$$ \sigma_ {m} = \frac {1}{2} (\sigma_ {1} + \sigma_ {3}) $$ is the average of the maximum and minimum principal stresses, and $\phi$ is the friction angle. For general states of stress the model is more conveniently written in terms of three stress invariants as $$ F = R _ {m c} q - p \tan \phi - c = 0, $$ where $$ R _ {m c} (\Theta , \phi) = \frac {1}{\sqrt {3} \cos \phi} \sin \left(\Theta + \frac {\pi}{3}\right) + \frac {1}{3} \cos \left(\Theta + \frac {\pi}{3}\right) \tan \phi , $$ $\phi$ is the slope of the Mohr-Coulomb yield surface in the $\pmb { p } { - } R _ { m c } q$ stress plane (see Figure 23.3.3–2), which is commonly referred to as the friction angle of the material and can depend on temperature and predefined field variables; c is the cohesion of the material; and Θ is the deviatoric polar angle defined as $$ \cos (3 \Theta) = \left(\frac {r}{q}\right) ^ {3}, $$ and $$ p = - \frac {1}{3} \operatorname{trace} (\boldsymbol {\sigma}) \quad \text { is the equivalent pressure stress }, $$ $$ q = \sqrt {\frac {3}{2} (\mathbf {S} : \mathbf {S})} \quad \text { is the Mises equivalent stress }, $$ $$ r = \left(\frac {9}{2} \mathbf {S} \cdot \mathbf {S}: \mathbf {S}\right) ^ {\frac {1}{3}} \quad \text { is the third invariant of deviatoric stress, and } $$ $$ \mathbf {S} = \boldsymbol {\sigma} + p \mathbf {I} \quad \text { is the deviatoric stress. } $$ The friction angle, $\phi ,$ controls the shape of the yield surface in the deviatoric plane as shown in Figure 23.3.3–2. The tension cutoff surface is shown for a meridional angle of $\Theta = 0 .$ . The friction angle range is $0 ^ { \circ } \leq \phi < 9 0 ^ { \circ }$ . In the case of $\phi = 0 ^ { \circ }$ the Mohr-Coulomb model reduces to the pressureindependent Tresca model with a perfectly hexagonal deviatoric section. In the case of $\phi \ : = \ : 9 0 ^ { \circ }$ the Mohr-Coulomb model reduces to the “tension cutoff” Rankine model with a triangular deviatoric section and $R _ { m c } = \infty$ (this limiting case is not permitted within the Mohr-Coulomb model described here). When using one-element tests to verify the calibration of the model, the output variables SP1, SP2, and SP3 correspond to the principal stresses $\sigma _ { 3 } , \sigma _ { 2 }$ , and $\sigma _ { 1 }$ , respectively. 

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Tension cutoff Rmc q Mohr-Coulomb φ c σt p Meridional plane


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Θ = 0 Mohr-Coulomb (φ = 20°) Tresca (φ = 0°) Rankine (φ = 90°) Θ = 4π/3 Θ = π/3 Drucker-Prager (Mises) Θ = 2π/3 Deviatoric plane

Figure 23.3.3–2 Mohr-Coulomb and tension cutoff surfaces in meridional and deviatoric planes. Isotropic cohesion hardening is assumed for the hardening behavior of the Mohr-Coulomb yield surface. The hardening curve must describe the cohesion yield stress as a function of plastic strain and, possibly, temperature and predefined field variables. In defining this dependence at finite strains, “true” (Cauchy) stress and logarithmic strain values should be given. An optional tension cutoff hardening (or softening) curve can be specified Rate dependency effects are not accounted for in this plasticity model. Input File Usage: Use the following options to specify the Mohr-Coulomb yield surface and cohesion hardening: \*MOHR COULOMB \*MOHR COULOMB HARDENING Abaqus/CAE Usage: Use the following options to specify the Mohr-Coulomb yield surface and cohesion hardening: Property module: material editor: Mechanical→Plasticity→Mohr Coulomb Plasticity Property module: material editor: Mechanical→Plasticity→Mohr Coulomb Plasticity: Cohesion # Rankine surface In Abaqus tension cutoff is modeled using the Rankine surface, which is written as $$ F _ {t} = R _ {r} (\Theta) q - p - \sigma_ {t} (\bar {\varepsilon_ {t}} ^ {p l}) = 0, $$ where $R _ { r } ( \Theta ) = ( 2 / 3 )$ , and $\sigma _ { t }$ is the tension cutoff value representing softening (or hardening) of the Rankine surface, as a function of tensile equivalent plastic strain, $\bar { \varepsilon _ { t } } ^ { p l }$ . Input File Usage: Use the following option to specify hardening or softening for the Rankine surface: \*TENSION CUTOFF Abaqus/CAE Usage: Use the following option to specify hardening or softening for the Rankine surface: Property module: material editor: Mechanical→Plasticity→Mohr Coulomb Plasticity: toggle on Specify tension cutoff; Tension Cutoff # Plastic behavior: flow potentials The flow potentials used for the Mohr-Coulomb yield surface and the tension cutoff surface are described below. # Plastic flow on the Mohr-Coulomb yield surface The flow potential, G, for the Mohr-Coulomb yield surface is chosen as a hyperbolic function in the meridional stress plane and the smooth elliptic function proposed by Menétrey and Willam (1995) in the deviatoric stress plane: $$ G = \sqrt {(\epsilon c | _ {0} \tan \psi) ^ {2} + (R _ {m w} q) ^ {2}} - p \tan \psi , $$ where $$ R _ {m w} (\Theta , e) = \frac {4 (1 - e ^ {2}) \cos^ {2} \Theta + (2 e - 1) ^ {2}}{2 (1 - e ^ {2}) \cos \Theta + (2 e - 1) \sqrt {4 (1 - e ^ {2}) \cos^ {2} \Theta + 5 e ^ {2} - 4 e}} R _ {m c} (\frac {\pi}{3}, \phi), $$ and $$ R _ {m c} (\frac {\pi}{3}, \phi) = \frac {3 - \sin \phi}{6 \cos \phi}, $$ $\psi$ is the dilation angle measured in the $\pmb { p } { - } R _ { m w } q$ plane at high confining pressure and can depend on temperature and predefined field variables; $c | _ { 0 }$ is the initial cohesion yield stress, $c | _ { 0 } = c | _ { \bar { \varepsilon } ^ { p l } = 0 } ;$ is the deviatoric polar angle defined previously; E is a parameter, referred to as the meridional eccentricity, that defines the rate at which the hyperbolic function approaches the asymptote (the flow potential tends to a straight line in the meridional stress plane as the meridional eccentricity tends to zero); and e is a parameter, referred to as the deviatoric eccentricity, that describes the “out-ofroundedness” of the deviatoric section in terms of the ratio between the shear stress along the extension meridian ( ) and the shear stress along the compression meridian $\begin{array} { r } { ( \Theta = { \frac { \pi } { 3 } } ) } \end{array}$ . A default value of is provided for the meridional eccentricity, . By default, the deviatoric eccentricity, e, is calculated as $$ e = \frac {3 - \sin \phi}{3 + \sin \phi}, $$ where is the Mohr-Coulomb friction angle; this calculation corresponds to matching the flow potential to the yield surface in both triaxial tension and compression in the deviatoric plane. Alternatively, Abaqus allows you to consider this deviatoric eccentricity as an independent material parameter; in this case you provide its value directly. Convexity and smoothness of the elliptic function requires that $1 / 2 < e \le 1$ . The upper limit, $e ~ = ~ 1 ~ ( \mathrm { o r } ~ \phi ~ = ~ 0 ^ { \circ }$ when you do not specify the value of e), leads to $R _ { m w } ( \Theta , e \mathrm { ~ = ~ }$ $\begin{array} { r } { { 1 } ) = R _ { m c } ( \frac { \pi } { 3 } , \phi ) } \end{array}$ , which describes the Mises circle in the deviatoric plane. The lower limit, $e = 1 / 2$ (or $\phi = 9 0 ^ { \circ }$ when you do not specify the value of e), leads to $R _ { m w } ( \Theta , e = 1 / 2 ) = 2 R _ { m c } ( \frac { \pi } { 3 } , \phi )$ and would describe the Rankine triangle in the deviatoric plane (this limiting case is not permitted within the Mohr-Coulomb model described here). This flow potential, which is continuous and smooth, ensures that the flow direction is always uniquely defined. A family of hyperbolic potentials in the meridional stress plane is shown in Figure 23.3.3–3, and the flow potential in the deviatoric stress plane is shown in Figure 23.3.3–4. 

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R_mw q dε^pl ψ εC|0 p

Figure 23.3.3–3 Family of hyperbolic flow potentials in the meridional stress plane. 

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Θ = 0 Rankine (e = 1/2) Θ = π/3 Menetrey-Willam (1/2 < e ≤ 1) Θ = 4π/3 Mises (e = 1) Θ = 2π/3

Figure 23.3.3–4 Menétrey-Willam flow potential in the deviatoric stress plane. Flow in the meridional stress plane can be close to associated when the angle of friction, , and the angle of dilation, , are equal and the meridional eccentricity, , is very small; however, flow in this plane is, in general, nonassociated. Flow in the deviatoric stress plane is always nonassociated. Input File Usage: Use the following option to allow Abaqus to calculate the value of e (default): \*MOHR COULOMB Use the following option to specify the value of e directly: \*MOHR COULOMB, DEVIATORIC ECCENTRICITY=e Abaqus/CAE Usage: Use the following option to allow Abaqus to calculate the value of e (default): Property module: material editor: Mechanical→Plasticity→Mohr Coulomb Plasticity: Plasticity: Deviatoric eccentricity: Calculated default Use the following option to specify the value of e directly: Property module: material editor: Mechanical→Plasticity→Mohr Coulomb Plasticity: Plasticity: Deviatoric eccentricity: Specify: e # Plastic flow on the Rankine surface A flow potential that results in a nearly associative flow is chosen for the Rankine surface and is constructed by modifying the Menétrey-Willam potential described earlier: $$ G _ {t} = \sqrt {\left(\epsilon_ {t} \sigma_ {t} | _ {0}\right) ^ {2} + \left(R _ {t} q\right) ^ {2}} - p, $$ where $$ R _ {t} (\Theta , e _ {t}) = \frac {1}{3} \frac {4 (1 - e _ {t} ^ {2}) \cos^ {2} \Theta + (2 e _ {t} - 1) ^ {2}}{2 (1 - e _ {t} ^ {2}) \cos \Theta + (2 e _ {t} - 1) \sqrt {4 (1 - e _ {t} ^ {2}) \cos^ {2} \Theta + 5 e _ {t} ^ {2} - 4 e _ {t}}}, $$ $\sigma _ { t } | _ { 0 }$ is the initial value of tension cutoff; $\epsilon _ { t }$ is the meridional eccentricity, similar to defined earlier; and $e _ { t }$ is the deviatoric eccentricity, similar to defined earlier. Abaqus uses values of and , for $\epsilon _ { t }$ and $\textstyle e _ { t } ,$ respectively. # Nonassociated flow Since the plastic flow is nonassociated in general, the use of this Mohr-Coulomb model generally requires the unsymmetric matrix storage and solution scheme in Abaqus/Standard (see “Defining an analysis,” Section 6.1.2). # Elements The Mohr-Coulomb plasticity model can be used with any stress/displacement elements other than onedimensional elements (beam, pipe, and truss elements) or elements for which the assumed stress state is plane stress (plane stress, shell, and membrane elements). # Output 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 are available for the Mohr-Coulomb plasticity model: PEEQ Equivalent plastic strain, $\begin{array} { r } { \bar { \varepsilon } ^ { p l } = \int \frac { 1 } { c } \pmb { \sigma } : d \varepsilon ^ { p l } } \end{array}$ , where c is the cohesion yield stress. PEEQT Tensile equivalent plastic strain, $\bar { \varepsilon _ { t } } ^ { p l }$ , on the tension cutoff yield surface. # Additional reference • Menétrey, Ph., and K. J. Willam, “Triaxial Failure Criterion for Concrete and its Generalization,” ACI Structural Journal, vol. 92, pp. 311–318, May/June 1995. # 23.3.4 CRITICAL STATE (CLAY) PLASTICITY MODEL Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE # References • “Material library: overview,” Section 21.1.1 • “Inelastic behavior,” Section 23.1.1 • \*CLAY PLASTICITY • \*CLAY HARDENING • \*POTENTIAL • \*SOFTENING REGULARIZATION • “Defining clay plasticity” in “Defining plasticity,” Section 12.9.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide • “Critical state models,” Section 4.4.3 of the Abaqus Theory Guide # Overview The clay plasticity model provided in Abaqus: • describes the inelastic behavior of the material by a yield function that depends on the three stress invariants, an associated flow assumption to define the plastic strain rate, and a strain hardening theory that changes the size of the yield surface according to the inelastic volumetric strain; • can have an isotropic or an anisotropic yield function; • requires that the elastic part of the deformation be defined by using the isotropic or orthotropic linear elastic material model (“Linear elastic behavior,” Section 22.2.1) or, in Abaqus/Standard, the porous elastic material model (“Elastic behavior of porous materials,” Section 22.3.1) within the same material definition (porous elasticity is supported only for isotropic yield functions); • allows for the hardening law to be defined by a piecewise linear form or, in Abaqus/Standard, by an exponential form; • may optionally include hardening in hydrostatic tension; and • can be used in conjunction with a regularization scheme for mitigating mesh dependence in situations where the material exhibits strain localization with increasing plastic deformation. # Yield surface The model is based on the yield surface $$ \frac {1}{\beta^ {2}} \left(\frac {p - p _ {t}}{a} - 1\right) ^ {2} + \left(\frac {\tilde {t}}{M a}\right) ^ {2} - 1 = 0, $$ where

$p = -\frac{1}{3} \text{trace } \sigma$	is the equivalent pressure stress;
$\tilde{t} =$	is the deviatoric stress measure;
$\frac{1}{2} \tilde{q} \left[ 1 + \frac{1}{K} - \left( 1 - \frac{1}{K} \right) \cos 3 \Theta \right]$
$\Theta$	is the deviatoric polar angle defined as $\cos 3 \Theta = \left( \frac{r}{q} \right)^{3}$ ;
$\tilde{q}$	is the Hill’s potential, as defined in “Anisotropic yield/creep,” Section 23.2.6;
$q = \sqrt{\frac{3}{2} \mathbf{S} : \mathbf{S}}$	is the Mises equivalent stress;
$r = (\frac{9}{2} \mathbf{S} : \mathbf{S} \cdot \mathbf{S})^{\frac{1}{3}}$	is the third stress invariant;
$M$	is a constant that defines the slope of the critical state line;
$\beta$	is a constant that is equal to 1.0 on the “dry” side of the critical state line ( $p < a + p_t$ ) but may be different from 1.0 on the “wet” side of the critical state line ( $\beta \neq 1.0$ introduces a different ellipse on the wet side of the critical state line; i.e., a tighter “cap” is obtained if $\beta < 1.0$ as shown in Figure 23.3.4–1);
$a = \frac{p_c - p_t}{(1 + \beta)}$	is a measure of the size of the yield surface (Figure 23.3.4–1);
$p_c$	is the yield stress in hydrostatic compression;
$p_t$	is the yield stress in hydrostatic tension; and
$K$	is the ratio of the flow stress in triaxial tension to the flow stress in triaxial compression and determines the shape of the yield surface in the plane of principal deviatoric stresses (the “II-plane”: see Figure 23.3.4–2); Abaqus requires that $0.778 \leq K \leq 1.0$ to ensure that the yield surface remains convex.

The user-defined parameters M, , and K can depend on temperature as well as other predefined field variables, . For the isotropic model, the expression for reduces to the Mises equivalent stress, . The model is described in detail in “Critical state models,” Section 4.4.3 of the Abaqus Theory Guide. Input File Usage: Use the following option to define the isotropic model: \*CLAY PLASTICITY Use the following options to define the orthotropic model: \*CLAY PLASTICITY \*POTENTIAL Abaqus/CAE Usage: Use the following option to define the isotropic model: Property module: material editor: Mechanical→Plasticity→Clay Plasticity Use the following options to define the orthotropic model: Mechanical→Plasticity→Clay Plasticity: Suboptions→Potential