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was determined. In uniaxial compression \begin{array} { r } { p = \frac { 1 } { 3 } \bar { \sigma } ^ { c r } } \end{array} ; therefore, the uniaxial compression test line has a slope of 1/3. This approach has several consequences. One is that the cohesion creep strain rate is a function of both q and p . This allows the determination of realistic material properties in cases in which, due to high hydrostatic pressures, q is very high. If one looks at the yield strength of the material in this region to be a composite of cohesion strength and friction strength, this model corresponds to cohesion-determined creep. As a result, there is a cone in p – q space inside which there is no cohesion creep.
Next consider the consolidation creep mechanism. In this case we wish to make creep dependent on the hydrostatic pressure above a threshold value of p _ { a } , with a smooth transition to the areas in which the mechanism is not active ( p < p _ { a } ) . Therefore, we define equivalent creep surfaces as constant pressure surfaces. In the p – q plane that translates into vertical lines. ABAQUS requires that consolidation creep properties be measured in a hydrostatic compression test. The effective creep pressure, \bar { p } ^ { c r } , is then the point on the p-axis with a relative pressure
Equation 4.4.4-2
\bar {p} ^ {c r} = p - p _ {a}.
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 will account for the fact that a positive pressure will produce negative (that is, compressive) volumetric creep components.
Creep flow rule
The creep flow rules are derived from creep potentials, G ^ { c r } , in such a way that
Equation 4.4.4-3
d \pmb {\varepsilon} ^ {c r} = \frac {d \bar {\pmb {\varepsilon}} ^ {c r}}{f ^ {c r}} \frac {\partial G ^ {c r}}{\partial \pmb {\sigma}},
where d \bar { \varepsilon } ^ { c r } is the equivalent creep strain rate, which must be work conjugate to the equivalent creep stress:
\begin{array}{l} d \bar {\varepsilon} ^ {c r} = \left| d \varepsilon_ {1 1} ^ {c r} \right| \quad \text { in the uniaxial compression case }, \\ = \left| d \varepsilon_ {v o l} ^ {c r} \right| \quad \text { in the volumetric compression case. } \\ \end{array}
Since d \varepsilon ^ { c r } is obviously work conjugate to { \pmb \sigma } , f ^ { c r } is a proportionality factor defined by
Equation 4.4.4-4
f ^ {c r} = \frac {1}{\bar {\sigma} ^ {c r}} \pmb {\sigma}: \frac {\partial G ^ {c r}}{\partial \pmb {\sigma}},
with
\bar { \sigma } ^ { c r } = \rvert \sigma _ { 1 1 } \lvert in the uniaxial compression case ;
= p in the volumetric compression case:
Cohesion creep
For the cohesion mechanism the creep potential is assumed to follow the same potential as the creep strain rate in the Drucker-Prager creep model (``Models for granular or polymer behavior,'' Section 4.4.2); 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 4.4.4-7:
Equation 4.4.4-5
G _ {s} ^ {c r} = \sqrt {(0 . 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \tan \beta) ^ {2} + q ^ {2}} - p \tan \beta ,
where d is the material cohesion.
Figure 4.4.4-7 Creep potentials: cohesion mechanism.
text_image
q Δε^cr β Δε^cr similar hyperboles m a t e r i a l p o i n t p_a p
The equivalent cohesion creep strain rate is then determined from the uniaxial law:
\Delta \bar {\varepsilon} _ {s} ^ {c r} = h _ {s} (\bar {\sigma} ^ {c r}, \bar {\varepsilon} _ {s} ^ {c r}, \theta , f ^ {\alpha}).
Equation 4.4.4-1, Equation 4.4.4-3, and Equation 4.4.4-5 produce the flow rule for this mechanism
\Delta \pmb {\varepsilon} _ {s} ^ {c r} = \frac {\Delta \overline {{\varepsilon}} _ {s} ^ {c r}}{f ^ {c r}} \left(\frac {q}{\sqrt {(0 . 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \tan \beta) ^ {2} + q ^ {2}}} \mathbf {n} + \frac {1}{3} \tan \beta \mathbf {I}\right),
where
\mathbf {n} = \frac {\partial q}{\partial \pmb {\sigma}} = \frac {3}{2} \frac {\pmb {S}}{q},
and
f ^ {c r} = \frac {\bar {\sigma} ^ {c r}}{\sqrt {(0 . 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \tan \beta) ^ {2} + (\bar {\sigma} ^ {c r}) ^ {2}}} - \frac {1}{3} \tan \beta .
The proportionality factor, f ^ { c r } , is not a constant in this model. Its expression indicates that it will become negative if
\bar {\sigma} ^ {c r} < 0. 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \frac {\frac {1}{3} (\tan \beta) ^ {2}}{\sqrt {1 - (\frac {1}{3} \tan \beta) ^ {2}}}.
It turns out that below this stress level, which for typical materials will be very low, the stress vector and the normal to the creep potential are pointing in opposite directions:
\pmb {\sigma}: \frac {\partial G _ {s} ^ {c r}}{\partial \pmb {\sigma}} < 0,
which is equivalent to
q - p \tan \beta < q - \frac {q ^ {2}}{\sqrt {(0 . 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \tan \beta) ^ {2} + q ^ {2}}}.
Thus, there is a small zone just outside the "no creep" cone for which this is the case. Consequently, creep data obtained within this zone should show a creep strain rate in the opposite direction from the applied stress at very low stress levels, which will usually not be the case. To overcome this difficulty, ABAQUS will modify the creep data entered such that f ^ { c r } \ge 0 . 1 . Therefore, you would not expect correspondence between calculated creep strains and measured creep properties in a region defined by
\bar {\sigma} ^ {c r} < 0. 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \frac {(0 . 1 + \frac {1}{3} \tan \beta) \tan \beta}{\sqrt {1 - (0 . 1 + \frac {1}{3} \tan \beta) ^ {2}}}.
This modification is usually not significant, since typical creep analyses have loads that are applied quickly, followed by long-term creep. Hence, the stress level for most of the analysis will usually be well beyond the modified zone.
An example of "slow" loading in which the approximation is visible is included in ``Verification of creep integration,'' Section 3.2.6 of the ABAQUS Benchmarks Manual. As is clear in the example, the effect of the approximation is small in spite of the fact that the load is ramped up over the step.
The equivalent cohesion creep strain rate is a function of both q and p through \bar { \boldsymbol { \sigma } } ^ { c r } . The creep potential is the von Mises circle in the deviatoric stress plane (the ¦-plane). Although creep flow is associated in the deviatoric stress plane, the use of a creep potential different from the equivalent creep surface implies that creep flow is nonassociated.
Consolidation creep
For the consolidation mechanism the creep potential is derived from the plastic potential of the cap zone (Figure 4.4.4-8):
Equation 4.4.4-6
G _ {c} ^ {c r} = \sqrt {(p - p _ {a}) ^ {2} + (R q) ^ {2}}.
Recall that this mechanism is active only if p \geq p _ { a } .
Figure 4.4.4-8 Creep potentials: consolidation mechanism.
text_image
q material point Δε^cr β similar ellipses p_a Δε^cr p
The equivalent consolidation creep strain rate is then determined from the uniaxial law
\Delta \bar {\varepsilon} _ {c} ^ {c r} = h _ {c} (\bar {p} ^ {c r}, \bar {\varepsilon} _ {c} ^ {c r}, \theta , f ^ {\alpha}).
Equation 4.4.4-3 and Equation 4.4.4-4 produce f ^ { c r } = 1 ; and Equation 4.4.4-2, Equation 4.4.4-3, and Equation 4.4.4-6 produce the flow rule for this mechanism:
\Delta \varepsilon_ {c} ^ {c r} = \frac {\Delta \bar {\varepsilon} _ {c} ^ {c r}}{G _ {c} ^ {c r}} (R ^ {2} q \mathbf {n} - \frac {1}{3} (p - p _ {a}) \mathbf {I}).
Note that there is an equivalent pressure stress, p¹, work conjugate of the equivalent consolidation creep strain, which is different from the effective creep pressure, \bar { p } ^ { c r } . Such equivalent pressure stress is given by
\bar {p} = \frac {R ^ {2} q ^ {2} + p (p - p _ {a})}{G _ {c} ^ {c r}}
and has the characteristic that it reduces to the pressure in a hydrostatic compression test.
The creep potential is the von Mises circle in the deviatoric stress plane (the ¦-plane). Creep flow is nonassociated in this mechanism.
This formulation is quite simplistic and ignores the effects of q on the creep function, h _ { c } . The two creep mechanisms operate independently from each other. This implies that h _ { s } does not depend on \bar { \varepsilon } _ { c } ^ { c r } and that h _ { c } does not depend on \bar { \mathcal { E } } _ { s } ^ { c r } . The only cross effects between both mechanisms are obtained through the dependency of p _ { a } on the volumetric creep from any of them.
4.4.5 Mohr-Coulomb model
The Mohr-Coulomb failure or strength criterion has been widely used for geotechnical applications. Indeed, a large number of the routine design calculations in the geotechnical area are still performed using the Mohr-Coulomb criterion.
The Mohr-Coulomb criterion assumes that failure is controlled by the maximum shear stress and that this failure shear stress depends on the normal stress. This can be represented by plotting Mohr's circle for states of stress at failure in terms of the maximum and minimum principal stresses. The Mohr-Coulomb failure line is the best straight line that touches these Mohr's circles ( Figure 4.4.5-1). Thus, the Mohr-Coulomb criterion can be written as
\tau = c - \sigma \tan \phi ,
where ¿ is the shear stress, ¾ is the normal stress (negative in compression), c is the cohesion of the material, and Á is the material angle of friction.
Figure 4.4.5-1 Mohr-Coulomb failure criterion.
text_image
τ (σ,τ) s = σ₁ - σ₃/2 c σ₁ σ₁ σ₃ σ₃ φ σₘ = σ₁ + σ₃/2
From Mohr's circle,
\tau = s \cos \phi ,
\sigma = \sigma_ {m} + s \sin \phi .
Substituting for \tau and \sigma , the Mohr-Coulomb criterion can be rewritten as
s + \sigma_ {m} \sin \phi - c \cos \phi = 0,
where
s = \frac {1}{2} (\sigma_ {1} - \sigma_ {3})
is half of the difference between the maximum and minimum principal stresses (and is, therefore, the maximum shear stress) and
\sigma_ {m} = \frac {1}{2} (\sigma_ {1} + \sigma_ {3})
is the average of the maximum and minimum principal stresses (the normal stress). Thus, unlike the Drucker-Prager criterion, the Mohr-Coulomb criterion assumes that failure is independent of the value of the intermediate principal stress. The failure of typical geotechnical materials generally includes some small dependence on the intermediate principal stress, but the Mohr-Coulomb model is generally considered to be sufficiently accurate for most applications. This failure model has vertices in the deviatoric stress plane (see Figure 4.4.5-2).
Figure 4.4.5-2 Mohr-Coulomb model in the deviatoric plane.
text_image
S₃ Mohr-Coulomb S₂ S₁ Drucker-Prager (Mises)
The constitutive model described here is an extension of the classical Mohr-Coulomb failure criterion. It is an elastoplastic model that uses a yield function of the Mohr-Coulomb form; this yield function includes isotropic cohesion hardening/softening. However, the model uses a flow potential that has a
hyperbolic shape in the meridional stress plane and has no corners in the deviatoric stress space. This flow potential is then completely smooth and, therefore, provides a unique definition of the direction of plastic flow.
Strain rate decomposition
An additive strain rate decomposition is assumed:
Equation 4.4.5-1
d \boldsymbol {\varepsilon} = d \boldsymbol {\varepsilon} ^ {e l} + d \boldsymbol {\varepsilon} ^ {p l},
where d" is the total strain rate, d \pmb { \varepsilon } ^ { e l } is the elastic strain rate, and d \varepsilon ^ { p l } is the inelastic (plastic) strain rate.
Elastic behavior
The elastic behavior is modeled as linear and isotropic.
Yield behavior
The Mohr-Coulomb criterion written above in terms of the maximum and minimum principal stresses can be written for general states of stress in terms of three stress invariants. These invariants are the equivalent pressure stress,
p = - \frac {1}{3} \operatorname{trace} (\boldsymbol {\sigma});
the Mises equivalent stress,
q = \sqrt {\frac {3}{2} (\mathbf {S} : \mathbf {S})},
where S is the stress deviator, defined as
\mathbf {S} = \boldsymbol {\sigma} + \mathbf {p I};
and the third invariant of deviatoric stress,
r = (\frac {9}{2} \mathbf {S} \cdot \mathbf {S}: \mathbf {S}) ^ {\frac {1}{3}}.
The Mohr-Coulomb yield surface is then written as
Equation 4.4.5-2
F = R _ {m c} q - p \tan \phi - c = 0,
where \phi ( \theta , f ^ { \alpha } ) is the friction angle of the material in the meridional stress plane, where µ is the
Mechanical Constitutive Theories
temperature and f ^ { \alpha } , \alpha = 1 , 2 . . : are other predefined field variables; c ( \bar { \varepsilon } ^ { p l } , \theta , f ^ { \alpha } ) represents the evolution of the cohesion of the material in the form of isotropic hardening (or softening); \bar { \varepsilon } ^ { p l } is the equivalent plastic strain, its rate defined by the plastic work expression
c \dot {\bar {\varepsilon}} ^ {p l} = \pmb {\sigma}: \dot {\pmb {\varepsilon}} ^ {p l};
and R _ { m c } is the Mohr-Coulomb deviatoric stress measure defined as
R _ {m c} (\Theta , \phi) = \frac {1}{\sqrt {3} \cos \phi} \sin \left(\Theta + \frac {\pi}{3}\right) + \frac {1}{3} \cos \left(\Theta + \frac {\pi}{3}\right) \tan \phi ,
where \Theta is the deviatoric polar angle (Chen and Han, 1988) defined as
\cos (3 \Theta) = \left(\frac {r}{q}\right) ^ {3}.
The friction angle of the material, \phi , also controls the shape of the yield surface in the deviatoric plane as shown in Figure 4.4.5-3. The range of values the friction angle can have is 0 ^ { \circ } \leq \phi < 9 0 ^ { \circ } . In the case of \phi = 0 ^ { \circ } the Mohr-Coulomb model reduces to the pressure-independent Tresca model with a perfectly hexagonal deviatoric section. In the case of \phi = 9 0 ^ { \circ } the Mohr-Coulomb model would reduce to the "tension cut-off" Rankine model with a triangular deviatoric section and R _ { m c } = \infty (this limiting case is not permitted within the Mohr-Coulomb model described here).
Figure 4.4.5-3 Mohr-Coulomb yield surface in meridional and deviatoric planes.
text_image
R_{mc}q φ c p Θ = π/3 Θ = 0 Mohr-Coulomb (φ = 20°) Tresca (φ = 0°) Rankine (φ = 90°) Θ = 4π/3 Drucker-Prager (Mises) Θ = 2π/3
Flow rule
Potential flow is assumed, so
Equation 4.4.5-3
d \varepsilon^ {p l} = \frac {d \bar {\varepsilon} ^ {p l}}{g} \frac {\partial G}{\partial \pmb {\sigma}},
where g can be written as
g = \frac {1}{c} \pmb {\sigma}: \frac {\partial G}{\partial \pmb {\sigma}}
and G is the flow potential, chosen as a hyperbolic function in the meridional stress plane and a smooth elliptic function in the deviatoric stress plane:
Equation 4.4.5-4
G = \sqrt {(\epsilon c | _ {0} \tan \psi) ^ {2} + (R _ {m w} q) ^ {2}} - p \tan \psi ,
where \psi ( \theta , f ^ { \alpha } ) is the dilation angle measured in the p { - } R _ { m w } q plane at high confining pressure; c | _ { 0 } = c | _ { \bar { \varepsilon } ^ { p l } = 0 } is the initial cohesion yield stress; and ² is a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). This flow potential, which is continuous and smooth in the meridional stress plane, ensures that the flow direction is defined uniquely in this plane. The function asymptotically approaches a linear flow potential at high confining pressure stress and intersects the hydrostatic pressure axis at 90°. A family of hyperbolic potentials in the meridional stress plane is shown in Figure 4.4.5-4.
Figure 4.4.5-4 Family of hyperbolic flow potentials in the meridional plane.
text_image
R_mw q d ε^pl ψ εC_l0 p
The flow potential is also continuous and smooth in the deviatoric stress plane (the ¦-plane); we adopt the deviatoric elliptic function used by Menétrey and Willam (1995):
R _ {m w} (\Theta , e) = \frac {4 (1 - e ^ {2}) \cos^ {2} \Theta + (2 e - 1) ^ {2}}{2 (1 - e ^ {2}) \cos \Theta + (2 e - 1) \sqrt {4 (1 - e ^ {2}) \cos^ {2} \Theta + 5 e ^ {2} - 4 e}} R _ {m c} (\frac {\pi}{3}, \phi),
where £ is the deviatoric polar angle defined previously, \begin{array} { r } { R _ { m c } ( \frac { \pi } { 3 } , \phi ) = ( 3 - \sin \phi ) / 6 } \end{array} cos \phi , and e is a
parameter that describes the "out-of-roundedness" of the deviatoric section in terms of the ratio between the shear stress along the extension meridian ( \Theta = 0 ) and the shear stress along the compression meridian \begin{array} { r } { ( \Theta = \frac { \pi } { 3 } ) } \end{array} . The elliptic function has the value \begin{array} { r } { R _ { m w } ( \Theta = 0 , e ) = R _ { m c } ( \frac { \pi } { 3 } , \phi ) / e } \end{array} along the extension meridian and has the value \begin{array} { r } { R _ { m w } \left( \Theta = \frac { \pi } { 3 } , e \right) = R _ { m c } \left( \frac { \pi } { 3 } , \phi \right) } \end{array} along the compression meridian; this ensures that the flow potential matches the yield surface at the triaxial compression and extension in the deviatoric plane provided that e is defined appropriately (see further discussion later). Although the elliptic function is defined only in the sector 0 \le \Theta \le \pi / 3 , the polar radius R _ { m w } ( \Theta , e ) extends to all polar directions 0 \leq \Theta \leq 2 \pi using the three-fold symmetry shown in Figure 4.4.5-5.
Figure 4.4.5-5 Menétrey-Willam flow potential in the deviatoric plane.
text_image
θ = 0 Rankine (e = 1/2) θ = π/3 Menetrey-Willam (1/2 < e ≤ 1) θ = 4π/3 Mises (e = 1) θ = 2π/3
By default, the out-of-roundedness parameter, e, is dependent on the friction angle \phi ; it is calculated by matching the flow potential to the yield surface in both triaxial tension and compression in the deviatoric plane:
e = \frac {3 - \sin \phi}{3 + \sin \phi}.
Alternatively, e can also be considered to be an independent material parameter; in this case the user can provide its value directly. Convexity and smoothness of the elliptic function requires that 1 / 2 < e \le 1 . The upper limit, e = 1 ( { \mathfrak { o r } } \phi = 0 ^ { \circ } ) , leads to \begin{array} { r } { R _ { m w } \left( \Theta , e = 1 \right) = R _ { m c } \left( \frac { \pi } { 3 } , \phi \right) } \end{array} , which describes the Mises circle in the deviatoric plane. The lower limit, e = 1 / 2 \ ( { \mathrm { o r } } \phi = 9 0 ^ { \circ } ) , leads to R _ { m w } \left( \Theta , e = 1 / 2 \right) = 2 R _ { m c } ( \textstyle { \frac { \pi } { 3 } } , \phi ) cos £ and would describe the Rankine triangle in the deviatoric plane (this limiting case is not permitted within the Mohr-Coulomb model described here).
Flow in the meridional stress plane can be close to associated when the angle of friction, \phi , and the angle of dilation, Ã, are equal and the eccentricity parameter, \epsilon , is very small; however, flow in this plane is, in general, nonassociated. Flow in the deviatoric stress plane is always nonassociated. Therefore, the use of this Mohr-Coulomb model generally requires the solution of nonsymmetric equations.