- Email: [email protected]
An improved method for numerical determination of SZW using FEM
Nuclear Engineering and Design 239 (2009) 1207–1211
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
An improved method for numerical determination of SZW using FEM Sanjeev Saxena a,∗ , N. Ramakrishnan b a b
Scientist, Advanced Materials and Processes Research Institute (AMPRI), Hoshangabad Road, Bhopal 462064, India Former Director, Advanced Materials and Processes Research Institute (AMPRI), Hoshangabad Road, Bhopal 462064, India
a r t i c l e
i n f o
Article history: Received 2 December 2008 Received in revised form 10 February 2009 Accepted 18 February 2009
a b s t r a c t The current investigation essentially comprises a number of 3D non-linear FEM simulations to study and correlate crack tip blunting with experimental results. The study attempts to establish a physically tangible procedure for numerical determination of stretch zone width (SZW). The proposed methodology explains the mechanism involved in the creation of stretched zone. This aspect was missing in its earlier method of determination as half the crack tip opening distance which in-turn determined using 45◦ line method. It is concluded that the proposed method of SZW determination using large deformation FEM analysis can reasonably simulate the process of blunting of the crack tip and can predict the material’s SZW and J-integral values. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Complex structures may have stresses in some regions that exceed the elastic limit. This has created need for a fracture criterion that would also include elastic–plastic to fully plastic behaviour. There is an increasing effort to use the stretch zone width (SZW) ahead of a fatigue pre-crack as a measure of the fracture toughness (J) of ductile materials in mode-I (Paranjpe and Banerjee, 1979; Mills, 1981; Amouzouvi and Bassim, 1982; Doig et al., 1984; Bassim et al., 1992; Bassim, 1995; Smith et al., 1995). The SZW method relies on the microscopic observation of the crack tip blunting and therefore J can be established with fewer specimens and fewer size restrictions. In the light of the foregoing it seems evident that the principle of this approach permits geometry independent evaluation of J, provided a high degree of precision in measuring the SZW can be achieved. SZW can be experimentally evaluated on fracture surface of tested standard fracture specimens using the scanning electron microscope (SEM). The problem in its experimental evaluation is in identifying the size of stretching zone on a blunted crack front (Tarafder et al., 2004). Although there have been enormous efforts to understand the SZW numerically. Works done in the past were mainly experimental based with a few attempts to evaluate SZW numerically (Yin et al., 1983; Kobayashi et al., 1985; Tai, 1996; Saxena and Ramakrishnan, 2007). Conventional way of numerical determination of SZW is based on the assessment of CTOD. Several definitions are available for the assessment of CTOD as shown in Fig. 1 (Meyers and Chawla, 1999). Among them the most commonly used method for CTOD assessment is
45◦ line method. It is to note that CTOD definitions given in Fig. 1 are same if the crack blunts in a semicircle. Also, conventionally SZW is numerically taken as 1/2CTOD with the assumption of semicircular blunting of the crack tip that is not always true. It can have different shape as the shape of blunted crack tip mainly depends upon the hardening characteristics of the material that varies with the material. This numerical approach of SZW calculation also does not account clearly the mechanism involved in the creation of stretched zone. The evaluation of initiation fracture toughness based on SZW measurement is getting considered (Roos et al., 1993; Eisele et al., 1994; Saxena and Ramakrishanan, 2007) as geometry independent fracture parameter. Thus it increases the importance of improving the existing numerical SZW determination method. In the current work deformation behaviour of CT specimens of Armco iron are simulated by finite element method (FEM) under mode-I loading. The current investigation essentially comprises a number of 3D non-linear FEM simulations to study and correlate crack tip blunting with experimental SZW. The ABAQUS software has been used in the analysis. The numerically predicted SZW values are quite comparable with that of experimental results. The proposed SZW method also explains the probable mechanism involved in the creation of stretched zone. The experimentally (Srinivas et al., 1991, 1994) obtained properties of Armco iron are used as the effective properties of the homogeneous continuum. The present study essentially pertains to ductile crack initiation and has not concerned with crack growth related aspects. 2. Numerical FEM model
∗ Tel.: +91 755 2457244 fax: +91 755 2457042. E-mail address: san [email protected] (S. Saxena). 0029-5493/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2009.02.014
To establish the methodology for numerical SZW determination, the experimental results of Armco iron at room temperature
1208
S. Saxena, N. Ramakrishnan / Nuclear Engineering and Design 239 (2009) 1207–1211
Table 1 Tensile test data for different grain sizes of Armco iron. Armco iron grain size (m)
Elastic modulus (E) (GPa)
Yield stress ( y ) (MPa)
Tensile strength (MPa)
n
Critical energy density (MJ/m3 )
38 78 118 252 420
196 196 196 196 196
208 189 180 158 151
299 296 296 268 271
0.3 108 0.29 92 0.28 81 0.26 64 0.255 59
Experimental blunting line slope (◦ )
Proposed method based blunting line slope (◦ )
45.25 43.13 43.54 40.58 40.20
42.36 41.57 41.17 38.64 38.82
Trend
↓
consideration of only one fourth of the specimen geometry for computational economy. The mesh was constructed with eight noded brick elements. In the mesh convergence study several finite element models having elements size near the crack tip varies from 100 m to 5 m were considered. After the mesh convergence study the element size near the crack front is stick to 10 m in the radial direction and 5 m along the blunted crack surface. Such type of initial mesh configuration near the crack tip region helps in keeping good aspect ratio of the elements near the crack tip, during its blunting. In the thickness direction ten elements are used. Generally the SZW in various engineering materials varies up to 250 m and in extreme cases it can be higher also. Therefore in the present analysis the dense mesh is provided up to 250 m to arrest the steep parameters variation near the crack front. Other than the SZW size, the convergence of mesh, the hour-glassing and mesh distortion during the blunting process are the criteria considered in the finalization of mesh size. The mesh is further coarsening in the region away from the crack front. A magnified view of the mesh morphology around the crack tip is shown in Fig. 3a and that of the corresponding deformed mesh in Fig. 3b. The material undergoes a large strain and rotation at the crack tip, which necessitated a constitutive framework based on finite deformation for the numerical simulation. The flow behaviour of
Fig. 1. Different CTOD assessment methods: (a) original crack tip line method; (b) 45◦ line method; (c) tangent line method.
were used (Srinivas et al., 1991, 1994). The tensile test data for different grain sizes of Armco iron used in the analysis are given in Table 1. To have better understanding of the variation of fracture parameter across the thickness 3D FEM model have been used (Fig. 2). The investigation was limited to CT specimen analysis subjected to mode-I type of loading. The symmetry in this case permits
Fig. 2. Finite element mesh (8-noded brick elements) for one-fourth 3D model of CT specimen (W = 100 mm; B = 50 mm; a/W = 0.55).
Fig. 3. View of the FEM mesh at the crack tip. (a) Initial condition; (b) deformed condition. CTOD, crack tip opening displacement; SZH, stretch zone height; SZW, stretch zone width.
S. Saxena, N. Ramakrishnan / Nuclear Engineering and Design 239 (2009) 1207–1211
1209
Fig. 5. Plastic energy density variation with X-co-ordinates (deformed) from the crack tip. Fig. 4. True stress–strain material variation for Armco iron (grain size = 118 m).
the material in plastic regime is assumed to follow the power law. f = Kεn
(1)
where f is the true flow stress, K is the strength coefficient, ε is the elastic–plastic true strain and n is the strain-hardening exponent. In the power law variation of material’s true stress–strain curve, the constant K is determined to match well the yield strength of the material. The other constant n is an average best fit on a log–log plot of true stress–strain curve and is expected to hold well for the full range of strain values. In Fig. 4 the true stress–strain curve for Armco iron (grain size = 118 m) material has been shown. The analyses were done using commercial FEM software ABAQUS version 6.6. 3. Proposed numerical SZW determination method In the fracture testing of compact tension specimen with the increase in load line displacement (LLD) maximum part of plastic energy dissipates near the crack front. In order to demarcate the highly deformed region from the other remaining region, it requires defining a critical energy density to quantify highly stretched region in the form of SZW. The critical energy density used to delineate the highly stretched region is defined on the material’s true stress–strain curve as the energy density integral up to critical strain (εc ) equals to strain-hardening exponent (n) for the specific case of power law variation of material’s true stress–strain curve (denoted by area shown as dark colour in Fig. 4 and as given by Eq. (2)):
ECrit =
εc
dε
Fig. 6. Variation of numerically determined SZW across the thickness at different LLD.
proposed SZW determination methodology. The figure also showed the experimental critical SZW value for Armco iron having grain size of 118 m. The length of experimental critical SZW line showed the region in depth direction observed for measuring critical SZW. The SZW is also determined by assessing CTOD by most commonly used 45◦ line method. In Fig. 7, the proposed methodology based SZW results near mid plane are compared with half CTOD results measured conventionally (45◦ line method). The stretch zone height (SZH) is also determined using the proposed method of
(2)
0
The variation of plastic energy density (PED) near the blunted crack front with respect to deformed X-co-ordinates from crack front with increase in LLD is shown in Fig. 5. The above variation is shown at a plane near to mid thickness. The region having energy density greater than the critical energy density delineates the stretched region ( in Fig. 5) denoted as numerically determined SZW and will keep on increasing with LLD. The above term ECrit (Eq. (2)) is defined using the total strains of the tensile test. As the elastic contribution is negligibly small compared to the plastic part and also as ABAQUS software provides only the plastic contribution, the two have been compared in Fig. 5. The same methodology is used to determine stretch zone at different planes in a 50 mm thick CT specimen. Fig. 6 shows across the thickness variation of numerically predicted SZW based on the proposed method at different load line displacement. The variation of numerically predicted SZW across the thickness resembles very well with the SZW variation generally observed in experimental results, thus validating the
Fig. 7. Predicted stretch zone size variation using 45◦ line method and proposed method.
1210
S. Saxena, N. Ramakrishnan / Nuclear Engineering and Design 239 (2009) 1207–1211
The procedure to numerically predict SZW can be summarized in three steps:
Fig. 8. Comparison of experimental with numerical (using Eq. (3) and SZW by proposed method) results.
SZW considering as Y-deformation of the region where the plastic energy density exceeds the critical energy density defined by Eq. (2). It is observed that SZH predicted by proposed energy based method compares well with 45◦ line method whereas SZW predicted by proposed method is on the higher side for a LLD. In conventional (45◦ line) method semicircular blunting shape is always assumed in numerical prediction of SZW whereas no such assumption is made in the proposed SZW method. It can be appreciated that in a particular case of Armco iron (grain size: 118 m) blunting shape angle works out to be lesser than 45◦ . The validation of proposed methodology of SZW determination is also done using the experimental results of fracture toughness and SZW value (Srinivas et al., 1991). Once the stretch zone width is predicted accurately, the numerical fracture toughness can easily be calculated using Eq. (3) as follows: J = mCTOD = m(2SZW)
(3)
where m turns out to be a material independent parameter ≈1.25 (Suresh et al., 1999) when using is the stress measure defined by integral average on the material’s true stress strain curve and CTOD is the crack tip opening displacement. The yield strength as a stress measure does not relate the behaviour of the material during strain hardening subsequent to yielding. The flow stress (average of yield to ultimate) although taking hardening into account, implicitly assumes the variation to be linear. To account for the non-linearity, Suresh et al. (1999) considered the stress measure () as given by Eq. (4) that is based on an integral average.
εc dε = 0 εc 0
dε
(4)
In Eq. (4) the numerator corresponds to the area under the true stress–strain curve up to a critical strain (εc ) corresponding to onset of necking also equals to strain-hardening exponent (n) for a specific case of power law variation of true stress–strain material curve. Since the denominator is the critical strain itself, the division yields the integral average of the true stress. For a grain size of Armco iron, Fig. 8 showed the comparison of experimental results with numerically determined fracture toughness results using SZW predicted by proposed method. The experimental variation of blunting slope matches well with numerical results using proposed SZW method. Similarly in other cases also it compares well with experimental results as shown in Table 1. In Fig. 8, it can also be appreciated that the numerically predicted fracture toughness values using Eq. (3) is on the conservative side as compared to experimental results. Thus it shows that using critical material flow value; the present method can predict well the experimental SZW and fracture toughness values.
(1) Plot the variation of plastic energy density with deformed Xco-ordinates w.r.t. crack tip, for every increase in load line displacement. (2) Define the critical energy density for the material. Critical energy density corresponds to the energy density integral up to critical strain equals to strain-hardening exponent (n) for a specific case of power law material hardening behaviour that is obtained by tensile specimen testing. This energy density is defined on a true stress–strain material variation. (3) Measure the region (deformed X-co-ordinates) exceeding the material’s critical energy density defined by Eq. (2) in the figure drawn in step 1. This deformed X-co-ordinate w.r.t. crack tip is SZW and it will be a region on blunted crack surface, where the energy density exceeds the critical energy density. 4. Comparison of existing and proposed numerical SZW method Conventionally, half of CTOD is considered as SZW. In the proposed method SZW is defined in terms of J-integral using critical energy density that corresponds to critical strain at the onset of necking in tensile test equals to strain-hardening coefficient (n) value for power law material hardening behaviour. This critical energy density corresponds to onset of necking in smooth tensile specimen test. Stretch zone width that appears, as pseudo crack growth in J–R curve has to be measured in the direction of crack growth. The existing numerical method of SZW assessment is based on measurement made in the direction normal to the crack growth direction whereas in the proposed numerical SZW method, the measurements are made in the direction of crack growth. In the existing numerical SZW method generally semicircular blunting shape is assumed whereas no such assumption in made in the proposed method. The existing numerical SZW method does not clearly accounts the probable mechanism involved, as it fails to predict the variation of SZW size over the thickness in standard fracture specimen. The proposed numerical SZW method provides full information of its variation across the fracture specimen thickness that resembles well with usually observed experimental results. 5. Conclusions The present study attempts at achieving an insight into defining the procedure for numerical determination of stretch zone width using a large deformation FEM analysis. The investigation essentially comprised of number of FEM simulations to study and correlate various ductile fracture parameters with experimental results. To establish the procedure of numerical determination of SZW, a detailed analysis of the zone of intense plastic deformation was carried out. An attempt has been made to establish a numerical procedure to determine SZW and is finally compared with experimental results. The region of high deformation at the crack tip is delineated using the critical energy density obtained by tensile test. The proposed improved numerical method of SZW determination accurately predicts the slope of blunting that compares well with experimental results without having limit of loading and SZW size. This methodology also explains the probable mechanism involved in the creation of stretch zone. The proposed improved method also provides full information in regards to variation of SZW across the specimen thickness that resembles well with the experimental observations. It is concluded that the proposed method of SZW determination can reasonably simulate the process of blunting of the crack tip and can predict the material’s SZW and J-integral values. In the
S. Saxena, N. Ramakrishnan / Nuclear Engineering and Design 239 (2009) 1207–1211
present work, though the methodology is demonstrated on Armco iron material the method can also be applied for other materials. References Amouzouvi, K.F., Bassim, M.N., 1982. Determination of fracture toughness from stretch zone width measurement in predeformed AISI Type 4340 steel. Mater. Sci. Eng. 55, 257–262. Bassim, M.N., Mathews, J.R., Hyatt, C.V., 1992. Evaluation of fracture toughness of HSLA80 steel at high loading rates using stretch zone measurements. Eng. Fract. Mech. 43 (2), 297–303. Bassim, M.N., 1995. Use of the stretch zone for the characterization of ductile fracture. J. Mater. Process. Technol. 54, 109–113. Doig, P., Smith, R.F., Flewitt, P.E.J., 1984. The use of stretch zone width measurements in the determination of fracture toughness of low strength steels. Eng. Fract. Mech. 19 (4), 653–664. Eisele, U., Herter, K.H., Schuler, X., 1994. Influence of the multiaxility of stress state on the ductile fracture behaviour of degraded piping components. In: Schwalbe, K.H., Berger, C. (Eds.), ECF 10: Structural Integrity: Experiments Models and Applications, vol. 1. Berlin, pp. 249–254. Kobayashi, Hideo, Nakamura, Haruo, Nakazawa, Hajime, 1985. Comparison of J1C test methods recommended by ASTM and JSME elastic-plastic fracture test methods: user’s experience. In: Wessel, E.T., Loss, F.J. (Eds.), ASTM STP 856. American Society for Testing and Materials, pp. 3–22. Meyers, M.A., Chawla, K.K., 1999. Mechanical Behaviour of Materials. Prentice-Hall, New Jersey, p. 368. Mills, W.J., 1981. On the relationship between stretch zone formation and the J integral for high strain-hardening materials. J. Test Eval. 9 (1), 56–62.
1211
Paranjpe, S.A., Banerjee, S., 1979. Interrelation of crack opening displacement and J-integral. Eng. Fract. Mech. 11, 43–53. Roos, E., Eisele, U., Silcher, U., 1993. Effect of stress state on the ductile fracture behavior of large-scale specimens. In: Hackett, E.M., Schwalbe, K.H., Dodds Jr., R.H. (Eds.), Constraint Effects in Fracture, ASTM STP 1171. American Society for Testing and Materials, Philadelphia, pp. 41–64. Saxena, Sanjeev, Ramakrishanan, N., 2007. Characterizing crack initiation load in circumferentially through-wall cracked elbows under bending load. Int. J. Pressure Vessels Piping 84, 493–501. Saxena, Sanjeev, Ramakrishnan, N., 2007. A comparison of micro, meso and macroscale FEM analysis of ductile fracture in a CT specimen (mode I). Comput. Mater. Sci. 39 (1), 1–7. Smith, J., Bassim, M.N., Liu, C.D., 1995. Effect of fatigue pre-cracking on stretch zone formation. Eng. Fract. Mech. 52 (3), 401–408. Srinivas, M., Malakondaiah, G., Armstrong, R.W., Rama Rao, P., 1991. Ductile fracture toughness of polycrystalline Armco Iron of varying grain size. Acta Mater. 39 (5), 807–816. Srinivas, M., Sundararajan, G., Malakondaiah, G., Rama Rao, P., 1994. An analysis of ductile fracture initiation toughness in iron, its binary alloys and nickel. In: Proceeding of Royal Society, 447 (A), London, pp. 237–251. Suresh, R.K.V., Ramakrishnan, N., Srinivas, M., Rama Rao, P., 1999. On the determination of JIC using the stretch zone width method. J. Test Eval. 27 (3), 211–217. Tai, W.H., 1996. Computer simulation of plastic damage evolution in cracked body of low-carbon steel. Eng. Fract. Mech. 54 (1), 141–146. Tarafder, M., Dey, S., Sash, B., Sivaprasad, S., Tarafder, S., 2004. Initiation fracture toughness of HSLA steel through automatic measurement of stretch zone. Mater. Sci. Technol. 20, 1531–1537. Yin, S.W., Gerbrands, R.A., Hartevelt, M., 1983. An investigation of the blunting line. Eng. Fract. Mech. 18 (5), 1025–1036.
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
An improved method for numerical determination of SZW using FEM Sanjeev Saxena a,∗ , N. Ramakrishnan b a b
Scientist, Advanced Materials and Processes Research Institute (AMPRI), Hoshangabad Road, Bhopal 462064, India Former Director, Advanced Materials and Processes Research Institute (AMPRI), Hoshangabad Road, Bhopal 462064, India
a r t i c l e
i n f o
Article history: Received 2 December 2008 Received in revised form 10 February 2009 Accepted 18 February 2009
a b s t r a c t The current investigation essentially comprises a number of 3D non-linear FEM simulations to study and correlate crack tip blunting with experimental results. The study attempts to establish a physically tangible procedure for numerical determination of stretch zone width (SZW). The proposed methodology explains the mechanism involved in the creation of stretched zone. This aspect was missing in its earlier method of determination as half the crack tip opening distance which in-turn determined using 45◦ line method. It is concluded that the proposed method of SZW determination using large deformation FEM analysis can reasonably simulate the process of blunting of the crack tip and can predict the material’s SZW and J-integral values. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Complex structures may have stresses in some regions that exceed the elastic limit. This has created need for a fracture criterion that would also include elastic–plastic to fully plastic behaviour. There is an increasing effort to use the stretch zone width (SZW) ahead of a fatigue pre-crack as a measure of the fracture toughness (J) of ductile materials in mode-I (Paranjpe and Banerjee, 1979; Mills, 1981; Amouzouvi and Bassim, 1982; Doig et al., 1984; Bassim et al., 1992; Bassim, 1995; Smith et al., 1995). The SZW method relies on the microscopic observation of the crack tip blunting and therefore J can be established with fewer specimens and fewer size restrictions. In the light of the foregoing it seems evident that the principle of this approach permits geometry independent evaluation of J, provided a high degree of precision in measuring the SZW can be achieved. SZW can be experimentally evaluated on fracture surface of tested standard fracture specimens using the scanning electron microscope (SEM). The problem in its experimental evaluation is in identifying the size of stretching zone on a blunted crack front (Tarafder et al., 2004). Although there have been enormous efforts to understand the SZW numerically. Works done in the past were mainly experimental based with a few attempts to evaluate SZW numerically (Yin et al., 1983; Kobayashi et al., 1985; Tai, 1996; Saxena and Ramakrishnan, 2007). Conventional way of numerical determination of SZW is based on the assessment of CTOD. Several definitions are available for the assessment of CTOD as shown in Fig. 1 (Meyers and Chawla, 1999). Among them the most commonly used method for CTOD assessment is
45◦ line method. It is to note that CTOD definitions given in Fig. 1 are same if the crack blunts in a semicircle. Also, conventionally SZW is numerically taken as 1/2CTOD with the assumption of semicircular blunting of the crack tip that is not always true. It can have different shape as the shape of blunted crack tip mainly depends upon the hardening characteristics of the material that varies with the material. This numerical approach of SZW calculation also does not account clearly the mechanism involved in the creation of stretched zone. The evaluation of initiation fracture toughness based on SZW measurement is getting considered (Roos et al., 1993; Eisele et al., 1994; Saxena and Ramakrishanan, 2007) as geometry independent fracture parameter. Thus it increases the importance of improving the existing numerical SZW determination method. In the current work deformation behaviour of CT specimens of Armco iron are simulated by finite element method (FEM) under mode-I loading. The current investigation essentially comprises a number of 3D non-linear FEM simulations to study and correlate crack tip blunting with experimental SZW. The ABAQUS software has been used in the analysis. The numerically predicted SZW values are quite comparable with that of experimental results. The proposed SZW method also explains the probable mechanism involved in the creation of stretched zone. The experimentally (Srinivas et al., 1991, 1994) obtained properties of Armco iron are used as the effective properties of the homogeneous continuum. The present study essentially pertains to ductile crack initiation and has not concerned with crack growth related aspects. 2. Numerical FEM model
∗ Tel.: +91 755 2457244 fax: +91 755 2457042. E-mail address: san [email protected] (S. Saxena). 0029-5493/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2009.02.014
To establish the methodology for numerical SZW determination, the experimental results of Armco iron at room temperature
1208
S. Saxena, N. Ramakrishnan / Nuclear Engineering and Design 239 (2009) 1207–1211
Table 1 Tensile test data for different grain sizes of Armco iron. Armco iron grain size (m)
Elastic modulus (E) (GPa)
Yield stress ( y ) (MPa)
Tensile strength (MPa)
n
Critical energy density (MJ/m3 )
38 78 118 252 420
196 196 196 196 196
208 189 180 158 151
299 296 296 268 271
0.3 108 0.29 92 0.28 81 0.26 64 0.255 59
Experimental blunting line slope (◦ )
Proposed method based blunting line slope (◦ )
45.25 43.13 43.54 40.58 40.20
42.36 41.57 41.17 38.64 38.82
Trend
↓
consideration of only one fourth of the specimen geometry for computational economy. The mesh was constructed with eight noded brick elements. In the mesh convergence study several finite element models having elements size near the crack tip varies from 100 m to 5 m were considered. After the mesh convergence study the element size near the crack front is stick to 10 m in the radial direction and 5 m along the blunted crack surface. Such type of initial mesh configuration near the crack tip region helps in keeping good aspect ratio of the elements near the crack tip, during its blunting. In the thickness direction ten elements are used. Generally the SZW in various engineering materials varies up to 250 m and in extreme cases it can be higher also. Therefore in the present analysis the dense mesh is provided up to 250 m to arrest the steep parameters variation near the crack front. Other than the SZW size, the convergence of mesh, the hour-glassing and mesh distortion during the blunting process are the criteria considered in the finalization of mesh size. The mesh is further coarsening in the region away from the crack front. A magnified view of the mesh morphology around the crack tip is shown in Fig. 3a and that of the corresponding deformed mesh in Fig. 3b. The material undergoes a large strain and rotation at the crack tip, which necessitated a constitutive framework based on finite deformation for the numerical simulation. The flow behaviour of
Fig. 1. Different CTOD assessment methods: (a) original crack tip line method; (b) 45◦ line method; (c) tangent line method.
were used (Srinivas et al., 1991, 1994). The tensile test data for different grain sizes of Armco iron used in the analysis are given in Table 1. To have better understanding of the variation of fracture parameter across the thickness 3D FEM model have been used (Fig. 2). The investigation was limited to CT specimen analysis subjected to mode-I type of loading. The symmetry in this case permits
Fig. 2. Finite element mesh (8-noded brick elements) for one-fourth 3D model of CT specimen (W = 100 mm; B = 50 mm; a/W = 0.55).
Fig. 3. View of the FEM mesh at the crack tip. (a) Initial condition; (b) deformed condition. CTOD, crack tip opening displacement; SZH, stretch zone height; SZW, stretch zone width.
S. Saxena, N. Ramakrishnan / Nuclear Engineering and Design 239 (2009) 1207–1211
1209
Fig. 5. Plastic energy density variation with X-co-ordinates (deformed) from the crack tip. Fig. 4. True stress–strain material variation for Armco iron (grain size = 118 m).
the material in plastic regime is assumed to follow the power law. f = Kεn
(1)
where f is the true flow stress, K is the strength coefficient, ε is the elastic–plastic true strain and n is the strain-hardening exponent. In the power law variation of material’s true stress–strain curve, the constant K is determined to match well the yield strength of the material. The other constant n is an average best fit on a log–log plot of true stress–strain curve and is expected to hold well for the full range of strain values. In Fig. 4 the true stress–strain curve for Armco iron (grain size = 118 m) material has been shown. The analyses were done using commercial FEM software ABAQUS version 6.6. 3. Proposed numerical SZW determination method In the fracture testing of compact tension specimen with the increase in load line displacement (LLD) maximum part of plastic energy dissipates near the crack front. In order to demarcate the highly deformed region from the other remaining region, it requires defining a critical energy density to quantify highly stretched region in the form of SZW. The critical energy density used to delineate the highly stretched region is defined on the material’s true stress–strain curve as the energy density integral up to critical strain (εc ) equals to strain-hardening exponent (n) for the specific case of power law variation of material’s true stress–strain curve (denoted by area shown as dark colour in Fig. 4 and as given by Eq. (2)):
ECrit =
εc
dε
Fig. 6. Variation of numerically determined SZW across the thickness at different LLD.
proposed SZW determination methodology. The figure also showed the experimental critical SZW value for Armco iron having grain size of 118 m. The length of experimental critical SZW line showed the region in depth direction observed for measuring critical SZW. The SZW is also determined by assessing CTOD by most commonly used 45◦ line method. In Fig. 7, the proposed methodology based SZW results near mid plane are compared with half CTOD results measured conventionally (45◦ line method). The stretch zone height (SZH) is also determined using the proposed method of
(2)
0
The variation of plastic energy density (PED) near the blunted crack front with respect to deformed X-co-ordinates from crack front with increase in LLD is shown in Fig. 5. The above variation is shown at a plane near to mid thickness. The region having energy density greater than the critical energy density delineates the stretched region ( in Fig. 5) denoted as numerically determined SZW and will keep on increasing with LLD. The above term ECrit (Eq. (2)) is defined using the total strains of the tensile test. As the elastic contribution is negligibly small compared to the plastic part and also as ABAQUS software provides only the plastic contribution, the two have been compared in Fig. 5. The same methodology is used to determine stretch zone at different planes in a 50 mm thick CT specimen. Fig. 6 shows across the thickness variation of numerically predicted SZW based on the proposed method at different load line displacement. The variation of numerically predicted SZW across the thickness resembles very well with the SZW variation generally observed in experimental results, thus validating the
Fig. 7. Predicted stretch zone size variation using 45◦ line method and proposed method.
1210
S. Saxena, N. Ramakrishnan / Nuclear Engineering and Design 239 (2009) 1207–1211
The procedure to numerically predict SZW can be summarized in three steps:
Fig. 8. Comparison of experimental with numerical (using Eq. (3) and SZW by proposed method) results.
SZW considering as Y-deformation of the region where the plastic energy density exceeds the critical energy density defined by Eq. (2). It is observed that SZH predicted by proposed energy based method compares well with 45◦ line method whereas SZW predicted by proposed method is on the higher side for a LLD. In conventional (45◦ line) method semicircular blunting shape is always assumed in numerical prediction of SZW whereas no such assumption is made in the proposed SZW method. It can be appreciated that in a particular case of Armco iron (grain size: 118 m) blunting shape angle works out to be lesser than 45◦ . The validation of proposed methodology of SZW determination is also done using the experimental results of fracture toughness and SZW value (Srinivas et al., 1991). Once the stretch zone width is predicted accurately, the numerical fracture toughness can easily be calculated using Eq. (3) as follows: J = mCTOD = m(2SZW)
(3)
where m turns out to be a material independent parameter ≈1.25 (Suresh et al., 1999) when using is the stress measure defined by integral average on the material’s true stress strain curve and CTOD is the crack tip opening displacement. The yield strength as a stress measure does not relate the behaviour of the material during strain hardening subsequent to yielding. The flow stress (average of yield to ultimate) although taking hardening into account, implicitly assumes the variation to be linear. To account for the non-linearity, Suresh et al. (1999) considered the stress measure () as given by Eq. (4) that is based on an integral average.
εc dε = 0 εc 0
dε
(4)
In Eq. (4) the numerator corresponds to the area under the true stress–strain curve up to a critical strain (εc ) corresponding to onset of necking also equals to strain-hardening exponent (n) for a specific case of power law variation of true stress–strain material curve. Since the denominator is the critical strain itself, the division yields the integral average of the true stress. For a grain size of Armco iron, Fig. 8 showed the comparison of experimental results with numerically determined fracture toughness results using SZW predicted by proposed method. The experimental variation of blunting slope matches well with numerical results using proposed SZW method. Similarly in other cases also it compares well with experimental results as shown in Table 1. In Fig. 8, it can also be appreciated that the numerically predicted fracture toughness values using Eq. (3) is on the conservative side as compared to experimental results. Thus it shows that using critical material flow value; the present method can predict well the experimental SZW and fracture toughness values.
(1) Plot the variation of plastic energy density with deformed Xco-ordinates w.r.t. crack tip, for every increase in load line displacement. (2) Define the critical energy density for the material. Critical energy density corresponds to the energy density integral up to critical strain equals to strain-hardening exponent (n) for a specific case of power law material hardening behaviour that is obtained by tensile specimen testing. This energy density is defined on a true stress–strain material variation. (3) Measure the region (deformed X-co-ordinates) exceeding the material’s critical energy density defined by Eq. (2) in the figure drawn in step 1. This deformed X-co-ordinate w.r.t. crack tip is SZW and it will be a region on blunted crack surface, where the energy density exceeds the critical energy density. 4. Comparison of existing and proposed numerical SZW method Conventionally, half of CTOD is considered as SZW. In the proposed method SZW is defined in terms of J-integral using critical energy density that corresponds to critical strain at the onset of necking in tensile test equals to strain-hardening coefficient (n) value for power law material hardening behaviour. This critical energy density corresponds to onset of necking in smooth tensile specimen test. Stretch zone width that appears, as pseudo crack growth in J–R curve has to be measured in the direction of crack growth. The existing numerical method of SZW assessment is based on measurement made in the direction normal to the crack growth direction whereas in the proposed numerical SZW method, the measurements are made in the direction of crack growth. In the existing numerical SZW method generally semicircular blunting shape is assumed whereas no such assumption in made in the proposed method. The existing numerical SZW method does not clearly accounts the probable mechanism involved, as it fails to predict the variation of SZW size over the thickness in standard fracture specimen. The proposed numerical SZW method provides full information of its variation across the fracture specimen thickness that resembles well with usually observed experimental results. 5. Conclusions The present study attempts at achieving an insight into defining the procedure for numerical determination of stretch zone width using a large deformation FEM analysis. The investigation essentially comprised of number of FEM simulations to study and correlate various ductile fracture parameters with experimental results. To establish the procedure of numerical determination of SZW, a detailed analysis of the zone of intense plastic deformation was carried out. An attempt has been made to establish a numerical procedure to determine SZW and is finally compared with experimental results. The region of high deformation at the crack tip is delineated using the critical energy density obtained by tensile test. The proposed improved numerical method of SZW determination accurately predicts the slope of blunting that compares well with experimental results without having limit of loading and SZW size. This methodology also explains the probable mechanism involved in the creation of stretch zone. The proposed improved method also provides full information in regards to variation of SZW across the specimen thickness that resembles well with the experimental observations. It is concluded that the proposed method of SZW determination can reasonably simulate the process of blunting of the crack tip and can predict the material’s SZW and J-integral values. In the
S. Saxena, N. Ramakrishnan / Nuclear Engineering and Design 239 (2009) 1207–1211
present work, though the methodology is demonstrated on Armco iron material the method can also be applied for other materials. References Amouzouvi, K.F., Bassim, M.N., 1982. Determination of fracture toughness from stretch zone width measurement in predeformed AISI Type 4340 steel. Mater. Sci. Eng. 55, 257–262. Bassim, M.N., Mathews, J.R., Hyatt, C.V., 1992. Evaluation of fracture toughness of HSLA80 steel at high loading rates using stretch zone measurements. Eng. Fract. Mech. 43 (2), 297–303. Bassim, M.N., 1995. Use of the stretch zone for the characterization of ductile fracture. J. Mater. Process. Technol. 54, 109–113. Doig, P., Smith, R.F., Flewitt, P.E.J., 1984. The use of stretch zone width measurements in the determination of fracture toughness of low strength steels. Eng. Fract. Mech. 19 (4), 653–664. Eisele, U., Herter, K.H., Schuler, X., 1994. Influence of the multiaxility of stress state on the ductile fracture behaviour of degraded piping components. In: Schwalbe, K.H., Berger, C. (Eds.), ECF 10: Structural Integrity: Experiments Models and Applications, vol. 1. Berlin, pp. 249–254. Kobayashi, Hideo, Nakamura, Haruo, Nakazawa, Hajime, 1985. Comparison of J1C test methods recommended by ASTM and JSME elastic-plastic fracture test methods: user’s experience. In: Wessel, E.T., Loss, F.J. (Eds.), ASTM STP 856. American Society for Testing and Materials, pp. 3–22. Meyers, M.A., Chawla, K.K., 1999. Mechanical Behaviour of Materials. Prentice-Hall, New Jersey, p. 368. Mills, W.J., 1981. On the relationship between stretch zone formation and the J integral for high strain-hardening materials. J. Test Eval. 9 (1), 56–62.
1211
Paranjpe, S.A., Banerjee, S., 1979. Interrelation of crack opening displacement and J-integral. Eng. Fract. Mech. 11, 43–53. Roos, E., Eisele, U., Silcher, U., 1993. Effect of stress state on the ductile fracture behavior of large-scale specimens. In: Hackett, E.M., Schwalbe, K.H., Dodds Jr., R.H. (Eds.), Constraint Effects in Fracture, ASTM STP 1171. American Society for Testing and Materials, Philadelphia, pp. 41–64. Saxena, Sanjeev, Ramakrishanan, N., 2007. Characterizing crack initiation load in circumferentially through-wall cracked elbows under bending load. Int. J. Pressure Vessels Piping 84, 493–501. Saxena, Sanjeev, Ramakrishnan, N., 2007. A comparison of micro, meso and macroscale FEM analysis of ductile fracture in a CT specimen (mode I). Comput. Mater. Sci. 39 (1), 1–7. Smith, J., Bassim, M.N., Liu, C.D., 1995. Effect of fatigue pre-cracking on stretch zone formation. Eng. Fract. Mech. 52 (3), 401–408. Srinivas, M., Malakondaiah, G., Armstrong, R.W., Rama Rao, P., 1991. Ductile fracture toughness of polycrystalline Armco Iron of varying grain size. Acta Mater. 39 (5), 807–816. Srinivas, M., Sundararajan, G., Malakondaiah, G., Rama Rao, P., 1994. An analysis of ductile fracture initiation toughness in iron, its binary alloys and nickel. In: Proceeding of Royal Society, 447 (A), London, pp. 237–251. Suresh, R.K.V., Ramakrishnan, N., Srinivas, M., Rama Rao, P., 1999. On the determination of JIC using the stretch zone width method. J. Test Eval. 27 (3), 211–217. Tai, W.H., 1996. Computer simulation of plastic damage evolution in cracked body of low-carbon steel. Eng. Fract. Mech. 54 (1), 141–146. Tarafder, M., Dey, S., Sash, B., Sivaprasad, S., Tarafder, S., 2004. Initiation fracture toughness of HSLA steel through automatic measurement of stretch zone. Mater. Sci. Technol. 20, 1531–1537. Yin, S.W., Gerbrands, R.A., Hartevelt, M., 1983. An investigation of the blunting line. Eng. Fract. Mech. 18 (5), 1025–1036.