Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 2, pp. 269-290, Warsaw 2008 THE INFLUENCE OF MATERIAL PROPERTIES ON THE Q-STRESS VALUE NEAR THE CRACK TIP FOR ELASTIC-PLASTIC MATERIALS Marcin Graba Kielce University of Technology, Faculty of Mechatronics and Machine Design, Kielce, Poland e-mail: mgraba@tu.kielce.pl In the paper, the values of the Q-stress determined for various elastic- plasticmaterials for single edge notched specimens in bending (SEN(B)) are presented. The influence of the yield strength, the work-hardening exponent and the crack length on the Q-parameter is tested. The nume- rical results are approximated by closed form formulas. Key words: Q-stres, J-Q theory, fracture mechanics 1. Introduction In 1968 J.W. Hutchinson (Hutchinson, 1968) published a fundamental work, which characterized stress fields in front of a crack for thenon-linearRamberg- Osgood (R-O) material in the form σij =σ0 ( J ασ0ε0Inr ) 1 1+n σ̃ij(θ,n) (1.1) where r and θ are polar coordinates of the coordinate system located at the crack tip, σij are components of the stress tensor, J is the J-integral, n – R-O exponent, α – R-O constant, σ0 – yield stress, ε0 – strain related to σ0 through ε0 =σ0/E.The functions σ̃ij(n,θ), In(n)mustbe foundbysolving the fourth order non-linear homogenous differential equation independently for the plane stress and plane strain (Hutchinson, 1968). Equation (1.1) is commonly called the ”HRR solution”. The HRR solution includes the first term of the infinite series only. The numerical analysis showns that results obtained using the HRR solution are different from the results obtained by the finite element method (FEM). To eliminate this difference, it is necessary tousemore terms in theHRRsolution. 270 M. Graba Fig. 1. The crack opening stress distribution for elastic-plastic materials, obtained using the HRR solution (own calculation); E=206000MPa, n=5, σ0 =315MPa, ν=0.3, ε0 =0.00153, θ=0 Li and Wang (1985) using two terms in the Airy function obtained the second term of the asymptotic expansion for the two materials described by n = 3 and n = 10. Next, they compared their results with the HRR fields and FEM results. Their analysis showed that using the two term solution to describe the stress field near the crack tip brings closer analytical results to theFEMresults.Two term solutionmuchbetter describes the stress field near the crack tip, and the value of the second term, which may not be negligible depending on the material properties and geometry of the specimen. Fig. 2. A comparison of the FEM results and HRR solution for plane stress and plane strain for a single edge notched specimen under bending (SEN(B)) (own calculation); a/W =0.5,W =40mm,E=206000MPa, n=5, σ0 =315MPa, ν=0.3, ε0 =0.00153, θ=0 The influence of material properties... 271 Yang et al. (1993) using the Airy function with the separate variables in the infinite series form, proposed that the stress field near the crack tip may be described by Eq. (1.2) in an infinite series form σij σ0 = +∞∑ k=1 Akr skσ̃ (k) ij (θ) (1.2) where k is the number of series terms, Ak is the amplitude for the k-th series term, r is thenormalized distance fromthe crack tip, sk is thepower exponent for the k-th series term, and σ̃ (k) ij is the ”stress” function. When limited to three termsonly,Eq. (1.2)maybewritten in the following form σij σ0 =A1r sσ̃ (1) ij (θ)+A2r tσ̃ (2) ij (θ)+ A22 A1 r2t−sσ̃ (3) ij (θ) (1.3) where the σ̃ (k) ij functions must be found by solving the fourth order non- linear homogenous differential equation independently for the plane stress and plane strain (Yang et al., 1993), s is the power exponent, which is identical to one in the HRR solution (s may be calculated as s = −1/(n+1)), t is the power exponent for the second term of the asymptotic expansion, which must be found numerically by solving the fourth order non-linear homogenous differential equation independently for the plane stress and plane strain (Yang et al., 1993), r is the normalized term of the asymptotic expansion, which must be found numerically by solving the fourth order non-linear homogenous differential equation independently for the plane stress and plane strain (Yang et al., 1993), r is the normalized distance from the crack tip calculated as r = r/(J/σ0), A1 is the amplitude of the first term of the infinite series evaluated as A1 =1/(αε0In) n+1, and A2 is the amplitude of the second term, which is calculated by fitting Eq. (1.3) to the numerical results of the stress fields close to the crack tip. Shih et al. (1993) proposed a simplified solution. They assumed that the FEM results are exact and computed the difference between the numerical and HRR results. They proposed that the stress field near the crack tip may be described by the following equation σij σ0 = ( J αε0σ0Inr ) 1 n+1 σ̃ij(θ,n)+Q ( r J/σ0 )q σ̂ij(θ,n) (1.4) where σ̂ij(θ,n) are functions evaluated numerically, 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. 272 M. Graba Fig. 3. The influence of the work hardening exponent on the power exponents s(k) (a) and the ”stress” functions σ̃ (k) ij (b) for three terms of the asymptotic solution (own graphs based on results presented in Yang et al. (1993)) Fig. 4. J-Q trajectories for a centrally cracked plate under 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 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 the ±π/2 angle. O’Dowd and Shih proposed only two terms to describe the stress field near the crack tip σij =(σij)HRR+Qσ0σ̂ij(θ) (1.5) The influence of material properties... 273 Fig. 5. J-Q trajectories for a single edge notched specimen under bending (SEN(B)): (a) plane stress; (b) plane strain (own calculation); W =40mm, a/W =0.5, σ0 =315MPa, ν =0.3,E=206000MPa, n=5 Fig. 6. A comparison of J-Q trajectories (a) and Q= f(log[J/(aσ0)]) trajectories (b) for CC(T) and SEN(B) specimen (own calculation); W =40mm, a/W =0.5, σ0 =315MPa, ν=0.3,E=206000MPa, n=5, r=2J/σ0 Toavoid the ambiguity during the calculation of the Q-stress,O’Dowdand Shih suggested,where the Q-stressmay be evaluated. Itwas assumed that the Q-stress should be computed at r = 2J/σ0 in the θ = 0 direction. O’Dowd and Shih postulated that for θ=0 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 (1.6) 274 M. Graba where (σθθ)FEM is the stress value calculated using FEM, and (σθθ)HRR is the stress value evaluated form the HRR solution. During analysis, O’Dowd and Shih showed that in the range of θ = ±π/4, the following relationships take place: Qσ̂θθ ≈Qσ̂rr, σ̂θθ/σ̂rr ≈ 1 and Qσ̂rθ ≈ 0 (because Qσ̂rθ ≪Qσ̂θθ). Thus, the Q-stress value determines the level of the hydrostatic stress. For the plane stress, the Q-parameter is equal to zero, but for the plane strain, the Q-parameter is, in most cases, smaller than zero. 2. Discussion about J-A2 and J-Q theory To describe the stress field near the crack tip for elastic-plastic materials, the HRR solution is most often used, Eq. (1.1). However, the results obtained are usually overestimated and the analysis is conservative. A lot of analyses, which were carried out in the nineties, proved that the multi-terms description using three terms of the asymptotic solution is better thanO’Dowd’s approach.The A2 amplitude,which is used in the J-A2 theory suggested by Yang et al. (1993) is nearly independent of the distance of its determination in contrast to the Q-stress, which depends on the place where it is calculated. But the J-A2 theory sometimes is very burdensome, because an engineer must solve a fourth order nonlinear differential equation to deter- mine the σ̃ (k) ij function and the power exponent t. Next, the engineer using FEM results calculates the A2 amplitude by fitting Eq. (1.3) to the numerical results. For using the O’Dowd approach, the engineer needs the Q-stress only (calculated numerically). That is why the O’Dowd approach is easier and more convenient to use in contrast to the J-A2 theory. The J-Q theory found application in European Engineering Programs, like SINTAP or FITNET. The Q-stress is applied for formulation of the fracture criterion and to the assessment of the fracture toughness of the structural component. Thus the O’Dowd theory has practical application in engineering issues. Sometimes, the application of the J-Q theory may be limited, because there is no value of the Q-stress for a given material and specimen. Using any fracture criterion, for example proposed by O’Dowd (1995), or another criterion, the engineer can estimate the fracture toughness quite fast, if the Q-stress is known. The literature does not announce a Q-stress catalogue and Q-stress valuea as a function of the external load, material properties or geometry of the specimen. In some papers, the engineer may find J-Q graphs The influence of material properties... 275 Fig. 7. The influence of the work-hardening exponent (a) and the yield strength (b) on Q-stress values for CC(T) specimens (own calculation); W =40mm, a/W =0.5, ν=0.3,E =206000MPa for a certain group of materials. The best solution would be a catalogue of J-Q graphs formaterials characterized by various yield strengths anddifferent work-hardening exponents. Such a catalogue should take into consideration the influence of the external load, kind of the specimen (SEN(B) specimen – bending, CCT specimen – tension) and geometry of the specimen, too. In the next part of this paper, values of the Q-stress will be determined for various elastic-plastic materials for single edge notched specimens under bending (SEN(B)). All results will be presented in a graphical form Q= f(J) and Q= f(log[J/(aσ0)]).Next, thenumerical resultswill be theapproximated by closed form formulas. 3. Details of numerical analysis In the numerical analysis, the single edge notched specimens in bending (SEN(B)) were used (Fig.8). Dimensions of the specimens satisfy the ASTM E 1820-05 standard requirements. Computations were performed for plane strain using small strain option. The relative crack length was a a/W = {0.05;0.20;0.50;0.70} where a is a crack length and the width of specimens W was equal to 40mm. The choice of the SEN(B) specimen was intentional, because the SEN(B) specimens are used in the laboratory test in order to determine the critical values of the J-integral, which may be treated as the fracture toughness, if some conditions are satisfied. 276 M. Graba Fig. 8. A single edge notched specimen under bending (SEN(B)) The computations were performed using ADINA SYSTEM 8.3. Due to symmetry, only a half of the specimen was modeled. The finite element mesh was filled with 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 was modeled using 36 semicircles. The first of themwas 20 times smaller than the last one. It alsomeans that the first finite element behind the crack tip is smal- ler 2000 times than thewidth of the specimen.The crack tipwasmodeled as a quarter of the arcwhose radiuswas equal to rw =5·10 −6m (0.000125W). The whole SEN(B) specimen was modeled using 323 finite elements and 1490 no- des.An example of the finite elementmodel for SEN(B) specimen is presented in Figure 9. Fig. 9. (a) The finite element model for the SEN(B) specimen; (b) the crack tip model of the SEN(B) specimen; (c) the contour for calculation of the J-integral The influence of material properties... 277 In FEM simulation, the deformation theory of plasticity and the vonMis- ses 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) for σ>σ0 (3.1) where α=1.The tensile properties ofmaterials whichwere used in the nume- rical analysis are presentedbelow inTable 1. InFEManalysis, the calculations were done for sixteenmaterials, which differed by the yield stress and thework hardening exponent. The J-integralwere calculatedusing twomethods.Thefirstmethod, called the ”virtual shift method”, uses the concept of the virtual crack growth to compute the virtual energy change. The second method is based on the J- integral definition J = ∫ C [ wdx2− t ( ∂u ∂x1 )] ds (3.2) where w is the strain energy density, t is the stress vector acting on the contour C drawn around the crack tip, u denotes the displacement vector and ds is the infinitesimal segment of the contour C. In the numerical analysis, 64 SEN(B) specimens were used, which differed by the crack length (different a/W) and material properties (different ratio σ0/E and values of the power exponent n). Table 1.Mechanical properties of thematerials used in the numerical analysis σ0 [MPa] E [MPa] ν ε0 =σ0/E α n σ̃θθ(θ=0) In 315 206000 0.3 0.00153 1 3 1.94 5.51 500 0.00243 5 2.22 5.02 1000 0.00485 10 2.50 4.54 1500 0.00728 20 2.68 4.21 4. Numerical results The analysis of the obtained results wasmade in the range J/σ0