Stress concentrations at grain boundaries due to anisotropic elastic material behavior

Transcription

1 2 WIT Press, ISBN Stress concentrations at grain boundaries due to anisotropic elastic material behavior A. Schick', C.-P. Fritzen', W. Floer*, Y.M. Hu\ U. Krupp^ & H.-J. Christ ^Institutfur Mechanik & Regelungstechnik, Universitat Siegen, Germany ' 'Institut fur Werkstofftechnik, Universitat Siegen, Germany Abstract Scanning electron microscopy (SEM) examinations on samples of the (3-titanium alloy LCB (low cost beta by TIMET) after fatigue tests revealed a strong influence of the crystallographic misorientation of adjacent grains on crack initiation and short crack growth. From that point of view the knowledge of the anisotropic elastic properties of individual grains and their influence on the stress distribution within the microstructure is of great importance to understand crack initiation and propagation mechanisms. For this purpose the microstructure of polycrystals at sites where cracks initiated was simulated by means of three dimensional finite element (FE) calculations, considering the elastic anisotropy of the grains. However, to ensure a realistic simulation, experimental data like the grain geometry and crystallographic orientation were used, which had been measured by the electron back-scattered diffraction (EBSD) technique. Beside this, the values of the elastic constants of the single-crystalline material were required. In this study a method was developed and applied to determine the three elastic constants of the anisotropic bcc structure of single-crystalline p-titanium (solution annealed LCB) from experimental data of polycrystals. The results of local strain measurements by ISDG (inter ferometric strain-displacement gauge) and EBSD evaluation of the microstructure as well as the Young's modulus and the Poisson's ratio of the polycrystal serve as input parameters for this method. The calculated anisotropic elastic constants allow the application of the above-mentioned FE-simulation to the microstructure of the polycrystalline alloy LCB. SEM examinations of the fatigued samples show that short cracks initiate exactly at the locations where the FE-simulation yielded the highest stresses.

2 2 WIT Press, ISBN Damage and Fracture Mechanics VI 1 Introduction Several studies show the strong influence of microstructural features, e.g. grain size, grain orientation and grain boundary geometry, on fatigue crack initiation and short crack propagation [e.g. 1,2]. The anisotropic elastic behavior of the grains inside a polycrystalline material leads to stress singularities at the grain boundaries which may be the reason for crack initiation during fatigue. The strength of these singularities depends on the elastic constants of the material and the misorientation of the adjacent grains, similar to the behavior which could be observed for composite materials in [3]. Several authors investigated stress concentrations in bicrystals with grain boundaries which run perpendicular and parallel to the load axis [e.g. 4,5]. In this study it is shown that the elastically anisotropic behavior is of practical importance for crack initiation in LCB during fatigue. As an example Figure 1 shows two types of cracks which were formed during fatigue testing: transgranular cracks which initiate in plastic deformed slip planes (Fig. la) and inter granular cracks where grain boundaries break nearly without plastic deformation (Fig. Ib) [6]. To investigate the influence of the elastic anisotropic behavior on crack initiation in LCB, the anisotropy of the elastic properties was determined and finite element calculations were applied to the regions of the surface, where cracks initiated during fatigue testing. Figure 1: SEM micrographs (channeling contrast) of different crack types (P-Ti alloy LCB): transgranular (a), inter granular (b).

3 2 WIT Press, ISBN Material parameter identification Damage and Fracture Mechanics VI 395 In order to carry out FE calculations the material parameters of LCB are needed. Due to the symmetry of the cubic structure of the bcc p-ti lattice of LCB, the stress-strain relationship in the crystalline co-ordinates x\ /, z*, the parameters of which are denoted by the superscript 1, is given by: 4 4 = 4 rl sj ^44 " sj_ (1) Three independent material parameters Su\ Su and ^44* characterize the elastic behavior of a cubic crystal. For commercial p-ti alloys these material parameters are not available, only in ref. [7] an anisotropy parameter of A = 1.7 for Ti-4V was published. In the following a method is introduced to obtain the elastic properties of the material from measurements on polycrystals. The elastic anisotropy behavior is characterized by the anisotropy parameter A: (2) which includes the ratio between the shear moduli 1/&/ and l/2(s\\*-su) [8]. A = \ represents isotropic material behavior. By tensor transformation the compliance matrix can be obtained in global specimen co-ordinates x, y, z. Especially for Sn one gets, taking the direction cosines between the tensile axis and the crystal co-ordinates /,, /%, h into account [8,9]: where (3) el _ O ~ (4) S^ > is equivalent to A > I and vice versa. S\i is the reciprocal value of the Young's modulus and it is a linear function of P. The Young's modulus can be measured for individual grains by applying the ISDG-technique which measures the displacement between two microhardness indents inside a grain using laser interferometry [1], Figure 2. By that, the calculation of the strain sc within different grains becomes possible. Assuming that the average value of the 'global applied' normal stress a = FIA is equal to

4 2 WIT Press, ISBN Damage and Fracture Mechanics VI t : ISDG F: EBSD Figure 2: Schematic representation how to determine the Young's modulus of an individual grain, the grain orientation and the Young's modulus of a poly crystal. that acting between the two indent markers, the calculation of the Young's modulus for the individual grain is possible (Fig. 2). To fulfil the demand for a uniform stress within the grains reasonably, specimens with grain sizes of up to 6 jum where produced by means of grain coarsening annealing. The orientation function F in eqn (3) was determined by EBSD measurements of the crystal orientation data of different grains. In this way one obtains from eqns (3) and (4): (5a),2+F^, (5b) where E( is the Young's modulus of grain /. Now Sn* can be calculated: ^n =^-^-1 p^p. (6) In fact not only two measurements were performed to calculate Sn. A Least- Square method was used to obtain an average value of all measurements. In this way, a value of 5, i for LCB of 4.3 x 1"^ m* IN resulted. As eqn (3) shows, the slope of the function (compare with Fig. 3) obeys the following equation: as,, dt (7)

5 2 WIT Press, ISBN Damage and Fracture Mechanics VI Figure 3: Graphical representation of eqn (3),.S\,(T), exemplary for LCB. The value of the slope of the curve - 2S* can be determined from the ISDG- and EBSP-measurements as -2S* = 3.68x 1~"m? IN. As seen in Figure 3, the slope of the function Sn(T) is positive (S* < ). Hence, the anisotropy parameter should be lower than 1. From eqn (7) we obtain &*/: In order to solve this equation, S^ needs to be known. To get upper and lower bounds of this parameter, limiting value concepts were introduced. From these theories elastic constants of polycrystals can be calculated from the corresponding constants of single-crystals. In the present study these models are used to estimate the single-crystal constants from the elastic parameters of the polycrystal (Young's modulus and Poisson's ratio). The most comfortable theories are those from Reuss [8,9] (average of single-crystal moduli along all crystal orientations, keeping the stress constant, lower bound of Young's modulus) and Voigt [8,9] (average of single-crystal constants along each crystal orientation, keeping the strain constant, upper bound of the Young's modulus). They are offirstorder. That means, the statistical distribution, interactions and surface distortion of the grains are not taken into consideration. Reuss calculated the Young's modulus pc,reuss and the Poisson's ratio VPC, Reuss ^8,Ret4SS (9) Analogously, the average values of the Voigt model are -PC, Voigt * PC, Voigt, 55,(1)

6 2 WIT Press, ISBN Damage and Fracture Mechanics VI where B = 1 - A. Eqn (9) as well as eqn (1) can be transformed to calculate S^ which has to be negative. One obtains upper and lower bounds of 12*. The lower the level of the anisotropy, the closer the bounds. The measured Young's modulus of 84.5 GPa and Poisson's ratio of.365 as well as the values Sn* and -2S * as given above serve as input parameters in this procedure. The results are presented in Table 1. An exemplary test for iron was carried out, too. The calculated single-crystal constants yield a good agreement to the known constants of iron [11] (Alcaic = 2.39; A^njn = 2.34). This cannot be expected for the values of LCB because parameter variations showed a strong interdependence between the solution and the parameter S\\\ While the value of SH* for iron is given, for LCB it was measured including a statistical uncertainty which could be reduced by performing further measurements. Table 1: Obtained single-crystal constants of LCB; E^ =.845x1" y^=.365, S,/=.43xlO-"m:/# -2^* =3.68x1""^ 4, k-. ] «. [,»- >/, /] A Reuss Voigt arithm. average S^ Figure 4 shows the three dimensional representation of the Young's modulus as it results from the calculated and measured anisotropy of LCB (a) and iron (b), respectively. Clearly the high level of the anisotropy of LCB is documented. The maximum Young's modulus is parallel the [1] axis of the bcc crystal. Figure 4: Three dimensional representation of the Young's modulus of LCB (a) and iron (b). (b)

7 2 WIT Press, ISBN Finite element simulation 3.1 Model Damage and Fracture Mechanics VI 399 The experimentally determined material parameters were introduced into a FEcalculation using the software package PATRAN / ABAQUS to simulate the stress distribution inside the grains due to elastic anisotropy. For the FE mesh SEM micrographs of real grain structures were used to define grain shapes (Fig. 5). The crystallographic orientation was given by EBSD measurements of each grain. Grain shape and orientation as well as the material parameters of different cracked grain boundaries were taken into a three-dimensional FE model (Fig. 5), because the observed effects are of three-dimensional nature (e.g. the boundary-layer effect in composite laminates is three-dimensional in nature and cannot be described by the classical two-dimensional lamination theory [3]). Load: A uniformly distributed tensile load of 6A/Pa was applied on the front surface. This value was also chosen as stress amplitude during the fatigue tests. Constraint: One corner node: %% =%y =%z = ; one corner node: w% =%y = ; rest: u* =. Mesh: In the center of the model, where real grains are simulated, the finest mesh is used which consists of three-dimensional 8-nodes anisotropic elements. Their orientations are taken from the EBSD-results. Therefore only the coordinates of the crystallites have to be defined and assigned to the corresponding elements. Due to the singular nature of the problem [3], a large number of elements is required. Stepped grain boundaries were preferred which allow an adaptation of the model to different situations and which limit the model size. The elements are larger on the left and right of the fine-meshed region and consist of three-dimensional isotropic 8-nodes elements. bearing SEM micrograph, EBSD measurement discretization cracked grain boundary region with fine mesh, simulated grains force isotropic environment Figure 5: Schematic representation of thefiniteelement model (11956 elements).

8 2 WIT Press, ISBN Damage and Fracture Mechanics VI 3.2 Results The calculation was carried out for different surface areas of fatigued samples containing intergranular and transgranular cracks. The simulations show at those grain boundaries, where cracks initiated in fatigue tests, significant high stresses at the rear surface of the model and very low stresses at the front of the model. Figure 6 shows an example of an intergranular crack and the appropriate rear surface of the model. The crack initiated between grain 87 and grain 88. Corresponding to this, the simulation yielded the highest principal major stress in the area of this grain boundary 87/88 (in Fig. 6: the darker the color, the higher the stress). Figure 6: SEM micrograph (a) and major principal stress distribution (b) of a intergranular cracked grain boundary. Figure 7: SEM micrograph (a) and major principal stress distribution (b) of a transgranular cracked grain boundary.

9 2 WIT Press, ISBN Damage and Fracture Mechanics VI 41 A similar situation was observed for transgranular cracks (Fig 7). The stresses, both principal major and maximum shear stresses, at those grain boundaries where the cracks initiated are at the models rear surface significantly higher than the stresses at other grain boundaries. On the other hand the simulation shows the lowest stresses at the same sites, just at the materials front surface. This means a steep gradient of the stress amplitude along the grain boundary into the bulk of the material. Plastic deformation starting from the grain boundary is promoted if suitable slip systems exist. In other cases the high stresses lead to intergranular cracking (Fig. 6). 4 Conclusions A combined experimental and theoretical method has been introduced to determine the elastic constants of single-crystals from those of polycrystals applying ISDG and EBSD measurements of individual grains. The theoretical algorithm was successfully demonstrated for iron and applied to the 3-titanium alloy LCB where it yielded a relatively strong anisotropy. The material parameters were used for FE calculations. Several situations from fatigue experiments were chosen where cracks initiated at grain boundaries. The results show generally the highest stresses exactly at those grain boundaries where intergranular as well as transgranular cracks were formed during fatigue tests. Concluding, the elastic anisotropy of the alloy LCB cannot be neglected while predicting initiation sites of cracks. Probably the crack growth rate will be influenced by the stresses due to the elastic anisotropy, too. This would lead to unregular crack growth rates, particularly during the starting-phase of crack propagation. References [1] Miller, K.J., de los Rios, E.R. The behavior of short fatigue cracks, EGF Publication I, [2] Suresh, S. Fatigue of Materials. Cambridge University Press: Cambridge, [3] Wang, S.S., Choi I. Boundary-Layer Effects in Composite Laminates. Journal of Applied Mechanics. 49, pp , [4] Peralta, P., Schober, A., Laird, C Elastic stresses in anisotropic bicrystals. Materials Science and Engineering A, 169, pp , [5] Chen, C.R., Li, S.X.. Wang, Z.G. Characteristics of strain and resolved shear stress in a bicrystal with the grain boundary perpendicular to the tensile axis. Materials Science and Engineering A, 247, pp , 1998.

10 2 WIT Press, ISBN Damage and Fracture Mechanics VI [6] Hu, Y.M., Floer, W., Krupp, U., Christ, H.-J. Application of EBSP Technique to Study Microstructurally Short Fatigue Crack Initiation and Growth in a Beta Titanium Alloy. Materials Science and Engineering, in print. [7] Ahlberg, L.A., Buck, O., Paton, N.E., Fischer, E.S. Effects of Hydrogen on Anisotropic Elastic Properties. Scripta Metallurgica, 13, pp , [8] Kuhn, H.-A. Anwendung von Grenzwertkonzepten und Phasenmischungsregeln auf die elastischen Eigenschqfen von Superlegierungen zwischen Raumtemperatur und 12 C. Doctoral Thesis, Erlangen, [9] Paufler, P., Schulze, G.E.R. Physikalische Grundlagen mechanischer Festkorpereigenschaften. Vieweg-Verlag: Braunschweig, [1] Im, S.-W. Untersuchung von Mikrorissen bei Wechselbeanspruchung durch Laser interferometrie. Doctoral Thesis, VDI-Fortschrittsberichte, 5, VDI-Verlag: Dusseldorf, 199. [11] Simmons, G., Wang, H. Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook. The M. I. T. Press: Cambridge, Massachusetts, London, 1971.