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The Anomalous Behavior of Material Overlapping in Elasticity and a Possible Cure
Adair Aguiar
University of São Paulo
Av. Trabalhador são-carlense , 400, Cx. P. 359, São Carlos, 13566-590, BR
Phone: (011-55-16) 3373-9463, Email: aguiarar@sc.usp.br
Abstract:
We present theoretical and numerical results concerning the imposition of the injectivity constraint on a
class of elastic solids. In classical linear elasticity, solutions to these problems are not injective inside
the solids, yielding the unrealistic behavior of material overlapping. To find a cure for this behavior, we
formulate a problem in this class as a constrained minimization problem, which consists of finding a
displacement field that minimizes the total potential energy of the solid subject to the constraint that the
associated deformation field be locally invertible. We show that the solution of this constrained
problem is everywhere injective. Another approach to impose the injectivity constraint consists of
assuming a proper nonlinear elastic behavior of the material. The material behavior near the corners of
the punch is such that, while still singular, prevents material overlapping. Away from these points, this
material behaves like the classical semi-linear material of John introduced in 1960. This work is part of
an ongoing collaboration with Professor Roger Fosdick in the area of Singularities and Constraints in
Elastostatics.
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