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Nonlinear Waves in Solid Continua with Finite Deformation

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DOI: 10.4236/ajcm.2015.53032    4,625 Downloads   5,181 Views   Citations

ABSTRACT

This work considers initiation of nonlinear waves, their propagation, reflection, and their interactions in thermoelastic solids and thermoviscoelastic solids with and without memory. The conservation and balance laws constituting the mathematical models as well as the constitutive theories are derived for finite deformation and finite strain using second Piola-Kirchoff stress tensor and Green’s strain tensor and their material derivatives [1]. Fourier heat conduction law with constant conductivity is used as the constitutive theory for heat vector. Numerical studies are performed using space-time variationally consistent finite element formulations derived using space-time residual functionals and the non-linear equations resulting from the first variation of the residual functional are solved using Newton’s Linear Method with line search. Space-time local approximations are considered in higher order scalar product spaces that permit desired order of global differentiability in space and time. Computed results for non-linear wave propagation, reflection, and interaction are compared with linear wave propagation to demonstrate significant differences between the two, the importance of the nonlinear wave propagation over linear wave propagation as well as to illustrate the meritorious features of the mathematical models and the space-time variationally consistent space-time finite element process with time marching in obtaining the numerical solutions of the evolutions.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Surana, K. , Knight, J. and Reddy, J. (2015) Nonlinear Waves in Solid Continua with Finite Deformation. American Journal of Computational Mathematics, 5, 345-386. doi: 10.4236/ajcm.2015.53032.

References

[1] Surana, K.S. (2014) Advanced Mechanics of Continua. CRC Press, Boca Raton.
[2] Engelbrecht, J. (1983) Nonlinear Wave Processes of Deformation in Solids. Pitman Publishing, London.
[3] Graham, R.A. (1993) Solids under High-Pressure Shock Compression. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4613-9278-1
[4] Zarembo, L.K. and Krasil’nikov, V.A. (1970) Nonlinear Phenomena in the Propagation of Elastic Waves in Solids. Soviet Physics Uspekhi, 13, 778-797.
http://dx.doi.org/10.1070/PU1971v013n06ABEH004281
[5] Fosdick, R., Ketema, Y. and Yu, J.H. (1997) A Non-linear Oscillator with History Dependent Force. International Journal of Non-Linear Mechanics, 33, 447-459.
http://dx.doi.org/10.1070/PU1971v013n06ABEH004281
[6] Lima, W.J.N. de and Hamilton, M.F. (2003) Finite-Amplitude Waves in Isotropic Elastic Plates. Journal of Sound and Vibration, 265, 819-839.
http://dx.doi.org/10.1016/S0022-460X(02)01260-9
[7] Gei, M., Bigoni, D. and Franceschini, G. (2004) Thermoelastic Small-Amplitude Wave Propagation in Nonlinear Elastic Multilayers. Mathematics and Mechanics of Solids, 9, 555-568.
http://dx.doi.org/10.1177/1081286504038675
[8] Lima, W.J.N. de and Hamilton, M.F. (2005) Finite Amplitude Waves in Isotropic Elastic Waveguides with Arbitrary Constant Cross-Sectional Area. Wave Motion, 41, 1-11.
http://dx.doi.org/10.1016/j.wavemoti.2004.05.004
[9] Renton, J.D. (1987) Applied Elasticity: Matrix and Tensor Analysis of Elastic Continua. Ellis Horwood, Chichester.
[10] Landau, L.D. and Lifshitz, E.M. (1986) Theory of Elasticity. Pergamon Press, New York.
[11] Engelbrecht, J., Berezovski, A. and Salupere, A. (2007) Nonlinear Deformation Waves in Solds and Dispersion. Wave Motion, 44, 493-500.
http://dx.doi.org/10.1016/j.wavemoti.2007.02.006
[12] Shariyat, M., Lavasani, S.M.H. and Khaghani, M. (2010) Nonlinear Transient Thermal Stress and Elastic Wave Propagation Analyses of Thick Temperature-Dependent FGM Cylinders, Using a Second-Order Point-Collocation Method. Applied Mathematical Modeling, 34, 898-918.
http://dx.doi.org/10.1016/j.apm.2009.07.007
[13] Yu, S.T.J., Yang, L., Lowe, R. and Bechtel, S.E. (2010) Numerical Simulation of Linear and Nonlinear Waves in Hypoelastic Solids by the CESE Method. Wave Motion, 47, 168-182.
http://dx.doi.org/10.1016/j.wavemoti.2009.09.005
[14] Berezovski, A., Berezovski, M. and Engelbrecht, J. (2006) Numerical Simulation of Nonlinear Elastic Wave Propagation in Piecewise Homogeneous Media. Materials Science and Engineering A, 418, 364-369.
http://dx.doi.org/10.1016/j.msea.2005.12.005
[15] Shariyat, M., Khaghani, M. and Lavasani, S.M.H. (2010) Nonlinear Thermoelasticity, Vibration, and Stress Wave Propagation Analyses of Thick FGM Cylinders with Temperature-Dependent Material Properties. European Journal of Mechanics A/Solids, 29, 378-391.
http://dx.doi.org/10.1016/j.euromechsol.2009.10.007
[16] Li, Y., Vandewoestyne, B. and Abeele, K.V.D. (2012) A Nodal Discontinuous Galerkin Finite Element Method for Nonlinear Elastic Wave Propagation. Journal of the Acoustical Society of America, 131, 3650-3663.
http://dx.doi.org/10.1121/1.3693654
[17] Shariyat, M. (2012) Nonlinear Transient Stress and Wave Propagation Analyses of the FGM Thick Cylinders, Employing a Unified Generalized Thermoelasticity Theory. International Journal of Mechanical Sciences, 65, 24-37.
http://dx.doi.org/10.1016/j.ijmecsci.2012.09.001
[18] Yu, Y.M. and Lim, C.W. (2013) Nonlinear Constitutive Model for Axisymetric Bending of Annular Graphene-Like Nanoplate with Gradient Elasticity Enhancement Effects. Journal of Engineering Mechanics, 139, 1025-1035.
http://dx.doi.org/10.1061/(ASCE)EM.1943-7889.0000625
[19] Nucera, C. and di Scalea, F.L. (2014) Nonlinear Semianalytical Finite-Element Algorithm for the Analysis of Internal Resonance Conditions in Complex Waveguides. Journal of Engineering Mechanics, 140, 502-522.
http://dx.doi.org/10.1061/(ASCE)EM.1943-7889.0000670
[20] Surana, K.S., Maduri, R. and Reddy, J.N. (2006) One Dimensional Elastic Wave Propagation in Periodically Laminated Composites Using h; p; k Framework and STLS Finite Element Processes. Mechanics of Advanced Materials and Structures, 13, 161-196.
http://dx.doi.org/10.1080/15376490500451809
[21] Surana, K.S. and Reddy, J.N. (2015) Mathematics of Computations and the Finite Element Method for Initial Value Problems. Book Manuscript in Progress.
[22] Surana, K.S., Ahmadi, A.R. and Reddy, J. (2002) The k-Version of Finite Element Method for Self-Adjoint Operators in BVP. International Journal of Computational Engineering Science, 3, 155-218.
http://dx.doi.org/10.1142/S1465876302000605
[23] Surana, K.S., Ahmadi, A.R. and Reddy, J. (2003) The k-Version of Finite Element Method for Non-Self-Adjoint Operators in BVP. International Journal of Computational Engineering Science, 4, 737-812.
http://dx.doi.org/10.1142/S1465876303002179
[24] Surana, K.S., Ahmadi, A.R. and Reddy J. (2004) The k-version of Finite Element Method for Nonlinear Operators in BVP. International Journal of Computational Engineering Science, 5, 133-207.
http://dx.doi.org/10.1142/S1465876304002307
[25] Winterscheidt, D. and Surana, K.S. (1993) p-Version Least-Squares Finite Element Formulation for Convection-Diffusion Problems. International Journal for Numerical Methods in Engineering, 36, 111-133.
http://dx.doi.org/10.1002/nme.1620360107
[26] Winterscheidt, D. and Surana, K.S. (1994) p-Version Least Squares Finite Element Formulation for Two-Dimensional, Incopressible Fluid Flow. International Journal for Numerical Methods in Fluids, 18, 43-69.
http://dx.doi.org/10.1002/fld.1650180104
[27] Bell, B.C. and Surana, K.S. (1994) A Space-Time Coupled p-Version Least-Squares Finite Element Formulation for Unsteady Fluid Dynamics Problems. International Journal for Numerical Methods in Engineering, 37, 3545-3569.
http://dx.doi.org/10.1002/nme.1620372008
[28] Bell, B.C. and Surana, K.S. (1996) A Space-Time Coupled p-Version Least Squares Finite Element Formulation for Unsteady Two-Dimensional Navier-Stokes Equations. International Journal for Numerical Methods in Engineering, 39, 2593-2618.
http://dx.doi.org/10.1002/(SICI)1097-0207(19960815)39:15<2593::AID-NME968>3.0.CO;2-2
[29] Surana, K.S., Reddy, J.N. and Allu, S. (2007) The k-Version of Finite Element Method for Initial Value Problems: Mathematical and Computational Framework. International Journal of Computational Methods in Engineering Science and Mechanics, 8, 123-136.
http://dx.doi.org/10.1080/15502280701252321
[30] Surana, K.S., Allu, S., Reddy, J.N. and Tenpas, P.W. (2008) Least Squares Finite Element Processes in hpk Mathematical Framework for Non-Linear Conservation Law. International Journal of Numerical Methods in Fluids, 57, 1545-1568.
http://dx.doi.org/10.1002/fld.1695
[31] Reddy, J.N. (2004) An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, New York.
http://dx.doi.org/10.1093/acprof:oso/9780198525295.001.0001
[32] Bathe, K.J. (1996) Finite Element Procedures. Prentice Hall, New Jersey.
[33] Riks, E. (1979) An Incremental Approach to the Solution of Snapping and Buckling Problems. International Journal of Solids and Structures, 15, 529-551.
http://dx.doi.org/10.1016/0020-7683(79)90081-7

  
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