**2. Approaches to problem solution and research methods**

Functional approach to development and creation of new machines, objects, and complex technical systems was studied by many researchers [3]. They state that any material object is characterized by a certain totality (matrix) of functions among which it is possible to single out useful, harmful, and neutral functions (**Figure 1**). Unlike a material approach, a functional approach is based on the fact that the product is made to perform a number of functions provided by corresponding material carriers (the cheapest ones or the ones with the least costly manufacturing steps).

This approach can also be applied to working technologies: the manufacturing process expressed through material carriers is to be minimized according to criteria taken into consideration—working time, cost price, and quality. Systemized data of this approach are presented in a number of papers, for example, [4].

The idea of modularity of technological processes (TP) and their functional orientation can develop in the following direction.

As all types of functions (useful, neutral, and harmful) are available in a final product, manufacturing steps are to be oriented in such a way that harmful

*Cutting Superhard Materials by Jet Methods (on Functional Approach) DOI: http://dx.doi.org/10.5772/intechopen.87094*

#### **Figure 1.**

of the structural condition and physical and mechanical properties of the materials of the processed workpieces. Processing of products made from laminated compounds is especially complicated as there is a danger of breakage of adhesion bonds in the "base-surface layer" plane. Workpieces made from such products include diamond carbide (DC) composite containing an upper (working) layer of polycrystal superhard composite on the basis of synthetic diamond (PCD) and a lower (supporting) layer from hard alloy based on tungsten carbide (HA), obtained by

*Recent Advancements in the Metallurgical Engineering and Electrodeposition*

It was demonstrated that, first of all, liquid blasting and laser blasting have good prospects in industrial use for cutting flat workpieces from polycrystal superhard materials (PSHM) and HA, as they allow creation of the cut surface with sufficient efficiency. However, the problems of provision of high quality of the worked

So, surfaces of products from HA and PSHM obtained by the mentioned methods are characterized by high roughness and essential deviation of the shape. In Ref. [2], it is shown that hybrid processing methods based on a combination of different ways of power and other flows impact on the material that make it possible to essentially reduce working hours necessary for production and improve the quality of processing. At the same time, analysis of the final product from the point of view of its useful functions enables improvement of the method and scientific substantiation of most rational ways of impact on the workpiece to

The solution to these problems can be found in the use of an innovative approach to development of hybrid working methods; its essence consists in provision of useful functions of the product on the basis of morphological analysis of variants of combination of power and energy flows generating a hybrid production

approach to creation of hybrid processes as a morphological combination of various ways of power and energy flows that impact on the worked piece when functions and properties of the final product are formed by totality of results of some technological transitions realized on micro-, meso-, or macro-levels of

**2. Approaches to problem solution and research methods**

this approach are presented in a number of papers, for example, [4].

orientation can develop in the following direction.

The idea of modularity of technological processes (TP) and their functional

As all types of functions (useful, neutral, and harmful) are available in a final

product, manufacturing steps are to be oriented in such a way that harmful

The purpose of the chapter consists in development of principles of functional

Functional approach to development and creation of new machines, objects, and complex technical systems was studied by many researchers [3]. They state that any material object is characterized by a certain totality (matrix) of functions among which it is possible to single out useful, harmful, and neutral functions (**Figure 1**). Unlike a material approach, a functional approach is based on the fact that the product is made to perform a number of functions provided by corresponding material carriers (the cheapest ones or the ones with the least costly manufacturing steps). This approach can also be applied to working technologies: the manufacturing process expressed through material carriers is to be minimized according to criteria taken into consideration—working time, cost price, and quality. Systemized data of

sintering in high-pressure apparatus [1].

achieve the maximum quality level.

process.

the product.

**170**

surfaces of the products have not yet been solved in full.

*Classification of the object functions: B, basic; S, secondary; D, derived (obtained without special provision); С, connected (accompany useful functions); I, independent.*

functions be weakened and useful ones, in their turn, be obtained in the minimum number of steps. Under these conditions a technological process can be considered prospective if weakening or complete elimination of harmful functions takes place along with creation of useful functions during the steps.

Analysis of typical products of mechanical engineering from the point of view of functional approach reveals that practically always creation of a particular useful consumer function *Fp* will go together with manifestation of neutral *Fn* and harmful *Fv* functions. Then a product having only useful (under certain conditions) functions is ideal from the point of view of operation:

$$F\_p = F\_{p\mathbf{z}}, F\_v = \mathbf{0}, F\_n \to \min,\tag{1}$$

where *Fpz* are the product useful functions having the following matrix of consumer properties

$$P = \sum\_{i=1}^{l} F\_{pi} + \sum\_{j=1}^{m} F\_{nj} + \sum\_{k=1}^{p} F\_{vk} \,. \tag{2}$$

Hence, an equation of restrictions (1) and optimization (2) makes it possible to choose the most rational material carriers of functions on the basis of morphological analysis and then to pass to material carriers in the technological process creating these functions.

As there is a functional interrelation between separate functions, that is,

$$F\_v = pF\_{p^\bullet} F\_n = qF\_p.\tag{3}$$

taking into account the fact that a function is created by a separate TP step in the form of transformation element *Wp*, Eq. (2) can be presented in the following form:

$$P = \sum\_{i=1}^{l} W\_{\text{p}} F\_{pi} + \sum\_{j=1}^{m} W\_{\text{n}} q F\_{pj} + \sum\_{k=1}^{p} W\_{\text{v}} p F\_{pk} \,. \tag{4}$$

Taking into consideration the fact that TP cannot be aimed at creation of harmful functions and properties in the product, minimization of the unnecessary functions is expressed by dependence:

$$P = \sum\_{i=1}^{l} W\_{\text{p}} F\_{pi} + \sum\_{j=1}^{m} \overline{W\_{p}} qF\_{pj} + \sum\_{k=1}^{p} \overline{W\_{p}} pF\_{pk},\tag{5}$$

or after transformation, in the following way

$$\mathbf{P} = \sum\_{i=1}^{l} F\_{pi} \left( W\_p + \overline{W\_p} q + \overline{W\_p} p \right). \tag{6}$$

is expedient to create morphological tables that may provide the basis for search of

Let the created product realize some totality of functions *F*1…*Fl*. To guarantee them the elements of condition of workpiece, *Еki* is to be formed in such a way

*Fl* ¼ *Сki* ¼ *СpiWpj* ¼ Σ *ЕkiWpj:*

steps, find causes of occurrence of harmful functions and eliminate them (or invert), and also determine the possibility for hybridization of the process to

As generation of assigned functions is a multivariant task (**Figure 2** and **Table 1**), logical relations obtained on the basis of morphological analysis can be simplified according to the known rules of Boolean algebra provided that restrictions be met (7). In this case differentiation or hybridization of operation may have new effects that are to be taken into account by corresponding weight

*Provision of the product functions Fpi by separate elements when they are generated by TP steps Wij.*

12 3 4 5 1 *Fp*<sup>1</sup> E2∩E3∩E4 *Wp*<sup>1</sup> ∗ *Wpj* þ *Wp*<sup>3</sup> — 2 *Fp*<sup>2</sup> E2∩E3+ E7∩E8 *Wp*<sup>4</sup> ∗*Wp*<sup>7</sup> þ *Wp*<sup>3</sup> ∗*Wp*<sup>5</sup> þ *Wp*<sup>2</sup> *Fv*<sup>1</sup>

**Material carriers—TP steps Probable harmful**

**functions**

**Element provision**

*Components of the product useful functions and their provision by TP.*

Application of this approach makes it possible to reveal rational sequence of TP

more rational variants of combination of technological actions.

*Cutting Superhard Materials by Jet Methods (on Functional Approach)*

*DOI: http://dx.doi.org/10.5772/intechopen.87094*

(**Figure 2**) that

meet condition (1).

coefficients (8).

**Figure 2.**

**No. п/п**

…

**Table 1.**

**173**

*L Fp*<sup>l</sup>

**Product functions**

Approaching of the totality of the product properties to the ideal implies transformation of summands *Wn q* and *Wvp* into zero, which is possible under the condition of absence of functional relation between useful and harmful properties of the product or under the condition that the process of obtaining useful properties due to a particular TP step is at the same time the inverse one as to the occurring harmful functions. Availability and interrelation of TP separate elements enable presentation of Eq. (6) in the following way:

$$P = \sum\_{i=1}^{l} F\_{pi} (W\_p + \overline{W\_p}q + \overline{W\_p}p + \overline{W}\_p(p+q)).\tag{7}$$

For totality of alternative variants of TP, the obtained equality is supplemented by quantitative signs of every function, the integral sum of which is equal to 1, then

$$I = \sum\_{i=1}^{l} F\_{pi} \left( b\_{ki} W\_{pi} - b\_{vi} \overline{W\_{pi}} p - b\_{ri} \overline{W}\_{pi} (p + q) \right). \tag{8}$$

under the condition that *bniWpiq* ¼ 0, where *bki* and *bvi* are the corresponding weight coefficients of each of the useful and harmful functions and *bri* is the weight coefficient of interaction of independent steps revealing reserves in improvement of output properties of the final product.

Describing the object by setting its initial condition *Сn*, as a totality of parameters characterizing the form and dimensions of the workpiece, its physical and mechanical properties, and final condition *Ck* via particular forms (dimensions, relative position of the surfaces, physical and mechanical properties, etc.), the technological transformation function *φ*<sup>0</sup> is presented as

$$\rho\_0: \left\{ \begin{array}{c} \text{C}\_{n1} \\ \text{C}\_{n2} \\ \vdots \\ \text{C}\_{nR} \end{array} \right\} \to \left\{ \begin{array}{c} \text{C}\_{k1} \\ \text{C}\_{k2} \\ \vdots \\ \text{C}\_{kT} \end{array} \right\} \tag{9}$$

where *CnR* is *r*-th elementary property of the workpiece; *CkT* is *t*-th elementary property of the product; and *R* and *T* are total number of parameters of the workpiece and the product, respectively. The function *φ*<sup>0</sup> = Σ*WiEj*, where *Ej* is the product separate elements creating its properties.

As the product separate functions expressed via obtaining parameters of geometric accuracy, condition, structure, etc. can be generated in different ways, it *Cutting Superhard Materials by Jet Methods (on Functional Approach) DOI: http://dx.doi.org/10.5772/intechopen.87094*

is expedient to create morphological tables that may provide the basis for search of more rational variants of combination of technological actions.

Let the created product realize some totality of functions *F*1…*Fl*. To guarantee them the elements of condition of workpiece, *Еki* is to be formed in such a way (**Figure 2**) that

$$F\_l = C\_{ki} = C\_{p i}\\\mathcal{W}\_{p j} = \Sigma \, E\_{ki} \mathcal{W}\_{p j} \dots$$

Application of this approach makes it possible to reveal rational sequence of TP steps, find causes of occurrence of harmful functions and eliminate them (or invert), and also determine the possibility for hybridization of the process to meet condition (1).

As generation of assigned functions is a multivariant task (**Figure 2** and **Table 1**), logical relations obtained on the basis of morphological analysis can be simplified according to the known rules of Boolean algebra provided that restrictions be met (7). In this case differentiation or hybridization of operation may have new effects that are to be taken into account by corresponding weight coefficients (8).


**Figure 2.**

Taking into consideration the fact that TP cannot be aimed at creation of harmful functions and properties in the product, minimization of the unnecessary func-

Approaching of the totality of the product properties to the ideal implies trans-

For totality of alternative variants of TP, the obtained equality is supplemented by quantitative signs of every function, the integral sum of which is equal to 1, then

under the condition that *bniWpiq* ¼ 0, where *bki* and *bvi* are the corresponding weight coefficients of each of the useful and harmful functions and *bri* is the weight coefficient of interaction of independent steps revealing reserves in improvement

Describing the object by setting its initial condition *Сn*, as a totality of parameters characterizing the form and dimensions of the workpiece, its physical and mechanical properties, and final condition *Ck* via particular forms (dimensions, relative position of the surfaces, physical and mechanical properties, etc.), the

*Wp qFpj* <sup>þ</sup><sup>X</sup>

*p*

*k*¼1

*Fpi Wp* <sup>þ</sup> *Wp <sup>q</sup>* <sup>þ</sup> *Wp <sup>p</sup>* � �*:* (6)

*Fpi Wp* <sup>þ</sup> *Wp <sup>q</sup>* <sup>þ</sup> *Wp <sup>p</sup>* <sup>þ</sup> *Wp*ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* � �*:* (7)

*Fpi bkiWpi* � *bviWpip* � *briWpi*ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* � �*:* (8)

*Wp pFpk,* (5)

tions is expressed by dependence:

*<sup>P</sup>* <sup>¼</sup> <sup>X</sup> *l*

presentation of Eq. (6) in the following way:

*<sup>P</sup>* <sup>¼</sup> <sup>X</sup> *l*

*<sup>I</sup>* <sup>¼</sup> <sup>X</sup> *l*

of output properties of the final product.

*i*¼1

technological transformation function *φ*<sup>0</sup> is presented as

*φ*<sup>0</sup> :

product separate elements creating its properties.

**172**

*Cn*<sup>1</sup> *Cn*<sup>2</sup> 9 >>>>= *Ck*<sup>1</sup> *Ck*<sup>2</sup> 9 >>>>=

>>>>;

(9)

8 >>>><

>>>>:

⋮ *CkT*

>>>>; !

As the product separate functions expressed via obtaining parameters of geometric accuracy, condition, structure, etc. can be generated in different ways, it

where *CnR* is *r*-th elementary property of the workpiece; *CkT* is *t*-th elementary property of the product; and *R* and *T* are total number of parameters of the workpiece and the product, respectively. The function *φ*<sup>0</sup> = Σ*WiEj*, where *Ej* is the

8 >>>><

>>>>:

⋮ *CnR*

*i*¼1

*i*¼1

or after transformation, in the following way

*<sup>P</sup>* <sup>¼</sup> <sup>X</sup> *l*

*i*¼1

*<sup>W</sup>*р*Fpi* <sup>þ</sup>X*<sup>m</sup>*

*Recent Advancements in the Metallurgical Engineering and Electrodeposition*

*j*¼1

formation of summands *Wn q* and *Wvp* into zero, which is possible under the condition of absence of functional relation between useful and harmful properties of the product or under the condition that the process of obtaining useful properties due to a particular TP step is at the same time the inverse one as to the occurring harmful functions. Availability and interrelation of TP separate elements enable

*Provision of the product functions Fpi by separate elements when they are generated by TP steps Wij.*


**Table 1.** *Components of the product useful functions and their provision by TP.*

The product properties are generated as a result of a number of manufacturing steps during which a complete or partial change of the initial properties takes place. Technological transformation of a workpiece into a product is achieved by purposeful total technological impacts *Wij*(*tk*) of material *So*(*tk*), energy *Еo*(*tk*), and information *Io*(*tк*) types which enable presentation of a scheme of output properties generation according to **Figure 3** and write down.

totality of different actions. In this case if the element geometric characteristics (e.g., flatness, accuracy of linear dimensions) are its output index, this process can be realized by different types of actions that more completely correspond to the properties of the workpiece elements. As actions providing the conditions for minimum error of the shape are to be taken without reinstallation of the workpiece and change of its position in the fixation and orientation device, such processes should

Let some element *Е<sup>m</sup>* of a product be obtained due to realization of discontinu-

will be required for the realization. However, if it is taken into account that a new

place, where *Rnj* is the field of creation of new types of tools; *Rsi* is the *i*-th totality of the known engineering solutions; and *ρ<sup>i</sup>* is the weight of the subset of the known engineering solutions, then the newly created tool can combine the means for

tool is created on the basis of the known ones, that is, expression *Rnj* <sup>¼</sup> <sup>⋂</sup>*<sup>ρ</sup><sup>i</sup>*

Let totalities of properties of two tools represent expressions

*k*1

9 >=

>;

*, I*<sup>2</sup> ¼

Then a hybrid tool obtained on the basis of the principle of morphological search and combination of properties will consist of *m* elements and *m* < *k* + *j*, as some of properties of initial tools can be combined. Thus, the index of hybridization

rational engineering solutions for a hybrid tool on the grounds of a totality of the required properties of the worked product and also the possibilities to achieve them

Consider specific features of application of this approach to creation of hybrid processes for working of diamond-bearing products intended to be used as tools. Such products are rather simple, the number of elements *Еі*, defining functions *Fj* of the latter is not large and, as a rule represents several surfaces making working

Conventionally, these products represent plates of various geometric forms and are made homogeneous or laminated depending on their purpose. They are mainly worked up by an abrasive tool (AT). The use of various abrasive wheels enables obtaining flat elements of surface (during cutting or profile ones during wheel

An alternative consists in application of laser working methods, in particular, laser cutting (LC), laser cutting with water cooling (LCC) or water-jet-guided laser (WJGL), and also a method using loose abrasive accelerated by a supersonic liquid flow—hydro-abrasive cutting (HAC). In this case the obtained surfaces can be of an arbitrary form determined by operational movement of the tool in relation to the

Peculiar features of application of these processes are discussed in [1, 3, 5]. As preliminary research has revealed, separate application of these methods is

HAC is known to generate a stressed condition at the obstacle—the worked

Let us consider specific features of LC, LCC, WJGL, and HAC from the point of

*ρ*11*S*<sup>2</sup>

8 ><

>:

*ρ*1*jS*<sup>2</sup>

<sup>11</sup> *ρ*21*S*<sup>2</sup>

<sup>1</sup>*<sup>j</sup> ρ*2*<sup>j</sup> S*2

*<sup>m</sup>* . This index provides the possibility to find

………

<sup>21</sup> *ρ<sup>k</sup>*1*S*<sup>2</sup>

<sup>2</sup>*<sup>j</sup> ρkjS*<sup>2</sup>

*k*1

9 >=

>;

*kj*

*kj*

*ij*þ*k*. It can be expected that *<sup>k</sup>* tools, accordingly,

*<sup>i</sup>*¼<sup>1</sup>*Rsi* takes

*:* (10)

*ij* and *П<sup>t</sup>*

*Cutting Superhard Materials by Jet Methods (on Functional Approach)*

*DOI: http://dx.doi.org/10.5772/intechopen.87094*

fundamentally different types of actions (**Figure 3**).

<sup>21</sup> *ρ<sup>k</sup>*1*S*<sup>1</sup>

<sup>2</sup>*<sup>j</sup> ρkjS*<sup>1</sup>

surfaces of the tool and a cutting element fastening plane.

be considered hybrid ones.

ous technological actions *П<sup>t</sup>*

*ρ*11*S*<sup>1</sup>

8 ><

>:

by available means.

*ρ*1*jS*<sup>1</sup>

<sup>11</sup> *ρ*21*S*<sup>1</sup>

<sup>1</sup>*<sup>j</sup> ρ*2*<sup>j</sup> S*1

of the created instrument is *kg* <sup>¼</sup> *<sup>k</sup>*þ*<sup>j</sup>*

periphery copying (**Figure 4a**).

worked surface (**Figure 4c**).

view of a functional approach.

surface; this state is described by components

not optimal.

**175**

………

*I*<sup>1</sup> ¼

$$\mathcal{W}\_{\vec{\eta}}(t\_k) = \mathcal{S}\_o(t\_k) \cup \mathcal{E}\_{\mathcal{O}}(t\_k) \cup I\_o(t\_k) \dots$$

Then, on the grounds of the condition that tool technological impacts on the product are to be performed at the levels from nano-areas to the product on the whole, and the product is a 3D object, to realize the totality of variants of technological impacts the morphological matrix will correspond to the following form:

$$A\_{3}^{\operatorname{II}} = \left\| \begin{array}{cccccccc} \boldsymbol{I} & \boldsymbol{I}\_{12}^{\operatorname{S}} \dots & \boldsymbol{I}\_{21}^{\operatorname{S}} & \boldsymbol{I}\_{22}^{\operatorname{S}} \dots & \boldsymbol{I}\_{11}^{\operatorname{S}} & \boldsymbol{I}\_{12}^{\operatorname{S}} \dots & \boldsymbol{I}\_{21}^{\operatorname{S}} & \boldsymbol{I}\_{22}^{\operatorname{s}} \dots & \boldsymbol{E}\_{11}^{\operatorname{S}} & \boldsymbol{E}\_{12}^{\operatorname{S}} \dots & \boldsymbol{E}\_{21}^{\operatorname{S}} & \boldsymbol{E}\_{22}^{\operatorname{S}} \dots \\ \boldsymbol{I}\_{11}^{\operatorname{s}} & \boldsymbol{I}\_{12}^{\operatorname{s}} \dots & \boldsymbol{I}\_{21}^{\operatorname{s}} & \boldsymbol{I}\_{22}^{\operatorname{s}} \dots & \boldsymbol{I}\_{11}^{\operatorname{s}} & \boldsymbol{I}\_{12}^{\operatorname{s}} \dots & \boldsymbol{I}\_{21}^{\operatorname{s}} & \boldsymbol{I}\_{22}^{\operatorname{s}} \dots & \boldsymbol{E}\_{11}^{\operatorname{S}} & \boldsymbol{E}\_{22}^{\operatorname{s}} \dots \\ \boldsymbol{I}\_{21}^{\operatorname{s}} & \boldsymbol{I}\_{12}^{\operatorname{s}} \dots & \boldsymbol{I}\_{21}^{\operatorname{s}} & \boldsymbol{I}\_{22}^{\operatorname{s}} & \boldsymbol{I}\_{21}^{\operatorname{s}} & \boldsymbol{I}\_{11}^{\operatorname{s}} & \boldsymbol{I}\_{22}^{\operatorname{s}} \dots & \boldsymbol{I}\_{21}^{\operatorname{s}} & \boldsymbol{E}\_{12}^{\operatorname{s}} \dots & \boldsymbol{E}\_{21}^{\operatorname{s}} & \boldsymbol{E}\_{22}^{\operatorname{s}} \dots \end{array}$$

where *П<sup>s</sup>* 11, *П<sup>s</sup>* 12, ...; *П<sup>s</sup>* 21, *П<sup>s</sup>* 22, ...; *П<sup>t</sup>* 11*,П<sup>t</sup>* <sup>12</sup>*,* …*;П<sup>t</sup>* 21*,П<sup>t</sup>* <sup>22</sup>*,* …*;* …*П<sup>v</sup>* 11, *П<sup>v</sup>* 12, ...; *П<sup>v</sup>* 21, *П<sup>v</sup>* 22, ...; ...—variants of discontinuous technological actions along the corresponding axes *s*, *t* and *v* of the coordinate system of *s, t, v; Н<sup>s</sup>* 11, *Н<sup>s</sup>* 12, ...; *Н<sup>s</sup>* 21, *Н<sup>s</sup>* <sup>22</sup>*, ...;. Н<sup>t</sup>* 11, *Н<sup>t</sup>* 12, ...; *Нt* 21, *Н<sup>t</sup>* 22, ...; *Н<sup>v</sup>* 11, *Н<sup>v</sup>* 12, ...; *Н<sup>v</sup>* 21, *Н<sup>v</sup>* 22, ...—different variants of continuous technological actions along the axes *s, t* and *v* of the coordinate system of *s, t, v*; *Е<sup>s</sup>* 11, *Е<sup>s</sup>* 12, ...; *Е<sup>s</sup>* 21, *Еs* 22, ...;—different variants of one-time technological actions.

The presence of variants of discontinuous technological actions makes it possible to consider processing of one element of the product in the form of a successive

**Figure 3.**

*Generation of the product properties during manufacturing steps by conventional (upper) and hybrid (lower) tools.*

*Cutting Superhard Materials by Jet Methods (on Functional Approach) DOI: http://dx.doi.org/10.5772/intechopen.87094*

totality of different actions. In this case if the element geometric characteristics (e.g., flatness, accuracy of linear dimensions) are its output index, this process can be realized by different types of actions that more completely correspond to the properties of the workpiece elements. As actions providing the conditions for minimum error of the shape are to be taken without reinstallation of the workpiece and change of its position in the fixation and orientation device, such processes should be considered hybrid ones.

Let some element *Е<sup>m</sup>* of a product be obtained due to realization of discontinuous technological actions *П<sup>t</sup> ij* and *П<sup>t</sup> ij*þ*k*. It can be expected that *<sup>k</sup>* tools, accordingly, will be required for the realization. However, if it is taken into account that a new tool is created on the basis of the known ones, that is, expression *Rnj* <sup>¼</sup> <sup>⋂</sup>*<sup>ρ</sup><sup>i</sup> <sup>i</sup>*¼<sup>1</sup>*Rsi* takes place, where *Rnj* is the field of creation of new types of tools; *Rsi* is the *i*-th totality of the known engineering solutions; and *ρ<sup>i</sup>* is the weight of the subset of the known engineering solutions, then the newly created tool can combine the means for fundamentally different types of actions (**Figure 3**).

Let totalities of properties of two tools represent expressions

$$I\_1 = \left\{ \begin{array}{ccccc} \rho\_{11}\mathbb{S}^1{}\_{11} & \rho\_{21}\mathbb{S}^1{}\_{21} & \rho\_{k1}\mathbb{S}^1{}\_{k1} \\ \dots & \dots & \dots \\ \rho\_{\frac{1}{2}}\mathbb{S}^1{}\_{\frac{1}{2}} & \rho\_{\frac{2}{2}}\mathbb{S}^1{}\_{\frac{2}{2}} & \rho\_{k\frac{1}{2}}\mathbb{S}^1{}\_{k\frac{1}{2}} \end{array} \right\}, I\_2 = \left\{ \begin{array}{ccccc} \rho\_{11}\mathbb{S}^2{}\_{11} & \rho\_{21}\mathbb{S}^2{}\_{21} & \rho\_{k1}\mathbb{S}^2{}\_{k1} \\ \dots & \dots & \dots \\ \rho\_{\frac{1}{2}}\mathbb{S}^2{}\_{\frac{1}{2}} & \rho\_{\frac{2}{2}}\mathbb{S}^2{}\_{\frac{2}{2}} & \rho\_{k\frac{1}{2}}\mathbb{S}^2{}\_{k\frac{1}{2}} \end{array} \right\}. \tag{10}$$

Then a hybrid tool obtained on the basis of the principle of morphological search and combination of properties will consist of *m* elements and *m* < *k* + *j*, as some of properties of initial tools can be combined. Thus, the index of hybridization of the created instrument is *kg* <sup>¼</sup> *<sup>k</sup>*þ*<sup>j</sup> <sup>m</sup>* . This index provides the possibility to find rational engineering solutions for a hybrid tool on the grounds of a totality of the required properties of the worked product and also the possibilities to achieve them by available means.

Consider specific features of application of this approach to creation of hybrid processes for working of diamond-bearing products intended to be used as tools. Such products are rather simple, the number of elements *Еі*, defining functions *Fj* of the latter is not large and, as a rule represents several surfaces making working surfaces of the tool and a cutting element fastening plane.

Conventionally, these products represent plates of various geometric forms and are made homogeneous or laminated depending on their purpose. They are mainly worked up by an abrasive tool (AT). The use of various abrasive wheels enables obtaining flat elements of surface (during cutting or profile ones during wheel periphery copying (**Figure 4a**).

An alternative consists in application of laser working methods, in particular, laser cutting (LC), laser cutting with water cooling (LCC) or water-jet-guided laser (WJGL), and also a method using loose abrasive accelerated by a supersonic liquid flow—hydro-abrasive cutting (HAC). In this case the obtained surfaces can be of an arbitrary form determined by operational movement of the tool in relation to the worked surface (**Figure 4c**).

Peculiar features of application of these processes are discussed in [1, 3, 5].

As preliminary research has revealed, separate application of these methods is not optimal.

Let us consider specific features of LC, LCC, WJGL, and HAC from the point of view of a functional approach.

HAC is known to generate a stressed condition at the obstacle—the worked surface; this state is described by components

The product properties are generated as a result of a number of manufacturing steps during which a complete or partial change of the initial properties takes place. Technological transformation of a workpiece into a product is achieved by purposeful total technological impacts *Wij*(*tk*) of material *So*(*tk*), energy *Еo*(*tk*), and information *Io*(*tк*) types which enable presentation of a scheme of output properties

*Wij*ð Þ¼ *tk So*ð Þ *tk* ∪*E0*ð Þ *tk* ∪*Io*ð Þ *tk :*

Then, on the grounds of the condition that tool technological impacts on the product are to be performed at the levels from nano-areas to the product on the whole, and the product is a 3D object, to realize the totality of variants of technological impacts the morphological matrix will correspond to the

<sup>11</sup> *H<sup>S</sup>*

<sup>11</sup> *H<sup>t</sup>*

<sup>11</sup> *H<sup>v</sup>*

*Generation of the product properties during manufacturing steps by conventional (upper) and hybrid*

11*,П<sup>t</sup>*

...; ...—variants of discontinuous technological actions along the corresponding axes

The presence of variants of discontinuous technological actions makes it possible to consider processing of one element of the product in the form of a successive

<sup>12</sup>… *H<sup>S</sup>*

<sup>12</sup>… *H<sup>t</sup>*

<sup>12</sup>… *H<sup>v</sup>*

<sup>12</sup>*,* …*;П<sup>t</sup>*

11, *Н<sup>s</sup>*

<sup>21</sup> *H<sup>s</sup>*

<sup>21</sup> *H<sup>t</sup>*

<sup>21</sup> *H<sup>v</sup>*

21*,П<sup>t</sup>*

12, ...; *Н<sup>s</sup>*

<sup>22</sup>… *ES*

<sup>22</sup>… *Et*

<sup>22</sup>… *Ev*

<sup>22</sup>*,* …*;* …*П<sup>v</sup>*

22, ...—different variants of continuous technological

21, *Н<sup>s</sup>*

<sup>11</sup> *ES*

<sup>11</sup> *Et*

<sup>11</sup> *Ev*

11, *П<sup>v</sup>*

<sup>22</sup>*, ...;. Н<sup>t</sup>*

<sup>12</sup>… *ES*

<sup>12</sup>… *Et*

<sup>12</sup>… *Ev*

11, *Е<sup>s</sup>*

12, ...; *П<sup>v</sup>*

11, *Н<sup>t</sup>*

<sup>21</sup> *ES* <sup>22</sup>…  

<sup>21</sup> *Et* <sup>22</sup>…

<sup>21</sup> *Ev* <sup>21</sup>…

> 21, *П<sup>v</sup>* 22,

12, ...;

12, ...; *Е<sup>s</sup>* 21,

generation according to **Figure 3** and write down.

<sup>21</sup> *П<sup>S</sup>*

<sup>21</sup> *П<sup>t</sup>*

<sup>21</sup> *П<sup>v</sup>*

21, *П<sup>s</sup>*

21, *Н<sup>v</sup>*

22, ...;—different variants of one-time technological actions.

<sup>22</sup>… *H<sup>S</sup>*

*Recent Advancements in the Metallurgical Engineering and Electrodeposition*

<sup>22</sup>… *H<sup>t</sup>*

<sup>22</sup> *H<sup>v</sup>*

22, ...; *П<sup>t</sup>*

actions along the axes *s, t* and *v* of the coordinate system of *s, t, v*; *Е<sup>s</sup>*

following form:

 

*П<sup>S</sup>* <sup>11</sup> *П<sup>S</sup>*

*Пt* <sup>11</sup> *П<sup>t</sup>*

*Пv* <sup>11</sup> *П<sup>v</sup>*

where *П<sup>s</sup>*

22, ...; *Н<sup>v</sup>*

<sup>12</sup>… *П<sup>S</sup>*

<sup>12</sup>… *П<sup>t</sup>*

<sup>12</sup>… *П<sup>v</sup>*

12, ...; *П<sup>s</sup>*

*s*, *t* and *v* of the coordinate system of *s, t, v; Н<sup>s</sup>*

12, ...; *Н<sup>v</sup>*

11, *П<sup>s</sup>*

11, *Н<sup>v</sup>*

*AП <sup>З</sup>* ¼

*Нt* 21, *Н<sup>t</sup>*

*Еs*

**Figure 3.**

**174**

*(lower) tools.*

characterized by the property to efficiently continue working only till the moment when the loss of jet energy due to friction against the surface of the funnel that appeared is comparable with the energy at which the cutting process ceases. In particular, in [1, 5] it is shown that during working of carbides (HA), the dimple depth *hl* increases due to manifestation of mechanisms of micro-cutting and

material flow stress; *u* is the feed rate; ε is the material specific energy; and *Dj* is the

On the other hand, due to flow energy losses, the real depth of the obtained

*N*2*N*<sup>3</sup>

A peculiar feature of cutting hard composite workpieces used in tools both separately and in the form of bases for the required layers of other materials consists in the fact that particles flowing on the surface cause local highly intensive loads resulting in some elastoplastic compressive macrodeformations in local volumes of the surface layer. These loads are mainly received by the carbide structure (for alloys containing cobalt ≤10%). Further pickup of the abrasive particle by liquid flow results in removal of compression load and partial elastic restoration of the deformed volume of the surface layer, that is, in occurrence of tensions in this local volume, which causes redistribution of tensions among the components of HA structure. In this case, at first carbide grain boundaries break, which results in appearance of microcracks in HA carbide grains themselves and plastic deformation along the dislocation mechanism of cobalt bundle [2, 3]. After that the boundaries

<sup>þ</sup> ð Þ <sup>1</sup> � *<sup>N</sup>*<sup>1</sup>

2 *dj*

<sup>1</sup>�*<sup>c</sup>* <sup>þ</sup> *Cf*ð Þ <sup>1</sup> � *<sup>N</sup>*<sup>1</sup>

<sup>π</sup>*u*ε*Dj* , where *с* is the process constant; *ma* is the abrasive mass

*f b*

*<sup>v</sup> ; vc* is the critical velocity of abrasive parti-

ffiffiffiffiffiffiffiffi <sup>2</sup>*pb=*<sup>ρ</sup> <sup>p</sup> <sup>þ</sup>*ma*

; σ is the

*,* (11)

deformation destruction, that is, the process is described by the expression

flow; *<sup>v</sup>* is the velocity of abrasive particles movement, *<sup>v</sup>* <sup>¼</sup> <sup>2</sup>*pbf <sup>b</sup>*

*Cutting Superhard Materials by Jet Methods (on Functional Approach)*

flow diameter corresponding to the diameter of the nozzle section.

ffiffiffiffiffiffiffiffiffiffi *mav*<sup>2</sup> 8σ*u*

between the carbide grains and the bundle and the bundle itself break.

of the jet and its drift from the theoretic axis to the side opposite to the feed movement. It is facilitated by selectivity of destruction caused by "nonrigidity" of the jet, due to which creation of specific destruction areas takes place (**Figure 5b**); their location and dimension determine the form of hydro-cutting front and deviation of the jet by angle α, whose value is determined by the relation of speed *vz* of jet penetration into the worked material and feed rate *s*. This phenomenon results in appearance of surface defects in the form of waviness and also in cut edge deviation

increases, and distortion of the jet as a "nonrigid" tool increases.

Hence, movement of the destruction area slows down when the cutting depth

Much higher rigidity of polycrystal superhard materials, including cubic boron nitride (PSHM, CBN) and diamond-bearing elements (DBE), causes the fact that a moving flow of abrasive grains slightly influences the worked material and the workpiece is not cut at a speed admissible by manufacturing conditions (**Figure 5a**). Obtaining the initial groove and creation of a cut or a channel result in distortion

The jet ability to selectively go about obstacles results in the fact that heterogeneity of the worked material and specific features of the power scheme may provoke availability of both bumpy elements of the surface and surface cleavages

dimple will be less and can be taken into account by the relation

r

*hl* ¼ *c*

2 ε*d*<sup>2</sup> *j mav ; N*<sup>3</sup> <sup>¼</sup> *<sup>u</sup>*

*<sup>v</sup> ; N*<sup>2</sup> ¼ <sup>π</sup>

*hl* ¼ *c*

ffiffiffiffiffiffiffi *mav*<sup>2</sup> 8σ*u*<sup>2</sup> q

where *<sup>N</sup>*<sup>1</sup> <sup>¼</sup> *vc*

from orthogonality.

(**Figure 5c**).

**177**

cles movement.

<sup>þ</sup> <sup>2</sup>*ma*ð Þ <sup>1</sup>�*<sup>c</sup> <sup>v</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.87094*

**Figure 4.** *Processing tool working surface shaped abrasive wheel (a), model of a jet device (b), a laser head for cutting the model (c).*

$$\begin{split} \boldsymbol{\sigma}\_{r} &= 2G \left( \frac{\partial U}{\partial r} + \frac{\mu \boldsymbol{\varepsilon}}{1 - 2\mu} \right); \boldsymbol{\sigma}\_{t} = 2G \left( \frac{U}{r} + \frac{\mu \boldsymbol{\varepsilon}}{1 - 2\mu} \right); \\ \boldsymbol{\sigma}\_{z} &= 2G \left( \frac{\partial H}{\partial \boldsymbol{z}} + \frac{\mu \boldsymbol{\varepsilon}}{1 - 2\mu} \right); \boldsymbol{\tau} = 2G \left( \frac{\partial H}{\partial \boldsymbol{z}} + \frac{\mu \boldsymbol{\varepsilon}}{1 - 2\mu} \right); \\ \boldsymbol{\varepsilon} &= \boldsymbol{\varepsilon}\_{r} + \boldsymbol{\varepsilon}\_{t} + \boldsymbol{\varepsilon}\_{z} = \frac{\partial U}{\partial r} + \frac{U}{r} \frac{\partial H}{\partial \boldsymbol{z}}; \boldsymbol{\varepsilon} = \frac{1 - 2\nu}{2(1 - \nu)G} (\boldsymbol{\sigma}\_{r} + \boldsymbol{\sigma}\_{t} + \boldsymbol{\sigma}\_{z}); \\ &= \frac{1 - 2\nu}{E} (\boldsymbol{\sigma}\_{r} + \boldsymbol{\sigma}\_{t} + \boldsymbol{\sigma}\_{z}) \begin{cases} (1 - 2\nu) \left[ \Delta U - \frac{U}{r^{2}} \right] + \frac{\partial \boldsymbol{\varepsilon}}{\partial r} = 0 \\ (1 - 2\nu) \Delta U + \frac{\partial \boldsymbol{\varepsilon}}{\partial r} = 0. \end{cases} \end{split}$$

where *U*(*t*) and *H*(*t*) are components of movements at a particular point of the surface that are determined as *U t*ðÞ¼� ð Þ <sup>1</sup>�2<sup>ν</sup> *po*ð Þ*<sup>t</sup> Dk* 2 � �<sup>2</sup> <sup>4</sup>*<sup>G</sup> ;* 0 < *Di* <sup>2</sup> <sup>&</sup>lt; *Dk* 2 *; H t*ðÞ¼� ð Þ <sup>1</sup>�2<sup>ν</sup> *po*ð Þ*<sup>t</sup> Dk* <sup>2</sup>*<sup>G</sup>* ; *G* and μ are the shift module and Poisson ratio of the worked material, respectively; <sup>ε</sup> is the volume deformation; <sup>Δ</sup> <sup>¼</sup> *<sup>d</sup>*<sup>2</sup> *dr*<sup>2</sup> <sup>þ</sup> *<sup>d</sup> <sup>r</sup>∂<sup>r</sup>* <sup>þ</sup> *<sup>d</sup>*<sup>2</sup> *<sup>∂</sup>z*<sup>2</sup> is the Laplace operator; and *<sup>р</sup>*<sup>0</sup> is the pressure at the obstacle *pi* <sup>¼</sup> <sup>0</sup>*,* <sup>5</sup>*ρv*<sup>2</sup> *<sup>i</sup>* <sup>þ</sup> <sup>ρ</sup>*v*<sup>2</sup> *<sup>i</sup> dQ*.

Intensification of the stressed condition contributes to development of initial defects and creation of a grid of microcracks actively joining under the action of abrasive particles when material particles come off the surface. HAC is

*Cutting Superhard Materials by Jet Methods (on Functional Approach) DOI: http://dx.doi.org/10.5772/intechopen.87094*

characterized by the property to efficiently continue working only till the moment when the loss of jet energy due to friction against the surface of the funnel that appeared is comparable with the energy at which the cutting process ceases.

In particular, in [1, 5] it is shown that during working of carbides (HA), the dimple depth *hl* increases due to manifestation of mechanisms of micro-cutting and deformation destruction, that is, the process is described by the expression *hl* ¼ *c* ffiffiffiffiffiffiffi *mav*<sup>2</sup> 8σ*u*<sup>2</sup> q <sup>þ</sup> <sup>2</sup>*ma*ð Þ <sup>1</sup>�*<sup>c</sup> <sup>v</sup>*<sup>2</sup> <sup>π</sup>*u*ε*Dj* , where *с* is the process constant; *ma* is the abrasive mass flow; *<sup>v</sup>* is the velocity of abrasive particles movement, *<sup>v</sup>* <sup>¼</sup> <sup>2</sup>*pbf <sup>b</sup> f b* ffiffiffiffiffiffiffiffi <sup>2</sup>*pb=*<sup>ρ</sup> <sup>p</sup> <sup>þ</sup>*ma* ; σ is the material flow stress; *u* is the feed rate; ε is the material specific energy; and *Dj* is the flow diameter corresponding to the diameter of the nozzle section.

On the other hand, due to flow energy losses, the real depth of the obtained dimple will be less and can be taken into account by the relation

$$h\_l = c\sqrt{\frac{m\_a v^2}{8\sigma u}} + \frac{(\mathbf{1} - N\_1)^2 d\_j}{\frac{N\_2 N\_3}{1 - \varepsilon} + C\_f (\mathbf{1} - N\_1)},\tag{11}$$

where *<sup>N</sup>*<sup>1</sup> <sup>¼</sup> *vc <sup>v</sup> ; N*<sup>2</sup> ¼ <sup>π</sup> 2 ε*d*<sup>2</sup> *j mav ; N*<sup>3</sup> <sup>¼</sup> *<sup>u</sup> <sup>v</sup> ; vc* is the critical velocity of abrasive particles movement.

A peculiar feature of cutting hard composite workpieces used in tools both separately and in the form of bases for the required layers of other materials consists in the fact that particles flowing on the surface cause local highly intensive loads resulting in some elastoplastic compressive macrodeformations in local volumes of the surface layer. These loads are mainly received by the carbide structure (for alloys containing cobalt ≤10%). Further pickup of the abrasive particle by liquid flow results in removal of compression load and partial elastic restoration of the deformed volume of the surface layer, that is, in occurrence of tensions in this local volume, which causes redistribution of tensions among the components of HA structure. In this case, at first carbide grain boundaries break, which results in appearance of microcracks in HA carbide grains themselves and plastic deformation along the dislocation mechanism of cobalt bundle [2, 3]. After that the boundaries between the carbide grains and the bundle and the bundle itself break.

Hence, movement of the destruction area slows down when the cutting depth increases, and distortion of the jet as a "nonrigid" tool increases.

Much higher rigidity of polycrystal superhard materials, including cubic boron nitride (PSHM, CBN) and diamond-bearing elements (DBE), causes the fact that a moving flow of abrasive grains slightly influences the worked material and the workpiece is not cut at a speed admissible by manufacturing conditions (**Figure 5a**).

Obtaining the initial groove and creation of a cut or a channel result in distortion of the jet and its drift from the theoretic axis to the side opposite to the feed movement. It is facilitated by selectivity of destruction caused by "nonrigidity" of the jet, due to which creation of specific destruction areas takes place (**Figure 5b**); their location and dimension determine the form of hydro-cutting front and deviation of the jet by angle α, whose value is determined by the relation of speed *vz* of jet penetration into the worked material and feed rate *s*. This phenomenon results in appearance of surface defects in the form of waviness and also in cut edge deviation from orthogonality.

The jet ability to selectively go about obstacles results in the fact that heterogeneity of the worked material and specific features of the power scheme may provoke availability of both bumpy elements of the surface and surface cleavages (**Figure 5c**).

<sup>σ</sup>*<sup>r</sup>* <sup>¼</sup> <sup>2</sup>*<sup>G</sup> <sup>∂</sup><sup>U</sup>*

<sup>σ</sup>*<sup>z</sup>* <sup>¼</sup> <sup>2</sup>*<sup>G</sup> <sup>∂</sup><sup>H</sup>*

<sup>ε</sup> <sup>¼</sup> <sup>ε</sup>*<sup>r</sup>* <sup>þ</sup> <sup>ε</sup>*<sup>t</sup>* <sup>þ</sup> <sup>ε</sup>*<sup>z</sup>* <sup>¼</sup> *<sup>∂</sup><sup>U</sup>*

<sup>¼</sup> <sup>1</sup> � <sup>2</sup><sup>ν</sup>

*H t*ðÞ¼� ð Þ <sup>1</sup>�2<sup>ν</sup> *po*ð Þ*<sup>t</sup> Dk*

**176**

**Figure 4.**

*model (c).*

*∂r*

*∂z*

*<sup>E</sup>* ð Þ <sup>σ</sup>*<sup>r</sup>* <sup>þ</sup> <sup>σ</sup>*<sup>t</sup>* <sup>þ</sup> <sup>σ</sup>*<sup>z</sup>*

of the surface that are determined as *U t*ðÞ¼� ð Þ <sup>1</sup>�2<sup>ν</sup> *po*ð Þ*<sup>t</sup> Dk*

material, respectively; <sup>ε</sup> is the volume deformation; <sup>Δ</sup> <sup>¼</sup> *<sup>d</sup>*<sup>2</sup>

operator; and *<sup>р</sup>*<sup>0</sup> is the pressure at the obstacle *pi* <sup>¼</sup> <sup>0</sup>*,* <sup>5</sup>*ρv*<sup>2</sup>

<sup>þ</sup> *με* 1 � 2μ

<sup>þ</sup> *με* 1 � 2μ

> 8 >><

> >>:

where *U*(*t*) and *H*(*t*) are components of movements at a particular point

Intensification of the stressed condition contributes to development of initial defects and creation of a grid of microcracks actively joining under the action of

abrasive particles when material particles come off the surface. HAC is

� �

*∂r* þ *U r ∂H ∂z*

*;* <sup>σ</sup>*<sup>t</sup>* <sup>¼</sup> <sup>2</sup>*<sup>G</sup> <sup>U</sup>*

*Processing tool working surface shaped abrasive wheel (a), model of a jet device (b), a laser head for cutting the*

*;* <sup>τ</sup> <sup>¼</sup> <sup>2</sup>*<sup>G</sup> <sup>∂</sup><sup>H</sup>*

*;* <sup>ε</sup> <sup>¼</sup> <sup>1</sup> � <sup>2</sup><sup>ν</sup>

ð Þ <sup>1</sup> � <sup>2</sup><sup>ν</sup> <sup>Δ</sup>*<sup>U</sup>* � *<sup>U</sup>*

<sup>2</sup>*<sup>G</sup>* ; *G* and μ are the shift module and Poisson ratio of the worked

ð Þ 1 � 2ν Δ*U* þ

*r*

*∂z*

� �

<sup>þ</sup> *με* 1 � 2μ � �

> <sup>þ</sup> *με* 1 � 2μ

2 1ð Þ � <sup>ν</sup> *<sup>G</sup>* ð Þ <sup>σ</sup>*<sup>r</sup>* <sup>þ</sup> <sup>σ</sup>*<sup>t</sup>* <sup>þ</sup> <sup>σ</sup>*<sup>z</sup>*

þ *∂ε <sup>∂</sup><sup>r</sup>* <sup>¼</sup> <sup>0</sup>

� �

*r*2

2 � �<sup>2</sup> <sup>4</sup>*<sup>G</sup> ;* 0 < *Di*

*dr*<sup>2</sup> <sup>þ</sup> *<sup>d</sup>*

*<sup>i</sup>* <sup>þ</sup> <sup>ρ</sup>*v*<sup>2</sup>

*∂ε <sup>∂</sup><sup>r</sup>* <sup>¼</sup> <sup>0</sup>*:* *;*

*;*

*,*

<sup>2</sup> <sup>&</sup>lt; *Dk* 2 *;*

*<sup>∂</sup>z*<sup>2</sup> is the Laplace

*<sup>r</sup>∂<sup>r</sup>* <sup>þ</sup> *<sup>d</sup>*<sup>2</sup>

*<sup>i</sup> dQ*.

� �

*Recent Advancements in the Metallurgical Engineering and Electrodeposition*

Thus, the possibilities of HAC process from the point of view of the functional approach can be presented by elements of **Table 2**. In this case the ability of the jet to destroy the obstacle with creation of vertical edges can be regarded as a useful function only for a restricted number of materials.

Then

$$\mathcal{W}\_{\mathcal{U}}(t\_k) = \mathcal{W}\_{\mathcal{I}}^{Fp\mathbb{1}}(t\_k) \cap \mathcal{W}\_{\mathcal{I}}^{Fv\mathbb{1}}(t\_k) \tag{12}$$

under the condition that *Fv*<sup>2</sup> and *Fn*<sup>1</sup> can be neglected and useful and harmful properties (functions) are manifested simultaneously. Corresponding transformations in the form of the process result, for example, depth *h* (the surface element

*h*j*WFp* ¼ *b*<sup>0</sup> þ *b*1*sk* þ *b*2*Ma;*

where *Ma* is the abrasive grains mass flow, *hm* is the thickness of the worked

It is obvious from the given relations that these two dependences are interconnected and increase of cutting depth *h* at a higher rate of contour feed *sk*

*Variants of combination of a water jet and a laser beam for performance of WJGL (a) or LCC (b–e).*

*<sup>W</sup>Fp* ¼ 0 and δj

More possibilities are provided by combined working of material by wate-jetguided laser (WJGL). In this case working variants correspond to schemes in **Figure 6**; in this case both WJGL (**Figure 6a**) and LCC (**Figure 6b**–**e**) can be

δj*WFv* ¼ *b*<sup>0</sup> þ *b*1*sk* þ *b*2*hm,* (13)

*<sup>W</sup>Fv* ¼ 0.

linear value) for HA, may represent regression equations in the form

*Cutting Superhard Materials by Jet Methods (on Functional Approach)*

material, and for other materials *h*j

*DOI: http://dx.doi.org/10.5772/intechopen.87094*

results in increase of waviness δ.

realized.

**Figure 6.**

**179**

#### **Figure 5.**

*Surface defects as manifestation of harmful functions of technological actions: (а) incomplete cutting; (b) surface waviness; (c) surface cleavages.*


#### **Table 2.**

*Provision of the product function by creation of element Еі by a technological action Wij(tk) at HAC.*

*Cutting Superhard Materials by Jet Methods (on Functional Approach) DOI: http://dx.doi.org/10.5772/intechopen.87094*

under the condition that *Fv*<sup>2</sup> and *Fn*<sup>1</sup> can be neglected and useful and harmful properties (functions) are manifested simultaneously. Corresponding transformations in the form of the process result, for example, depth *h* (the surface element linear value) for HA, may represent regression equations in the form

$$h|\_{W^{\mathbb{F}}} = b\_0 + b\_1 s\_k + b\_2 \mathcal{M}\_{a\flat}$$

$$|\mathfrak{S}|\_{W^{\mathbb{F}}} = b\_0 + b\_1 s\_k + b\_2 h\_m \tag{13}$$

where *Ma* is the abrasive grains mass flow, *hm* is the thickness of the worked material, and for other materials *h*j *<sup>W</sup>Fp* ¼ 0 and δj *<sup>W</sup>Fv* ¼ 0.

It is obvious from the given relations that these two dependences are interconnected and increase of cutting depth *h* at a higher rate of contour feed *sk* results in increase of waviness δ.

More possibilities are provided by combined working of material by wate-jetguided laser (WJGL). In this case working variants correspond to schemes in **Figure 6**; in this case both WJGL (**Figure 6a**) and LCC (**Figure 6b**–**e**) can be realized.

**Figure 6.** *Variants of combination of a water jet and a laser beam for performance of WJGL (a) or LCC (b–e).*

Thus, the possibilities of HAC process from the point of view of the functional approach can be presented by elements of **Table 2**. In this case the ability of the jet to destroy the obstacle with creation of vertical edges can be regarded as a useful

*Fp1*ð Þ *tk* <sup>∩</sup>*W1*

*Fv1*ð Þ *tk* (12)

function only for a restricted number of materials.

*W1i*ð Þ¼ *tk W1*

*Recent Advancements in the Metallurgical Engineering and Electrodeposition*

*Surface defects as manifestation of harmful functions of technological actions: (а) incomplete cutting;*

**Useful** *Fpi* **Harmful** *Fvk* **Neutral** *Fnj*

cutting modes

PSHM Weak High Complete Up to 50–60 PCD Absent Absent Insignificant Up to 50–75

*Provision of the product function by creation of element Еі by a technological action Wij(tk) at HAC.*

**Waviness δ, μm Surface**

**cleavage**

**Workpiece heat, Т (°С)**

Minimum Up to 50–60

Then

**Figure 5.**

**Table 2.**

**178**

**Worked material**

*(b) surface waviness; (c) surface cleavages.*

**Functions** *F*<sup>0</sup>

**the product** *h***, mm**

**Obtaining of orthogonal edge of**

HA *Fp1* depends on cutting modes *Fvk* depends on

The action of a water jet or a laser beam can be both simultaneous and successive. In this case different variants of laser blasting and possibilities of variation of the form of jet nozzle flow section, as shown in [6], provide good prospects in combination of methods of actions on the worked piece and, consequently, in meeting principle (10).

Application of thermo-hydrodynamic jets typical of WJGL or LCC causes heating of the workpiece whose temperature field can be described by equation [7]

$$T(x,y,z,t) = \frac{P}{\pi \frac{1}{2}\rho c} \int\_0^l \frac{\left[\left(4\alpha \tau + A^2\right)\left(4\alpha \tau + B^2\right)a\tau\right]^{1/2}}{\times \left[e^{-\frac{r^2}{4\alpha}} - h\*(\pi \alpha \tau)^{1/2} \operatorname{erfc}\left(\frac{x}{2(a\tau)^{1/2}} + h\*(a\tau)^{1/2}\right) \times e^{h\*z + h\*^{1/2}a\tau}\right]} \frac{d\tau}{dt}$$

where *t* is the time from the moment of beginning of thermal impulse action, ρ, с, λ, α ¼ <sup>λ</sup> *ср*, and *h\** are density, specific heat capacity, heat conductivity coefficient, temperature conductivity of the workpiece material, and coefficient of heat transfer from the surface of the workpiece, respectively;α ¼ <sup>λ</sup> *ср А* and *В* are the bigger and smaller half axes of beam elliptic section; and *P* = π*qAB* is the power of the lasing emitter. Integral equation of heat energy balance in an arbitrary area ω ⊂ Ω according to [4] will take the form Ð *w ∂e <sup>∂</sup><sup>t</sup> dv* <sup>¼</sup> <sup>Ð</sup> *<sup>w</sup> gdv* � ∮ *<sup>∂</sup><sup>w</sup> qT* ! <sup>þ</sup> *cg*ρ*gT vf* !*; n* ! � �*ds:*

Thus, it is possible to create a table of provision of the product with the function via formation of its separate element or a totality of elements. This table is analogous to the one considered above (e.g., for WJGL; **Table 3**), but it reflects the peculiar feature of each of the variants in **Figure 6**. It should be noted that the given tables just illustrate the approach and draw attention to the most important func-

PCD Nonlinear medium-intensive Essential Absent Is observed in

*Provision of the product function by creation of element* Еі *by a technological action Wij(tk)—WJGL.*

**Useful** *Fpi* **Harmful** *Fvk* **Neutral** *Fnj*

**δ, μm**

**Thermodestruction**

**Cracking** *l***, mm**

Minimum Exists Absent

Minimum Minimum Is observed in

**Variation of porosity** *P*

destructed zone

destructed zone

On the basis of **Table 3**, technological action for obtaining a particular element

Corresponding transformations for LB for different materials include four equa-

*<sup>W</sup>Fp* ¼ *b*<sup>0</sup> þ *b*1*sk* þ *b*2*T* þ *b*3*Qv* þ …*;*

*<sup>W</sup>Fv* ¼ *b*<sup>0</sup> þ *b*1*sk* þ *b*2*T* þ *b*3*Qv* þ *b*4*hl* þ …*,*

*<sup>W</sup>Fv* ¼ *b*<sup>0</sup> þ *b*1*sk* þ *b*2*T* þ *b*3*Qv* þ *b*4*hl* þ …

where *Т* is the impulse energy, *Q <sup>v</sup>* is the liquid discharge out of the nozzle, and *hl*

Analogous tables, including the ones with extended features of functions *F*<sup>0</sup>

The obtained dependences (12) and (14) make it possible to determine the totality of variants for carrying out the process of working of a particular element of the surface, and the most optimal method can be searched on the basis of morpho-

such as carbon-carbon composites, multilayer aircraft panels, etc. [9–11].

**3. Used equipment, workpieces, and succession of experimental**

Cut workpieces of HA T5K10, PSHM on the basis of CBN "borsinit" represented plates of the size 12.7 � 12.7 mm, of the thickness up to 3.5 mm, DC workpieces were two-layer composites of the size 12.7 � 12.7 mm with an upper

This approach is applicable not only to the processing of superhard materials. It can also be successfully used for the treatment of special difficult-to-cut materials,

*P*j*WFv* ¼ *b*<sup>0</sup> þ *b*1*sk* þ *b*2*T* þ *b*3*Qv* þ *b*4*hl* þ … (15)

*Fv1*ð Þ *tk* <sup>∩</sup> *W1*

*Fv2*ð Þ *tk* <sup>∩</sup>*W1*

*n1*ð Þ *tk :* (14)

, are

tions, while a totality of such functions may be much bigger.

*Fp1*ð Þ *tk* <sup>∩</sup>*W1*

*W2i*ð Þ¼ *tk W1*

δj

*l*j

is the depth of the groove in the worked piece.

created for all the types of working.

logical analysis.

**research**

**181**

tions now (according to the number of components):

*h*j

can be presented as

**Worked material**

**Table 3.**

**Functions** *F*<sup>0</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.87094*

HA Intensive, linearly depending on *t* (number of cuts *N*)

PSHM Intensive, decreasing with increase of groove depth *hl*

**Obtaining of orthogonal edge of the product (depth** *h***, mm)**

*Cutting Superhard Materials by Jet Methods (on Functional Approach)*

Taking into account the boundary conditions in the cutting zone

$$\begin{aligned} c\rho \frac{dT}{dt} - \lambda \Delta T &= \frac{(T - R\_0)kP}{\pi AB} \exp \times \\ \times \left[ -2\left(\frac{(\varkappa - vt)^2}{a^2} + \left(\frac{\mathcal{V}}{b}\right)^2\right) \right] \times \exp\left(-kz\right) \end{aligned} \qquad \lambda \frac{dT}{dz}\Big|\_{z=0} = a(T)(T - T\_p),$$

temperature distribution across the sample section is obtained; its analysis reveals the following (**Figure 7**): temperature distribution across the surface is determined by conditions of coolant outflow and may considerably vary with the change of the flow shape; when the thickness of the sample increases, considerable reduction of the jet's ability to perform the work of destruction can be observed, as at the same amount of rejected heat, the amount of delivered heat constantly decreases.

**Figure 7.**

*Changes in border radius zone temperature field (T = 850°C) as a function of time for different processing methods LC (*�*, <sup>Ж</sup>,* ●*), LCC (*▲*), WJGL (*♦*,* ■*) (а) and cutting depth in materials (HA,* ♦*; PSTM,* ■*) depending on the number of cuts (b).*


**Table 3.**

The action of a water jet or a laser beam can be both simultaneous and successive. In this case different variants of laser blasting and possibilities of variation of the form of jet nozzle flow section, as shown in [6], provide good prospects in combination of methods of actions on the worked piece and, consequently, in

Application of thermo-hydrodynamic jets typical of WJGL or LCC causes heating of the workpiece whose temperature field can be described by equation [7]

*erfc <sup>z</sup>*

*ср*, and *h\** are density, specific heat capacity, heat conductivity coefficient,

where *t* is the time from the moment of beginning of thermal impulse action, ρ,

temperature conductivity of the workpiece material, and coefficient of heat transfer

smaller half axes of beam elliptic section; and *P* = π*qAB* is the power of the lasing emitter. Integral equation of heat energy balance in an arbitrary area ω ⊂ Ω

� exp ð Þ �*kz*

temperature distribution across the sample section is obtained; its analysis reveals the following (**Figure 7**): temperature distribution across the surface is determined by conditions of coolant outflow and may considerably vary with the change of the flow shape; when the thickness of the sample increases, considerable reduction of the jet's ability to perform the work of destruction can be observed, as at the same amount of

*Changes in border radius zone temperature field (T = 850°C) as a function of time for different processing methods LC (*�*, <sup>Ж</sup>,* ●*), LCC (*▲*), WJGL (*♦*,* ■*) (а) and cutting depth in materials (HA,* ♦*; PSTM,* ■*)*

*w ∂e <sup>∂</sup><sup>t</sup> dv* <sup>¼</sup> <sup>Ð</sup>

Taking into account the boundary conditions in the cutting zone

<sup>π</sup>*AB* exp �

<sup>2</sup>ð Þ *ατ* <sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> *<sup>h</sup>* <sup>∗</sup> ð Þ *ατ* <sup>1</sup>*=*<sup>2</sup> !

*<sup>w</sup> gdv* � ∮ *<sup>∂</sup><sup>w</sup> qT*

λ *dT dz* � � � � *z*¼0

" #

� *eh* <sup>∗</sup> *<sup>z</sup>*þ*<sup>h</sup>* <sup>∗</sup> <sup>2</sup>*ατ*

!*; n*

*ds:*

*ср А* and *В* are the bigger and

! <sup>þ</sup> *cg*ρ*gT vf*

¼ *α*ð Þ *T T* � *Tp* � �*,*

! � �

*dt ,*

*y*2 <sup>4</sup>*ατ*þ*B*<sup>2</sup> <sup>4</sup>*ατ* <sup>þ</sup> *<sup>A</sup>*<sup>2</sup> � � <sup>4</sup>*ατ* <sup>þ</sup> *<sup>B</sup>*<sup>2</sup> � �*ατ* � �1*=*<sup>2</sup> �

*e* ð Þ *<sup>x</sup>*�*v t*ð Þ �*<sup>z</sup>* <sup>2</sup> <sup>4</sup>*ατ*þ*A*<sup>2</sup>

<sup>4</sup>*ατ* � *<sup>h</sup>* <sup>∗</sup> ð Þ *πατ* <sup>1</sup>*=*<sup>2</sup>

*Recent Advancements in the Metallurgical Engineering and Electrodeposition*

� *<sup>e</sup>*� *<sup>z</sup>*<sup>2</sup>

from the surface of the workpiece, respectively;α ¼ <sup>λ</sup>

meeting principle (10).

*P* π 1 <sup>2</sup> ρ*c* ð*l* 0

according to [4] will take the form Ð

� �<sup>2</sup> ð Þ *<sup>x</sup>* � *vt* <sup>2</sup>

*dt* � *λΔ<sup>T</sup>* <sup>¼</sup> ð Þ *<sup>T</sup>* � *<sup>R</sup>*<sup>0</sup> *kP*

" # !

*<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*

*b* � �<sup>2</sup>

rejected heat, the amount of delivered heat constantly decreases.

*T x*ð Þ¼ *; y; z; t*

с, λ, α ¼ <sup>λ</sup>

*c*ρ *dT*

**Figure 7.**

**180**

*depending on the number of cuts (b).*

*Provision of the product function by creation of element* Еі *by a technological action Wij(tk)—WJGL.*

Thus, it is possible to create a table of provision of the product with the function via formation of its separate element or a totality of elements. This table is analogous to the one considered above (e.g., for WJGL; **Table 3**), but it reflects the peculiar feature of each of the variants in **Figure 6**. It should be noted that the given tables just illustrate the approach and draw attention to the most important functions, while a totality of such functions may be much bigger.

On the basis of **Table 3**, technological action for obtaining a particular element can be presented as

$$\mathcal{W}\_{2i}(\mathbf{t}\_k) = \mathcal{W}\_1^{Fp1}(\mathbf{t}\_k) \cap \mathcal{W}\_1^{Fv1}(\mathbf{t}\_k) \cap \mathcal{W}\_1^{Fv2}(\mathbf{t}\_k) \cap \mathcal{W}\_1^{v1}(\mathbf{t}\_k). \tag{14}$$

Corresponding transformations for LB for different materials include four equations now (according to the number of components):

$$h|\_{W^{\text{Fr}}} = b\_0 + b\_1 s\_k + b\_2 T + b\_3 Q\_v + \dots;$$

$$\delta|\_{W^{\text{Fr}}} = b\_0 + b\_1 s\_k + b\_2 T + b\_3 Q\_v + b\_4 h\_l + \dots;$$

$$d|\_{W^{\text{Fr}}} = b\_0 + b\_1 s\_k + b\_2 T + b\_3 Q\_v + b\_4 h\_l + \dots$$

$$P|\_{W^{\text{Fr}}} = b\_0 + b\_1 s\_k + b\_2 T + b\_3 Q\_v + b\_4 h\_l + \dots \tag{15}$$

where *Т* is the impulse energy, *Q <sup>v</sup>* is the liquid discharge out of the nozzle, and *hl* is the depth of the groove in the worked piece.

Analogous tables, including the ones with extended features of functions *F*<sup>0</sup> , are created for all the types of working.

The obtained dependences (12) and (14) make it possible to determine the totality of variants for carrying out the process of working of a particular element of the surface, and the most optimal method can be searched on the basis of morphological analysis.

This approach is applicable not only to the processing of superhard materials. It can also be successfully used for the treatment of special difficult-to-cut materials, such as carbon-carbon composites, multilayer aircraft panels, etc. [9–11].
