Stiff phase nucleation in a phase-transforming bar due to the collision of non-stationary waves

Abstract

We deal with a new phase nucleation in a phase-transforming bar caused by a collision of two non-stationary waves. We consider an initial stage of dynamical process in the finite bar before the moment of time when the waves emerged due to new phase nucleation reach the ends of the bar. The model of a phase-transforming bar with trilinear stress–strain relation is used. The problem is formulated as a scale-invariant initial value problem with additional restrictions in the form of several inequalities involving the problem parameters. We consider the particular limiting case where the stiffness of a new phase inclusion is much greater than the stiffness of the initial phase and obtain the asymptotic solution in the explicit form. In particular, the domains of existence of the solution in the parameter space are constructed.

Appendix: The motion of the phase boundary at a near-critical speed

Assume that the phase boundary speed $V$ is subcritical and close to its critical value:

$$\begin{aligned} V= -1+\delta ^2, \end{aligned}$$

(135)

where $\delta >0$ is a small parameter. Let us try to find the possible value for the rate of loading

$$\begin{aligned} {v}={v}_0 +O(\delta ) \end{aligned}$$

(136)

such that the problem solution can possess this property. According to (42) and (43) one has

$$\begin{aligned} \varepsilon _{\mathrm{e}}= & {} {\varepsilon _{\mathrm{e}}}_{({-2})}\delta ^{-2} +O(\delta ^{-1}) \equiv \frac{2{v}_{0}\left( c_{\mathrm{3}}^2-1\right) +z}{c_{\mathrm{3}}^2+1}\delta ^{-2} +O(\delta ^{-1}), \end{aligned}$$

(137)

$$\begin{aligned} \varepsilon _{\mathrm{i}}= & {} {\varepsilon _{\mathrm{i}}}_{(0)} +O(\delta ) \equiv \frac{4{v}_{0}-z}{c_{\mathrm{3}}^2+1} +O(\delta ). \end{aligned}$$

(138)

Substituting expansions (137) and (138) into Eq. (45) and taking coefficients at $\delta ^{-2}$, one gets:

$$\begin{aligned} \left( c_3^2-1\right) {\varepsilon _{\mathrm{i}}}_{(0)}{\varepsilon _{\mathrm{e}}}_{({-2})}+z{\varepsilon _{\mathrm{e}}}_{({-2})}=0. \end{aligned}$$

(139)

From the above equation it follows that

$$\begin{aligned} {\varepsilon _{\mathrm{i}}}_{(0)}=-\frac{z}{c_3^2-1}. \end{aligned}$$

(140)

Taking into account Eq. (138), one can see that

$$\begin{aligned} {v}_{0} = -\frac{z}{2\left( c_{\mathrm{3}}^2-1\right) }. \end{aligned}$$

(141)

Consider the case $c_3 \rightarrow \infty$. Substituting Eq. (44) into (141) now yields

$$\begin{aligned} v_0\rightarrow \frac{\varepsilon _3^\mathrm{m}}{2}. \end{aligned}$$

(142)