Nonuniformity of stressstrain state by sections of forged piece
1. Introduction
Improving the competitiveness of engineering products related to enhancement of extrusion technologies using computer modelling of the material behaviour that allows production of highquality products (Aliev et al., 2001; Favrot et al., 1997, Ryabicheva, 2012).
It is well known that a wide range of complexshaped parts with flanges and spherical cavities are applied in machinebuilding and operating at variable loadings and high wear conditions. This is why the mentioned parts are produced of compact materials by various types of extrusion. The extrusion techniques are less used for production parts from powder materials due to presence of residual porosity and density variation. The extrusion technologies for parts with spherical cavities producing in the automotive industry have studied insufficiently. Production of parts may be carried out using various deformation schemes by selection the optimal initial shape and porosity of billets, as well as the deformation temperature. The most common process flowsheet for parts from powder materials is the scheme involving pressing of billet (compact), sintering and subsequent final stamping to obtain the necessary accuracy and density (Ryabicheva et al., 2011).
Finite element simulation is the most effective way for determination of optimal process variables of forming operations. However, simulation of extrusion of porous billets from powder materials with taking into account dependences of mechanical properties from porosity, thermal and strain rate deforming conditions does not allow to estimate the convergence of finite element method (Awrejcewicz et al., 2004; Awrejcewicz & Pyryev, 2009). Mathematical formulation of the nonlinear coupled thermal plasticity problem makes necessary implementation of advanced solution methods for systems of linear algebraic equations (Awrejcewicz et al., 2007).
This work aims on improvement a quality of automotive parts based on a theoretical analysis of the stressstrain state, temperature fields and density distribution during radialdirect extrusion of porous powder billets.
2. The mathematical model of radial direct extrusion of porous powder billets
In this chapter mathematical modelling has been conducted on the basis of plasticity theory of porous bodies and focused on sequential solving the following problems:
construction a system of differential equations of the nonlinear coupled thermal plasticity problem for threedimensional model of billetstamp system on the basis of the laws of plasticity theory of porous bodies with taking into account density distribution and other singularities of deformable porous body (Awrejcewicz et al., 2007; Ryabicheva & Orlova, 2012; Segal et al., 1981);
application of a finite element method for solving of nonlinear coupled thermal plasticity problem (Awrejcewicz et al., 2007, Lienhard IV & Lienhard V, 2003);
proving the stability of the finite element solution for the examined class of problems (Lienhard IV & Lienhard V, 2003; Awrejcewicz et al., 2007; Awrejcewicz & Pyryev, 2009);
formulation of the method and solving of physically nonlinear coupled thermal plasticity problem for a threedimensional billetstamp model of radialdirect extrusion of porous powder billets and examine the influence of temperature and deformation fields’ coupling on simulation results (Awrejcewicz et al., 2007; Awrejcewicz& Pyryev, 2009; Ryabicheva, 2012);
verification of the results of finite element simulation by experimental investigation of radialdirect extrusion of porous powder billets (Ryabicheva et al., 2012).
The plastic potential is considered as a function of stress tensor components corresponding to smooth, convex, closed surface into the stress space (Shtern et al., 1982; Ryabicheva & Orlova, 2012). This potential may be presented in the following way (Shtern, 1981; Segal et al., 1981):
where
θ  is the porosity;
m  is the parameter characterizing the degree of imperfection of the contacts in the powder billet and defining different resistance of a porous body during its testing in tension and compression. The rate of volume change resulting from the plastic deformation is presented by the expression (Shtern et al., 1982; Segal et al., 1981):
where σ_{0}  is the flow stress of hard phase, which is a function of accumulated deformation ω and is determined by a hardening curve of powder material at uniaxial tension.
A flow stress of hard phase may be expressed as the function
where γ  is the shape changing rate.
The value of accumulated deformation ω is renewed by solving of differential equation (Skorokhod, 1973; Shtern et al., 1982; Segal et al., 1994):
where W  is the equivalent strain rate:
The finite element method presented as a series of procedures has used for determination of distributions of stress and strain intensity, as well as density in the volume of porous billet. The first procedure is triangulation of plastically deformed body or transition from a continuum billet to its finite element counterpart. Such simulation requires implementation of extremal requirement for the functional (Shtern et al., 1982, Segal et al., 1981):
where
The first integral in (6) is the total rate of energy dissipation, the second integral  is the power of the external stresses. For a porous body, which deforms plastically, the dissipation function
where
The stressstrain state of porous powder billet and density distribution at radial direct extrusion may be calculated using dependences (1)  (7) and specific mathematical approaches with implementation of a Hilbert space (Awrejcewicz et al., 2007) and advanced solution method for a system of linear algebraic equations obtained by finite element discretization of a volume of porous powder billet (Awrejcewicz et al., 2007, Awrejcewicz & Pyryev, 2009, Ryabicheva et al., 2012).
The technology of radial direct extrusion of forged pieces from water atomized steel powder Ancorsteel^{®} 150 HP with a spherical cavity and small flange with the ratio D_{flange} /D_{out} = 1.1 has been considered.
Temperature changes by sections of billet were determined using a heat conduction law. The analysis of interaction of contact surfaces has been conducted during each loading step, so, for elements inside of billet and in contact with tool surfaces heat conduction is determined only. The Fourier heat conduction differential equation was implemented for calculation of temperature field (Lienhard IV & Lienhard V, 2003):
where
k_{T}  is the summary heat conduction coefficient;
C  is the specific heat capacity;
ρ  is the density of material;
T  is the temperature, K;
τ  is the loading time step.
The minimum of heat conduction functional is related to each loading step (Segal et al., 1981; Lienhard IV & Lienhard V, 2003):
The outward heat transfer between medium and surface of the billet is carrying out by convective heat transfer. The boundary conditions of the third kind have implemented on the surface of the billet (Lienhard IV & Lienhard V, 2003; Ryabicheva et al., 2012):
where
The predictorcorrector method and Arbitrary Lagrangian Eulerian (ALE) formulation were implemented for more effective solving of nonlinear coupled thermal plasticity problem and prevention of gradual distortion of mesh due to severe plastic deformations during radialdirect extrusion. The transfinite mapping method is used to create an initial mesh and remeshing (Wisselink, 2000; Stoker, 1999).
3. Initial data for modelling
Two variants of radial direct extrusion, which are different by a shape of initial billet, have been considered. The first variant of billet is the bushing with a hole and the second is the bushing with a hole and a spherical cavity in the upper butt end. The input data for simulation: the strain rate is 0.5 m/s, deformation temperature interval is 1100  900 °C, friction coefficient 0.2, initial porosity of powder billet is 15 %. The dimensions of billet for extrusion by the first variant: the outer diameter D_{init} is 27 mm, hole diameter 9 mm, height 31 mm, diameter of forged piece 28 mm, flange diameter 30.8 mm, height 26 mm, hole diameter 8.5 mm, die preheating temperature 200 °C. Material of stamp is die steel 5HNV GOST 5950  2000. The finite element model of the billetstamp system at the beginning and the end of the extrusion is presented in Fig. 1.
Analysis of the stressstrain state and temperature field were performed in sections of billet, as shown in Fig. 2. The effective method to reduce a nonuniformity of the stressstrain state during extrusion is formation of relieving coneshaped cavity in the initial billet (Fig. 4). According to the recommendations (Ryabicheva et al., 2011), relieving cavity for reducing of significant nonuniformity of stressstrain state in the upper end of the billet has made. The simulation scheme is presented in Fig. 3. It was assumed for simulation that the cavity depth is equal to the radius of the sphere, the cone angle was equal to 15°, 30° and 40°.
Drawings of initial porous powder billets are presented in Fig. 4.
4. Computer simulation of radialdirect 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 nonuniformity of stressstrain 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.
Nonuniformity of stressstrain state is a significant cause of nonuniformity of the temperature field (Fig. 6). The temperature rises up to 1200 °C due to the thermal effect of plastic deformation 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.
Nonuniformity of temperature field and stressstrain 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 flowthrough flaw and then leads to loss of plastic equilibrium and cracking of products. The stages of flowthrough 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.
The stress state near the defect is significantly nonuniform (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 concentration factor a_{σ} with taking into account the structure and plastic properties of powder material (Skorokhod, 1985; Ryabicheva, 2012):
where
Stress concentration factor on the surface of forged piece a_{σ} = 2.5  3.0. After closure of the fold stress concentrator formed at its end, leading to increase in stress concentration factor a_{σ} 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.
The nonuniformity coefficients of stress σ_{inh} and deformations e_{inh} were implemented for estimation the nonuniformity of stressstrain state:
where
N  is the number of finite elements inside the model.
In case of uniform deformation the values of σ_{inh} and e_{inh} are asymptotically approaching zero.
The results of analysis of nonuniformity of stressstrain state by sections of forged piece confirms that the highest nonuniformity of stressstrain state has been observed in section OC, which corresponds to retraction of surface layers of the metal during formation of flowthrough flaw (Table 1).



OA  0.29  0.31 
OB  0.33  0.36 
OC  0.41  0.56 
Thus, conditions that are leading to formation of defects have established by numerical simulation of extrusion of the porous powder billet with a hole.
5. Computer simulation of radialdirect 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 coneshaped relieving cavity on the nonuniformity of stressstrain 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 nonuniformity 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 flowthrough 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.
Thus, implementation of compacts with the relieving cavity and α =15° was not ensured decreasing of nonuniformity of stressstrain 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 nonuniformity of stressstrain state.
Increasing the angle α up to 30° reduces nonuniformity of stressstrain state on 30 %. The largest and smallest σ_{i} and e_{i} differ by 1.7 times and 2.1 times, respectively (Fig. 10). Parameters of stressstrain state are distributed more uniformly by sections. Consequently, a flowthrough flaw was not formed in forged pieces during extrusion.
The results of analysis of nonuniformity of stressstrain 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 
The nonuniformity of stressstrain 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 nonuniformity of stressstrain 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.
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.
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 stressstrain state during radialdirect extrusion of forged pieces at D_{flange}/D_{out} = 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 stressstrain state parameters at α = 40° in three sections of forged piece are presented on Fig. 12. Retrieved reduction of nonuniformity of stressstrain state by 610% 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 nonuniformity analysis of stressstrain state by sections of forged piece with relieving cavity angle α = 40° are presented in Table 3. The nonuniformity of stressstrain 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 stressstrain state.
Improvement of the uniformity of stressstrain 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 radialdirect extrusion of cylindrical billets with generatrix inclination angle α within 30  40° has shown that the presence of relieving cavity increases the uniformity of the temperature field and stressstrain state. This helps to reduce the stress intensity and stress concentration factor k_{σ} at the edges of the cavity by 3  4 times.



OA  0.08  0.01 
OB  0.06  0.01 
OC  0.08  0.03 
As a result, the probability of flowthrough 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 D_{out} = 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/cm^{3} during extrusion of billets without relieving cavity at ratio D_{flange}/D_{out} 1.1  1.3 (Fig. 14, a) was reached in the volume of metal adjacent to the surface of forged piece at D_{flange}/D_{out} = 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 adjacent to the surface of sphere and in the flange. A density close to 7.83 g/cm^{3} has obtained in the billet at α = 15° only at the ratio D_{flange}/D_{out} = 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 D_{flange}/D_{out} = 1.1  1.2 at different initial density while increasing angle α to 30° (Fig. 14, c). However, if D_{flange} / D_{out} = 1.3 the density of compact material obtained. The high density has obtained at the cavity angle α = 40° for D_{flange}/D_{out} = 1.1  1.2 (Fig. 13, d). It is rather difficult to change the volume of forged piece at any density for D_{flange}/D_{out} = 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 D_{flange}/D_{out} = 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^{3} that indicates a possibility to obtain the equidence and highstrength details.
6. Determination of extrusion force
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_{i}^{ave} on the contact surface of upper punch and powder billet or function for σ_{i} may be written in the following way (Wagoner & Chenot, 2001):
where F  is the area of contact surface of upper punch and powder billet.
The expression (13) for radialdirect extrusion of forged piece with spherical cavity and central hole with accounting the modelling results may be transformed to:
where D_{flange}  is the outer diameter of flange;
D_{hole}  is the diameter of hole.
Values of extrusion force calculated by formula (14) at different inclination angles of generatrix of relieving cavity for D_{flange}/D_{out} = 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 D_{sf} = 20 mm).
P, kN  
0  133.1  92.0 
15  118.2  81.7 
30  103.7  71.6 
40  84.5  58.4 
D_{flange}/D_{out}  P, kN  
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 relative error 7  9 %.
7. Conclusions
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 nonuniformity of the stressstrain 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 nonuniformity of the stressstrain state, temperature field and maximum density indicate a possibility to obtain highquality products.
A highquality details with a spherical cavity and the ratio D_{flange} /D_{out} = 1.2 may be obtained from powder billets with 15 % initial porosity and relieving cavity with generatrix inclination angle 40°. Details with the ratio D_{flange} / D_{out} = 1.3 may not be produced from such billets due to the presence of cracks and nonuniformity into the flange. Highquality details with the ratio D_{flange}/D_{out} = 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/cm^{3} and the equidensity of flange observed in forged pieces at the ratio D_{flange}/D_{out} = 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 D_{flange}/D_{out} = 1.3 and density 7.79  7.81 g/cm^{3} 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 %.
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