Maximum mechano-damage power release-based phase-field modeling of mass diffusion in damaging deformable solids

Abstract

We formulate in three space dimensions a phase-field theory of mass diffusion in a damaging deformable solid matrix without making use of thermal quantities. The approach relies on three fundamental postulates: the diffusing species mass balance, the maximum mechano-damage power release principle, and an energy imbalance inequality. The solid is modelled mechanically as a Cauchy continuum. The kinematics of the continuum is given in terms of the diffusing species concentration and the solid matrix placement and damage fields. The formulation is based on the multiplicative decomposition of the total deformation gradient into an intercalation and an elastic deformation gradients. Constitutively, the model relies on three free energy functionals, i.e., chemical, damage, and strain energies. Damage is the phase field of the formulation, as non-locality is ensured by the dependence of the damage energy upon the damage gradient. Aimed at giving an insight into the properties of the model, the results of finite-element dynamic simulations are reported for a mechanically confined one-dimensional continuum.

Appendix A

The balance of voids, which allows to relate the fields $\rho _{0}^{R}$ and $\rho _{0}^{C}$, reads as

$$\begin{aligned} \int \limits _{\mathscr {S}_{0}}\rho _{0}^{R}\ \mathrm {d}V=\int \limits _{\mathscr {S}}\rho _{0}^{C}\ \mathrm {d}v\ , \end{aligned}$$

(82)

where $\mathscr {S}_{0}\subseteq \mathscr {B}_{0}$ is an arbitrary sub-body of $\mathscr {B}_{0}$ and $\mathscr {S}=\chi (\mathscr {S}_{0})\subseteq \mathscr {B}$ is the current shape of $\mathscr {S}_{0}$.

Using the change of variable $x=\chi (X)$ allows to write

$$\begin{aligned} \int \limits _{\mathscr {S}}\rho _{0}^{C}\ \mathrm {d}v=\int \limits _{\mathscr {S}_{0}}\rho _{0}^{C}\det \mathsf {F}\ \mathrm {d}V\ , \end{aligned}$$

(83)

which, along with Eq. (82) and following localization, implies the following point-wise equality

$$\begin{aligned} \rho _{0}^{R}=\rho ^{C}\det \mathsf {F}\ . \end{aligned}$$

(84)

Remark that an analogous result holds for the molar densities of the diffusing species, namely

$$\begin{aligned} \rho _{\mathrm {s}}^{R}=\rho _{\mathrm {s}}^{C}\det \mathsf {F}\ . \end{aligned}$$

(85)