10.8 Stress_Model: Cap Model (Geomaterials)

Following common usage in geomechanics, compressive stresses and strains are taken as positive in the following. The following hyperelastic stored energy function is employed:

(1)

where

(2)

= elastic compressibility index, and elastic shear modulus defined by the expression:

(3)

where material parameter which describes the variation of the shear modulus with the elastic volumetric strain. In Eq. 1, are parameters corresponding to the strain-free reference state at which:

(4)

and

(5)

with

(6)

The elasto-plasticity is defined in terms of a yield function and a failure criterion as illustrated on Fig. 10.8.1.

1. Yield Function The yield function is of the following form:

(7)

where M = material parameter (= slope of critical state line), and preconsolidation stress = scalar plastic state variable describing the size of the yield function. An associative flow rule is used, and the following hardening rule is employed:

(8)

where plastic volumetric strain rate; compressibility index for virgin loading. The cap yield function plots in stress space as an ellipsoid as in the modified Cam_clay theory (Roscoe and Burland, 1968).

2. Failure Criterion The failure criterion is of the following form:

(9)

and a non-associative flow rule is used. The failure criterion plots in stress space as a circular cone of the Drucker-Prager (1952) type.

In Eqs. 7 and 9, a = attraction = c/tan where c = cohesion, and friction angle. The slope M can be selected to fit the corresponding Mohr-Coulomb criterion by setting:

: external cone (10a)

: internal cone (10b)

Figure 10.8.1 Cap Model

Notes/

(1) The compressibility indices and are related to the slopes of the virgin loading and unloading-reloading curves on the plot as measured in one-dimensional or hydrostatic consolidation tests as:

(11)

where

(12)

for virgin compression and swelling, respectively; void ratio at the reference mean stress .

(2) The following variations of the elastic bulk and shear moduli with the mean stress p can be derived from Eqs. 1 and 3:

(13)

where

(14)

Thus, with a suitable choice of parameters, the elastic shear modulus can be made either a constant or a linear function of p.

References / Bibliography

1. Borja, R.I., C. Tamagnini, and A. Amorosi, "Coupling Plasticiy and Energy-Conserving Elasticity Models for Clays", J. Geotech. Geoenvironmental Eng., ASCE, Vol. 123, No. 10, (1997), pp. 948–957.

2. Drucker, D.C. and V. Prager, "Soil Mechanics and Plastic Analysis or Limit Design", Quarterly of Applied Mathematics, Vol. 10, (1952), pp. 157–165.

3. Roscoe, K.H. and J.H. Burland, "On the Generalized Stress-Strain Behavior of Wet Clays," Engineering Plastaity, J. Heyman and F.A. Leckie, Eds., Cambridge University Press, London, England, (1968), pp. 535–609.

4. Scofield, A. and P. Wroth, Critical State Soil Mechanics, McGraw-Hill, Inc., New York, NY, (1968).

CAP

Material_name = CAP

Material_set_number = mset , etc...

The following data are used to describe the Cap model.

Note Variable Name Type Default Description

• Keywords Read Method

Material_set_number integer [1] Material set number Numat

(1) Hyperelastic_case integer [0] Hyperelastic free energy function

Solid_mass_density real [0.0] Solid mass density

Shear_modulus real [0.0] Shear modulus

Bulk_modulus real [0.0] Bulk modulus B

Friction_angle real [0.0] Friction angle > 0.0

Cohesion real [0.0] Cohesion c 0.0

Dilation_angle real [0.0] Dilation angle 0.0

Internal_cone list [on] Internal cone option

on / off

Tension_cutoff list [off] Tension cutoff option

on / off

(cont’d)

Notes/

(1) Only applicable to finite deformation case (see Section 9.2.1).

(cont'd)

Note Variable Name Type Default Description

(2) Porosity real [] Porosity n₀

(2) Void_ratio real [] Void ratio e₀

Ref_mean_stress real [0.0] Reference mean stress

(3) OCR real [1.0] Overconsolidation ratio OCR 1

(4) Compression_index real [0.0] Compression index C_c

(4) Swelling_index real [0.0] Swelling index C_s < C_c

Variable_shear_modulus real [0.0] Variable shear modulus coefficient

(cont’d)

Notes/

(2) Either the porosity or the void_ratio must be specified.

(3) The preconsolidation mean stress at the reference state is computed as:

(4) The compression index C_c and the swelling index C_s are defined as the slope of virgin loading and unloading-reloading curves, respectively, on the plot as measured in one-dimensional or hydrostatic consolidation tests, where void ratio; mean stress, viz.,

(cont'd)

Note Variable Name Type Default Description

(5) Initial_stress

initial_stress_11 real [0.0] Component 11 ()

initial_stress_22 real [0.0] Component 22 ()

initial_stress_33 real [0.0] Component 33 ()

initial_stress_12 real [0.0] Component 12 ()

initial_stress_23 real [0.0] Component 23 ()

initial_stress_31 real [0.0] Component 31 ()

(6) Fluid_mass_density real [0.0] Mass density (fluid phase)

(6) Fluid_bulk_modulus real [0.0] Fluid bulk modulus

Notes/

(5) Tensile stresses are positive. If all are equal to zero, set internally equal to .

(6) Only applicable to porous media models.

EXAMPLE

Define_Material_Model /

name = "cam_clay" /

number_of_material_sets = 1

stress_model /

material_name = cap /

material_type = nonlinear

material_set_number = 1 /

shear_modulus = 8.174e+02 /

bulk_modulus = 2.180e+03 /

friction_angle = 40.0 /

dilation_angle = 30.0 /

void_ratio = 0.889 /

ref_mean_stress = 30.0 /

OCR = 1 /

variable_shear_modulus = 0.0 /

compression_index = 0.40 /

swelling_index = 0.06 /

internal_cone = off

Notes . .