1. General piecewise-linear yield criterion can be presented in the form of a linear equation relating the principal normal stresses \({{\sigma }_{1}}\), \({{\sigma }_{2}}\), \({{\sigma }_{3}}\) of symmetric stress tensor:

$${{a}_{1}}{{\sigma }_{1}} + {{a}_{2}}{{\sigma }_{2}} - {{a}_{3}}{{\sigma }_{3}} = b,$$
(1.1)

where \({{a}_{1}}\), \({{a}_{2}}\), \({{a}_{3}}\), \(b\) are the constitutive constants.

A significant simplification for equations of the theories of plasticity can be achieved by special numeration for the axes of the main trihedron of the stress tensor s: we enumerate the main axes in such a way that for the current stress state, the corresponding principle normal stresses \({{\sigma }_{1}}\), \({{\sigma }_{2}}\), \({{\sigma }_{3}}\) are arranged in decreasing order

$${{\sigma }_{1}} \geqslant {{\sigma }_{2}} \geqslant {{\sigma }_{3}}.$$
(1.2)

Intermediate principal normal stress plays a special role in plasticity theories [1–6]. Its influence on the yield of metals and the deformation of loosely coupled media can often be neglected. Therefore, we assume that \({{a}_{2}} = 0\). We divide equation (1.1) by \({{a}_{1}}\) and adopt the following notations

$$\bar {a} = \frac{{{{a}_{3}}}}{{{{a}_{1}}}},\quad \bar {b} = \frac{b}{{{{a}_{1}}}},$$

after which the piecewise-linear yield condition used below is reduced to the form

$${{\sigma }_{1}} - \bar {a}{{\sigma }_{3}} = \bar {b}.$$
(1.3)

The Coulomb–Mohr medium is one of the variants of the piecewise linear yield criterion, which perfectly simulates the mechanical behavior of dry sands, soils, granular media, i.e. loose materials with a granular, porous or granular structure [4]. In terms of principal stresses \({{\sigma }_{1}}\), \({{\sigma }_{2}}\), \({{\sigma }_{3}}\), the Mohr–Coulomb yield criterion for bulk media with internal friction and adhesion is formulated as follows [7, 8]:

$$\frac{{{{\sigma }_{1}} - {{\sigma }_{3}}}}{2} = c\cos \gamma - \sin \gamma \frac{{{{\sigma }_{1}} + {{\sigma }_{3}}}}{2},$$
(1.4)

where, c, \(\gamma \) are the constitutive constants. The Coulomb–Mohr criterion (1.4) can also be reduced to the following form, which is equivalent to form of (1.3):

$${{\sigma }_{1}} - a{{\sigma }_{3}} = 2k.$$
(1.5)

Here, the material constants a and k are related to the coefficient of adhesion and the angle of internal friction \(\gamma \) by the following relations:

$$a = \frac{{1 - \sin \gamma }}{{1 + \sin \gamma }},\quad k = \frac{{c\cos \gamma }}{{1 + \sin \gamma }}.$$

2. In the kinematics of ideally plastic solids, it is convenient to operate with increments of the displacement vector \(du\) and strain tensor dε. We orient the unit basis vectors \({\mathbf{l}}\), \({\mathbf{m}}\), \({\mathbf{n}}\) along the principal axes of the stress tensor (and tensor dε).

The increment of the displacement vector \(du\) can be represented as an expansion in vectors of a local orthonormal basis \({\mathbf{l}}\), \({\mathbf{m}}\), \({\mathbf{n}}\)

$$d{\mathbf{u}} = {\mathbf{l}}d{{u}_{{\left\langle 1 \right\rangle }}} + {\mathbf{m}}d{{u}_{{\left\langle 2 \right\rangle }}} + {\mathbf{n}}d{{u}_{{\left\langle 3 \right\rangle }}}.$$
(2.1)

We set the spectral representation of the strain tensor increment de in the form

$$d\boldsymbol{\varepsilon} = \mathbf{l} \otimes \mathbf{l}(d{{\varepsilon }_{1}}) + \mathbf{m} \otimes \mathbf{m}(d{{\varepsilon }_{2}}) + \mathbf{n} \otimes \mathbf{n}(d{{\varepsilon }_{3}}),$$
(2.2)

where \({\mathbf{l}}\), \({\mathbf{m}}\), \({\mathbf{n}}\) is an orthonormal basis of eigenvectors common both for the stress tensor σ and for the strain tensor increment dε; \(d{{\varepsilon }_{1}}\), \(d{{\varepsilon }_{2}}\), \(d{{\varepsilon }_{3}}\) are the principal increments of (plastic) deformation (eigenvalues of the tensor dε).

Let us further introduce a special numeration for the axes of the principal trihedron in such a way that the following inequalities as well as (1.2) are valid

$$d{{\varepsilon }_{1}} \geqslant d{{\varepsilon }_{2}} \geqslant d{{\varepsilon }_{3}}.$$
(2.3)

By virtue of the associated plastic flow rule

$$d{{\varepsilon }_{1}} = d\lambda ,\quad d{{\varepsilon }_{2}} = 0,\quad d{{\varepsilon }_{3}} = - \bar {a}d\lambda \quad (d\lambda \geqslant 0),$$
(2.4)

systems of ordered principal stresses and principal strain increments are consistent in the presence of a constitutive constraint

$$\bar {a} \geqslant 0.$$

The concept of asymptotic directors of the increment of the deformation tensor dε and its representation in terms of asymptotic directors ''l, ''n are considered in articles [7, 8], as well as in an earlier publication [9]. In particular, the dyadic representation of the strain tensor increment dε has the form

$$d\boldsymbol{\varepsilon} = \mathbf{l}(d{{\varepsilon }_{2}}) + (d{{\varepsilon }_{1}} - d{{\varepsilon }_{3}}){\text{sym}}{\kern 1pt} (''\mathbf{l} \otimes ''\mathbf{n}).$$
(2.5)

The angle between the asymptotic directors ''l, ''n is calculated using the Lode kinematic parameter

$$\cos ''\iota = - \nu ,$$
(2.6)

where

$$\nu = \frac{{2d{{\varepsilon }_{2}} - d{{\varepsilon }_{1}} - d{{\varepsilon }_{3}}}}{{d{{\varepsilon }_{1}} - d{{\varepsilon }_{3}}}}.$$
(2.7)

The flow on the facet of the piecewise-linear plasticity criterion obeys the kinematic constraint \(d{{\varepsilon }_{2}} = 0\) following from the associated flow rule.

In terms of notations of field theory, the system of differential equations of kinematics with respect to the physical components of the displacement vector increment \(d{{u}_{{\left\langle 1 \right\rangle }}}\), \(d{{u}_{{\left\langle 3 \right\rangle }}}\) has the following form:

$$\begin{gathered} ( - {{\kappa }_{1}} + \mathbf{n} \cdot \boldsymbol{\nabla} )d{{u}_{{\left\langle 1 \right\rangle }}} + ( - {{\kappa }_{3}} + \mathbf{l} \cdot \boldsymbol{\nabla} )d{{u}_{{\left\langle 3 \right\rangle }}} = 0, \\ \mathop {\sin }\nolimits^2 \frac{{''\iota }}{2}{\kern 1pt} ((\mathbf{l} \cdot \boldsymbol{\nabla} )d{{u}_{{\left\langle 1 \right\rangle }}} + {{\kappa }_{1}}d{{u}_{{\left\langle 3 \right\rangle }}}) + \mathop {\cos }\nolimits^2 \frac{{''\iota }}{2}{\kern 1pt} ((\mathbf{n} \cdot \boldsymbol{\nabla} )d{{u}_{{\left\langle 3 \right\rangle }}} + {{\kappa }_{3}}d{{u}_{{\left\langle 1 \right\rangle }}}) = 0; \\ \end{gathered} $$
(2.8)

it belongs to the hyperbolic type; the characteristic directions coincide with the directions of the conjugate directors ''l, ''n.

Conjugate directors indicate directions in the plane orthogonal to the second principal axis of the tensor dε that are orthogonal to the directions of asymptotic directors ''l, ''n. The director ''l is orthogonal to the asymptotic director ''n, and the director ''n is orthogonal to ''l.

3. For instantaneous elongation of the elements of the characteristic lines, we find

$$\begin{gathered} ''\mathbf{l} \cdot (d\varepsilon ) \cdot ''\mathbf{l} = 0, \\ ''\mathbf{n} \cdot (d\varepsilon ) \cdot ''\mathbf{n} = 0, \\ \end{gathered} $$
(3.1)

that is, during the flow, the linear elements perpendicular to the directions of the asymptotic directors ''l, ''n do not undergo instantaneous elongation, i.e. material fibers oriented along the directors ''l, ''n do not instantly elongate or shorten.

First of all, instead of physical components \(d{{u}_{{\left\langle 1 \right\rangle }}}\), \(d{{u}_{{\left\langle 3 \right\rangle }}}\), the displacement vector increment du relative to the basis l, \(n\), we introduce its orthogonal projections onto the conjugate directions ''l, ''n: \(d{{u}_{{\left\langle {\bar {1}} \right\rangle }}}\), \(d{{u}_{{\left\langle {\bar {3}} \right\rangle }}}\).

Differentiations along isostatic \({{d}_{1}}\), \({{d}_{3}}\) and conjugate directions \({{\bar {d}}_{1}}\), \({{\bar {d}}_{3}}\) are related by the formulas [10]

$$\begin{gathered} {{d}_{1}} = \frac{1}{{2\sin \frac{{''\iota }}{2}}}\left( {{{{\bar {d}}}_{1}} + {{{\bar {d}}}_{3}}} \right), \\ {{d}_{3}} = \frac{1}{{2\cos \frac{{''\iota }}{2}}}\left( {{{{\bar {d}}}_{3}} - {{{\bar {d}}}_{1}}} \right). \\ \end{gathered} $$
(3.2)

Differential relations along characteristic lines (compare with [11])

$$\begin{gathered} {{{\bar {d}}}_{1}}d{{u}_{{\left\langle {\bar {1}} \right\rangle }}} - \frac{{\cos ''\iota d{{u}_{{\left\langle {\bar {1}} \right\rangle }}} + d{{u}_{{\left\langle {\bar {3}} \right\rangle }}}}}{{\sin ''\iota }}{{{\bar {d}}}_{1}}\left( {\theta - \frac{{''\iota }}{2}} \right) = 0, \\ {{{\bar {d}}}_{3}}d{{u}_{{\left\langle {\bar {3}} \right\rangle }}} + \frac{{d{{u}_{{\left\langle {\bar {1}} \right\rangle }}} + \cos ''\iota d{{u}_{{\left\langle {\bar {3}} \right\rangle }}}}}{{\sin ''\iota }}{{{\bar {d}}}_{3}}\left( {\theta + \frac{{''\iota }}{2}} \right) = 0, \\ \end{gathered} $$
(3.3)

where θ is the angle between some fixed direction in the flow plane and the eigenvector l.

For a piecewise linear yield criterion, the angle \(''\iota \) is constant. After a series of transformations, the flow kinematics relations (3.3) acquire the following final form:

$$\begin{gathered} {{{\bar {d}}}_{1}}d{{u}_{{\left\langle {\bar {1}} \right\rangle }}} - (\mathop {\bar {a}}\limits_1 d{{u}_{{\left\langle {\bar {1}} \right\rangle }}} + \mathop {\bar {a}}\limits_3 d{{u}_{{\left\langle {\bar {3}} \right\rangle }}}){{{\bar {d}}}_{1}}\theta = 0, \\ {{{\bar {d}}}_{3}}d{{u}_{{\left\langle {\bar {3}} \right\rangle }}} + (\mathop {\bar {a}}\limits_3 d{{u}_{{\left\langle {\bar {1}} \right\rangle }}} + \mathop {\bar {a}}\limits_1 d{{u}_{{\left\langle {\bar {3}} \right\rangle }}}){{{\bar {d}}}_{3}}\theta = 0, \\ \end{gathered} $$
(3.4)

where

$$\mathop {\bar {a}}\limits_1 = \frac{{1 - \bar {a}}}{{2\sqrt {\bar {a}} }},\quad \mathop {\bar {a}}\limits_3 = \frac{{1 + \bar {a}}}{{2\sqrt {\bar {a}} }}.$$

It is curious to note that the constitutive constraint

$$\bar {a} \geqslant 0$$

and the inequality of irreversibility do not limit the sign of the dilation rate, i.e. during the flow, the medium can both loosen and contract irreversibly (in the most precise sense of the word).