Numerical Simulation of Orthogonal Cutting using the Material Point Method

Transcription

1 Numerical Simulation of Orthogonal Cutting using the Material Point Method John A. Nairn Wood Science and Engineering, Oregon State University, Corvallis, OR 97330, USA Tel: Fax: Abstract A material oint method simulation of orthogonal cutting that can simulate cutting into steady-state chi curling is described. The modeling used ductile fracture mechanics using cohesive zone in the cutting ath. Robust simulations required a new mechanism to dam kinetic energy artifacts associated with dynamic crack roagation. The simulations dislayed two regimes crack-ti touching, where the tool reaches the crack ti, and lastic bending, where the tool is searated from the crack ti by a ga. The simulations were comared to analytical models that were revised to account for rubbing forces and hardening laws. Key words: A. Cutting, B. Material Point Method, C. Comutational Mechanics, D. Cohesive Zones. Introduction Several recent aers advocate modeling of orthogonal cutting as a ductile fracture mechanics roblem for a crack roagating in the direction of the tool ti [,, 3, 4, 5, 6, 7]. Some analytical modeling [, 6] and exeriments [4, 5, 8] show the fracture mechanics view can hel interret exeriments and exlain some roblems of classic cutting models based solely on lasticity and friction [, ]. A suggestion of this new aroach is that cutting exeriments can be used to measure the toughness of ductile materials. The concet is that extraolation of cutting forces to zero deth of cut should have non-zero intercet equal to the material s fracture toughness. The challenge is to devise the best exerimental methods for getting reliable extraolations in the resence or large amounts of work due to lasticity and friction. Recommendation of such exerimental rotocols should be guided by modeling. The current analytical modeling has been limited to basic material roerties, simle yielding models (such as elastic-lastic, and simle frictional contact. This aer s goal is to develo a numerical model for orthogonal cutting with the otential to handle more realistic material roerties (such as large-strain constitutive laws, arbitrary lasticity and contact laws, and more realistic secimen geometries and boundary conditions. The numerical modeling of orthogonal cutting through to steady-state chi curling and wraing involves large strains (close to 00% shear strains are seen in calculations, large dislacements and rotations, dynamic contact both between the tool and the cut material and between layers of a curling chi, and evolution of an exlicit crack. The finite element method (FEM has been used for orthogonal cutting (e.g., [9]. In general, rior FEM models have assumed the rocess is dominated by lasticity and friction and ignored crack roagation in the ath of the tool. Most have been for small amounts of Prerint submitted to Elsevier November 8, 04

2 chi formation, rather then comlete chi formation into the steady-state cutting regime. A drawback of many FEM models is severe mesh distortion requiring adative remeshing [9]. Furthermore, FEM lasticity and friction models cannot model significant tool movement without addition of a searation criterion (e.g., stress or stain limits []. Such ad hoc criteria may not rovide a rigorous fracture mechanics simulation of cutting. A alternative numerical method, called the material oint method (MPM [0, ], seems well suited to simulations of orthogonal cutting roblems. It can handle large strains and deformations without remeshing, dynamically model all contact situations [, 3], and imlement exlicit cracks [4, 5] for ductile fracture mechanics. This aer resents an MPM model of orthogonal cutting through to steady-state cutting with chi curling. The simulations worked well, but required develoment of a new MPM daming scheme, called PIC Daming, that aears esecially effective at daming vibrations caused by kinetic energy released in dynamic crack roagation. The simulations were verified by comarison to analytical models, but the models had to be modified to account for rubbing forces on the bottom of the tool and to handle elastic-lastic materials with linear hardening. The cutting simulations dislayed two cutting regimes chi shearing, where the tool ti touches the crack ti, and lastic bending, where the chi is bent but the tool ti is dislaced from the crack ti. The shearing occurred for thin cuts followed by a sudden transition to lastic bending with a dro in cutting forces for thicker cuts. The simulations could handle both regimes and the transition between the two regimes. The exlicit crack roagation was handled by using a cohesive zone model along the cutting ath.. Numerical Methods The material oint method (MPM discretizes the object into articles and uses a background grid for solution of the momentum equation [0]. The geometry and discretization for the MPM model are shown schematically in Fig.. The simulations varied the deth of cut (h. The secimen s width, base deth, and re-existing crack length were scaled to deth of cut as wh, bh, and ah, resectively, using scaling factors w, b, and a. The width and crack length factors were set to w = 30 and a = 8, which rovided enough cutting length to achieve steady state cutting as well as comlete chiing curling. The base factor was set to b = 3 (as exlained below. The tool rake angle ( was varied while the clearance angle was ket constant at =5. The secimen s entire bottom edge was held at zero dislacement in both the x and y directions. The inset shows the background grid and MPM articles near the initial notch ti (at lower resolution then used in simulations. The secimen filled the grid with four articles er cell. The tool also used four article er cell, but sheared those articles to conform to the tool shae. This inset shows articles drawn at 60% of their size. The actual MPM articles, as quadrilateral regions, comletely filled sace in both the secimen and the tool. Because cutting simulations result in large deformations and large-scale rotations, it was crucial to use a hyer-elastic, lastic material model for the secimen [6]. The elastic resonse of the isotroic material was modeled as a neo-hookean material with elastic strain energy given by: W = K Ä J e ä ln J e + G Tr B e 3 T Here J e = det F e is determinant of the elastic deformation gradient, B e = F e F e is the deviatoric art of the left Cauchy-Green strain tensor, F e = F e /J /3 e is the deviatoric art of the elastic deformation (

3 α h dh ah θ wh Figure : A schematic drawing of the numerical model for orthogonal cutting to a deth of cut h with initial crack length ah, base deth dh, and total width wh. The tool has rake angle and clearance angle. The inset shows MPM background grid and discretization into material oints in the cut material and the rigid tool. All simulations used more articles, or higher resolution, then deicted in the inset. gradient, and K and G are the low-strain bulk and shear moduli of the material. The lastic resonse was modeled by yielding when the magnitude of the deviatoric stress reaches r ksk = f ( ( 3 where f ( is any hardening law that deends on cumulative lastic strain found by integrating d = /3kd" k where d" is the incremental lastic strain tensor. Any hardening law can be used, but all simulations here comared to analytical models and thus were limited to linear hardening with f ( = y0 + E, where y0 is the initial yield stress and E is the lastic modulus. Material roerties were set by selecting a tangent modulus E T as the sloe of a tensile stress-strain curve in the lastic region, from which lastic modulus is E = EE T /(E E T where E is the low-strain elastic modulus. The lastic resonse was imlemented using return maing methods [7]. Crack roagation was modeling using the CRAMP algorithm for addition of exlicit cracks to MPM simulations [4]. Because it was difficult to control crack roagation and direction from the comlex and large lastic deformation rocesses near the crack ti, the crack roagation was modeled using cohesive zones. In brief, an exlicit crack of length ah was inserted at the start of the calculations. To hel in starting the cutting rocess, a distance h at the beginning of the crack was inserted as a traction-free crack. The remaining length of the crack incororated cohesive traction laws as described for MPM simulations by Nairn [5]. Most simulations used a cubic traction law: = 7 4 c c c (3 where c is the cohesive stress and c is the critical crack oening dislacement [8]. The area under this law is the toughness J c = 9 c c /6. A simle aroach to mixed-mode loading is a decouled failure criterion [9]: = J I( n + J II( t J Ic J IIc (4 3

4 where J I ( n and J II ( t are areas under the mode I and II cohesive laws u to normal ( n and tangential ( t crack oening dislacements and J Ic and J IIc are toughnesses in ure mode I and II. To minimize the number of simulation arameters, all simulations here used the same cohesive laws in mode I and II (J Ic = J IIc = J c and Ic = IIc = c ; i.e., failure occurs when total energy release rate J c = J I ( n +J II ( t. The fraction of mode I energy released (J I ( n /J c was outut during crack growth to evaluate the mode-mixity of the cutting rocess. The tool was modeled as a rigid material. The article centers were aligned to conform to the tool s rake and clearance angles and the initial article domains (squares were sheared to arallelograms to have domain edges exactly match the tool edges (see Fig.. Although aligning article centers always heled the calculations, shearing the domains only influences the results when using shae function methods that account for domain deformation (e.g., convected article domain integration or CPDI [0]. All calculations here using uniform, generalized interolation methods (or ugimp [] where the integration domain remains a square, but translates with the articles; ugimp was used for all simulations because it was more efficient and accounting for domain shearing (with CPDI had very little effect. The rigid articles in the tool ignored the crack lanes used by the CRAMP algorithm [4], which allowed the tool to interact with both the to and bottom surfaces of the crack though contact mechanics. In other words, this secial material could be inside a crack and serve to wedge oen the crack including touching the crack ti when material roerties allowed it. The contact was modeled using Coulomb friction using MPM multimaterial contact methods []. Recent work on contact methods has emhasized the imortance of accurately calculating the contact normals [3, ]. This issue could be handled rigorously for these cutting simulations because all contact normals could be redetermined. The normal (from cut material into the tool on the tool s to surface was set to ˆn =(sin, cos and on the bottom surface was set to ˆn =(0,... PIC Daming The most imortant MPM change that led to robust and stable simulations for a wide variety of secimen geometries, material roerties, and cohesive law roerties was to introduce a new method for udating article velocity and osition. The MPM time ste solves the momentum equation on the background grid or (n+ i = (n i + m (n i a (n i t (5 Here subscrit i indicates a nodal value, subscrits (n and (n + denote time ste n and and the udated results, is momentum, a is acceleration, and m is mass. Once the grid udate is done, these results are used to udate article velocity and osition. The standard methods used by MPM codes is a FLIP method (for Full Lagrangian Imlicit Particle [] where article velocity, v, is udated using the grid acceleration: X v (n+,flip = v (n + a (n g! t where a (n g! = a (n i S (n i (6 Here subscrit indicates a article quantity and subscrit g! indicates extraolation of a grid result to the article location using the MPM shae function (S i []. An alternative udate scheme used in article methods is the PIC method (for Particle In Cell [3] that extraolates grid velocity directly to the article: X v (n+,pic = v (n+ g! = v (n g! + a(n g! t where v (n g! = i 4 i (n i m (n i S (n i (7

5 0 th st nd Figure : Simulation of orthogonal cutting using 00% PIC udate for velocity by various udates for osition. The 0 th udate used Eq. (9, which is the standard MPM udate when using FLIP udate for velocity. The st and nd udates used Eq. (0 to first or second order in t, resectively. The FLIP method was referred because PIC can cause numerical diffusion resulting in higher viscosity and heat conduction than exected []. But, Stomakhin et al. [4] suggests that MPM simulations can be imroved by combining FLIP and PIC simulations and this suggestion led to significant imrovement in cutting simulations. But two questions remained why does addition of PIC hel and is it reasonable to include it in MPM simulations? My roosal is that use of PIC in MPM simulations is better described as a new form are artificial daming alied to the referred FLIP methods. This new form of daming seems to be articularly effective for simulations involving crack roagation. The numerical diffusion that is a concern for fluid dynamics modeling [] may actually be a benefit to crack roagation modeling where the numerical diffusion damens kinetic energy caused by increments in crack growth. In real materials, energy released by crack growth is absorbed by crack ti rocesses. But, in comutational mechanics, dynamic crack roagation injects kinetic energy into the system that is not absorbed by standard material models. The addition of PIC daming seems to rovide an effective daming mechanism for that unrealistic kinetic energy. A new FLIP velocity udate with PIC daming can be written as v (n+ = v (n + a (n g! PIC v (n v (n g! t (8 where PIC =( / t is a PIC daming coefficient. The term varies from, which gives a ure FLIP velocity udate, to 0, which gives a ure PIC velocity udate (cf. Eqs. (6 and (7. This general equation has the standard form for daming where the acceleration is reduced by a term roortional to velocity. A unique feature of MPM exloited here is that there are two velocities current article velocity (v (n (n and the velocity extraolated from the grid to the article (v g!. In well behaved simulations, these two velocities remain close such that the standard udate roceeds by FLIP methods. When MPM encounters noise (such as kinetic energy injected into the system by roagating an existing crack, that noise is sometimes reflected in velocity variations within a background cell that can cause differences between these two velocities. PIC daming can be viewed as daming out such local velocity errors. Another advantage of viewing PIC as a daming mechanism is that it rovides guidance for how to modify the osition udate as well. All current MPM codes use first order FLIP osition udate of x (n+ = x (n + v (n+ g! t (9 When Stomakhin et al. [4] added artial PIC udates, they did not change this osition udate, but 5

6 they only used 5-0% PIC and may not have needed a change. When using more PIC daming, it is vital to modify the osition udate as well. By integrating v (n+ g! dt from 0 to t using the midoint rule, a general second order FLIP osition udate is derived as: x (n+ = x (n + v (n+ g! t a (n g! + PIC v (n v (n g! ( t (0 Figure shows the results of cutting simulations after the onset of chi curling when using full PIC daming ( = 0 but varying the osition udate method. The three osition udates were the standard MPM osition (0 th order or Eq. (9, the modified udate in Eq. (0 to st order (which includes the PIC term but not the a (n g!, and the full nd order Eq. (0. This figure lots glyhs at each material oint that were deformed from their initial square to a arallelogram defined by the deformation gradient on the article. A good simulation should fill sace with no gas. Clearly, when PIC daming is used to alter the velocity udate, the osition udate must change as well. This fact is shown by the very oor results when using 0 th order. When PIC daming is used, a first order udate in not enough. This observation is illustrated by the st order diagram in Fig.. By this method, the articles near the edges, which are the articles with the most deformation, do not udate well. All issues are fixed by the nd order udate where the deformed glyhs reveal highly accurate tracking of article deformations including those that are highly deformed on the edges of the curling chi (see Fig.... Steady State Cutting Simulations A goal of the simulations was to achieve steady-state cutting with chi curling; all simulations used the oen-source MPM software NairnMPM [5]. The cutting seed was set to m/sec. Alying this seed at the start of the simulation, however, caused inertial effects. To minimize these effects, the tool seed was gradually increased from zero to m/sec over the first 30% of the total simulation time. During this ram hase, the simulations was damed using a variant of a Nose-Hoover thermostat [6, 7]. This thermostat adds daming in a feedback mechanism based on total kinetic energy. One change made from a rior MPM imlementation [7] was to evaluate total kinetic energy from grid masses and velocities rather than article masses and velocity. Like the PIC daming discussed above, the grid kinetic energy was less rone to velocity "errors" and gave more reliable daming. The feedback daming was turned off once the tool reached full seed. The PIC daming, however, was left on throughout the entire simulations and used = 0. The intent of the simulation was to model a thin cut off a bulk material. The first aroach was to use silent boundary conditions that are meant to absorb stress waves and thereby mimic simulations of an infinite medium [8]. Unfortunately, these boundary conditions did not achieve their goal in the cutting geometry. Instead, both x and y dislacements on the bottom were fixed, but such conditions require sufficient deth to avoid edge effects. Figure 3 shows average cutting force as a function of the deth, dh, between the cutting lane and the rigid boundary conditions. As long as d 3, the forces were constant and thus all simulations used d = 3. To determine resolution required, the article size was varied. Figure 3 shows average cutting force as a function of article size for deth of cut of 0.9 mm. The cutting force continued to decline. As a comromise for simulation time and resolution, all simulations used 5 articles through the thickness of the chi, which corresonds to 60 µm article when deth of cut is 0.9 mm. Combining all simulation strategies, Fig. 4 shows horizontal,, and vertical, F t, force of the tool on the cut material. The forces were determined by summing the momentum changes imosed by the contact algorithm on the rigid tool [3]. The cut material had K = 980 MPa, G = 376 MPa, y0 = 5 MPa, E T = 00 MPa, and density = g/cm 3. The cubic cohesive law had G c = 000 J/m, 6

7 articles in chi Cutting Force (N 6.4 Cutting Force (N 7 6 Deth of cut = 0.9 mm Base Deth (d Particle Size (µm Figure 3: A. The cutting force as a function of deth of material below the cut lane. B. The cutting force as a function of article size used in the MPM discretization. The deth of cut was 0.9 mm Cutting Force (N F t Time (ms Figure 4: The cutting force, and negative of the transverse force, F t, as a function of time for deth of cut of 0.9 mm and frictionless contact. All other material roerties are listed in the text of the aer. The steady state forces were measured by averaging the forces in the constant regime (e.g., t > 4 ms. 7

8 t = ms t = ms t = 3 ms t = 4 ms t = 5 ms t = 6 ms t = 7 ms t = 8 ms Figure 5: Snashots of a cutting simulation showing initiation and formation a a comlete chi. The shades of gray show equivalent stress from low (lightest gray = 0 MPa to high (black = 50 MPa. The simulation details are given in text of the aer. The steady-state cutting rocess started at t = 4 ms. c = 40 MPa, and c = mm. These material roerties have low strain tensile modulus of E = 000 MPa and are similar to roerties for olyethylene. The deth of cut was 0.9 mm and the contact was frictionless. The total times for such simulations varied from about hour to 4 hours deending on rocessor seed and number of rocessors used in arallel calculations. The forces initially oscillated during the ram u hase, but once reaching constant seed, the cutting roceeded by steady state conditions. The forces reorted for all simulation results are an average of the forces within the steady state region. These tyical curves show stable and noise-free cutting forces. Simulations without PIC daming had similar forces, but higher noise and frequent instabilities. In other words, PIC daming stabilized the results without changing the outut forces. Figure 5 shows sna shoots of the cutting rocess with shades of gray indicating the equivalent stress. 3. Analytical Modeling Whether one views analytical models as a tool for verifying the simulations or vice versa, simulations were comared to existing and revised analytical models for cutting that include fracture energy in the crack roagation lane. Figure 6A defines the cutting forces. and F t are forces alied by the tool and corresond to forces measured by a tool instrumented with a biaxial force gage. The tool contacts both the chi being removed and the material below the tool resulting in normal force N and rubbing force F b n. If the contact is undergoing frictional sli by Coulomb friction, these normal forces induce shearing forces S = µn and F b s = µf b n where µ is the coefficient of friction. The friction law follows: S =( µf b n sin +(F t F b n cos = µn = µ ( µf b n cos (F t F b n sin ( which can be rearranged and solved for F t : F t = Z +( µzf b n where Z = µ tan + µ tan 8 (

9 α A N B h S F t Φ F t b F n b Figure 6: A. The cutting force,, and transverse force, F t, that would be measured with an instrumented tool having rake angle. These forces can be resolved into normal forces on the cut material (N and Fn b and corresonding shear forces when contact has friction (S and Ft b. The direction of all force arrows indicates a ositive force in the modeling equations. B. When the tool touches the crack ti, the chi is fully lastic with a sli lane angle. Equation ( has several uses. In many analytical models, it is used to eliminate F t and thereby derive a rediction for alone, but this use does not eliminate F b n. As a consequence, many textbook models of cutting simly ignore the rubbing forces. This aroach has two roblems. First, simulations show that rubbing forces are non-negligible. Second, when F b n is ignored, Eq. ( redicts that F t is roortional to, but that rediction disagrees with exeriments [4, 8]. Williams et al. [6] noted this issue and added an adhesion term to the friction law (i.e., S = G a + µn. Although an adhesion term is easily imlemented in simulations, all simulations here assumed Coulomb friction, which means rior models had to be revised to include rubbing forces before comaring to simulations. When those forces are needed in a model, Eq. ( can be used to determine F b n from simulations results for and F t. Figure 6B shows a tool touching the ti with comlete yielding of the chi at sli lane angle. Atkins [] suggests a total energy balance that accounts for fracture energy in the cutting rocess (which was modified here to include rubbing forces: dx = y sin (hbdx+s cos( dx + F b s dx + G c bdx (3 where dx is an increment of crack advancement, y is shear yield strength, is shear strain, h is cut deth, b is width, and G c is toughness. The first term on the right is lastic work. The shear yield strength and tensile yield strength are related by y = y /Y, where Y = for Tresca yield criterion or 3 by yield criterion in Eq. (. By geometry, the shear strain is = cot + tan( []. The remaining terms are frictional work on the chi, on the rubbing surface, and fracture work, resectively. Eliminating S, F b s and (by minimizing cutting force [6], the analytical model becomes: b = G c + µf n b b +»! yh Y µg Z + + Z c sec + (4 Y h y (cos + µ sin The last term in the square root differs from rior models [, 6], but those differences are small. If contact is frictionless, the result is articularly simle (linear in h and indeendent of rubbing forces: b = G c + yh (sec tan (when µ = 0 (5 Y 9

10 A M/(bh σ y A k 0 Figure 7: The curve shows moment-curvature relation for a beam with linear hardening. A is the area between the loading curve and the elastic unloading curve. A is the area between the alied moment (the horizontal dashed line and the momentcurvature relation. When the chi is not fully lastic (or is elastic, the tool will not touch the crack ti and the modeling needs a lastic bending analysis. Williams et al. [6] derives such an analysis resulting in: bg c ( sin F t cos = + A (k 0 (6 bg c bg c F t bg c y0k 0 E 0 = A (k 0 bg c (7 where = 0.64 is a crack root rotation correction, E 0 is lain strain modulus, A (k 0 and A (k 0 are areas associated with the moment-curvature (M-ale relation for bending the chi (see Fig. 7, and k 0 = ale/ale y where ale y = y0 /(he 0 is the curvature at the onset of yielding. Although Williams et al. [6] ignores rubbing forces, they can be included by using Eq. ( to give: where k 0 is found by solving A (k 0 y0k 0 = + bg c E 0 ( sin Z cos F + A (k 0 + ( µzf b n cos c bg = c bg c bg c sin Z cos Ç Z + ZA (k 0 + ( µzf å b n ( sin bg c bg c (8 (9 Williams et al. [6] analyzes elastic-lastic materials. The analysis can be extended to hardening materials by finding areas A and A from moment-curvature relation for a hardening material. For the linear hardening law mentioned above, the moment curvature relation is derived to be: 8 < 6 bh k 0 y0 k 0 ale Å M = : 4 bh y0 + E ã (k0 + k 3E 0 k 0 > 0 3k 0 (0 0

11 40 30 α=5 α=5 /(b G c 0 α=30 α= σ y0 h/( E G c Figure 8: The simulated, dimensionless cutting forces (symbols for an elastic-lastic lastic material with frictionless contact for various rake angles and as a function of dimensionless deth of cut. The dashed lines are analytical models for the cutting simulations. The required areas for k 0 A (k 0 b A (k 0 b are: = h y0 E 0 = h y0 E 0 + E 3E 0 + E 3k 0 3E 0 (k 0 + k0 E ñ E 0 k 0 + Ç 4 k 0 åô (k 0 + k 0 k 0 åç + 3k 0 ( ( For k 0 <, A /b = h y0 k 0 /(6E0 and A = 0. This revised analysis reduces to Williams et al. [6] for both elastic (k 0 < and lastic (k 0 bending when F b n = E = Results The symbols in Fig. 8 show cutting force as a function of deth of cut for various rake angles and for an elastic lastic material (E T = 0 and other roerties same as in the Steady State Cutting Simulations section with frictionless contact (µ = 0. All lots in this aer use the scaling suggested by Williams et al. [6] for dimensionless forces (F/(bG c and dimensionless deth of cut (h y0 /(E0 G c. For the roerties used here, dimensionless deth of cut was 0.39h or actual cuts ranged from 0. mm to 7 mm. At low deth of cut, the simulations linearly increased in force. At some critical deth of cut, which deended on rake angle, the cutting force droed dramatically to a low value and thereafter varied slowly with deth of cut. The dramatic dro corresonded to a transition from the tool touching the crack ti to lastic bending with the tool removed from the crack ti and that transition occurred in a very narrow deth range. For examle, Fig. 9 shows the abrut change from touching to not touching

12 .0 mm. mm Figure 9: The tool ti/crack ti relation for an elastic-lastic lastic material with frictionless contact for rake angle = 30 during steady-state cutting. The left side is for deth of cut of.0 mm while the right side is for. mm. between deth of cuts.0 and. mm (or 0.78 and 0.9 in dimensionless units when = 30. The black regions indicate lastic yielding and thus the transition corresonded to lastic collase in the chi to incomlete yielding. Simulations with the tool touching the crack ti were challenging and needed to resolve comlex contact scenarios around the crack ti. These simulations were not stable until the addition of PIC daming, but with that daming, MPM rovides a good numerical tool for roblems involving crack-ti contact. The dashed lines in Fig. 8 show redictions by analytical models, but that modeling does not redict the transition from touching to lastic bending. For comarisons, the modeling curves used the crack ti touching model (Eq. (5 for µ = 0 u to deth of cut where simulations showed transition to lastic bending and thereafter used the lastic bending model. The lastic bending analysis needed inut of normal rubbing forces; the simulation results for this force are lotted in Fig. 0. The rubbing forces were low during tool ti touching (the increase for very thin deths of cut is discussed below, but increased raidly just before the transition to lastic bending. In the lastic bending regime, the rubbing forces were roughly linear with deth of cut and increased significantly with rake angle. The lastic bending calculations in Fig. 8 used the linear fits to F b n indicated in Fig. 0. The lastic bending model has similar trends to simulations results, but does not agree as well with simulations as does the crack ti touching model. One issue is how to redict the transition between cutting regimes? Here the transition from touching to bending was determined by simulations. One aroach used in analytical modeling is to consider both models and assume the failure mode is the one with the lower force [6]. This aroach disagrees with simulations. The dotted line extension for the = 30 simulations shows the bending model extended to lower deth of cut. At all deths of cut simulated, the minimum analytical model force is the bending model. The simulations showed, however, that for thin deths of cut, the tool reached the crack ti causing the force to be much higher than the bending model. Figure shows the effect of coefficient of friction for an elastic-lastic material (E T = 0 and rake angle of = 30. The symbols are simulations and the dashed lines combine the touching model in Eq. (4 with a lastic bending model. The rubbing forces needed by the analytical models were taken from simulation F b n as a function of deth of cut. The simulations and analytical model agree well for all values of coefficient of friction for the initial region corresonding to crack ti touching. The transition to lastic bending occurred at thicker cuts when friction was added. In the lastic bending

13 . F n b /(b Gc α=45 α=30 α=5 0. α= σ y0 h/( E G c Figure 0: The simulated, dimensionless rubbing forces (symbols for an elastic-lastic lastic material with frictionless contact for various rake angles and as a function of dimensionless deth of cut. The solid lines are linear fits to the rubbing forces for dimensionless deth of cut greater than 0.8. regime, the models were again below the simulation results, but had similar trends for friction effects. No lastic bending mode for µ = 0.6 is shown, because the analytical model does not work when Z > 0 or µ>tan = In dynamic frictional sliding, the tangential stress is S = min(s 0, µn where S 0 is shear force corresonding to stick conditions and µn is shear force during sliding. The simulations can combine sli and stick frictional contact while the analytical model assumes S is always µn, and thus only alies during comlete frictional sliding. All simulations with an elastic-lastic material showed a large and raid transition from tool ti touching to lastic bending regime, but no such transitions are seen in exeriments on real materials [4, 8]. Either the simulations are unrealistic or an elastic-lastic material does not reresent real materials well. To test for the later hyothesis, simulations were run for frictionless contact at a rake angle of = 30 as a function of the hardening modulus, E T ; the results are in Fig.. Increasing E T shifted the transition to lower deth of cut and significantly reduced the magnitude of the transition. By the time E T reached 5 MPa, which is only.5% of the elastic modulus of E = 000 MPa, the force dro at the transition was nearly gone. Realistic modeling for cutting of most materials will need to account for hardening roerties of those materials. When E T = 00 MPa, all simulations were in the lastic bending regime. Figure 3 comares simulations (symbols to lastic bending theory (dashed lines for E T = 00 MPa, frictionless contact, and various rake angles. The rubbing forces needed as inut for the lastic bending theory, were found from simulations results. The simulations and bending theory have similar trends. The cutting force increased as the rake angle decreased and it increased raidly for thin cuts, but then leveled off for deeer cuts. The differences are that the simulation forces are 0% to 90% higher then the bending model and simulation forces continued to rise while the model redicts a maximum in force followed by a decrease for dee cuts. Although either the simulation or the bending model may be wrong, an alternative exlanation is that they are both correct but are solving different roblems. The analytical modeling was based on small-strain, one-dimensional, elasticity theory and crack growth used fracture 3

14 μ=0.6 μ=0.3 /(b G c μ= σ y0 h/( E G c Figure : The simulated, dimensionless cutting forces (symbols for an elastic-lastic lastic material as a function of the coefficient of friction for rake angle = 30 and as a function of dimensionless deth of cut. The dashed lines are analytical models for the cutting simulations. /(b G c E T =5 E T =0 E T =5 E T =0 E T =50 E T = σ y0 h/( E G c Figure : The simulated, dimensionless cutting forces (symbols for a linear hardening material as a function tangent modulus (E T for rake angle = 30 and as a function of dimensionless deth of cut. The dotted lines connect the oints and are not analytical modeling results. 4

15 6 5 4 α=5 α=30 α=45 /(b G c 3 α=45 α=30 α= σ y0 h/( E G c Figure 3: The simulated, dimensionless cutting forces (symbols for a linear hardening material with E T = 00 MPa for various rake angles and as a function of dimensionless deth of cut. The dashed lines are analytical models for the cutting simulations. mechanics with a fixed value of G c. The simulations used large-strain, two-dimensional lasticity theory and crack growth used a cohesive law with a fixed value of G c but otentially variable cohesive stress ( c or shae of the law (other than cubic. Because simulation shear strains reached 80%, one should exect differences between small-strain and large-strain material models. Perhas the force eak in the analytical model is an artifact of small-strain assumtions? A similar eak was never seen in simulations even when carried out to very dee cuts. The 0% to 90% difference in magnitudes may be due to fracture modeling. Figure 4 shows simulations comared to modeling for E T = 00 MPa, frictionless contact, rake angle = 30, fracture toughness G c = 000 J/m, but with different values for the cohesive stress. The simulation results were above or below models by changing the cohesive stress. Additional simulations looked at varying the cohesive law from a cubic law to a triangular law and also showed differences in magnitude and curvature of the results. Clearly, it would be ossible vary the cohesive stress and shae of the cohesive low to achieve an exact match between theory and modeling. Figure 5 shows the effect of cohesive stress on simulations and modeling for both touching and lastic bending; these simulations reeated the simulations in Fig., but changed cohesive stress from c = 40 MPa to c = 0 MPa The lower cohesive stress imroved the agreement in the lastic bending regime and retained the good agreement of the touching regime. A significant affect of cohesive stress is that it reduced the thickness need to convert to lastic bending. In summary, simulation results were sensitive to cohesive law details. Although failure was always simulated as occurring at a constant total J c, the outut did rovide the mode mixity at failure. Figure 6 lots scatter diagram of mode I ercent for all elastic lastic simulations in Figs. (8 and ( as a function of Z, which was a good indicator of mode mixity. The solid symbols are for all results in the lastic bending regime, where failure modes were always more than 90% mode I and indeendent of Z. The sread of the symbols was caused by slow decrease in mode I character as the deth of cut increased. The oen symbols are for touching mode and they were correlated with Z. Increasing Z by either decreasing rake angle or increasing friction both caused similar decreases in mode I character. The mode I character was also affected by the cohesive stress. 5

16 5 4 σ c = 40 /(b G c 3 σ c = 0 σ c = 0 Theory σ y0 h/( E G c Figure 4: The simulated, dimensionless cutting forces (symbols for a linear hardening material with E T = 00 MPa a function of the cohesive stress ( c for rake angle = 30 and as a function of dimensionless deth of cut. The dashed line is the analytical model for the cutting simulations μ=0.6 0 /(b G c 5 0 μ=0.3 5 μ= σ y0 h/( E G c Figure 5: The simulated, dimensionless cutting forces (symbols for an elastic-lastic lastic material with cohesive stress c = 0 MPa as a function of the coefficient of friction for rake angle = 30 and as a function of dimensionless deth of cut. The dashed lines are analytical models for the cutting simulations 6

17 00 J I /J c (% Touching Bending Z Figure 6: The fraction mode I content in the released energy during steady state cutting as a function of Z. The filled symbols are for simulations in the lastic bending regime. The oen symbols are for simulations with the tool touching the crack ti. Figure 7: Snashots of a long-time cutting simulations for deth of cut 0.5 mm, rake angle = 5, and hardening materials with E T = 00 MPa. The arrow indicates the direction of cut. As c decreased from 40 MPa to 0 MPa, the mode I character of the lastic bending regime decreased from over 90% to about 75%. A goal of these simulations was to model steady-state cutting including chi curling, when it occurs. Figure 7 show a long-time simulation in the lastic bending regime (E T = 00 MPa for rake angle = 5 and deth of cut of 0.5 mm. The simulations modeled chi curling well. The modeling of chi curling requires dynamic modeling of self contact as the chi curves over and reaches the material surface. Fortunately, MPM automatically models self contact, but this free contact modeling can only model stick contact conditions. The modeling of frictional sliding, as done between tool and the cut material, uses different multimaterial methods that are caable of modeling friction sliding. For this secific roblem, self contact by stick is sufficient because as their chi curls, there should be little relative motion between the layers of the chi. The simulations described here were robust and able to vary many arameters, thus roviding otential for many future alications. Two challenges for future work are simulating very thin cuts 7

18 and simulating tool sharness effects. All successful simulations here were for deths of cut 0. mm or higher. Because exeriments extend to thinner cuts [4, 8], it would be useful to simulate that regime. The increase in rubbing forces for thin cuts (see Fig. 0 may be onset of an artifact that causes instability. One ossibility for the instability may be the cohesive law methods. The cohesive law used had a critical oening dislacement of c = mm, which is similar in magnitude to the thinnest cuts ossible. Perhas alternative fracture mechanics methods are needed to simulate very thin cuts? Modeling of tool sharness requires changing the discretization of the tool and enough articles to resolve the radius of curvature at the tool ti. Setting u such a simulation is easy, but the first simulations attemts did not work well. The main issue is likely finding accurate normals around the tool ti. With a shar ti, the normals could be exactly secified, but with a blunt ti, they need to calculated within the simulation. Perhas modeling of tool sharness effects requires further imrovements in MPM contact methods? 5. Conclusions This aer shows that the material oint method (MPM is caable of robust simulations of orthogonal cutting that include extensive amounts of tool advance such that the simulations reach steady-state cutting rocesses with chi formation and curling. The develoment of stable MPM simulations required some imortant MPM changes. The most imortant features were high-strain material models, accurate modeling of tool shae and its contact normals, and the use of PIC daming. The simulations modeled tool advance by ductile fracture mechanics with a cohesive zone along the crack roagation ath. Although cohesive zone laws are common in numerical modeling of crack roagation, they may not rovide a fundamental fracture mechanics analysis. Cohesive zones were used here because they heled control roagation and not because they were judged the best way to model crack roagation. The simulation results showed that some results (i.e., lastic bending regime are sensitive to the shae of the cohesive law even when total toughness is constant. All numerical modeling should recede with caution when interreting results that rely on cohesive laws. Desite concerns over use of cohesive laws, the MPM simulations rovide a robust tool for studying new cutting roblems that are beyond the caabilities of analytical models. Two examles are veneer eeling of logs and laning of wood. These roblems require new material models and accounting for comlex geometries in the tool set us. Acknowledgement This material is based uon work suorted by the National Institute of Food and Agriculture, United States Deartment of Agriculture, under McIntire-Stennis account #986, roject #OREZ-WSE-849- U. References [] A. Atkins, Modelling metal cutting using modern ductile fracture mechanics: Quantitative exlanations for some longstanding roblems, International Journal of Mechanical Sciences 45 ( ( [] A. Atkins, Rosenhain and Sturney revisited: The tear chi in cutting interreted in terms of modern ductile fracture mechanics, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 8 (0 (

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