Detecting precursors of localization by strain-field analysis

- Method is developed to detect the onset of localization prior to loss of stability.
- Emerging patterns in strain-fields are detected using Minkowski functionals.
- Appropriate kinematic enrichment of the macroscopic description is proposed.
- Proposed method is applicable to both experimental as well as numerical data.
- Method is applicable to multi-scale models with embedded localization zones.

Localization in the deformation field, even though initiated locally at the microscopic scale, leads upon increased deformation to fracture at the macroscopic scale, thereby violating the separation of length scales. Localization and damage can be accounted for in macroscopic modeling by appropriate enrichments at that level, however doing so requires (i) detecting the onset of localization prior to its actual occurrence and (ii) quantifying the kinematical characteristics of the localization band. This paper serves that goal. A methodology is developed to analyze the evolution of strain- and displacement-fields during deformation. A key ingredient in this analysis is the use of the Minkowski functionals (also known as intrinsic volumes, quermass integrals, or curvature integrals) from integral geometry, to detect emerging patterns in thresholded strain- and displacement-fields. Doing so, the onset of localization in the microstructure is detected as the emergence of a correlated and narrow pattern of high strains, prior to the actual loss of material stability. Furthermore, the developed localization band is characterized in terms of a weak displacement discontinuity, incorporating the width and direction of the band. The developed methodology uses kinematical fields only, and is therefore applicable to both numerical and experimental deformation-field data. For illustration purposes, numerical data from a finite-element simulation of a deformed voided microstructure is used, without any loss of generality.