Measurements of plastic displacement gradient components in three dimensions using marker particles and synchrotron X-ray absorption microtomography

Abstract

A universal method is presented for characterising the three-dimensional (3D) plastic displacement gradient field in bulk materials that contain particles or voids observable by X-ray tomography. Millimetre sized samples are investigated by absorption contrast microtomography using high intensity X-rays from a synchrotron source. The positions of dispersed marker particles are determined as a function of imposed strain. The particle diameter can be in the micrometre range, and the volume fraction can be less than 1%. The method is demonstrated by evaluation of compression deformation of a cylindrical Al specimen containing W marker particles. By interpolating the displacement gradient components determined at each particle on a 30×30×30 μm³ grid, 3D maps of the displacement gradient components are obtained with a resolution of 10⁻² in each component. Limitations of the method are discussed, and the potential for application in materials science is outlined.

Introduction

The local plastic strain is likely to deviate from the imposed (macroscopic) plastic strain in materials that contain voids or cracks, in materials such as composites that contain second phases with elastic or plastic properties unlike those of the matrix, and in graded materials with gradients of elastic or plastic properties. Even in single phase polycrystalline metal aggregates the elastic and plastic properties of the individual grains depend on crystal orientation, with the result that the plastic strain state within a grain deviates from the externally imposed strain state. At external plastic strains larger than 10 or 20%, the original grains subdivide into smaller crystal orientation domains, a process that reflects the variation of plastic strain within an original grain.

Methods to assess local plastic strains primarily focus on surface characterisation. Scratches, etched patterns or grids deposited on the surface of samples have been used to measure grain boundary shear and to measure the relative creep rates of the deforming phases in multiphase materials [1], [2]. A recent version of such techniques is the grid distortion technique described by Fisher et al. involving deposition of a regular array of marker particles on a surface and measurement by optical diffraction [3], [4], [5], [6]. In addition, irregularly distributed etch pits have been used to assess plastic strain distributions within grains on the surface of polycrystalline samples [7]. The spatial resolution of these methods depends on the marker grid scale, it is on the micrometre scale when photolithography methods are used for grid deposition. Photogrammetric methods can be adapted to the problem of assessing surface topology [8], [9].

Laser speckle pattern techniques have been developed for the measurement of local elastic and plastic strains [10]. Although the strain resolution is excellent, the spatial resolution of current laser speckle techniques is of the order of 0.1 mm [11]. These techniques have been applied to strain mapping during the deformation of polycrystalline metals and ceramics as well as biological materials.

Unfortunately, the extension of surface strain mapping methods to strain fields in three dimensions has been limited. Wire or grid markers have been used for studies of deformation processes but the scale of the internal grids is usually coarser than 0.1 mm, insufficient to detect localised phenomena [12], [13]. In addition, sample preparation usually involves welding or brazing, which can alter the properties of the material in the vicinity of the markers. The perturbing influences of the sample preparation method and the lack of spatial resolution limit the applicability of such three-dimensional (3D) strain mapping methods.

Recently, high energy synchrotron X-ray diffraction has been introduced as a probe for examining the local crystal orientation of individual grains embedded in a polycrystal [14]. The hard X-rays can penetrate millimetres to centimetres of material. Under favourable conditions a simultaneous description can be given of the position, volume, crystal orientation and elastic strain of up to several hundred grains [15], [16], [17], [18]. Furthermore, for coarse-grained materials, the topology of the grain boundaries can be mapped [18]. This diffraction method, known as 3DXRD, can also be used to probe crystal rotation in individual grains during straining [14]. However, crystal rotation is not a direct manifestation of plastic strain; it is a manifestation of the coupling between microscopic shear deformation mechanisms and constraints imposed either on the sample or on an individual grain by its neighbours. Furthermore, the 3DXRD method is inherently limited to crystalline materials and, at present, is limited to modest deformations, up to about 20%.

The objective of this article is to present a universal method for mapping the plastic displacement gradient tensor in bulk materials that contain particles or voids observable by X-ray tomography, with a spatial resolution in the micrometre range. X-rays from a synchrotron source are used as a probe and imaging is performed by means of X-ray absorption contrast microtomography. To enable the microtomographic measurements, marker particles with strong absorption contrast are added to the matrix prior to deformation. The particle trajectories are identified as functions of imposed strain and the local plastic deformation gradient tensor is revealed by subsequent analyses. The method is verified by a study of a compacted Al powder specimen in which W particles have been added as markers.