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Theoretical and Applied Mechanics

Mechanics of Solids
^ On the Domain of Validity of Linearized Elasticity
D. E. Carlson*
University of Illinois
dec@uiuc.edu

It is widely recognized that the validity of the linearized theory of elasticity rests on the assumption that the deformation be "small" in some appropriate sense. Careful elasticians know that the basic equations of the linearized theory are implied by the nonlinear theory under the assumption that the displacement gradient be small. Here, we note that justification of the substitution of the linearized constitutive equation into the stress equations of motion or equilibrium requires that the nondimensionalized second gradient of the displacement also be small. We show by example that the resulting limitation on the domain of validity of the linearized theory can be severe.

^ Nematic Optical Elastomers
E. Fried,* D. E. Carlson, R. D. James (Univ. of Minnesota)
National Science Foundation, CMS 96-10286
e-fried@uiuc.edu, dec@uiuc.edu

Optical elastomers are a novel class of materials formed by the cross-linking of polymeric liquid crystals. These materials possess properties of both scientific and technological importance, with observed phenomena including stress-induced optical switching, stress-induced shifts in phase-transition temperatures, piezoelectricity, spontaneous macroscopic shape changes accompanying phase transitions, and nonmonotonic (or discontinuous) stress–strain relations. This research combines experiment, theory, and computation with the aim of developing an understanding of the optomechanical response of nematic optical elastomers under biaxial loading conditions.

^ Compressed Thin-Film Diaphragms
G. Gioia,* A. M. Cuitiño (Rutgers University)
National Science Foundation CMS-9820823
ggioia@uiuc.edu

This project involves experimental, theoretical, and computational research into the mechanics of anisotropically compressed thin-film diaphragms. We have carried out experiments showing that in lieu of the well-behaved folding patterns that are obtained in the isotropic case, the anisotropic case leads to microstructures, that is, fine folding patterns that lack a length scale associated with the boundary conditions. We have clarified this behavior by means of theoretical work, proving that the observed microstructures are traceable to the spinodal structure of the underlying energy function. We have also estimated the regularizing effect of bending and verified conclusions both computationally and experimentally.

^ Shear-Pin Load Cell Analysis
J. W. Phillips,* B. E. Soltz
Caterpillar; National Center for Supercomputing Applications
jwp@uiuc.edu

Shear pins are often used as load cells in cranes, tractors, and other heavy equipment to monitor payloads and other structural loading. Shear pins require calibration in order to be useful, but some recent results indicate that there is some discrepancy between lab calibrations and field test results. We are attempting to identify the sources of disagreement and provide guidelines for proper calibration.

^ A Generalized Formulation of Linearized Elastodynamics
D. A. Tortorelli,* D. E. Carlson
University of Illinois
dtortore@uiuc.edu, dec@uiuc.edu

The conventional theory of linearized elastodynamics addresses the case of motions that have small displacement gradients with respect to a reference configuration of the elastic body that is unstressed and at rest. Here, we develop a theory of much wider applicability in which the linearization is with respect to a reference configuration that is loaded and in rigid motion.

^ On the Theory of Thermoelasticity in the Presence of Internal Constraints
D. A. Tortorelli,* D. E. Carlson
University of Illinois
dtortore@uiuc.edu

Generally, in internally constrained nonlinear thermoelasticity, it is hypothesized that the dependent fields are given by constitutive functions of appropriate independent fields to within indeterminate reaction fields. The assumption that the reactions do not contribute to the rate of entropy production in processes that satisfy the internal constraints leads to the form of the reactions. Here, we show that by a slight reorganization of the conventional development, it is possible to infer the existence of the reaction fields (except for the one corresponding to the heat flux). Furthermore, the forms of the reactions are obtained, as are the usual results for the constitutive functions.


Summary of Engineering Research