10.12 Stress_Model: Isotropic Hypo Plastic Model

HYPO_PLASTIC

Material_name = HYPO_PLASTIC

Material_set_number = mset , etc...

The incremental constitutive equations for the hypoplastic model are written as:

where

= effective stress tensor

= deviatoric stress

= mean effective stress

The hypoplastic incremental equations are written as:

where

= strain

= shear strain

= volumetric deformation

The bulk and shear moduli depend on the current mean effective stress p according to:

where p₁ is a reference mean stress, and n the power exponent.

The constitutive equations must be integrated between t_k and t_k+1. Integration for the mean stress is straightforward and gives:

if

if

To integrate the deviatoric equation it is assumed that the shear strain tensor varies linearly between t_k and t_k+1. Then:

where is the rate of increase of , assumed constant during the time interval from t_k to t_k+1.

Integration of this equation gives:

The equations are valid either for initial loading or for unloading-reloading. The only difference is that for unloading-reloading the values of G₁ and B₁ are replaced by G₁^u and B₁^u, respectively.

Initial loading and unloading-reloading states are identified by the following conditions:

Initial loading: ,

Unloading:

Reloading: ,

Where p_max is the maximum mean effective stress reached during the loading history, up to time t_k.

Note Variable Name Type Default Description

• Keywords Read Method

Material_set_number integer [1] Material set number Numat

Mass_density real [0.0] Mass density

Shear_modulus real [0.0] Shear modulus G₁

Bulk_modulus real [0.0] Bulk modulus B₁

Unloading_shear_modulus real [0.0] Unloading shear modulus G₁^u

Unloading_bulk_modulus real [0.0] Unloading bulk modulus B₁^u

Ref_mean_stress real [0.0] Ref mean stress p₁

Power_exponent real [0.0] Power exponent n

(1) 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 ()

(2) Solid_mass_density real [0.0] Mass density (Solid Phase)

(2) Fluid_mass_density real [0.0] Mass density (Fluid Phase)

(2) Fluid_bulk_modulus real [0.0] Fluid bulk modulus

(2) Porosity real [0.0] Porosity

• List Read Method

Material data must follow in the form:

< m, G₁ (m), B₁ (m), G₁^u (m), B₁^u (m), p₁ (m), n (m), (m), (m), (m), (m), Pf(m) >

< (Stres(i, m), i = 1, 6) >

< terminate with a blank record >.

Notes/

(1) Tensile stresses are positive.

(2) Only applicable to porous media models.

Notes . .