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On a Possible Mechanism of Shear Deformation in Nanoscale Cutting
On a Possible Mechanism of Shear Deformation in Nanoscale Gutting Toyoshiro Inamura’, Nobuhiro Takezawal, Yasuhiro Kumaki’, Toshio Sata Nagoya Institute of Technology, Gokiso-cho, Shouwa-ku, Nagoya 466, Japan * Toyota Technological Institute 2-12-1, Hisakata, Tempaku-ku, Nagoya 468, Japan Received on January 13,1994
Summary
Based on the method of transformation from an atomic model to a corresponding continuum model. the stress and strain distributions in iianoscale cutting ha.ve been evaluated. The results show that a workpiece is subjected to concentrated compressive and shear strain at the primary shear zone, though the area along the rake face of the tool is strained tensilely. The results also show that the interior of the workpiece is: however, exposed to high, almost constant compressive stress. -4possible mechanism of these different stress and strain distributions is discussed as well as its interpretation on a macroscale. Keywords: Simulation, ?YIicromachining,Cutting
Introduction Atomic-model-based simulation of nanoscale cutting has been carried out to analyze the mechanism of nanoscale cutting[2]-[8]. However, the purpose of studies in this approach by the present authors is not only to clarify the mechanism of the nanoscale cutting itself, but also to investigate the relationship between the mechanics of microscale cutting and that of macroscale cutting. In relation to the latter, the authors believe that some of the apparent differences between the mechanism of nanoscale cutting and that. of macroscale cutting can be attributed to the differences in viewpoints between microscale and macroscale for a single event. If this is true, studies on microscale cutting can serve to offer us new microscopic ltnowledge on the fundamental mechanism of cutting which hitherto has been studied only from the niacroscopic point of view. This paper is a continued report on the mechanics of nanoscale cutting examined on the basis of the above idea. In our previous paper[6], we reported a method of ‘equivalent’ transformation from an atomic model to a corresponding continuum model. This enables us to evaluate stress and strain distributions in nanoscale cutting which itself is simulated using atomic models of tool and worlcpiece such that it cannot produce such distributions by itself. The result thus obtained was, however, an unexpected one with regard to the stress distribution such that there was almost no concentrated shear stress in the primary shear zone and that maximum shear stress was considerably lower than the theoretical yield shear stress of the workpiece material. In this report, after a brief explanation of the physical meanings of the transformation method, we esamine the details of the stress and strain distributions in nanoscale cutting together with the microscopic mechanism of deformation which produces the result mentioned above. Then we also discuss, based on the Prandtl-Reuss formula, how this mechanics is ‘viewed’or ‘interpreted’ in the framework of the plastic deformation law on a macroscale.
At omic/cont inuum transformation The method of ‘equivalent’ transformation froiii an atomic model to a corresponding continuum one should be mentioned briefly. For details of the method, tlie reader should refer to [6]. The term ‘equivalent’ means that total potential energy within an arbitrary domain in an atomic niodel is equal to strain energy within tlie same doniain in tlie corresponding continuum niodel. Under this condition, displacement u(rq)of atom q at position rq = (zply,,), for q = 1,2,. . -,and displacement n(rp)at rp = (z,], yp) in the continuum niodel are assumed to be related to each other by Annals of the ClRP Vol. 43/1/1994
4Tp)
=
J,
4 - p 1
r*)4rq)dr,,=
.z W(Tp1 Q ’
Tq)’lL(T,,)
I
(1) where w(rprT,,)is a weight function. the value of which is always positive and satisfies Cw(r,, T,,) = 1 in the do9
main SZ. This equation (1) implies that the displacement u(rP)at r p in the continuum model is given as a weighted mean of displacements u(rq)of the surrounding atoms in the doniain R whose center IS at rp., This relation will be justified from various points of view; it represents a general forin of interpolation which is obviously needed when the point rp is not at an atom position; it represents a process to average out atomic random vibration and to filter strain energy; it n!eans that the value at a specified point in the continuum model should be regarded as the expected value for an atom at that point under the condition that atoms take their positions according to some probability distribution. Although it seems that each of the above interpretations gives some clues for determining the proper shape of a weight function; the problem is still unsolved as yet. In this paper, we merely use a twodimensional normal distribution function with the variance a2= a2 (‘a’ represents the lattice constant), taking the radius of the circle domain R to be 3a = 3a. Then the strain components can be obtained, according to the standard definition, by differentiating the resulting displacement n(r,) with regard to the X and Y coordinate values. The above equation (1) combined with demand for energy conservation between the two models gives the following relation between the stress components flij ( T ~ at ) rp in tlie continuum model and the stress components aij(r,,)of atom q at T,,: (2) where the atomic stress coniponents az,(r,,)are defined as those acting on the surface of the atomic volume associating with the atom q. It should be noted that the stress components f l j , ( r P )in the continuum model are the values associating with the sbecified point r,,! whereas the stress components aij(rg)are tlie values associating with the atom q whose spatial position does not have a deterministic meaning. Then equation (2) implies that tlie stress components aij(rP)are the espected values for tlie atom q whose stochastic position is represented by the r P ) . These atomic stress compoweight function w(rq1 nents aij(rq)can be evaluated by applying Born’s formula to the atomic arrangement concerned[l]. To convert u(r,,) and aij,(r,,)into ;ii(rp)and r ; . ( r) respectively, we first discretize tlie continuum iiiocfel atj(rq) =
w(rq1 rp)atj(rp)drp
1
&
47
.. .. .. ..... (a) ma>cinium principal strain (incremental). I
(a) before a specified tool advancement.
.... .. .....
i
(b) niaviinum principal stress. Fig2 Direction and inagnitude of the mzcviinum principal stress and strain in the workpiece during cutting. (b) after a specified tool advancement. Fig. 1 Simulated deformation of the workpiece crystal. the finite-element method and compose the weight function to fit the cliscretized field. This procedure gives the discrete forms of equations (1) and (2)’ which are written with matrices and vectors. Then ii r,) can be obtained by a sirri le matrix-vector multip ication derived from equation (17, while ~ ‘ ( r ,can ) be obtained by solving a linear equation derived from equation (2). For details of the procedure, the reader should refer to [i].
\
Stress/St rain analysis The atomic-model-based simulation carried out in this study is the same as that reported in our previous paper[6]. The deformation of the atomic model during cutting is shown in Figs.1 (a) and (b) as that before and after a certain advancement of the tool. In these figures, lines are drawn to indicate atoms at the same height before cutting. These figures show that atoms under the relief face of the tool undergo severe shearing distortion, and that distortion of the atomic arrangement in the socalled ‘primary shear zone’ does not show simple shearing action. The latter is especially so in the deformation around the tool tip, where the atomic arrangement seems to be distorted in some unstable manner. It is also seen in these figures that atoms in the uncut area begin to swell up. However, the mechanism which causes these deformations is not clear in these figures. The results transformed into the values in the corresponding continuum model are shown in Fig.2 as to the maximum incremental principal strain distribution in (a) and the corresponding mnxinium principal stress distribution in (b). The incremental strain has been computed based on the results in Figs.1 (a) and (b), while the results in Fig.:! (b) have been obtained based on Fig.1 (a). In these figures: short lines are drawn so that their directions and lengths represent the directions and magnitudes of the principal stress and strain values, respectively. It is seen in Fig.2 (a) that the workpiece material is subjected to high compressive strain around the ‘primary shear zone’. However, it is remarliable that the area
48
along the rake face of the tool is strained tensilely so as to hamper the motion of the workpiece atoms. This is due to the adhesive force between the tool and the worlipiece. On the other hand, the stress distribution in Fig2 (b) is somewhat different from what is expected from the strain distribution in Fig.2 (a). In Fig2 (b), the most noticeable features are ‘eyelike’distributions found in the uncut area as well as in the area just ahead of the tool tip and below the finished surface. These unusual stress distributions seem to be a symptom of buckling deformation initiating in these areas. It is also seen in this figxire that the stress distribution in the ‘primary shear zone’ is very complex! being composed of the combined compressive components of the stress nearly in the direction of the inaximuni principal strain in Fig.2 (a), which is due to the force acting so as to push up the chip and that in the horizontal direction, wliich.is due to the force from the tool. The magnitude of the compressivestress coinponeiits is almost the same everywhere and is about IOGPa. which is very high: though the value itself depends on the weight function used. The difference between the stress and strain distributions observed above is also seen in the case between the maximum shear strain distribution in Fig.3 (a) and Fig.4 (a) and the maximum shear stress distribution in Fig.3 (11) and Fig.4 (b). Among these figures, Figs.3 (a) and (b) show the directions of the strain and stress values, respectively, while Figs.4 (a) ancl (b) sIio\v their magnitudes. It is seen in Fig.3 (a) ancl Fig.4 (a) that the ’primary shear zone’: which appears in Fig.4 (a) as a highly strained area indicated with dark shading, is, in fact, subjected to shear strain not in the direction of the shear zone but in the horizontal direction. Then, from there to the uncut area, the direction of the shear strain split into two flows, one going upward to the free surface and the other one going downward. This change of flow direction seems to cause a. lunip or a wavy pattern on the free surface in the uncut area. On the other hand, the maximum shear stress in Fig.3 (b) and Fig.4 (I)) eshibits a very complicated and alinost unexpected result. The most striking feature of Fig.4 (b) is that there is almost no concentrated shear stress in the ‘primary shear zone’. The magnitude of the stress value around there is around 1-3GPa and is
I
(aj nnsiinuni shear strain (incremental j
i
1
(a)maximum shear stra.in (incremental).
i
(bj maximum shear stress.
(b) maximum shear stress.
Fig.3 Direction of maxiinum shear stress and strain in the workpiece during cutting.
Fig.4 Magnitude of the maximum shear stress and strain in the workpiece during cutting.
lower than the theoretical yield shear stress, though it is higher than the value observed in macroscale cutting. In addition, the result in Fig.3 (b) shows that the direction of the maximum shear stress chmges in a very complex manner. This result, of course, corresponds to that ofthe maximum principal stress in Fig2 (b) a i d it should be reiterated that the distortion of the flow pattern in Fig.3 (b) is related to the formation of an ‘eyelike‘ pattern in Fig2 (b). Then the apparent difference between the stress and strain distributions found in a wide area in Fig.2, Fig3 and Fig.4 seems to be attributable to the anisotropic property of the monocrystal copper used in this study. Indeed, the theory of plastic deformation of a monocrystal states that plastic deformation is caused by shearing which occurs when one of the shear stress components resolved in the directions of slip planes of the crystal structcre exceeds a critical value. However, it is not easy to ascertain whether or not this theory holds in this case because the directions of slip pla.iies are not clear for the distorted atomic arrangement in Fig.1 and because the direction of the maximum shear strain in Fig.3 (a) cannot be guaranteed to represent the direction of the slip plane. On the contrary, it rather seems that the above theory does not hold in the primary shear zone because the theory demands. in any case: a concentrated shear stress component in the direction of the slip plane in this area, but because there are no concentrated stress coniponeiits either with regard to tlie iiiaximuiii principal stress or with regard to the innsinium shear stress, there is no mechanism to produce that concenbrated shear stress component in that area. To ascertain the valiclity of this rather negative coiijecture, we focus on the angle 0 between the direction of the mayimum shear stress 7, am1 the direction of the maximum shear strain Y,,~. Because the constitutive equation on a macroscale insists that the angle 0 should be zero and because the theory of plnstic cleforination of a monocrystal also requires a sufficient value of the shear stress component in tlie direction of the , means maximum sheas strain, the value 0 = ~ / 4 which a shear defoiinatioii without any s h e x stress, generally cannot occur, or if it does, cannot produce a large value of ( ~ ~within ~ 7 the ~ framework ) of these theories. Then we have computed the product value: ( T , ~ ~xT f~( 0~)~by)
taking the function f ( 0 ) to be in the form as shown in Fig.5. The result is also shown in Fig3 with the iiidicatioii of the area. of high values with dark shading. This figure shows that.: in fact, the primary shear zone includes many such dark areas of ‘straiige’ mechanics. Althoiigh the detailed discussion of the mechanism of this ’strange’ mechanics must be left to future studies, it seeins that the mechanism is related to a kind of buckling incluced by instability due to severe compression because we can see a dark spot also in the uncut area where budding deformation is clearly developing due to the compression. The above discussion shows that the micromechanics of shear deformation in nanoscale cutting is not simply what we expect. Next we exaniine whether or not this mechanics produces unreasonable results in the franiework of the plastic deformation theory on a macroscale. If it. does not produce apparently unreasonable results. then we examine how the mechanics is ’viewed’or ‘interpreted‘ on a macroscale. To this end, we apply the Plandtl-Reuss formula to our case to compute the increinented plastic strain components ckfj and the incremental elastic strain components d&fj as follows:
where aij is stress deviation, 5 effective stress, & effective incremeiital strainl d&ij total incremental strain, all of which are given based on the results of simulation. The above ecluakions (3) and (4) are, of course, valid only for areas of plastic deformation. so we first have distinguished those areas from areas of elastic deformation by pulling bad; tlie tool a little and by checking whether or not the incremental strain diminishes. Then if the niechanics in nanoscale cutting looks unreasonable when viewed through tlie macroscale law: the strain distributions computed on the basis of equations (3) and (4) will become unreasonable ones. Because it is not easy to see this in the form of strain, we have restored the plastic and elastic deforiiiations based on these strain components, respectively, and have shown the results in Fig.G, where areas of elastic deformation are restored based on the strain coniponents given by Hoolie’s law. In this figure, the total deformation restored on the basis of the
49
Conclusions
Fig.5 Area of 'strange' mec1ia.nics where shear deformation occurs with almost no shear stress. (area with dark shading corresponds to those areas.)
(1) A method of transformation froni an atomic model to an eqi.iivalent continuum model is presented together with its various interpretations. i2) Workpiece material is subjected to concentrated compressive and shear strain at the primary shear zone during cutting. However, the area along the rake face of a tool is subjected to tensile strain due to the adhesive force between the tool and workpiece. (3) Stress distribution in the workpiece during cutting is very comples with regard to its direction, indicating the initiation of a kind of buckling deformation. The interior region of the workpiece: including the area of primary shear? is not exposed to any concentrated shear stress but esposed to relatively constant and high compressive stress. (4) The difference between tlie stress and strain distributions in the workpiece during cutting cannot be explained either by the theory of plastic deformation on a macroscale or by the theory of plastic deforniation for a monocrystal. The mechanism of deformation at tlie primai-y shear zone seems to be related to bucliling due to severe compression in that area. ( 3 ) The deforination of the worlipiece at the primary shear zone is nothing but shear plastic deformation by a yield shear stress, when viewed in the framework of tlie Plandtl-Reuss formula. This may be the result of tlie fact that a phenomenological law on a macroscale will include a variety of microscopic mechanisms of deformat ion.
Acknowledgment Fig.6 Elastic and plastic deformations restored so as to hold the Plandtl-Reuss formula. total incremental strain is also shown with light shading. It is seen in Fig.6 that the concentrated plastic shear deformation does occur in the primary shear zone though tlie elastic and plastic deformations occur in almost opposite directions to each other, contrary to our observation on a macroscale. However, it can be said that the Plandtl-Reuss formula can produi:e considerably reasonable deforination, especially for plastic deformation in the primary shear zone. Then we have examined how the concentrated shear deformation in the primary shear zone is interpreted in the framework of the Plandtl-Reuss formula. Among tlie cluantities appearing in the right-hand side of equation (3), the stress deviation and effective stress r do not exhibit any concentration in that zone, as easily understood from the results in Fig.2 and Fig.4. Thus the origin of the strain concentration resides in tlie concentration of the effective incremental strain dr, which in turn is related to the effective incremental stress d r by da = HI&, with H' being the tangential slope of the stress-strain curve. Then because d r does not eshibit any concentration, the origin of the concentration is attributed to a. very small value of H', which means that the material can deform at some coilstant stress value. This is the phenomenon which we call plastic deformation caused by a yield stress. After all, the strain concentration at the primary shear zone is interpreted as ordinary shear plastic deformation in the framework of the Plandtl-Reuss formula, though the fact that the stress value in that area is almost the same as in the other area cannot be explained in this framework. This latter fact might be viewed such that the workpiece material in that area is 'weaker' than in the other uea.
50
The authors express their sincere gratitude to M.4TSUURA MACHINERY Co. and OSAWA FUND for their financial support during this study. This study is also supported by a. Chnt-in- Aid for Scientific Research froni the Ministry of Education, Science and Culture.
References
[l] Hashimoto, &I., Ishida, M., Doyama, M., 1984, Atomic
Studies of Grain Boundary Segregation in Fe-P and Fe-B Allovs-1. -4toniic Structure and Stress Distribution, .Act; hiIetal., 32, 1-27. 121 Hoover. W. G., Belali, J. F., DeGroot, 4.J., Hoover, C. G., Stowers, I. F., 1991, Molecular Dynamics Modeling of Indentation and Cutting, Thrust Area Report FY90, LLNL. [3] Ilmwa, N., Shimada, S., Tanaka, H., Ohmori, G., 1991. An Atomistic Analysis of Nanoinetric Chip Removal as Affected by Tool-Work Interaction in Diamond Turning, Annals of the CIRP, 40, l, 551-554. [4]Inaniura, T.,Suzuli. H., Talcezawa, N., 1991, Cutting Experiments in a Computer Using Atomic Models of Copper Crystal and a Diamond Tool, Int. J. JSPE, 25'4. 259-266. 1992, Atomic-Scale Cut[5] Inamura, T., Talcezawa, ?I., ting in a Computer Using Crystal Models of Copper and Diamond, Annals of tlie CIRP, 41, 1, 121-124. [GI Inamura, T., Talcezawa, N., Kunial
1
Summary
Based on the method of transformation from an atomic model to a corresponding continuum model. the stress and strain distributions in iianoscale cutting ha.ve been evaluated. The results show that a workpiece is subjected to concentrated compressive and shear strain at the primary shear zone, though the area along the rake face of the tool is strained tensilely. The results also show that the interior of the workpiece is: however, exposed to high, almost constant compressive stress. -4possible mechanism of these different stress and strain distributions is discussed as well as its interpretation on a macroscale. Keywords: Simulation, ?YIicromachining,Cutting
Introduction Atomic-model-based simulation of nanoscale cutting has been carried out to analyze the mechanism of nanoscale cutting[2]-[8]. However, the purpose of studies in this approach by the present authors is not only to clarify the mechanism of the nanoscale cutting itself, but also to investigate the relationship between the mechanics of microscale cutting and that of macroscale cutting. In relation to the latter, the authors believe that some of the apparent differences between the mechanism of nanoscale cutting and that. of macroscale cutting can be attributed to the differences in viewpoints between microscale and macroscale for a single event. If this is true, studies on microscale cutting can serve to offer us new microscopic ltnowledge on the fundamental mechanism of cutting which hitherto has been studied only from the niacroscopic point of view. This paper is a continued report on the mechanics of nanoscale cutting examined on the basis of the above idea. In our previous paper[6], we reported a method of ‘equivalent’ transformation from an atomic model to a corresponding continuum model. This enables us to evaluate stress and strain distributions in nanoscale cutting which itself is simulated using atomic models of tool and worlcpiece such that it cannot produce such distributions by itself. The result thus obtained was, however, an unexpected one with regard to the stress distribution such that there was almost no concentrated shear stress in the primary shear zone and that maximum shear stress was considerably lower than the theoretical yield shear stress of the workpiece material. In this report, after a brief explanation of the physical meanings of the transformation method, we esamine the details of the stress and strain distributions in nanoscale cutting together with the microscopic mechanism of deformation which produces the result mentioned above. Then we also discuss, based on the Prandtl-Reuss formula, how this mechanics is ‘viewed’or ‘interpreted’ in the framework of the plastic deformation law on a macroscale.
At omic/cont inuum transformation The method of ‘equivalent’ transformation froiii an atomic model to a corresponding continuum one should be mentioned briefly. For details of the method, tlie reader should refer to [6]. The term ‘equivalent’ means that total potential energy within an arbitrary domain in an atomic niodel is equal to strain energy within tlie same doniain in tlie corresponding continuum niodel. Under this condition, displacement u(rq)of atom q at position rq = (zply,,), for q = 1,2,. . -,and displacement n(rp)at rp = (z,], yp) in the continuum niodel are assumed to be related to each other by Annals of the ClRP Vol. 43/1/1994
4Tp)
=
J,
4 - p 1
r*)4rq)dr,,=
.z W(Tp1 Q ’
Tq)’lL(T,,)
I
(1) where w(rprT,,)is a weight function. the value of which is always positive and satisfies Cw(r,, T,,) = 1 in the do9
main SZ. This equation (1) implies that the displacement u(rP)at r p in the continuum model is given as a weighted mean of displacements u(rq)of the surrounding atoms in the doniain R whose center IS at rp., This relation will be justified from various points of view; it represents a general forin of interpolation which is obviously needed when the point rp is not at an atom position; it represents a process to average out atomic random vibration and to filter strain energy; it n!eans that the value at a specified point in the continuum model should be regarded as the expected value for an atom at that point under the condition that atoms take their positions according to some probability distribution. Although it seems that each of the above interpretations gives some clues for determining the proper shape of a weight function; the problem is still unsolved as yet. In this paper, we merely use a twodimensional normal distribution function with the variance a2= a2 (‘a’ represents the lattice constant), taking the radius of the circle domain R to be 3a = 3a. Then the strain components can be obtained, according to the standard definition, by differentiating the resulting displacement n(r,) with regard to the X and Y coordinate values. The above equation (1) combined with demand for energy conservation between the two models gives the following relation between the stress components flij ( T ~ at ) rp in tlie continuum model and the stress components aij(r,,)of atom q at T,,: (2) where the atomic stress coniponents az,(r,,)are defined as those acting on the surface of the atomic volume associating with the atom q. It should be noted that the stress components f l j , ( r P )in the continuum model are the values associating with the sbecified point r,,! whereas the stress components aij(rg)are tlie values associating with the atom q whose spatial position does not have a deterministic meaning. Then equation (2) implies that tlie stress components aij(rP)are the espected values for tlie atom q whose stochastic position is represented by the r P ) . These atomic stress compoweight function w(rq1 nents aij(rq)can be evaluated by applying Born’s formula to the atomic arrangement concerned[l]. To convert u(r,,) and aij,(r,,)into ;ii(rp)and r ; . ( r) respectively, we first discretize tlie continuum iiiocfel atj(rq) =
w(rq1 rp)atj(rp)drp
1
&
47
.. .. .. ..... (a) ma>cinium principal strain (incremental). I
(a) before a specified tool advancement.
.... .. .....
i
(b) niaviinum principal stress. Fig2 Direction and inagnitude of the mzcviinum principal stress and strain in the workpiece during cutting. (b) after a specified tool advancement. Fig. 1 Simulated deformation of the workpiece crystal. the finite-element method and compose the weight function to fit the cliscretized field. This procedure gives the discrete forms of equations (1) and (2)’ which are written with matrices and vectors. Then ii r,) can be obtained by a sirri le matrix-vector multip ication derived from equation (17, while ~ ‘ ( r ,can ) be obtained by solving a linear equation derived from equation (2). For details of the procedure, the reader should refer to [i].
\
Stress/St rain analysis The atomic-model-based simulation carried out in this study is the same as that reported in our previous paper[6]. The deformation of the atomic model during cutting is shown in Figs.1 (a) and (b) as that before and after a certain advancement of the tool. In these figures, lines are drawn to indicate atoms at the same height before cutting. These figures show that atoms under the relief face of the tool undergo severe shearing distortion, and that distortion of the atomic arrangement in the socalled ‘primary shear zone’ does not show simple shearing action. The latter is especially so in the deformation around the tool tip, where the atomic arrangement seems to be distorted in some unstable manner. It is also seen in these figures that atoms in the uncut area begin to swell up. However, the mechanism which causes these deformations is not clear in these figures. The results transformed into the values in the corresponding continuum model are shown in Fig.2 as to the maximum incremental principal strain distribution in (a) and the corresponding mnxinium principal stress distribution in (b). The incremental strain has been computed based on the results in Figs.1 (a) and (b), while the results in Fig.:! (b) have been obtained based on Fig.1 (a). In these figures: short lines are drawn so that their directions and lengths represent the directions and magnitudes of the principal stress and strain values, respectively. It is seen in Fig.2 (a) that the workpiece material is subjected to high compressive strain around the ‘primary shear zone’. However, it is remarliable that the area
48
along the rake face of the tool is strained tensilely so as to hamper the motion of the workpiece atoms. This is due to the adhesive force between the tool and the worlipiece. On the other hand, the stress distribution in Fig2 (b) is somewhat different from what is expected from the strain distribution in Fig.2 (a). In Fig2 (b), the most noticeable features are ‘eyelike’distributions found in the uncut area as well as in the area just ahead of the tool tip and below the finished surface. These unusual stress distributions seem to be a symptom of buckling deformation initiating in these areas. It is also seen in this figxire that the stress distribution in the ‘primary shear zone’ is very complex! being composed of the combined compressive components of the stress nearly in the direction of the inaximuni principal strain in Fig.2 (a), which is due to the force acting so as to push up the chip and that in the horizontal direction, wliich.is due to the force from the tool. The magnitude of the compressivestress coinponeiits is almost the same everywhere and is about IOGPa. which is very high: though the value itself depends on the weight function used. The difference between the stress and strain distributions observed above is also seen in the case between the maximum shear strain distribution in Fig.3 (a) and Fig.4 (a) and the maximum shear stress distribution in Fig.3 (11) and Fig.4 (b). Among these figures, Figs.3 (a) and (b) show the directions of the strain and stress values, respectively, while Figs.4 (a) ancl (b) sIio\v their magnitudes. It is seen in Fig.3 (a) ancl Fig.4 (a) that the ’primary shear zone’: which appears in Fig.4 (a) as a highly strained area indicated with dark shading, is, in fact, subjected to shear strain not in the direction of the shear zone but in the horizontal direction. Then, from there to the uncut area, the direction of the shear strain split into two flows, one going upward to the free surface and the other one going downward. This change of flow direction seems to cause a. lunip or a wavy pattern on the free surface in the uncut area. On the other hand, the maximum shear stress in Fig.3 (b) and Fig.4 (I)) eshibits a very complicated and alinost unexpected result. The most striking feature of Fig.4 (b) is that there is almost no concentrated shear stress in the ‘primary shear zone’. The magnitude of the stress value around there is around 1-3GPa and is
I
(aj nnsiinuni shear strain (incremental j
i
1
(a)maximum shear stra.in (incremental).
i
(bj maximum shear stress.
(b) maximum shear stress.
Fig.3 Direction of maxiinum shear stress and strain in the workpiece during cutting.
Fig.4 Magnitude of the maximum shear stress and strain in the workpiece during cutting.
lower than the theoretical yield shear stress, though it is higher than the value observed in macroscale cutting. In addition, the result in Fig.3 (b) shows that the direction of the maximum shear stress chmges in a very complex manner. This result, of course, corresponds to that ofthe maximum principal stress in Fig2 (b) a i d it should be reiterated that the distortion of the flow pattern in Fig.3 (b) is related to the formation of an ‘eyelike‘ pattern in Fig2 (b). Then the apparent difference between the stress and strain distributions found in a wide area in Fig.2, Fig3 and Fig.4 seems to be attributable to the anisotropic property of the monocrystal copper used in this study. Indeed, the theory of plastic deformation of a monocrystal states that plastic deformation is caused by shearing which occurs when one of the shear stress components resolved in the directions of slip planes of the crystal structcre exceeds a critical value. However, it is not easy to ascertain whether or not this theory holds in this case because the directions of slip pla.iies are not clear for the distorted atomic arrangement in Fig.1 and because the direction of the maximum shear strain in Fig.3 (a) cannot be guaranteed to represent the direction of the slip plane. On the contrary, it rather seems that the above theory does not hold in the primary shear zone because the theory demands. in any case: a concentrated shear stress component in the direction of the slip plane in this area, but because there are no concentrated stress coniponeiits either with regard to tlie iiiaximuiii principal stress or with regard to the innsinium shear stress, there is no mechanism to produce that concenbrated shear stress component in that area. To ascertain the valiclity of this rather negative coiijecture, we focus on the angle 0 between the direction of the mayimum shear stress 7, am1 the direction of the maximum shear strain Y,,~. Because the constitutive equation on a macroscale insists that the angle 0 should be zero and because the theory of plnstic cleforination of a monocrystal also requires a sufficient value of the shear stress component in tlie direction of the , means maximum sheas strain, the value 0 = ~ / 4 which a shear defoiinatioii without any s h e x stress, generally cannot occur, or if it does, cannot produce a large value of ( ~ ~within ~ 7 the ~ framework ) of these theories. Then we have computed the product value: ( T , ~ ~xT f~( 0~)~by)
taking the function f ( 0 ) to be in the form as shown in Fig.5. The result is also shown in Fig3 with the iiidicatioii of the area. of high values with dark shading. This figure shows that.: in fact, the primary shear zone includes many such dark areas of ‘straiige’ mechanics. Althoiigh the detailed discussion of the mechanism of this ’strange’ mechanics must be left to future studies, it seeins that the mechanism is related to a kind of buckling incluced by instability due to severe compression because we can see a dark spot also in the uncut area where budding deformation is clearly developing due to the compression. The above discussion shows that the micromechanics of shear deformation in nanoscale cutting is not simply what we expect. Next we exaniine whether or not this mechanics produces unreasonable results in the franiework of the plastic deformation theory on a macroscale. If it. does not produce apparently unreasonable results. then we examine how the mechanics is ’viewed’or ‘interpreted‘ on a macroscale. To this end, we apply the Plandtl-Reuss formula to our case to compute the increinented plastic strain components ckfj and the incremental elastic strain components d&fj as follows:
where aij is stress deviation, 5 effective stress, & effective incremeiital strainl d&ij total incremental strain, all of which are given based on the results of simulation. The above ecluakions (3) and (4) are, of course, valid only for areas of plastic deformation. so we first have distinguished those areas from areas of elastic deformation by pulling bad; tlie tool a little and by checking whether or not the incremental strain diminishes. Then if the niechanics in nanoscale cutting looks unreasonable when viewed through tlie macroscale law: the strain distributions computed on the basis of equations (3) and (4) will become unreasonable ones. Because it is not easy to see this in the form of strain, we have restored the plastic and elastic deforiiiations based on these strain components, respectively, and have shown the results in Fig.G, where areas of elastic deformation are restored based on the strain coniponents given by Hoolie’s law. In this figure, the total deformation restored on the basis of the
49
Conclusions
Fig.5 Area of 'strange' mec1ia.nics where shear deformation occurs with almost no shear stress. (area with dark shading corresponds to those areas.)
(1) A method of transformation froni an atomic model to an eqi.iivalent continuum model is presented together with its various interpretations. i2) Workpiece material is subjected to concentrated compressive and shear strain at the primary shear zone during cutting. However, the area along the rake face of a tool is subjected to tensile strain due to the adhesive force between the tool and workpiece. (3) Stress distribution in the workpiece during cutting is very comples with regard to its direction, indicating the initiation of a kind of buckling deformation. The interior region of the workpiece: including the area of primary shear? is not exposed to any concentrated shear stress but esposed to relatively constant and high compressive stress. (4) The difference between tlie stress and strain distributions in the workpiece during cutting cannot be explained either by the theory of plastic deformation on a macroscale or by the theory of plastic deforniation for a monocrystal. The mechanism of deformation at tlie primai-y shear zone seems to be related to bucliling due to severe compression in that area. ( 3 ) The deforination of the worlipiece at the primary shear zone is nothing but shear plastic deformation by a yield shear stress, when viewed in the framework of tlie Plandtl-Reuss formula. This may be the result of tlie fact that a phenomenological law on a macroscale will include a variety of microscopic mechanisms of deformat ion.
Acknowledgment Fig.6 Elastic and plastic deformations restored so as to hold the Plandtl-Reuss formula. total incremental strain is also shown with light shading. It is seen in Fig.6 that the concentrated plastic shear deformation does occur in the primary shear zone though tlie elastic and plastic deformations occur in almost opposite directions to each other, contrary to our observation on a macroscale. However, it can be said that the Plandtl-Reuss formula can produi:e considerably reasonable deforination, especially for plastic deformation in the primary shear zone. Then we have examined how the concentrated shear deformation in the primary shear zone is interpreted in the framework of the Plandtl-Reuss formula. Among tlie cluantities appearing in the right-hand side of equation (3), the stress deviation and effective stress r do not exhibit any concentration in that zone, as easily understood from the results in Fig.2 and Fig.4. Thus the origin of the strain concentration resides in tlie concentration of the effective incremental strain dr, which in turn is related to the effective incremental stress d r by da = HI&, with H' being the tangential slope of the stress-strain curve. Then because d r does not eshibit any concentration, the origin of the concentration is attributed to a. very small value of H', which means that the material can deform at some coilstant stress value. This is the phenomenon which we call plastic deformation caused by a yield stress. After all, the strain concentration at the primary shear zone is interpreted as ordinary shear plastic deformation in the framework of the Plandtl-Reuss formula, though the fact that the stress value in that area is almost the same as in the other area cannot be explained in this framework. This latter fact might be viewed such that the workpiece material in that area is 'weaker' than in the other uea.
50
The authors express their sincere gratitude to M.4TSUURA MACHINERY Co. and OSAWA FUND for their financial support during this study. This study is also supported by a. Chnt-in- Aid for Scientific Research froni the Ministry of Education, Science and Culture.
References
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Studies of Grain Boundary Segregation in Fe-P and Fe-B Allovs-1. -4toniic Structure and Stress Distribution, .Act; hiIetal., 32, 1-27. 121 Hoover. W. G., Belali, J. F., DeGroot, 4.J., Hoover, C. G., Stowers, I. F., 1991, Molecular Dynamics Modeling of Indentation and Cutting, Thrust Area Report FY90, LLNL. [3] Ilmwa, N., Shimada, S., Tanaka, H., Ohmori, G., 1991. An Atomistic Analysis of Nanoinetric Chip Removal as Affected by Tool-Work Interaction in Diamond Turning, Annals of the CIRP, 40, l, 551-554. [4]Inaniura, T.,Suzuli. H., Talcezawa, N., 1991, Cutting Experiments in a Computer Using Atomic Models of Copper Crystal and a Diamond Tool, Int. J. JSPE, 25'4. 259-266. 1992, Atomic-Scale Cut[5] Inamura, T., Talcezawa, ?I., ting in a Computer Using Crystal Models of Copper and Diamond, Annals of tlie CIRP, 41, 1, 121-124. [GI Inamura, T., Talcezawa, N., Kunial
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