and Goodier 1970; Jaeger and Cook 1969), the stress
done with the following basic components (Wood
invariants are:
1990):
Elastic properties to define the recoverable de-
Ii = invariants of normal stress:
formation;
A mathematical surface to define the yield
I1 = σx + σy + σz = σ1 + σ2 + σ3
(1)
boundary between elastic behavior and plastic
material behavior;
(
)
I2 = - σy σz + σz σx + σx σy + τyz2
A plastic flow potential to mathematically de-
(2)
fine the plastic deformation (also called a plas-
+τzx + τxy = - (σ2 σ3 + σ3σ1 + σ1σ2 )
2
2
tic flow law); and
A hardening/softening rule defining the move-
ment (expansion or contraction) of the yield
I3 = σx σy σz + 2τyz τzx τxy - σx τyz2
surface during plastic deformation.
(3)
2
2
-σy τzx - σz τxy = σ1σ2 σ3
A good overview of the development of plasticity
theory and constitutive modeling of soil is given in
Ji = invariants of deviatoric or shear stress:
Scott (1985). Schofield (Schofield and Wroth 1968)
extended plasticity theory to the critical state concept,
J1 = sx + sy + sz
(4)
defining either contractile or dilatant deformation of
porous material as a function of its specific volume or
void ratio. In critical state theory this rule is devel-
(
) + (σz - σx )2 + (σx - σy )
J 2 = 1 σy - σz
2
2
6
oped around the concept of a "critical state," where
the plastic shearing deformation occurs at a constant
+τyz2 + τzx 2 + τxy2
volume. Perhaps the most famous critical state
model, the Camclay model, was developed based on
= 1 (σ2 - σ3 ) + (σ3 - σ1 ) + (σ1 - σ2 )
the behavior of clays. The concepts are equally appli-
2
2
2
(5)
6
cable to defining the shearing and volumetric behav-
ior of granular materials such as granular soils or
(
)
snow (Wood 1990). Although the concepts are appli-
= I12 + 3I2 / 3
cable for both cohesive and granular materials, the
behavior of the granular materials has not been ex-
plored as thoroughly, particularly regarding the influ-
J3 = sx sy sz + 2τyz τ zx τxy - s x τyz2
(6)
ence of deviatoric stress on the yield surface, which
-sy τzx 2 - sz τxy2
is less clearly defined in soils but may take on a much
different shape than the yield surface of metals.
Wood (1990) considers this a difference in detail
(
)
= 2I13 + 9I1I2 + 27I3 / 27
rather than a difference in concept.
The critical state class of model used in this study is
the crushable foam model, specifically designed for
where the components of stress deviation si are
highly compressible materials, is which a characteristic
of fresh snow. The modified DruckerPrager cap
si = σi - p for i = x, y, z, and i = 1, 2, 3.
(7)
model also has the features of a critical state model
(i.e. regions of constant volume shear deformation, and
Model parameters are defined in either the pres-
compactivedilatant flow). Both models use non-
suredeviatoric plane (also called the meridinal, pq,
associative flow (i.e. the flow potential is not associ-
or pt plane) or the pressurevolume plane (vp or v
ated with the yield surface), except on the cap surface
ln p plane) according to the following definitions.
of the DruckerPrager model.
1. Mean (total) normal stress, also called equiva-
lent pressure stress, or the octahedral normal stress,
Yield surface
which determines uniform compression or dilation:
The yield surface for both material models is de-
scribed in terms of stress invariant functions of the
p = σoct = 1 I1 .
(8)
normal σi and shear stress τij, where i and j are direc-
3
tions x, y, and z and represent principal stresses when
6
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