Conservation laws and canonical forms in the Stroh formalism of anisotropic elasticity
Author:
Gary A. Hatfield
Journal:
Quart. Appl. Math. 54 (1996), 739-758
MSC:
Primary 73C02; Secondary 35Q72, 73B40
DOI:
https://doi.org/10.1090/qam/1417237
MathSciNet review:
MR1417237
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Abstract: We find and classify all first-order conservation laws in the Stroh formalism. All possible non-semisimple degeneracies are considered. The laws are found to depend on three arbitrary analytic functions. In some instances, there is an “extra” law which is quadratic in $\nabla u$. Separable and inseparable canonical forms for the stored energy function are given for each type of degeneracy and they are used to compute the conservation laws. The existence of a real Stroh eigenvector is found to be a necessary and sufficient condition for separability. The laws themselves are stated in terms of the Stroh eigenvectors.
J. D. Eshelby, The force on an elastic singularity, Philos. Trans. Roy. Soc. London Ser. A 244, 57–112 (1951)
J. R. Rice, A path-independent integral and the approximate analysis of strain concentrations by notches and cracks, J. Appl. Mech. 35, 376–386 (1968)
J. K. Knowles and Eli Sternberg, On a class of conservation laws in linearized and finite elasto-statistics, Arch. Rat. Mech. Anal. 44, 187–211 (1972)
E. Noether, Invariante Variationprobleme, Göttigner Nachrichten, Mathematisch physikalische klasse 2, 235–257 (1918)
E. Bessel-Hagen, Über die Erhaltungssätze der Elektrodynamik, Math. Ann. 84, 258–276 (1921)
B. Budiansky and J. R. Rice, Conservation laws and energy release rates, J. Appl. Mech. 40, 201–203 (1973)
D. G. B. Edelen, Aspects of variational arguments in the theory of elasticity: Fact and folklore, Internat. J. Solids and Structures 17, 729–740 (1981)
P. J. Olver, Conservation laws in elasticity I. General results, Arch. Rat. Mech. Anal. 85, 119–129 (1984)
P. J. Olver, Conservation laws in elasticity II. Linear homogeneous isotropic elastostatics, Arch. Rat. Mech. Anal. 85, 131–160 (1984)
P. J. Olver, Errata to [9], Arch. Rat. Mech. Anal. 102, 385–387 (1988)
P. J. Olver, Conservation laws in elasticity III. Planar linear anisotropic elastostatics, Arch. Rat. Mech. Anal. 102, 167–181 (1988)
G. Caviglia and A. Morro, Conservation laws in anisotropic elastostatics, Internat. J. Engrg. Sci. 26, 393–400 (1988)
J. D. Eshelby, W. T. Read, and W. Shockley, Anisotropic elasticity with applications to dislocation theory, Acta Metal. 1, 251–259 (1953)
A. N. Stroh, Dislocation and cracks in anisotropic elasticity, Philos. Mag. 3, 625–646 (1958)
A. N. Stroh, Steady state problems in anisotropic elasticity, J. Math. Phys. 41, 77–103 (1962)
G. A. Hatfield, Conservation laws in anisotropic elasticity, Ph.D. Thesis, University of Minnesota, 1994
C.-S. Yeh, Y.-C. Shu, and K.-C. Wu, Conservation laws in anisotropic elasticity I. Basic framework, Proc. Roy. Soc. London Ser. A 443, 139–151 (1993)
T. C. T. Ting, On anisotropic elastic materials that possess three identical Stroh eigenvalues as do isotropic materials, Quart. Appl. Math. 52, 363–375 (1994)
P. J. Olver, Canonical anisotropic elastic moduli in Modern Theory of Anisotropic Elasticity and Applications (J. Wu, T. C. T. Ting, and D. Barnett, eds.), SIAM, Philadelphia, 1990, pp. 325–339
M. E. Gurtin, The linear theory of elasticity in Mechanics of Solids, Vol. 2 (C. Truesdell, ed.), Springer-Verlag, New York, 1973, pp. 1–295
P. J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics 107, Springer-Verlag, New York, 1986
P. Chadwick and G. D. Smith, Foundations of the theory of surface waves in anisotropic elastic materials, Adv. Appl. Mech. 17, 51–262 (1977)
T. C. T. Ting, The Stroh formalism and certain invariances in two-dimensional anisotropic elasticity in Modern Theory of Anisotropic Elasticity and Applications (J. Wu, T. C. T. Ting, and D. Barnett, eds.), SIAM, Philadelphia, 1990, pp. 3–32
T. C. T. Ting and C. Hwu, Sextic formalism in anisotropic elasticity for almost non-semisimple matrix N, Internat. J. Solids and Structures 24, 65–76 (1988)
D. M. Barnett and J. Lothe, Synthesis of the sextic and the integral formalism for dislocation, Green’s function and surface waves in anisotropic elastic solids, Phys. Norv. 7 (1973)
A. S. Lodge, The transformation to isotropic form of the equilibrium equations for a class of anisotropic elastic solids, Quart. J. Mech. Appl. Math. 8, 211–225 (1955)
P. J. Olver, Canonical elastic moduli, J. Elasticity 19, 189–212 (1988)
N. B. Alphutova, A. B. Movchan, and S. A. Nazarov, On algebraic equivalence of planar problems in anisotropic and orthotropic elasticity, Vestnik Leningrad. Universiteta. Mekhanika, No. 3, 1991 (Russian)
G. W. Milton and A. B. Movchan, A correspondence between plane elasticity and the two-dimensional real and complex dielectric equations in anisotropic media, Proc. Roy. Soc. London Ser. A 450, 293–317 (1995)
T. C. T. Ting, Effects of change of reference coordinates on the stress analysis of anisotropic elastic materials, Internat. J. Solids and Structures 18, 139–152 (1982)
P. J. Olver, The equivalence problem and canonical forms for quadratic lagrangians, Adv. Appl. Math. 9, 226–257 (1988)
Max Jodeit, Jr. and P. J. Olver, On the equation $f = M grad g$, Proc. Roy. Soc. Edinburgh 116A, 341–358 (1990)
K.-C. Wu, Representations of stress intensity factors by path-independent integrals, J. Appl. Mech. 35, 780–785 (1989)
C.-S. Yeh, Y.-C. Shu, and K.-C. Wu, Conservation laws in anisotropic elasticity II. Extension and application to thermoelasticity, Proc. Roy. Soc. London Ser. A 443, 153–161 (1993)
J. D. Eshelby, The force on an elastic singularity, Philos. Trans. Roy. Soc. London Ser. A 244, 57–112 (1951)
J. R. Rice, A path-independent integral and the approximate analysis of strain concentrations by notches and cracks, J. Appl. Mech. 35, 376–386 (1968)
J. K. Knowles and Eli Sternberg, On a class of conservation laws in linearized and finite elasto-statistics, Arch. Rat. Mech. Anal. 44, 187–211 (1972)
E. Noether, Invariante Variationprobleme, Göttigner Nachrichten, Mathematisch physikalische klasse 2, 235–257 (1918)
E. Bessel-Hagen, Über die Erhaltungssätze der Elektrodynamik, Math. Ann. 84, 258–276 (1921)
B. Budiansky and J. R. Rice, Conservation laws and energy release rates, J. Appl. Mech. 40, 201–203 (1973)
D. G. B. Edelen, Aspects of variational arguments in the theory of elasticity: Fact and folklore, Internat. J. Solids and Structures 17, 729–740 (1981)
P. J. Olver, Conservation laws in elasticity I. General results, Arch. Rat. Mech. Anal. 85, 119–129 (1984)
P. J. Olver, Conservation laws in elasticity II. Linear homogeneous isotropic elastostatics, Arch. Rat. Mech. Anal. 85, 131–160 (1984)
P. J. Olver, Errata to [9], Arch. Rat. Mech. Anal. 102, 385–387 (1988)
P. J. Olver, Conservation laws in elasticity III. Planar linear anisotropic elastostatics, Arch. Rat. Mech. Anal. 102, 167–181 (1988)
G. Caviglia and A. Morro, Conservation laws in anisotropic elastostatics, Internat. J. Engrg. Sci. 26, 393–400 (1988)
J. D. Eshelby, W. T. Read, and W. Shockley, Anisotropic elasticity with applications to dislocation theory, Acta Metal. 1, 251–259 (1953)
A. N. Stroh, Dislocation and cracks in anisotropic elasticity, Philos. Mag. 3, 625–646 (1958)
A. N. Stroh, Steady state problems in anisotropic elasticity, J. Math. Phys. 41, 77–103 (1962)
G. A. Hatfield, Conservation laws in anisotropic elasticity, Ph.D. Thesis, University of Minnesota, 1994
C.-S. Yeh, Y.-C. Shu, and K.-C. Wu, Conservation laws in anisotropic elasticity I. Basic framework, Proc. Roy. Soc. London Ser. A 443, 139–151 (1993)
T. C. T. Ting, On anisotropic elastic materials that possess three identical Stroh eigenvalues as do isotropic materials, Quart. Appl. Math. 52, 363–375 (1994)
P. J. Olver, Canonical anisotropic elastic moduli in Modern Theory of Anisotropic Elasticity and Applications (J. Wu, T. C. T. Ting, and D. Barnett, eds.), SIAM, Philadelphia, 1990, pp. 325–339
M. E. Gurtin, The linear theory of elasticity in Mechanics of Solids, Vol. 2 (C. Truesdell, ed.), Springer-Verlag, New York, 1973, pp. 1–295
P. J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics 107, Springer-Verlag, New York, 1986
P. Chadwick and G. D. Smith, Foundations of the theory of surface waves in anisotropic elastic materials, Adv. Appl. Mech. 17, 51–262 (1977)
T. C. T. Ting, The Stroh formalism and certain invariances in two-dimensional anisotropic elasticity in Modern Theory of Anisotropic Elasticity and Applications (J. Wu, T. C. T. Ting, and D. Barnett, eds.), SIAM, Philadelphia, 1990, pp. 3–32
T. C. T. Ting and C. Hwu, Sextic formalism in anisotropic elasticity for almost non-semisimple matrix N, Internat. J. Solids and Structures 24, 65–76 (1988)
D. M. Barnett and J. Lothe, Synthesis of the sextic and the integral formalism for dislocation, Green’s function and surface waves in anisotropic elastic solids, Phys. Norv. 7 (1973)
A. S. Lodge, The transformation to isotropic form of the equilibrium equations for a class of anisotropic elastic solids, Quart. J. Mech. Appl. Math. 8, 211–225 (1955)
P. J. Olver, Canonical elastic moduli, J. Elasticity 19, 189–212 (1988)
N. B. Alphutova, A. B. Movchan, and S. A. Nazarov, On algebraic equivalence of planar problems in anisotropic and orthotropic elasticity, Vestnik Leningrad. Universiteta. Mekhanika, No. 3, 1991 (Russian)
G. W. Milton and A. B. Movchan, A correspondence between plane elasticity and the two-dimensional real and complex dielectric equations in anisotropic media, Proc. Roy. Soc. London Ser. A 450, 293–317 (1995)
T. C. T. Ting, Effects of change of reference coordinates on the stress analysis of anisotropic elastic materials, Internat. J. Solids and Structures 18, 139–152 (1982)
P. J. Olver, The equivalence problem and canonical forms for quadratic lagrangians, Adv. Appl. Math. 9, 226–257 (1988)
Max Jodeit, Jr. and P. J. Olver, On the equation $f = M grad g$, Proc. Roy. Soc. Edinburgh 116A, 341–358 (1990)
K.-C. Wu, Representations of stress intensity factors by path-independent integrals, J. Appl. Mech. 35, 780–785 (1989)
C.-S. Yeh, Y.-C. Shu, and K.-C. Wu, Conservation laws in anisotropic elasticity II. Extension and application to thermoelasticity, Proc. Roy. Soc. London Ser. A 443, 153–161 (1993)
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© Copyright 1996
American Mathematical Society
