# Multi-scale Modeling of Partially Stabilized Zirconia with Applications to TRIP-Matrix Composites

- 570 Downloads

## Abstract

The understanding of how the microstructure influences the mechanical response is an essential pre-requisite for materials tailored to match specific requirements. The aim of this chapter is to further this understanding in the context of Mg-PSZ-TRIP-steel composites on three different scales using a set of methods ranging from phase-field simulations over micromechanics to continuum constitutive modeling. On the microscale, using a Ginzburg-Landau type phase-field model the effects of cooling- and stress-induced martensitic phase transformation in MgO-PSZ is clearly distinguished. Additionally with this method the role of energy barrier in variant selection and the effect of residual stress contributing to the stability of the tetragonal phase are also investigated. On the mesomechanical scale, an analytical 2D model for the martensitic phase transformation and self-accommodation of inclusions within linear elastic materials has been successfully developed. The influences of particle size and geometry, chemical driving force, temperature and surface energy on the \(t \rightarrow m\) transformation are investigated in a thermostatic approach. On the continuum scale, a continuum material model for transformation plasticity in partially stabilized zirconia ceramics has been developed. Nonlinear hardening behavior, hysteresis and monoclinic phase fraction during a temperature cycle are analyzed. Finally, The mechanical properties of a TRIP steel matrix reinforced by ZrO\(_2\) particles are analyzed on representative volume elements. Here the mechanical properties of the composite as function of volume fraction of both constituents and the strength of the interface are studied.

## 21.1 Introduction

### 21.1.1 Aims and Scopes of the Present Work

One central aim of the Collaborative Research Center SFB799 “TRIP-matrix composites” was the development of a particle reinforced composite, composed of a TRIP-steel metallic matrix and ceramic particles of partially stabilized zirconia PSZ. The underlying idea was to exploit the phase transformation capability of both constituents in order to enhance and to optimize the mechanical properties by making such a composite.

In particular, the combination of the strain-induced phase transformation in the TRIP steel and the stress-induced transformation in PSZ offers the opportunity to compensate local stress concentrations at considerably high plastic deformations. This reinforcing effect has been verified by several experiments with these metal-matrix-composites MMC under monotonous [1] loading.

In complementation to the fabrication and characterization of these MMC, a thorough theoretical-numerical modeling of the composite material was necessary to understand and to simulate the phase transformation and the deformation behavior of both constituents.

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.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**.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.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

Partially stabilized zirconia (PSZ) is widely used because of its enhanced fracture toughness and nonlinear stress-strain behavior. These favorable mechanical properties of PSZ result from a solid state phase transformation at regions of high stress concentration (e.g. crack tips). This effect, known as transformation toughening, was first reported by Garvie et al. [2] and was extensively investigated by [3, 4, 5]. Generally, some conditions have to be fulfilled for transformation toughening. The existence of a metastable phase in the material is required, which can be achieved either by microstructural parameters such as grain size or by changing the chemical composition. The martensitic (instantaneous) transformation from the metastable parent phase to the stable resultant phase has to be stress-induced.

*t*-phase) embedded coherently in the grains of a polycrystalline cubic matrix material (see Fig. 21.2). These precipitates can transform into the monoclinic (

*m*-phase) [7] triggered either by temperature or stress, resulting in the formation of multiple, partially self-accommodating variants. The \(t \rightarrow m\) phase transformation, if unconstrained, is accompanied by a volume dilatation of about 4% and a shear strain of about 8%. The increase in volume induces (residual) compressive stresses in the cubic (

*c*-phase) matrix leading to a shielding effect at stress concentrations, which contributes to the toughness of the material.

## 21.2 Micromechanical Phase-Field Approach

Phase-field is considered to be a powerful mathematical computational tool in simulations involving interface kinetics. In the past decades PF approach has been successfully established in various fields for materials science understanding such as: solidification, solid-state phase transformation, precipitate evolution and coarsening kinetics, grain growth, martensitic phase transformation (MPT) and also in damage and crack growth phenomena.

For past few decades there has been active research towards the direction of modelling partially stabilized zirconia (PSZ) materials. Wang et al. [8] was one of the early study on PSZ for \(c \rightarrow t\) phase transformation involving Ginzburg-Landau (GL) phenomenological theory based PF model. Later in [9] the authors simulated alternating band structure formed by self-organized orientation variants of \(t\)-phase particles. In [10] the first three-dimensional model for generic \(c \rightarrow t\) MPT was presented. Mamivand et al. [11] reported the first work on anisotropic and inhomogeneous PF modeling for \(t \rightarrow m\) phase transformation in zirconia ceramics. The work discussed the simulation results based on different initial and boundary conditions in comparison to experimental observations. Further the authors [12] incorporated the effect of stress and temperature to capture the forward \(t \rightarrow m\) and reverse \(m \rightarrow t\) transformation to model pseudo-elastic behavior in polycrystalline zirconia.

A comprehensive work on non-conserved type GL-based phase-field models for generic martensitic phase transformation was developed in a series of three papers from Levitas et al. [13, 14, 15]. Levitas et al. used a \(2-3-4\) or higher order polynomial for approximating the Gibbs energy and effective strain transition from austenite to any martensitic variant. This work principally relies on the phenomenological GL phase-field model developed by Levitas et al. with \(2-4-6\) type Landau potential.

### 21.2.1 Phase-Field Method

*L*is the positive kinetic coefficient, and \(\beta \) is a positive gradient energy coefficient. The interface energy contribution provided by Levitas and Preston [13] was used.

### 21.2.2 Model Setup

Lattice transformation | Transformation strain \(\varvec{\epsilon }_{ij}\) |
---|---|

\(c\, \rightarrow {{t}}_1\) | \( \varvec{\epsilon }_{\text {tr}}^{t1} = \begin{bmatrix} \quad -0.0007 &{} \quad 0 \\ \quad 0 &{} \quad 0.0197 \end{bmatrix}\) |

\(c \rightarrow {{t}}_2\) | \( \varvec{\epsilon }_{\text {tr}}^{t2} = \begin{bmatrix} \quad 0.0197 &{} 0 \\ 0 &{} \quad -0.0007 \end{bmatrix}\) |

\({{t}} \rightarrow {{m}}_+\) | \( \varvec{\epsilon }_{\text {tr}}^{m+} = \begin{bmatrix} \quad 0.012479 &{} \quad 0.079614 \\ \quad 0.079614 &{} \quad 0.019139 \end{bmatrix}\) |

\(t \rightarrow {{m}}_-\) | \( \varvec{\epsilon }_{\text {tr}}^{m-} = \begin{bmatrix} \quad 0.012479 &{} \quad -0.079614 \\ \quad -0.079614 &{} \quad 0.019139 \end{bmatrix}\) |

Elastic stiffness (in GPa) of \(c\)-phase , \(t\)-phase and \(m\)-phase [19]

Phases | \( \mathbf{E} _{11} \) | \( \mathbf{E} _{22} \) | \( \mathbf{E} _{33} \) | \( \mathbf{E} _{44} \) | \( \mathbf{E} _{55} \) | \( \mathbf{E} _{66} \) | \( \mathbf{E} _{12} \) | \( \mathbf{E} _{13} \) | \( \mathbf{E} _{16} \) | \( \mathbf{E} _{23} \) | \( \mathbf{E} _{26} \) | \( \mathbf{E} _{36} \) | \( \mathbf{E} _{45} \) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

\(c\)-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 |

Since the specific domain setup differs in some cases of our simulations provided in the following subsections, they are discussed in detail in the corresponding subsections of selected results. Overall in common, we assume anisotropic elastic behaviour in both elastic and phase transformation domain in our simulations. The effective transformation strain and elastic constants at a material point inside the phase transformation domain are evaluated as a function of \(\phi \) [16]. The description of anisotropy is necessary to capture variant orientation relationship and the effects of various external loading directions on MPT. The material parameters used in the model are listed in Tables 21.1 and 21.2.

### 21.2.3 Selected Results and Discussion

#### 21.2.3.1 Phase Stability and Energy Barriers

On the thermodynamic perspective of zirconia at ambient temperature, it is clear that the global minimum is at \(m\)-phase and the global maximum is at \(t\)-phase . So theoretically the metastable \(t\)-phase always tends to transform to stable \(m\)-phase . But in almost all commercial PSZ ceramics (refer MgO-ZrO\(_2 \)micrographs from the book of Hannink et al. [18]) the \(t\)-phase is observed to be stable at ambient temperature. Multiple factors may cause such a stabilization, which include: stabilizer doping such as MgO, presence of residual stresses from prior \(c \rightarrow t\) transformation, but also defects like dislocations and grain boundaries. Later in this work, the effect of residual stress is investigated. Here in this section we compare the potential functions from Mamivand et al. [11] and Levitas and Preston [14] commonly used in literature for modelling \(t \rightarrow m\) transformations. We investigate the capabilities of these functions for such a stabilization.

The \(2-4-6\) potential used by Mamivand et al. [11] for approximating the Gibbs energy contribution defines energy barrier just by the analytical function. The barrier is then levered by the enthalpy difference between the parent and product phase irrespective of temperature, see Fig. 21.4a. Even at ambient temperature the approximated Gibbs energy landscape provides an energy barrier considering only thermodynamic contribution by pure zirconia, which is in contradiction to true physical behaviour. In the case of Levitas and Preston [14] based formulation for temperatures below \(M_s\) the function doesn’t exhibit any barrier for transformation.

At high temperature just above \(M_s\) (see Fig. 21.4d) the energy landscape calculated based on Mamivand’s potential formulation shows local minima at the parent and product phase. For the same, a global maximum or energy barrier is visible at order parameter \(\phi \approx 0.1\) (see \(T < M_s\)). But as the temperature increases this decreases the barrier, and after crossing the \(T_0\) there remains an intermediate local minimum which is neither parent nor product phase, the local minimum is close to order parameter \(\phi \approx 0.1\), see Fig. 21.4b, c. In contrast, Levitas type potential used in this work has no intermediate minimum rather provides a barrier between the parent and product phase for temperatures above \(M_s\).

Based on Levitas et al. formulation, utilized in this work for zirconia ceramics at any temperature, a global/local minima is retained at the product phase. And for *T* > \(M_s\) a local minimum is also retained in parent phase, see Fig. 21.4b, d. Thus the potential function used in this work results in a proper representation of zirconia ceramics material behaviour from a pure thermodynamic stand point.

#### 21.2.3.2 Variant Selection by Energy Barriers

Input parameters used in \(t \rightarrow m\) simulation

Description | Symbol | Value | Unit |
---|---|---|---|

\(m\)-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 |
| 2 | m\(^3\)/Js |

Material parameter |
| 6 | – |

Figure 21.6a–c show a cooling induced martensitic transformation in a single crystal, rotated by \(45^{\circ }\) around ‘b’-axis (see Fig. 21.5a) with isothermal condition below \(M_s\) at 1250 K. A surface plot inside the PTD shows the evolution of order parameter \(\phi \) where the color legend represents, red being \(m\) \(^+\), blue as \(m\) \(^-\) and green as \(t\).

In cooling induced case there is no intermediate energy barrier between parent and product phase for transformation (see Fig. 21.5b) since the temperature is below \(M_s\) and here product \(m\)-phase is stable. So after initialization an almost homogeneous nucleation process takes place where all possible nucleation sites of all possible variants are preferred to grow because of the adequate thermodynamic driving force (see Fig. 21.6a–d). In the numerical simulation, such condition will lead to different microstructure arrangements for different initialization. The evolved microstructures would be of mixed patterns where junction planes are parallel or orthogonal to ‘c’-axis, which could be observed within a single grain (both orientation scenarios specified by Hannink et al. [18]). In Fig. 21.6c a large quantity of junction plane between two \(m\)-phase variants are orthogonal to ‘c’-axis direction and there are small amount of junction planes (upper right, lower left and lower right) oriented parallel to ‘c’-axis. Figures of both such lamella directions of twin formation have already been presented in the work of Hannink et al. [18] within a single \(t\)-phase lentil.

Simulation parameters here remain the same as cooling induced microstructure formation case, except the operating temperature being 1310 K above \(M_s\) and with \(\sigma _{\text {app}}\,=\,\)1 GPa compression along ‘b’-axis (see Fig. 21.5a). An initial superimposed noise with a range confined within the barrier of the Gibbs enthalpy landscape (see Fig. 21.5b) is applied.

Since the pure thermodynamic driving force is not adequate to trigger the transformation, there would not be martensitic evolution at all. As the compressive stress is superimposed additionally to the thermodynamic contribution, depending on the orientation of crystal relative to the applied stress some variants are preferred to grow by decreasing the energy barrier and some are obstructed by increasing the barrier. In other words, the energy landscape is skewed such that some variants have energy barrier and others don’t, see Fig. 21.9a. This becomes clear by comparison between solid blue curve where no external stress is applied, and dashed red curve after application of external stress. Here in dashed red curve one variant experiences a barrier and the other doesn’t. In this example (Fig. 21.6d–f) the \(m\) \(^+\)—red nucleation sites are preferred. At the initial stage, \(m\) \(^+\) red variant nucleates and grows and meanwhile \(m\) \(^-\) blue variant vanishes because of the energy barrier. Additionally, by superimposing normal stress \(\sigma _{\text {app}}\) the driving force exceeds the minimum driving force required for transformation above \(M_s\) and triggers the transformation. Initially only \(m\) \(^+\) variant lamellae grow such that they increase the strain energy. As the lamellae reach the grain boundary or imperfections, this piles up stress and triggers the \(m\) \(^-\) blue self accommodating variant thus reducing a part of the strain energy gained.

According to the investigation on MgO-ZrO\(_2 \)by Kelly and Ball [17] the potential twinning plane/junction plane for twin related variants is either [100]\(_m\)/‘a’-axis or [001]\(_m\)/‘c’-axis, based on our model base axis orientation in Fig. 21.3. The resulting junction plane [001]\(_m\)/‘c’-axis (see Fig. 21.6) is consistent with the experimental observations of [17, 18, 23, 24]. It is clear that among the two possible orientation scenarios specified by Hannink et al. [18] between \(m\)-phase and \(t\)-phase , junction plane parallel to ‘c’-axis, twin-related variants retain some untransformed \(t\)-phase , which is also consistent with our observations. But the reason for possible conditions under which such oriented structure could be reproduced has not been discussed yet before.

#### 21.2.3.3 Origin and Effect of Residual Stresses

The probable initial existence of residual stresses [25, 26] in the \(t\)-phase matrix as a result of the \(c \rightarrow t\) transformation and their effect on \(t \rightarrow m\) transformation is also not well understood yet. Such a residual stress is not yet considered in modeling for \(t \rightarrow m\) . In almost all commercial ceramics \(t\)-phase is stable at ambient temperature. Multiple factors may cause such a stabilization, here we look into the possibility and the effects of residual stress present prior to \(t \rightarrow m\) transformation. In order to evaluate the peak residual stress which could be expected during \(c \rightarrow t\) transformation, we simulate \(c \rightarrow t\) transformation inside a \(c\)-phase matrix. The evolution of average pressure inside the \(t\)-phase lentil during transformation is tracked. This pressure is later used as an initial condition to mimic presence of residual stress during \(t \rightarrow m\) transformation.

A simple model for single lentil setup within a square phase transformation domain is chosen with a size such that an average size of tetragonal lentil could be accommodated. This phase transformation domain is placed within an large elastic domain with \(c\)-phase . The transformation domain is initialized with \(c\)-phase and a circular seed of \(t\)-phase is placed at the centre of the phase transformation domain. The initialization and boundary conditions are set similar to those of model for \(t \rightarrow m\) . The domain is allowed to transform from cubic to tetragonal (\(c \rightarrow t\) ) by undercooling at 1300 K without any external mechanical loading. The thermodynamic functions for evaluating Gibbs enthalpy values are taken from [11, 21]. We assume anisotropic elastic behavior for the whole domain. The elastic constants of respective phases are provided in Table 21.2. Based on the lattice constants from [8] one can evaluate the transformation strain (see Table 21.1), the critical temperature \(M_s^{c\rightarrow t}\) = 1423 K and equilibrium temperature \(T_0^{c\rightarrow t}\) = 584 K acquired from [27]. All other parameters are similar to those of previously explained model setup and listed in Tables 21.2 and 21.3.

In order to evaluate the mean in-plane pressure on a multi lentil setup, a similar setup like a single lentil setup is choose. This setup represents a periodically placed RVE. The placement of the initial seeds are arranged such that they represent a proper microstructure. The seed at the center is replaced with a noise where an equal possibility is given to both variants to nucleate and grow. Because of the stress state of the neighboring \(t\)-phase lentils a selective nucleation of red variant takes place which is more favorable. The peak average pressure experienced by these \(t\)-phase lentils are tracked and plotted in Fig. 21.8. In this multi lentil setup the resulting mean in-plane pressure inside the lentils is 0.35 GPa which is larger than that of the single lentil simulation case. This gives a clear evidence of residual stress from prior \(c \rightarrow t\) transformation.

As consequence of result from single and multi lentil simulations we choose \(\approx \)0.3 GPa as the initial pressure inside the lentil which is also consistent with the FEM based investigation on tetragonal inclusion in a cubic matrix by [26].

*T*= 1250 K\(\,<\,M_s\). Figure 21.9b shows the microstructure formed by stress induced transformation at 1250 K. It is clear that the residual stress from the \(c \rightarrow t\) transformation contributed to the stability of \(t\)-phase . As the operating temperature decreases the residual stress required to introduce a barrier for transformation increases (see Fig. 21.10). By this it becomes clear that residual stress is not the only contribution involved in the \(t\)-phase stability.

## 21.3 Mesomechanical Model

### 21.3.1 Transformation Criterion for a Single Precipitate Embedded in an Infinite Matrix

In this section we extend the transformation criterion developed in [6] in order to account for the pressure sensitivity of the material. This is done in a plane-strain setting based on a number of assumptions.

#### 21.3.1.1 Working Hypotheses

- 1.
The elastic tensors of the cubic matrix and tetragonal precipitate (inclusion) are assumed to be isotropic and identical.

- 2.
The inclusion is modeled as having a rectangular cross-section with width

*B*, height*H*and aspect ratio \(\alpha =H/B\) in its untransformed state. - 3.
The pseudo-twin structure after \(t \rightarrow m\) transformation is modeled as a stack of equal-thickness lamellae, each of which carries a strain of \(\tilde{\tilde{\varepsilon }}_{11}=\tilde{\tilde{\varepsilon }}_{22}=2\%\) resulting in a relative volume change of \(\tilde{\tilde{\varepsilon }}_\mathrm {vol}=4\%\) and shear transformation strain \(\tilde{\tilde{\varepsilon }}_{12}=\pm 8\%\) (specified in the crystallographic coordinate system

^{1}). Note that, while all lamellae are assumed to have the same thickness, the actual value of this thickness as well as the number of lamellae 2*k*are part of the solution and the corresponding effective shear strain of the inclusion is denoted by \(\tilde{\varvec{\varepsilon }}^{(k)}:=\left\langle \tilde{\tilde{\varepsilon }}\right\rangle _{\mathrm {I}}\), where \(\left\langle \cdot \right\rangle _{\mathrm {I}}\) is the averaging operator over the domain of the inclusion. Specifically, \(\tilde{\varepsilon }_{11}^{(k)}=\tilde{\tilde{\varepsilon }}_{11}\), \(\tilde{\varepsilon }_{22}^{(k)}=\tilde{\tilde{\varepsilon }}_{22}\) resulting in \(\tilde{\varepsilon }_\mathrm {vol}^{(k)}=\tilde{\tilde{\varepsilon }}_\mathrm {vol}=: \tilde{\varepsilon }_\mathrm {vol}\) and \(\tilde{\varepsilon }_{12}^{(k)}=\left\langle \tilde{\tilde{\varepsilon }}_{12}\right\rangle _{\mathrm {I}}\). - 4.
As the specific lamellae arrangement is not part of the solution, we estimate the elastic energy contribution resulting from the “zig-zag” at the inclusion boundary by assuming strictly alternating configuration (see Fig. 21.12).

- 5.We assume that phase transformation occurs when the Gibbs-enthalpy is equal in the transformed \(\left( G^\mathrm {m}\right) \) and untransformed \(\left( G^\mathrm {t}\right) \) states, i.e.,where \(\Delta E_\mathrm {el}\) is the difference in the elastic strain energy, \(\Delta E_\mathrm {ch}\) is the difference in the chemical part of the bulk enthalpy and \(\Delta E_\mathrm {sur}\) is the difference in surface energy.$$\begin{aligned} \Delta G=G^\mathrm {m}-G^\mathrm {t}=\Delta E_\mathrm {el} + \Delta E_\mathrm {ch} + \Delta E_\mathrm {sur} {\mathop {=}\limits ^{!}} 0, \end{aligned}$$(21.5)

#### 21.3.1.2 Energetic Contributions

#### 21.3.1.3 The Transformation Criterion

*k*. This ambiguity is resolved by choosing the number of lamellae such that it minimizes the Gibbs free enthalpy in the transformed state, i.e.,

*k*, (21.26) can be rewritten e.g. as a criterion for the applied far-field stress at fixed temperature,

### 21.3.2 Uniaxial Loading

In this section we apply the transformation criterion to uniaxial loading conditions in order to investigate the tension-compression asymmetry predicted by the model as well as geometric effects.

#### 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

*B*and aspect ratio \(\alpha \) of the inclusions and increases with increasing surface energy \(\beta _{\mathrm {I/I}}\).

The sensitivity to the inclusion size and shape is particularly pronounced; for \(\alpha =5\) a change in *B* from 29 to 24 nm results in a change in transformation stress of \(250\%\) in relative terms (see Fig. 21.14). A similar effect is achieved by changing the aspect ratio from 6 to 4 (see Fig. 21.15). Further, it should be noted that the transformation stress abruptly changes at certain values of *B*. This effect is due to the discrete nature of the optimization problem (21.27) and a direct consequence of our assumptions concerning the post-transformation microstructure; every jump of the transformation stress corresponds to a change in number of lamellae *k* and therefore to a change in microstructure.

## 21.4 Homogenization Within an Infinite Grain

*K*and shear modulus \(\mu \). To complete the formulation, assumptions concerning post-transformation behavior of \(\mathbb {C}^\mathrm {I}\) are required, specifically

- 1.
the elastic properties of the monoclinic and tetragonal phase are identical,

- 2.
as long as \(\left| \tilde{\varepsilon }_{12}^{(k)} \right| < \tilde{\tilde{\varepsilon }}_{12}\), the inclusions have no resistance to shear parallel to the c-axis, i.e. in the \(\varvec{e}_1\otimes \varvec{e}_2\)-direction.

- 1.before the onset of transformation (\(f_\mathrm {m}\left( \left\langle \varvec{\sigma }\right\rangle _{\mathcal {B}_{M}}, T\right) = 0\))$$\begin{aligned} \left\langle \mathbb {C}\right\rangle \left( \left\langle \varvec{\sigma }\right\rangle _{\mathcal {B}_{M}}, T, \hat{B}\right) = \mathbb {C}, \end{aligned}$$(21.42)
- 2.after the onset of transformation (\(f_\mathrm {m}\left( \left\langle \varvec{\sigma }\right\rangle _{\mathcal {B}_{M}}, T\right) = f_\text {t}\))where$$\begin{aligned}&\left\langle \mathbb {C}\right\rangle \left( \left\langle \varvec{\sigma }\right\rangle _{\mathcal {B}_{M}}, T, \hat{B}\right) = \nonumber \\&\begin{bmatrix} K + \left[ \frac{4}{3} - Z_{1}\left( \varphi , f_{\text {t}}\right) \right] \mu &{} K - \left[ \frac{2}{3} - Z_{1}\left( \varphi , f_{\text {t}}\right) \right] \mu &{} 0\\ K - \left[ \frac{2}{3} - Z_{1}\left( \varphi , f_{\text {t}}\right) \right] \mu &{} K + \left[ \frac{4}{3} - Z_{1}\left( \varphi , f_{\text {t}}\right) \right] \mu &{} 0\\ 0 &{} 0 &{} \left[ 1- Z_{2}\left( \varphi , f_{\text {t}}\right) \right] \mu \end{bmatrix} , \end{aligned}$$(21.43)$$\begin{aligned} Z_{1}\left( \varphi , f_{\text {t}}\right)&:= Z\left( f_{\text {t}}\right) \sin ^2(2\varphi ),&Z\left( f_{\text {t}}\right) \cos ^2(2\varphi ),\end{aligned}$$(21.44)$$\begin{aligned} Z_{2}\left( \varphi , f_{\text {t}}\right)&:= Z\left( f_{\text {t}}\right) = \frac{f_\mathrm {t}}{1-\left[ 1-f_\mathrm {t}\right] P_1},&\cos \varphi = \varvec{e}_{1}\cdot \bar{\varvec{e}}_1 \end{aligned}$$(21.45)
- 3.after the transformation shear reaches its maximum value \(\tilde{\tilde{\varepsilon }}_{12}\),$$\begin{aligned} \left\langle \mathbb {C}\right\rangle \left( \left\langle \varvec{\sigma }\right\rangle _{\mathcal {B}_{M}}, T, \hat{B}\right) = \mathbb {C}. \end{aligned}$$(21.46)

## 21.5 Continuum Mechanics Approach

A pragmatic engineering approach to phase transition is a phenomenological modeling based on non-linear constitutive laws in the framework of continuum mechanics. The fundamentals are outlined e.g. in [34]. In particular for partially stabilized zirconia (PSZ), such a model was developed by Sun et al. [29]. Based on the concept of representative volume element (RVE) and the Hill-Rice internal variable theory [35], this model provides a set of constitutive equations for the inelastic deformations caused by tetragonal-monoclinic \(t \rightarrow m\) phase transformation as function of monoclinic volume fraction. The model is restricted to a material point only. The authors of [29] did not realize an implementation of their model into a numerical tool to solve a boundary value problem for applications to real structures of PSZ. Therefore, in the present work, the Sun model was implemented into the finite element code ABAQUS [36] to allow simulations of the TRIP-matrix composite as will be reported in Chap. 22.

Due to missing quantitative data for the model parameters Sun et al. [29] introduced instead of this an additional hardening term in the transformation condition, which is limited to the special case of proportional mechanical loading under isothermal conditions. Another weakness of this model is the assumption and averaging of homogeneously distributed microscopic quantities over the RVE. Therefore, in Mehlhorn et al. [37] the basic concept of the Sun model has been extended to capture not only the mechanical but as well the thermally induced phase transformation and thermal expansion to simulate thermomechanical processes. Moreover, the influence of the size of transformable tetragonal particles in the cubic matrix has been incorporated. The basic assumptions and the specific formulation of the model within a thermodynamic framework will be presented in the following.

### 21.5.1 Constitutive Model for Phase Transformation in PSZ

#### 21.5.1.1 Homogenization of PSZ Material

*matrix*, contains two crystallographic phases: the untransformable cubic zirconia and transformable, tetragonal particles embedded in the cubic phase. The second component, denoted as

*inclusions*, contains monoclinic zirconia particles, which are generated by phase transformation from their metastable tetragonal parents when the RVE is sufficiently high loaded.

*T*is assumed to be homogeneously distributed in the RVE. We denote the RVE domain with \(\mathcal {B}_{R}\), the matrix and inclusion subdomains with \(\mathcal {B}_{M}\) and \(\mathcal {B}_{I}\), and their volumes with \(V_{R}\), \(V_{M}\) and \(V_{I}\), respectively. Thus, the relative volume fraction of transformed material is the basic internal variable calculated by

*A*describes the strength of the constraint imposed on the transformed monoclinic inclusions by the surrounding elastic matrix. The matrix stress \(\varvec{\sigma }^{M}\) is related to the macroscopic stress \(\varvec{\Sigma }\) acting on the RVE, via the elastic stiffness \(\mathbb {C}\) and the amount \(f_\mathrm {m}\) of transformed phase, and can be calculated by an Eshelby approach (see e.g. [38]) and the Mori-Tanaka homogenization scheme.

#### 21.5.1.2 Thermodynamic State Potentials

*T*and the actual state of inelastic deformation represented by the monoclinic volume fraction \( f_\mathrm {m}\) and the transformation strain \( \left\langle \varvec{\epsilon }^{\mathrm {ps}} \right\rangle _{\mathcal {B}_{I}}\).

*q*is the volume specific heat of transformation of zirconia and \(T^{*}\) is the tetragonal-monoclinic equilibrium temperature of zirconia.

*h*(

*r*) may be chosen as constant in the

*C*differs depending on whether forward or reverse transformation occurs.

### 21.5.2 Numerical Results

#### 21.5.2.1 Particle Size Dependent Surface Energy Change

*r*of the currently active transforming particles varies with the monoclinic volume fraction \(f_\mathrm {m}\) during the process of phase transformation, (21.62) is studied for three different size distribution functions \(h_{i}(r_\mathrm {min,}i,r_\mathrm {max})\) with \(i=1,2,3\). For the upper limit of the particle size range a typical PSZ particle radius of \(r_\mathrm {max}=1\cdot 10^{-7}\,\mathrm{m}\) is chosen. The values of \(r_\mathrm {min}\) are taken as following fractions: \(r_\mathrm {min,1}/r_\mathrm {max}=0{.}95\), \(r_\mathrm {min,2}/r_\mathrm {max}=0{.}5\), and \(r_\mathrm {min,3}/r_\mathrm {max}=0{.}25\). Figure 21.18 gives a graphical representation of (21.62), using these values. It is obvious that a narrow size distribution (as \(r_\mathrm {min}/r_\mathrm {max} = 0{.}95\)) is very close to a constant particle size of the original Sun model, resulting in a slight dependence of the radius

*r*on the transformed volume fraction \(f_\mathrm {m}\). The wider the distribution function \(h(f_\mathrm {m})\) is (i.e. with smaller particle size ratios \(r_\mathrm {min}/r_\mathrm {max}\)), the stronger is the nonlinear dependence of

*r*on \(f_\mathrm {m}\).

*r*(

*f*) on the volume specific surface energy change \(\Delta \varphi _{{R}}^{\mathrm {sur}}\) (21.58) is illustrated in Fig. 21.19. For a narrow particle size distribution with ratio \(r_\mathrm {min}/r_\mathrm {max}=0{.}95\), the surface energy change \(\Delta \varphi _{{R}}^{\mathrm {sur}}\) grows almost linear with \(f_\mathrm {m}\) similar as in the original Sun model. For smaller ratios \(r_\mathrm {min}/r_\mathrm {max}\), the extended material model shows a strong nonlinear increase of \(\Delta \varphi _{{R}}^{\mathrm {sur}}\), especially if \(f_\mathrm {m}\rightarrow f_\mathrm {m}^\mathrm {max}\), as it can be seen for \(r_\mathrm {min}/r_\mathrm {max}=0{.}25\) in Fig. 21.19. This means, \(\Delta \varphi _{{R}}^{\mathrm {sur}}\) acts as a transformation barrier, preventing very small particles from transforming even under high thermal or mechanical loading.

#### 21.5.2.2 Temperature-Induced Phase Transformation

In order to demonstrate the ability of the extended material model to reproduce the hysteresis strain-temperature behavior of PSZ ceramics, a cooling-heating cycle \(1373\,\mathrm{K}\) – \(293\,\mathrm{K}\) – \(1373\,\mathrm{K}\) is numerically simulated. The required model parameters associated with the changes in chemical energy (*q* and \(T^*\), see (21.57)), are taken from literature: heat of transformation \(q=2.82\,\mathrm{J/m}^{3}\) and the phase equilibrium temperature \(T^*=1447\,\mathrm{K}\). The difference in specific surface energies between the tetragonal and monoclinic phase was set to \(\Delta \varphi ^{{\text {sur}}{(t\rightarrow m)}}=0.36\,\mathrm{J/m}^{2}\), see [37]. Since no values for the dissipation parameter \(D_0\) and the amount of transformable tetragonal material \( f_\mathrm {m}^\mathrm {max}\) were available, they were estimated in order to obtain physically meaningful results. Moreover, a variation of these parameters is performed to study their influence on the material model behavior. \(D_0\) was specified to the values 10, 20, and 30 \(\mathrm{MPa}\). \( f_\mathrm {m}^\mathrm {max}\) was set to 0.15, 0.25 and 0.35, respectively. For all remaining model parameters the values published by Sun et al. [29] are used.

Figure 21.20 shows the numerically obtained strain-temperature curves \(E_{11}-T\) and the corresponding phase evolution \(f_\mathrm {m}\)–*T* during the cooling-heating cycle for different sets of model parameters. As it can be seen, the typical strain hysteresis loops of PSZ ceramics are predicted by the material model, caused by a tetragonal-to-monoclinic transformation on cooling and a reverse transformation on heating. In each diagram, the influence of particle size distribution is included by varying the ratio \((r_\mathrm {min,i},r_\mathrm {max})\). It can be seen in all diagrams, that smaller size ratios lead to a considerably nonlinear strain-temperature behavior and rounded transition curves. The influence of \(D_0\) on the strain-temperature curves can be observed in Fig. 21.20a–d. \(D_0\) governs the size of the strain hysteresis between cooling and heating. In contrast, \( f_\mathrm {m}^\mathrm {max}\) influences the total transformation strain and hence the length of the temperature interval in which transformation occurs, see Fig. 21.20e–h.

These results demonstrate the feasibility of the extended material model, which forms a solid basis for simulations of structures and composites made of PSZ. Unfortunately, it was not possible to identify the required parameters for the type of MgO-stabilized ZrO\(_2\) manufactured in the CRC799.

## 21.6 Simulations of ZrO\(_2 \)-Particle Reinforced TRIP-Steel Composite

In order to assist the development of particle reinforced composites manufactured by a powder metallurgical process route from TRIP-steel and partially stabilized ZrO\(_2 \) ceramics particles, accompanying numerical simulations have been carried out. The mechanical properties of such a composite material are quite complex as they arise from the properties of its individual components, their volume content, and the properties of the interface between them. As explained in the previous sections, PSZ can undergo a stress-triggered phase transformation. This can lead to an additional toughening effect compared to non-transformable ceramics as observed in [41]. The TRIP-steel exhibits a deformation induced phase transformation from the austenitic parent phase to martensite. By combining the two materials using TRIP steel as matrix and PSZ as strengthening particles, an elasto-viscoplastic particle-reinforced composite is created with the capability of phase transformation in each component.

### 21.6.1 Unit Cell Model of the Composite

A well established method to investigate the mechanical response of composites is a parameter study using a suitable mechanical cell model of the composite, which is simulated by means of the finite element method, see for example Mishnaevsky [42].

In this work, this approach has been applied to study the effective stress-strain behavior of this particulate TRIP steel-ZrO\(_2 \) composite. Details can be found in the publications of Mehlhorn, Prüger et al. [43, 44, 45]. The influence of the volume content of ZrO\(_2 \) particles and the interface properties on the overall response of the composite is investigated. Three different interface types are considered: (i) perfectly bonded, (ii) not bonded, and (iii) cohesive law, respectively, The calculations of the material responses are performed using a finite element analysis of unit cells of the composites under tensile, compressive and biaxial loading. Here, selected results will be reported.

Numerical simulations of composites require proper constitutive equations for both constituents. Here, the Sun model [29] as explained above is employed for the PSZ ceramics. For modeling the viscoplastic deformation and martensitic phase transformation of the TRIP steel, the constitutive law developed by Prüger [46] is applied. It describes the strain-induced transformation from a fully austenitic microstructure (\(\gamma \)) to martensite (\(\alpha '\)) under thermal and/or mechanical loading. Both material models were available as Fortran routines implemented via UMAT interface into the finite element software ABAQUS [47]. More information about the used material parameters for the PSZ and the specific TRIP-steel can be found in [43, 44, 45], and in Chap. 22.

*a*with a single spherical ZrO\(_2 \) particle placed in its center.

The mechanical model exhibits a triple symmetry with respect to geometry and loading. Therefore the use of one-eighth of the RVE is admissible, and a corresponding FEM discretization is elaborated. Although the unit cell was numerically simulated under various stress triaxialities, only the results for uniaxial loading are reported here. Regarding the interface between the components, two limiting cases are discussed here: the perfectly bonded connection and the non-bonded, frictionless movable contact. An optimal composite possesses a high energy absorption capacity and exhibits pronounced phase transformation in the ZrO\(_2 \) ceramic and the TRIP steel. The macroscopic true stress and true strain tensors \(\mathbf {\Sigma }\) and \(\mathbf {E}\) are used in order to evaluate the mechanical work according to \(W = \int \nolimits _{0}^{\bar{\mathbf {E}}} \!\mathbf {\Sigma }:\mathrm {d}\mathbf {E} \), where \(\bar{\mathbf {E}}\) denotes the considered deformation stage. Because the elastic strains are small, *W* equals approximately the energy absorption for sufficiently large total strains. In order to quantify the relative change in energy absorption capacity, this energy is related to those values \( W_{\mathrm {hom}} \) obtained for a unit cell made only of TRIP steel.

During deformation an inhomogeneous distribution of the volume fractions of the monoclinic zirconia and the martensite develop in the ceramic and the TRIP steel, respectively. Therefore the averages of \(f_{\alpha '}\) and \(f_{\text {m}}\) over the corresponding volumes are used. The simulation is stopped, when the maximum principle stress in the PSZ reaches its ultimate tensile strength \(\sigma _{\mathrm {cr}}^{\mathrm {t}}=1600\,\mathrm{MPa}\).

### 21.6.2 Results and Discussion

Energy absorption capacity for the composite with perfectly bonded and non-cohesive interface in uniaxial loading

| 0.05 | 0.05 | 0.10 | 0.10 | 0.20 | 0.20 |
---|---|---|---|---|---|---|

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 |

In case of the **perfectly bonded interface**, the stress-strain curves show a distinct dependence on the volume fraction of ZrO\(_2 \) ceramic *f*, as depicted in Fig. 21.22a. It can be observed that an increasing *f* leads to higher yield stresses and strain hardening rates compared to the unreinforced TRIP steel \((f=0)\). As consequence, a pronounced increase in the energy absorption *W* of the composite is obtained. Comparing the values given in Table 21.5, the ratio \({W}/{W^{\mathrm {hom}}}\) increases up to more than \(35\%\) (for same macroscopic equivalent strain). Due to the strong interface, load is transferred from the matrix to the reinforcement during deformation of the composite. Therefore high stresses occur in the ceramic, which reduces the maximal attainable strain with increasing zirconia content *f*. Regarding the phase transformation behavior, a higher volume fraction of zirconia *f* enhances the tendency to phase transformation in zirconia as well as in TRIP steel (Fig. 21.22b). The phase transformation capacity in the PSZ component is saturated to the maximum of \( 35\% \) in a smaller strain interval. The tendency to phase transformation in the TRIP steel \(f_\alpha '\) increases at higher zirconia content, but is limited due to failure of the ceramic.

In case of a **non-cohesive interface**, both the initial yield stress and the strain hardening rate tend to decrease with higher zirconia content *f*, see Fig. 21.23a. Figure 21.23b shows that the development of martensite is considerably higher than in the case of the perfectly bonded interface. At the end of deformation nearly 30% of martensite has evolved. Because of the non-cohesive interface, no tensile stresses are transferred from the TRIP steel matrix to the zirconia inclusion. Thus, no phase transformation is seen in zirconia. Moreover, the area of the load bearing cross-section consists of TRIP steel only and is the smaller the higher the zirconia content becomes, which reduces macroscopic yield stress. After debonding, the particle acts partially like a void. However, no softening is observed in the macroscopic stress-strain response because of the hardening behavior of the TRIP steel and the locking effect caused by the particle.

## 21.7 Conclusions

Based on the work of Levitas and Preston [13] for generic martensitic transformation, a phase-field model for MgO-ZrO\(_2 \) material was implemented [16]. The potential function used in this work results in a proper representation of transformation behavior of zirconia ceramics from a pure thermodynamic stand point. In the simulations different patterns of microstructures were found for cooling induced and stress induced transformation. These patterns are consistent with experimental observations by Hannink et al. [18]. It is evident that the presence of an energy barrier plays a key role in variant selection and the transformation path taken. By which, in stress induced case a sequential growth of lamellae was visible. In contrast, the cooling induced case is categorized with an almost homogeneous nucleation where all variants are preferred to evolve. Additionally on a single crystal level the simulations showed that, in the stress induced case, microstructure with junction planes parallel to the ‘c’-axis is formed because of variant selection. It was shown that residual stresses inside \(t\)-phase lentils from \(c \rightarrow t\) transformation have a magnitude of \({\approx }0.3\) GPa and contribute to the stability of the *t*-phase. Also the magnitude of stress required for introducing energy barrier increases with decreasing temperature below \(M_s\).

A mesomechanical, two-dimensional model for Mg-PSZ with an energetic transformation criterion based on the analytical solution for a rectangular inclusion in an infinite matrix has been successfully developed. Using this model the influence of individual parameters such as size, geometry and surface energies on the transformation initiation and resulting microstructure can be efficiently studied. It predicts that the stability of the tetragonal inclusions deteriorates as the inclusions grow in size and aspect-ratio. Further, the tension-compression asymmetry of the transformation behavior known from experiments is captured correctly. A homogenization approach based on the Mori-Tanaka method predicts the transformation to be auto-catalytic within a grain.

A continuum material model for transformation plasticity in partially stabilized zirconia ceramics has been further developed to account for (i) particle size dependent phase transformation behavior, (ii) temperature dependent phase transformation, and (iii) thermoelastic deformation. These more physically based features lead to a nonlinear hardening behavior and smoothly rounded hysteresis curves for the strain and the generated monoclinic phase fraction during a temperature cycle. The influence of the tetragonal particle size distribution on phase transformation could be predicted qualitatively quite well.

Finally, the mechanical properties of a TRIP steel matrix reinforced by ZrO_{2} particles are analyzed, taking the phase transformation in both constituents into account. The influence of the volume content and the interface properties of ZrO_{2} particles on the overall response of the composite is investigated. Material variants with three different zirconia contents and two different interface types, perfectly bonded and non-cohesive, respectively, are considered. The calculations of the material responses are performed using a finite element analysis of representative volume elements of the composites under tensile, compressive and biaxial loading. The results indicate that the enrichment of the TRIP steel with zirconia particles leads to a significant strengthening effect provided the interface has cohesive properties.

## Footnotes

- 1.
If nothing else is specified, all tensor components in this work are referred to a coordinate system with orthonormal basis \((O,\{\varvec{e}_{1},\, \varvec{e}_{2}\})\), where \(\varvec{e}_{1}\) is aligned along the tetragonal c-axis.

## Notes

### Acknowledgements

The authors gratefully acknowledge the the German Research Foundation or Deutsche Forschungsgemeinschaft (DFG) for supporting this work in subproject C4, and was created as part of the Collaborative Research Center TRIP-Matrix-Composites (Project number 54473466—CRC 799). We appreciate the contributions of former colleagues Dr.-Ing. Uwe Mühlich, Dr.-Ing. Stefan Prüger, and Lars Mehlhorn to the achievements.

## References

- 1.S. Decker, L. Krüger, S. Richter, S. Martin, U. Martin, Steel Res. Int.
**83**(6), 521 (2012)Google Scholar - 2.R.C. Garvie, R.H.J. Hannink, R.T. Pascoe, Nature
**258**(5537), 703 (1975)Google Scholar - 3.A.G. Evans, A.H. Heuer, J. Am. Ceram. Soc.
**63**(5–6), 241 (1980)Google Scholar - 4.A.G. Evans, N. Burlingame, M. Drory, W.M. Kriven, Acta Metall.
**29**(2), 447 (1981)Google Scholar - 5.F.F. Lange, J. Mater. Sci.
**17**(1), 240 (1982)Google Scholar - 6.T. Hensl, U. Mühlich, M. Budnitzki, M. Kuna, Acta Mater.
**86**, 361 (2014)Google Scholar - 7.R.C. Garvie, J. Phys. Chem.
**69**(4), 1238 (1965)Google Scholar - 8.Y. Wang, H. Wang, L.Q.Q. Chen, A.G. Khachaturyan, J. Am. Ceram. Soc.
**76**(12), 3029 (1993)Google Scholar - 9.Y. Wang, H.Y.Y. Wang, L.Q.Q. Chen, A.G. Khachaturyan, J. Am. Ceram. Soc.
**78**(3), 657 (1995)Google Scholar - 10.Y. Wang, A.G. Khachaturyan, Acta Mater.
**45**(2), 759 (1997)Google Scholar - 11.M. Mamivand, M.A. Zaeem, H.E. Kadiri, L.Q. Chen, Acta Mater.
**61**(14), 5223 (2013)Google Scholar - 12.M. Mamivand, M. Asle Zaeem, H. El Kadiri, Int. J. Plast.
**60**, 71 (2014)Google Scholar - 13.V. Levitas, D. Preston, Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations I Austenite<–> martensite. Phys. Rev. B
**66**(13), 1–9 (2002). https://doi.org/10.1103/PhysRevB.66.134206 - 14.V. Levitas, D. Preston, Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. II. Multivariant phase transformations and stress space analysis. Phy. Rev. B
**66**(13), 1–15. https://doi.org/10.1103/PhysRevB.66.134207 - 15.V. Levitas, D.L. Preston, D.W. Lee, Phys. Rev. B
**68**(13), 1 (2003)Google Scholar - 16.M.K. Rajendran, M. Kuna, M. Budnitzki, Undercooling versus stress induced martensitic phase transformation: the case of MgO—partially stabilized zirconia. Comput. Mater. Sci.
**174**, 109460 (2019). https://doi.org/ 10.1016/j.commatsci.2019.109460Google Scholar - 17.P.M. Kelly, C.J. Ball, J. Am. Ceram. Soc.
**69**(3), 259 (1986)Google Scholar - 18.R.H.J. Hannink, P.M. Kelly, B.C. Muddle, J. Am. Ceram. Soc.
**83**(3), 461 (2000)Google Scholar - 19.X.S. Zhao, S.L. Shang, Z.K. Liu, J.Y. Shen, J. Nucl. Mater.
**415**(1), 13 (2011)Google Scholar - 20.H. Lukas, S. Fries, B. Sundman,
*Computational Thermodynamics—The CALPHAD Method*(Cambridge University Press, 2007)Google Scholar - 21.D. Pavlyuchkov, G. Savinykh, O. Fabrichnaya, J. Eur. Ceram. Soc.
**34**(5), 1397 (2014)Google Scholar - 22.L.Q. Chen, W. Yang, Phys. Rev. B
**50**(21), 15752 (1994)Google Scholar - 23.G.K. Bansal, A.H. Heuer, Acta Metall.
**20**(11), 1281 (1972)Google Scholar - 24.G.K. Bansal, A.H. Heuer, Acta Metall.
**22**(4), 409 (1974)Google Scholar - 25.R.C. Garvie, J. Phys. Chem.
**82**(2), 218 (1978)Google Scholar - 26.C.R. Chen, S.X. Li, Q. Zhang, Mater. Sci. Eng. A
**272**(2), 398 (1999)Google Scholar - 27.W.E. Lee, M. Rainforth,
*Ceramic Microstructures Property Control by Processing*(Chapman & Hall, 1994)Google Scholar - 28.R. Garvie, M. Swain, J. Mater. Sci.
**20**, 1193 (1985)Google Scholar - 29.Q.P. Sun, K.C. Hwang, S.W. Yu, J. Mech. Phys. Solids
**39**(4), 507 (1991)Google Scholar - 30.C. Wang, M. Zinkevich, F. Aldinger, J. Am. Ceram. Soc.
**89**(12), 3751 (2006)Google Scholar - 31.I.W. Chen, Y.H. Chiao, Acta Metall.
**31**(10), 1627 (1983)Google Scholar - 32.T. Mori, K. Tanaka, Acta Metall.
**21**(5), 571 (1973)Google Scholar - 33.Y. Benvensite, Mech. Mater.
**6**(2), 147 (1987)Google Scholar - 34.F.D. Fischer, Q.P. Sun, K. Tanaka, Appl. Mech. Rev.
**49**(6), 317 (1996)Google Scholar - 35.J.R. Rice, J. Mech. Phys. Solids
**19**(6), 433 (1971)Google Scholar - 36.Abaqus,
*Abaqus, Online documentation*, 6th edn. (Dassault Systems, 2014)Google Scholar - 37.L. Mehlhorn, U. Mühlich, M. Kuna, Adv. Eng. Mater.
**15**(7), 638 (2013)Google Scholar - 38.D. Gross, T. Seelig,
*Bruchmechanik—Mit einer Einführung in die Mikromechanik*, 4th edn. (Springer, Berlin Heidelberg, 2007)Google Scholar - 39.A. Puzrin, G. Houlsby, Int. J. Plast.
**16**(9), 1017 (2000)Google Scholar - 40.T. Mura, Mechanics of elastic and inelastic solids, in
*Micromechanics of Defects in Solids*(Springer, 1991)Google Scholar - 41.D.B. Marshall, J. Am. Ceram. Soc.
**69**(3), 173 (1986)Google Scholar - 42.L.L. Mishnaevsky, Acta Mater.
**52**(14), 4177 (2004)Google Scholar - 43.L. Mehlhorn, S. Prüger, S. Soltysiak, U. Mühlich, M. Kuna, Steel Res. Int.
**82**(9), 1022 (2011)Google Scholar - 44.S. Prüger, L. Mehlhorn, S. Soltysiak, M. Kuna, Comput. Mater. Sci.
**64**, 273 (2012)Google Scholar - 45.S. Prüger, L. Mehlhorn, U. Mühlich, M. Kuna, Adv. Eng. Mater.
**15**(7), 542 (2013)Google Scholar - 46.S. Prüger, Thermomechanische Modellierung der dehnungsinduzierten Phasenumwandlung und der asymmetrischen Verfestigung in einem TRIP-Stahlguss. Ph.D. thesis, TU Bergakademie Freiberg, 2016Google Scholar
- 47.Hibbitt, Karlsson, Sorenson,
*ABAQUS: Version 6.7 Documentation*(HKS, 2009)Google Scholar

## Copyright information

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.