Abstract
A new spectral element (SE) is formulated to analyse wave propagation in anisotropic inhomogeneous beam. The inhomogeneity is considered in the longitudinal direction. Due to this particular pattern of inhomogeneity, the governing partial differential equations (PDEs) have variable coefficients and an exact solution for arbitrary variation of material properties, even in frequency domain, is not possible to obtain. However, it is shown in this work that for exponential variation of material properties, the equations can be solved exactly in frequency domain, when the same parameter governs the variation of elastic moduli and density. The SE is formed using this exact solution as interpolating polynomial. As a result a single element can replace hundreds of finite elements (FEs), which are essential for all wave propagation analysis and also for accurate representation of the inhomogeneity. The developed element is used for eliciting several advantages of the gradation, including mode selection, mode blockage and smoothening of stress waves.
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References
Caviglia G, Morro A (1992) Inhomogeneous Waves in Solids and Fluids, World Scientific, Singapore
Chakraborty A, Gopalakrishnan S, Reddy JN (2003) A new beam finite element for the analysis of functionally graded materials. Int J Mech Sc 45(3):519–539
Chakraborty A, Gopalakrishnan S (2003) Various Numerical Techniques for Analysis of Longitudinal Wave Propagation in Inhomogeneous One-dimensional Waveguides. Acta Mechanica 162(1–4):1–27
Chakraborty A, Gopalakrishnan S (2003) A spectrally formulated finite element for wave propagation analysis in Functionally Graded Beams. Int J Sol Str 40(10):2421–2448
Doyle JF (1997) Wave Propagation in Structures. Springer Verlag, New York
Golub G, Loan CV (1989) Matrix Computations. The Johns Hopkins University Press, Baltimore
Gong SW, Lam KY, Reddy JN (1999) The elastic response of functionally graded cylindrical shells to low-velocity impact. Int J Impact Engineering 22:397–417
Han X, Liu GR, Xi ZC, Lam KY (2002) Characteristics of waves in a functionally graded cylinder. Int J Numer Meth Eng 53:653–676
Han X, Liu GR, Lam KY (2002) Transient waves in plates of functionally graded materials. Int J Numer Meth Eng 52:851–865
Kuhlemeyer RL, Lysmer J (1973) Finite element accuracy for wave propagation problems. J Soil Mech Foundations Division, ASCE 99(SM5):421–427
Liu GR, Tani J (1994) Surface waves in functionally gradient piezoelectric plates. ASME Journal of vibration and acoustics 116: 440–448
Liu GR, Han X, Lam KY (1999) Stress wave in functionally gradient materials and its use for material characterization. Composite Part B30:383–394
Ohyoshi T, Sui GJ, Miuro K (1996) Using of stacking model of the linearly inhomogeneous layers elements. Proceedings of the ASME aerospace division 52:101–106
Praveen GN, Reddy JN (1998) Nonlinear Transient Thermoelastic Analysis of Functionally Graded Ceramic-Metal Plates. Int J Solids Struct 35(33):4457–4476
Reddy JN, Chin CD (1998) Thermomechanical Analysis of Functionally Graded Cylinders and Plates. J Thermal Stresses 26(1):593–626
Reddy JN (2000) Analysis of functionally graded plates. Int J Numer Meth Eng 47:663–684
Roy Mahapatra D, Gopalakrishnan S (2003) A spectral finite element model for analysis of axial-flexural-shear coupled wave propagation in laminated composite beams. Composite Struct 59(1):67–88
Beskos DE, Narayana GV (1983) Dynamic response of frameworks by numerical Laplace transform. Comp Meth Appl Mech Engg 37:289–307
Narayanan GV, Beskos DE (1982) Numerical operational methods for time-dependent linear problems. Int J Numer Meth Eng 18:1829–1854
Tsepoura KG, Papargyri-Beskou S, Polyzos D, Beskos DE (2002) Static and dynamic analysis of a gradient-elastic bar in tension. Arch Appl Mech 72:483–497
Papargyri-Beskou S, Polyzos D, Beskos DE (2003), Dynamic analysis of gradient elastic flexural beams. Struct Eng Mech 15:705–716
Pipkins DS, Atluri SN (1993) Non-linear analysis of wave propagation using transform methods. Comp Mech 11:207–227
Cheng FY (2000) Matrix Analysis of Structural Dynamics: Applications and Earthquake Engineering. M Dekker, New York
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Chakraborty, A., Gopalakrishnan, S. A spectral finite element for axial-flexural-shear coupled wave propagation analysis in lengthwise graded beam. Comput Mech 36, 1–12 (2005). https://doi.org/10.1007/s00466-004-0637-2
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DOI: https://doi.org/10.1007/s00466-004-0637-2