Some Concepts from Geo and Soil Mechanics

S−φS^f_visc− ∂Ψ^s

∂E

: ˙E+

p^f− ∂Ψ^s

∂(Jφ)

· d^s(Jφ) dt −

S^s−∂Ψ^s

∂T

· d^sT

dt ≥0. (3.58) SinceE,JφandT can vary independently and their time derivative can have arbitrary sign and absolute values, all terms in brackets in equation (3.58) have to vanish. One obtains the following constitutive equations for the skeleton phase of a porous medium:

S =φS^f_visc+∂Ψ^s(E, Jφ, T)

∂E , p^f = ∂Ψ^s(E, Jφ, T)

∂(Jφ) , S^s = ∂Ψ^s(E, Jφ, T)

∂T . (3.59)

Alternatively, it is also possible to chooseE,m^f andT as independent variables. Starting from equation (3.50) instead of equation (3.54) one can derive [66] after analogous calculations the following alternative set of constitutive equations:

S =φS^f_visc+∂Ψ^s(E, m^f, T)

∂E , g^f = ∂Ψ^s(E, m^f, T)

∂m^f , S = ∂Ψ^s(E, m^f, T)

∂T . (3.60)

A third alternative can be formulated with the definition

G^s = Ψ^s−p^fJφ (3.61)

as

S =φS^f_visc+∂G^s(E, p^f, T)

∂E , Jφ=−∂G^s(E, p^f, T)

∂p^f , S^s =−∂G^s(E, p^f, T)

∂T .

(3.62) Further forms and derivations can be found in [66].

Remark 3.5 The energy terms in the constitutive equations(3.59) include the termJφand not solely the porosityφ. This is due to the fact, that the constitutive laws are formulated depending on materialquantities. The porosity, however, is a spatial quantity. Considering equation(1.1) and (2.11)it becomes clear, thatJφcan be interpreted as material porosity, denoting the ratio of current fluid volume to material volume

Jφ· dΩ0 = dΩ^f_t. (3.63)

3.5.2. Some Concepts from Geo and Soil Mechanics

In this section, selected terminologies originating from geo and soil mechanics are introduced.

Terzaghi’s principle of effective stress in Section 3.5.2.1 is well-known in this field. Porous materials are classically characterized by the Biot moduli, described in Section 3.5.2.2. Lastly, drained and undrained conditions are commented in Section 3.5.2.3.

3.5.2.1. Terzaghi’s Principle of Effective Stress

Terzaghis principle of effective stress will be introduced in this section, as it is often referred to, especially in classical soil mechanics (see e.g. [45, 73, 220]). The comments given here are a compact version of the explanations in [66, sec. 3.4.1].

The effective stress S^eff of a porous medium is defined as the sum of the total stress of the mixture and the hydrostatic pressure

S^eff =S+p^fJC⁻¹. (3.64)

The effective stress defines the mechanical state of loading, which actually induces macroscopic deformation in case of a (nearly) incompressible skeleton phase. The density of the skeleton phase is assumed to be constant and the change of volume is uniquely defined by the change of porosity. Then, it follows from the balance of mass of the skeleton phase (3.12), that

Jφ=J+φ0 −1. (3.65)

Using the identity

d^sJ

dt =JC⁻¹ : ˙E, (3.66)

and neglecting viscous effects, one obtains from the dissipation inequality (3.54) Φ^s = S+p^fJC⁻¹

: ˙E−S^sd^sT

dt − d^sΨ^s

dt ≥0, (3.67)

and finally the following constitutive relations, similar to equation (3.59) S^eff =S+p^fJC⁻¹ = ∂Ψ(E, T)

∂E , S = ∂Ψ(E, T)

∂T . (3.68)

From the first constitutive equation in (3.68) one can conclude that the effective stress can be modeled as derivative of a strain energy function in the same way as in classical elastodynamics.

Due to these relations, the complexity of systems, where Terzaghi’s principle of effective stress is applicable, is reduced significantly.

3.5.2.2. Linear Poroelasticity: Biot Modulus and Biot Tangent

Next to the known material elasticity tensor, which gives the material response to macroscopic strain, there are commonly two additional stiffness measures: the Biot modulus and the Biot tangent. In linear theory those moduli characterize the stress response due to change of porosity and pore pressure and were also used for postulating non-linear constitutive laws (see e.g. [48, 55]). Even though they do not play a central role in this thesis, they will be shortly reviewed in the following, as they are often referred to in the literature. The definitions given here are a short summary of [66, sec. 4.1.2].

As infinitesimal changes of state are considered, the distinction between different strain and stress measures is dropped here. The constitutive equation (3.62) then gives for the isothermal,

non-viscous case

dσij =Cijkldεkl−bijdp^f, (3.69a) d(Jφ) = (bB)_ij dεij + dp^f

N . (3.69b)

Here, the forth-order tensor

Cijkl= ∂²G^s

∂εij∂εkl

(3.70) denotes the material elasticity tensor andG^sthe free enthalpy of the skeleton, see equation (3.61).

The symmetric, second-order tensor

(bB)_ij =− ∂²G^s

∂εij∂p^f (3.71)

denotes the Biot tangent. It relates changes of porosity with changes of strains as well as changes of the pore pressure with changes of stress at a constant strain state. Second, the scalar value

1

N =− ∂²G^s

(∂p^f)² (3.72)

is the inverse Biot modulus. It gives the behavior of the pore pressure at variations of the porosity.

Both quantities are constant in case of linear theory. In the special case of an incompressible matrix, the volume change of the porous medium = εiiis uniquely determined by the change of material porosity:

d= d(Jφ). (3.73)

Then, it immediately follows from equation (3.69b) (bB)_ij =δij, 1

N = 0, (3.74)

with the Kronecker deltaδij. Equation (3.69a) then gives

d(σij +p^fδij) =Cijkldkl, (3.75) which is consistent with the results obtained from the considerations of the effective stress in equation (3.68).

3.5.2.3. Drained and Undrained Conditions

A categorization, which is common in geo and soil mechanics, are so-called drained and undrained conditions. Both terminologies can be applied to a whole problem setting as to a boundary. The behavior of a porous medium is said to be fully drained, if the duration of the consolidation process is short compared to the time scale of the problem considered [242]. This means that flow and pressure variations can be neglected. So, in fact, a pure solid problem is investigated. If a fluid boundary is assumed to be drained, it represents a free outflow at reference pressure (most often zero pressure). The other extreme case areundrained conditions. There, it

is assumed that the loading is so fast and/or the permeability is so low, that there is hardly any fluid flowing. Then, the fluid will instantly carry a part of the load. From a macroscopic point of view, this case can also be written as pure solid problem with modified material parameters, see e.g. [242, chap. 2.8]. In general, the stiffness is increased due to the resistance of the pore fluid. In this context, it is distinguished between undrained and drained moduli. A problem with impermeable boundaries, i.e. no fluid flux over the boundaries, is also called undrained situation.

Im Dokument A Computational Approach to Coupled Poroelastic Media Problems  (Seite 44-47)