Hypo-Plasticity

In hypoplasticity a direct relation is used between strain rates and effective stress rates. Rigid body rotations (objectivity) are treated elsewhere (see the section on memory). The effective stress tensor $\sigma_{ij}$ follows from the total stress tensor $\sigma_{ij}$ minus any pore pressures (see groundflow). The Masin law is tuned to clays. The Wolffersdorff law is tuned to sands. The Niemunis visco law describes time dependent soil behaviour.

Masin law

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The law proposed by MASIN [10] and [11] is used.

The constitutive equation in rate form reads:

$$\mathaccent''7017{\mbox{\bf T}}={\boldsymbol{\mathcal L}}:\mbox{\bf D}+ f_d \mbox{\bf N}\Vert\mbox{\bf D}\Vert$$ (5)

where D is the Euler's stretching tensor, T is the Cauchy stress tensor and

$${\boldsymbol{\mathcal L}} =3 f_s \left(c_1{\boldsymbol{\mathcal I}}+c_2a^2\hat{\mbox{\bf T}}\otimes\hat{\mbox{\bf T}}\right) = \boldsymbol{\mathcal L}:\left(-Y\frac{\mbox{\bf m}}{\Vert\mbox{\bf m}\Vert}\right) \hat{\mbox{\bf T}} =\frac{\mbox{\bf T}}{\operatorname{tr}\mbox{\bf T}}$$ N (6)

$\boldsymbol{1}$ is the second-order identity tensor and $\boldsymbol{\mathcal I}$ is the fourth-order identity tensor, with components:

$$\left(\mathcal{I}\right)_{ijkl}=\frac{1}{2}\,\left(1_{ik}1_{jl}+1_{il}1_{jk}\right)$$ (7)

The functions $f_s(\operatorname{tr}{\mbox{\bf T}})$ (barotropy factor) and $f_d(\operatorname{tr}{\mbox{\bf T}},e)$ (pyknotropy factor) are given by:

$$f_s =-S_{i} \frac{{\rm tr}\mbox{\bf T}}{\lambda^\ast}\left(3+a^2-2^\alpha a\sqrt{3}\right)^{-1} f_d =\left[-\ \frac{2{\rm tr}\mbox{\bf T}}{3sp_r}\exp\left(\frac{\ln\left(1+e\right)-N}{\lambda^\ast}\right)\right]^\alpha$$ (8)

where $p_r$ is the reference stress for the parameter $N$, typically taken as 1 kPa, and the factor $S_i$ is a function of sensitivity $s$:

$$S_{i}=\frac{s-k(s-s_f)}{s}$$ (9)

The scalar function $Y$ and the second-order tensor m are given, respectively, by:

$$Y=\left(\frac{\sqrt{3}a}{3+a^2}-1\right)\frac{ \left(I_1I_2+9I_3\... ...t)\left(1-\sin^2\varphi_c\right) }{8I_3\sin^2\varphi_c}+\frac{\sqrt{3}a}{3+a^2}$$ (10)

in which:

$$I_1 ={\rm tr} <tex2html_comment_mark>1 I_2 =\frac{1}{2}\left[\mbox{\bf T}:\mbox{\bf T}-\left(I_1\right)^2\right] <tex2html_comment_mark>2 I_3 =\det$$

and

$$=-\ \frac{a}{F}\left[ \hat{\mbox{\bf T}}+\hat{\mbox{\bf T}}^\ast-... ...}}-1} {\left(F/a\right)^2+\hat{\mbox{\bf T}}:\hat{\mbox{\bf T}}}\right) \right]$$ (11)

in which:

$$\hat{\mbox{\bf T}}^\ast =\hat{\mbox{\bf T}}-\frac{\boldsymbol{1}}{3} F =\sqrt{\frac{1}{8}\tan^2\psi+\frac{2-\tan^2\psi}{2+\sqrt{2}\tan\psi\cos3\theta}}- \frac{1}{2\sqrt{2}}\tan{\psi}$$ (12)

$$\tan\psi =\sqrt{3}\Vert\hat{\mbox{\bf T}}^\ast\Vert \cos3\theta =-\sqrt{6} \frac{{\rm tr}\left(\hat{\mbox{\bf T}}^\ast\cdot\hat{\... ...st\right)} {\left(\hat{\mbox{\bf T}}^\ast:\hat{\mbox{\bf T}}^\ast\right)^{3/2}}$$ (13)

Finally, the scalars $a$, $\alpha$, $c_1$ and $c_2$ are given as functions of the material parameters $\varphi _c$, $\lambda^*$, $\kappa^*$ and $r$ by the following relations:

$$a =\frac{\sqrt{3}\left(3-\sin{\varphi_c}\right)}{2\sqrt{2}\sin{\varphi_c}} <tex2html_comment_mark>3 \alpha =\frac{1}{\ln2}\,\ln\left[\frac{\lambda^\ast-\kappa^\ast S_{i}} {\lambda^\ast+\kappa^\ast S_{i}}\left(\frac{3+a^2}{a\sqrt{3}}\right)\right]$$ (14)

$$c_1 =\frac{2\left(3+a^2-2^\alpha a\sqrt{3}\right)}{9rS_{i}} <tex2html_comment_mark>4 c_2 =1+(1-c_1)\frac{3}{a^2}$$ (15)

Evolution of the state variables $e$ (void ratio) and $s$ (sensitivity) is governed by

$$\dot{e}=\left(1+e\right)\operatorname{tr}$$ (16)

$$\dot{s}=-\frac{k}{\lambda^\ast}(s-s_f)\sqrt{\left(\dot{\epsilon}_v\right)^2+ \frac{A}{1-A}\left(\dot{\epsilon}_s\right)^2}$$ (17)

where $\dot{\epsilon}_v=\operatorname{tr}$D and $\dot{\epsilon}_s=\sqrt{2/3}\Vert{\rm dev}\ {\mbox{\bf D}}\Vert$.

The basic hypoplastic model requires five constitutive parameters, namely $\varphi _c$, $\lambda ^\ast$, $\kappa ^\ast$, $N$ and $r$, state is characterised by the Cauchy stress T and void ratio $e$.

An extended model allows us to take into account the effects of meta-stable structure of natural clays. This extension requires three additional parameters ($k$, $A$, $s_f$), and one additional state variable $s$. A basic model without the structure effects is recovered if $s=s_f=1$ and $A\neq1$. The $s$ should be always greater or equal to 1.

Table 1: Typical parameters of the hypoplastic model for clays.

	$\varphi _c$	$\lambda^*$	$\kappa^*$	$N$	$r$	$k$	$A$	$s_f$
London clay	22.6$^\circ$	0.11	0.016	1.375	0.4	-	-	-
Pisa clay	21.9$^\circ$	0.14	0.0075	1.56	0.3	0.4	0.1	1

The basic law parameters should be specified in group_materi_plasti_hypo_masin. The extended parameters for the structure should be specified in group_materi_plasti_hypo_masin_structure. The hypoplastic history variables, $e$ for this basic model, and $e$ and $s$ for the extended model, should be initialised with materi_plasti_hypo_history. As an alternative to specify the $e$ you can specify the ${\rm OCR}$ at the start of the calculation in group_materi_plasti_hypo_masin_ocr (which is used to determine the initial $e$ via $e = exp( N - \lambda^* ~ ln(\vert OCR\vert) - \lambda^* ~ ln( \vert p / p_r \vert ) ) - 1.$).

Wolffersdorff law

The law proposed by WOLFFERSDORFF [18] is used.

$$\dot{\sigma}_{ij} = L_{ijkl} \dot{\epsilon}_{ij} + f_d N_{ij} \... ...epsilon}_{kl} } = L_{ijkl} ( d_{kl} - f_d Y m_{kl} \vert\vert d \vert\vert )$$

Here the part with $L_{ijkl}$ gives a linear relation between strain rates and stress rates and the part with $N_{ij}$ gives a nonlinear relation. The constitutive tensors $L_{ijkl}$ and $f_d N_{ij}$ are functions of the effective stress tensor $\sigma_{ij}$ and void ratio $e$. In the above $d$ denotes the strain rate tensor $\epsilon$, $Y$ denotes the degree of nonlinearity $Y = \vert\vert L^{-1} : N \vert\vert$ and the flowrule $m$ is defined by $m = - ( L^{-1} : N ) ^ {\rightarrow}$ where a ${\rightarrow}$ denotes euclidian normalisation.

$$L_{ijkl} = f_b f_e \displaystyle \frac{1}{\hat{\sigma}_{mn}\hat{\sigma}_{mn}} \hat{L_{ijkl}}$$

$$N_{ij} = f_b f_e \, \displaystyle \frac{F\,a}{\hat{\sigma}_{kl}\hat{\sigma}_{kl}} \left( \,\hat{\sigma}_{ij} \,+ \, \hat{\sigma}^*_{ij} \, \right)$$

$${\rm and} \quad \hat{\sigma}_{ij} = \sigma_{ij} / (\sigma_{mn}\de... ... - \frac{1}{3}\,\delta_{ij} \quad , \quad I_{ijkl} = \delta_{ik} \delta_{jl} ~,$$

$$a = \displaystyle \frac{\sqrt{3}(3-\sin\varphi_c)}{2\sqrt{2}\sin\varphi_c}$$

$$F \,=\, \sqrt{\displaystyle \frac{1}{8}\tan^2\psi+\frac{2-\tan^2\... ... {2+\sqrt{2}\tan\psi\cos 3\theta}}-\displaystyle \frac{1}{2\sqrt{2}}\tan\psi ~,$$

$$\quad \tan\psi=\sqrt{3}\sqrt{\hat{\sigma}^*_{ij}\hat{\sigma}^*_{i... ...}^*_{ki}} {\left[ \hat{\sigma}^*_{mn}\hat{\sigma}^*_{mn} \right]^{3/2}} \quad .$$

For the $\hat{L_{ijkl}}$ above we have:

$$\hat{L}_{ijkl} = \left( F^2\,I_{ijkl} + a^2 \, \hat{\sigma}_{ij}\hat{\sigma}_{kl} \right) \\$$

For $\hat{\sigma}^*_{ij}=0$ is $F=1$.

The scalar factors $f_b$, $f_e$ and $f_d$ take into account the influence of mean pressure and density:

$$f_b = \displaystyle \frac{h_s}{n}\,\left(\displaystyle \frac{e_{i0}}{e_... ...tyle \frac{e_{i0}-e_{d0}}{e_{c0}-e_{d0}}\;\right)^{\alpha} \right]^{-1} \quad ,$$

$$f_d = \,\left(\displaystyle \frac{e-e_d}{e_c-e_d}\;\right)^{\alpha} \quad .$$

and $f_e = \left( \frac{e_c}{e} \right) ^{\beta}$.

Three characteristic void ratios - $e_i$ (during isotropic compression at the minimum density), $e_c$ (critical void ratio) and $e_d$ (maximum density) - decrease with mean stress:

$$\displaystyle \frac{e_i}{e_{i0}} = \frac{e_c}{e_{c0}} = \frac{e_d... ...} = \exp \left[ - \left(-\frac{\sigma_{ij}\delta_{ij}}{h_s} \right)^n \right]$$

The range of admissible void ratios is limited by $e_i$ and $e_d$. The model parameters can be found in Tab. 2. They correspond to Hochstetten sand from the vicinity of Karlsruhe, Germany [18].

Table 2: Basic hypoplastic parameters of Hochstetten sand.

$\varphi$ [$^{\circ}$]	$h_s$ [MPa]	$n$	$e_{c0}$	$e_{d0}$	$e_{i0}$	$\alpha$	$\beta$
33	1000	0.25	0.95	0.55	1.05	0.25	1.0

The basic law parameters should be specified in group_materi_plasti_hypo_wolffersdorff. The hypoplastic history variables should be initialised with materi_plasti_hypo_history.

Visco law

For visco hypoplasticity with intergranular strains the stress rate reads:

$$\dot{\sigma}_{ij} = M_{ijkl}\dot{\epsilon}_{kl} - L_{ijkl} \dot{\epsilon}^{vis}_{kl}$$

For visco hypoplasticity the $L_{ijkl}$ reads:

$$L_{ijkl} = f_b \hat{L}_{ijkl}$$

where

$$f_b = \frac{ - \sigma_{kk} }{ ( 1 + a^2/3 ) \kappa }$$

where $\kappa$ is a user specified material constant $\kappa$ (= Butterfield's swelling index upon isotropic unloading), and $a$ relates to the user specified residual (=critical) friction angle $\varphi _c$ as:

$$a = \frac{\sqrt{3}(3-\sin\varphi_c )}{2\sqrt{2}\sin\varphi_c }$$

The pressure normalised stiffness is:

$$\hat{L}_{ijkl} = F^2\,I_{ijkl} + a^2 \, \hat{\sigma}_{ij}\hat{\sigma}_{kl} + b^2 ( I_{ijkl} - \frac{1}{3} I_{ikjl} )$$

where

$$b^2 = \frac{ (1+\frac{1}{3}a^2)(1 - 2 \nu) }{ 1+\nu } - 1$$

For visco hypoplasticity the $M_{ijkl}$ reads:

$$M_{ijkl} = [ \rho^{\chi} m_T + (1-\rho^{\chi})m_R ] L_{ijkl} +$$

$$+ \left\{ \begin{array}{lll} \rho^{\chi} (1-m_T) L_{ijmn} \hat{S}_{... ...l} & {\rm for} & \hat{S}_{ij}\dot{\epsilon}_{ij} \leq 0 \\ \end{array} \right.$$

where $\hat{S}$ intergranular strains are the same as in the formulation without viscosity.

The viscosity strain rate is assumed to be:

$$\dot{\epsilon}^{vis}_{ij} = D_r \hat{m}_{ij} ( \frac{1}{OCR} ) ^ {\frac{1}{I_v}}$$

where the normalised flow rule $\hat{m}_{ij}$ is

$$\hat{m}_{ij} = \frac{ {m}_{ij} }{ \sqrt{ {m}_{ij} {m}_{ij} } }$$

with

$${m}_{ij} = - \left[ \frac{F^2}{a^2} ( \hat{\sigma}_{ij} + \hat{\s... ...gma}_{ij}^* - \hat{\sigma}_{ij} \hat{\sigma}_{kl} \hat{\sigma}_{kl}^* \right]$$

The over-consolidation ratio ${\rm OCR}$ appearing in the expression for the viscous creep rate is a function of the effective stress $\sigma_{ij}$ and of the void ratio $e$

$${\rm OCR} = \frac{ p_e }{ {p_e}^+ }$$

wherein the void ratio is hidden in the equivalent pressure $p_e$ and ${p_e}^+$ is a special stress invariant.

The equivalent pressure $p_e$ is calculated from

$$\ln \left( \frac{ 1 + e_{e0} }{ 1 + e } \right) = \lambda \ln \left( \frac{ p_e }{ p_{e0} } \right)$$

with a user specified material constant $\lambda$ (= Butterfield's first compression index) and also user-specified reference parameters $e_{e0}$, $p_{e0}$ which describe any pair of the void ratio and the effective pressure registered upon an isotropic $D_r$-isotach, i.e. during an isotropic first (= virgin) compression test with a constant volumetric rate of deformation equal to $-\sqrt{3} D_r \frac{\lambda}{\lambda - \kappa}$.

The stress invariant ${p_e}^+$ is calculated using

$${p_e}^+ = \begin{cases}\frac{p}{\beta_R - 1} \left[ \beta_R \sqrt... ... p ( 1 + { \eta }^2 ) \frac{ 1 + \beta_R }{ 2 } & \text{otherwise} \end{cases}$$

wherein

$$\eta = q/(M p) ~~$$and$$~~~ M = \frac{6 F \sin\varphi_c}{3- \sin \varphi_c }$$

where $p = -\sigma_{kk}/3$ and $q = \sqrt{ \frac32 \sigma^*_{kl} \sigma^*_{kl} }$ are the popular Roscoe's stress invariants. and $\beta _R$ (= flattening factor for the Rendulic's cap) are the user supplied material constants.

You can specify an initial value of the void ration $e_0$ in -hyhis0 with control_reset_dof. Then the ${\rm OCR}$ can be calculated with the above equations. As an alternative you can specify the ${\rm OCR}$ at the start of the calculation in group_materi_plasti_hypo_niemunis_visco_ocr; then the initial void ratio will be calculated as follows: $p_e^+$ will be determined from the equation above, then $p_e$ is determined from $p_e = {\rm OCR} p_e^+$ and then the initial void ratio $e_0$ is determined from $e_0 = (1+e_e0) * {(p_e/p_{e0})}^{- \lambda} - 1.$ (reference: Niemunis communications). Application of the specified OCR is triggered by control_materi_plasti_hypo_niemunis_visco_ocr_apply.

User parameters should be specified in group_materi_plasti_hypo_niemunis_visco.

Cohesion extension

A simplistic approach to include cohesion is used here. Instead of feeding the real effective stress state $\sigma_{ij}$ into the hypoplastic law, an alternative effective stress state $\sigma_{ij}^c$ is used. Cohesion is modeled by subtracting in each of the normal stress components a value $c$ representing cohesion: $\sigma_{11}^c = \sigma_{11} - c$, $\sigma_{22}^c = \sigma_{22} - c$ and $\sigma_{33}^c = \sigma_{33} - c$. The shear stresses are not altered: $\sigma_{12}^c = \sigma_{12}$, etc.

The cohesion value should be specified in group_materi_plasti_hypo_cohesion.

Intergranular strains extension

In order to take into account the recent deformation history, an additional tensorial state variable $S_{ij}$¹ is introduced.

Denoting the normalized magnitude of $S_{ij}$

$$\rho = \displaystyle \frac{\sqrt{S_{ij}S_{ij}}}{R}$$

(R is a material parameter) and the direction of $S_{ij}$

$$\hat{S}_{ij} = \displaystyle \frac{S_{ij}}{\sqrt{S_{kl}S_{kl}}}$$

( $\hat{S}_{ij}=0$ for $S_{ij}=0$), the evolution equation for the intergranular strain tensor reads:

$$\dot{S}_{ij} = \left\{ \begin{array}{lll} ( I_{ijkl}-\rho^{\beta... ...for} & \hat{S}_{ij}\dot{\epsilon}_{ij} \leq 0 \\ \end{array} \right. \quad ,$$

where $\dot{S}_{ij}$ is the objective rate of intergranular strain. Rigid body rotations are treated elsewhere (see the section on memory). From the evolution equation (3.2.4) it follows that $\rho$ must remain between 0 and 1.

The general stress-strain relation is now written as

$$\dot{\sigma}_{ij} = M_{ijkl}\dot{\epsilon}_{kl} \quad .$$

The fourth order tensor $M_{ijkl}$ represents the incremental stiffness and is calculated from the hypoplastic tensors $L_{ijkl}$ and $N_{ij}$ which may be modified by scalar multipliers $m_T$ and $m_R$, depending on $\rho$ and on the product $\hat{S}_{ij}\dot{\epsilon}_{ij}$:

$$M_{ijkl} = [ \rho^{\chi} m_T + (1-\rho^{\chi})m_R ] L_{ijkl} +$$

$$+ \left\{ \begin{array}{lll} \rho^{\chi} (1-m_T) L_{ijmn} \hat{S}_{... ...l} & {\rm for} & \hat{S}_{ij}\dot{\epsilon}_{ij} \leq 0 \\ \end{array} \right.$$

$\chi$ and $\gamma$ are additional material parameters.

An example intergranular parameters can be found in Tab. 3.

Table 3: Example of Intergranular hypoplastic parameters.

$R$	$m_R$	$m_T$	$\beta _x$	$\chi$	$\gamma$
$1\cdot 10^{-4}$	5.0	2.0	0.50	6.0	6.0

The intergranular parameters should be specified in group_materi_plasti_hypo_strain_intergranular. Additionally you need to include materi_strain_intergranular in the initialisation part.

The additional parameter gamma is very important only for the accumulation of permanent displacements or pore pressures in cyclic or dynamic analysis with small strains. For monotonic loading or higher strains gamma is not very important. And thus for such monotonic loading or higher strains you should take $\gamma = \chi$.

Pressure dependent initial void ratio extension

You can correct the initial void ratio $e_{0}$, as specified in the initial value for the history variable in the node_dof records, for the initial pressure to obtain a corrected initial void ratio $e$.

$$\frac{e}{e_{0}} = \exp \left[ - \left(-\frac{\sigma_{ij}\delta_{ij}}{h_s} \right)^n \right]$$

See the basic law description for the parameters $h_s$ and $n$. The $\sigma_{ij}$ denotes the effective stress tensor (total stresses minus any groundflow pressure). This pressure dependent initial void ratio correction can be activated by control_materi_plasti_hypo_pressure_dependent_void_ratio. After the initial void ratio has been established, the development of the void ratio is governed by volumetric compression or extension of the granular skeleton.