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Dynamic response of a nanobeam induced by ramp-type heating and subjected to a moving load

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Abstract

The problem considered is that of an isotropic, thermoelastic nanobeam subjected to a moving concentrated loading. The solution of the problem is presented in the context of generalized theory of nonlocal thermoelasticity. Modal analysis along with Laplace’s transform technique is employed to obtain analytical solution of the governing partial differential equations. Effect of velocity of the moving load, nonlocal parameter and ramping-time parameter on dynamic deflection, temperature and flexural moment of the nanobeam has been investigated. Numerical results obtained from the present study are presented graphically and discussed.

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Abbreviations

a :

Internal characteristic length

A = bh :

Area of nanobeam cross-section

\(b\,\,( - b/2 \le y \le + b/2)\) :

Width of nanobeam

C E :

Specific heat at constant strain

E :

Young’s modulus

\(e = \partial u/\partial x\) :

Normal strain

EI :

Flexural rigidity

e 0 :

Material constant

f(x,t):

Varying ramp-type function

h(−h/2 ≤ z≤h/2):

Thickness of nanobeam

I = bh 3/12:

Inertia moment of nanobeam cross-section

K :

Thermal conductivity

L(0 ≤ x ≤ L):

Length of nanobeam

M :

Flexural moment

M T :

Moment of nanobeam due to the presence of thermal effects

q(x,t):

External moving load

Q 0 :

Constant strength of external moving load

s :

Laplace’s variable

t 0 :

Ramp-type parameter

T 0 :

Environment temperature

u :

Axial displacement

w :

Lateral deflection of nanobeam in the \(z\) direction at some position \(x\)

\(\alpha_{T} = \alpha_{t} /\left( { 1- 2v} \right)\) :

Stress-temperature modulus

α t :

Thermal expansion

\(\xi = (e_{0} a)^{2}\) :

Nonlocal parameter

δ (·):

Dirac’s function

v :

Poisson’s ratio

ρ :

Material density

\(\upsilon\) :

Constant speed of external moving load

θ = T−T 0 :

Thermodynamical temperature

θ 0 :

Thermal constant

σ x :

Nonlocal normal stress

τ q :

Phase lag of heat flux

τ θ :

Phase lag of gradient of temperature

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    Ahmed E. Abouelregal

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Ashraf M. Zenkour

  3. Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt

    Ashraf M. Zenkour

Authors
  1. Ahmed E. Abouelregal
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  2. Ashraf M. Zenkour
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Correspondence to Ashraf M. Zenkour.

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Abouelregal, A.E., Zenkour, A.M. Dynamic response of a nanobeam induced by ramp-type heating and subjected to a moving load. Microsyst Technol 23, 5911–5920 (2017). https://doi.org/10.1007/s00542-017-3365-1

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