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Read e-book online Amplification of Nonlinear Strain Waves in Solids (Series on PDF

By Alexey V. Porubov

ISBN-10: 9812383263

ISBN-13: 9789812383266

ISBN-10: 9812794298

ISBN-13: 9789812794291

A remedy of the amplification of nonlinear pressure waves in solids. It addresses difficulties concurrently: the sequential analytical attention of nonlinear pressure wave amplification and choice in wave publications and in a medium; and the demonstration of using even specific analytical recommendations to nonintegrable equations in a layout of numerical simulation of unsteady nonlinear wave approaches. The textual content contains a number of particular examples of the stress wave amplification and choice as a result of the effect of an exterior medium, microstructure, relocating element defects, and thermal phenomena. The volume's major beneficial properties are: nonlinear types of the stress wave evolution in a rod subjected by way of a number of dissipative/active components; and an analytico-numerical technique for recommendations to the governing nonlinear partial differential equations with dispersion and dissipation. The paintings can be compatible for introducing readers in mechanics, mechanical engineering and utilized arithmetic to the idea that of lengthy nonlinear pressure wave in one-dimensional wave courses. it may even be priceless for self-study through pros in all components of nonlinear physics.

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Additional info for Amplification of Nonlinear Strain Waves in Solids (Series on Stability, Vibration and Control of Systems, Series a, 9)

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16) allows us to conclude that solution may now exist only when / = g = 0. Then one can get the algebraic Mathematical tools for the governing equations analysis 41 equations for the solution parameters equating to zero coefficients at the terms p' pk, k = 0 + 3, and (Ap + Bf/2: p' p3 :cA + 2d = 0 p' p2 : 9cAB -VA + +15dB = 0 p' p : 9cAB - 2VA + +12dB = 0 p' (Ap + Bf'2 : 12cB3 - AVB2 + dA(Bg2 - Ag3) = 0 :N = 0 One can obtain the following solution of these equations: A = 2d 2dei . , B = , i = 1 -=- 3, V = c c 1 — -cB, 2 where e* are the real roots of Eq.

The first of them is u=-^-+B. 16). Then equating to zero coefficients at each order of p+C and p' one can derive the algebraic equations on A, B, C, phase velocity V and the Weierstrass function parameters g2, 93(p + C)-4:(g2C-g3-AC3)2(6Af-3A2c-6d) 3 (p + C ) - : (g2C -Q3- 3 = 0, 2 4C )(12C - g2)(UAf - 9A2c - 12d) = 0, (p + C)-2 : 4(g2C - g3 - 4C3)(2bB - V + 48ACf + 3c(12A2C 2 2 2 B )-48Cd)-3(12C -g2) (2f-2Ac-d) - = 0, (p + C ) - 1 : 4(g2C - g3 - 4C 3 )(10/ - 4 / + 3Ac + 6d) - (12C2 g2)(2Bb -V + 24ACf + 3(B2 - 12A2C)c - 12Ad) = 0, p° : 4(12C 2 - g2)qA2 = N, + 24AC/ + 6(B2 - \2A2C)c - \2Cd = 0, (p + C):2Bb-V (p + C)2:2Af p\p + d + 2A2c = 0, + C)-3 : (g2C -g3- 4C3)(g + 2Bf - Ab - 3ABc) = 0, p'(p + C)-2 : (12C 2 - g2)(2Ab -g-2Bf + 6ABc) = 0, p' : 2Ab + g + 2Bf + 6ABc = 0.

2. Below only those results are shown that were obtained using both numerical methods. We have tried various shapes of the initial localized pulses, rectangular, Gaussian distribution etc. Influence of the fifth-order dispersive term. First the 5th-order KdV equation was studied. 5. We found that the rectangular initial pulse splits into a sequence of solitary waves when the coefficients of dispersive terms, d and / , are of opposite sign. For both coefficients positive the initial rectangular profile is dispersed without formation of any localized waves.

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Amplification of Nonlinear Strain Waves in Solids (Series on Stability, Vibration and Control of Systems, Series a, 9) by Alexey V. Porubov

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