Constitutive Models Relations between Stress and Strain
3.11 Critical State Models for Soils
consist of a two phase mixture of particles and water. They exhibit very complex behavior in
response to stress, and a number of different constitutive theories are used to
model them in fact, entire books are devoted to critical
state soil models. Here, we outline a
basic soil model known as `Cam-clay,’ developed at Cambridge (UK). It uses many of the concepts that are used
to model plastic deformation of metals, so you will find it helpful to review
Sections 3.6 and 3.7 before reading this one.
Features of the behavior of soils
1. Soils cannot withstand significant tensile stress: we
therefore focus on their response to combined pressure and shear loading.
2. The behavior of a soil is very sensitive to its water
content. Two types of experiment are
conducted on soils: in a `drained’ test, water is allowed to escape from the
specimen as it is compressed (so the water pressure is zero); in an `undrained’
test, the volume of the specimen (water + soil particles) is held fixed. In the latter test, the water pressure can
be measured by means of a manometer connected to the pressurized cell.
3. Under combined pressure and shear loading, soil
behaves like a frictional material. In a
drained test, the solid can support shear stresses without excessive deformation, where M is a material property (analogous to
friction coefficient). If the shear
stress reaches ,
the material collapses usually by shearing along one or more discrete
shear planes parallel to the maximum resolved shear stress. If the same test is conducted on an undrained
specimen, shear occurs earlier, because the water supports part of the
hydrostatic pressure. In this case
shear failure occurs when ,
where p is the applied hydrostatic
pressure, and is the water pressure.
4. If subjected to loads below those required to cause
catastrophic collapse, soils show a complicated behavior that resembles that of
a strain hardening metal except that soils compact in response to
combined shear and pressure, whereas metals do not. In addition, the strain
hardening occurs only as a result of the compaction: shear strain does not
increase the solid’s strength. For
example, the figure shows the response of a soil sample to a test where in
which specimen is subjected to a constant pressure, together with a steadily
increasing shear stress. The soil
accumulates a permanent shear strain, and also compacts. The strength of the solid increases up to
the limiting shear stress ,
at which point the compaction reaches a steady state, and the specimen
continues to deform at constant shear stress.
3.11.2 Constitutive equations for
constitutive equations for Cam-clay are very similar to the rate independent
plastic equations in Section 3.6. The
main concepts are,
1. Strain rate
decomposition into elastic and
decomposition into contributions from
the water pressure (or `pore pressure’) and from the pressure supported by the
soil particles. The pore pressure must
be calculated by modeling fluid seepage through the soil.
stress-strain law, which specifies
the elastic part of the strain in terms of stress;
4. A yield
criterion, which determines the magnitude of the plastic strain rate, given
the stresses and the resistance of the material to flow. Unlike metals, the yield criterion for a soil
is a function of the hydrostatic stress, or pressure, in addition to shear
stress. The yield criterion is expressed in terms of a State Variable which characterizes the resistance of the material
to flow (analogous to yield stress).
5. A Plastic flow
rule that specifies the ratios of the plastic strain components under
6. A hardening
law, which specifies how the state variable evolves with plastic straining
7. The yield surface, flow rule and hardening law also
define a critical state criterion for the solid. The critical state criterion specifies the
combination of stresses that lead to unconstrained collapse of the solid.
These are discussed in more
We assume small strains, so
shape changes are characterized by .
The strain is decomposed into elastic and plastic parts as
that the soil is subjected to a stress . The pressure is (note the negative sign). In general, part of this pressure is
supported by the water in the soil; while the rest is supported by the soil
particles themselves. The pressure is
decomposed into two parts
is the contribution to the pressure from the
soil particles and is the contribution from the water.
using the constitutive equation in a boundary value problem, the water pressure
must be calculated as a separate problem, in addition to solving the usual
mechanical field equations. Here, we
outline briefly a simple approximate
description of fluid seepage through a soil.
More general treatments are also available, which include nonlinear
versions of the flow law, finite strain effects, as well as the effects of
fluid absorption by the soil particles to form a gel, the tendency of soil to
absorb fluid due to capillarity, and the effects of partial soil saturation.
1. Fluid seepage through the soil is driven by gravity
and by fluid pressure variations. The
driving force is quantified by the piezometric
head, defined as
z is the height above some arbitrary
datum, is the fluid pressure (compressive
pressure is positive); is the fluid density; and g is the acceleration due to gravity.
2. The volume of material flowing through unit area of
solid in the direction per unit time obeys Darcy’s law
k is a material parameter, known as
the permeability of the medium.
The fluid itself
may be compressible, with bulk modulus
The fluid can be
absorbed in cavities in the soil. The
volume fraction of cavities n is
is the cavity volume in a volume of soil
(including both cavities and soil particles) dV.
5. At time t=0
the solid starts with some cavity volume fraction ;
this volume fraction evolves as the solid is deformed. Usually the dominant contribution to the
cavity volume change occurs as a result of plastic compaction of the soil (more
sophisticated treatments include an elastic contribution). The cavity volume fraction after the solid is
subjected to an infinitesimal plastic strain is
6. At time t=0,
a (possibly zero) fluid pressure acts on the solid; and for t>0 the values of either fluid
pressure, or volumetric flow rate must be specified on the boundary of the
Finally, the rate
of change of fluid pressure follows from conservation of fluid volume as
The elastic strains are
related to the stresses using the standard linear elastic stress-strain
law. The elastic strain is related to
where are the components of the elastic compliance
tensor. For the special case of an
isotropic material with Young’s modulus and Poisson’s ratio
criterion and critical state surface
yield criterion specifies the stresses that are required to cause plastic flow
in the soil. The concept is identical to
the yield criterion used in metal plasticity, except that, unlike metals,
hydrostatic pressure can cause yield in a soil.
The yield criterion is
is the pressure exerted by the stresses on the
where is the Von-Mises effective stress;
a material property, whose physical significance was described in Sect
3.10.1. Usually M<1.
4. a is
a state variable that quantifies the current yield strength of the soil. At
time t=0, the soil has some finite
which subsequently evolves with plastic straining, as described below.
yield criterion is sketched in principal stress space on the figure it resembles a football (if you are American or a rugby ball to the rest of the world) with
its axis parallel to the line . The shape of the football depends on M for M=1
it is a sphere; for M<1 it is
stretched parallel to its axis. The size
of the yield locus is determined by a.
yield criterion, together with the hardening law, also define a critical state surface, which
determines the stress where unrestricted shear deformation can occur at
constant shear stress (i.e. with zero hardening). If the stresses lie inside the critical state
surface (this is known as the `wet’ side of critical state), the material shows
stable strain hardening behavior. If the
stresses lie outside the critical state surface (known as the `dry’ side of the
critical state), the material softens with plastic straining, and so violates
the Drucker stability condition. Under
these conditions the material is unstable, and plastic strain tends to localize
into shear bands.
The critical state surface
for Camclay is
The material is stable for and unstable for . The critical state surface is sketched in the
figure: it is a cone, which cuts through the fattest part of the yield surface.
flow law specifies the plastic strain components under a multiaxial state of
stress. Like metal plasticity, the
Cam-clay model bases the flow law on the yield criterion, so that
where is a dimensionless constant that depends on
the increment of stress applied to the solid, and is proportional to the
plastic strain magnitude. The procedure to calculate is discussed in more detail below: if the
stress state lies inside the critical state surface ( ), then can be expressed in terms of the stress
increment and the fluid pressure increment applied
to the solid as
Here for and for ,
and c is a material property
governing the hardening rate, as defined below.
These expressions are valid only if the stress lies inside the critical
state surface. If the stress lies on or
outside the critical state surface then cannot be determined from the stress
soil becomes stronger if it is compacted to crush the soil particles
together. This is described in the
constitutive law by making the state variable a evolve with plastic straining in some appropriate way. A simple hardening law that captures the main
features of experiments is
c is a dimensionless material
property, which determines the hardening rate.
Notice that, in this law, hardening occurs only as a result of
compaction, and not as a result of shear deformation.
the plastic stress-strain relation
When using the constitutive
equation, the formulas outlined in the preceding sections must be combined to
predict the plastic strain resulting from an increment in stress and fluid pressure increment . This is done as follows:
Check the yield
criterion. If ,
the plastic strain is zero .
2. Check to see whether the stresses lie inside the
critical state surface. If the material behaves like a stable, strain
hardening plastic solid, and the plastic strain increment can be calculated by
following steps 3-5 below.
3. Check for elastic unloading. The solid will unload elastically, with ,
if the stress increment brings the stress below yield. This is the case
4. If plastic strain does occur, the yield criterion must
be satisfied throughout plastic straining.
This requires that
is straightforward to show that
The hardening law
and plastic flow rule give
combining 3-5 leads to
the stress state is at yield and also lies on the critical state the material behaves like a perfectly plastic
solid (with constant flow stress). In
1. The solid unloads elastically, with if
2. If the solid deforms
plastically, the stress state must satisfy
In this case the plastic
strain cannot be determined from the stress increment: any is admissible.
In a situation where the total strain of the solid is prescribed, the
plastic strain increment can be determined by first solving for the elastic
strain increment, and subtracting it from the total strain.
If the stress lies outside the critical state and is
at yield , the material softens. In this case it is
impossible to distinguish unambiguously between elastic unloading and plastic
flow accompanied by strain softening.
The deformation in this regime usually consists of intense plastic
shearing along one or more discrete shear bands, while the rest of the material
unloads elastically. Boundary value
problems with material behavior in the unstable regime are generally ill-posed
and cannot be solved uniquely. However, attempting to load a soil past the
critical state usually results in catastrophic collapse (such as a landslide),
so detailed solutions to boundary value problems in this regime are not of much
practical interest. The critical state
surface can be used as a failure criterion to avoid collapse.
3.11.3 Application of the critical state equations to
simple 2D loading
The constitutive equations for soils are complicated,
and a simple 2D example helps to interpret them. To this end, consider a solid
subjected to a 2D stress state of the form ,
as illustrated in the picture. Assume that the specimen is drained, so that the
water pressure . In addition, assume that at time t=0
the solid has strength .
this loading, the yield surface can be plotted in 2D, as a graph of the
critical combinations of p and q that cause yield, as shown in the picture
below. The yield surface is an ellipse,
with semi-axes a and Ma.
The critical state surface is a straight line with slope M.
can now examine the behavior of the solid as it is loaded. Consider first the response to a constant
pressure on the `wet’ side of critical state ,
together with a steadily increasing shear stress q. In this case
1. The solid first reaches yield when
2. If the shear stress is raised beyond yield, the solid
will deform plastically. Since the flow
law is derived from the yield criterion, the plastic strain direction is normal
to the yield surface. That is to say, if
the solid experiences a plastic shear strain and volumetric strain ,
the vector ( ) is normal to the yield surface, as shown in
3. On the `wet’ side of
critical state, the volumetric plastic strain component dv is always compressive.
This means the solid compacts, and its strength increases (recall that and during compaction).
4. As the yield surface
expands, the volumetric strain component associated with an increase in shear
stress decreases (remember that we assume a constant
pressure). The hardening rate
therefore decreases with strain, until the stress reaches the critical state. At this point ,
so there is no further hardening.
Next, consider behavior on the `dry’ side of
critical state. In this case
1. The solid first reaches yield when
2. As before, the direction of the plastic strain is
normal to the yield surface.
3. Notice that on the `dry’
side of critical state, the volumetric plastic strain component dv is always dilatational. This means its strength decreases with
plastic straining, as shown in the figure (recall that and during dilation).
4. The yield surface contracts
during plastic straining, and this process continues until the stress reaches
the critical state. At this point the
solid continues to deform at a constant shear stress.
3.11.4 Typical values of material properties for soils
Soil properties are highly variable, and for
accurate predictions you will need to measure directly the properties of the
soil you are intending to model. In
addition, soil models that are used in practice are somewhat more sophisticated
than the simplified version given here. As a rough guide, material properites
estimated from data in D.M. Wood, “Soil Behavior and Critical State Soil
Mechanics,” Cambridge University Press, Cambridge, 1990 are listed in the table
Water bulk modulus
Critical state constant