Material Derivatives

One of the main specialties of porous media mechanics compared to classical continuum mechanics is the presence of two phases within one domain. Although mathematically sound, the simultaneous presence of skeleton and fluid phase at every point within the continuum can seem rather non-intuitive or unusual at first glance. A simple example is the effect of incompressibility.

Even if structure and fluid phase are assumed to behave incompressible, the determinant of the deformation gradient is not necessarily equal to 1, as one with a traditional solid continuum mechanics background could expect. For instance, the volume can change due to in- and outflow without microscopic volume change, see Figure 3.2. Instead, incompressibility implies a direct relation between porosity and the Jacobian determinantJ, see Section 4.4.1 for details.

Figure 3.2: Macroscopic deformation without deformation of the microscopic structure.

Skeleton and fluid phases can deform independently, despite the fact that they could have oc-cupied the same material volume in the initial configuration (see Figure 3.3). In this section,

ϕ(t)

fluid phase skeleton phase

ϕf(t) fluid phase

skeleton phase

Figure 3.3: Independent deformation of skeleton phaseϕ(black) and fluid phaseϕ^f(grey), taken from [243].

a mathematical specialty arising from this continuum approach is considered, namely the oc-currence of two material time derivatives. In classical continuum mechanics, the material time derivative is defined to follow the Lagrangean observer. Thus it describes the change in time of a physical particle’s property. In a way, this is the time derivative with the most evident physical interpretation, when thinking of quantities as the density or the temperature in solid mechanics, for instance. As porous media consist of two types of phases at least, two different material time derivatives have to be distinguished: the material time derivative with respect to the skeleton, following the skeleton particle andthe material time derivative with respect to the fluid phase, following the fluid particle. The total time derivative with respect to the skeleton phase d^s/dtof a material quantity(•) (X, t)is given as

d^s(•) (X, t)

dt = ∂(•) (X, t)

∂t X

, (3.1)

and the total time derivative of a spatial quantity(•) (x, t)as d^s(•) (x(X, t), t)

dt = ∂(•) (x, t)

∂t x

+∇(•)·v^s, (3.2)

using the velocity of the skeletonv^s = ∂x/∂t|_X. The Lagrangean observer of the porous con-tinuum is defined to follow the skeleton phase. Hence, this material derivative is the same as the material derivative in classical solid mechanics. With respect to the fluid, the material co-ordinatesX of the skeleton can be interpreted as an independently moving material configura-tion, as the movement of the skeleton is different from the movement of the fluid. In this con-textX X^f, t

is depending both on time and the material coordinatesX^f associated with the fluid phase. Such a setting is similar to a description based on an ALE formulation. Concerning

porous media, the observer is not arbitrary, but the Lagrangean observer of the skeleton phase.

For the fluid, the basic idea yet stays the same. The similarity is even more evident in practice, as in ALE formulations, the mesh displacement is often determined by solving an elastostatic prob-lem. The material derivative with respect to the fluid phase of a spatial quantity(•) (x(X, t), t) can be written as

Consequently, the velocity of the fluid is given as v^f = d^fx

Solving (3.4) for the second summand, and substituting this expression in (3.3) yields d^f(•) (x, t) where ∂x(X, t)/∂t|_X = v^s has been utilized. Note, that this equation and its derivation is equivalent to the fundamental ALE equation (2.6), if the skeleton coordinatesX are interpreted as reference coordinates of the fluid and thereforev^sas grid velocity.

The material derivative of an integral quantity can be evaluated as d^π Therein, the chain rule and the material derivative of the Jacobian determinant

d^πJ

dt =J∇·v^π, (3.7)

were used, with the indexπ = s,f denoting the respective phase.

Remark 3.1 One has to keep in mind, that both time derivative with respect to skeleton and fluid phase can - and actually will - be applied to any type of quantity, disregarding if the quantity itself belongs to the skeleton or the fluid. The physical interpretation of a material derivative only holds, if time derivative and quantity match, hence for instance for the material time derivative with respect to the fluid phase of a quantity of the fluid (e.g. its density). If this is not the case, a clear physical interpretation becomes elusive.

3.3. Conservation of Mass

The balance of mass for both phases will be derived in the following. In the absence of sources or sinks, the mass in the domainΩtoccupied by the porous medium at timethas to be conserved,

i.e. d^s

whereρ^s andρ^f denote the averaged microscopic density of the respective phase. Equation (3.8) is the balance of mass of skeleton, (3.10) the balance of mass of the fluid. As φ and (1−φ) denote the volume fraction of the fluid and skeleton phase, respectively,ρ^s(1−φ)andρ^fφ can be interpreted as macroscopic densities. Applying equation (3.6) the local balance of mass can be formulated as The balance of mass of the skeleton (3.10) is preferably written in the material configuration, yielding

Jρ^s(1−φ) =ρ^s₀(1−φ0) = m^s₀, (3.12) whereρ^s₀,φ0andm^s₀denote theinitial skeleton density, porosity, and mass, respectively. Alterna-tively, the balance of mass is often written in terms of macroscopic masses or densities. Inserting equation (3.2) into equation (3.11) gives

∂(ρ^fφ)

The term beneath the brackets follows from the product rule. Using equation (3.7) one can re-formulate equation (3.13) to obtain the following alternative form of the continuity equation:

1 J

d^sm^f

dt +∇·w^f = 0, (3.14)

with thelocal fluid mass

m^f =ρ^fφJ, (3.15)

and therelative fluid mass flux

w^f =ρ^fφ v^f −v^s

. (3.16)

Introducing thematerial relative fluid mass flux

W^f =JF⁻¹·w^f, (3.17)

the continuity equation in material configuration follows as d^sm^f

dt +∇0 ·W^f = 0. (3.18)

This form of the balance of mass will be used for some thermodynamic considerations in Sec-tion 3.5. The equaSec-tions that actually will be used in the final non-linear system are the equa-tions (3.11) and (3.12).

Remark 3.2 The pull-back ofw^f in equation(3.17) is not defined as the pull-back of line ele-ments in equation(2.9)but via the Piola transform as used for area elements in equation(2.12). This definition is more suitable asw^f andW^f are representing fluxes. From equation(3.17) it follows that∇·w^f =∇₀ ·W^f, i.e. the Piola transform conserves the flux between material and spatial configuration.

Remark 3.3 Note that even when assuming the density to be constant, there will still be a tran-sient term in the continuity equation of the fluid (3.11), as the porosity can and will change in time due to finite deformations. Therefore, the macroscopic continuity equation for the fluid phase shows characteristics of a compressible flow, even if the microscopic flow is assumed to be incompressible.

3.4. Balance of Linear Momentum

The balance of momentum for the whole porous medium in the spatial configuration can be

written as Z

ρbˆ−ρ^s(1−φ)a^s−ρ^fφa^fdΩt+ Z

σ·n dΓt=0, (3.19)

wherea^sanda^f denote theaveraged microscopic skeletonandfluid acceleration, respectively.

The macroscopic Cauchy stress tensorσrepresents the loading state of the whole porous medium, i.e. both phases. The body forces per spatial unit volumeˆbrefer to themacroscopic total density given by

ρ=ρ^s(1−φ) +ρ^fφ. (3.20)

Equation (3.19) seems to be intuitive, as it resembles the balance of linear momentum (2.29) known from classical elastodynamics. However, its derivation starting from the microscopic equations is non-trivial. An approach based on volume averaging is sketched in Appendix A.1.1.

After application of Gauss’ divergence theorem, the local balance of momentum in spatial con-figuration can be obtained as

∇ ·σ+ρ^s(1−φ)

ˆb−a^s

+ρ^fφ

bˆ−a^f

=0. (3.21)

Transforming this equation to the material frame leads to

∇₀·(F ·S) +Jρ^s(1−φ)

ˆb−a^s

+Jρ^fφ

bˆ−a^f

=0. (3.22)

An alternative form of the local balance of momentum in the material configuration is obtained by introducing the conservation of mass of the skeleton phase (3.12) as

∇₀·(F ·S) +m^s₀

bˆ−a^s

+Jρ^fφ

ˆb−a^f

=0. (3.23)

In this form the nature of equation as the balance of linear momentum of the mixture is evi-dent. Two acceleration terms account for the inertia of the skeleton and the fluid respectively.

The other contributions, i.e. the stress contribution and the body forces, are not written sepa-rately for the two phases. The body force is applied on the whole mixture and distributed by the corresponding mass fraction. The stress and the deformation gradient are written with re-spect to the macroscopic, averaged deformation. Hence, the constitutive law for the stress tensor will include both structural and fluid contributions. Note, that this stress tensor differs from the effective stress, which is a quantity commonly used in soil mechanics, see Section 3.5.2.1.

Im Dokument A Computational Approach to Coupled Poroelastic Media Problems  (Seite 34-39)