Solitary waves in auxetic rods with quadratic nonlinearity: Exact analytical solutions and numerical simulations

Abstract

Special soliton analytical solutions are seeked for the so-called double dispersion equation (or Porubov's equation), which describes the propagation of longitudinal waves in an elastic rod. Using the F-expansion method, a new interesting class of traveling solitary waves is obtained. As a byproduct, the results obtained by other authors for some special cases are regained. Some numerical simulations are also performed. Splitting of various initial pulses during propagation into a sequence of solitary waves is considered. The dependence of the amplitude and the velocity of the solitary waves on Poisson's ratio is discussed in detail. Collisions between some obtained solitary waves are also presented. The simulation results are compared with the obtained exact analytical solutions-the latter reflect the perfect balance between the nonlinearity and dispersion. The comparison indicates that the obtained exact solutions are useful for describing more complicated wave fields studied by numerical simulations. It also indicates that the numerical results for auxetic materials are in better agreement with theoretical solutions than in the case of ordinary materials. Furthermore, one can conclude that the Secant pulse generates solitary waves closer to the analytical predictions than a Gaussian pulse. This is similar to the case of variational method, where the Secant trial function is also more proper than the Gaussian one.

Temperature Variability of Poisson’s Ratio and Its Influence on the Complex Modulus Determined by Dynamic Mechanical Analysis

“…Although the input data remains the same throughout the experimental testing, this may not be true for the intrinsic characteristics of the tested specimen. For example, it is known that Poisson's ratio has a relevant role in the dynamic properties of materials [13][14][15][16][17][18] and is sensible to any temperature changes in the material [19,20]. Such fact is even more relevant when this change occurs at relatively low temperature, e.g., glass/rubbery transition in polymers.…”

Propagation of Ultrashort Pulses in Nonlinear Media

“…Using the relations (41), (42) and equating to zero the coefficients of F i (ξ ) F j (ξ ) (i = 0, 1; j = 0, ±1, ±2, • • •) we obtain a system of algebraic equations for a 0 , a i , b i , k and c. Solving these equations yields the final result for u in the form (40). In [46] we apply this procedure to solve analytically different nonlinear PDEs.…”

“…For the sake of simplicity we assume |V | = V . Then using the formalism presented above we obtain (for the modulus number m = 1, see [46]) the solution…”

“…As usual F here denotes Fourier Transform and F −1 denotes its inverse transform. Calculations in (45) and (46) are performed effectively by the fast algorithm FFT [47]. The time partial derivatives of the envelope function in both the linear and nonlinear operatorL andNcan be easily checked by multiplying the Fourier coefficients U (ξ , ω k ) by powers of −iω k corresponding to the order of derivative and then performing the Inverse Fourier Transform.…”

Auxetics and other systems of “negative” characteristics

“…Wave propagation in auxetics is the subject of the following four papers. T. Bui Dinh, Van Cao Long, and Krzysztof W. Wojciechowski present exact analytical solutions and numerical simulations of solitary waves in auxetic rods with quadratic nonlinearity . Using the, so called, F‐expansion method, a new class of traveling solitary waves is obtained.…”