The missing term in the decomposition of finite deformation

•Third term is added to the multiplicative decomposition of the deformation gradient.•The third term is consistent with lattice accommodation of dislocations.•Elasticity solutions illustrate importance of this term for single dislocations.•The multiscale nature of the three term decomposition is explored.•Implications for crystal plasticity are considered. In finite inelasticity, the gradient of total deformation is typically split into a product of two terms: a reversible (elastic) term whose strain vanishes upon load removal at some scale, and an irreversible (plastic) component that remains. In this work it is argued that this decomposition represents a limiting case for which defects are absent from the volume of interest, with compatible elastic distortion associated with externally applied stress and plastic deformation associated with history of dislocation glide through the element. An additional (third) term should be incorporated in the multiplicative decomposition when applied to an element of material of any realistic volume, accounting for local lattice distortion due to defects within. In the limiting case that this volume approaches a few lattice spacings, the probability of interior defects tends towards zero, but a very small volume element containing a few defects, or a larger element containing a large density of defects, requires a third term in the multiplicative decomposition to represent contributions of defects to residual lattice distortion. Physical experiments and reported atomic and continuum calculations support these theoretical arguments. The magnitude of distortion from the “missing” third term is estimated analytically using elasticity solutions for straight dislocations. Advances to crystal plasticity theory involving a three-term decomposition are suggested.