Plastic drain
====================

In this paragraph, we give the expression for plastic deformation rates, distinguishing between the so-called general case where the stress state is located on a "regular" zone at the edge of the reversibility domain and the case where it is at the top of one of the cones.




General form of the rule of normality
------------------------

In space :math:`(\sigma ,A)`, inequalities [:ref:`éq 3.2- 3.2-2  <Ref522954170>`], [:ref:`éq 3.2- 3.2-3  <Ref522954228>`], [], [], [:ref:`éq 3.2- 3.2-4  <Ref522954242>`], [:ref:`éq 3.2- 3.2-5  <Ref522954255>`], define a convex domain that we will note :math:`{C}_{(\sigma ,A)}`. We'll note :math:`{\Psi }_{c}` the indicative function of this convex:


.. _RefEquation 4.1‑ 4.1-1:

:math:`{\psi }_{c}(\sigma ,A)\mathrm{=}\mathrm{\{}\begin{array}{c}0\text{si}(\sigma ,A)\mathrm{\in }{C}_{(\sigma ,A)}\\ \mathrm{\infty }\text{sinon}\end{array}` eq 4.1-4.1-1

When the border of the reversibility domain is reached, irreversible plastic deformations develop, according to the classical theory of plasticity.

For a standard material, [:ref:`bib4 <bib4>`] the flow law verifies the principle of maximum plastic work, which results in the equation:


.. _Ref523040299:

:math:`({\dot{\varepsilon }}^{p},\dot{\alpha })\mathrm{\in }\mathrm{\partial }{\Psi }_{c}` **eq 4.1‑** 4.1-2

where :math:`\partial {\Psi }_{c}` notes the subdifferential of the :math:`{\Psi }_{c}` function. We recall [:ref:`bib3 <bib3>`] that the subdifferential of a convex function at a point :math:`x` is the set of vectors :math:`z` such that: :math:`f({x}^{\text{*}})\ge f(x)+\langle z,{x}^{\text{*}}-x\rangle \forall {x}^{\text{*}}`

It is then easy to see that [:ref:`éq 4.1- 4.1-2  <Ref523040299>`] results in:


.. _Ref523040324:

:math:`{\Psi }_{c}({\sigma }^{\text{*}},{A}^{\text{*}})\ge {\Psi }_{c}(\sigma ,A)+{\dot{\varepsilon }}^{p}({\sigma }^{\text{*}}-\sigma )+\dot{\alpha }({A}^{\text{*}}-A)\forall {\sigma }^{\text{*}}\mathrm{et}{A}^{\text{*}}` **eq 4.1‑** 4.1-3

Given the definition of the characteristic function, it is easy to see that [:ref:`          éq 4.1- 4.1-3  <Ref523040324>`] is equivalent to:


.. _Ref523040439:

 :math:`{\dot{\varepsilon }}^{p}\sigma +\dot{\alpha }A\ge {\dot{\varepsilon }}^{p}{\sigma }^{\text{*}}+\dot{\alpha }{A}^{\text{*}}\forall {\sigma }^{\text{*}}\mathrm{et}{A}^{\text{*}}\in {C}_{(\sigma ,A)}` **eq 4.1‑** 4.1-4

In other words, the plastic flow is such that the :math:`(\sigma ,A)` couple achieves the maximum amount of plastic dissipation among the admissible thermodynamic forces.




Expression of plastic flow in the current part
--------------------------------------------------------

When the function :math:`f` is differentiable at the point in question :math:`(\sigma ,A)` the rule of normality is simply written

:math:`{\dot{\mathrm{\varepsilon }}}^{p}=\dot{\lambda }\frac{\partial f}{\partial \sigma }` **eq 4.2‑** 4.2-1

:math:`\dot{\alpha }=\dot{\lambda }\frac{\partial f}{\partial A}` **eq 4.2‑** 4.2-2

:math:`\dot{\lambda }` and :math:`f` verifying Kuhn-Tucker conditions:

:math:`\begin{array}{c}\dot{\lambda }\ge 0\\ f\le 0\\ \dot{\lambda }\mathrm{.}f=0\end{array}\}` **eq 4.2‑** 4.2-3

The work hardening variable is linked to the plastic multiplier by the law of work hardening. Using plastic work, you can write: :math:`\dot{\kappa }f=\sigma {\dot{\varepsilon }}^{p}`.

If :math:`f` is a homogeneous function of order 1 with respect to the tensor variable :math:`\sigma`, we have :math:`\sigma \frac{\mathrm{\partial }f}{\mathrm{\partial }\sigma }\mathrm{=}f`, which leads to equality: :math:`\dot{\lambda }\mathrm{=}\dot{\kappa }` and therefore finally to the equations:

:math:`{\dot{\varepsilon }}_{c}^{p}\mathrm{=}\dot{{\kappa }_{c}^{p}}\frac{\mathrm{\partial }{f}_{\mathit{comp}}}{\mathrm{\partial }\sigma }` **eq 4.2‑** 4.2-4

:math:`{\dot{\mathrm{\varepsilon }}}_{t}^{p}=\dot{{\mathrm{\kappa }}_{t}^{p}}\frac{\partial {f}_{\mathrm{trac}}}{\partial \sigma }` **eq 4.2‑** 4.2-5




Expression of plastic flow at the top of a cone
--------------------------------------------------------

We give two presentations of the same result. The first presentation uses the theory of generalized standard materials and sub-differentials, the second part of an equality posed a priory on plastic work.




Demonstration by the general theory of standard materials
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Domain :math:`{C}_{(\sigma ,A)}` consists of two cones. The function :math:`{\Psi }_{c}` cannot be differentiated either at the intersection of these two cones, or at the top of each of these cones. When point :math:`(\sigma ,\mathrm{A})` belongs to the intersection of the two cones, the previous equations remain valid, with the precision that the plastic deformations of compression and traction develop at the same time. This case called "multi-criteria" is also treated in [:ref:`bib4 <bib4>`]. Here we will only deal with the case where :math:`(\sigma ,A)` is at the top of a cone, and we will choose the most frequent case of the top of the tension cone, knowing that the case of the top of the compression cone is treated in exactly the same way.

The criteria are rewritten using the variables :math:`{\sigma }^{\mathrm{eq}}` and :math:`{\sigma }_{H}`, which are more practical in analytical developments.

:math:`{f}_{\text{trac}}(\sigma ,{A}_{t})=\frac{\sqrt{2}}{\mathrm{3d}}{\sigma }^{\text{eq}}+\frac{c}{d}{\sigma }_{H}-{f}_{t}^{\text{'}}+{A}_{t}\le 0` 4.3.1-1

:math:`{f}_{{}_{\text{trac}}}^{H}(\sigma ,{A}_{t})=\frac{c}{d}{\sigma }_{H}-{f}_{t}^{\text{'}}+{A}_{t}\le 0` **4.3.1-2**

So we are considering a case where:

:math:`\begin{array}{}{\sigma }^{\text{eq}}=0\\ \frac{c}{d}{\sigma }_{H}-{f}_{t}^{\text{'}}+{A}_{t}=0\end{array}\}` **4.3.1-3**

Starting from [:ref:`éq 4.1- 4.1-4  <Ref523040439>`], we will calculate the plastic dissipation to be the maximum of :math:`{\dot{\varepsilon }}^{p}{\sigma }^{\text{*}}+\dot{\alpha }{A}^{\text{*}}` for all the :math:`{\sigma }^{\text{*}},{A}^{\text{*}}\in {C}_{(\sigma ,A)}` couples

 :math:`{D}^{p}\text{=}\underset{{\sigma }^{\text{*}},{A}^{\text{*}}\in {C}_{(\sigma ,A)}}{\mathrm{Max}}({\dot{\varepsilon }}^{p}{\sigma }^{\text{*}}+\dot{\alpha }{A}^{\text{*}})` **4.3.1-4**

By writing while this maximum is finite and reached when :math:`{\sigma }^{\text{*}}\mathrm{=}\sigma` and :math:`{A}^{\text{*}}=A`, we will find conditions on :math:`{\dot{\varepsilon }}^{p}` and :math:`\dot{\alpha }`. In fact, the finite character will suffice.

.. code-block:: text

    By using the decomposition of the partly isotropic and deviatory tensors, and the particular shape of the work-hardening variables, we can easily find:

:math:`{\dot{\varepsilon }}^{p}{\sigma }^{\text{*}}+\dot{\alpha }{A}^{\text{*}}={\dot{\tilde{\varepsilon }}}^{p}{s}^{\text{*}}+3{\sigma }_{H}^{\text{*}}{\dot{\varepsilon }}_{H}^{p}+{\dot{{\kappa }_{t}}}^{p}{A}_{t}^{\text{*}}` 4.3.1-5

Let us then consider the set :math:`{\Sigma }_{1}` of zero-trace constrained vectors whose equivalent Von Mises stress is 1: :math:`{\Sigma }_{1}=\text{{}\sigma ,{\sigma }^{\mathrm{eq}}=\mathrm{1,}\text{trace}(\sigma )=0\text{}}`

:math:`(\sigma ,A)\in {C}_{(\sigma ,A)}\iff \{\begin{array}{}\sigma ={\sigma }^{\text{eq}}{s}_{1}+{\sigma }_{H}I\\ {s}_{1}\in {\Sigma }_{1}\\ \frac{\sqrt{2}}{\mathrm{3d}}{\sigma }^{\text{eq}}+\frac{c}{d}{\sigma }_{H}-{f}_{t}^{\text{'}}+{A}_{t}\le 0\\ \frac{c}{d}{\sigma }_{H}-{f}_{t}^{\text{'}}+{A}_{t}\le 0\end{array}` **4.3.1-6**

In other words, the "direction" of the constraint deviator is arbitrary for a :math:`(\sigma ,A)\in {C}_{(\sigma ,A)}` couple.

.. code-block:: text

    So we can write:

:math:`{D}^{p}=\underset{\underset{\underset{\{\begin{array}{c}\frac{\sqrt{2}}{\mathrm{3d}}{\sigma }^{{\mathrm{eq}}^{\text{*}}}+\frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\\ \frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\end{array}}{}}{{\sigma }^{{\mathrm{eq}}^{\text{*}}},{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}},{s}_{1}^{\text{*}}\in {\Sigma }_{1}}}{\mathrm{Max}}({\sigma }^{{\text{eq}}^{\text{*}}}\underset{{s}_{1}\in {\Sigma }_{1}}{\text{Max}}{\dot{\tilde{\varepsilon }}}^{p}{s}_{1}^{\text{*}}+3{\sigma }_{H}^{\text{*}}{\dot{\varepsilon }}_{H}^{p}+{\dot{\kappa }}_{t}^{p}{A}_{t}^{\text{*}})`**4.3.1-7**

It is clear that the maximum of :math:`{\dot{\tilde{\varepsilon }}}^{p}{s}_{1}^{\text{*}}` is reached when :math:`{s}_{1}^{\text{*}}` is "parallel" to :math:`{\dot{\tilde{\varepsilon }}}^{p}` and we then have: :math:`\underset{{s}_{1}\in {\Sigma }_{1}}{\text{Max}}{\dot{\tilde{\varepsilon }}}^{p}{s}_{1}^{\text{*}}=\frac{2}{3}{\dot{\tilde{\varepsilon }}}_{\mathrm{eq}}^{p}`.

[:ref:`éq 4.3.1-7 <éq 4.3.1-7>`] can therefore be written as:

 **4.3.1-8** :math:`\begin{array}{cccc}{D}^{p}\text{=}& \underset{\underset{\underset{\{\begin{array}{c}\frac{\sqrt{2}}{\mathrm{3d}}{\sigma }^{{\mathrm{eq}}^{\text{*}}}+\frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\\ \frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\end{array}}{}}{{\sigma }^{{\mathrm{eq}}^{\text{*}}},{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}}}{\mathrm{Max}}(\frac{2}{3}{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}{\sigma }^{{\mathrm{eq}}^{\text{*}}})& \text{+}& \underset{\underset{\underset{\{\begin{array}{c}\frac{\sqrt{2}}{\mathrm{3d}}{\sigma }^{{\mathrm{eq}}^{\text{*}}}+\frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\\ \frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\end{array}}{}}{{\sigma }^{{\mathrm{eq}}^{\text{*}}},{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}}}{\mathrm{Max}}(3{\sigma }_{H}^{\text{*}}\dot{{{\varepsilon }_{H}}^{p}})\\ & & & \\ & & \text{+}& \underset{\underset{\underset{\{\begin{array}{c}\frac{\sqrt{2}}{\mathrm{3d}}{\sigma }^{{\mathrm{eq}}^{\text{*}}}+\frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\\ \frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\end{array}}{}}{{\sigma }^{{\mathrm{eq}}^{\text{*}}},{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}}}{\mathrm{Max}}({\dot{\kappa }}_{t}^{p}{A}_{t}^{\text{*}})\end{array}\}`

Like :math:`{\dot{\tilde{\varepsilon }}}_{\mathrm{eq}}^{p}\ge 0`, for the first term we have:

:math:`\underset{\underset{\underset{\{\begin{array}{c}\frac{\sqrt{2}}{\mathrm{3d}}{\sigma }^{{\mathrm{eq}}^{\text{*}}}+\frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\\ \frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0\end{array}}{}}{{\sigma }^{{\mathrm{eq}}^{\text{*}}},{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}}}{\mathrm{Max}}(\frac{2}{3}{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}{\sigma }^{{\mathrm{eq}}^{\text{*}}})\text{=}\frac{2}{3}{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}(\frac{-\mathrm{3d}}{\sqrt{2}}{A}_{t}^{\text{*}}+\frac{\mathrm{3d}}{\sqrt{2}}{f}_{t}^{\text{'}}\frac{\mathrm{3c}}{\sqrt{2}}{\sigma }_{H}^{\text{*}})`**4.3.1-9**

[:ref:`éq 4.3.1-9 <éq 4.3.1-9>`] reported in [:ref:`éq 4.3.1-8 <éq 4.3.1-8>`] gives:

:math:`{D}^{p}=\sqrt{2}d{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}{f}_{t}^{\text{'}}+\underset{\underset{\frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0}{{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}}}{\mathrm{Max}}({\sigma }_{H}^{\text{*}}(3{\dot{{\varepsilon }_{H}}}^{p}-\sqrt{2}c{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}))+\underset{\underset{\frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0}{{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}}}{\mathrm{Max}}({A}_{t}^{\text{*}}({\dot{\kappa }}_{t}^{p}-\sqrt{2}d{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}))` **4.3.1-10**

So let's say:

:math:`m=3{\dot{{\varepsilon }_{H}}}^{p}-\sqrt{2}c{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}`, :math:`n={\dot{\kappa }}_{t}^{p}-\sqrt{2}d{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}`, and :math:`q=\sqrt{2}d{\dot{\tilde{{\varepsilon }_{\mathrm{eq}}}}}^{p}{f}^{\text{'}}`

With these notations, [:ref:`éq 4.3.1-10 <éq 4.3.1-10>`] becomes:

:math:`{D}^{p}=q+\underset{\underset{\frac{c}{d}{\sigma }_{H}^{\text{*}}-{f}_{t}^{\text{'}}+{A}_{t}^{\text{*}}\le 0}{{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}}}{\mathrm{Max}}(m{\sigma }_{H}^{\text{*}}+n{A}_{t}^{\text{*}})` **4.3.1-11**

This is a "simplex" problem. The domain of :math:`{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}` is represented on the [:ref:`Figure4.3.1‑a <Figure4.3.1‑a>`].


.. image:: images/Object_296.svg
    :width: 416
    :height: 239


.. _RefImage_Object_296.svg:


.. _Ref522937448:

**Figure** **4.3.1-a**

As the :math:`{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}` domain extends to :math:`-\infty` for both :math:`{\sigma }_{H}^{\text{*}}` and :math:`{A}_{t}^{\text{*}}`, for :math:`{D}^{p}` to be over, :math:`m` and :math:`n` must be positive. The maximum of :math:`m{\sigma }_{H}^{\text{*}}+n{A}_{t}^{\text{*}}` is reached for a :math:`{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}` couple located on the edge of the domain of :math:`{\sigma }_{H}^{\text{*}},{A}_{t}^{\text{*}}`.

So we have: :math:`{D}^{p}=q+n{f}_{t}^{\text{'}}+\underset{{\sigma }_{H}^{\text{*}}}{\mathrm{Max}}({\sigma }_{H}^{\text{*}}(m-n\frac{c}{d}))`

For :math:`{D}^{p}` to be over, you need to:

:math:`m=n\frac{c}{d}`

Using the definitions of :math:`m` and :math:`n`, this relationship gives:

:math:`3\dot{{\mathrm{\varepsilon }}_{H}^{p}}=\frac{c}{d}{\dot{\mathrm{\kappa }}}_{t}^{p}` **eq 4.3.1-12**

In addition, the constraints :math:`m\ge 0` and :math:`n\ge 0` give:

:math:`3\dot{{\mathrm{\varepsilon }}_{H}^{p}}\ge \sqrt{2}c{\dot{\tilde{\mathrm{\varepsilon }}}}_{{}^{\text{eq}}}^{p}` **eq 4.3.1-13**

and

:math:`{\dot{\mathrm{\kappa }}}_{t}^{p}\ge \sqrt{2}d{\dot{\tilde{\mathrm{\varepsilon }}}}_{{}^{\text{eq}}}^{p}` **eq 4.3.1-14**

these last two inequalities being equivalent due to [:ref:`éq 4.3.1-11 <éq 4.3.1-11>`].

The equations [:ref:`éq 4.3.1-11 <éq 4.3.1-11>`] and [:ref:`éq 4.3.1-12 <éq 4.3.1-12>`] define the plastic flow at the top of one of the cones in the reversibility domain.




Demonstration through plastic work
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The starting point is to consider that compared to the developments made at regular points, it is essentially the relationships [:ref:`éq 4.2-1 <éq 4.2-1>`] and [:ref:`éq 4.2-2 <éq 4.2-2>`], called rules of normality, that can no longer be written. However, the [:ref:`éq 4.2-1 <éq 4.2-1>`] relationship implies :math:`{\dot{\kappa }}_{t}^{p}{f}_{t}({\kappa }_{t}^{p})=\sigma \dot{{\varepsilon }^{p}}` equality, which can be maintained.

So we will start from the equation:

:math:`{\dot{\kappa }}_{t}^{p}{f}_{t}({\kappa }_{t}^{p})=\sigma \dot{{\varepsilon }^{p}}` **eq 4.3.2-1**

We use the partly isotropic and deviatory decomposition of the tensors and find:

:math:`{\dot{\kappa }}_{t}^{p}{f}_{t}({\kappa }_{t}^{p})=\dot{{\tilde{\varepsilon }}^{p}}s+3{\sigma }_{H}\dot{{\varepsilon }_{H}}` **eq 4.3.2-2**

At the top of the pull cone, we have the relationships [:ref:`éq 4.3.1-3 <éq 4.3.1-3>`], which, given in [:ref:`éq 4.3.2-2 <éq 4.3.2-2>`], give, also using [:ref:`éq 3.5- 3.5-2  <Ref523041101>`]:

:math:`{\dot{\kappa }}_{t}^{p}{f}_{t}({\kappa }_{t}^{p})=3\frac{d}{c}{f}_{t}({\kappa }_{t}^{p})\dot{{\varepsilon }_{H}}` **eq 4.3.2-3**

And so we find the relationship [:ref:`éq 4.3.1-12 <éq 4.3.1-12>`]:. :math:`3\dot{{\varepsilon }_{H}}=\frac{c}{d}{\dot{\kappa }}_{t}^{p}`




Set of behavioral equations (summary)
------------------------------------------------

We note :math:`H` the elasticity matrix:

:math:`H=\left[\begin{array}{cccccc}\mathrm{\lambda }+\mathrm{2\mu }& \mathrm{\lambda }& \mathrm{\lambda }& 0& 0& 0\\ \mathrm{\lambda }& \mathrm{\lambda }+\mathrm{2\mu }& \mathrm{\lambda }& 0& 0& 0\\ \mathrm{\lambda }& \mathrm{\lambda }& \mathrm{\lambda }+\mathrm{2\mu }& 0& 0& 0\\ 0& 0& 0& \mathrm{2\mu }& 0& 0\\ 0& 0& 0& 0& \mathrm{2\mu }& 0\\ 0& 0& 0& 0& 0& \mathrm{2\mu }\end{array}\right]`

With:

:math:`\mathrm{\lambda }=\mathrm{\nu  }\frac{E}{(1+\mathrm{\nu  })(1-\mathrm{2\nu  })}` and :math:`\mathrm{\mu }=\frac{E}{2(1+\mathrm{\nu  })}`, and :math:`K=\frac{\mathrm{3\lambda }+\mathrm{2\mu }}{3}`

The constraint-deformity relationships are finally written as:


.. _Ref524402687:

:math:`\sigma \mathrm{=}H(\varepsilon \mathrm{-}{\varepsilon }_{c}^{p}\mathrm{-}{\varepsilon }_{t}^{p})` **eq 4.4‑** 4.4-1

For a regular point of the compression cone:


.. _Ref524402720:

:math:`{f}_{\mathrm{comp}}(\sigma ,{A}_{c})=\frac{\sqrt{2}}{\mathrm{3b}}{\sigma }^{\mathrm{eq}}+\frac{a}{b}{\sigma }_{H}-\phi {f}_{c}^{\text{'}}+{A}_{c}\le 0` **eq 4.4‑** 4.4-2

:math:`{\dot{\mathrm{\kappa }}}_{c}^{p}{f}_{\mathrm{comp}}=0;{\dot{\mathrm{\varepsilon }}}_{c}^{p}={\dot{\mathrm{\kappa }}}_{c}^{p}\frac{\partial {f}_{\mathrm{comp}}}{\partial \sigma }` **eq 4.4‑** 4.4-3

For a regular point of the traction cone:

 :math:`{f}_{\mathrm{trac}}(\sigma ,{A}_{t})=\frac{\sqrt{2}}{\mathrm{3d}}{\sigma }^{\mathrm{eq}}+\frac{c}{d}{\sigma }_{H}-\phi {f}_{t}^{\text{'}}+{A}_{t}\le 0` **eq 4.4‑** 4.4-4

:math:`{\dot{\mathrm{\kappa }}}_{t}^{p}{f}_{\mathrm{trac}}=0;{\dot{\mathrm{\varepsilon }}}_{t}^{p}={\dot{\mathrm{\kappa }}}_{t}^{p}\frac{\partial {f}_{\mathrm{trac}}}{\partial \sigma }` **eq 4.4‑5**

For a point at the top of the compression cone:


.. _Ref524402755:

:math:`s=0` **eq 4.4‑** **6**

:math:`{f}_{{}_{\text{comp}}}^{H}(\sigma ,{A}_{c})=\frac{a}{b}{\sigma }_{H}-\phi {f}_{c}^{\text{'}}+{A}_{c}=0` **eq 4.4‑7**

:math:`3{\dot{\mathrm{\varepsilon }}}_{cH}^{p}=\frac{a}{b}{\dot{\mathrm{\kappa }}}_{c}^{p}` **eq 4.4‑8**


.. _Ref524402766:

:math:`3{\dot{\mathrm{\varepsilon }}}_{cH}^{p}\ge \sqrt{2}a{\dot{\tilde{\mathrm{\varepsilon }}}}_{c\mathrm{eq}}^{p}` **eq 4.4‑** **9**

For a point at the top of the traction cone:


.. _Ref524402782:

:math:`s=0` **eq 4.4‑** **10**

:math:`{f}_{{}_{\text{trac}}}^{H}(\mathrm{\sigma },{A}_{t})=\frac{c}{d}{\mathrm{\sigma }}_{H}-{f}_{t}^{\text{'}}+{A}_{t}=0` **eq 4.4-11**


.. _Ref524748147:

:math:`3{\dot{\mathrm{\varepsilon }}}_{tH}^{p}=\frac{c}{d}{\dot{\mathrm{\kappa }}}_{t}^{p}` **eq 4.4‑** **12**


.. _Ref524748094:

:math:`3{\dot{\mathrm{\varepsilon }}}_{tH}^{p}\ge \sqrt{2}c{\dot{\tilde{\mathrm{\varepsilon }}}}_{t\mathrm{eq}}^{p}` **eq 4.4‑** **13**