3. Description of law CJS#
3.1. Partition of deformations#
The overall deformation increment is divided into three parts, relating to each of the mechanisms involved:
\({\dot{\varepsilon }}_{\text{ij}}={\dot{\varepsilon }}_{\text{ij}}^{e}+{\dot{\varepsilon }}_{\text{ij}}^{\text{ip}}+{\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}\)
where \({\dot{\varepsilon }}_{\text{ij}}^{e}\), \({\dot{\varepsilon }}_{\text{ij}}^{\text{ip}}\), and \({\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}\) are the increments of elastic deformation, isotropic plastic deformation, and deviatory plastic deformation, respectively.
3.2. Elastic mechanism#
The elastic part of the law is of the hypoelastic type, whose general expression is:
\({\dot{\varepsilon }}_{\text{ij}}^{e}=\frac{{\dot{s}}_{\text{ij}}}{2G}+\frac{{\dot{I}}_{1}}{9K}{\delta }_{\text{ij}}\)
where \({I}_{1}\) is the first stress invariant: \({I}_{1}=\text{tr}(\sigma )\), \(s\) is the deviating part of the stress tensor, and where \(K\) and \(G\) are the elastic volume strain modulus and shear modulus respectively. These depend on the state of stresses according to:
\(K={K}_{o}^{e}{(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{n}\), \(G={G}_{o}^{e}{(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{n}\)
\({K}_{o}^{e}\), \({G}_{o}\), \({P}_{a}\), and \(n\) are model parameters. \({P}_{a}\) is a reference pressure equal to -100 kPa.
3.3. Isotropic plastic mechanism#
The corresponding load surface \({f}^{i}\) is, in the space of the main stresses, a plane perpendicular to the hydrostatic axis, i.e.:
\({f}^{i}(\sigma ,{Q}_{\text{iso}})=-\frac{({I}_{1}+{Q}_{\text{init}})}{3}+{Q}_{\text{iso}}\)
where \({Q}_{\text{iso}}\) is the thermodynamic force that depends on the internal variable \(q\) according to:
\({\dot{Q}}_{\text{iso}}={K}^{p}\dot{q}={K}_{o}^{p}{(\frac{{Q}_{\text{iso}}}{{P}_{a}})}^{n}\dot{q}\)
\({K}_{o}^{p}\), \({P}_{a}\) and \(n\) are the parameters of the plastic deviatory mechanism (\({P}_{a}\) and \(n\) are identical to those of the elastic mechanism). The rule of normality makes it possible to express the evolution of the plastic deformation and of the work-hardening variable as a function of the evolution of the plastic multiplier \({\lambda }^{i}\):
\({\dot{\varepsilon }}_{\text{ij}}^{\text{ip}}={\dot{\lambda }}^{i}\frac{\partial {f}^{i}}{\partial {\sigma }_{\text{ij}}}=-\frac{1}{3}{\dot{\lambda }}^{i}{\delta }_{\text{ij}}\) and \(\dot{q}=-{\dot{\lambda }}^{i}\frac{\partial {f}^{i}}{\partial {Q}_{\text{iso}}}=-{\dot{\lambda }}^{i}\)
Taking into account the second equation, the law of work hardening can also be put in the form:
\({\dot{Q}}_{\text{iso}}=-{\dot{\lambda }}^{i}{K}_{o}^{p}{(\frac{{Q}_{\text{iso}}}{{P}_{a}})}^{n}\)
3.4. Deviatory plastic mechanism#
The load surface of this second plastic mechanism is a convex surface with ternary symmetry defined by the equation:
\({f}^{d}(\sigma ,R,X)={q}_{\text{II}}h({\theta }_{q})+R({I}_{1}+{Q}_{\text{init}})\)
with \({q}_{\text{ij}}={s}_{\text{ij}}-{I}_{1}{X}_{\text{ij}}\)
\({q}_{\text{II}}=\sqrt{{q}_{\text{ij}}{q}_{\text{ij}}}\)
\(h({\theta }_{q})={(1+\gamma \text{cos}(3{\theta }_{q}))}^{1/6}={(1+\mathrm{\gamma }\sqrt{\text{54}}\frac{\text{det}(q)}{{q}_{\text{II}}^{3}})}^{1/6}\).
The scalar \(R\) and the tensor \(X\) represent respectively the mean radius and the center of the load surface in the deviatory plane.
\(s\), \(q\), and \(X\) are deviatory tensors. \(\gamma\) is a parameter that reflects the asymmetric behavior of soils in compression and extension. \(\theta\) is the Lode angle.
This load surface evolves according to two types of work hardening: isotropic work hardening and kinematic work hardening.
Note:
3.4.1. isotropic work hardening#
The isotropic work-hardening law is written as follows:
\(\dot{R}=\frac{A{R}_{m}^{2}\dot{r}}{{({R}_{m}+Ar)}^{2}}\)
The thermodynamic force \(R\) is a function of \(r\) whose evolution is given by:
\(\dot{r}=-{\dot{\lambda }}^{d}\frac{\partial {f}^{d}}{\partial R}{(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{\text{-}1\text{.}5}=-{\dot{\lambda }}^{d}({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{\text{-}1\text{.}5}\)
By direct integration of the law of work-hardening, it comes:
\(R=\frac{A{R}_{m}r}{{R}_{m}+Ar}\), or also \(r=\frac{R{R}_{m}}{A({R}_{m}-R)}\)
The law of work hardening can therefore also be expressed by:
\(\dot{R}=-{\dot{\lambda }}^{d}A{(1-\frac{R}{{R}_{m}})}^{2}({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-1\text{.}5}={\dot{\lambda }}^{d}{G}^{R}(\sigma ,R)\)
with \({G}^{R}(\sigma ,R)=-A{(1-\frac{R}{{R}_{m}})}^{2}({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-1\text{.}5}\)
and where \({R}_{m}\) (which is the mean radius of the elastic domain at break) and \(A\) are parameters of the model.
3.4.2. kinematic work hardening#
The law of kinematic work hardening is given by:
\({\dot{X}}_{\text{ij}}=\frac{1}{b}{\dot{\alpha }}_{\text{ij}}\)
The thermodynamic force \(X\) is a function of the variable \(\alpha\) whose non-linear evolution is given by:
\({\dot{\alpha }}_{\text{ij}}=-{\dot{\lambda }}^{d}\left[\text{dev}(\frac{\partial {f}^{d}}{\partial {X}_{\text{ij}}})-({I}_{1}+{Q}_{\text{init}})\phi {X}_{\text{ij}}\right]{(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-1\text{.}5}\)
The term \(-({I}_{1}+{Q}_{\text{init}})\phix\) makes it possible to obtain non-linear kinematic work hardening, reflecting the limitation of the evolution of the load surface.
Taking into account \(\frac{\partial {f}^{d}}{\partial {X}_{\text{ij}}}=\frac{\partial {f}^{d}}{\partial {q}_{\text{kl}}}\frac{\partial {q}_{\text{kl}}}{\partial {X}_{\text{ij}}}=-(I+{Q}_{{\text{init}}_{1}})\frac{\partial {f}^{d}}{\partial {q}_{\text{ij}}}\), and asking: \({Q}_{\text{ij}}=\text{dev}(\frac{\partial {f}^{d}}{\partial {q}_{\text{ij}}})\), he finally comes up for the law of work hardening:
\({\dot{X}}_{\text{ij}}={\dot{\lambda }}^{d}\frac{1}{b}({Q}_{\text{ij}}+\phi {X}_{\text{ij}})({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-1\text{.}5}={\dot{\lambda }}^{d}{G}_{\text{ij}}^{X}(\sigma ,X)\)
with \({G}_{\text{ij}}^{X}(\sigma ,X)=\frac{1}{b}({Q}_{\text{ij}}+\phi {X}_{\text{ij}})({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-1\text{.}5}\).
where \(\phi\) a function that limits the evolution of \(X\) and is a model parameter.
The tensor \(Q\) is calculated using the formula:
\({Q}_{\text{ij}}=\frac{1}{h{(\theta )}^{5}}\left[(1+\frac{\gamma }{2}\text{cos}(3\theta ))\frac{{q}_{\text{ij}}}{{q}_{\text{II}}}+\frac{\gamma \sqrt{\text{54}}}{{\mathrm{6q}}_{\text{II}}^{2}}\text{dev}(\frac{\partial \text{det}(q)}{\partial {q}_{\text{ij}}})\right]\)
The preceding expression is obtained in the following way. We have:
\(\frac{\partial {f}^{d}}{\partial {q}_{\text{ij}}}=h({\theta }_{q})\frac{\partial {q}_{\text{II}}}{\partial {q}_{\text{ij}}}+{q}_{\text{II}}\frac{\partial h({\theta }_{q})}{\partial {q}_{\text{ij}}}\)
where \(\frac{\partial {q}_{\text{II}}}{\partial {q}_{\text{ij}}}\) and \(\frac{\partial h({\theta }_{q})}{\partial {q}_{\text{ij}}}\) are respectively given by:
\(\frac{\partial {q}_{\text{II}}}{\partial {q}_{\text{ij}}}=\frac{{q}_{\text{ij}}}{{q}_{\text{II}}}\)
\(\frac{\partial h({\theta }_{q})}{\partial {q}_{\text{ij}}}=\frac{1}{6h{({\theta }_{q})}^{5}}\frac{\partial }{\partial {q}_{\text{ij}}}(1+\gamma \sqrt{\text{54}}\frac{\text{det}(q)}{{q}_{\text{II}}^{3}})=\frac{-\gamma \text{cos}(3{\theta }_{q})}{\mathrm{2h}({\theta }_{q}{)}^{5}}\frac{{q}_{\text{ij}}}{{q}_{\text{II}}^{2}}+\frac{\gamma \sqrt{\text{54}}}{\mathrm{6h}({\mathrm{\theta }}_{q}{)}^{5}{q}_{\text{II}}^{3}}\frac{\partial \text{det}(q)}{\partial {q}_{\text{ij}}}\)
From where
\(\frac{\partial {f}^{d}}{\partial {q}_{\text{ij}}}=\frac{1}{h{({\theta }_{q})}^{5}}\left[(1+\frac{\gamma }{2}\text{cos}(3{\theta }_{q}))\frac{{q}_{\text{ij}}}{{q}_{\text{II}}}+\frac{\gamma \sqrt{\text{54}}}{{\mathrm{6q}}_{\text{II}}^{2}}(\frac{\partial \text{det}(q)}{\partial {q}_{\text{ij}}})\right]\)
For its part, function \(\phi\) is given by:
\(\varphi ={\varphi }_{o}h({\theta }_{s}){Q}_{\text{II}}\)
where \({Q}_{\text{II}}=\sqrt{{Q}_{\text{ij}}{Q}_{\text{ij}}}\) and \(h({\theta }_{s})={(1+\gamma \text{cos}(3{\theta }_{s}))}^{1/6}={(1+\gamma \sqrt{\text{54}}\frac{\text{det}(s)}{{s}_{\text{II}}^{3}})}^{1/6}\). The term \({\phi }_{o}\) is expressed in terms of the breaking characteristics of the material.
3.4.3. Law of evolution of the plastic deviatory mechanism#
In granular materials, volume variation may occur for purely deviatory loading. This variation in volume is linked to the discontinuous aspect of the material and to the kinematic conditions that result during loading. This particular phenomenon does not make it possible to define deviatory plastic deformations on the basis of the rule of normality alone. This is why the deviatory plastic mechanism is not associated. There is therefore a potential function controlling the evolution of deformations:
\({\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}={\dot{\lambda }}^{d}{G}_{\text{ij}}^{d}\)
The potential function is defined based on the following kinematic condition:
\({\dot{\varepsilon }}_{v}^{\text{dp}}=-\beta (\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}}-1)\frac{\mid {s}_{\text{ij}}{\dot{e}}_{\text{ij}}^{\text{dp}}\mid }{{s}_{\text{II}}}\)
where \(\mathrm{\beta }\) is a model parameter and \({s}_{\text{II}}^{c}\) represents the characteristic stress state. A surface, which is identical in shape to the load surface in the stress space, separates the contracting states from the expanding states. This surface, called characteristic, has the equation:
\({f}^{c}={s}_{\text{II}}^{c}h({\theta }_{s})+{R}_{c}({I}_{1}+{Q}_{\text{init}})\)
where \({R}_{c}\) is a parameter corresponding to the mean radius of this characteristic surface. The kinematic condition can also be in the form:
\(\begin{array}{}{\dot{\varepsilon }}_{v}^{\text{dp}}+\beta (\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}}-1)\frac{\mid {s}_{\text{ij}}{\dot{e}}_{\text{ij}}^{\text{dp}}\mid }{{s}_{\text{II}}}={\dot{\varepsilon }}_{v}^{\text{dp}}+\beta (\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}}-1)\frac{\mid {s}_{\text{ij}}{\dot{e}}_{\text{ij}}^{\text{dp}}\mid }{{s}_{\text{ij}}{\dot{e}}_{\text{ij}}^{\text{dp}}}\frac{{s}_{\text{ij}}{\dot{e}}_{\text{ij}}^{\text{dp}}}{{s}_{\text{II}}}\\ ={\dot{\varepsilon }}_{v}^{\text{dp}}+\frac{{\beta }^{\text{'}}}{{s}_{\text{II}}}{s}_{\text{ij}}{\dot{e}}_{\text{ij}}^{\text{dp}}\\ ={{\dot{\varepsilon }}_{v}^{\text{dp}}+\frac{{\beta }^{\text{'}}}{{s}_{\text{II}}}{s}_{\text{ij}}\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}=0\end{array}\)
where \({\beta }^{\text{'}}=\beta (\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}}-1)\text{signe}({s}_{\text{ij}}{\dot{\varepsilon }}_{\text{ij}}^{\text{dp}})\).
It is then possible to try to express this kinematic condition using a \(n\) tensor in the form:
\({\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}{n}_{\text{ij}}=0\)
that is to say, after decomposition of each term into deviatory and hydrostatic parts:
\({\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}{n}_{\text{ij}}=({\dot{e}}_{\text{ij}}^{\text{dp}}+\frac{1}{3}{\dot{\varepsilon }}_{v}^{\text{dp}}{\delta }_{\text{ij}})({n}_{1}{s}_{\text{ij}}+{n}_{2}{\delta }_{\text{ij}})={n}_{1}{s}_{\text{ij}}{\dot{e}}_{\text{ij}}^{\text{dp}}+{n}_{2}{\mathrm{dt}\varepsilon }_{v}^{\text{dp}}=0\)
We deduce the relationship \(\frac{{n}_{1}}{{n}_{2}}=\frac{{\beta }^{\text{'}}}{{s}_{\text{II}}}\), which added to the normalization condition \(n:n=1\), leads to the expressions:
\({n}_{1}=\frac{\frac{{\beta }^{\text{'}}}{{s}_{\text{II}}}}{\sqrt{{\beta }^{\text{'}2}+3}}\) and \({n}_{2}=\frac{1}{\sqrt{{\beta }^{\text{'}2}+3}}\), which is \({n}_{\text{ij}}=\frac{{\beta }^{\text{'}}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}}+{\delta }_{\text{ij}}}{\sqrt{{\beta }^{\text{'}2}+3}}\)
The law of evolution in \({\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}\) must be such that the kinematic condition is satisfied. It is therefore proposed to take the projection of \({\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}\) on the deformation hypersurface of normal \(n\), i.e.:
\({\dot{\varepsilon }}_{\text{ij}}^{\text{dp}}={\dot{\lambda }}^{d}(\frac{\partial {f}^{d}}{\partial {\sigma }_{\text{ij}}}-(\frac{\partial {f}^{d}}{\partial {\sigma }_{\text{kl}}}{n}_{\text{kl}}){n}_{\text{ij}})={\dot{\lambda }}^{d}{G}_{\text{ij}}^{d}\)
with \({G}_{\text{ij}}^{d}=\frac{\partial {f}^{d}}{\partial {\sigma }_{\text{ij}}}-(\frac{\partial {f}^{d}}{\partial {\sigma }_{\text{kl}}}{n}_{\text{kl}}){n}_{\text{ij}}\).
Moreover, for the calculation of the potential, it can be noted that:
\(\begin{array}{}\frac{\partial {f}^{d}}{\partial {\sigma }_{\text{ij}}}=\frac{\partial {f}^{d}}{\partial {q}_{\text{kl}}}\frac{\partial {q}_{\text{kl}}}{\partial {\sigma }_{\text{ij}}}+R{\delta }_{\text{ij}}\\ \text{=}\left[\text{dev}(\frac{\partial {f}^{d}}{\partial {q}_{\text{kl}}})+\frac{1}{3}\frac{\partial {f}^{d}}{\partial {q}_{\text{mm}}}{\delta }_{\text{kl}}\right]\left[{\delta }_{\text{ik}}{\delta }_{\text{jl}}-{\delta }_{\text{ij}}(\frac{1}{3}{\delta }_{\text{kl}}+{X}_{\text{kl}})\right]+R{\delta }_{\text{ij}}\\ \text{=}{Q}_{\text{kl}}{\delta }_{\text{ik}}{\delta }_{\text{jl}}-{\delta }_{\text{ij}}(\frac{1}{3}{Q}_{\text{kl}}{\delta }_{\text{kl}}+{Q}_{\text{kl}}{X}_{\text{kl}})+\frac{1}{3}\frac{\partial {f}^{d}}{\partial {q}_{\text{mm}}}\left[{\delta }_{\text{ik}}{\delta }_{\text{jl}}{\delta }_{\text{kl}}-{\delta }_{\text{ij}}(\frac{1}{3}{\delta }_{\text{kl}}{\delta }_{\text{kl}}+{\delta }_{\text{kl}}{X}_{\text{kl}})\right]+R{\delta }_{\text{ij}}\\ \text{=}{Q}_{\text{ij}}-({Q}_{\text{kl}}{X}_{\text{kl}}-R){\delta }_{\text{ij}}\end{array}\)
3.4.4. Fracture surface#
The failure state results from the non-linear nature of the work-hardening laws and from the existence of limit values associated with the work hardening variables \(R\) and \(X\). The limit of \(R\), noted \({R}_{m}\), is reached when \(r\) approaches infinity. The \({X}_{\text{ij}}\) limit is reached when \({\dot{X}}_{\text{ij}}\) becomes zero.
Under these conditions:
\({Q}_{\text{ij}}=\phi {X}_{\text{ij}}\) and \({Q}_{\text{II}}=\phi {X}_{\text{II}\text{lim}}\Rightarrow {X}_{\text{II}\text{lim}}=\frac{1}{{\phi }_{o}h({\theta }_{s})}\)
In the state of rupture we therefore have [Figure 3.4.4-a]:
\({q}_{\text{II}}=\frac{{s}_{\text{II}}+{I}_{1}{X}_{\text{II}\text{lim}}\text{cos}\alpha }{\text{cos}({\theta }_{s}-{\theta }_{q})}\)
By replacing this expression and the value of \(R\) at break, in the equation of the load area at break, we obtain the equation of a limit envelope for the load surfaces:
\({f}^{r}={s}_{\text{II}}h({\theta }_{s})+{R}_{r}({I}_{1}+{Q}_{\text{init}})=0\)
with \({R}_{r}=\frac{\text{cos}\alpha }{{\phi }_{o}}+\frac{h({\theta }_{s})}{h({\theta }_{q})}{R}_{m}\text{cos}({\theta }_{s}-{\theta }_{q})\), mean radius of the envelope, which is determined from the mechanical characteristics at breakage of the material. The value of \({\phi }_{o}\) can then be deduced from this:
\({\phi }_{o}=\frac{\text{cos}\alpha }{{R}_{r}-\frac{h({\theta }_{s})}{h({\theta }_{q})}{R}_{m}\text{cos}({\theta }_{s}-{\theta }_{q})}\)
with \(\text{cos}\alpha =\frac{{q}_{\text{II}}^{2}-{s}_{\text{II}}^{2}-{({I}_{1}{X}_{\text{II}})}^{2}}{2{s}_{\text{II}}{I}_{1}{X}_{\text{II}}}\)
Figure 3.4.4-a: Representation of fracture, characteristic and load surfaces in the deviatory plane
In addition, \({R}_{r}\) is related to the maximum friction angle and depends on the mean stress and the relative density. To take into account the dependence of the maximum friction angle on the average stress and the relative density, we consider the relationship:
\({R}_{r}={R}_{c}+\mu \text{ln}(\frac{3{p}_{c}}{{I}_{1}+{Q}_{\text{init}}})\)
where \({R}_{c}\) and \(\mu\) are model parameters. \({p}_{c}\) is the critical mean stress, that is, the minimum mean stress (it is negative with our sign convention) known by the material throughout its history. It depends on the initial relative density according to the classical notion of a critical line in plane \((e,\text{ln}\mid p\mid )\):
\({p}_{c}={p}_{\text{co}}\text{exp}(-c{\varepsilon }_{v})\)
where \({p}_{\text{co}}\) is the initial critical pressure and \(1/c\) is the slope of the critical state line in the \((\mid {\varepsilon }_{v}\mid ,\text{ln}\mid p\mid )\) plane.
3.5. Model prioritization#
3.5.1. Summary description of the three levels CJS#
From the complete description of the model given above, three levels of increasing complexity are deduced, the characteristics of which are summarized in the following table:
Elastic mechanism |
Isotropic plastic mechanism |
Deviatory plastic mechanism |
|
CJS1 |
linear |
not activated |
activated, perfect plasticity |
CJS2 |
nonlinear |
activated |
activated, isotropic work hardening |
CJS3 |
nonlinear |
activated |
activated, kinematic work hardening |
Table 3.5.1-1: The different mechanisms used by the different levels of the CJS model
3.5.2. Parameter review CJS#
In addition, we can also summarize the correspondence between the different levels of the model and the parameters associated with each of them:
Table 3.5.2-1: Summary of the various parameters according to levels CJS
In Code_Aster, the elastic parameters of model CJS (\({K}_{o}^{e}\) and \({G}_{o}\)) are directly taken into account in the elastic characteristics of the material, i.e. through the Young’s modulus \(E\) and the Poisson’s ratio \(\mathit{NU}\).
In Code_Aster, the user does not explicitly indicate the CJS level he has chosen. In fact, it is the choice of the various parameters that determines the corresponding level. To summarize, we have the following logic tests that are integrated into the code:
if \(n=0\) then level CJS1,
if (\(n\ne 0\) and \(A\ne 0\)) then level CJS2,
if (\(n\ne 0\) and \(A=0\)) then level CJS3.
Note:
The user must set the value of \({P}_{a}\) equal to \(\mathrm{-}100\mathit{kPa}\) depending on the units selected. Also, for CJS3, the value of \({p}_{\mathit{co}}\) must be negative.
3.5.3. Correspondence with cohesion and angle of friction#
Soil mechanics are used to using the concepts of cohesion Cohesion \(c\), angle of friction \(\varphi\) and angle of expansion: \(\psi\). These parameters are used in Mohr Coulomb’s law.
Level 1 of law CJS makes it possible to find very similar behavior by making the following choice of parameters:
\({(\frac{1-\gamma }{1+\gamma })}^{1/6}=\frac{3-\mathrm{sin}(\varphi )}{3+\mathrm{sin}(\varphi )}\)
\({R}_{m}\mathrm{=}\frac{2\sqrt{\frac{2}{3}}\text{sin}(\varphi ){(1\mathrm{-}\gamma )}^{1\mathrm{/}6}}{3\mathrm{-}\mathrm{sin}(\varphi )}\)
\({Q}_{\mathit{init}}\mathrm{=}\mathrm{-}3c\text{cotan}(\varphi )\)
\(\beta \mathrm{=}\frac{\mathrm{-}2\sqrt{6}\text{sin}\psi }{3–\mathrm{sin}\psi }\)