Strain Solitons in Solids and How to Construct Them

by Samsonov; Alexander M.

• ISBN13: 9780849306846
• ISBN10: 0849306841
• Format: Hardcover
• Copyright: 2001-01-18
• Publisher: Chapman & Hall/

Summary

Although the theory behind solitary waves of strain shows that they hold significant promise in nondestructive testing and a variety of other applications, an enigma has long persisted-the absence of observable elastic solitary waves in practice. Inspired by this apparent contradiction, Strain Solitons in Solids and How to Construct Them refines the existing theory, explores how to construct a powerful deformation pulse in a waveguide without plastic flow or fracture, and proposes a direct method of strain soliton generation, detection, and observation.The author focuses on the theory, simulation, generation, and propagation of strain solitary waves in a nonlinearly elastic, straight cylindrical rod under finite deformations. He introduces the general theory of wave propagation in nonlinearly elastic solids and shows, from first principles, how its main ideas can lead to successful experiments. In doing so, he develops a new approach to solving the corresponding doubly dispersive equation (DDE) with dissipative terms, leading to new explicit and exact solutions. He also shows that the method is applicable to a variety of nonlinear problems.First discovered in virtual reality, nonlinear waves and solitons in solids are finally moving into the genuine reality of physics, mechanics, and engineering. Strain Solitons in Solids and How to Construct Them shows how to balance the mathematics of the problem with the application of the results to experiments and ultimately to generating and observing solitons in solids.

Author Biography

Alexander M. Samsonov is a Professor, Senior Research Fellow, and Head of the Research Group for Nonlinear Wave Theory in the Theoretical Department of the Ioffe Physico-Technical Institute of the Russian Academy of Sciences in St. Petersburg, Russia

Table of Contents

Nonlinear waves in elastic solids	1	(23)
Basic definitions	1	(11)
Wave terminology	3	(4)
Deformation and strains	7	(3)
Stresses	10	(2)
Physical and geometrical sources of nonlinearity	12	(7)
Elastic potentials and moduli	16	(3)
Compressibility, dispersion and dissipation in wave guides	19	(4)
A mathematical description of the general deformation wave problem	23	(26)
Action functional and the Lagrange formalism	23	(3)
Coupled equations of long wave propagation	26	(1)
One-dimensional quasi-hyperbolic equation	27	(10)
Derivation of the Doubly Dispersive Equation (DDE)	28	(3)
Refinement of the derivation of the DDE	31	(5)
Equations for wave in non-uniform highly nonlinear wave guide	36	(1)
Main assumptions and 2-D coupled equations	37	(7)
Waves in a wave guide embedded in an external medium	44	(5)
Direct methods and formal solutions	49	(30)
Nonlinear hyperbolic and evolution equations	49	(4)
Travelling wave solutions to the KdV and the DDE equations	52	(1)
Conservation laws	53	(3)
The Hamiltonian structure	54	(2)
Some notices in critical points analysis for an o.d.e.	56	(4)
New approach to a solution for an autonomous dissipative nonlinear equation	60	(2)
A general theorem of reduction	62	(2)
Dissipative equations with polynomial nonlinearity	64	(7)
Square and cubic polynomial nonlinearities	66	(4)
Physical interpretation of additional conditions for the travelling wave existence	70	(1)
Elliptic function solutions to higher order problems	71	(3)
Example of a nonlinear reaction-diffusion problem	74	(5)
Discontinuous solutions	75	(1)
Bounded periodical and solitary wave solutions	76	(1)
Autosoliton solution	76	(1)
Periodic bounded solutions in case Δ > 0	77	(2)
Nonlinear strain waves in elastic wave guides	79	(42)
Features of longitudinal waves in a rod	79	(5)
Experiments in nonlinear waves in solids	84	(19)
Strain soliton observation	93	(4)
Why is it a soliton?	97	(1)
Reflection of a strain soliton	98	(5)
Solitons in inhomogeneous rods	103	(8)
Solitons in the non-uniform rod	105	(3)
Solitons in the inhomogeneous rod	108	(3)
Experiments in soliton propagation in the non-uniform rod	111	(10)
Nonlinear waves in complex wave guides	121	(52)
Longitudinal nonlinear waves in an elastic plate	121	(7)
Generation and observation of the strain soliton in a plate	124	(4)
Longitudinal waves in rods embedded in a surrounding medium	128	(22)
Kinks in a rod in full contact with a surrounding medium	129	(2)
Waves in a rod in sliding contact with a surrounding medium	131	(1)
Statement and the solution to the problem	132	(1)
External stresses on the rod lateral surface	133	(2)
Derivation of strain-displacement relationships inside the rod	135	(1)
A nonlinear evolution equation for longitudinal strain waves along the rod and its solution	136	(2)
The influence of the external medium on the propagation of the strain soliton along the rod	138	(2)
A numerical simulation of unsteady strain wave propagation	140	(4)
Applications of the theory	144	(5)
Conclusions	149	(1)
Nonlinear waves in the layer upon the elastic half space	150	(23)
Physical background	150	(2)
Basic equations	152	(2)
The layer and the half-space contact with slippage	154	(7)
The full contact of the layer and the half-space	161	(6)
On various mathematical models of nonlinear waves in a layered medium	167	(4)
On physical experiments in waves in a layered medium	171	(2)
Numerical simulation of the solitary waves in solids	173	(32)
Numerical simulation of non-stationary deformation waves	174	(6)
Parametrisation of the strain solitary wave	178	(2)
Solitary waves in a homogeneous rod	180	(6)
Solitary waves in a nonuniform rod	186	(3)
Solitary waves in complex rods	189	(16)
Strain solitons in a rod with the periodically varied cross section	194	(4)
An example of classification using soliton transformation	198	(7)
Conclusive remarks and tentative applications	205	(4)
Bibliography	209	(14)
Appendix	223	(6)
Index	229