Investigation of a great number of physical systems shows that a Landau free energy of the form F(φ)=Hφ+(A/2)φ2+(B/3)φ3+(C/4)φ4 describes a first-order phase transition in an internal or external field H. To study the formation of static domain walls in these systems we include a spatial gradient (Ginzburg) term of the scalar order parameter φ. From the variational derivative of the total free energy we obtain a static equilibrium condition. By solving this equation exactly for different physical parameters and boundary conditions, we obtained different quasi-one-dimensional soliton-like solutions. These solutions correspond to three different types of domain walls between the two different phases which are created in the system. In addition, we obtain soliton lattice (domain wall array) solutions, calculate their formation energy and the asymptotic interaction between the solitons. By introducing certain transformations, we show that the solutions obtained here can be used to study domain walls in other ...

Landau theory of domain walls for one-dimensional asymmetric potentials

The interaction of two plane symmetric shocks in a solid sample induces a significant increase of both the pressure and the temperature in the central zone where the incident compressive pulses cross each other. In iron samples, such loading conditions may produce typical structural defects (twins, dislocations) and phase transitions that can be revealed by posttest examination of the recovered targets. We have used two high-power laser beams to irradiate simultaneously both surfaces of thin iron foils. The recovered samples have been sectioned and observed in optical microscopy. A very dense twin distribution in the central zone has confirmed the pressure amplification due to the interaction of the incident shocks. The occurrence of a phase transition has been inferred from the presence of short characteristic twins. Spall fraction has been observed near both irradiated surfaces, and additional damage has been evidenced at the center of the samples. Numerical tools have been adapted to simulate the experiments. Computations have provided estimates of the stress histories inside the samples, and the ability of simple twin, phase change, and spall models has been tested to predict the observed results.

Interaction of two laser shocks inside iron samples

In this work, the spectral stability of a class of solutions of the one-dimensional nonlinear reaction–diffusion equation with spatial symmetry and asymmetry is studied. The solutions are first obtained using the $$G'/G$$
 -expansion method, and it is shown that this technique generates only kink and antikink type solutions. The essential spectra of the perturbed differential operator at the equilibrium states are obtained. The Evans function is used to determine the point spectrum using Lie midpoint method and Magnus method over a large range of spatial distance. It is shown that for a symmetric potential well, the stationary kink and antikink solutions connecting the stable equilibrium (vacuum) states of the system are stable. Interestingly, it is found that for a potential with broken symmetry, the stable stationary kink/antikink solutions transform into stable traveling kink/antikink solutions. This partially answers the question (Alfimov and Medvedeva in Phys Rev E 84:056606, 2011) on the qualitative features of the potential that are decisive for the existence/nonexistence of traveling kink/antikink solutions. The region of stability of the traveling kink/antikink solutions in the parameter space of the potential is identified. An interesting exchange of wave speeds between the kink and antikink solutions through the stationary solution state corresponding to the symmetric potential is brought out.

Spectral stability of one-dimensional reaction–diffusion equation with symmetric and asymmetric potential

Effects of shock waves (generated by a nanosecond laser pulse in plates of Armco-iron) on structural changes are analysed. Localisation of processes of martensitic transformation and twinning – for various values of laser pulse duration – is studied both experimentally and numerically. A proposed model accounts for interaction of shock wave propagation and structure changes. Realisation of martensitic transformation and twin formation influences wave front modification. A stress amplitude decrease with increasing distance from a microcrater determines, together with the pulse duration, a character of spatial localisation of structural changes. Numerical results are compared with experimental data and serve as a basis for additional interpretation of phenomena.

A.A. Kalin

Papers

Evolution of a laser-generated shock wave in iron and its interaction with martensitic transformation and twinning