in fig1 , a tool wz engaging a workpiece we is shown in schematic perspective in the form of a peripheral milling cutter which is operated in counter rotation with the angle speed ω . a machining force or cutting force f is divided into an active force f a and a passive force f p based on a working plane ae . the active force f a in the working plane ae is thereby power - determining for the chip formation . the passive force f p vertical to the working plane ae is however not involved during the machining . the active force f a is divided into a cutting force f c based on the cutting direction , into a feed force f f based on the feed direction : f =√ { square root over ( f a 2 + f p 2 )}=√{ square root over ( f c 2 + f f 2 + f p 2 )} ( 1 ). a cutting angle φ is determined by the contact width ae of the tool wz and projects from the cutting point — that is the entry of the blade of the tool wz into the workpiece we — to the exit point of the blade from the workpiece we . the cutting force law by kienzle has dominated as the method of calculation for the performance - guided cutting force f c . the specific cutting force k c is thereby the ratio of the cutting force f c to the machining cross section a . this machining cross section occurring during a machined withdrawal results as a function of a machining width b and a machining thickness h : the specific cutting force k c is the part of the cutting force f c which acts on 1 mm 2 of the machining cross section a . however , k c is not constant , but is influenced by different factors . the specific cutting force k c is a material - dependent machining value which hardly depends on the machining width b , but nearly exclusively on the machining thickness h or t and a feed f . if the chip removal is larger , the necessary specific cutting force k c will become smaller . the machining thickness h or t changes hereby during the milling . one proceeds then from a mean machining thickness h m for determining the cutting force f c . when the insertion depth increases thereby , the removing width decreases . the value of the specific cutting force k c sinks thereby . kienzle was the first to express this relation by a power law : k c = k c1 . 1  h − zc in n / mm 2 ( 3 ). the specific cutting force k c1 . 1 indicates the cutting force k c based on a machining width b of 1 mm and a machining thickness h of 1 mm . this dependence is generally represented as a straight line in the double - logarithmic scale . an exponent z c represents the gradient of the straight line k c = f ( h ) in this coordinate system . if the equation ( 3 ) is introduced into the cutting force formula ( 2 ) by kienzle , the following results : the exponent 1 − z c describes here the increase value of the specific cutting force k c1 . 1 . the kienzle machining force model valid up to now can only be used in a limited manner for the machining of hard and brittle materials , which is particularly due to the erratic material breakage during the machining . this known approach was expanded as follows : according to martelotti , the following is valid for the current differential cutting force df cutting for a tooth or a blade : with a ( specific ) machining thickness t and a differential db of the cutting depth or the machining width b . the cutting force df is thereby the sum of two cutting forces — namely a cutting force cutting the material df cutting and a deformation force df edge on the blade . it is assumed here that the angle of twist ( λ ) ( fig4 ) of the tool wz is zero : d { right arrow over ( f )}={ right arrow over ( k )} edge db +{ right arrow over ( k )} cutting tdb ( 6b ), with a specific force k edge on the blade and a specific cutting force k cutting . fig2 shows the tool wz being inserted into a workpiece we or the torques resulting thereby during the machining in a schematic cross - section . the tool wz is shown on the left at the beginning of the insertion into the workpiece we , whereby a cutting width a of the tool wz changes depending on the feed s per tooth s t and the mean cutting force thus increases with time . the cutting width a on the right is constant and the cutting force is periodic . the ( specific ) machining thickness t for every point of the blade can be calculated as a function of the angular position φ of the point and the feed per tooth s t : with a current insertion angle φ which determines the position on the blade point when it is in contact with the workpiece we . the feed s t per tooth results from : with a speed v , the angular speed ω and a tooth number n f of the milling cutter . the differential db of the cutting depth can be determined as follows according to fig3 , where the tool wz which is inserted into the workpiece we can be seen in a schematic plan view or in a side view : with a milling cutter diameter d , and angle of twist λ and the insertion angle φ . accordingly , the point on the blade delays behind the associated end point of the blade by the angle of twist λ . the delay angle δ thereby results from : with the cutting depth or the machining width b and the milling cutter diameter d . the delay angle δ can be seen in combination with fig4 , where a tool wz formed as an end milling cutter is shown in a schematic side view . the delay angle δ is thereby used to classify the cutting geometry as type 1 or type 2 : with an entry and exit angle φ 1 and φ 2 ( fig3 ). type 1 is used when the entire blade contacts the workpiece we for a certain time ; type 2 is used when the entire blade contacts the workpiece we only for a moment , or when any parts of the blade do not have any contact at all with the workpiece we . the cutting force f according to equation ( 6b ) results as : the following cutting force f c based on the tooth number n f of the milling cutter results thereby : in this formula , all cutting forces f i are calculated at one point of time , as all differential parts of the cutting force f c are calculated for every cut . the change of the cutting force vector (( f ( t )) is expressed as the multiplication of the functions in the time : { right arrow over ( f )}( t )={ right arrow over ( k )} edge ( t ) h ( t )+{ right arrow over ( k )} cutting ( t ) a ( t ) ( 14 ). the specific cutting forces k edge ( t ) and k cutting ( t ) are multiplied with the scalar function or the contact length h ( t ) of the blade with the workpiece we and the machining cross section a ( t ). the time variable can be replaced by the insertion angle θ of the fixed point on the peripheral milling cutter and by the angular speed ω of the tool we : { right arrow over ( f )} ( θ )= { right arrow over ( k )} edge ( θ ) h ( θ )+ { right arrow over ( k )} cutting ( θ ) a ( θ ) ( 15b ). the functions h ( t ) and a ( t ) will be calculated separately in the following . fig5 shows the contact length h 1 of the first blade in the gradient of the insertion angle θ in the cartesian coordinate system . the following equation results for the contact length h 1 ( θ ): with the milling cutter d , the angle of twist λ , the insertion angle θ and l 1 nd l 2 as functions of the insertion angle θ . these two values l 1 , l 2 are , as can be seen in the following tables , calculated for every milling phase of the insertion angle θ . every blade passes through three phases according to the following table from the entry angle φ 1 to the exit angle φ 2 : phase a , in which the tool wz engages in the workpiece we and every rotation dφ enlarges the contact length h of the blade with the workpiece we . phase b , where the contact length h of the blade with the workpiece we is constant . phase c , where the contact length h reduces . type 1 type 2 phase l 1 ( θ ) l 2 ( θ ) l 1 ( θ ) l 2 ( θ ) für e 1 & lt ; θ ≦ e 2 - phase a φ 1 θ φ 1 θ für e 2 & lt ; θ ≦ e 3 - phase b θ − δ θ θ − δ θ für e 3 & lt ; θ ≦ e 4 - phase c θ − δ φ 2 θ − δ φ 2 the values of e 1 , e 2 and e 3 can be taken from the following table . for the arbitrary blade the following contact length h n is obtained according to equation ( 16 ): with the milling cutter diameter d , the angle of twist λ , the insertion angle θ , an angle ξ between the blades , the number n of the blade and l 1 and l 2 as functions not only of the insertion angle θ , but also the number n of the blade . the angle ξ between the blades is thereby : with the tooth number n f of the milling cutter . for the entire contact length results : with the insertion angle θ , the number n of the blade and the contact length h n of the nth blade with the workpiece we . the contact length h ( θ ) of each blade for a milling cutter with 4 teeth is shown in fig6 . in fig7 is shown the machining cross section a n of the first blade with an insertion angle θ in the workpiece we in perspective . the entry angle φ 1 and the exit angle φ 2 are thus 30 ° or π / 2 . for the machining cross section a 1 results : with the feed s t per tooth , the milling cutter diameter d , the angle of twist λ , the insertion angle θ and l 1 and l 2 as functions of the insertion angle θ . these two values l 1 , l 2 have to be calculated for every milling phase of the insertion angle θ . the machining cross section a 1 for a blade n and a cutting geometry of type 1 is shown in fig8 for the three phases a , b , c according to the above table . the machining cross section a n can be determined for any arbitrary blade n according to the following equation : with the feed s t per tooth , the milling cutter diameter d , the angle of twist λ , the insertion angle θ , the angle ξ between the blades , the number n of the blade and l 1 and l 2 as functions of θ and n . as the final machining cross section results : with the tooth number n f of the milling cutter , the insertion angle θ , the number n of the blade and the machining cross section a n at the nth blade . in fig9 is accordingly shown the machining cross section a ( θ ) for a milling cutter with 4 blades ( angle ξ between the blades = 90 °). so as to compare the obtained data with experimental data , the cutting force has to be divided based on the x -, y -, z - directions into f x , f y and f z : with the machining cross section a , the contact length h of the blade with the workpiece we , the insertion angle θ , the components of the specific cutting force k cx , k cy , k cz and the components of the specific cutting force k ex , k ey , k ez on the blade . the specific cutting force is not to be divided in x -, y -, z - directions , but in t -, r -, z - directions tangentially , radially and axially consistent with the tool . the new functions a r ( θ ) and h r ( θ ) are used for this : the functions a r ( θ ) and h r ( θ ) are indeed the product of the rotary matrix r ( θ ) or of the machining cross section a and the contact length h : as rotary matrix r n ( θ ) for every blade is obtained : with the insertion angle θ , the angle ξ between the blades and the number n of the blades . for all blades for the rotated machining cross section a r the following is valid : with the rotary matrix r n for the nth blade , the machining cross section a n at the nth blade , the insertion angle θ , the tooth number n f of the milling cutter and the number n of the blade . with the rotary matrix r n for the nth blade , the contact length h n of the nth blade , the insertion angle θ , the tooth number n f of the milling cutter and the number n of the blade . so as to simplify the calculations , s 1 ( θ ), s 2 ( θ ), s 3 ( θ ) und s 4 ( θ ) are defined : with the machining cross section a n at the nth blade , the contact length h n of the nth blade , the insertion angle θ , the tooth number n f of the milling cutter , the number n of the blade and the angle ξ between the blades . if the equations ( 29a - d ) are introduced into equation ( 24 ), the following results : if the second matrix in equation ( 30 ) is described as letter j , the following results : during the calculation of equation ( 32 ) results the problem that three parameters and six unknown variables are present . to solve this problem , it is assumed that two successive points have the same specific cutting forces . the specific cutting forces for every interval are calculated as such : the matrix j has to be supplemented by the square matrix . after the mean specific cutting forces have been determined , the semi - empirical cutting forces f x , f y , f z can be calculated with their help and can be compared to experimental data if necessary : by means of the process modeling established above ( fig1 ), the course of the specific cutting force components of a 4 - cutting milling head is shown in the course of the rotation . according to the above mechanistic model , the tangential cutting force f t ( φ ), the radial cutting force f r ( φ ), and the axial cutting force f a ( φ ) are determined as a function of the changing machining cross section ah ( φ ) and the cutting depth a : with the force coefficients or specific cutting forces k tc , k rc , k ac , by the shear or machining process in the tangential , radial and axial direction , and with the specific cutting forces k te , k re , k ae , which indicate the material removal at the edge of the blade . due to the process modeling shown above , a tool wz was determined , whose blades comprise a cutting angle φ smaller than 10 ° and in particular between 4 ° to 6 ° and accordingly perform a chiseling effect on the hard material , so that a very controlled material breakage can be produced . the tool wz comprises , at least in the region of the blades a crystalline diamond coating and in particular a so - called nanocrystalline multi or double layer with a layer thickness of 23 μm plus 3 μm and a grain size of 20 nm to 100 nm . their layer thickness varies between 15 nm and 30 nm . the tool wz designed as a milling cutter preferably has an angle of twist δ larger than 35 °, and particularly from 40 ° to 41 °. for the machining of ceramics , the tool wz is operated as a milling cutter with a cutting width ae from 0 . 01 mm to 0 . 2 mm , and in particular with a cutting width ae of 0 . 02 mm . furthermore , the tool wz is constructed in such a manner that a cutting depth ap of 0 . 5 mm to 4 mm and in particular 2 mm can be realized . furthermore , the tool wz is constructed in such a manner that it can be operated with a cutting speed of 50 to 150 m / min . it is obvious that the diameter d of a tool wz formed as an end milling cutter or a radius miller can vary within a large range ; with the machining of ceramics however , a diameter d of about 6 mm has proved to be particularly suitable . with a tool wz formed as a radius cutter , a corner radius of 0 . 2 mm to 0 . 6 mm is preferably provided thereby . with regard to the machining method and the machining type , a feed s t per tooth in the magnitude of 0 . 06 mm and a rotational speed between 2650 rpm to 7800 rpm have proved to be particularly suitable as parameters in addition to the above described cutting speed v .