2. Introduction to the Barcelona model: behavior under isotropic loading

Isotropic loading paths are visualized by varying the net pressure \(p''=-\sigma_m''=-(\sigma_m+p_g)\) and the capillary pressure \(p_c=p_g-p_l\) in order to characterize the behavior of the soil in its elastic and plastic phases. To do this, we denote the void index \(e=d\Omega^p/d\Omega^s\) (ratio between the volume of the pores \(d\Omega^p\) and the volume of the solid skeleton \(d\Omega^s\) in the VER), for which we admit the additive decomposition between the elastic and plastic parts \(e^e\) and \(e^p\), respectively.

2.1. At constant capillary pressure

Initially, we consider a loading at constant capillary pressure starting from an initial state for which the soil is very loose (i.e. with high porosity), typically \(e>1\). Its elastic limit under compression is almost zero. By increasing the net pressure, the incremental relationship is then observed schematically:

\[of = de^e + de^p = -\ lambda (p_c)\ cfrac {dp "} {p"} :label: plastic_compression\]

The line represented by the previous relationship in plane \((\ln p'',e)\) is similar to the consolidation curve for saturated soils. Here, however, slope \(\lambda(p_c)\) is a decreasing function of capillary pressure. It is proposed in the Barcelona model in the form:

\[\]

lambda (p_c) =lambda_0 ((1-r)exp (-beta p_c) +r)

label:

variation_plastic_coefficient

where \(\lambda_0,r\) and \(\beta\) are constants.

At one point on the consolidation curve, a net pressure discharge makes it possible to observe reversible behavior, which can be written schematically:

\[of = de^e =-\ kappa\ cfrac {dp "} {p"} :label: elastic_compression\]

Unlike \(\lambda(p_c)\), the \(\kappa\) slope in the \((\ln p'',e)\) plane is constant. The Fig. 2.1 shows the appearance of the answers obtained by compression_plastique, variation_coefficient_plastique, and compression_elastique.

Fig. 2.1 Variation of the void index with net pressure for two capillary pressures \(p_{c2}>p_{c1}\).

The area of elasticity, at constant capillary pressure, represented by the Barcelona model is defined by:

\[\]

left| p » -frac {p_ {con} (p_c, e^p) -kp_c} {2}right|-frac {p_ {con} (p_c, e^p) +kp_p) +kp_p) +kp_c} {2}leq 0,quadtext {with}quad

p_ {con} (p_c, e^p) = p_rleft (cfrac {p_ {con} (0, e^p)} {p_r}right) ^cfrac {lambda_0-lambda_0-kappa} {lambda (p_c) -kappa}

label:

limits_compression

\(k p_c\) represents the isotropic traction limit, which increases proportionally with capillary pressure via the parameter \(k\). The isotropic compression limit is \(p_{con}(p_c,e^p)\), an increasing and non-linear function of capillary pressure, where \(p_r\) is a parameter. The zero capillary pressure value \(p_{con}(0,e^p)\) is obtained by the following reasoning.

For this purpose, consider a loading path with zero capillary pressure in which the net pressure increases. When it reaches \(p_{con}(0,e^p)\), loading causes a plastic variation in the void index. Using compression_plastique and compression_elastique, we get:

\[de^p = de-de^e = - (\ lambda_0-\ kappa)\ frac {dp_ {con} (0, e^p)} {p_ {con} (0, e^p)}\ Longrightarrow p_ {p)}}\ Longrightarrow p_ {con} (\ con} (0, e^p)} {\ lambda_0-\ kappa}\ right) :label: definition_ecrouissage_1\]

\(p_{con}(0,0)\) is the initial consolidation pressure (isotropic compression limit) at zero capillary pressure.

2.2. At constant net pressure

We now consider a loading at constant net pressure. When capillary pressure reaches Once the maximum value has previously been reached and continues to increase, a plastic variation in the void index is generated, which can be written schematically:

\[of = de^e + de^p = -\ lambda_s\ cfrac {dp_c} {p_c+p_ {atm}} :label: capillary_plastic_compression\]

with \(\lambda_s\) constant. In the case of a capillary pressure discharge from the line represented by the line represented by the relationship compression_plastique_capillaire in the plane \((\ln p_c,e)\), the response is reversible and represented by the relationship:

\[of = de^e = -\ kappa_s\ cfrac {dp_c} {p_c+p_ {atm}} :label: capillary_elastic_compression\]

with \(\kappa_s\) and \(p_{atm}\) two constants. The latter, being usually evaluated [bib2] __ as being atmospheric pressure, has the sole advantage of defining compression_elastique_capillaire in saturated conditions where \(p_{cap}=0\).

The area of elasticity, at constant net pressure, represented by the Barcelona model is defined by:

\[p_c-s_0 (e^p)\ leq 0 :label: capillary_limit\]

where the threshold \(s_0(e^p)\) is written, in accordance with compression_plastique_capillaire and compression_elastique_capillaire:

\[de^p = de-de^p = - (\ lambda_s-\ kappa_s)\ cfrac {ds_0 (e^p)} {s_0 (e^p) +p_ {atm}}\ Longrightarrow s_0 (e^p) =\ left (e^p) =\ left (e^p) =\ left (-\ frac {e^p) =\ left (-\ frac {e^p) =\ left (^p} {\ lambda_s-\ kappa_s}\ right) - p_ {atm} :label: definition_ecrouissage_2\]

By combining limites_compression and limite_capillaire, the elasticity domain under isotropic loading is delimited on the Fig. 2.2. It gets bigger when the plastic void index \(e_p\) decreases (i.e. when the pores close).

Fig. 2.2 Elasticity domains under initial isotropic loading and for \(e^p<0\).

Note:

The Fig. 2.2 shows that \(p_{con}(p_c,e^p)\) is an increasing function of capillary pressure \(p_c\). To be exact, this is only true when \(p_{con}(0,e^p)>p_r\) in accordance with limites_compression -b. For \(p_{con}(0,e^p)<p_r\) on the other hand, \(p_{con}(p_c,e^p)\) decreases with \(p_c\). There is no experimental justification for this decrease. It is an artifact of the Barcelona model.

2.3. General behavior under isotropic loading

The Section 2.1 and Section 2.2 now make it possible to summarize the behavior equations of the Barcelona model under isotropic loading as follows:

• Elastic behavior:

\[de^e = -\ kappa\ cfrac {dp "} {p"} -\ kappa_s\ cfrac {dp_c} {p_c+p_ {atm}} :label: elasticite_incrementale_1\]

• The elasticity domain:

\[\]

left| p » -frac {p_ {con} (p_c, e^p) -kp_c} {2}right|-frac {p_ {con} (p_c, e^p) +kp_p) +kp_p) +kp_c} {2}leq 0,quad

p_c-s_0 (e^p)leq 0 :label: isotropic_elasticity_domain

• Work hardening functions:

\[p_ {con} (p_c, e^p) = p_r\ left (\ cfrac {p_ {con} (0,0)\ exp\ left (-\ cfrac {e^p} {\ lambda_0-\ lambda_0-\ kappa 0-\ kappa-\ kappa}\ right)} {p_r}\ right) ^\ cfrac {\ lambda_0-\ kappa} {\ lambda (p_c) -\ kappa},\ quad s_0 (e^p) =\ left (s_0 (0) +p_ {atm}\ right)\ exp\ left (-\ frac {e^p} {\ lambda_s-\ kappa_s-\ kappa_s}\ right) - p_ {atm} :label: work_working_functions\]

• The flow of the plastic void index (here written outside the two « corners » of the elasticity domain, see Fig. 2.2):

\[\]

begin {align}

&de^p < 0quadtext {si}quad p » -p_ {con} (p_c, e^p) = 0,quad dp » -frac {partial p _ {con}} {partial p_c} dp_c >0vphantom {big {(}}}\ &de^pgeq 0quadtext {si}quad p »+kp_c = 0,quad dp »+kdp_c = 0vphantom {Big {(}}}\ &de^p < 0quadtext {si}quad p_c = s_0 (e^p),quad dp_c > 0vphantom {Big {(}}}\ &de^p = 0quadtext {otherwise}vphantom {Big {(}}} end {align} :label: isotropic_flow

Having addressed the Barcelona model in the case of isotropic loading, his behavioral equations in a three-dimensional formulation can now be developed in Section 3.

Note:

The void index \(e\) can be deduced, in the case that can be linearized all along, from the volume deformation \(\varepsilon_v\) by the following relationship:

with \(e_0\) the initial void index. Likewise, the relationship equivalence_deformation_indice_des_vides_1 is assumed to be true on the parts elastic and plastic of the void index and of volume deformation, which leads to:

In particular, this last relationship has been used since Section 3.