김경종
18 KiB
In this equation \beta = \beta ( \theta , f ^ { \alpha } ) is a user-specified constant that can be a function of temperature µ and other predefined field variables f ^ { \alpha } , \alpha = 1 , 2 . . . . This constant is used to modify the shape of the yield surface on the "wet" side of critical state, so the elliptic arc on the "wet" side of critical state has a different curvature from the elliptic arc used on the "dry" side: \beta = 1 on the "dry" side of critical state, while \beta < 1 in most cases on the "wet" side, as shown in Figure 4 . 4 . 3 { - } 4 . a ( \theta , f ^ { \alpha } ) defines the hardening of the plasticity model, and is the point on the p-axis at which the elliptic arcs of the yield surface intersect the critical state line, as indicated in Figure 4.4.3-4. M ( \theta , f ^ { \alpha } ) is the slope of the critical state line in the p { - } t plane (the ratio of t to p at critical state); and t = q / g _ { \mathrm { i } } , where g is used to shape the yield surface in the ¦ plane, and is defined as
g = \frac {2 K}{1 + K + (1 - K) (r / q) ^ {3}},
where K ( \theta , f ^ { \alpha } ) is a user-defined constant. If K = 1 , the yield surface does not depend on the third stress invariant, and the ¦-plane section of the yield surface is a circle: this choice gives the original form of the Cam-clay model. The effect of different values of K on the shape of the yield surface in the ¦-plane is shown in Figure 4.4.3-2.
Figure 4.4.3-5 Cam-clay surfaces in the deviatoric plane.
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S₃ t = ½ q [1 + ½ K - (1 - ½ K) (½ q)³] Curve | K a | 1.0 b | 0.8 S₁ S₂
To ensure convexity of the yield surface, 0 . 7 7 8 \leq K \leq 1 . 0 .
Associated flow is used with the modified Cam-clay plasticity model. The size of the yield surface is defined by a: the evolution of this variable, therefore, characterizes the hardening or softening of the material. It is observed experimentally that, during plastic deformation,
d e = - \lambda d (\ln p),
where ¸ is a constant. Integrating this equation, and using Equation 4.4.3-1, Equation 4.4.1-2, and Equation 4.4.1-4, we obtain
a = a _ {0} \exp \left[ (1 + e _ {0}) \frac {1 - J ^ {p l}}{\lambda - \kappa J ^ {p l}} \right],
Equation 4.4.3-4
where a _ { 0 } defines the position of a at the beginning of the analysis--the initial overconsolidation of the material. The value of a _ { 0 } can be given directly in the *CLAY PLASTICITY option or can be computed as
a _ {0} = \frac {1}{2} \exp \left(\frac {e _ {1} - e _ {0} - \kappa \ln p _ {0}}{\lambda - \kappa}\right),
where p _ { 0 } is the initial value of the equivalent pressure stress, and e _ { 1 } is the intercept of the virgin consolidation line with the void ratio axis in a plot of void ratio versus equivalent pressure stress, shown in Figure 4.4.3-6.
Figure 4.4.3-6 Assumed soil response in pure compression (exponential hardening/softening case).
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| In p | e, voids ratio | |------|---------------| | 0 | 0 | | 1 | -1 | | 2 | -2 | | 3 | -3 | | 4 | -4 | | 5 | -5 | | 6 | -6 | | 7 | -7 | | 8 | -8 | | 9 | -9 | | 10 | -10 | | 11 | -11 | | 12 | -12 | | 13 | -13 | | 14 | -14 | | 15 | -15 | | 16 | -16 | | 17 | -17 | | 18 | -18 | | 19 | -19 | | 20 | -20 |The evolution of the yield surface can alterthe yield stress in hydrostatic compression, p _ { c } , vely be defined as a piecewise linear function re and the corresponding volumetric plastic strain \varepsilon _ { \mathrm { v o l } } ^ { p l } (Figure 4.4.3-7):
p _ {c} = p _ {c} (\varepsilon_ {\mathrm{vol}} ^ {p l}).
The evolution parameter, a, is then given by
a = \frac {p _ {c}}{(1 + \beta)}.
Note that the volumetric plastic strain axis has an arbitrary origin: \varepsilon _ { \mathrm { v o l } } ^ { p l } | _ { 0 } is the position on this axis corresponding to the initial state of the material, thus defining the initial hydrostatic pressure, p _ { c } | _ { 0 } ; and, hence, the initial yield surface size, a _ { 0 } .
Figure 4.4.3-7 Piecewise linear hardening/softening curve.
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| -ε_vol^pl | p_c |
|---|---|
| 0 | p_c |
| ε_vol^pl | p_c |
| -ε_vol^pl | p_c |
ABAQUS checks that the initial effective stress state lies inside or on the initial yield surface. At any material point where the yield function is violated, a _ { 0 } is adjusted so that Equation 4.4.3-3 is satisfied exactly (and, hence, the initial stress state lies on the yield surface).
4.4.4 Drucker-Prager/Cap model for geological materials
The modified Drucker-Prager/Cap plasticity model in ABAQUS is intended for geological materials that exhibit pressure-dependent yield. The yield surface includes two main segments: a shear failure surface, providing dominantly shearing flow, and a "cap," which intersects the equivalent pressure stress axis (Figure 4.4.4-1).
Figure 4.4.4-1 Modified Drucker-Prager/Cap model: yield surfaces in the p { - } t plane.
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Transition surface, F_t Shear failure, F_s α (d+p_a tanβ) Cap, F_c d+p_a tanβ β d p_a R (d+p_a tanβ) p_b p
There is a transition region between these segments, introduced to provide a smooth surface. The cap 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 and transition yield surfaces.
The model uses associated flow in the cap region and nonassociated flow in the shear failure and transition regions. The model has been extended to include creep, with certain limitations that are outlined in this section. The creep behavior is envisaged as arising out of two possible mechanisms: one dominated by shear behavior and the other dominated by hydrostatic compression.
Strain rate decomposition
A linear strain rate decomposition is assumed, so
d \pmb {\varepsilon} = d \pmb {\varepsilon} ^ {e l} + d \pmb {\varepsilon} ^ {p l} + d \pmb {\varepsilon} ^ {c r},
where d" is the total strain rate, d \pmb { \varepsilon } ^ { e l } is the elastic strain rate, d \varepsilon ^ { p l } is the inelastic (plastic) time-independent strain rate, and d \pmb { \varepsilon } ^ { c r } is the inelastic (creep) time-dependent strain rate.
Elastic behavior
The elastic behavior can be modeled as linear elastic or by using the porous elasticity model including tensile strength, described in ``Porous elasticity,'' Section 4.4.1. If creep has been defined, the elastic behavior must be modeled as linear.
Plastic behavior
The yield/failure surfaces used with this model are written in terms of the three stress invariants: the equivalent pressure stress,
p = - \frac {1}{3} \mathrm{trace} (\pmb {\sigma});
the Mises equivalent stress,
q = \sqrt {\frac {3}{2} (\mathbf {S} : \mathbf {S})};
and the third invariant of deviatoric stress,
r = (\frac {9}{2} \mathbf {S} \cdot \mathbf {S}: \mathbf {S}) ^ {\frac {1}{3}},
where S is the stress deviator, defined as
\mathbf {S} = \pmb {\sigma} + p \mathbf {I}.
We also define the deviatoric stress measure
t = \frac {q}{2} \left[ 1 + \frac {1}{K} - \left(1 - \frac {1}{K}\right) \left(\frac {r}{q}\right) ^ {3} \right],
where K ( \theta , f ^ { \alpha } ) is a material parameter that may depend on temperature, µ, and other predefined fields f ^ { \alpha } , \alpha = 1 , 2 , \ldots . This measure of deviatoric stress is used because it allows matching of different stress values in tension and compression in the deviatoric plane, thereby providing flexibility in fitting experimental results and a smooth approximation to the Mohr-Coulomb surface. Since r / q = 1 in uniaxial tension, t = q / K in this case; since r / q = - 1 in uniaxial compression, t = q in that case. When K = 1 , the dependence on the third deviatoric stress invariant is removed; and the Mises circle is recovered in the deviatoric plane: t = q . Figure 4.4.4-2 shows the dependence of t on K. To ensure convexity of the yield surface, 0 . 7 7 8 \leq K \leq 1 . 0 .
Figure 4.4.4-2 Typical yield/flow surfaces in the deviatoric plane.
radar
| Curve | K |
|---|---|
| a | 1.0 |
| b | 0.8 |
With this expression for the deviatoric stress measure, the Drucker-Prager failure surface is written as
F _ {s} = t - p \tan \beta - d = 0,
where \beta ( \theta , f ^ { \alpha } ) is the material's angle of friction and d ( \theta , f ^ { \alpha } ) is its cohesion (see Figure 4.4.4-1).
The cap yield surface has an elliptical shape with constant eccentricity in the meridional ( p-t) plane (Figure 4.4.4-1) and also includes dependence on the third stress invariant in the deviatoric plane (Figure 4.4.4-2). The cap surface hardens or softens as a function of the volumetric plastic strain: volumetric plastic compaction (when yielding on the cap) causes hardening, while volumetric plastic dilation (when yielding on the shear failure surface) causes softening. The cap yield surface is written as
F _ {c} = \sqrt {(p - p _ {a}) ^ {2} + \left[ \frac {R t}{(1 + \alpha - \alpha / \cos \beta)} \right] ^ {2}} - R (d + p _ {a} \tan \beta) = 0,
where R ( \theta , f ^ { \alpha } ) is a material parameter that controls the shape of the cap, \alpha ( \theta , f ^ { \alpha } ) is a small number that is defined below, and p _ { a } is an evolution parameter that represents the volumetric plastic strain driven hardening/softening. The hardening/softening law is a user-defined piecewise linear function relating the hydrostatic compression yield stress, p _ { b } , and the corresponding volumetric inelastic (plastic and/or creep) strain, p _ { b } = p _ { b } ( \varepsilon _ { \mathrm { v o l } } ^ { i n } , \theta , f ^ { \alpha } ) (Figure 4.4.4-3), where \varepsilon _ { \mathrm { v o l } } ^ { i n } = \varepsilon _ { \mathrm { v o l } } ^ { p l } + \varepsilon _ { \mathrm { v o l } } ^ { c r } .
Figure 4.4.4-3 Typical Cap hardening.
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| x | p_b |
|---|---|
| 0 | 0 |
| 1 | 0.5 |
| 2 | 1 |
| 3 | 1.5 |
| 4 | 2.5 |
| 5 | 3.5 |
| 6 | 5 |
| 7 | 7 |
| 8 | 10 |
The evolution parameter, p _ { a } , , is defined as
p _ {a} = \frac {p _ {b} - R d}{(1 + R \tan \beta)}.
The parameter ® is a small number (typically 0.01 to 0.05) used to define a smooth transition surface between the shear failure surface and the cap:
F _ {t} = \sqrt {(p - p _ {a}) ^ {2} + \left[ t - (1 - \frac {\alpha}{\cos \beta}) (d + p _ {a} \tan \beta) \right] ^ {2}} - \alpha (d + p _ {a} \tan \beta) = 0.
Flow rule
Plastic flow is defined by a flow potential that is associated on the cap and nonassociated on the failure yield surface and transition yield surfaces. The nonassociated nature of these surfaces stems from the shape of the flow potential in the meridional plane. The flow potential surface in the meridional plane is shown in Figure 4.4.4-4.
Figure 4.4.4-4 Modified Drucker-Prager/Cap model: flow potential in the p { - } t plane.
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Similar ellipses Gₐ (Shear failure) G꜀ (cap) d + pₐ tanβ (1 + α - α secβ) (d + pₐ tanβ) R (d + pₐ tanβ) p
It is made up of an elliptical portion in the cap region that is identical to the cap yield surface:
G _ {c} = \sqrt {(p - p _ {a}) ^ {2} + \left[ \frac {R t}{1 + \alpha - \alpha / \cos \beta} \right] ^ {2}}
and another elliptical portion in the failure and transition regions that provides the nonassociated flow component in the model:
G _ {s} = \sqrt {[ (p _ {a} - p) \mathrm{tan} \beta ] ^ {2} + \left[ \frac {t}{1 + \alpha - \alpha / \mathrm{cos} \beta} \right] ^ {2}}.
The two elliptical portions, G _ { c } and G _ { s } , form a continuous and smooth potential surface.
Nonassociated flow implies that the material stiffness matrix is not symmetric, so the unsymmetric matrix scheme should be used. In ABAQUS/Standard this requires the use of UNSYMM=YES on the *STEP option. However, 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 will give an acceptable convergence rate: in such cases the UNSYMM parameter may not be needed.
Creep model
Classical "creep" behavior of materials that also exhibit plastic behavior according to the modified Drucker-Prager/Cap model can be defined through the *CAP CREEP option.
The creep behavior in such materials is intimately tied to the plasticity behavior (through the definition of creep flow potentials and test data), so it is necessary to have the plasticity options *CAP
PLASTICITY and *CAP HARDENING present as part of the material behavior definition. The elastic part of the behavior must be linear.
The rate-independent part of the plastic behavior is limited by the following restrictions:
® = 0--that is, no transition zone is allowed;
K=1--that is, no third stress invariant effects are taken into account.
In such a case, the deviatoric stress measure t is equal to the Mises equivalent stress, q, and the yieldt surface has a von Mises (circular) section in the deviatoric stress plane.
Creep behavior
The built-in ABAQUS creep laws or uniaxial laws defined through user subroutine CREEP can be used. The integration of the creep strain rate is first attempted explicitly, as described in
``Rate-dependent metal plasticity (creep),'' Section 4.3.4. The integration is done by the backward Euler method (as described in ``Rate-dependent metal plasticity (creep),'' Section 4.3.4) if the stability limit is exceeded, a geometrically nonlinear analysis is being performed, or plasticity becomes active.
In this model we assume the existence of two separate and independent creep mechanisms. One is a cohesion mechanism, which operates similarly to the Drucker-Prager creep model described in
``Models for granular or polymer behavior,'' Section 4.4.2. The other is a consolidation mechanism, which operates similarly to the cap zone plasticity. We then have
d \varepsilon^ {c r} = d \varepsilon_ {s} ^ {c r} + d \varepsilon_ {c} ^ {c r},
where d \pmb { \varepsilon } _ { s } ^ { c r } is the creep strain rate due to the cohesion mechanism and d \pmb { \varepsilon } _ { c } ^ { c r } is the creep strain rate due to the consolidation mechanism.
As described above, the cap surface hardens or softens as a function of the volumetric plastic strain and volumetric creep strain: volumetric inelastic compaction (when yielding on the cap or creeping through the consolidation mechanism) causes hardening, while volumetric plastic dilation (when yielding on the shear failure surface or creeping through the cohesion mechanism) causes softening. The separation between the two yield surfaces and the dominant regions for the two creep mechanisms are defined by the evolution parameter, p _ { a } , which relates to the user-defined hydrostatic compression yield stress, p _ { b } = p _ { b } ( \varepsilon _ { \mathrm { v o l } } ^ { p l } + \varepsilon _ { \mathrm { v o l } } ^ { c r } , \theta , f ^ { \alpha } ) (Figure 4.4.4-3).
The cohesion mechanism is active for all stress states that have a positive equivalent creep stress as explained below. The consolidation mechanism is active for all stress states in which the pressure is larger than p _ { a } . Figure 4.4.4-5 illustrates the active regions in this formulation.
Figure 4.4.4-5 Regions of activity of cohesion and consolidation creep mechanisms.
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cohesion and consolidation creep (d + p_a tanβ) cohesion creep no creep consolidation creep β p_a p R (d + p_a tanβ)
We adopt the notion of the existence of creep isosurfaces (or equivalent creep surfaces) of stress points that share the same creep "intensity," as measured by an equivalent creep stress. Consider the cohesion creep mechanism first. When the material plastifies, the equivalent creep surface should coincide with the yield surface; therefore, we define the equivalent creep surfaces by homogeneously scaling down the yield surface. In the p – q plane that translates into parallels to the yield surface, as depicted in Figure 4.4.4-6. ABAQUS requires that cohesion creep properties be measured in a uniaxial compression test.
Figure 4.4.4-6 Equivalent creep stress for cohesion creep.
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q 1 yield surface β 3 material point equivalent creep surface σ cr no creep p
The equivalent creep stress, { \bar { \sigma } } ^ { c r } , is determined as the intersection of the equivalent creep surface with the uniaxial compression curve. As a result,
Equation 4.4.4-1
\bar {\sigma} ^ {c r} = \frac {(q - p \tan \beta)}{(1 - \frac {1}{3} \tan \beta)},
where \beta ( \theta , f ^ { \alpha } ) is the material angle of friction. Figure 4.4.4-6 shows how the equivalent creep stress