A convective model for poro-elastodynamics with damage and fluid flow towards Earth lithosphere modelling

Thermodynamically consistent model for poroelastic rocks towards tectonic and volcanic processes and earthquakes

“…The parallel combination of Maxwellian viscoelastic rheology and Stokes fluidic rheology is called Jeffreys' rheology, used in particular in [31] and a combination with plasticity in [47,49]. The overall mechanical rheological model behind (7a-d) with (10), (14), and ( 15) is schematically depicted in Figure 1.…”

“…Here, let us only briefly mention that, in the semicompressible variant, a rigorous mathematical analysis of the system (7c-g) with (39) with S V and ξ augmented by the multipolar terms as mentioned above can be performed by merging and modified the results available for an anisothermal model with damage [44] with the isothermal diffusion model [49]. The essential point is to have boundedness of the velocity gradient ∇v(t) in space at particular time instants t and sufficiently smooth initial conditions.…”

“…Let us still remark that an attempt to formulate the model [31] thermodynamically consistently has been done in [45] in terms of displacements, but not all time-derivatives have been objective and the energy balance was thus corrupted. An improved variant using the velocity-strain formulation and convective times derivatives was devised in [49], although only isothermal and not fully objective. Several aspects of the model have been devised in [44] towards solid-liquid phase transformation as occurs in freezing of water and melting of ice in a simplified semi-compressible variant (like in Sect.…”

The Stefan problem in a thermomechanical context with fracture and fluid flow

“…The stress/velocity formulation bears a generalization for a nonquadratic stored energy ϕ = ϕ(Σ) as a function of an auxiliary stress, also called a "proto-stress", cf. Remark 8 or [40]. The actual stress will be then S = Cϕ (Σ) with the fixed elastic tensor C playing an auxiliary role.…”

“…Let us note that we used the convective time derivative in (40), which is related with the attribute of the "volume-fraction" variable χ as an intensive variable taking values in [0, 1]. In contrast to it, χ in (39d) in the dimension J/m 3 is an extensive variable and can be summed up with ϑ, giving w which should be transported as w in (36).…”