Engineering Transactions, 39, 3-4, pp. 461-538, 1991

Generalized Strain and Stress Measures: Critical Survey and New Results

A. Curnier
Ecole Polytechnique Federale de Lauisanne, Lausanne

L. Rakotomanana
Ecole Polytechnique Federale de Lauisanne, Lausanne

Four basic principles: objectivity, isotropy, consistency and regularity are proposed to res the concepts of generalized strain and (more originally) of generalized stress. These principles are used to derive two general representations of the corresponding strain and stress functions. Based on a materiaI definition of conjugacy, each candidate strain is then placed in one-to-one correspondence with a certain conjugate stress and vice versa. Besides the classical strain-stress pairs already current in the literature, an interesting family of new strains and conjugate stresses is disclosed in the process. The main contributions of this paper, however, are to demonstrate the superiority of a particular class of strain and stress measures, herein called "congruent", and to reveal the coexistence of different definitions of conjugacy, which is a source of confusion.

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C.A.TRUESDELL, The mechanical foundations of elasticity and fluid dynamics, J. Rat. Mech. Anal., 1, 125-300, 1952.

C.A.TRUESDELL and R.TOUPIN, The classical field theories, Encyclopedia of Physics, III/1, Springer Verlag, Berlin 1960.

Z.KARNI and M.REINER, The general measure of deformation, in: Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, 217-227, Macmillan, New York 1964.

B.R.SETH, Generalized strain measure with applications to physical problems, in: Second-Order Effects in Elasticity, Plasticity and Fluid dynamics, 162-172, Macmillan, New York 1964.

D.B.MACVEAN, Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren, Zeit. Ang. Math. Phys., 19, 157-185, 1968.

R.HILL, On constitutive inequalities for simple materials, I, II, J. Mech. Phys. Solids, 16, 229-242, 315-322, 1968.

R.W.OGDEN, On stress rates in solid mechanics with applications to elasticity theory, Proc. Camb. Phil. Soc., 75, 303-319, 1974.

J.E.FITZGERALD, A tensorial Hencky measure of strain and strain rate for finite deformations, in: Developments in Theoretical and Applied Mechanics, 10, 635- 648, 1980.

S.NEMAT-NASSER, On finite deformation elasto-plasticity, Int. J. Sol. Struct., 18, 10, 857-872, 1982.

S.N.ATLURI, Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely deformed solids, with applications to plates and shells, I, Comp. and Struct., 18, 1, 93-116, 1984.

Z-H.GUO and R.N.DUBEY, Basic aspects of Hill's method in solid mechanics, SM Arch., M. Nijhoff Pub!., 353-380, 1984.

C.A.TRUESDELL and W.NOLL, The nonlinear field theories of mechanics, Encyclopedia of Physics, III/3, Springer Verlag, Berlin 1965.

R.HILL, Aspects of invariance in solid mechanics, in: Adv. in Appl. Mech., 18, 1-75, Academic Press, New York 1978.

C.VALLEE, Lois de comportement elastique isotropes en grandes diformations, Int. J. Eng. Sci., 16, 451-457, 1978.

H.PELZER, Zur Kinetischen Theorie der Kautschuk-Elastizitlit, Monatshefte fur Chemie, 71, 444-447, 1938.

M.MOONEY, A theory of large elastic deformation, J. Appl. Phys., 11, 582-592, 1940.

F.T.WALL, Statistical thermodynamics of rubber, J.Chem. Phys. 10, 132-134, 1942.

R.S.RIVIN, Large elastic deformations of isotropic materials, I-IV, Phil. Trans. Roy. Soc. London, 240-241, 459-525, 379-397, 1948.

T.C.DOYLE and J.L.ERICKSEN, Nonlinear elasticity, in: Adv. in Appl. Mech., 4, 53-115, Academic Press, New York 1956.

F.T.WALL, Statistical thermodynamics of rubber. II, J.Chem. Phys., 10, 485-488, 1942.

M.REINER, Examination of strain-tensors, in: Recent Progress in Applied Mechanics, The Folke Odquist Volume, Wiley, 475-486, 1967.

J.H.WEINER, Statistical mechanics of elasticity, Wiley, 1983.

C.A.TRUESDELL, Geometric interpretation for reciprocal deformation tensors, Quart. Appl. Math., 15, 434-435, 1956.

J.E.MARSDEN and T.J.R.HUGHES, Mathematical foundations of elasticity, Prentice Hall, 1983.

A.HOGER and D.E.CARLSON, Determination of the stretch and rotation in the polar decomposition of the deformation gradient, Quart. Appl. Math., 113-117, April 1984.

K.WEISSENBERG, La mecanique des corps deformables, Arch. Sci. Phys, Natur., 17, 45-103, 106-171, 1935.

J.SERRIN, Mathematical principles of classical fluid mechanics, Encyclopedia of Physics, VIII/1, Springer Verlag, Berlin 1959.

J.J.MOREAU, Lois de l'elasticite en grandes diformations, 12, Seminaire d'Analyse Convexe, Montpellier 1979.

Z-H.GUO, Rates of stretch tensors, J. Elast., 14, 263-267, 1984.

A.HOGER and D.E.CARLSON, On the derivative of square root of a tensor and Guo's rate theorems, J. Elast., 14, 329-336, 1984.

M.E.GURTIN, An introduction to continuum mechanics, Academic Press, 1981.

R.W.ODGEN, Nonlinear elastic deformations, Ellis Horwood, 1984.

C.C.WANG and C.A.TRUESDELL, lntroduction to rational elasticity, Nordhoff Intern. Publ., 1973.

M.E.GURTIN and K.SPEAR, On the relationship between the logarithmic strain rate and the stretching tensor, Int. J. Sol. Struct., 437-444, 1983.

J.P.BOEHLER, Application of tensor functions in solid mechanics, Lecture Notes, CISM, Udine 1984.

H.ZIEGLER and D.MCVEAN, On the notion of an elastic solid, in: Recent Progress in Applied Mechanics, The Folke Odqvist Volume, 561-572, Wiley, 1967.

L.E.MANSFIELD, Linear algebra with geometric applications, M. Dekker Corpor., 1976.

W.T.KOITER, On the complementary energy theorem in nonlinear elasticity theory, in: Trends in Applications of Pure Mathematics to Mechanics, Pitman Publ., 1975.

H.BUFLER, On the work theorems for finite and incremental elastic deformations with discontinuous fields: a unified treatment of different versions, Comp. Math. Appl. Mech. Eng., 36, 95-124, 1983.

R.W.OGDEN, On non-uniqueness in the traction boundary-value problem for a compressible elastic solid, Quart. Appl. Math., 337-344, October 1984.

F.D.MURNAGHAN, Finite deformation of the an elastic solid, Amer. J. Math., 591, 235-260, 1937.

S.S.ANTMAN and J .E.OSBORN, The principle of virtual work and integral laws of motion, Arch. Rat. Mech. Anal., 69, 231-262, 1979.

W.NOLL, La micanique classique, basee sur un axiome d'objectivite, Colloque internationale sur La Methode Axiomatique dans les Mecaniques Classiques et Nouvelles, 47-56, Paris 1963.

E.GREEN and R.S.RIVLIN, On Cauchy's equations of motion, J. Appl. Math. Phys., 15, 290-292, 1964.

M.JAMMER, Concepts of force., A study in the foundations of dynamics, Harvard University Press, 1957.

P.GERMAIN, La methode des puissances virtuelles en micanique des milieux continus, J. Mecanique, 12, 2, 235-274, 1973.

T.BELYTSCHKO, An overview of semi-discretization and time integration procedures, in: Computational Methods for Transient Analysis, 2-65, 1983.

H.COHEN and C.C.WANG, A note on hyperelasticity, Arch. Rat. Mech. Anal., 85, 213-236, 1984.

L.ANAND, On H.Hencky's approximate strain-energy function for moderate deformations, J. Appl. Mech., ASME Trans., 46, 78-82, 1979.

J.C.SIMO and K.S.PISTER, Remarks on rate constitutive equations for finite deformation problems: computational implications, Comp. Math. Appl. Mech. Eng., 46, 201-215, 1984.

S.W.KEY and E.D.KRIEG, On the numerical implementation of inelastic time­dependent and time-independent, finite strain constitutive equations in structural mechanics, Camp. Meth. Appl. Mech. Eng., 33, 431-452, 1982.

Z-H.GUO and R.N.DUBEY, Spins in deforming continuum, SM Arch., 9, 53-61, M. Nijhoff Publ., 1984.

A.J.M.SPENCER, Theory of invariants, in: Continuum Physics I, Academic Press, 1971.

J.P.BOEHLER, Lois decomportement anisotrope des milieux continua, J. Mecanique, 17, 2, 153-190, 1978.

A.J.M.SPENCER, Application of tensor functions in solid mechanics, Lecture Notes, CISM, Udine 1984.

C.C.WANG, A new representation theorem for isotropic functions, Part I, Arch. Rat. Mech. Anal., 36, 166-197, 1970.

J.J.TELEGA, Some aspects of invariant theory in plasticity. Part I. New results relative to representation of isotropic and anisotropic tensor Junctions, Arch. Mech., 36, 147-162, 1984.

E.W.BILLlNGTON, Referential stress tensor, Acta Mecha.nica, 55, 263-266, 1985.

T.C.TING, Determination of C112, c-1/2 and more general isotropic tensor functions of C, J. Elast., 15, 319-323, 1985.

M.KLEIBER and B.RANIECKI, Elastic-plastic materials at finite strains, in: Plasticity Today, Modeling Methods and Application, Elsevier, 1985.

A.ROGER, The material time derivative of logarithmic strain, Int. J. Solids and Struct., 22, 9, 1019-1032, 1986.

D.E.CARLSON and A.HOGER, The derivative of a tensor-valued Junction of a ten­sor, Quart. Appl. Math., 44, 3, 409-423, 1986.

E.CHU, Aspects of strain measures and strain rates, Acta Mechanica, 59, 103-112, 1986.

Y.MA and C.S.DESAI, Alternative definition of finite strains, J. Engng. Mech., 116, 4, 901-919, 1990.

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