김경종
381 lines
21 KiB
Markdown
381 lines
21 KiB
Markdown
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Unloading measurements in these tests are useful to calibrate the elasticity, particularly in cases where the initial elastic region is not well defined.
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The hydrostatic compression test stress-strain curve gives the evolution of the hydrostatic compression yield stress, $p _ { b } ( \varepsilon _ { v o l } ^ { p l } )$ , required for the cap hardening curve definition.
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The friction angle, $\beta ,$ , 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 $\beta$ and the intersection with the q-axis gives $\pmb { d } .$ For more details on the calibration of $\beta$ and $d ,$ see the discussion on calibration in “Extended Drucker-Prager models,” Section 23.3.1.
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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.
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# Abaqus/Standard creep model
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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, $p _ { b }$ , 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” in “Linear elastic behavior,” Section 22.2.1).
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Creep behavior defined for the modified Drucker-Prager/Cap model is active only during soils consolidation, coupled temperature-displacement, and transient quasi-static procedures.
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# Creep formulation
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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 23.3.2–5 shows the regions of applicability of the creep mechanisms in p–q space.
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# Equivalent creep surface and equivalent creep stress for the cohesion creep mechanism
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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 23.3.2–6.
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<details>
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<summary>text_image</summary>
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cohesion and consolidation creep
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(d+pₐ,tanβ)
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cohesion creep
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no creep
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consolidation creep
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β
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pₐ
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R(d+pₐ,tanβ)
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</details>
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Figure 23.3.2–5 Regions of activity of creep mechanisms.
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<details>
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<summary>text_image</summary>
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q
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1
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yield surface
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β
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3
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material point
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equivalent creep
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surface
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σcr
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no creep
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p
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</details>
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Figure 23.3.2–6 Equivalent creep stress for cohesion creep.
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Abaqus/Standard requires that cohesion creep properties be measured in a uniaxial compression test. The equivalent creep stress, $\bar { \sigma } ^ { c r }$ , is determined as follows:
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$$
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\bar {\sigma} ^ {c r} = \frac {(q - p \tan \beta)}{(1 - \frac {1}{3} \tan \beta)}.
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$$
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Abaqus/Standard also requires that $\bar { \sigma } ^ { c r }$ be positive. Figure 23.3.2–6 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.
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Equivalent creep surface and equivalent creep stress for the consolidation creep mechanism
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Next, consider the consolidation creep mechanism. In this case we wish to make creep dependent on the hydrostatic pressure above a threshold value of ${ \dot { p } } _ { a }$ , with a smooth transition to the areas in which the mechanism is not active $\left( p < p _ { a } \right)$ . 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, $\bar { p } ^ { c r }$ , is then the point on the p-axis with a relative pressure of $\bar { p } ^ { c r } = p - p _ { a }$ . 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/Standard account for the fact that a positive pressure will produce negative (that is, compressive) volumetric creep components.
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# Creep flow
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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,” Section 23.3.1); that is, a hyperbolic function:
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$$
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G _ {s} ^ {c r} = \sqrt {(0 . 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \tan \beta) ^ {2} + q ^ {2}} - p \tan \beta .
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$$
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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 23.3.2–7. The cohesion creep potential is the von Mises circle in the deviatoric stress plane (the -plane).
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Abaqus/Standard protects for numerical problems that may arise for very low stress values. See “Drucker-Prager/Cap model for geological materials,” Section 4.4.4 of the Abaqus Theory Guide, for details.
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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 23.3.2–8):
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$$
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G _ {c} ^ {c r} = \sqrt {[ p - p _ {a} ] ^ {2} + [ R q ] ^ {2}}.
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$$
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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,” Section 3.2.6 of the Abaqus Benchmarks Guide.
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<details>
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<summary>text_image</summary>
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q
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Δε^cr
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β
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Δε^cr
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similar hyperboles
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material point
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p_a
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p
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</details>
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Figure 23.3.2–7 Cohesion creep potentials in the $_ { p - q }$ plane.
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<details>
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<summary>text_image</summary>
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q
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material point
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Δε^cr
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β
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similar
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ellipses
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p_a
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Δε^cr
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p
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</details>
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Figure 23.3.2–8 Consolidation creep potentials in the $_ { p - q }$ plane.
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# Nonassociated flow
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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,” Section 6.1.2). 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.
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# Specifying creep laws
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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.
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# User subroutine CREEP
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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 “solutiondependent 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.
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Input File Usage: Use either or both of the following options:
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\*CAP CREEP, MECHANISM=COHESION, LAW=USER
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$* { \mathrm { C A P ~ C R E E P } } , { \mathrm { M E C H A N I S M } } { = } { \mathrm { C O N S O L I D A T I O N } } , { \mathrm { L A W } } { = } { \mathrm { U S E R } }$
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Abaqus/CAE Usage: Define one or both of the following:
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Property module: material editor: Mechanical→Plasticity→Cap Plasticity:
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Suboptions→Cap Creep Cohesion: Law: User
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Suboptions→Cap Creep Consolidation: Law: User
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“Time hardening” form of the power law model
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With respect to the cohesion mechanism, the power law is available
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$$
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\dot {\bar {\varepsilon}} ^ {c r} = A (\bar {\sigma} ^ {c r}) ^ {n} t ^ {m},
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$$
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where
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$\dot { \bar { \varepsilon } } ^ { c r }$ is the equivalent creep strain rate;
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gcr $\bar { \boldsymbol { \sigma } } ^ { c r }$ is the equivalent cohesion creep stress;
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$\pmb { t }$ is the total or the creep time; and
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A, n, and m are user-defined creep material parameters specified as functions of temperature and field variables.
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In using this form of the power law model with the consolidation mechanism, $\bar { \sigma } ^ { c r }$ can be replaced by $\bar { p } ^ { c r }$ , the effective creep pressure, in the above relation.
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Input File Usage: Use either or both of the following options:
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\*CAP CREEP, MECHANISM=COHESION, LAW=TIME
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$^ { * } \mathrm { C A P } \mathrm { C R E E P } , \mathrm { M E C H A N I S M { = } C O N S O L I D A T I O N , L A W { = } T I M E }$
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Abaqus/CAE Usage: Define one or both of the following:
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Property module: material editor: Mechanical→Plasticity→Cap Plasticity:
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Suboptions→Cap Creep Cohesion: Law: Time
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Suboptions→Cap Creep Consolidation: Law: Time
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“Strain hardening” form of the power law model
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As an alternative to the “time hardening” form of the power law, as defined above, the corresponding “strain hardening” form can be used. For the cohesion mechanism this law has the form
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$$
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\dot {\bar {\varepsilon}} ^ {c r} = \left(A (\bar {\sigma} ^ {c r}) ^ {n} [ (m + 1) \bar {\varepsilon} ^ {c r} ] ^ {m}\right) ^ {\frac {1}{m + 1}}.
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$$
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In using this form of the power law model with the consolidation mechanism, $\bar { \sigma } ^ { c r }$ can be replaced by $\bar { p } ^ { c r }$ , the effective creep pressure, in the above relation.
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For physically reasonable behavior A and n must be positive and $- 1 < m \leq 0$
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Input File Usage: Use either or both of the following options:
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$* { \mathrm { C A P ~ C R E E P , ~ M E C H A N I S M = C O H E S I O N , L A W = S T R A I N } }$
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$* \mathrm { C A P } \mathrm { C R E E P } , \mathrm { M E C H A N I S M = C O N S O L I D A T I O N } , \mathrm { L A W = S T R A I N }$
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Abaqus/CAE Usage: Define one or both of the following:
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Property module: material editor: Mechanical→Plasticity→Cap Plasticity:
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Suboptions→Cap Creep Cohesion: Law: Strain
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Suboptions→Cap Creep Consolidation: Law: Strain
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# Singh-Mitchell law
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A second cohesion creep law available as data input is a variation of the Singh-Mitchell law:
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$$
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\dot {\bar {\varepsilon}} ^ {c r} = A e ^ {(\alpha \bar {\sigma} ^ {c r})} (t _ {1} / t) ^ {m},
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$$
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where $\dot { \bar { \varepsilon } } ^ { c r }$ , t, and $\bar { \sigma } ^ { c r }$ are defined above and $A , \alpha , t _ { 1 }$ , 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, $0 . 0 < m \le 1 . 0$ , and $t _ { 1 }$ should be small compared to the total time.
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In using this variation of the Singh-Mitchell law with the consolidation mechanism, $\bar { \sigma } ^ { c r }$ can be replaced by $\bar { p } ^ { c r }$ , the effective creep pressure, in the above relation.
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Input File Usage: Use either or both of the following options:
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$* { \mathrm { C A P ~ C R E E P , ~ M E C H A N I S M = C O H E S I O N , L A W = S I N G H M } }$
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$* \mathrm { C A P } \mathrm { C R E E P } , \mathrm { M E C H A N I S M { = } C O N S O L I D A T I O N , L A W { = } S I N G H M }$
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Abaqus/CAE Usage: Define one or both of the following:
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Property module: material editor: Mechanical→Plasticity→Cap Plasticity:
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Suboptions→Cap Creep Cohesion: Law: SinghM
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Suboptions→Cap Creep Consolidation: Law: SinghM
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# Time-dependent behavior
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In the “time hardening” 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.
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Input File Usage: Use one of the following options:
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\*CAP CREEP, TIME=TOTAL (default)
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\*CAP CREEP, TIME=CREEP
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Abaqus/CAE Usage: Specifying the time type is not supported in Abaqus/CAE.
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# Numerical difficulties
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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 to avoid such difficulties in the calculation of creep strain increments.
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# Creep integration
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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,” Section 23.2.4.
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# Initial conditions
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The initial stress at a point can be defined (see “Defining initial stresses” in “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). 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 23.3.2–1), 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.
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# Elements
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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).
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# Output
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In addition to the standard output identifiers available in Abaqus (“Abaqus/Standard output variable identifiers,” Section 4.2.1, and “Abaqus/Explicit output variable identifiers,” Section 4.2.2), the following variables have special meaning in the cap plasticity/creep model:
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PEEQ , the cap position.
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PEQC 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 $\begin{array} { r } { \bar { \varepsilon } ^ { p l } = \int _ { 0 } ^ { t } \sqrt { \frac { 2 } { 3 } \dot { \varepsilon } ^ { p l } : \dot { \varepsilon } ^ { p l } } d t } \end{array}$ where $\dot { \varepsilon } ^ { p l }$ is the corresponding rate of plastic flow. The total volumetric inelastic strain is defined as $\begin{array} { r } { \varepsilon _ { \mathrm { v o l } } ^ { i n } = \int _ { 0 } ^ { t } \dot { \varepsilon } _ { k k } ^ { p l } d t + \mathbf { \bar { \int } } _ { 0 } ^ { t } \dot { \varepsilon } _ { k k } ^ { c r } d t } \end{array}$ ％．
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CEEQ Equivalent creep strain produced by the cohesion creep mechanism, defined as $\int \frac { \bar { \pmb { \sigma } } { : } d \pmb { \varepsilon } ^ { c r } } { \bar { \pmb { \sigma } } ^ { c r } }$ g:dec where $\begin{array} { r } { \bar { \sigma } ^ { c r } = \frac { \dot { ( } q - p \tan \beta ) } { ( 1 - \frac { 1 } { 3 } \tan \beta ) } } \end{array}$ is the equivalent creep stress.
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CESW Equivalent creep strain produced by the consolidation creep mechanism, defined as $\int ^ { \frac { \pmb { \sigma } : d \pmb { \varepsilon } ^ { c r } } { \hat { p } } }$ , where $\begin{array} { r } { \bar { p } = \frac { \mathbf { \dot { R } } ^ { 2 } \mathbf { \Phi } q ^ { 2 } + p \mathbf { \Phi } ( p - \mathbf { \dot { p } } _ { a } ) } { G _ { c } ^ { c r } } } \end{array}$ is the equivalent creep pressure. Gr
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<!-- source-page: 379 -->
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# 23.3.3 MOHR-COULOMB PLASTICITY
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Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
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# References
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• “Material library: overview,” Section 21.1.1
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• “Inelastic behavior,” Section 23.1.1
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• \*MOHR COULOMB
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• \*MOHR COULOMB HARDENING
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• \*TENSION CUTOFF
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• “Defining Mohr-Coulomb plasticity” in “Defining plasticity,” Section 12.9.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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# Overview
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The Mohr-Coulomb plasticity model:
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• is used to model materials with the classical Mohr-Coloumb yield criterion;
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• allows the material to harden and/or soften isotropically;
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• uses a smooth flow potential that has a hyperbolic shape in the meridional stress plane and a piecewise elliptic shape in the deviatoric stress plane;
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• is used with the linear elastic material model (“Linear elastic behavior,” Section 22.2.1);
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• can be used with the Rankine surface (tension cutoff) to limit load carrying capacity near the tensile region; and
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• can be used for design applications in the geotechnical engineering area to simulate material response under essentially monotonic loading.
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# Elastic behavior
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The elastic part of the response is specified as described in “Linear elastic behavior,” Section 22.2.1. Linear isotropic elasticity is assumed.
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# Plastic behavior: yield criteria
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The yield surface is a composite of two different criteria: a shear criterion, known as the Mohr-Coulomb surface, and an optional tension cutoff criterion, modeled using the Rankine surface.
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# Mohr-Coulomb surface
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The Mohr-Coulomb criterion assumes that yield occurs when the shear stress on any point in a material reaches a value that depends linearly on the normal stress in the same plane. The Mohr-Coulomb model is based on plotting Mohr’s circle for states of stress at yield in the plane of the maximum and
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<!-- source-page: 380 -->
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minimum principal stresses. The yield line is the best straight line that touches these Mohr’s circles (Figure 23.3.3–1).
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<details>
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<summary>text_image</summary>
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τ
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φ
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C
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S = σ₁ - σ₃ / 2
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σ₁ σ₁ σ₃ σ₃
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σₘ = σ₁ + σ₃ / 2
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(compressive stress)
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</details>
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Figure 23.3.3–1 Mohr-Coulomb yield model.
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Therefore, the Mohr-Coulomb model is defined by
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$$
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\tau = c - \sigma \tan \phi ,
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$$
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where $\sigma$ is negative in compression. From Mohr’s circle,
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$$
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\tau = s \cos \phi ,
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$$
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$$
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\sigma = \sigma_ {m} + s \sin \phi .
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$$
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Substituting for $\tau$ and $\sigma ,$ multiplying both sides by $\phi ,$ and reducing, the Mohr-Coulomb model can be written as
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$$
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s + \sigma_ {m} \sin \phi - c \cos \phi = 0,
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$$
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where
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$$
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s = \frac {1}{2} (\sigma_ {1} - \sigma_ {3})
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$$
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is half of the difference between the maximum principal stress, $\sigma _ { 1 }$ , and the minimum principal stress, $\sigma _ { 3 }$ (and is, therefore, the maximum shear stress),
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