Phase structural ordering kinetics of secondphase formation in the vicinity of a crack
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Abstract
The formation of a second phase in presence of a crack in a crystalline material is modelled and studied for different prevailing conditions in order to predict and a posteriori prevent failure, e.g. by delayed hydride cracking. To this end, the phase field formulation of Ginzburg–Landau is selected to describe the phase transformation, and simulations using the finite volume method are performed for a wide range of material properties. A sixth order Landau potential for a single structural order parameter is adopted because it allows the modeling of both first and second order transitions and its corresponding phase diagram can be outlined analytically. The elastic stress field induced by the crack is found to cause a spacedependent shift in the transition temperature, which promotes a secondphase precipitation in vicinity of the crack tip. The spatiotemporal evolution during nucleation and growth is successfully monitored for different combinations of material properties and applied loads. Results for the secondphase shape and size evolution are presented and discussed for a number of selected characteristic cases. The numerical results at steady state are compared to meanfield equilibrium solutions and a good agreement is achieved. For materials applicable to the model, the results can be used to predict the evolution of an eventual secondphase formation through a dimensionless phase transformation in the cracktip vicinity for given conditions.
Keywords
Phase transformation Mode I crack Phase field method Ginzburg–Landau formulation Precipitation kinetics1 Introduction
Most metallic materials experience some type of interaction with their surrounding environment, which often results in changes in the material morphology that may induce deterioration of their mechanical and physical properties. For many practical applications such changes may derogate a material function and its usefulness, which may eventually lead to failure. If stresses are applied to a structure subjected to a corrosive environment, cracks that would not occur in the absence of one of these two controlling conditions will appear, even for virtually inert material. Thus, the combination of these factors may lead to failure, which is commonly referred to as stress corrosion cracking (Jones 1992).
Hydrogen embrittlement (HE) can be considered to be a type of stress corrosion, since it commonly concerns a ductile metal undergoing a brittle fracture as a result of the combination of applied stresses and a corrosive environment. HE can also occur in mechanically loaded materials that already contain hydrogen emanating from either the manufacturing process or from earlier exposure to hydrogen. For some cases of HE, hydride formation plays a central role in the detoriation process. A hydride is a brittle nonmetallic phase that may cause the embrittlement of metallic materials such as titanium and zirconiumbased alloys and reduce their load bearing capabilities (Coleman and Hardie 1966; Coleman et al. 2009; Chen et al. 2004; Luo et al. 2006). For metal based components exposed to hydrogenrich environments, such as fuel cladding materials in nuclear power reactors or components in rocket engines, there is an impending risk of hydrides forming, which could lead to the so called delayed hydride cracking (DHC). This is a subcritical crack growth mechanism (Coleman et al. 2009; Singh et al. 2004; Coleman 2007; Northwood and Kosasih 1983) that can severely reduce the component lifetime and jeopardize its integrity.
Hydride formation occurs as a result of a combination of complex mechanisms, including for instance simultaneous hydrogen diffusion, hydride precipitation and material deformation (Varias and Massih 2002). For some metals, including zirconium and titanium, such formation has been specifically observed in the vicinity of stress concentrators, following the increased hydrogen transport toward regions of high hydrostatic stresses (Birnbaum 1976; Takano and Suzuki 1974; Grossbeck and Birnbaum 1977; Shih et al. 1988; Cann and Sexton 1980). It is also known that the transformation is not only driven by changes of concentration of species, such impurities and alloying elements, but mechanical stresses can per se promote the precipitation of a second phase (Birnbaum 1984; Allen 1978; Varias and Massih 2002). This type of transformation can be beneficial for instance for certain ceramic materials, which may experience an increase in fracture toughness following the transformation. For such cases, the crack propagation may be impeded following the transformation of metastable phase particles into stable particles with increased volume at the crack tip (Hutchinson 1989; Evans and Cannon 1986). Thus, the resulting transformationinduced compressive stresses in front of the cracktip may obstruct its opening and, consequently, limit the crack growth. However, for the case of transition metals and hydrogen, such beneficial transformation is rarely observed. In fact, for the hydride forming materials the effect is rather the opposite, leading to embrittlement and reduction of the fracture toughness.
Over the years, numerous models have been developed to describe the formation of a second phase at a flaw tip in a variety of crystalline materials (Varias and Massih 2002; Deschamps and Bréchet 1998; GómezRamírez and Pound 1973; Boulbitch and Korzhenevskii 2016; Léonard and Desai 1998; Hin et al. 2008; Massih 2011a; Bjerkén and Massih 2014; Jernkvist and Massih 2014; Jernkvist 2014). Among those are the models resting on the phasefield approach based on Ginzburg–Landau theory, see e.g. Provatas and Elder (2010). Such modeling has found many applications, especially in the areas of magnetic field theory where it has been found useful for providing predictive models (Cyrot 1973; Berger 2005; BarbaOrtega et al. 2009; Cao et al. 2013; Gonçalves et al. 2014). But more importantly for the present application it has also proven to be an efficient methodology to model and predict microstructure evolution in materials. The review paper by Chen (2002) and the references therein give a thorough account of applications for which phasefield modeling has been successfully used to predict the microstructural evolution. For phase field theory applied to solid material microstructure modeling order parameters are used to describe the evolution of a material state (Desai and Kapral 2009). Typically, two different order parameters may be identified to describe different type of phase transformations: the nonconserved structural order parameter corresponding to a diffusionlesstype transformation and the conserved composition order parameter which represents a diffusional transformation. With this in mind, Hohenberg and Halperin (1977) defined three types of models based on the use of these order parameters individually (model A and B, respectively) and their coupling (model C).
In the material, the solid solution may be defined as a disordered material state and crystal structures with a lower degree of symmetry are considered ordered. Under stress field conditions, such as stresses induced by the presence of flaws, the evolution of the order parameters are used to model a secondphase formation. One such study was reported by Massih (2011a), who presented a general setup with coupled conserved and nonconserved field variables in the presence of cracks and dislocations in an elastic solid. Further, in a paper by Boulbitch and Korzhenevskii (2016), a nonconserved order parameter is used to study quasistatic phase transformation in the process zone of a propagating crack. These works constitute a suitable base reference to study the secondphase formation in presence of a crack. In the present paper, the effect of stress concentration in the presence of a crack on the secondphase formation kinetics in crystalline materials is modeled. This work is an extension of the Massih’s and Bjerkén’s works (Bjerkén and Massih 2014; Massih 2011b), where precipitation kinetics at dislocations is studied by using a scalar nonconserved order parameter (Model A), accounting for microstructural reordering. The timedependent Ginzburg–Landau (TDGL) equation is solved numerically to capture the different spatiotemporal state changes of the material in the vicinity of the crack tip. This is achieved by assessing the spatiotemporal fluctuations of the structural order parameter for different sets of material parameters, loads and phenomenological coefficients, making of this paper a full parametrical study, usually undone in the literature. The driving force of this equation, which gives the rate of change the structural order parameter, is the functional derivative of the total free energy with respect to the nonconserved order parameter (Provatas and Elder 2010). To include the effect of the crack to the modeling, a mechanical equilibrium condition is added to the free energy formulation for the system at hand. The mechanical equilibrium is solved analytically reducing the number of equations to be solved to one, the TDGL equation, as in Boulbitch and Korzhenevskii (2016). To this end, a small perturbation from a stressed reference configuration under known stress conditions due to the presence of the crack is assumed. A sixthorder Landau potential energy is incorporated in the structural free energy of the system in order to model first and second order transitions. Moreover, the total free energy of the system takes into account a coupling between the dilatation of the second phase and the order parameter (Boulbitch and Tolédano 1998). The effect of hydrogen concentration is not considered in the model.
The paper is organized as follows: first, the model employed to study the formation of the second phase in presence of the crack is described in Sect. 2. This description is followed by a theoretical steadystate analysis, accompanied by a phase diagram, in Sect. 3. Thereafter, the methodology for the numerical simulations is presented in Sect. 4. The following section, Sect. 5, demonstrates the results obtained for different cases and parameters which contains comparisons between theoretical steadystate and longtime numerical data and the influence of the interface energy on the solutions. Finally, in Sect. 6, the findings are summarized and the conclusions are stated.
2 Model description
To gain insight behind the second phase nucleation around a crack tip, in the present study we solve Eq. (18) for different sign combinations of \(\alpha _{0}\) and \(\beta \). This aims to investigate the importance of temperature induced solubility variation for different materials, while the interaction between displacement field and order parameter are varied.
3 Steadystate analysis
When \(\alpha _0<0\), the parameter A is always negative. This means that the transition temperature \(T_c\) becomes higher than the material temperature T, regardless of the distance from the crack. Thus, the whole material is expected to transform into the second phase. If instead \(\alpha _0 > 0\), a secondphase region may form in the proximity of the crack tip with the size of the zone depending on the value of \(\kappa \,\text {sgn}(\beta )\). For this case the transition temperature is higher than the material temperature only locally, which explains the limited local transformation. For \(\beta >0\), only second order transitions between the phases may take place as the stability lines of the phases coincides (at \(A=0\)), regardless of the value of \(\kappa \). For negative \(\beta \), the regions of stability of the individual phases overlap, where a phase either may be stable or metastable. The stability limits for phase I (\(A=0\)) and II (\(A=1/4\kappa \)) are indicated in the figure. Between these limits the transition line \(A=\frac{3}{16\,\kappa }\) can be identified, which represents a situation at which both phases are equally stable and only first order transitions are expected.

a pure secondphase area, II,

an area with second phase and metastable solid solution, I* + II,

an area with solid solution with metastable second phase, I + II*, and

a pure solid solution, I.
4 Numerical method
To numerically solve the TDGL equation, presented in Sect. 2, we use the opensource partial differential equation solver package FiPy (Guyer et al. 2009), which is based on a standard finite volume approach. For the modeling, the 2ddomain illustrated in Fig. 1 is discretized using a square mesh consisting of \(1000\times 1000\) equallysized square elements with the dimensionless side length corresponding to \(\varDelta \tilde{l}=0.2\). This setup, combined with a LUfactorization solution scheme, was found to yield well converged results and the system is large enough such that the boundary conditions do not affect the results. The size of the time step is deduced from a convergence study, which revealed that a time step \(\varDelta \tau =0.1\) is sufficient to ensure a stable solution.
At the boundary, the gradient of \(\varPhi \) is prescribed to be perpendicular to the boundary such that \(\nabla \varPhi \cdot \mathbf n =0\), where \(\mathbf n \) is a unit vector perpendicular to the boundary.
As initial condition the order parameter, \(\varPhi _{init}\) was randomly set throughout the entire domain, with values in the range \([5 \times 10^{5}, 1 \times 10^{4}]\).
5 Results and discussion
5.1 Temperature higher than bulk transition temperature: \(T >T_{c_0}\)
If the temperature \(T >T_{c_0}\), no phase transition is expected in an uncracked material, and implies that \(\alpha =\alpha _0>0\). The introduction of a loaded crack results in a shift of \(\alpha \), see Eqs. (11), (14), whose magnitude depends on the position relative to the crack tip. In general, the closer to the crack tip, the higher the stress, and the larger the shift, which inevitably implies that the driving force for the phase transformation is larger.
Other cases where \(\kappa \,\text {sgn}(\beta )< 3/16\) are also studied. Similar evolution patterns are obtained as for \(\kappa \,\text {sgn}(\beta )=1\), i.e. an initial order parameter peak rises followed by a limited space expansion. Yet, the evolution have other characteristic values, e.g. the maximum values of \(\varPhi _{peak}\) and \(\tilde{w}\), respectively, vary, see Fig. 4.
5.2 Temperature lower than bulk transition temperature: \(T <T_{c_0}\)
5.3 Comparisons of evolution characteristics
In order to quantitatively describe the evolution behavior, some characteristics measurements are defined: \(\varPhi _{max}\) denotes the maximum value of \(\varPhi _{peak}, \tau _{mp}\) the time to reach this value, \(\tilde{w}_{ss}\) steadystate width, and \(\tau _{95\%ss}\) required for the system to reach 95% of \(\tilde{w}_{ss}\). Results for the presented cases are summarized in Table 1.
For the situation where \(\alpha _0>0\), three different types of evolution patterns are obtained relating to different intervals of \(\kappa \,\text {sgn}(\beta )\): \(\beta>0, \beta <0 \wedge \kappa >3/16\) and \(\beta<0 \wedge \kappa <3/16\), respectively. Results from the simulations of the corresponding cases are presented in the following paragraphs, and in the last paragraph results for \(\alpha _0<0\) are discussed.
For all investigated cases where \(\alpha _0>0\) and \(\beta <0\), it is found that \(\varPhi _{max}, \tilde{w}_{ss}\) and \(\tau _{95\%ss}\) decrease with increasing \(\kappa \), see Table 1. In contrast, the time \(\tau _{mp}\) that characterizes the initial stage of the precipitation is approximately the same (\(\approx \)5) and not exceeding 20% of \(\tau _{95\%ss}\) for any of the considered cases. The latter observations are also true for \(\beta >0\). Further, larger \(\varPhi _{max}\) are reached for negative \(\beta \) than for positive \(\beta \) with same value of \(\kappa \).
It is also found that in cases where \(\kappa \,\text {sgn}(\beta )<3/16\), \(w_{ss}\) increases with increasing \(\kappa \,\text {sgn}(\beta )\). It can be deduced that there is an approximate linear relation between the two with \(\text {d}w_{ss}/\text {d}\tau _{95\%ss}\approx \rho _0/50\). However if \(\beta >0\), all choices of \(\kappa \) result in similar values of the final width \(w_{ss}\) and the time to reach 95% of \(w_{ss}\), respectively. Thus, as been mentioned earlier, the precipitation process is not influenced by the value of \(\kappa \), and it can be concluded that a Landau potential of lower order than what is used in this study may be sufficient to simulate the evolution if \(\beta >0\).
Summary of the characteristic values {\(\varPhi _{max}\), \(\tau _{mp}\), \(\tilde{w}_{ss}\) and \(\tau _{95\%ss}\)} for different combinations of \(\kappa \,\text {sgn}(\beta )\) and \(\alpha _0\)
\(\alpha _0>0\)  

\(\kappa \,\text {sgn}(\beta )\)  \(4\)  \(2\)  \(1\)  \(1/4\)  \(1/9\) 
\(\varPhi _{max}\)  1.12  1.37  1.69  2.70  3.66 
\(\tau _{mp}\)  4.6  4.6  4.6  4.6  4.4 
\(\tilde{w}_{ss}/\rho _0\)  1.55  1.67  1.92  *  − 
\(\tau _{95\%ss}\)  36.5  40.5  56.0  *  − 
\(\alpha _0>0\)  

\(\kappa \,\text {sgn}(\beta )\)  1  2  4  
\(\varPhi _{max}\)  1.29  1.13  0.977  
\(\tau _{mp}\)  4.9  4.8  4.7  
\(\tilde{w}_{ss}/\rho _0\)  1.40  1.39  1.38  
\(\tau _{95\%ss}\)  28.5  29.5  29.0 
\(\alpha _0<0\)  

\(\kappa \,\text {sgn}(\beta )\)  \(1\)  1  
\(\varPhi _{max}\)  1.84  1.50  
\(\tau _{mp}\)  2.3  2.4  
\(\tilde{w}_{ss}/\rho _0\)  −  −  
\(\tau _{95\%ss}\)  10.9  11.7 
5.4 Comparison with analytic local steadystate solutions
Here, the numerical results are compared with the analytically deduced local steadystate solutions that are presented in Sect. 3, which from now on is referred to as the analytical solutions. This aims at investigating the possibilities and limitations of using these analytical results to predict the evolution pattern for different configurations.
In the analytical study, the Laplacian term in Eq. (18) is neglected, roughly implying that the solution in each point in space is unaware of its neighbors. The Laplacian term not only governs the smoothening of the interface between disordered matrix and second phase, but it also affects the peak shape of the \(\varPhi \)profile. In addition, it influences the nonzero values of \(\varPhi \) along the crack surfaces.
Figure 9a presents the numerically steadystate and analytical solutions for the case with \(\alpha _0<0\) and \(\kappa \,\text {sgn}(\beta )=1\), where it is found that the curves match very well. However, an exception is found very close to the crack tip where the analytical solution goes to infinity and the effect of the interface energy on the numerical results is pronounced. The same observations can be done for all studied cases for which \(\alpha _0<0\) and, \(\alpha _0>0\) with \(3/16<\kappa \,\text {sgn}(\beta )<0\).
In Fig. 9b, steadystate profiles for the case with \(\alpha _0>0\) and \(\kappa \,\text {sgn}(\beta )=1\) are displayed. A match is found between the numerical and analytical results although differences subsist in the direct vicinity of the crack tip and at the interface between second phase and solid solution. The location corresponding to the stability limits of phase I and II are included, as well as the transition line between region I*+ II and I + II* given by the phase diagram in Fig. 2. It can be seen that the smooth interface is located about the line that indicates the first order transition between globally stable phases. Similar observations can be done for all studied cases for which \(\alpha _0>0\) with \(\kappa \,\text {sgn}(\beta )<3/16\) and \(\alpha _0>0\) with \(\beta >0\). In the latter case however, the smooth interface is approximately centered at \(\tilde{x}=\rho _0\), which corresponds to the second order transition location for the analytical solution.
The strong resemblances of the curve appearance indicate the numerical steadystate shape and approximate size of the second phase can be predicted by using the analytic local steadystate solutions for all cases. However, the assumption that transformation takes place wherever \(\varPhi \ne 0\) is disputable since that numerical solution renders a larger precipitate than the analytical solution is considered, see e.g. Fig. 9b. Indeed, the numerical result display an interface with a certain thickness whereas it is sharp in the analytical solution. The intersection between the curves ahead the crack tip in Fig. 9b corresponds to an apparent inflexion point of the interface profile obtained numerically, and the same observation is made for all cases where \(\alpha _0>0\) and \(\kappa \,\text {sgn}(\beta ) < 3/16\). The smoothness of the interface depends on the interface energy and thus cannot directly be predicted from the analytical solution. Additionally, the steadystate solutions display possible metastable phases. The stability analysis was performed only in a point wise way. However, the metastable phases might not appear if the total energy of the system and the temporal evolution of the spacedependent field are considered.
5.5 Influence of stresses on interface width
5.6 Effect of material properties and load
In the presented model, the coefficients \(\alpha _0, \beta _0\) and \(\gamma \) in the Landau potential Eq. (3) are material dependent, but they are not related to the elastic parameters, neither is the interface energy coefficient g. Instead changes in the elastic properties are introduced in the model via the shift in the transition temperature as seen in Eq. (15) and the value of \(\beta \), cf. Eq. (12). The strength of the interaction between the displacement field and the field parameter represented by \(\xi \) also affects the evolution of a precipitate. For a stiffer material (larger \(\varLambda \)), \(\beta \) has a lower value than for a more compliant material, which corresponds to an increased \(\kappa \). In the case of \(\beta >0\), this means that the phase transition mode is still of second order (see Fig. 2) but a higher transition rate is achieved, cf. Table 1. However, if \(\varLambda \) is reduced such that there is change in sign of \(\beta \) to a negative value, first order transitions may be possible. The same argument can be made for a material with a less pronounced interaction between the displacement field and the order parameter, i.e. a lower \(\xi \). Additionally, if \(K_I \) increases, \(r_0\) [see Eq. (13)] and \(\rho _0\) also increase, which leads to a stronger influence of the crack on its surrounding area.
In the case of diffusionless phase transformation, the approach in the present work may be useful. If all necessary material properties and the stress intensity factor are known, the steady state of the secondphase precipitation can be deduced by using the phase diagram outlined in Fig. 2. Size, shape, growth rate, time to complete transformation, and to some extent the smoothness of the interface, can be thereafter be calculated using the dimensionless results presented in this work.
5.7 Further remarks
The presented work, which includes a full parametric study, allows capturing all different scenarii corresponding to different combinations of load, phenomenological coefficients and material properties in time and space for a phase transformation induced by the presence of a crack in a elastic structure. This type of study is usually omitted in the literature. In addition, the microstructure evolution of a material is traditionally modeled by considering two coupled aspects: the phase kinetics and the mechanical equilibrium, and, usually, they are numerically solved separately e.g. Bair et al. (2017), Ma et al. (2006), Thuinet et al. (2013). Here, an analytical solution for the mechanical equilibrium in plane strain for an elastic structure containing a crack is found through the use of the Irwin’s analytical solutions and is directly incorporated into the TDGL equation as in Boulbitch and Korzhenevskii (2016) in order to account for the presence of a fixed crackinduced stress. Thus, only one equation has to be solved rendering the model timeefficient.
A large number of models in the literature, which are based on phase field theory, and the present one employ a phenomenological Landau potential, which derives from a power series in the order parameter. However, these models are not fully quantitative. This limits the study visavis the effects of temperature transient and temperature gradient (Shi and Xiao 2015), and our understanding of the meaning of the Landau coefficients. An attempt was made by Shi and Xiao (2015) to relate the Landau potential coefficients to physical quantities but the study still lies on several arbitrary simplifying assumptions.
For most engineering materials, the structural changes are accompanied by a concentration redistribution of species, and coupled evolution laws for structural and concentration order parameters may be the tool to successfully capture such behaviour.
In the case of hydride forming metals, there is a lack of some of the necessary material data and the phase transition kinetics are not fully mapped. Nevertheless, abinitio calculations such as reported in Olsson et al. (2014, 2015) may contribute to fill in the gaps, as well as recently performed experiments, cf. Maimaitiyili et al. (2015), Maimaitiyili et al. (2016). Further experiments that are especially designed for capturing phase transformation induced by a crack, are recommended.
6 Summary and conclusions
A phase field approach is used to investigate the formation of a second phase at the vicinity of a crack tip in an isotropic and linear elastic material. The tempospatial evolution of the microstructure is computed by numerically solving the TDGL equation. To capture the phase transitions, we use a sixth order Landau potential for a single structural order parameter, which represents the degree of ordering of the crystal structure.
For the considered systems the phase diagram for meanfield thermodynamic equilibrium is derived, clearly showing the existence of possible metastable phases and first order transitions, which might not emerge by considering the total energy of the system and the temporal evolution of the spacedependent field. The driving force for the phase transformation at the crack tip can be attributed to the phase transition temperature being locally shifted as a result of the crack induced stress field, which effectively acts as quenching in the crack tip vicinity. Different phase transformation scenarios are simulated by using a wide range of combinations of parameters, which represent varying material properties and stress levels. It is found that close to the crack tip, the driving force is always large enough to induce precipitation within a confined area through a second order transformation. The subsequent evolution pattern depends on the parameter set at hand, and a shift to first order transition during growth of the precipitate is observed occasionally. However, the presence of any metastable phases cannot be revealed from the calculations.
The complete steadystate solutions, with the exception of the shape of the smooth interface, is found to be accurately predicted by using the meanfield solution for each location. The interface width is found not to scale with the interface gradient energy coefficient because of the inhomogeneous stress field around the crack. Finally, the evolution of the order parameter is studied following quenching of the whole material. It is shown that the formation of the second phase is enhanced by the presence of the crack, despite that the entire system undergoes transformation simultaneously.
Once the diffusionless phasetransformation model parameters of an applicable material system are known, the presented results will allow to predict the kinetics of precipitations, e.g hydride formation, in the cracktip vicinity. Thus, this will contribute to the failure risk quantification of structures avoiding the use of expensive experiments.
Notes
Acknowledgements
The simulations in this work were performed using computational resources provided by the Swedish National Infrastructure for Computing (SNIC) at LUNARC, Lund University. The authors are grateful to professor Ali R. Massih for many valuable discussions.
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