Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 1, pp. 23-46, Warsaw 2012 50th anniversary of JTAM THE INFLUENCE OF MATERIAL PROPERTIES AND CRACK LENGTH ON THE Q-STRESS VALUE NEAR THE CRACK TIP FOR ELASTIC-PLASTIC MATERIALS FOR CENTRALLY CRACKED PLATE IN TENSION Marcin Graba Kielce University of Technology, Faculty of Mechatronics and Machine Design, Kielce, Poland e-mail: mgraba@tu.kielce.pl In the paper, values of the Q-stress determined for various elastic-plastic materials for centre crackedplate in tension (CC(T)) are presented. The influence of the yield strength, the work-hardening exponent and the crack length on the Q-parameterwas tested. The numerical results were approximated by closed form formulas. This paper is a continuation of the catalogue of the numerical solutions presented in 2008, which pre- sents Q-stress solutions for single edge notch specimens in bending – SEN(B). Both papers present full numerical results and their approxi- mation for two basic specimenswhich are used to determine in the labo- ratory tests the fracture toughness – J-integral, and both specimens are proposed by FITNET procedure used to idealize the real components. Key words: fracture mechanics, cracks, Q-stress, stress fields, HRR so- lution, FEM, J-integral, O’Dowd theory 1. Introduction – theoretical backgrounds about J-Q theory The stress fieldnear crack tip for thenon-linearRamberg-Osgood (R-O)mate- rial was described in 1968 byHutchinsonson, who published the fundamental work for fracture mechanics. The presented by Hutchinson solution, now cal- led “the HRR solution”, includes the first term of the infinite series only. The numerical analysis shows that the results obtained using the HRR solution are different from the results obtained using the finite elementmethod –FEM (Fig.1). To eliminate this difference, it is necessary to use more terms in the HRR solution. 24 M. Graba Fig. 1. Comparison of FEM results and HRR solution for the plane stress and plane strain for a single edge notched specimen in bending (SEN(B)) and centrally cracked plate in tension (CC(T)); E=206000MPa, n=5, ν=0.3, σ0 =315MPa, ε0 =σ0/E=0.00153, a/W =0.50,W =40mm, θ=0 First it was done by Li and Wang (1985), who using two terms in the Airy function, obtained the second term of the asymptotic expansion only for two different materials, described by the R-O exponent equal to n = 3 and n = 10. Their analysis shows that the two term solution much better describes the stress field near the crack tip, and the value of the second term, which may not to be negligible depends on the material properties and the specimen geometry. Amoreaccurate solutionwasproposedbyYang et al. (1993),whousingthe Airy functionwith the separatevariables proposed that the stressfieldnear the crack tip may be described by an infinite series form. The proposed by them solution is currentlyusedwith only three termsof the asymptotic solution, and it is often called “J-A2 theory”. Yang et al. (1993) conducted full discussion about their idea. They showed that the multi-terms description, which uses three termsof the asymptotic solution is better than theHutchinson approach. The A2 amplitude, which is used in the J-A2 theory suggested byYang et al. (1993) isnearly independentof thedistanceof thedetermination, butusingthe J-A2 theory in engineering practice is sometimes very burdensome, because an engineer must know the σ̃ (k) ij function and the power exponent t, which are to be calculated by solving a fourth order differential equation, and next using FEM results, the engineer must calculate the A2 amplitude. The simplified solution for describing the stress field near the crack tip for elastic plastic materials was proposed by O’Dowd and Shih (1991, 1992). That conceptwas discussedbyShih et al. (1993). They assumed that theFEM results are exact and computed the difference between the numerical andHRR The influence of material properties and crack length... 25 results.Theyproposed that the stress fieldnear the crack tipmaybedescribed by the following equation σij =σ0 ( J αε0σ0In(n)r ) 1 n+1 σ̃ij(θ,n)+σ0Q ( r J/σ0 )q σ̂ij(θ,n) (1.1) where r and θ are polar coordinates of the coordinate system located at the crack tip, σij are the components of the stress tensor, J is the J-integral, n is the R-O exponent, α is the R-O constant, σ0 – the yield stress, ε0 – strain related to σ0 through ε0 =σ0/E, σ̂ij(θ;n) are functions evaluated numerical- ly, q is the power exponent whose value changes in the range (0;0.071), and Q is a parameter, which is the amplitude of the second term in the asymptotic solution. The functions σ̃ij(n,θ), In(n) must be found by solving the fourth order non-linear homogenous differential equation independently for the plane stress and plane strain (Hutchinson, 1968) or these functions may be found using the algorithm and computer code presented in Gałkiewicz and Graba (2006). O’Dowd and Shih (1991, 1992) tested the Q-parameter in the range J/σ0 < r < 5J/σ0 near the crack tip. They showed, that the Q-parameter weakly depends on the crack tip distance in the range of ±π/2.Theyproposed only two terms to describe the stress field near the crack tip σij =(σij)HRR+Qσ0σ̂ij(θ) (1.2) where (σij)HRR( is the first term of Eq. (1.1) and it is the HRR solution. To avoid the ambiguity during the calculation of the Q-stress, O’Dowd and Shih (1991, 1992) suggested that the Q-stress should be computed at the distance from the crack tip which is equal to r=2J/σ0 for the direction θ = 0. They postulated that for the θ = 0 direction the function σ̂θθ(θ = 0) is equal to 1. That is why the Q-stress may be calculated from the following relationship Q= (σθθ)FEM − (σθθ)HRR σ0 for θ=0 and rσ0 J =2 (1.3) where (σθθ)FEM is the stress value calculated usingFEMand (σθθ)HRR is the stress evaluated form theHRR solution (these are the opening crack tip stress components). During analysis, O’Dowd and Shih (1991, 1992) showed that the Q-stress value determines the level of the hydrostatic stress. For a plane stress, the Q-parameter is equal to zero or it is close to zero, but for a plane strain, the Q-parameter is in themost cases smaller than zero (Fig.2).The Q-stress value for a plane strain depends on the external loading and distance from the crack tip – especially for large external loads (Fig.2b). 26 M. Graba Fig. 2. J-Q trajectoriesmeasured at six distances near the crack tip for centrally cracked plate in tension (CC(T)): (a) plane stress, (b) plane strain (own calculation); W =40mm, a/W =0.5, σ0 =315MPa, ν =0.3,E=206000MPa, n=5 2. Engineering aspects of J-Q theory, fracture criteria based on the O’Dowd approach Using theO’DowdandShihtheory todescribe the stressfieldnear thecrack tip for elastic-plastic materials, the difference between the HRR solution (Hutch- nison, 1968) and the results obtained using the finite element method (FEM) can be eliminated. O’Dowd’s theory is quite simple to use in practice, because in order to describe the stress field near the crack tip, wemust know onlyma- terial properties (yield stress, work hardening exponent), J-integral and the Q-stress value, which may be evaluated numerically or determined using the approximation presented in literature, for example Graba (2008). O’Dowd’s approach is easier and more convenient to use in contrast to J-A2 theory, which was proposed byYang et al. (1993). Based on the J-Q theory, O’Dowd (1995) proposed the following fracture criterion JC = JIC ( 1− Q σc/σ0 )n+1 (2.1) where JC is the real fracture toughness for a structural element characterised by a geometrical constraint defined by Q-stress (whose value is usually is smaller than zero), JIC is the fracture toughness for the plane strain condition for Q = 0 and σc is the critical stress according to the Ritchie-Knott-Rice hypothesis (Ritchie et al., 1973). Proposed by O’Dowd fracture criterion was discussed by Neimitz et al. (2007), where the authors proposed another form. They modified O’Dowd’s The influence of material properties and crack length... 27 formulas (Eq. (2.1)), by replacing the critical stress σc bymaximum opening stress σmax, which must be evaluated numerically using the large strain for- mulation. The proposed by Neimitz et al. (2007) formulas have the following form JC = JIC ( 1− Q σmax/σ0 )n+1 (2.2) For a single edge notch in bending (SEN(B)), Neimitz et al. (2007) – using the finite element method and the large strain formulation – estimated the maximumopening stress σmax for several materials (different R-O exponents, different yield stresses) and for several crack lengths. The J-Q theory foundapplication inEuropeanEngineeringPrograms, like SINTAP (1999) or FITNET (2006). The Q-stresses are applied for construc- tion of the fracture criterion and to assess the fracture toughness of structural components. The real fracture toughness KCmat may be evaluated using the formula proposed by Ainsworth and O’Dowd (1994). They showed that the increase in fracture in both the brittle and ductile regimesmay be represented by an expression of the form KCmat = { Kmat for βLr > 0 Kmat[1+α(−βLr) k] for βLr < 0 (2.3) where Kmat is the fracture toughness for the plane strain condition obtained using FITNETprocedures, and β is the parameter calculated using the follo- wing formula β= { T/(Lrσ0) for elastic materials Q/Lr for elastic-plastic materials (2.4) where Lr is the ratio of the actual external load P and the limit load P0 (or the reference stress), which may be calculated using FITNET procedures (FITNET, 2006). The constants α and k, which are occurring in Eq. (2.3), arematerial and temperature dependent (Table 1). Sherry et al. (2005a,b) proposed procedu- res to calculate the constants α and k. Thus O’Dowd’s theory has practical application to engineering issues. Sometimes, the J-Q theorymaybe limited, because there is novalue of the Q-stress for a given material and specimen. Using any fracture criterion, for example that proposedbyO’Dowd (1995) or another one, Eq. (2.3) (FITNET, 2006) or that presented by Neimitz et al. (2007) (see Eq. (2.2)), or presented byNeimitz et al. (2004), an engineer can estimate the fracture toughness quite fast, if the Q-stress is known. 28 M. Graba Table 1. Some values of the α and k parameters from Eq. (2.3) (SINTAP, 1999; FITNET, 2006) Material Temperature Fracture mode α k A533B (steel) −75◦C cleavage 1.0 1.0 A533B (steel) −90◦C cleavage 1.1 1.0 A533B (steel) −45◦C cleavage 1.3 1.0 Low Carbon Steel −50◦C cleavage 1.3 2.0 A515 (steel) +20◦C cleavage 1.5 1.0 0.0 1.0 ASTM 710 Grade A +20◦C ductile 0.6 1.0 1.0 2.0 Literature does not announce the Q-stress catalogue and Q-stress value as functionsof the external load,material properties or geometry of the specimen. Thenumerical analysis shown inGraba (2008) indicates that the Qparameter depends onmaterial properties, specimengeometry and external load. In some papers, an engineermay find J-Q graphs for a certain group ofmaterials. The best solutionwill be thecatalogue of J-Qgraphs formaterials characterisedby various yield strengths, different work-hardening exponents. Such a catalogue should take into consideration the influence of the external load, kind of the specimen (SEN(B) specimen–bending,CC(T)– tension or SEN(T)– tension) and its geometry. For SEN(B) specimens, such a catalogue was presented in Graba (2008),whopresented Q-stress values for specimenswithpredominance of bending for different materials and crack lengths. In the literature, there is no similar catalogue for specimens with predominance of tension. That is why, in the next parts of the paper, values of the Q-stress will be determined for various elastic-plastic materials for a centrally cracked plate in tension (CC(T)). TheCC(T) specimen is the basic structural elementwhich is used in the FITNET procedures (FITNET, 2006) to the modelling of real structures. All results will be presented in a graphical form – the Q= f(J) graphs. Next, the numerical results will be approximated by closed form formulas. 3. Details of numerical analysis In the numerical analysis, the centrally cracked plate in tension (CC(T)) was used (Fig.3). Dimensions of the specimens satisfy the standard requirement which is set up in FEM calculation L ­ 2W , where W is the width of the specimen and L is themeasuring length of the specimen. Computations were The influence of material properties and crack length... 29 performed for a plane strain using small strain option. The relative crack length was a/W = {0.05,0.20,0.50,0.70} where a is the crack length. The width of specimens W was equal to 40mm(for this case, themeasuring length L­ 80mm).All geometrical dimensions of theCC(T) specimen are presented in Table 2. Fig. 3. Centrally cracked plate in tension (CC(T)) Table 2. Geometrical dimensions of the CC(T) specimen used in numerical analysis width W [mm] measuring total relative crack crack length length length length 4W [mm] 2L [mm] a/W a [mm] 40 160 176 0.05 2 0.20 8 0.50 20 0.70 28 The choice of the CC(T) specimen was intentional, because the CC(T) specimens are used in the FITNETprocedures (FITNET, 2006) formodelling of real structural elements. Also in the FITNET procedures, the limit load and stress intensity factors for CC(T) specimens are presented. However in the EPRI procedures (Kumar et al., 1981), the hybridmethod for calculation of the J-integral, crack opening displacement (COD) or crack tip opening displacement (CTOD) are given. Also some laboratory tests in order to de- termine the critical values of the J-integral may be done using the CC(T) specimen, see for example Sumpter and Forbes (1992). 30 M. Graba Computations were performed using ADINA SYSTEM 8.4 (ADINA, 2006a,b). Due to the symmetry, only a quarter of the specimenwasmodelled. The finite elementmeshwas filledwith 9-node plane strain elements. The size of the finite elements in the radial direction was decreasing towards the crack tip, while in the angular direction the size of each element was kept constant. The crack tip region wasmodelled using 36 semicircles. The first of themwas 25 times smaller than the last one. It also means that the first finite element behind the crack tip was smaller 2000 times than the width of the specimen. The crack tip was modelled as a quarter of the arc whose radius was equal to rw = (1-2.5) · 10 −6m. Figure 4 presents exemplary finite element model for CC(T) specimen. Fig. 4. (a) The finite element model for CC(T) specimen used in the numerical analysis (due to the symmetry, only a quarter of the specimen was modelled); (b) the crack tip model used for the CC(T) specimen In the FEM simulation, the deformation theory of plasticity and the von Misses yield criterion were adopted. In the model, the stress-strain curve was approximated by the relation ε ε0 = { σ/σ0 for σ¬σ0 α(σ/σ0) n for σ>σ0 where α=1 (3.1) The tensile properties formaterials which were used in the numerical analysis are presented below in Table 3. In the FEM analysis, calculations were done The influence of material properties and crack length... 31 for sixteenmaterials, which differedby the yield stress and thework hardening exponent. Table 3. Mechanical properties of the materials used in numerical analysis (σ0 – yield stress,E –Young’s modulusl ν – Poisson’s ratio, ε0 – strain corre- sponding the yield stress, α – constant in the power law relationship, n work hardening exponent used in Eq. (3.1)) σ0 [MPa] E [MPa] ν ε0 =σ0/E α n 315 206000 0.3 0.00153 1 3 500 0.00243 5 1000 0.00485 10 1500 0.00728 20 The J-integral was estimated using the “virtual shiftmethod”. It uses the concept of virtual crack growth to compute virtual energy change (ADINA, 2006a,b). In the numerical analysis, 64 CC(T) specimens were used, which differed by the crack length (different a/W) and material properties (different ratios σ0/E and values of the power exponent n). 4. Numerical results – analysis of J-Q trajectories for CC(T) specimens The analysis of the results obtained by the finite elementmethod showed that in the range of distance from the crack tip J/σ0 < r < 6J/σ0, the Q-stress decreases if the distance from the crack tip increases (Fig.5). If the external load increases, the Q-stress decreases and the difference between the Q-stress calculated in the followingmeasurement points (distance r from the crack tip) increases (Fig.5). For the sake of the fact that the Q-parameter, which is used in the fracture criterion, is calculated at a distance equal to r=2J/σ0 (which was proposed byO’Dowd and Shih (1991, 1992)), it is necessary to carry out full analysis of the obtained results at this distance from the crack tip. Assessing the influence of the crack length on the Q-stress value, it is necessary to notice that if the crack length decreases, then the Q-stress re- aches a greater negative value for the same J-integral level – see Fig.6. For CC(T) specimens characterised by a short crack, the J-Q curves reach faster the saturation level than for CC(T) specimens characterised by normative 32 M. Graba Fig. 5. “The J-Q family curves” for CC(T) specimen calculated at six distances r for plane strain (W =40mm, a/W =0.50, n=10, ν =0.3, E=206000MPa, σ0 =1000MPa, ε0 =σ0/E=0.001485); (a) whole loading spectrum, (b) magnified portion of the graph Fig. 6. The influence of the crack length on the J-Q trajectories for CC(T) specimen characterised byW =40mm, n=10, ν=0.3, E=206000MPa, σ0 =1000MPa, ε0 =σ0/E=0.00485 (plane strain at the distance from the crack tip r=2J/σ0) (a/W =0.50) and long (a/W =0.70) cracks. Itmay be noticed that for short cracks, faster changes of the Q-parameter are observed if the external load increases (see the graphs in Appendices). As shown in Fig.7, if the yield stress increases, the Q-parameter incre- ases too, and it reflects for all CC(T) specimens with different crack lengths a/W . For smaller yield stresses, the J-Q trajectories shape up lower, and fa- ster changes of the Q-parameter are observed if the external load is increases The influence of material properties and crack length... 33 (Fig.7). Comparing the J-Q trajectories for different values of σ0/E, it is ob- served that the biggest differences are characterised formaterials with a small work-hardening exponent (n=3 for strongly work-hardening materials) and the smallest for materials characterised by large work-hardening exponents (n = 20 for weakly work-hardening materials) – see the graphs in Appendi- ces. If the crack length increases, this difference somewhat increases too. For smaller yield stresses, the J-Q curves for CC(T) specimens reach the satura- tion level for bigger external loads than the J-Q curves for CC(T) specimens characterised by large yield stresses. Fig. 7. The influence of the yield stress on J-Q (a) and Q= f(log[J/(aσ0)]) (b) trajectories for CC(T): W =40mm, a/W =0.50, n=10, ν=0.3, E=206000MPa (plane strain for the distance from the crack tip r=2J/σ0) Figures 8and9present somegraphsof the J-Q trajectorieswhich showthe influenceof theworkhardening exponent n on the Q-stress value and J-Q cu- rves. If the yield stress decreases, the differences between the J-Q trajectories characterised for materials described by different work-hardening exponents are bigger. For CC(T) specimens, ambiguous behaviour of the J-Q trajecto- ries depending of thework-hardening exponent is observed in comparisonwith SEN(B) specimens, which was presented in Graba (2008). In most cases (dif- ferent relative crack lengths a/W , different yield stresses (σ0/E ­ 0.00364)), if the work-hardening exponent is smaller (strongly work-hardening mate- rials) than the Q-stress value increases (Fig.9b). For small yield stresses (0.00153 ¬ σ/E ¬ 0.00200), if the external load increases, then the Q-stress value decreases if the work-hardening exponent decreases (Fig.8a). For mate- rials characterised by the yield stress σ0/E =0.00243, the difference between the J-Q trajectories are small. Mutual intersecting and overlapping of the trajectories are observed too (Fig.9a). 34 M. Graba Fig. 8. The influence of the work-hardening exponent on J-Q (a) and Q= f(log[J/(aσ0)]) (b) trajectories for CC(T):W =40mm, a/W =0.20, ν =0.3, E=206000MPa, σ0 =315MPa, ε0 =σ0/E=0.00153 (plane strain for the distance from the crack tip r=2J/σ0) Fig. 9. The influence of the work-hardening exponent on J-Q trajectories for CC(T): W =40mm, ν=0.3,E =206000MPa and (a) a/W =0.50, σ0 =500MPa, ε0 =σ0/E=0.00243, (b) a/W =0.70, σ0 =1000MPa, ε0 =σ0/E=0.00485 (plane strain for the distance from the crack tip r=2J/σ0) 5. Approximation of the numerical results for CC(T) specimens All the obtained in the numerical analysis results were used to create a catalo- gue of the J-Q trajectories for different specimens (characterised by different loading application, crack length) and different materials. The presented in the paper results are complementary with the directory presented in 2008 for SEN(B) specimens (Graba, 2008). The current paper gives full numerical re- The influence of material properties and crack length... 35 sults for specimens with predominance of tension. The previous paper, which wasmentioned above, gave numerical results and their approximation for spe- cimens with predominance of bending. The presented numerical computations provided the J-Q catalogue and universal formula (5.1) which allows one to calculate the Q-stress for CC(T) specimens and take into consideration all the parameters influencing the value of the Q-stress. All results were presented in the Q= f(log[J/(aσ0)]) graph forms (for example see Fig.8b and Fig.10). Fig. 10. The influence of the work-hardening exponent on Q= f(log[J/(aσ0)]) trajectories for SEN(B) specimen: W =40mm, ν=0.3,E=206000MPa, (a) a/W =0.50, σ0 =500MPa, ε0 =σ0/E=0.00243 and (b) a/W =0.70, σ0 =1000MPa, ε0 =σ0/E=0.00485; which were used in the procedure of approximation Next, all graphs were approximated by simple mathematical formulas ta- king thematerial properties, external load and geometry of the specimen into consideration. All the approximations were made for the results obtained at thedistance r=2J/σ0. Each of the obtained trajectories Q= f(log[J/(aσ0)]) was approximated by the third order polynomial in the form Q(J,a,σ0)=A+B log J aσ0 +C ( log J aσ0 )2 +D ( log J aσ0 )3 (5.1) where the A,B,C,D coefficients depend on thework-hardening exponent n, yield stress σ0 and crack length a/W . The rank of the fitting of formula (5.1) to numerical results for the worst case was equal R2 = 0.94 for the crack length a/W = 0.05. For other crack lengths a/W = {0.20,0.50,0.70}, the rank of the fitting of formula (5.1) satisfied the condition R2 ­ 0.99. For different work hardening exponents n, yield stresses σ0 and ratios a/W , 36 M. Graba which were not included in the numerical analysis, the coefficients A, B, C and D may be evaluated using the linear or quadratic approximation. The results of numerical approximation using formula (5.1) for CC(T) specimens (all coefficients and the rank of the fitting) are presented in Tables 4-7. Table 4. Coefficients of equation (5.1) for CC(T) specimen with the crack length a/W =0.05 n A B C D R2 σ0 =315MPa, σ0/E =0.00153 3 −1.79540 −0.16046 0.00270 −0.05173 0.979 5 −1.84658 −0.29915 −0.12998 −0.07354 0.982 10 −1.84196 −0.61308 −0.41668 −0.13219 0.961 20 −1.74217 −0.60418 −0.44835 −0.13965 0.939 σ0 =1000MPa, σ0/E =0.00485 3 −1.54832 −0.56730 −0.33063 −0.11413 0.982 5 −1.72656 −0.63071 −0.33882 −0.10721 0.986 10 −1.49156 −0.10931 −0.03536 −0.05032 0.989 20 −1.60795 −0.40632 −0.28062 −0.10566 0.996 σ0 =500MPa, σ0/E =0.00243 3 −1.61802 −0.35121 −0.26183 −0.12290 0.991 5 −1.74621 −0.47823 −0.33828 −0.12560 0.980 10 −1.79245 −0.70894 −0.52808 −0.16144 0.969 20 −1.74847 −0.77333 −0.63058 −0.18304 0.934 σ0 =1500MPa, σ0/E =0.00728 3 −1.33418 −0.24308 −0.05234 −0.04542 0.951 5 −1.51558 −0.28024 −0.05331 −0.03969 0.970 10 −1.52391 −0.19560 −0.02477 −0.03644 0.981 20 −1.59474 −0.30780 −0.10917 −0.05471 0.984 Table 5. Coefficients of equation (5.1) for CC(T) specimen with the crack length a/W =0.20 n A B C D R2 σ0 =315MPa, σ0/E =0.00153 3 −3.40016 −2.97172 −1.63678 −0.33292 0.995 5 −2.81279 −2.11444 −1.22345 −0.26738 0.990 10 −2.23934 −1.40529 −0.89907 −0.21962 0.999 20 −2.13638 −1.32808 −0.84394 −0.20595 1.000 The influence of material properties and crack length... 37 σ0 =1000MPa, σ0/E =0.00485 3 −3.65130 −3.39214 −1.55538 −0.26722 0.973 5 −1.67933 −0.35103 −0.11710 −0.05261 0.995 10 −1.49619 −0.12541 −0.07218 −0.05963 0.997 20 −1.53751 −0.18282 −0.11915 −0.07285 0.991 σ0 =500MPa, σ0/E =0.00243 3 −2.27413 −1.18967 −0.58590 −0.13421 0.992 5 −2.29981 −1.38659 −0.81232 −0.19438 0.997 10 −2.42665 −1.88288 −1.19659 −0.27781 0.998 20 −2.57462 −2.29845 −1.48745 −0.33839 0.997 σ0 =1500MPa, σ0/E =0.00728 3 −1.27982 0.01996 0.14503 0.01481 0.989 5 −1.41550 −0.01120 0.11191 −0.00153 0.994 10 −1.54844 −0.25319 −0.11256 −0.06390 0.993 20 −1.67907 −0.45441 −0.25880 −0.10071 0.994 Table 6. Coefficients of equation (5.1) for CC(T) specimen with the crack length a/W =0.50 n A B C D R2 σ0 =315MPa, σ0/E =0.00153 3 −3.85021 −2.64950 −1.05024 −0.17336 0.990 5 −2.54684 −1.13625 −0.51015 −0.11358 0.997 10 −2.24656 −0.98456 −0.50146 −0.11605 0.997 20 −3.18413 −2.55066 −1.29798 −0.24468 0.996 σ0 =1000MPa, σ0/E =0.00485 3 −3.42176 −2.52110 −0.90895 −0.13071 0.977 5 −1.63674 0.01781 0.19411 0.01940 0.994 10 −1.68070 −0.08345 0.05407 −0.02520 0.997 20 −1.88835 −0.39917 −0.13029 −0.06196 0.996 σ0 =500MPa, σ0/E =0.00243 3 −3.55938 −2.47891 −0.95900 −0.15488 0.983 5 −0.96124 1.06111 0.56236 0.05471 0.998 10 −1.61943 −0.06732 −0.04079 −0.04818 0.999 20 −2.39669 −1.42289 −0.78360 −0.17829 0.999 38 M. Graba σ0 =1500MPa, σ0/E =0.00728 3 −1.27394 −0.01699 0.04887 −0.01944 0.997 5 −1.57516 −0.02788 0.20193 0.03045 0.994 10 −1.94130 −0.43244 0.01182 −0.00341 0.994 20 −2.07357 −0.56398 −0.06157 −0.02087 0.995 Table 7. Coefficients of equation (5.1) for CC(T) specimen with the crack length a/W =0.70 n A B C D R2 σ0 =315MPa, σ0/E =0.00153 3 −3.39313 −1.86453 −0.70116 −0.12257 0.991 5 −1.86720 −0.11810 −0.07379 −0.05049 0.998 10 −3.70437 −2.86445 −1.35357 −0.24114 0.997 20 −5.11211 −4.87495 −2.27736 −0.37934 0.997 σ0 =1000MPa, σ0/E =0.00485 3 −2.93370 −1.54934 −0.40487 −0.04995 0.986 5 −2.16067 −0.26326 0.17639 0.02853 0.997 10 −2.42945 −0.26930 0.25072 0.04251 0.998 20 −2.39733 −0.11232 0.32626 0.05005 0.998 σ0 =500MPa, σ0/E =0.00243 3 −6.77352 −5.84374 −2.12683 −0.28448 0.981 5 −2.14513 −0.20561 0.07296 −0.00700 0.997 10 −2.09694 −0.47913 −0.18855 −0.06352 0.998 20 −2.21086 −0.88306 −0.48750 −0.12529 0.999 σ0 =1500MPa, σ0/E =0.00728 3 −2.02517 −0.78494 −0.17842 −0.02840 0.992 5 −2.05092 −0.46715 0.02955 0.00524 0.989 10 −2.10385 −0.19800 0.21559 0.03379 0.992 20 −2.31937 −0.32264 0.19251 0.03093 0.996 Figure 11 presents the comparison of the numerical results and their ap- proximation for J-Q trajectories for several cases of the CC(T) specimens. Appendices A-D attached to the paper present in a graphical form (Figs.12- 15) all numerical results obtained for CC(T) specimens in plain strain. All results are presented using the J-Q trajectories for each analyzed case. The influence of material properties and crack length... 39 Fig. 11. Comparison of the numerical results and their approximation for J-Q trajectories for CC(T) specimens:W =40mm, a/W =0.50, E=206000MPa, ν=0.3 and (a) σ0 ∈{315,500}MPa, n∈{5,10}, (b) σ0 ∈{1000,1500}MPa, n∈{10,20} 6. Conclusions In the paper, values of the Q-stress were determined for various elastic-plastic materials for centrally cracked plate in tension (CC(T)). The influence of the yield strength, the work-hardening exponent and the crack length on the Q- parameterwas tested.Thenumerical resultswere approximatedbyclosed form formulas. In summary, itmay be concluded that the Q-stress depends on geo- metry and the external load. Different values of the Q-stress are obtained for a centrally cracked plane in tension (CC(T)) and different for the SEN(B) specimen, which was characterised by the same material properties (see Ap- pendices of this paper andAppendices in Graba (2008)). The Q-parameter is a function of thematerial properties; its value depends on thework-hardening exponent nand theyield stress σ0. If the crack lengthdecreases, then Q-stress reaches greater negative value for the same external load. The presented in the paper catalogue of the Q-stress values and J-Q trajectories for specimenswith predominance of tension (CC(T) specimens) is complementary with the numerical solution presented inGraba (2008), which gave J-Q trajectories for specimens with predominance of bending (SEN(B) specimens)).Both papersmaybe quite useful for solving engineering problems in which the fracture toughness or stress distribution near the crack tip must be quite fast estimated. 40 M. Graba Appendix A. Numerical results for CC(T) specimen in plane strain with the crack length a/W =0.05 (distance from the crack tip r=2J/σ0) Fig. 12. The influence of the yield stress on J-Q trajectories for CC(T) specimens with the crack length a/W =0.05 for different power exponents in R-O relationship: (a) n=3, (b) n=5, (c) n=10, (d) n=20 (W =40mm, ν=0.3, E =206000MPa) The influence of material properties and crack length... 41 Appendix B. Numerical results for CC(T) specimen in plane strain with the crack length a/W =0.20 (distance from the crack tip r=2J/σ0) Fig. 13. The influence of the yield stress on J-Q trajectories for CC(T) specimens with the crack length a/W =0.20 for different power exponents in R-O relationship: (a) n=3, (b) n=5, (c) n=10, (d) n=20 (W =40mm, ν=0.3, E =206000MPa) 42 M. Graba Appendix C. Numerical results for CC(T) specimen in plane strain with the crack length a/W =0.50 (distance from the crack tip r=2.J/σ0) Fig. 14. The influence of the yield stress on J-Q trajectories for CC(T) specimens with the crack length a/W =0.50 for different power exponents in R-O relationship: (a) n=3, (b) n=5, (c) n=10, (d) n=20 (W =40mm, ν=0.3, E =206000MPa) The influence of material properties and crack length... 43 Appendix D. Numerical results for CC(T) specimen in plane strain with the crack length a/W =0.70 (distance from the crack tip r=2J/σ0) Fig. 15. The influence of the yield stress on J-Q trajectories for CC(T) specimens with the crack length a/W =0.70 for different power exponents in R-O relationship: (a) n=3, (b) n=5, (c) n=10, (d) n=20 (W =40mm, ν=0.3, E =206000MPa) Acknowledgements The support of Kielce University of Technology, Faculty of Mechatronics and Machine Design through grant No. 1.22/8.57 is acknowledged by the author of the paper. 44 M. Graba References 1. 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Sumpter J.D.G., Forbes A.T., 1992, Constraint based analysis of shallow cracks inmild steel,TWI/EWI/IS International Conference on Shallow Crack FractureMechanics Test and Application, M.G.Dawes, Edit., Cambridge, UK, paper 7 21. Yang S., Chao Y.J., Sutton M.A., 1993, Higher order asymptotic crack tip in a power-law hardeningmaterial,Engineering Fracture Mechanics, 45, 1, 99-120 Wpływ stałych materiałowych i długości pęknięcia na rozkład naprężeń Q przed wierzchołkiem pęknięcia w materiałach sprężysto-plastycznych dla płyty z centralną szczeliną poddanej rozciąganiu Streszczenie Wpracy przedstawione zostaływartości naprężeń Qwyznaczone dla szereguma- teriałów sprężysto-plastycznych dla płyt z centralną szczeliną na wskroś poddawa- nych rozciąganiu (CC(T)). Omówiony został wpływ granicy plastyczności i wykład- nika umocnienia na wartość naprężeń Q, a także wpływ długości pęknięcia. Wyniki 46 M. Graba obliczeń numerycznych aproksymowano formułami analitycznymi. Rezultaty pracy stanowią podręczny katalog krzywych J-Q dla próbek CC(T) – próbek z przewagą rozciągania,możliwydowykorzystaniawpraktyce inżynierskiej.Prezentowanewyniki są kontynuacją katalogu zaprezentowanego w roku 2008], który zawierał numerycz- ne rozwiązania i ich aproksymacje dla próbek z przewagą zginania (próbki SEN(B)). Oba elementy konstrukcyjne (próbki CC(T) i SEN(B)) często są wykorzystywane do wyznaczania odporności na pękanie w warunkach laboratoryjnych, a w analizie inżynierskiej stosuje się je jako uproszczenie złożonego obiektu konstrukcyjnego, co zalecane jest w procedurach FITNET. Manuscript received August 13, 2010; accepted for print April 4, 2011