r7.02.01 Rate of energy release in linear thermoelasticity

Summary:

The calculation of the energy restoration rate \(G\) by the \(\theta\) method is presented in 2D or 3D for a linear thermo-elastic problem. We explain how field \(\theta\) is introduced and how the energy return rate is calculated with the CALC_G operator.

Mechano-reliable studies to assess the probability of the rupture initiating, in addition, require its derivative with respect to a domain variation driven by another field. The implementation of this option in the code is detailed.

The calculation of \(G\) is available both for a meshed crack (classical finite elements) and for a non-meshed crack (enriched finite elements: method X- FEM). The calculation of its derivative is only available in 2D, for a meshed crack.

• 1. Calculation of the energy return rate using the \(\theta\) method in linear thermoelasticity
  • 1.1. Behavioral relationship
  • 1.2. Potential energy and equilibrium relationships
  • 1.3. Lagrangian expression for the energy return rate
• 2. Discretization of the energy return rate
  • 2.1. Method \(\theta\) in dimension 2
  • 2.2. Method \(\theta\) in dimension 3
  • 2.3. Choice of discretization of G in dimension 3
  • 2.4. G implantation in linear thermoelasticity
• 3. Introduction of field \(\theta\)
  • 3.1. Conditions to be met
  • 3.2. Choice of field \(\theta\) in dimension 3
  • 3.3. Choice of field \(\mathrm{\theta }\) in dimension 2
  • 3.4. Alternative method: field \(\theta\) defined according to the number of layers
  • 3.5. Alternative method
• 4. Bibliography