Nonuniform Stress Field Determination Based on Deformation Measurement

Abstract

In this investigation, we demonstrate a technique that, under certain circumstances, will determine stresses associated with a nonuniform deformation field without knowing the detailed constitutive behavior of the deforming material. This technique is based on (1) a detailed deformation measurement of a domain (currently in 2D) and (2) the observation that for isotropic materials, the strain and the stress, which form the so-called work-conjugate pair, are co-axial, or their eigenvectors share the same directions. The particular measures for strain and stress chosen in this study are the Lagrangian (or Green-Lagrangian) strain and the second Piola–Kirchhoff stress. The deformation measurement provides the field of the principal stretch orientation θλ and since the Lagrangian strain and the second Piola–Kirchhoff stress are co-axial, the principal stress orientation θs of the second Piola–Kirchhoff stress is then determined. The Cauchy stress is related to the second Piola–Kirchhoff stress through the deformation gradient tensor, which can be measured experimentally. We then show that the principal stress orientation θσ of the Cauchy stress is the sum of the principal stretch orientation θλ and the local rigid-body rotation θq, which is determinable by the deformation gradient through polar decomposition. Such a relationship is valid for finite deformations. With the principal stress orientation θσ known, the equation of equilibrium, now in terms of the two principal stresses, σ1 and σ2, and θσ, can be solved numerically with appropriate traction boundary conditions. The stresses determined using this technique obviously satisfy the equation of equilibrium, in contrast to those obtained from a constitutive model with input from deformation measurement. The technique and the associated numerical scheme are verified and validated through two virtual test cases representative of the simply-connected and multiply-connected domains, where exact solutions are available. The technique is then applied to an experimental case of nonuniform deformation of a polyvinyl chloride (PVC) sheet with a circular hole and subject to uniaxial tension. In this case, the associated stress field is also determined through a constitutive model of hyperelasticity, the generalized neo-Hookean (GNH) model, calibrated for the particular PVC sheet. Limitations and restrictions of the technique and the associated numerical scheme, as well as possible extensions will be discussed.