Discrete dislocation modelling of near threshold fatigue crack propagation

discrete ordinates(do)模型公式英文版Discrete Ordinates (DO) Model FormulaThe Discrete Ordinates (DO) model is a numerical method used in radiation transport calculations, particularly in the field of computational fluid dynamics. It solves the radiative transfer equation, which governs the propagation of radiation energy through a medium. The DO model discretizes the angular domain, allowing for the computation of radiation intensity at various directions.The basic DO model formula can be expressed as:I(s,Ω)= I0(s,Ω)e-∫s0κ(s')ds' + ∫s0κ(s')∫Ω'4πp(Ω'→Ω)I(s',Ω')e-∫ss'κ(s'')ds''ds'where:I(s,Ω) is the radiation intensity at position s and direction Ω.I0(s,Ω) is the incident radiation intensity at position s and direction Ω.κ(s) is the absorption coefficient at position s.p(Ω'→Ω) is the probability of a photon being scattered from direction Ω' to direction Ω.The integrals represent the accumulation of radiation intensity along the path from the source to the point of interest.The DO model uses a finite number of discrete ordinates (or directions) to approximate the angular dependence of the radiation intensity. This approximation allows for efficient numerical solutions, especially in complex geometries where analytical solutions are not feasible.The DO model is widely used in various applications such as combustion modeling, solar radiation analysis, and radiation heat transfer in participating media. It provides a computationally efficient means to model radiation transport in complex systems.中文版离散坐标（DO）模型公式离散坐标（DO）模型是一种用于辐射传输计算的数值方法，特别是在计算流体动力学领域。

Hydraulic Fracturing:Basic Concepts and Numerical ModellingPanos PapanastasiouDepartment of Civil and Environmental Engineering University of Cyprus1991-2002 in Schlumberger Cambridge ResearchHydraulic fracturing z Petroleum engineering–stimulate oil and gas reservoirs,cuttings re-injectionz Environmental engineering–waste disposal in shallow formations,cleaning up contaminated sitesz Geotechnical engineering–injection of grout, dam construction2InitialsOutlinez Basic fracturing theory: controlling parameters–fracture opening, propagation, modes, initiation, closurez Perforating for fracturingz Fracture geometry–deviated and horizontal wellbores–tortuosity and multiple fracturesz Hydraulic fracturing modeling–physical processes, geometrical models, height growth, net-pressure z Fracturing weak formations–elastoplastic modeling, experiments on soft rocks (DelFrac)3Initials4Initials Why hydraulic fracturing in Petroleum engineering?z Bypass near-wellbore formation damage–chemical incompatibilityz Extend a conductive path deep into the formation–increase area exposure to flowz Reservoir management tool–change flow, fewer wells, wellplacement, IVF, frac&pack, screen-lesscompletion5Initials Fracture Openingminz Fracture opens if the net pressure–p net=p f -σmin > 0z Fracture opening–w(x)=4 p net (L 2-x 2)1/2/E’–E’=E/(1-ν2) is the plane strain modulusz maximum width for x=0–for constant height: W=4 p net L/E’–for radial fracture: W=8 p net R/ (πE’)z singular stress at the crack tip, for x=L–σyy =p net [x/(x 2-L 2)1/2-1]Fracture Propagationz The stresses ahead of the crack tip are singular characterized by the stress intensity factor K I–σij= [K I/ (2πr )1/2] f(θ) + ...–example: an elliptical crack, K I = p net L 1/2z A crack will propagate if–K I = K IC–K IC is a material parameter called fracture toughness.Typical values for rocks are 0.1 -2 MPa m 1/2p L6InitialsFracture ModesI. opening mode II. sliding mode III. tearing modetensile fractures hydraulic fractures drilling induced faultsshear fractures andturning of fracturesnear wellboresplitting of thecrack front,multiple fractures7Initials8Initials Fracture Initiation in Open Holesz Fracture initiation at lower pressures–large contrast between insitu stresses–high pore pressure, e.g. eject at low rates prior pressurization –preexisting flaws and natural fracturesσHp is the formation pressure T is the tensile strength9Initials z Breakdown pressure–Pb = 3σh -σH -p+T –p is the formation pressure –T is the tensile strength –no fluid penetration, upper boundz Closure stress–ISIP in low permeability formationstimeBHPPressure vs Time AnalysisFracture Initiation and ClosurePerforated Cased Holes10InitialsNear Wellbore Fracture Geometry13InitialsExperiments in Delft Fracturing Consortium (1997)Deviated and Horizontal Wells14Initials Fracture Tortuosityz Gradual or sharp fracture re-orientation to the preferred plane resultsin width restriction near the wellz Tortuosity occurs–in high differential stress fields –in deviated wells–in long perforated intervals and in phased perforations –in reservoirs with natural fracturesz Problems–near-wellbore friction resulting in pressure drop –premature screen-out due to proppant bridgingσhMultiple Fracturesz Propagation of multiple fractures away from the wellbore areaz Multiples occur–in multiple or long perforated intervals with phased perforations–in deviated wells where the separation between fractures is large compared to the fracture height Array–in reservoirs with natural fracturesz Problems–increase treating net-pressure–reduced fracture widths: increase screenout–increased leakoff: lower efficiency–Reduced fracture length15InitialsHorizontal Wellbores16InitialsModelling Hydraulic Fracture Propagationz Optimize the treatment (pumping schedule, proppant stages)–increase well production–reduce costz Control where the fracture is growing–avoid fracturing near layers with different content: oil, gas, water–create long fractures in some layersz Predict the response during treatmentz Post-evaluation of the treatment17Initials18InitialsPhysical Processes in Hydraulic FracturingzzFluid leakoff in the formationzRock deformationzFracture propagationzProppant transportp wICPressure Loading on Fracture Surfacesz Pressure drop: dp/dx=12 μq/w3z Net pressure p net=p f -σh gives K I(+) >0 Array z Closure stress over fluid-lag gives K I(-) <0z Fracture propagates when K I(+) + K I(-)K ICz Fracture toughness K IC is small butan apparent fracture toughness19Initials2D MODELS20InitialsFracture Profiles in Layered Formations•Fracture may not penetrate deep tothe optimum length•Fracture may connect several payzones separated by shale layers•Fracture may grow in non-productivelayers•Problems with proppant placement•Indirect Vertical Fracturing (IVF) forsand control21InitialsFracture tipz High net-pressures (p net=p frac-σmin)–high apparent fracture toughness: due to scale effect, confiningpressure, heterogeneities and plasticity–flow behaviour near the tip: fluid-lag, rock dilation–underestimation of the closure stress (σ)3,max1,minσσp frachigh shear stress:plastic zonefluid lagcohesive zone22Initials23InitialsElasto-plastic HF modelz Fluid-flow in the fracture–Newtonian viscous fluid, lubrication theory:dp/dx=12 μq/w3z Rock deformation–Mohr-Coulomb flow theory of plasticityz Fracture propagation–Cohesive modelz Finite element analysis–fully coupled solution, specialcontinuation algorithm–meshing/remeshingFluid-flowBoundary conditions24InitialsRock deformation25Initials26InitialsPropagation criterionInitial solution27InitialsNumerical Implementation z Finite Element Discretization–8 node element–6 node interface elementsz Finite Differences for Fluid Flowz Coupling of fluid-flow with rock-deformation –Arc-length algorithm based on volume controlz Meshing/ remeshing28Initials29InitialsPlastic fractures are wider and shorter than the elastic fractures Fluid-lag is smaller in the plastic fracture30InitialsApparent fracture toughness ishigher for the plastic fracture31Initials32InitialsPlastic fracture closes first nearthe tipFracture is open at zero net-pressureand closes at negative values33InitialsStresses after fracturingThe rock formation is more stable after fracturingThe closure stress on proppant is lower near the wellboreDislocation model z Position and strength of super-dislocations–zero stress intensity factor at the cracktip–stresses satisfy Mohr-Coulomb yieldcriterion at dislocations–total crack-opening-displacement ismaximizedPapanastasiou and Atkinson (2000)34InitialsFracturing Weak Rocksz Rock dilation: narrower fracture near the tip is a wronghypothesisz Plastic yielding: plays a shielding mechanism increasingthe effective fracture toughness–contrast of insitu stresses–effective Young’s modulus, rock strength–pumping parameters: flow rate x viscosityz Fractures are wider and shorter, compared with the elasticmodel35Initialsz Break in slope in pressure decline does not reflect true fracture closure, for soft rocks. Closure stress might be underestimated z Use apparent toughness and unloading modulus in the HFsimulators.z Closure stress on proppant–higher near the tip and lower near the wellbore–more uniformly distributed with increasing fracture length z Formation Stability–decrease of stresses after fracturing–reduced risk of formation failure36Initials37Initials Experimental Set-up (DelFrac)pumpσhpressure transducerσfracturecclampslvdtAcoustic transducerRough fracture surface with dry tipfluid frontfracture tip38Initials39Initials Pressure and width in plaster experimentscσh σ,cσh σ,Closure in plasterμconfining stressμconfining stressStrong Plaster:Weak Plaster40InitialsDelft Fracturing Consortium Resultsz Optimum completion configuration–Perforations with 0/180 gave best link-up, but perforation spacing must be very small for link-up–Fractures at unfavourable perfs propagated only at low stress contrast;optimum phasing different for high/low stress contrast–Perforation orientation has little impact–Multiple fractures may be induced even for good perforation link-upz Fracture pressure has a strong positive influence on fracture geometryz Fracturing in the annulus is important in practice. It has a large impact on pressure and geometry.41Initials11/9/2006。

Fast, spatial domain potential field modelingGabriel Strykowski*, KMS (National Survey and Cadastre), Copenhagen, 2400, DENMARKFabio Boschetti, and Franklin G. Horowitz, CSIRO Exploration and Mining, Perth, 6009, AUSTRALIASummaryWe develop a technique based upon a polynomial approximation of the Green’s function for gravitational attraction of horizontal laminae, to improve the performance of spatial domain potential field modeling. We present a numerical example with good accuracy and speed.IntroductionIn this contribution we propose an alternative method to the traditional spatial domain technique of prism integration in potential field modeling. The advantages and disadvantages of prism integration as compared to e.g. Fourier methods or wavelet methods are well known. Our aim is to improve the main disadvantages of the space domain methods in order to make them an attractive alternative to FFT based techniques. Our long-range goal is to enhance the flexibility in the complexity of source distribution that can be handled, as well as to significantly improve the computational speed. However, these goals must be achieved without compromising the accuracy of the computed potential field signal. The first numerical experiments reported here were successful in all respects. Our proposal bears some resemblance to the method of stacked horizontal lamina of Talwani and Ewing (1960), see e.g. Blakely (1996), except that the lamina are used to discretize the Green's function and not to parameterize directly the anomalous source. Conceptually, the lamina, which are rectangular in shape, describe the horizontal cross-section of an infinite column of unit mass density. The mass column is offset horizontally with respect to the gravity station. We derive expressions for all three components of the gravitational attraction vector due to the lamina as a function of the vertical variable.A major speedup comes from approximating the depth/height dependence of these "exact" expressions by the sum of a polynomial and a step function. This model is valid within a specified depth/height span. Prior to modeling, the coefficients of the polynomial approximation and the parameters of the step function can be computed for different horizontal offsets and stored in a table. Except for a zone near the gravity station, the transition between these models as a function of the horizontal offset is smooth and suitable for interpolation. Through our first numerical experiment we gained some experience about the relation between the "sparseness" of the table, the maximal polynomial degree of the model, and the accuracy of the model response as compared to the "exact" prism formula. One important aspect of the method is that the geometrical configuration between the gravity station and the mass density location, including the orientation (e.g. the local direction of the vertical) is contained in the tabulated model parameters. Thus, it does not matter whether a digital elevation model is given locally on a regular Cartesian grid, or regionally in geographical coordinates on a sphere. The data are used as they preexist. All the geometrical aspects are included in the tabulated model. For a spherical Earth approximation, the model describes the attraction of a distant mass column (the distance expressed as the length of an arc along the periphery of a sphere) as a function of the height/depth at the location of the mass column.There are many long-range aspects of the proposed method. However, just to indicate immediate objectives, the method can be used for direct/inverse/mixed potential field modeling problems in which the source is parameterized on a regular grid (either Cartesian- or spherical). The gravity station can be offset in the horizontal with respect to these grid points. By discretizing the Green's function, we do all the "heavy computations" (such as the geometrical setup, the fixed grid spacing, accounting for the sphericity of the Earth, etc.) prior to modeling.TheoryFollowing the notation of eq. (9.4) in Blakely (1996), the vertical gravitational acceleration ),,(zyxg z is:>@³³³???ccccccccccy xzzzzyyxxy dx dzzzyxz dzyx2/3222),,(,,gUJ,where J is the gravitational constant and U is the mass density. The dependence on x?and y?can be subdivided into a finite number of rectangular elements yx?K?(the grid spacing) centered in discrete grid points j i y x??,. Within each such area, and at each depth, physical properties, e.g. 7, are assumed constant. Consequently, the assumption is 'z iji j77¦¦|.Inserting this into the above equation yields:¦¦³ c |?ij ij ijz z z z G z z d z y x '',,g UJwhere>@³³' c ' c '' c c c cc {c 22222/3222)'(G y y y y x x x x ij j j i i z z y y x x y d x d z z z z One notices immediately, that¦¦|ijijijz G z y x *,,g UJwhere * denotes a one dimensional convolution in z ?.Thus, with a complicated vertical mass density distribution(e.g. borehole data), one could use a vertical Fourier domain convolution, instead of summing many individual prism responses. Hence, the fundamental design of the method can be used for a substantial speedup as compared to the traditional prism integration methods for complex structures. Notice that the speed is gained without restricting the complexity of the model mass distribution.In some applications (e.g. terrain corrections) a constant 7is adequate. Thus, ¦¦³?c |ij z ijz z d z z G z y x ',,g UJ.We have derived the exact vector expressions for the gravitational attraction of a rectangular horizontal lamina,i.e. >@',','z z G z z G z z G z ijy ijx ij. In the spherical case,the local vertical direction varies between the gravity station and the mass element. Thus, the full vector is required even to model only the vertical component of the attraction. Referring to equation (14.196) in (Spiegel, 1968)and equation (3.3.49) in (Abramowitz and Stegun, 1972),we get:z z Y z z Y z z X z z X z ij y x y x z z G ? ?? ? c c c c c 21212221arctanwhere 2/1x x X i ' , 2/2x x X i ' , 2/1y y Y i ' ,and 2/2y y Y i ' .The formulae for the horizontal components is a slight generalization of example 3.2.3 in (Blakely, 1996). Also,equation (14.241) in (Spiegel, 1968) has been used. The resulting expressions are:2121'ln'22222222Y Y X X x ijy z z y x z z x z z x x z z G ? c c c c c c cand2121'ln'22222222Y Y X X y ij x z z y x z z y z z y y z z G c c c c c c c cMethodLooking at the above expressions not much seems to be gained in terms of the computational efficiency, e.g. as compared to the three-dimensional homogenous prism formula in (Blakely, 1996). However, the advantage lies in using the "exact" expression for computing the attraction for a number of discrete values of z ', z z z c {', and,subsequently, to approximate these values by a simpler model. A good choice is a model that is easily integrable with respect to z c , such as a polynomial. The resulting integral is also a polynomial and there is a simple relation between the coefficients. What matters is that the "exact"expression is adequately approximated within the vertical spanz z z ? r 'r max max . Increasing the polynomialdegree until the approximation is adequate can do this. Our experience is that for large horizontal offsets, a low degree polynomial (n=6) gave a fit to within 0.1% - 0.2%. For the near zone, with offsets about half of the grid spacing, the character of 'z z Gij changes from discontinuous to continuous (e.g. Figure 1a for fct | 0.5). Here, a polynomial of degree 37 yields the same quality of approximation. We argue that it is worth going up to this degree of approximation because it is only necessary for a narrow transition zone.Figure 1a) First numerical experiment. ij G for different horizontal offsets.y fct y x off off b ,0,.Figure 1b) the function ij G is approximated by ij f , as the sum of a polynomial n ijp and a step function ijs .Returning to the problem of modeling 'z z G ij , the exact values at a suite of discrete depths (we used 1000 in the numerical example of this paper) were approximated by a function '''z z s z z p z z f ij n ij ij where 'z z p n ijis a polynomial of order n , and 'z z s ijis astep function (see Figure 1b).The polynomial coefficients and step function parameters are entered into a table, one entry per tabulation location.The first numerical experimentOne purpose of the first numerical experiment was to demonstrate that the gravitational attraction of a homogenous rectangular prism can, in fact, be reconstructed accurately from the tabulated parameters of 'z z f ij . Another purpose was to contrast the computational speed of this technique with that of computing the attraction of a rectangular prism. Finally, we investigated choosing locationsj i y x , for sparse, butadequate storage of the parameters of 'z z f ij .We have applied the technique to the spherical case, i.e. the heights/depths of the prisms refer to a spherical surface and not to the plane. We chose m z 4000max . The horizontal offset between the mass column and the gravity station is defined by a local spherical rectangle. We use km R E 6371 as the standard radius of the Earth, and the grid spacing ? K ? y xm m 27801575K , whichcorresponds to 00025.0025.0K K O I at the latitude 055 I (where: 1- geographical latitude, O -geographical longitude). The intent was to construct the equivalent of the Hammer zones for terrain correction computations (e.g. Heiskanen and Moritz, 1967) using our new parameterization of the Green's function.The parameters of 'z z f ij were stored in tables for four different zones. Each zone has different horizontal spacing of tabulated locations and is used for approximating ij G at different distance intervals from the gravity station. In the experiment we chose a uniform spacing between table values in each zone and expressed it as a fraction/multiple of y x ? K ? . The factors were 0.1 for Zone 1; 0.5 for Zone 2; 1.0 for Zone 3, and 3.0 for Zone 4. In this way, the Green's function close to the gravity station (Zone 1) is tabulated more densely than further away in Zone 2 to Zone 4. The number of zones can be chosen arbitrarily. The computer programs we have written can be used to find a more appropriate (sparse) way of storing these values.On figure 2 we show the results for Zone 1 (extending to a distance of 4170 m in y-direction and 3936 m in y-direction), for 1000 random prism models. The models included both random offset locations and random heights/depths of the prisms. As seen from Figure 2 the difference between the responses computed by the two methods is negligible. A typical relative difference between the two values is of the order of 0.1%-0.5%. However,there were some larger outliers (up to 15%). These outliers can be handled by a more appropriate tabulation (denser tabulation in the transition zone), so that a uniform accuracy of approximation can be obtained for all offsetdistances.Figure 2: First numerical experiment. Vertical gravitational attraction of 1000 random models in Zone 1 (see Table 1) (a) The exact prism formula. (b) Using tabulated parameters ofijf. (c) The difference between (a) and (b).The outliers in Figure 2c are because we used the nearest tabulated offset value in Figure 2b instead of the true offset, which is used in Figure 2a. This introduces a misfit, which can be minimized by a denser sampling of the table values (especially in the transition zone). In the distant zones the relative accuracy is of the same order, but without the outliers. Instead of nearest table value, one can use linear interpolation between the coefficients of the polynomial of 'z z f ij (the value of the step function is zero).We found that, once the table is computed (prior to modeling), the proposed method is three times faster. However, this number is probably very conservative as we are in the process of optimizing the new technique. Also, the main advantage of the new technique is not necessarily that the computation is faster for one prism, but in the computational shortcuts for a "many prism" modeling scenario (e.g. the convolution example in the Theory section can be utilized).ConclusionsWe propose a new efficient method of potential field modeling equivalent to spatial domain prism integration. The method is based on a new way of parameterizing the Green's function, which resembles the method of stacked horizontal lamina of Talwani and Ewing (1960). Preliminary numerical experiments indicate that the computation of the gravitational response of a single prism is faster by (at least) factor 3. We anticipate, once the method is fully optimized, obtaining a speedup factor of (5-10) for individual prisms. The main advantages of the method lie, however, in the possibility of handling "many prism" modeling problems associated with complex mass density distributions. Concerning the accuracy of the gravitational signal computed by the new method we have achieved a relative accuracy of 0.1%-0.5%. The outliers in the relative values can be improved to the same degree of accuracy by a more appropriate construction of tables.ReferencesAbramowitz, M. and I.A. Stegun, Handbook of mathe-matical functions. Dover Publ., Inc., N.Y., 9th ed., 1972.Blakely, R. J., Potential Theory in Gravity & Magnetic Applications, Cambridge University Press, 1996.Heiskanen, W.A., and H. Moritz, Physical Geodesy, W.H. Freeman and Co., San Francisco and London, 1967. Spiegel, M. R., Mathematical Handbook of Formulas and Tables. Schaum's Outline Series, McGraw-Hill Book Co., 1968.Talwani, M., and Ewing, M., Rapid computation of gravitational attraction of three-dimensional bodies of arbitrary shape. Geophysics, 25, 203-225, 1960. AcknowledgementsThis study was conducted during Gabriel Strykowski's research stay at the Division of Exploration and Mining, CSIRO, Western Australia. We thank Dr. Peter Hornby, CSIRO, Perth, for valuable discussions and suggestions. The Danish Natural Science Research Council and Inge Lehmann's Legat contributed with financial support.。

REVIEW ARTICLEAdvances in discrete element modelling of underground excavationsCarlos Labra ÆJerzy Rojek ÆEugenio On˜ate ÆFrancisco ZarateReceived:5November 2007/Accepted:6May 2008/Published online:17July 2008ÓSpringer-Verlag 2008Abstract The paper presents advances in the discrete element modelling of underground excavation processes extending modelling possibilities as well as increasing computational efficiency.Efficient numerical models have been obtained using techniques of parallel computing and coupling the discrete element method with finite element method.The discrete element algorithm has been applied to simulation of different excavation processes,using dif-ferent tools,TBMs and roadheaders.Numerical examples of tunnelling process are included in the paper,showing results in the form of rock failure,damage in the material,cutting forces and tool wear.Efficiency of the code for solving large scale geomechanical problems is also shown.Keywords Coupling ÁDiscrete element method ÁFinite element method ÁParallel computation ÁTunnelling1IntroductionA discrete element algorithm is a numerical technique which solves engineering problems that are modelled as a large system of distinct interacting bodies or particles that are subject to gross motion.The discrete element method (DEM)is widely recognized as a suitable tool to model geomaterials [1,2,4,8].The method presents important advantages in simulation of strong discontinuities such as rock fracturing during an underground excavation or rock failure induced by a tunnel excavation.It is difficult to solve such problems using conventional continuum-based procedures such as the finite element method (FEM).The DEM makes possible the simulation of different excavation processes [5,7]allowing the determination of the damage of the rock or soil,or evaluation of cutting forces in rock excavation with roadheaders or TBMs.Different possibil-ities of DEM applications in simulation of tunnelling process are shown in the paper.Examples include new developments like evaluation of tool wear in rock cutting processes.The main problem in a wider use of this method is the high computational cost required by the simulations first of all due to large number of discrete elements usually required.Different strategies are possible in addressing this problem.This paper will present two approaches:parall-elization and coupling the DEM and FEM.Parallelization techniques are useful for the simulation of large-scale problems,where the number of particles involved does not allow the use of a single processor,or where the single processor calculation would require an extremely long time.A shared memory parallelization of the DEM algorithm is presented in the paper.A high per-formance code for the simulation of tunnel construction problems is described and examples of the efficiency of thebra ÁE.On˜ate ÁF.Zarate International Center for Numerical Methods in Engineering,Technical University of Catalonia,Gran Capitan s/n,08034Barcelona,Spaine-mail:clabra@ E.On˜ate e-mail:onate@ F.Zaratee-mail:zarate@J.Rojek (&)Institute of Fundamental Technological Research,PolishAcademy of Sciences,Swietokrzyska 21,00049Warsaw,Poland e-mail:jrojek@.plActa Geotechnica (2008)3:317–322DOI 10.1007/s11440-008-0071-2code for solving large-scale geomechanical problems are shown in the paper.In many cases discontinuous material failure is localized in a portion of the domain,the rest of it can be treated as continuum.Continuous material is usually modelled more efficiently using the FEM.In such problems coupling of the discrete element method with the FEM can provide an optimum solution.Discrete elements are used only in a portion of the analysed domain where material fracture occurs,while outside the DEM subdomainfinite elements can be bining these two methods in one model of rock cutting allows us to take advantages of each method. The paper presents a coupled discrete/finite element tech-nique to model underground excavation employing the theoretical formulation initiated in[5]and further devel-oped in[6].2Discrete element method formulationThe discrete element model assumes that material can be represented by an assembly of distinct particles or bodies interacting among themselves.Generally,discrete elements can have arbitrary shape.In this work the formulation employing cylindrical(in2D)or spherical(in3D)rigid particles is used.Basic formulation of the discrete element formulation using spherical or cylindrical particles wasfirst proposed by Cundall and Strack[1].Similar formulation has been developed by the authors[5,7]and implemented in the explicit dynamic code Simpact.The code has a lot of original features like modelling of tool wear in rock cut-ting,thermomechanical coupling and other capabilities not present in commercial discrete element codes.Translational and rotational motion of rigid spherical or cylindrical elements is described by means of the Newton–Euler equations of rigid body dynamics:M D€r D¼F D;J D_X D¼T Dð1Þwhere r D is the position vector of the element centroid in a fixed(inertial)coordinate frame,X D is the angular veloc-ity,M D is the diagonal matrix with the element mass on the diagonal,J D is the diagonal matrix with the element moment of inertia on the diagonal,F D is the vector of resultant forces,and T D is the vector of resultant moments about the element central axes.Vectors F D and T D are sums of all forces and moments applied to the element due to external load,contact interactions with neighbouring spheres and other obstacles,as well as forces resulting from damping in the system.Equations of motion(1)are inte-grated in time using the central difference scheme.The overall behaviour of the system is determined by the cohesive/frictional contact laws assumed for the inter-action between contacting rigid spheres(or discs in2D).The contact law can be seen as the formulation of the material model on the microscopic level.Modelling of rock or cohesive zones requires contact models with cohesion allowing tensile interaction force between particle.In the present work the simplest of the cohesive models,the elastic perfectly brittle model is used.This model is char-acterized by linear elastic behaviour when cohesive bonds are active:r¼k n u n;s¼k t u tð2Þwhere r and s are the normal and tangential contact force, respectively,k n and k t are the interface stiffness in the normal and tangential directions and u n and u t the normal and tangential relative displacements,respectively. Cohesive bonds are broken instantaneously when the interface strength is exceeded in the tangential direction by the tangential contact force or in the normal direction by the tensile contact force.The failure(decohesion)criterion is written as:r R n;j s j R t;ð3Þwhere R n and R t are the interface strengths in the normal and tangential directions,respectively.Breakage of cohe-sive bonds allows us to simulate fracture of material and its propagation.In the absence of cohesion the frictional contact is assumed with the Coulomb friction model.3Coupling the DEM and FEMIn the present work the so-called explicit dynamic formu-lation of the FEM is used.The explicit FEM is based on the solution of discretized equations of motion written in the current configuration in the following form:M F€r F¼F ext FÀF int Fð4Þwhere M F is the mass matrix,r F is the vector of nodal displacements,F F ext and F F int are the vectors of external loads and internal forces,respectively.Similarly to the DEM algorithm,the central difference scheme is used for time integration of(4).It is assumed that the DEM and FEM can be applied in different subdomains of the same body.The DEM and FEM subdomains,however,do not need to be disjoint—they can overlap each other.The common part of the subdomains is the part where both discretization types are used with gradually varying contribution of each modelling method.This idea follows that used for molecular dynamics coupling with a continuous model in[9].The coupling of DEM and FEM subdomains is provided by additional kinematical constraints.Interface discrete elements are constrained by the displacementfield of overlapping interfacefinite elements.Making use of thesplit of the global vector of displacements of discrete ele-ments,r D ,into the unconstrained part,r DU ,and the constrained one,r DC ,r D ={r DU ,r DC }T ,additional kine-matic relationships can be written jointly in the matrix notation as follows:v ¼r DC ÀNr F ¼0;ð5Þwhere N is the matrix containing adequate shape functions.Additional kinematic constraints (5)can be imposed by the Lagrange multiplier or penalty method.The set of equations of motion for the coupled DEM/FEM system with the penalty coupling is as follows"M F 0000"M DU 0000"M DC 0000"J D26643775€r F €r DU €r DC _X D 8>><>>:9>>=>>;¼"F ext F À"F int F þN T k DF v "F DU"F DC Àk DF v "T D 8>><>>:9>>=>>;ð6Þwhere k DF is the diagonal matrix containing on its diagonal the values of the discrete penalty function,and globalmatrices "M F ;"M DU ;"M DC and "J D ;and global vectors"F int F ;"F ext F ;"F DU ;"F DC and "T D are obtained by aggregation of adequate elemental matrices and vectors taking into account appropriate contributions from the discrete and finite element parts.Equation (6)can be integrated in time using the standard central difference scheme.4Application of DEM to simulation of tunnelling process Fracture of rock or soil as well as interaction between a tunnelling machine and rock during an excavation process can be simulated by means of the DEM.This kind of analysis enables the comparison of the excavation process under different conditions.4.1Simulation of tunnelling with a TBMSimplified models of a tunnelling process must be used due to a high computational cost of a full-scale simulation in this case.We assume that the TBM is modelled as a cylinder with a special contact model for the tunnel face is adopted.Figure 1presents a simplified tunnelling process.The rock sample,with a diameter of 10m and a length of 7m,is discretized with randomly generated and densely com-pacted 40,988spheres.Discretization of the TBM geometry employs 1,193rigid triangular elements.Tunnelling pro-cess has been carried out with prescribed horizontal velocity 5m/h and rotational velocity of 10rev/min.Rock properties of granite are used,and the microscopic DEM parameters corresponding to the macroscopic granite properties are obtained using the methodology described in [10].A special condition is adopted to eliminate the spherical particles in the face of the tunnel.Each particle,which is in contact with the TBM and lacks cohesive contacts with other particles,is removed from the model.Thus,the advance of the TBM and the absorption of the material in the shield of the TBM is modelled.Figure 1a,c presents the displacement of the TBM and the elimination of the rock material.The area affected by the loss of cohesive contacts,resulting in material failure is shown in Fig.2.This loss of cohesion can be considered as damage ,because it produces the change of the equivalent Young modulus.4.2Simulation of linear cutting test of single disccutter Simulation of the linear cutting test was performed.A rock sample with dimensions of 13591095cm is repre-sented by an assembly of randomly generated and densely compacted 40,449spherical elements of radii ranging from 0.08to 0.60cm.The granite properties are assumed in the simulation and appropriate DEM parameters are evaluated.The disc cutter is treated as a rigid body and the parameters describing its interaction with the rock are as follows:contact stiffness modulus k n =10GPa,Coulomb friction coefficient l =0.8.The velocity of the disc cutter is assumed to be 10m/s.Fig.1Simulation of TBM excavation:Evolution and elimination ofmaterialFig.2Simulation of TBM excavation:Damage over tunnel surfaceFigure 3a shows the discretization of the disc cutter.Only the area of the cutter ring in direct interaction with the rock is discretized with discrete elements due to the com-putational cost reasons.The whole model is presented in Fig.3b.The evolution of the normal cutting force during the process is depicted in Fig.4a.The values of the forces should be validated,because the boundary condition can affect the results.The evolution of the wear,using the for-mulation presented in [5],can be seen in Fig.4b.The elimination of the discrete elements,where the wear exceed the prescribed limit,permit the modification of the disc cutter shape,which leads to a change of the interaction forces.In the present case,a low value of the wear constant is considered,in order to maintain the initial tool shape.Accumulated wear indicates the areas where the removal of the tool material is most intensive.An acceleration of the wear process using higher values of the wear constant is required in order to obtain in a short time considered in the analysis the amount of wear equivalent to real working time.5High performance simulationsOne of the main problems with the DEM simulation is the computational cost.The contact search,the force calcula-tion for each contact,and the large number of elements necessary to resolve a real life problem requires a high computational effort.High performance computation,and parallel implementation could be necessary to run simu-lations with large number of time steps.The advances of the computer capabilities during last years and the use of multiprocessors techniques enable the use of parallel computing methods for the discrete element analysis of large scale real problems.A shared memory parallel version of the code is tested.The main idea is to make a partition of the mesh of particles and use each processor for the contact calculation at different parts of the mesh.The partition process is performed using a special-ized library [3].The calculation of the cohesive contacts requires most of the computational cost.A special structure for the database,and the dynamic load balance is used in order to obtain a good performance for the simulations.Two different structures for the contact data are used in order to have a good management of the information.The first data structure is created for the initial cohesive contacts,where a static array can be used.The other data structureisFig.3Linear cutting test simulation:a cutter ring with partial discretization;b full discretized model0.014Table 1Times for different number of processors Time (s)versus processors 124Total404.31272.93156.85Static contacts (per step)0.12790.06920.0351Dynamic contacts (per step)0.00590.00570.0055Time integration (per step)0.04260.03570.0344Speed up1.001.842.58designed for the dynamic contacts,occurring in the process of rock fragmentation,and the interaction between differ-ent bodies.The management of this kind of contact is completely dynamic,and it is not necessary to store vari-ables with the history information.Table 1presents the times of parallel simulations of a tunnelling process,which was described earlier.The main computational cost is due to the cohesive contacts evalu-ation.The results shown in the table confirm that a good speed-up has been achieved.6DEM and DEM/FEM simulation of rock cutting A process of rock cutting with a single pick of a roadheader cutter-head has been simulated using discrete and hybrid discrete/finite element models.In the hybrid DEM/FEM model discrete elements have been used in the part of rock mass subjected to fracture,while the other part have been discretized with finite elements.In both models the tool is considered rigid,assuming the elasticity of the tool is irrelevant for the purpose of modelling of rock fracture.Figure 5presents results of DEM and DEM/FEM sim-ulation.Both models produce similar failures of rock during cutting.Cutting forces obtained using these two models are compared in Fig.6.Both curves show oscilla-tions typical for cutting of brittle rock.In both cases similar values of amplitudes are observed.Mean values of cutting forces agree very well.This shows that combined DEM/FEM simulation gives similar results to a DEM analysis,while being more efficient numerically—computation time has been reduced by half.7Conclusions •Discrete element method using spherical or cylindrical rigid particles is a suitable tool in modelling of underground excavation processes.•Use of the model in a particular case requires calibra-tion of the discrete element model using available experimental results.•Discrete element simulations of real engineering prob-lems require large computation time and memory resources.•Efficiency of discrete element computation can be improved using technique of parallel computations.Parallelization makes possible the simulation of large problems.•The combination of discrete and finite elements is an effective approach for simulation of underground rock excavation.Acknowledgments The work has been sponsored by the EU project TUNCONSTRUCT (contract no.IP 011817-2)coordinated by Prof.G.Beer (TU Graz,Austria).References1.Cundall PA,Strack ODL (1979)A discrete numerical method for granular assemblies.Geotechnique29:47–65Fig.5Simulation of rock cutting:a DEM model,b DEM/FEM model2.Campbell CS(1990)Rapid granularflows.Annu Rev Fluid Mech2:57–923.Karypis G,Kumar V(1998)A fast and high quality multilevelscheme for partitioning irregular graphs.SIAM J Sci Comput 20:359–3924.Mustoe G(ed)(1992)Eng Comput9(2).Special issue5.On˜ate E,Rojek J(2004)Combination of discrete element andfinite element methods for dynamic analysis of geomechanics put Methods Appl Mech Eng193:3087–3128 6.Rojek J(2007)Modelling and simulation of complex problems ofnonlinear mechanics using thefinite and discrete element meth-ods(in Polish).Habilitiation Thesis,Institute of Fundamental Technological Research Polish Academy of Sciences,Warsaw7.Rojek J,On˜ate E,Zarate F,Miquel J(2001)Modelling of rock,soil and granular materials using spherical elements.In:2nd European conference on computational mechanics ECCM-2001, Cracow,26–29June8.Williams JR,O’Connor R(1999)Discrete element simulationand the contact problem.Arch Comp Meth Eng6(4):279–304 9.Xiao SP,Belytschko T(2004)A bridging domain method forcoupling continua with molecular put Methods Appl Mech Eng193:1645–166910.Zarate F,Rojek J,On˜ate E,Labra C(2007)A methodology todetermine the particle properties in2d and3d dem simulations.In:ECCOMAS thematic conference on computational methods in tunnelling EURO:TUN-2007,Vienna,Austria,27–29August。

210IEEE SIGNAL PROCESSING LETTERS,VOL.6,NO.8,AUGUST1999Noise Modeling for Nearfield Array Optimization Thushara D.Abhayapala,Student Member,IEEE,Rodney A.Kennedy,Member,IEEE,and Robert C.Williamson,Member,IEEEAbstract—In this letter,an exact series representation for a nearfield spherically isotropic noise model is introduced.The proposed noise model can be utilized effectively to apply well-established farfield array processing algorithms for nearfield applications of sensor arrays.A simple array gain optimization is used to demonstrate the use of the new noise model.Index Terms—Nearfield beamforming,noise modeling,opti-mization,reverberation.I.I NTRODUCTIONN EARFIELD sensor array design is important in telecon-ferencing and speech acquisition applications[1],[2]. Most array processing literature deals with situations where the desired source and the noise sources are assumed to be in the farfield of the array;this considerably simplifies the design problem.In most stochastic optimization techniques, the noise correlation matrix plays an integral part of the design. Infixed beamformer design,the noisefield is assumed to be known,and usually modeled by either white Gaussian noise or farfield spherically isotropic noise which results from a uniform distribution of noise sources over all directions in the farfield.For nearfield applications of sensor arrays such as tele-conferencing,the noisefield consists of undesirable nearfield sound sources as well as reverberation caused by the desired and noise ing the source-image method[3],we can visualize reverberation with point image-sources.In an average size room,some or allfirst-order reflected image-sources will be in the nearfield of the array while multiply reflected ones will be in the farfield.Due to absorption by walls,a multiply reflected reverberant noise source contributes less power compared tofirst order reflected ones.Thus,the overall noisefield is mainly due to nearfield noise sources, and an assumption of farfield spherically isotropic noise or white Gaussian noise is a very crude approximation.In[4],the farfield spherically isotropic noise was used to model the effect of reverberation without considering the effect of nearfield noise sources.Manuscript received October26,1998.The associate editor coordinating the review of this manuscript and approving it for publication was Prof.H. Messer-Yaron.T.D.Abhayapala and R.A.Kennedy are with the Telecommunications En-gineering Group,Research School of Information Sciences and Engineering, The Institute of Advanced Studies,Australian National University,Canberra, ACT0200,Australia(e-mail:Thushara.Abhayapala@.au;Rod-ney.Kennedy@.au).R. C.Williamson is with the Department of Engineering,Faculty of Engineering and Information Technology,Australian National University, Canberra,ACT0200,Australia(e-mail:Bob.Williamson@.au). Publisher Item Identifier S1070-9908(99)05800-9.As an alternative,in this paper we model the noisefield with uniformly distributed sources over all directions in the nearfield at afixed distance from the array origin and call it nearfield spherically isotropic noise.This noise model can be utilized effectively to apply any signal processing criterion based on isotropic type noise correlation to nearfield applica-tions.In our simulation example in Section IV we will show that a design based on this nearfield noise model performs better than one based on a farfield noise model in a more realistic mixed farfield-nearfield noisefield.II.G AIN O PTIMIZATION FOR AN A RBITRARY A RRAYAs motivation for the theoretical development,we consider a simple array optimization technique as applied to a nearfield array.The array gain is a key array performance indicator,definedbytotal noise powerreceivedsensors,arbitrarily placed in a boundedregionatand thefrequency.Thus,the power received from the desiredlocationwhere-element columnvector and defining asquare(3) leads to the matrixformulation.By assuming the nearfield spherically isotropic noisefield, i.e.,having uniformly distributed noise sources on a sphere ofradiusABHAYAPALA et al.:NEARFIELD ARRAY OPTIMIZATION211 ,whereand the integration is over all directions.We definethe,and(1)becomes a ratio of quadraticformswhichmaximizesand are commonly knownas the source correlation matrix and noise correlation matrix,respectively.The optimum array weights are[5](4)between two sensors due to nearfieldisotropic noisefield.We write the wavefield at the sensorlocationfor using the spherical harmonic expansion[6,p.30]as(7)where areintegers;andand are the half integer order Bessel functions ofthefirst and second kind,respectively;whereare the associated Legendre functions.It isknownthatwhich encloses the pair of sensors.Another formof(9)can be derived using the relationship[6,p.27]are the Legendre functions.Combining(9)and(10),we write the correlation between twosensorsas,the angle between the twosensors,and the distancesto the twosensorsand appear as separate factors.This property of the series expansion(11)facilitates thecalculation of correlation between two sensors for varioussensor orientations.This can be significant advantage over thethree-dimensional integral representation(4)when all pairs ofsensors in a large array must be considered.For the simple case of a line array through theorigin,[8,p.208],thecorrelation between two sensors for a line arrayis.Exploiting therelation[6,page30],wefind(11)reducesto(14)which is a well-known result for farfield spherically isotropicnoisefields[10,p.49].For the simple case of a lineararray with half wavelength spacings,the observed noise isuncorrelated between sensors;this fact is readily evident from(14).212IEEE SIGNAL PROCESSING LETTERS,VOL.6,NO.8,AUGUST1999Fig.1.Response of the optimum array based on nearfield noise model (solid line)to sources at three and 30wavelengths from the array origin.Also shown is the response of the farfield noise model based optimum array (dashed line).IV.S IMULATION E XAMPLEWe now present a design example to demonstrate the use of nearfield isotropic noise modeling for nearfield beamforming.Our demonstration is based on the simple array gain optimiza-tion technique outlined in Section II,however this noise model can be applied to wide class of optimization methods.The design is for a double-sided linear array of nine sensorswith an intersensor spacingofis the wavelength.Suppose the desired source is in the nearfieldatfrom the array origin,on the broadside of the array.We calculate the optimum weight vector (6),with the noise correlationmatrixfor nearfield isotropic noise (12)at a sphere ofradius .We approximate the infinite series in (12)by the first 21terms.Generally these series expansions are convergent and are readily approximated by finite number of terms depending on the array configuration and the desired operating distance.Fig.1shows the responses of the resulting array (solidline)to a nearfield sourceatfrom the array origin and to a farfield sourceat.Also shown is the response of an optimum array designed using farfield isotropic noisemodel (14)(dashed).The nearfield noise model based design provides a better directional array gain in the nearfield and simultaneously provides similar farfield noise rejection when compared with the farfield noise model based design.For both design methods,the power received from a sourceat at the look direction is about 25dB less than that of the desired sourceat .The tradeoff for using the nearfield noise model is the better directional gain at the expense of slightly wider main lobe width.V.C ONCLUSIONIn this letter,we have introduced an exact series rep-resentation for nearfield/farfield isotropic noise field,which may be useful in sensor array applications in the nearfield.While the model has only been demonstrated here for a small line array,it is generally applicable to more complex sensor geometries.More important,this result allows the application of well-established farfield array processing algorithms for the nearfield applications.R EFERENCES[1]J.L.Flanagan,D.A.Berkeley,G.W.Elko,J.E.West,and M.M.Sondhi,“Autodirective microphone systems,”Acustica ,vol.73,pp.58–71,1991.[2] F.Khalil,J.P.Jullien,and A.Gilloire,“Microphone array for soundpickup in teleconference systems,”J.Audio Eng.Soc.,vol.42,pp.691–700,Sept.1994.[3]J.B.Allen and D.A.Berkley,“Image method for efficiently simulatingsmall-room acoustics,”J.Acoust.Soc.Amer.,vol.65,pp.943–950,Apr.1979.[4]J.G.Ryan and R.A.Goubran,“Nearfield beamforming for micro-phone arrays,”in Proc.IEEE Int.Conf.Acoustics,Speech,and Signal Processing.,May 1997,pp.363–366.[5] D.K.Cheng and F.I.Tseng,“Gain optimization for arbitrary arrays,”IEEE Trans.Antennas Propagat.,vol.AP-13,pp.973–974,Nov.1965.[6] D.Colton and R.Kress,Inverse Acoustic and Electromagnetic ScatteringTheory .Berlin,Germany:Springer-Verlag,1997.[7]T.Jonsson and J.Yngvason,Waves and Distributions .Singapore:World Scientific,1995.[8]H.B.Dwight,Tables of Integrals and Other Mathematical Data .NewYork:Macmillan,1966.[9]G.N.Watson,Ed.,A Treatise on the Theory of Bessel Functions .Cambridge,U.K.:Cambridge Univ.Press,1958.[10] D.H.Johnson and D.E.Dudgeon,Array Signal Processing .Engle-wood Cliffs,NJ:Prentice-Hall,1993.。

Journal of Materials Processing Technology 153–154(2004)442–449Semi-solid die casting process with three steps die systemP.K.Seo a ,∗,K.J.Park b ,C.G.Kang caDepartment of Mechanical and Precision Engineering,Graduate School,Pusan National University,Pusan 609-735,South Koreab NSC (Net Shape Concurrent)Industry,Namyang-Dong,Jinhae 645-480,Kyongnam,South Koreac School of Mechanical Engineering,Pusan National University,Pusan 609-735,South KoreaAbstractIn automobile suspension part,knuckle as functional component is very complex shape and required high strength.To manufacture the high strength component like knuckle with semi-solid die casting process,there are required reheating condition,casting plan,die design,injection condition,defect analysis and measurement of mechanical properties.In this study,aluminum knuckle with semi-solid die casting process was developed to replace the conventional steel components.Semi-solid die casting was established to lead the laminar flow and pressure transfer to the end point of product and to avoid oxide skin and shrinkage porosity.Three steps die system was made up two-step moving die and biscuit cutting system.And that was suitable for center-positioned gate and designed to hold the similar filling length and applying pressure.Aluminum knuckle component made by three steps die system designed by casting simulation.Also,injection condition was established to remove oxide skin and liquid segregation.After controlled above process parameters,tensile strength,yield strength,elongation and microstructure were investigated to verify the real automobile applicability.©2004Published by Elsevier B.V .Keywords:Semi-solid die casting;Three steps die system;Injection condition;Mechanical properties1.IntroductionSemi-solid die casting of Al alloy is suitable for compli-cated large parts of near net shape without defect and excel-lent mechanical properties in comparison with conventional casting process [1–3].Die design for the product with high quality in semi-solid die casting is required to prevent mi-cro porosity,inflow of oxide skin,and liquid segregation.Therefore,shape and size of runner,position and size of overflow and air vent,and arrangement of heating line are very important.Micro porosity,incomplete filling,turbulent flow,and liquid segregation are occurred during semi-solid die casting [4–7].To prevent these defects in advance,it is very effective to use computer simulation.Part and die design by computer simulation has many advantages compared to the conven-tional methods that were performed by designer’s experi-ences and trial and errors.It is very difficult to predict the flow behavior inside the cavity because semi-solid material has thixotropic,pseudoplastic and visco-plastic characteris-∗Corresponding author.E-mail addresses:pkseo92@pusan.ac.kr (P.K.Seo),kjpark@nscind.co.kr (K.J.Park),cgkang@pusan.ac.kr (C.G.Kang).tic [8–11].Therefore,in this study,semi-solid forming simu-lation is carried out using Ostwald–de Waele viscosity model which is suitable for the flow behavior of semi-solid material [12,13].To develop automobile component by semi-solid die casting,part design,die design,optimal injection con-dition and evaluation of microstructure and mechanical properties in manufactured component are necessary.Partic-ularly,knuckle,which is a suspension component,is com-plex shape and required excellent mechanical properties.In this study,to manufacture aluminum knuckle by semi-solid die casting,three steps die system with center-positioned gate was designed by computer simulation and forming experiment was carried out.Microstructure and mechanical properties are inspected throughout the whole product.2.Die design for semi-solid die casting 2.1.Three steps die systemIt is necessary to reheat the billet by electromagnetic stir-ring for the globular microstructure and the rheological be-havior in the semi-solid die casting.During the reheating process,oxide skin is formed in the periphery of the billet.If this oxide skin is entranced inside the cavity,fatal defects0924-0136/$–see front matter ©2004Published by Elsevier B.V .doi:10.1016/j.jmatprotec.2004.04.041P .K.Seo et al./Journal of Materials Processing Technology 153–154(2004)442–449443may be occurred.Therefore,the tapered plunger tip is very useful because the oxide skin is induced between sleeve and plunger tip and then remains as biscuit in the sleeve.In the general die casting,the gate is mainly positioned in the lower part of product.In the case of the center-positioned gate,the air,which is mixed with molten metal,remains inside the cavity and forms porosities because the density of air is much lower than that of molten metal.But,in the case of semi-solid die casting,the flow inside the cavity can become laminar and air rarely mix with material due to the high vis-cosity of the reheated material.Therefore,if overflows and air vents are properly designed in the product positioned be-low the gate,the excellent product without oxide skin and porosities can be formed.In the case of center-positioned gate,moving die is not separated from fixed die due to the interference of the biscuit part and the cutting of gate is very difficult after pulling out the product.In case the shape of final filling position is complex and the filling distance is far from the gate,pressure transfer at the final filling position is very difficult.To solve these problems,three steps die system is proposed in this study.Three steps die system is that fixed plate and biscuit cutting system are added to the general die system.When the fixed plate separated from fixed die under locking fixed die and moving die,biscuit part is gone out of the sleeve.And then,biscuit part is cut by drop of the cutter using hydraulic apparatus.Finally,moving die is separated from fixed die and the product is pulled out from moving die.By using these three steps process,oxide skin can be removed in the sleeve and biscuit is cut in die separating process.Knuckle is an automobile suspension part with complex shape and pressure transfer is difficult to the final filling region.In this study,three steps die system suitable for manufacturing an aluminum knuckle is designed and manufactured by using computer simulation.2.2.Simulation for semi-solid die castingTo change a material from steel to aluminum in auto-mobile suspension component,the shape that was applied to a steel component should be modified with a new shape because the stiffness of aluminum is much lower thanthatFig.1.3D model for a knuckle by semi-sold die casting (model 1:2.875kg,model 2:2.408kg)and the change of injection speed to the stroke displacement.of steel.Therefore,in this study,two models satisfied with static and dynamic load condition are designed to decide a model suitable for the semi-solid die casting.In one model,the dimension of each position is increased by the same as stiffness of steel component and in the other model,the thickness is the same throughout the whole component ex-cept the assembly position.Fig.1(a)and (b)show the three-dimensional (3D)mod-eling of two aluminum knuckles to substitute for a steel knuckle weighed about 4.5kg.Model 1weighs 2.875kg and model 2does 2.408kg.In case of model 2,the weight re-duction ratio is 46%as compared with a conventional steel component.Fig.1(c)shows the injection condition to find out the formability in models 1and 2by using semi-solid die casting simulation.Unlike an ordinary die casting,plunger speed should be decreased when the material passes through the gate to prevent the turbulent flow inside the cavity.For the filling and solidification analysis,rheological behavior of semi-solid material should be considered.The viscosity of semi-solid material has drastically changed in semi-solid state (f s =50–55%).This characteristic of semi-solid material is called ‘thixotropic’.The dependency of viscosity on shear rate must be recognized to establish the rheology model of semi-solid material.The viscos-ity decreases as shear rate increases in liquid region of aluminum.It is generally known that the rheology model of filling analysis is Ostwald–de Waele model and Newtonian model.But,filling analysis is performed with Ostwald–de Waele model in this study,because it is reported that theoretical results have a good agreement with experimental value in Ostwald–de Waele model rather than Newtonian model.Ba-sic governing equations of MAGMAsoft which is adopted for control volume method are continuity equation,Navier Stoke’s equation,energy equation and volume of fluid (VOF)method as liquid forming process.The rheology equation of Ostwald–de Waele model is as follows [14]η=ρm ˙γn −1(1)where ηis the apparent dynamic viscosity (Pa s);ρthe den-sity (kg/m 3);m the Ostwand–de Waele coefficient (m 2/s);˙γthe shear rate (1/s);and n the Ostwald–de Waele exponent.444P .K.Seo et al./Journal of Materials Processing Technology 153–154(2004)442–449Table 1Properties used to simulate semi-solid die casting of a knuckle by using A356alloy ParametersSymbol UnitValuesSolidus temperature T s ◦C 547Liquidus temperature T l ◦C617Latent heatQ kJ/kg 430Initial billet temperature T b ◦C 583Initial die temperatureT d ◦C250Heat transfer coefficient between die and dieh d W/m 2K 1000Heat transfer coefficient between material and die h m W/m 2K Temperature dependant Number of control volumes Model 1–EA 1802568Model 2–EA 1807014Number of metal cells Model 1–EA 60504Model 2–EA52112The number of rectangular elements formed by dividing the perpendicular coordinate is 1,802,568and the number of metal cells at the die cavity is 60,504.The thermophys-ical properties such as specific heat capacity,density,solid fraction,thermal conductivity and heat transfer coefficient,which are needed in filling analysis,are given as a function of temperature.These thermophysical properties provided by MAGMAsoft are used.Table 1shows the properties of A356alloy used for semi-solid die casting simula-tion.Fig.2(a)shows the change of solidification rate according to the solidification time.Final solidification time is 45s in model 1and 38s in model 2.In the real manufacturing process,cycle time is decreased by 7s.Hot spot is isolatedly solidified region and solidification shrinkage may occur in hot spot region as shown in Fig.2(b)and (c).Compared model 1and model 2,hot spot is more widely distributed and product region is solidified later than biscuit region in model 1.It is seen that more solidification shrinkage may occur and the transfer of pressure is more difficult in model 1.Therefore,in this study,model 2with fewer defects in view of formability and comparatively short cycle time is selected to apply to the semi-solid die casting with three-step die system,although two models are satisfied with the static and dynamic stressanalysis.Fig.2.The change of solidification rate according to the solidification time and hot spot distribution in models 1and 2to predict the occurrence of shrinkage.2.3.Analysis of injection conditionIn the semi-solid die casting process,irrelevant injection condition is directly connected with the product with defects due to the flow behavior and the variation of temperature inside the cavity.If the injection speed is very fast,oxide skin around the billet after reheating may remain inside the product.If the injection speed is very low,liquid segregation because of the difference of velocity between solid phase and liquid phase may occur and shrinkage porosity form due to the delay of finally filling time.For that reason,to manufacture the satisfactory product without defects,it is very important to find out the injection condition suitable for the semi-solid die casting.In this study,the velocity is varied to the stroke displacement at the position of runner and gate and then the velocity distribution is inspected throughout the computational analysis.Fig.3shows the change of injection condition (Exp.Nos.1–4)to the stroke displacement in model 2.The velocity until the sleeve is fully filled is 1.2m/s to prevent the drastic decrease of temperature and changed from 0.3to 1.0m/s at the runner and changed from 0.2to 0.5m/s after passing through the gate.Fig.4shows the velocity distribution at 30%filled to the injection condition.In the Exp.No.1as shown in Fig.4(a),P .K.Seo et al./Journal of Materials Processing Technology 153–154(2004)442–449445Fig. 3.The change of injection speed to the stroke displacement in model 2.incomplete filling region is occurred due to the high velocity as 1.0m/s.Air at the incomplete filling region is entranced with the reheated material and is not gone out of the cav-ity.Therefore,porosities are widely distributed inside the product after final filling.In Exp.No.2,the velocity at the runner is 0.5m/s and after this is decreased as 0.2m/s.Be-cause the velocity is slow,the final filling time is delayed and complete filling is difficult.In Exp.No.3,the velocity is abruptly decreased from 1.2to 0.3m/s at the runner.But,metal front can be scat-tered because the velocity is abruptly increased as 0.5m/s in passing through the gate.In Exp.No.4,the velocity at the center of runner is higher than in Exp.Nos.2and 3,and is lower than in Exp.No.1.The time in passing through the gate is 103m s.It is shorter than in Exp.Nos.2and 3(about 205m s)but is longer than in Exp.No.1(64m s).Veloc-Fig.4.The comparison of velocity distribution in 30%filled to the injection speed.ity gradually decreased from center position to cavity wall.Therefore,it is considered that if a knuckle is formed by the injection condition of Exp.No.4,semi-solid material flows smoothly without air entrapment.Fig.5shows the variation of velocity to the injection con-dition at each point in the runner.P1is the position con-nected with biscuit,P2is the middle position of the runner and P3is positioned around the gate as shown in Fig.5(a).In Exp.No.1,after the velocity increases to about 900cm/s,abruptly decreases to about 400cm/s.Maximum difference of velocity at measuring points is about 300cm/s and then metal front may be scattered inside the cavity.In Exp.No.2,the difference of velocity is low,but the final filling time is about 0.9s.Therefore,because the metal front may be solidified in the final filling region,incomplete filling is pre-dicted.In Exp.No.3as shown in Fig.5(c),the velocity is faster in the product region than in the runner.Final filling time is about 0.5s and the difference of velocity is small.In Exp.No.4,the velocity is about 450cm/s at the runner.After passing through the gate,the velocity at the position 1remains 200cm/s until final filling,but slowly increases by 300cm/s at the positions 2and pared with Exp.No.2,the velocity distribution at the product part is similar but the velocity at the runner is faster in Exp.No.4than in Exp.No.2and final filling time is longer in Exp.No.2than in Exp.No.4.Fig.6shows the variation of velocity to the injection con-dition at each point in the gate after passing through the run-ner.P4is the upper position,P5is the middle position and P6is the lower position at the gate as shown in Fig.6(a).In Exp.No.1,the velocity distribution is fluctuated and then the semi-solid material may be dispersed and occur turbu-446P .K.Seo et al./Journal of Materials Processing Technology 153–154(2004)442–449Fig.5.The variation of velocity to the injection condition at each point in the runner.lent flow in the product part.In Exp.No.2as shown in Fig.6(b),the velocity is uniform at the P5.But it is almost 0.0m/s after 0.7s at the P6and slowly increases by 250cm/s at the P4.These points out that the lower product region is fully filled at 0.7s and after that time,the upperproduct Fig.6.The variation of velocity to the injection condition at each point in the gate.region is finally filled for about 0.22s.Fig.6(c)shows the velocity distribution in Exp.No.3.It takes 0.3s from 0.2to 0.5s after passing through the gate to completely fill the cavity.Although the temperature drops a little because the time is very short to fill the cavity,the flow behavior is notP .K.Seo et al./Journal of Materials Processing Technology 153–154(2004)442–449447enough to the semi-solid die casting due to the agitated ve-locity distribution.Fig.6(d)shows the velocity distribution in Exp.No.4.The velocity is uniform at the P5similar to the Exp.No.2.But,the semi-solid material with higher tem-perature is filled in the final filling region,because the final filling time is 0.7s,which is shorter than in Exp.No.2.As shown above computational analysis,the small change of plunger velocity has an enormous effect on the velocity of semi-solid material inside the cavity.To optimize the injection condition of the semi-solid die casting was very difficult.In this study,the injection condition of Exp.No.4is estimated as suitable and applied to the real semi-solid die casting with three-step die system.3.Forming experiment and resultsBased on the casting simulation and the injection condi-tion,three-step die system was designed and manufactured and forming experiment was carried out in 840t die casting machine.As shown in Fig.7(a),three-step die system for aluminum knuckle is composed of fixed plate,fixed die,bis-cuit cutting system,moving die,and moving plate.Fig.7(b)Fig.7.Die structure for the knuckle component by three-step semi-solid die casting:(a)3D solid modeling,(b)photograph of die casting machine and three-step diesystem.Fig.8.Injection speed to the stroke displacement and applying pressure condition.shows the photograph of setting up three-step die system with die casting machine.Fig.8shows the set up value and the measuring value to the injection speed and applying pressure.Injection speed to the stroke displacement is set up on Exp.No.4as shown in Fig.3.After the injection speed is decreased from 1.2to 0.6m/s at the runner,it holds 0.23m/s until final filling.It is seen that the set up value accord with the measuring value.Applying pressure is input as 1600bar after 0.045s.Applying pressure is fluctuated until about 0.4s,and then it holds 1600bar during 38s,which is the time fully solidified.Fig.9shows microstructures after T6heat treatment at the final filling region as indicated in Fig.10.In positions A–C,pressure transfer and complete filling is difficult be-cause filling distance is very long.But,incomplete filling region,coalesced silicon crystals and liquid segregation are not observed.Throughout the whole region,the microstruc-ture to the position makes no wide difference,primary ␣is globurized and finely circular silicon crystals distribute all around among primary ␣.Fig.10shows the solid fraction and silicon size by the image analysis to the position.Average solid fraction is 63%and it is very uniform to the position.Therefore,it is seen448P .K.Seo et al./Journal of Materials Processing Technology 153–154(2004)442–449Fig.9.Microstructures at positionsA–C.Fig.10.Solid fraction and mean silicon size at each position.that liquid segregation,which is the one of the main defects,is not occurred throughout the product.Silicon size is from 3.2to 4.6␮m and average value is 3.8␮m.It is reported that coalesced silicon is formed when eutectic phase isincom-Fig.12.Ultimate tensile strength and elongation of knuckle at each position (T6heat treatment condition:3h at 530◦C and 8h at 160◦C).Fig.11.Equivalent diameter and mean roundness at each position.pletely remelt during reheating process [15].Therefore,it is seen that the reheating condition applied to this experiment is proper to semi-solid die casting.Fig.11shows the equivalent diameter and mean round-ness to the position by image analysis.Equivalent diame-ter is the diameter of circle with the same area equal to the area of globule.Equivalent diameter is 72␮m as av-erage and has a little difference from 26to 98␮m to the position.If mean roundness is 1,globule is complete cir-cle.Mean roundness is 1.8as average and has no wide difference.Fig.12(a)shows the distribution of ultimate tensile strength,yield strength and elongation to the position af-ter T6heat treatment (3h at 530◦C and 8h at 160◦C).UTS and YS are respectively 348and 257MPa as av-erage and show the uniform distribution over all po-sitions.Elongation is 7.5%as average and has a little difference from 6.5to 9.4%to the position.Aluminum knuckle by semi-solid die casting has excellent mechan-ical properties sufficient to an automobile suspension component.P.K.Seo et al./Journal of Materials Processing Technology153–154(2004)442–4494494.ConclusionsTo develop an aluminum knuckle by using semi-solid die casting with three steps die system,die design by computer simulation,the establishment of injection condition,forming experiment,the observation of microstructure,and the mea-surement of mechanical properties were carried out.Based on these results,followings are summarized:(1)To develop knuckle component by semi-solid die cast-ing,lightweight aluminum model and three steps die system have been proposed and manufactured by die design applying casting simulation.(2)By the results offlow behavior to the injection condition,it is seen that plunger speed has an enormous effect on the velocity distribution at the runner and gate.Injection condition suitable for semi-solid die casting with the center-positioned gate has been found consideringfinal filling time and the temperature of metal front.(3)From the observation of microstructure at each posi-tion of fabricated part,solid fraction,silicon size,and mean roundness were uniform over all positions.And, ultimate tensile strength,yield strength,and elongation were sufficient to an automobile suspension component. AcknowledgementsThis study has been supported by the Engineering Research Center for Net Shape and Die Manufacturing (ERC/NSDM),which arefinanced jointly by the Korea Science and Engineering Foundation(KOSEF)and the Ministry of Science and Technology(MOST).The au-thors would like to express their deep gratitude to the ERC/NSDM,KOSEF,and MOST.References[1]M.Garat,L.Maenner,Ch.Sztur,State of the art of thixocasting,in:G.L.Chiarmetta,M.Rosso(Eds.),Proceedings of the SixthInternational Conference on Semi-Solid Processing of Alloys and Composites,Turin,2000,pp.187–194.[2]M.Ozawa,M.Hara,K.Kawano,T.Kaneuchi,H.Sakuragi,Devel-opment of aluminum alloy suspension parts by new semi-solid die casting,SOKEIZAI42(12)(2001)13–18.[3]H.V.Atkinson,D.Liu,Development of high performance aluminiumalloys for thixoforming,in:Y.Tsutsui,M.Kiuchi,K.Ichikawa(Eds.), Proceedings of the Seventh International Conference on Semi-Solid Processing of Alloys and Composites,Tsukuba,2002,pp.51–56.[4]M.Suery,A.Zavaliangos,Key problems in rheology of semi-solidalloys,in:G.L.Chiarmetta,M.Rosso(Eds.),Proceedings of the Sixth International Conference on Semi-Solid Processing of Alloys and Composites,Turin,2000,pp.129–135.[5]K.Young,P.Eisen,SSM(semi-solid metal)technological alterna-tives for different applications,in:G.L.Chiarmetta,M.Rosso(Eds.), Proceedings of the Sixth International Conference on Semi-Solid Processing of Alloys and Composites,Turin,2000,pp.97–102.[6]P.K.Seo,Y.I.Son,C.G.Kang,The effect of plunger tip shape onthe formability in semi-solid die casting process,Trans.Mater.Proc.11(4)(2002)312–322.[7]P.K.Seo,C.G.Kang,Y.I.Son,The effect of velocity control methodon the part characteristic in semi-solid die casting,Trans.Korean Soc.Mech.Eng.A26(10)(2002)2034–2043.[8]M.C.Flemings,Behavior of metal alloys in the semisolid state,Metall.Trans.A22A(1991)947–981.[9]P.Kumar,C.Martin,S.Brown,Shear rate thickeningflow behaviorof semisolid slurries,Metall.Trans.A24A(1993)1107–1116. 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Discrete linear objects in dimension n :the standard modelEric Andres *IRCOM-SIC,SP2MI,BP 30179,F-86962Futuroscope Cedex,FranceReceived 15May 2002;received in revised form 4September 2002;accepted 17October 2002AbstractA new analytical description model,called the standard model,for the discretization of Eu-clidean linear objects (point,m -flat,m -simplex)in dimension n is proposed.The objects are defined analytically by inequalities.This allows a global definition independent of the number of discrete points.A method is provided to compute the analytical description for a given lin-ear object.A discrete standard model has many properties in common with the supercover model from which it derives.However,contrary to supercover objects,a standard object does not have bubbles.A standard object is ðn À1Þ-connected,tunnel-free and bubble-free.The standard model is geometrically consistent.The standard model is well suited for modelling applications.Ó2003Elsevier Science (USA).All rights reserved.Keywords:Discrete geometry;Digitization;Dimension n ;Simplex;m -flat1.IntroductionWhen working in discrete geometry,aside from considering an object simply as a set of discrete points,the problem of defining discrete geometrical objects arises.A discrete 2D line segment can be defined as 8-connected,4-connected or even discon-nected as a dotted line.There is not a unique way of defining a discrete object or of digitizing a Euclidean object.This problem has been around for 40years and many different discrete object definitions have been proposed.One can say thatauthors Graphical Models 65(2003)92–111*Fax:+33-5-49-49-6570.E-mail address:andres@sic.univ-poitiers.fr.1524-0703/03/$-see front matter Ó2003Elsevier Science (USA).All rights reserved.doi:10.1016/S1524-0703(03)00004-3E.Andres/Graphical Models65(2003)92–11193 have followed three main approaches to define discrete geometrical objects:an algo-rithmic approach,a topological approach,and a more recent analytical approach followed in this paper.In the algorithmic approach[1,10,13,16,21–24,34]a discrete object is the result of a generation algorithm.Historically,thefirst approach that has been used,it has shown a number of limitations.It is often difficult to control the properties of the so defined discrete objects.For instance,the discrete objects might not be geometrically consistent:the edge of a3D triangle is typically not nec-essary a3D line segment or the3D triangle is not a piece of3D plane[21,22].It is also difficult to propose generation algorithms for discrete objects in dimension high-er than three.Except for n-dimensional lines[34],to the best authors knowledge,no discrete object,in dimensions higher than three,has been algorithmically defined.In the topological approach,a discrete object is typically defined as a class of objects verifying local properties,often topological in nature[18–20,25,28].While it is,by definition,easier to obtain the desired properties,it is difficult to be sure with such an approach,that the class of objects defined by a given set of properties is not larger than what is initially expected.A third,more recent approach,defines a discrete ob-ject by a global analytical definition[2–5,7,9,17,18,25,30,32].This approach has many advantages such as providing a compact definition(independent on the num-ber of points forming the discrete object),a global control of the discrete object.It has also an advantage that is not immediately visible when one is not familiar with this approach.It allows a good control of the local topological properties of the ob-ject.The many links with mathematical morphology are also an interesting property of some analytically defined models such as the supercover model[7,19,26,29,31,33]. One of the main advantages is that it is relatively easy to define discrete objects in an arbitrary dimension[3,4,7,30,32].The standard model introduced in the following pages is analytically defined.A new analytical description model for all linear objects in dimension n(discrete points,m-flats,and geometrical simplices)is presented in this paper.The analytical model is called the standard model.The names derives from the name given by Francßon[18]toðnÀ1Þ-connected analytical discrete3D planes(see also[4]for gen-eral details on discrete analytical hyperplanes).To the best authors knowledge,it is thefirst time that a discrete model is proposed that defines a large class of discrete objects in arbitrary dimensions.The standard model is called a discrete analytical model because the discrete objects(points,m-flats,simplices)are defined analytically by inequalities.The analytical definition is independent of the number of discrete points of the object.For instance,a3D standard triangle is defined by17or less in-equalities independently of its size.The model we propose has many interesting properties.The model is geometri-cally consistent:for instance,the vertices of a3D standard polygon are3D standard points,the edges of a3D standard polygon are3D standard line segments and the 3D standard polygon is a piece of a3D standard plane.It has been shown that the standard model is in fact a0-discretization of Brimkov et al.[12]and therefore isðnÀ1Þ-connected and tunnel-free.In3D,ðnÀ1Þ-connectivity in our notations corresponds to the classical6-connectivity.Contrary to the supercover model,from which it derives,the standard objects are bubble-free.One of the problems of the su-percover model is that it is not topologically consistent.A supercover m -flat is always ðn À1Þ-connected but sometimes it has simple points (located on so-called bubbles on the object).This makes the model difficult to use in practice [14,15].For instance,a supercover of a Euclidean n D point can be composed of any 2i discrete points,06i 6n .A standard m -flat is almost identical to the supercover m -flat,it remains ðn À1Þ-connected and tunnel-free,except for the simple points in the bubbles that are removed.The standard digitization of a n D Euclidean point is always composed only of one discrete point.Finally,the standard model has a very important property in the framework of discrete modelling:S t ðF [G Þ¼S t ðF Þ[S t ðG Þ.This means that,for instance,the definition of the standard 3D polygon is sufficient to define the standard model of an arbitrary Euclidean polygonal 3D object.The definition of the standard model is derived from the supercover model [2,5–7,14,15,31].A standard object is obtained by a simple rewriting process of the in-equalities defining analytically a supercover object [7].A supercover linear object is defined by a set of inequalities ‘‘P n i ¼1a i X i 6a 0.’’The simple points in the bubbles are points that verify ‘‘P n i ¼1a i X i ¼a 0.’’In order to remove the simple points,and thus bubbles,some of the inequalities need simply to be rewritten into ‘‘P n i ¼1a i X i <a 0.’’The selection of inequalities that are modified is based on an ori-entation convention.Depending on the orientation of the half-space,the corre-sponding inequality is modified or not.In Section 2,we introduce our notations and the principal properties of the super-cover model on which the standard model is based.In Section 3the standard model is introduced and defined.We start,in Section 3.1,by explaining why such a ‘‘heavy’’mathematical machinery is necessary to define ðn À1Þ-connected discrete objects.We show in particular why a classical,misleading,approach does not work.In Sec-tion 3.2,we explain the basic ideas behind the standard model.In Section 3.3,we introduce the orientation convention that forms the basis of the definition of the standard model.The standard model is defined for all linear primitives in dimension n in Section 3.4.The properties of the standard primitives,especially the tunnel-free-ness and the ðn À1Þ-connectivity,are presented in Section 3.5.In Section 4,we ex-amine the different classes of standard linear objects to see how the definition is translated in practice and how the different inequalities defining the objects are estab-lished.Conclusion and several perspectives are presented in Section 5.2.Preliminaries2.1.Basic notations in discrete geometryMost of the following notations correspond to those given by Cohen and Kauf-man in [14,15]and those given by Andres in [7].We provide only a short recall of these notions.Let Z n be the subset of the n D Euclidean space R n that consists of all the integer coordinate points.A discrete (resp.Euclidean )point is an element of Z n (resp.R n ).A discrete (resp.Euclidean )object is a set of discrete (resp.Euclidean )points.A discrete94 E.Andres /Graphical Models 65(2003)92–111inequality is an inequality with coefficients in R from which we retain only the integer coordinate solutions.A discrete analytical object is a discrete object defined by a fi-nite set of discrete inequalities .An m -flat is a Euclidean affine subspace of dimension m .Let us consider a set P of m þ1linearly independent Euclidean points P 0;...;P m .We denote A m ðP Þthe m -flat induced by P (i.e.,the m -flat containing P ).We denote S m ðP Þthe geometrical simplex of dimension m in R n induced by P (i.e.,the convex hull of P ).For S ¼S m ðP Þa geometrical simplex,we denote S ¼A m ðP Þthe corre-sponding m -flat.For an n -simplex S ¼S n ðP Þ,we denote E ðS ;P i Þthe half-space of boundary A n À1ðP n P i Þthat contains P i (see Fig.1).We denote p i the i th coordinate of a point or vector p .Two discrete points p and q are k -neighbours ,with 06k 6n ,if j p i Àq i j 61for 16i 6n ,and k 6n ÀP ni ¼1j p i Àq i j .The voxel V ðp Þ&R n of a discrete n D point p is definedby V ðp Þ¼½p 1À12;p 1þ12 ÂÁÁÁ½p n À12;p n þ12 .For a discrete object F ,V ðF Þ¼S p 2F V ðp Þ.We denote r n the set of all the permutations of f 1;...;n g .Let us de-note J nm the set of all the strictly growing sequences of m integers all between 1and n :J n m ¼f j 2Z m j 16j 1<j 2<ÁÁÁ<j m 6n g .This defines a set of multi-indices .Let us consider an object F in the n -dimensional Euclidean space R n ,with n >1.The orthogonal projection is defined by:p i ðF Þ¼fðq 1;...;q i À1;q i þ1;...;q n Þj q 2R n gfor 16i 6n ;p j ðF Þ¼ðp j 1 p j 2 ÁÁÁ p j m ÞðF Þfor j 2J n m :The orthogonal extrusion is defined bye j ðF Þ¼p À1j ðp j ðF ÞÞfor j 2J n m :Example.Let us consider the set of points P ¼f P 0ð0;0;0Þ;P 1ð9;1;1Þ;P 2ð3;8;4Þg .The corresponding simplex T ¼S 2ðP Þis a 3D triangle.The orthogonal projection p 2ðT Þ¼S 2ðp 2ðP ÞÞ¼S 2ðfð0;0Þ;ð9;1Þ;ð3;4ÞgÞis a 2D triangle.The orthogonal ex-trusion e 2ðT Þ¼fð0;t ;0Þ;ð9;t ;1Þ;ð3;t ;4Þj t 2R g is a 3D Euclidean object defined by threehalf-spaces.Fig.1.Triangle T ¼S 2ðf P 0;P 1;P 2gÞ,edge S 1ðf P 0;P 1gÞ,straight line A 1ðf P 0;P 1gÞ,and half-space E ðf P 0;P 1;P 2g ;P 2Þ.E.Andres /Graphical Models 65(2003)92–11195We define an axis arrangement application r j,for j2J nm,by: r j:R n!R nx!ðx rjð1Þ;x rjð2Þ;...;x rjðnÞÞ;where the permutation r j2r n is defined byr j¼for16i6m;r jðj iÞ¼i else for m<i6n;r jðk rÞ¼iso that k r<k rþ1and k r¼j s for all16r6nÀm and for all16s6m.The axis arrangement application has been specifically designed so that it verifies the twofollowing properties:p jðFÞ¼pð1;2;...;mÞðrÀ1j ðFÞÞand e jðFÞ¼r jðeð1;2;...;mÞðrÀ1jðFÞÞÞfor allF in R n and j2J nm.Example.Let us consider the5D point Pð1;2;3;4;5Þand j¼ð2;4Þ2J52.Thecorresponding axis arrangement application is defined by rð2;4Þ:x!ðx3;x1;x4;x2;x5Þand rÀ1ð2;4Þ:x!ðx2;x4;x1;x3;x5Þ.The orthogonal projection verifies pð2;4ÞðPÞ¼pð1;2ÞðrÀ1ð2;4ÞðPÞÞ¼pð1;2Þð2;4;1;3;5Þ¼ð1;3;5Þ.The orthogonal extrusion verifies eð2;4Þð1;3;5Þ¼rð2;4Þðeð1;2ÞðrÀ1ð2;4ÞðPÞÞÞ¼rð2;4Þðeð1;2Þð2;4;1;3;5ÞÞ¼rð2;4ÞðpÀ1ð1;2Þð1;3;5ÞÞand therefore eð2;4Þð1;3;5Þ¼rð2;4Þðfðt;u;1;3;5Þjðt;uÞ2R2gÞ¼fð1;t;3;u;5Þjðt;uÞ2R2g.2.2.Geometric properties of the supercoverA discrete object G is a cover of a Euclidean object F if F&VðGÞand 8p2G;VðpÞ\F¼£.The supercover SðFÞof a Euclidean object F is defined by SðFÞ¼f p2Z n j VðpÞ\F¼£g(see Fig.2a).SðFÞis by definition a cover of F.It is easy to see that if G is a cover of F,then G&SðFÞ.The supercover of FcanFig.2.Supercover definitions.96 E.Andres/Graphical Models65(2003)92–111be defined in different ways:S ðF Þ¼ðF ÈB 1ð12ÞÞ\Z n ¼f p 2Z n j d 1ðp ;F Þ612g (see Fig.2b)where B 1ðr Þif the ball centered on the origin,of radius r for the distance d 1.This links the supercover to mathematical morphology [7,26,29,31].The supercover has many properties.Let us consider two Euclidean objects F andG ,and a multi-index j 2J n m ,then:S ðF Þ¼S a 2F S ða Þ,S ðF [G Þ¼S ðF Þ[S ðG Þ,ifF &G ,then S ðF Þ&S ðG Þ.These properties are well known [14,15].The following properties are more recent and are useful in the framework of this paper:S ðF ÂG Þ¼S ðF ÞÂS ðG Þ,r j ðS ðF ÞÞ¼S ðr j ðF ÞÞ,p j ðS ðF ÞÞ¼S ðp j ðF ÞÞ,and e j ðS ðF ÞÞ¼S ðe j ðF ÞÞ¼r j ðZ m ÂS ðp j ðF ÞÞÞ[7].Definition 1(Bubble ).A k -bubble,with 16k 6n ,is the supercover of a Euclidean point that has exactly k half-integer coordinates.A half-integer is a real l þ12,with l an integer.A k -bubble is formed of 2k discrete points.A 2-bubble can be seen in Fig.2a (marked by the black circle).The two white dots are what we call here ‘‘simple’’points.This corresponds to an extension of the notion of simple points that fits a supercover simplex.A point P belonging to the supercover simplex S is said to be a simple point if it is a simple point for S with the classical definition given in Section 2.1.Definition 2(Bubble-free ).The cover of an m -flat is said to be bubble-free if it has no k -bubbles for k >m .The cover of a simplex S is said to be bubble-free if S is bubble-free.There are two types of bubbles in the supercover of an m -flat F .The k -bub-bles,for k 6m ,are discrete points that are part of all the covers of F .If we remove any of these points,the discrete object is not a cover anymore.In the k -bubbles,for k >m ,there are discrete points that are ‘‘simple’’points.The aim of this paper is to propose discrete analytical objects that are bubble-free and ðn À1Þ-connected by removing some of the simple points.In Fig.2a,by removing one of the two simple points,we obtain a bubble-free,1-connected discrete 2D line segment.Lemma 1.A discrete point p belongs to a k -bubble,k >m ,of the supercover of an m -flat F if and only if there exists a point a 2F with k half-integer coordinates such that p 2S ða Þ.The proof of this lemma is obvious.3.Standard modelThe aim of this paper is to propose a new cover class,called the standard cover.The standard cover is so far only defined for linear objects in all dimensions.The dis-crete analytical model has been designed to conserve most of the properties of theE.Andres /Graphical Models 65(2003)92–11197supercover,to be bubble-free andðnÀ1Þ-connected.The supercover model has al-most all the properties we are looking for:tunnel-freeness,ðnÀ1Þ-connectivity,sta-bility for union,etc.The only property that is missing is the bubble-freeness.Some supercover objects have simple points.The model is therefore not topologically con-sistent and this is a problem for several applications such as,for instance,polygonal-ization.For this reason several attempts have been made to modify the supercover discretization by modifying the definition of a pixel[14,15,27].We show in the fol-lowing section that such attempts cannot work.In our approach,presented in Sec-tion3.2,we explain how,by studying the analytical description of linear objects,it is possible to remove selectively the simple points in the supercover model while pre-serving the modeling properties.In the section that follow the standard model and its properties are introduced.3.1.What does not work with the classical approachSeveral unsuccessful attempts have been made to define discrete objects that have supercover type modeling properties with bubble-freeness andðnÀ1Þ-connectivity properties[14,15,27].All these ideas basically modify,in various ways,the definition of a voxel in order to avoid bubbles.We give here a simple such example and show why it does not work that way(see[14]for some other examples).In Fig.3,the pixel definition has been changed.A pixel is now formed of the SW vertex(black disk),the two corresponding edges(bold edges)and itsÕinterior.The three other vertices and two other edges do not belong to the pixel.This definition derives thatSp2Z n VðpÞ¼R n with VðpÞ\VðqÞ¼£for p¼q:The discretization of a discreteline is necessarily bubble-free.However,as we see in Fig.3,the discretised line x1Àx2¼0is not1-connected.In fact,it has been shown as early as in1970[27],that no change in the definition of the pixel or voxel can lead to a correct solution.This means that a simple pixel definition modification avoids bubbles but creates primi-tives that are not topologically consistent.This makes such a model useless for ap-plications such as polygonalization.Tunnel-freeness property is also lost with such an approach.3.2.Standard model approach:a modification of the supercover definitionThe discrete analytical description of the supercover of a linear convex is defined as intersection of half-spaces defined by discrete inequalities P n i ¼1a i x i 6a 0[2,5–7].A linear concave object is simply considered as union of convexes.The orientation of each half-space is checked with an orientation convention and depending on it,its inequality ‘‘P n i ¼1a i x i 6a 0’’remains unchanged or is replaced by ‘‘P n i ¼1a i x i <a 0.’’Let us give a simple example,the 2D straight line D :3x 1À7x 2¼0shown in Fig.4,to illustrate why and how this works.The general case in dimension n works ex-actly in the same way.The supercover of the Euclidean line D is described by the two inequalities S ðD Þ¼fðx 1;x 2Þ2Z 2j À563x 1À7x 265g .A bubble occurs only when the straight line D contains half-integer coordinate points.We have then (and only then)discrete points verifying on one side 3x 1À7x 2¼À5and on the other side 3x 1À7x 2¼5.All these points are simple points.Removing the points on one side only leads to a discrete straight line that is 1-connected,separating,1-minimal and bubble-free.This can be done simply by replacing a ‘‘6’’by a ‘‘<’’for one of the two inequalities in the supercover analytical description.In the case of Fig.4,we have S t ðD Þ¼fðx 1;x 2Þ2Z 2j À563x 1À7x 2<5g .The change is based on an orien-tation convention.Opposing half-spaces such as ‘‘3x 1À7x 265’’and ‘‘À3x 1þ7x 265’’have a different orientation in this convention and thus only one of them will have its Õ‘‘6’’changed into ‘‘<.’’This ensures that only one simple point for the 2D line will be removed.3.3.Orientation conventionThe standard model,contrary to the supercover,is not unique [7,9].For instance,in example of Fig.4,one of two possible simple points can be removed.Each selec-tion leads to another standard model definition.It depends on the orientation con-vention selection.One orientation convention per dimension R m ,m >0,isrequired.Fig.4.Standard and supercover straight line.The black points belong to both line.The white point be-longs only to the supercover.E.Andres /Graphical Models 65(2003)92–11199This selection must then remain unchanged for all the primitives handled.The selec-tion of an orientation convention per dimension has to be coherent with the operator p .The property S t ðp j ðF ÞÞ¼p j ðS t ðF ÞÞfor the operator p should be verified.If this is not the case,the modelling properties would not be verified (such as S t ðF [G Þ¼S t ðF Þ[S t ðG Þ,etc.).In general,with arbitrary orientation conventions there is no reason for this property to be verified.We propose a set of orientation conventions,denoted O n and called the basic orientation conventions.The basic ori-entation conventions verify the above mentioned property.Definition 3(Standard orientation ).Let us consider a discrete analytical half-space E :P n i ¼1C i X i 6B and the basic orientation convention O n .We say that E has a standard orientation if:•C 1>0;•or if C 1¼0and C 2>0;•...•or if C 1¼ÁÁÁ¼C n À1¼0and C n >0:If E has not a standard orientation,then we say that E has a supercover orientation.We consider from now on,without loss of generality,only the basic orientation conventions for all n >0.All the standard primitives are defined with these basic ori-entation conventions.The basic orientation conventions are coherent with respect to the operators p .After p j ,for j 2J nm ,the orientation convention O n in R n becomesO n Àm in R n Àm .3.4.Standard model definitionAll the elements required to define the standard discretization model of linear ob-jects in R n are available.Definition 4(Standard model ).Let F be a linear Euclidean object in R n whose su-percover is described analytically by a finite set of inequalities F k :P n i ¼1C i ;k X i 6B k .The standard model S t ðF Þof F ,for the basic orientation convention O n ,is the discrete object described analytically by a finite set of discrete inequalities F 0k ob-tained by substituting each inequality F k by F 0k defined as follows:•If F k has a standard orientation,then F 0k :P n i ¼1C i ;k X i <B k ;•else F 0k :P n i ¼1C i ;k X i 6B k .This definition is algorithmically easy to set up.Once a discrete analytical descrip-tion of an object is available,the transition from the supercover model to the stan-dard model and vice versa is trivial.3.5.Geometric properties of the standard modelIn this section,some properties of the standard model are presented.These prop-erties are very important for the derivation of our model description.Let us consider100 E.Andres /Graphical Models 65(2003)92–111a Euclidean linear object F of topological dimension m in R n .We have by definition S t ðF Þ&S ðF Þeven more precisely,if p 2S ðF Þn S t ðF Þ,then d 1ðp ;F Þ¼12.A stan-dard object is a supercover object from which some discrete points have been re-moved.These points are all at a distance 1from the Euclidean primitive.We have S t ðF Þ¼S ðF Þif no point,with at least m þ1half-integer coordinates,belongs to the boundary of F .The differences between the supercover of F and the standard model of F are located in the k -bubbles of F ,for k >m .Fig.4illustrates this in di-mension 2.One of the immediate consequences of this last property,is that the stan-dard model remains a cover:F &V ðS t ðF ÞÞ.That is why the standard model is also sometimes called standard cover [33,31].The standard model retains most of the set properties of the supercover.It is easy to deduce from Definition 4,that if we consider two Euclidean linear objects F and G in R n ,then:S t ðF [G Þ¼S t ðF Þ[S t ðG Þ;S t ðF \G Þ&S t ðF Þ\S t ðG Þ;F &G )S t ðF Þ&S t ðG Þ;S t ðF ÂG Þ¼S t ðF ÞÂS t ðG Þ;S t ðp j ðF ÞÞ¼p j ðS t ðF ÞÞ;S t ðe j ðF ÞÞ¼e j ðS t ðF ÞÞ:The first property ensures that we will be able to construct complex discrete ob-jects out of basic elements such as simplices.These last properties are characteristic of correct orientation conventions.The properties are only verified if the orientation conventions are defined for all dimensions lower or equal to n and if they are coher-ent with respect to the operator p .This is the case for the basic orientation conven-tions O k ,for k 6n .It is important to notice that,in general,S t ðF Þ¼S a 2F S t ða Þ.This property of the supercover is not conserved.We have S t ðF [G Þ¼S t ðF Þ[S t ðG Þfor a union of a finite number of objects.This comes simply from the fact that the standard model is not defined for an analytical description that has an infinite number of discrete in-equalities.One simple example for that is given by the 2D line D :x 1Àx 2¼0:The standard model of the line is S t ðD Þ:À16x 1Àx 2<1while S a 2F S t ða Þ:À1<x 1Àx 2<1.One of the main properties of the standard model concerns the connectivity and the tunnel-freeness.Theorem 2(Connectivity and tunnel-freeness).Let F be a Euclidean linear object of topological dimension m in R n .Its standard model S t ðF Þis ðn À1Þ-connected and tunnel-free.The standard model is a particular case of k -discretizations as introduced by Brim-kov et al.in [12].It is shown that the standard model is in fact a 0-discretization (The-orem 3in [12])and that 0-discretizations are ðn À1Þ-connected and tunnel-free E.Andres /Graphical Models 65(2003)92–111101(Proposition 3in [11]and Theorem 4in [12]).Another property proved in [12,31,33]is that the standard model minimizes the Hausdorffdistance with the Euclidean object.4.Des cription ofs tandard primitivesWe will examine now the discrete analytical description of the different classes of standard linear primitives (half-space,point,m -flat and m -simplex)and how they can be computed.Our purpose here is to propose a discretization scheme that can be used in practical applications.As stated in Definition 4,every analytical description of a standard linear primitive is based on the analytical description of a standard half-space.That is the one we present first.We deduce from it the discrete analytical for-mulas describing a standard point,m -flat and m -simplex in the sections that follow.4.1.Standard half-spaceThe standard half-space is given by:Proposition 3(Standard half-space).Let us consider a Euclidean half-space E :P n i ¼1C i X i 6B .The standard model S t ðE Þof E ,according to an orientation convention,isanalytically described by:•If E has a standard orientation,then S t ðE Þ¼p 2Z n X n i ¼1C i p i (<B þP n i ¼1j C i j 2);•elseS t ðE Þ¼p 2Z n X n i ¼1C i p i (6B þP n i ¼1j C i j 2):The proposition is an immediate extension to dimension n of results on the super-cover [2,4,5,7]and of Definition 4.4.2.Standard pointThe analytical description of a standard point can easily be deduced from the one of the standard half-space.It is however interesting to notice that the standard dis-cretization of a Euclidean point is always composed of one and only one discrete point contrary to what happens with a supercover discretization of a Euclidean point that can be formed of 2k points,06k 6n (in case of a k -bubble).Proposition 4(Standard point).Let us consider a Euclidean point a 2R n and the basic orientation convention O n .The standard model S t ða Þof a is the discrete point102 E.Andres /Graphical Models 65(2003)92–111。

Institut für Materialprüfung, Werkstoffkunde10100m (nm)(m)853-S111 Microstructural Mechanics Abt. 10regime1-9526-W808ae Notched tensile specimen with elastic-plastic property gradientAbt. 10526-W848ae Notched Tensile-SpecimenAbt. 10CUT 1CUT 1CUT 2Fazit: ReduzierteSpannungenRohr-Rohr-VerbindungenCVD Beschichtung -Querschliff100 µmσσD-Parameter bei εges.=0,6%B1B2B3B4B5σσReal MicrostructureStrain [%]%45.0=ε%337.0=εRandomly distributed fibersAl/46vol.%B f 50µmFe/50vol.%Cu20µm526-W874e Matricity ModelAbt. 10Crack inside an Al/20vol.%SiC-CompositeFEM-Model and Calculated Crack Path Simulated Crack Path in the Microstructure853-Z300ePath Simulation10-90m (nm)(m)Experimental and100908070Load / kNE60B E60A15 NiCuMoNb 5 E60A and E60B T= 90°C100908070Load / kN15 NiCuMoNb 5 E60A and E60B T= 90°C6050403020106050403020E60B E60A0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Change of Cross SectionNotch radius 2 mmT - Direction100Notch radius 8 mmT - Direction0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Change of Cross Section ∆D / mmResults of the Notched Tensile SpecimenNotched Tensile Specimens841-B572 Abt. 10500 450 400J-Integral / N/mm15 NiCuMoNb 5 E60A and E60B CT25 TL - SpecimensT=90°CE60A350 300 250 200 150 100 50 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0E60BCrack Growth ∆a / mmCrack Growth Resistance Curves at T = 90 °CCrack Growth Resistance Curves841-B573 Abt. 1050045015 NiCuMoNb 5 Dmax- Distribution500450Frequency of Particles40015 NiCuMoNb 5 D - Distribution minFrequency of Particles40035030025020015010050350300E60B2502023 Cu-ParticlesE60B2023 Cu-Particles200150E60A1321 Cu-ParticlesE60A1321 Cu-Particles10050005101520253035404550005101520253035404550Particle Size / nmParticle Size / nmFrequency Distribution of Cu-Precipitates (E60A, E60B)Cu-Precipitates841-B581 Abt. 1070 60 501321 ParticlesNumber of Particles15 NiCuMoNb 5 E60A and E60B D max vs Dmin180E60B E60A2023 Particles16014015 NiCuMoNb 5 E60A and E60B Area Distribution E60BE60A: 1321 Particles E60B: 2023 Particles120Dmax / nm40 30 20 10 0 0 5 10 15 20 25 30 35 401008060E60A402000102030405060708090100Dmin / nmParticle Area / nm2Dmax vs Dmin and Area Distribution of Cu-Precipitates (E60A, E60B)Cu-Precipitates841-B582 Abt. 10FE-Discretization and Boundary Conditions of CT-Specimen with Ic = 0.1 mmElements(QU8)lcNodal PointsFE-Discretization841-B584 Abt. 10Rousselier - ModelΦ= 1σvf σkDfe σv f σo σk D – von Mises Equivalent Stress – Void Volume Fraction – Yield Limit – Material Dependent Constant – Material Independent Constant σH – Hydrostatic StressσH (1 - f) σkσo = 0fo = 0.002 fc = 0.05 lc = 0.1 mm D =2 σk = 445 MPaInitial Void Volume Fraction Critical Void Volume Fraction Average Particle Distance Integration Constant StressRousselier-Model841-B585 Abt. 10Specimen Shape Standard Tensile Specimen Notched Tensile Specimen ρ = 2 mm Notched Tensile Specimen ρ = 8 mmCalculation axial-sym. axial-sym.Particle Distance lc1 = 0,05 lc2 = 0,10 lc1 = 0,05 lc2 = 0,10 lc1 = 0,05 lc2 = 0,10lc1 = 0,05 lc2 = 0,10 lc2 = 0,15 lc2 = 0,20Initial Void Volumef03 = 0,003 ) 2 f01 = 0,001 - f03 = 0,003 ) f01 = 0,001 - f05 = 0,003 ) 1 f04 = 0,004 u. f05 = 0,005 ) f01 = 0,001 - f05 = 0,005 ) f01 = 0,001 - f05 = 0,003 ) 1 f04 = 0,004 u. f05 = 0,005 ) f01 = 0,001 - f05 = 0,005 ) f01 = 0,001 - f03 = 0,003 ) 2 f01 = 0,001 - f03 = 0,003 ) 2 f03 = 0,003 - f05 = 0,005 ) 2 f05 = 0,005 )2 2 2 2 2 1axial-sym.CT-Specimen TPB-Specimen SECT-Specimen CCP-Specimen1 22D / Plane Strain 2D / Plane Strain 2D / Plane Strain 2D / Plane Strainlc2 = 0,10 lc2 = 0,10 lc2 = 0,10f02 = 0,002 2) f02 = 0,002 2) f02 = 0,002 2)) E60A ) E60A and E60BPerformed FE-Calculations841-B583 Abt. 1080070060015 NiCuMoNb 5 E60A T= 90°CT - Directionl c2 = 0,1 mm800 700 600Stress σ / MPa15 NiCuMoNb 5 E60B T= 90°CT - Directionl c2 = 0,1 mmExperimentStress σ / MPa5004003002001000 0.00500 400 300 200 100Experimentf = 0,003 o3 f = 0,002 o2 f = 0,001 o1fo3 = 0,003 fo2 = 0,002 fo1 = 0,001FE-CalculationsFE-CalculationsTensile Specimen A1110.050.100.150.200.250.30Tensile Specimen B111 0 0.00 0.05 0.10 0.150.200.250.30Strain ε / m/mStrain ε / m/mExperimental and Calculated Technical Yield CurvesYield Curves841-B586 Abt. 1010015 NiCuMoNb 5 f = 0,005 FE-Calculations 90 E60A o5 f = 0,004 o4 T= 90°C f = 0,003 80 Fracture o3 f = 0,002100908070Load / kNFracture15 NiCuMoNb 5 E60B T = 90°C70Load F / kNo2f = 0,001 o1Experimentsf60605040302050Experiments40f o4 f = 0.005 o5f o3 = 0.004f = 0.001 o1 = 0.002 o2 = 0.003FE-Calculations302010Notch Radius 2l = 0.1 mm c20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Change of Cross Section ∆D / mmT - Direction100Notch Radius 2 mml = 0.1 mm c2T - Direction0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Change of Cross Section ∆D / mmComparison of Experimental and Calculated Tensile Specimens, Notch Radius 2 mm (E60A, E60B)Notch Radius 2 mm841-B587 Abt. 1010015 NiCuMoNb 5 90 E60A T= 90°C 80 70 60 50 40 30 20 10 Notch Radius 8 mml = 0.1 mm c2100FE-Calculationsf = 0.005 o5 f = 0.004 o4 f = 0.003 o3 f = 0.002 o2 f = 0.001 o1908070Load / kN15 NiCuMoNb 5 E60B T = 90°Cfo5Load F / kN= 0.005 f = 0.004 o4 f = 0.003 o3 f = 0.002 o2 f = 0.001 o1FE-Calculations6050403020100ExperimentsExperiments T - DirectionNotch Radius 8 mml = 0.1 mm c20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Change of Cross Section ∆D / mmT - Direction0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Change of Cross Section ∆D / mmComparison of Experimental and Calculated Tensile Specimens, Notch Radius 8 mm (E60A, E60B)Notch Radius 8 mm841-B588 Abt. 1070l = 0.1 mm 15 NiCuMoNb 5 c2 Experiment E60A / T = 90°C 60 CT25 TL - Specimen FE-Calculationsa = 26.337 mm / W = 50.5 mm a/W = 0.52215 NiCuMoNb 5 E60B / T = 90°C 60 CT25 TL - Specimen a = 28.89 mm / W = 50.1 mma/W = 0.57770l = 0.1 mm c250Load F / kNfo1 = 0.001 fo2 = 0.00250Load / kN40fo3 = 0.00340302010fo2f = 0.001 o1 = 0.002f = 0.003 o330FE-Calculations20Experiment100 0.00.51.01.52.02.50 0.00.51.01.52.02.5Crack Opening COD / mmCrack Opening COD / mmComparison of Experimental and Calculated Crack Opening Behaviour for the CT-Specimen (E60A, E60B)Load-Crack Opening Relation841-B589 Abt. 1080070015 NiCuMoNb 5 E60A / T = 90°CCCP-Specimenlc2 = 0.1 mmf = 0.002800a/W = 0.5 / 2W=100 mm o270015 NiCuMoNb 5 E60B T= 90°Clc2 = 0.1 mmf = 0.002 o2600J-Integral / N/mm500a/W = 0.5 / W = 50 mmJ-Integral / N/mmFE-Calculations600CCP-Specimen a/W = 0.5 / 2W = 100 mm SECT-Specimena/W = 0.5 / W = 50 mmSECT-Specimen TPB-Specimen a/W = 0.5 / W = 50 mm CT-Specimen mm a/W = 0.522 / W = 50.5FE-Calculations500400400TPB-Specimen a/W = 0.5 / W = 50 mm CT-Specimen mm a/W = 0.577 / W = 50.1 Experiment (CT-Specimen)300300200Experiment (CT-Specimen)2001001000.00.51.01.52.02.53.03.54.000 0.00.51.01.52.02.53.03.54.0Crack Growth ∆a / mmCrack Growth ∆a / mmCrack Growth Resistances (E60A, E60B)Crack Growth Resistance and Variation of Specimen Geometry841-B592 Abt. 10600 500J-Integral / N/mm15 NiCuMoNb 5 E60A / T = 90°C CT25 TL - Specimena = 26.337 mm / W = 50.5 mm a/W = 0.522l = 0.1 mm c2600 500J-Inetgral / N/mm15 NiCuMoNb 5 E60B / T = 90°C CT25 TL - Specimena = 28.89 mm / W = 50.1 mm a/W = 0.577l = 0.1 mm c2fo1 = 0.001 fo2 = 0.002400 300 200 100 0 0.0FE-CalculationsExperiment400 300 200FE-Calculationsf = 0.001 o1 f = 0.002 o2 f = 0.003 o3fo3 = 0.003Experiment 100 0 0.00.51.01.52.02.53.03.54.00.51.01.52.02.53.03.54.0Crack Growth ∆a / mmCrack Growth ∆a / mmComparison of Experimental and Calculated Crack Growth Resistance Curves for the CT-Specimen (E60A, E60B)Crack Growth Resistance Curves841-B590 Abt. 10600 500Material 15 NiCuMoNb 5 States E60A and E60B T = 90°C CT25 TL - Specimen FE-Simulationsf lo2= 0,002 = 0,075 mmcJ-Integral / N/mm400 300 200 100 0a = 26,337 mm / W = 50,5 mm a/W = 0,522State E60Aa = 28.89 mm / W = 50,1 mm a/W = 0,577State E60BExperiments0.00.51.0Crack Elongation ∆ a / mm1.52.02.53.03.54.0Crack Growth Resistance Curves773-S902 Abt. 10Plane TS LT LSDmax [ µm] E60A: 1.2 – 56.4 E60B: 1.2 – 73.6 E60A: 1.2 – 60.5 E60B: 1.2 – 78.7 E60A: 1.2 – 130.2 E60B: 1.2 – 87.5Dmin [ µm] E60A: 0.6 – 13.9 E60B: 0.6 – 10.0 E60A: 0.6 – 16.0 E60B: 0.6 – 11.4 E60A: 0.6 – 9.90 E60B: 0.6 – 10.3From these data the initial void volume fraction fo and the mean particle distance Ic can be determined: Plane TS LT LS fo [-] E60A: 0.6064 ·10-3 E60B: 0.6486 ·10-3 E60A: 0.6849 ·10-3 E60B: 0.6121 ·10-3 E60A: 0.6703 ·10-3 E60B: 0.6439 ·10-3 lc [mm] E60A: 0.049 E60B: 0.043 E60A: 0.050 E60B: 0.044 E60A: 0.053 E60B: 0.046Volumes and Distances in Case of Spherical Precipitates: fo, Ic841-B577 Abt. 1018016014015 NiCuMoNb 5 E60A Relation D maxto DminLS-Plane180160140DmaxDminDmax / µm100806040200Dmax / µm12012010080604020Dmean = (Dmax +Dmin ) / 2Dmin1291 Particles15 NiCuMoNb 5 E60B Relation Dmaxto Dmin1628 ParticlesLS-PlaneDmaxDmean = (Dmax +Dmin ) / 205101520005101520Dmin / µmDmin / µmLength/Width of Nonmetallic Inclusions (E60A, E60B)Inclusions841-B576 Abt. 10100090080070015 NiCuMoNb 5 E60A and E60B Dmin- Distribution E60B1628 Particles1000 900 80015 NiCuMoNb 5 E60A and E60B - Distribution D max1803 ParticlesLT-PlaneFrequency600FrequencyLS-Plane700 600 500 400 300 E60B E60A4003002001000E60ADmin500Dmax200 100 0051015202530051015202530Particle Size / µmParticle Size / µmFrequency Distribution of Nonmetallic InclusionsMnS-Inclusions are not Spherical841-B578 Abt. 1015 NiCuMoNb 5T = 90°C, E60Bf0=0.13% lc=0.1mmElongated Particles taken into account, Improved evaluation software Crack Growth Resistance for E60B773-S901 Abt. 10Precipitation Induced Aging of SteelsMonte Carlo SimulationInitial StateDislocation - TheoryInteraction Dislocation – Small ParticlesDamage MechanicsVoid Formation Large particlesTemperature, Time Aged StateStress σ / MPa800 700 15 N iC uM oN b 5 E60A and E60B T= 90°CB 112 B111 Mechanical Behaviour B113Damage Behaviour15 NiCuMoNb 5 T = 90°C, E60B f0=0.13% lc=0.1mmStress/MPa600 500 400 300 200 100800 600 400 200 0E 60BAged State Initial StateA112 A111 A 113E60A0.1 0.2 Strain / 0.050.3T - D irection0 0.00Micro 0.100.150.200.25Strain ε / m/mNanoMicroMacroSimulation Levels841-B503 Abt. 10Numerically Derived Crack Resistant Curves for Different Strength IncreasesCrack Resistant Curves – Worst Case526-W607e Abt. 10HierarchicalParameter Link to bridge:。