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Nonlinear dynamics of SMA-fiber-reinforced composite beams subjected to a primary/secondary-resonance excitation

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Abstract

In this paper, nonlinear free vibration and primary/secondary resonance analyses of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are investigated. The simplified Brinson constitutive model and cosine phase transformation kinetics are utilized to simulate the behavior of the SMA materials and calculate the recovery stress. In order to predict the behavior of the smart laminated beam, Euler–Bernoulli beam theory and nonlinear von-Kármán strain field are employed. Two types of micromechanical models, namely Voigt and Reuss models are considered. The Galerkin procedure together with the elliptic function and multi timescales method is adopted to obtain analytical solutions for the nonlinear free vibration and primary/secondary response phenomena. Numerical results reveal that some of the geometrical and physical parameters such as the SMA volume fraction, the amount of prestrain in the SMA fiber, orientation of composite fiber, vibration amplitude and temperature are important factors affecting the free vibration characteristic in the pre/post-buckled region, and primary and secondary resonance of the laminated beams reinforced with SMA fibers. The analytical solutions and results are reported for the first time and can serve as benchmark for researchers to validate their numerical and analytical methods in the future.

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Abbreviations

a :

Central amplitude of the beam

A 11 :

Extensional stiffness coefficient

b :

Width of the composite beam

B 11 :

Extension-bending coupling coefficient

D 11 :

Bending stiffness coefficient

E 11 :

Young’s modulus in 1 direction

E 22 :

Young’s modulus in 2 direction

E s :

Young’s modulus of SMA fiber

E m :

Young’s modulus of matrix

F :

Amplitude of excitation

G 12 :

Modulus of rigidity

G s :

Modulus of rigidity of SMA fiber

G m :

Modulus of rigidity of matrix

h :

Total thickness of the beam

I :

Area moment of inertia

k :

ith Layer, starting from the bottom

K :

Kinetic energy

L :

Length of the beam

m :

Cosine of the ply angle

M :

Moment resultant

n :

Sine of the ply angle

N :

Force resultant

\({\overline{Q}_{ij}}\) :

Reduced stiffness matrix

T :

Temperature

T 0 :

Reference temperature

t :

Time

\({\tilde{t}_0}\) :

Fast timescale

t 1 :

Slow timescale

u :

Axial displacement

U :

Strain energy

V m :

Volume fraction of the matrix

V s :

Volume fraction of the SMA fiber

w :

Lateral displacement

x, z :

Beam coordinates in the x, z directions

ε x :

Axial strain

κ :

Curvature

ω L :

Linear fundamental frequency

ω NL :

Nonlinear fundamental frequency

λ :

Detuning parameter

ψ :

Frequency of excitation

\({\Delta T_{\rm Cr}}\) :

Critical buckling temperature difference

υ 12 :

Poisson’s ratio

ρ :

Density of composite beam

α s :

Thermal expansion coefficient of the SMA

α m :

Thermal expansion coefficient of the matrix

ξ :

Total martensite volume fraction

ξ S :

Stress induced martensite volume fraction

ξ S0 :

Initial Stress induced martensite volume fraction

ξ T :

Temperature induced martensite volume fraction

ξ T0 :

Initial Temperature induced martensite volume fraction

Θ :

Thermal coefficient of expansion

E M :

Martensite Young’s modulus

E A :

Austenite Young’s modulus

ε L :

Maximum residual strain

ε 0 :

Prestrain

A s :

Austenite start temperature

A f :

Austenite finish temperature

C A :

Stress influence coefficient

σ r :

Recovery stress obtained through SMA activion

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Authors and Affiliations

  1. Thermoelasticity Center of Excellence, Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

    H. Asadi, M. Bodaghi, M. Shakeri & M. M. Aghdam

Authors
  1. H. Asadi
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  2. M. Bodaghi
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  3. M. Shakeri
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  4. M. M. Aghdam
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Correspondence to M. M. Aghdam.

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Asadi, H., Bodaghi, M., Shakeri, M. et al. Nonlinear dynamics of SMA-fiber-reinforced composite beams subjected to a primary/secondary-resonance excitation. Acta Mech 226, 437–455 (2015). https://doi.org/10.1007/s00707-014-1191-4

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  • DOI: https://doi.org/10.1007/s00707-014-1191-4

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