10.9 Stress_Model: Multi-Yield Elasto-Plastic
Models (Geomaterials)
The following hyperelastic stored energy function with uncoupled volumetric and deviatoric parts is employed:
where
In the small deformation case, the form of the material
constitutive tangent moduli tensor 
is given as follows:
in which 
= plastic modulus; 
and 
= symmetric
second-order tensors such that gives the direction of
plastic deformations, is the outer normal to
the active yield surface; 
= fourth-order tensor
of elastic moduli, assumed isotropic for the particular class of material
models implemented.
The plastic potential is selected such that:
Several material models have been implemented and may be selected by specifying the value of the control parameter Plasticity_sub_type as follows:
(1) Plasticity_sub_type = 1 to 5: pressure non-sensitive materials.
The yield function in this case is of the Von Mises type with:
where 
is the deviatoric
stress tensor, i.e.,
is the coordinate of
the center of the yield surface in the deviatoric stress space; and 
is the size of the yield
surface.
(2) Plasticity_sub_type = 8: pressure sensitive materials.
The yield function in this case is of the Drucker-Prager / Mohr_Coulomb type with (see Ref. [5]):
where
, 
attraction 
. The function 
determines the shape
of the cross-section on the deviatoric plane and
in which
and
material
parameter. For a Drucker-Prager circular
cone: 
, whereas for a round-cornered Mohr-coulomb cone:
where
friction angle.
Several different plastic potential functions may be selected by specifying the value of the Plastic_potential_code as follows:
• Plasticity_potential_code = 0:
The plastic potential in this case is selected as follows:
in which 
normalized stress
ratio, viz.,
with 
dilation stress ratio,
and 
dilation parameter
(see Ref. [6]) as:
in which 
damage_rate, and 
cumulative plastic
shear distortions, viz.,
with
plastic shear
distortion rate.
• Plastic_potential_code = 1:
The plastic potential in this case is selected as follows:
1. Compactive phase: 
2. Dilative phase: 
(loading case)
(unloading case)
where 
outer normal to the
yield surface.
As previously, 
dilatational parameter,
which is scaled according to the level of confinement as follows:
with
reference mean normal
stress.
A collection of nested yield
surfaces may be used. This allows for
the adjustment of the plastic hardening rule to any experimental hardening
data; for example, data obtained from axial or simple shear tests. It is assumed that the yield surfaces are all
similar, and that a plastic modulus (
) is associated with each one.
Several different plastic hardening rules may be selected by specifying the value of Plasticity_sub_type, as indicated below:
• Plasticity_sub_type= 1: Isotropic hardening rule
The yield surfaces in this case do not change position, but merely increase in size as loading proceeds.
• Plasticity_sub_type = 2: Isotropic hardening/softening rule
This case is a generalization of the
previous model in which softening starts to occur when the outermost yield
surface is reached. At this point, the
elasto-plastic shear modulus is set to be 
, and remains constant until .LE.
. Thereafter 
.
• Plasticity_sub_type = 3: Kinematic hardening rule
In this case, the yield surfaces do not change size, but are translated in stress space by the stress point.
• Plasticity_sub_type = 4: Kinematic
hardening/Isotropic softening rule
In this case a combination of
kinematic/isotropic hardening laws is used.
A nonlinear isotropic hardening/softening model is adopted in which a
saturation hardening/softening function of the exponential type is used as
follows:
where r = 
= reduction strength
ratio (r > 0), and = reduction_rate
are material
parameters; 
cumulative plastic
shear distortions.
• Plasticity_sub_type= 5: Kinematic hardening/Isotropic softening rule
The particular material model implemented in that option assumes cyclic degradation of the material properties according to the rule:
as observed in cyclic strain-controlled simple
shear soil tests; 
Shear stress
amplitude; 
Shear strain amplitude;
Number of cycles.
• Plasticity_sub_type = 8: Kinematic
hardening rule
A purely kinematic hardening is adopted for that model. The dependence of the moduli on the effective mean normal stress is assumed of the following form:
where 
, and 
power exponent, a
material constant (see e.g., Ref. [9]). The following may be used as estimates:
for cohesionless soils 
, and 
for cohesive soils.
References / Bibliography
1. Prevost, J.H.,"Plasticity Theory for Soil Stress-Strain Behavior," J. Eng. Mech. Div., ASCE, Vol. 104, No. EM5, (1978), pp. 1177-1194.
2. Prevost, J.H., and T.J.R. Hughes,"Finite Element Solution of Elastic-Plastic Boundary Value Problems," J. Appl. Mech., ASME, Vol. 48, No. 1, (1981), pp. 69-74.
3. Prevost, J.H.,"Nonlinear Transient Phenomena in Elastic-Plastic Solids," J. Eng. Mech. Div., ASCE, Vol. 108, No. EM6, (1982), pp. 1297-1311.
4. Prevost, J.H.,"Localization of
Deformations in Elastic-Plastic Solids," Int. J. Num. Meth.
5. Prevost, J.H.,"A Simple Plasticity
Theory for Frictional Cohesionless Soils," Int. J. Soil Dyn. Earth.
6. Popescu, R., and J.H. Prevost,
"Centrifuge Validation of a Numerical Model
for Dynamic Soil Liquefaction," Soil
Dyn. and Earth.
7. Prevost, J.H. and C.M. Keane, "Shear
Stress-Strain Curve Generation from Simple Material Parameters," J. Geotech.
8. Hayashi, H., M. Honda, T. Yamada, and F.
Tatsuoka, "Modeling of Nonlinear Stress Strain Relations of Sands for
Dynamic Response Analysis," Proceedings,
10th WCEE,
9. Richart, F.E., J.R. Hall and R.D. Woods, Vibrations of Soils and Foundations, Prentice-Hall, (1970).
MULTI_YIELD
Material_name = MULTI_YIELD Max_number_of_yield_surfaces = Nys_max
Material_set_number = mset , etc...
The maximum number of yield surfaces for all materials in the set must be provided following the material name.
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:
Mass_density real [0.0] Mass density 
(2) Shear_modulus real [0.0] Shear modulus 
(2) Bulk_modulus real [0.0] Bulk modulus 
Activation_time real [0.0] Time at which nonlinearities are activated.
(3) 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 (
)
(4) Solid_mass_density real [0.0] Mass density (Solid Phase) 
(4) Fluid_mass_density real [0.0] Mass density (Fluid Phase) 
(4) Fluid_bulk_modulus real [0.0] Fluid bulk modulus 
(4) Porosity real [0.0] Porosity 
Number_of_yield_surfaces integer [Nys_max] Number of yield surfaces 
0 and
Nys_max
Yield_type list [*] Yield function type
Mises
Drucker_Prager
Mohr_Coulomb
(cont'd)
(cont'd)
Note Variable Name Type Default Description
Plasticity_sub_type integer [3] Plasticity material sub-type 1 and 8
Principal_anisotropy integer [2] Principal cross-anisotropy direction.
• Plasticity_sub_type = 4
Reduction_ratio real [0.0] Reduction strength ratio
Reduction_rate real [0.0] Reduction
rate 
• Plasticity_sub_type = 8
Internal_cone list [on] Internal cone option (only applicable
to
on / off Drucker-Prager yield function type)
Plastic_potential_code integer [0] Plastic potential code 0 and 1
= 0: standard; =1: enhanced.
(5) Ref_mean_stress real [0.0] Reference mean normal stress 
> 0.0
Power_exponent real [0.0] Power exponent 
> 0.0
Cohesion real [0.0] Cohesive coefficient 
0.0
Friction_angle_comp real [0.0] Ultimate friction angle in compression
> 0.0
Friction_angle_ext real [
] Ultimate
friction angle in extension
> 0.0
Dilation_angle_comp real [0.0] Dilation angle in compression 
Dilation_angle_ext real [0.0] Dilation angle in extension 
(6) Dilatational_parameter_Xpp real [1.0] Dilatational parameter 
0.0
Max_dilatational_Xpp real [Xpp] Maximum dilatational parameter
0.0
Dilatational_ratio real [1.0] Dilatational ratio
Damage_rate real [0.0] Damage rate 
(cont'd)
(cont'd)
Note Variable Name Type Default Description
• Shear Stress-Strain Generation Data
Number_of_generation_pts integer [100] Number of generation points
Stress_driven list [on] Stress / strain driven option
on / off
Shear stress-strain generation data must follow
• List Read Method
Material data must follow in the form:
< m, Nys(m), IHyper(m),
G(m), B(m), (m), (m), (m), (m), Pf(m), cpt(m) >
if (Plasticity_sub_type le 5) then
< 
_max(m), 
_max(m), (m), xl(m), xu(m) >
if (Plasticity_sub_type eq 8) then
< c(m), p1(m), 
(m), 
_c(m), _e(m), Xpp(m), Xpp_comp(m), Xpp_ext(m) >
< 
_c(m), _e(m), 
(m), Slope(m), max_c(m), max_e(m),
_c(m), x1_c(m), xu_c(m), _e(m), x1_e(m), xu_e(m) >
< Stres(i, m), i = 1, 6) >
< terminate with a blank record >.
Notes/
(1) Only applicable to finite deformation case (see Section 9.2.1).
(2)---- For Plasticity_sub_type 
and 
are the elastic shear and bulk moduli at the reference mean
stress 
(see Note 5).
(3) Tensile stresses are positive.
(4) Only applicable to porous media models.
(5) The dependence of the elastic shear and bulk moduli on the (effective) mean normal stress is assumed of the following form:
(6) See
Ref. [6] for details.
Shear Stress-Strain Data Generation:
For the shear stress-strain curve
generation, given G1 = maximum
shear modulus, 
= maximum shear
stress, and 
= maximum shear
strain, two options are available as follows:
Option
1: Let y = 
/ (G1) and x = 
/ , then
where the parameter y1 is determined by requiring that at x =1, y = ymax as detailed in Ref. [7].
Option 2: Let y = / and x = / 
with 
/ G1,
then:
with
where xi , x1, and xu are material parameters as detailed in Ref [8].
•
Plasticity_sub_type = 1 to 5
Max_shear_stress real [0.0] Maximum shear stress > 0.0
Max_shear_strain real [0.06] Maximum shear strain > (/G1)
(7) Coefficient_alpha real [0.0] Generation coefficient 
0.0
Coefficient_x1 real [0.30] Generation coefficient x1 0.0
Coefficient_xu real [1.0] Generation coefficient xu 0.0
Notes/
(7) If = 0.0, the generation
option 1 is used by default.
For Plasticity_sub_type = 8, the shear
stress-strain data are generated at the reference mean normal stress p1. The
maximum shear stress at the reference (effective) mean normal stress p1 is computed as follows: Let: 
= (effective) vertical stress; 
= (effective) horizontal stress; and
following common usage in geotechnical engineering, assume that compressive
stresses are counted as positive. Then the mobilized friction angle 
is computed as:
sin = 
; a = c / tan c =
attraction
Note
that in the above expression > 0 is positive in
compressioon, and < 0 is negative in
extension. Let:
p
= 
= mean stress; q =
= shear stress
Initially
p1 = 
v (1 + 2K0) / 3 ; q1 = v (1 - K0) = 3p1(1
- K0) / (1 + 2K0)
and
at failure (ultimate state), the maximum shear stress (max) is
computed as max = |qmax|, with:
qmax = 2sinmax (a + a1) / (1 - sinmax (2S + 1/3)); a1
= p1 - Sq1
where
S = slope of axial stress path followed in the test (see Note 9); max = c in compression tests
and max = -e in extension tests,
respectively.
Note Variable Name Type Default Description
•
Plasticity_sub_type = 8
(8) Lateral_stress_coefficient real [v/(1-v)] Coefficient of lateral stress K0 0.0
(9) Axial_stress_path_slope real [0.0] Slope of axial stress path 0.0
Max_shear_strain_comp real [0.06] Max shear strain in compression maxc 0.0
Max_shear_strain_ext real [maxc] Max shear strain in extension maxe 0.0
(10) Coefficient_alpha_comp real [0.0] Generation coefficient in compression c 0.0
Coefficient_x1_comp real [0.30] Generation coefficient in compression x1c 0.0
Coefficient_xu_comp real [1.00] Generation coefficient in compression xuc 0.0
Coefficient_alpha_ext real [c] Generation
coefficient in extension e 0.0
Coefficient_x1_ext real [0.30] Generation coefficient in extension x1e 0.0
Coefficient_xu_ext real [1.0] Generation coefficient in extension xue 0.0
Notes/
(8) If K0 = 0.0, set
internally equal to elastic K0 = 
/ (1 - ), with = Poisson's ratio:
= (3B1 - 2G1)
/ 2(3B1 + G1) ; K0 = (3B1 - 2G1)
/ (3B1 + 4G1)
(9) In conventional drained axial compression/extension soil tests, Slope = Dp/Dq = 1/3.
(10) If c = 0.0, the
generation option 1 is used by default.
EXAMPLE
Stress_model /
material_name = multi_yield /
max_number_yield_surfaces = 20 /
material_type = nonlinear
Material_set_number = 1 /
shear_modulus = 3.00E7 /
bulk_modulus = 2.00E7 /
mass_density = 2.65E3 /
fluid_mass_density = 1.E3 /
fluid_bulk_modulus = 1.0E9 /
porosity = 0.43 /
plasticity_sub_type = 8 /
ref_mean_stress = 2.0E5 /
power_exponent = 0.5 /
number_yield_surfaces = 20 /
dilation_angle_comp = 30.0 /
dilation_angle_ext = 30.0 /
dilatational_parameter_Xpp = 1.0 /
friction_angle_comp = 30.0 /
friction_angle_ext = 30.0 /
lateral_stress_coefficient = 1.0 /
axial_stress_path_slope = 0.33 /
max_shear_strain_comp = 0.05 /
max_shear_strain_ext = 0.03 /
initial_stress_11 = -2.E5 /
initial_stress_22 = -2.E5 /
initial_stress_33 = -2.E5
Notes . .
Notes . .