**4. Computer simulation of radial-direct extrusion of forged piece with a spherical cavity and flange from cylindrical billet with axial hole**

Finite element simulation of extrusion of forged piece from cylindrical billet with a hole has shown clearly that maximum intensities of stress and deformation observed in the surface layers of the spherical cavity by sections of forged piece OA, OB and OC. These values were decreased gradually while increasing the distance from the surface of the forged piece (Fig. 5). The maximum values of the studied variables and dramatic non-uniformity of stress-strain state have been observed in the section OC, where the highest probability of defects formation established. The intensity of stress reduced to 5.28 times while growing the distance from the surface of the forged piece, the intensity of deformation decreased to 1.6 times. At a distance of 1.8 - 2.0 mm from the surface of the cavity there is a maximum of values due to overcooling of the metal in the area of transition of spherical cavity to the hole.

Non-uniformity of temperature field and stress-strain state of forged piece creates conditions for the formation of the fold at a distance of 2 - 3 mm from the edge of the hole, which is gradually transformed into the flow-through flaw and then leads to loss of plastic equilibrium and cracking of products. The stages of flow-through flaw defect evolution (Fig. 7, a) to the fold (Fig. 7, b, c) and crack (Fig. 7, d) have been observed. The retraction of the surface layer inside of forged piece around the flaw transforms it to the fold. Later, under pressure from a punch, the cavity of fold collapses, edges sharpening and becoming stress concentrators with

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(a) (b) (c) (d)

The stress state near the defect is significantly non-uniform (Fig. 8) and in the layers of metal adjacent to the surface of the fold, the stress intensity was about 160 - 240 MPa and the stress intensity corresponds to the lower edge of the spherical cavity surface within 180 - 250 MPa.

Estimation of stress concentration, considering the influence of temperature and strain rate conditions on properties of deformable material has provided using technical stress concen‐ tration factor aσ with taking into account the structure and plastic properties of powder

> max , *<sup>i</sup> i*

Stress concentration factor on the surface of forged piece a<sup>σ</sup> = 2.5 - 3.0. After closure of the fold stress concentrator formed at its end, leading to increase in stress concentration factor aσ up to 4.0 - 6.0, resulting formation of cracks into the forged piece. This promotes the evolution of folds in a failure, and also causes deterioration of the spherical surface in the region of its transition into the inner hole. This cracking is accompanied by stress relaxation, which leads

<sup>=</sup> (11)

s

s

*a*s

max - is the highest intensity of stress near the defect;


to reduction of stress down to 70 MPa after crack propagation.

**Figure 7.** Evolution of flow-through flaw to a fold at extrusion of powder billets with cylindrical hole.

material (Skorokhod, 1985; Ryabicheva, 2012):

where *σ<sup>i</sup>*

*σi*

following initiation and propagation of crack and formation of failure.

**Figure 5.** Distribution of the intensity of stress and intensity of deformation during extrusion of billet with a cylindrical hole in the sections: 1 - OA; 2 - OB; 3 - OC.

Non-uniformity of stress-strain state is a significant cause of non-uniformity of the temperature field (Fig. 6). The temperature rises up to 1200 °C due to the thermal effect of plastic deforma‐ tion while increasing the distance from the surface of the billet. The overcooled layer is formed at the points of contact with the forging tool due to heat transfer by conduction and convection.

**Figure 6.** The temperature field of billet at extrusion of cylindrical powder billets with a hole.

Non-uniformity of temperature field and stress-strain state of forged piece creates conditions for the formation of the fold at a distance of 2 - 3 mm from the edge of the hole, which is gradually transformed into the flow-through flaw and then leads to loss of plastic equilibrium and cracking of products. The stages of flow-through flaw defect evolution (Fig. 7, a) to the fold (Fig. 7, b, c) and crack (Fig. 7, d) have been observed. The retraction of the surface layer inside of forged piece around the flaw transforms it to the fold. Later, under pressure from a punch, the cavity of fold collapses, edges sharpening and becoming stress concentrators with following initiation and propagation of crack and formation of failure.

state have been observed in the section OC, where the highest probability of defects formation established. The intensity of stress reduced to 5.28 times while growing the distance from the surface of the forged piece, the intensity of deformation decreased to 1.6 times. At a distance of 1.8 - 2.0 mm from the surface of the cavity there is a maximum of values due to overcooling

**Figure 5.** Distribution of the intensity of stress and intensity of deformation during extrusion of billet with a cylindrical

Non-uniformity of stress-strain state is a significant cause of non-uniformity of the temperature field (Fig. 6). The temperature rises up to 1200 °C due to the thermal effect of plastic deforma‐ tion while increasing the distance from the surface of the billet. The overcooled layer is formed at the points of contact with the forging tool due to heat transfer by conduction and convection.

**Figure 6.** The temperature field of billet at extrusion of cylindrical powder billets with a hole.

of the metal in the area of transition of spherical cavity to the hole.

hole in the sections: 1 - OA; 2 - OB; 3 - OC.

126 Computational and Numerical Simulations

**Figure 7.** Evolution of flow-through flaw to a fold at extrusion of powder billets with cylindrical hole.

The stress state near the defect is significantly non-uniform (Fig. 8) and in the layers of metal adjacent to the surface of the fold, the stress intensity was about 160 - 240 MPa and the stress intensity corresponds to the lower edge of the spherical cavity surface within 180 - 250 MPa.

Estimation of stress concentration, considering the influence of temperature and strain rate conditions on properties of deformable material has provided using technical stress concen‐ tration factor aσ with taking into account the structure and plastic properties of powder material (Skorokhod, 1985; Ryabicheva, 2012):

$$a\_{\sigma} = \frac{\sigma\_i^{\text{max}}}{\sigma\_i},\tag{11}$$

where *σ<sup>i</sup>* max - is the highest intensity of stress near the defect;

*σi* - is the intensity of stress under the given deforming conditions.

Stress concentration factor on the surface of forged piece a<sup>σ</sup> = 2.5 - 3.0. After closure of the fold stress concentrator formed at its end, leading to increase in stress concentration factor aσ up to 4.0 - 6.0, resulting formation of cracks into the forged piece. This promotes the evolution of folds in a failure, and also causes deterioration of the spherical surface in the region of its transition into the inner hole. This cracking is accompanied by stress relaxation, which leads to reduction of stress down to 70 MPa after crack propagation.

**Figure 8.** The stress state during the evolution of fold to crack.

The non-uniformity coefficients of stress σinh and deformations einh were implemented for estimation the non-uniformity of stress-strain state:

$$\sigma\_{\rm inh} = \frac{\sum\_{j=1}^{N} \sqrt{\left(\sigma\_i^{\rm ave} - \sigma\_i^j\right)^2}}{\sigma\_i^{\rm ave} N}, \quad e\_{\rm inh} = \frac{\sum\_{j=1}^{N} \sqrt{\left(e\_i^{\rm ave} - e\_i^j\right)^2}}{e\_i^{\rm ave} N},\tag{12}$$

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

**Section of billet σinh einh**

**Table 1.** Non-uniformity of stress-strain state by sections of forged piece

OA 0.29 0.31 OB 0.33 0.36 OC 0.41 0.56

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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

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

In this case, the intensity of stress and deformation values during extrusion of billet with

**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.

generatrix inclination angle of relieving cavity 15° are lower than without it.

**relieving cavity**

decreased (Fig. 9, b).

to 16 mm.

where *σ<sup>i</sup> ave* - is the average stress intensity in the volume of billet;

*ε*˙*i ave* - is the average intensity of deformation in the volume of billet;

*σi j* - is the stress intensity into a finite element j;

*ε*˙*i j* - is the intensity of deformation into a finite element j;

N - is the number of finite elements inside the model.

In case of uniform deformation the values of σinh and einh are asymptotically approaching zero.

The results of analysis of non-uniformity of stress-strain state by sections of forged piece confirms that the highest non-uniformity of stress-strain state has been observed in section OC, which corresponds to retraction of surface layers of the metal during formation of flowthrough flaw (Table 1).

Thus, conditions that are leading to formation of defects have established by numerical simulation of extrusion of the porous powder billet with a hole.


**Table 1.** Non-uniformity of stress-strain state by sections of forged piece

The non-uniformity coefficients of stress σinh and deformations einh were implemented for

( ) ( ) 2 2 1 1 , ,


*e e*

å å (12)

*N N ave j j ave i i i i*

*i i*

In case of uniform deformation the values of σinh and einh are asymptotically approaching zero. The results of analysis of non-uniformity of stress-strain state by sections of forged piece confirms that the highest non-uniformity of stress-strain state has been observed in section OC, which corresponds to retraction of surface layers of the metal during formation of flow-

Thus, conditions that are leading to formation of defects have established by numerical

*e N e N*

*j j inh ave inh ave*

= =

 s

= =



s

s

estimation the non-uniformity of stress-strain state:

**Figure 8.** The stress state during the evolution of fold to crack.

128 Computational and Numerical Simulations



simulation of extrusion of the porous powder billet with a hole.

N - is the number of finite elements inside the model.

s

where *σ<sup>i</sup>*

*ε*˙*i ave*

*σi j*

*ε*˙*i j* *ave*

through flaw (Table 1).
