**5. Computer simulation of radial-direct extrusion of forged piece with the spherical cavity and flange from cylindrical compact with axial hole and relieving cavity**

The effect of the generatrix inclination angle α, radius of sphere R and size of cone-shaped relieving cavity on the non-uniformity of stress-strain state and temperature field has been investigated. The angle α was equal to 15°, 30°, 40° and sphere's radius have changed from 6 to 16 mm.

As a result of implementation the relieving cavity with α = 15°, the non-uniformity of stressstrain state decreased, in compare with extrusion of billet without the cavity, but was not completely eliminated (Fig. 9). The maximum stress intensities in the surface layers of the spherical cavity of forged piece for all three sections have found (Fig. 9, a). The intensity of stress decreases at increasing of the distance from the cavity surface, especially in the most dangerous section OC down to 52 MPa. Intensity of deformation maximized at the distance of 1.9 - 2.4 mm from the surface, indicating the risk of flow-through flaw formation, and then also decreased (Fig. 9, b).

In this case, the intensity of stress and deformation values during extrusion of billet with generatrix inclination angle of relieving cavity 15° are lower than without it.

**Figure 9.** The distribution of the intensity of stress and intensity of deformation during extrusion of billet with reliev‐ ing cavity (α = 15°): 1 - is the section OA; 2 - is the section OB; 3 - is the section OC.

Thus, implementation of compacts with the relieving cavity and α =15° was not ensured decreasing of non-uniformity of stress-strain state to an appropriate level. Consequently, in the transition region of spherical cavity in the hole during the final extrusion step a flaw is formed, but was not developed into a fold as the result of decreasing the non-uniformity of stress-strain state.

Dependences of the intensity of stress by layers of powder material for various radii of spherical cavity R and different values of angle α are presented on Fig. 11 according to the

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131

**Figure 11.** Dependences of maximum intensity of stress from the angle α and radius R: 1 - R = 16 mm; 2 - R = 14 mm; 3

Analysis of dependences has shown that intensity of stress reaches a minimum at α = 30 - 40°. The angle α is close to 40° at increasing of R, and angle α is close to 30° while decreasing

Thus, to obtain the most uniform stress-strain state during radial-direct extrusion of forged pieces at Dflange/Dout = 1.1, the billet with relieving cavity (Fig. 4, b) and α = 30 - 40° may be

To verify the validity of this conclusion the distribution of stress-strain state parameters at α = 40° in three sections of forged piece are presented on Fig. 12. Retrieved reduction of nonuniformity of stress-strain state by 6-10% at extrusion, according to (9), has compared with

The maximal values of the intensity of deformations are lower and shifted to deeper layers of forged piece at 2.3 - 4.0 mm. The results of the non-uniformity analysis of stress-strain state by sections of forged piece with relieving cavity angle α = 40° are presented in Table 3. The non-uniformity of stress-strain states is lower for all three sections, in compare with

Therefore, extrusion of porous powder billets with relieving cavity having a generatrix

Improvement of the uniformity of stress-strain state by using billets with relieving cavity allows obtaining a more uniform temperature field by the section of forged piece (Fig. 13).

Thus, the comparative analysis of the radial-direct extrusion of cylindrical billets with generatrix inclination angle α within 30 - 40° has shown that the presence of relieving cavity

inclination angle within 30 - 40° provides a uniform stress-strain state.

of R. This means that the range of permissible values of angle α is within 30 - 40°.

modelling results.

recommended.

extrusion at α = 30°.

extrusion at α = 30°.


Increasing the angle α up to 30° reduces non-uniformity of stress-strain state on 30 %. The largest and smallest σ<sup>i</sup> and ei differ by 1.7 times and 2.1 times, respectively (Fig. 10). Parameters of stress-strain state are distributed more uniformly by sections. Consequently, a flow-through flaw was not formed in forged pieces during extrusion.

The results of analysis of non-uniformity of stress-strain state by sections of forged piece during extrusion of billets with relieving cavity at α = 15° and α = 30° are presented in Table 2.


**Table 2.** Evaluation of non-uniformity of stress-strain state at various values of the inclination angle of relieving cavity generatrix

The non-uniformity of stress-strain state in the sections OB and OC decreases with increasing of inclination angle of the relieving cavity generatrix that improving quality of forged piece, but does not completely eliminates defects. Further reduction of non-uniform‐ ity of stress-strain state by increasing the angle α was confirmed by simulation of radial direct extrusion of billets with spherical cavity at inclination angle α = 5 - 45° and radius of sphere R = 6 - 16 mm.

**Figure 10.** The distribution of the intensity of stress – (a) and deformation – (b) during extrusion of billets with reliev‐ ing cavity (α = 30°): 1 - is the section OA; 2 - is the section OB; 3 - is the section OC.

Dependences of the intensity of stress by layers of powder material for various radii of spherical cavity R and different values of angle α are presented on Fig. 11 according to the modelling results.

Thus, implementation of compacts with the relieving cavity and α =15° was not ensured decreasing of non-uniformity of stress-strain state to an appropriate level. Consequently, in the transition region of spherical cavity in the hole during the final extrusion step a flaw is formed, but was not developed into a fold as the result of decreasing the non-uniformity of

Increasing the angle α up to 30° reduces non-uniformity of stress-strain state on 30 %. The

of stress-strain state are distributed more uniformly by sections. Consequently, a flow-through

The results of analysis of non-uniformity of stress-strain state by sections of forged piece during extrusion of billets with relieving cavity at α = 15° and α = 30° are presented in Table 2.

> OA 0.25 0.15 0.13 0.17 OB 0.28 0.10 0.14 0.12 OC 0.33 0.13 0.23 0.15

**Section of billet <sup>σ</sup>inh einh**

**Table 2.** Evaluation of non-uniformity of stress-strain state at various values of the inclination angle of relieving cavity

The non-uniformity of stress-strain state in the sections OB and OC decreases with increasing of inclination angle of the relieving cavity generatrix that improving quality of forged piece, but does not completely eliminates defects. Further reduction of non-uniform‐ ity of stress-strain state by increasing the angle α was confirmed by simulation of radial direct extrusion of billets with spherical cavity at inclination angle α = 5 - 45° and radius

**Figure 10.** The distribution of the intensity of stress – (a) and deformation – (b) during extrusion of billets with reliev‐

ing cavity (α = 30°): 1 - is the section OA; 2 - is the section OB; 3 - is the section OC.

differ by 1.7 times and 2.1 times, respectively (Fig. 10). Parameters

**α = 15° α = 30° α = 15° α = 30°**

stress-strain state.

generatrix

of sphere R = 6 - 16 mm.

largest and smallest σ<sup>i</sup>

130 Computational and Numerical Simulations

and ei

flaw was not formed in forged pieces during extrusion.

**Figure 11.** Dependences of maximum intensity of stress from the angle α and radius R: 1 - R = 16 mm; 2 - R = 14 mm; 3 - R = 12 mm; 4 - R = 10 mm; 5 - R = 8 mm; 6 - R = 6 mm.

Analysis of dependences has shown that intensity of stress reaches a minimum at α = 30 - 40°. The angle α is close to 40° at increasing of R, and angle α is close to 30° while decreasing of R. This means that the range of permissible values of angle α is within 30 - 40°.

Thus, to obtain the most uniform stress-strain state during radial-direct extrusion of forged pieces at Dflange/Dout = 1.1, the billet with relieving cavity (Fig. 4, b) and α = 30 - 40° may be recommended.

To verify the validity of this conclusion the distribution of stress-strain state parameters at α = 40° in three sections of forged piece are presented on Fig. 12. Retrieved reduction of nonuniformity of stress-strain state by 6-10% at extrusion, according to (9), has compared with extrusion at α = 30°.

The maximal values of the intensity of deformations are lower and shifted to deeper layers of forged piece at 2.3 - 4.0 mm. The results of the non-uniformity analysis of stress-strain state by sections of forged piece with relieving cavity angle α = 40° are presented in Table 3. The non-uniformity of stress-strain states is lower for all three sections, in compare with extrusion at α = 30°.

Therefore, extrusion of porous powder billets with relieving cavity having a generatrix inclination angle within 30 - 40° provides a uniform stress-strain state.

Improvement of the uniformity of stress-strain state by using billets with relieving cavity allows obtaining a more uniform temperature field by the section of forged piece (Fig. 13).

Thus, the comparative analysis of the radial-direct extrusion of cylindrical billets with generatrix inclination angle α within 30 - 40° has shown that the presence of relieving cavity

**Figure 12.** The distribution of the intensity of stress and intensity of deformation at extrusion of billets with relieving cavity (α = 40°): 1 - is the section OA; 2 - is the section OB; 3 - is the section OC.

adjacent to the surface of sphere and in the flange. A density close to 7.83 g/cm3

**Figure 13.** The temperature field of the billet at extrusion of billet with relieving cavity at α = 40°.

In two other cases, it decreases by the volume of flange, but with a smaller gradient. A high density obtained for Dflange/Dout = 1.1 - 1.2 at different initial density while increasing angle α to 30° (Fig. 14, c). However, if Dflange / Dout = 1.3 the density of compact material obtained. The high density has obtained at the cavity angle α = 40° for Dflange/Dout = 1.1 - 1.2 (Fig. 13, d). It is rather difficult to change the volume of forged piece at any density for Dflange/Dout = 1.3. Pores and cracks were appeared on spherical surface of the metal due to increased loosening at radial

Simulation of the density distribution at Dflange/Dout = 1.3 during extrusion of billets with 10 % initial porosity and inclination angle of relieving cavity generatrix 40° shown the density variation within 7.79 - 7.81 g/cm<sup>3</sup> that indicates a possibility to obtain the equidence and high-

One of the most important problems during development of metal forming technologies for powder billets is determination of deforming force that is necessary for reasonable choice of pressing equipment. The analytical expression of extrusion force P in polar coordinates for

may be written in the following way (Wagoner & Chenot, 2001):

2

p

 sf

,

0 0

*r i i*

*P rdrd dr rd*

 f

= = s

where F - is the area of contact surface of upper punch and powder billet.

*F*

ave on the contact surface of upper punch and powder billet

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133

òò ò ò (13)

in the billet at α = 15° only at the ratio Dflange/Dout = 1.1 (Fig. 14, b).

flow of metal in the gap and on the side of flange.

**6. Determination of extrusion force**

known average intensity of stress σ<sup>i</sup>

strength details.

or function for σ<sup>i</sup>

has obtained

increases the uniformity of the temperature field and stress-strain state. This helps to reduce the stress intensity and stress concentration factor kσ at the edges of the cavity by 3 - 4 times.


**Table 3.** Evaluation of non-uniformity of stress-strain state at the generatrix inclination angle α = 40°

As a result, the probability of flow-through flaw formation and risk of crack propagation during extrusion have diminished rapidly.

Analysis of the density changing by the volume of forged piece was simulated for extrusion of billet with the initial porosity 15 %, outer diameter Dout = 27 mm and diameter of hole 9 mm. The density variation and equidensity at different conditions of radial direct extrusion of forged piece have been investigated.

The most difficult is to ensure equidensity at extrusion of flange of forged piece due to tensile stresses. Therefore, density distribution is presented by section OA (Fig. 14) where the highest probability of defects formation occurs.

The maximum density of 7.77 g/cm3 during extrusion of billets without relieving cavity at ratio Dflange/Dout 1.1 - 1.3 (Fig. 14, a) was reached in the volume of metal adjacent to the surface of forged piece at Dflange/Dout = 1.1 that does not corresponding to the density of compact material. This is due to the increase of tensile stress, leading to a tightening of the surface layer of the metal forging deeper. Moreover, the greater a flange, the more decrease in density of metal Computer Modelling of Radial-Direct Extrusion of Porous Powder Billets http://dx.doi.org/10.5772/57142 133

**Figure 13.** The temperature field of the billet at extrusion of billet with relieving cavity at α = 40°.

adjacent to the surface of sphere and in the flange. A density close to 7.83 g/cm3 has obtained in the billet at α = 15° only at the ratio Dflange/Dout = 1.1 (Fig. 14, b).

In two other cases, it decreases by the volume of flange, but with a smaller gradient. A high density obtained for Dflange/Dout = 1.1 - 1.2 at different initial density while increasing angle α to 30° (Fig. 14, c). However, if Dflange / Dout = 1.3 the density of compact material obtained. The high density has obtained at the cavity angle α = 40° for Dflange/Dout = 1.1 - 1.2 (Fig. 13, d). It is rather difficult to change the volume of forged piece at any density for Dflange/Dout = 1.3. Pores and cracks were appeared on spherical surface of the metal due to increased loosening at radial flow of metal in the gap and on the side of flange.

Simulation of the density distribution at Dflange/Dout = 1.3 during extrusion of billets with 10 % initial porosity and inclination angle of relieving cavity generatrix 40° shown the density variation within 7.79 - 7.81 g/cm<sup>3</sup> that indicates a possibility to obtain the equidence and highstrength details.

### **6. Determination of extrusion force**

increases the uniformity of the temperature field and stress-strain state. This helps to reduce the stress intensity and stress concentration factor kσ at the edges of the cavity by 3 - 4 times.

**Figure 12.** The distribution of the intensity of stress and intensity of deformation at extrusion of billets with relieving

OA 0.08 0.01 OB 0.06 0.01 OC 0.08 0.03

As a result, the probability of flow-through flaw formation and risk of crack propagation

Analysis of the density changing by the volume of forged piece was simulated for extrusion of billet with the initial porosity 15 %, outer diameter Dout = 27 mm and diameter of hole 9 mm. The density variation and equidensity at different conditions of radial direct extrusion of

The most difficult is to ensure equidensity at extrusion of flange of forged piece due to tensile stresses. Therefore, density distribution is presented by section OA (Fig. 14) where the highest

Dflange/Dout 1.1 - 1.3 (Fig. 14, a) was reached in the volume of metal adjacent to the surface of forged piece at Dflange/Dout = 1.1 that does not corresponding to the density of compact material. This is due to the increase of tensile stress, leading to a tightening of the surface layer of the metal forging deeper. Moreover, the greater a flange, the more decrease in density of metal

during extrusion of billets without relieving cavity at ratio

**Section of billet σinh einh**

**Table 3.** Evaluation of non-uniformity of stress-strain state at the generatrix inclination angle α = 40°

cavity (α = 40°): 1 - is the section OA; 2 - is the section OB; 3 - is the section OC.

during extrusion have diminished rapidly.

forged piece have been investigated.

132 Computational and Numerical Simulations

probability of defects formation occurs.

The maximum density of 7.77 g/cm3

One of the most important problems during development of metal forming technologies for powder billets is determination of deforming force that is necessary for reasonable choice of pressing equipment. The analytical expression of extrusion force P in polar coordinates for known average intensity of stress σ<sup>i</sup> ave on the contact surface of upper punch and powder billet or function for σ<sup>i</sup> may be written in the following way (Wagoner & Chenot, 2001):

$$P = \iint\_{F} \sigma\_{i} r dr d\phi = \int dr \int\_{0}^{2\pi} \sigma\_{i} r d\phi \,\prime \,\tag{13}$$

where F - is the area of contact surface of upper punch and powder billet.

α, ° σ*<sup>i</sup>*

Dflange/Dout σ*<sup>i</sup>*

**Table 5.** The extrusion force at different flange size, α= 40° (Dsf = 20 mm)

generatrix inclination angle within 30 - 40°.

angle of relieving cavity within 30 - 40° and 10 % initial porosity.

The highest density of 7.80 - 7.83 g/cm3

and density 7.79 - 7.81 g/cm3

relative error 7 - 9 %.

**7. Conclusions**

*ave*

**Table 4.** Extrusion force at different inclination angles of generatrix of relieving cavity for Dflange/Dout = 1.1

*ave*

1.1 84.5 58.4 1.2 106.2 89.6 1.3 127.4 124.5

The results of computer modelling and laboratory experiments are well concordant with

In this chapter the results of computer modelling of radial direct extrusion of forged piece with the spherical cavity and small flange from a cylindrical billet with a porosity of 15 % and axial hole have shown a high non-uniformity of the stress-strain state, temperature field and density distribution by sections of forged piece that leads to appearing of defects are reported. It has been shown, among other, how the smallest non-uniformity of the stress-strain state, temper‐ ature field and maximum density indicate a possibility to obtain high-quality products.

A high-quality details with a spherical cavity and the ratio Dflange /Dout = 1.2 may be obtained from powder billets with 15 % initial porosity and relieving cavity with generatrix inclination angle 40°. Details with the ratio Dflange / Dout = 1.3 may not be produced from such billets due to the presence of cracks and non-uniformity into the flange. High-quality details with the ratio Dflange/Dout = 1.3 may be made from billets of 10 % initial porosity and relieving cavity with

at the ratio Dflange/Dout = 1.1 - 1.2, obtained from billets with generatrix inclination angle of relieving cavity within 30 - 40° and initial porosity of 15 %. Forged pieces with Dflange/Dout = 1.3

The simulation and experimental results are well concordant with relative error 7 - 9 %.

 133.1 92.0 118.2 81.7 103.7 71.6 84.5 58.4

, MPa P, kN

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135

, MPa P, kN

and the equidensity of flange observed in forged pieces

may be produced from powder billets with generatrix inclination

**Figure 14.** Density distribution by the section OA of forged piece (a) - with no relieving cavity; (b, c, d) - with the reliev‐ ing cavity (generatrix inclination angle α is 15, 30, 40°, respectively): 1 - Dflange / Dout = 1.1; 2 - Dflange / Dout =1.2; 3 - Dflange / Dout = 1.3.

The expression (13) for radial-direct extrusion of forged piece with spherical cavity and central hole with accounting the modelling results may be transformed to:

$$P = \int\_{D\_{\rm hole}}^{D\_{\rm flong}} \frac{1}{2} dD \int\_{0} \sigma\_{i} D \, d\rho = \frac{\sigma\_{i}^{\rm av} \pi \{D\_{\rm flang}^{2} - D\_{\rm hole}^{2}\}}{4},\tag{14}$$

where Dflange - is the outer diameter of flange;

Dhole - is the diameter of hole.

Values of extrusion force calculated by formula (14) at different inclination angles of generatrix of relieving cavity for Dflange/Dout = 1.1 and height of the cavity 6 mm are presented in the Table 4. The dependence of extrusion force from the relative flange size is presented in the Table 5 (α = 40°, diameter of sphere Dsf = 20 mm).

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**Table 4.** Extrusion force at different inclination angles of generatrix of relieving cavity for Dflange/Dout = 1.1


**Table 5.** The extrusion force at different flange size, α= 40° (Dsf = 20 mm)

The results of computer modelling and laboratory experiments are well concordant with relative error 7 - 9 %.

### **7. Conclusions**

The expression (13) for radial-direct extrusion of forged piece with spherical cavity and central

**Figure 14.** Density distribution by the section OA of forged piece (a) - with no relieving cavity; (b, c, d) - with the reliev‐ ing cavity (generatrix inclination angle α is 15, 30, 40°, respectively): 1 - Dflange / Dout = 1.1; 2 - Dflange / Dout =1.2; 3 -

> *σi ave*

Values of extrusion force calculated by formula (14) at different inclination angles of generatrix of relieving cavity for Dflange/Dout = 1.1 and height of the cavity 6 mm are presented in the Table 4.

The dependence of extrusion force from the relative flange size is presented in the Table 5 (α

*<sup>π</sup>*(*Dflange* <sup>2</sup> <sup>−</sup>*Dhole*

<sup>2</sup> )

<sup>4</sup> , (14)

hole with accounting the modelling results may be transformed to:

2*π σi Ddφ* =

*P* = *∫ Dhole*

where Dflange - is the outer diameter of flange;

= 40°, diameter of sphere Dsf = 20 mm).

Dhole - is the diameter of hole.

134 Computational and Numerical Simulations

Dflange / Dout = 1.3.

*Dflange* 1 <sup>2</sup> *dD ∫* 0 In this chapter the results of computer modelling of radial direct extrusion of forged piece with the spherical cavity and small flange from a cylindrical billet with a porosity of 15 % and axial hole have shown a high non-uniformity of the stress-strain state, temperature field and density distribution by sections of forged piece that leads to appearing of defects are reported. It has been shown, among other, how the smallest non-uniformity of the stress-strain state, temper‐ ature field and maximum density indicate a possibility to obtain high-quality products.

A high-quality details with a spherical cavity and the ratio Dflange /Dout = 1.2 may be obtained from powder billets with 15 % initial porosity and relieving cavity with generatrix inclination angle 40°. Details with the ratio Dflange / Dout = 1.3 may not be produced from such billets due to the presence of cracks and non-uniformity into the flange. High-quality details with the ratio Dflange/Dout = 1.3 may be made from billets of 10 % initial porosity and relieving cavity with generatrix inclination angle within 30 - 40°.

The highest density of 7.80 - 7.83 g/cm3 and the equidensity of flange observed in forged pieces at the ratio Dflange/Dout = 1.1 - 1.2, obtained from billets with generatrix inclination angle of relieving cavity within 30 - 40° and initial porosity of 15 %. Forged pieces with Dflange/Dout = 1.3 and density 7.79 - 7.81 g/cm3 may be produced from powder billets with generatrix inclination angle of relieving cavity within 30 - 40° and 10 % initial porosity.

The simulation and experimental results are well concordant with relative error 7 - 9 %.
