2. Implementing the model#

2.1. Effective constraints and net constraints#

It should be noted that the formulation THM resides in the distinction between effective constraint \(\mathrm{\sigma }\text{'}\) and total constraint \(\mathrm{\sigma }\), such as:

(2.1)#\[ d\ mathrm {\ sigma} =d\ mathrm {\ sigma}\ text {'} +d\ mathrm {\ pi}\]

with \(\mathrm{\pi }\) hydraulic stress such as \(d\mathrm{\pi }=-b(\mathit{dPg}-S\mathit{dPc})\)

where S is liquid saturation and Pg is gas pressure.

In the case of a net stress formulation, we have:

(2.2)#\[ d\ mathrm {\ sigma} =d\ stackrel {~} {\ mathrm {\ sigma}} +d\ mathrm {\ pi}\]

with \(\mathrm{\pi }\) the hydraulic stress which is then defined as \(d\mathrm{\pi }=-b(\mathit{dPg})\).

It is this formulation that is used in the case of the GonfElas law, which differs from the other laws of behavior available in unsaturated THM. This allows this law to have ratings consistent with those of the Barcelona law.

2.2. Programming the law#

The law is programmed incrementally on the mean stress (applied to net constraints) which gives:

\[\]

: label: eq-4

mathrm {Delta}stackrel {~} {{mathrm {sigma}} _ {m}} = {K} _ {0}mathrm {Delta} {mathrm {epsilon}} {epsilon}} {epsilon}} _ {epsilon}} _ {V} +bmathrm {Delta}mathit {PG} {delta} {delta} {delta} {delta} {delta} {} {delta} {} {mathrm} {delta} {delta} {{}} {mathrm {Delta} {

With:

\[\]

: label: eq-5

stackrel {~} {{mathrm {sigma}}} _ {m}} =frac {1} {3}mathit {Tr} (stackrel {~} {mathrm {sigma}})

and by introducing the swelling pressure function in saturated and unsaturated:

(2.3)#\[\begin{split} \ mathit {PG} (\ mathit {Pc}) =\ {\ begin {array} {cc}} A\ left (\ frac {\ sqrt {\ mathrm {\ pi}}}} {2\ sqrt {{\ mathrm {\ mathrm {\ mathrm {\ beta}}}}}\ right)\ mathit {Erf}\ left (\ frac {\ mathit {Pc}}}} {A}\ sqrt {{\ mathrm {\ beta}}} _ {m}}}\ right) +\ frac {1} {2 {\ mathrm {\ beta}} _ {m}}\ left (\ begin {array} {c} {c} {c} {c}} 1- {e} ^ {e} ^ {- {e} ^ {- {e} ^ {- - {\ mathrm {\ beta}}} _ {m}} {\ left (\ frac {\ array}} {c} {c}} {c}} {A}\ right)} ^ {2}}\\\ phantom {\ rule {2em} {0ex}}\ end {array}\ right) &\ text {si} S<1\\ mathit {Pc}\ text {Pc}\ text {if}\\ phantom {si} S=1\ hfill &\ phantom {\ array} {2em} {0ex}}\ end {array}}\end{split}\]

with \(\mathit{Erf}(x)=\underset{0}{\overset{x}{\int }}{e}^{-{\mathrm{\chi }}^{2}}d\mathrm{\chi }\).

2.3. Material data and identification#

The material parameters specific to the law and to be entered in DEFI_MATERIAU are:

  • BETAM: dimensionless material parameter corresponding to \({\beta }_{m}\) of the law above.

  • PREF: homogeneous parameter at a pressure corresponding to \(A\) of the law above.

The identification of \({\beta }_{m}\) is done by looking for the inflation pressure. Let \({P}_{\mathit{gf}}({\mathit{Pc}}_{0})\) be the swelling pressure found by the model when we re-saturate a sample in a locked deformation test and starting from \({\mathit{Pc}}_{0}\) suction. Remember that at saturation, \(\mathit{Pc}=0\), which implies that:

(2.4)#\[ {P} _ {\ mathit {gf}}} ({\ mathit {Pc}}} _ {0}) =\ underset {{\ mathit {Pc}} _ {0}} {\ overset {0} {\ int}} {\ int}}} ({\ int}}} b\ left} b\ left (1+\ frac {\ mathit {Pc}}} {A}\ right) {e} ^ {- {\ mathrm {\ int}}} {\ int}}} b\ left (1+\ frac {\ mathit {Pc}} {A}\ right) {e} ^ {- {\ mathrm {\ int}}} {\ int}}} b\ left (1+\ frac {\ mathit {Pc}} {_ {m} {\ left (\ frac {s} {A}\ right)} ^ {2}}\ mathit {dPC}\]

After integration, we obtain:

(2.5)#\[\begin{split} \ frac {{P} _ {\ mathit {gf}}} ({\ mathit {gf}}} ({\ mathit {Pc}}} _ {0})} {A} =\ frac {\ mathrm {\ pi}}}} {\ mathit {\ pi}}}} {\ mathit {\ pi}}}} {\ mathrm {\ beta}}} {A} =\ frac {\ mathrm {\ pi}}}} {\ mathrm {\ pi}}}} {2\ sqrt {\ mathit {Pc}}} {2\ sqrt {\ mathit {Pc}}} {2\ sqrt {\ mathit {Pc}}}}} _ {0}} {A}\ sqrt {{\ mathrm {\ mathrm {\ beta}}} _ {m}}\ right) +\ frac {1} {2 {\ mathrm {\ beta}}} _ {m}}}\ {m}}}}\ left (\ left (\ left (\ fr) ac {{\ mathit {Pc}} _ {0}} {0}} {0}} {A}\\ right)} ^ {2}}\\ phantom {\ rule {2em} {0ex}}}\ end {array}\ right)\end{split}\]

The expected swelling pressure corresponds to the restoration path between the dry state (\({P}_{c}=\mathrm{\infty }\)) and the saturated state, i.e. \({P}_{\mathit{gf}}={P}_{\mathit{gf}}(\mathrm{\infty })\). We know that \(\mathrm{Erf}(\infty )=1\) and therefore: \(\frac{{P}_{\mathit{gf}}}{A}=\frac{\sqrt{\mathrm{\pi }}}{2\sqrt{{\mathrm{\beta }}_{m}}}+\frac{1}{2{\mathrm{\beta }}_{m}}\)

An identification of the coefficient \({\mathrm{\beta }}_{m}\) is deduced from this.

2.4. Internal output variables#

There are no internal variables in the output.