Abstract
Material frame indifference implies that the solution in non-linear elasticity theory for a connected body rigidly rotated at its border is a rigid, stress-free, deformation. If the same problem is considered within linear elasticity theory, considered as an approximation to the true elastic situation, one should expect that if the angle of rotation is small, the body still undergoes a rigid deformation while the corresponding stress, though not zero, remains consistently small. Here, we show that this is true, in general, only for homogeneous bodies. Counterexamples of inhomogeneous bodies are presented for which, whatever small the angle of rotation is, the linear elastic solution is by no means a rigid rotation (in a particular case it is an “explosion”) while the stress may even become infinite. If the same examples are re-interpreted as problems in an elasticity theory based upon genuinely linear constitutive relations which retain their validity also for finite deformations, it is shown that they would deliver constraint reaction forces that are not in equilibrium in the actual, deformed, state. This furnishes another characterization of the impossibility of an exact linear constitutive theory for elastic solids with zero residual stress.
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References
Ball, J.: Convexity condition and existence theory in non-linear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)
Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33(4), 24 (2007)
De Giorgi, E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Unione Mat. Ital. 1, 135–137 (1968)
Fosdick, R., Royer-Carfagni, G.: Multiple natural states for an elastic isotropic material with polyconvex stored energy. J. Elast. 60, 223–231 (2000)
Fosdick, R., Royer-Carfagni, G.: The constraint of local injectivity in linear elasticity theory. Proc. R. Soc. Lond. A 457, 2167–2187 (2001)
Fosdick, R., Serrin, J.: On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids. J. Elast. 9, 83–89 (1979)
Fosdick, R., Freddi, F., Royer-Carfagni, G.: Bifurcation instability in linear elasticity with the constraint of local injectivity. J. Elast. 90(1), 99–126 (2008)
Heinonen, J., Kilpelainen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Oxford University Press, New York (1993)
Horgan, C.O., Chan, A.M.: The pressurized hollow cylinder or disk problem for functionally graded isotropic materials. J. Elast. 55, 43–59 (1999)
Lekhnitskii, S.G.: Anisotropic Plates. Gordon & Breach, New York (1968)
Morrey, X. Jr.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)
Podio-Guidugli, P.: De Giorgi’s counterexample in elasticity. Q. Appl. Math. 34, 411–419 (1977)
Podio-Guidugli, P.: The Piola-Kirchhoff stress may depend linearly on the deformation gradient. J. Elast. 17, 183–187 (1987)
Podio-Guidugli, P.: A primer in elasticity. J. Elast. 58, 1–104 (2000)
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Freddi, F., Royer-Carfagni, G. From Non-Linear Elasticity to Linearized Theory: Examples Defying Intuition. J Elasticity 96, 1–26 (2009). https://doi.org/10.1007/s10659-009-9191-7
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DOI: https://doi.org/10.1007/s10659-009-9191-7
Keywords
- Non-linear elasticity theory
- Linear elasticity theory
- Irregular solutions
- Anisotropic elasticity
- Composite materials
- Functionally graded materials