Microscale constitutive relations

grain node

interface hydraulic node shared hydraulic DOF p^h

Fig. 5.9: Shared hydraulic degrees of freedom p^hfor nodes with equal fluid pressure to merge the diffusive grain flow and the interface channel flow.

The number of degrees of freedom p^h is equal to the number of hydraulic interface nodes for which the degrees of freedom are taken into account in (5.53).

Moreover, all condensed degrees of freedom p^b in (5.97) are related to nodes for which a shared degree of freedom is defined. This means that in practice, a reassembly of system of equations (5.97) on the degrees of freedom p^hcan be used to merge the diffusive flow problem and the interface flow problem. Reassembling (5.97) and addition of (5.53) will lead to the combined system of equations for degrees of freedom p^h:

G + H^?hh H^?hp H^?ph H^?pp

  p^h

∇p^REV



=P $ + q^?h m^{?dif f}



(5.100) with [H^∗] and {q^h} the reassembled equivalent of [H] and {q^b} in (5.97) and [G]

andP $ the contribution of (5.53). Subscript []^p refers to the prescribed macro degrees of freedom. This system of equations is again a system of equations of degrees of freedom with the connectivity of the interface hydraulic nodes. A sub-sequent condensation following the procedure of Section 5.2 gives the relations between the macro gradient of pressure and the macroscale fluid mass flux of Equations (5.63) and (5.64). Moreover; as long as the diffusive fluid flow prop-erties of the material (the components of the permeability tensor) are constant, the condensation has to be performed only once at the beginning of the com-putation and the result can be used in all identical microstructures throughout the computation. The same approach might be followed for the mechanical part of computations, although it should be noted that this only applies to purely elastic deformation under small strain assumption and is not in line with the en-visioned multiscale approach in which the choice of constitutive behaviour of the micromechanical components is not restricted by a linearity condition.

5.4 Microscale constitutive relations

For the constitution of the micromechanical REV, the constitutive behaviour of the different components (solid grains, cohesive interfaces and liquid phase) are developed.

5.4.1 Solid constitutive law

As the principle of the microscale model is based on the concept of material de-gradation (softening, damage,...) and the major part of the deformation to take place in the interfaces between grains, the grains themselves don’t need sophist-icated constitutive behaviour. Although it is very well possible to introduce any

continuous constitutive relation in the grains, only a simple linear elastic con-stitutive equation is introduced here. This allows (small) strains in the grains.

An isotropic, linear elastic relation is therefore defined for the solid grains as

σij = 2µεij+ λεkkδij (5.101)

where ε_ij is the small-strain tensor defined as ε_ij = ¹₂(^∂u_∂xⁱ

j +^∂u_∂x^j

i). The Lam´e parameters µ and λ are used as model parameters, σ_ij is the Cauchy stress, and δ_ij the Kronecker delta.

Alternatively, this can be written in matrix form as



or taking the variation of the stress state as a linear function of the variation of the deformation gradient tensor;



5.4.2 Fluid constitutive law

Fluid compressibility is taken into account using the fluid compressibility modulus k^w;

ρ˙^w=ρ^w

k^wp˙ (5.104)

with ˙ρ^w the rate of change of the current fluid density and ˙p as a function of the rate of change in pressure p. Integration and the definition of a reference pressure p⁰ with corresponding reference density ρ^w₀, gives the formulation of the fluid density as

The cohesive forces are described by a damage law, relating the relative normal and tangential displacement to the normal and tangential cohesive forces respect-ively. In a first effort, the inter-dependency between the components is reduced as much as possible to facilitate a proper evaluation of the micromechanical re-sponse. Therefore a decoupling between normal and tangential components is adopted for the relation between relative displacement and cohesive forces. The damage law is characterized by three parameters;

• T^maxthe maximum cohesion

• D(t) the history of relative displacement between the interface boundaries

5.4 Microscale constitutive relations 61

• δ^c the critical relative displacement at which complete decohesion is ob-tained

The history parameter D(t) at time t is defined relative to the critical relative displacement δ^c for the normal and tangential displacement separately:

D_t^t= max 1 δ^c_t max

0≤τ ≤t(|∆u^τ_t|) , D_t⁰



if 0 ≤|∆u^t_t| ≤δ_t^c

= 1 if δ_t^c<|∆u^t_t|

D^t_n= max 1 δ^c_n max

0≤τ ≤t(|∆u^τ_n|) , D_n⁰



if ∆u^t_n ≤δ_n^c

= 1 if δ_n^c<|∆u^t_n|

(5.106)

This defines D^t at time t as a parameter with values between D⁰ > 0 (initial state) and 1 (complete decohesion). The parameter D⁰ serves here as a damage history for the initial configuration (geological history, effects of excavation of the sample, ...), but can also be seen as a control parameter for the stiffness of an

’undamaged’ material. D⁰is always larger than 0, since D⁰= 0 would introduce infinite stiffness. Although this would be physically correct as it represents a non-active interface, it requires reformulation of the finite element system of equations to avoid infinite stiffness terms and ill-posed systems of equations. Taking a very small (but non-zero) initial state parameter D⁰ can serve as an alternative to obtain non-active interfaces, as the very high interface stiffness will serve as a penalization of the relative interface displacement.

Using the state parameter D^t the constitutive laws for the normal and tan-gential interface cohesion is defined as follows:

T_n^t = _δ¹_c

n

 1 D_n^t − 1

T_n^max∆u^t_n if ∆u^t_n> 0

= _δ¹c n

 1 D_n^t − 1

T_n^max∆u^t_n+ κ∆^t_n² if ∆u^t_n≤ 0 (5.107)

T_t^t = T_t^max

1

D^t_t − 1_∆ut t

δ_t^c (5.108)

where κ is a penalization term to avoid grain inter-penetration. The penalization is applied as a function of the negative interface opening squared to avoid an incremental non-linearity around ∆un = 0. The law describes a classical damage law with a linear softening branch and linear unloading-reloading (Figure 5.10).

The use of the separate interface laws for normal and tangential stress/strain relations implies a decoupling between normal and tangential components, thereby leaving some coupling aspects like interface friction out of consideration. The lack of friction in the interface does however not exclude the mean stress dependency of the response. As a combined effect of the asymmetry introduced by the pen-alization of the negative normal interface openings and the close contact of all grains, a mean stress dependency in the failure criterion is obtained and relatively high effective internal friction angles can be obtained. Some results of this stress-dependent behaviour is demonstrated in the modelling of the COx mechanical response in Section 14.2.

D⁰t D^tt 1 Δt/δt

Ttmax

Tt

Δ^tt/δt

D⁰n D^tn 1 Δn/δn

Tnmax

Tn

Δ^tn/δn

T^tt

T^tn

-Ttmax

Fig. 5.10: Schematization of the damage laws for the interface cohesive forces Tnand Tt.

5.4.4 The hydraulic constitutive law

The compressibility of the fluid phase is considered to be small with respect to the spatial variation of the fluid pressure, which corresponds to the local assumption of the REV. Given that variation of the fluid phase density is only dependent on local fluid pressure through the fluid compressibility k^w, the fluid density is considered constant throughout the REV.

The change in fluid phase density ˙ρ^w is directly related to the change in fluid pressure as:

˙

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)