6. Material parameters and internal variables#

6.1. Material parameters#

The mandatory \(E\) and \(\nu\) parameters under the ELAS keyword are not used by law CAM_CLAY. The ELAS keyword can therefore be avoided if the user does not need to enter \(\alpha\) or \(\rho\).

The data specific to the Cam_Clay model are:

  • The elastic shear modulus \(\mu\),

  • The critical slope \(M\),

  • The porosity associated with initial pressure and linked to the initial void index: \(n=\frac{{e}_{0}}{1+{e}_{0}}\)

  • Initial compressibility \({K}_{\text{cam}}\),

  • Tolerated traction pressure \({P}_{\text{trac}}\), (must be negative)

  • The elastic swelling coefficient: \(\kappa\) (which leads to \({k}_{0}\)),

  • The plastic compressibility coefficient: \(\lambda\) (which leads to \(k\)),

  • The initial critical pressure \({P}_{\text{cr}0}\) such that \({P}_{\text{cr}0}-{P}_{\text{trac}}\) is equal to half of the consolidation pressure,

Note 1:

The amount of data is low, which makes the model very simple. One of the most visible limitations of the model is the assumption of the alignment of critical points on a line with a slope \(M\). Moreover, this is the expression of the concept of internal friction. We can also interpret the quantity \(M\) by relating it to the Coulomb angle of internal friction by the relationship: \(\text{sin}\varphi =\frac{\mathrm{3M}}{6+M}\). Now we know that for very cohesive materials, this angle varies when the mean stress decreases. Moreover, we note that for a calibration of \(M\) on a triaxial test at a certain average stress, we simulate with this model the triaxials made with an average stress that is not too different but we cannot correctly estimate the plastic bearings for a wide range of confinement pressures (cf. [bib2]). It is therefore necessary to reset \(M\) for several mean stress ranges.

Note 2:

The increase in stresses is linked to the increase in volume deformations according to one or other of the laws of behavior:

With Cam_Clay:

\(\mathrm{\Delta P}=({k}_{0}{P}^{-}+{K}_{\text{cam}}){\mathrm{\Delta \epsilon }}_{v}\)

\(\text{tr}(\mathrm{\Delta \sigma })=3({k}_{0}{P}^{-}+{K}_{\text{cam}}){\mathrm{\Delta \epsilon }}_{v}\) with \({k}_{0}=\frac{1+{e}_{0}}{\kappa }\) where \({e}_{0}=\frac{n}{1-n}\); \(n\) is the porosity and it is a material data.

In elasticity:

\(\text{tr}(\mathrm{\Delta \sigma })=\frac{E}{(1-\mathrm{2\nu })}\text{tr}(\mathrm{\Delta \epsilon })={\mathrm{3K\Delta \epsilon }}_{v}\)

The analogy between the hydrostatic part of Cam_Clay and linear elasticity in the initial state allows you to write:

\(\frac{(1+{e}_{0}){P}^{-}}{\kappa }+{K}_{\text{cam}}=\frac{E}{3(1-\mathrm{2\nu })}\)

\(E\) and \(\nu\) are not material data but rather \(\mu\) the shear modulus: \(\mu =\frac{E}{2(1+\nu )}\)

This is the same as writing the following equality by eliminating \(E\):

\(\frac{(1+{e}_{0}){P}^{-}}{\kappa }+{K}_{\text{cam}}=\frac{\mathrm{2\mu }}{3}\frac{(1+\nu )}{(1-\mathrm{2\nu })}\) or \(\frac{(1+\nu )}{(1-\mathrm{2\nu })}=\frac{3(1+{e}_{0}){P}^{-}+{\mathrm{3K}}_{\text{cam}}\kappa }{2\text{μκ}}\)

and we find the expression for \(\nu\):

\(\nu =\frac{3(1+{e}_{0}){P}^{\text{-}}+3{K}_{\mathrm{cam}}\kappa -2\mu \kappa }{6(1+{e}_{0}){P}^{\text{-}}+6{K}_{\mathrm{cam}}\kappa -2\mu \kappa }\)

at the start of the calculation, \({P}^{-}\) corresponds to the initial constraints field.

we can then deduce \(E\) from \(\nu\): \(E=\mathrm{2\mu }(1+\nu )\)

the following conditions must be verified:

\(0<\nu =\frac{3(1+{e}_{0}){P}^{-}+{\mathrm{3K}}_{\text{cam}}\kappa -2\text{μκ}}{6(1+{e}_{0}){P}^{-}+{\mathrm{6K}}_{\text{cam}}\kappa +2\text{μκ}}\le 0\text{.}5\) and \(E>0\)

if one or the other of the two conditions is not satisfied, an alarm message warns the user of the inconsistency of the parameters provided.

Note 3:

*If \({P}_{\text{trac}}\) is given null:

two possibilities for \({K}_{\text{cam}}\):

1- \({K}_{\text{cam}}\) positive (zero initial constraints are allowed)

2- \({K}_{\text{cam}}\) null (constraints must be initialized)

*If \({P}_{\text{trac}}\) is negative:

only one possibility for \({K}_{\text{cam}}\):

  • \({K}_{\text{cam}}\) positive how to satisfy the \({k}_{0}{P}_{\text{trac}}+{K}_{\text{cam}}>0\) relationship

(you cannot initialize the constraints and give a zero value to \({K}_{\text{cam}}\))

6.2. Internal variables#

\({V}_{1}\): critical pressure \({P}_{\text{cr}}\)

\({V}_{2}\): plastic condition

\({V}_{3}\): confinement constraint \(P\)

\({V}_{4}\): equivalent stress \(Q\)

\({V}_{5}\): volume plastic deformation \({\epsilon }_{v}^{p}\)

\({V}_{6}\): equivalent plastic deformation \({\epsilon }_{\text{eq}}^{p}\)

\({V}_{7}\): void index \(e\)