress as residual stress does not affect the FPZ for the cases stud- the function of cohesive energy. ied. However, the tensile residual stress both reduces the It can be seen that with the increase of the cohesive en- size of the plastic zone and rotates the plastic zone back- ergy 0, the cleavage fracture toughness increases for both ward for all 0, as shown in Fig. 15b. We can also observe with and without residual stresses, as shown in Fig. 14a. that with the increase of 0, the plastic zone size increases Figure 14b shows that the normalized cleavage fracture for both with and without residual stress cases, which can toughness tends to converge to the case without residual explain the increasing cleavage fracture toughness showed stress with increasing 0. Note that the cleavage frac- in Fig. 14a. ture toughness with residual stress was normalized by the The maximum cohesive stress is an another important toughness without residual stress effect. Figure 15 shows parameter in cohesive zone model. In the present study, the effect of the residual stress on the plastic zone for dif- the effect of residual stress on the cleavage fracture tough- ferent 0, and the length of the FPZ, which measures the ness for three maximum cohesive stresses, i.e. σ max/σ 0 = distance between the point where all traction is lost and 2.8, 3.0 and 3.3, was studied. Residual stress field with 21 where the peak stress is first attained. α11 = 0.004 and α22 = 0.002, i.e. RsField3, was used.

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 600 X. B. REN et al.

Fig. 15 Effect of the residual stress on (a) the length of FPZ, and (b) plastic zone size of the surrounding materials for εp = 1%. For each cohesive energy case, the left plot of plastic zone is the case without residual stress and the right one shows the case with residual stress. E/σ 0 = 450, ν = 0.3; n = 0.1; σ max = 3σ 0; α11 = 0.004, α22 = 0.002.

increases. When the maximum cohesive stress is smaller, the energy needed to initiate a crack is less, and a fully developed plastic zone cannot be formed. It has been shown that plastic dissipation only becomes significant compared 21 to 0 when fully developed plastic zone can form. Thus, a stronger effect of residual stress on the cleavage fracture toughness can be expected for higher σ max.

Effect of applied T-stress In this study, no real structural component was considered. It is thus interesting to investigate the effect of residual stress on the cleavage fracture toughness for different geometry constraint levels characterized by T-stress.The outer boundary condition for the MBL model is governed Fig. 16 Cleavage fracture toughness as a function of maximum by the elastic K-field and a T-stress under small-scale- σ cohesive stress for both with and without residual stress cases. E/ 0 yielding condition. For mode I loading, K is the amplitude = 450, ν = 0.3; n = 0.1; = 100 N/mm; α = 0.004, α = 0.002. 0 11 22 of the singular stress field, while the T is a non-singular stress term, acting parallel to the crack plane. Geometry Cohesive energy 0 was fixed to be 100 N/mm. The rela- constraint effects on fracture behaviour can be investi- tionship between the cleavage fracture toughness and the gated by utilizing the T-stress.32 In the current study, T- maximum cohesive stress is showed in Fig. 16. stress with the value of T/σ 0 = –0.5, –0.25, 0 and 0.5 was It can be seen that with the increase of σ max, the reduc- studied. The same residual stress field as previous with tion of cleavage fracture toughness due to residual stress α11 = 0.004 and α22 = 0.002, i.e. RsField3, was used. The

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 RESBrittle 601

It can be seen that the negative T-stress enlarges the plastic zone. However, the presence of tensile residual stress significantly decreases the size of the plastic zone and rotates the plastic zone backward slightly. The positive T-stress shrinks the plastic zone, and the residual stress further rotates the plastic zone backward. We thus can conclude that the tensile residual stress influences the cleavage fracture toughness similar way as the positive T-stress.

CONCLUSIONS In this paper the effect of residual stresses on cleavage fracture toughness has been investigated. The damage mechanics-based cohesive zone model was utilized to simulate the FPZ. The MBL model simulations were Fig. 17 Effect of residual stresses on the cleavage fracture performed under mode I, plane strain conditions, and σ σ = ν = = = toughness for different T/ 0. E/ 0 450, 0.3; n 0.1; 0 the remote boundary conditions of the model is gov- 100 N/mm, σ = 3σ ; α = 0.004, α = 0.002. max 0 11 22 erned by elastic K-field and T-stress. Residual stresses were introduced into the FE model by the eigenstrain method. Cleavage fracture was assumed to occur when the opening stress of the first integration point of the first cohesive element ahead of the crack tip dropped to zero. Far-field contour J-integral has been employed to quan- tify the cleavage fracture toughness. Cohesive zone model with a bilinear TSL was employed to study the effect of residual stresses on cleavage fracture behaviour. The introduction of a small fictitious viscosity in the TSL in combination with the use of a small step- increment in the simulations improved the convergence rate, and its effect on the results is negligible. Results show that residual stresses affect both the length of the FPZ and surrounding material plasticity. Local compressive residual stress enhances the cleavage fracture toughness while positive residual stresses have oppo- Fig. 18 Effect of residual stress on size of plastic zone for different site influences. The compressive residual stress enlarges geometry constraint when εp = 1% at crack initiation. N represents the plastic zone significantly while tensile residual stresses the case without residual stress and W denotes the case with shrink the plastic zone and rotate the plastic zone back- residual stress. E/σ 0 = 450, ν = 0.3; n = 0.1; 0 = 100 N/mm, σ max = 3σ 0; α11 = 0.004, α22 = 0.002. ward. When the welds are geometrically similar, the effect of residual stresses on the cleavage fracture toughness is a linear function of the size of the weld. The dominant cleavage fracture toughness was plotted as the function of cohesive parameters 0 and σ max also play an important T/σ 0 in Fig. 17. role on the effect of residual stresses on the cleavage frac- It can be seen that with the increase of the T-stress,the ture toughness. With the increase of cohesive energy, the cleavage fracture toughness decreases for both with and effect of residual stresses on the cleavage toughness de- without residual stresses. As expected, with the increase of creases. The reduction of the toughness caused by resid- the T-stress the crack-tip constraint increases and thus the ual stresses increases with the increase of the maximum plastic zone shrinks. Similar results were also reported by cohesive stress. Tvergaard and Hutchinson33 in their study on the effect The effect of residual stresses on the cleavage fracture of T-stress on mode I crack growth resistance in a ductile toughness becomes weaker for higher geometry con- solid. It is interesting to observe that with the increase straint configuration. It has been found that residual of the T-stress the effect of the residual stress decreases. stresses show similar behaviour as the T-stress. When com- Figure 18 shows the effect of the residual stress on plastic bining the residual stresses with T-stress, the superposition zone size for different T-stress. principle can be applied. For higher geometry constraint

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 602 X. B. REN et al. configuration, the effect of tensile residual stress becomes finite element method and reliability of estimated values. smaller. However, for lower geometry constraint case, the Trans. Japan Weld. Res. Inst. 4, 123–131. combined effect can induce a significant reduction of the 12 Hill, M. R. and Nelson, D. V. (1995) The inherent strain cleavage fracture toughness. method for residual stress determination and its application to a long welded joint. ASME Press. Vessels Pip. 318, Though wide range of parameters have been investi- 343–352. gated in this paper, the contribution of plasticity from 13 Hill, M. R. and Nelson, D. V. (1996) Determining residual the surrounding materials and component geometry con- stress through the thickness of a welded plate. ASME Press. straint are believed to play a dominant role in the effect Vessels Pip. Div. (Publication) PVP 327, 29–36. of residual stresses on the cleavage fracture toughness. 14 Mochizuki, M., Hayashi, M. and Hattori, T. (1999) Residual Further experimental verification will be of interest in stress analysis by simplified inherent strain at welded pipe future. junctures in a pressure vessel. J. Press. Vessel Technol. 121, 353–357. 15 de Borst, R. (2003) Numerical aspects of cohesive-zone Acknowledgements models. Eng. Fract. Mech. 70, 1743–1757. 16 Cornec, A., Scheider, I. and Schwalbe, K. (2003) On the The funding from the Research Council of Norway practical application of the cohesive model. Eng. Fract. Mech. through the ‘STORFORSK’ project No.167397/V30 is 70, 1963–1987. greatly acknowledged. The first author would like to ac- 17 Barenblatt, G. (1962) The mathematical theory of equilibrium knowledge Dr. Erling Østby and Dr. Sigmund Kyrre As˚ cracks in brittle fracture. Adv. Appl. Mech. 7, 55–129. of SINTEF Materials and Chemistry, Dr. Junhua Zhao 18 Dugdale, D. (1960) Yielding of steel sheets containing slits. J. and Dr. Junyan Liu of Norwegian University of Sci- Mech. Phys. Solids 8, 100–104. ence and Technology, for their valuable comments and 19 Hillerborg, A., Modeer, M. and Petersson, P. (1976) Analysis of crack formation and crack growth in concrete by means of discussions. fracture mechanics and finite elements. Cement Concr. Res. 6, 773–782. 20 Olden, V., Thaulow, C., Johnsen, R., Østby, E. and Berstad, REFERENCES T. (2008) Application of hydrogen influenced cohesive laws in the prediction of hydrogen induced stress cracking in 25% Cr 1 Lei, Y., O’dowd, N.P. and Webster, G. (2000) Fracture duplex stainless steel. Eng. Fract. Mech. 75, 2333– mechanics analysis of a crack in a residual stress field. Int. J. 2351. Fract. 106, 195–216. 21 Tvergaard, V. and Hutchinson, J. (1992) The relation 2 Xu, W. and Burdekin, F. (1998) Effects of residual stresses on between crack growth resistance and fracture process constraint and fracture behaviour of wide plates. Proc.: Math. parameters in elastic–plastic solids. J. Mech. Phys. Solids (UK) Phys. Eng. Sci. 454, 2505–2528. 40, 1377–1397. 3 Liu, J., Zhang, Z.L. and Nyhus, B. (2008) Residual stress 22 Needleman, A. (1987) A continuum model for void nucleation induced crack tip constraint. Eng. Fract. Mech. 75, 4151– by inclusion debonding. J. Appl. Mech. 54, 525–531. 4166. 23 Xu, X. and Needleman, A. (1994) Numerical simulations of 4 Ren, X.B., Zhang, Z.L. and Nyhus, B. (2009) Effect of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, residual stresses on the crack-tip constraint in a modified 1397–1434. boundary layer model. Int. J. Solids Struct. 46, 2629–2641. 24 Chaboche, J., Feyel, F. and Monerie, Y. (2001) Interface 5 Gao, X. and Dodds, Jr. R. (2001) An engineering approach to debonding models: A viscous regularization with a limited assess constraint effects on cleavage fracture toughness. Eng. rate dependency. Int. J. Solids Struct. 38, 3127–3160. Fract. Mech. 68, 263–283. 25 Pezzotta, M. and Zhang, Z.L. (2010) Effect of thermal 6 Panontin, T. and Hill, M. (1996) The effect of residual mismatch induced residual stresses on grain boundary stresses on brittle and ductile fracture initiation predicted by micro-cracking of Titanium Diboride ceramics. J. Mater. Sci. micromechanical models. Int. J. Fract. 82, 45, 382–391. 317–333. 26 Rice, J.R. (1968) A path independent integral and the 7 Mirzaee-Sisan, A., Truman, C., Smith, D. and Smith, M. approximate analysis of strain concentration by cracks and (2007) Interaction of residual stress with mechanical loading notches. J. Appl. Mech. 35, 379–386. in a ferritic steel. Eng. Fract. Mech. 74, 2864–2880. 27 Lei, Y. (2005) J-integral evaluation for cases involving 8 Ritchie, R., Knott, J. and Rice, J. (1973) On the relationship non-proportional stressing. Eng. Fract. Mech. 72, between critical tensile stress and fracture toughness in mild 577–596. steel. J. Mech. Phys. Solids 21, 395–410. 28 Ueda, Y. and Fukuda, K. (1989) New measuring method of 9 Neimitz, A., Graba, M. and Ga kiewicz, J. (2007) An three-dimensional residual stresses in long welded joints alternative formulation of the Ritchie–Knott–Rice local using inherent strains as parameters– L method. J. Eng. fracture criterion. Eng. Fract. Mech. 74, 1308–1322. Mater. Technol. 111, 1–8. 10 ABAQUS Version 6.7 User’s Manual. Hibbitt, Karlsson and 29 Ueda, Y., Nakacho, K. and Yuan, M. (1991) Application of Sorenson Inc. 2007. FEM to theoretical analysis, measurement and prediction of 11 Ueda, Y., Fukuda, K., Nakacho, K. and Endo, S. (1975) A welding residual stresses. Trans. JWRI(Japan Welding Research new measuring method of residual stresses with the aid of Institute)(Japan) 20, 97–107.

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603 RESBrittle 603

30 Bouchard, P. J. and Withers, P. J. (2006) Identification of 32 Gao, X., Shih, C., Tvergaard, V. and Needleman, A. (1996) residual stress length scales in welds for fracture assessment. Constraint effects on the ductile-brittle transition in small ECF16–16th European Conference of Fracture.Springer, scale yielding. J. Mech. Phys. Solids 44, Alexabdroupolis, Greece, pp. 163–176. 1255–1282. 31 Du, Z. and Hancock, J. (1991) The effect of non-singular 33 Tvergaard, V. and Hutchinson, J. (1994) Effect of T-stress on stresses on crack-tip constraint. J. Mech. Phys. Solids 39, mode I crack growth resistance in a ductile solid. Int. J. Solids 555–567. Struct. 31, 823–833.

c 2011 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 34, 592–603