Abstract
An approach to the solution of three-dimensional static problems for a transversely isotropic (rectilinear anisotropy) body is expounded and the solutions for piezoceramic canonical bodies are systematized. The result of the study is explicit analytical solutions of three-dimensional problems. Bodies are examined whose boundary surface is the coordinate surfaces in coordinate systems that permit the separation of the variables in the three-dimensional Laplace equation. The stress concentration in bodies near necks, cavities, inclusions, and cracks is investigated. The stress intensity factors of the force field and electric induction near elliptic and parabolic cracks are determined. The contact interaction of a piezoceramic half-space with elliptic and parabolic dies is studied. The bodies are under various mechanical, thermal, and electric loads
Similar content being viewed by others
REFERENCES
V. A. Bezhanyan, “Plane adhesive-contact problem of electroelasticity for a half-plane,” in: Partial Differential Equations in Applied Problems [in Russian] (1986), pp. 17-21.
V. A. Bezhanyan and A. F. Ulitko, “Vector boundary-value problems of electroelasticity for piezoceramic cylinders,” Izv. AN Arm. SSR, Ser. Mekh., No. 6, 16-20 (1984).
V. A. Bezhanyan and A. F. Ulitko, “Adhesive-contact electroelastic problem for a half-space,” Dop. AN URSR, Ser. A, Fiz.-Mat. Tekhn. Nauky, No. 6, 16-19 (1986).
V. A. Bezhanyan and A. F. Ulitko, “Adhesive-contact problem for a piezoceramic half-space covered with an electrode under a die,” Dop. AN URSR, Ser. A, Fiz-Mat. Tekhn. Nauky, No. 6, 35-39 (1990).
L. V. Belokopytova, O. A. Ivanenko, and L.A. Fil'shtinskii, “Coupled electric and mechanical fields in piezoelastic bodies with cuts or inclusions,” Dinam. Prochn. Mashin, No. 34, 16-21 (1981).
L. V. Belokopytova and L. A. Fil'shtinskii, “Two-dimensional boundary-value problem of electroelasticity for piezoceramic medium with cuts,” Prikl. Mat. Mekh., 43, No. 1, 138-143 (1979).
D. Berlinkur, D. Kerran, and G. Jaffe, “Piezoceramic electric and piezomagnetic materials and their application in transducers,” in: W. P. Mason (ed.), Physical Acoustics, Principles and Methods, Vol. 1, Part A, Methods and Devices, Academic Press, New York-London (1964).
V. A. Boriseiko, “Coupled electroelastic vibrations of a thick-walled piezoceramic sphere in a compressible fluid,” Tepl. Napryazh. Élem. Constr., 12, 111-115 (1972).
V. A. Boriseiko, V. T. Grinchenko, and A. F. Ulitko, “Electroelastic relations for piezoceramic shells of revolution,” Prikl. Mekh., 12, No. 2, 26-33 (1976).
V. A. Boriseiko, V. S. Martynenko, and A. F. Ulitko, “Electroelastic relations for piezoceramic meridionally polarized shells of revolution,” Prikl. Mekh., 15, No. 12, 36-42 (1979).
I. A. Vekovishcheva, “Theory of bending of thin piezoelectric plates,” Izv. AN Arm. SSR, Mekh., 25, No. 4, 30-37 (1972).
I. A. Vekovishcheva, “Strain distribution and electric field in an electroelastic half-space with a direct piezoelectric effect,” Prikl. Mekh., 9, No. 12, 48-52 (1973).
I. A. Vekovishcheva, “Theory of bending of thin piezoelectric plates. Derivation of natural boundary conditions,” Izv. AN SSSR, Mekh. Tverd. Tela, 25, No. 6, 108-113 (1974).
L. A. Galin, Contact Problem in the Theory of Elasticity [in Russian], Gostekhizdat, Moscow (1955).
I. P. Get'man and Yu. A. Ustinov, “The theory of inhomogeneous electroelastic plates revisited,” Prikl. Mat. Mekh., 43, No. 5, 923-932 (1979).
I. A. Glozman, Piezoceramics [in Russian], Énergiya, Moscow (1972).
E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press (1931).
V. T. Grinchenko, Equilibrium and Steady-State Vibrations of Finite Elastic Bodies [in Russian], Naukova Dumka, Kiev (1978).
V. T. Grinchenko, A. F. Ulitko, and N. A. Shul'ga, Electroelasticity, Vol. 5 of the five-volume series Mechanics of Coupled Fields in Structural Members [in Russian], Naukova Dumka, Kiev (1989).
A. N. Guz and F. G. Makhort, Acoustoelectromagnetoelasticity, Vol. 3 of the five-volume series Mechanics of Coupled Fields in Structural Members [in Russian], Naukova Dumka, Kiev (1988).
A. A. Erofeev, Piezoelectronic Automatic Devices [in Russian], Mashinostroenie, Leningrad (1982).
V. E. Zhirov, “Electroelastic equilibrium of a piezoceramic plate,” Prikl. Mat. Mekh., 41, No. 6, 1114-1121 (1977).
S. A. Kaloerov, A. I. Baeva, and Yu. A. Glushchenko, “Two-dimensional electroelastic problem for a multiply connected piezoelectric body with cavities and plane cracks,” Teor. Prikl. Mekh., 32, 64-79 (2001).
V. G. Karnaukhov and I. F. Kirichok, “Refined theory of laminated viscoelastic piezoceramic shells taking heat generation into account,” Prikl. Mekh., 21, No. 6, 53-60 (1985).
V. G. Karnaukhov and I. F. Kirichok, Electrothermoviscoelasticity, Vol. 4 of the five-volume series Mechanics of Coupled Fields in Structural Members [in Russian], Naukova Dumka, Kiev (1988).
A. S. Kosmodamianskii and V. N. Lozhkin, “Generalized plane stress state of thin piezoelectric plates,” Prikl. Mekh., 11, No. 5, 45-53 (1975).
A. S. Kosmodamianskii and V. N. Lozhkin, “Asymptotic electroelastic analysis of a thin piezoelectric layer,” Prikl. Mekh., 14, No. 5, 3-8 (1978).
B. A. Kudryavtsev, “Mechanics of piezoelectric materials,” Itogi Nauki Tekhn., Ser. Mekh. Deform. Tverd. Tela, 11, VINITI, Moscow, 5-66 (1978).
B. A. Kudryavtsev, V. Z. Parton, and V. I. Rakitin, “Fracture mechanics of piezoelectric materials: A rectilinear tunnel crack in the interface with a conductor,” Prikl. Mat. Mekh., 39, No. 1, 149-159 (1975).
B. A. Kudryavtsev, V. Z. Parton, and V. I. Rakitin, “Fracture mechanics of piezoelectric materials: An axisymmetric crack in the interface with a conductor,” Prikl. Mat. Mekh., 39, No. 2, 352-362 (1975).
B. A. Kudryavtsev, V. Z. Parton, and N. A. Senik, “Piezoceramic shell polarized across the thickness. Electrode-free surfaces,” Probl. Prochn., No. 1, 79-83 (1985).
W. Cady, Piezoelectricity, Dover, New York (1964).
V. N. Lozhkin and L. N. Oleinik, “Stress state of a thin piezoelectric plate with an elliptic opening,” Mekh. Tverd. Tela, No. 8, 127-130 (1976).
V. N. Lozhkin and L. N. Oleinik, “Elastic equilibrium of an arbitrarily anisotropic piezoelectric body,” Mat. Fiz., No. 24, 98-101 (1978).
A. I. Lur'e, Three-Dimensional Elastic Problems [in Russian], Gostekhizdat, Moscow (1955).
V. V. Modarskii and Yu. A. Ustinov, “Symmetric vibrations of piezoelectric plates,” Izv. AN Arm. SSR, Ser. Mekh., 29, No. 5, 51-58 (1976).
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1, McGraw-Hill, New York (1953).
H. Neuber, Kerbspannungslehre (Theory of Notch Stress Concentration), Springer Verlag, Berlin-Heidelberg-New York-Tokyo (1985).
W. Nowacki, Electromagnetic Effects in Solids [Russian translation], Mir, Moscow (1986).
V. V. Panasyuk, E. A. Andreikiv, and V. Z. Parton, Fundamentals of Fracture Mechanics, Vol. 1 of the four-volume series Fracture Mechanics and Strength of Materials [in Russian], Naukova Dumka, Kiev (1988).
V. Z. Parton and B. A. Kudryavtsev, Electroelasticity of Piezoceramic and Electroconductive Bodies [in Russian], Nauka, Moscow (1988).
A. H. Passos Morgado, Yu. N. Podil'chuk, and I. V. Lebedeva, “Investigation of stress state of a piezoceramic body containing an external elliptical crack,” Int. Appl. Mech., 35, No. 1, 33-40 (1999).
A. H. Passos Morgado, Yu. N. Podil'chuk, and I. V. Lebedeva, “Stress analysis of a piezoceramic body with a spheroidal inclusion,” Teor. Prikl. Mekh., 29, 28-42 (1999).
Yu. N. Podil'chuk, Boundary-Value Static Problems for Elastic Bodies, Vol. 1 of the six-volume series Three-Dimensional Problems in the Theory of Elasticity and Plasticity [in Russian], Naukova Dumka, Kiev (1984).
Yu. N. Podil'chuk, “Exact analytical solutions of three-dimensional boundary-value static problems for a transversely isotropic canonical body (review),” Prikl. Mekh., 33, No. 10, 3-30 (1997).
Yu. N. Podil'chuk, “Representation of the general solution of the static electroelastic equations for a transversely isotropic piezoceramic body in terms of harmonic functions,” Prikl. Mekh., 34, No. 7, 20-26 (1998).
Yu. N. Podil'chuk, “Electroelastic equilibrium of transversely isotropic piezoceramic media with cavities, inclusions, and cracks,” Prikl. Mekh., 34, No. 10, 109-119 (1998).
Yu. N. Podil'chuk, “On the stress state of a piezoceramic medium with a tunneled elliptical recess,” Int. Appl. Mech., 35, No. 7, 713-721 (1999).
Yu. N. Podil'chuk, “The electroelasticity problem for piezoceramic media with bilateral hyperbolic tunnel cavities,” Int. Appl. Mech., 35, No. 8, 804-811 (1999).
Yu. N. Podil'chuk, “The action of a concentrated force and a concentrated electric charge in an infinite transversally isotropic piezoceramic medium,” Int. Appl. Mech., 36, No. 4, 492-500 (2000).
Yu. N. Podil'chuk, “The stress state of a transversally isotropic piezoceramic body with a parabolic crack in a uniform heat flow,” Int. Appl. Mech., 36, No. 8, 1037-1046 (2000).
Yu. N. Podil'chuk and I. V. Lebedeva, “Equilibrium of a piezoceramic cylinder with an elliptic crack,” Prikl. Mekh., 34, No. 8, 40-48 (1998).
Yu. N. Podil'chuk and I. V. Lebedeva, “Equilibrium of piezoceramic cylinder with spheroidal cavity,” Int. Appl. Mech., 35, No. 5, 477-487 (1999).
Yu. N. Podil'chuk and A. H. Passos Morgado, “Representation of the general solution of the static thermoelectroelastic equations for a transversely isotropic piezoceramic body in terms of harmonic functions,” Teor. Prikl. Mekh., 29, 42-51 (1999).
Yu. N. Podil'chuk and A. H. Passos Morgado, “Stress distribution in a transversally isotropic piezoceramic body with an elliptic crack in a uniform heat flow,” Int. Appl. Mech., 36, 2, 203-215 (2000).
Yu. N. Podil'chuk and A. H. Passos Morgado, “The thermoelectroelastic state of a transversally isotropic piezoceramic body with a spheroidal cavity in a uniform heat flow,” Int. Appl. Mech., 36, No. 9, 1187-1197 (2000).
Yu. N. Podil'chuk and T. M. Proshchenko, “General electroelastic problem for a transversely isotropic one-sheet hyperboloid of revolution,” Teor. Prikl. Mekh., 32, 16-27 (2001).
Yu. N. Podil'chuk and T. M. Proshchenko, “Stress concentration in a piezoceramic medium near a hyperboloidal recess under pure shear,” Teor. Prikl. Mekh., 35, 20-28 (2002).
Yu. N. Podil'chuk and V. F. Tkachenko, “Contact electroelastic problem for a flat elliptic die,” Teor. Prikl. Mekh., 28, 40-51 (1998).
Yu. N. Podil'chuk and V. F. Tkachenko, “Thermoelastic contact problem for a heated elliptic die forced into a transversely isotropic piezoceramic half-space,” Teor. Prikl. Mekh., 30, 37-53 (1999).
Yu. N. Podil'chuk and V. F. Tkachenko, “Contact electroelasticity problem for a nonplane elliptical die,” Int. Appl. Mech., 35, No. 6, 544-554 (1999).
Yu. N. Podil'chuk and V. F. Tkachenko, “The contact problem of electroelasticity for a flat punch with parabolic cross-section,” Int. Appl. Mech., 35, No. 11, 1096-1103 (1999).
Yu. N. Podil'chuk and V. F. Tkachenko, “The equilibrium of a piezoceramic cylindrical body with a parabolic crack,” Int. Appl. Mech., 36, No. 3, 348-357 (2000).
I. B. Polovinkina and A. F. Ulitko, “The theory of equilibrium of piezoceramic cracked bodies,” Tepl. Napryazh. Élem. Konstr., 18, 10-17 (1978).
T. M. Proshchenko, “Electroelastic problem for a piezoceramic medium with a tunnel elliptic inclusion,” Teor. Prikl. Mekh., 34, 57-62 (2001).
L. A. Galin (ed.), Development of Contact Problems in the USSR [in Russian], Nauka, Moscow (1976).
V. A. Rvachev, “On a wedge-shaped die forced into an elastic half-space,” Prikl. Mat. Mekh., 23, No. 1, 169-171 (1959).
A. F. Ulitko, The Method of Vector Eigenfunctions in Three-Dimensional Elastic Problems [in Russian], Naukova Dumka, Kiev (1979).
A. F. Ulitko, “On some features in the formulation of boundary-value problems of electroelasticity,” in: Current Problems of Mechanics and Aviation [in Russian] (1982), pp. 290-300.
Yu. A. Ustinov, “Homogeneous solutions and the limit passage from three to two dimensions for electroelastic plates with variable properties,” in: Proc. 10th All-Union Conf. on the Theory of Shells and Plates (Kutaisi, 1975) [in Russian], Vol. 1, Tbilisi (1975), pp. 286-295.
I. Yu. Khoma, “On the equations of the generalized theory of piezoelectric shells,” Prikl. Mekh., 17, No. 12, 115-118 (1981).
I. Yu. Khoma, “Developing the generalized theory of thermopiezoelastic shells,” Prikl. Mekh., 19, No. 12, 65-71 (1983).
I. Yu. Khoma, “General solution of the equilibrium equations for nonthin transversely isotropic piezoceramic plates polarized across the thickness,” Teor. Prikl. Mekh., 35, 185-193 (2002).
N. A. Shul'ga and A. M. Bolkisev, Vibrations of Piezoelectric Bodies [in Russian], Naukova Dumka, Kiev (1990).
B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics, Academic Press, New York (1971).
N. Bugdayci and D. B. Bogy, “A two-dimensional theory for piezoelectric layers used in electromechanical transducers,” Int. J. Solids Struct., 17, No. 12, 1179-1202 (1981).
R. L. Chowdhurry, “On an axisymmetric boundary value problem for an elastic dielectric half-space,” Int. J. Solids Struct., 18, No. 3, 263-271 (1982).
M. Y. Chung and T. C. T. Ting, “Piezoelectric solid with an elliptic inclusion or hole,” Int. J. Solids Struct., 33, No. 23, 3343-3361 (1996).
M. C. Dökmeci, “On the higher order theories of piezoelectric crystal surfaces,” J. Math. Phys., 15, No. 12, 2248-2252 (1974).
R. D. Mindin, “Equations of high frequency vibrations of thermopiesoelectric crystal plates,” Int. J. Solids Struct., 10, No. 6, 625-637 (1974).
Y. C. Pan and T. W. Chou, “Point force solution for an infinite transversally isotropic solid,” Trans. ASME, Ser. E, J. Appl. Mech., 98, 608-612 (1976).
Yu. N. Podil'chuk, “Exact analytical solutions of three-dimensional static thermoelastic problems for a transversally isotropic body in curvilinear coordinate systems,” Int. Appl. Mech., 37, No. 6, 728-761 (2001).
Yu. N. Podil'chuk and T. M. Proshchenko, “Stress concentration in a transversally isotropic piezoceramic cylinder near a hyperboloidal neck,” Int. Appl. Mech., 37, No. 8, 1034-1045 (2001).
Yu. N. Podil'chuk and L. N. Tereshchenko, “A magnetoelastic field in a ferromagnetic with an elliptic inclusion,” Int. Appl. Mech., 38, No. 5, 585-593 (2002).
Singh Bridj Mohan and S. Dhaliwal Ranjit, “Mixed boundary-value problems of steady-state thermoelasticity and electrostatics,” J. Term. Stress., 1, No. 1, 1-11 (1978).
H. A. Sosa, “Plane problems in piezoelectric media with defects,” Int. Solids Struct., 28, No. 4, 491-505 (1991).
H. Sosa and N. Khutoryansky, “New developments concerning piezoelectric materials with defects,” Int. J. Solids Struct., 33, No. 23, 3399-3414 (1996).
Wang Zi-Kung and Chen Geng-Chao, “A general solution and the application of space axisymmetric problem in piezoelectric material,” Appl. Math. Mech. Engl. Ed., 15, No. 7, 615-626 (1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Podil'chuk, Y.N. Exact Analytical Solutions of Static Electroelastic and Thermoelectroelastic Problems for a Transversely Isotropic Body in Curvilinear Coordinate Systems. International Applied Mechanics 39, 132–170 (2003). https://doi.org/10.1023/A:1023953313612
Issue Date:
DOI: https://doi.org/10.1023/A:1023953313612