**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:


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):

$$F = \frac{\tau^2}{\bar{\varrho}^2} + (1 + m)^2 \frac{\left(p + \frac{m}{m+1}\bar{\rho}\sigma\_s\sqrt{\psi}\right)^2}{\bar{\psi}} - \bar{\rho}\sigma\_{s'} \tag{1}$$

where *<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>3</sup> *<sup>σ</sup>ij δij* - is the medium pressure;

*τ* = (*σij* − *pδij* )(*σij* − *pδij* )- is the intensity of shear stress;

*φ* =(1−*θ*)<sup>2</sup> , *ψ*<sup>=</sup> <sup>2</sup> 3 (1−*θ*)<sup>2</sup> *<sup>θ</sup>* - are porosity functions; θ - is the porosity;

This work aims on improvement a quality of automotive parts based on a theoretical analysis of the stress-strain state, temperature fields and density distribution during radial-direct

**2. The mathematical model of radial direct extrusion of porous powder**

In this chapter mathematical modelling has been conducted on the basis of plasticity theory

**1.** construction a system of differential equations of the nonlinear coupled thermal plasticity problem for three-dimensional model of billet-stamp 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,

**2.** application of a finite element method for solving of nonlinear coupled thermal plasticity

**3.** 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);

**4.** formulation of the method and solving of physically nonlinear coupled thermal plasticity problem for a three-dimensional billet-stamp model of radial-direct 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,

**5.** verification of the results of finite element simulation by experimental investigation of

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):

> *m <sup>m</sup>* <sup>+</sup> <sup>1</sup> *<sup>ρ</sup>*¯*σ<sup>s</sup> <sup>ψ</sup>*)

2

*<sup>ψ</sup>* <sup>−</sup>*ρ*¯*σs*, (1)

radial-direct extrusion of porous powder billets (Ryabicheva et al., 2012).

( *p* +

)- is the intensity of shear stress;

*<sup>F</sup>* <sup>=</sup> *<sup>τ</sup>* <sup>2</sup> *φ*

+ (1 + *m*)<sup>2</sup>

*<sup>θ</sup>* - are porosity functions;


of porous bodies and focused on sequential solving the following problems:

problem (Awrejcewicz et al., 2007, Lienhard IV & Lienhard V, 2003);

extrusion of porous powder billets.

120 Computational and Numerical Simulations

2012; Segal et al., 1981);

2012);

where *<sup>p</sup>* <sup>=</sup> <sup>1</sup>

*τ* = (*σij* − *pδij*

*φ* =(1−*θ*)<sup>2</sup>

<sup>3</sup> *<sup>σ</sup>ij δij*

, *ψ*<sup>=</sup> <sup>2</sup> 3

)(*σij* − *pδij*

(1−*θ*)<sup>2</sup>

**billets**

*ρ*¯ =1−*θ*- is the relative density;

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):

$$e \sim \frac{2(1+m)^2}{\nu}p + \frac{2m(1+m)\sigma\_0}{\sqrt{\nu}},\tag{2}$$

where σ0 - 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 *<sup>σ</sup>* <sup>=</sup>*σ*<sup>0</sup> <sup>+</sup> *<sup>K</sup><sup>ω</sup>* 0.5, where K - is the hardening coefficient. The rate of accumulating deformation in hard phase of porous body was determined on the basis of postulate of uniqueness of the dissipation function formulated by Skorokhod V.V. (Skorokhod, 1973):

$$
\rho \alpha = \sqrt{1 - \theta} \left( \frac{m}{1 + m} \sqrt{\mu \nu} + \frac{\sqrt{(1 + m)^2 \nu^2 + e^2 \nu}}{1 + m} \right), \tag{3}
$$

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):

$$\frac{d\alpha}{dt} = \mathbf{W}\_{\prime} \tag{4}$$

where W - is the equivalent strain rate:

$$
\Delta W = \frac{1}{\sqrt{1-\Theta}} \sqrt{\psi e^2 + \varphi \gamma^2}.\tag{5}
$$

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 contin‐ uum billet to its finite element counterpart. Such simulation requires implementation of extremal requirement for the functional (Shtern et al., 1982, Segal et al., 1981):

$$J(\mathbf{v}\_i(\mathbf{x})) = \int\_{\Omega} D(e\_{ij}(\mathbf{V}\_i))d\Omega + \int\_{\partial\Omega\_p} p\_i \mathbf{v}\_i d\langle \partial\Omega \rangle\_{\prime} \tag{6}$$

ρ - is the density of material;

Lienhard IV & Lienhard V, 2003):

The minimum of heat conduction functional is related to each loading step (Segal et al., 1981;

222 <sup>222</sup> , *<sup>T</sup>*

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

The predictor-corrector 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

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 Dinit 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 billet-stamp system at the beginning and the end of the extrusion

( <sup>∂</sup>*<sup>T</sup>*

¶¶¶ è ø òòò (9)

Computer Modelling of Radial-Direct Extrusion of Porous Powder Billets

http://dx.doi.org/10.5772/57142

123

<sup>∂</sup>*<sup>τ</sup>* ′). (10)

*TTT Q k dV xyz* æ ö ¶¶¶ = ++ ç ÷

surface of the billet (Lienhard IV & Lienhard V, 2003; Ryabicheva et al., 2012):

(*T*<sup>0</sup> <sup>−</sup>*Tпр*) <sup>=</sup> <sup>−</sup>*<sup>λ</sup>* ′

*V*

*α* ′



remeshing (Wisselink, 2000; Stoker, 1999).

**3. Initial data for modelling**

is presented in Fig. 1.

<sup>∂</sup>*<sup>τ</sup>* ′ - is the temperature gradient.

T - is the temperature, K; τ - is the loading time step.

where *α* ′

*λ* ′

∂*T*

where *D*(*eij* (*Vi* )) - is the dissipative function;

*pi* - is the stress vector on the surface of the processed billet;

*vi* - is the velocity vector on the surface of the processed billet.

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 *D*(*eij* (*Vi* )) may be presented by the following expression (Shtern et al., 1982):

$$D\left(e\_{ij}\left(V\_i\right)\right) = \frac{\sqrt{\gamma^2 \phi + e^2 \varphi}}{\sqrt{1 - \theta}} \tau\_s + \frac{p\_0 e}{\sqrt{1 - \theta}},\tag{7}$$

where *Vi* <sup>=</sup>*v*(*x*), *eij* <sup>=</sup> <sup>1</sup> <sup>2</sup> ( <sup>∂</sup>*vi* ∂ *xj* + ∂*vj* ∂ *xi* ), *p*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>2</sup> <sup>3</sup> *<sup>τ</sup><sup>s</sup> <sup>ψ</sup> m* <sup>1</sup> <sup>+</sup> *<sup>m</sup>* , *τs*- is the shear yield stress.

The stress-strain 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 Dflange /Dout = 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 tem‐ perature field (Lienhard IV & Lienhard V, 2003):

$$k\_T \left(\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial \mathbf{y}^2} + \frac{\partial^2 T}{\partial \mathbf{z}^2}\right) dV = \mathbf{C} \rho \frac{\partial T}{\partial \mathbf{z}} dV\_\prime \tag{8}$$

where

kT - is the summary heat conduction coefficient;

C - is the specific heat capacity;

ρ - is the density of material;

T - is the temperature, K;

*J*(v*<sup>i</sup>*

where *D*(*eij*

*pi*

*vi*

*D*(*eij* (*Vi* (*Vi*

122 Computational and Numerical Simulations

where *Vi* <sup>=</sup>*v*(*x*), *eij* <sup>=</sup> <sup>1</sup>

Ryabicheva et al., 2012).

been considered.

where

(*x*))= *∫* Ω *D*(*eij* (Vi

)) - is the dissipative function;



( ( ))

<sup>2</sup> ( <sup>∂</sup>*vi* ∂ *xj* + ∂*vj* ∂ *xi*

perature field (Lienhard IV & Lienhard V, 2003):

kT - is the summary heat conduction coefficient;

C - is the specific heat capacity;

))*d*Ω + *∫*

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

)) may be presented by the following expression (Shtern et al., 1982):

), *p*<sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>2</sup>

222

*xyz*

2 2 <sup>0</sup> , 1 1 *ij i <sup>s</sup> <sup>e</sup> p e De V* gf

+ = + - -

 y t

> q

<sup>1</sup> <sup>+</sup> *<sup>m</sup>* , *τs*- is the shear yield stress.

q

<sup>3</sup> *<sup>τ</sup><sup>s</sup> <sup>ψ</sup>*

The stress-strain 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,

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 Dflange /Dout = 1.1 has

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 tem‐

> <sup>222</sup> , *<sup>T</sup> TTT T <sup>k</sup> dV C dV*

æ ö ¶¶¶ ¶ ç ÷ ++ = ¶¶¶ ¶ è ø

r

t

*m*

∂Ω*<sup>p</sup> pi* v*i* *d*(∂Ω), (6)

(7)

(8)

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):

$$\mathbf{Q} = \iiint\limits\_{V} k\_T \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) dV \,\tag{9}$$

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):

$$
\alpha' \{ T\_0 - T\_{mp} \} = -\lambda' \left( \frac{\partial T}{\partial \tau'} \right). \tag{10}
$$

where *α* ′ - is the heat-transfer coefficient;

*λ* ′ - is the heat conduction coefficient;

∂*T* <sup>∂</sup>*<sup>τ</sup>* ′ - is the temperature gradient.

The predictor-corrector 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).
