An investigation of die profile effect on die wear of plane strain extrusion using incremental slab method and finite element analysis

Abstract

In this research work, a new incremental slab method analysis has been derived to calculate extrusion force and its die pressure distribution during an extrusion process, while a die of arbitrary shape is used. In addition, a computational algorithm has been presented based on the proposed analysis method and Archard’s wear model to investigate the effect of die profile on its working life. Three different die profiles including optimum curved, optimum constant angle, and cylindrical shaped are numerically evaluated. The results revealed that the predicted extrusion loads and die pressure distributions through the implementation of the applied method are in an acceptable agreement with FEM analysis results. Moreover, it has been demonstrated that the maximum wear depth on all die profiles is located at the die exit area, which indicates the predominant effect of the material velocity profile. It is also found that the die life of the two optimum dies would be the same, but the least working life estimated is the working lifetime of the cylindrical die profile.

Appendix

Derivation of Eqs. (1) and (2):

The stress field included some terms in x-direction as follows:

Friction on the upper/lower interface:

$$ 2\mu {P}_v(wdr)\cos \left(\alpha \right) $$

(31)

Normal stress on upper/lower interface:

$$ 2{P}_v(wdr)\sin \left(\alpha \right) $$

(32)

Radial stress (σ_r):

$$ {\int}_{-\alpha}^{\alpha}\cos \left(\theta \right){\sigma}_r wrd\theta $$

(33)

Radial stress (σ_r + dσ_r):

$$ {\int}_{-\alpha}^{\alpha}\cos \left(\theta \right)\left({\sigma}_r+d{\sigma}_r\right)w\left(r+ dr\right) d\theta $$

(33)

Friction on lateral interfaces:

$$ 2\upmu {P}_h rdr{\int}_{-\alpha}^{\alpha}\cos \left(\theta \right) dr $$

(35)

Considering force balance criteria in the x-direction and by substituting Eqs. (31) through (35) into Eq. (36), the differential relation of Eq. (37) is obtained and expressed as follows:

$$ \sum f(x)=0 $$

(36)

$$ 2\mu {P}_v(wdr)\cos \left(\alpha \right)+2{P}_v(wdr)\sin \left(\alpha \right)-2{\sigma}_r wr\sin \left(\alpha \right)+2\left({\sigma}_r+d{\sigma}_r\right)\left(r+ dr\right)w\sin \left(\alpha \right)+4\upmu {P}_hr\sin \left(\alpha \right) dr=0 $$

(37)

Equation (1) is derived by simplifying and disregarding second differential terms of Eq. (37). A similar method was employed to derive Eq. (2) while friction terms (μP_v and μP_h) were replaced by mk.

Derivation of Eq. (4):

State of stress during a plane strain extrusion process is defined as Eqs. (38) and (39) under the assumption of coulomb friction and constant friction factor, respectively.

$$ {\sigma}_{i.j}=\left[\begin{array}{ccc}{\sigma}_r& {\mu P}_h& -{\mu P}_v\\ {}{\mu P}_h& -{P}_h& 0\\ {}-{\mu P}_v& 0& -{P}_v\end{array}\right] $$

(38)

$$ {\sigma}_{i.j}=\left[\begin{array}{ccc}{\sigma}_r& mk& - mk\\ {} mk& -{P}_h& 0\\ {}- mk& 0& -{P}_v\end{array}\right] $$

(39)

To utilize the Tresca yield criteria, principle stresses are needed [1], while the state of stress presented by Eqs. (38) and (39) contains shearing stress terms (−μP_h, −μP_v, and mk). In order to simplify the relation of yield stress to the imposed stresses on material, the shearing terms have been disregarded. A similar assumption was made by Panteghini [39] in which an upper bound and a slab method analysis were developed to study the three-dimensional plate drawing process. The comparison between the results pointed out that the new analytical method is in promising agreement with FE and experimental results. Therefore, based on this approach, the state of stress would be expressed as Eq. (40).

$$ {\sigma}_{i.j}\approx \left[\begin{array}{ccc}{\sigma}_1& 0& 0\\ {}0& {\sigma}_2& 0\\ {}0& 0& {\sigma}_3\end{array}\right]=\left[\begin{array}{ccc}{\sigma}_r& 0& 0\\ {}0& -{P}_h& 0\\ {}0& 0& -{P}_v\end{array}\right] $$

(40)

Then, using the Tresca yield criteria leads to:

$$ {\sigma}_y={\sigma}_1-{\sigma}_3={\sigma}_r+{P}_v $$

(41)

Equation (4) was obtained by substituting Eq. (3) into Eq. (41).

Derivation of Eq. (25):

The parameters of Eq. (24) can be expressed as [2]:

dV = dzdA, dp = σ_ndA & dL = udt (42)

By substituting Eq. (41) into Eq. (24) and dividing both sides of it by dA, Eq. (42) obtained and expressed as:

$$ dz=k\frac{\sigma_nu}{H}\ dt $$

(43)

Equation (43) can be rewritten by integral operation as Eq. (25), while σ_n, u, and H parameters were assumed to be constant during the extrusion process.

Derivation of Eq. (29):

The presented relation for material velocity in the deformation zone of any shape by Farzad and Ebrahimi is:

$$ {\dot{u}}_{r_n}=-\left(\frac{V_0{t}_0}{\sin \left({\alpha}_n\right)}\right)\ \frac{\cos \left({\theta}_n\right)}{r_n} $$

(44)

At the upper/lower die-material interface, the θ_n parameter is constant and equal to the local element’s angle, α_n. Thus, the material velocity can be expressed as:

$$ {\dot{u}}_{r_n}=-{V}_0{t}_0\ \frac{\cot \left({\alpha}_n\right)}{r_n} $$

(45)

Derivation of Eq. (30):

The material velocity on the lateral die interface varies as the intended point’s angle over the range of −α_n to α_n changes. So, an average of the velocity is needed.

$$ {\dot{\overline{u}}}_{r_n}=\frac{1}{\alpha_n-\left(-{\alpha}_n\right)}{\int}_{-{\alpha}_n}^{\alpha_n}-\left(\frac{V_0{t}_0}{\sin \left({\alpha}_n\right)}\right)\ \frac{\cos \left({\theta}_n\right)}{r_n}d{\theta}_n $$

$$ {\dot{\overline{u}}}_{r_n}=-\frac{1}{2{\alpha}_n}\left(\frac{V_0{t}_0}{r_n\sin \left({\alpha}_n\right)}\right)\ \Big(\sin \left({\theta}_n\right)\left|\genfrac{}{}{0pt}{}{\alpha_n}{-{\alpha}_n}\Big)\right. $$

(46)

Then:

$$ {\dot{\overline{u}}}_{r_n}=-\frac{V_0{t}_0}{r_n{\alpha}_n} $$

(47)

The radial position of each point, r_n, can be expressed based on semi-thickness of the material on its bow (according to Ref [6]), t_n, as follows:

$$ {r}_n=\frac{t_n}{\sin \left({\alpha}_n\right)} $$

(48)

Substituting Eq. (48) into Eq. (47) leads to obtain Eq. (30) for an average value of material velocity on the lateral die-work piece interface.