Near Net Shape Manufacturing of CuCr Vacuum Switching Contacts ...

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$Refractory metals in furnace construction and engineering$

WS 15/4 17th Plansee Seminar 2009, Vol. 3 Feist, Oberbreyer et al. q Drucker-Prager shear-failure surface d 1 tan β elastic domain (E, ) p a elliptical cap-surface p b R (d + p a tan ß) p q a = d + p a tanß (a) (b) Figure 1: Combined DRUCKER-PRAGER-cap plasticity model: (a) yield surface in the hydrostatic-deviatoric stress-space, (b) evolution of the yield surface with respect to density. compaction processes as die compaction. However, the cap surface is not suited for describing the response under stress-states with lower values of triaxiality. Such stress states take place in the green compact especially during the ejection phase but might occur also during compaction with more complex toolgeometries. Such stress states might lead to the formation of discontinuities and cracks in the powderbody being compacted and hence should be avoided. Thus, for the simulation of the compaction process it is desirable to be able to assess such stress-states in order to identify critical regions or process stages. To account for respective stress states the cap-surface is combined with a shear-failure surface fs. In the most simple case the well-known cone-shaped DRUCKER-PRAGER yield-surface (Fig. 1a) given as q ¡0 p ¡ ¡ fs = q − p tanβ − d (3) can be used with d and β as the cohesive strength and the angle of internal friction of the material, respectively. With fc and fs considered to intersect at pa, ellipse-axis qa in (1) can be expressed as qa = d + pa tanβ. (4) It has to be noted that the employed shear-failure surface represents a valid representation of the mechanical response only for stress states with positive triaxiality values. However, it does not properly account for three-dimensional tensile stress states which is out of scope of the employed model in the context of powder compaction. With respect to the hardening law it is assumed that the cap-surface fc exhibits hardening representing the behavior of the metallic powder under hydrostatic compression. To this end, the cap-apex value pb (2) is related to the logarithmic volumetric strain εv = ε el v + ε pl v consisting of a (reversible) elastic part ε el v and a (irreversible) plastic part ε pl v . The total volumetric strain εv in turn can be expressed in terms of the current material density ρ as � � ρ εv = ln ρ0 (5) (6) ¡th
Feist, Oberbreyer et al. 17th Plansee Seminar 2009, Vol. 3 WS 15/5 with ρ0 as the reference density of the material (i.e. the density of the powder in loose packing). Since the contribution of the elastic part in (5) is of a few orders less than the one from the plastic part, in general it can be neglected, such that Hardening of the cap-surface can then be formally defined as � � ρ εv = ln ≈ ε ρ0 pl v . (7) pb = fh (εv) (8) where fh represents a C 1 -continuous, monotonously increasing hardening function appropriate for describing the compaction behavior of the powder under consideration. In general, power or exponential type functions can be successfully employed. It can be seen from experiments that the cap-surface will not keep its shape under compressive loading, such that the cap-eccentricity parameter R in (1) also has to be made dependent on the volumetric strain. Furthermore, compaction of granular material is accompanied by a significant increase in the strength of the material defined by the shear-failure parameters d and β in (3). Hence, parameters R, d and β are formally related to the volumetric strain and the density of the material as R = fR (εv) d = fd (εv) β = fβ (εv). (9) Appropriate evolution functions must be chosen and their coefficients adapted in order to fit experimental results for the particular powder under consideration. The evolution of the combined DRUCKER-PRAGERcap surface based on evolution-equations (8) and (9) are shown for an exemplary powder in Fig. 1b. Together, the convex cap-surface fc = 0 and the shear-failure surface fs = 0 confine the elastic domain (Fig. 1a). For stress-states within this domain isotropic elastic behavior expressed by the constants of elasticity E and ν is assumed to be valid. In order to be able to numerically capture the elastic springback taking place during the ejection phase of a green compact it is also necessary to describe the densitydependence of the modulus of elasticity E again formally expressed in terms of the volumetric strain as Constitutive modeling of sintering E = fE (εv). (10) Sintering is a heat-treatment process conducted at high-temperatures that causes the powder-particles within the porous green compact to be bonded together. Since the volume of the pores between the powder-particles is reduced during this process, the material shrinks and consequently its density is increased. A constitutive model describing the mechanical behavior of a solid being sintered for usage in the context of simulations of the PM process-chain must be able to reproduce both evolution of density and shrinkage as shown for two different CuCr-alloys by results from sintering experiments in Fig. 2. Hence, it is sufficient to again resort to a purely phenomenological model, which however is derived from a micromechanical description of inter-particle interactions [4, 5].
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Near Net Shape Manufacturing of CuCr Vacuum Switching Contacts ...

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