Uniqueness, continuous dependence and reciprocity theorems in thermoelastic relaxed micromorphic continuum

A mixed initial-boundary value problem is formulated for linear thermo-elastic relaxed micromorphic continuum with asymmetric micro-distortion tensor, micro-dislocation tensor and temperature field. First, Lagrange identity involving two admissible processes at different instants for the considered mixed initial-boundary value problem is established. Using this identity, four theorems are proved: (i) uniqueness of solution without recourse either to an energy conservation law or to any boundedness assumptions on the thermoelastic coefficients, (ii) continuous dependence of solution of the problem on loads, (iii) continuous dependence of solution of the problem on the initial data, and (iv) reciprocity relation. The use of positive definiteness conditions on the thermoelastic coefficients is also avoided.

Automated summary

A mixed initial-boundary value problem is formulated for linear thermo-elastic relaxed micromorphic continuum with asymmetric micro-distortion tensor, micro-dislocation tensor and temperature field. The study establishes Lagrange identity and proves four theorems regarding uniqueness of solution, continuous dependence of solution on loads and initial data, and reciprocity relation. The use of positive definiteness conditions on the thermoelastic coefficients is avoided in the analysis.