Phase-field elasticity model based on mechanical jump conditions

Abstract

Computational models based on the phase-field method typically operate on a mesoscopic length scale and resolve structural changes of the material and furthermore provide valuable information about microstructure and mechanical property relations. An accurate calculation of the stresses and mechanical energy at the transition region is therefore indispensable. We derive a quantitative phase-field elasticity model based on force balance and Hadamard jump conditions at the interface. Comparing the simulated stress profiles calculated with Voigt/Taylor (Annalen der Physik 274(12):573, 1889), Reuss/Sachs (Z Angew Math Mech 9:49, 1929) and the proposed model with the theoretically predicted stress fields in a plate with a round inclusion under hydrostatic tension, we show the quantitative characteristics of the model. In order to validate the elastic contribution to the driving force for phase transition, we demonstrate the absence of excess energy, calculated by Durga et al. (Model Simul Mater Sci Eng 21(5):055018, 2013), in a one-dimensional equilibrium condition of serial and parallel material chains. To validate the driving force for systems with curved transition regions, we relate simulations to the Gibbs-Thompson equilibrium condition (Johnson and Alexander, J Appl Phys 59(8):2735, 1986).

Appendices

Appendix

1.1 Transformation of stresses and strains in Voigt notation

In the Voigt notation, the strains and stresses can be written as

$$\begin{aligned} \varvec{\sigma }^v&=\begin{pmatrix} \sigma _{11},&\sigma _{22},&\sigma _{33},&\sigma _{23},&\sigma _{13},&\sigma _{12} \end{pmatrix}^T, \end{aligned}$$

(79)

$$\begin{aligned} \varvec{\epsilon }^v&=\begin{pmatrix} \epsilon _{11},&\epsilon _{22},&\epsilon _{33},&2\epsilon _{23},&2\epsilon _{13},&2\epsilon _{12} \end{pmatrix}^T. \end{aligned}$$

(80)

Then, the transformation (10) becomes $\varvec{\sigma }^v_B=\varvec{M}^v_\sigma \varvec{\sigma }^v$, with the transformation matrix

$$\begin{aligned} \varvec{M}^v_\sigma \!=\! \left( \begin{array}{llllll} n_1n_1 &{}\, n_2n_2 &{}\, n_3n_3 &{}\, 2n_2n_3 &{}\, 2n_1n_3 &{}\, 2n_1n_2 \\ t_1t_1 &{}\, t_2t_2 &{}\, t_3t_3 &{}\, 2t_2t_3 &{}\, 2t_1t_3 &{}\, 2t_1t_2 \\ s_1s_1 &{}\, s_2s_2 &{}\, s_3s_3 &{}\, 2s_2s_3 &{}\, 2s_1s_3 &{}\, 2s_1s_2 \\ t_1s_1 &{}\, t_2s_2 &{}\, t_3s_3 &{}\, t_2s_3+t_3s_2 &{}\, t_1s_3+t_3s_1 &{}\, t_1s_2+t_2s_1 \\ n_1s_1 &{}\, n_2s_2 &{}\, n_3s_3 &{}\, n_2s_3+n_3s_2 &{}\, n_1s_3+n_3s_1 &{}\, n_1s_2+n_2s_1 \\ n_1t_1 &{}\, n_2t_2 &{}\, n_3t_3 &{}\, n_2t_3+n_3t_2 &{}\, n_1t_3+n_3t_1 &{}\, n_1t_2+n_2t_1 \end{array}\right) . \end{aligned}$$

(81)

An analogue transformation for the strain can be written as $\varvec{\epsilon }^v_B=\varvec{M}^v_\epsilon \varvec{\epsilon }^v$, with the transformation matrix

$$\begin{aligned} \varvec{M}^v_\epsilon \!=\! \left( \begin{array}{llllll} n_1n_1 &{}\, n_2n_2 &{}\, n_3n_3 &{}\, n_2n_3 &{}\, n_1n_3 &{}\, n_1n_2 \\ t_1t_1 &{}\, t_2t_2 &{}\, t_3t_3 &{}\, t_2t_3 &{}\, t_1t_3 &{}\, t_1t_2 \\ s_1s_1 &{}\, s_2s_2 &{}\, s_3s_3 &{}\, s_2s_3 &{}\, s_1s_3 &{}\, s_1s_2 \\ 2t_1s_1 &{}\, 2t_2s_2 &{}\, 2t_3s_3 &{}\, t_2s_3+t_3s_2 &{}\, t_1s_3+t_3s_1 &{}\, t_1s_2+t_2s_1 \\ 2n_1s_1 &{}\, 2n_2s_2 &{}\, 2n_3s_3 &{}\, n_2s_3+n_3s_2 &{}\, n_1s_3+n_3s_1 &{}\, n_1s_2+n_2s_1 \\ 2n_1t_1 &{}\, 2n_2t_2 &{}\, 2n_3t_3 &{}\, n_2t_3+n_3t_2 &{}\, n_1t_3+n_3t_1 &{}\, n_1t_2+n_2t_1 \end{array}\right) . \end{aligned}$$

(82)

We reorder the components of the strain and stress vectors and define

$$\begin{aligned} \varvec{\epsilon }^{\alpha }_B&:={\Big ( \underbrace{\epsilon _{nn},\epsilon _{ns},\epsilon _{nt}}_{\varvec{\epsilon }_n}, \underbrace{\epsilon _{tt},\epsilon _{ss},\epsilon _{ts}}_{\varvec{\epsilon }_t}\Big )}^T= {\left( \varvec{\epsilon }_n,\varvec{\epsilon }_t\right) }^T,\end{aligned}$$

(83)

$$\begin{aligned} \varvec{\sigma }^{{\alpha }}_B&:={\Big ( \underbrace{\sigma _{nn},\sigma _{ns},\sigma _{nt}}_{\varvec{\sigma }_n}, \underbrace{\sigma _{tt},\sigma _{ss},\sigma _{ts}}_{\varvec{\sigma }_t}\Big )}^T ={\left( \varvec{\sigma }_n, \varvec{\sigma }_t\right) }^{T}. \end{aligned}$$

(84)

Due to this reformulation, we permute the rows of the previous matrices $\varvec{M}^v_\sigma$ and $\varvec{M}^v_\epsilon$, but the named properties remain

$$\begin{aligned} \varvec{M}_\sigma&\!=\! \left( \begin{array}{llllll} n_1n_1 &{}\, n_2n_2 &{}\, n_3n_3 &{}\, 2n_2n_3 &{}\, 2n_1n_3 &{}\, 2n_1n_2 \\ n_1t_1 &{}\, n_2t_2 &{}\, n_3t_3 &{}\, n_2t_3+n_3t_2 &{}\, n_1t_3+n_3t_1 &{}\, n_1t_2+n_2t_1 \\ n_1s_1 &{}\, n_2s_2 &{}\, n_3s_3 &{}\, n_2s_3+n_3s_2 &{}\, n_1s_3+n_3s_1 &{}\, n_1s_2+n_2s_1 \\ t_1t_1 &{}\, t_2t_2 &{}\, t_3t_3 &{}\, 2t_2t_3 &{}\, 2t_1t_3 &{}\, 2t_1t_2 \\ s_1s_1 &{}\, s_2s_2 &{}\, s_3s_3 &{}\, 2s_2s_3 &{}\, 2s_1s_3 &{}\, 2s_1s_2 \\ t_1s_1 &{}\, t_2s_2 &{}\, t_3s_3 &{}\, t_2s_3+t_3s_2 &{}\, t_1s_3+t_3s_1 &{}\, t_1s_2+t_2s_1 \\ \end{array}\right) ,\end{aligned}$$

(85)

$$\begin{aligned} \varvec{M}_\epsilon&\!=\! \left( \begin{array}{llllll} n_1n_1 &{}\, n_2n_2 &{}\, n_3n_3 &{}\, n_2n_3 &{}\, n_1n_3 &{}\, n_1n_2 \\ 2n_1t_1 &{}\, 2n_2t_2 &{}\, 2n_3t_3 &{}\, n_2t_3+n_3t_2 &{}\, n_1t_3+n_3t_1 &{}\, n_1t_2+n_2t_1 \\ 2n_1s_1 &{}\, 2n_2s_2 &{}\, 2n_3s_3 &{}\, n_2s_3+n_3s_2 &{}\, n_1s_3+n_3s_1 &{}\, n_1s_2+n_2s_1 \\ t_1t_1 &{}\, t_2t_2 &{}\, t_3t_3 &{}\, t_2t_3 &{}\, t_1t_3 &{}\, t_1t_2 \\ s_1s_1 &{}\, s_2s_2 &{}\, s_3s_3 &{}\, s_2s_3 &{}\, s_1s_3 &{}\, s_1s_2 \\ 2t_1s_1 &{}\, 2t_2s_2 &{}\, 2t_3s_3 &{}\, t_2s_3+t_3s_2 &{}\, t_1s_3+t_3s_1 &{}\, t_1s_2+t_2s_1 \\ \end{array}\right) . \end{aligned}$$

(86)

This follow the transformations of stresses and strains as used in Eqs. (20) and  (21).

Driving force contributions of VT and RS model

The main assumption of the VT scheme [1] is that the strains of overlapping phases are equal. In a two phase system with $\phi _{\alpha }$ and $\phi _{\beta }$, the corresponding volume fractions follow Eq. (1) for the stress $\varvec{\sigma }^{VT}$. The energy density results accordingly to [9–11]

$$\begin{aligned} f_{el}^{VT}(\varvec{\phi }, \varvec{\epsilon })&= f_{el}^\alpha (\varvec{\epsilon }) h(\phi ) + f_{el}^\beta (\varvec{\epsilon }) (1-h(\phi )) \nonumber \\&= \frac{1}{2} \left( \langle \varvec{\epsilon } - \tilde{\varvec{\epsilon }}^\alpha , \varvec{\mathcal {C}}^\alpha (\varvec{\epsilon } - \tilde{\varvec{\epsilon }}^\alpha ) \rangle h(\phi ) \right. \nonumber \\&\quad \left. + \,\langle \varvec{\epsilon } - \tilde{\varvec{\epsilon }}^\beta , \varvec{\mathcal {C}}^\beta (\varvec{\epsilon } - \tilde{\varvec{\epsilon }}^\beta ) \rangle (1-h(\phi )) \right) . \end{aligned}$$

(87)

Using Eq. (4), the corresponding contribution of the driving force for the VT model is given by

$$\begin{aligned} \Delta f_{el}^{VT}(\varvec{\phi }, \varvec{\epsilon })&= \frac{1}{2}\left( \langle \varvec{\epsilon } - \tilde{\varvec{\epsilon }}^\beta , \varvec{\mathcal {C}}^\beta (\varvec{\epsilon } - \tilde{\varvec{\epsilon }}^\beta ) \rangle \right. \nonumber \\&\quad \left. - \langle \varvec{\epsilon } - \tilde{\varvec{\epsilon }}^\alpha , \varvec{\mathcal {C}}^\alpha (\varvec{\epsilon } - \tilde{\varvec{\epsilon }}^\alpha ) \rangle \right) \frac{\partial h(\phi )}{\partial \phi }. \end{aligned}$$

(88)

The assumption of the RS model is that the stresses are equal for the two overlapping phases. The system variables are now changed from strains to stresses. The corresponding elastic potential of phase $\alpha$ for such a system can be derived by changing the system variable in Eq. (87) with the Legendre transformation to the form

$$\begin{aligned} f_{el}^{RS}(\varvec{\phi }, \varvec{\sigma }) = f_{el}^{VT}(\varvec{\phi }, \varvec{\epsilon }) - \frac{\partial f_{el}^{VT}(\varvec{\phi }, \varvec{\epsilon })}{\partial \varvec{\epsilon }} \varvec{\epsilon }. \end{aligned}$$

(89)

This results in

$$\begin{aligned} f_{el}^{RS}&= \frac{1}{2} \left\langle \varvec{\sigma }^{RS}, \left( \varvec{\mathcal {S}}^\beta (1-h(\phi )) - \varvec{\mathcal {S}}^\alpha h(\phi ) \right) \varvec{\sigma }^{RS} \right\rangle \nonumber \\&\quad - \left\langle \varvec{\sigma }^{RS}, \tilde{\varvec{\epsilon }}^\alpha h(\phi ) + \tilde{\varvec{\epsilon }}^\beta (1-h(\phi )) \right\rangle , \end{aligned}$$

(90)

with $\varvec{\sigma }^{RS}$ given in Eq. (2). The corresponding contribution of the driving force for the RS model can be expresses as

$$\begin{aligned} \Delta f_{el}^{RS}(\varvec{\phi }, \varvec{\sigma })&= \left( \frac{1}{2}\left\langle \varvec{\sigma }^{RS}, \left( \varvec{\mathcal {S}}^\alpha - \varvec{\mathcal {S}}^\beta \right) \varvec{\sigma }^{RS} \right\rangle \right. \nonumber \\&\quad \left. + \left\langle \varvec{\sigma }^{RS}, \tilde{\varvec{\epsilon }}^\alpha - \tilde{\varvec{\epsilon }}^\beta \right\rangle \right) \frac{\partial h(\phi )}{\partial \phi }. \end{aligned}$$

(91)