Application of Sharafutdinov's Ray Transform in Integrated Photoelasticity

We explain the main concepts centered around Sharafutdinov's ray transform, its kernel, and the extent to which it can be inverted. It is shown how the ray transform emerges naturally in any attempt to reconstruct optical and stress tensors within a photoelastic medium from measurements on the state of polarization of light beams passing through the strained medium. The problem of reconstruction of stress tensors is crucially related to the fact that the ray transform has a nontrivial kernel; the latter is described by a theorem for which we provide a new proof which is simpler and shorter as in Sharafutdinov's original work, as we limit our scope to tensors which are relevant to Photoelasticity. We explain how the kernel of the ray transform is related to the decomposition of tensor fields into longitudinal and transverse components. The merits of the ray transform as a tool for tensor reconstruction are studied by walking through an explicit example of reconstructing the $\sigma_{33}$-component of the stress tensor in a cylindrical photoelastic specimen. In order to make the paper self-contained we provide a derivation of the basic equations of Integrated Photoelasticity which describe how the presence of stress within a photoelastic medium influences the passage of polarized light through the material

The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.Comment: 48 pages, 1 figur

Failure analysis of a cracked plate based on endochronic plastic theory coupled with damage

An anisotropic model of damage mechanics for ductile fracture incorporating the endochronic theory of plasticity is presented in order to take into account material deterioration during plastic deformation. An alternative form of endochronic (internal time) theory which is actually an elasto-plastic damage theory with isotropic-nonlinear kinematic hardening is developed for ease of numerical computation. Based on this new damage model, a finite element algorithm is formulated and then employed to characterize the fracture of thin aluminum plate containing a center crack. A new criterion termed as Y R -Criterion is proposed to define both the crack initiation angle and load. Experiments have been conducted to verify the validity of the proposed damage model and it is found that the theoretical crack initiation loads correspond closely with the measured values.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42776/1/10704_2004_Article_BF00034511.pd