Clarification request: LinearIsotropicElasticity rate_form with temperature-dependent elastic moduli

[zhangjunzheng136]

Hi NEML2 developers,

I would like to ask for clarification about the intended semantics of LinearIsotropicElasticity with rate_form = true, especially when the elastic moduli depend on temperature.

I am coupling NEML2 with MOOSE for a thermo-mechanical problem. In my original model, I used LinearIsotropicElasticity in rate form:

[elasticity]
  type = LinearIsotropicElasticity
  coefficients = 'interpolated_E 0.3'
  coefficient_types = 'YOUNGS_MODULUS POISSONS_RATIO'
  rate_form = true
  strain = 'state/internal/Ee'
  stress = 'state/S'
[]

[integrate_stress]
  type = SR2BackwardEulerTimeIntegration
  variable = 'state/S'
  time = 'forces/t'
  rate = 'state/S_rate'
[]

Here interpolated_E depends on temperature, and both temperature and elastic strain can change during the same time step.
In this setup, the stress update appears to behave like

S_rate = C(T) : Ee_rate

followed by time integration. However, if the intended constitutive relation is the current elastic relation

S = C(T) : Ee

then the material time derivative would be

S_rate = C(T) : Ee_rate + C_dot(T) : Ee

The second term can be important when both temperature and elastic strain vary.
In my thermo-mechanical benchmark, the rate-form version produced MPa-level stress differences compared with COMSOL. When I changed the NEML2 model to use the current elastic relation directly,

Ep = old_Ep + dt * Ep_rate
Ee = E_total - Ep
S = C(T) : Ee

with

[elasticity]
  type = LinearIsotropicElasticity
  coefficients = 'interpolated_E 0.3'
  coefficient_types = 'YOUNGS_MODULUS POISSONS_RATIO'
  rate_form = false
  strain = 'state/internal/Ee'
  stress = 'state/S'
[]

the discrepancy was greatly reduced.

So should I use the rate form of LinearIsotropicElasticity with some other configurations, or only when C_dot(T)=0 ?

[reverendbedford]

That model is right for what it is, essentially a hypoelastic constitutive assumption that the stress rate is linearly proportional to the strain rate. As you've noted, it won't exactly recover the standard Hooke's law with temperature dependent elastic moduli (except in the constant temperature case).

That said, it's not wrong it's just a different choice. Kind of like how hypoelastic models don't recover hyperelastic plasticity formulations except in special cases. If you want to exactly recover the classical Hooke's law behavior you can easily adopt either of your solutions (integrate the strain or, alternatively, include the additional C_dot term). Both should be easy to do in NEML2!

[zhangjunzheng136]

I have got it, thank you for your guidance.