Fatigue life prediction for a cracked notched element under symmetric load condition

Transcription

1 Fatigue life prediction for a cracked notched element under symmetric load condition M.E. Biancolini, C. Brutti, G. Cappellini & M. D'Ulisse Dep. of Mechanical Engineering Rome University "Tor Vergata", Italy Abstract In this paper the crack growth emerging from a notch is studied for round bar under symmetric load condition. Modifying and synthesising some models found in literature, a method for fatigue life prediction is outlined and verified. The model takes into account the elastic-plastic crack growth and the change of crack shape during propagation. Overall life is divided in two parts:firsta surface emerging crack propagates in the 3 D stressed region, then a Paris law stable propagation is assumed. A detailed 3D FEM model was developed in order to evaluate SIF vs. crack length, an elliptical crack shape was assumed imposing the proper eccentricity growth. The same FEM model was then exploited to evaluate the extension of plastic zone at notch tip. Fatigue life prediction for a given geometry was then carried out by means of a simple numerical framework, showing a good agreement with experimental data ranging from low cycle to high cycle fatigue. 1 Introduction In structural engineering, in general, and in machine design, in particular, it's very important to evaluate the life under fatigue loads. In the traditional approach this is performed using limit stress and safety coefficient. More recently many research efforts were devoted to define reliable procedures able to evaluate initiation and growth of cracks. As the machine elements are generally notched, this procedure has to take into account not only the material properties and the external loads but also the stress gradient and its features.

2 334 Damage and Fracture Mechanics VI This approach normally named "Damage tolerance design" is a very powerful tool to make more reliable structures and to know, during their operating life, the real state regarding to the remaining cycles before the collapse, in order to schedule the periodical inspections necessary for monitoring the defects evolution. Linear Elastic Fracture Mechanics (LEFM) is a very powerful tool to predict fatigue crack growth and critical crack size but for notched components only elastic-plastic fracture Mechanics (EPFM) gives corrects results in the first stage of crack growth. To apply this approach to actual component design is very difficult, because closed form solution to evaluate critical dimension and growing law, are available only for simple geometry. Therefore numerical methods for stress and strain fields prediction (FEM, BEM) are widely exploited together with crack law model [9]. Lin and Smith [1] have proposed a numerical method suitable to plate with elliptical crack, successively improved for arbitrary crack in notched and smooth bar [2]. Ahmad and Yates proposed an elastic-plastic model for notched bar with circumferential crack [8], based on a closed form solution for the stress intensity factor proposed by Yates [6]. A detailed numerical and experimental analysis was carried out by Carpinteri [3] in order to investigate the shape evolution of elliptical cracks in notched and smooth specimens. In this paper an elastic-plastic model is proposed for fatigue life prediction of notched bar with elliptical crack under symmetric load condition. We used the model of Ahmad and Yates for the estimation of the first part of fatigue life, when the crack propagates in the notch influenced region. The extension of plastic zone was evaluated numerically by a simplified axisymmetric FEM model. The first step of long crack propagation study was performed achieving the curve AK vs. crack length by a detailed 3D FEM model. Shape evolution was imposed as observed experimentally by Carpinteri. Then basing on this result the Paris law was integrated. The results are then compared with the experimental data published by Ahmad et al. [7] showing a good agreement. 2 Numerical model 2.1 Short crack growth Ahmad and Yates [8] observed that a little crack immersed in the plastic zone at the notch root, behaves as a crack linked to the microstucture of the material. Even with little applied load, a plastic zone arises at notch root; then in the first part of operating life of the notched specimen, the crack belongs completely to this plastic zone. In this condition of full plasticity around the crack, LEFM is not yet a reliable tool; therefore Paris-Erdogan law, tailored for smooth specimens and long cracks, predicts inaccurately the growth rate of the crack.

3 Damage and Fracture Mechanics VI 335 Basing on experimental results obtained for smooth medium steel specimen under low cycle fatigue load, Hobson [4] proposed a correlation for flaw growth valid for the first stage influenced by material microstructure: ^ = C(d_a,) (1) dn where a, is the surface crack length, d is the distance between microstructural barriers, C is a function of applied load and material properties. NFFM theory proposed by Ahmad and Yates is derived from Hobson theory for short cracks with two simple transformations. The surface crack length a^ is replaced with the crack length in the specimen c, and microstructural barrier distance d is replaced with An, the extent of plastic zone at notch root. Equation (1) becomes: c) (2) To take into account that the growth rate is greater in the notch influenced region, Ahmad and Yates proposed to use the Smith and Miller correction for crack length [5]: e = 1.69\c I (3) V P valid when the crack is in the region influenced by the stress field at the notch for e ~ D when the crack go out from this region c > Q.llJDp. This further correction lead to: ^ -- _a);witho=c+e (4) The coefficient C can be derived experimentally by high strain fatigue testing on smooth specimens. Experimental results for an HY80 steel are well correlated (95% confidence) with the following formula for C: C = %" (5) where C/ and /// are material dependent parameters, &<, is the notch root strain.

4 336 Damage and Fracture Mechanics VI 2.2 Long crack growth The Paris-Erdogan law (eq. 6) was assumed valid when the crack become long enough to emerge from the notch plasticity influenced zone. Furthermore this equation can be applied if AK is greater than the threshold value; otherwise the crack does not propagate. When the crack length reaches the critic value the propagation becomes unstable producing the failure of the structure. The parameters C, m and AKu, are material features, together with AK% that measures the fracture toughness. " (6) As it is well known in the LEFM AK depends from the load, the crack shape and size and from the specimen geometry according with the following equations: AX"y=yAo-V^7 (7) where a is a reference crack length, Ac is the amplitude of applied cyclic stress; Y is the dimensionless stress intensity factor and is a function of geometry and of the type of applied load. Normally the crack initiation yields an elliptic crack generated from the surface and propagating through the specimen. In order to predict elliptic crack growth two approach are available. The first one considers directly the variation of AK along the crack front, the greater are the local values of AK the greater are the local speed of crack front. For this reason fatigue growth produces a variation in crack shape, the ellipse becomes flat due to the fact that in radial direction K values are lower than near the boundary. To predict this variation it is necessary to develop a FEM model able to compute the K along the crack front and to modify the elements shapes according to real crack propagation. This can be achieved remeslung the model at every load step. The second method available has a lower numerical cost, for this reason was adopted in this work, and it is based on an assumed evolution of ellipse eccentricity. To predict crack evolution, Paris law has to be integrated only in radial direction, because the evolution in other directions is implicitly assumed. 2.3 Plastic zone extent As in the model chosen a parameter very important is the extent of plastic zone near the notch, a specific study was performed in order to evaluate the plastic zone size for the notched bar. An axisymmetric FEM model was developed; it is enough detailed because the non symmetric crack shape was taken into account in the calculation of the stress intensity factor. In figure 1 the FEM model and the strain resulting for a particular configuration are shown. For the same configuration the evolution of plastic zone size A/? for different loads are plotted in figure 2. The results show a good agreement with values found in literature.

5 Damage and Fracture Mechanics VI 337 Figure 1: Axisymmetric FEM model, and plastic strain contour map. 2.4 Numerical evaluation of SIF vs. crack length In order to calculate the stress intensity factor along the crack boundary for different crack lengths, a detailed 3D FEM model was developed. The physical model has two symmetry plane, for this only a quarter of the bar was modelled as shown in figure 3. The mesh is optimised in order to manage different crack lengths and shapes. Remeshing is not necessary because the shape evolution was assumed a priori. 0.8 An(C) : C,Load. ~*~ FEM analisys ""*~ Reference values Figure 2: Extent of the plastic zone vs. applied load.

6 338 Damage and Fracture Mechanics VI Figure 3: 3D FEM model In order to choose the better method of calculation of KI by means of the FEM analysis, a preliminary comparison was performed for a simple geometry. Three methods were tested using the same mesh; the first one consists in an energetic approach in which the Griffith energy is obtained by the closure work as the product of closure reaction at the crack tip and the displacement of the same points computed after an increment da of the crack length. - 1 Fu (g) If LEFM hypothesis are satisfied K is then correlated to G as follow: (9) The second method consists in the substitution of nodal displacement in the Westergaard equation to obtain KI. Also the third method is based on substitution of nodal displacement, but using the quarter point shifting technique. This technique consists in the shifting of the midside node in the quarter point position, in such a manner the shape functions of the isoparametric elements become singular and the order of the singularity is the same of the Westergaard functions. This artifice changes the element in a special one that captures very well, even with a coarse mesh, the high gradient in displacement field [9]. K can be evaluated by substitution of quarter point displacement in the Westergaard equations for displacement field: The errors relative to theoretical value for the numerical test are about 16% for simple substitution, 7% using quarter point element and 4% using the energetic approach; all the calculation are performed using the same mesh. (10)

7 Damage and Fracture Mechanics VI 339 The energetic method is the most suitable when the crack shape evolution is assumed a priori, because the only price to pay to the high precision of this method is the request of performing two analyses; if the evolution from minimum to maximum crack length is divided in N step, only N+1 calculation are required. In fact the complete curve KI vs crack length was computed by means of the energetic approach post processing a series of analysis performed with different crack advance. To reach the same accuracy with a single run method, a substitution method could be used but with a finer mesh: on the contrary a better result can be achieved with special element and the quarter point technique. The local approach is able to predict the value of K for each nodal position on the crack front but only the value in the radial direction was taken into account for the growing law, the growing in other directions derives directly from the assumed shape. The mesh used for calculation is shown in figure 3. FEM model results are computed for a greater than 0.5 mm, the values for the zero limit of a are taken from Erjian Si [10] as Y=0.74K,,values near zero and a=0.5 are connected by a straight line. The results are exposed in figure 4 together with circumferential crack ones, assuming the geometric data of the application described in the last paragraph and imposing that the expression adopted for dimensionless FIS is: (H) where a is the crack length, for circumferential crack or ellipse semi axis for elliptical crack, <j^ is the nominal stress referred to the net area of the uncracked specimen Figure 4: FIS vs. crack length. 7^ refers to circumferential crack, 7/ for elliptic crack, c crack length or minor ellipse semi axis, Jnet section diameter

8 340 Damage and Fracture Mechanics VI 2.5 Overall procedure Starting from the pre-processed value of SIF vs. crack length and from the plastic zone size vs. load, the crack evolution was divided in two parts. Assuming for each part the proper crack growth law. The initiation stage is handled with the theory for short cracks previously exposed, while for the propagation stage the Paris law was integrated. ^ total1* initiation 1*propagation \ *- ) ^ A/7 - a ( Integration limits as chosen assuming that in the first part of fatigue life, the initial crack length corresponds to total roughness value; the transition crack length between initiation and propagation stage was assumed equal to the plastic zone extent; the final crack length correspond to the critical crack size and depends on the applied load. 3 Results and Discussion The procedure now exposed was applied, in order to compare the results with experimental data, the experimental results were published by Ahmad et al. [7]. 3.1 Geometry features and load condition The application regards a circular notched bar, with nominal diameter d^l2 mm. The semi circular notch deep is D=J mm, notch radius is p=l mm, the overall FEM model length is 1=25 nun and correspond, due to the symmetry condition, to a 50 mm bar. Net section elastic stress concentration factor (theoretic) is Ay=2.3. The bar is subjected to a fatigue load, applied as a symmetric push pull cycle. 3.2 Material The test was performed for a medium strength steel with a bainitic structure HY80, Standard Number The nominal composition is 2.8 % Ni, 1.4% Cr, 0.4% Mo, 0.15% C. Stress-strain behaviour under cyclic load was represented by the Ramberg-Osgood equation: v \l N (14) with E=2070 MPa, ^=77 0 MPa, A^=0.072 The parameter C in eq. (5) become C = ^6^^ where ^ is the strain ( %) at notch root. For fatigue life in the stage of stable propagation, Paris-Erdogan law was assumed with the following parameters = OA/C",C = I

9 3.3 Results Damage and Fracture Mechanics VI 341 The numerical model was applied for this example, first performing a series of FEM 3D linear analysis, in order to obtain FIS vs. Crack length for this geometry, and recording the results; then performing a series of non linear FEM axisymmetric analysis, this geometry was studied in order to obtain the plastic zone extent vs. applied load s WO 1'10" 1-10* 1'10" t'10* I'lO/ ~**~ Elastic-plastic model ~* Elastic model Experimental Figure 5: Fatigue cycle life vs. notch root strain amplitude This results were then imported in a worksheet where the growth law was integrated for different load values. The results are shown in figure 5 together with the experimental results, showing a very good agreement; in the same figure the results of a full elastic model are plotted too; it is clear a slower evolution predicted by such model, compared with the ones elastic-plastic. Furthermore the elastoplastic procedure proposed in this paper seems to get results in good agreement with the experimental data available. 4 Conclusions In this paper was exposed a numerical approach for the study of crack propagation for round notched bar under symmetric load condition.

10 342 Damage and Fracture Mechanics VI Overall life is divided in two parts: an initiation stage in which a surface crack propagates in the notch root, followed by a stable propagation stage governed by the Paris law. For the first stage the elastoplastic model proposed by Ahmad and Yates was improved and generalised performing a numerical evaluation of the plastic region at notch root by a non-linear FEM analysis, easily applicable to different notch shapes for which a closed form solution is not available. For the propagation stage, a detailed 3D FEM model was developed in order to evaluate SIF vs. crack length; an elliptical crack shape was assumed imposing the proper eccentricity growth. Starting from this preprocessed results, the integration of Paris law can easily performed. The numerical example developed, using the proposed procedure, is in good agreement with the experimental data available in literature. This confirms that the way chosen is very promising in order to evaluate correctly the fatigue life of notched elements with small cracks. References [1] X. B. Lin and R.A. Smith, A numerical prediction of fatigue crack growth for a surface defect Fa fig. and Ft-act. ofeng. Mat. and Struct., 1995, 18, 247. [2] X. B. Lin and R.A. Smith, Fatigue growth simulation for cracks in notched and unnotched round bars Int. J. Mech, Sci., 1998, 40, 405. [3] A. Carpinteri Elliptical-arc surface cracks in round bars Fatigue and Fracture of Engineriing Materials and Structures, 1992, 15, [4] P.O. Hobson, M.W. Brown, E.R. de los Rios (1986) Two phases of short crack growth in a medium carbon steel Behaviour of short fatigue crack (edited by K.J. Miller and E.R. de los Rios) EOF publ.l, MEP, Inst. Mech. Eng. London pp [5] R. A. Smith and K. J. Miller. The growth of fatigue cracks from circular notches. Int. Journ. of Fracture 9, 1973, pp [6] J. R. Yates (1991). A simple approximation for the stress intensity factor of crack at notch. Journal of strain analysis , 1. [7] H. Y. Ahmad, M. P. Clode and J. R. Yates. Predicting Notch Fatigue Lifetimes. Fatigue '96. [8] H. Y. Ahamad, J. R. Yates (1994). An elastic-plastic model for fatigue crack growth at notches. Fatig. Fract. Engng. Mater. Struct. 17(6) pp [9] "Fracture Mechanics". T.L. Anderson, CRC Press Boca Ratmn, [10] Erjian Si (1990). Stress intensity factors for surface cracks emanating from the circumferential notch root in notched round bars. Eng. Fract. Mech. 37, 4, pp