Sintering Technology

Randall M. German, Gary L. Messing, Robert G. Cornwall

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780824797751
• eBook ISBN: 9781000148602
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

Based on the sintering conference held at the Pennsylvania State University, USA, this text presents advances in the application of sintering to the most important industrial materials. It offers results on both solid-state and microphase sintering as well as microstructure evolution, and introduces new applications, processes, materials and solutions to technical problems.

FE-Simulation of Compaction and Solid State Sintering of Cemented Carbides

Jan Brandt

Department of Solid Mechanics University of Linköping, S-581 83 Linköping, Sweden Aug 15 1995

Abstract

The process of hard metal products compaction and sintering (solid state) is simulated with a material model CAP88+SINT95 implemented in the Finite Element (FE) package LS-DYNA2D.

1. Introduction

We have further developed the FE material model for compaction simulation CAP88, from Sandvik Coromant AB, to also describe solid state sintering.

The model for sintering is visco-elastic, with a ”build in shrinkage motor” describing the sintering activity. The model is primarily aimed at sintering without external load (that is not HIP-ing), but a specimen is generally not stress-free in its interior after the compaction operation, so considering the stress field in the creep response is quite important.

The material model has been implemented in the FE-code LS-DYNA2D (Hallquist, 1990), that uses explicit time integration, which proved quite robust for the compaction part.

2. Stress and Deformation update

The equation of motion is recast in a Galerkin form, which then is discretized by the finite element procedure:

$M\ddot{U}=F^{ext}-F^{\mathrm{int}}$

(1)

$\ddot{U}=M^{-1}\left(F^{ext}-F^{int}\right)$

(2)

$F^{int}={\displaystyle \sum _{elem{e}_i}{F}^{int,e_i}}$

(3)

$F^{int,e_i}={\displaystyle {\int }_{elem{e}_i}B\sigma dV}$

(4)

$\dot{∊}=B{\dot{U}}^{e_i}$

(5)

In the above formulas M is the diagonal mass matrix, U is the nodal displacement vector, F^ext is the external nodal (applied) force vector and F^int is the internal resisting nodal force vector. σ and ϵ are stress and strain tensors and B is the strain-displacement matrix. An overdot stands for time derivative. σ, ϵ and B all pertain to the one integration point of an element. U^e_i, is the nodal displacement vector of one element and F^int,e_i is the corresponding internal force vector. The internal state variable d is defined in (12).

The rational for the state, U, σ, d, update from time t to t + Δt and U from t - Δt/2 to t + Δt/2 is:

1. update $\dot{U}$, U according to the central difference scheme:

$\dot{U}\left(t+\Delta t/2\right)=\dot{U}\left(t-\Delta t/2\right)+\ddot{U}\left(t\right)\Delta t$

(6)

$U\left(t+\Delta t\right)=U\left(t\right)+\dot{U}\left(t+\Delta t/2\right)\Delta t$

(7)

2. calculate σ, d at t + Δt according to section 3, $\dot{ϵ}(t+\Delta t/2\text{)}$ according to (5) is used

3. calculate F^int at t + Δt according to (4) and (3)

4. calculate Ü at t + Δt according to (2)

5. update the time t ← t + Δt and go back to 1.

3. Material Model

The choice of mode for a certain element integration point e at an instant of time t is based on the temperature T:

1 T < T^sint, compaction mode: elasto-plastic, CAP-type; or tensile fracture, smeared crack bands

2 T ≥ T^sint, sinter mode: visco-elastic, ”sinter stress” and mechanical stress cause a Norton-type of creep

Except for the fracture submode we can state a unified description of the stress update:

$\Delta {∊}^{total}=\Delta {∊}^{elastic}+\Delta {∊}^{inelastic}+\Delta {∊}^{thermal}$

(8)

$T<T^{sint},\Delta {∊}^{inelastic}=\Delta {∊}^{plastic},\text{fcn}\text{of}\sigma \text{,}\Delta {∊}^{total},d$

(9)

$T\ge T^{sint},\Delta {∊}^{inelastic}=\Delta {∊}^{creep},\text{fcn}\text{of}\sigma ,\Delta t,d,L,T,\dot{T}$

(10)

$\Delta \sigma =D\Delta {∊}^{elastic}$

(11)

$d\equiv \rho \text{current}/\rho \text{fully}\text{compacted}$

(12)

$L\equiv \text{average}(\text{WC})\text{particle}\text{size}$

(13)

D is the stiffness matrix. In the sequel we shorten superscripts according to t for total, e for elastic, i for inelastic, th for thermal, pl for plastic, cr for creep, s for sint, as in ϵ^t, ϵ^e, ϵⁱ, ϵ^th, ϵ^cr and T^s.

In the case of tensile fracture during the powder compaction there is a degradation of the load carrying capacity after the tensile failure stress σ^u has been reached:

$\Delta \sigma =\overline{D}\Delta {∊}^t,\Delta \sigma <0,\text{for}\text{a}\Delta {∊}^t\text{that}\text{further}\text{opens}\text{a}\text{crack}$

(14)

3.1 Compaction mode

The yield curve of the studied WC-Co powder at three different degrees of compaction is shown in Fig. 1. J₁ is the first invariant of the stress tensor (15) that is proportional to the pressure. J₂ is the second invariant of the stress deviator tensor (16) that is proportional to the von Mises stress.

Figure 1 Volumetric plastic strain along stress paths; yield curves

3.2 Tensile fracture mode

D is based on: thin crack bands; experimentally measured crack opening energies; and the smearing out of crack bands over the (otherwise elastically responding) elements.

3.3 Sinter mode

A model for unloaded sintering (Ågren et al., 1995) has been extended to also model the influence of a mechanical stress field. We adopt the following notations: repeated indices implicate summation over the whole range of that index as in ${∊}_{kk}={\displaystyle {\sum }_{\text{k}=1,3}{∊}_{kk};}{\delta }_{ij}=1\text{for}\text{i=j}$, otherwise O.

${\sigma }_m\equiv \frac{J_1}{3}\equiv \frac{1}{3}{\sigma }_{kk}\text{(mean}\text{stress)}$

(15)

${\sigma }_{ij}^{’}\equiv {\sigma }_{ij}-{\sigma }_m{\delta }_{ij}\left(\text{stress}\text{deviator}\right)$

(16)

${∊}_m^i\equiv \frac{1}{3}{∊}_{kk}^i\left(\text{mean}\text{inelastic}\text{strain}\right)$

(17)

${∊}_{ij}^{i’}\equiv {∊}_{ij}^i-{∊}_m^i{\delta }_{ij}\left(\text{inelastic}\text{strain}\text{deviator}\right)$

(18)

$f_p=1-d\left(f_p\text{is}\text{volume}\text{fraction}\text{of}\text{pores}\right)$

(19)

$\dot{d}=-3{\dot{∊}}_m^{i}d$

(20)

The model tells how Δϵⁱ or d (12) shall be updated during a time increment Δt given d,T,$\dot{T}$. In a more elaborate version of the model also L is updated. We postpone the introduction of a mechanical stress field a little.

${\dot{∊}}_m^i=-k_1{D}_{eff}\left(T,{\dot{T}}^{aver}\right)f_{bl}\left(T,t-t^{iso}\right){\sigma }^s$

(21)

$D_{eff}\left(T,{\dot{T}}^{aver}\right)=\left[D_v\left(T\right)+\omega D_s\left(T\right)\right]\left[1+f_{s2}\left(T\right){\dot{T}}^{aver}\right]f_k\left(d\right)$

(22)

${\sigma }^s=f_{s1}\left(d\right)\text{γ/}L$

(23)

Where D_x are various diffusion coefficients of (WC) particles, ω is a blending factor and k₁ represents the self diffusivity of the binder (Co). f_bl is a blocking factor (due to friction between carbide particles) which is closely represented as an exponential decay by the time elapsed, t−t^iso, after isothermal (T^iso) conditions started f_k(d) is a factor on D_eff that represents the state of densification and the initial distribution of binder. F_s1(d) is a factor...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Preface
• Table of Contents
• Contributors
• Neck Growth
• Small Particle Sintering
• Cermets and Non-oxide Ceramics
• Liquid Phase Sintering
• Alumina-Sintering and Grain Growth
• Grain Growth
• Novel and Reactive Sintering Processes
• Constrained Densification
• Index