Abstract. 1 Introduction

Transcription

1 Fracture of high-strength steel notched bars: a fractographic and numerical study J. Toribio Department of Materials Science, University ofla Coruna, ETSI Caminos, Campas deelvina, 15192, La Coruna, Spain Abstract This paper deals with the fracture of high-strength pearlitic steel bars subjected to multiaxial stress states produced by notches of very different geometries. The research includes a fractographic study of the microscopic modes of fracture by scanning electron microscopy, and a numerical analysis by thefiniteelement method to compute the distribution of continuum mechanics variables in the samples at the fracture instant. The results demonstrate that fracture takes place when the distortional part of the strain energy density reaches a critical value over a critical distance characteristic of the microstructure of the material. 1 Introduction Notch-like defects (i.e., with root radius different from zero) are very frequent in structural components [1] due to the particular working conditions (e.g. anchorages for prestressed concrete). They generate a triaxial stress distribution near the notch (cf. cracked ones), which allows a detailed analysis of the influence of stress state and triaxiality on ductile failure [2] and microscopic mechanisms of fracture [3]. The final aim of this paper is the establishment of a fracture criterion for notched samples of high-strength pearlitic steel, to provide insight into the factors promoting the mechanical failure of this steel under triaxial stress states produced by notches and into the relationship between the macroscopic continuum mechanics variables and the microscopic parameters affecting fracture.

2 702 Localized Damage 2 Experimental programme A high strength pearlitic steel was used, with a yield strength of 600 MPa and a ultimate tensile strength of 1200 MPa. It presents a coarse pearlitic microstructure, with a pearlite interlamellar spacing of 0.3 nm, an average size of the cleavage facet of 75 jam, and an average pearlite colony size of about 15 jim. Fracture tests under tension loading were performed on axisymmetric notched specimens with a circumferentially-shaped notch (Fig. 1). Four notch geometries were used with different depths and radii, in order to achieve very different stress states in the vicinity of the notch tip and thus very distinct constraint situations, thus allowing an analysis of the influence of such factors on the micromechanical fracture processes. Macroscopic results of the fracture tests may be summarized as follows. Geometries A, B and C presented a macroscopically brittle fracture behaviour; geometry D is the only one that presented a macroscopically ductile behaviour with clear decrease in load. These macroscopically brittle or ductile behaviours are defined on the basis of the load-displacement curve for the fracture tests: brittle if there is no decrease in load and ductile when the curve presents a significant load decrease after the point of maximum load (cf. [4]). Fracture was assumed to occur at the maximum load point for all geometries, which represents the failure situation or condition of instability under load control. Therefore the fracture instant will be associated with such a maximum load point throughout this paper, which represents the exactfinalfractureinstant for geometries A, B and C, and the failure or instability condition for geometry D. B -D/2- R/D = 0.03 R/D =0.05 R/D =0.36 R/D = 0.40 A/D = 0.10 A/D=0.39 A/D = 0.10 A/D = 0.39 Figure 1: Axisymmetric notched geometries.

3 Localized Damage Fractographic study Fractographic analysis of all fractured specimens was carried out by means of scanning electron microscopy (SEM). Three conventional microscopic fracture modes were observed: micro-void coalescence (MVC) (zone in which thefractureprocess initiates), cleavage-like (C) (zone corresponding to brittle fracture in an unstable manner), and shear lip (L) (zone of final fracture when it is clearly ductile). The summary of the microscopical analyses is given in Fig. 2. In all cases, the process initiates by MVC. The propagation mechanism is by cleavage if fracture is unstable and brittle whereas it is by MVC in the case of fully ductile behaviour with slow fracture propagation. When the macroscopic mode offractureis peripheral and plane, the general micro-mechanism is cleavage-like (A, B and C). When the macroscopic mode offractureis central with cup and cone shape, the predominantfractureprocess is by MVC (D). It has been demonstrated that the cleavage plane of pearlite consists of {100} planes of ferrite [5]. Then, cleavage fracture occurs through the equally-oriented ferrite in adjacent pearlite colonies, which requires lower energy than in the case of colonies with different orientations of the ferrite. In addition, the size of these cleavage fracture units increases as the prior austenite grain size increases [6], because there is a larger amount of equally-oriented ferrite which is created from the same austenite grain. The average size of the cleavage facets is about 75 ^m, which represents the basic fracture unit [7] when it is typically brittle. MVC Figure 2: Fractographic analysis (MVC: micro-void coalescence, C: cleavage-like, L: shear lip).

4 704 Localized Damage 4 Numerical study In a fracture test the macroscopic external variables (force, displacement,...) can be measured and the microscopic modes of fracture observed by SEM analysis. To find the distribution of macroscopic internal variables in the continuum mechanics sense (stress, strain, strain energy density,...) at any time, and particularly at the moment offracture,thefiniteelement method (FEM) with an elastic-plastic code was applied, using a Mises yield surface with an incompressible constitutive equation (classical plasticity). The constitutive equation of the material (relationship between equivalent stress and strain) was introduced into the computer program by means of a Ramberg-Osgood equation. The external load was applied step by step, in the form of nodal displacements. An improved Newton-Raphson method was adopted, modifying the tangent stiflhess matrix at each step. The finite elements used in the computations were isoparametric with second-order interpolation (eight-node quadrilaterals and six-node triangles). The elastic-plastic FEM computations for all geometries in the form of load vs. displacement curve were compared with the experimental results of the fracture tests. The agreement was very good, since the numerical results were always within the band of experimental scatter. Therefore the computer FEM analysis was seen to be adequate to model the distribution of internal variables (stress-strain state) in the samples at any stage of the fracture test, and particularly at the fracture instant (maximum load point). 5 Fracture criterion On the basis of the elastoplasticfiniteelement results, two candidate variables will be considered for the formulation of the fracture criterion, and their distribution at the fracture instant analyzed : (1) The strain energy density, defined as: co= j <or d (1) <D> where 0 and e are respectively the stress and strain tensors and is the inner product of the two tensors. It has been used by Guillemot [8] and Sih [9] as the critical variable for fracture processes (strain energy density criterion). (2) The effective or equivalent stress in the Von Mises sense, whose expression is: o = (3 a'* ^/2) 1/2 (2) where 0' is the stress deviatoric tensor. This variable is a direct function of the distortional part of the strain energy density, i.e., the component associated with shape changes in the material.

5 Localized Damage 705 Figs. 3 and 4 show respectively the distribution of strain energy density and equivalent stress at the fracture instant, calculated by the elastic-plastic FEM analysis. The plastic zone (line of equivalent stress equal to the yield strength of the material, i.e., 0.6 GPa) clearly exceeds the fracture zone, and seems to have no influence on the fracture process which develops by cleavage. In the matter of the fracture criterion, both the strain energy density and the equivalent stress could be the governing variables, so a more detailed analysis is required to elucidate which is the relevant one. The two distributions are consistent with the fractographic analysis, since the fracture is peripheral (maximum at the notch tip) in geometries A, B and C, and central (maximum at the center of the section) in geometry D. Figure 3: Distribution of co at the fracture instant. Figure 4: Distribution of a at the fracture instant. To establish the criterion, it is necessary to know the average values of the governing variables over the fracture zone or critical region located at the notch tip. The depth of the critical region (x<.) will be several times but never less than the size of the cleavage

6 706 Localized Damage facet (xcl=75 ^m), related to the prior austenite grain size [6] and the basic unit of brittle fracture in pearlitic steels [7]. Given the small size of the critical area, one simple method to compute the average with a relative error less than 1%, consists of substituting the average value by the value at the median point. To compute this average value, the distributions co (r,z) and a (r,z) in the fracture zone are required (r and z being the cylindrical coordinates). To this end, afirstorder polynomial interpolation (Ci r z + C/2 r + GS z + C4> was performed in the criticalfiniteelement that located just at the notch tip on the basis of the numerical results at the Gauss's points at the fracture instant. Table 1. Critical values of the strain energy density co (in MPa) Geometry xc = XCL xc = 2 XCL xc = 3 XCL A B C D Average Variation ±7 28% ±5 20% ±7 32% Table 2. Critical values of the equivalent stress a (in MPa) Geometry XG = XCL XG = 2 XCL XQ = 3 XCL A B C D Average ± Variation 6% 4% 7% Tables 1 and 2 show the average values at thefractureinstant of the strain energy density and the effective or equivalent stress. Averages are calculated over critical sizes (xc) equal to one, two and three times the size of the cleavage facet (XCL). The equivalent stress is seen to be the relevant variable to establish the criterion, since it remains constant for the different sample geometries, i.e., under different triaxial stress states. The variations of strain energy

7 Localized Damage 707 density are too high for it to be considered a parameter describing fracture under different stress states. The optimum distance for the fracture criterion to be applied is two cleavage facets (minimal variations of the critical equivalent stress), although the results are not substantially different using one and three cleavage facets. A simple general criterion can thus be stated on the basis of the distortional part of the strain energy density. This criterion is formulated as follows: Fracture will take place when the distortional part of the strain energy density (or, accordingly, the effective or equivalent stress in the Von Mises sense) reaches a critical value over a critical region characteristic of the microstructure of the material. In mathematical form: <G> = GC over x^ (3) where < a > is the average equivalent stress at thefractureinstant, GC the critical equivalent stress of the material and xc the depth of the critical region, the two latter being characteristic parameters of the material, one related to the material toughness and the other to microstructural features. For the pearlitic steel used in this research, the characteristics of strength and microstructure (parameters of the material) are: GC = 1270 ± 10 MPa (4) XG = 2 XCL = 150 uim (5) which can be obtained by fracture tests on notched bars and numerical analysis to compute the distribution of continuum mechanics variables at the fracture instant. This fracture criterion has been successfully applied to the modelling of the fracture process in hydrogen environment of notched samples of the same highstrength steel as used in this work [10]. 6 Conclusions A relationship was found between the macroscopic and the microscopic fracture modes: when the fracture is peripheral and plane, the main mechanism consists of cleavage (initiated by MVC) whereas when the fracture is central with cup and cone shape, the fracture process is by MVC in the whole fracturearea. The distributions of macroscopic internal variables at the fracture instant, obtained by numerical methods, were consistent with the SEM fractographic analysis, since the maximum value of the macroscopic continuum mechanics variables was achieved in the area where microscopic fracture initiated. A fracture criterion was formulated in the following way: fracture will take place when the distortional part of the strain energy density

8 708 Localized Damage (or, accordingly, the effective or equivalent stress in the von Mises sense) reaches a critical value over a critical region characteristic of the microstructure of the material. Acknowledgements The financial support of this work by the Spanish DGICYT (Grant UE94-001) and Xunta de Galicia (Grants XUGA 11801A93 and XUGA 11801B95) is gratefully acknowledged. In addition, the author would like to thank Professor M. Elices (Polytechnical University of Madrid, Spain) for his encouragement and assistance, to Dr. A.M. Lancha (CIEMAT, Madrid, Spain) for the SEM fractographic analysis and to Mr. J. Monar (Nueva Montana Quijano Co., Santander, Spain) for providing the steel used in the experimental programme. References 1. Elices, M. Fracture of steels for reinforcing and prestressing concrete, in Fracture Mechanics of Concrete (eds. G.C. Sih and A. DiTommaso), pp Martinus Nijhoff, Dordrecht, The Netherlands, Hancock, J.W. & Brown, D.K. On the role of strain and stress state in ductile failure. J. Mech. Phys. Solides, 1983, 31, Beremin, F.M. Influence de la triaxialite des contraintes sur la rupture par dechirement ductile et la rupture fragile par clivage d'un atier doux. J. Mecanique Appliquee, 1980, 4, Landes, J.D. The effect of constraint on fracture safe design, in Structural Integrity: Experiments-Models-Applications/EOF 10 (eds. K.H. Schwalbe and C. Berger), pp EMAS, West Midlands, Alexander, D.J. & Bernstein, I.M. The cleavage plane of pearlite. Me tall. Trans., 1982, ISA, Park, Y.J. & Bernstein, I.M. The process of crack initiation and effective grain size for cleavage fracture in pearlitic eutectoid steel, Metall. Trans., 1979,10A, Lewandowski, J.J. & Thompson, A.W. Effects of the prior austenite grain on the ductility of fully pearlitic eutectoid steel, Metall. Trans., 1986, 17A, Guillemot, L.F. Criterion for crack initiation and spreading, Engng. Fracture Mech., 1976, 8, Sih, G.C. Mechanics and physics of energy density theory, Theor. Appl. Fracture Mech., 1985, 4, Toribio, J. & and Elices, M. The role of local strain rate in the hydrogen embrittlement of round-notched samples, Corros. Sci., 1992, 33,