3.1.7.9. Pressure Independent Multi Yield

Code Developed by: UC San Diego (Dr. Zhaohui Yang):

PressureIndependMultiYield material is an elastic-plastic material in which plasticity exhibits only in the deviatoric stress-strain response. The volumetric stress-strain response is linear-elastic and is independent of the deviatoric response. This material is implemented to simulate monotonic or cyclic response of materials whose shear behavior is insensitive to the confinement change. Such materials include, for example, organic soils or clay under fast (undrained) loading conditions.

During the application of gravity load (and static loads if any), material behavior is linear elastic. In the subsequent dynamic (fast) loading phase(s), the stress-strain response is elastic-plastic (see MATERIAL STAGE UPDATE below). Plasticity is formulated based on the multi-surface (nested surfaces) concept, with an associative flow rule. The yield surfaces are of the Von Mises type.

The command to create the material object is:

function

nDmaterial PressureIndependMultiYield $tag $nd $rho $refShearModul $refBulkModul $cohesi $peakShearStra <$frictionAng=0. $refPress=100. $pressDependCoe=0. $noYieldSurf=20 <$r1 $Gs1 …> >

Table 3.1.7.1 Arguments

Argument	Type	Description
$tag	integer	A positive integer uniquely identifying the material among all nDMaterials.
$nd	integer	Number of dimensions, 2 for plane-strain, and 3 for 3D analysis.
$rho	float	Saturated soil mass density.
$refShearModul (Gr)	float	Reference low-strain shear modulus, specified at a reference mean effective confining pressure refPress of p’r (see below).
$refBulkModul (Br)	float	Reference bulk modulus, specified at a reference mean effective confining pressure refPress of p’r (see below).
$cohesi (c)	float	Apparent cohesion at zero effective confinement.
$peakShearStra (γmax)	float	An octahedral shear strain at which the maximum shear strength is reached, specified at a reference mean effective confining pressure refPress of p’r (see below).
$frictionAng (Φ)	float	Friction angle at peak shear strength in degrees. (optional: default is 0.0).
$refPress (p’r)	float	Reference mean effective confining pressure at which Gr and Br and γmax are defined, optional (default is 100. kPa).
$pressDependCoe (d)	float	A positive constant defining variations of G and B as a function of instantaneous effective confinement p’(default is 0.0). see notes 4 and 5 below.
$noYieldSurf	integer	Number of yield surfaces, optional (must be less than 40: default is 20). The surfaces are generated based on the hyperbolic relation defined in Note 2 below.
$r $Gs	float	Instead of automatic surfaces generation (Note 2), you can define yield surfaces directly based on desired shear modulus reduction curve. To do so, add a minus sign in front of noYieldSurf, then provide noYieldSurf pairs of shear strain (γ) and modulus ratio (Gs) values. For example, to define 10 surfaces: … -10γ1Gs1 … γ10Gs10 …

Note

1. The friction angle \(\phi\) and cohesion c define the variation of peak (octahedral) shear strength \(\tau_f\) as a function of current effective confinement \({p^t}_i\):

\[\tau_f = \frac{2 \sqrt{2} sin \phi}{3 - sin \phi}{p^t}_i + \frac{2 \sqrt{2}}{3}c\]

2. Automatic surface generation: at a constant confinement \(p^t\), the shear stress \(\tau\) (octahedral) - shear strain \(\gamma\) (octahedral) nonlinearity is defined by a hyperbolic curve (backbone curve):

\[\tau = \frac{G \gamma}{1 + \frac{\gamma}{\gamma_r}\left ( \frac{{p^t}_r}{p^t} \right)^d}\]

where \(\gamma_r\) satisfies the following equation at \({p^t}_r\)

\[\tau_f = \frac{2 \sqrt{2} sin \phi}{3 - sin \phi}{p^t}_r + \frac{2 \sqrt{2}}{3}c = \frac{G_r \gamma_{max}}{1 + \gamma_{max}/\gamma_r}\]

3. (User defined surfaces) The user specified friction angle \(\phi = 0\). cohesion c will be ignored. Instead, c is defined by \(c=\sqrt 3 \sigma_m / 2\), where \(\sigma_m\) is the product of the last modulus and strain pair in the modulus reduction curve. Therefore, it is important to adjust the backbone curve so as to render an appropriate c.

If the user specifies \(\gamma\) > 0, this \(\phi\) will be ignored. Instead, \(\phi\) is defined as follows:

\[sin \phi = \frac{3 (\sqrt 3 \sigma_m - 2c)/{p^t}_r}{6 + (\sqrt 3 \sigma_m - 2c)/{p^t}_r}\]

If the resulting \(\phi <0\), we set \(\phi =0\) and \(c=\sqrt 3 \sigma_m/2\).

Also remember that improper modulus reduction curves can result in strain softening response (negative tangent shear modulus), which is not allowed in the current model formulation. Finally, note that the backbone curve varies with confinement, although the variation is small within commonly interested confinement ranges. Backbone curves at different confinements can be obtained using the OpenSees element recorder facility

4. $presDependCoef d defines variations of G and B as a function of instantaneous effective confinement :math:p^t as follows:

\[ \begin{align}\begin{aligned}G = G_r \left ( \frac{p^t}{{p^t}_r} \right)^d\\B = B_r \left ( \frac{p^t}{{p^t}_r} \right)^d\end{aligned}\end{align} \]

5. If \(\phi = 0.0\), d is reset to 0.0.

6. OUTPUT The following information may be extracted for this material at a given integration point, using the OpenSees Element Recorder facility “stress”, “strain”, “backbone”, or “tangent”.

   • For 2D problems, the stress output follows this order: \(\sigma_{xx}\), \(\sigma_{yy}\), \(\sigma_{zz}\), \(\sigma_{xy}\),:math:eta_r, where \(\eta_r\) is the ratio between the shear (deviatoric) stress and peak shear strength at the current confinement \((0<=\eta_r<=1.0)\). The strain output follows this order: \(\epsilon_{xx}\), \(\epsilon_{yy}\), \(\epsilon_{xy}\)

   • For 3D problems, the stress output follows this order: \(\sigma_{xx}\), \(\sigma_{yy}\), \(\sigma_{zz}\), \(\sigma_{xy}\),:math:sigma_{yz}, \(\sigma_{zx}\), \(\eta_r\) and the strain output follows this order: \(\epsilon_{xx}\), \(\epsilon_{yy}\), \(\epsilon_{zz}\), \(\gamma_{xy}\), \(\gamma_{yz}\), \(\gamma_{zx}\)

   • The “backbone” option records (secant) shear modulus reduction curves at one or more given confinements. The specific recorder command is as follows:

recorder Element –ele $eleNum -file $fName -dT $deltaT material $GaussNum backbone $p1 <$p2 …>

where p1, p2, … are the confinements at which modulus reduction curves are recorded. In the output file, corresponding to each given confinement there are two columns: shear strain γ and secant modulus Gs. The number of rows equals the number of yield surfaces.

** SUGGESTED PARAMETER VALUES **

Parameters	Soft Clay	Medium Clay	Stiff Clay
rho	1.3 ton/m3 or 1.217x10-4(lbf)(s2)/in4	1.5 ton/m3 or 1.404x10-4(lbf)(s2)/in4	1.8 ton/m3 or 1.685x10-4(lbf)(s2)/in4
refShearModul	1.3x104kPa or 1.885x103psi	6.0x104kPa or 8.702x104psi	1.5x105kPa or 2.176x104 psi
refBulkModu	6.5x104kPa or 9.427x103psi	3.0x105kPa or 4.351x104psi	7.5x105kPa or 1.088x105psi
cohesi	18kPa or 2.611psi	37kPa or 5.366psi	75kPa or 10.878psi
peakShearStra (at p’r=80 kPa or 11.6 psi)	0.1	0.1	0.1
frictionAng	0	0	0
pressDependCoe	0	0	0

** Pressure Dependent Multi Yield Examples **