Thermal-Chemical-Mechanical Coupling and Material Evolution in Finitely Deforming Solids With Advancing Fronts of Reactive Fluids

A new stabilized method is presented for coupled thermo-chemo-mechanical processes in reactive fluid-solid mixtures. The phenomena are coupled through locally homogenized mixture theory where chemo-mechanical coupling is induced via Lagrange multiplier that emanates from maximization of entropy production inequality. The balance of energy for the mixture is obtained by adding together the energy balance for the fluid and solid constituents. In the mixture theory model the kinematics of constituents is represented via an independent set of balance laws for each of the interacting constituents. A significant feature of the mixture model is the interactive force field in the momentum balance equations that couples the constituents implicitly at the level of the governing system of equations. The constitutive relations for the constituents in the mixture model are based on maximization of the rate of entropy production. Since each constituent is not discretely modeled and the interactive effects are mathematically coupled at the local continuum level, the resulting system serves as a physics-based reduced-order model for the complex microstructure of the material system.

Through the stated set of governing and constitutive equations, there are several coupled phenomena that we are able to capture. Specifically, there are three categories of coupling effects: (i) thermo-mechanical effects, (ii) chemo-mechanical effects, and (iii) thermo-chemical effects. The thermo-mechanical effects include the thermal expansion of the solid, captured via the Cauchy stress tensor dependency on the temperature implied by the Helmholtz free energy function; the effect of the rate of deformation on the temperature, through the coupling term in the balance of energy equation; and the dependency of the mechanical properties of the solid on the temperature. Chemo-mechanical effects include the expansion of the solid associated with the extent of the reaction, captured via the dependency of the Cauchy stress tensor on the extent of reaction implied by the Helmholtz free energy function; and the dependency of the mechanical properties on the reaction state. Thermo-chemical effects include the heat generated by the exothermic reaction along the reaction zone, which is a moving heat source captured by the heat source term dependency on the mass reaction rate; the dependency of the rate of reaction on temperature since the reaction rate constant is a function of temperature; and the dependency of thermal properties of the solid on the reaction state.

Evolving nonlinearity and coupled thermo-mechanical effects and chemo-mechanical effects give rise to spatially localized phenomena, namely boundary layers, shear bands, and steep gradients that appear at the reaction fronts. For large reaction rates, the balance of mass of the fluid becomes a singularly perturbed equation (reaction-dominated), which may exhibit boundary and/or internal layers. Likewise, for large reaction rates and/or low diffusivity, the balance of linear momentum for the fluid constituent also becomes a singularly perturbed PDE. Presence of these features in the solution requires stable numerical methods, and we present a variational multiscale (VMS)-based stabilized finite element method for the initial-boundary value problem. Mathematical attributes of the method are investigated via a range of numerical test cases that involve diffusion of chemically reacting fluids through nonlinear thermoelastic solids. Enhanced stabilization features and higher spatial accuracy of the models and the methods are highlighted.

References

[1] R. Hall, H. Gajendran, A. Masud, Diffusion of chemically reacting fluids through nonlinear elastic solids: mixture model and stabilized methods, Math. Mech. Solids. 20 (2015) 204–227.

[2] H. Gajendran, R.B. Hall, A. Masud, K.R. Rajagopal, Chemo-mechanical coupling in curing and material-interphase evolution in multi-constituent materials, Acta Mech. 229 (2018) 3393–3414.

[3] M. Anguiano, H. Gajendran, R.B. Hall, K.R. Rajagopal, A. Masud, Chemo-mechanical coupling and material evolution in finitely deforming solids with advancing fronts of reactive fluids, Acta Mech. 231 (2020) 1933–1961.