Finite-Element Modeling of Titanium Powder Densification
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HIGH-temperature powder compaction is a well-established method to create complex shapes with good mechanical properties from Ti-6Al-4V, which are the work-horse titanium alloys in the aerospace[1,2] and biomedical industries.[3–6] Powder densification at increased temperature is accelerated by the application of an external stress,[7–10] through deformation at contact points between powders, which is controlled by the power-law creep mechanism with the constitutive equation e_ x ¼ C rnx
½1
where e_ x is the steady-state (secondary) uniaxial strain rate, rx is the uniaxial stress, C is a constant incorporating an Arrhenius temperature dependence (C = 4.8 9 107 MPa2.8s1 for Ti-6Al-4V at 1293 K (1020 C)[11,12]), and n is the creep stress exponent (n = 2.8 for Ti-6Al-4V[12]). Based on Eq. [1], equations predicting densification kinetics (density vs time) can then be derived[7–9,13–16] for an assembly of spherical powders with initial relative density q0 subjected to an external stress r in a uniaxial die pressing experiment. The densification rates q_ for initial stage densification (relative density q < 90 pct, where deformation of powders at contact points and increasing coordination number are important) and final stage densification (q > 90 pct, considering the shrinkage of individual pores in a matrix) are respectively given as follows: 1=3 1 q0 n1=2 q0 Bi r n ½2 q_ I ¼ 3:06 C 3 q q0 q2n2=3 BING YE, Postdoctoral Fellow, and DAVID C. DUNAND, Professor, are with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208. Contact e-mail: [email protected] MARC R. MATSEN, Boeing Technical Fellow, is with Boeing Research and Technology, The Boeing Company, Seattle, WA 98124. Manuscript submitted January 25, 2011. Article published online August 3, 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
3 q ð1 qÞ 3Bf r n q_ II ¼ C 2 2n ð1 ð1 qÞ1=n Þn
½3
where q0 is the initial powder density and the constants Bi and Bf take into account compaction geometry (Bi = 1.1 and Bf = 1.8 for uniaxial die pressing[15]).When thermally cycled between its allotropic a- and b- phases, Ti-6Al-4V exhibits transformation mismatch plasticity[11,12,17–20] with a constitutive equation given by e_ x ¼
4 5n DV 1 rx 3 4 n þ 1 V Dt r0
½4
where e_ x is the average uniaxial strain rate during thermal-cycling transformation-mismatch plasticity; DV/V (=0.96 pct) is the volume mismatch between the allotropic a and b phases[11]; Dt is the period of the thermal cycles spanning two transformations on heating and cooling, respectively; and r0 is the average internal stress generated during the phase transformation (r0 = 7.4 MPa for Ti-6Al-4V[11]). Transformation mismatch plasticity (Eq. [4]) is a special case of a creep-type equation (Eq. [1]) with creep stress exponent n = 1 and the other parameters in Eq. [4] merged as a constant C, with a value of 1.045 9 105 MPa1s1[11,16] for Ti-6Al4V for thermal cycling 1133 K to 1293 K (860 C to 1020
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