김경종
417 lines
21 KiB
Markdown
417 lines
21 KiB
Markdown
<!-- source-page: 351 -->
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<details>
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<summary>scatter</summary>
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| p | q |
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| ---- | ---- |
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| 0.0 | 0.0 |
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| 0.2 | 0.2 |
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| 0.4 | 0.4 |
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| 0.6 | 0.6 |
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| 0.8 | 0.8 |
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| 1.0 | 1.0 |
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</details>
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Figure 23.3.1–8 Yield surface in meridional plane.
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<details>
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<summary>text_image</summary>
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-σ₃
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σ₁= σ₂≥ σ₃
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-σ₁
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-σ₂
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</details>
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a
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<details>
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<summary>text_image</summary>
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-σ₁
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σ₁≥σ₂=σ₃
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-σ₂
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-σ₃
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</details>
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b
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Figure 23.3.1–9 a) Triaxial compression and b) tension.
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$$
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r ^ {3} = - \big (\sigma_ {1} - \sigma_ {3} \big) ^ {3},
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$$
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so that
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$$
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t = q = \sigma_ {1} - \sigma_ {3}.
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$$
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The triaxial compression results can, thus, be plotted in the meridional plane shown in Figure 23.3.1–8.
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# Linear Drucker-Prager model
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Fitting the best straight line through the triaxial compression results provides $\beta$ and d for the linear Drucker-Prager model.
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Triaxial tension data are also needed to define K in the linear Drucker-Prager model. Under triaxial tension the specimen is again confined by pressure, after which the pressure in one direction is reduced. In this case the principal stresses are $0 \geq \sigma _ { 1 } \geq \sigma _ { 2 } = \sigma _ { 3 }$ (Figure 23.3.1–9b).
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<!-- source-page: 352 -->
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The stress invariants are now
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$$
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p = - \frac {1}{3} (\sigma_ {1} + 2 \sigma_ {3}),
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$$
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$$
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q = \sigma_ {1} - \sigma_ {3},
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$$
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and
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$$
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r ^ {3} = (\sigma_ {1} - \sigma_ {3}) ^ {3},
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$$
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so that
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$$
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t = \frac {q}{K} = \frac {1}{K} \left(\sigma_ {1} - \sigma_ {3}\right).
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$$
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Thus, K can be found by plotting these test results as q versus p and again fitting the best straight line. The triaxial compression and tension lines must intercept the p-axis at the same point, and the ratio of values of q for triaxial tension and compression at the same value of p then gives K (Figure 23.3.1–10).
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<details>
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<summary>text_image</summary>
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q
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Best fit to triaxial
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compression data
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β
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d
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p
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q_c
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q_t
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q_t / q_c = K
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Best fit to triaxial tension data
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</details>
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Figure 23.3.1–10 Linear model: fitting triaxial compression and tension data.
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# Hyperbolic model
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Fitting the best straight line through the triaxial compression results at high confining pressures provides $\beta$ and $d ^ { \prime }$ for the hyperbolic model. This fit is performed in the same manner as that used to obtain $\beta$ and
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<!-- source-page: 353 -->
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d for the linear Drucker-Prager model. In addition, hydrostatic tension data are required to complete the calibration of the hyperbolic model so that the initial hydrostatic tension strength, $p _ { t } | _ { 0 }$ , can be defined.
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# General exponent model
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Given triaxial data in the meridional plane, Abaqus provides a capability to determine the material parameters ${ \pmb a } ,$ b, and $p _ { t }$ required for the exponent model. The parameters are determined on the basis of a “best fit” of the triaxial test data at different levels of confining stress. A least-squares fit which minimizes the relative error in stress is used to obtain the “best fit” values for a, b, and $p _ { t }$ . The capability allows all three parameters to be calibrated or, if some of the parameters are known, only the remaining parameters to be calibrated. This ability is useful if only a few data points are available, in which case you may wish to fit the best straight line ( ) through the data points (effectively reducing the model to a linear Drucker-Prager model). Partial calibration can also be useful in a case when triaxial test data at low confinement are unreliable or unavailable, as is often the case for cohesionless materials. In this case a better fit may be obtained if the value of $p _ { t }$ is specified and only a and b are calibrated.
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The data must be provided in terms of the principal stresses $\sigma _ { 1 } ( = \ \sigma _ { 2 } )$ and $\sigma _ { 3 }$ , where $\sigma _ { 1 }$ is the confining stress and $\sigma _ { 3 }$ is the stress in the loading direction. The Abaqus sign convention must be followed such that tensile stresses are positive and compressive stresses are negative. One pair of stresses must be entered from each triaxial test. As many data points as desired can be entered from triaxial tests at different levels of confining stress.
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If the exponent model is used as a failure surface (perfect plasticity), the Drucker-Prager hardening behavior does not have to be specified. The hydrostatic tension strength, $p _ { t }$ , obtained from the calibration will then be used as the failure stress. However, if the Drucker-Prager hardening behavior is specified together with the triaxial test data, the value of $p _ { t }$ obtained from the calibration will be ignored. In this case Abaqus will interpolate $p _ { t }$ directly from the hardening data.
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<table><tr><td>Input File Usage:</td><td>Use both of the following options:*DRUCKER PRAGER, SHEAR CRITERION=EXPONENT FORM, TEST DATA*TRIAXIAL TEST DATA</td></tr><tr><td>Abaqus/CAE Usage:</td><td>Property module: material editor: Mechanical→Plasticity→DruckerPrager: Shear criterion: Exponent Form, toggle on Use Suboption Triaxial Test Data, and select Suboptions→Triaxial Test Data</td></tr></table>
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# Matching Mohr-Coulomb parameters to the Drucker-Prager model
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Sometimes experimental data are not directly available. Instead, you are provided with the friction angle and cohesion values for the Mohr-Coulomb model. In that case the simplest way to proceed is to use the Mohr-Coulomb model (see “Mohr-Coulomb plasticity,” Section 23.3.3). In some situations it may be necessary to use the Drucker-Prager model instead of the Mohr-Coulomb model (such as when rate effects need to be considered), in which case we need to calculate values for the parameters of a Drucker-Prager model to provide a reasonable match to the Mohr-Coulomb parameters.
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<!-- source-page: 354 -->
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The Mohr-Coulomb failure model is based on plotting Mohr’s circle for states of stress at failure in the plane of the maximum and minimum principal stresses. The failure line is the best straight line that touches these Mohr’s circles (Figure 23.3.1–11).
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<details>
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<summary>text_image</summary>
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τ
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S = σ₁ - σ₃ / 2
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φ
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C
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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 23.3.1–11 Mohr-Coulomb failure 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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<!-- source-page: 355 -->
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$$
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\sigma_ {m} = \frac {1}{2} (\sigma_ {1} + \sigma_ {3})
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$$
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is the average of the maximum and minimum principal stresses, and $\phi$ is the friction angle. Thus, the model assumes a linear relationship between deviatoric and pressure stress and, so, can be matched by the linear or hyperbolic Drucker-Prager models provided in Abaqus.
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The Mohr-Coulomb model assumes that failure is independent of the value of the intermediate principal stress, but the Drucker-Prager model does not. 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 model has vertices in the deviatoric plane (see Figure 23.3.1–12).
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<details>
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<summary>text_image</summary>
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S₃
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Mohr-Coulomb
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S₁
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S₂
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Drucker-Prager
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</details>
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Figure 23.3.1–12 Mohr-Coulomb model in the deviatoric plane.
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The implication is that, whenever the stress state has two equal principal stress values, the flow direction can change significantly with little or no change in stress. None of the models currently available in Abaqus can provide such behavior; even in the Mohr-Coulomb model the flow potential is smooth. This limitation is generally not a key concern in many design calculations involving Coulomb-like materials, but it can limit the accuracy of the calculations, especially in cases where flow localization is important.
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# Matching plane strain response
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Plane strain problems are often encountered in geotechnical analysis; for example, long tunnels, footings, and embankments. Therefore, the constitutive model parameters are often matched to provide the same flow and failure response in plane strain.
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<!-- source-page: 356 -->
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The matching procedure described below is carried out in terms of the linear Drucker-Prager model but is also applicable to the hyperbolic model at high levels of confining stress.
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The linear Drucker-Prager flow potential defines the plastic strain increment as
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$$
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d \varepsilon^ {p l} = d \bar {\varepsilon} ^ {p l} \frac {1}{(1 - \frac {1}{3} \tan \psi)} \frac {\partial}{\partial \pmb {\sigma}} (t - p \tan \psi),
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$$
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where $d \bar { \varepsilon } ^ { p l }$ is the equivalent plastic strain increment. Since we wish to match the behavior in only one plane, we can take , which implies that . Thus,
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$$
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d \varepsilon^ {p l} = d \bar {\varepsilon} ^ {p l} \frac {1}{(1 - \frac {1}{3} \tan \psi)} \left(\frac {\partial q}{\partial \pmb {\sigma}} - \tan \psi \frac {\partial p}{\partial \pmb {\sigma}}\right).
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$$
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Writing this expression in terms of principal stresses provides
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$$
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d \varepsilon_ {1} ^ {p l} = d \bar {\varepsilon} ^ {p l} \frac {1}{(1 - \frac {1}{3} \tan \psi)} \left(\frac {1}{2 q} (2 \sigma_ {1} - \sigma_ {2} - \sigma_ {3}) + \frac {1}{3} \tan \psi\right),
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$$
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with similar expressions for $d \varepsilon _ { 2 } ^ { p l }$ and $d \varepsilon _ { 3 } ^ { p l }$ . Assume plane strain in the 1-direction. At limit load we must have ${ d \varepsilon _ { 1 } } ^ { p l } = 0$ , which provides the constraint
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$$
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\sigma_ {1} = \frac {1}{2} (\sigma_ {2} + \sigma_ {3}) - \frac {1}{3} \tan \psi q.
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$$
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Using this constraint we can rewrite q and p in terms of the principal stresses in the plane of deformation, $\sigma _ { 2 }$ and $\sigma _ { 3 }$ , as
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$$
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q = \frac {3 \sqrt {3}}{2 \sqrt {9 - \tan^ {2} \psi}} (\sigma_ {2} - \sigma_ {3}),
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$$
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and
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$$
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p = - \frac {1}{2} (\sigma_ {2} + \sigma_ {3}) + \frac {\tan \psi}{2 \sqrt {3 (9 - \tan^ {2} \psi)}} (\sigma_ {2} - \sigma_ {3}).
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$$
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With these expressions the Drucker-Prager yield surface can be written in terms of $\sigma _ { 2 }$ and $\sigma _ { 3 }$ as
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$$
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\frac {9 - \tan \beta \tan \psi}{2 \sqrt {3 (9 - \tan^ {2} \psi)}} (\sigma_ {2} - \sigma_ {3}) + \frac {1}{2} \tan \beta (\sigma_ {2} + \sigma_ {3}) - d = 0.
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$$
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The Mohr-Coulomb yield surface in the plane is
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$$
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\sigma_ {2} - \sigma_ {3} + \sin \phi (\sigma_ {2} + \sigma_ {3}) - 2 c \cos \phi = 0.
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$$
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<!-- source-page: 357 -->
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By comparison,
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$$
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\sin \phi = \frac {\tan \beta \sqrt {3 (9 - \tan^ {2} \psi)}}{9 - \tan \beta \tan \psi},
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$$
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$$
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c \cos \phi = \frac {\sqrt {3 (9 - \tan^ {2} \psi)}}{9 - \tan \beta \tan \psi} d.
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$$
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These relationships provide a match between the Mohr-Coulomb material parameters and linear Drucker-Prager material parameters in plane strain. Consider the two extreme cases of flow definition: associated flow, $\psi = \beta _ { ; }$ , and nondilatant flow, when $\psi = 0$ . For associated flow
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$$
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\tan \beta = \frac {\sqrt {3} \sin \phi}{\sqrt {1 + \frac {1}{3} \sin^ {2} \phi}} \quad \mathrm{and} \quad \frac {d}{c} = \frac {\sqrt {3} \cos \phi}{\sqrt {1 + \frac {1}{3} \sin^ {2} \phi}},
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$$
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and for nondilatant flow
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$$
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\tan \beta = \sqrt {3} \sin \phi \quad \text { and } \quad \frac {d}{c} = \sqrt {3} \cos \phi .
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$$
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In either case $\sigma _ { c } ^ { 0 }$ is immediately available as
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$$
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\sigma_ {c} ^ {0} = \frac {1}{1 - \frac {1}{3} \tan \beta} d.
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$$
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The difference between these two approaches increases with the friction angle; however, the results are not very different for typical friction angles, as illustrated in Table 23.3.1–1.
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Table 23.3.1–1 Plane strain matching of Drucker-Prager and Mohr-Coulomb models.
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<table><tr><td rowspan="2">Mohr-Coulomb friction angle, $\phi$ </td><td colspan="2">Associated flow</td><td colspan="2">Nondilatant flow</td></tr><tr><td>Drucker-Prager friction angle, $\beta$ </td><td> $d/c$ </td><td>Drucker-Prager friction angle, $\beta$ </td><td> $d/c$ </td></tr><tr><td> $10^{\circ}$ </td><td> $16.7^{\circ}$ </td><td>1.70</td><td> $16.7^{\circ}$ </td><td>1.70</td></tr><tr><td> $20^{\circ}$ </td><td> $30.2^{\circ}$ </td><td>1.60</td><td> $30.6^{\circ}$ </td><td>1.63</td></tr><tr><td> $30^{\circ}$ </td><td> $39.8^{\circ}$ </td><td>1.44</td><td> $40.9^{\circ}$ </td><td>1.50</td></tr><tr><td> $40^{\circ}$ </td><td> $46.2^{\circ}$ </td><td>1.24</td><td> $48.1^{\circ}$ </td><td>1.33</td></tr><tr><td> $50^{\circ}$ </td><td> $50.5^{\circ}$ </td><td>1.02</td><td> $53.0^{\circ}$ </td><td>1.11</td></tr></table>
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“Limit load calculations with granular materials,” Section 1.15.4 of the Abaqus Benchmarks Guide, and “Finite deformation of an elastic-plastic granular material,” Section 1.15.5 of the Abaqus Benchmarks Guide, show a comparison of the response of a simple loading of a granular material
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<!-- source-page: 358 -->
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using the Drucker-Prager and Mohr-Coulomb models, using the plane strain approach to match the parameters of the two models.
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# Matching triaxial test response
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Another approach to matching Mohr-Coulomb and Drucker-Prager model parameters for materials with low friction angles is to make the two models provide the same failure definition in triaxial compression and tension. The following matching procedure is applicable only to the linear Drucker-Prager model since this is the only model in this class that allows for different yield values in triaxial compression and tension.
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We can rewrite the Mohr-Coulomb model in terms of principal stresses:
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$$
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\sigma_ {1} - \sigma_ {3} + (\sigma_ {1} + \sigma_ {3}) \sin \phi - 2 c \cos \phi = 0.
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$$
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Using the results above for the stress invariants ${ p , q , }$ and r in triaxial compression and tension allows the linear Drucker-Prager model to be written for triaxial compression as
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$$
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\sigma_ {1} - \sigma_ {3} + \frac {\tan \beta}{2 + \frac {1}{3} \tan \beta} (\sigma_ {1} + \sigma_ {3}) - \frac {1 - \frac {1}{3} \tan \beta}{1 + \frac {1}{6} \tan \beta} \sigma_ {c} ^ {0} = 0,
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$$
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and for triaxial tension as
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$$
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\sigma_ {1} - \sigma_ {3} + \frac {\tan \beta}{\frac {2}{K} - \frac {1}{3} \tan \beta} (\sigma_ {1} + \sigma_ {3}) - \frac {1 - \frac {1}{3} \tan \beta}{\frac {1}{K} - \frac {1}{6} \tan \beta} \sigma_ {c} ^ {0} = 0.
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$$
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We wish to make these expressions identical to the Mohr-Coulomb model for all values of $( \sigma _ { 1 } , \sigma _ { 3 } )$ . This is possible by setting
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$$
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K = \frac {1}{1 + \frac {1}{3} \tan \beta}.
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$$
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By comparing the Mohr-Coulomb model with the linear Drucker-Prager model,
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$$
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\tan \beta = \frac {6 \sin \phi}{3 - \sin \phi},
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$$
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$$
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\sigma_ {c} ^ {0} = 2 c \frac {\cos \phi}{1 - \sin \phi},
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$$
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and, hence, from the previous result
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$$
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K = \frac {3 - \sin \phi}{3 + \sin \phi}.
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$$
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These results for $\beta , K .$ and $\sigma _ { c } ^ { 0 }$ provide linear Drucker-Prager parameters that match the Mohr-Coulomb model in triaxial compression and tension.
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<!-- source-page: 359 -->
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The value of K in the linear Drucker-Prager model is restricted to $K \geq 0 . 7 7 8$ for the yield surface to remain convex. The result for K shows that this implies $\phi \leq 2 2 ^ { \circ }$ . Many real materials have a larger Mohr-Coulomb friction angle than this value. One approach in such circumstances is to choose $K =$ and then to use the remaining equations to define $\beta$ and $\sigma _ { c } ^ { 0 }$ . This approach matches the models for triaxial compression only, while providing the closest approximation that the model can provide to failure being independent of the intermediate principal stress. If is significantly larger than 22°, this approach may provide a poor Drucker-Prager match of the Mohr-Coulomb parameters. Therefore, this matching procedure is not generally recommended; use the Mohr-Coulomb model instead.
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While using one-element tests to verify the calibration of the model, it should be noted that the Abaqus output variables SP1, SP2, and SP3 correspond to the principal stresses $\sigma _ { 3 } , \sigma _ { 2 }$ , and $\sigma _ { 1 }$ , respectively.
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# Creep models for the linear Drucker-Prager model
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Classical “creep” behavior of materials that exhibit plasticity according to the extended Drucker-Prager models 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 Drucker-Prager plasticity and Drucker-Prager hardening must be included in the material definition.
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Creep and plasticity can be active simultaneously, in which case the resulting equations are solved in a coupled manner. To model creep only (without rate-independent plastic deformation), large values for the yield stress should be provided in the Drucker-Prager hardening definition: the result is that the material follows the Drucker-Prager model while it creeps, without ever yielding. When using this technique, a value must also be defined for the eccentricity, since, as described below, both the initial yield stress and eccentricity affect the creep potentials. This capability is limited to the linear model with a von Mises (circular) section in the deviatoric stress plane $( K = 1 ; \mathrm { i } . \mathsf { e } _ { \cdot } ,$ no third stress invariant effects are taken into account) and can be combined only with linear elasticity.
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Creep behavior defined by the extended Drucker-Prager 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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The creep potential is hyperbolic, similar to the plastic flow potentials used in the hyperbolic and general exponent plasticity models. If creep properties are defined in Abaqus/Standard, the linear Drucker-Prager plasticity model also uses a hyperbolic plastic flow potential. As a consequence, if two analyses are run, one in which creep is not activated and another in which creep properties are specified but produce virtually no creep flow, the plasticity solutions will not be exactly the same: the solution with creep not activated uses a linear plastic potential, whereas the solution with creep activated uses a hyperbolic plastic potential.
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# Equivalent creep surface and equivalent creep stress
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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. When the material plastifies, it is desirable to have the equivalent creep surface coincide with the yield surface; therefore, we define the equivalent
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<!-- source-page: 360 -->
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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 23.3.1–13.
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<details>
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<summary>text_image</summary>
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q
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yield surface
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β
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material point
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equivalent creep
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surface
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σ̄cr
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no creep
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p
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</details>
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Figure 23.3.1–13 Equivalent creep stress defined as the shear stress.
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Abaqus/Standard requires that creep properties be described in terms of the same type of data used to define work hardening properties. The equivalent creep stress, , is then determined as follows:
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qcr $\bar { \sigma } ^ { c r } = \frac { ( q - p \tan \beta ) } { ( 1 - \frac 1 3 \tan \beta ) }$ if creep is defined in terms of the uniaxial compression stress, $\sigma _ { c } ;$
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$= { \frac { \left( q - p \tan \beta \right) } { \left( 1 + { \frac { 1 } { 3 } } \tan \beta \right) } }$ if creep is defined in terms of the uniaxial tension stress , $\sigma _ { t } \mathbf { ; }$
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$= \left( q - p \tan \beta \right)$ if creep is defined in terms of the cohesion,d.
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Figure 23.3.1–13 shows how the equivalent point is determined when the material properties are in shear, with stress d. A consequence of these concepts is that there is a cone in p–q space inside which creep is not active since any point inside this cone would have a negative equivalent creep stress.
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# Creep flow
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The creep strain rate in Abaqus/Standard is assumed to follow from the same hyperbolic potential as the plastic strain rate (see Figure 23.3.1–6):
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$$
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G ^ {c r} = \sqrt {(\epsilon \bar {\sigma} | _ {0} \tan \psi) ^ {2} + q ^ {2}} - p \tan \psi ,
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||
$$
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||
|
||
where
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|
||
$$
|
||
\psi (\theta , f _ {i})
|
||
$$
|
||
|
||
is the dilation angle measured in the p–q plane at high confining pressure;
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