We seek a solution in the form of a traveling wave *α*(*x*-*u*_{0}*t*), *u*(*x*-*u*_{0}*t*), *w*(*x*-*u*_{0}*t*), *ρ*_{1}(*x*-*u*_{0}*t*), *P*(*x*-*u*_{0}*t*). Then

$${u}^{\prime}\mathrm{(}u\text{-}{u}_{0}\mathrm{)}=\text{-}\frac{1}{\mathrm{(}1\text{-}\alpha \mathrm{)}{\rho}_{s}}{P}^{\prime}\text{\hspace{1em}(13)}$$(13)

$${\alpha}^{\prime}\mathrm{(}u\text{-}{u}_{0}\mathrm{)}=\mathrm{(}1\text{-}\alpha \mathrm{)}{u}^{\prime}\text{\hspace{1em}(14)}$$(14)

$$\text{-}{u}_{0}{{\rho}^{\prime}}_{1}+\mathrm{(}{\rho}_{1}w{\mathrm{)}}^{\prime}=0\text{\hspace{1em}(15)}$$(15)

where the prime denotes derivative with respect to corresponding traveling wave variables.

The first integrals of Eqs. (13)–(15) are:

$$\alpha =1\text{-}\frac{{C}_{1}}{u\text{-}{u}_{0}}\text{\hspace{1em}(16)}$$(16)

$$P=\mathrm{(}{C}_{2}\text{-}{C}_{1}{\rho}_{s}u\mathrm{)}\text{\hspace{1em}(17)}$$(17)

$$\alpha {\rho}_{e}\mathrm{(}u\text{-}{u}_{0}\text{-}\frac{K{P}^{\prime}}{B\mathrm{(}1\text{-}\alpha \mathrm{)}{\rho}_{e}}\mathrm{)}={C}_{3}\text{\hspace{1em}(18)}$$(18)

Transforming Eq. (18) with the help of Eqs. (16) and (17) and using the variable $$\overline{u}=u\text{-}{u}_{0}$$, we obtain the following equation containing the rate of the second phase:

$$\frac{d\overline{u}}{d\eta}=\frac{{C}_{3}\text{-}\mathrm{(}\overline{u}\text{-}{C}_{1}\mathrm{)}{\rho}_{e}}{\overline{u}\text{-}{C}_{1}}\text{\hspace{1em}(19)}$$(19)

where $$d\eta =\frac{d\xi}{K{\rho}_{s}}.$$ Next, we consider the case *ρ*_{e}=*AP* [17], for which

$$\frac{d\overline{u}}{d\eta}=\frac{\mathrm{(}\overline{u}\text{-}{\overline{u}}_{1}\mathrm{)}\mathrm{(}\overline{u}\text{-}{\overline{u}}_{2}\mathrm{)}}{\overline{u}\text{-}{C}_{1}},\text{\hspace{1em}(20)}$$(20)

where $${\overline{u}}_{1},$$ and $${\overline{u}}_{2}$$ denote the velocities of the second phase on the boundary of the localized zone. Integration of this equation leads to:

$$\mathrm{(}\frac{{C}_{1}\text{-}{\overline{u}}_{2}}{{\overline{u}}_{1}\text{-}{\overline{u}}_{2}}\mathrm{)}\mathrm{ln}\mathrm{(}\overline{u}\text{-}{\overline{u}}_{2}\mathrm{)}\text{-}\mathrm{(}\frac{{C}_{1}\text{-}{\overline{u}}_{1}}{{\overline{u}}_{2}\text{-}{\overline{u}}_{1}}\mathrm{)}\mathrm{ln}\mathrm{(}\overline{u}\text{-}{\overline{u}}_{1}\mathrm{)}=d\eta +C\text{\hspace{1em}(21)}$$(21)

To determine the constants involved in Eqs. (20) and (21), the following boundary conditions are used:

$$\begin{array}{c}\overline{u}\mathrm{(}0\mathrm{)}={\overline{u}}_{1},\text{\hspace{0.17em}}\overline{u}\mathrm{(}L\mathrm{)}={\overline{u}}_{2}\text{,\hspace{0.17em}}{\overline{u}}^{\prime}\mathrm{(}0\mathrm{)}=0,\text{\hspace{0.17em}}{\overline{u}}^{\prime}\mathrm{(}L\mathrm{)}=0,\text{\hspace{0.17em}}\alpha \mathrm{(}0\mathrm{)}={\alpha}_{1},\text{\hspace{0.17em}}\\ \alpha \mathrm{(}L\mathrm{)}={\alpha}_{2}\end{array}\text{\hspace{1em}(22)}$$(22)

Then, by Eq. (22), the first integrals will take the form:

$$\begin{array}{l}\mathrm{(}1\text{-}{\alpha}_{1}\mathrm{)}{\overline{u}}_{1}={C}_{1}\\ {P}_{1}={C}_{2}\text{-}{C}_{1}{\rho}_{s}{\overline{u}}_{1}\\ \text{-}{\overline{u}}_{1}^{2}{C}_{1}\rho +\mathrm{(}{C}_{1}^{2}\rho +{C}_{2}\mathrm{)}{\overline{u}}_{1}\text{-}{C}_{3}\text{-}{C}_{1}{C}_{2}=0\\ \mathrm{(}1\text{-}{\alpha}_{2}\mathrm{)}{\overline{u}}_{2}={C}_{1}\\ {P}_{2}={C}_{2}\text{-}{C}_{1}{\rho}_{s}{\overline{u}}_{2}\\ \text{-}{\overline{u}}_{2}^{2}{C}_{1}\rho +\mathrm{(}{C}_{1}^{2}\rho +{C}_{2}\mathrm{)}{\overline{u}}_{2}\text{-}{C}_{3}\text{-}{C}_{1}{C}_{2}=0\end{array}\text{\hspace{1em}(23)}$$(23)

Returning to Eqs. (21) and (23) by using the variable *u*, we construct the speed plotted for the second phase for the case *u*_{1}>*u*_{2} and *α*_{1}<*α*_{2} with respect to coordinate *x* at various times (Figure 1). This clearly shows that a kind of “shock transition” occurs. Consequently, there are areas of the deformable material, which are not involved in the plastic deformation. This is confirmed by experimental observation [5]. The speed of the containment chamber may be defined as $${u}_{0}=\frac{\mathrm{(}{\alpha}_{1}\text{-}1\mathrm{)}{u}_{1}+\mathrm{(}1\text{-}{\alpha}_{2}\mathrm{)}{u}_{2}}{{\alpha}_{1}\text{-}{\alpha}_{2}}.$$ If *u*_{1}=0 and *u*_{2}=*u*_{*}, where *u*_{*} is the velocity of the traverse beam, the values of the marginal rate of localization exceeds the rate of the traverse beam in the testing machine, which is also consistent with the experiment. The case *u*_{1}<*u*_{2} and *α*_{1}>*α*_{2} also allows the existence of shock transition. Note that similar relationships were obtained in Ref. [18] for fixed dynamical structures, and in Ref. [19] for a shock wave in an ideal gas.

Figure 1 Velocity of the second phase with respect to the spatial coordinates at various points of time (1–*t*=0, 2–*t*=1, 3–*t*=2).

We define the width of the shock transition with a relation having the following form: $$l=\frac{{u}_{1}\text{-}{u}_{2}}{\mathrm{max}\mathrm{(}\frac{du}{dx}\mathrm{)}}.$$ The evaluation of this magnitude shows that it has the value of ∼10 μm, which coincides with the characteristic length scales of heterogeneity observed in the experiment. Also note that the free path of dislocation motion in materials is of the same order of magnitude [20].

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.