김경종
365 lines
23 KiB
Markdown
365 lines
23 KiB
Markdown
<!-- source-page: 361 -->
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$$
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\bar {\sigma} | _ {0} = \bar {\sigma} | _ {\bar {\varepsilon} ^ {p l} = 0, \dot {\bar {\varepsilon}} ^ {p l} = 0}
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$$
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E
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is the initial yield stress taken from the user-specified Drucker-Prager hardening data; and
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is a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the creep potential tends to a straight line as the eccentricity tends to zero).
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Suitable default values are provided for , as described below. This creep potential, which is continuous and smooth, ensures that the creep flow direction is always uniquely defined. The function approaches the linear Drucker-Prager flow potential asymptotically at high confining pressure stress and intersects the hydrostatic pressure axis at 90°. A family of hyperbolic potentials in the meridional stress plane was shown in Figure 23.3.1–6. The creep potential is the von Mises circle in the deviatoric stress plane (the -plane).
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The default creep potential eccentricity is , which implies that the material has almost the same dilation angle over a wide range of confining pressure stress values. Increasing the value of provides more curvature to the creep potential, implying that the dilation angle increases as the confining pressure decreases. Values of that are significantly less than the default value may lead to convergence problems if the material is subjected to low confining pressures, because of the very tight curvature of the creep potential locally where it intersects the p-axis. 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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If the creep material properties are defined by a compression test, numerical problems may arise for very low stress values. Abaqus/Standard protects for such a case, as described in “Models for granular or polymer behavior,” Section 4.4.2 of the Abaqus Theory Guide.
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# Nonassociated flow
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The use of a creep potential different from the equivalent creep surface implies that the material stiffness matrix is not symmetric; therefore, the unsymmetric matrix storage and solution scheme should be used (see “Defining an analysis,” Section 6.1.2). If the difference between $\beta$ and $\psi$ is not large and the region of the model in which 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 and the unsymmetric matrix scheme may not be needed.
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# Specifying a creep law
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The definition of creep behavior in Abaqus/Standard is completed by specifying the equivalent “uniaxial behavior”—the creep “law.” In many practical cases the creep “law” is defined through user subroutine CREEP because creep laws are usually of very complex form to fit experimental data. Data input methods are provided for some simple cases, including two forms of a power law model and a variation of the Singh-Mitchell law.
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# User subroutine CREEP
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User subroutine CREEP provides a very general capability for implementing viscoplastic models in Abaqus/Standard in which the strain rate potential can be written as a function of the equivalent stress
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and any number of “solution-dependent state variables.” When used in conjunction with these material models, the equivalent creep stress, $\bar { \boldsymbol { \sigma } } ^ { c r }$ , is 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: \*DRUCKER PRAGER CREEP, LAW=USER
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Suboptions→Drucker Prager Creep: Law: User
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“Time hardening” form of the power law model
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The “time hardening” form of the power law model is
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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, defined so that $\dot { \bar { \varepsilon } } ^ { c r } = | \dot { \varepsilon } _ { 1 1 } ^ { c r } |$ if the equivalent creep stress is defined in uniaxial compression, $\dot { \bar { \varepsilon } } ^ { c r } = \dot { \varepsilon } _ { 1 1 } ^ { c r }$ if defined in uniaxial tension, and $\dot { \bar { \varepsilon } } ^ { c r } = \dot { \gamma } ^ { c r } / \sqrt { 3 }$ if defined in pure shear, where $\gamma ^ { c r }$ is the engineering shear creep strain;
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gcr $\bar { \boldsymbol { \sigma } } ^ { c r }$ is the equivalent creep stress;
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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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Input File Usage: \*DRUCKER PRAGER CREEP, LAW=TIME
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Suboptions→Drucker Prager Creep: 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:
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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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For physically reasonable behavior A and n must be positive and $- 1 < m \leq 0$
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Input File Usage: \*DRUCKER PRAGER CREEP, LAW=STRAIN
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Suboptions→Drucker Prager Creep: Law: Strain
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# Singh-Mitchell law
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A second 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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Input File Usage: \*DRUCKER PRAGER CREEP, LAW=SINGHM
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Suboptions→Drucker Prager Creep: 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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$$
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\begin{array}{l} * D R U C K E R P R A G E R C R E E P, T I M E = T O T A L (d e f a u l t) \\ * D R U C K E R P R A G E R C R E E P, T I M E = C R E E P \\ \end{array}
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$$
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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 $1 0 ^ { - 2 7 }$ , 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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When we need to study the behavior of a material that has already been subjected to some work hardening, Abaqus allows you to prescribe initial conditions for the equivalent plastic strain, $\bar { \varepsilon } ^ { p l }$ , by specifying the conditions directly (see “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1).
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For more complicated cases initial conditions can be defined in Abaqus/Standard through user subroutine HARDINI.
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<table><tr><td>Input File Usage:</td><td>Use the following option to specify the initial equivalent plastic strain directly:*INITIAL CONDITIONS, TYPE=HARDENINGUse the following option in Abaqus/Standard to specify the initial equivalent plastic strain in user subroutine HARDINI:*INITIAL CONDITIONS, TYPE=HARDENING, USER</td></tr></table>
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<table><tr><td>Abaqus/CAE Usage:</td><td>Use the following options to specify the initial equivalent plastic strain directly:Load module:Create Predefined Field:Step:Initial, choose Mechanical for theCategoryandHardeningfor theTypes for Selected StepUse the following options in Abaqus/Standard to specify the initial equivalent plastic strain in user subroutineHARDINI:Load module:Create Predefined Field:Step:Initial, choose Mechanical for theCategoryandHardeningfor theTypes for Selected Step;Definition:User-defined</td></tr></table>
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# Elements
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The Drucker-Prager models can be used with the following element types: plane strain, generalized plane strain, axisymmetric, and three-dimensional solid (continuum) elements. All Drucker-Prager models are also available in plane stress (plane stress, shell, and membrane elements), except for the linear Drucker-Prager model with creep.
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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 for the Drucker-Prager plasticity/creep model:
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PEEQ Equivalent plastic strain.
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For the linear Drucker-Prager plasticity model PEEQ is defined as $\bar { \varepsilon } ^ { p l } | _ { 0 } +$ $\textstyle \int _ { 0 } ^ { t } { \dot { \varepsilon } } ^ { p l } d t$ ; where $\bar { \varepsilon } ^ { p l } | _ { 0 }$ is the initial equivalent plastic strain (zero or user-specified; see “Initial conditions”) and $\dot { \varepsilon } ^ { p l }$ is the equivalent plastic strain rate.
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For the hyperbolic and exponential Drucker-Prager plasticity models PEEQ is defined as $\bar { \varepsilon } ^ { \bar { p } l } | _ { 0 } + \int \frac { \pmb { \sigma } { : } d \pmb { \varepsilon } ^ { p l } } { \sigma ^ { 0 } }$ g:depl g0 , where $\bar { \varepsilon } ^ { p l } | _ { 0 }$ is the initial equivalent plastic strain and $\sigma ^ { 0 }$ is the yield stress.
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CEEQ Equivalent creep strain, $\int \dot { \bar { \varepsilon } } ^ { c r } d t$ .
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# 23.3.2 MODIFIED DRUCKER-PRAGER/CAP MODEL
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Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
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# References
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• “Inelastic behavior,” Section 23.1.1
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• “Material library: overview,” Section 21.1.1
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• “Rate-dependent plasticity: creep and swelling,” Section 23.2.4
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• “CREEP,” Section 1.1.1 of the Abaqus User Subroutines Reference Guide
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• \*CAP PLASTICITY
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• \*CAP HARDENING
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• \*CAP CREEP
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• “Defining cap 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 modified Drucker-Prager/Cap plasticity/creep model:
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• is intended to model cohesive geological materials that exhibit pressure-dependent yield, such as soils and rocks;
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• is based on the addition of a cap yield surface to the Drucker-Prager plasticity model (“Extended Drucker-Prager models,” Section 23.3.1), which provides an inelastic hardening mechanism to account for plastic compaction and helps to control volume dilatancy when the material yields in shear;
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• can be used in Abaqus/Standard to simulate creep in materials exhibiting long-term inelastic deformation through a cohesion creep mechanism in the shear failure region and a consolidation creep mechanism in the cap region;
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• can be used in conjunction with either the elastic material model (“Linear elastic behavior,” Section 22.2.1) or, in Abaqus/Standard if creep is not defined, the porous elastic material model (“Elastic behavior of porous materials,” Section 22.3.1); and
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• provides a reasonable response to large stress reversals in the cap region; however, in the failure surface region the response is reasonable only for essentially monotonic loading.
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# Yield surface
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The addition of the cap yield surface to the Drucker-Prager model 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
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by providing softening as a function of the inelastic volume increase created as the material yields on the Drucker-Prager shear failure surface.
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The yield surface has two principal segments: a pressure-dependent Drucker-Prager shear failure segment and a compression cap segment, as shown in Figure 23.3.2–1. The Drucker-Prager failure segment is a perfectly plastic yield surface (no hardening). Plastic flow on this segment produces inelastic volume increase (dilation) that causes the cap to soften. On the cap surface plastic flow causes the material to compact. The model is described in detail in “Drucker-Prager/Cap model for geological materials,” Section 4.4.4 of the Abaqus Theory Guide.
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<details>
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<summary>text_image</summary>
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Transition
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surface, F_t
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Shear failure, F_s
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α(d+p_a tanβ)
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β
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Cap, F_c
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d+p_a tanβ
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p_a
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R(d+p_a tanβ)
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p_b
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t
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d
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p
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</details>
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Figure 23.3.2–1 Modified Drucker-Prager/Cap model: yield surfaces in the p–t plane.
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# Failure surface
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The Drucker-Prager failure surface is written as
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$$
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F _ {s} = t - p \tan \beta - d = 0,
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$$
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where $\beta ( \theta , f _ { i } )$ and $d ( \theta , f _ { i } )$ represent the angle of friction of the material and its cohesion, respectively, and can depend on temperature, , and other predefined fields $f _ { i } , i = 1 , 2 , 3 \dots$ . The deviatoric stress measure t is defined as
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$$
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t = \frac {1}{2} q \left[ 1 + \frac {1}{K} - \left(1 - \frac {1}{K}\right) \left(\frac {r}{q}\right) ^ {3} \right];
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$$
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<!-- source-page: 367 -->
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and
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$$
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\begin{array}{l} p = - \frac {1}{3} \operatorname{trace} (\boldsymbol {\sigma}) \quad \text { is the equivalent pressure stress }, \\ q = \sqrt {\frac {3}{2} \mathbf {S} : \mathbf {S}} \quad \text { is the Mises equivalent stress }, \\ r = \left(\frac {9}{2} \mathbf {S}: \mathbf {S} \cdot \mathbf {S}\right) ^ {\frac {1}{3}} \quad \text { is the third stress invariant, and } \\ \mathbf {S} = \boldsymbol {\sigma} + p \mathbf {I} \quad \text { is the deviatoric stress. } \\ \end{array}
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$$
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$K ( \theta , f _ { i } )$ is a material parameter that controls the dependence of the yield surface on the value of the intermediate principal stress, as shown in Figure 23.3.2–2.
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<details>
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<summary>text_image</summary>
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S₃
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S₁
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S₂
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a
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b
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</details>
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$$
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t = \frac {1}{2} q \left[ 1 + \frac {1}{K} - \left(1 - \frac {1}{K}\right) \left(\frac {r}{q}\right) ^ {3} \right]
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$$
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$$
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\begin{array}{c c} \text {Curve} & K \\ \hline a & 1. 0 \\ b & 0. 8 \end{array}
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$$
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Figure 23.3.2–2 Typical yield/flow surfaces in the deviatoric plane.
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The yield surface is defined so that K is the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression. implies that the yield surface is the von Mises circle in the deviatoric principal stress plane (the -plane), so that the yield stresses in triaxial tension and compression are the same; this is the default behavior in Abaqus/Standard and the only behavior available in Abaqus/Explicit. To ensure that the yield surface remains convex requires $0 . 7 7 8 \leq K \leq 1 . 0$ .
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# Cap yield surface
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The cap yield surface has an elliptical shape with constant eccentricity in the meridional (p–t) plane (Figure 23.3.2–1) and also includes dependence on the third stress invariant in the deviatoric plane (Figure 23.3.2–2). The cap surface hardens or softens as a function of the volumetric inelastic strain: volumetric plastic and/or creep compaction (when yielding on the cap and/or creeping according to the consolidation mechanism, as described later in this section) causes hardening, while volumetric
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plastic and/or creep dilation (when yielding on the shear failure surface and/or creeping according to the cohesion mechanism, as described later in this section) causes softening. The cap yield surface is
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$$
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F _ {c} = \sqrt {[ p - p _ {a} ] ^ {2} + \left[ \frac {R t}{(1 + \alpha - \alpha / \cos \beta)} \right] ^ {2}} - R (d + p _ {a} \tan \beta) = 0,
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$$
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where $R ( \theta , f _ { i } )$ is a material parameter that controls the shape of the cap, $\alpha ( \theta , f _ { i } )$ is a small number that we discuss later, and $p _ { a } ( \varepsilon _ { \mathrm { v o l } } ^ { p l } + \varepsilon _ { \mathrm { v o l } } ^ { c r } )$ is an evolution parameter that represents the volumetric inelastic strain driven hardening/softening. The hardening/softening law is a user-defined piecewise linear function relating the hydrostatic compression yield stress, $p _ { b }$ , and volumetric inelastic strain (Figure 23.3.2–3):
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$$
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p _ {b} = p _ {b} (\varepsilon_ {\mathrm{vol}} ^ {i n} | _ {0} + \varepsilon_ {\mathrm{vol}} ^ {p l} + \varepsilon_ {\mathrm{vol}} ^ {c r}).
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$$
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<details>
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<summary>line</summary>
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| x | p_b |
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| --- | --- |
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| 0 | 0 |
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| 1 | 0.5 |
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| 2 | 1 |
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| 3 | 1.5 |
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| 4 | 2.5 |
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| 5 | 3.5 |
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| 6 | 5 |
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| 7 | 7 |
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| 8 | 10 |
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</details>
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Figure 23.3.2–3 Typical Cap hardening.
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The volumetric inelastic strain axis in Figure 23.3.2–3 has an arbitrary origin: $\varepsilon _ { \mathrm { v o l } } ^ { i n } | _ { 0 } ( = \varepsilon _ { \mathrm { v o l } } ^ { p l } | _ { 0 } + \varepsilon _ { \mathrm { v o l } } ^ { c r } | _ { 0 } )$ is the position on this axis corresponding to the initial state of the material when the analysis begins, thus defining the position of the cap $\left( \boldsymbol { p } _ { b } \right)$ in Figure 23.3.2–1 at the start of the analysis. The evolution parameter $p _ { a }$ is given as
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$$
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p _ {a} = \frac {p _ {b} - R d}{(1 + R \tan \beta)}.
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$$
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The parameter is a small number (typically 0.01 to 0.05) used to define a transition yield surface,
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<!-- source-page: 369 -->
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$$
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F _ {t} = \sqrt {[ p - p _ {a} ] ^ {2} + \left[ t - (1 - \frac {\alpha}{\cos \beta}) (d + p _ {a} \tan \beta) \right] ^ {2}} - \alpha (d + p _ {a} \tan \beta) = 0,
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$$
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so that the model provides a smooth intersection between the cap and failure surfaces.
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# Defining yield surface variables
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You provide the variables d, , $R , \varepsilon _ { v o l } ^ { i n } \big | _ { 0 } , \alpha ,$ , and K to define the shape of the yield surface. In Abaqus/Standard $0 . 7 7 8 \leq K \leq 1 . 0 .$ , while in Abaqus/Explicit K = 1 ( ). If desired, combinations of these variables can also be defined as a tabular function of temperature and other predefined field variables.
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Input File Usage: \*CAP PLASTICITY
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Cap Plasticity
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# Defining hardening parameters
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The hardening curve specified for this model interprets yielding in the hydrostatic pressure sense: the hydrostatic pressure yield stress is defined as a tabular function of the volumetric inelastic strain, and, if desired, a function of temperature and other predefined field variables. The range of values for which $p _ { b }$ is defined should be sufficient to include all values of effective pressure stress that the material will be subjected to during the analysis.
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Input File Usage: \*CAP HARDENING
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Cap Plasticity: Suboptions→Cap Hardening
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# Plastic flow
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Plastic flow is defined by a flow potential that is associated in the deviatoric plane, associated in the cap region in the meridional plane, and nonassociated in the failure surface and transition regions in the meridional plane. The flow potential surface that we use in the meridional plane is shown in Figure 23.3.2–4: it is made up of an elliptical portion in the cap region that is identical to the cap yield surface,
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$$
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G _ {c} = \sqrt {[ p - p _ {a} ] ^ {2} + \left[ \frac {R t}{(1 + \alpha - \alpha / \cos \beta)} \right] ^ {2}},
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$$
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and another elliptical portion in the failure and transition regions that provides the nonassociated flow component in the model,
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$$
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G _ {s} = \sqrt {\left[ (p _ {a} - p) \tan \beta \right] ^ {2} + \left[ \frac {t}{(1 + \alpha - \alpha / \cos \beta)} \right] ^ {2}}.
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$$
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The two elliptical portions form a continuous and smooth potential surface.
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<!-- source-page: 370 -->
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<details>
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<summary>text_image</summary>
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Similar
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ellipses
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G_s (Shear failure)
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G_c (cap)
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(1+α-α secβ)(d+p_a tanβ)
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d+p_a tanβ
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R(d+p_a tanβ)
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p
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t
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</details>
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Figure 23.3.2–4 Modified Drucker-Prager/Cap model: flow potential in the p–t plane.
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# Nonassociated flow
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Nonassociated flow implies that the material stiffness matrix is not symmetric and the unsymmetric matrix storage and solution scheme should be used in Abaqus/Standard (see “Defining an analysis,” Section 6.1.2). 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 rate of convergence; in such cases the unsymmetric matrix scheme may not be needed.
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# Calibration
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At least three experiments are required to calibrate the simplest version of the Cap model: a hydrostatic compression test (an oedometer test is also acceptable) and either two triaxial compression tests or one triaxial compression test and one uniaxial compression test (more than two tests are recommended for a more accurate calibration).
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The hydrostatic compression test is performed by pressurizing the sample equally in all directions. The applied pressure and the volume change are recorded.
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The uniaxial compression test involves compressing the sample between two rigid platens. The load and displacement in the direction of loading are recorded. The lateral displacements should also be recorded so that the correct volume changes can be calibrated.
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Triaxial compression experiments are performed using a standard triaxial machine where a fixed confining pressure is maintained while the differential stress is applied. Several tests covering the range of confining pressures of interest are usually performed. Again, the stress and strain in the direction of loading are recorded, together with the lateral strain so that the correct volume changes can be calibrated.
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