Computational homogenized macroscale response

ρ^w= ρ^w p˙

k^w (5.109)

Given an initial fluid density ρ⁰_wat reference pressure p⁰, the closed-form expres-sion of the fluid density is

ρw(p) = ρ⁰_wexp p − p⁰ kw



(5.110)

5.5 Computational homogenized macroscale response

In this section, the homogenization approach for obtaining the macroscale re-sponse from the REV under enforced deformation is discussed.

5.5.1 Homogenization of stress

The Hill-Mandel macro homogeneity condition [Hill, 1965, Mandel, 1972] is used as starting point for deriving the micro-to-macro transition of the REV averaged stress and its tangent operator. This condition requires the average microscale work to be equal to the macroscale work. Under small strain assumption the work conjugation of the stress tensor σ and strain tensor^∂u_∂xⁱ

j are naturally obtained and the macro homogeneity condition can be written in a virtual work formulation

W^{M ?}= σ_ij^M∂u^{M ?}_i

∂xj

= σ^REV_ij ∂u^{REV ?}_i

∂xj

= 1

V^REV Z

Ω^REV

W^m?dΩ (5.111)

As the interfaces introduce discontinuities in the microscale displacement field, the microscale virtual work can not directly be related to σ_ij^m∂u_∂x^?i

j and an equival-ent, continuous microscale displacement field ˆui needs to be introduced for this

5.5 Computational homogenized macroscale response 63

purpose. Around an interface, a zone of arbitrary width l is defined (see Figure 5.11), hereafter referred to as the contact zone.

Δu

l n

ε_m

εm

ε_m

reference state current state

Fig. 5.11: Discontinuous interface with interface contact zone

Given a vector ~l = l~n (with interface normal vector ~n as its orientation) to characterize the cross section of the cohesive zone, the deformation of the domain by a certain strain field ^∂u_∂x^ti

j and interface relative displacement ∆u^t_i defines the updated length of this vector as:

l_i^t=

l

Z

0

(δij+∂u^t_i

∂x_j)dl_j⁰+ ∆u^t_i= l_i⁰+

l

Z

0

∂u^t_i

∂x_jdl_j⁰+ ∆u^t_i (5.112) An equivalent strain of the contact zone can be defined by the introduction of an equivalent displacement field ˆu_i:

∂ ˆu_i

∂xj

= 1 l

Z

~l

∂u_i

∂xj

dl + ∆uinj



(5.113) or in terms of a small strain tensor:

ˆ εij =1

l

Z

~l

εijdl +1

2(∆uinj+ ∆ujni)



(5.114)

For small enough ~l this is equivalent to

ˆ

ε_ij = ε_ij+ 1

2l(∆u_in_j+ ∆u_jn_i) (5.115)

Outside the contact zone, the strain remains unchanged and ˆε_ij = ε_ij. With an integration of the strain over the reference domain, the discontinuities can be taken into account by considering ˆu_i, which leads to:

∂u^REV_i

∂x_j = 1 V⁰

Z

Ω⁰

∂ ˆui

∂x_jdΩ (5.116)

Note that for lim

l→0 displacement fields ˆu_i and u_i converge, although this means that in the contact zone ˆε → ∞ with l → 0. Nevertheless, the introduction of the equivalent displacement field allows to apply Gauss theorem to transform the domain integral over deformation into a boundary integral over displacements (5.116), which holds for both displacement fields ui and ˆui for any l:

∂u^REV_i

Stress continuity is guaranteed over the REV domain under small strain as-sumption (no interface opening). This means that for small enough l, the stress state σ_ijcan be assumed constant over the cross section of the contact zone. This means that the Hill-Mandel condition requires

σ_ij^M∂u^M_i

Gauss divergence theorem and equilibrium equation (5.3) (stress and displace-ment fields σ and ~ˆu are continuous and the internal boundaries drop out for ˆui) allows to write

Using the periodic boundary conditions (4.8) and (4.11), the virtual displacement of the external boundary is reduced to

1

with ~v^∗ the virtual ”change” of the periodicity vector ~y given by v_i^?=∂u^?REV_i

∂x_j yj (5.121)

Substitution of ~v^∗allows to take the virtual part of the displacement outside the integral;

For the Hill-Mandel condition (5.111) to be satisfied, the integral in the right hand side of equation (5.122) needs to hold

σ_ij^REV = 1 V^REV

Z

Γ^F

¯tiyjdΓ (5.123)

Gauss theorem and periodic boundary conditions allow rewriting once more into σ_ij^REV = 1

V^REV Z

Ω⁰

σ^m_ijdΩ (5.124)

5.5 Computational homogenized macroscale response 65

which provides the definition of the homogenized stress that corresponds to the Hill-Mandel macro-homogeneity condition under small-strain assumptions. In case of finite element discretization of the REV domain for solving the boundary value problem, the macro stress tensor is easily obtained from Equation (5.123) when it is written as the sum of the nodal reaction forces on the follow boundary:

σ_ij^REV = 1 V^REV

X

Γ^F

f_i^FyjdΓ (5.125)

5.5.2 Homogenization of fluid mass flux

For the fluid part of the balance equations given in (3.5), the same approach for establishing the macro-homogeneity can be followed. In other works this method is used by ¨Ozdemir [2009] for thermomechanical problems and by Massart and Selvadurai [2012, 2014] for fluid fluxes. The macro-homogeneity condition for the hydraulic balance equations reads:

M˙^REVp^?REV − m^REV_i ∂p^?REV

with ˙M and m_i on the microscale yet to be defined. As the microscale problem is solved under steady state conditions, validated by the separation of scales, in this case separating the timescales of the hydraulic storage ˙M [fluid mass/time]

and the flux ~m [fluid mass/time] at the microlevel, this reduces temporarily to m^REV_i ∂p^?REV

with mi the microscale fluid mass flux. Gauss divergence theorem and the equi-librium of the system allows to write

m^REV_i ∂p^?REV

where q = miniis the normal outward mass flux over the REV. The (fluctuation of the) virtual pressures are constrained by the boundary conditions (4.12) and (4.14) which allows rewriting into

m^REV_i ∂p^?REV

From this, the definition of m^REV_i is found as the integral of the microscale boundary mass flux q^F over the follow boundary:

m^REV_i = 1 V^REV

Z

Γ^F

q^FyidΓ (5.130)

For the finite element discretization of the REV boundary value problem, this integral can easily be computed as the sum of the fluxes in the nodes on the

follow boundary:

The microscale outward flux on the follow boundary q^Fis the assembly of interface fluxes $ and diffusive grain fluxes. This summation is identical to the summation in 5.57 for the total flux over the follow boundaries.

Moreover, applying once more the divergence theorem shows that the macro-scale mass flux is the average of the micromacro-scale mass flux

m^REV_i = 1 V^REV

Z

Ω

midΩ (5.132)

5.5.3 Homogenization of fluid mass

To obtain the macroscopic fluid content M a direct approach is followed by defin-ing the total amount of fluid in the REV as the amount of fluid in the grain pore space plus the amount of fluid in the interfaces, for which the volume is defined by the integration of the hydraulic opening over the interfaces. The interface volume V^int is much smaller than the REV volume V^REV = V^REV,0 and therefore, the amount of fluid is computed as

M = 1

With ρ^wconstant over the REV and the interface volume defined by the integral of the hydraulic interface aperture over the 1D pore network S^int, this can be rewritten as

with ¯ϕ the REV average of the grain porosity. To obtain the fluid mass storage term ˙M , a finite difference approximation is made over the time interval ∆t. This time interval is taken to be the same as the macroscale time step. For a time step

∆t from t − ∆t to t, the fluid storage term ˙M^t is obtained as follows:

M˙^t≈ M^t− M^t−∆t

∆t (5.135)

The specific fluid mass M^t depends on the density of the fluid and the relative volume it occupies. As the REV pore volume only changes due to the interface volume, the rate of change in specific fluid mass M is found to be dependent˙ on the rate of change of fluid density ˙ρ^w and the rate of change of interface volume ˙V^int. This means that the fluid storage has a rheological part as well as a geometrical part. With the assumption of constant grain pore volume (not to be confused with incompressibility of the grains!), the geometrical component of the fluid storage is relatively small. The effect of this definition is investigated in Part III.

6. COMPUTATIONAL HOMOGENIZATION BY STATIC

In document Multiscale modelling of the hydromechanical behaviour of argillaceous rocks (Page 94-99)