A complete set of potential functions consisting of three scalar functions is presented for coupled displacement-temperature equations of motion and heat equation for an arbitrary x3-convex domain containing a linear thermoelastic transversely isotropic material, where the x3-axis is parallel to the axis of symmetry of the material. The proof of the completeness theorem is based on a retarded logarithmic potential function, retarded Newtonian potential function, repeated wave equation, the extended Boggio's theorem for the transversely isotropic axially convex domain, and the existence of a solution for the heat equation. It is shown that the solution degenerates to a set of complete potential functions for elastodynamics and elastostatics under certain conditions. In a special case, the number of potential functions is reduced to one, and the required conditions are discussed. Another special case involves the rotationally symmetric configuration with respect to the axis of symmetry of the material.
Issue Section:
Research Papers
References
1.
Eubanks
, R. A.
, and Sternberg
, E.
, 1954
, “On the Axisymmetric Problem of Elasticity Theory for a Medium With Transverse Isotropy
,” J. Rat. Mech. Anal.
, 3
, pp. 89
–101
.2.
Gurtin
, M. E.
, 1972
, The Linear Theory of Elasticity, Mechanics of Solids II
, C. A.
Truesdell
, ed., Springer-Verlag
, Berlin
, pp. 1
–295
.3.
Love
, A. E. H.
, 1944
, A Treatise on the Mathematical Theory of Elasticity
, Dover Publications Inc.
, New York
.4.
Michell
, J. H.
, 1900
, “The Stress in an Aeolotropic Elastic Solid With an Infinite Plane Boundary
,” Proc. London Math. Soc.
, 32
, pp. 247
–258
.10.1112/plms/s1-32.1.2475.
Elliott
, H. A.
, 1948
, “Three Dimensional Stress Distribution in Hexagonal Aeolotropic Crystals
,” Proc. Cambridge Philos. Soc.
, 44
(4
), pp. 522
–533
.10.1017/S03050041000245316.
Hu
, H. C.
, 1953
, “On the Three Dimensional Problems of the Theory of Elasticity of a Transversely Isotropic Body
,” Acad. Sci. Sin.
, 2
, pp. 145
–151
.7.
Nowacki
, W.
, 1954
, “The Stress Function in Three Dimensional Problems Concerning an Elastic Body Characterized by Transversely Isotropy
,” Bull. Acad. Pol. Sci.
, 2
, pp. 21
–25
.8.
Lekhnitskii
, S. G.
, 1981
, Theory of Elasticity of an Anisotropic Body
, Mir
, Moscow
.9.
Eskandari-Ghadi
, M.
, 2005
, “A Complete Solution of the Wave Equations for Transversely Isotropic Media
,” J. Elasticity
, 81
, pp. 1
–19
.10.1007/s10659-005-9000-x10.
Eskandari-Ghadi
, M.
, and Pak
, R. Y. S.
, 2009
, “Axisymmetric Body-Force Fields in Elastodynamics of Transversely Isotropic Media
,” ASME J. Appl. Mech.
, 76
, p. 061016
.10.1115/1.313086611.
Wang
, M. Z.
, and Wang
, W.
, 1995
, “Completeness and Nonuniqueness of General Solutions of Transversely Isotropic Elasticity
,” Int. J. Solids Struct.
, 32
, pp. 501
–513
.10.1016/0020-7683(94)00114-C12.
Lodge
, A. S.
, 1955
, “The Transformation to Isotropic Form of the Equilibrium Equations for a Class of Anisotropic Elastic Solids
,” Q. J. Mech. Appl. Math.
, 8
, pp. 211
–225
.10.1093/qjmam/8.2.21113.
Carlson
, D. E.
, 1972
, Linear Thermoelasticity, Mechanics of Solids II
, C. A.
Truesdell
, ed., Springer-Verlag
, Berlin
, pp. 297
–345
.14.
Green
, A. E.
, and Naghdi
, P. M.
, 1993
, “Thermoelasticity Without Energy Dissipation
,” J. Elasticity
, 31
, pp. 189
–208
.10.1007/BF0004496915.
Green
, A. E.
, and Naghdi
, P. M.
, 1991
, “A Re-Examination of the Basic Postulates of Thermomechanics
,” Proc. R. Soc. London, Ser. A
, 432
, pp. 171
–194
.10.1098/rspa.1991.001216.
Biot
, M. A.
, 1954
, “Theory of Elasticity and Consolidation for a Porous Anisotropic Solid
,” J. Appl. Phys.
, 26
, pp. 182
–185
.10.1063/1.172195617.
Biot
, M. A.
, 1956
, “Thermoelasticity and Irreversible Thermodynamics
,” J. Appl. Phys.
, 27
, pp. 240
–253
.10.1063/1.172235118.
Deresiewicz
, H.
, 1958
, “Solution of the Equations of Thermoelasticity
,” Proceedings of the 3rd U.S. National Congress of Theoretical and Applied Mechanics, Brown University
, Providence, RI, June 11–14, pp. 287
–291
.19.
Zorski
, H.
, 1958
, “Singular Solutions for Thermoelastic Media
,” Bull. Acad. Pol. Sci.
, 6
, pp. 331
–339
.20.
Nowacki
, W.
, 1967
, “On the Completeness of Stress Functions in Thermoelasticity
,” Bull. Acad. Pol. Sci.
, 15
, pp. 583
–591
.21.
Eubanks
, R. A.
, and Sternberg
, E.
, 1956
, “On the Completeness of the Boussinesq-Papkovich Stress Functions
,” J. Rat. Mech. Anal.
, 5
, pp. 735
–746
.22.
Mindlin
, R. D.
, 1963
, “Note on the Galerkin and Papkovich Stress Functions
,” Bull. Am. Math. Soc.
, 42
, pp. 373
–376
.10.1090/S0002-9904-1936-06304-423.
Sternberg
, E.
, and Eubanks
, R. A.
, 1957
, “On Stress Functions for Elastokinetics and the Integration of the Repeated Wave Equation
,” Q. Appl. Math.
, 15
, pp. 149
–153
.24.
Truesdell
, C. A.
, 1959
, “Invariant and Complete Stress Functions for General Continua
,” Arch. Rat. Mech. Anal.
, 4
, pp. 1
–29
.10.1007/BF0028137625.
Sternberg
, E.
, 1960
, “On the Integration of the Equation of Motion in the Classical Theory of Elasticity
,” Arch. Rat. Mech. Anal.
, 6
, pp. 34
–50
.10.1007/BF0027615226.
Sternberg
, E.
, 1960
, “On Some Recent Developments in the Linear Theory of Elasticity
,” Structural Mechanics
, Goodier
and Hoff
, eds., Pergamon Press, Inc.
, Elmsford, NY
.27.
Sternberg
, E.
, and Gurtin
, M. E.
, 1962
, “On the Completeness of Certain Stress Functions in the Linear Theory of Elasticity
,” Proceedings of the Fourth U.S. National Congress on Applied Mechanics
, Berkeley, CA, June 18–21, pp. 793
–797
.28.
Gurtin
, M. E.
, 1962
, “On Helmholtz's Theorem and the Completeness of the Papkovich-Neuber Stress Functions for Infinite Domains
,” Arch. Rat. Mech. Anal.
, 9
, pp. 225
–233
.10.1007/BF0025334629.
Stippes
, M.
, 1969
, “Completeness of Papkovich Potentials
,” Q. Appl. Math.
, 26
, pp. 477
–483
.30.
Carlson
, D. E.
, 1983
, “A Note on the Solutions of Boussinesq, Love, and Maeguerre in Axisymmetric Elasticity
,” J. Elasticity
, 13
, pp. 345
–349
.10.1007/BF0004300131.
Tran-Cong
, T.
, 1989
, “On the Completeness of the Papkovich-Neuber Solution
,” Q. Appl. Math.
, 47
, pp. 645
–659
.32.
Tran-Cong
, T.
, 1995
, “On the Completeness and Uniqueness of Papkovich-Neuber and the Non-Axisymmetric Boussinesq, Love and Burgatti Solutions in General Cylindrical Coordinates
,” J. Elasticity
, 36
, pp. 227
–255
.10.1007/BF0004084933.
Pak
, R. Y. S.
, and Eskandari-Ghadi
, M.
, 2007
, “On the Completeness of a Method of Potentials in Elastodynamics
,” Q. Appl. Math.
, 65
, pp. 789
–797
.34.
Eskandari-Ghadi
, M.
, and Pak
, R. Y. S.
, 2008
, “Elastodynamics and Elastostatics by a Unified Method of Potentials
,” J. Elasticity
, 92
, pp. 187
–194
.10.1007/s10659-008-9156-235.
Verruijt
, A.
, 1969
, “The Completeness of Biot’s Solution of the Coupled Thermoelastic Problem
,” Q. Appl. Math.
, 26
, pp. 485
–490
.36.
Kellogg
, O. D.
, 1953
, Foundation of Potential Theory
, Dover Publications Inc.
, New York
.37.
Phillips
, H. B.
, 1933
, Vector Analysis
, John Wiley & Sons Inc.
, New York
.38.
Morse
, P. M.
, and Feshbach
, H.
, 1953
, Methods of Theoretical Physics
, McGraw-Hill Book Company Inc.
, New York
.39.
Raoofian Naeeni
, M.
, Eskandari-Ghadi
, M.
, Ardalan
, A. A.
, Rahimian
, M.
, and Hayati
, Y.
, 2013
, “Analytical Solution of Coupled Thermoelastic Axisymmetric Transient Waves in a Transversely Isotropic Half-Space
,” ASME J. Appl. Mech.
, 80
(2
), p. 024502
.10.1115/1.400778640.
Pekeris
, C. L.
, 1955
, “The Seismic Surface Pulse
,” Proc. Natl. Acad. Sci. USA
, 41
, pp. 629
–639
.10.1073/pnas.41.9.62941.
Fedorov
, F. I.
, 1968
, Theory of Elastic Waves in Crystals
, Plenum Press
, New York
.42.
Apostol
, T. M.
, 1957
, Mathematical Analysis: A Modern Approach to Advanced Calculus
, Addison-Wesley Publishing Company Inc.
, Boston, MA
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