The modified Drucker-Prager/Cap plasticity/creep model:
is intended to model cohesive geological materials that exhibit
pressure-dependent yield, such as soils and rocks;
is based on the addition of a cap yield surface to the Drucker-Prager
plasticity model (Extended Drucker-Prager Models),
which provides an inelastic hardening mechanism to account for plastic
compaction and helps to control volume dilatancy when the material yields in
shear;
can be used in
Abaqus/Standard
to simulate creep in materials exhibiting long-term inelastic deformation
through a cohesion creep mechanism in the shear failure region and a
consolidation creep mechanism in the cap region;
provides a reasonable response to large stress reversals in the cap
region; however, in the failure surface region the response is reasonable only
for essentially monotonic loading.
The addition of the cap yield surface to the Drucker-Prager model serves two
main purposes: it bounds the yield surface in hydrostatic compression, thus
providing an inelastic hardening mechanism to represent plastic compaction; and
it helps to control volume dilatancy when the material yields in shear by
providing softening as a function of the inelastic volume increase created as
the material yields on the Drucker-Prager shear failure surface.
The yield surface has two principal segments: a pressure-dependent
Drucker-Prager shear failure segment and a compression cap segment, as shown in
Figure 1.
The Drucker-Prager failure segment is a perfectly plastic yield surface (no
hardening). Plastic flow on this segment produces inelastic volume increase
(dilation) that causes the cap to soften. On the cap surface plastic flow
causes the material to compact. The model is described in detail in
Drucker-Prager/Cap model for geological materials.
Failure Surface
The Drucker-Prager failure surface is written as
where
and
represent the angle of friction of the material and its cohesion, respectively,
and can depend on temperature, ,
and other predefined fields .
The deviatoric stress measure t is defined as
and
is the equivalent pressure stress,
is the Mises equivalent stress,
is the third stress invariant, and
is the deviatoric stress.
is a material parameter that controls the dependence of the yield surface on
the value of the intermediate principal stress, as shown in
Figure 2.
The yield surface is defined so that K is the ratio of
the yield stress in triaxial tension to the yield stress in triaxial
compression.
implies that the yield surface is the von Mises circle in the deviatoric
principal stress plane (the -plane), so that the
yield stresses in triaxial tension and compression are the same; this is the
default behavior in
Abaqus/Standard and
the only behavior available in
Abaqus/Explicit.
To ensure that the yield surface remains convex requires
.
Cap Yield Surface
The cap yield surface has an elliptical shape with constant eccentricity in
the meridional (p–t) plane (Figure 1)
and also includes dependence on the third stress invariant in the deviatoric
plane (Figure 2).
The cap surface hardens or softens as a function of the volumetric inelastic
strain: volumetric plastic and/or creep compaction (when yielding on the cap
and/or creeping according to the consolidation mechanism, as described later in
this section) causes hardening, while volumetric plastic and/or creep dilation
(when yielding on the shear failure surface and/or creeping according to the
cohesion mechanism, as described later in this section) causes softening. The
cap yield surface is
where
is a material parameter that controls the shape of the cap,
is a small number that we discuss later, and
is an evolution parameter that represents the volumetric inelastic strain
driven hardening/softening. The hardening/softening law is a user-defined
piecewise linear function relating the hydrostatic compression yield stress,
,
and volumetric inelastic strain (Figure 3):
The volumetric inelastic strain axis in
Figure 3
has an arbitrary origin:
is the position on this axis corresponding to the initial state of the material
when the analysis begins, thus defining the position of the cap
()
in
Figure 1
at the start of the analysis. The evolution parameter
is given as
The parameter
is a small number (typically 0.01 to 0.05) used to define a transition yield
surface,
so that the model provides a smooth intersection between the cap and failure
surfaces.
Defining Yield Surface Variables
You provide the variables d, ,
R, ,
,
and K to define the shape of the yield surface. In
Abaqus/Standard
,
while in
Abaqus/ExplicitK= 1 ().
If desired, combinations of these variables can also be defined as a tabular
function of temperature and other predefined field variables.
Defining Hardening Parameters
The hardening curve specified for this model interprets yielding in the hydrostatic pressure
sense: the hydrostatic pressure yield stress is defined as a tabular function of the
volumetric inelastic strain, and, if desired, a function of temperature and other
predefined field variables. The range of values for which is defined should be sufficient to include all values of effective
pressure stress that the material is subjected to during the analysis. If the value of the
volumetric inelastic strain becomes greater than the last user-specified value, Abaqus/Explicit extrapolates the stress-strain relationship based on the slope at the last point of the
curve.
Plastic Flow
Plastic flow is defined by a flow potential that is associated in the
deviatoric plane, associated in the cap region in the meridional plane, and
nonassociated in the failure surface and transition regions in the meridional
plane. The flow potential surface that we use in the meridional plane is shown
in
Figure 4:
it is made up of an elliptical portion in the cap region that is identical to
the cap yield surface,
and another elliptical portion in the failure and transition regions that
provides the nonassociated flow component in the model,
The two elliptical portions form a continuous and smooth potential surface.
Nonassociated Flow
Nonassociated flow implies that the material stiffness matrix is not symmetric and the
unsymmetric matrix storage and solution scheme should be used in Abaqus/Standard (see Defining an Analysis). If the
region of the model in which nonassociated inelastic deformation is occurring is confined,
it is possible that a symmetric approximation to the material stiffness matrix gives an
acceptable rate of convergence; in such cases the unsymmetric matrix scheme may not be
needed.
Calibration
At least three experiments are required to calibrate the simplest version of the Cap model: a
hydrostatic compression test (an odometer test is also acceptable) and either two triaxial
compression tests or one triaxial compression test and one uniaxial compression test (more
than two tests are recommended for a more accurate calibration).
The hydrostatic compression test is performed by pressurizing the sample
equally in all directions. The applied pressure and the volume change are
recorded.
The uniaxial compression test involves compressing the sample between two
rigid platens. The load and displacement in the direction of loading are
recorded. The lateral displacements should also be recorded so that the correct
volume changes can be calibrated.
Triaxial compression experiments are performed using a standard triaxial
machine where a fixed confining pressure is maintained while the differential
stress is applied. Several tests covering the range of confining pressures of
interest are usually performed. Again, the stress and strain in the direction
of loading are recorded, together with the lateral strain so that the correct
volume changes can be calibrated.
Unloading measurements in these tests are useful to calibrate the
elasticity, particularly in cases where the initial elastic region is not well
defined.
The hydrostatic compression test stress-strain curve gives the evolution of
the hydrostatic compression yield stress, ,
required for the cap hardening curve definition.
The friction angle, ,
and cohesion, d, which define the shear failure dependence
on hydrostatic pressure, are calculated by plotting the failure stresses of the
two triaxial compression tests (or the triaxial compression test and the
uniaxial compression test) in the pressure stress (p)
versus shear stress (q) space: the slope of the straight
line passing through the two points gives the angle
and the intersection with the q-axis gives
d. For more details on the calibration of
and d, see the discussion on calibration in
Extended Drucker-Prager Models.
R represents the curvature of the cap part of the yield
surface and can be calibrated from a number of triaxial tests at high confining
pressures (in the cap region). R must be between 0.0001
and 1000.0.
Abaqus/Standard Creep Model
Classical “creep” behavior of materials that exhibit plasticity according to
the capped Drucker-Prager plasticity model can be defined in
Abaqus/Standard.
The creep behavior in such materials is intimately tied to the plasticity
behavior (through the definitions of creep flow potentials and definitions of
test data), so cap plasticity and cap hardening must be included in the
material definition. If no rate-independent plastic behavior is desired in the
model, large values for the cohesion, d, as well as large
values for the compression yield stress, ,
should be provided in the plasticity definition: as a result the material
follows the capped Drucker-Prager model while it creeps, without ever yielding.
This capability is limited to cases in which there is no third stress invariant
dependence of the yield surface ()
and cases in which the yield surface has no transition region
().
The elastic behavior must be defined using linear isotropic elasticity (see
Defining Isotropic Elasticity).
Creep behavior defined for the modified Drucker-Prager/Cap model is active
only during soils consolidation, coupled temperature-displacement, and
transient quasi-static procedures.
Creep Formulation
This model has two possible creep mechanisms that are active in different
loading regions: one is a cohesion mechanism, which follows the type of
plasticity active in the shear-failure plasticity region, and the other is a
consolidation mechanism, which follows the type of plasticity active in the cap
plasticity region.
Figure 5
shows the regions of applicability of the creep mechanisms in
p–q space.
Equivalent Creep Surface and Equivalent Creep Stress for the Cohesion Creep Mechanism
Consider the cohesion creep mechanism first. We adopt the notion of the
existence of creep isosurfaces of stress points that share the same creep
“intensity,” as measured by an equivalent creep stress. Since it is desirable
to have the equivalent creep surface coincide with the yield surface, we define
the equivalent creep surfaces by homogeneously scaling down the yield surface.
In the p–q plane the equivalent creep
surfaces translate into surfaces that are parallel to the yield surface, as
depicted in
Figure 6.
Abaqus/Standard
requires that cohesion creep properties be measured in a uniaxial compression
test. The equivalent creep stress, ,
is determined as follows:
Abaqus/Standard
also requires that
be positive.
Figure 6
shows such an equivalent creep stress. A consequence of these concepts is that
there is a cone in p–q space inside
which creep is not active. Any point inside this cone would have a negative
equivalent creep stress.
Equivalent Creep Surface and Equivalent Creep Stress for the Consolidation Creep Mechanism
Next, consider the consolidation creep mechanism. In this case we wish to
make creep dependent on the hydrostatic pressure above a threshold value of
,
with a smooth transition to the areas in which the mechanism is not active
().
Therefore, we define equivalent creep surfaces as constant hydrostatic pressure
surfaces (vertical lines in the p–q
plane).
Abaqus/Standard
requires that consolidation creep properties be measured in a hydrostatic
compression test. The effective creep pressure, ,
is then the point on the p-axis with a relative pressure
of .
This value is used in the uniaxial creep law. The equivalent volumetric creep
strain rate produced by this type of law is defined as positive for a positive
equivalent pressure. The internal tensor calculations in
Abaqus/Standard
account for the fact that a positive pressure will produce negative (that is,
compressive) volumetric creep components.
Creep Flow
The creep strain rate produced by the cohesion mechanism is assumed to
follow a potential that is similar to that of the creep strain rate in the
Drucker-Prager creep model (Extended Drucker-Prager Models);
that is, a hyperbolic function:
This creep flow potential, which is continuous and smooth, ensures that
the flow direction is always uniquely defined. The function approaches a
parallel to the shear-failure yield surface asymptotically at high confining
pressure stress and intersects the hydrostatic pressure axis at a right angle.
A family of hyperbolic potentials in the meridional stress plane is shown in
Figure 7.
The cohesion creep potential is the von Mises circle in the deviatoric stress
plane (the -plane).
The creep strain rate produced by the consolidation mechanism is assumed
to follow a potential that is similar to that of the plastic strain rate in the
cap yield surface (Figure 8):
The consolidation creep potential is the von Mises circle in the
deviatoric stress plane (the -plane). The
volumetric components of creep strain from both mechanisms contribute to the
hardening/softening of the cap, as described previously. For details on the
behavior of these models refer to
Verification of creep integration.
Nonassociated Flow
The use of a creep potential for the cohesion mechanism different from the
equivalent creep surface implies that the material stiffness matrix is not
symmetric, and the unsymmetric matrix storage and solution scheme should be
used (see
Defining an Analysis).
If the region of the model in which cohesive inelastic deformation is occurring
is confined, it is possible that a symmetric approximation to the material
stiffness matrix will give an acceptable rate of convergence; in such cases the
unsymmetric matrix scheme may not be needed.
Specifying Creep Laws
The definition of the creep behavior is completed by specifying the
equivalent “uniaxial behavior”—the creep “laws.” In many practical cases the
creep laws are defined through user subroutine
CREEP because creep laws are usually of complex form to fit
experimental data. Data input methods are provided for some simple cases. To
avoid drawbacks of the time hardening and strain hardening forms, it is
recommended that you use the time power law model rather than the time
hardening form and the power law model rather than the strain hardening form,
as discussed below.
User Subroutine CREEP
User subroutine
CREEP provides a general capability for implementing
viscoplastic models in which the strain rate potential can be written as a
function of the equivalent stress and any number of “solution-dependent state
variables.” When used in conjunction with these materials, the equivalent
cohesion creep stress, ,
and the effective creep pressure, ,
are made available in the routine. Solution-dependent state variables are any
variables that are used in conjunction with the constitutive definition and
whose values evolve with the solution. Examples are hardening variables
associated with the model. When a more general form is required for the stress
potential, user subroutine
UMAT can be used.
Time Hardening Form
With respect to the cohesion mechanism, the "time hardening" form is
available
where
is the equivalent creep strain rate;
is the equivalent cohesion creep stress;
t
is the total or the creep time; and
A, n, and
m
are user-defined creep material parameters specified as functions of
temperature and field variables.
In using this form with the consolidation mechanism,
can be replaced by ,
the effective creep pressure, in the above relation.
Strain Hardening Form
For the cohesion mechanism the "strain hardening" form is
In using this form with the consolidation mechanism,
can be replaced by ,
the effective creep pressure, in the above relation.
For physically reasonable behavior A and
n must be positive and .
Time Power Law Model
The time power law model has the following form:
where
and
are defined above; and ,
,
,
and
are material parameters.
The model is equivalent to the time hardening form. It is
recommended that you use the time power law model when the value of the
parameter
is very small ().
In this case the equivalent time power law model is obtained by setting
,
keeping the parameters
and
unchanged, and setting
to an arbitrary value greater than zero (typically,
is set to one).
Power Law Model
The power law model has the following form:
where ,
and
are defined above; and ,
,
and
are material parameters.
This model is equivalent to the strain hardening form. It
is recommended that you use the power law model when the value of the parameter
is very small ().
In this case the equivalent power law model is obtained by setting
,
keeping the parameters
and
unchanged, and setting
to an arbitrary value greater than zero (typically,
is set to one).
Singh-Mitchell Law
A second cohesion creep law available as data input is a variation of the
Singh-Mitchell law:
where ,
t, and
are defined above and A, ,
,
and m are user-defined creep material parameters specified
as functions of temperature and field variables. For physically reasonable
behavior A and
must be positive, ,
and
should be small compared to the total time.
In using this variation of the Singh-Mitchell law with the consolidation
mechanism,
can be replaced by ,
the effective creep pressure, in the above relation.
Time-Dependent Behavior
In the time hardening form, the time power law model, and
the Singh-Mitchell law model, the total time or the creep time can be used. The
total time is the accumulated time over all general analysis steps. The creep
time is the sum of the times of the procedures with time-dependent material
behavior. If the total time is used, it is recommended that small step times
compared to the creep time be used for any steps for which creep is not active
in an analysis; this is necessary to avoid changes in hardening behavior in
subsequent steps.
Numerical Difficulties
Depending on the choice of units for the creep laws described above, the
value of A may be very small for typical creep strain
rates. If A is less than 10−27, numerical
difficulties can cause errors in the material calculations. Therefore, use
another system of units or use the time power law or power law model to avoid
such difficulties in the calculation of creep strain increments.
Creep Integration
Abaqus/Standard
provides both explicit and implicit time integration of creep and swelling
behavior. The choice of the time integration scheme depends on the procedure
type, the parameters specified for the procedure, the presence of plasticity,
and whether or not a geometric linear or nonlinear analysis is requested, as
discussed in
Rate-Dependent Plasticity: Creep and Swelling.
Initial Conditions
The initial stress at a point can be defined (see Defining Initial Stresses). If such a stress point lies outside the initially defined cap or transition yield
surfaces and under the projection of the shear failure surface in the
p–t plane (illustrated in Figure 1), Abaqus will try to adjust the initial position of the cap to make the stress point lie on the
yield surface and a warning message will be issued. If the stress point lies outside the
Drucker-Prager failure surface (or above its projection), an error message will be issued
and execution will be terminated.
Elements
The modified Drucker-Prager/Cap material behavior can be used with plane
strain, generalized plane strain, axisymmetric, and three-dimensional solid
(continuum) elements. This model cannot be used with elements for which the
assumed stress state is plane stress (plane stress, shell, and membrane
elements).
Equivalent plastic strains for all three possible yield/failure surfaces
(Drucker-Prager failure surface - PEQC1, cap surface - PEQC2, and transition surface - PEQC3) and the total volumetric inelastic strain (PEQC4). For each yield/failure surface, the equivalent plastic strain
is
where
is the corresponding rate of plastic flow. The total volumetric inelastic
strain is defined as
CEEQ
Equivalent creep strain produced by the cohesion creep mechanism, defined as
where
is the equivalent creep stress.
CESW
Equivalent creep strain produced by the consolidation creep mechanism,
defined as ,
where
is the equivalent creep pressure.