ICF13 Beijing (China) 2013 Vol. B
13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- New strategy for identification parameters of a micromechanical model coupled with ductile damage Jean-Claude Rakotoarisoaa,c , Donné Razafindramary b,c, Akrum Abdul-Latifb,c,* a Université d’Antsiranana Madagascar b Université Paris 8 c Laboratoire d’Ingénierie des Systèmes Mécaniques et des Matériaux (LISMMA), Supméca, 3, rue Fernand Hainaut - 93407 St Ouen Cedex – France *e-mail: aabdul@iu2t.univ-paris8.fr Abstract A low-cyclic fatigue micromechanical model proposed recently [1] for emphasizing the concept of damage induced anisotropy is used. The solution of these nonlinear constitutive relations is an important topic since it requires an important computational time. With a high nonlinearity due to damage, the identification of model parameters represents consequently an important subject. In fact, a combination of the genetic algorithm (for the global optimization) with pattern search algorithm (for the local optimization) is proposed. A comparative study is conducted under complex cyclic loadings showing the ability of the proposed approach in calibrating model parameters. Keywords Low-cyclic fatigue, parameters identification, global minimum and local minimum optimization 1. Introduction Despite the existence of increasingly powerful computers, the progress in the constitutive equations development is continuous and can be conducted via computational optimization process. Several types of modes like micromechanical approaches are proposed for describing mechanical complex phenomena. Non-linear responses under cyclic loading, for example, make the related resolutions very expensive in computing time and in memory capacity. Numerically, it has been recently reported that the algorithm of Burlisch–Stöer gives the best compromise between computational time and precision compared to other well-known algorithms. For a given model, the identification of parameters is an important issue and should be as accurate as possible to describe efficiently the material behavior. In fact, the use of reliable optimization algorithms is to minimize the difference between the model prediction and experimental behavior. Different methods have been developed to resolve this type of problem. They can, in general, be divided into two major groups: the first one which converges quickly is for local optimization. However, its major disadvantage is the possibility of converging towards local minimums. The Pattern Search algorithm is part of this group. It does not require the gradient calculation of the objective function and accepts parallel computing on different computer processors. The second group is formed by the evolutionary method based on the evolution of individuals. The genetic algorithm is a part of this group which is related to the global minimum convergence. However, it is slow because it requires several evaluations of the objective function. Therefore, this study highlights the concept of damage induced anisotropy via the used model. Numerical solutions of these nonlinear constitutive equations require normally an important computational time. Therefore, a new strategy of model parameters calibration is considered. In fact, a hybrid approach is used, in this paper, to exploit the benefits of these groups of algorithms. Hence, a combination of the genetic algorithm (GA) with pattern search algorithm (PSA) is proposed. The basic idea of this approach is to look for the global minimum with the GA, then move to the local
13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- minimum search via the PSA to improve the obtained result. Since the identification of the model parameters is of particular interest in this paper, therefore, several applications are presented. They deal with cyclic plasticity with damage induced anisotropy behavior of polycrystals and its related phenomena under complex history of loading path conditions. To conduct the identification process, the two methods (GA&PSA) are complementary and have different applications. The GA optimizes globally the model parameters leaving the field to the local method, PSA, to determine the final values of these parameters. Then, in order to evaluate the performances of the proposed method, a comparative study is performed under complex cyclic loadings showing the ability of such strategy to identify the model parameters. 2. Employed Micromechanical Model The used micromechanical model utilizes three operating levels which are: microscale (slip system), mesoscale (granular) and macroscale (overall). The theoretical formulation of the developed model is presented in detail in [1]. However, a short description of the main features of the model equations at the overall level is illustrated below. ∑∑ = ⊗ Σ= 3 1 * i i i i p p (1) ∑ = = ⊗ 3 1 i i i Q p p (2) ∑ = + = Σ ⊗ 3 1 * ( ) i i i i Q H p p (3) + + + + + = ia jb ka lb ijkl p Q Q Q Q (4) + =D D p T (5) o T dR I D R: = − (6) o T T d R D p D p R: + = + + & & & (7) R E R E M e d e d & & & & + + Σ= : : (8) e e o e T e e o e T e e o e T R E E E p R E E D E p R E E D E p M D : : : : : : 2 1 : : : 2 1 & & & & ∂ ∂ − ∂ ∂ − ∂ ∂ =− + + + (9)
13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- To determine the overall damage tensor D (Eq. 5), the spectral decomposition concept of stress tensor is adopted (Eqs. 1-4). * iΣ is the ith principal strain and pi the i th corresponding to the unit principal direction of eigenvalue and eigenvector of Σ*. The symbol ⊗ represents the tensor product. The 4th order positive spectral projection tensor + P given in (Eq. 4) is determined by equations (Eq. 2) and (Eq. 3). According to (Eq. 5), the damage is considered to be entirely active when all the eigenvalues are positive in the three principal directions; whereas, it becomes fully passive once the eigenvalues are negative, i.e., depending on the + P configuration. Hence, + P allows verifying naturally the complexity of the damage activation/deactivation phenomenon whatever the applied loading path. The overall rigidity tensor for a damaged material d R and its evolution d R& are defined respectively by equations (Eq. 6) and (Eq. 7), where o R is the classical 4 th order rigidity tensor for an initially isotropic material. As recently proposed [1], the overall stress tensor evolution coupled with damage activation/deactivation phenomenon is mathematically described by (Eq. 8). In (Eq. 9), the second term in the right-hand side depends explicitly on the eigenvectors variation during cyclic loading. Thus, when the loading is applied according to laboratory reference axes, the principal vectors coincide with the latter. In this case, these vectors are constant, i.e., their characteristics vary neither with respect to time, nor according to the deformation. Hence, the second and third terms in the right-hand side of (Eq. 9) vanishes. As a result, (Eq. 8) has the advantage to successfully treat a great number of loading types especially the multiaxial ones. 3. Algorithms of optimization The identification process is to find numerically a set of model coefficients, which correlates the best possible predictions and experimental results. It is based on minimizing the difference between the recorded model response and the given experimental result. Such a difference can never be zero. However, the rule states that when the difference is smaller, the set of coefficients is better. In this work, identification (calibration) of the model parameters is to find a search space where these values should minimize the gap between experimental results and predictions. Solving this problem is realized by minimizing the following function: ∑ = = N n n F P F P 1 ( ) ( ) , (10) ( ) ( ) ∫ − − − = 1 0 exp exp 1 0 1 ( ) t t sim T sim n V V DV V dt t t F P . (11) where, P: Model parameters, N: number of tests, [t0, t1]: time interval of the test n, Vexp-Vsim: difference between observed experiments and their simulations for the test n, D: weighting matrix of the test n. The complexities of search space are the minimum function using radically different methods of resolutions. As a first approximation, the deterministic method is suitable for search in small
13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- state-space; whereas for complex and large search state-space, this requires rather a method of stochastic search (genetic algorithm, pattern search ...). The difficulties of these problems via conventional optimization methods give rapidly this family of algorithms able to handle large combinatorial problems with mixed variable. It is more interested for solving practical problems by a general classification of optimization problems and solved methods [2,3]. Briefly, genetic algorithms are adaptive heuristic search algorithm based on the evolutionary idea of natural selection and genetic. Moreover, they are a part of evolutionary computing, a rapidly growing area of artificial intelligence. The strategy proposed in this work is to utilize a combination of the GA with the PSA. We will briefly describe these two algorithms. 3.1 Genetic Algorithm This algorithm starts with the creation of the initial population of individuals and terminates with the convergence towards the best individuals of population giving therefore the optimized solution. The transition from one generation to another is accomplished by applying the following process: (i) mechanism of evaluation, (ii) selection and (iii) modification, up to obtaining a stopping criterion. The structure of this algorithm is given by the flowchart [4] (figure 1). Figure 1 Structure of the genetic algorithm Each individual of a given population is defined by a chain of genes that correspond to the different parameters to be identified. To avoid the difficulties that may arise in the binary coding and decoding of individual, a real coding GA is used [4]. The values of each parameter are bounded by Random initialization of the population Evaluation of the "fitness" of each individual Reached tolerance Recopying the population Classification of individuals from best to worst New population creation by crossover and mutation of individuals of the previous population End optimization Yes no
13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- an interval [xmin, xmax] normally determined in this type of modeling according to mechanical bases. The main reason to establish these limits is to make the search process more efficient by reducing its space. The initial population is produced by: - Initial solutions (individuals) proposed based on the expert opinion and on experimental observations; - Solutions chosen randomly in the search space. This allows, in fact, to start the search from various solutions of the search space incorporating expert opinion. Different numerical tests should be conducted and led to choice of stochastic operators as follows: - Selection type elitism, which allows to highlight the best individuals in the population. These are the most developed individuals which will participate in the improvement. Such a technique has the advantage of faster convergence to the best individuals to the detriment of individuals which seem less appropriate and could provide elements for the creation of new individuals. - Crossover scattered, which is cut individuals into several portions (2 or 3 portions) to obtain new individuals - Adapt feasible mutation which randomly generates directions that are adaptive compared to the last generation successful or not. The feasible region is limited by the constraints. A pitch length is selected along each direction in such a manner that the bounds constraints are satisfied. 3.2. Pattern Search Algorithm Pattern search is a direct search method. This method is employed for solving optimization problems that does not require any information about the gradient of the objective function. The pattern search begins at the initial point xo. At the first iteration the mesh size is 1 and the GPS (Generalized Pattern Search) algorithm adds the pattern vectors to the initial point xo to compute the following mesh points. The algorithm computes the objective function at the mesh points using the following approach: n i n i xm x v = + Δ (12) / ( ) min(( )) 1 i i j n f xm f xmj x xm = = + (13) Where i xm : the mesh points, nx : the current point, iv : the pattern vector and nΔ : the current mesh size. A pattern is a set of vectors {vi} that the PSA utilizes to define which points to search at each iteration. The set {vi} is determined by the number of independent variables in the objective function. For example, if there are three independent variables in the optimization problem, the default for a 2N positive basis consists of the following pattern vectors:
13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- [ ] [ ] [ ] [ ] [ ] [ ] − = = − = − = = = 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 6 5 4 3 2 1 v v v v v v (14) For a more description of this, see Kolda, Lewis, and Torczon [5]. To accelerate the convergence, we stop the iteration as soon as it finds a mesh point whose fitness value is smaller than that of the current point. After a successful poll, the algorithm multiplies the current mesh size by 2. If none of the mesh points has a smaller objective function value at current point, so the poll is unsuccessful, the algorithm does not change the current point at the next iteration and multiplies the current mesh size by 0.5. In our application the criteria for stopping the pattern search are the number of objective function evaluations. 4. Numerical applications The evaluation of the proposed new strategy for identification is carried out through the description of the elastic-inelastic cyclic behavior of a polycrystal under uniaxial tension-compression (TC), biaxial tension-torsion with 90° out-of-phase angle (TT90). Our polycrystal is a random orientation distribution of 40 grains of a single-phase FCC. Initially, a database for both cyclic loading (TC and TT90) is numerically made up to final damaging of this grains distribution using the coefficients summed up in (Table 1) Table 1 Coefficients used to create the database Elastic-Inelastic parameters Model parameters E (MPa) ν α z K b s Ko (MPa) Qs (MPa) h1=h2=… =h5 h6 Cg (MPa) ag coefficients 215000 0.32 1 20 50 13 240 256 1 2.29 95100 10 Damage parameters Model parameters Ss s o ws d1 d2 0γ coefficients 2 0.95 1 1 1.4 1.25 Such a database is considered as an experimental one. Thereafter, the identification process of model parameters is started by setting the damage parameters changing six key parameters related to the inelastic behavior (bs, ko, Qs, h6, C g and ag). Therefore, the identification process by the global minimum optimization concept is made through the genetic algorithm. After several tests on the population size, a population of 600 individuals is employed to optimize 6 coefficients of the model. Several iterations are made obtaining several families of model parameters followed by a local identification using the pattern search algorithm. The optimized model parameters are summed up
13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- in (Table 2). Table 2 Initial and optimized coefficients Elastic-inelastic parameters Model parameters E (MPa) ν α z K b s ko (MPa) Qs (MPa) h1=h2=… =h5 h6 Cg (MPa) ag Initial 215000 0.32 1 20 50 24.5 340 100 1 1.7 75100 6 Optimized 215000 0.32 1 20 50 14.5 267.5 397.6 1 6.3 33830 8.71 Damage parameters Model parameters Ss s o ws d1 d2 0γ Initial 2 0.95 1 1 1.4 1.25 optimized 2 0.95 1 1 1.4 1.25 One of the main reasons which promotes this association is that both algorithms are parallelizable as shown in [2,3]. This allows gaining in computation time by exploiting computer multiprocessor. The experimental recorded fatigue lives are 38, and 14 cycles for TC and TT90, respectively. The model predicts these lives faithfully giving therefore 38, and 12 cycles in TC and TT90, respectively. Figure 2 Evolution of the overall stress during TC and TT90 up to the final fracture Figure 2 represents the typical evolutions of the maximum overall stress (pick stress value for each cycle) versus cyclic time using the same maximum von-Mises equivalent macro-strain for the two cyclic loading paths (TC and TT90). The predicted responses describe properly the experimental
13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- results. Note that these predicted results are computed with the optimized model parameters summarized in table 2. 5. Conclusion The objective of this work is to propose a new strategy to optimize the identification parameters of a micromechanical model coupled with damage [1]. Hence, a combination of genetic algorithm with pattern search algorithm is developed. The genetic algorithm optimizes globally the model parameters; whereas the pattern search algorithm, considered as a local method, has a role to determine the final values of these model coefficients. This model is tested under different cyclic loading complexities. It is recognized that this combination shows its ability to optimize the identification process. Consequently, the predicted responses describe faithfully the experimental results. Acknowledgements The authors are grateful to the AUF and SCAC of the Embassy France in Madagascar for supplying their financial support. References [1] Abdul-Latif, A., and Mounounga T., B. S., (2009), Damage Deactivation Modeling under Multiaxial Cyclic Loadings for Polycrystals, International Journal of Damage Mechanics, 18, 177-198. [2] Dréo J., Pétrowski A., Siarry P,.and Taillard E. (2003), Métaheuristiques pour l’Optimisation Difficile, Eyrolles, ISBN : 2-212-11368-4. [3] Smith R.E., Perelson A.S., and Forrest S., (1993), Searching for diverse, cooperative populations with genetic algorithms, Evolutionary Computation, 1(2), pp.127–149. [4] Christophe Bontemps, Principes Mathématiques et Utilisations des Algorithmes Génétiques, Extrait de cours, Novembre 1995. [5] Kolda, Tamara G., Robert Michael Lewis, and Virginia Torczon. "A generating set direct search augmented Lagrangian algorithm for optimization with a combination of general and linear constraints." Technical Report SAND2006-5315, Sandia National Laboratories, August 2006.
13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Advanced Assessment of Ductile Tearing in Nuclear Reactor Pressure Vessel Steel Using X-ray Tomography Michael Daly1*, Fabien Leonard1, John K Sharples2, Andrew H Sherry1 1 Dalton Nuclear Institute, The University of Manchester, Pariser Building - G Floor, Sackville Street, Manchester, M13 9PL, UK 2 AMEC Technical Services, Walton House, Birchwood Park, Warrington, Cheshire, WA3 6AT, UK * Corresponding author: michael.daly@postgrad.manchester.ac.uk Abstract Reactor pressure vessels (RPV) are manufactured from medium strength low allow ferritic steel specifically selected of its high toughness and weldability. The ability of the RPV to withstand crack propagation is crucial to maintaining the operational safety of the reactor plant. Current generations of RPV steels operate at sufficiently high temperatures to ensure that the material remains ductile during its service life. Furthermore, new materials are engineered to exhibit greater ductility and fracture toughness throughout their operating life. Therefore understanding and being able to predict the ductile fracture behaviour is critical for assuring the safety of RPV steels during operating conditions. This paper presents the results of an experimental programme aimed at using 3D X-ray tomography to quantify the volume fraction of ductile voids in tested pre-cracked specimens manufactured from A508 Class 3 RPV steel. The results indicate a high concentration of voids very close to the fracture surface and voids extending 3.6mm below the crack. The data and experimental methodology could be used to calibrate predictive mechanistically based models such as the Gurson-Tvergaard-Needlman (GTN). Keywords Ductile, Tearing, Steel, X-ray, Tomography 1. Introduction The mechanism of ductile fracture is characterised by the nucleation, growth and coalescence of voids at initiating particles. These particles are categorised as inclusions and second phase particles, and in ferritic steel are most often manganese sulphide (MnS) inclusions and metallic carbide particles (MnC). The voids form at these particles within the volume of high plastic strain and triaxial stresses ahead of a crack-tip or stress concentrator. Two nucleating processes have been observed [1]: voids formation by either decohesion of the interface between the matrix and the inclusion/particle, or by cracking of the inclusion/particle itself. Voids then grow under the influence of increasing plastic strain and high hydrostatic stress within the material. A crack will propagate once neighbouring voids coalesce and/or reach a critical size producing a macroscopic flaw. The coalescence of the voids can be considered as the final stage in the crack growth mechanism. The larger particles nucleate voids at lower stresses and strains [2] . Smaller particles will start contributing to void nucleation when the material is subjected to greater plastic deformation. The nucleation of these smaller voids at proximity to smaller particles, often between larger voids or
13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- microcracks where intense shear bands are present, may result in a void sheeting effect further contributing to void coalescence [3, 4]. The metallurgical characteristics of the microstructure, including the size and distribution of the initiating particles which can often concentrate close to or on the grain boundaries will contribute to the nucleation and coalescence process. The distribution of these particles may also be uneven within the material with banding regions of greater concentration of particles or varying grain sizes [5]. There exists a range of mechanistically based models that have been developed to describe the ductile fracture process. One of these is the Gurson-Tvergaard-Needleman (GTN) model [6] which characterises failure by defining a material yield function which depends greatly on the stress states and on material specific characteristics. These characteristics need to be calibrated to enable a simulation of ductile crack growth. 1.1 The Gurson Tvergaard Needleman The GTN model assumes the material is homogeneous and behaves as a continuum with an idealised void volume fraction distribution. Crucially, the model takes into consideration both the strain softening effects of void nucleation, growth and coalescence as well as the competing effect of the matrix hardening behaviour to define a material yielding function Φ, defined as (Eq.1): Φ σ e,σ m,σ, f * = σ e σ 2 +2q 1 f * cosh 3q 2 σ m 2σ - 1+q 3 f * 2 =0 (1) Where: σ e = macroscopic Von Mises Stress σ m = macroscopic mean stress σˉ = flow stress for the matrix material f∗ = current void fraction The values for q1, q2 and q3 were introduced by Tvergaard and Needleman to better simulate the experimental observations. These are often taken as q1 = 1.5, q2 = 1.0 and q3 = q1 2. The rate of void growth is related to the plastic part of the strain rate tensor ε k p k and the void nucleation rate is related to the equivalent plastic strain rate, p eq ε& in (Eq. 2): f*= fgrowth+ fnucleation= !1- f$ε k p k+Λε e p q (2) The first term expresses the growth rate of existing voids assuming the matrix material is incompressible and the second term defines the quantity of new voids that have nucleated as a result of the increasing plastic strain. The scaling coefficient, is characterised by (Eq. 3):
13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Λ= fN s N√2π exp,- 1 2 εe p q-εN SN 2. (3) Where: f N= volume fraction of void nucleating particles sN = standard deviation ε N = mean value ε eq P = equivalent plastic strain. An additional feature of the GTN model, introduced by Tvergaard and Needleman, was to take into consideration the initial void fraction f0, a critical void volume fraction for coalescence fc, and a critical void fraction that corresponds to the failure of the matrix, fF. f*=/ f for f ≤fc fcfu *- f c fF- fc ! f- fc$ for f >fc 3 (4) Where: fc = critical void volume fraction (typically fc = 0.15 for carbon steel) fF = actual void volume fraction at final fracture fu * = modified void volume fraction (typically f u * = 1/q1) The distribution of the initiating particles as well as their void volume fraction are key microstructural features that are needed to accurately calibrate the GTN model. These material specific parameters are usually calibrated using metallographic observations of the non-fractured material and the material volume around fractured test specimens from carefully controlled experiments. The fracture tests can be performed using a range of specimens introducing different levels of constraint and stress states. The highly constrained pre-cracked compact test (CT) specimen is frequently used to measure fracture toughness and will be used and discussed throughout this paper. Previous experiments have shown that void volume fractions (VVF) may vary by material but also by specimen types. Work by Kerry et al [7] on a high strength and low toughness aluminium alloy AL2024-T351 have shown that there is a difference in the distribution of the void volume fraction below the fracture surface for notched tensile specimens when compared with CT specimens. Using 3D X-ray tomography Taylor et al demonstrated that the CT specimens exhibited a higher concentration of voids close to the fracture surface when compared with that measured close to the fracture surface in notched tensile specimens. On the other hand, the voids extended further below the fracture surface in notched tensile specimens than was observed in CT specimens. Further work has recently been performed by Daly et al [8] with respect to an A508 Class 3 RPV ferritic steel to quantify the void volume fraction using 2D optical micrographs. Similar
13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- observations showed a higher concentration of voids for pre-cracked specimens than for notched tensile specimens. Additionally, a greater analysis of the area below the fracture surface has shown that the void volume fraction can extend to a few millimetres below the fracture surface with large clusters of voids extending up to 3.5mm below the fracture surface. The aim of this paper is to extend the observations made in this previous work by using 3D X-ray tomography to further quantify the void volume fraction below the fracture surface in pre-cracked CT specimens of A508 Class 3 RPV ferritic steel. The methodology and observations will be discussed as well as its implications for the calibration and application of the GTN model. 2. Experimental 2.1 Material The material used throughout this experiment was an A508 Class 3 ferritic steel. The specimens were extracted from the outer ring of an upright wedge-shaped block originating from a larger ring forging. All the specimens were extracted from the same location and in the same orientation. The chemical composition (wt%) of the ferritic steel was evaluated using spectographic analysis and the results are indicated in Table 1. Table 1: Chemical composition in wt% of A508 Class 3 steel. 2.2 Mechanical testing The tensile properties of the material were determined using standard round-bar test specimens oriented in the hoop direction. Three tensile specimens were tested on a Zwick 1464 at room temperature using a strain rate of 0.025% s-1 according to BS EN ISO 6892 procedure [9]. Ten fracture toughness tests were performed according to the ESIS P2-92 [10] standard using CT specimens with standard dimensions of thickness, B = 25mm, width, W = 50mm and a crack length to specimen width ratio, a/W = 0.53. Specimens were 20% side-grooved following fatigue pre-cracking. Tests were performed using both the unloading compliance and the multi specimen methods. Out of the ten tested CT specimens, two were left intact in order to preserve the crack tip for analysis. 2.3 Metallographic analysis The cracked and parent material was imaged using optical and scanning electron microscopes. The parent material was imaged to characterise the general microstructure of the ferritic steel with a specific interest on grain size and inclusion/particle type and distribution. The cracked specimens were analysed to characterise the ductile fracture mechanism and distribution of voids below the C Si Mn P S Cr Mo Ni Al Co Cu Sn Ti V 0.18 0.23 1.3 <0.005 <0.005 0.25 0.55 0.81 0.02 0.01 0.04 0.005 <0.01 0.01
13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- fracture surface. For the parent material, metallographic sections were taken to view the material in the axial-radial plane. The fractured sections were machined through the tested specimen halves in the region where plain strain fracture was expected to take place. The metallographic sections were progressively ground and polished to a mirror finish of 0.25 µm using diamond paste and etched using colloidal silica and 2% Nital. 2.4 X-ray Tomography Analysis The test samples for X-ray tomography imaging were machined below the fracture surface of three CT specimens using electrical discharge machining (EDM). The samples were approximately 0.5mm in diameter and 12mm in length and were extracted at regular intervals starting at the pre-cracked region but before the initiation of ductile tearing. The remaining specimens were extracted from below the ductile crack path and beyond the crack arrest point. The sections were extracted as close as possible to the region where plane strain was expected to take place with the greatest amount of ductile tearing damage. The surfaces of these small cylinders were lightly polished to remove any rust or scaling resulting from the EDM. The top 4mm very close to the fracture surface of these specimens were scanned at the Henry Moseley X-ray Imaging Facility at The University of Manchester using the Nikon Metrology 225/320 kV Custom Bay system equipped with a 225 kV static multi-metal anode source and a PerkinElmer 2000 × 2000 pixels 16-bit amorphous silicon flat panel detector. The scanning was performed with a molybdenum target using a voltage of 80 kV and a current of 130 µA. The data acquisition was carried out with an exposure time of 1000 ms with no filtration. The number of projections was set to 3,142 and the number of frames per projection was 1. The entire volume was reconstructed at full resolution with a voxel size of 2.0 µm along the x, y, and z directions. The data processing was performed with Avizo® Fire 7.0 software. An edge preserving smoothing filter was applied to the raw data to reduce image noise in each data set. Standard data processing was used to determine the void size distribution whereas a methodology similar to [11] was employed to determine the void to fracture surface distance and evolution of void volume fraction. 2.5 Quantification of ductile tearing damage Using the Avizo Fire data, the void volume fraction was estimated by measuring the voxel counts of metallic voxels against the count of porous voxels below the fracture surface. The VVF was calculated for each Regions Of Interest (ROI). A ROI of 100µm in height was utilised to divide the specimens into smaller cylinder regions which were comparable to the units used in Daly et al [8]. The VVF was calculated for the specimens originating below the pre-cracked surface as well as the region below the ductile tearing surface and beyond the crack arrest.
13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 3. Results The results from the three tensile tests at room temperature are summarised in Table 2. The average yield stress was 446 MPa and the ultimate tensile stress was 594 MPa. The fracture toughness properties of the A508 Class 3 steel are illustrated as a J R-curve in Figure 1 which includes data from both the unloading compliance tests and monotonically loaded tests presented together. The data from both test types are in agreement and the initiation toughness, measured by the intersection of the blunting line including 0.2 mm tearing and the power-law curve fit to the data is ~ 475 kJ/m2. The specimens for tomography analysis were extracted from test samples B, G and C as these specimens were subjected to the most ductile tearing. Table 2: Tensile Test results Figure 1: J R-curve for A508 Class 3 material tested in the hoop-radial direction at 23oC. Figure 2 illustrates the general upper bainitic microstructure of the ferritic steel under the optical microscope and SEM respectively. The average grain size was estimated at 11µm. But the microstructure is interspersed with clusters of very small grains and regions where very large grains are present. Figure 2: General microstructure of the bainitic steel under optical microscope. Figure 3Figure 4 illustrate selected microstructural observations of voids in the material. Microvoids 200µm
13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- were observed to initiate and grow by the decohesion of carbides from the matrix. Larger voids, in some cases, were shown to nucleate at large particles. These larger voids were observed to be present well below the fracture surface and ahead of the crack tip. Figure 3: Large macroscopic voids with an inclusion Figure 4: Microvoids nucleating at proximity of carbides 3.1 X-ray Tomography The X-ray tomography images of Figure 5 shows the typical distribution of voids below the fracture surface for a specimen extracted from the fracture surface. Voids as small as 10µm in diameter could be resolved with a high degree of confidence. Figure 6 shows a render of the range of shapes and sizes of voids observed. Some voids have dumbbells morphologies possibly indicating coalescence. The ability to use the X-ray tomography technique enabled the imaging of voids in their entirety. Furthermore, this technique demonstrated the ability to visualise and quantify voids and in some cases, clusters of voids up to 3.6mm below the fracture surface. Figure 5: 3D tomographic image of the ferritic steel samples and the void distribution below the fracture surface. 50µm 4µm
13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- Figure 6: Magnified 3D tomographic images showing the range of sizes and shapes of voids quantified below the fracture surface. 3.2 Quantification results The variation of the void volume fraction as a function of distance below the fracture surface in the CT specimens is illustrated in Figure 7. The data were calculated by quantifying the average VVF for cylindrical cells of 100µm in height and starting from the fracture surface. The VVF for each specimen location (0mm, 0.5mm…) was averaged over all three specimens (B, C and G) to obtain an average VVF for the first 100µm below the fracture surface and every 100µm down to 3.6mm. The presence of voids ahead of the final crack tip was also taking into consideration. Samples B and G had crack extensions of approximately 2mm, the extractions beyond the crack front were averaged separately and labeled as “beyond crack tip” on the plot. Figure 7: VVF as a function of distance below the fracture surface The following observations can be made: • The void volume fraction is highest close to the fracture surface and reduces to zero as a function of distance below the crack. A maximum VVF of 9.75 × 10-3 is measured
13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- for the 2mm specimens. The VVF remains relatively high up to 800µm below the fracture surface. • Secondary peaks are observed well below the fracture surface especially for the specimens extracted below extensive ductile tearing. The peaks are often representative of clusters of voids or very large voids at depths ranging from 1mm to 3.6mm. • Pre-cracking of the fracture toughness specimens produces very little observable ductile tearing damage. • At 2mm of ductile tearing, the ductile damage is immediately quantifiable very close to the fracture surface. For the other specimens, voids only become visible after 250µm below the fracture surface and peaks at a ratio of 7.60 × 10-3. • The data captured within the volume of material at proximity to the crack initiation and blunting (0.5mm) produce some of the highest VVF values with a maximum of 7.60 × 10-3 at 350µm below the crack surface. • Voids have been imaged and quantified beyond the crack tips indicating ductile tearing damage ahead of the crack path. 4. Discussion Taylor et al [7] quantified the void volume fraction below the fracture surface in failed CT specimens of AL2024-T351 aluminium alloy using optical and X-ray tomography. A critical void volume fraction ƒf of approximately 1.0×10 -2 was calculated for the aluminium alloy which compares favourably with the results obtained for the RPV ferritic steel of ƒf = 9.75×10 -3. It is worth noting that the peaks within the void volume fraction data may disproportionately increase the average volume fraction well below the fracture surface for the three test specimens. But as the material should be considered as a continuum with an even distribution of initial voids and initiating particles, the average void volume fraction over a large number of tests should be representative of the material’s bulk and fracture characteristics. On the other hand, the ductile damage extends further beyond the crack surface in the ferritic steel than in the aluminium alloy. The aluminium alloy exhibits a sharp reduction in the VVF which reaches ƒ = 0 at 300µm below the fracture surface. The ferritic steel exhibits ductile tearing damage up to 3.6 mm below the fracture surface identified by large voids and clusters of voids. The extent of the ductile damage was equally observed by previous work from Daly et al [8] using optical imaging analysis. The substantial extent of the ductile damage was further observed in an equivalent HY100 ferritic steel. The work from Everett et al [2] identified voids below the fracture surface of fractured notched tensile specimens using a synchrotron source with equivalent resolutions. The distribution of voids deep below the fracture surface was attributed to microstructural banding and larger MnS inclusions. These larger MnS inclusions preferentially promoted the nucleation of voids at relatively low strains.
13th International Conference on Fracture June 16–21, 2013, Beijing, China -10- The effect of the plastic strain field produced from the propagating crack may provide sufficient strains at inclusions and particles to nucleate and grow relatively large voids deep below the fracture surface. A correlation of the experimental work with a finite element analysis is required to attain an estimate of the gradient of strains ahead and below the crack during propagation. Finally, the tomographic images suffered from a blurring effect as a result of the X-ray beam hardening and X-ray spot being too large in comparison with the void sizes that were being imaged. Consequently, a substantial number of voids were only partially resolved and quantified since the pixel colours of these voids matched other metallic areas of the specimens and needed to be culled to obtain a reproducible and systematic quantifying tool. Further work will aim to use a synchrotron facility that will reduce such imaging artefacts and will ultimately increase the overall void volume fraction within the specimens. 5. Conclusions This paper has described preliminary work undertaken to characterise the ductile fracture properties and fracture mechanism in A508 Class 3 steel using X-ray tomography analysis. The main conclusions from the work are as follows: • The mechanical and fracture toughness properties have been quantified in the hoop direction. The average yield stress is 446 MPa and the initiation toughness defined by the 0.2 mm blunting line is ~ 475 kJ/m2. • The ductile fracture mechanism was identified to occur by the decohesion of the matrix from inclusions and second phase particles. • The ductile damage was successfully imaged and quantified in 3D using X-ray laboratory sources to image voids of approximately 10µm in diameter and larger below the crack surface of compact test specimens. • A relatively high VVF was quantified for specimens extracted at a 2mm crack extension but high concentrations of voids were also observed at 350µm and intermittently until 3600µm for specimens extracted along the length of the crack path. The results differ from experiments carried out with aluminium alloys where the VVF reached the background level at 300µm below the fracture surface. • As a result of the limitations of the X-ray machine and software, a proportion of the voids were not quantified which has produced a VVF lower than is actually present in the material. • The distribution of the voids deep below the fracture surface is attributed to microstructural banding and larger inclusions requiring lower strains. 6. Acknowledgements The authors are grateful to AMEC in Risley for their support in the use of their material testing and
13th International Conference on Fracture June 16–21, 2013, Beijing, China -11- microscopy equipment. And, Fabien Leonard and Tristan Lowe from the University of Manchester Henry Moseley X-ray Imaging Facility for the use of the machines and their expertise. 7. References [1] R. H. Van Stone, T. B. Cox, J. R. Low and J. A. Psioda, "Microstructural aspects of fracture by dimpled rupture," International Metals Reviews , vol. 30, no. 4, pp. 157-179, 1985. [2] R. K. Everett, K. E. Simmonds and A. B. Geltmacher, "Spatial distribution of voids in HY-100 Steel by X-ray Tomography," Scripta Materialia, vol. 44, pp. 165-169, 2001. [3] T. Pardeon and J. W. Hutchinson, "An Extended Model for Void Growth and Coalescence," Journal of the Mechanics and Physics of Solids, vol. 48, pp. 2467-2512, 2000. [4] V. Tvergaard, "Ductile Fracture by Cavity Nucleation Between Larger Voids," Journal of the Mechanics and Physics of Solids, vol. 30, pp. 265-286, 1982. [5] C. I. A. Thomson, M. J. Worswick, A. K. Pilkey and D. J. Loyd, "Void Coalescence Within Periodic Clusters of Particles," Journal of the Mechanics and Physics of Solids, vol. 51, pp. 127-146, 2003. [6] Z. L. Zhang, "A Complete Gurson Model," Non Linear Fracture and Damage Mechanics, pp. 223-248, 2001. [7] K. L. Taylor and A. H. Sherry, "The characterization and interpretation of ductile fracture mechanisms in AL2024-T351 using X-ray and focused ion beam tomography," Vols. 60 (2012) 1300-1310, 2012. [8] M. Daly, J. K. Sharples and A. H. Sherry, "Advanced Assessment of the Integrity of Ductile Components," in ASME Pressure Vessels and Piping Division Conference, Toronto, 2012. [9] British Standards, ""Metallic materials. Tensile Testing. Method of test at ambient temperature" BS EN ISO 6892-1:2009". [10] European Structural Integrity Society, "ESIS Standard No. P2-92: Procedure for determining the fracture behaviour of materials," ESIS, 1992. [11] F. Leonard, J. Stein, A. Wilkinson and P. Withers, "3D Characterisation of Void Distribution in Resin Film Infused Composites," in Conferene on Industrial Computed Tomography, 2012.
13th International Conference on Fracture June 16–21, 2013, Beijing, China 3D synchrotron laminography assessment of damage evolution in blanked dual phase steels Mouhcine Kahziz1,2,*, Thilo Morgeneyer2, Matthieu Mazière2, Lukas Helfen3, Eric Maire4, Olivier Bouaziz1,2 1 ArcelorMittal Research S.A., voie Romaine, F-57239 Maizières-lès-Metz, France 2 Mines ParisTech, Centre des Matériaux, UMR CNRS 7633, BP 87, 91003 Evry Cedex, France 3 Institute for Synchrotron Radiation – ANKA, Forschungszentum Karlsruhe, D-76021 Karlsruhe, Germany 4 Université de Lyon, INSA-Lyon, MATEIS CNRS UMR 5510, 20 avenue Albert Einstein, 69621 Villeurbanne, France * Corresponding author: mouhcine.kahziz@mines-paristech.fr Abstract The mechanical performance of automotive structures made of advanced high strength steels (AHSS) is often seen reduced by the presence of cut-edges. Here an attempt is made to gain insight into the initial damage state and the damage evolution during loading of a cut-edge. This is assessed in 3D and in-situ by synchrotron laminography observation during simultaneous tensile and bending loading of a cut-edge produced by stamping. Laminography is a technique that allows to observe regions of interest in thin sheetlike objects. It is found for the DP600 laboratory steel grade that the fracture zone is very rough and that needle voids from the surface and in the material bulk follow ferrite-martensite flow lines. During loading the needle voids grow from the fracture zone surface and coalesce with voids in the bulk. The needle cracks coalesce with the burnish zone though narrow zones, called void sheets. The formed cracks are inclined by 45° compared to the load direction. Keywords: Dual phase steels, cutting edges, X-ray laminography, damage 1. Introduction Advanced High Strength Steels (AHSS) grades remain the most widely used and developed materials in the automotive industry in order to reduce the “weight” of structural parts. Among these AHSS grades, dual phase (DP) steels with their ferrite-martensite composite microstructure present a good compromise between strength and formability. DP steels consist of a ferritic matrix containing a hard martensitic second phase in the form of islands. They are produced by controlled cooling from the austenite phase (in hot-rolled products) or from the two-phase ferrite plus austenite phase (for continuously annealed cold-rolled and hot-dip coated products) to transform some austenite to ferrite before a rapid cooling transforms the remaining austenite to martensite. However, the forming processes could affect the mechanical behavior of these grades. Some observations have shown that the cutting step tended to alter the good mechanical properties of this grade [1,2]. These studies have shown that the cutting process of DP sheets affects the adjacent material that extends into the bulk region of the sheet. This affected zone is characterized by a hardening and microstructural deformation which leads to local decohesion of ferritic and martensitic phases [1,11]. This drop in mechanical performance can significantly reduce the properties and then the use of AHSS. While ductile fracture mechanisms of this steel and its base materials (i.e. ferrite and austenite separately) have been discussed in the past [5,6,7,8], the damage mechanisms of DP cut-edges are not well known. This study aims at offering knowledge of the microstructural initial state of a cutedge and the evolution of damage from using three dimensions in-situ X-ray synchrotron laminography adapted to 3D observation of regions of interest sheet-like specimen and to identify the damage mechanisms leading to the crack formation initiating from the cut-edge. X-ray synchrotron laminography has been used in the present study to visualize for the first time -1-
13th International Conference on Fracture June 16–21, 2013, Beijing, China damage evolution from a cut-edge during in-situ tensile bending test. The method, in contrast to computed tomography which is for axisymetric objects, allows to image in three dimensions a region of interest (for instance the crack tip) inside a sheet-like sample without cutting it. The laminography set-up used here is located at the ID19 beam line at the European Synchrotron Radiation Facility (ESRF) in Grenoble (France). The acquisition was performed with a voxel size of 0.778 μm3. Applications in the study of damage of an aluminum grade can be found in the literature [12]. More details about the laminography technique are also given elsewhere [12,13]. 2. Material and experiment 2.1. Studied material The material used in this study is a laboratory dual phase steel with a ultimate tensile strength of approximately 600 MPa and a fracture strain of around 17 % at the as-received condition. The material was supplied as a 0.8mm thick cold rolled sheet. The chemical composition and microstructure are given in figure1. (a) C Mn Si Cr 0.08 0.8 0.23 0.68 (b) Figure 1: (a) DP microstructure visualized by scanning electron microscopy (after nital 0.2% etching). (b) Chemical composition of DP600 steel (weight %) The martensite islands appear to be aligned along the rolling direction. In the following the rolling direction will be referred to as L, the long transverse direction as T and the short transverse direction as S. 2.2. The cut-edge profile The shearing, which is a cutting in a straight line over the entire width of the sheet by the action of a moving blade perpendicular to the plane of the sheet, is the most widely used and least expensive process for separating metal panels. In this study, we assume that the shearing and punching have the same effects on the cut edge as the 2D descriptions of these processes are identical. Figure 2 shows an optical micrograph of the polished surface of a sample after the cutting process. The sheared surface profile is characterized by the existence of 4 characteristic zones: rollover, fracture, burnish and burr (figure2). This observation is consistent with the results found in the literature [1,3]. -2-
13th International Conference on Fracture June 16–21, 2013, Beijing, China Figure 2: optical micrograph of a DP600 cut-edge profile (after 0.2% nital etching) The proportion of these 4 characteristic zones are heavily dependent on cutting parameters such as the material nature, clearance, cutting edge radius and cutting speed [3,4,9]. The fracture zone displays the highest damage and presents a high roughness. Some zones of decohesion at the ferritemartensite interfaces can be observed in the fracture zone and are aligned along the flow lines [1]. 2.3. Experiment The experimental technique used in this study was synchrotron radiation computed laminography (SRCL). It is a non-destructive technique similar to synchrotron radiation computed tomography (SRCT), for three dimensions imaging of objects that are extended in two dimensions. It provides a unique opportunity to observe internal damage mechanisms in three dimensions during extended crack propagation in sheet materials [12]. Unlike SRCT which is especially adapted to compact or one-dimensionally elongated objects which stay in the field of view of the detector system under rotation, SRCL is optimized to image regions of interest (ROIs) out of flat, sheet-like specimens. For this, the specimen rotation axis is inclined at an angle of θ < 90° with respect to the beam direction (θ = 90° corresponds to the case of CT). For sheet-like specimens, this enables a relatively constant average X-ray transmission over the entire scanning range of 360°, which allows reliable projection data to be acquired [12]. Although the 3D Fourier domain of the specimen is not sampled completely [13], which leads to imaging artefacts, these artefacts are often less disruptive than the ones produced by (limited-angle) CT [13]. The sample geometry shown in figure3 (a) was used. A hole with a radius of 5 mm was punched out from a sheet of DP steel and an elongated crack was machined up to one edge. The loading was achieved perpendicular to the crack, via a two-screw displacement-controlled wedging device that controls the specimen notch crack mouth opening displacement (CMOD) similar to the one used in Ref. [13,14]. To avoid the sample buckling and out-of-plane motion, an anti-buckling device was used. The entire rig was mounted in a dedicated plate that was removed from the SRCL rotation stage between loading steps. The loading was applied via stepwise increases in the CMOD, one turn of the screw corresponding to 1 mm of CMOD. 3 scans were performed before any loading in order to map and image the initial state of the cut-edge. After each loading step, a scan of the ROI containing the crack tip was carried out. -3-
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