NONLOCAL ANALYSIS OF DYNAMIC INSTABILITY OF MICRO-AND NANO-RODS

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1 NONLOCAL ANALYSIS OF DYNAMIC INSTABILITY OF MICRO-AND NANO-RODS Andrzej Tylikowski, Warsaw University of Technology Narbutta Warsaw Poland Abstract. The dynamic stability problem is solved for onedimensional structures subjected to time-dependent deterministic or stochastic axial forces. The stability analysis of structures under time-dependent forces strongly depends on dissipation energy. The simplest model of viscous damping with constant coefficient was commonly assumed in previous papers despite the fact that there are other more sophisticated theories of energy dissipation according to which different engineering constant have different dissipative properties. The paper is concerned with the stochastic parametric vibrations of micro- and nano-rods based on the Eringen s nonlocal elasticity theory and Euler-Bernoulli beam theory. The asymptotic instability, and almost sure asymptotic instability criteria involving a damping coefficient, structure and loading parameters are derived using Liapunov s direct method. Using the appropriate energylike Liapunov functional sufficient conditions for the asymptotic instability, and the almost sure asymptotic instability of undeflected form of beam are derived. The nonlocal Euler-Bernoulli beam accounts for the scale effect, which becomes significant when dealing with short micro- and nano- rods. From obtained analytical formulas it is clearly seen that the small scale effect decreases the dynamic instability region. Instability regions are functions of the axial force variance, the constant component of axial force and the damping coefficient. Keywords: Nonlocal continuum mechanics, Stochastic parametric vibrations, Dynamic instability, Energy-like functional, Liapunov method 1. INTRODUCTION Dynamics stability of distributed systems (continua) has been an object of considerable attention over the past half of century. Numerous papers are available on isotropic and laminated beams, shafts, plates and shells under periodic and random forces. Most of papers have applied finite dimensional or modal approximations in analysis of vibration and stability. The Liapunov direct method is a quite different approach and can be successfully used to analyze continuous systems described by partial differential equations. A significant advantage is offered by the method in that the equations of motion do not have to be solved in order to examine the stability. An application of nonlocal continuum model to representative problems of nanotechnology was demonstrated by Peddieson, Buchanan and McNitt (3). A model, based on nonlocal continuum mechanics, was applied by Sudak (3) to solve the buckling of multiwalled nested carbon nanotubes. The detailed study on the flexural wave dispersion in single-walled nanotubes on the basis of beam models in a wide range of wave numbers was presented by Wang and Hu (5). Wang and Varadan (6) show that the vibration analysis results based on nonlocal continuum mechanics are in agreement with the experimental reports in the field. Studies of wave properties of single- and double-walled nanotubes were performed by Lu et. al (7). The influence of thermal effects on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory was examined by Benzair et. al (8). The vibrational characteristics of double-walled nanotubes with initial stresses were examined by Heiriche, Tounsi and Benzair (8) using nonlocal Bernoulli - Euler beam model. Dynamic stability of doublewalled carbon nanotubes treated as co-axial Euler beams with different bending stiffnesses was analysed via Liapunov method by Tylikowski (6). Based on the Donnel-Vlasov shell theory, a double - elastic shell model was presented by Tylikowski (8) for the parametric vibrations of double-walled carbon nanotubes under time-dependent membrane forces of thermal origin.. PROBLEM FORMULATION The theory of nonlocal continuum mechanics assumes that the stresses at a given reference point are functions of the strain state of all points in the body. In this way, the internal length scale enters into constitutive equations as a material parameter. Adopting Eringen s (1983) nonlocal elasticity a uniaxial stress state is described by the following equation σ (e o a) σ,xx = Eε (1) where σ - the uniaxial stress, ε - the axial strain, E - Young s modulus e o is a constant appropriate to the material, and a denotes the internal characteristic length. Calculating the bending moment in a given crossection of the beam we have EJW,XX + (e o a) M,XX = M ()

2 where W is the beam transverse displacement, and EJ is the beam bending stiffness. In order to derive the equation of the beam transverse motion we start from the dynamics equations M,XX = ρaw,tt + β ρaw,t + N X W,XX (3) where T - time, ρ - mass density, A - cross-section area, β - coefficient of viscous damping, N X - total axial force. Parametric vibrations of structures forces strongly depend on dissipation energy. The simplest model of viscous damping with constant coefficient was commonly assumed in previous papers despite the fact that there are other more sophisticated theories of energy dissipation according to which different engineering constant have different dissipative properties. We calculate the second order derivative of expressions in Eq. () with respect to X and we eliminate M substituting Eq. (3) EJW,XXXX +(e o a) ( ρaw,ttxx +β ρaw,txx +N X W,XXXX ) = ρaw,tt +β ρaw,t +N X W,XX (4) Rearranging we have ρaw,tt + β ρaw,t + N X W,XX + EJW,XXXX + (e o a) [ ρaw,ttxx + β ρaw,txx + N X W,XXXX ] = X (, l) (5) Introducing dimensionless coordinates T = tk t = tl ρa/ej X = xl W = wl (6) and denotations v = w,t β = β k t ǫ = (e o a/l) f(t) + f o = N X kt /ρa (7) we obtain the linear partial differential equation with time - dependent coefficients w,tt +β w,t + ( f(t)+f o ) w,xx +w,xxxx ǫ [ w,ttxx +β w,txx + ( f(t)+f o ) w,xxxx ] = x (, 1) (8) Due to the simply supported edges we have the following boundary conditions w(, t) = w(1, t) = w,xx (, t) = w,xx (1, t) = (9) The transverse motion of the beam is described by a linear uniform equation with the trivial solution w = v = () corresponding to the undisturbed state. The trivial solution is called almost sure asymptotically stable (Kozin, 197) if P { lim t w(., t) = } = 1 (11) where w(., t) is a measure of disturbed solution w from the equilibrium state, and P is a probability measure. 3. ALMOST-SURE INSTABILITY ANALYSIS In order to examine the in stability of trivial solution w = v = we construct the Liapunov functional as a sum of modified kinetic energy and the elastic energy of the beam using the approach described by Tylikowski (1991) V = 1 1 ( v + βvw + β w + ǫ ( v,x + βv,x w,x + β w,x) + w,xx f o ( w,x + ǫw,xx) ) dx (1) The functional (1) is called "the best" as it gives the greatest almost sure stability domain. If the classical condition for static buckling (f o < π /(1+ǫπ )) is fulfilled, the functional (1) is positive-definite and the measure of distance can be chosen as the square root of the functional w(., t) = V (13) If the trajectories of the forces are physically realizable ergodic processes the classical calculus is applied to calculation of the time-derivative of Eq. (1). Upon differentiation with respect to time, substituting dynamic equation (8) and using beam boundary conditions (9) we obtain the time-derivative of functional in the form dv dt = βv + U (14)

3 where an auxiliary functional U is defined as follows U = 1 ( β vw + β 3 w + ǫ ( β v,x w,x + β 3 w,x ) f(t) ( )( ) ) βw + v w,xx ǫw,xxxx dx (15) In order to find a function λ satisfying inequality U λv (16) we look for a stationary point of functional U λv, which is equivalent to inequality (16) for the second order functionals. Solving the Euler auxiliary variational problem δ(u λv ) = we can find the appropriate function λ. In the case of simply supported edges described by boundary conditions (9) there exists a solution of Eq. (8) in the form of infinite series w(x, t) = G n (t)sin nπx (17) Therefore, its time-derivatives has the form v(x, t) = H n (t)sin nπx where the infinite sequence of functions G mn, H mn n = 1,,... is not known. Due to the orthogonality of series present in Eq. (17), and Eq. (18) the value of functionals can be calculated as a sum of the suitable quadratic terms V = U = V n U n where V n and U n are calculated from Eq. (1) and Eq.(15) for a single term of the expansion, respectively. If λ n, which satisfies a single term inequality, is known dv n dt + βv n = U n λ n V n then the following chain of inequalities is true ( ) U = U n λ n V n min λ n V = λv (),,... Therefore, the function λ can be effectively calculated in the form β + f(t)n π / λ = min,,... β + n π [ ] (3) n π /(1 + ǫn π ) f o Substituting the n-th terms of expansions (17)-(18) into inequality (14) we obtain the second order quadratic inequality with respect to the two variables G n, H n. The inequality solution is equivalent to finding the smallest root of the second order algebraic equation. Using a property of function λ in equality (14) leads to the first order differential inequality the solution of which has the form V(t) V()exp [ ( 1 t t ) ] λ(s)ds β t. (4) Therefore, the sufficient criterion of the asymptotic instability has the form 1 β lim t t t λ(s)ds. If the process f satisfies an ergodic property guaranteeing the equality of time and assemble averages with probability one the sufficient condition of the almost sure asymptotic instability can be written as follows (18) (19) () (1) (5) β Eλ (6) where E denotes the mathematical expectation.

4 4. RESULTS Based on the formulation obtained above, instability domains of single-walled nanotubes are discussed here. The instability domains of single-walled carbon nanotube are obtained for the Gaussian process as well as the harmonic process with uniformly distributed phase in the range ( π). The process amplitude A f is used to calculate the effective value A f /. In order to determine the instability domains discrete values of random force f are chosen and λ n is computed via Eq. (3). Then we choose the smallest value. This is accomplished for various values of ǫ by choosing variance Df and varying β until the inequality (6) is satisfied. The mathematical expectation is calculated by means of the known probability density of parametric excitation f. The variance Df depends on the probability distribution of time-dependent component of axial force. Instability domains are situated above drawn lines and are determined for changing values of the dimensionless scale parameter ǫ =.,.5,.1,.5,.4. The scale parameters are calculated for e o a = nm nm and l = nm. Instability boundaries are shown on the plane β - Df for different Gaussian process f o Ε. Ε.5 Ε. Ε.5 Ε Figure 1. Changes of instability domains of single-nanotube with dimensionless scale parameter ǫ. values of scale parameter ǫ and the zero value of constant component of axial force.the instability domains increase as the small scale effect increase. The increase of instability domains is rather significant. In Fig. the negligible influence Gaussian and harmonic processes f o Ε. Ε Figure. Small scale effect on instability domains with Gaussian process (continuous lines) and harmonic process (broken lines). of the force probability distribution on instability domain is shown. The comparison is made for the equal variances of both processes. The variance of harmonic process is equal to the square of efective value Df = A /. The behaviour of instability boundaries for both class of parametric excitation is qualitatively the same. The influence of small scale

5 parameter on instability regions in the near critical region, where the compressive forse is close to the critical Timoshenko force (f o = 7.7), is presented. in Fig. 3. Comparison with Fig. 1 and Fig. 4 shows that the instability regions are highly sensitive in the near critical region on changes of small scale parameter. Figure 4 corresponds to a rather high stretching of the beam (f o = 15) Gaussian process f o Ε. Ε.5 Ε. Ε.5 Ε Figure 3. Small scale effect on instability domains with near critical compresive constant force. 4 Gaussian process f o 15 Ε. Ε.5 Ε. Ε.5 Ε Figure 4. Small scale effect on instability domains with stretching constant force. 5. CONCLUSIONS The direct Liapunov method has been effectively applied to a high order partial differential equation with timedependent coefficient.the influence of the nonlocal scale parameter, damping coefficient, the constant component and the variance of axial parametric excitation on the dynamic instability domains are shown. The nonlocal Euler-Bernoulli beam accounts for the scale effect, which becomes significant when dealing with short micro- and nano- rods. From obtained analytical formulas and numerical calculations it is clearly seen that the small scale effect increases the dynamic instability region. The influence of going from the Gaussian process to the harmonic one has negligible effects.

6 6. ACKNOWLEDGEMENTS The financial support granted by Warsaw University of Technology is greatly appreciated. 7. REFERENCES Bezair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N., and Boumia, L., 8, "The termal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", J. of Physics D: Applied Physics,Vol. 41, pp Eringen., A., C., 1983, "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. of Applied Physics, Vol. 54, pp Heirreche, H., Tounsi, and Benzair, A., 8, "Scale effect on wave propagation of double-walled carbon nanotubes with initial axial loading", Nanotechnology, Vol. 19, p Kozin, F., 197, "Stability of stochastic dynamical systems", Lecture Notes in Mathematics, Vol.94, pp Lu, P., Lee, H.,P.,Lu, C.,Zhang, P.,Q., 7, "Application of nonlocal beam models for carbon nanotubes", International J. of Solids and Structures, Vol. 44, Peddieson, J., Buchanan, G.,R., and McNitt, R.,P., 3, "Application of nonlocal continuum models to nanotechnology", International J. of Engineering Sciences, Vol. 41, pp Sudak, L.,J., 3, "Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics", J. of Applied Physics, Vol.94, pp Tylikowski, A., 1991, "Stochastic Stability of Continuous Systems", PWN, Warszawa, p. 6,(in Polish). Tylikowski, A., 6, "Dynamic stability of carbon nanotubes", Mechanics and Mechanical Engineering International J., Vol., pp Tylikowski, A., 8, "Instability of thermally induced vibrations of carbon nanotubes", Archive of Applied Mechanics, Vol.78, pp Wang, L., and Hu, H., 5, "Flexural wave propagation in single-walled carbon nanotubes", Physical Review B, Vol. 71, p Wang, Q., and Varadan, V., K., 6, "Vibration of carbon nanotubes studied using nonlocal continuum mechanics", Smart Materials and Structures, Vol. 15, pp RESPONSIBILITY NOTICE The author is solely responsible for the printed material included in this paper.