Bounds on the volume fraction of 2-phase, 2-dimensional elastic bodies and on (stress, strain) pairs in composites

TL;DR

This paper derives sharp bounds on the volume fraction and stress-strain pairs in 2D elastic composites using boundary measurements, advancing material characterization techniques.

Contribution

It introduces new bounds based on boundary data for two-phase elastic bodies, linking measurable quantities to internal material properties.

Findings

Bounds on volume fraction derived from boundary measurements.

Inequalities for stress and strain pairs in composites.

Conditions for sharpness of bounds identified.

Abstract

Bounds are obtained on the volume fraction in a two-dimensional body containing two elastically isotropic materials with known bulk and shear moduli. These bounds use information about the average stress and strain fields, energy, determinant of the stress, and determinant of the displacement gradient, which can be determined from measurements of the traction and displacement at the boundary. The bounds are sharp if in each phase certain displacement field components are constant. The inequalities we obtain also directly give bounds on the possible (average stress, average strain) pairs in a two-phase, two-dimensional, periodic or statistically homogeneous composite