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The Constraint of Local Injectivity in Linear Elasticity Theory

Roger Fosdick and Gianni Royer-Carfagni
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 457, No. 2013 (Sep. 8, 2001), pp. 2167-2187
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/3067499
Page Count: 21
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The Constraint of Local Injectivity in Linear Elasticity Theory
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Abstract

There are problems in classical linear elasticity theory whose known solutions must be rejected because they predict unacceptable deformation behaviour, such as the interpenetration of material regions. What has been missing is a proper account of the constraint that allowable deformations must be injective. This type of constraint is highly nonlinear and non-convex, even within the classical linear theory, and it is expected to give rise to the existence of an appropriate constraint reaction field. We propose to determine the displacement field u(·) : B → Rn (n = 2, 3) of an elastic body B ⊂ Rn such that the potential energy is minimized subject to the constraint that the deformation y = f(x) = x + u(x), x ∈ B, is locally invertible, i.e. det(1 + ▽u) > 0 in B. In linear elasticity theory, the strain energy (assumed positive definite) is a quadratic function of ▽u and, in the context of plane problems where the dimension n = 2, the constraint is properly closed, which allows us to prove, at least in this case, an existence theorem for such minimizers in W1,2 ( B). We then investigate the form of the corresponding Euler-Lagrange equations in dimension n = 2 or 3 and characterize the associated constraint reaction (Lagrange multiplier) field. Finally, we review an example problem from linear elasticity theory for an aeolotropic disk whose classical solution supports a subregion of material interpenetration, and we give an alternative solution within the constrained theory which avoids this unacceptable behaviour. The subregion of the disk in which the Lagrange multiplier constraint reaction field is active is determined, as is the field itself. We find that the existence of a constraint reaction field is essential if overlapping of the material is to be avoided. Correspondingly, when the constraint of local invertibility is applied we see that the body becomes significantly stiffer in response to the same loads. Though, perhaps, only coincidental, the existence of a constraint reaction field is remindful of the fact that the force between the atoms of a substance becomes strongly repelling when the separation distance between them is made relatively small.

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