The linear approach based on the conviction that result of the net effect is a sum of individual effects, and that the response is directly proportional to the effect, was ruling in science for many centuries. Linear mathematical models imply unambiguous determinism as the consequence is uniquely determined by the reason. All classical laws in physics are linear. The processes described by the laws are stable, valid for the systems near equilibrium and well reproduced in experiments. However, high-rate processes can decline the system state from equilibrium. Under nonequilibrium conditions the linear approach fell down, response to the effect begin to retard from the effect and can be spread over space. The behavior of nonequilibrium systems can be very complicated. The dynamic complexity can arise due to multiple interactions of an open system with its surrounding. Impossible to separate an influence of the individual factor among all the rest effects because of close-loops formed in the system. Retarding responses to the effects lead to the system instability and fluctuations. The occurrence of oscillations and instabilities reduce the control opportunity and obstruct the system study.
New theoretical approach to nonequilibrium transport developed on the base of nonequilibrium statistical mechanics and cybernetic physics proposes a way of structurization of the system under dynamic external loading. New structure elements have intermediate sizes between macro- and microscopic scales. Being the information carriers the mesoscopic structure introduces the internal close-loop into the system. So, the self-organization and self-regulation should be included into the mathematical model of an open system far from equilibrium to reveal indeterminacy effects. So, as long as physicists would use differential rigid models for high-rate processes, the gap between fundamental science and practice will not overcome.
Results of experimental research on the shock loading of solid materials [1] had demonstrated that dependences of the waveforms and threshold of structure instability on strain-rate, target thickness and state of the material structure were not described in the framework of the conventional continuum mechanics approach. New concept of shock-wave processes in condensed matter has been proposed on base of nonlocal theory of nonequilibrium transport which allowed a transition from the elastic medium reaction to the hydrodynamic one depending on the rate and duration of the loading. A new mathematical model of elastic-plastic wave has been constructed to describe the elastic precursor relaxation and the plastic front formation taking into account the changing material properties during the wave propagation [2-3]. Analysis of experimental waveforms has shown that for the shock-induced processes it is incorrect to divide components of stress and strain into elastic and plastic parts. The model allowed accounting of inertial medium properties, changing of its structure together with changing of its mechanical properties under short-duration loading.
Problem on an elastic-plastic plane wave propagation induced by a shock at the velocity V0 in semi-space full of condensed matter is considered. When a loading stress is below an elastic limit an elastic waveform propagate in a matter without mechanical energy dissipation into heat. If the stress exceeds an elastic limit, the wave propagation is followed by the irreversible mass transport, the shear relaxation and the energy loss. In solids a two-wave front is forming under shock loading: elastic precursor and retarding plastic flow. The elastic precursor is going at the constant longitudinal sound velocity C and relaxing up to its stationary value. In the reference connected to the elastic precursor ς = t/tR - x/CtR, ξ = x/L (tR is the loading time, and L is a target width) mass and momentum transport equations in the linear approximation with respect to the small parameter V0,/C<<1 can be reduced to a one integral equation for the mass velocity v(ς). The model parameters τ = tr/tR, θ = tm/tR determine relative relaxation and retardation typical times. If CtR/L<<1, the derived equation describes the wave front, and the parameters define the front evolution.
First in the elastic precursor, the stress is proportional to the strain. When the stress exceeds an elastic limit, the stress begins depend on the strain-rate. Unlike the concept of the ideal elastic-plastic wave, in the integral model the loading stops on the elastic precursor top, and the retarding plastic front results from the medium inertia. An explicit approximate solution to the equation corresponds to all the set of experimental dependences between stress, strain and strain-rate in the wide range of the shock velocities and target widths. The wave front evolution during its propagation is described using the control theory of adaptive systems. The minimal integral entropy production inside the waveform can be chosen as the goal functional. The generalized entropy production σ(ξ)=0 for the elastic waves, σ>0 for the dissipative processes and σ<0 for the synergetic structure formation in the transition regime. The structure formation at mesoscale both in the shear form and rotations had been found out in experiments on the shock loading of different metals [1]. The model parameters as the control ones define the structure evolution by the speed-gradient algorithm [4] that introduces a close-loop into the system and makes it complete and self-regulated.
The energy exchange between the scale levels of dynamic straining is determined by the experimentally measured in real time values: the mass velocity variation and the velocity loss for the structure formation. An internal criterion for a transition of the deformed material into a structure unstable state has been determined.
- Meshcheryakov Yu.I., Divakov A.K., Zhigacheva N.I., Makarevich I.P., Barakhtin B.K. (2008). Dynamic structures in shock-loaded copper. Rev. B. 78. pp. 64301-64316.
- A.Khantuleva. (2003). The shock wave as a nonequilibrium transport process. High-pressure Compression of Solids VI: Old Paradigms and New Challenges, Springer, P. 215-254.
- A. Khantuleva. (2013). Nonlocal theory of nonequilibrium transport processes. S.-Petersburg: S.-Petersburg State University, 276 p.
- Fradkov A.L. (2005) Application of cybernetic methods in physics. Physics.-Uspekhi, 48(2), pp. 103-127.
