21.1 Introduction
21.1.1 Aims and Scopes of the Present Work
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At the microscale level it was needed to simulate the actual kinetics of the transformation process inside of single tetragonal phase particles in PSZ. This task could be best accomplished by using the phase-field method.
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To study the transformation conditions of an ensemble of tetragonal lentils in polycrystalline PSZ ceramics, a semi-analytical thermostatic approach was applied at the mesoscale.
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In order to enable quantitative strength analysis of structures made of PSZ and MMC, a phenomenological constitutive law at the macroscale was further developed and implemented in a FEM-environment.
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To support the development of tailored particle MMC TRIP-matrix composites, representative volume elements on the composite level were simulated, which allow to predict the mechanical properties of the composite as function of volume fraction of both constituents and the strength of the interface.
21.1.2 Introduction to Partially Stabilized Zirconia
21.2 Micromechanical Phase-Field Approach
21.2.1 Phase-Field Method
21.2.2 Model Setup
Lattice transformation | Transformation strain \(\varvec{\epsilon }_{ij}\) |
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\(c\, \rightarrow {{t}}_1\) | \( \varvec{\epsilon }_{\text {tr}}^{t1} = \begin{bmatrix} \quad -0.0007 &{} \quad 0 \\ \quad 0 &{} \quad 0.0197 \end{bmatrix}\) |
\(c \rightarrow {{t}}_2\) | \( \varvec{\epsilon }_{\text {tr}}^{t2} = \begin{bmatrix} \quad 0.0197 &{} 0 \\ 0 &{} \quad -0.0007 \end{bmatrix}\) |
\({{t}} \rightarrow {{m}}_+\) | \( \varvec{\epsilon }_{\text {tr}}^{m+} = \begin{bmatrix} \quad 0.012479 &{} \quad 0.079614 \\ \quad 0.079614 &{} \quad 0.019139 \end{bmatrix}\) |
\(t \rightarrow {{m}}_-\) | \( \varvec{\epsilon }_{\text {tr}}^{m-} = \begin{bmatrix} \quad 0.012479 &{} \quad -0.079614 \\ \quad -0.079614 &{} \quad 0.019139 \end{bmatrix}\) |
Phases |
\( \mathbf{E} _{11} \)
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\( \mathbf{E} _{22} \)
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\( \mathbf{E} _{33} \)
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\( \mathbf{E} _{44} \)
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\( \mathbf{E} _{55} \)
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\( \mathbf{E} _{66} \)
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\( \mathbf{E} _{12} \)
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\( \mathbf{E} _{13} \)
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\( \mathbf{E} _{16} \)
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\( \mathbf{E} _{23} \)
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\( \mathbf{E} _{26} \)
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\( \mathbf{E} _{36} \)
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\( \mathbf{E} _{45} \)
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\(c\)-phase | 390 | 390 | 390 | 60 | 60 | 60 | 162 | 162 | 0 | 162 | 0 | 0 | 0 |
\(t\)-phase | 327 | 327 | 264 | 59 | 59 | 59 | 100 | 62 | 0 | 62 | 0 | 0 | 0 |
\(m\)-phase | 361 | 408 | 258 | 100 | 81 | 126 | 142 | 55 | −21 | 196 | 31 | −18 | −23 |
21.2.3 Selected Results and Discussion
21.2.3.1 Phase Stability and Energy Barriers
21.2.3.2 Variant Selection by Energy Barriers
Description | Symbol | Value | Unit |
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\(m\)-phase start temperature | \(M_s^{{{t}}\rightarrow {{m}}}\) | 1305 | K |
\(m\)-phase equilibrium temperature | \(T_0^{{{t}}\rightarrow {{m}}}\) | 1367 | K |
Gradient energy coefficient | \(\beta \) | 5\(\,\times \,10^{-11}\) | J/m |
Kinetic coefficient | L | 2 | m\(^3\)/Js |
Material parameter | a | 6 | – |
21.2.3.3 Origin and Effect of Residual Stresses
21.3 Mesomechanical Model
21.3.1 Transformation Criterion for a Single Precipitate Embedded in an Infinite Matrix
21.3.1.1 Working Hypotheses
21.3.1.2 Energetic Contributions
21.3.1.3 The Transformation Criterion
21.3.2 Uniaxial Loading
21.3.2.1 Orientation Dependence of the Transformation Stress
21.3.2.2 Sensitivity with Respect to the Inclusion Size, Aspect Ratio and Interfacial Energy
21.4 Homogenization Within an Infinite Grain
21.5 Continuum Mechanics Approach
21.5.1 Constitutive Model for Phase Transformation in PSZ
21.5.1.1 Homogenization of PSZ Material
21.5.1.2 Thermodynamic State Potentials
21.5.2 Numerical Results
21.5.2.1 Particle Size Dependent Surface Energy Change
21.5.2.2 Temperature-Induced Phase Transformation
21.6 Simulations of ZrO\(_2 \)-Particle Reinforced TRIP-Steel Composite
21.6.1 Unit Cell Model of the Composite
21.6.2 Results and Discussion
f | 0.05 | 0.05 | 0.10 | 0.10 | 0.20 | 0.20 |
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Interface | Bonded | Non-cohesive | Bonded | Non-cohesive | Bonded | Non-cohesive |
\(W/W_\mathrm {hom}\) | 1.06 | 0.92 | 1.16 | 0.85 | 1.37 | 0.72 |