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The use of a virtual configuration in formulating constitutive equations for residually stressed elastic materials. (English) Zbl 0857.73007
The authors present a method by which one can utilise data obtained from standard destructive experiments to derive constitutive equations that describe the mechanical behaviour of residually stressed elastic materials. In contrast to previous methods, the one given here leads to a constitutive equation that is an explicit function of the residual stress, and includes only material parameters that are required to describe the stress free material. The authors first obtain the constitutive equation for a special class of those residually stressed bodies which can be cut up into a finite number of parts, each of which is entirely free of residual stress. Then they introduce the idea of pointwise stress free configurations to obtain the constitutive equation for a more general class of residually stressed bodies. The derivation for this latter class is based on the assumption that each infinitesimally small neighbourhood of the body has an associated stress free configuration called its ‘virtual configuration’. It also requires that the constitutive equation for the stress free or natural material is known and invertible.
The authors go on to illustrate the use of their theory by considering two particular bodies where the undeformed configuration is a thick walled incompressible spherical shell which supports a residual stress. One is an example of the special class and the other an example of the more general class, and in both cases the natural material is a Mooney-Rivlin material. They conclude with a discussion of the relationship between appropriate virtual configurations and destructive experiments. In particular, the use of the theory in improving the design of such experiments and the use of the experiments in obtaining data required to formulate the appropriate constitutive equation are considered.

74A20 Theory of constitutive functions in solid mechanics
74B99 Elastic materials
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[1] R.M. Fisher and J.Z. Duan, Influence of substrate on cracking of vapour-deposited thin films due to residual stress. In: B.M. DeKoven, A.J. Gellman and R. Rosenberg (eds),Interfaces Between Polymers, Metals, and Ceramics: materials Research Society Symposium Proceedings. Materials Research Society (1989) pp. 299–304.
[2] H. Nagayoshi, H. Morinaka, K. Kamisako, K. Kuroiwa, T. Shimada and Y. Tarui, Residual stress of a-Si1-xNx: H films prepared by afterglow plasma chemical vapor deposition technique.Japanese Journal of Applied Physics 13 (Part 2) (1992) L867-L869. · doi:10.1143/JJAP.31.L867
[3] H.T. Hahn, Residual stresses in polymer matrix composite laminates.Journal of Composite Materials 10 (1976) 266–278. · doi:10.1177/002199837601000401
[4] R. Tandon and D.J. Green, The effect of crack growth stability induced by residual compressive stresses on strength variability.Journal of Materials Research 7 (1992) 765–771. · doi:10.1557/JMR.1992.0765
[5] T.A. Harris, M.A. Ragen and R.F. Spitzer, The effect of hoop and material residual stresses on the fatigue life of high speed rolling bearings.Tribology Transactions 35 (1992) 194–198. · doi:10.1080/10402009208982109
[6] Y.C. Fung,Biomechanics: Motion, Flow, Stress, and Growth. Springer-Verlag (1990). · Zbl 0743.92007
[7] J.M. Guccione, A.D. McCulloch and L.K. Waldman, Passive material properties of intact ventricular myocardium.ASME Biomech. Eng. 113 (1991) 42–55. · doi:10.1115/1.2894084
[8] V. Hauk, P. Holler and E. Macherauch, Measuring techniques of residual stresses – present situation and future aims. In: E. Macherauch and V. Hauk (eds),Residual Stresses in Science and Technology. DGM Informationsgesellschaft (1987) pp. 231–242.
[9] G.A. Matzkanin, Nondestructive evaluation of residual stress in composites and advanced materials: a state-of-the-art review. In: E. Macherauch and V. Hauk (eds),Residual Stresses in Science and Technology. DGM Informationsgesellschaft (1987) pp. 101–108.
[10] E. Macherauch and K.H. Kloos, Origin, measurement and evaluation of residual stresses. In: E. Macherauch and V. Hauk (eds),Residual Stresses in Science and Technology. DGM Informationsgesellschaft (1987) pp. 3–26.
[11] B.E. Johnson and A. Hoger, The dependence of the elasticity tensor on residual stress.Journal of Elasticity 33 (1993) 145–165. · Zbl 0812.73024 · doi:10.1007/BF00705803
[12] A.L. Cauchy, Sur l’equilibre et le mouvement interieur des corps consideres comme des masses continues.Ex. de Math 4 (1829) 293–319.
[13] A. Hoger, On the determination of residual stress in an elastic body.Journal of Elasticity 16 (1986) 303–324. · Zbl 0616.73033 · doi:10.1007/BF00040818
[14] R.S. Marlow, On the stress in an internally constrained elastic material.Journal of Elasticity 27 (1992) 97–131. · Zbl 0757.73006 · doi:10.1007/BF00041645
[15] A. Hoger, Residual stress in an elastic body: a theory for small strains and arbitrary rotations.Journal of Elasticity 31 (1993) 1–24. · Zbl 0811.73018 · doi:10.1007/BF00041621
[16] A. Hoger, The constitutive equation for finite deformations of a transversely isotropic hyperelastic material with residual stress.Journal of Elasticity 33 (1993) 107–118. · Zbl 0799.73016 · doi:10.1007/BF00705801
[17] A. Hoger, The elasticity tensor of a transversely isotropic hyperelastic material with residual stress.Journal of Elasticity (in press). · Zbl 0868.73020
[18] M.E. Gurtin,Introduction to Continuum Mechanics. Academic Press (1984). · Zbl 0559.73001
[19] K. Taiamizawa and T. Matsuda, Kinematics for bodies undergoing residual stress and its applications to the left ventricle.Journal of Applied Mechanics 57 (1990) 321–329. · doi:10.1115/1.2891992
[20] K. Hayashi and K. Taiamizawa, Stress and strain distributions and residual stresses in arterial walls. In: Fung, Hayashi and Seguchi (eds),Progress and New Directions in Biomechanics. Mita Press (1989).
[21] R.S. Rivlin and D.W. Saunders, Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber.Phil. Trans. Roy. Soc. A243 (1951) 251–288. · Zbl 0042.42505 · doi:10.1098/rsta.1951.0004
[22] R.S. Rivlin, A uniqueness theorem in the theory of highly-elastic materials.Proc. Cambridge Phil. Soc. 44 (1948) 595–597. · Zbl 0032.08904 · doi:10.1017/S0305004100024610
[23] C. Truesdell and W. Noll, The non-linear field theories of mechanics.Handbuch der Physik III/3. Springer-Verlag (1965). · Zbl 0779.73004
[24] J.L. Ericksen, Inversion of a perfectly elastic spherical shell.Z. angew. Math. Mech. 35 (1955) 382–385. · Zbl 0065.17802 · doi:10.1002/zamm.19550350909
[25] P.C. Chen and Y. Oshida, Residual stress analysis of a multi-layer thin film structure by destructive (curvature) and non-destructive (X-ray) methods. In: B.M. DeKoven, A.J. Gellman, and R. Rosenberg (eds),Interfaces Between Polymers, Metals, and Ceramics: Materials Research Society Symposium Proceedings. Materials Research Society (1989) pp. 363–368.
[26] W.H. Chu and M. Mehregany, A study of residual stress distribution through the thickness of p+ silicon films.IEEE Transactions on Electron Devices 40 (Number 7) (1993) 1245–1250. · doi:10.1109/16.216428
[27] J.H. Omens, H.A. Rockman and J.W. Covell, Passive ventricular mechanics in tight-skin mice.American Journal of Physiology 266 (Number 3 Part 2) (1994) H1169-H1176.
[28] R. Skalak, S. Zargaryan, R.K. Jain, P. Netti and A. Hoger, Compatibility and the genesis of residual stress by volumetric growth.J. Theor. Biology (in press). · Zbl 0858.92005
[29] J.A. Blume, Compatibility conditions for a left Cauchy-Green strain field.Journal of Elasticity 21 (1989) 271–308. · Zbl 0725.73041 · doi:10.1007/BF00045780
[30] A. Hoger, On the residual stress possible in an elastic body with material symmetry.Archive for Rational Mechanics and Analysis 88 (Number 3) (1985) 271–289. · Zbl 0571.73011 · doi:10.1007/BF00752113
[31] M.E. Gurtin, The linear theory of elasticity.Handbuch der Physik VIa/2 Springer-Verlag (1972). · Zbl 0317.73002
[32] R.D. Cook and W.C. Young,Advanced Mechanics of Materials. Macmillan (1985).
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