# Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

# Discontinuity Surfaces in Elasto-Visco-Plastic Media

• Yury A. Rossikhin
• Marina V. Shitikova
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_105-1

## Definitions

Surfaces of weak and strong discontinuity in elasto-visco-plastic media are the transient waves carrying the jumps in the field parameters on the wave front; in so doing the magnitudes of plastic deformations should be continuous on the surface.

## Preliminary Remarks

Propagation and attenuation of weak and strong discontinuities in a viscoelastoplastic medium have been investigated under the Mises yield condition in Bykovtsev and Verveiko (1966), wherein it has been shown that two types of waves could exist in such a medium: irrotational and equivoluminal, their velocities of propagation have been calculated, and relationships describing the variation in wave intensity during propagation have been derived.

In the present entry, the behavior of strong and weak discontinuities in the same sort of a medium under the Tresca yield condition and the maximal reduced stress condition will be analyzed.

An acceleration wave could be interpreted as an isolated surface, on which the stresses, principle stress directional cosines, and velocities are continuous, while certain of the partial derivatives have discontinuities (see details in the entry “Ray Expansion Theory”). The symbols [ ] will be used hereafter to indicate a difference in values of a certain variable on opposite sides of the surface of discontinuity, i.e., $$\left [a\right ]=a^{+} -a^{-}$$.

The surface of strong discontinuity could be interpreted as a limiting layer with the thickness Δh at Δh → 0, within which the displacement velocities and stresses change their values from a+ to a monotonically and continuously; in so doing the values of stresses within the layer should be finite in order the magnitudes of plastic deformations should be continuous on the surface Σ.

## Problem Formulation

Assume that in a viscoelastoplastic body that generalizes the model of the Bingham material, the deformations are small and consist of two parts, elastic and plastic (Bykovtsev and Verveiko, 1966):
\displaystyle \begin{aligned} e_{ij} =e_{ij}^{e} +e_{ij}^{p}. \end{aligned}
(1)
The elastic strain tensor is associated with the stresses by the Hooke’s law
\displaystyle \begin{aligned} \sigma _{ij} =\lambda e_{kk}^{e} \delta _{ij} +2\mu e_{ij}^{e}, \end{aligned}
(2)
and the plastic strain velocity tensor $$\varepsilon ^p_{ij} ={\dot e}^p_{ij}$$ with the stress tensor by the Tresca yield condition (Bykovtsev and Ivlev, 1998; Rossikhin, 1969) (Fig. 1)
\displaystyle \begin{aligned} \left|\left(\sigma _{i} -\eta \varepsilon _{i}^{p} \right)-\left(\sigma _{j} -\eta \varepsilon _{j}^{p} \right)\right|=k , \end{aligned}
(3)
where λ and μ are Lame constants, η is the viscosity coefficient, and k is the yield limit of the material.
Assuming that the stress-strain condition corresponds to an edge of the plasticity prism yields
\displaystyle \begin{aligned} \sigma _{i} -\eta \varepsilon _{i}^{p} =\sigma _{j} -\eta \varepsilon _{j}^{p} =\sigma _{k} -\eta \varepsilon _{k}^{p} \pm k, \end{aligned}
(4)
where i = 1, j = 2, and k = 3.
The strains are expressed in terms of the displacements ui by means of the Cauchy formula
The quantities $$\varepsilon _{ij}^{p}$$ are associated with the $$\varepsilon _{i}^{p}$$ as follows:
\displaystyle \begin{aligned} \varepsilon _{ij}^{p} =\varepsilon _{1}^{p} l_{i} l_{j} +\varepsilon _{2}^{p} m_{i} m_{j} +\varepsilon _{3}^{p} n_{i} n_{j}, \end{aligned}
(6)
where li, mi, and ni are the directional cosines for the principal stresses and velocities of the strains $$\varepsilon _{i}^{p}$$.
With allowance for (4), Eq. (6) could be rewritten as
\displaystyle \begin{aligned} \eta \varepsilon _{ij}^{p} =s_{ij} +kn_{i} n_{j} -1/3\;k\delta _{ij}, \end{aligned}
(7)
where
\displaystyle \begin{aligned} s_{ij} =\sigma _{ij} -1/3\;\sigma _{kk} \delta _{ij}. \end{aligned}
(8)
From (1) to (8), it could be obtained
\displaystyle \begin{aligned} \dot \sigma _{ij} &=\lambda v_{k,k} \delta _{ij}\\ & \quad +\mu \left[v_{i,j} +v_{j,i} -\frac{2}{\eta } \left(s_{ij} +kn_{i} n_{j} -\frac 13 \delta _{ij} \right)\right], \end{aligned}
(9)
where a dot over a letter indicates a time derivative, while a subscript following a comma labels the space differentiation with respect to the corresponding coordinate.
The components of the stress and displacement velocities tensors must satisfy the equation of motion
\displaystyle \begin{aligned} \sigma _{ij,j} =\rho \dot v_{i}, \end{aligned}
(10)
where ρ is the density.
The following system must be added to (9) and (10):
\displaystyle \begin{aligned} l_{i} l_{j} +m_{i} m_{j} +n_{i} n_{j} =\delta _{ij} , \end{aligned}
(11)
\displaystyle \begin{aligned} \sigma _{ij} =\sigma _{1} l_{i} l_{j} +\sigma _{2} m_{i} m_{j} +\sigma _{3} n_{i} n_{j}. \end{aligned}
(12)

Thus, relationships (9)–(12) determine the dynamic behavior of the viscoelastoplastic material.

Using (7) and (8) and considering that the stresses are continuous at the surface of discontinuity yield
\displaystyle \begin{aligned} \left[e_{ij}^{p} \right]=0. \end{aligned}
(13)
If the condition of the maximal reduced stress is taken as the plasticity condition (Fig. 2)
\displaystyle \begin{aligned} \left| \left(\sigma _{i} -\eta \varepsilon _{i}^{p} \right)-\sigma \right|=k, \end{aligned}
(14)
then considering, for example, the edge B1, on which the following relations are valid
\displaystyle \begin{aligned} \sigma _{1} - 1/2\;\sigma _{2} -1/2\;\sigma _{3} -\eta \varepsilon _{1}^{p} =k, \end{aligned}
(15)
\displaystyle \begin{aligned} -\sigma _{2} +1/2\;\sigma _{1} +1/2\;\sigma _{3} +\eta \varepsilon _{2}^{p} =k, \end{aligned}
(16)
then after certain manipulations from (6) and (9), it could be found
\displaystyle \begin{aligned} \eta \varepsilon _{ij}^{p} =3/2\;s_{ij} +k(m_{i} m_{j} -l_{i} l_{j} ). \end{aligned}
(17)

Thus, different plasticity conditions, (3) and (14), correspond to different expressions for $$\varepsilon _{ij}^{p}$$, i.e., to (7) and (17), respectively.

It was shown in Bykovtsev and Verveiko (1966) that in a viscoelastoplastic medium, there could exist two types of acceleration waves and two types of shock waves, the propagation velocities of which are
\displaystyle \begin{aligned} \rho c^{2} =\lambda +2\mu , \quad \rho c^{2} =\mu . \end{aligned}
(18)

It could be shown that the assumption that the stress tensor is bounded implies the values of plastic deformation of the wave surface are continuous.

## Attenuation Equations for the Acceleration and Shock Waves

In order to write the attenuation equations for the acceleration waves and shock waves, let us differentiate (9) with respect to xi and (10) with respect to t and sum over repeating subscripts. Taking then the differences in the resulting expressions on opposite sides of the wave surface yields
\displaystyle \begin{aligned} \left[\dot\sigma _{ik,k} \right]=\left(\lambda +\mu \right)\left[v_{k,ki} \right] +\mu \left[v_{i,kk} \right]-\mu \left[\varepsilon _{ik,k}^{p} \right], \end{aligned}
(19)
\displaystyle \begin{aligned} \left[\dot\sigma _{ik,k} \right]=\rho \left[\ddot{v_{i} } \right] . \end{aligned}
(20)
The second-order kinematic conditions of compatibility have the form (Rossikhin, 1969)
\displaystyle \begin{aligned} \left[\dot v_{i,j} \right]&=\left(-cL_{i} +\frac{\delta \lambda _{i} }{\delta t} \right)\nu _{j} -c\lambda _{i,\alpha } x_{j,\beta } g^{\alpha \beta }, \end{aligned}
(21)
\displaystyle \begin{aligned} \left[v_{i,jk} \right]&=L_{i} \nu _{j} \nu _{k} +g^{\alpha \beta } \lambda _{i,\alpha } \left(\nu _{j} x_{k,\beta } +\nu _{k} x_{j,\beta } \right)\\ &\quad -\lambda _{i} g^{\alpha \beta } b_{\alpha \sigma } g^{\sigma \tau } x_{j,\beta } x_{k,\tau }, \end{aligned}
(22)
\displaystyle \begin{aligned} \left[\ddot v_{i} \right]&=c^{2} L_{i} -2c\frac{\delta \lambda _{i} }{\delta t} , \end{aligned}
(23)
where λi and Li characterize the discontinuities in the first- and second-order derivatives of the velocities, respectively; gαβ is the contravariant matrix tensor of the wave surface; bαβ are the coefficients of the second quadratic form for this surface; xi and yα are the Cartesian and curvilinear coordinates of the surface, respectively; α, β, σ = 1, 2, and δδt indicates the δ-derivative (Thomas, 1961), νi are components of the unit vector normal to the wave surface; and c is the velocity of its propagation.
Eliminating the quantities $$\left [\dot \sigma _{ik,k} \right ]$$ from (19) to (20) with allowance for (21)–(23), it could be found
\displaystyle \begin{aligned}\begin{array}{rcl} {} \left(\rho c^{2} -\mu \right)L_{i} -\left(\mu +\lambda \right)L_{k} \nu _{k} \nu _{i} -2\rho c\frac{\delta \lambda _{i} }{\delta t} +2\mu \lambda _{i} \varOmega \\ -\left(\lambda +\mu \right)g^{\alpha \beta } \lambda _{k,a} x_{i,\beta } \nu _{k} -\left(\lambda +\mu \right)g^{\alpha \beta } \lambda _{k,a} \nu _{i} x_{k,\beta } \\ +\left(\lambda +\mu \right)\lambda _{k} g^{\alpha \beta } g^{\sigma \tau } b_{\alpha \sigma } x_{i,\tau } x_{k,\beta } +\mu \left[\varepsilon _{ik,k}^{p} \right]=0, \end{array} \end{aligned}
(24)
where Ω is the average curvature of the surface.
Transforming (24) for irrotational and equivoluminal waves (Bykovtsev and Verveiko, 1966) yields
\displaystyle \begin{aligned} \rho c\frac{\delta \omega }{\delta t} =\left(\lambda +2\mu \right)\omega \varOmega +\mu \left[\varepsilon _{ik,k}^{p} \right]\nu _{i} , \end{aligned}
(25)
\displaystyle \begin{aligned} \frac{\delta \lambda _{i} }{\delta t} =c\lambda _{i} \varOmega+c \left(\left[\varepsilon _{ik,k}^{p} \right] -\left[\varepsilon _{lj,j}^{p} \right]\nu _{l} \nu _{i} \right). \end{aligned}
(26)
When deriving relationships (25) and (26), which define the changes of the parameters of the shock waves during their propagation when the deformed state ahead of the wave front is known, it was assumed that νlxl,α = 0, since the vectors xl,α and νl are orthogonal and, moreover,
\displaystyle \begin{aligned} \nu _{j,\alpha } =-g^{\sigma \tau } b_{\sigma \tau } x_{j,\alpha }, \quad \frac{\delta \nu _{i} }{\delta t} =0, \\ g^{\alpha \beta } g^{\sigma \tau } b_{\alpha \sigma } x_{j,\tau } x_{j,\beta } =2\varOmega,\\ \lambda _{i} \nu _{i} =\omega, \quad \lambda _{k} =\omega \nu _{k} . \end{aligned}
(27)

### Tresca Yield Condition

In order to find an equation for the discontinuities $$\left [\varepsilon _{ij,k}^{p} \right ]$$ using the Tresca yield condition, it is necessary to differentiate Eq. (7) with respect to xk and to write it in discontinuities as
\displaystyle \begin{aligned} \eta \left[\varepsilon _{ij,k}^{p} \right]=\left[s_{ij,k} \right]+k\left(\left [n_{i,k} \right]n_{j} +\left[n_{j,k} \right]n_{i} \right) . \end{aligned}
(28)
To determine the values of discontinuities $$\left [n_{i,k} \right ]$$ in terms of $$\left [\sigma _{ij,k} \right ]$$, let us differentiate (11) and (12) with respect to xk and take the difference of the values on opposite sides of the surface Σ. The result is
\displaystyle \begin{aligned} \left[\left(l_{i} l_{j} \right),_{k} \right]+\left[\left(m_{i} m_{j} \right),_{k} \right] +\left[\left(n_{i} n_{j} \right),_{k} \right]=0, \end{aligned}
(29)
\displaystyle \begin{aligned} \left[\sigma _{ij,k} \right]&=\left[\sigma _{1,k} \right]l_{i} l_{j} +\left[\left(l_{i} l_{j} \right),_{k} \right]\sigma _{1} \\ &\quad +\left[\sigma _{2,k} \right]m_{i} m_{j} +\left[\left(m_{i} m_{j} \right),_{k} \right]\sigma _{2} \\ &\quad +\left[\sigma _{3,k} \right]n_{i} n_{j} +\left[\left(n_{i} n_{j} \right),_{k} \right]\sigma _{3} . \end{aligned}
(30)
Using the kinematic conditions of compatibility on the surface of discontinuity
\displaystyle \begin{aligned} \left[l_{i,k} \right]=a_{i} \nu _{k} , \quad \left[m_{i,k} \right]=b_{i} \nu _{k}, \\ \left[n_{i,k} \right]=c_{i} \nu _{k} , \quad \left[\sigma _{i,k} \right]=A_{i} \nu _{k} . \end{aligned}
(31)
and considering that
\displaystyle \begin{aligned} \left[\sigma _{ij,k} \right]=\mu _{ij} \nu _{k} ,\quad \mu _{ij} =-\frac{\lambda }{c} \omega \delta _{ij} -\frac{\mu }{c} \left(\lambda _{i} \nu _{j} +\lambda _{j} \nu _{i} \right) , \end{aligned}
(32)
the equalities (29) and (30) could be rewritten as
\displaystyle \begin{aligned} a_{i} l_{j} +a_{j} l_{i} +b_{i} m_{j} +b_{j} m_{i} +c_{i} n_{j} +c_{j} n_{i} =0, \end{aligned}
(33)
\displaystyle \begin{aligned} &-\frac{\lambda }{c} \omega \delta _{ij} -\frac{\mu }{c} \left(\lambda _{i} \nu _{j} +\lambda _{j} \nu _{i} \right)\\ &=A_{1} l_{i} l_{j} {+} A_{2} m_{i} m_{j} {+} A_{3} n_{i} n_{j} +\left(a_{i} l_{j} +a_{j} l_{i} \right)\sigma _{1} \\ &\quad +\left(b_{i} m_{j} +b_{j} m_{i} \right)\sigma _{2} +\left(c_{i} n_{j} +c_{j} n_{i} \right)\sigma _{3} , \end{aligned}
(34)
wherein the values Ai, ai, bi, and ci could be found from (32).
For strong discontinuities, the attenuation equation for irrotational and equivoluminal waves could be written in the following form (Bykovtsev and Verveiko, 1966):
\displaystyle \begin{aligned} \rho c\frac{\delta \omega }{\delta t} &=\left(\lambda +2\mu \right)\varOmega +\mu \left[\varepsilon _{ij}^{p} \right] \nu _{i} \nu _{j} , \end{aligned}
(35)
\displaystyle \begin{aligned} \frac{\delta \left[v_{i} \right]}{\delta t} &=c\varOmega \left[v_{i} \right]+c\left(\left[\varepsilon _{ij}^{p} \right] \nu _{i} -\left[\varepsilon _{lj}^{p} \right]\nu _{l} \nu _{j} \nu _{i} \right) . \end{aligned}
(36)
From the difference in (7) written on the opposite sides of the wave surface with due account for $$a^{-} =a^{+} -\left [a\right ]$$, the following expression for the Tresca condition could be obtained:

### Maximal Reduced Stress Yield Condition

To find expressions for $$\left [\varepsilon _{ij}^{p} \right ]$$ while using the maximal reduced stress yield condition, it is necessary to write relationship (17) on different sides of the wave surface Σ and take the difference of the obtained expressions, resulting in
The set of Eqs. (11) and (12) written behind the front of the wave surface
\displaystyle \begin{aligned} &l_{i}^{-} l_{j}^{-} +m_{i}^{-} m_{j}^{-} +n_{i}^{-} n_{j}^{-} =\delta _{ij}, \end{aligned}
(39)
\displaystyle \begin{aligned} &\sigma _{ij}^{-} =\sigma _{1}^{-} l_{i}^{-} l_{j}^{-} +\sigma _{2}^{-} m_{i}^{-} m_{j}^{-} +\sigma _{3}^{-} n_{i}^{-} n_{j}^{-} \end{aligned}
(40)
should be added to (37) and (38).
Assuming that the stressed state ahead of the wave front is known, from (39) and (40), a system of equations for determining the unknowns $$\left [l_{i} \right ],\left [m_{i} \right ],\left [n_{i} \right ],\left [\sigma _{i} \right ]$$ in terms of the known quantities $$\sigma _{{ }_{ij} }^{+}$$ and $$\left [\sigma _{ij} \right ]$$ could be obtained:
\displaystyle \begin{aligned} &\left(l_{i}^{+} -\left[l_{i} \right]\right)\left(l_{j}^{+} -\left[l_{j} \right]\right)+\left(m_{i}^{+} -\left[m_{i} \right]\right)\left(m_{j}^{+} -\left[m_{j} \right]\right)\\ &+\left(n_{i}^{+} -\left[n_{i} \right]\right) \left(n_{j}^{+} -\left[n_{j} \right]\right)=\delta _{ij} , \end{aligned}
(41)
\displaystyle \begin{aligned} &\sigma _{ij}^{+} -\left[\sigma _{ij} \right]\\ &=\left(\sigma _{1}^{+} -\left[\sigma _{1} \right]\right)\left(l_{i}^{+} -\left[l_{i} \right]\right)\left(l_{j}^{+} -\left[l_{j} \right]\right) \\ &\quad +\left(\sigma _{2}^{+} -\left[\sigma _{2} \right]\right) \left(m_{i}^{+} -\left[m_{i} \right]\right)\left(m_{j}^{+} -\left[m_{j} \right]\right) \\ &\quad +\left(\sigma _{3}^{+} -\left[\sigma _{3} \right]\right)\left(n_{i}^{+} -\left[n_{i} \right]\right) \left(n_{j}^{+} -\left[n_{j} \right]\right). \end{aligned}
(42)

## Numerical Examples

### An Irrotational Spherical Wave

As an example of propagation of waves of weak discontinuity, attenuation of an irrotational spherical wave in a space that is extended uniformly toward the σ3-axis could be considered. Considering that σ1 = σ2 = 0, σ3≠0, n1 = n2 = 0, and n3 = 1, from (32), it follows
\displaystyle \begin{aligned} c_{1} =-\frac{2\mu \omega }{\sigma _{3} c} \nu _{2} \nu _{1}, \quad c_{2} =-\frac{2\mu \omega }{\sigma _{3} c} \nu _{3} \nu _{2}, \quad c_{3} =0 . \end{aligned}
(43)
Considering (43) and substituting (31) into (28) yield
\displaystyle \begin{aligned} \left[\varepsilon _{ij,j}^{p} \right] \nu _{i} =-\frac{4\mu \omega }{c} \left[\frac{1}{3} +\frac{k}{\sigma _{3} } \nu _{3}^{2} \left(1- \nu _{3}^{2} \right)\right]. \end{aligned}
(44)
In deriving this relationship, it has been assumed that
\displaystyle \begin{aligned} {\left[\dot \sigma _{ij} \right]} =\lambda \lambda _{k} \nu _{k} \delta _{ij} +\mu \left(\lambda _{i} \nu _{j} +\lambda _{j} \nu _{i} \right). \end{aligned}
Now substituting (44) in Eq. (25) yields
\displaystyle \begin{aligned} \rho c\frac{\delta \omega }{\delta t} &=\left(\lambda +2\mu \right)\omega \varOmega\\ &\quad -\frac{4\omega \mu ^{2} }{c} \left[\frac{1}{3} +\frac{k}{\sigma _{3} } \nu _{3}^{2} \left(1- \nu _{3}^{2} \right)\right]. \end{aligned}
(45)
From Thomas (1961), it is known that
\displaystyle \begin{aligned} \varOmega =\frac{\varOmega _{0} -K_{0} ct}{1-2\varOmega _{0} ct+K_{0}^{2} c^{2} t^{2} }, \end{aligned}
(46)
where K0 and Ω0 are the Gaussian and average curvatures of the wave surface at t = 0.
For a spherical wave, it is valid that $$\varOmega =-R_{0}^{-1}$$ and $$K_{0} =R_{0}^{-2}$$, and thus integrating (46) with due account for (45) yields
\displaystyle \begin{aligned} \omega =\omega _{0} \frac{R_{0} }{R_{0} +ct} \exp \left\{-\frac{4\mu ^{2} }{\eta \rho c^{2} } \left[\frac{1}{3} +\frac{k}{\sigma _{3} } v_{3}^{2} \left(1-v_{3}^{2} \right)\right]t\right\}, \end{aligned}
(47)
where ω0 is the magnitude of ω at t = 0.
Let us compare (47) with the result obtained in Bykovtsev and Verveiko (1966) for the Mises yield condition. The expression found in Bykovtsev and Verveiko (1966) for ω has the form
\displaystyle \begin{aligned} \omega =\omega _{0} \frac{R_{0} }{R_{0} +ct} \exp \left\{-\frac{4\mu ^{2} }{\eta \rho c^{2} } \left[\frac{1}{3} -\frac{\sqrt{3} }{4} \frac{k}{\sigma _{3} } \left(-3 \nu _{3}^{4} +2 \nu _{3}^{2} +1\right)\right]t\right\}.\end{aligned}
(48)
In Fig. 3a, b, the dashed lines (Tresca condition) and solid lines (Mises condition) show ωω0 as a function of μ = 4μ2tηρc2 at $$\nu _3^2=0$$ and 1, respectively, while in Fig. 4 the $$\nu _3^2$$-dependence of ωω0 is presented.

Reference to Figs. 3 and 4 shows that in the direction of the principal stress, ω varies identically for both yield conditions, while for other magnitudes of $$\nu _{3}^{2}$$, the value of ω obtained with the Tresca condition decays more rapidly than that with the Mises condition. With the increase in σ3, the difference in the attenuation laws vanishes.

### An Irrotational Plane Wave

Let us consider the attenuation of an irrotational plane wave in a uniformly deformable space, assuming that the material deforms plastically on both sides of the wave surface, and the wave propagates in the z-direction; in so doing the coordinate axes coincide with the directions of the principal stresses. Then
\displaystyle \begin{aligned} &\left[\sigma _{12} \right]=\left[\sigma _{13} \right]=\left[\sigma _{23} \right]=0, \quad \left[\sigma _{23} \right]\ne 0, \\ &\nu _{1} =\nu _{2} =0, \quad \nu _{3} =1. \end{aligned}
(49)
Writing (11) and (12) behind the wave front as
\displaystyle \begin{aligned} \sigma _{ij}^{-} l_{i}^{-} =\sigma _{1}^{-} l_{i}^{-} , \ \ \sigma _{ij}^{-} m_{i}^{-} =\sigma _{2}^{-} m_{i}^{-} , \ \ \sigma _{ij}^{-} n_{i}^{-} =\sigma _{3}^{-} n_{i}^{-} , \end{aligned}
(50)
yields
\displaystyle \begin{aligned} \left[\sigma _{1} \right]=\left[\sigma _{11} \right], \quad \left[\sigma _{2} \right]=\left[\sigma _{22} \right], \quad \left[\sigma _{3} \right]=\left[\sigma _{33} \right], \end{aligned}
(51)
with $$l_{2}^{-} =l_{3}^{-} =0$$, l1 = 1, $$m_{1}^{-} =m_{3}^{-} =0$$, $$m_{2}^{-} =1$$, $$n_{1}^{-} =n_{2}^{-} =0$$, and $$n_{3}^{-} =1$$, i.e., the directional cosines for the principal stresses remain continuous when passing through the surface of discontinuity.
As a consequence, from (37) and (38), it follows, respectively, for the Tresca condition
\displaystyle \begin{aligned} \eta \left[\varepsilon _{ij}^{p} \right]\nu _{i} \nu _{j} =\left[s_{ij} \right]\nu _{i} \nu _{j}, \end{aligned}
(52)
and for the maximal reduced stress condition
Substituting (52) or (53) in (35), two expressions for ω under the Tresca and maximal reduced stress conditions could be obtained as a result of integration
\displaystyle \begin{aligned} \omega =\omega _{0} \exp \left(-\frac{4}{3} \frac{\mu ^{2} }{\eta \rho c^{2} } t\right), \end{aligned}
(54)
\displaystyle \begin{aligned} \omega =\omega _{0} \exp \left(-2\frac{\mu ^{2} }{\eta \rho c^{2} } t\right), \end{aligned}
(55)
where
\displaystyle \begin{aligned} \left[\sigma _{ij} \right]=-\frac{\lambda }{c} \left[v_{k} \right] \nu _{k} \delta _{ij} -\frac{\mu }{c} \left(\left[v_{i} \right]\nu _{j} +\left[v_{j} \right]\nu _{i} \right). \end{aligned}
From (54) and (55), it is seen that in the second case, ω is attenuated more rapidly than in the first, what is illustrated in Fig. 5, where a dashed line and dashed-dot line refer to the Tresca and maximal reduced stress conditions, respectively.
The comparison of relationships (54) and (55) with the formula obtained in Bykovtsev and Verveiko (1966) for ω by means of the Mises condition
$$\displaystyle \begin{gathered} {} \omega =\omega _{0} \exp \left\{-\frac{4}{3} \frac{\mu ^{2} t}{\eta \rho c^{2} } \left[1-\frac{\sqrt{2} k}{\left(s_{ij}^{+} s_{ij}^{+} \right)^{1/2} } +\frac{3}{2} \frac{\sqrt{2} k\left(s_{ij}^{+} \nu _{i} \nu _{j} \right)^{2} } {\left(s_{ij}^{+} s_{ij}^{+} \right)^{3/2} } \right]\right\} \end{gathered}$$
(56)
shows that in contrast to (54) and (55), the expression obtained for ω with the Mises condition (56) depends on the stressed state ahead of the front of wave surface. With the variation of $$\left (s_{ij}^{+} s_{ij}^{+} \right )^{1/2}$$ from $$\sqrt {2} k$$ to , the expression for ω changes, remaining within the limits from $$\omega _{0} \exp \left [-\frac {2\mu ^{2} \left (s_{ki}^{+} \nu _{i} \nu _{k} \right )^{2} }{\eta \rho c^{2} 2k^{2} } t\right ]$$ to $$\omega _{0} \exp \left (-\frac {4}{3} \frac {\mu ^{2} t}{\eta \rho c^{2} } \right )$$.

Since in the example under consideration $$\left (s_{ki}^{+} \nu _{i} \nu _{k} \right )^{2} \le \left (s_{ij}^{+} s_{ij}^{+} \right )^{2}$$, then under the Mises condition, ω cannot attenuate more rapidly than under the maximal reduced stress condition.

## Conclusion

In the present entry, the propagation and decay of weak and strong discontinuities in a viscoelastoplastic medium have been investigated under three different types of yield condition: Mises condition of plasticity, Tresca condition, and the maximal reduced stress condition. It has been shown that two types of waves could exist in such a medium: irrotational and equivoluminal waves. The decay in such a medium occurs faster than in an elastic material, although the waves propagate with velocities of elastic waves. Decay factors depend on the stressed state ahead of the front of the wave surface and on the propagation direction. The comparison of the characteristic values for three types of yield condition has been carried out.

## References

1. Bykovtsev GI, Ivlev DD (1998) Theory of plasticity. Vladivostok, Dal’naukaGoogle Scholar
2. Bykovtsev GI, Verveiko ND (1966) Wave propagation in a viscoelastoplastic medium (in Russian). Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela 4:111–123Google Scholar
3. Rossikhin YA (1969) Wave propagation in a viscoelastoplastic medium. Sov Appl Mech 5(5):512–517. https://doi.org/10.1007/BF00887338
4. Thomas TY (1961) Plastic flow and fracture in solids. Academic Press, New York/London