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A Finite Element Simulation of Longitudinal Impact Waves in Elastic Rods

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Materials with Complex Behaviour II

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 16))

Abstract

In this chapter, wave propagation in a thin rod struck by a rigid mass is considered and a finite element simulation of the system is developed. Both cases of free–free and fixed-free rods are considered. Though impact generates a propagating stress wave in both cases, the free–free rod is going to have a rigid-body motion. The analytical equations of motion are presented and the corresponding finite element equations are derived. A numerical scheme is constructed and solutions are obtained using Newmark implicit integration method and Newton–Raphson iterative technique. Solutions include time histories of displacement, velocity, stress, and contact force. The contact force is calculated, according to St. Venant’s impact model. Numerical results of the simulation are compared to traditional analytical results. A simulated visualization of the propagation of the stress wave in the rod is presented, which enhances the understanding of this complicated physical phenomenon. The achieved results are accurate enough to have confidence in using this model for practical applications in wave propagation simulation and analysis.

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Abbreviations

A :

Cross-sectional area (m2)

c :

Wave propagation velocity (m/s)

E :

Young’s modulus (N/m2)

F :

Contact force (N)

\( \{ f\} \) :

Global force vector (N)

\( \{ f\}_{el} \) :

Force vector for element in contact with the rigid mass (N)

\( \left[ K \right] \) :

Global stiffness matrix (N/m)

l :

Length (m)

L :

Lagrangean (J)

m :

Mass of the rod (kg)

\( m_{0} \) :

Mass of the rigid mass (kg)

\( \left[ M \right] \) :

Global mass matrix (kg)

\( \left[ N \right] \) :

Finite element shape functions (m/m)

q :

Displacement of the rigid mass (m)

\( \dot{q} \) :

Velocity of the rigid mass (m/s)

t :

Time (s)

\( t_{c} \) :

Contact period (s)

T :

Kinetic energy (J)

u :

Displacement of the rod at position x(m)

\( \{ U\} \) :

Nodal displacement vector (m)

\( \{ \dot{U}\} \) :

Velocity vector (m/s)

\( \{ \mathop U\limits^{..} \} \) :

Acceleration vector (m/s2)

\( \left\{ U \right\}_{el} \) :

Nodal displacement vector element in contact with the rigid mass (m)

\( U_{s} \) :

Strain energy (J)

\( \upsilon_{0} \) :

Initial velocity of the rigid mass (m/s)

W :

Work done (J)

x :

Position in the rod (m)

ε:

Strain in the rod (m/m)

Π :

Potential energy (J)

ρ :

Density (kg/m3)

σ :

Stress in the rod (N/m2)

τ:

Time for the wave to travel across the rod from one end to the other end (s)

\( \left( {} \right)_{N} \) :

Value at the previous time step (N = 0,1,2,…)

\( \left( {} \right)_{N + 1} \) :

Value at the current time step (N = 0,1,2,…)

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Author information

Authors and Affiliations

  1. Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, P.O. Box 21544, Alexandria, Egypt

    Hesham A. Elkaranshawy

  2. Department of Mechanical Engineering Technology, Yanbu Industrial College, Yanbu, Saudi Arabia

    Nasser S. Bajaba

Authors
  1. Hesham A. Elkaranshawy
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  2. Nasser S. Bajaba
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Corresponding author

Correspondence to Hesham A. Elkaranshawy .

Editor information

Editors and Affiliations

  1. Faculty of Mechanical Engineering, Department of Applied Mechanics, Technical University of Malaysia, Johor, Malaysia

    Andreas Öchsner

  2. Faculdade de Engenharia da Universidade, Departamento de Engenharia Mecânica, Universidade do Porto, Rua Dr. Roberto Frias s/n, Porto, 4200-465, Portugal

    Lucas F. M. da Silva

  3. Magdeburg, Institut für Mechanik, Otto-von-Guericke-Universität, Universitätsplatz 2, Magdeburg, 30106, Sachsen-Anhalt, Germany

    Holm Altenbach

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Elkaranshawy, H.A., Bajaba, N.S. (2012). A Finite Element Simulation of Longitudinal Impact Waves in Elastic Rods. In: Öchsner, A., da Silva, L., Altenbach, H. (eds) Materials with Complex Behaviour II. Advanced Structured Materials, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22700-4_1

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