김경종
385 lines
22 KiB
Markdown
385 lines
22 KiB
Markdown
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The functional dependence $\bar { \sigma } ( \bar { \varepsilon } ^ { p l } , \dot { \bar { \varepsilon } } ^ { p l } , \theta , f _ { i } )$ includes hardening as well as rate-dependent effects. The material data can be input either directly in a tabular format or by correlating it to static relations based on yield stress ratios.
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Rate dependence as described here is most suitable for moderate- to high-speed events in Abaqus/Standard. Time-dependent inelastic deformation at low deformation rates can be better represented by creep models. Such inelastic deformation, which can coexist with rate-independent plastic deformation, is described later in this section. However, the existence of creep in an Abaqus/Standard material definition precludes the use of rate dependence as described here.
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When using the Drucker-Prager material model, Abaqus allows you to prescribe initial hardening by defining initial equivalent plastic strain values, as discussed below along with other details regarding the use of initial conditions.
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# Direct tabular data
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Test data are entered as tables of yield stress values versus equivalent plastic strain at different equivalent plastic strain rates; one table per strain rate. Compression data are more commonly available for geological materials, whereas tension data are usually available for polymeric materials. The guidelines on how to enter these data are provided in “Rate-dependent yield,” Section 23.2.3.
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Input File Usage: \*DRUCKER PRAGER HARDENING, RATE=
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Suboptions→Drucker Prager Hardening: toggle on Use strain-rate-dependent data
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# Yield stress ratios
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Alternatively, the strain rate behavior can be assumed to be separable, so that the stress-strain dependence is similar at all strain rates:
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$$
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\overline {{\sigma}} = \sigma^ {0} (\bar {\varepsilon} ^ {p l}, \theta , f _ {i}) R (\dot {\bar {\varepsilon}} ^ {p l}, \theta , f _ {i}),
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$$
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where $\sigma ^ { 0 } ( \bar { \varepsilon } ^ { p l } , \theta , f _ { i } )$ is the static stress-strain behavior and $R ( \dot { \bar { \varepsilon } } ^ { p l } , \theta , f _ { i } )$ is the ratio of the yield stress at nonzero strain rate to the static yield stress (so that $R ( 0 , \theta , f _ { i } ) = 1 . 0 )$ .
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Two methods are offered to define R in Abaqus: specifying an overstress power law or defining the variable R directly as a tabular function of $\dot { \bar { \varepsilon } } ^ { p l }$ .
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# Overstress power law
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The Cowper-Symonds overstress power law has the form
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$$
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\dot {\bar {\varepsilon}} ^ {p l} = D (R - 1) ^ {n} \quad \mathrm{for} \quad \bar {\sigma} \geq \sigma^ {0},
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$$
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where $D ( \theta , f _ { i } )$ and $n ( \theta , f _ { i } )$ are material parameters that can be functions of temperature and, possibly, of other predefined field variables.
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Input File Usage: Use both of the following options: \*DRUCKER PRAGER HARDENING \*RATE DEPENDENT, TYPE=POWER LAW
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Suboptions→Drucker Prager Hardening; Suboptions→Rate Dependent: Hardening: Power Law
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# Tabular function
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When R is entered directly, it is entered as a tabular function of the equivalent plastic strain rate, $\dot { \bar { \varepsilon } } ^ { p l }$ ; temperature, ; and predefined field variables, $f _ { i }$ .
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Input File Usage: Use both of the following options: \*DRUCKER PRAGER HARDENING \*RATE DEPENDENT, TYPE=YIELD RATIO
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Suboptions→Drucker Prager Hardening; Suboptions→Rate Dependent: Hardening: Yield Ratio
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# Johnson-Cook rate dependence
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Johnson-Cook rate dependence has the form
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$$
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\dot {\bar {\varepsilon}} ^ {p l} = \dot {\varepsilon} _ {0} \mathrm{exp} \left[ \frac {1}{C} (R - 1) \right] \quad \mathrm{for} \quad \bar {\sigma} \geq \sigma^ {0},
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$$
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where $\dot { \varepsilon } _ { 0 }$ and C are material constants that do not depend on temperature and are assumed not to depend on predefined field variables.
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Input File Usage: \*DRUCKER PRAGER HARDENING \*RATE DEPENDENT, TYPE=JOHNSON COOK
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Suboptions→Drucker Prager Hardening; Suboptions→Rate Dependent: Hardening: Johnson-Cook
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# Stress invariants
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The yield stress surface makes use of two invariants, defined as the equivalent pressure stress,
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$$
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p = - \frac {1}{3} \mathrm{trace} (\pmb {\sigma}),
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$$
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and the Mises equivalent stress,
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$$
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q = \sqrt {\frac {3}{2} (\mathbf {S} : \mathbf {S})},
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$$
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where is the stress deviator, defined as
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$$
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\mathbf {S} = \pmb {\sigma} + p \mathbf {I}.
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$$
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In addition, the linear model also uses the third invariant of deviatoric stress,
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$$
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r = (\frac {9}{2} \mathbf {S} \cdot \mathbf {S}: \mathbf {S}) ^ {\frac {1}{3}}.
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$$
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# Linear Drucker-Prager model
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The linear model is written in terms of all three stress invariants. It provides for a possibly noncircular yield surface in the deviatoric plane to match different yield values in triaxial tension and compression, associated inelastic flow in the deviatoric plane, and separate dilation and friction angles.
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# Yield criterion
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The linear Drucker-Prager criterion (see Figure 23.3.1–1a) is written as
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$$
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F = t - p \tan \beta - d = 0,
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$$
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where
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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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$\beta ( \theta , f _ { i } )$ is the slope of the linear yield surface in the p–t stress plane and is commonly referred to as the friction angle of the material;
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$\pmb { d }$ is the cohesion of the material; and
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K(0,fi) is the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression and, thus, controls the dependence of the yield surface on the value of the intermediate principal stress (see Figure 23.3.1–2).
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In the case of hardening defined in uniaxial compression, the linear yield criterion precludes friction angles $\beta > 7 1 . 5 ^ { \circ } \left( \tan \beta > 3 \right)$ , which is unlikely to be a limitation for real materials.
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When , , which implies that the yield surface is the von Mises circle in the deviatoric principal stress plane (the -plane), in which case the yield stresses in triaxial tension and compression are the same. To ensure that the yield surface remains convex requires $0 . 7 7 8 \leq K \leq 1 . 0$ .
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The cohesion, $d ,$ of the material is related to the input data as
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<details>
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<summary>radar</summary>
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| Curve | K |
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|-------|-----|
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| a | 1.0 |
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| b | 0.8 |
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</details>
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Figure 23.3.1–2 Typical yield/flow surfaces of the linear model in the deviatoric plane.
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$$
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\begin{array}{l} d = (1 - \frac {1}{3} \tan \beta) \sigma_ {c} \quad \text {if hardening is defined by the uniaxial compression yield stress,} \sigma_ {c}; \\ = (\frac {1}{K} + \frac {1}{3} \tan \beta) \sigma_ {t} \quad \mathrm{ifhardeningisdefinedbytheuniaxialtensionyieldstress,} \sigma_ {t}; \\ = d \quad \text { if hardening is defined by the cohesion, } d = \frac {\sqrt {3}}{2} \tau (1 + \frac {1}{K}). \\ \end{array}
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$$
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# Plastic flow
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G is the flow potential, chosen in this model as
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$$
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G = t - p \tan \psi ,
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$$
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where $\psi ( \theta , f _ { i } )$ is the dilation angle in the p–t plane. A geometric interpretation of $\psi$ is shown in the p–t diagram of Figure 23.3.1–3. In the case of hardening defined in uniaxial compression, this flow rule definition precludes dilation angles $\psi > 7 1 . 5 ^ { \circ } ( \tan \psi > 3 )$ . This restriction is not seen as a limitation since it is unlikely this will be the case for real materials.
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For granular materials the linear model is normally used with nonassociated flow in the $_ { p - t }$ plane, in the sense that the flow is assumed to be normal to the yield surface in the -plane but at an angle $\psi$ to the t-axis in the $_ { p - t }$ plane, where usually $\psi < \beta .$ , as illustrated in Figure 23.3.1–3. Associated flow results from setting $\psi = \beta$ . The original Drucker-Prager model is available by setting $\psi = \beta$ and $K = 1$ . Nonassociated flow is also generally assumed when the model is used for polymeric materials. $\mathrm { I f } \psi = 0 $ , the inelastic deformation is incompressible; if $\dot { \psi } \geq 0$ , the material dilates. Hence, $\psi$ is referred to as the dilation angle.
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<details>
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<summary>text_image</summary>
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t
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dε^pl
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ψ
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β
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β
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d
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hardening
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p
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</details>
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Figure 23.3.1–3 Linear Drucker-Prager model: yield surface and flow direction in the p–t plane.
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The relationship between the flow potential and the incremental plastic strain for the linear model is discussed in detail in “Models for granular or polymer behavior,” Section 4.4.2 of the Abaqus Theory Guide.
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Input File Usage: \*DRUCKER PRAGER, SHEAR CRITERION=LINEAR
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Shear criterion: Linear
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# Nonassociated flow
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Nonassociated flow implies that the material stiffness matrix is not symmetric; therefore, the unsymmetric matrix storage and solution scheme should be used in Abaqus/Standard (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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# Hyperbolic and general exponent models
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The hyperbolic and general exponent models available are written in terms of the first two stress invariants only.
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# Hyperbolic yield criterion
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The hyperbolic yield criterion is a continuous combination of the maximum tensile stress condition of Rankine (tensile cutoff) and the linear Drucker-Prager condition at high confining stress. It is written as
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$$
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F = \sqrt {l _ {0} ^ {2} + q ^ {2}} - p \tan \beta - d ^ {\prime} = 0,
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$$
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where $l _ { 0 } = d ^ { \prime } | _ { 0 } - p _ { t } | _ { 0 }$ tan $\beta$ and
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<table><tr><td> $p_{t}|_{0}$ </td><td>is the initial hydrostatic tension strength of the material;</td></tr><tr><td> $d'(\bar{\sigma})$ </td><td>is the hardening parameter;</td></tr><tr><td> $d'|_{0}$ </td><td>is the initial value of $d'$ ; and</td></tr><tr><td> $\beta(\theta, f_{i})$ </td><td>is the friction angle measured at high confining pressure, as shown in Figure 23.3.1–1(b).</td></tr></table>
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The hardening parameter, $d ^ { \prime } ( \bar { \sigma } )$ , can be obtained from test data as follows:
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$d ^ { \prime } = \sqrt { l _ { 0 } ^ { 2 } + { \sigma _ { c } } ^ { 2 } } - \frac { \sigma _ { c } } { 3 } \tan \beta$ ifardeningiseld $\sigma _ { c } ;$
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$= \sqrt { l _ { 0 } ^ { 2 } + { \sigma _ { t } } ^ { 2 } } + \frac { \sigma _ { t } } { 3 } \tan \beta$ if hardening isfdbythuiaxialtesionyeldstres, $\sigma _ { t } ;$
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$= \sqrt { l _ { 0 } ^ { 2 } + d ^ { 2 } }$ if hardening is defined by the cohesion,d.
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The isotropic hardening assumed in this model treats $\beta$ as constant with respect to stress as depicted in Figure 23.3.1–4.
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<details>
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<summary>text_image</summary>
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hardening
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β
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q
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p
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l₀/tanβ
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l₀/tanβ
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l₀/tanβ
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</details>
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Figure 23.3.1–4 Hyperbolic model: yield surface and hardening in the $_ { p - q }$ plane.
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Input File Usage: \*DRUCKER PRAGER, SHEAR CRITERION=HYPERBOLIC
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Shear criterion: Hyperbolic
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# General exponent yield criterion
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The general exponent form provides the most general yield criterion available in this class of models. The yield function is written as
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$$
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F = a q ^ {b} - p - p _ {t} = 0,
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$$
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where
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<!-- source-page: 347 -->
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$$
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a (\theta , f _ {i}) \text { and } b (\theta , f _ {i})
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$$
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are material parameters that are independent of plastic deformation; and
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$$
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p _ {t} (\bar {\sigma})
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$$
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is the hardening parameter that represents the hydrostatic tension strength of the material as shown in Figure 23.3.1–1(c).
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$p _ { t } ( \bar { \sigma } )$ is related to the input test data as
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$$
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\begin{array}{l} p _ {t} = a \sigma_ {c} ^ {b} - \frac {\sigma_ {c}}{3} \quad \mathrm{ifhardeningisdefinedbytheuniaxialcompressionyieldstress,} \sigma_ {c}; \\ = a \sigma_ {t} ^ {b} + \frac {\sigma_ {t}}{3} \quad \text {if hardening is defined by the uniaxial tension yield stress,} \sigma_ {t}; \\ = a d ^ {b} \quad \text { if hardening is defined by the cohesion, } d. \\ \end{array}
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$$
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The isotropic hardening assumed in this model treats a and b as constant with respect to stress, as depicted in Figure 23.3.1–5.
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<details>
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<summary>text_image</summary>
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(\frac{p_t}{a})^{1/b} \quad q \quad \text{hardening} \quad p_t
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</details>
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Figure 23.3.1–5 General exponent model: yield surface and hardening in the p–q plane.
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The material parameters a and b can be given directly. Alternatively, if triaxial test data at different levels of confining pressure are available, Abaqus will determine the material parameters from the triaxial test data, as discussed below.
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Input File Usage: \*DRUCKER PRAGER, SHEAR CRITERION=EXPONENT FORM
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Drucker Prager: Shear criterion: Exponent Form
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# Plastic flow
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G is the flow potential, chosen in these models as a hyperbolic function:
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$$
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G = \sqrt {(\epsilon \bar {\sigma} | _ {0} \tan \psi) ^ {2} + q ^ {2}} - p \tan \psi ,
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$$
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where
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$$
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\psi (\theta , f _ {i})
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$$
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is the dilation angle measured in the p–q plane at high confining pressure;
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<!-- source-page: 348 -->
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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 flow 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. The value of will depend on the yield stress used.
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This flow potential, which is continuous and smooth, ensures that the 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 $9 0 ^ { \circ }$ . A family of hyperbolic potentials in the meridional stress plane is shown in Figure 23.3.1–6. The flow potential is the von Mises circle in the deviatoric stress plane (the -plane).
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<details>
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<summary>text_image</summary>
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dε^pl
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ψ
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q
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|εσ|0
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p
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</details>
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Figure 23.3.1–6 Family of hyperbolic flow potentials in the $_ { p - q }$ plane.
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For the hyperbolic model flow is nonassociated in the $_ { p - q }$ plane if the dilation angle, , and the material friction angle, $\beta ,$ are different. The hyperbolic model provides associated flow in the $_ { p - q }$ plane only when $\beta = \psi$ and $d ^ { \prime } | _ { 0 } / \tan \beta - p _ { t } | _ { 0 } = \epsilon \bar { \sigma } | _ { 0 }$ . A default value of $\epsilon = ( d ^ { \prime } | _ { 0 } - p _ { t } | _ { 0 }$ tan $\beta ) / ( \bar { \sigma } | _ { 0 }$ tan $\beta )$ is assumed if the flow potential is used with the hyperbolic model, so that associated flow is recovered when $\psi = \beta$ .
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For the general exponent model flow is always nonassociated in the $_ { p - q }$ plane. The default flow potential eccentricity is $\epsilon \ : = \ : 0 . 1$ , 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 flow potential, implying that the dilation angle increases more rapidly 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 flow potential locally where it intersects the p-axis.
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<!-- source-page: 349 -->
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The relationship between the flow potential and the incremental plastic strain for the hyperbolic and general exponent models is discussed in detail 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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Nonassociated flow implies that the material stiffness matrix is not symmetric; therefore, the unsymmetric matrix storage and solution scheme should be used in Abaqus/Standard (see “Defining an analysis,” Section 6.1.2). If the difference between $\beta$ and $\psi$ in the hyperbolic model is not large and if 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. In such cases the unsymmetric matrix scheme may not be needed.
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# Progressive damage and failure
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In Abaqus/Explicit the extended Drucker-Prager models can be used in conjunction with the models of progressive damage and failure discussed in “Damage and failure for ductile metals: overview,” Section 24.2.1. The capability allows for the specification of one or more damage initiation criteria, including ductile, shear, forming limit diagram (FLD), forming limit stress diagram (FLSD), and Müschenborn-Sonne forming limit diagram (MSFLD) criteria. After damage initiation, the material stiffness is degraded progressively according to the specified damage evolution response. The model offers two failure choices, including the removal of elements from the mesh as a result of tearing or ripping of the structure. The progressive damage models allow for a smooth degradation of the material stiffness, making them suitable for both quasi-static and dynamic situations.
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<table><tr><td>Input File Usage:</td><td>Use the following options:</td></tr><tr><td></td><td>* DAMAGE INITIATION</td></tr><tr><td></td><td>* DAMAGE EVOLUTION</td></tr></table>
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<table><tr><td>Abaqus/CAE Usage:</td><td>Property module: material editor: Mechanical→Damage for DuctileMetals→damage initiation type: specify the damage initiation criterion:Suboptions→Damage Evolution: specify the damage evolution parameters</td></tr></table>
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# Matching experimental triaxial test data
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Data for geological materials are most commonly available from triaxial testing. In such a test the specimen is confined by a pressure stress that is held constant during the test. The loading is an additional tension or compression stress applied in one direction. Typical results include stress-strain curves at different levels of confinement, as shown in Figure 23.3.1–7. To calibrate the yield parameters for this class of models, you need to decide which point on each stress-strain curve will be used for calibration. For example, if you wish to calibrate the initial yield surface, the point in each stress-strain curve corresponding to initial deviation from elastic behavior should be used. Alternatively, if you wish to calibrate the ultimate yield surface, the point in each stress-strain curve corresponding to the peak stress should be used.
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<!-- source-page: 350 -->
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<details>
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<summary>text_image</summary>
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-σ₃
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-σ₁
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-σ₂
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σ₃
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• points chosen to define
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shape and position of
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yield surface
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increasing
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confinement
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ε₃
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</details>
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Figure 23.3.1–7 Triaxial tests with stress-strain curves for typical geological materials at different levels of confinement.
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One stress data point from each stress-strain curve at a different level of confinement is plotted in the meridional stress plane (p–t plane if the linear model is to be used, or p–q plane if the hyperbolic or general exponent model will be used). This technique calibrates the shape and position of the yield surface, as shown in Figure 23.3.1–8, and is adequate to define a model if it is to be used as a failure surface (perfect plasticity). The models are also available with isotropic hardening, in which case hardening data are required to complete the calibration. In an isotropic hardening model plastic flow causes the yield surface to change size uniformly; in other words, only one of the stress-strain curves depicted in Figure 23.3.1–7 can be used to represent hardening. The curve that represents hardening most accurately over the range of loading conditions anticipated should be selected (usually the curve for the average anticipated value of pressure stress).
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As stated earlier, two types of triaxial test data are commonly available for geological materials. In a triaxial compression test the specimen is confined by pressure and an additional compression stress is superposed in one direction. Thus, the principal stresses are all negative, with $0 \geq \sigma _ { 1 } = \sigma _ { 2 } \geq \sigma _ { 3 }$ (Figure 23.3.1–9a). In the preceding inequality $\sigma _ { 1 } , \sigma _ { 2 }$ , and $\sigma _ { 3 }$ are the maximum, intermediate, and minimum principal stresses, respectively.
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The values of the stress invariants are
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|
||
$$
|
||
p = - \frac {1}{3} (2 \sigma_ {1} + \sigma_ {3}),
|
||
$$
|
||
|
||
$$
|
||
q = \sigma_ {1} - \sigma_ {3},
|
||
$$
|
||
|
||
and
|