text: The deformation field at the slip-band is due to three-dimensional elastic and plastic strains where the concentrated shear of the slip band tip deforms the grain in its vicinity. The elastic strains describe the stress concentration ahead of the slip band, which is important as it can affect the transfer of plastic deformation across grain boundaries. An understanding of this is needed to support the study of yield and inter/intra-granular fracture. The concentrated shear of slip bands can also nucleate cracks in the plane of the slip band, and persistent slip bands that lead to intragranular fatigue crack initiation and growth may also form under cyclic loading conditions. To properly characterise slip bands and validate mechanistic models for their interactions with microstructure, it is crucial to quantify the local deformation fields associated with their propagation. However, little attention has been given to slip bands within grains (i.e., in the absence of grain boundary interaction).
  The long-range stress field (i.e., the elastic strain field) around the tip of a stress concentrator, such as a slip band, can be considered a singularity equivalent to that of a crack. This singularity can be quantified using a path independent integral since it satisfies the conservation laws of elasticity. The conservation laws of elasticity related to translational, rotational, and scaling symmetries were derived initially by Knowles and Sternberg  from the Noether's theorem. Budiansky and Rice introduced the J-, M-, L-integral and were the first to give them a physical interpretation as the strain energy-release rates for mechanisms such as cavity propagation, simultaneous uniform expansion, and defect rotation, respectively. When evaluated over a surface that encloses a defect, these conservation integrals represent a configurational force on the defect. That work paved the way for the field of Configurational mechanics of materials, with the path-independent J-integral now widely used to analyse the configurational forces in problems as diverse as dislocation dynamics, misfitting inclusions, propagation of cracks, shear deformation of clays, and co-planar dislocation nucleation from shear loaded cracks. The integrals have been applied to linear elastic and elastic-plastic materials and have been coupled with processes such as thermal  and electrochemical  loading, and internal tractions. Recently, experimental fracture mechanics studies have used full-field in situ measurements of displacements  and elastic strains  to evaluate the local deformation field surrounding the crack tip as a J-integral.
  Slip bands form due to plastic deformation, and the analysis of the force on a dislocation considers the two-dimensional nature of the dislocation line defect. General definitions of the Peach–Koehler configurational force (𝑃)  (or the elastic energy-momentum tensor ) on a dislocation in the arbitrary 𝑥, 𝑥, 𝑥 coordinate system, decompose the Burgers vector (𝑏) to orthogonal components. This leads to the generalised definition of the J-integral in equations below. For a dislocation pile-up, the J-integral is the summation of the Peach–Koehler configurational force of the dislocations in the pile-up (including out-of-plane, 𝑏 ).
  𝐽 = ∫ 𝑃 𝑛 𝑑𝑆 = ∫(𝑊 𝑛− 𝑇 𝑢) 𝑑𝑆
  𝐽 = 𝑅 𝐽,   𝑖,𝑗,𝑘=1,2,3
  where 𝑆 is an arbitrary contour around the dislocation pile-up with unit outward normal 𝑛, 𝑊 is the strain energy density, 𝑇 = 𝜎 𝑛 is the traction on 𝑑𝑆, 𝑢 are the displacement vector components, 𝐽 is 𝐽-integral evaluated along the 𝑥 direction, and 𝑅 is a second-order mapping tensor that maps 𝐽 into 𝑥 direction. This vectorial 𝐽-integral leads to numerical difficulties in the analysis since 𝐽 and, for a three-dimensional slip band or inclined crack, the 𝐽 terms cannot be neglected.