Formulation and Identification of an Anisotropic Constitutive Model for Describing the Sintering of Stainless Steel Powder Compacts

B06-01-3 2006 POWDER METALLURGY World Congress

Alexandre Vagnona, Didier. Bouvardb, Georges Kapelskic

Institut National Polytechnique de Grenoble, Laboratoire GPM2, UMR CNRS 5010, BP 46, 38402 Saint Martin d’Hères, FRANCE [email protected], [email protected], [email protected]

Abstract

Anisotropic constitutive equations for sintering of metal powder compacts have been formulated from a linear viscous transversely-isotropic model in which an anisotropic sintering stress has been introduced to describe free sintering densification kinetics. The identification of material parameters defined in the model, has been achieved from thermo- mechanical experiments performed on 316L stainless steel warm-compacted powder in a dilatometer allowing controlled compressive loading.

Keywords : Sintering, compacts, anisotropy, modelling, constitutive equation

1. Introduction M is the structure tensor ( M = v 3 ⊗ v 3 ), where v 3 is a 3 3 The finite element simulation of the macroscopic behaviour unit vector in the anisotropy direction). For a transversely of a complex green component during sintering can allow isotropic material, αi parameters can be expressed with only predicting its dimensional changes. Most constitutive 5 independent material parameters: models proposed for this purpose are isotropic. They are not relevant for describing the behaviour of metal powder α1 = F tr(σ ) + (D − F) tr(M 3σ ) (5) compacts, which is usually strongly anisotropic due to prior α = (D − F)tr(σ ) + (C + A − 2D − 4G)tr(M σ ) (6) die compaction. This anisotropy can be assumed to be of 2 3 the transversely isotropic type with the pressing direction as α3 = A − F (7) the singular direction. In this paper, anisotropic constitutive α = 2G − A + F (8) equations for sintering have been formulated from a linear 4 viscous transversely-isotropic model in which an anisotropic It is usual to replace these parameters to more comprehensive sintering stress has been introduced to describe free ones, such as viscosities and viscous Poisson’s ratios: vp vp sintering densification kinetics. To identify the material 1 1 − υ − υ 1 A = C = D = AT F = TT (9) parameters defined in the model, an extensive set of G = ηT ηA ηA ηT ηAT thermo-mechanical experiments has been performed on Thus ηA and ηT are the viscosities in axial and transverse 316L stainless steel warm-compacted powder in a dilatometer vp direction respectively, ηAT is the shear viscosity, νAT and allowing controlled compressive loading. vp νTT are viscous Poisson’ratios. For describing the free sintering deformation, we add a sintering stress tensor σs 2. Formulation of Constitutive Equation to the external stress tensor. The final expression of the constitutive equation is thus In classical isotropic constitutive equation proposed for v s v D = α1 I + α 2 M 3 + α 3 (σ + σ ) sintering [1-2], the viscous stain rate tensor Dij is written: (10) ′ s s v σ ij (σ m −σ s ) D = + δ +α 4 (M 3 (σ + σ ) + (σ + σ )M 3 ) ij 2G 3K ij ’ s where σ ij is the stress deviator tensor, σ m is the mean stress, The components of the sintering stress tensor σ can be σ s is the sintering stress, δij is the unit tensor, K and G are expressed from the free sintering strain rate in axial and the bulk and shear viscosities, respectively. To formulate an transverse directions, ε and ε , and from the viscosity equation describing transverse isotropy we start from the A T general tensorial representation of Boehler [3] : parameters (ηA and ηT). To have a complete model we D = α I +α M +α σ +α (M σ +σ M ) should add a thermo-elastic strain rate. The elastic part can 1 2 3 3 4 3 3 be neglected whereas the thermal term is of course

© Korean Powder Metallurgy Institute - 64 - absolutely required. It can be assumed to be isotropic (thus n1 (T) 1 1 1 1 T εT (T,χ ) = Ω (T).(χ ) (20) a single material parameter is required) or transversely T isotropic if one wants to take into account the anisotropy observed during heating (then two thermal expansion The deformation due to packing pore closing has been coefficients are required). supposed to be isotropic during heating. The model finally includes 8 material parameters, which Viscosity changes as function of T and χ have been fitted are a priori functions of the temperature, T, the actual with the following relation: relative density, ρ and the green density, ρ 0. They have γ A,T (T) been identified from the experiments performed on stainless ηA,T = φA,T (T).(χ) steel warm-pressed compacts of relative green density 0.9. Viscous Poisson’s ratios, which are difficult to measure, are assumed to be constant and equal to 0.3. 3. Model Parameter Identification As a control of model parameter identification, the strains during a standard sintering cycle has been calculated with The standard sintering conditions for 316L steel powder the equations described above and compared with compacts have been chosen to be the following : heating dilatometry measures (Fig. 1). The adjustment is rather rate 30°C/min, delubrication plateau at 700°C during 15 satisfactory. In particular the drop in axial direction due to min, sintering plateau at 1250°C during 30min, 10% H2 / contact pore closed is correctly described. The deformation 90% Ar atmosphere. Dilatometry measurements have been during sintering plateau is not perfectly modelled, which performed in axial and transverse directions (Fig. 1). It results in a significant difference in final axial strain. appears that density changes due to thermal expansion are larger than the changes due to sintering. Thus the actual relative density seems not to be the right parameter to 3.E-02 characterize the progress of sintering. Hence we define the 1200 ρ 3 = 0,897 following parameter for the description of sintering 0 2.E-02 advancement: 900 캜 χ = ρ – α∆T – ρ0 5.E-03 where α is the bulk thermal expansion coefficient and ∆T is 600 -5.E-03 Strain the difference between the actual temperature and room Temperature temperature. AX exp. Temperature (Temperature 300 ) AX simul. eq(2) Dilatometry tests also proved that the final anisotropy TRV exp -2.E-02 was mainly due to the axial shrinkage suddenly occurring TRV simul. eq(2) around 1000° (Fig. 1), which has been interpreted as the 0 -3.E-02 effect of contact pore closing [4]. The deformation is thus 0 2000 4000 6000 8000 10000 supposed to be due to two mechanisms: Time (s) - Mechanism 1: packing pore shrinkage, which is almost isotropic ; Fig. 1. Comparison of experimental and predicted - Mechanism 2: contact pore closing, which results only in strains during free sintering test. axial shrinkage. Then the sintering advancement parameter, χ, is divided in two terms, χ = χ1 + χ2, where χ1 describes packing pore 4. References changes and χ2 describes contact pore changes. The free sintering strain rate in axial direction is next 1. V. V.Skorokhod, E. Olevskii, M. B. Shtern, Powder expressed as the addition of the contributions of both Metall. Ceram., 363, 3,1993, 21-26 mechanisms, whereas it is assumed that only Mechanism 1 2. E. Olevsky, Mat. Sci. Eng. Report 1998; R23:2 contributes to the transverse strain rate: 3. J. P. Boehler (ed.), Applications of Tensor Functions in Solid Mechanics, 1987, Springer-Verlag, New-York 1 1 2 2 εA = ε A (χ ,T) + ε A (χ ,T) (16) 4. A. Vagnon, O. Lame, D. Bouvard, M. Di Michiel, D. 1 1 εT = ε T (χ ,T) (17) Bellet, G. Kapelski, Acta Mater., 54, 2, 2006, 513-522 An analytical expression has been fitted to each function by using the strain rates changes deduced from dilatometry measurements: 5. Acknowledgements

n1 (T) 1 1 1 1 A ε A (T,χ ) = Ω (T).(χ ) (18) This work has been achieved with the financial support of A the European Commission through PM-MACH Growth n 2 (T) 2 2 2 2 A Project (N° GRD1-2001-40775). ε A (T,χ ) = Ω (T).(χ ) (19) A