Multi-phase-field modeling of grain growth in polycrystalline titanium under magnetic field and elastic strain

Abstract

A two-dimensional constitutive model was developed to simulate grain boundary motion in polycrystalline titanium exposed simultaneously to magnetic field and elastic strain based on the thermodynamic laws. The multi-scale coupled finite element and multi-phase-field simulations were used to investigate the simultaneous effects of the driving forces arising from the magnetic field and elastic strain energy on microstructure evolution of titanium bicrystalline and polycrystalline samples. The multi-phase-field approach was employed to implement the kinetic relations of grain boundary migration at the mesoscale level. On the other hand, the equilibrium equations were implemented on a macroscale level by the finite element method. Based on the simulation results, the magnetically induced driving force overrides the elastic strain driving force and causes texture evolution toward orientations that contain less magnetic stored energy when the microstructure is exposed to a magnetic field of sufficient strength. Additionally, applying an elastic strain before annealing reduces the time required for magnetic field annealing by accelerating the microstructure evolution. The mean grain size and desired texture grow rapidly when the magnetic field strength and elastic strain are simultaneously increased.

Appendix: Numerical implementation

A semi-concurrent multi-scale time-integration method was used for the numerical implementation of the constitutive equations using Thamboraja and Jamshidian [2]. This multi-scale coupled MPF and finite element computational method is implemented in Abaqus standard finite element software by writing a UMAT subroutine. The details of the numerical algorithm for the time integration method are as follows:

The index λ represents the integration point of finite elements, where $\lambda =1,2,\dots ,{\lambda }_{el}$ and ${\lambda }_{el}$ represent the total number of integration points of the finite elements. The index $k=\mathrm{1,2},\dots ,\Omega$ is used to represent the grid points of RVE, where Ω represents the total number of grid points.

The quantity of a variable at the grid points of the RVE is the mesoscale quantity, whereas a quantity at finite element integration points is a macroscale quantity. The quantity # at the integration point of elements λ is represented as ${\#}^{\uplambda }$. The quantity # is displayed as $\#^{\lambda ,\kappa }$ in the K-the point of the grid from the RVE attached to the integration point of the λ finite elements.

In the numerical algorithm process, we only track species at a grid point at that grid point and its nearby vicinity [2, 44]. The ${A}_{p}$ list represents a set containing species, which satisfy condition ${0<\xi }_{i}\le 1$ at the grid point and its nearest neighboring points. In addition, each member of the ${A}_{p}$ set is unique.

In this paragraph, the discussion is limited to the RVE attached to the integration point of the finite elements λ. The grid point number K in the RVE is labeled as${ G}^{ \lambda ,\kappa }$, which is in position $(x_{1},{{x}}_{2},{{x}}_{3})$ in the reference configuration. Grid points in positions $(x_{1}+z, {{x}}_{2},{{x}}_{3})$, $(x_{1}-z, {{x}}_{2},{{x}}_{3})$, $(x_{1}, {{x}}_{2}+ z, {{x}}_{3})$, $(x_{1}, {{x}}_{2}- z , {{x}}_{3}),$ and$(x_{1}, {{x}}_{2}, {{x}}_{3}+ z)$, $(x_{1}, {{x}}_{2}, {{x}}_{3}- z)$ are known in the reference configuration as the neighboring grid points of ${G}^{ \lambda ,\kappa }$ where Z represents the uniform spacing of the grid. The grid point index for each of the neighboring grid points of ${G}^{ \lambda ,\kappa }$ is a member of the ${Z}^{ \lambda ,\kappa }$ set. Thus, the set ${Z}^{ \lambda ,\kappa }$ has six members, and the member number J from the set ${Z}^{ \lambda ,\kappa }$ with $J=\mathrm{1,2},\dots ,6$ is represented as${ Z}_{j}^{\lambda ,\kappa }$. Labels for each grid point, their coordinates in the reference configuration, and tags for their neighboring grid points are obtained from an external file.

In the time marching method, t donates the current time and $\Delta t>0$ presents the time step and τ = t + ∆t. The Euler method is used to temporally integrate the grain growth equations.

The algorithm used for the time integration method is as follows:

Start the loop on all points of finite element integration points $\lambda =1,{\lambda }_{el}$.

• Given macroscale quantities: ${\overline{\mathbf{F}} }^{\lambda }\left(t\right),{\overline{\mathbf{T}} }^{\lambda }\left(\tau \right),{\overline{\theta }}^{\lambda }\left(t\right),{\overline{\theta }}^{\lambda }\left(\tau \right),{\overline{\mathbf{T}} }^{\lambda }\left(t\right)$, H.

• Macroscale quantities to be updated: $\left\{{\overline{\mathbf{T}} }^{\lambda }\left(\tau \right)\right\}$.

  Start the loop on all points of the grid of $k=1,\Omega$.

• Given mesoscale quantities: $\left\{{\xi }_{i}^{\lambda ,k}\left(t\right),{A}_{p}^{\lambda ,k}\left(t\right),{Z}^{\lambda ,\kappa }\right\}$.

• Mesoscale quantities to be updated: $\left\{{\xi }_{i}^{\lambda ,k}\left(\tau \right),{A}_{p}^{\lambda ,k}\left(\tau \right)\right\}$.

• Step 1: Determining the deformation gradient tensor at the mesoscale F(t) and F(τ):

  $${\mathbf{F}}\left( t \right) = {\overline{\mathbf{F}}}^{\lambda } \left( t \right) \quad {\text{and}} \quad{ }{\mathbf{F}}\left( \tau \right) = {\overline{\mathbf{F}}}^{\lambda } \left( \tau \right).$$

• Step 2: Specifying the temperature on the mesoscale θ(t) and θ(τ)

  $$\theta \left( t \right) = \overline{\theta }^{\lambda } \left( t \right)\quad {\text{and}}\quad { }\theta \left( \tau \right) = \overline{\theta }^{\lambda } \left( \tau \right).$$

• Step 3: Calculating the strain tensor at the mesoscale ${\mathbf{E}}_{i}\left(t\right)$ and ${\mathbf{E}}_{i}\left(\tau \right)$ for each $i\epsilon {A}_{\xi }$ species:

  $${\mathbf{E}}_{i} \left( t \right) = \frac{1}{{2\left\{ {\left( {{\mathbf{F}}_{i} \left( t \right)} \right)^{T} {\mathbf{F}}_{i} \left( t \right) - {\mathbf{I}}} \right\}}},$$
  $${\mathbf{E}}_{i} \left( \tau \right) = \frac{1}{{2\left\{ {\left( {{\mathbf{F}}_{i} \left( \tau \right)} \right)^{T} {\mathbf{F}}_{i} \left( \tau \right) - {\mathbf{I}}} \right\}}}.$$

• Step 4: Determining the set ${A}_{\xi }$ of species, which are present at the grid point and its immediate neighbors:

  $$A_{\xi } = \left[ { \cup_{j = 1}^{6} A_{p}^{{\lambda ,Z_{j}^{\lambda ,k} }} \left( t \right)} \right] \cup A_{p}^{\lambda ,k} \left( t \right).$$

  ${A}_{\xi }=1$ indicates no inter-species exchange or transfer and therefore step 11 should be performed after updating the set ${A}_{p}^{\lambda ,k}\left(\tau \right)={A}_{p}^{\lambda ,k}\left(t\right)$ in ${\xi }_{i}^{\lambda ,k}\left(\tau \right)={\xi }_{i}^{\lambda ,k}\left(t\right)$ for all $i \in {A}_{p}^{\lambda ,k}\left(t\right)$ species.

• Step 5: Calculating the free energies ${\psi }_{i}^{e}\left(t\right)$، ${\psi }_{i}^{M}\left(t\right)$، ${\psi }_{i}^{\theta }\left(t\right)$ on the mesoscale for each $i\epsilon {A}_{\xi }$ species:

  $$\psi_{i}^{e} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}{\overline{\mathbf{E}}}_{i} :{\text{c}}_{i} \left[ {{\overline{\mathbf{E}}}_{i} } \right],$$
  $$\psi_{i}^{M} \left( t \right) = \frac{1}{2}\mu_{0} H^{2} X_{i} ,$$
  $$\psi_{i}^{\theta } \left( t \right) = c_{i} \left[ {\left( {\theta \left( t \right) - \theta_{0} } \right) - \theta \left( t \right)\ln (\theta \left( t \right)/\theta_{0} } \right].$$

• Step 6: Calculating the driving forces for ${f}_{i}^{\xi }\left(t\right)$ inter-species exchange for each $i\epsilon {A}_{\xi }$ species:

  $$f_{i}^{\xi } \left( t \right) = f_{i}^{m} \left( t \right) + f_{i}^{e} \left( t \right) + f_{i}^{\theta } \left( t \right) + f_{i}^{M} \left( t \right),$$
  $$f_{i}^{m} \left( t \right) = - \mathop \sum \limits_{{s \in A_{\xi } }} \frac{{\epsilon_{is}^{\xi } }}{2}\nabla^{2} \xi_{i}^{\lambda ,k} \left( t \right) - \mathop \sum \limits_{{s \in A_{\xi } }} \omega_{is}^{\xi } \xi_{s}^{\lambda ,k} \left( t \right) \quad {\text{with}} \quad s \ne i,$$
  $$f_{i}^{e} \left( t \right) = \frac{{g^{\prime}\left( {\xi_{i}^{\lambda ,k} \left( t \right)} \right)\left[ {\mathop \sum \nolimits_{{s \in A_{\xi } }} g\left( {\xi_{s}^{\lambda ,k} \left( t \right)} \right)\left( {\psi_{s}^{e} \left( t \right) - \psi_{i}^{e} \left( t \right)} \right)} \right]}}{{\left[ {\mathop \sum \nolimits_{{s \in A_{\xi } }} g\left( {\xi_{s}^{\lambda ,k} \left( t \right)} \right)} \right]^{2} }},$$
  $$f_{i}^{\theta } \left( t \right) = \frac{{g^{\prime}\left( {\xi_{i}^{\lambda ,k} \left( t \right)} \right)\left[ {\mathop \sum \nolimits_{{s \in A_{\xi } }} g\left( {\xi_{s}^{\lambda ,k} \left( t \right)} \right)\left( {\psi_{s}^{\theta } \left( t \right) - \psi_{i}^{\theta } \left( t \right)} \right)} \right]}}{{\left[ {\mathop \sum \nolimits_{{s \in A_{\xi } }} g\left( {\xi_{s}^{\lambda ,k} \left( t \right)} \right)} \right]^{2} }},$$
  $$f_{i}^{M} \left( t \right) = \frac{{g^{\prime}\left( {\xi_{i}^{\lambda ,k} \left( t \right)} \right)\left[ {\mathop \sum \nolimits_{{s \in A_{\xi } }} g\left( {\xi_{s}^{\lambda ,k} \left( t \right)} \right)\left( {\psi_{s}^{M} \left( t \right) - \psi_{i}^{M} \left( t \right)} \right)} \right]}}{{\left[ {\mathop \sum \nolimits_{{s \in A_{\xi } }} g\left( {\xi_{s}^{\lambda ,k} \left( t \right)} \right)} \right]^{2} }}.$$

  It is worth noting that ${\xi }_{s}^{\lambda ,k}\left(t\right)=0$ if $i\notin {A}_{p}^{\lambda ,k}\left(t\right)$. The finite difference method is used to calculate the second-order gradient of the MPF variables ${\nabla }^{2}{\xi }_{s}^{\lambda ,\kappa }(t)$ [44].

  $$\nabla^{2} \xi_{s}^{\lambda ,\kappa } \left( t \right) = \frac{{\left[ {\mathop \sum \nolimits_{j = 1}^{6} \xi_{s}^{{\lambda ,z_{j}^{\lambda ,\kappa } }} \left( t \right)} \right] - 6\xi_{s}^{\lambda ,\kappa } \left( t \right)}}{{z^{2} }}.$$

• Step 7a: Calculating the total driving force for the inter-species exchange (${f}_{pq}\left(t\right))$ for the species $p,q\in {A}_{\xi }$ with p < q:

  $$f_{pq} \left( t \right) = f_{p}^{\xi } \left( t \right) - f_{q}^{\xi } \left( t \right).$$

• Step 7b: Calculating the inter-species transfer rate ${\Delta \xi }_{pq}$ for $p,q\in {A}_{\xi }$ species with p < q:

  If $\left|{f}_{pq}(t)\right|>{f}_{pq}^{\xi ,c},$ then, we have:

  $$\Delta \xi_{pq} = L_{pq}^{\xi } \left( t \right)f_{pq} \left( t \right)\Delta t.$$

  where the coefficients of mobility ${\widehat{{\varvec{L}}}}_{{\varvec{p}}{\varvec{q}}}^{{\varvec{\xi}}}({\varvec{\theta}}({\varvec{t}}))$ $=$ ${{\varvec{L}}}_{{\varvec{p}}{\varvec{q}}}^{{\varvec{\xi}}}({\varvec{t}})$.

  If $\left|{{\varvec{f}}}_{{\varvec{p}}{\varvec{q}}}({\varvec{t}})\right|<{{\varvec{f}}}_{{\varvec{p}}{\varvec{q}}}^{{\varvec{\xi}},{\varvec{c}}}$, then, we have:

  $$\Delta \xi_{pq} = 0.$$

  The set ${A}_{\xi }$ contains the quantities ${\Delta \xi }_{pq}\ne 0$ for the species $p,q\in {A}_{\xi }$ with p < q. If ${A}_{\xi }=\varnothing$, then ${\xi }_{i}^{\lambda ,k}\left(\tau \right)={\xi }_{i}^{\lambda ,k}\left(t\right)$ for each species is $i \epsilon { A}_{p}^{\lambda ,k}\left(t\right)$; then, the set ${A}_{p}^{\lambda ,k}\left(\tau \right)={A}_{p}^{\lambda ,k}\left(t\right)$ is updated and then we proceed with step 11.

• Step 8: Updating the VF of the species (${\xi }_{i}^{\lambda ,k}\left(\tau \right))$ for each species $i\in {A}_{\xi }$:

  $$\xi_{i}^{\lambda ,k} \left( \tau \right) = \xi_{i}^{\lambda ,k} \left( t \right) + \mathop \sum \limits_{p < q} K_{ipq} \Delta \xi_{pq} , \quad p,q \in A_{\xi } .$$

  If ${\xi }_{i}^{\lambda ,k}\left(\tau \right)>1$, then ${\xi }_{i}^{\lambda ,k}\left(\tau \right)=1$ and if ${\xi }_{i}^{\lambda ,k}\left(\tau \right)<0$, then ${\xi }_{i}^{\lambda ,k}\left(\tau \right)=0$.

• Step 9: Updating the set ${A}_{p}^{\lambda ,k}\left(\tau \right)$ of the species, which satisfy the following conditions:

  $$0 < \xi_{i}^{\lambda ,k} \left( \tau \right) \le 1, \quad i \in A_{\xi } .$$

• Step 10: Ensuring that the constraint $\sum_{i\epsilon {A}_{p}^{\lambda ,k}(\tau )}{\xi }_{i}^{\lambda ,k}\left(\tau \right)=1$ is always satisfied by substituting:

  $$\xi_{i}^{\lambda ,k} \left( \tau \right) \quad {\text{with}} \quad \frac{{\xi_{i}^{\lambda ,k} \left( \tau \right)}}{{\mathop \sum \nolimits_{{s \in A_{p}^{\lambda ,k} \left( \tau \right)}} \xi_{s}^{\lambda ,k} \left( \tau \right)}}, \quad i \in A_{p}^{\lambda ,k} \left( \tau \right).$$

• Step 11: Updating Cauchy stress at mesoscale, ${\mathrm{T}}^{\lambda ,k}(\tau )$:

  $${\mathbf{T}}^{{{\uplambda },{\text{k}}}} \left( {\uptau } \right) = \left( {{\text{det}}{\mathbf{F}}\left( {\uptau } \right)} \right)^{ - 1} {\mathbf{F}}\left( {\uptau } \right){\mathbf{T}}^{*} \left( {\uptau } \right)\left( {{\mathbf{F}}\left( {\uptau } \right)} \right)^{{\text{T}}} .$$

  where ${\mathrm{T}}^{*}\left(\tau \right)$ is the second Piola- Kirchhoff stress on the mesoscale:

  $${\mathbf{T}}^{*} \left( {\uptau } \right) = \frac{{\mathop \sum \nolimits_{{i \in A_{p}^{\lambda ,k} \left( \tau \right)}} g\left( {\xi_{i}^{\lambda ,\kappa } \left( \tau \right)} \right)\left\{ {C_{i} \left[ {{\mathbf{E}}\left( \tau \right) - \lambda_{i} \left( {\theta \left( \tau \right) - \theta_{0} } \right){\mathbf{I}}} \right]} \right\}}}{{\mathop \sum \nolimits_{{i \in A_{p}^{\lambda ,k} \left( \tau \right)}} g\left( {\xi_{i}^{\lambda ,\kappa } \left( \tau \right)} \right)}}.$$

The end of the loop is on all grid points of the RVE.

• Step A: Updating Cauchy stress at macro scale using Eq. 19:

  $${\overline{\mathbf{T}}}^{\lambda } \left( \tau \right) = \frac{1}{{\Omega }}\mathop \sum \limits_{\kappa = 1}^{{\Omega }} {\mathbf{T}}^{\lambda ,k} \left( {\uptau } \right)$$

• Step B: Determining the Jacobin matrix for the finite-element code Abaqus/Standard for Newton–Raphson iterations [2, 44].

The end of the loop is on finite element integration points.